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+{"text":"\\section{Introduction}\nOne of the elemental forms of document processing includes classification. \nSince the last couple of years, it is in demand because of the increasing \navailability of data in digital format which has resulted into the  requirement of systematization of  \nthat data. Manual organization of huge data can be tedious if strict time \nconstraints are set, increasing the necessity of automated document classification. The contexts of words in the documents play a very important role in deciding the category of the document. The human brain is very effective in consideration of contexts in the incoming information for taking the appropriate action. \n\n\\smallskip\n\nThe principles of HTM theory can be used to meet the requirements of organizing of data.  \nHTM \ntakes inspiration from the mammalian brain which has been evolving over \nmillions of years and is able to process data efficiently. As HTM is \nbiologically plausible, it is based on simple rules and not complex mathematics. HTM theory is being developed by a US based company called Numenta, Inc.\n\\smallskip\n\n\\section{Related Work}\nSome of the conventional methods for text\/document classification are mentioned below:\n\\subsection{Naive Bayes}\nThe Naive Bayes classifier is a probabilistic classifier and is based on the Bayes theorem. It works well with small samples of data. The posterior probability of a particular document belonging to various classes is calculated. The document is assigned to the class with the highest posterior probability. The Naive Bayes classifier assumes strong independence between the features. This is a major limitation of this classifier and hence has low performance in cases where the features are correlated \\cite{comparison}.\n\n\\subsection{Support Vector Machines}\nSupport Vector Machines (SVMs) are supervised machine\nlearning algorithms. In case of a multi class problem, first\nthe problem has to be decomposed into two separate class\nproblems as SVM can work only with binary classification\nproblem. They will probably give poor results when total\nnumber of samples are very less than the total number of\nfeatures. In comparison with decision making classifier and\nlogistic regression, SVM takes more time for computation \\cite{comparison}.\n\n\\subsection{K-Nearest Neighbour}\nK-Nearest Neighbour (KNN) is used for classification of objects by calculating the distance of training samples from each object. KNN classification is a simple and widely used approach for text classification. However, it is computationally intensive and\nclassification time is high \\cite{comparison}. Also, it is difficult to find the ideal value of k \\cite{knn}.\n\n\n\\subsection{Convolutional Neural Network}\nConvolutional Neural Network (CNN) works well with static text classifications. CNN is a type of feed forward neural network, comprising of neurons with trainable weights and biases. CNN comprises of a number of convolutional layers with nonlinear activation functions like ReLU or tanh applied to the results. CNN suffers from the limitations of the requirement of large data and big processing power to be able to predict accurately \\cite{cnn}.\n\nHTM theory is primarily used for Classification, Prediction and Anomaly Detection purposes. One of its application for Classification is mentioned below:\n\n\\subsection{Land - forms classification}\nAs HTM based models have a common learning algorithm, it can be used for classifying images. HTM theory has been used for classifying different land-forms like trees, roads, buildings and farms using the images obtained from satellites. The framework used achieved an accuracy of 90.4\\%,\\cite{perea2009application}  which is at par with the conventional machine learning techniques for image classification.\n\nSince HTM theory can be used for image classification purposes, it can hold a promise to classify text\/documents.\n\n\\section{Overview of Hierarchical Temporal Memory}\nHTM is a theory which seeks to apply the \nstructural as well as algorithmic properties of the neocortex to machine \nlearning problems \\cite{hawkins2010hierarchical}. The neocortex proves to be the center of intelligence in the \nmammalian brain. It is responsible for processing complex activities such as \ncommunication, planning and prediction. Structurally, neocortex is a 2 mm thick \ntissue divided into a number of different regions. A region is a network of \ninterconnected neurons \\cite{hawkins2010hierarchical}. This attributes to the presence of input connections \nfrom different sensory organs \\cite{hawkins2006hierarchical,onintelligence} like eyes, ears etc. The term \"Hierarchical\" in the theory is owing to \nthe fact that, HTM network contains a hierarchy of levels arranged in the \npyramid-like structure. These levels are present in the form of regions that \nare again composed of columns which finally consist of neurons. These neurons need \nnot be physically arranged in a hierarchy, but are logically arranged in the \nhierarchical format. The lower levels in hierarchy represent data having lower \nabstraction\/complexity. As we go higher in the hierarchy, the data abstraction stored in the memory increases. Time plays a crucial role in the \nway data is stored in mammalian brain. \"Temporal\" implies that the HTM network takes \ninto consideration the sequence of the incoming data. A continuous stream of \ninput data is aptly learned as spatial and temporal sequences.\n\\smallskip\n\nA remarkable property of the neocortex is that the input from all the sensory \norgans is processed in the same manner. Hence, it has a common learning algorithm \nfor inputs from all types of sensory organs \\cite{hawkins2010hierarchical}. \n\n\\begin{figure*} [htb!] \n\\includegraphics[width=1\\textwidth]{process_htm_final.png}\n\\caption{\\sl System Architecture Diagram\n\\label{fig:implementation}}\n\\end{figure*} \n\n\n\n\\subsection{Structure of a Neuron}\n\n\nInside the mammalian brain, neurons play a central role in information handling. Some \nrelevant parts of the neuron for our study are mentioned below. \n\n\\subsubsection{Proximal Dendrites}\nProximal dendrites are in close proximity to the cell body. The proximal \ndendrites are connected directly to the inputs from the sensory \norgans. \n\n\n\\subsubsection{Distal Dendrites}\nDistal dendrites are the ones that are afar from the cell body. The distal \ndendrites have connections with various other neurons in the neocortex. Majority of the \nconnections to the axon are from distal dendrites as compared to the \nconnections made by proximal dendrites \\cite{synapse}. \n\n\n\\subsubsection{Synapse}\nA synapse is a connection between an axon of one neuron and dendrite of the \nother. The ongoing process of breaking and reforming these synapses between \ncells results in learning of new data and thus gradually forgetting the old \none.\\smallskip\n\nThere is a permanence value associated with every synapse and a threshold \nlinked with every neuron. Thus, for a neuron to get activated, the total number \nof synapses with permanence values higher than the threshold value must be more \nthan the stimulus threshold.\n\n\\subsection{Sparse Distributed Representation}\nThough the neocortex contains billions of neurons in highly interconnected \nmanner, only a tiny fraction of them are active for a particular input \\cite{sdr1}. Hence, only small percentage of active neurons are responsible for \nrepresenting the input information. This is called as Sparse Distributed Representation (SDR). \nEven though single activated neuron has the potential to convey some meaning, \nthe full information can only be conveyed when it is interpreted within the \ncontext of other neurons.\nAs the information is spread across a tiny percentage of the active bits, SDRs are more noise tolerant than dense representations, making them ideal for text processing.\n\n\\subsection{Spatial Pooler}\nHTM includes two important parts - Spatial Pooler (SP) and Temporal Pooler (TP). Spatial Pooler, also known as Pattern Memory, has been emphasized in this study.\n\\smallskip\n\nThe neurons in the neocortex are arranged in columns, which represent features of the input. Every neuron in a particular column, which represent different context for an input, is \nconnected to specified number of bits in the input bit array. The selection of \nbits to be connected to the neurons in a particular column is random. The bits \nwhich are connected to a particular column are known as a potential pool of \nthat particular column. Connections between input bit and the column neuron is \ncalled as a synapse. Every synapse has a value associated with it known as \npermanence value similar to that of a mammalian brain. Permanence value is always in the range of 0 and 1. There is a threshold value associated with synapse's permanence.\n\nIf the permanence value of a synapse associated to an input bit is greater than the threshold, the activation of the column of neurons \nis influenced by the input bit. The permanence value of a synapse is adjusted in the learning phase.\n\n\nThe main role of SP in HTM is finding spatial patterns in the input data. It is decomposed into three stages:\n\\medskip\n\\subsubsection{Overlap}\nIn this stage, overlap score of each column is calculated. Overlap score is \n the count of active bits in the potential pool of a particular \ncolumn having permanence value greater than the threshold. \n\\smallskip\n\\subsubsection{Inhibition}\nThe columns are sorted according to their overlap scores from highest to lowest. A particular fraction (in our study, 0.5\\%, Table \\ref{spatial_pooler_paramters}, $NumActiveColumnsPerInhArea$) of the top columns is selected (also called as active columns or the winning columns) for the learning phase. Rest other columns are inhibited from learning.\n\\smallskip\n\\subsubsection{Learning}\nDuring Learning, the permanence value of the synapses in the potential pool of the winning \ncolumns is incremented (by synPermActiveInc, Table \\ref{spatial_pooler_paramters}) or decremented (by synPermInactiveDec, Table \\ref{spatial_pooler_paramters}). When the active column is connected to an \nactive bit then the permanence value of the synapse corresponding to that \nactive bit is incremented. However, when the active column is connected to an\ninactive bit then the permanence value of the synapse corresponding to that \ninactive bit is decremented. This is the result of column expecting that bit to be active. The synapse permanence is decremented as a \npunishment. \n\n\\section{Implementation}\n\nThe flowchart in figure \\ref{fig:implementation} is our high-level \narchitecture diagram for document categorization. As the mammalian brain requires electrical \nsignals for learning, the learning algorithm i.e., Spatial Pooler also requires \nbit patterns for processing. So, to convert text into bit arrays, Latent Semantic \nIndexing (LSI) technique is used, which converts semantically similar sentences \ninto similar bit arrays. These bit arrays (which need not be sparse)\nare fed to the Spatial Pooler where it simulates the working of neurons in \nthe brain and gives SDR as the output. The \nactive bits in the SDR represent the neurons \nwhich get activated in the Spatial Pooler. Since semantically similar \ntext belong to the same category, it is easy to classify the \ntext into different categories.\n\\medskip\n\\subsection{Latent Semantic Indexing}\nAs HTM theory is modelled after the mammalian brain, its input also should be in \naccordance with the input format received by the brain. The brain receives \ninput in the form of electrical signals which correspond to bit arrays. Latent Semantic Indexing(LSI) helps in determining hidden \nfeatures in documents \\cite{papadimitriou1998latent}. Thus the technique is used to extract the \ncontextual-usage meaning of words from the documents\\cite{lsa3}. \nThe LSI framework consists of 3 steps which are mentioned below.\n\\smallskip\n\n\n\\subsubsection{Preprocessing of input data}\nIn the initial step, the input text is tokenized and stopwords are removed from \nevery document of the corpus\\footnote{We have used the wikipedia language corpus as it \nincludes a large vocabulary which is useful for generic datasets. The corpus \nwill change if the dataset is in a language other than English or contains a \nlarge number of words which are not present in the vocabulary.}. Each term in \nthe text is then represented as a tuple containing term-id and term frequency. \nA matrix is created in which the rows denote the unique terms and the \ncolumns denote the documents. Every cell denotes the term count in the corresponding document.\nThe matrix of term-frequency counts obtained from the term document matrix is \nthen modified using the TF-IDF technique so as to give more weight to rare \nterms compared to common terms across documents and also to frequently occurring \nterms in a particular document.   \nThe formula for weighing each term can be represented as,\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1\\linewidth]{LSA_with_bold_new.png}\n\\caption{\\sl Latent Semantic Indexing framework\n\\label{fig:lsa_framework}}\n\\end{centering}\n\\end{figure} \n\n\n\\begin{gather}\n  Document Term Weight = f_{t,d} \\times \\ln(N \/ n_t) \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l}\n    f_{t,d} & count of term~$t$ in document~$d$ \\\\       \n    N & the total count of documents \\\\\n    n_t & the count of documents having term~$t$ \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\smallskip\n\n\nThe term-document matrix gets modified to contain weights of each term \nin a given document. The dimensionality reduction of this matrix is done using \nSingular Value Decomposition (SVD).\n\\smallskip\n\\subsubsection{Singular Value Decomposition}\nLSA uses SVD for generating the vectors of a particular text \\cite{lsa1, lsa2}. The matrix $X$(term-document) is used to calculate two matrices. These are,\n\n\\begin{gather}\n  Y = X^TX \\\\\n  Z = XX^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l}\n    $X$ & term - document matrix \\\\\n\t$Y$ & document - document matrix \\\\       \n    $Z$ & term - term matrix \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nAfter finding eigenvectors of $Y$ and $Z$ matrices, we get left singular matrix, $L$ \nand right singular matrix, $R$ respectively. Thus, term - document matrix, $X$, \n is divided into unique combination of three matrices as follows -\n\n\\eat{\n\\begin{gather}\n X = L \\Sigma R^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\multirow{2}{*}{$L$} & is the left singular vector matrix representing \\\\\n  & weights of a term for corresponding concepts. \\\\\n  \\multirow{2}{*}{$R^T$} & is the transpose of the right singular vector matrix \n\\\\\n  & representing  weights of documents belonging to particular concepts. \\\\\n\\smallskip\n\\multirow{2}{*}{$\\Sigma$} & is the diagonal matrix representing \\\\\n  & weights of the concepts found in the text\\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n}\n\n\\begin{gather}\n\\label{4}\n X = L \\Sigma R^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  $L$ & Term - Concept weight matrix \\\\\n  R^T & Concept - Document weight matrix \\\\\n  \\Sigma & Diagonal matrix representing concept weights \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n$\\Sigma$ is calculated by taking the square root of the eigenvalues of matrix $Y$. \n\\smallskip\n\nTo reduce the dimensionality of the matrices in equation \\ref{4}, top $k$ concepts are selected and thus \nmatrix $X$ is approximated as,\n\n\\begin{gather}\n X_k = L_k \\Sigma_k R_k^T\n\\end{gather}\n\n\nIn our study, $k$ is taken to be 400 in order to consider top 400 concepts. This \nmarks the end of the training phase. \n\nIn the testing phase, after generating \nweight matrix using the Term Frequency - Inverse Document Frequency (TF - IDF) \nmodel, input text gets converted into a query matrix, $Q$. This matrix $Q$ is then \nmultiplied with matrices $L_K$ and $\\Sigma_k$ to generate new query vectors calculated as follows:\n\n\\begin{gather}\n  New Query Vectors = Q L_k \\Sigma_k\n\\end{gather}\n\n\\subsubsection{Extraction of top features}\n\nThe query vectors are converted into bit arrays of size 400. \nThe indices of the top 40 features from the query vectors represent the '1's in the bit arrays and the indices of the remaining features represent '0's.\n\n\\subsection{Spatial Pooler}\nThe bit arrays from the LSI encoder are then passed to the Spatial Pooler for \nlearning. The Spatial Pooler gives similar Sparse Distributed Representations \n(SDRs) for similar input text. The major parameters of the Spatial Pooler which \nsignificantly affect the accuracy of our model are mentioned in Table \\ref{spatial_pooler_paramters}.\n\n\\medskip\n\n\\begin{table}[htbp]\n\\caption{Spatial Pooler Parameters}\n\\label{spatial_pooler_paramters}\n\\begin{center}\n\n\\begin{tabular}{|l|c|}\n\t\\hline\n\t\\textbf{Parameters} & \\textbf{Values}\\\\\n\t\\hline\n    inputDimensions & 400\\\\\n\t\\hline\n    columnDimensions & 20000\\\\\n\t\\hline\n    potentialRadius & 200\\\\\n\t\\hline\n    numActiveColumnsPerInhArea & 100\\\\\n\t\\hline\n    synPermActiveInc & 0.01\\\\\n\t\\hline\n    synPermInactiveDec & 0.008\\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\smallskip\nThe active indices of the SDR are then fed to the Classifier.\n\n\\subsection{Classifier}\nIn order to predict target class labels, a sequence of N-dimensional SDRs is \nassigned to a set of k class labels. The Classifier makes \nuse of a single layer feed forward neural network. In the figure \n\\ref{fig:classifier}, the number of output neurons is equal to the number of \npredefined categories. The number of input neurons is equal to the number of \nbits in any SDR.\n\n\\begin{figure}  \n\\includegraphics[width=0.9\\linewidth]{sdr_classifier.png}\n\\caption{\\sl Classifier\n\\label{fig:classifier}}\n\\end{figure} \n\nThe algorithmic description of the classifier is as follows,\n\\smallskip\n\\subsubsection{Matrix Initialisation}\nSince all classes have an equal chance of occurrence before learning, all \nvalues in the weight matrix are initialised to zero. \n\\smallskip\n\\subsubsection{Inference}\nIn this phase, the predicted class probabilities for each input pattern are \ncalculated. The calculations include two steps as mentioned below.\n\\smallskip\n\\setlength{\\parindent}{4ex}\n\n\\textit{i) Weighted sum :} Weighted sum of the input is calculated for each output neuron to determine the activation levels of the neuron. \nActivation level of an output neuron can be determined by the summation of the product of all the input bits to the input layer neurons with the weights of its corresponding connections to the output layer neuron. \nThe formula for the activation level being,\n\n\n\\begin{gather}\n a_j = \\sum_{i=1}^{N} w_{ij} \\times x_i \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  a_j & activation level of the $j^{th}$ output layer neuron \\\\\n  n & number of input layer neurons\\\\\n  \\eat{w_{ij} & weight of the connection from the $i^{th}$ input layer  \n\\\\ & neuron to the $j^{th}$ output layer neuron \\\\}  \n   \\multirow{2}{*}{$w_{ij}$} & weight of the connection from the $i^{th}$ input \nneuron \\\\\n  & to the $j^{th}$ output neuron. \\\\\n  x_i & Input bit value. It is either 0 or 1. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\par\n\n\\setlength{\\parindent}{4ex}\n\n\\textit{ii) Softmaxing the activation levels :} The probability distribution of \nthe categories is calculated by exponentiating and normalizing the activation \nlevels of the neurons in the output array using the softmax function. The \nformula for the probability distribution being,\n\n\\begin{gather}\n P\\left[ C_k | x, w\\right] = y_k = \\frac{e^{a_k}}{\\sum_{i=1}^{k} e^{a_i}} \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  y_k & Probability of predicting the category index $k$. \\\\\n  k & Number of predefined categories. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\par\n\n\\subsubsection{Learning}\nDuring each iteration, the classifier makes a prediction of the category index \nof a given SDR. This prediction is of the form of the probability distribution \nover different category indexes.\nThe connection weights are updated to learn and improve the prediction results. Connection weights are adjusted only for the active bits. \nThe Connection Weights are determined using maximum likelihood estimation (MLE) on independent \ninput SDRs. Since, the SDRs are independent of each other, they would satisfy \nthe following equation.\n\n\\begin{gather}\n P\\left[ z^1, z^2, ..., z^t\\right] =  \\prod_{t} P\\left( z^t | x^t, w\\right) \\\\\n x^t = (x^{t}_1, x^{t}_2, x^{t}_3, ..., x^{t}_N) \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  z^t & actual category index of $t^{th}$ SDR. \\\\\n  x^t & Sparse Distributed Representation. \\\\\n  x^{t}_1 & first bit of the $t^{th}$ SDR. \\\\\n  w & Connection weights. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nA value of $w$ is selected so that likelihood gets maximized. The loss function to select $w$ \\cite{neuneier2012train}, is as follows : \n\n\\begin{gather}\n L = -\\ln \\left(  \\prod_{t} p\\left(y_t | x_t, w\\right)\\right) \\\\\n= -\\Sigma_{t}\\ln P\\left(y_t | x_t, w\\right)\n\\end{gather}\n\nGradient descent is used is to minimize the loss function.\n\n\\begin{gather}\n\\frac{\\partial L}{\\partial w_{ij}} = \\frac{\\partial L}{\\partial a_j} \\times \n\\frac{\\partial a_j}{\\partial w_{ij}} \\\\\n=\\left(y_j - z_j\\right)x_i \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  y_j & Predicted probability of $j^{th}$ category index. \\\\\n  z_j & Actual probability of $j^{th}$ category index. \\\\\n  x_i & Input bit value to the $i^{th}$ input neuron. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nError in connection weight between $i^{th}$ input neuron to the $j^{th}$ output neuron is,\n\n\\begin{gather}\nError_{ij}\\ for\\ active\\ input\\ bits = y_j - z_j\n\\end{gather}\n\n\\begin{gather}\nupdate_{ij} = \\alpha \\times error_{ij} \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\alpha & Learning rate. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nThe value $update_{ij}$ is used to update the connection weight between the \n$i^{th}$ input neuron to the $j^{th}$ output neuron using the formula,\n\n\\begin{gather}\nw_{new_{ij}} = w_{old_{ij}} + update_{ij}\n\\eat{\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\alpha & Learning rate. \\\\\n  \\end{tabular}\\nonumber}\n\\end{gather}\n\nBut, if we just want to update connection weights for the bits which are active \nwe multiply $update_{ij}$ with $x_i$.\n\n\\begin{gather}\nw_{new_{ij}} = w_{old_{ij}} + update_{ij} \\times x_i\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  w_{new_{ij}} & updated weight of the connection from $i^{th}$ input \\\\\n  &  neuron to the $j^{th}$ output neuron \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nThe output layer neuron with the highest probability represents the category index of \nthe input text.\n\n\\section{Results}\nMany experiments were performed to test the accuracy and performance of \nour model. We selected two standard datasets for document classification,\nnamely, 20 Newsgroup dataset from the sklearn dataset repository and Movie \nReviews dataset from the NLTK corpus repository. The datasets were \nsplit into train set and test set in the ratio 9:1. The classification framework used in this study gives comparable accuracies with the models mentioned in the table \\ref{tp_rate} on the same datasets. \n\\smallskip\n\n\\begin{table}[htbp]\n\\caption{TRUE POSITIVE RATE}\n\\label{tp_rate}\n\\begin{center}\n\n\\begin{tabular}{ | m{6.5em} | m{2cm}| m{2cm} | } \n\n\\hline\n\\bfseries Classification Techniques & \\bfseries 20 newsgroup & \\bfseries Movie \nReviews \\\\ \n\\hline\nSVM\\cite{svmmr} & ---- & 84.40\\% \\\\ \n\\hline\nDecision Trees\\cite{decisiontreesmr} & ---- & 61.10\\% \\\\ \n\\hline\nNaive Bayes\\cite{naivebayes20,naivebayesmr} & 86.00\\% & 62.35\\% \\\\ \n\\hline\nB-Tree\\cite{btree20} & 82.64\\% & ---- \\\\ \n\\hline\nBayesian Networks \\cite{bayes_net}& 78.58\\% & ---- \\\\\n\\hline\nHTM & 83.19\\% & 73.60\\% \\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\smallskip\n\n\\section{Conclusion and Future Scope}\n\nThis paper puts forward the results of using the Hierarchical Temporary Memory \nmodel for document categorization. The results prove that the HTM model gives \nan accuracy comparable to the conventional techniques used for text \nclassification. \nThe number of columns and the SDR sparsity has a significant effect on the \nperformance of the spatial pooler. As per our model, The optimal values of the \nnumber of columns was 20,000 and the sparsity was 0.5\\%.\n\nThe main advantages of this model are: a limited number of parameters, can be \ntrained on small\\cite{synapse} corpus and faster training.\n\nIn future, we plan to modify the encoding process of our model and also \nincorporate the Temporal Pooler which can help to increase the accuracy of the \nmodel.\n\n\\section{Acknowledgement}\nWe are grateful to Mr. Nikhil Malhotra of Maker's Lab, Tech Mahindra Ltd. and Mr. Satish Kumbhar of College of Engineering, Pune, for guiding us through the research.\n\n\\nocite{*}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{s.1}}\n\n\nThe construction of diffusions on infinite-dimensional manifolds and the study of the regularity properties of their induced measures have been a topic of great interest for at least the past 50 years; see for example \\cite{DalettskiSnaiderman1969,Kuo1971,Kuo1972,Elworthy1975,FdlP1995,ADK2003,AM2006,Malliavin08}, although many other references exist.  The purpose of the present paper is to construct diffusions on a general class of infinite-dimensional nilpotent Lie groups, and to show that the associated heat kernel measures are quasi-invariant under  appropriate translations.  We demonstrate that this class of groups is quite rich. We focus here on the elliptic setting, but comment that, as nilpotent groups are standard first models for studying hypoellipticity, examples of infinite-dimensional versions of such spaces are important for the more general study of hypoellipticity in infinite dimensions.  This is an area of great interest, and is of particular relevance in the study of stochastic PDEs and their applications \\cite{AiraultMalliavin2004,BakhtinMattingly2007,BaudoinTeichmann2005,HairerMattingly2011,Malliavin1990,MattinglyPardoux2006}. The present paper studies heat kernel measures for elliptic diffusions on these spaces, which is a necessary precursor to understanding the degenerate case.\n\n\n\\subsection{Main results}\nLet $(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ denote an abstract Wiener\nspace, where $\\mathfrak{g}$ is a Banach space equipped with a\ncentered non-degenerate Gaussian measure $\\mu$ and associated\nCameron--Martin Hilbert space $\\mathfrak{g}_{CM}$.  We will assume\nthat $\\mathfrak{g}_{CM}$ additionally carries a nilpotent Lie algebra\nstructure $[\\cdot,\\cdot]: \\mathfrak{g}_{CM}\\times \\mathfrak{g}_{CM}\n\\rightarrow \\mathfrak{g}_{CM}$, and we will further assume that\nthis Lie bracket is Hilbert-Schmidt.  We will call such a space an {\\it\nabstract nilpotent Lie algebra}. Via the standard\nBaker-Campbell-Hausdorff-Dynkin formula, we may then equip\n$\\mathfrak{g}_{CM}$ with an explicit group operation under which\n$\\mathfrak{g}_{CM}$ becomes an infinite-dimensional group.  When\nthought of as a group, we will denote this space by $G_{CM}$.  \nWe equip $G_{CM}$ with the left-invariant Riemannian metric which\nagrees with the inner product on $\\mathfrak{g}_{CM}\\cong T_eG_{CM}$, and we denote the\nRiemannian distance by $d$.  \nNote that, despite the\nuse of the notation $\\mathfrak{g}$, we do not assume that the Lie bracket\nstructure extends to $\\mathfrak{g}$, and so this space is not necessarily a\nLie algebra or a Lie group.  Still, when $\\mathfrak{g}$ is playing a role\ntypically played by the group, we will denote $\\mathfrak{g}$ by $G$.\n\nIf $\\mathfrak{g}$ were a Lie algebra (and thus carried an associated group\nstructure), we could construct a Brownian motion on\n$G$ as the solution to the Stratonovich stochastic differential equation\n\\[ \\delta g_t = g_t\\delta B_t := L_{g_t*}\\delta B_t, \\text{ with }\n\tg_0=\\mathbf{e}=(0,0), \\]\nwhere $L_x$ is left translation by $x\\in G$ and $\\{B_t\\}_{t\\ge0}$ is a\nstandard Brownian motion on $\\mathfrak{g}$ (as a Banach space) with $\\mathrm{Law}(B_1)=\\mu$.\nIn finite dimensions, the solution to this stochastic differential equation may be obtained\nexplicitly as a formula involving the Lie bracket.  In particular, for $t>0$\nand $n\\in\\mathbb{N}$, let $\\Delta_n(t)$ denote the simplex in\n$\\mathbb{R}^n$ given by\n\\[ \\{s=(s_1,\\cdots,s_n)\\in\\mathbb{R}^n: 0<s_1<s_2<\\cdots<s_n<t\\}. \\]\nLet $\\mathcal{S}_n$ denote the permutation group on $(1,\\cdots,n)$,\nand, for each $\\sigma\\in \\mathcal{S}_n$, let $e(\\sigma)$ denote the\nnumber of ``errors'' in the ordering\n$(\\sigma(1),\\sigma(2),\\cdots,\\sigma(n))$, that is, $e(\\sigma)=\\#\n\\{j<n: \\sigma(j)>\\sigma(j+1)\\}$.\nThen the Brownian motion on $G$ could be written as \n\\begin{equation}\n\\label{e.ibm}\ng_t = \\sum_{n=1}^{r-1} \\sum_{\\sigma\\in\\mathcal{S}_n}  \n\t\\left( (-1)^{e(\\sigma)}\\bigg\/ n^2 \n\t\\begin{bmatrix} n-1 \\\\ e(\\sigma) \\end{bmatrix}\\right) \n\t\\int_{\\Delta_n(t)} \n\t[ [\\cdots[\\delta B_{s_{\\sigma(1)}},\\delta B_{s_{\\sigma(2)}}],\\cdots], \n\t\\delta B_{s_{\\sigma(n)}}],\n\\end{equation}\nwhere this sum is finite under the assumed nilpotence.  \nAn obstacle to the development of a general theory of stochastic differential equations on infinite-dimensional Banach spaces is the lack of smoothness of the norm in a general Banach space which is necessary to define a stochastic integral on it.\nStill, in Section \\ref{s.mult}, we prove a general result to define a class of iterated stochastic integrals with respect to Brownian motion on the Banach space $\\frak{g}$ that includes the expression above.  Additionally, we show that one may make\nsense of the above expression when the Lie bracket on $\\mathfrak{g}_{CM}$ does not necessarily extend to $\\mathfrak{g}$. Thus we are able to define a ``group Brownian motion'' $\\{g_t\\}_{t\\ge0}$ on $G$ via (\\ref{e.ibm}). We let $\\nu_t:=\\mathrm{Law}(g_t)$ be the heat kernel measure on $G$.\n\nIn particular, the integrals above are defined as a limit of stochastic integrals on finite-dimensional subgroups $G_\\pi$ of $G_{CM}$. We show that these $G_\\pi$ are nice  in the sense that they approximate $G_{CM}$ and that there exists a uniform lower bound on their Ricci curvatures.  \n\nUsing these results, we are able to prove the following main theorem.  \n\n\n\\begin{thm}\n\\label{t.quasi}\nFor $h\\in G_{CM}$, let $L_h,R_h:G_{CM}\\rightarrow G_{CM}$ denote left and\nright translation by $h$, respectively. Then $L_h$ and $R_h$ define measurable\ntransformations on $G$, and for all $T>0$, $\\nu_t\\circ L_h^{-1}$ and $\\nu_t\\circ\nR_h^{-1}$ are absolutely continuous with respect to $\\nu_t$.  Let \n\\[ J_t^l(h,\\cdot) := \\frac{d(\\nu_t\\circ L_h^{-1})}{d\\nu_t} \\qquad \\text{ and }\n\t\\qquad J_t^r(h,\\cdot) := \\frac{d(\\nu_t\\circ R_h^{-1})}{d\\nu_t} \\]\nbe the Radon-Nikodym derivatives, $k$ be the uniform lower bound on the Ricci\ncurvatures of the finite-dimensional approximation groups $G_\\pi$ \nand \n\\[ c(t) := \\frac{t}{e^t-1}, \\qquad \\text{ for all } t\\in\\mathbb{R}, \\]\nwith the convention that $c(0)=1$.  Then, for all $p\\in[1,\\infty)$,\n$J_t^l(h,\\cdot),J_t^r(h,\\cdot)\\in L^p(\\nu_t)$ and both satisfy the estimate\n\\[ \\|J_t^*(h,\\cdot)\\|_{L^p(\\nu_t)} \\le \\exp \\left( \\frac{c(kt)(p-1)}{2t}\n\td(\\mathbf{e},h)^2\\right), \\]\nwhere $*=l$ or $*=r$.\n\\end{thm}\n\nThe fact that one may define a measurable left or right action on $G$ by an\nelement of $G_{CM}$ is discussed in Section \\ref{s.mga}.  The lower bound on\nthe Ricci curvature is proved in Proposition \\ref{p.Ric}.\n\n\n\\subsection{Discussion}\n\nThe present paper builds on the previous work in \\cite{DriverGordina2008} and\n\\cite{Melcher2009}, significantly generalizing\nthese previous works in several ways.\nIn particular, the paper \\cite{Melcher2009} considered\nanalogous results for ``semi-infinite Lie groups'', which are \ninfinite-dimensional nilpotent Lie groups\nconstructed as extensions of finite-dimensional nilpotent Lie groups\n$\\mathfrak{v}$ by an\ninfinite-dimensional abstract Wiener space (see Example \\ref{ex.ext}).  \nAt several points in the analysis\nthere, the fact that $\\mathrm{dim}(\\mathfrak{v})<\\infty$ was used in a\ncritical way.  In particular, it was used to show that the stochastic\nintegrals defining the Brownian motion on $G$ as in equation\n(\\ref{e.ibm}) were well-defined.  \nIn the present paper, we have removed this restriction, as well as removing the\n``stratified'' structure implicit in the construction as a Lie group extension.\n\nAgain, we note that, despite the\nuse of the notation $\\mathfrak{g}$, it is not assumed that the Lie bracket\nstructure on $\\mathfrak{g}_{CM}$ extends to $\\mathfrak{g}$, and so $\\mathfrak{g}$ itself is not necessarily a\nLie algebra or Lie group.  In \\cite{DriverGordina2008} and \\cite{Melcher2009}, it\nwas assumed that the Lie bracket was a continuous map defined on\n$\\mathfrak{g}$.  However, it turns out that\nthe group construction on $\\mathfrak{g}_{CM}$ is  the only\nnecessary structure for the subsequent analysis.\nAs is usual for the infinite-dimensional setting,\nwhile the heat kernel measure is itself supported on the larger space\n$\\mathfrak{g}$, its critical analysis depends more on the structure of $\\mathfrak{g}_{CM}$.\nStill, as was originally done in \\cite{DriverGordina2008} and then in \\cite{Melcher2009},\none may instead define an abstract nilpotent Lie algebra starting \nwith a continuous nilpotent bracket\n$[\\cdot,\\cdot]:\\mathfrak{g}\\times\\mathfrak{g}\\rightarrow\\mathfrak{g}$.  For\nexample, in the event that $\\mathfrak{g}=W\\times\\mathfrak{v}$ where\n$\\mathfrak{v}$ is a finite-dimensional Lie algebra and\n$[\\cdot,\\cdot]:\\mathfrak{g}\\times\\mathfrak{g}\\rightarrow\\mathfrak{v}$, it is\nwell-known that this implies that the restriction of the bracket to\n$\\mathfrak{g}_{CM}=H\\times\\mathfrak{v}$ is Hilbert-Schmidt.  (For\nany continuous bilinear $\\omega:W\\times W\\rightarrow K$ where $K$ is a Hilbert\nspace, one has that $\\|\\omega\\|_{(H^{\\otimes 2})^*\\otimes K}<\\infty$; this\nfollows for example from Corollary 4.4 of \\cite{Kuo75}.)\nMore generally, in order for the subsequent theory to make sense, one would\nnaturally need\nto require that $\\mathfrak{g}_{CM}$ be a Lie subalgebra of $\\mathfrak{g}$, that is, for\nthe restriction of the Lie bracket to $\\mathfrak{g}_{CM}$ to preserve\n$\\mathfrak{g}_{CM}$.  As the proofs in the sequel rely strongly on the bracket\nbeing Hilbert-Schmidt, it would be then necessary to add the Hilbert-Schmidt hypothesis as it \ndoes not follow immediately if one only assumes a continuous bracket on\n$\\mathfrak{g}$ which preserves $\\mathfrak{g}_{CM}$.  \n\n\n\n\nAdditionally, the spaces studied in the present paper are well-designed for the study of\ninfinite-dimensional hypoelliptic heat kernel measures, and there has already\nbeen progress on proving quasi-invariance and stronger smoothness properties\nfor these measures in the simplest case of a step two Lie algebra with\nfinite-dimensional center; see \\cite{BGM2013} and\n\\cite{DEM2013}. More generally, the paper \\cite{Pickrell2011} explores related interesting lines of inquiry for heat kernel measures on infinite-dimensional groups, largely in the context of groups of maps from manifolds to Lie groups.\n\n\n\n{\\it Acknowledgements. } The author thanks Bruce Driver and Nathaniel Eldredge for\nhelpful conversations during the writing of this paper.\n\n\n\n\\section{Iterated It\\^o integrals}\n\\label{s.mult}\n\nRecall the standard construction of abstract Wiener spaces.  \nSuppose that $W$ is a real separable Banach space and $\\mathcal{B}_{W}$ is\nthe Borel $\\sigma$-algebra on $W$.\n\n\\begin{defn}\n\\label{d.2.1} \nA measure $\\mu$ on $(W,\\mathcal{B}_{W})$ is called a (mean zero,\nnon-degenerate) {\\em Gaussian measure} provided that its characteristic\nfunctional is given by\n\\[\n\\hat{\\mu}(u) := \\int_W e^{iu(x)} d\\mu(x)\n\t= e^{-\\frac{1}{2}q(u,u)}, \\qquad \\text{ for all } u\\in W^*,\n\\]\nfor $q=q_\\mu:W^*\\times W^*\\rightarrow\\mathbb{R}$ a symmetric, positive\ndefinite quadratic form.\nThat is, $q$ is a real inner product on $W^*$.\n\\end{defn}\n\n\n\n\\begin{thm}\n\\label{t.2.3}\nLet $\\mu$ be a Gaussian measure on $W$.  \nFor $w\\in W$, let\n\\[ \n\\|w\\|_H := \\sup\\limits_{u\\in W^*\\setminus\\{0\\}}\\frac{|u(w)|}{\\sqrt{q(u,u)}},\n\\]\nand define the {\\em Cameron--Martin subspace} $H\\subset W$ by\n\\[ H := \\{h\\in W : \\|h\\|_H < \\infty\\}. \\]\nThen $H$ is a dense subspace of $W$, and there exists a unique inner\nproduct $\\langle\\cdot,\\cdot\\rangle_H$ on $H$ such that $\\|h\\|_H^2\n= \\langle h,h\\rangle_H$ for all $h\\in H$, and $H$ is a separable\nHilbert space with respect to this inner product.  For any $h\\in\nH$, $\\|h\\|_W \\le C \\|h\\|_H$ for some $C<\\infty$.  \n\\end{thm}\n\nAlternatively, given $W$ a\nreal separable Banach space and $H$ a real separable Hilbert space \ncontinuously embedded in $W$ as a dense subspace, then for each $w^*\\in W^*$ there\nexists a unique $h_{w^*}\\in H$ such that $\\langle h,w^*\\rangle = \\langle h,\nh_{w^*}\\rangle_H$ for all $h\\in H$.  Then $W^*\\ni w^*\\mapsto h_{w^*}\\in\nH$ is continuous, linear, and one-to-one with a dense range\n\\begin{equation}\n\\label{e.H*} \nH_* := \\{h_{w^*}:w^*\\in W^*\\},\n\\end{equation}\nand $W^*\\ni w^*\\mapsto h_{w^*}\\in\nW$ is continuous. A Gaussian measure on $W$ is a Borel\nprobability measure $\\mu$ such that, for each $w^*\\in W^*$, the random\nvariable $w\\mapsto\\langle w,w^*\\rangle$ under $\\mu$ is a centered Gaussian\nwith variance $\\|h_{w^*}\\|_H^2$.\n\n\n\n\nSuppose that $P:H\\rightarrow H$ is a finite rank orthogonal projection\nsuch that $PH\\subset H_*$. Let $\\{h_j\\}_{j=1}^m$ be an orthonormal basis for\n$PH$.  Then we may extend $P$ to a (unique) continuous operator\nfrom $W\\rightarrow H$ (still denoted by $P$) by letting\n\\begin{equation}\n\\label{e.proj}\nPw := \\sum_{j=1}^m \\langle w, h_j\\rangle_H  h_j\n\\end{equation}\nfor all $w\\in W$.  \n\\begin{nota}\n\\label{n.proj}\nLet $\\mathrm{Proj}(W)$ denote the collection of finite rank projections\non $W$ such that $PW\\subset H_*$ and $P|_H:H\\rightarrow H$ is an orthogonal\nprojection, that is, $P$ has the form given in equation\n\\eqref{e.proj}.\n\\end{nota}\n\nLet $\\{B_t\\}_{t\\ge0}$ be a Brownian motion on $W$ with variance determined by\n\\[\n\\mathbb{E}\\left[\\langle B_s,h\\rangle\n\t\t_{H} \n\t\t\\langle B_t,k\\rangle\n\t\t_H\\right] \n\t= \\langle h,k \\rangle_{H} \\min(s,t),\n\\]\nfor all $s,t\\ge0$ and $h,k\\in H_*$, where $H_*$ is as in (\\ref{e.H*}).  Note that for any $P\\in\\mathrm{Proj}(W)$, $PB$ is a Brownian motion on $PH$. In the rest of this section, we will verify the existence of martingales defined as certain iterated stochastic integrals with respect to $B_t$.\n\n\nThe following is Proposition 4.1 of \\cite{Melcher2009}.  Note that again this\nwas stated in the context where $H=\\mathfrak{g}_{CM}$ was a ``semi-infinite Lie algebra'', but \na brief inspection of the proof shows that this is a general statement about stochastic\nintegrals on Hilbert spaces.\n\n\\begin{prop}\n\\label{p.int1}\nLet $\\{P_m\\}_{m=1}^\\infty\\subset\\mathrm{Proj}(W)$ such that $P_m|_H\\uparrow\nI_H$.  Then, for $\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$ a continuous \nmapping, let\n\\begin{align*}\nJ_n^m(\\xi)_t &:= \\int_{\\Delta_n(t)} \\langle P_m^{\\otimes n} \\xi(s), dB_{s_1}\n\t\t\\otimes\\cdots\\otimes dB_{s_n}\n\t\t\\rangle_{H^{\\otimes n}} \\\\\n\t&= \\int_{\\Delta_n(t)} \\langle \\xi(s), dP_m B_{s_1}\n\t\\otimes\\cdots\\otimes dP_m B_{s_n}\n\t\\rangle_{H^{\\otimes n}}. \n\\end{align*}\nThen $\\{J_n^m(\\xi)_t\\}_{t\\ge0}$ is a continuous $L^2$-martingale, \nand there exists a\ncontinuous $L^2$-martingale $\\{J_n(\\xi)_t\\}_{t\\ge0}$ such that\n\\begin{equation} \n\\label{e.Jnm}\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t|J_n^m(\\xi)_\\tau-J_n(\\xi)_\\tau|^2 \\right] = 0\n\\end{equation}\nand  \n\\begin{equation} \n\\label{e.xi}\n\\mathbb{E}|J_n(\\xi)_t|^2\\le\n\t\\|\\xi\\|^2_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\\end{equation}\nfor all $t<\\infty$.  The process\n$J_n(\\xi)$ is well-defined independent of the choice of increasing\northogonal projections $\\{P_m\\}_{m=1}^\\infty$ into $H_*$, and so will be denoted by\n\\[ J_n(\\xi)_t \n\t= \\int_{\\Delta_n(t)} \\langle \\xi(s), dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\n\t\\rangle_{H^{\\otimes n}}.\n\\]\n\\end{prop}\n\n\\iffalse\n\\begin{proof} \nNote first that, \n\\[ J_n^m(\\xi)_t = \\sum_{i_1,\\ldots,i_n=1}^m \n\t\\int_{\\Delta_n(t)} \\langle\\xi(s),h_{i_1}\\otimes\\cdots\\otimes\n\th_{i_n}\\rangle_{H^{\\otimes n}}\n\tdB_{s_1}^{i_1}\\cdots dB_{s_n}^{i_n} \n\\]\nwhere $\\{B^i\\}_{i=1}^m$ are independent real\nvalued Brownian motions.  Let $\\xi_{i_1,\\ldots,i_n} :=\n\\langle\\xi,h_{i_1}\\otimes\\cdots\\otimes h_{i_n}\\rangle$.  Then\n\\[ |\\xi_{i_1,\\ldots,i_n}(s) |^2\\le \\|\\xi(s)\\|^2\n\t_{H^{\\otimes n}} \\]\nand $\\xi_{i_1,\\ldots,i_n} \\in L^2(\\Delta_n(t))$.  Thus, $J_n^m(\\xi)_t$ is \ndefined as a\n(finite dimensional) vector-valued multiple Wiener-It\\^o integral, see for\nexample \\cite{Ito51,Shigekawa04}.\n\nNow note that\n\\begin{align*} \ndJ&_n^m(\\xi)_t \n\t= \\int_{\\Delta_{n-1}(t)} \\langle \\xi(s_1,\\ldots,s_{n-1},t),\n\t\tdP_mB_{s_1}\\otimes\\cdots\\otimes dP_mB_{s_{n-1}} \\otimes dP_mB_t\n\t\t\\rangle_{H^{\\otimes n}} \\\\\n\t&= \\sum_{i=1}^m \\int_{\\Delta_{n-1}(t)} \\langle \\xi(s_1,\\ldots,s_{n-1},t),\n\t\tdP_mB_{s_1}\\otimes\\cdots\\otimes dP_mB_{s_{n-1}} \\otimes h_i\n\t\t\\rangle_{H^{\\otimes n}} dB_t^i.\n\\end{align*}\nThus, the quadratic variation $\\langle J_n^m(\\xi) \\rangle_t$ is given by\n\\[ \\sum_{i=1}^m \\int_0^t \n\t\\bigg|\\int_{\\Delta_{n-1}(\\tau)}\n\t\\langle \\xi(s_1,\\ldots,s_{n-1},\\tau),\n\tdP_mB_{s_1}\\otimes\\cdots\\otimes\n\tdP_mB_{s_{n-1}}\\otimes h_i \\rangle\n\t_{H^{\\otimes n}}\\bigg|^2 d\\tau,\n\\]\nand\n\\begin{align*}\n\\mathbb{E}&|J_n^m(\\xi)_t|^2 \n\t= \\mathbb{E} \\langle J_n^m(\\xi) \\rangle_t \\\\\n\t&= \\sum_{i_1=1}^m \\int_0^t \n\t\t\\mathbb{E}\\bigg[\\sum_{i_2=1}^m \\int_0^{\\tau_1}\n\t\t\\bigg|\\int_{\\Delta_{n-2}(\\tau_2)}\n\t\t\\langle \\xi(s_1,\\ldots,s_{n-2},\\tau_2,\\tau_1),\n\t\tdP_mB_{s_1}\\otimes\\cdots \\\\\n\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \n\t\t\\cdots \\otimes dP_mB_{s_{n-2}}\n\t\t\\otimes h_{i_2}\\otimes h_{i_1}\\rangle\n\t\t_{H^{\\otimes n}}\\bigg|^2d\\tau_2 \\bigg]d\\tau_1.\n\\end{align*}\nIterating this procedure $n$ times gives\n\\begin{align*}\n\\mathbb{E}|J_n^m(\\xi)_t|^2 \n\t&= \\sum_{i_1,\\ldots,i_n=1}^m \\int_{\\Delta_n(t)} \n\t\t\\left|\\langle \\xi(\\tau_1,\\cdots,\\tau_n), \n\t\th_{i_1}\\otimes\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2 d\\tau_1\\cdots d\\tau_n \\\\\n\t&\\notag\n\t= \\int_{\\Delta_n(t)} \\|P_m^{\\otimes n} \\xi(s) \\|\n\t\t_{H^{\\otimes n}}^2\n\t\\le \\|\\xi \\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2,\n\\end{align*}\nand thus, for each $n$, $J_n^m(\\xi)_t$ is bounded uniformly in $L^2$\nindependent of $m$.  \n\n\nNow, for $P\\in\\mathrm{Proj}(W)$, let \n$\nJ_n^P(\\xi)_t\n\t:= \\int_{\\Delta_n(t)} \\langle P^{\\otimes n} \\xi(s), d B_{s_1}\n\t\\otimes\\cdots\\otimes d B_{s_n}\n\t\\rangle_{H^{\\otimes n}}. \n$\nFor $P,Q\\in\\mathrm{Proj}(W)$, a similar argument to the above implies that \n\\begin{equation}\n\\label{e.gub}\n\\mathbb{E}|J_n^P(\\xi)_t - J_n^Q(\\xi)_t|^2 \t\n\t= \\int_{\\Delta_n(t)} \\|P^{\\otimes n} \\xi(s) - Q^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}}\\,ds.\n\\end{equation}\nIn particular, take $P=P_m$ and $Q=P_\\ell$ with $\\ell\\le m$, and note that\n\\begin{align}\n\\notag\n&\\|P_m^{\\otimes n} \\xi(s) - P_\\ell^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}} \\\\\n\t&\\notag\\quad= \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\sum_{j=1}^n \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-P_\\ell)\\otimes P_\\ell^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\label{e.gab}\n\t\\quad\\le n  \\sum_{j=1}^n \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-P_\\ell)\\otimes P_\\ell^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\notag\n\t\\quad= n \\sum_{j=1}^n \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=\\ell+1}^m\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\ell \n\t\t\\left|\\langle \\xi(s), h_{i_1}\\otimes\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2 \n\t\\rightarrow 0,\n\\end{align}\nas $\\ell,m\\rightarrow\\infty$, for all $s\\in\\Delta_n(t)$, since \n$\\|\\xi\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2<\\infty$.\nThus, equation (\\ref{e.gub}) implies that\n\\[ \\lim_{\\ell,m\\rightarrow \\infty}\n\t\\mathbb{E}\\left|J_n^m(\\xi)_t - J_n^\\ell(\\xi)_t \\right|^2\n\t= 0, \\]\nby dominated convergence, and $\\left\\{J_n^m(\\xi)_t\\right\\}_{m=1}^\\infty$ \nis Cauchy in $L^2$.  \nSince the space of continuous $L^2$-martingales is complete in the\nnorm $M\\mapsto \\mathbb{E}|M_t|^2$, there exists a continuous martingale\n$\\{J_n(\\xi)_t\\}_{t\\ge0}$ such that\n\\[\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}|J_n^m(\\xi)_t-J_n(\\xi)_t|^2 = 0. \n\\]\nCombining this with Doob's\nmaximal inequality proves equation (\\ref{e.Jnm}).\n\n\nTo see that $J_n(\\xi)_t$ is independent of the choice of basis, suppose now that\n$\\{h'_j\\}_{j=1}^\\infty\\subset H_*$ is another orthonormal\nbasis for $H$, and let $P_m':W\\rightarrow H_*$ \nbe the corresponding orthogonal projections.\nConsider the inequality (\\ref{e.gab}) with $P_\\ell$ replaced by $P_m'$.\nWriting $P_m-P_m'=(P_m-I) + (I-P_m')$, and \nconsidering terms for each fixed $j$, we have \n\\begin{align*}\n& \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-I)\\otimes(P_m')^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&= \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=m+1}^\\infty\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\infty\n\t\t\\bigg|\\langle \\xi(s), h_{i_1}\\otimes\n\t\t\\cdots \\otimes h_{i_j} \n\t\t\\otimes P_m' h_{i_{j+1}}\n\t\t\\otimes\\cdots\\otimes P_m' h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\le \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=m+1}^\\infty\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\infty\n\t\t\\left|\\langle \\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2\n\t\\rightarrow 0,\n\\end{align*}\nas $m\\rightarrow\\infty$.  Similarly,\n\\begin{align*}\n&\\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(I-P_m')\\otimes(P_m')^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\quad= \\sum_{i_1,\\ldots,i_n=1}^\\infty\t\n\t\t\\bigg|\\langle P_m^{\\otimes j-1} \\otimes \n\t\t(I-P_m')\\otimes(P_m')^{n-j-1}\\xi(s),  h_{i_1}'\\otimes\n\t\t\\cdots\\otimes h_{i_n}' \\rangle\n\t\t_{H^{\\otimes n}} \\bigg|^2 \n\t\\rightarrow 0,\n\\end{align*}\nas $m\\rightarrow\\infty$.  Thus,\n\\[ \\lim_{m\\rightarrow\\infty} \\|P_m^{\\otimes n} \\xi(s) - (P_m')^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}} = 0, \\]\nfor each $s\\in\\Delta_n(t)$. \nThus, for $J_n^{m'}(\\xi)_t := J_n^{P_m'}(\\xi)_t$,\nusing equation (\\ref{e.gub}) with $P=P_m$ and $Q=P_m'$ shows that\n\\[ \\lim_{m\\rightarrow\\infty} \\mathbb{E}|J_n^m(\\xi)_t \n\t- J_n^{m'}(\\xi)_t |^2 = 0, \\]\nagain by dominated convergence.  \n\\end{proof}\n\\fi\n\n\n\n\nNow we may use this result to define stochastic integrals taking values in\nanother Hilbert space $K$.\n\n\\begin{prop}\n\\label{p.int2}\nLet $K$ be a Hilbert space and $F\\in L^2(\\Delta_n(t),(H^{\\otimes\nn})^*\\otimes K)$ be a continuous map.  That is, $F:\\Delta_n(t)\\times H^{\\otimes\nn}\\rightarrow K$ is a map continuous in $s$ and linear on $H^{\\otimes n}$ such that\n\\[ \\int_{\\Delta_n(t)} \\|F(s)\\|_2^2\\,ds \n\t= \\int_{\\Delta_n(t)} \\sum_{j_1,\\ldots,j_n=1}^\\infty\n\\|F(s)(h_{j_1}\\otimes\\cdots\\otimes h_{j_n})\\|_K^2\\,ds <\\infty. \\]\nThen\n\\[ J_n^m(F)_t := \\int_{\\Delta_n(t)} F(dP_m B_{s_1}\n\t\\otimes\\cdots\\otimes dP_m B_{s_n}) \n\\]\nis a continuous $K$-valued $L^2$-martingale, and there exists a continuous $K$-valued $L^2$-martingale\n$\\{J_n(F)_t\\}_{t\\ge0}$ such that\n\\begin{equation}\n\\label{e.intF}\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\|J_n^m(F)_\\tau-J_n(F)_\\tau\\|_{K}^2 \\right] = 0,\n\\end{equation}\nfor all $t<\\infty$.  The martingale $J_n(F)$ is well-defined independent of \nthe choice of orthogonal projections, \nand thus will be denoted by\n\\[\nJ_n(F)_t = \\int_{\\Delta_n(t)} F(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}).\n\\]\n\\end{prop}\n\n\\begin{proof}\nLet $\\{e_j\\}_{j=1}^{\\infty}$ be an orthonormal basis of $K$. \nSince $\\langle F(s)(\\cdot), e_j \\rangle$ is linear on\n$H^{\\otimes n}$, for each $s$ there exists\n$\\xi_j(s)\\in H^{\\otimes n}$ such that\n\\begin{equation}\n\\label{e.xij} \n\\langle \\xi_j(s), k_1\\otimes\\cdots\\otimes k_n \\rangle \n\t= \\langle F(s)(k_1\\otimes\\cdots\\otimes k_n), e_j \\rangle. \n\\end{equation}\nIf $\\xi_j:\\Delta_n(t)\\rightarrow H^{\\otimes n}$ \nis defined by equation (\\ref{e.xij}), \nthen clearly $\\xi_j\\in L^2(\\Delta_n(t),H^{\\otimes n})$ and in particular\n\\begin{align*}\n\\|F\\|_{L^2(\\Delta_n(t)\\times H^{\\otimes n},K)}^2\n\t&=  \\sum_{j=1}^\\infty \\|\\xi_j\\|_{L^2(\\Delta_n(t),H^{\\otimes n})} <\\infty.\n\\end{align*}\nThus, for $J_n(\\xi_j)$ as defined in Proposition \\ref{p.int1},\n\\begin{align*}\n\\mathbb{E}\\left[ \\sum_{j=1}^\\infty |J_n(\\xi_j)_t|^2\\right] \n\t&\\le \\frac{t^n}{n!}\\mathbb{E}\\left[ \\int_{\\Delta_n(t)} \\sum_{j=1}^\\infty |\\langle \\xi_j(s),\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\\rangle_{H^{\\otimes n}}|^2\n\t\t\\right] \\\\\n\t&= \\frac{t^n}{n!}\\sum_{j=1}^\\infty \\|\\xi_j\\|^2_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\t<\\infty,\n\\end{align*}\nand so we may write\n\\begin{align*} \n\\sum_{j=1}^\\infty J_n(\\xi_j)_t e_j \n\t&= \\sum_{j=1}^\\infty \\int_{\\Delta_n(t)} \n\t\t\\langle \\xi_j(s),\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\\rangle_{H^{\\otimes n}} e_j  \\\\\n\t&= \\int_{\\Delta_n(t)} \\sum_{j=1}^\\infty\n\t\t\\langle F(s)(\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}),e_j\\rangle_K e_j \\\\\n\t&= \\int_{\\Delta_n(t)} \n\t\tF(s)(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}).\n\\end{align*}\nThus, taking $J_n(F)_t := \\sum_{j=1}^\\infty J_n(\\xi_j)_t e_j$,\nwe also have that \n\\begin{align*}\n\\mathbb{E}\\|J_n(F)_t - J_n^m(F)_t\\|_K^2\n\t&= \\mathbb{E}\\left[\\sum_{j=1}^\\infty\n\t\t|J_n(\\xi_j)_t-J_n^m(\\xi_j)_t|^2\\right]\n\t\\rightarrow 0\n\\end{align*}\nas $m\\rightarrow\\infty$ by (\\ref{e.Jnm}) and dominated convergence since\n\\[ \\mathbb{E}|J_n(\\xi_j)_t-J_n^m(\\xi_j)_t|^2 \n\t\\le 4\\|\\xi_j\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2 \\]\nby (\\ref{e.xi}).  Then equation (\\ref{e.intF}) holds by Doob's maximal\ninequality.\n\\end{proof}\n\nNote that the preceding results then imply that one may define the above\nstochastic integrals with respect to {\\it any} increasing sequence of\northogonal projections -- that is, we need not require that the projections\nextend continuously to $W$.\n\n\n\\begin{prop}\n\\label{p.bad}\nLet $V$ be an arbitrary finite-dimensional subspace of $H$, and let\n$\\pi:H\\rightarrow V$ denote orthogonal projection onto $V$.  Then for any\nHilbert space $K$ and  $F\\in L^2(\\Delta_n(t),(H^{\\otimes\nn})^*\\otimes K)$ a continuous map, the\nstochastic integral \n\\[ J_n^\\pi(F)_t := \\int_{\\Delta_n(t)} F(d\\pi B_{s_1}\n\t\\otimes\\cdots\\otimes d\\pi B_{s_n}) \n\\]\nis well-defined, and $\\{J_n^\\pi(F)_t\\}_{t\\ge0}$\nis a continuous $K$-valued $L^2$-martingale.  Moreover, if $V_m$ is an increasing sequence of\nfinite-dimensional subspaces of $H$\nsuch that the corresponding orthogonal projections $\\pi_m\\uparrow I_H$, then \n\\begin{equation*} \n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\|J_n^{\\pi_m}(F)_\\tau-J_n(F)_\\tau\\|^2 \\right] = 0,\n\\end{equation*}\nwhere $J_n(F)$ is as defined in Proposition \\ref{p.int2}.\n\\end{prop}\n\n\\begin{proof}\nFirst consider the case that $K=\\mathbb{R}$, and thus $F(s)=\\langle\n\\xi(s),\\cdot\\rangle$ for a continuous $\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$.\nSince $\\pi^{\\otimes n}\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$, the definition\nof $J_n^\\pi(\\xi)=J_n(\\pi^{\\otimes n}\\xi)$ follows from Propostion \\ref{p.int1}.  Moreover, by equation\n(\\ref{e.xi}), \n\\begin{align*}\n\\mathbb{E}|J_n^{\\pi_m}(\\xi)_t - J_n(\\xi)_t|^2\n\t&= \\mathbb{E}|J_n(\\pi_m^{\\otimes n}\\xi)_t - J_n(\\xi)_t|^2 \n\t= \\mathbb{E}|J_n(\\pi_m^{\\otimes n}\\xi-\\xi)_t|^2 \\\\\n\t&\\le \\|\\pi_m^{\\otimes n}\\xi-\\xi\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\t\\rightarrow 0\n\\end{align*}\nas $m\\rightarrow\\infty$.  Now the proof for general $F$ follows just as in\nProposition \\ref{p.int2}.\n\\end{proof}\n\n\n\n\n\\section{Abstract nilpotent Lie algebras and groups}\n\\label{s.prelim}\n\n\n\n\\begin{defn}\n\\label{d.semi}\nLet $(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ be an abstract\nWiener space such that $\\mathfrak{g}_{CM}$ is equipped with a nilpotent\nHilbert-Schmidt Lie bracket.  Then we will call\n$(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ an {\\it abstract nilpotent Lie\nalgebra}.\n\\end{defn}\n\n\nThe Baker-Campbell-Hausdorff-Dynkin formula implies that\n\\[ \\log(e^A e^B) = A+B+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_A^{n_1} \\mathrm{ad}_B^{m_1} \\cdots\n\t\t\\mathrm{ad}_A^{n_k} \\mathrm{ad}_B^{m_k} A,\n\\]\nfor all $A,B\\in\\mathfrak{g}_{CM}$, where \n\\begin{equation*} \na_{n,m}^k := \\frac{(-1)^k}{(k+1)m!n!(|n|+1)}, \n\\end{equation*}\n$\\mathcal{I}_k := \\{(n,m)\\in\\mathbb{Z}_+^k\\times\\mathbb{Z}_+^k : \nn_i+m_i>0 \\text{ for all } 1\\le i\\le k \\}$, and for each multi-index\n$n\\in\\mathbb{Z}_+^k$,\n\\[ n!= n_1!\\cdots n_k! \\quad \\text{ and } \\quad |n|=n_1+\\cdots+n_k; \\]\nsee, for example, \\cite{duiskolk}.  If $\\mathfrak{g}_{CM}$ is nilpotent of step\n$r$, then\n\\[ \\mathrm{ad}_A^{n_1} \\mathrm{ad}_B^{m_1} \\cdots\n\t\t\\mathrm{ad}_A^{n_k} \\mathrm{ad}_B^{m_k} A = 0 \\quad\n\\text{if } |n|+|m|\\ge r. \\]\nfor $A,B\\in\\mathfrak{g}_{CM}$.  \nSince $\\mathfrak{g}_{CM}$ is simply connected and nilpotent, \nthe exponential map is a global diffeomorphism (see, for\nexample, Theorems 3.6.2 of \\cite{Varadarajan} or 1.2.1 of \\cite{CorGrn90}). \nIn particular, we may view $\\mathfrak{g}_{CM}$ as both a Lie algebra and Lie group, and \none may verify that\n\\begin{align}\n\\label{e.mult}\ng\\cdot h &= g+h+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_g^{n_k} \\mathrm{ad}_h^{m_k} g\n\\end{align}\ndefines a group structure on $\\mathfrak{g}_{CM}$.  Note that $g^{-1}=-g$ and \nthe identity $\\mathbf{e}=(0,0)$.\n\n\\begin{defn}\nWhen we wish to emphasize the group structure on $\\mathfrak{g}_{CM}$, we will\ndenote $\\mathfrak{g}_{CM}$ by $G_{CM}$.\n\\end{defn}\n\n\n\n\n\\begin{lem}\n\\label{l.cts}\nThe Banach space topology on $\\mathfrak{g}_{CM}$ makes $G_{CM}$ into a topological group.\n\\end{lem}\n\\begin{proof}\nSince $\\mathfrak{g}_{CM}$ is a topological vector space, \n$g\\mapsto g^{-1}=-g$ and $(g_1,g_2)\\mapsto g_1+g_2$ are continuous by\ndefinition.  The map $(g_1,g_2)\\mapsto [g_1,g_2]$ is continuous in \nthe $\\mathfrak{g}_{CM}$ topology by the boundedness of the Lie bracket.\nIt then follows from (\\ref{e.mult}) that $(g_1,g_2)\\mapsto g_1\\cdot g_2$\nis continuous as well.\n\\end{proof}\n\n\n\n\\label{s.mga}\\subsection{Measurable group actions on $G$}\n\nAs discussed in the introduction, given a Hilbert-Schmidt Lie bracket on\n$\\mathfrak{g}_{CM}$ and a subsequently defined group operation on $G_{CM}$,\none may define a measurable action on $G$ by left or right multiplication by\nan element of $G_{CM}$.  \n\nIn particular, let $\\{e_n\\}_{n=1}^\\infty$ be an orthonormal basis of $\\mathfrak{g}_{CM}$.  \nFor now, fix $n$ and consider the mapping $\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}^*$ given by $h\\mapsto \\langle\n\\mathrm{ad}_h\\cdot,e_n\\rangle$.  Then this is a continuous linear map on $\\mathfrak{g}_{CM}$\nand in the usual way we may make the identification of\n$\\mathfrak{g}_{CM}^*\\cong \\mathfrak{g}_{CM}$ so that we\ndefine the operator $A_n:\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}$ given by\n\\[ \\langle A_nh, k\\rangle = \\langle \\mathrm{ad}_h k,e_n\\rangle; \\]\nin particular, $A_nh=\\mathrm{ad}_h^*e_n$.  Note that, for any $h,k\\in \\mathfrak{g}_{CM}$\n\\[ \\langle A_n^*h,k\\rangle  \n\t= \\langle A_nk,h\\rangle \n\t= \\langle \\mathrm{ad}_k^* e_n,h\\rangle\n\t= \\langle e_n , \\mathrm{ad}_k h \\rangle\n\t= - \\langle e_n , \\mathrm{ad}_h k \\rangle\n\t= - \\langle \\mathrm{ad}_h^* e_n,k\\rangle \\]\nand thus $A_n^*=-A_n$.\nNow  fix $h\\in G_{CM}=\\mathfrak{g}_{CM}$.  Then for $\\mathrm{ad}_h:\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}$ we may write\n\\[\n\\mathrm{ad}_h k = \\sum_n \\langle \\mathrm{ad}_hk, e_n\\rangle e_n\n\t= \\sum_n \\langle A_nh,k\\rangle e_n.\n\\]\nSince each $\\langle A_nh,\\cdot\\rangle\\in \\mathfrak{g}_{CM}^*$ has a measurable linear\nextension to $G$ such that $\\|\\langle\nA_nh,\\cdot\\rangle\\|_{L^2(\\mu)} = \\|A_nh\\|_{\\mathfrak{g}_{CM}}$ \n(see, for example, Theorem 2.10.11 of \\cite{Bogachev1998}), we may extend\n$\\mathrm{ad}_h$ to a measurable linear transformation from $G=\\mathfrak{g}$\nto $G_{CM}=\\mathfrak{g}_{CM}$ (still denoted by $\\mathrm{ad}_h$) given by\n\\[ \n\\mathrm{ad}_h g := \\sum_n \\langle A_nh,g\\rangle e_n. \\]\nNote that here we are using the fact that\n\\begin{align*} \\sum_n \\|\\langle A_nh,\\cdot\\rangle\\|_{L^2(\\mu)}^2\n\t= \\sum_n \\| A_nh\\|_{H}^2\n\t&= \\sum_{n,m} \\langle A_nh,e_m\\rangle^2 \\\\\n\t&= \\sum_{n,m} \\langle \\mathrm{ad}_{h}e_m,e_n\\rangle^2 \n\t\\le \\|h\\|^2 \\|[\\cdot,\\cdot]\\|_{HS}^2\n\\end{align*}\nwhich implies that\n\\[ \\sum_n \\langle A_nh,g\\rangle ^2 < \\infty \\quad g\\text{-a.s.} \\]\nSimilarly, we may define\n\\[ \\mathrm{ad}_g h := -\\mathrm{ad}_h g\n\t= - \\sum_n \\langle A_nh,g\\rangle e_n. \\]\n\nIn a similar way, note that we may write, for $h,k\\in G_{CM}$ and $m< r$,\n\\begin{align*} \n\\mathrm{ad}_h^m k \n\t&= \\sum_{\\ell_1}\\cdots \\sum_{\\ell_m} \\left(\\prod_{b=1}^{m-1} \n\t\t\\langle A_{\\ell_{b}}h, e_{\\ell_{b+1}}\\rangle  \\right)\\langle\n\t\tA_{\\ell_m}h,k\\rangle e_{\\ell_1} \\\\\n\t&= (-1)^m \\sum_{\\ell_1}\\cdots \\sum_{\\ell_m} \\left(\\prod_{b=1}^{m-1} \n\t\t\\langle A_{\\ell_{b}}e_{\\ell_{b+1}},h \\rangle  \\right)\\langle\n\t\tA_{\\ell_m}k,h\\rangle e_{\\ell_1}.\n\\end{align*}\nand thus for $h,k,x\\in G_{CM}$ and $n+m< r$\n\\begin{multline*} \n\\mathrm{ad}_k^n \\mathrm{ad}_h^m x \n\t= (-1)^n \\sum_{j_1}\\cdots \\sum_{j_n}\n\t\t\\sum_{\\ell_1}\\cdots\\sum_{\\ell_m} \n\t\t\\left(\\prod_{a=1}^{n-1} \n\t\t\t\\langle A_{j_{a}}e_{j_{a+1}},k \\rangle  \\right)\n\t\t\\langle A_{j_n}e_{\\ell_1},k\\rangle \\\\\n\t\t\\left(\\prod_{b=1}^{m-1} \n\t\t\t\\langle A_{\\ell_{b}}h, e_{\\ell_{b+1}} \\rangle  \\right)\n\t\t\\langle A_{\\ell_m}h,x\\rangle e_{j_1}.\n\\end{multline*}\n\n\nMore generally for $|n|+|m|<r$\n\\begin{align*} \n&\\mathrm{ad}_k^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \n\t\t\\mathrm{ad}_k^{n_s} \\mathrm{ad}_h^{m_s} k \\\\\n\t&= (-1)^{|n|} \\sum_{j^1_1}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},k \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},k\\rangle \\\\\n\t&\\qquad\\qquad\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\end{align*}\nThus, for $h\\in G_{CM}$ and $|n|+|m|<r$, we may define a measurable action on $G$ given by\n\\begin{align*} \nG\\ni g\\mapsto\\,&\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \n\t\t\\mathrm{ad}_g^{n_s} \\mathrm{ad}_h^{m_s} g \\\\\n\t&:= (-1)^{|n|} \\sum_{j^1_1}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},g \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},g\\rangle \\\\\n\t&\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\in G_{CM}.\n\\end{align*}\n(For $|n|+|m|\\ge r$, we define this mapping to be 0, which is certainly\nmeasurable.)\nAgain, we are using that \n\\begin{multline*} \n\\sum_{j^1_1}\\Bigg\\| \\sum_{j^1_2}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},\\cdot \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},\\cdot\\rangle \\\\\n\t\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1} \\Bigg\\|_{L^2(\\mu)}^2 <\\infty.\n\\end{multline*}\nThis holds by straightforward but tedious computations --- in fact, iterative applications of\nCauchy-Schwarz combined with the fact that $\\sum_{n,m} \\|A_ne_m\\|^2 =\n\\|[\\cdot,\\cdot]\\|_{HS}^2<\\infty$. Thus,  \n\\begin{multline*} \n\\sum_{j^1_1}\\Bigg(\\sum_{j^1_2}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},g \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},g\\rangle \\\\\n\t\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\Bigg)^2 <\\infty ,\n\\end{multline*}\n$g$-a.s.~and $\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \\mathrm{ad}_g^{n_s}\n\\mathrm{ad}_h^{m_s} g $ as given above is defined a.s.\nThus, we have the following result.\n\n\n\n\\begin{prop}\n\\label{p.gp}\nFor $h\\in G_{CM}$, the mapping $G\\ni g\\mapsto g\\cdot h\\in G$ defined\nanalogously to (\\ref{e.mult}) is a measurable right group action by\n$G_{CM}$ on $G$, and similarly for the left action $g\\mapsto h\\cdot g$.\n\\end{prop}\n\n\\subsection{Examples of abstract nilpotent Lie algebras}\n\n\n\\begin{example}[Free nilpotent Lie algebras]\n\\label{ex.free}\nStarting with an abstract Wiener space $(W,H,\\mu)$, one may construct in the\nstandard way the abstract free\nnilpotent Lie algebra of step $r$ with generators $\\{h_i\\}_{i=1}^\\infty$ an\northonormal basis of $H$.  See for example Section 0 of \\cite{Gromov1996}.\n\\end{example}\n\n\n\\begin{example}[Heisenberg-like algebras]\n\\label{ex.heis}\nLet $(W_i,H_i,\\mu_i)$ for $i=1,2$ be abstract Wiener spaces.  Then for any\n$\\omega:H_1\\times H_1\\rightarrow H_2$ a Hilbert-Schmidt map, we may define a\nLie bracket on $\\mathfrak{g}_{CM}=H_1\\times H_2$ by\n\\[ [(h_1,h_2),(h_1',h_2')] := (0,\\omega(h_1,h_1')), \\]\nand $\\mathfrak{g}=W_1\\times W_2$ may be thought of as an abstract\nHeisenberg-like algebra as in \\cite{DriverGordina2008}.\n\\end{example}\n\nThese abstract Heisenberg-like algebras are central extensions of\none abstract Wiener space by\nanother abstract Wiener space.  The next example generalizes this\nconstruction.\n\n\\begin{example}[Extensions of Lie algebras]\n\\label{ex.ext}\nLet $\\mathfrak{v}$ and $\\mathfrak{h}$ be Lie\nalgebras, and let $\\mathrm{Der}(\\mathfrak{v})$ denote the set of derivations\non $\\mathfrak{v}$; that is, $\\mathrm{Der}(\\mathfrak{v})$ consists of all\nlinear maps $\\rho:\\mathfrak{v}\\rightarrow\\mathfrak{v}$ satisfying Leibniz's\nrule:\n\\[ \\rho([X,Y]_\\mathfrak{v}) = [\\rho(X),Y]_\\mathfrak{v} +\n\t[X,\\rho(Y)]_\\mathfrak{v}. \\]\nNow suppose there are a linear mapping \n$\\alpha: \\mathfrak{h} \\rightarrow \\mathrm{Der}(\\mathfrak{v})$ \nand a skew-symmetric bilinear mapping\n$\\omega:\\mathfrak{h}\\times \\mathfrak{h}\\rightarrow\\mathfrak{v}$, \nsatisfying, for all $X,Y,Z\\in \\mathfrak{h}$,\n\\begin{equation}\n\\label{b1} \n\\tag{B1}\n[\\alpha_X,\\alpha_Y] - \\alpha_{[X,Y]_\\mathfrak{h}} \n\t= \\mathrm{ad}_{\\omega(X,Y)} \n\\end{equation}\nand\n\\begin{equation}\n\\label{b2} \n\\tag{B2}\n\\sum_{\\text{cyclic}}\\left(\\alpha_X \\omega(Y,Z) -\n\t\\omega([X,Y]_\\mathfrak{h},Z)\\right)= 0.  \n\\end{equation}\nThen, one may verify that, for $X_1+V_1,X_2+V_2\\in \\mathfrak{h}\\oplus\n\\mathfrak{v}$,\n\\[ [X_1+V_1, X_2+V_2]_\\mathfrak{g} :=  [X_1,X_2]_\\mathfrak{h} + \\omega(X_1,X_2) \n\t+ \\alpha_{X_1}V_2 -\\alpha_{X_2}V_1 + [V_1,V_2]_\\mathfrak{v} \\]\ndefines a Lie bracket on $\\mathfrak{g}:=\\mathfrak{h}\\oplus\\mathfrak{v}$, \nand we say $\\mathfrak{g}$ is an extension of $\\mathfrak{h}$ over \n$\\mathfrak{v}$. \nThat is, $\\mathfrak{g}$ is the Lie algebra with ideal\n$\\mathfrak{v}$ and quotient algebra\n$\\mathfrak{g}\/\\mathfrak{v}=\\mathfrak{h}$.  The associated exact sequence is\n\\[ 0\\rightarrow\\mathfrak{v}\\overset{\\iota_1}{\\longrightarrow}\\mathfrak{g}\n\t\\overset{\\pi_2}{\\longrightarrow}\\mathfrak{h}\\rightarrow0,\\] \nwhere $\\iota_1$ is\ninclusion and $\\pi_2$ is projection.  In fact,  these are the \nonly extensions of\n$\\mathfrak{h}$ over $\\mathfrak{v}$\n(see, for example, \\cite{AMR00}).\n\n\nNow suppose that $(W,H,\\mu)$ is a real abstract Wiener space, and\n$(\\mathfrak{v},\\mathfrak{v}_{CM},\\mu^0)$ is an abstract nilpotent Lie algebra.\nMotivated by the previous discussion, we may\nconsider $H$ as an abelian Lie algebra and construct extensions of\n$H$ over $\\mathfrak{v}_{CM}$.\nIn this case, we need  a linear mapping\n$\\alpha:H\\rightarrow \\mathrm{Der}(\\mathfrak{v}_{CM})$\nand a skew-symmetric bilinear mapping\n$\\omega:H\\times H\\rightarrow\\mathfrak{v}_{CM}$,\nsuch that $\\omega$ and\n$\\alpha:H\\times\\mathfrak{v}_{CM}\\rightarrow\\mathfrak{v}_{CM}$ \nare both Hilbert-Schmidt and together $\\omega$ and $\\alpha$\nsatisfy (\\ref{b1}) and (\\ref{b2}), which in\nthis setting become\n\\begin{equation*}\n[\\alpha_X,\\alpha_Y] = \\mathrm{ad}_{\\omega(X,Y)}\n\\end{equation*}\nand\n\\begin{equation*}\n\\alpha_X \\omega(Y,Z) + \\alpha_Y \\omega(Z,X) + \\alpha_Z \\omega(X,Y) = 0, \n\\end{equation*}\nfor all $X,Y,Z\\in H$.  Then we may define a Lie algebra structure\non $\\mathfrak{g}_{CM}= H\\oplus \\mathfrak{v}_{CM}$ via the Lie bracket\n\\begin{equation*} \n[(X_1,V_1), (X_2,V_2)]_{\\mathfrak{g}_{CM}} := (0, \\omega(X_1,X_2)\n\t+ \\alpha_{X_1}V_2 - \\alpha_{X_2}V_1 ). \n\\end{equation*}\nAgain, these are in fact the only extensions of $H$\nover $\\mathfrak{v}_{CM}$.  \n\\end{example}\n\nCombining Examples \\ref{ex.heis} and \\ref{ex.ext} shows that these\nconstructions are iterative, in that one may construct new abstract nilpotent\nLie algebras as Lie algebra extensions of another abstract nilpotent Lie\nalgebra.  The next example builds on the previous one to give one precise way\nto construct some Lie algebra extensions.\n\n\n\\begin{example}\n\\label{ex.beta}\nLet $\\beta:H\\rightarrow \\mathfrak{v}_{CM}$ be any Hilbert-Schmidt map, and for\n$X,Y\\in H$ and $V\\in \\mathfrak{v}_{CM}$ define\n\\[ \\omega(X,Y)  := [\\beta(X),\\beta(Y)]_{\\mathfrak{v}_{CM}} \\] \n\\[ \\alpha_X V := \\mathrm{ad}_{\\beta(X)} V = [\\beta(X),V]_{\\mathfrak{v}_{CM}}. \\] \nThen the Lie bracket on $\\mathfrak{g}_{CM}$ is given by\n\\begin{multline*}\n[(X,V),(Y,U)]  \\\\\n\t= (0,[\\beta(X),\\beta(Y)]_{\\mathfrak{v}_{CM}} \n\t+ [\\beta(X),U]_{\\mathfrak{v}_{CM}} - [\\beta(Y),V]_{\\mathfrak{v}_{CM}} \n\t+ [V,U]_{\\mathfrak{v}_{CM}}). \n\\end{multline*}\nNote that, if $\\mathfrak{v}_{CM}$ is nilpotent of step $r$, then\n$\\mathfrak{g}_{CM}$ will automatically be nilpotent of step $r$. \n\\end{example}\n\n\nThe examples above demonstrate that the space of abstract nilpotent Lie\nalgebras is quite rich, and there are many natural examples with a straightforward construction.  This\nsignificantly improves the results of \\cite{Melcher2009}, which studied \nheat kernel measures on nilpotent extensions of abstract\nWiener spaces over {\\it finite-dimensional} nilpotent Lie algebras.\nFor example, the restriction $\\mathrm{dim}(\\mathfrak{v}_{CM})<\\infty$\ntrivializes Example \\ref{ex.beta}.  We elaborate in the following remark.\n\n\n\\begin{remark}\nReturning to Example \\ref{ex.beta}, since $\\beta$ is linear and continuous, we have the decomposition\n\\[ H = \\mathrm{Nul}(\\beta) \\oplus \\mathrm{Nul}(\\beta)^\\perp, \\] \nwhere $\\mathrm{dim}(\\mathrm{Nul}(\\beta)^\\perp) \\le\n\\mathrm{dim}(\\mathfrak{v}_{CM})$.  Thus, for $X,Y\\in H$ we may write\n$X=X_1+X_2, Y=Y_1+Y_2\\in \\mathrm{Nul}(\\beta) \\oplus \\mathrm{Nul}(\\beta)^\\perp$\nand\n\\begin{align*} \n[(X_1+X_2,0),(Y_1+Y_2,0)] \n\t&= [ \\beta(X_1+X_2),\\beta(Y_1+Y_2)] \n\t= [\\beta(X_2),\\beta(Y_2)], \n\\end{align*}\nand $\\omega$ is a map on $\\mathrm{Nul}(\\beta)^\\perp\\times\n\\mathrm{Nul}(\\beta)^\\perp$.  \nThus, $[\\mathrm{Nul}(\\beta),\\mathrm{Nul}(\\beta)]=\\{0\\}$ and similarly\n$[\\mathrm{Nul}(\\beta),\\mathfrak{v}]=\\{0\\}$.  So\n\\[ \\mathfrak{g}_{CM}= H\\oplus\\mathfrak{v}_{CM} = \\mathrm{Nul}(\\beta) \\oplus\n\t\\mathrm{Nul}(\\beta)^\\perp\\oplus\\mathfrak{v}_{CM}, \\]\nand, in particular, when $\\mathrm{dim}(\\mathfrak{v}_{CM})<\\infty$, $\\mathfrak{g}_{CM}$ \nis just an extension of the finite-dimensional vector space\n$\\mathrm{Nul}(\\beta)^\\perp$ by the finite-dimensional Lie algebra\n$\\mathfrak{v}_{CM}$, and the construction is not truly infinite-dimensional.\n\\end{remark}\n\n\n\n\n\\subsection{Properties of $\\mathfrak{g}_{CM}$}\n\n\nThis section collects some results for\ntopological and geometric properties of $\\mathfrak{g}_{CM}$ that we'll require for\nthe sequel.  \n\n\\begin{prop}\n\\label{p.normest}\nFor all $m\\ge2$, $[[[\\cdot,\\cdot],\\ldots],\\cdot]:\\mathfrak{g}_{CM}^{\\otimes m}\n\\rightarrow \\mathfrak{g}_{CM}$ is Hilbert-Schmidt.\n\\end{prop}\n\n\\begin{proof}\nFor $m=2$, this follows from the definition of $G$.  Now assume the statement holds for all $m=\\ell$, and \nconsider $m=\\ell+1$.  Writing $[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell+1}]\\in \n\\mathfrak{g}_{CM}$ in terms of the orthonormal basis\n$\\{e_j\\}_{j=1}^\\infty$ and using \nH\\\"older's inequality gives\n\\begin{align*}\n\\|[[[\\cdot,&\\cdot],\\ldots],\\cdot]\\|_2^2\n\t= \\|[[[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes \\ell+1}\n\t\t\\otimes\\mathfrak{g}_{CM}} \\\\\n\t&= \\sum_{i_1,\\ldots,i_{\\ell+1}=1}^\\infty \n\t\t\\|[[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}],h_{i_{\\ell+1}}]\\|\n\t\t^2 \\\\\n\t&= \\sum_{i_1,\\ldots,i_{\\ell+1}=1}^\\infty \\left\\|\\sum_{j=1}^\\infty \n\t\t[e_j,h_{i_{\\ell+1}}]\n\t\t\\langle e_j, [[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\rangle\n\t\t\\right\\|^2 \\\\\n\t&\\le \\sum_{i_1,\\dots,i_{\\ell+1}=1}^\\infty \\left(\\sum_{j=1}^\\infty \n\t\t\t\\| [e_j,h_{i_{\\ell+1}}] \\|^2\\right)\n\t\t\\left(\\sum_{j=1}^\\infty \n\t\t|\\langle e_j,[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\rangle|^2\\right) \\\\\n\t&= \\left(\\sum_{i_{\\ell+1}=1}^\\infty \\sum_{j=1}^\\infty \n\t\t\t\\| [e_j,h_{i_{\\ell+1}}] \\|^2\\right)\n\t\t\\left(\\sum_{i_1,\\dots,i_\\ell=1}^\\infty \n\t\t\\|[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\|^2\\right)\n\t\t\\\\\n\t&\\le \\|[\\cdot,\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes 2}\n\t\t\\otimes\\mathfrak{g}_{CM}}^2 \\|[[[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes \\ell}\n\t\t\\otimes\\mathfrak{g}_{CM}}^2\n\\end{align*}\nand the last line is finite by the induction hypothesis.\n\\end{proof}\n\nNext we will recall that the flat and geometric topologies on\n$\\mathfrak{g}_{CM}$ are equivalent.  First we set the following notation.\n\n\\begin{notation}\nFor $g\\in G_{CM},$ let $L_g:G_{CM}\\rightarrow G_{CM}$ and $R_g:G_{CM}\\rightarrow G_{CM}$\ndenote left and right multiplication by $g$, respectively.  As $G_{CM}$\nis a vector space, to each $g\\in G_{CM}$ we can associate the tangent\n\tspace $T_g G_{CM}$ to $G_{CM}$ at $g$, which is naturally isomorphic to $G_{CM}$.\nFor $f:G_{CM}\\rightarrow\\mathbb{R}$ a Frech\\'{e}t smooth function and  \n\t$v,x\\in G_{CM}$ and $h\\in\\mathfrak{g}_{CM}$, let\n\\[ f'(x)h := \\partial_h f(x) = \\frac{d}{dt}\\bigg|_{0}f(x+th), \\]\n\tand let $v_x \\in T_x G_{CM}$ denote the tangent vector\nsatisfying $v_xf=f'(x)v$.  If $\\sigma(t)$ is any smooth curve in\n$G_{CM}$ such that $\\sigma(0) = x$ and $\\dot{\\sigma}(0)=v$ (for example,\n$\\sigma(t) = x+tv$), then\n\\[ L_{g*} v_x = \\frac{d}{dt}\\bigg|_0 g\\cdot \\sigma(t). \\]\n\\end{notation}\n\n\\begin{notation}\n\\label{n.length}\nLet $C^1([0,1],G_{CM})$ denote the collection of $C^1$-paths\n$g:[0,1]\\rightarrow G_{CM}$.  The length of $g$ is defined as \n\\[ \\ell_{CM}(g) := \\int_0^T \\|L_{g^{-1}(s)*}g'(s)\\|_{\\mathfrak{g}_{CM}}\\,ds.\n\\]\nThe Riemannian distance between $x,y\\in G_{CM}$ is then defined as\n\\begin{align*} d(x,y) := \\inf\\{\\ell_{CM}(g): g\\in C_h^1([0,1],G_{CM}) \\text{ such that }\n\tg(0)=x \\text{ and } g(1)=y \\}. \n\\end{align*}\n\\end{notation}\n\nThe following proposition was proved as Corollary 4.13 in\n\\cite{Melcher2009}.  This was proved under the conditions that\n$\\mathfrak{g}_{CM}$ was a ``semi-infinite Lie algebra'', that is,\nunder the assumption that $\\mathfrak{g}_{CM}$ was a nilpotent Lie algebra\nextension (as in Example \\ref{ex.ext}) \nof a finite-dimensional nilpotent Lie algebra $\\mathfrak{v}$ by an\nabstract Wiener space.  However, a cursory inspection of the proofs\nthere will show that they only depended on the fact that the Lie\nbracket was Hilbert-Schmidt, and not on the ``stratified'' structure\nof $\\mathfrak{g}_{CM}=H\\times\\mathfrak{v}$ or the fact that the image of the Lie bracket\nwas a finite-dimensional subspace.\n\n\n\n\\begin{prop}\n\\label{p.length}\nThe topology on $G_{CM}$ induced by $d$ is equivalent to the Hilbert\ntopology induced by $\\|\\cdot\\|_{\\mathfrak{g}_{CM}}$.\n\\end{prop}\n\n\n\nWe may find a uniform lower bound on the Ricci curvature of all\nfinite-dimensional subgroups of $G_{CM}$.  Finite-dimensional subgroups\nof $G_{CM}$ may be obtained by taking the Lie algebra\ngenerated by any finite-dimensional subspace of $\\mathfrak{g}_{CM}$\n(which is again necessarily finite-dimensional by the nilpotence\nof the bracket) and endowing it with the standard group operation via the\nBaker-Campbell-Hausdorff-Dynkin formula.   An analogue of the following proposition was\nproved as Proposition 3.23 and Corollary 3.24 in \\cite{Melcher2009}.\nIt follows directly from the form of the Ricci curvature on nilpotent\ngroups (endowed with a left invariant metric) and the assumption\nthat the Lie bracket is Hilbert-Schmidt.  This proof is essentially the same as in\n\\cite{Melcher2009}, but it is quite brief and so is included for completeness.\n\n\\begin{prop}\n\\label{p.Ric}\nLet \n\\[ k := -\\frac{1}{2} \\sup \\left\\{ \n\t\t\\|[\\cdot,X]\\|^2_{\\mathfrak{g}_{CM}^*\\otimes\\mathfrak{g}_{CM}}   \n\t\t: \\,\\|X\\|_{\\mathfrak{g}_{CM}}=1 \\right\\}. \\]\nThen $k>-\\infty$ and $k$ is the largest constant such that\n\\[ \\langle \\mathrm{Ric}^\\pi X,X\\rangle_{\\mathfrak{g}_\\pi} \\ge \n\tk \\|X\\|^2_{\\mathfrak{g}_\\pi}, \n\t\\quad \\text{ for all } X \\in\\mathfrak{g}_\\pi, \\]\nholds uniformly for all $\\mathfrak{g}_\\pi$ finite-dimensional Lie subalgebras\nof $\\mathfrak{g}_{CM}$.\n\\end{prop}\n\n\\begin{proof}\nFor $\\mathfrak{g}$ any nilpotent Lie algebra with orthonormal basis\n$\\Gamma$, \n\\begin{align*}\n\\langle \\mathrm{Ric}\\, X,X\\rangle \n\t&= \\frac{1}{4}\\sum_{Y\\in\\Gamma} \\|\\mathrm{ad}^*_Y X\\|^2 \n\t\t- \\frac{1}{2}\\sum_{Y\\in\\Gamma} \\|\\mathrm{ad}_Y X\\|^2 \n\t\\ge - \\frac{1}{2}\\sum_{Y\\in\\Gamma} \\|[Y,X]\\|^2\n\\end{align*}\nfor all $X\\in\\mathfrak{g}$.  Thus, for $\\mathfrak{g}_\\pi$ any\nfinite-dimensional Lie algebra\n\\[ \\langle \\mathrm{Ric}^\\pi X,X\\rangle_{\\mathfrak{g}_\\pi} \\ge \n\tk_\\pi \\|X\\|^2_{\\mathfrak{g}_\\pi}, \n\t\\quad \\text{ for all } X \\in\\mathfrak{g}_\\pi, \\]\nwhere\n\\begin{equation}\n\\label{e.pah}\n k_\\pi :=  - \\frac{1}{2}\\sup \\left\\{ \n\t\t\\|[\\cdot,X]\\|^2_{\\mathfrak{g}_\\pi^*\\otimes\\mathfrak{g}_\\pi}\n \t\t:\\, \\|X\\|_{\\mathfrak{g}_\\pi}=1 \\right\\} \n\t\\ge - \\frac{1}{2}\\|[\\cdot,\\cdot]\\|_2^2 > -\\infty.\n\\end{equation}\nTaking the infimum of $k_\\pi$ over all $\\mathfrak{g}_\\pi$ completes the proof.\n\\end{proof}\n\n\n\\section{Brownian motion on $G$}\n\\label{s.BM}\nSuppose that $B_t$ is a smooth curve in $\\mathfrak{g}_{CM}$ with\n$B_0=0$, and consider the differential equation\n\\[ \\dot{g}_t = g_t \\dot{B}_t \n\t:= L_{g_t*}\\dot{B}_t, \\quad \\text{ with } g_0=\\mathbf{e}. \\]  \nThe solution $g_t$ may be written as follows (see\n\\cite{Strichartz87}):  For $t>0$, let $\\Delta_n(t)$ denote the simplex in\n$\\mathbb{R}^n$ given by\n\\[ \\{s=(s_1,\\cdots,s_n)\\in\\mathbb{R}^n: 0<s_1<s_2<\\cdots<s_n<t\\}. \\]\nLet $\\mathcal{S}_n$ denote the permutation group on $(1,\\cdots,n)$, and, for each\n$\\sigma\\in \\mathcal{S}_n$, let $e(\\sigma)$ denote the number of ``errors'' in the\nordering $(\\sigma(1),\\sigma(2),\\cdots,\\sigma(n))$, that is, $e(\\sigma)=\\#\n\\{j<n: \\sigma(j)>\\sigma(j+1)\\}$.  Then\n\\begin{multline}\n\\label{e.ode}\ng_t = \\sum_{n=1}^r \\sum_{\\sigma\\in \\mathcal{S}_n} \n\t\\left( (-1)^{e(\\sigma)}\\bigg\/ n^2 \n\t\\begin{bmatrix} n-1 \\\\ e(\\sigma) \\end{bmatrix}\\right) \\times \\\\\n\t\\int_{\\Delta_n(t)} [ \\cdots[\\dot{B}_{s_{\\sigma(1)}},\n\t\\dot{B}_{s_{\\sigma(2)}}],\\ldots,] \\dot{B}_{s_{\\sigma(n)}}]\\, ds,\n\\end{multline}\nwhere the $n=1$ term is understood to be $\\int_0^t dB_s = B_t$.\nUsing this as our motivation, we first explore stochastic integral\nanalogues of equation (\\ref{e.ode}) where the smooth curve $B$ is replaced by\nBrownian motion on $\\mathfrak{g}$.\n\n\n\n\n\n\n\n\\subsection{Brownian motion and finite-dimensional approximations}\n\n\nWe now return to the setting of an abstract Wiener space\n$(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ endowed with a nilpotent\nHilbert-Schmidt Lie bracket on $\\mathfrak{g}_{CM}$.\nAgain, let $B_t$ denote Brownian motion on $\\mathfrak{g}$.\nBy equation (\\ref{e.ode}), the solution to the Stratonovich\nstochastic differential equation\n\\[ \\delta g_t = L_{g_t*} \\delta B_t, \\quad \\text{ with } g_0=\\mathbf{e}, \\]\nshould be given by\n\\begin{equation}\n\\label{e.gt}\ng_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n} c^\\sigma_n \\int_{\\Delta_n(t)} \n\t[ [\\cdots[\\delta B_{s_{\\sigma(1)}},\\delta B_{s_{\\sigma(2)}}],\\cdots], \n\t\\delta B_{s_{\\sigma(n)}}],\n\\end{equation}\nfor coefficients $c_n^\\sigma$ determined by equation (\\ref{e.ode}).\n\nTo understand the integrals in (\\ref{e.gt}), consider the following heuristic\ncomputation.\nLet $\\{M_n(t)\\}_{t\\ge0}$ denote the process in $\\mathfrak{g}^{\\otimes n}$ \ndefined by\n\\[ \nM_n(t) := \\int_{\\Delta_n(t)} \\delta B_{s_1}\\otimes\\cdots\\otimes \\delta\n\tB_{s_n}.\n\\]\nBy repeatedly applying the definition of the Stratonovich integral, \nthe iterated Stratonovich integral $M_n(t)$ \nmay be realized as a linear combination of iterated It\\^o integrals:\n\\[ M_n(t) = \\sum_{m=\\lceil n\/2\\rceil}^n \\frac{1}{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m} I^n_t(\\alpha), \\]\nwhere\n\\[ \\mathcal{J}_n^m := \\left\\{(\\alpha_1,\\ldots,\\alpha_m)\\in\\{1,2\\}^m :\n\t\\sum_{i=1}^m \\alpha_i = n \\right\\}, \\]\nand, for $\\alpha\\in\\mathcal{J}_n^m$, $I_t^n(\\alpha)$ is the iterated \nIt\\^o integral\n\\[ I_t^n(\\alpha) = \\int_{\\Delta_m(t)} dX^1_{s_1}\\otimes\\cdots\\otimes \n\tdX^m_{s_m} \\]\nwith\n\\[ dX_s^i = \\left\\{ \\begin{array}{cl} dB_s & \\text{if } \\alpha_i=1 \\\\\n\t\\sum_{j=1}^\\infty h_j \\otimes h_j \\, ds & \\text{if } \\alpha_i=2\n\t\\end{array} \\right. ; \\]\ncompare with Proposition 1 of \\cite{BenArous89}.  This change from multiple\nStratonovich integrals to multiple It\\^o integrals may also be recognized as a\nspecific case of the Hu-Meyer formulas \\cite{HuMeyer88-2,HuMeyer88-1},\nbut we will compute more explicitly to verify that our integrals are well-defined.\n\n\nDefine $F_1:\\mathfrak{g}_{CM}\\to\\mathfrak{g}_{CM}$ by $F_1(k)=k$, and for $n\\in\\{2,\\cdots,r\\}$ define $F_n:\\mathfrak{g}_{CM}^{\\otimes n}\n\\rightarrow\\mathfrak{g}_{CM}$ by\n\\begin{equation}\n\\label{e.1Fn}\n F_n(k_1\\otimes\\cdots\\otimes k_n) \n\t:= [ [ [\\cdots[k_1,k_2],k_3],\\cdots],k_n]. \n\\end{equation}\nFor each fixed $n$ and $\\sigma\\in\\mathcal{S}_n$, define $F_n^\\sigma:\\mathfrak{g}_{CM}^{\\otimes n}\n\\rightarrow\\mathfrak{g}_{CM}$ by\n\\begin{equation}\n\\label{e.Fn}\n\\begin{split}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_n) \n\t&:= F_n(k_{\\sigma(1)}\\otimes\\cdots\\otimes k_{\\sigma(n)}) \\\\\n\t&= [ [\\cdots[k_{\\sigma(1)},k_{\\sigma(2)}],\\cdots], \n\t\tk_{\\sigma(n)}].\n\\end{split}\n\\end{equation}\nThen we may write\n\\[\ng_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n} \n\tc^\\sigma_n F^\\sigma_n (M_n(t))\n\t= \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m} F^\\sigma_n (I^n_t(\\alpha)),\n\\]\npresuming we can make sense of the integrals $F_n^\\sigma(I_t^n(\\alpha))$.\n\nFor each $\\alpha$, let $p_\\alpha=\\#\\{i:\\alpha_i=1\\}$ and $q_\\alpha=\\#\\{i:\n\\alpha_i=2\\}$ (so that $p_\\alpha+q_\\alpha=m$ when $\\alpha\\in\\mathcal{J}_n^m$),\nand let \n\\[ \\mathcal{J}_n := \\bigcup_{m=\\lceil n\/2\\rceil}^n \\mathcal{J}_n^m. \\]\nThen, for each $\\sigma\\in\\mathcal{S}_n$ and $\\alpha\\in\\mathcal{J}_n$, \n\\[ F_n^\\sigma(I_t^n(\\alpha))\n\t= \\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t\t(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_{p_\\alpha}}),\n\\]\nwhere $\\hat{F}_n^{\\sigma,\\alpha}$ and $f_\\alpha$ are as follows.\n\nThe map $\\hat{F}_n^{\\sigma,\\alpha}:\\mathfrak{g}^{\\otimes p_\\alpha}\n\\rightarrow\\mathfrak{g}$ is defined by\n\\begin{multline}\n\\label{e.Fhat}\n\\hat{F}_n^{\\sigma,\\alpha}(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}) \\\\\n\t:= \\sum_{j_1,\\ldots,j_{q_\\alpha}=1}^\\infty \n\t\tF_n^{\\sigma'}(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\n\t\t\\otimes h_{j_1}\\otimes h_{j_1}\n\t\t\\otimes\\cdots\\otimes h_{j_{q_\\alpha}}\\otimes h_{j_{q_\\alpha}}), \n\\end{multline}\nfor $\\{h_j\\}_{j=1}^\\infty$ an orthonormal basis of $\\mathfrak{g}_{CM}$ and\n$\\sigma'=\\sigma'(\\alpha)\\in\\mathcal{S}_n$ given by \n$\\sigma'=\\sigma\\circ\\tau^{-1}$, for any $\\tau\\in\\mathcal{S}_n$ such that\n\\begin{multline*} \n\\tau(dX^1_{s_1}\\otimes\\cdots\\otimes dX^m_{s_m}) \\\\\n\t= \\sum_{j_1,\\cdots,j_{q_\\alpha}=1}^\\infty dB_{s_1}\\otimes\\cdots\n\t\\otimes dB_{s_{p_\\alpha}}\\otimes h_{j_1}\\otimes h_{j_1}\\otimes\\cdots\n\t\\otimes h_{j_{q_\\alpha}}\\otimes h_{j_{q_\\alpha}} ds_1\\cdots\n\tds_{q_\\alpha}.\n\\end{multline*}\n\nTo define $f_\\alpha$, first consider the polynomial of order $q_\\alpha$,\nin the variables $s_i$ with $i$ such that $\\alpha_i=1$ and in the variable $t$, given by evaluating the integral\n\\begin{equation}\n\\label{e.fprime} \nf_\\alpha'( (s_i:\\alpha_i=1),t)\n\t= \\int_{\\Delta'_{q_\\alpha}(t)} \\prod_{i: \\alpha_i=2} ds_i,\n\\end{equation}\nwhere $\\Delta'_{q_\\alpha}(t)=\\{s_{i-1}<s_i<s_{i+1}: \\alpha_i=2\\}$ with \n$s_0=0$ and $s_{m+1}=t$.  \nThen $f_\\alpha$ is $f_\\alpha'$ with the variables reindexed by the\nbijection $\\{i: \\alpha_i=1\\}\\rightarrow\\{1,\\ldots,p_\\alpha\\}$ that\nmaintains the natural ordering of these sets.\n(For example, for $\\alpha=(1,2,1,2)\\in\\mathcal{J}_6^4$,\n\\[ f_\\alpha'(s_1,s_3,t)=\\int_{\\{s_1<s_2<s_3,s_3<s_4<t\\}} ds_2\\,ds_4\n\t= (t-s_3)(s_3-s_1) , \\]\nso that $f_\\alpha(s_1,s_2,t)=(t-s_2)(s_2-s_1)$.) \n\nThis explicit realization of $f_\\alpha$ is not critical to the sequel.\nIt is really only necessary to know that \n$f_\\alpha$ is a polynomial of order $q_\\alpha$ in\n$s=(s_1,\\ldots,s_{p_\\alpha})$ and $t$, and thus may be written as\n\\[ f_\\alpha(s,t) = \\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a \n\t\\tilde{f}_{\\alpha,a}(s), \n\\]\nfor some coefficients $b_\\alpha^a\\in\\mathbb{R}$ and polynomials \n$\\tilde{f}_{\\alpha,a}$ of degree $q_\\alpha-a$ in $s$.  \nNow, if $\\hat{F}_n^{\\sigma,\\alpha}$ is\nHilbert-Schmidt on $\\mathfrak{g}_{CM}^{\\otimes p_\\alpha}$, then\n\\[ \\int_{\\Delta_{p_\\alpha}(t)} \\left\\|\\tilde{f}_{\\alpha,a}(s)\n\t\\hat{F}_n^{\\sigma,\\alpha} \\right\\|_2^2\\, ds\t\n\t= \\left\\|\\tilde{f}_{\\alpha,a}\\right\\|^2_{L^2(\\Delta_{p_\\alpha}(t))}\n\t\t\\left\\|\\hat{F}_n^{\\sigma,\\alpha} \\right\\|_2^2\n\t<\\infty,\n\\]\nand \n\\begin{align}\n\\label{e.a}\nF_n^\\sigma(I_t^n(\\alpha))\n\t&= \\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n(\\tilde{f}_{\\alpha,a}\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t\n\\end{align}\nmay be understood in the sense of the limit integrals in \nProposition \\ref{p.int2}.  (In particular, if $\\alpha_m=1$, then \n$f_\\alpha=f_\\alpha(s)$ does not depend on $t$, and Proposition \\ref{p.int2} \nimplies that $F_n^\\sigma(I_t^n(\\alpha))$ is a $\\mathfrak{v}$-valued \n$L^2$-martingale.)\n\nThe above computations show that, if for all $n$, $\\sigma\\in\\mathcal{S}_n$,\nand $\\alpha\\in\\mathcal{J}_n$,\n$\\hat{F}_n^{\\sigma,\\alpha}$ is Hilbert-Schmidt, then we may rewrite (\\ref{e.gt}) as\n\\[ g_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}  \n\t\\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n(\\tilde{f}_{\\alpha,a}\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t,\n\\]\nwhere $J_n$ is as defined in Proposition \\ref{p.int2}.  \nThe next proposition shows that  $\\hat{F}_n^{\\sigma,\\alpha}$ is \nHilbert-Schmidt as desired, and thus $g_t$ in (\\ref{e.gt}) is well-defined. \n\n\n\\begin{prop}\n\\label{p.HS}\nLet $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$, and \n$\\alpha\\in\\mathcal{J}_n$.\nThen $\\hat{F}_n^{\\sigma,\\alpha}:\\mathfrak{g}_{CM}^{\\otimes p_\\alpha}\n\\rightarrow \\mathfrak{g}_{CM}$ is Hilbert-Schmidt.\n\\end{prop}\n\n\\begin{proof}\nFor the whole of this proof, all sums will be taken over an orthonormal basis\nof $\\mathfrak{g}_{CM}$.\n\nNow, for $F_n$ and $F_n^\\sigma$ as defined in equations (\\ref{e.1Fn}) and\n(\\ref{e.Fn}), we may write\n\\begin{align*} \nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\\otimes h_1\\otimes\n\t\th_1\\otimes\\cdots\\otimes h_{q_\\alpha}\\otimes h_{q_\\alpha}) \n\t= F_n(A_{\\sigma(1)}\\otimes\\ldots\\otimes A_{\\sigma(n)})\n\\end{align*}\nwhere\n\\[ \nA_b := \\left\\{\\begin{array}{ll}\n\t\tk_b & \\text{if } b=1,\\ldots,p_\\alpha \\\\\n\t\th_{\\lceil (b-p_\\alpha)\/2\\rceil} & \\text{if } b=p_\\alpha + 1,\\ldots,n\n\t\\end{array} \\right. .\n\\] \n\\iffalse\n\\[ \nA_b := \\left\\{\\begin{array}{ll}\n\t\tk_b & \\text{if } b=1,\\ldots,p_\\alpha \\\\\n\t\th_{\\lceil \\ell\/2\\rceil} & \\text{if } b=p_\\alpha + \\ell\n\t\t\t\\text{ for } \\ell=1,\\ldots,2q_\\alpha\n\t\\end{array} \\right. .\n\\]\n\\fi\n\nIf $q_\\alpha=0$, then $p_\\alpha=n$, each $A_{\\sigma(j)}=k_i$ for some\n$i=1,\\ldots,n$, and\n\\begin{align*}\n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t&= \\sum_{k_1,\\ldots,k_n} \n\t\t\\|F_n^\\sigma(k_1\\otimes\\cdots\\otimes k_n)\\|_{\\mathfrak{g}_{CM}}^2 \\\\\n\t&= \\sum_{k_1,\\ldots,k_n} \n\t\t\\|F_n(k_1\\otimes\\cdots\\otimes k_n)\\|_{\\mathfrak{g}_{CM}}^2\n\t= \\|[ [[\\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes\n\tn}\\otimes\\mathfrak{g}_{CM}}^2\n\\end{align*}\nwhich is finite by Proposition \\ref{p.normest}.\n\n\nNow, if $q_\\alpha=1$, let $N=N(\\sigma)$ denote the second\n$j$ such that $\\sigma(j)\\in\\{n-1,n\\}$; that is,\n\\begin{align*}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes& k_{n-1}\\otimes h_1\\otimes\n\t\th_1) \\\\\n\t&= F_n(A_{\\sigma(1)}\\otimes \\cdots\\otimes A_{\\sigma(N-1)}\\otimes h_1 \\otimes\n\t\tA_{\\sigma(N+1)}\\otimes\\ldots\\otimes A_{\\sigma(n)}) \\\\\n\t&= [[[[ [ \\ldots[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N-1)}]\n\t\t,h_1],A_{\\sigma(N+1)}],\\ldots],A_{\\sigma(n)}]\n\\end{align*} \nwhere exactly one of $A_{\\sigma(1)},\\ldots,A_{\\sigma(N-1)}$ is $h_1$ and,\nfor all $j>N$, $A_{\\sigma(j)}=k_i$ for some \n\\[ i\\in I := I(\\sigma):= \\{ \\sigma(j):j=N+1,\\ldots,n\\} \\subseteq\n \\{1,\\ldots,p_\\alpha\\}= \\{1,\\ldots,n-2\\}. \\]\nThus, writing\n\\[ \\mathcal{A}(h_1, k_i : i\\in I^c) \n\t:= [ [ \\ldots[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N-1)}], \\]\nwe have that\n\\begin{multline*}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{n-1}\\otimes h_1\\otimes h_1) \\\\\n\t= \\sum_{e_1}\n\t\t\\,\\langle \\mathcal{A}(h_1,k_i:i\\in I^c),e_1\\rangle_{\\mathfrak{g}_{CM}}\n\t\t[ [ \\ldots[e_1,h_1],A_{\\sigma(N+1)}],\\ldots,A_{\\sigma(n)}],\n\\end{multline*}\nand so\n\\begin{align*}\n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t&= \\sum_{k_1,\\ldots,k_{n-1}}\\left\\|\n\t\t\\sum_{h_1} \\,F_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{n-1}\\otimes h_1\\otimes h_1)\n\t\t\\right\\|_{\\mathfrak{g}_{CM}}^2 \\\\\n\t&\\le \\sum_{k_1,\\ldots,k_{n-1}} \\left( \\sum_{h_1,e_1} \n\t\t|\\langle \\mathcal{A}(h_1,k_i :i\\in I^c),e_1\\rangle\n\t\t_{\\mathfrak{g}_{CM}}|^2 \\right) \\\\\n\t&\\qquad\\qquad\\qquad\\quad\\times\n\t\t\\left( \\sum_{h_1,e_1} \n\t\t\\|[ [[e_1,h_1],A_{\\sigma(N+1)}],\\ldots,A_{\\sigma(n)}]\\|\n\t\t_{\\mathfrak{g}_{CM}}^2 \\right) \\\\\n\t&= \\left(\\sum_{k_i:i\\in I^c,h_1,e_1} |\\langle \\mathcal{A}(h_1,k_i :i\\in I^c),e_1\\rangle\n\t\t_{\\mathfrak{g}_{CM}}|^2 \\right) \\\\\n\t&\\qquad\\qquad\\qquad\\quad \\times \n\t\t\\left( \\sum_{k_i:i\\in I,h_1,e_1} \n\t\t\\|[ [\\ldots[[e_1,h_1],A_{\\sigma(N+1)}],\\ldots],A_{\\sigma(n)}]\\|\n\t\t_{\\mathfrak{g}_{CM}}^2 \\right) \\\\\n\t&\\le \\|[ [ \\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|\n\t\t_{(\\mathfrak{g}_{CM}^*)^{\\otimes N-1}\\otimes\\mathfrak{g}_{CM}}^2\n\t\t\\|[ [ \\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|\n\t\t_{(\\mathfrak{g}_{CM}^*)^{\\otimes n-N+1}\\otimes\\mathfrak{g}_{CM}}^2.\n\\end{align*}\n \nNow more generally when $q_\\alpha\\ge2$, we may similarly ``separate'' the\npairs of $h_j$'s as above.  More precisely, define\n\\[ \\Phi(b) := \\Phi_\\alpha(b) := \\left\\{  \\begin{array}{ll} \n\tb & \\text{if } b=1,\\ldots, p_\\alpha \\\\\n\t\\lceil\\frac{b-p_\\alpha}{2}\\rceil+p_\\alpha & \\text{if } b=p_\\alpha+1,\\ldots,n\n\\end{array}\\right. .\\]\nLet $N_0=1$, and set\n\\[ \\Omega_1^j := \\{ \\Phi(\\sigma(\\ell)) : \\ell=N_0,\\ldots,j-1\\} \\quad\n\\text{ and } \\quad N_1 := \\min\\{ j>N_0:\n\t\\Phi(\\sigma(j))\\in\\Omega_1^j\\},  \\]\n\\[ \\Omega_2^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_1,\\ldots,j-1\\} \\quad\n\\text{ and } \\quad N_2 := \\min\\{ j>N_1:\n\t\\Phi(\\sigma(j))\\in\\Omega_2^j\\}. \\]\nSimilarly, we define \n\\[ \\Omega_{2m+1}^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_{2m},\\ldots,j-1 \\}, \n\t \\]\n\\[ N_{2m+1} := \\min\\left\\{ j>N_{2m}: \\Phi(\\sigma(j))\\in \\bigcup_{i=0}^{m-1}\n\t\\Omega_{2i+1}^{N_{2i+1}} \\cup \\Omega_{2m+1}^j \\right\\}, \\]\n\\[ \\Omega_{2m}^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_{2m-1},\\ldots,j-1 \\}, \\text{\n\tand}  \n\\]\n\\[ N_{2m} := \\min\\left\\{ j>N_{2m-1}: \\Phi(\\sigma(j))\\in\n\t\\bigcup_{i=1}^{m-1} \\Omega_{2m}^{N_{2m}}\\cup \\Omega_{2m}^j\\right\\}.\n\\]\n\n\nThen there is an $M<q_\\alpha$ such that the sets\n$\\{A_{\\sigma(N_i)},\\ldots,A_{\\sigma(N_{i+1}-1)}\\}_{i=0}^{M-1}$ and\n$\\{A_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\}$ separate the $h_j$'s in the sense that \nno $h_j$ is repeated inside any one of these sets, and moreover the union of\n``even'' sets contains exactly one copy of each $h_j$ and similarly with the \n``odd'' sets. We can write\n\\begin{align*}\nF_n^\\sigma&(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\\otimes h_1\\otimes\n\t\th_1\\otimes\\cdots\\otimes h_{q_\\alpha}\\otimes h_{q_\\alpha}) \\\\\n\t&= \\sum_{e_1,\\ldots,e_M} \\bigg\\{\n\t\t\\langle e_1,[ [[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t&\\qquad \\times\n\t\t\\left(\\prod_{i=1}^{M-1} \\langle e_{i+1},\n\t\t[ [[e_i,A_{\\sigma(N_i)}],\\ldots],A_{\\sigma(N_{i+1}-1)}]\\rangle\\right)\n\t\t[ [[e_M,A_{\\sigma(N_M)}],\\ldots],A_{\\sigma(n)}]\\bigg\\}.\n\\end{align*}\nIf $M$ is even, then\n\\begin{multline*} \n\\mathcal{A} =\n\t\\langle e_1,[[[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t\\times \\left(\\prod_{i=1}^{M\/2-1} \\langle e_{2i+1},\n\t[\n\t[[e_{2i},A_{\\sigma(N_{2i})}],\\ldots],A_{\\sigma(N_{2i+1})-1}]\\rangle\\right)\n\t[ [ [e_M, A_{\\sigma(N_M)}],\\ldots], A_{\\sigma(n)}]\n\\end{multline*}\nis a function of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and $k_i$ for $i\\in I$ some subset\nof $\\{1,\\ldots,n\\}$, and \n\\[ \\mathcal{B} = \n\t\\prod_{i=1}^{M\/2}\\langle e_{2i},\n\t\t[[[e_{2i-1},A_{\\sigma(N_{2i-1})}],\\ldots],A_{\\sigma(N_{2i})-1}]\\rangle \\]\nis a function of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and $k_i$ for $i\\in\nI^c$.  Thus,\n\\[ \n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t\\le \\left(\\sum_{k_i:i\\in I,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t \\|\\mathcal{A}\\|_{\\mathfrak{g}_{CM}}^2\\right)\n\t\t\\left(\\sum_{k_i:i\\in I^c,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t|\\mathcal{B}|^2\\right) \\]\nwhich is finite again by Proposition \\ref{p.normest}.\nSimilarly, if $M$ is odd,\n\\begin{multline*} \n\\mathcal{A} = \\langle\n\t\te_1,[[[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t\\times \\prod_{i=1}^{(M-1)\/2} \\langle e_{2i+1},\n\t\t[ [[e_{2i},A_{\\sigma(N_{2i})}],\\ldots],A_{\\sigma(N_{2i+1})-1}]\\rangle \n\\end{multline*}\nand\n\\[ \n\\mathcal{B} = \\left(\\prod_{i=1}^{(M-1)\/2}\\langle e_{2i},\n\t\t[ [ [e_{2i-1},A_{\\sigma(N_{2i-1})}],\\ldots],A_{\\sigma(N_{2i})-1}]\\rangle\\right)\n\t\t[ [ [e_M, A_{\\sigma(N_M)}],\\ldots],A_{\\sigma(n)}] \n\\]\nare both functions of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and some $k_i$, and\n\\[ \n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t\\le \\left(\\sum_{k_i:i\\in I,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t |\\mathcal{A}|_{\\mathfrak{g}_{CM}}^2\\right)\n\t\t\\left(\\sum_{k_i:i\\in I^c,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t\\|\\mathcal{B}\\|_{\\mathfrak{g}_{CM}}^2\\right) \\]\nis again finite by Proposition \\ref{p.normest} and this completes the proof.\n\\end{proof}\n\\iffalse\nif $M$ is even, get odd number of sets\n\\[ \n\t\\{A_{\\sigma(N_0)},\\ldots,A_{\\sigma(N_1-1)},\n\tA_{\\sigma(N_2)},\\ldots,A_{\\sigma(N_3-1)},\\ldots,  \n\tA_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\} \n\\]\n\\[ \\{A_{\\sigma(N_1)},\\ldots,A_{\\sigma(N_2-1)},\n\tA_{\\sigma(N_3)},\\ldots,A_{\\sigma(N_4-1)},\\ldots,  \n\tA_{\\sigma(N_{M-1})},\\ldots,A_{\\sigma(N_M)-1}\\} \\]\nif $M$ is odd, get even number of sets\n\\[ \\{A_{\\sigma(N_0)},\\ldots,A_{\\sigma(N_1-1)},\n\tA_{\\sigma(N_2)},\\ldots,A_{\\sigma(N_3-1)},\\ldots,  \n\tA_{\\sigma(N_{M-1})},\\ldots,A_{\\sigma(N_M-1)}\\} \\]\n\\[ \\{A_{\\sigma(N_1)},\\ldots,A_{\\sigma(N_2-1)},\n\tA_{\\sigma(N_3)},\\ldots,A_{\\sigma(N_4-1)},\\ldots,  \n\tA_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\} \\]\n\\fi\n\nPropositions \\ref{p.int2} and \\ref{p.HS} now allow us to make the following definition.\n\n\n\\begin{defn}\n\\label{d.BM}\nA {\\em Brownian motion} on $G$ is the continuous $G$-valued process defined by \n\\[g_t = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\t\\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t\t(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_{p_\\alpha}}), \\]\nwhere \n\\[ c_n^\\sigma=(-1)^{e(\\sigma)}\\bigg\/n^2\\begin{bmatrix} n-1 \\\\ e(\\sigma) \n\t\\end{bmatrix}, \\]\n$\\hat{F}_n^{\\sigma,\\alpha}$ is as defined in equation (\\ref{e.Fhat}), \nand $f_\\alpha$ is as defined below equation (\\ref{e.fprime}).\nFor $t>0$, let $\\nu_t=\\mathrm{Law}(g_t)$ be the {\\em heat kernel measure at\ntime $t$}, a probability measure on $G$.\n\\end{defn}\n\n\n\n\\begin{prop}[Finite-dimensional approximations]\n\\label{p.approx}\nFor $G_\\pi$ a finite-dimensional Lie subgroup of\n$G_{CM}$, let $\\pi$ denote orthogonal projection of\n$G_{CM}$ onto\n$G_\\pi$ and let $g^\\pi_t$ be the continuous process on $G_\\pi$\ndefined by\n\\[ g_t^\\pi = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t(d\\pi B_{s_1}\\otimes\\cdots\\otimes d\\pi B_{s_{p_\\alpha}}), \\]\nwhere the stochastic integrals are defined as in Proposition \\ref{p.bad}.\nThen $g_t^\\pi$ is Brownian motion on $G_\\pi$.  In particular, for\n$G_\\ell=G_{\\pi_\\ell}$ an increasing sequence of finite-dimensional Lie\nsubgroups such that the associated orthogonal projections $\\pi_\\ell$ are\nincreasing to $I_{\\mathfrak{g}_{CM}}$, let\n$g^\\ell_t=g_t^{\\pi_\\ell}$.  Then, for all $t<\\infty$,\n\\begin{equation} \n\\label{e.b}\n\\lim_{\\ell\\rightarrow\\infty}\\mathbb{E}\n\t\\left\\|g^\\ell_t-g_t\\right\\|_\\mathfrak{g}^2 = 0. \n\\end{equation}\n\\end{prop}\n\\begin{proof}\nFirst note that $g_t^\\pi$ solves the Stratonovich equation \n$\\delta g_t^\\pi = L_{g_t^\\pi*}\\delta \\pi B_t$ with $g_0^\\pi=\\mathbf{e}$, see\n\\cite{BenArous89,Castell93,Baudoin04} where $\\langle \\pi B\\rangle_t$ is a\nstandard $\\mathfrak{g}_\\pi$-valued Brownian motion. \nThus, $g_t^\\pi$ is a $G_\\pi$-valued\nBrownian motion.  \n\n\n\nBy equation (\\ref{e.a}) and its preceding discussion,\n\\[ g_t^\\ell = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n^\\ell(\\tilde{f}_\\alpha\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t, \\]\nand thus, to verify (\\ref{e.b}), it suffices to show that\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\\|\\pi_{\\ell}B_t- B_t\\|_\\mathfrak{g}^2 =\n0 \\]\nand\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E} \n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t\\right\\|^2\n\t = 0, \\]\nfor all $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$ and\n$\\alpha\\in\\mathcal{J}_n$.\n\nSo let $\\mu_t=\\mathrm{Law}(B_t)$.  Then it is known that, if $V$ is a finite-dimensional subspace\nof $\\mathfrak{g}_{CM}$ and $\\pi_V$ is the orthogonal projection from\n$\\mathfrak{g}_{CM}$ to $V$, then $\\pi_V$ admits a $\\mu_t$-a.s.~unique\nextension to $\\mathfrak{g}$.  Moreover, if $V_n$ is an increasing sequence of\nfinite-dimensional subspaces, then \n\\[ \\lim_{n\\rightarrow\\infty} \\mathbb{E}\\|\\pi_{V_n}B_t- B_t\\|_\\mathfrak{g}^2 = 0;  \\]\nsee for example Section 8.3.3 of \\cite{Stroock2011}.\n\nBy Proposition \\ref{p.HS},\n$\\hat{F}_n^{\\sigma,\\alpha}$ is Hilbert-Schmidt, and recall that\n$\\tilde{f}_\\alpha$ is a deterministic polynomial function in $s$.  Thus\n$J_n^\\ell(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ and\n$J_n(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ are \n$\\mathfrak{g}_{CM}$-valued martingales as defined in Proposition \\ref{p.bad},\nand Proposition \\ref{p.bad} gives the desired convergence as well (in\n$\\mathfrak{g}_{CM}$ and thus in $\\mathfrak{g}$).\n\\end{proof}\n\n\\begin{remark}\nIn fact, for each of the stochastic integrals $J_n(\\tilde{f}_\\alpha\n\\hat{F}_n^{\\sigma,\\alpha})$, it is possible to prove the stronger convergence\nthat, for all $p\\in[1,\\infty)$,\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_\\tau -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_\\tau\\right\\|^p\n\t\\right] = 0, \\]\nfor all $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$ and\n$\\alpha\\in\\mathcal{J}_n$. Again, Proposition \\ref{p.bad} gives the limit for $p=2$ and thus for\n$p\\in[1,2]$.  \nFor $p>2$, Doob's maximal inequality implies it suffices to show that \n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\\right\\|^p  = 0.  \\] \nSince each $J_n^\\ell(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ and\n$J_n(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ has chaos expansion\nterminating at degree $n$, a theorem of Nelson (see Lemma 2 of\n\\cite{Nelson73b} and pp. 216-217 of \\cite{Nelson73c}) implies that,\nfor each $j\\in\\mathbb{N}$, there exists $c_j<\\infty$ such that \n\\[ \\mathbb{E}\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\t\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\t\\right\\|^{2j}\n\t\\le c_j \\left(\\mathbb{E}\\left\\|\n\t\tJ_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\t\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\t\\right\\|^2\\right)^j. \\]\n\\end{remark}\n\nIn a similar way, one may prove the following convergence for the Brownian\nmotions under right translations by elements of $G_{CM}$.\n\n\\begin{prop}\n\\label{p.rconv}\nFor any $y\\in G_{CM}$, \n\\[ \\lim_{\\ell\\rightarrow\\infty}\n\t\\mathbb{E}\\|g_t^\\ell y -g_t y\\|^2_{\\mathfrak{g}} = 0. \\]\nwhere $g_ty$ is the measurable right group action of $y\\in G_{CM}$ on $g_t\\in\nG$, as in Proposition \\ref{p.gp}.\n\\end{prop}\n\n\\iffalse\n\\begin{proof}\nNote that $g_t=B_t+I_t$ where $B_t$ is $\\mathfrak{g}$-valued Brownian motion\nand $I_t$ is a finite sum of stochastic integrals taking values in\n$\\mathfrak{g}_{CM}$.  Thus,\n\\begin{align*}\ng_t\\cdot h &= g_t+h+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_{g_t}^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_{g_t}^{n_k} \\mathrm{ad}_h^{m_k} g_t\n\\end{align*}\nFor any $x,y\\in W$, \n\\begin{align*} \n\\mathrm{ad}_{x+y}&^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_{x+y}^{n_k} \\mathrm{ad}_h^{m_k} (x+y) \\\\\n\t&= (\\mathrm{ad}_x + \\mathrm{ad}_y)^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t(\\mathrm{ad}_{x} +\\mathrm{ad}_{y})^{n_k} \\mathrm{ad}_h^{m_k} (x+y)\n\t\t\\\\\n\t&= \\sum_{\\ell}\n\t\t\\mathrm{ad}_x^{\\ell_1^1}\\mathrm{ad}_{y}^{\\ell_2^1} \\cdots\n\t\t\\mathrm{ad}_x^{\\ell_{n_1-1}^1}\\mathrm{ad}_{y}^{\\ell_{n_1}^1} \\mathrm{ad}_h^{m_1}\\cdots\n\t\t\\mathrm{ad}_x^{\\ell_1^k}\\mathrm{ad}_{y}^{\\ell_2^k} \\cdots\n\t\t\\mathrm{ad}_x^{\\ell_1^k}\\mathrm{ad}_{y}^{\\ell_2^k} \\mathrm{ad}_h^{m_k} (x+y)\n\\end{align*}\n\n\\end{proof}\n\\fi\n\n\n\\begin{remark}\n\tNote that, while the present paper focuses on the case where $\\mu$ is non-degenerate and $B$ is Brownian motion on $G$, the above construction and finite-dimensional approximations would all follow with essentially no modification if one considered instead a Gaussian measure $\\mu$ whose support was, for example, a subspace $\\frak{h}$ of $\\frak{g}$ such that $\\frak{h}$ generates the span of $\\frak{g}$ via the Lie bracket.\n\\end{remark}\n\n\n\\subsection{Quasi-invariance and log Sobolev}\n\\label{s.hki}\n\n\n\nWe are now able to prove Theorem \\ref{t.quasi}, which states that the heat kernel measure $\\nu_t =\n\\mathrm{Law}(g_t)$ is\nquasi-invariant under left and right translation by elements of $G_{CM}$ and\ngives estimates for the Radon-Nikodym derivatives of the ``translated'' measures. Given the results so far, the proof could be given as an application of Theorem 7.3 and Corollary 7.4 of \\cite{DriverGordina2009}. However, we provide here a full proof for the reader's convenience.\n\n{\\it Proof of Theorem \\ref{t.quasi}.}\nFix $t>0$ and $\\pi_0$ an orthogonal projection onto a finite-dimensional\nsubspace $G_0$ of $\\mathfrak{g}_{CM}$.  Let $h\\in G_0$, and\n$\\{\\pi_n\\}_{n=1}^\\infty$ be an increasing sequence of projections such that\n$G_0\\subset \\pi_nG_{CM}$ for all $n$ and $\\pi_n|_{G_{CM}}\\uparrow\nI_{G_{CM}}$.  Let $J^{n,r}_t(h,\\cdot)$ denote the Radon-Nikodym derivative of\n$\\nu_t^n\\circ R_h^{-1}$ with respect to $\\nu_t^n$. Then for each $n$ and for\nany $q\\in[1,\\infty)$, we have the following integrated Harnack inequality\n\\[ \\left(\\int_{G_n} \\left(J^{n,r}_t(h,g)\\right)^q\n\td\\nu_t^n(g)\\right)^{1\/q} \\le\n\\exp\\left(\\frac{(q-1)k}{2(e^{kt}-1)}d_n(e,h)^2\\right)\n\\]\nwhere $k$ is the uniform lower bound on the Ricci curvature as in Proposition\n\\ref{p.Ric} and $d_n$ is Riemannian distance on $G_n$;\nsee for example Theorem 1.6 of \\cite{DriverGordina2009}.\n\n\nBy Proposition \\ref{p.rconv}, we have that for any $f\\in C_b(G)$, the class of bounded continuous\nfunctions on $G$\n\\begin{equation}\n\\label{e.5.7}\n\\begin{split}\n\\int_{G} f(gh) d \\nu_{t}(g)\n\t&= \\mathbb{E}[f(g_th)] \\\\\n\t&= \\lim_{n\\rightarrow\\infty} \\mathbb{E}[f(g_t^nh)]\n\t= \\lim_{n\\rightarrow\\infty}\\int_{G_{n}} (f\\circ i_n)(gh) \\,\nd\\nu_t^{n}(g),\n\\end{split}\n\\end{equation}\nwhere $ i_n:G_n\\rightarrow G$ denotes the inclusion map.  Note that for any $n$\n\\begin{align*}\n\\int_{G_n} |(f\\circ i_n)(gh)|\\,d\\nu_t^n(g)\n\t&= \\int_{G_n} J^{n,r}_t(h,g)|(f\\circ i_n)(g)|d\\nu_t^n(g) \\\\\n\t&\\le \\|f\\circ i_n\\|_{L^{q'}(G_n,\\nu_t^n)} \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d_n(e,h)^2\n\t\t\\right),\n\\end{align*}\nwhere $q'$ is the conjugate exponent to $q$.\nAllowing $n\\rightarrow\\infty$ in this last inequality yields\n\\begin{equation}\n\\label{e.c}\n\\int_G |f(gh)|\\,d\\nu_t(g)\n\t\\le \\|f\\|_{L^{q'}(G,\\nu_t)} \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right),\n\\end{equation}\nby equation \\eqref{e.5.7} and the fact that\nthe length of a path in $G_{CM}$\ncan be approximated by the lengths of paths in the finite-dimensional\nprojections.  That is, for any\n$\\pi_0$ and $\\varphi\\in C^1([0,1],G_{CM})$ with\n$\\varphi(0)=\\mathbf{e}$, there exists an increasing sequence\n$\\{\\pi_n\\}_{n=1}^\\infty$ of orthogonal projections such that $\\pi_0\\subset \\pi_n$,\n$\\pi_n|_{\\mathfrak{g}_{CM}}\\uparrow I_{\\mathfrak{g}_{CM}}$, and \n\\[ \\ell_{CM}(\\varphi) = \\lim_{n\\rightarrow\\infty}\n\t\\ell_{G_{\\pi_n}}(\\pi_n\\circ\\varphi). \\]\nTo see this, let $\\varphi$ be a path in $G_{CM}$.  Then one may show that\n\\begin{align*}\n\\ell_{G_{\\pi_n}}(\\pi_n\\circ\\varphi)\n\t&=\\int_0^1 \\left\\|\\pi_n\\varphi'(s) + \\sum_{\\ell=1}^{r-1}\n\t\tc_\\ell \\mathrm{ad}_{\\pi_n\\varphi(s)}^\\ell \\pi_n\\varphi'(s)\n\t\t\\right\\|_{\\mathfrak{g}_{CM}}\\,ds \n\\end{align*}\nfor appropriate coefficients $c_\\ell$; see for example Section 3 of\n\\cite{Melcher2009}.\nThus, we have proved that \\eqref{e.c} holds for $f\\in C_b(G)$ and $h\\in\n\\cup_{\\pi} G_\\pi$.  As this union is dense in $G$ by\nProposition \\ref{p.length},\ndominated convergence along with the continuity of $d(e,h)$ in $h$ implies\nthat \\eqref{e.c} holds for all $h\\in G_{CM}$.\n\nSince the bounded continuous functions are dense in $L^{q'}(G,\\nu_t)$ (see for\nexample Theorem A.1 of \\cite{Janson1997}), the inequality in (\\ref{e.c}) implies that the\nlinear functional $\\varphi_h:C_b(G)\\rightarrow\\mathbb{R}$ defined by\n\\[ \\varphi_h(f) = \\int_G f(gh)\\,d\\nu_t(g) \\]\nhas a unique extension to an element, still denoted by\n$\\varphi_h$, of $L^{q'}(G,\\nu_t)^*$ which satisfies the bound\n\\[ |\\varphi_h(f)| \\le \\|f\\|_{L^{q'}(G,\\nu_t)}\n\t\\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right) \\]\nfor all $f\\in L^{q'}(G,\\nu_t)$.  Since $L^{q'}(G,\\nu_t)^*\\cong L^q(G,\\nu_t)$, there\nthen exists a function $J_t^r(h,\\cdot)\\in\nL^q(G,\\nu_t)$ such that\n\\begin{equation}\n\\label{e.d}\n\\varphi_h(f) = \\int_G f(g)J_t^r(h,g)\\,d\\nu_t(g),\n\\end{equation}\nfor all $f\\in L^{q'}(G,\\nu_t)$, and\n\\[ \\|J_t^r(h,\\cdot)\\|_{L^q(G,\\nu_t)}\n\t\\le \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right). \\]\n\nNow restricting (\\ref{e.d}) to $f\\in C_b(G)$, we may rewrite this equation as\n\\begin{equation}\n\\label{e.last}\n\\int_G f(g)\\,d\\nu_t(gh^{-1})\n\t= \\int_G f(g) J_t^r(h,g)\\,d\\nu_t(g).\n\\end{equation}\nThen a monotone class argument (again use Theorem A.1 of\n\\cite{Janson1997}) shows that (\\ref{e.last}) is valid for all\nbounded measurable functions $f$ on $G$.  Thus,\n$d(\\nu_t\\circ R_h^{-1})\/d\\nu_t$ exists and is given by $J_t^r(h,\\cdot)$, which is in\n$L^q$ for all $q\\in(1,\\infty)$ and satisfies the desired bound.\n\nA parallel argument gives the\nanalogous result for $d(\\nu_t\\circ L_h^{-1})\/d\\nu_t$.  Alternatively, one\ncould use the right translation invariance just proved along with the facts that $\\nu_t$\ninherits invariance under the inversion map $g\\mapsto g^{-1}$ from its\nfinite-dimensional projections and that $d(e,h^{-1})=d(e,h)$.\n\\hfill$\\square$\n\n\nThe following also records the straightforward fact that the heat kernel measure does not charge $G_{CM}$.\n\n\\begin{prop}\nFor all $t>0$, $\\nu_t(G_{CM})=0$.\n\\end{prop}\n\n\\begin{proof}\nThis follows trivially from the fact that $g_t$ is the sum of a Brownian motion $B_t$ on\n$\\mathfrak{g}$ with a finite sequence of stochastic integrals taking values in\n$\\mathfrak{g}_{CM}$.\n\\end{proof}\n\nThus, $G_{CM}$ maintains its role as a dense subspace of $G$ of\nmeasure 0 with respect to the distribution of the ``group Brownian\nmotion''.  \n\n\\begin{defn}\n\\label{d.cyl} \nA function $f:G\\rightarrow\\mathbb{R}$ is said to be a\n{\\it (smooth) cylinder function} if $f=F\\circ\\pi$ for some\nfinite-dimensional projection $\\pi$ and\nsome (smooth) function $F:G_\\pi\\rightarrow\\mathbb{R}$.  Also, $f$ is a \n{\\it cylinder polynomial} if $f=F\\circ\\pi$ \nfor $F$ a polynomial function on $G_\\pi$.\n\\end{defn}\n\n\n\\begin{thm}\n\\label{t.logsob}\nGiven a cylinder polynomial $f$ on $G$, let\n$\\nabla f:G\\rightarrow\\mathfrak{g}_{CM}$ be the gradient of $f$,\nthe unique element of $\\mathfrak{g}_{CM}$ such that\n\\[ \\langle\\nabla f(g), h \\rangle_{\\mathfrak{g}_{CM}} = \\tilde{h}f(g)\n\t:= f'(g)(L_{g*}h_\\mathbf{e}), \\]\nfor all $h\\in\\mathfrak{g}_{CM}$.  \nThen for $k$ as in Proposition \\ref{p.Ric},\n\\[ \\int_G (f^2\\ln f^2)\\,d\\nu_t -\n\t\t\\left(\\int_G f^2\\,d\\nu_t \\right)\\cdot\\ln\\left(\\int_G\n\t\tf^2\\,d\\nu_t\\right)\n\t\\le 2\\frac{1-e^{-kt}}{k} \\int_G \\|\\nabla f\\|_{\\mathfrak{g}_{CM}}^2\n\t\t\\,d\\nu_t. \\]\n\\end{thm}\n\n\\begin{proof}\nFollowing the method of Bakry and Ledoux applied to $G_P$ (see Theorem 2.9 of \n\\cite{Driver96} for the case needed here) shows that\n\\[ \\mathbb{E}\\left[\\left(f^2\\ln f^2\\right)\\left(g^\\pi_t\\right)\\right] \n\t\t- \\mathbb{E}\\left[f^2\\left(g^\\pi_t\\right)\\right] \n\t\t\\ln\\mathbb{E}\\left[f^2\\left(g_t^\\pi\\right)\\right]\n\t\\le 2 \\frac{1 - e^{-k_\\pi t}}{k_\\pi} \\mathbb{E}\\left\\|(\\nabla^\\pi\n\t\tf)\\left(g^\\pi_t\\right)\\right\\|^2_{\\mathfrak{g}_\\pi},\n\\]\nfor $k_\\pi$ as in equation (\\ref{e.pah}).  Since the function $x\\mapsto\n(1-e^{-x})\/x$ is decreasing and $k\\le k_\\pi$ for all finite-dimensional\nprojections $\\pi$, \nthis estimate also holds with $k_\\pi$ replaced with $k$.  Now applying\nProposition \\ref{p.approx} to pass to the limit as $\\pi\\uparrow I$ gives \nthe desired result.\n\\end{proof}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{comment}\nThroughout this paper, the term ``knot\" is used both for a knot and a link. That is, a knot in this paper may have more than one component. In the classical knot theory, knots are treated as combinatorial\nobjects, for example as equivalence classes of polygons in the\nthree-dimensional space or as equivalence classes of diagrams that are related by Reidemeister moves.\nAiming at classifying knots, many knot invariants have been\nintroduced, for instance various polynomials, or integer valued invariants such as the bridge index, the braid index of the genus \\cite{BZ, cromwell2004} .\nWhile the classification of all knots is wide open, many knot families have been classified.\nFor example the 2-bridge knots have been completely classified in the 1950s\nby the work of Schubert~\\cite{schubert}.\nThe latter has stimulated the development of the theory of $3$-manifolds.\n\\end{comment}\n\nIn his famous paper \\cite{milnor}, Milnor showed how a knot invariant may be related to geometric properties of the space curve representing a knot by providing a lower bound of the\ntotal curvature of an embedded closed curve explicitly in terms of its bridge index of the knot the curve represents. In a similar spirit, Langer and Singer~\\cite{LS} raised a question concerning the equilibrium shape (a physical property) of a closed springy wire and the knot invariants of the knot the closed wire represents. Gallotti and Pierre-Louis~\\cite{gallotti} conjectured that the equilibrium shapes within knot types $K$ where braid index $\\qopname\\relax o{braid}(K)$ and\nbridge index $\\qopname\\relax o{bridge}(K)$ coincide, will be the $\\qopname\\relax o{bridge}(K)$-times covered circle. This conjecture has recently been proven in the case of 2-bridge knots~\\cite{GRvdM}. For the sake of convenience, we shall call a knot type  whose bridge index and braid index coincide a {\\em BB knot}.\n\nHeuristically, it is plausible that the equilibrium shape\nof a very thin springy wire should be, if topologically possible,\nvery close to a $k$-times covered circle.\nRecalling the F\\'ary--Milnor inequality~\\cite{milnor},\nthe minimum value would be\n$k=\\qopname\\relax o{bridge}(K)$.\nOn the other hand, a configuration of a wire that passes\n$\\qopname\\relax o{bridge}(K)$-times around a circle constitutes in\nfact a braid presentation which requires that $k\\ge \\qopname\\relax o{braid}(K)$\nwhich by $\\qopname\\relax o{braid}(K)\\ge\\qopname\\relax o{bridge}(K)$ implies $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)$. The left of Figure \\ref{experiment} shows an experiment of the equilibrium state of the figure eight knot realized by a closed springy wire, which is clearly not close to a multiply covered circle. Notice that the figure eight knot is not a BB knot since its bridge index is 2 and braid index is 3. On the other hand, our numerical simulations\nindicate that an equilibrium state of a BB knot $K$ is\nclose to a $\\qopname\\relax o{bridge}(K)$-times covered circle as shown in Figure \\ref{fig:sims}.\n\nThe class of BB knots might also be of potential interest to scientists looking for potential candidates of knot types that can be constructed via molecular knots.\nIn~\\cite{micheletti} the authors give a list of knots that either have been constructed or could potentially be constructed as different molecular knots. At least in the symmetric cases, we observe that there seems to be a prevalence for BB knots, in particular among knot classes with larger crossing number.\n\nIn this light, it is a natural question to investigate the\nfamily of all knot or link classes $K$ with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)$. This is the main focus of this paper.\nAs a first step towards a complete classification (which might or might\nnot be feasible) we searched the database KnotInfo \\cite{knotinfo} and found that there are 182 one component BB knots with crossing number up to $12$ (they are listed in Table~\\ref{table}).\nNext, we consider certain families of classes\n(namely torus knots\/links, 2-bridge knots\/links, Montesinos links,\nConway algebraic knots)\nand identify (infinite) subfamilies of BB knots within. \n\nMoreover, there are a number of interesting questions\nconcerning the number $\\mathcal B_{n}$ of BB knots with crossing number $n$.\nDoes $\\mathcal B_{n}$ grow exponentially?\nWhat about the ratio of $\\mathcal B_{n}$ over the number of \\emph{all} knots\nwith with crossing number $n$~?\n\n\\begin{figure}[!h]\n\\label{experiment}\n\\includegraphics[scale=.18,trim=20 0 180 0,clip]{figure8_1}\n\\quad%\n\\rotatebox[origin=c]{-20}{\\reflectbox{\\includegraphics[scale=.36]{bribra\/4_1.png}}}\n\\caption{Left: An experiment with a springy wire belonging to the figure-eight class (Wire model manufactured by \\textsc{why knots}, Aptos, in 1980, photographed by Bernd Bollwerk, Aachen);\nRight: A  local minimizer produced by numerical simulation featuring nearly the same shape (see~\\cite{BR} for details).\n}\n\\end{figure}\n\n\nBefore we proceed to the next section, we need to clarify a couple of key terms and concepts used in this paper. First,  in what follows we will use the word ``knot'' to indicate both a knot type or a particular knot embedding. What we mean will depend on the context and should be clear to the reader. We will use the term ``knot class'' when we talk about the set of embeddings of a knot type with certain geometric restrictions.\nMoreover, the term ``knot\" in this paper is also used for the commonly used term ``link\" in other literature. That is, a knot in this paper may have more than one component. When there is a need, we shall make this clear by specifying the number of components in the knot being discussed. For example, the BB knots listed in Table \\ref{table} are all knots with one component. Second, in the definition of BB knots, the knots are un-oriented (since orientation plays no role to the physical property of interest that leads to the definition of BB knots). However our approach to this problem requires the use of the braid index defined for oriented knots. We would like to point out the connection between these two definitions to avoid confusion. Let $K$ be an un-oriented knot, $\\vec{K}$ be an oriented knot corresponding to $K$ (that is, each component in $K$ has been assigned an orientation) and $\\qopname\\relax o{braid}(\\vec{K})$ be the braid index of $\\vec{K}$ (as an oriented knot). Then un-oriented braid index $\\qopname\\relax o{braid}(K)$ of $K$, which is what we have used in the above in the definition of BB knots, is defined as the minimum of $\\vec{K}$ where $\\vec{K}$ runs over all possible orientation assignments to the components of $K$. It is known (a rather obvious fact) that $\\qopname\\relax o{braid}(\\vec{K})\\ge \\qopname\\relax o{bridge}(K)$. Thus if $\\qopname\\relax o{braid}(\\vec{K})= \\qopname\\relax o{bridge}(K)$ for some $\\vec{K}$,  then we must have $\\qopname\\relax o{braid}(\\vec{K})= \\qopname\\relax o{braid}(K)$. Of course, if $k$ has only one component then for any choice of orientation $\\qopname\\relax o{braid}(\\vec K)=\\qopname\\relax o{braid} (K)$.\n\n\n\nWe organize the rest of the paper as follows: In Sections \\ref{sec:bend} and \\ref{numerics} we discuss BB knots made of thin springy wires (so called \\emph{elastic knots}) and consider the  bending energy of elastic knots in a numerical experiment. \nIn Section \\ref{knotfamilies} we identify BB knots among several knot families. In Section \\ref{numberofknots} we show that the number of BB knots with a given crossing number grows exponentially as a function of the crossing number\nand state a few questions.\n\n\\section{Bending energy minimizers within knot classes}\\label{sec:bend}\n\nThe aim of this section is to explain how the BB knots appear in an elementary model\nfor the experiment described in the introduction.\nFor convenience, we will present here a simplified version\nof the model of \\emph{elastic knots} which is described in Section~\\ref{numerics}\nbelow.\n\n\\subsection{A variational problem}\n\nNeglecting all other physical forces such as friction, shear, and twist,\nwe will assume that the behavior of that springy wire is only affected\nby the bending energy of its centerline $\\gamma:\\mathbb{R}\/\\mathbb{Z}\\to\\mathbb{R}^{3}$\n\\[ E_{\\mathrm{bend}}(\\gamma)=\\int_{\\gamma}(\\kappa(s))^{2} ds \\]\nwhere $\\kappa(s)$ denotes the curvature of $\\gamma$ at $s$\nwhile $s$ is an arc-length parameter. Without loss of generality\nwe may assume that $\\gamma$ is parametrized by arc-length which gives\n$E_{\\mathrm{bend}}(\\gamma)=\\|\\gamma''\\|_{L^{2}}^{2}$.\n\nOur aim would be to find global minimizers of $E_{\\mathrm{bend}}$ within a given knot class $K$.\nAs $E_{\\mathrm{bend}}$ is not invariant under scaling, we will consider only unit length embeddings\nwhich leads to considering the set\n\\[ \\mathscr C(K) = \\left\\{\\gamma\\in W^{2,2}(\\mathbb{R}\/\\mathbb{Z},\\mathbb{R}^{3}) \\middle|\n|\\gamma'(s)|\\equiv1, \\gamma(0)=0_{\\mathbb{R}^{3}},\\gamma\\in K\\right\\}. \\]\nHere we fix $\\gamma(0)$ just for technical reasons.\n\nWe will always assume that $K$ is a non-trivial tame knot class,\nfor the other cases are uninteresting.\nThe global minimizer of $E_{\\mathrm{bend}}$ within the unknot class is\njust the circle.\nAs wild knots are not $C^{1}$ and $W^{2,2}\\hookrightarrow C^{1,1\/2}$\nby the Sobolev embedding theorem, $\\mathscr C(K)=\\emptyset$ for wild $K$.\n\n\\subsection{Minimizers}\n\nIt turns out that, except for the trivial knot class, there are no such minimizers:\nAny potential minimizer within $K$ would be $C^{1}$ (due to the Sobolev embedding\n$W^{2,2}\\hookrightarrow C^{1,1\/2}$) and embedded (since it is a knot).\nAccording to~\\cite{DEJvR} %\nthere is a $C^{1}$-neighborhood consisting of knotted curves belonging to $K$ as well.\nIn particular, this neighborhood contains a $W^{2,2}$-neighborhood\nwith the same property which demonstrates that $\\gamma$ is not only\na global minimizer of $E_{\\mathrm{bend}}$ within $K$ but also a local minimizer\nof $E_{\\mathrm{bend}}$ with respect to \\emph{all} curves.\nNow, by Langer and Singer's seminal result~\\cite{LS},\nthe circle is the only curve with the latter property.\nThis fact reflects the observation that there are indeed points of\nself-contact present in physical models.\n\nNevertheless we can consider a minimal sequence $(\\gamma_{k})_{k\\in\\mathbf N}$ with respect to $E_{\\mathrm{bend}}$\nin $\\mathscr C(K)$,\n{\\em i.e.}, $\\gamma_{k}\\in\\mathscr C(K)$ for all $k\\in\\mathbf N$ with $E_{\\mathrm{bend}}(\\gamma_{k})\\to\\inf_{\\mathscr C(K)}E_{\\mathrm{bend}}$ as $k\\to\\infty$.\nBy the fact that $E_{\\mathrm{bend}}(\\gamma)=\\|\\gamma''\\|_{L^{2}}^{2}$\nand $\\gamma(0)=0_{\\mathbb{R}^{3}}$ for all $\\gamma\\in\\mathscr C(K)$\nwe deduce that this minimal sequence is uniformly bounded in $W^{2,2}$.\nBeing a reflexive space, bounded sets in $W^{2,2}$\nare weakly compact.\nTherefore we may extract a subsequence converging to some limit curve $\\gamma_{0}$\nwith respect to the weak $W^{2,2}$-topology.\nAny such limit curve belongs to the weak $W^{2,2}$-closure\nof $\\mathscr{C}(K)$ which will be denoted by $\\overline{\\mathscr{C}(K)}$.\n\nPotentially there can be many of those limit curves, depending on the\nminimal sequence.\nAny of these limit curves has the following properties:\nIt is not embedded (due to~\\cite{LS}, unless $K$ is trivial), so it\nbelongs to the weak $W^{2,2}$-\\emph{boundary} of $\\mathscr C(K)$.\nIts bending energy provides\na lower bound on the bending energy of any curve in $\\mathscr C(K)$.\nThis is due to the fact that $E_{\\mathrm{bend}}$ is lower \\emph{semi}continuous with respect to weak $W^{2,2}$-convergence.\n\n\\subsection{The case of BB knots}\n\nRecall that the F\\'ary--Milner inequality~\\cite{milnor} bounds the\ntotal curvature $\\int\\kappa(s)ds$ of a curve belonging to $K$ by $2\\pi \\qopname\\relax o{bridge}(K)$ such that\nwe have for $\\gamma\\in\\mathscr C(K)$\n\\[ 2\\pi \\qopname\\relax o{bridge}(K)<\\int_{\\gamma}\\kappa(s)ds=\\|\\gamma''\\|_{L^{1}}. \\]\nAs $\\gamma\\mapsto{}\\|\\gamma''\\|_{L^{1}}$ is not continuous with respect to weak\n$W^{2,2}$-convergence it is not clear whether this lower bound\nalso holds for $\\overline{\\mathscr C(K)}$.\nIn fact, using the existence of \\emph{alternating} quadrisecants\nestabished by Denne~\\cite{denne}, this has been\nproven~\\cite[Appendix]{GRvdM} for $\\qopname\\relax o{bridge}(K)=2$. In the following\nwe \\emph{assume} that it also holds for higher-order bridge indices.\nInvoking the Cauchy--Schwarz inequality, the limit curve $\\gamma_{0}\\in\\overline{\\mathscr C(K)}$ satisfies\n\\[ 2\\pi \\qopname\\relax o{bridge}(K)\\le\\int_{\\gamma_{0}}\\kappa(s)ds=\\|\\gamma_{0}''\\|_{L^{1}}\n\\le\\|\\gamma_{0}''\\|_{L^{2}} = \\left(E_{\\mathrm{bend}}(\\gamma_{0})%\n\\right)^{1\/2}. \\]\nUsing a braid representation we can construct\n$C^{2}$-smooth (even $C^{\\infty}$-smooth) curves belonging to $K$ inside the standard\n$\\varrho$-torus in $\\mathbb{R}^{3}$, {\\em i.e.}, the uniform neighborhood of\nwidth $\\varrho\\in(0,1)$ of the unit circle,\nsuch that each disk of the $\\varrho$-torus is intersected $\\qopname\\relax o{braid}(K)$-times.\nTranslating and rescaling,\nwe obtain a family $(\\beta_{\\varrho})_{\\varrho\\in(0,1)}\\subset\\mathscr C(K)$ such that \n$\\beta_{\\varrho}$ converges (with respect to the $W^{2,2}$-norm) to a $\\qopname\\relax o{braid}(K)$-times\ncovered circle of length one as $\\varrho\\searrow0$.\nOf course, the latter has bending energy $(2\\pi \\qopname\\relax o{bridge}(K))^{2}$.\nAs $\\gamma_{0}$ is an $E_{\\mathrm{bend}}$-minimizer within $\\overline{\\mathscr C(K)}$,\nwe arrive, for any $\\varrho\\in(0,1)$, at\n\\[ (2\\pi \\qopname\\relax o{bridge}(K))^{2}\\le\\left(\\int_{\\gamma_{0}}\\kappa(s)ds\\right)^{2}=\\|\\gamma_{0}''\\|_{L^{1}}^{2}\n\\le\\|\\gamma_{0}''\\|^2_{L^{2}} = E_{\\mathrm{bend}}(\\gamma_{0}) \\le E_{\\mathrm{bend}}(\\beta_{\\varrho})\n\\xrightarrow{\\varrho\\searrow0} (2\\pi \\qopname\\relax o{braid}(K))^{2}. \\]\nHere the condition $\\qopname\\relax o{braid}(K)=\\qopname\\relax o{bridge}(K)$ comes into play for it implies that\nall terms in the previous line are equal. Equality in the Cauchy--Schwarz inequality implies\nthat the integrand is constant a.e. Therefore the curvature of the minimizer\n$\\gamma_{0}$ must be equal to $2\\pi \\qopname\\relax o{bridge}(K)$ a.e.\n\n\\subsection{Caveat}\n\nIt is important to note that the latter condition does \\emph{not}\nimply that $\\gamma_{0}$ is a $\\qopname\\relax o{bridge}(K)$-times covered circle.\nIndeed, that would be wrong.\nIn case $\\qopname\\relax o{bridge}(K)=2$ one can rigorously prove~\\cite{GRvdM} that\nany $E_{\\mathrm{bend}}$ minimizer $\\gamma_{0}$ within $\\mathscr C(K)$\nconsists of two circles that either coincide or tangentially meet in\nprecisely one point.\nUp to isometries and reparametrization, the set of those minimizers can be\nparametrized by the angle between the two circles.\nIn the general situation we would expect a number of $\\qopname\\relax o{bridge}(K)$ circles\ntangentially meeting in (at least) one point.\n\nHowever, adding a positive thickness restriction to $\\mathscr C(K)$ or a penalty term to $E_{\\mathrm{bend}}$\nacts like a choice criterion that selects the $\\qopname\\relax o{bridge}(K)$-times covered circle from that family of $E_{\\mathrm{bend}}$-minimizers.\nThis will be outlined below in Section~\\ref{numerics}.\n\n\\section{BB knots realized by elastic wires}\\label{numerics}\n\n\n\\subsection{Elastic knots}\n\nIn Section~\\ref{sec:bend} above we considered the problem\nto minimize $E_{\\mathrm{bend}}$ in the class $\\mathscr C(K)$ of curves\nin the knot class~$K$. In order to do so, we looked at a\nminimal sequence of curves and extracted a subsequence\nthat converges to some limit curve $\\gamma_{0}$.\nThe problem is that these limit curves are not unique.\nFor instance, in the case of 2-bridge torus knots ({\\em i.e.}, the knots $T(n,2)$) we\nobtain an entire one-parameter family of minimizers.\nWhich of them is the most ``realistic'' one,\n{\\em i.e.}, one that would be observed in physical experiments?\n\nTo answer this question,\nwe need to incorporate the fact that we are looking at\nphysical ropes that have some very small thickness in our simulation model.\nThis can be done by either imposing a constraint to the space of curves~\\cite{gallotti,vdM:eke3}, \nor by adding a penalty term to the bending energy~\\cite{sossinsky,GRvdM,vdM:meek}.\nWe will discuss the latter approach, {\\em i.e.}, we shall adopt the following model\n\\[ E_{\\vartheta} = E_{\\mathrm{bend}}+\\vartheta\\mathcal R, \\qquad\\vartheta>0, \\]\nwhere $\\mathcal R$ denotes a self-avoiding functional\nsuch as the ropelength, {\\em i.e.}, the quotient of length over thickness~\\cite{Buck,DEJvR,GM99}.\nNow the minimization problem is well defined~\\cite{GRvdM,vdM:meek},\n{\\em i.e.}, for any $\\vartheta>0$ there is a curve $\\gamma_{\\vartheta}\\in\\mathscr C(K)$\nsuch that $E_{vartheta}(\\gamma_{\\vartheta})=\\inf_{\\mathscr C(K)} E_{\\vartheta}$.\nIn particular, for $\\vartheta\\in(0,1)$ we have that\n\\[ \\|\\gamma_{\\vartheta}''\\|_{L^{2}}^{2}\n\\le E_{\\vartheta}(\\gamma_{\\vartheta})\n= \\inf_{\\mathscr C(K)} E_{\\vartheta}\n\\le E_{\\vartheta}(\\gamma_{1})\n\\le E_{\\mathrm{bend}}(\\gamma_{1}) + \\mathcal R(\\gamma_{1}) \\]\nis uniformly bounded, %\nso we can extract a subsequence as detailed in Section~\\ref{sec:bend}.\nThe limit curve $\\gamma_{0}$ is referred to as an \\emph{elastic knot} for the knot class $K$~\\cite{GRvdM,vdM:meek}.\nNote that, unless $K$ is trivial, $\\gamma_{0}$ is not embedded.\n\nThere are other names for related concepts in the literature, namely\n``stiff knot'' (Gallotti and Pierre-Louis~\\cite{gallotti}),\nand ``normal form'' (Sossinsky who even thought of a complete classification of\nknot classes by this concept, see~\\cite{sossinsky} and references therein).\n\nNote that, a priori, we cannot expect either the minimizers\n$(\\gamma_{\\vartheta})_{\\vartheta>0}$ or the elastic knot $\\gamma_{0}$\nto be unique. A posteriori, one can show that the elastic knot\nfor the 2-bridge torus knot classes is (up to isometries) the doubly covered circle~\\cite{GRvdM}.\nThe proof relies on a generalization of the crookedness estimate in Milnor's proof~\\cite{milnor}.\nCurrently there are no rigorous results concerning elastic knots\nfor other knot classes. However, the $\\qopname\\relax o{braid}(K)$-times covered circle\nis a candidate for an elastic knot for any BB class~$K$.\nVice versa, one might speculate that the BB classes\nare the only ones whose elastic knots are several times\ncovered circles.\n\n\n\\subsection{Simulation of elastic knots}\n\nPhysical and numerical simulations related to elastic knots\nhave been carried out so far\nby Avvakumov and Sossinsky~\\cite{sossinsky}, Gallotti and Pierre-Louis~\\cite{gallotti},\n Gerlach et al.~\\cite{GRvdM}, as well as by one of the authors and colleagues recently in~\\cite{BR,BRR}.\nThe recently launched web application \\textsc{knotevolve}~\\cite{knotevolve} allows for carrying out a large variety of new experiments.\n\nNumerically, it is a challenging problem to approximate elastic knots as we face two forces which push the elastic knots in different directions. The experiments carried out in~\\cite{BR} shed some light on the energy landscape.\nFor instance, it is unlikely that the configuration shown in\nFigure~\\ref{experiment} is an elastic knot for the figure-eight class; see also the discussion in~\\cite[Section~6.3]{GiRvdM}.\n\n\nIn Figure~\\ref{fig:sims} we show some simulation results for BB knots with braid index up to 3. To this end, we applied the algorithm introduced in~\\cite{bartels13,BRR,BR,knotevolve}\nto the  regularization parameter $\\vartheta{}=10^{-4}$\nand initial configurations from the Knot Server~\\cite{knotserver}.\nFor the BB knots with braid index 3, these simulation results still show visible deviation from the\nthree-times covered circle, in spite of \nthe relatively small regularization parameter.\n\n\n\n\\newcommand{\\knot}[3][.2]{%\n\\fbox{\\begin{minipage}[b][38mm]{.22\\textwidth}\n\\includegraphics[scale=#1]{bribra\/#2_#3.png}\\par\\vfill\n\\includegraphics[scale=#1]{bribra\/#2_#3f.png}\n\\end{minipage}}\\makebox[0cm][r]{\\raisebox{1ex}{$#2_{#3}$\\ }}}\n\\newcommand{\\sknot}[3][.15]{%\n\\fbox{\\begin{minipage}[b][25mm]{.17\\textwidth}\n\\includegraphics[scale=#1]{bribra\/#2_#3.png}\\par\\vfill\n\\includegraphics[scale=#1]{bribra\/#2_#3f.png}\n\\end{minipage}}\\makebox[0cm][r]{\\raisebox{1ex}{$#2_{#3}$\\ }}}\n\n\n\\begin{figure}%\n\\knot3{1}\\hfill\n\\knot5{1}\\hfill\n\\knot7{1}\\hfill\n\\knot9{1}\\medskip\n\n\\sknot[.19]8{5}\\hfill\n\\sknot8{10}\\hfill\n\\sknot[.19]8{16}\\hfill\n\\sknot[.17]8{17}\\hfill\n\\sknot8{18}\\medskip\n\n\\knot8{19}\\hfill\n\\knot8{20}\\hfill\n\\knot8{21}\\hfill\n\\knot9{16}\n\n\\caption{Approximations of elastic knots for all thirteen BB classes\nwith crossing number at most nine.\nEach curve is shown from top and front view;\ncolors correspond to local curvature.\nThe knots in the first row as well as $8_{19}$ are torus knots.}\\label{fig:sims}\n\n\\end{figure}\n\n\n\n\n\\section{BB knot identification in knot families}\n\\label{knotfamilies}\n\nIn this section we will consider knot families including the torus knots, the 2-bridge knots, the alternating Montesinos knots, the non-alternating Montesinos knots, and the Conway algebraic knots. For the torus knots, the 2-bridge knots and the alternating Montesinos knots  we are able to identify all knots within these families that have equal bridge index and braid index. For the non-alternating Montesinos knots and the Conway algebraic knots, we have some partial results. The results for the torus knots and the 2-bridge knots are known, we decide to state them here for the sake of completeness. We would like to point out  that the  proof for the case of 2-bridge knots is new.\n\n\\subsection{Torus knots} \\label{A}\nEvery chiral pair of torus knots has a representative that can be presented by a pair of positive integers $p$, $q$ such that $p\\ge q\\ge2$ and is denoted by $T(p,q)$, with the number of components in  $T(p,q)$ given by $\\gcd(p,q)$.  It is well known that $\\qopname\\relax o{braid}(T(p,q))=\\qopname\\relax o{bridge}(T(p,q))=q$ \\cite{Mu2,S2} or \\cite{Sch2007} for a more recent proof. That is, every torus knot has equal bridge index and braid index. It is known that the crossing number of $T(p,q)$ is $(q-1)p$ \\cite{Mu2} hence the number of torus knots with a given crossing number $n$ is at most of the order of $n$. In other words, the number of torus knots will not contribute to the exponential growth of knots with equal bridge index and braid index. It is worthwhile for us to point out that when a torus knot has more than one component, it is assumed that all components are assigned parallel orientations. One component BB torus knots with crossing number up to 12 are listed in Table \\ref{table} marked with a superscript $^t$. \n\n\\subsection{ 2-bridge knots} \n\nThis case is easy to deal with, since the only knots that can arise a 2-braids are the $T(n,2)$ torus knots. Thus we have by default the following theorem:\n\n\\begin{theorem}\\label{torustwobridge} \nLet $K=B(\\alpha,\\beta)$ be a 2-bridge knot. Then $K$ is a BB knot if and only if $K$ is a  $T(n,2)$ torus knot.\n\\end{theorem}\n\n\nIn the following we give a second proof of the above theorem, introducing a method that will help us to deal with Montesinos knots in the next subsection and will be used in the proof of Theorem \\ref{exp_thm}. To explain this method we need to describe the family of 2-bridge knots in more detail.\nIt is known that every 2-bridge knot can be represented by an alternating diagram associated with two co-prime positive integers $0<\\beta<\\alpha$ in the following way. A vector $(a_1,a_2,...,a_n)$\nis called a {\\it standard continued fraction decomposition} of $\\frac{\\beta}{\\alpha}$ if $n$ is odd and all $a_i>0$ and\n$$\n\\frac{\\beta}{\\alpha}=\\frac{1}{a_1+\\frac{1}{a_{2}+\\frac{1}{.....\\frac{1}{a_n}}}}. \n$$\nIt may be necessary to allow $a_n=1$ in order to guarantee that the length of the vector $(a_1,a_2,...,a_n)$ is odd. Under these conditions the standard continued fraction expansion of $\\frac{\\beta}{\\alpha}$ is unique. An oriented 2-bridge knot (also called a {\\em 4-plat} or a {\\em rational knot}), denoted by $\\vec{K}=\\vec{B}(\\alpha,\\beta)$, is then presented by the {\\em standard diagram} as  shown in Figure \\ref{2bridgeone} using the vector $(a_1,a_2,...,a_n)$. Furthermore, without loss of generality for a standard diagram we can assign the component corresponding to the long arc at the bottom of Figure \\ref{2bridgeone} the orientation as shown, since 2-bridge knots are known to be invertible. When a 2-bridge knot has two components, there are two choices for the orientation of the other component. Usually the two different orientations of the other component lead to two different oriented 2-bridge knots \\cite{BZ, cromwell2004}  which may have different braid indices. For example, the braid index of the two-bridge link in Figure \\ref{2bridgeone} is ten, however if we re-orient one of the two components the braid index changes to nine, see Theorem \\ref{2bridge_theorem} below.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=1]{Figures2018\/fig11}\n\\caption{The 2-bridge knot $\\vec{B}(17426,5075)$ given by $(3,2,3,3,1,2,3,4,4)$.}\n\\label{2bridgeone}\n\\end{figure}\n\nSince all crossings corresponding to a given $a_i$ have the same crossing sign under the given orientation, we will define a signed vector $(b_1,b_2,...,b_n)$ where $b_i =\\pm a_i$ with its sign given by the crossing sign of the crossings corresponding to $a_i$. For example, for $\\vec{K}=\\vec{B}(17426,5075)$ with the orientation shown in Figure \\ref{2bridgeone} we obtain the signed vector $(3,2,3,3,-1,-2,-3,4,-4)$. In \\cite{DEHL2018} the following theorem was established:\n\n\\begin{theorem}\\label{2bridge_theorem} \\cite{DEHL2018} \nLet $\\vec{K}=\\vec{B}(\\alpha,\\beta)$ be an oriented 2-bridge link diagram with signed vector  $(b_1,b_2,...,b_{2k+1})$ in the standard form, then its braid index is given by\n\\begin{equation}\\label{2bridgeformula}\n\\textbf{b}(K)=1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}.\n\\end{equation}\n\\end{theorem}\n\nThe above theorem allows the computation of the braid index of any 2-bridge link using a minimal diagram. The computation of the braid index of an oriented 2-bridge link was first given by Murasugi \\cite{Mu} using a different method depending on a continued fraction expansion of $\\beta\/\\alpha$ using only even integers. That the two methods are equivalent was shown in \\cite{DEHL2018}. The formulation of Theorem \\ref{2bridge_theorem} allows us to prove Theorem \\ref{torustwobridge} in a way that can be generalized to Montesinos knots, see the next subsection.\n\n\n\\begin{proof}\nThe only if part is trivially true since every torus knot has equal bridge index and braid index as we discussed in Subsection \\ref{A}. Now, if $K=B(\\alpha,\\beta)$ is a BB knot, then we must have $\\qopname\\relax o{braid}(K)=2$ hence there exists an oriented version $\\vec{K}=\\vec{B}(\\alpha,\\beta)$ such that $\\qopname\\relax o{braid}(\\vec{K})=2$. By (\\ref{2bridgeformula}) we have \n$$\n2=1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2},\n$$\nwhere  $(b_1,b_2,...,b_{2k+1})$ is the signed vector of $\\vec{K}=\\vec{B}(\\alpha,\\beta)$. Notice that $\\mbox{\\rm sign}(b_1)=\\mbox{\\rm sign}(b_2)$ if $b_2\\not=0$. Furthermore, if $b_3>0$ then $b_2$ is even hence $b_2>1$. A proof of this can be found in \\cite{DEHL2018}, a reader can also prove this directly by considering the Seifert circle decomposition of $\\vec{K}$. We will prove the theorem in two separate cases:  $\\mbox{\\rm sign}(b_1)=1$ and $\\mbox{\\rm sign}(b_1)=-1$. \n\nCase 1. $\\mbox{\\rm sign}(b_1)=1$. If $\\mbox{\\rm sign}(b_{2k+1})=1$, then\n$$\n0=\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\ge \\frac{b_{2}}{2}\\ge 0,\n$$\nhence $b_2=0$. So $k=0$ and $K=T(n,2)$ for some integer $n\\ge 2$. On the other hand, if $\\mbox{\\rm sign}(b_{2k+1})=-1$, then \n\\begin{eqnarray*}\n&&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&\\ge &\n2+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k-1}\\frac{|b_{2j+1}|}{2}> 2,\n\\end{eqnarray*}\nwhich is a contradiction hence this case is not possible.\n\nCase 2. $\\mbox{\\rm sign}(b_1)=-1$. If $b_2=0$, then \n\\begin{eqnarray*}\n2&=&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&= &\n1+\\frac{|b_{1}|}{2},\n\\end{eqnarray*}\nhence $b_1=-2$ and $K$ is the Hopf link that is the mirror image of $T(2,2)$ discussed in Case 1 if we ignore the orientation. If $b_2\\not=0$, then $b_2<0$ and it is necessary that $b_1=-1$ and $b_3<0$ in this case (again this can be observed by considering the Seifert circle decomposition of $K$). It follows that \n\\begin{eqnarray*}\n2&=&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&\\ge &\n1+\\frac{1+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\frac{1+|b_{3}|}{2}.\n\\end{eqnarray*}\nThus we must have $b_3=-1$ and $\\mbox{\\rm sign}(b_{2k+1})=-1$. However $b_3=-1$ implies either $b_4=0$ or $b_4>0$. Since $b_4>0$ leads to $\\qopname\\relax o{braid}(\\vec{K})>2$, we must have $b_4=0$, hence $k=1$ and the signed vector of $K$ is of the form $(-1,-(n-2),-1)$ for some $n>2$. This is the mirror image of the torus knot $T(n,2)$ discussed in Case 1 above if we ignore the orientation.\n\\end{proof}\n\n\\subsection {Alternating Montesinos knots} We now consider the set of all alternating Montesinos knots, a family that is much larger than the family of 2-bridge knots. In general, a Montesinos knot $K=M(\\beta_1\/\\alpha_1,\\ldots, \\beta_k\/\\alpha_s,\\delta)$ is a knot with a diagram as shown in Figure \\ref{Montesinos}, where each diagram within a topological circle (which is only for the illustration and not part of the diagram) is a rational tangle $A_j$ that corresponds to some rational number $\\beta_j\/\\alpha_j$ with $|\\beta_j\/\\alpha_j|<1$ and $1\\le j\\le s$ for some positive integer $s\\ge 2$, and $\\delta$ is an integer that stands for an arbitrary number of half-twists, see Figure \\ref{Montesinos}.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=.4]{Figures2018\/fig14}\n\\caption{A diagram depicting a general Montesinos knot with $s$ rational tangles and $\\delta$ horizontal half-twists. The arrows indicate the potential orientation assignments if the knot is to be oriented.}\n\\label{Montesinos}\n\\end{figure}\n\nThe bridge index of a general Montesinos knot (alternating or nonalternating) is known and given by the following theorem.\n\n\\begin{theorem} \\cite{BoZi} \n\\label{bridgeMknot}\nLet $K=M(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be a Montesinos knot with $|\\beta_j\/\\alpha_j|<1$, $1\\le j\\le s$, then \n$\\qopname\\relax o{bridge}(K)=s$.\n\\end{theorem} \n\nAn explicit formula for the braid index of any alternating Montesinos knot is a rather new result \\cite{DEHL2018}. The discussion from here to (and including) Theorem \\ref{Montesinos_formula} is modified from \\cite{DEHL2018}, as it is needed in order to understand the concepts and the formulas used in Theorem \\ref{Montesinos_formula}. \n\nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an oriented Montesinos knot. Following \\cite{DEHL2018}, we will use the following conventions on $\\vec{K}$.\nIf $\\vec{K}$ is alternating then all fractions $\\beta_j\/\\alpha_j$ have the same sign and this is matched by the sign of $\\delta$ representing the $|\\delta|$ half-twists. As in the case of 2-bridge knots the sign of $\\delta$ and the $\\beta_j\/\\alpha_j$ should not be confused with the sign of individual crossings. For example, the signs of the crossings represented by $\\delta$ may not coincide with the sign of $\\delta$. The sign of the crossings represented by $\\delta$ depends on the orientations of the two strings in the $\\delta$-half twists. Since a knot and its mirror image have the same braid index, for alternating Montesinos knots we only need to consider the case  $\\beta_j\/\\alpha_j>0$ for each $j$. \n\n\n\nand that the crossings in the tangle diagrams are as chosen in the standard drawings of 2-bridge knots as shown in Figure \\ref{tangle}. The conclusion we reach will then be applicable to the case $\\beta_j\/\\alpha_j<0$ for each $j$ that mirrors the Montesinos knots discussed below. Furthermore, if $s=2$,  the a Montesinos knot is actually a 2-bridge knot and thus we only need to consider the case $s\\ge 3$. \nFor our purpose, we can always orient the top long strand in a Montesinos knot diagram from right to left as shown in Figure \\ref{Montesinos} since reversing the orientations of all components in a knot does not change its braid index. \n\nWe will use a standard drawing for each rational tangle $A_j$ which is given by the continued fraction of the rational number $\\beta_j\/\\alpha_j$ and contains an odd number of positive entries, exactly like what we did in the case of 2-bridge knots. That is, we assume that $0<\\beta_j<\\alpha_j$ and $\\beta_j\/\\alpha_j$ has a continued fraction decomposition of the form $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$. The four strands that entering\/exiting each tangle are marked as NW, NE, SW and SE. One example is shown at the left of Figure \\ref{tangle}. Here we note that $a_{2q_j+1}^j$ is allowed to equal one if needed to ensure that the vector $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$ has odd length. We have\n$$\n\\frac{\\beta_j}{\\alpha_j}=\\frac{1}{a_1^j+\\frac{1}{a_2^j+\\frac{1}{.....\\frac{1}{a_{2q_j+1}^j}}}}. \n$$\nThe closure of a rational tangle is obtained by connecting its NW and SW end points by a strand and connecting its NE and SE end points with another strand (as shown at the left side of Figure \\ref{tangle}). This closure is called the denominator $D(A_j)$ of the rational tangle $A_j$. Notice that $D(A_j)$ results in a normal standard diagram of  the oriented 2-bridge knot $\\vec{B}(\\alpha_j, \\beta_j)$ given by the vector $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$ (as shown at the right side of Figure \\ref{tangle}). Finally, we define $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ by $|b^j_m|=a^j_m$ with its sign matching the signs of the corresponding crossings in $\\vec{K}$ under the given orientation. Notice that $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ may be different from that defined for 2-bridge knots since the orientations of the strands here are inherited from $\\vec{K}$. We will use the notation $A_j(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ to denote the tangle $A_j$ and the signed vector associated with it that is inherited from the orientation of $\\vec{K}$. Notice further that in a standard Montesinos knot diagram, each rational tangle has at the bottom a vertical row of $a_1$ twists corresponding to the condition $|\\beta_j\/\\alpha_j|<1$. (Tangles with $|\\beta_j\/\\alpha_j|\\ge1$ end with a row of horizontal twists on the right, and these twists can be combined via flypes with the $\\delta$ half twists to reduce the tangle to $|\\beta_j\/\\alpha_j| \\mod(1)$.)\n\n\\begin{figure}[htb!]\n\\includegraphics[]{fig15x.pdf}\n\\caption{Left: A standard drawing of the rational tangle $56\/191 = A(3,2,2,3,3)$; Right: The denominator $D(A(3,2,2,3,3))$ is a standard 2-bridge knot diagram of the 2-bridge knot $K(191,56)$.}\n\\label{tangle}\n\\end{figure}\n\nLet us now consider the Seifert circle decomposition of $\\vec{K}$ by first examining how the arcs of Seifert circles entering and exiting each $A_j(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ might look. Figure \\ref{decomp} lists all eight possibilities for these arcs. Small Seifert circles within each tangle are not shown in Figure \\ref{decomp}. Observing (from Figure \\ref{tangle}) that the SW and SE strands meet at the last crossing in $b_{1}^j$, therefore if these two strands belong to two different Seifert circles, then they must have parallel orientation. Thus (vi) and (viii) are not possible. Furthermore, since we have assigned the top long arc in the Montesinos knot diagram the orientation from right to left, (iii) is not possible either. We say that $A_j$ is of {\\em Seifert Parity 1} if it decomposes as (i) in Figure \\ref{decomp}, of {\\em Seifert Parity 2} if it decomposes as (ii) or (iv) in Figure \\ref{decomp} and of {\\em Seifert Parity 3} if it decomposes as (v) or (vii) in Figure \\ref{decomp}. Note that the Seifert Parity of a tangle $A_j$ is not a property of the tangle alone, but depends on the structure of the diagram that contains $A_j$. (It should not be confused with the the term {\\em parity of a tangle}, which denotes how the arcs in a tangle are connected.)\n\n\\begin{figure}[htb!]\n\\includegraphics{fig16x}\n\\caption{Of the eight cases listed, (iii), (vi) and (viii) are not possible.}\n\\label{decomp}\n\\end{figure}\n\n\\medskip\n\\begin{remark}\\label{bj_sign}{\\em \nBy assigning the appropriate orientations to the two strands at SW and SE in the bottom portion of Figure \\ref{tangle}, one can relate the sign of $b_1^j$ to the Seifert Parity of $A_j$ as follows: $b_1^j<0$ if $A_j$ is of Seifert Parity 1 or 2 and  \n$b_1^j>0$ if $A_j$ is of Seifert Parity 3.}\n\\end{remark}\n\n\\begin{comment}\nThus, it becomes clear that the Seifert circle decomposition of $\\vec{K}$ contains the following: the Seifert circle(s) that contain the top and bottom long strands in $\\vec{K}$ (called {\\em huge} Seifert circle(s)), the Seifert circles that do not contain these long strands, but contain strands that enter\/exit one or more tangles (these will be called {\\em large} Seifert circles), and Seifert circles that are completely within a tangle (and these will be called {\\em medium} Seifert circles if they share crossings with a large Seifert circle, and {\\em small} Seifert circles otherwise). Notice that small Seifert circles are generated by anti-parallel half twists given by a single entry $b_{i}^j$ of a tangle $A_j$. $\\vec{K}$ can be classified into one of the following three classes. \n\\end{comment}\n\nLet us call a Seifert circle in the Seifert circle decomposition of $\\vec{K}$ a {\\em long} Seifert circle if it contains either the top long strand or\/and the bottom long strand. $\\vec{K}$ can be partitioned into the following three classes. \n\nClass M1. The top long strand and the bottom long strand belong to two different long Seifert circles, and the bottom long strand also has the orientation from right to left. Notice that $\\vec{K}$ is of Class M1 if and only if every $A_j$ is of Seifert Parity 1 and all crossings in $\\delta$ (if there are any) have negative signs.\n\nClass M2. The top long strand and the bottom long strand belong to two different long Seifert circles and the bottom long strand has the orientation from left to right. Notice that $\\vec{K}$ is of Class M2 if and only if every $A_j$ is of Seifert Parity 2 (more precisely case (ii) in Figure \\ref{decomp}) and in this case $\\delta=0$.\n\nClass M3. The top long strand and the bottom long strand belong to the same long Seifert circle. Notice that $\\vec{K}$ is of Class M3 if and only if all crossings in $\\delta$ (if there are any) have positive signs and at least one tangle is of Seifert Parity 3 where we consider the horizontal half twists given by $\\delta$ as an additional tangle besides the $A_j$ tangles.\n\n\\noindent\nLet $i=1$, 2 or 3 be the Seifert Parity type of $A_j$ and define \n\n\\begin{eqnarray}\n\\Delta_1(A_j)&=&\\Delta_3(A_j)=\\frac{(-1+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\Delta(A_j), \\label{Deltaoneorthree}\\\\\n\\Delta_2(A_j)&=&\\frac{(2+\\mbox{\\rm sign}(b^j_1)+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\Delta(A_j),\\label{Deltatwo}\n\\end{eqnarray}\nwhere \n$$\n\\Delta(A_j)=\\sum_{b^j_{2m}>0,1\\le m\\le q_j}b^j_{2m}\/2+\\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2.\n$$\n\nThe braid index for an oriented and alternating Montesinos knot $\\vec{K}$ is  given by the following theorem.\n\n\\begin{theorem}\\label{Montesinos_formula} \\cite{DEHL2018} \nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)=\\vec{M}(A_1,A_2,\\ldots, A_s,\\delta)$ be an oriented and alternating Montesinos knot with a normal standard diagram and the signed vector $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ for $A_j$, then we have\n\\begin{eqnarray*}\n\\qopname\\relax o{braid}(\\vec{K})&=&2+\\sum_{1\\le j\\le s}\\Delta_1(A_j)\\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M1}};\\\\%\n\\qopname\\relax o{braid}(\\vec{K})&=&1+\\sum_{1\\le j\\le s}\\Delta_2(A_j) \\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M2}};\\\\%\n\\qopname\\relax o{braid}(\\vec{K})&=&\\Delta_0(\\vec{K})+\\sum_{A_j \\in \\Omega_2}\\Delta_2(A_j)+\\sum_{A_j \\in \\Omega_3}\\Delta_3(A_j) \\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M3}},%\n\\end{eqnarray*}\nwhere $\\Omega_2$, $\\Omega_3$ are the sets of $A_j$'s that have Seifert Parity 2 and 3 respectively, $\\Delta_0(\\vec{K})=\\eta+\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}$ and $\\eta=\\vert \\Omega_3\\vert$. \n\\end{theorem}\n\nUsing Theorem \\ref{Montesinos_formula} we can now identify the alternating Montesinos knots where the bridge index equals the braid index.\n\n\\begin{theorem}\n\\label{braidequalsbridgeMknot}\nLet $ \\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an oriented and alternating Montesinos knot. Then $\\qopname\\relax o{braid}( \\vec{K})=\\qopname\\relax o{bridge}( K)=s$ if and only if the following conditions hold:\n\n(i) $\\vec{K}$ is of Class M3.\n\n(ii) $\\eta\\ge \\delta+2$ where $\\eta=\\vert \\Omega_3\\vert$ is the number of tangles $A_j$ with Seifert Parity 3.\n\n(iii) If $A_j$  has Seifert Parity 2 then $A_j=(-1,-s_j,-1)$ for some $s_j\\ge 0$. If $A_j$  has Seifert Parity 3 then $A_j=(s_j)$ for some $s_j> 0$.\n\\end{theorem}\n\n\\begin{proof}\n``$\\Longrightarrow$\": Assume that $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ is an oriented alternating Montesinos knot with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(\\vec{K})=s$.\n\nIf $\\vec{K}$ is of Class M1 then every $A_j$ has Seifert Parity 1 and there are at least two $A_j$'s with $\\Delta_1(A_j)=0$ since $\\Delta_1(A_j)\\ge 0$ for each $j$.\nBy Remark \\ref{bj_sign}, $\\mbox{\\rm sign}(b^j_1)=-1$. Thus \n$\\Delta_1(A_j)=0$ is possible only if  $\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$ in (\\ref{Deltaoneorthree}).\nHowever then $\\Delta_1(A_j)\\ge \\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2>0$ and therefore $\\Delta_1(A_j)=0$ is not possible.\n\nSimilarly if $\\vec{K}$ is of Class M2 then there exists at least one tangle $A_j$ with $\\Delta_2(A_j)=0$. This is only possible if $\\mbox{\\rm sign}(b^j_1)=\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$ in (\\ref{Deltatwo}).\nHowever in such a case \\newline\n$\\Delta_2(A_j)\\ge \\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2>0$ therefore $\\Delta_2(A_j)=0$ is not possible either. This proves that $\\vec{K}$ must of Class M3 and therefore all tangles must have Seifert Parity 2 or 3.\n\nLet $\\eta$ be the number of tangles $A_j$ with Seifert Parity 3 and $s-\\eta$ be the number of tangles $A_j$ with Seifert Parity 2. We note that the contribution $\\Delta_2(A_j)$ of a tangle with Seifert Parity 2 to the braid index is always an integer (it is just one less than the braid index of the corresponding two bridge link, see Theorem \\ref{2bridgeformula}) and therefore we know that a tangle of Seifert parity 2 must have $\\Delta_2(A_j)\\ge1$. We have\n$$\ns=\\qopname\\relax o{braid}(\\vec{K})\\ge \\Delta_0(\\vec{K})+s-\\eta +\\sum_{A_j \\in \\Omega_3}\\Delta_3(A_j)  \\ge s+\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}.\n$$\nThis implies that $0=\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}$, hence $\\eta\\ge \\delta+2$.\nMoreover we must have $\\Delta_2(A_j)=1$ for each $A_j$ with Seifert Parity 2 and $\\Delta_3(A_j)=0$ for each $A_j$ with Seifert Parity 3.\n\nIf $A_j$ has Seifert Parity 2 and $\\Delta_2(A_j)=1$, then $\\mbox{\\rm sign}(b^j_1)=-1$ by Remark \\ref{bj_sign} and equation (\\ref{Deltatwo}) becomes\n$$\n1=\\Delta_2(A_j)=\\frac{|b^j_1|}{2}+\\frac{(1+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\sum_{b^j_{2m}>0,1\\le m\\le q_j}b^j_{2m}\/2+\\sum_{b^j_{2m+1}<0,0< m\\le q_j}|b^j_{2m+1}|\/2\\ge \\frac{|b^j_1|}{2}.\n$$\nThus $|b^j_1|\\le 2$. If $b^j_1=-2$ then we must have $\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$, however $1=\\Delta_2(A_j)$ is only possible if $q_j=0$ and $A_j=(-2)$, which can be written as $(-1,0,-1)$ since $(2)$ and $(1,0,1)$ are both continued fraction decompositions of $1\/2$. If $b^j_1=-1$ then $2q_j+1\\ge 3$ and we must have $\\mbox{\\rm sign}(b^j_{2q_j+1})<0$ hence $b^j_{2q_j+1}=-1$ as well. In addition we can see that both $\\mbox{\\rm sign}(b^j_2)$ and $\\mbox{\\rm sign}(b^j_3)$ are negative. Thus we must have  $2q_j+1= 3$, that is, $A_j=(-1,-s_j,-1)$ for some integer $s_j>0$. \n\nIf $A_j$ has Seifert Parity 3, then $\\Delta_3(A_j)=0$. $b^j_1>0$ by Remark \\ref{bj_sign}. If $2q_j+1\\ge 3$ then one can easily check that $\\mbox{\\rm sign}(b^j_2)>0$, which would lead to $\\Delta_3(A_j)>0$. Thus $2q_j+1=1$ and $A_j=(s_j)$ for some integer $s_j>0$. \nThus we have shown that conditions (i), (ii) and (iii) are satisfied.\n\n``$\\Longleftarrow$\": This is straight forward by Theorem \\ref{Montesinos_formula}.\n \\end{proof}\n \n\n\\begin{definition}\\label{v_h}{\\em \nLet us call a tangle $A_j$ corresponding to a rational number of the form $\\beta\/\\alpha=1\/s_j$ for some $s_j\\ge 2$ a {\\em vertical} tangle, and a tangle $A_j$ corresponding to a rational number of the form $\\beta\/\\alpha=(s_j+1)\/(s_j+2)$ for some $s_j\\ge 1$ a {\\em horizontal} tangle. \n}\n\\end{definition}\n\nNote that the standard continued fraction decompositions of a vertical and a horizontal tangle are of the form $(s_j)$ and $(1,s_j,1)$ respectively. Using Theorem \\ref{braidequalsbridgeMknot}, we can then identify all BB knots in the set of all un-oriented alternating Montesinos knots as stated in the following theorem.\n \n\\begin{theorem}\n\\label{unorientedM_BBknots}\nLet ${K}={M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an un-oriented alternating Montesinos knot. Then $\\qopname\\relax o{braid}({K})=\\qopname\\relax o{bridge}( K)=s$ if and only if $K$ or its mirror image satisfies the following conditions:\n\n(i) every $A_j$ is either a vertical tangle or a horizontal tangle;\n\n(ii) $\\eta\\ge \\delta+2$ where $\\eta$ is the number of $A_j$'s that are vertical.\n\\end{theorem}\n\nThe key observation one needs to make in the proof of Theorem \\ref{unorientedM_BBknots} is that under condition (ii), we can oriented $K$ so that the resulting $\\vec{K}$ belongs to Class M3. We leave the details to the reader.\n \nThe BB alternating Montesinos knots with one component and crossing number up to 12 are listed in Table \\ref{table} indicated by a superscript $^\\dagger$.\n \n\n\\subsection{Non alternating Montesinos knots}\n\nA classification of Montesinos knots (including both the alternating and non-alternating cases) can be found in \\cite{BZ}. A non alternating Montesinos knot will have a minimum diagram when there are no integral twists on the right, that is, $\\delta=0$. We have the following:\n\n\\begin{theorem}\n\\label{braidequalsbridgeMknotnonalt}\nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,0)$ be an oriented Montesinos knot. If $\\vec{K}$ belongs to Class M3, $A_j=(-1,-s_j,-1)$ for some integer $s_j\\ge 0$ if it has Seifert Parity 2 and $A_j=(s_j)$ for some integer $s_j\\ge 2$ if it has Seifert Parity 3, then $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(\\vec{K})$.\n\\end{theorem}\n\n\\begin{proof} Let $\\vec{K}$ be an oriented Montesinos knot satisfying conditions (i) and (ii). \nIt is known that $ \\qopname\\relax o{braid}(\\vec{K})\\le \\gamma(\\vec{K})$ \\cite{Ya}, where $\\gamma(\\vec{K})$ denotes the number of Seifert circles in the Seifert circle decomposition of $\\vec{K}$. Thus we have $\\qopname\\relax o{bridge}(K)\\le \\qopname\\relax o{braid}(\\vec{K})\\le \\gamma(\\vec{K})$, and the result of the theorem follows if we can show that $\\gamma(\\vec{K})=s$ since $s=\\qopname\\relax o{bridge}(K)$ by Theorem \\ref{bridgeMknot}.\nEach tangle $A_j$ with Seifert Parity 2 contributes a Seifert circle (which is contained within the tangle $A_j$).\n Since each Seifert circle that is not completely contained within a tangle (this includes the large Seifert circle) must contain two vertical arcs: one each from a tangle of case (v) and from a tangle of case (vii) as defined in Figure \\ref{decomp} hence each tangle $A_j$ of Seifert Parity 3 also contributes one Seifert circle. It follows that $\\vec{K}$ has exactly $s$ Seifert circles.\n\\end{proof}\n\n\\begin{corollary}\\label{non_un_cor}\nLet ${K}={M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,0)$ be a Montesinos knot with $s\\ge 3$ (that is not necessarily alternating). If each $A_j$ or its mirror image is either a vertical or a horizontal tangle (as defined in Definition \\ref{v_h}) and at least two of them are vertical, then $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}({K})$.\n\\end{corollary}\n\nIf $K$ satisfies the conditions in the corollary, then we can oriented it in a way that it satisfies the conditions of Theorem \\ref{braidequalsbridgeMknotnonalt}. The details are left to the reader.\n\n\\begin{remark}{\\em\nWe note that the proof of Theorem \\ref{braidequalsbridgeMknotnonalt} does not depend on whether $\\vec{K}$ is alternating, thus it also works for an oriented alternating Montesinos knot $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ where $\\delta=0$. However this method is not powerful enough to prove Theorem \\ref{braidequalsbridgeMknot} since the Seifert circle decomposition of $\\vec{K}$ contains $s+\\delta$ Seifert circles when $\\vec{K}$ satisfies the conditions of Theorem \\ref{braidequalsbridgeMknot}. That is, $\\vec{K}$ (in its standard diagram form) does not minimize the number of Seifert circles. In \\cite{DEHL2018} we explain an algorithm that converts $\\vec{K}$ to a non alternating and non minimal diagram $D$ that minimizes the number of Seifert circles via so called reduction moves.}\n\\end{remark}\n\nAll non alternating Montesinos knots with one component and up to 12 crossings that satisfy the condition of \nCorollary \\ref{non_un_cor} are listed in Table \\ref{table} and are marked with a superscript $^\\ddagger$. It turns out that this is a complete list of one component BB non alternating Montesinos knots with crossing number up to 12. In fact, we conjecture that this is generally true. See Conjecture \\ref{conjecture1} at the end of the paper.\n\n\\subsection{Knots using Conway basic polyhedra}\n\nIn \\cite{Con67} the concept of Conway basic polyhedra was introduced. Here we will only concentrate on one class of such polyhedra.\nThe Conway basic polyhedra $n^*$ for $n=2k$ and $k\\ge 3$ can be thought of as taking a regular $k$-gon into which we inscribe a second regular $k$-gon such that its vertices touch the midpoints of the sides of the original $k$-gon. Into the smaller $k$-gon we inscribe a third $k$-gon in the same way. The left of Figure \\ref{eightstar} illustrate the case of $k=4$. This will form a knot  diagram $D$ with $n$ crossings if we make the diagram alternating. Moreover the writhe $w(D)=0$ (with proper orientation in case of a link), three Seifert circles and braid index 3. For example $8^*$ is the knot $8_{18}$ (the right of Figure \\ref{eightstar}) and $10^*$ is the knot $10_{123}$. \nWe note that for $n\\not\\equiv 0\\mod(3)$ we obtain a knot diagram and for $n\\equiv 0\\mod(3)$ we obtain a 3-component knot diagram, for example  for $k=3$ we obtain the Borromean rings ($6_2^3$ in Rolfsen notation). Note that when he first introduced these concepts, Conway used two symbols for the six crossing diagram ($k=3$) denoted by $6^{*}$ and $6^{**}$.  The two symbols denote isomorphic graphs that were introduced to make the obtained notation for knots to be nicer. However they are essentially the same basic polyhedron with alternative way to insert tangles. Conway developed a shorthand notation where the two symbols $6^{**}$ and $6^{*}$ are omitted. If there is an initial dot in the beginning of the symbol, then it is $6^{**}$. If there are more than one tangle symbols separated by dots without the basic polyhedron symbol, then it is meant to be $6^{**}$. If there is also a dot in the beginning of the symbol, then it is $6^{*}$.\nIf the tangle substitutions are simple then there is a natural way to view these diagrams as a 3 string braids without increasing the crossing number and we obtain a braid of three strings and bridge number three.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=.15]{Figures2018\/eightstar.pdf}\\qquad\n\\includegraphics[scale=.15]{Figures2018\/eighteighteen.pdf}\n\\caption{Left: The Conway basic polyhedra $8^*$. Right: The knot $8_{18}$. }\n\\label{eightstar}\n\\end{figure}\n\nUsing the $6^{*}$, $6^{**}$, $8^{*}$ and $10^{*}$ diagrams, we are able to identify all one component BB knots with crossing number up  to 12 that can be constructed using Conway basic polyhedra. They are listed in \nTable \\ref{table} and are marked with a superscript $^*$. We would like to point out that \nwhile the above knot construction using Conway basic polyhedra can be generalized, the generalized construction may not yield a BB knot.\nOne such example is $n^*$ with $n = 3k$ that can be thought of as inscribing four $k$-gons into each other. The simplest example is $9^*$ which forms the knot $9_{40}$. It has a diagram with 4 Seifert circles and its braid index is four, but its bridge number is only three.\n\n\\subsection{ Arborescent knots or Conway algebraic knots.}\n\nConway algebraic knots are a super family of the Montesinos knots \\cite{BoSi, Con67}.\nThe following describes Conway algebraic knots with bridge and braid index three.\nFirst, they are represented in the Conway notation as $[(a_1;b_1)(a_2;b_2)]$ with $|a_i|\\ge 2$ and $|b_i|\\ge 2$ \\cite{Con67}. These Conway algebraic knots may have one or more components, depending on the parity of $a_i$ and $b_i$. For example, if $a_1$, $a_2$ are both even and $b_1$, $b_2$ are both odd then the knot  has only one component. In the case of multiple components we need to assume that the components are  oriented such that $a_i$ and $b_i$ represent parallel oriented half twists. These knots can be either alternating or non-alternating. In the non alternating case they can also be denoted by $[(a_1;b_1)-(a_2;b_2)]$, $[(a_1;b_1)(a_2-1,1;-b_2)]$, $[(a_1-1,1;-b_1+1,-1)(a_2;b_2)]$ or  $[(-a_1+1,-1;b_1-1,1)(a_2;b_2)]$. The braid index of such a Conway algebraic knot is at most $3$ since the standard diagram given by the Conway notation has 3 Seifert circles in its Seifert circle decomposition. Since these are not 2-bridge knots, their bridge index is at least 3. Thus if $K$ is a Conway algebraic knot described here, then we have $\\qopname\\relax o{braid}(K)=\\qopname\\relax o{bridge}(K)=3$.  \n\n\\subsection{Summary of BB knots with crossing number up to 12.}\n\nTable \\ref{table} lists all one component BB knots with crossing number up to 12. The superscriptions are used in the table as follows. $^t$: torus knots; $^\\dagger$: alternating Montesinos knots; $^\\ddagger$: non alternating Montesinos knots; $^*$: knots constructed from Conway basic polyhedra; $^\\flat$: Conway algebraic knots. The symbols used for the knots follow the historical notation for knots with at most 10 crossings \\cite{rolfsen2003knots} and the notation created for the program Knotscape, developed by Morwen and Jim Hoste \\cite{hoste1999knotscape} for knots with more than 10 crossings, where the letters a and n denote alternating and non-alternating knots, respectively. We would like to point out that a few BB knots in Table \\ref{table} belonging to several families discussed above. For example the knot $8_{19}$ is the torus knot $T(4,3)$, the non alternating Montesinos knot corresponding to $[3;3;2;-]$, as well as the knot constructed using the $8^*$ Conway basic polyhedron. Another example is the knot $10_{124}$: it is the torus knot $T(5,3)$ and also the non alternating Montesinos knot $[5;3;2;-]$.\n\n\\begin{center}\n\\begin{longtable}{|llllllll|}\n\\hline\\hline\n $(3_1)^t$&&&&&&&\\\\ \\hline\n $(5_1)^t$&&&&&&&\\\\ \\hline\n $(7_1)^t$&&&&&&&\\\\ \\hline\n $(8_{5})^\\dagger$&$(8_{10})^\\dagger$&$(8_{16})^\\dagger$&$(8_{17})^*$&$(8_{18})^*$&\n $(8_{19})^{t*\\ddagger}$&$(8_{20})^\\ddagger$&$(8_{21})^\\ddagger$\\\\ \\hline\n $(9_1)^t$&$(9_{16})^\\dagger$&&&&&&\\\\ \\hline\n $(10_{46})^\\dagger$&$(10_{47})^\\dagger$&\n $(10_{48})^\\dagger$&$(10_{62})^\\dagger$&$(10_{64})^\\dagger$&$(10_{79})^\\flat$&\n$(10_{82})^*$&$(10_{85})^*$\\\\ \\hline\n$(10_{91})^*$&$(10_{94})^*$&\n$(10_{99})^*$&$(10_{100})^*$&$(10_{104})^*$&$(10_{106})^*$&\n$(10_{109})^*$&$(10_{112})^*$\\\\ \\hline\n$(10_{116})^*$&$(10_{118})^{t*}$&$(10_{123})^*$&$(10_{124})^{t\\ddagger}$&\n$(10_{125})^\\ddagger$&$(10_{126})^\\ddagger$&$(10_{127})^\\ddagger$&$(10_{139})^\\ddagger$\\\\ \\hline\n$(10_{141})^\\ddagger$&$(10_{143})^\\ddagger$&$(10_{148})^\\flat$&$(10_{149})^\\flat$&\n$(10_{152})^\\flat$&$(10_{155})^*$&$(10_{157})^*$&$(10_{159})^*$\\\\ \\hline\n$(10_{161})^*$&&&&&&&\\\\ \\hline\n$(11a_{44})^\\dagger$&$(11a_{47})^\\dagger$&$(11a_{57})^\\dagger$&$(11a_{231})^\\dagger$&\n$(11a_{240})^\\dagger$&$(11a_{263})^\\dagger$&$(11a_{338})^\\dagger$&$(11a_{367})^t$\\\\ \\hline\n$(11n_{71})^\\ddagger$&$(11n_{72})^\\ddagger$&$(11n_{73})^\\ddagger$&$(11n_{74})^\\ddagger$&\n$(11n_{75})^\\ddagger$&$(11n_{76})^\\ddagger$&$(11n_{77})^\\ddagger$&$(11n_{78})^\\ddagger$\\\\ \\hline\n$(11n_{81})^\\ddagger$&&&&&&&\\\\ \\hline\n$(12a_{146})^\\dagger$&$(12a_{167})^\\dagger$&$(12a_{369})^\\dagger$&$(12a_{576})^\\dagger$&\n$(12a_{692})^\\dagger$&$(12a_{801})^\\dagger$&$(12a_{805})^*$&\n$(12a_{815})^\\flat$\\\\ \\hline\n$(12a_{819})^*$&$(12a_{824})^\\flat$&$(12a_{835})^\\dagger$&\n$(12a_{838})^\\dagger$&$(12a_{850})^*$&$(12a_{859})^*$&$(12a_{864})^*$&\n$(12a_{869})^*$\\\\ \\hline\n$(12a_{878})^\\dagger$&$(12a_{898})^*$&$(12a_{909})^*$&\n$(12a_{916})^*$&$(12a_{920})^*$&$(12a_{981})^*$&$(12a_{984})^*$&\n$(12a_{999})^*$\\\\ \\hline\n$(12a_{1002})^*$&$(12a_{1011})^*$&$(12a_{1013})^*$&\n$(12a_{1027})^\\dagger$&$(12a_{1047})^*$&$(12a_{1051})^*$&$(12a_{1114})^*$&\n$(12a_{1120})^*$\\\\ \\hline\n$(12a_{1168})^*$&$(12a_{1176})^*$&$(12a_{1191})^*$&\n$(12a_{1199})^*$&$(12a_{1203})^*$&$(12a_{1209})^*$&$(12a_{1210})^*$&\n$(12a_{1211})^*$\\\\ \\hline\n$(12a_{1212})^*$&$(12a_{1214})^\\dagger$&$(12a_{1215})^*$&\n$(12a_{1218})^*$&$(12a_{1219})^*$&$(12a_{1220})^*$&$(12a_{1221})^*$&\n$(12a_{1222})^*$\\\\ \\hline\n$(12a_{1223})^*$&$(12a_{1225})^*$&$(12a_{1226})^*$&\n$(12a_{1227})^*$&$(12a_{1229})^*$&$(12a_{1230})^*$&$(12a_{1231})^*$&\n$(12a_{1233})^\\dagger$\\\\ \\hline\n$(12a_{1235})^*$&$(12a_{1238})^*$&$(12a_{1246})^*$&\n$(12a_{1248})^*$&$(12a_{1249})^*$&$(12a_{1250})^*$&$(12a_{1253})^*$&\n$(12a_{1254})^*$\\\\ \\hline\n$(12a_{1255})^*$&$(12a_{1258})^*$&$(12a_{1260})^*$&\n$(12a_{1283})^\\dagger$&$(12a_{1288})^\\flat$&$(12n_{113})^\\flat$&$(12n_{114})^\\flat$&\n$(12n_{190})^\\flat$\\\\ \\hline\n$(12n_{191})^\\flat$&$(12n_{233})^\\ddagger$&$(12n_{234})^\\ddagger$&\n$(12n_{235})^\\ddagger$&$(12n_{242})^\\ddagger$&$(12n_{344})^\\flat$&$(12n_{345})^\\flat$&\n$(12n_{417})^*$\\\\ \\hline\n$(12n_{466})^\\ddagger$&$(12n_{467})^\\ddagger$&$(12n_{468})^\\ddagger$&\n$(12n_{472})^\\ddagger$&$(12n_{570})^\\ddagger$&$(12n_{571})^\\ddagger$&$(12n_{574})^\\ddagger$&\n$(12n_{604})^\\flat$\\\\ \\hline\n$(12n_{640})^*$&$(12n_{647})^*$&$(12n_{666})^*$&\n$(12n_{674})^\\flat$&$(12n_{675})^\\flat$&$(12n_{679})^\\flat$&$(12n_{683})^*$&\n$(12n_{684})^*$\\\\ \\hline\n$(12n_{688})^\\flat$&$(12n_{707})^*$&$(12n_{708})^*$&\n$(12n_{709})^*$&$(12n_{721})^\\ddagger$&$(12n_{722})^\\ddagger$&$(12n_{725})^\\ddagger$&\n$(12n_{747})^*$\\\\ \\hline\n$(12n_{748})^*$&$(12n_{749})^*$&$(12n_{750})^*$&\n$(12n_{751})^*$&$(12n_{767})^*$&$(12n_{820})^*$&$(12n_{821})^*$&\n$(12n_{822})^*$\\\\ \\hline\n$(12n_{829})^*$&$(12n_{830})^*$&$(12n_{831})^*$&\n$(12n_{850})^*$&$(12n_{882})^*$&$(12n_{887})^*$&$(12n_{888})^\\flat$&       \n \\\\\\hline\\hline\n \\caption{List of one component BB knots with crossing number up to 12. \\label{table}}\n\\end{longtable}\n\\end{center}\n\n\\vspace{-0.7in}\n\\section{The number of BB knots with a given crossing number}\n\\label{numberofknots}\n\nTable \\ref{table2} summarizes the number of one component BB knots with crossing numbers up to 12. By separating the knots with odd crossing numbers and the knots with with even crossing number, we observe that the number of one component BB knots with a given crossing number $n$ increases as the crossing number increases and it seems that there are more BB knots in the case of an even crossing number greater than six; and (ii) the percentage of BB knots among all knots with the same crossing number decreases as the crossing number increases. What are the general behaviors of these numbers and percentages? More specifically, if we let $\\mathcal{K}_n$ and $\\mathcal{B}_n$ be the numbers of knots and BB knots with crossing number $n$ respectively, then how do $\\mathcal{B}_n$ and $\\mathcal{B}_n\/\\mathcal{K}_n$ behave? While it is \nplausible that $\\lim_{n\\to\\infty}\\mathcal{B}_n\/\\mathcal{K}_n=0$, we do not have a proof of it. However, we can prove that $\\lim_{n\\to\\infty}\\mathcal{B}_n=\\infty$ and does so exponentially. This is stated in Theorem \\ref{exp_thm}.\n\n\n\\begin{table}[!hb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \n\\hline  crossing \\# &3&4&5&6&7&8&9&10&11&12\\\\\n\\hline \\# of knots &1&1&2&3& 7&21&49&165&552&2176\\\\\n\\hline \\# of BB knots &1&0&1&0&1&8&2&33&17&119\\\\\n\\hline $\\approx$ percentage &100\\%&0\\%&50\\%&0\\%& 14.3\\%&38.1\\%&4.1\\%&20.0\\%&3.1\\%&5.5\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Number of one component BB knots with respect to the crossing number. \\label{table2}}\n\\end{table}\n\n\\begin{theorem}\\label{exp_thm}\nThe number $\\mathcal{B}_n$ grows exponentially with $n$. Moreover the number of one component BB knots with crossing number $n$ also grows exponentially with $n$.\n\\end{theorem}\n\\begin{proof}\nIn fact, we will prove that the number of BB knots with braid index 3 grows exponentially with the crossing number $n$. Assume that we only generate alternating 3-braids. Then a word representing such a braid can be generated by two symbols $\\sigma_1$ and $\\sigma_2^{-1}$. Using these two symbols there are $2^n$ words of length $n$ each of which representing an alternating braid with $n$ crossings.\nIf we close such a braid we obtain an alternating knot or link diagram with $n$ crossings and at most 3 components.\nNow we need to estimate how many of these diagrams represent distinct knots of $n$ crossings. There are several points we need to consider:\n\n(i) If $w$ is a braid word then any cyclic permutation of this word will yield the same link. Thus up to cyclic permutation there are at least $\\frac{2^n}{n}$ different words.\n\n(ii) Switching between $\\sigma_1$ and $\\sigma_2^{-1}$ in a braid word will result in the same knot. Thus up to cyclic permutation and exchange of $\\sigma_1$ and $\\sigma_2^{-1}$ there are $\\frac{2^{n-1}}{n}$ different words (at least).\n\n(iii) The diagram generated by the braid closure is a reduced alternating knot diagram if $\\sigma_1$ and $\\sigma_2^{-1}$ both occur in the word more than once. \n\n\n(iv) Using the classification theorem in \\cite{Bir93}, we see that a knot diagram obtained by closing a braid satisfying condition (iii) above admits flypes only if the braid is of the form $\\sigma_1^u \\sigma_2^{-1}\\sigma_1^z\\sigma_2^{-v}$, $\\sigma_2^{-u}\\sigma_1\\sigma_2^{-z}\\sigma_1^{v}$, $\\sigma_1^u\\sigma_2^{-v}\\sigma_1^z\\sigma_2^{-1}$ or  $\\sigma_2^{-u}\\sigma_1^v\\sigma_2^{-z}\\sigma_1^{1}$  for some positive integers $u$, $z$, and $v$. We note that $\\sigma_1^u \\sigma_2^{-1}\\sigma_1^z\\sigma_2^{-v}$ is flype equivalent to $\\sigma_1^u\\sigma_2^{-v}\\sigma_1^z \\sigma_2^{-1}$ and $\\sigma_2^{-u}\\sigma_1\\sigma_2^{-z}\\sigma_1^{v}$ is flype equivalent to $\\sigma_2^{-u}\\sigma_1^v\\sigma_2^{-z}\\sigma_1^{1}$.\n\nIt is easy to see that the number of braids satisfying condition (iii) and also admitting flypes is bounded above by $4\\cdot {{n-2}\\choose{2}}<n^3$. Furthermore, the number of words up to cyclic permutation and exchange of $\\sigma_1$ and $\\sigma_2^{-1}$ with at least two each of $\\sigma_1$ and $\\sigma_2^{-1}$ in the word is at least $2^{n-5}\/n$. The alternating knots obtained by closing braids satisfying condition (iii) and that do not admit flypes are all distinct, thus there exists a constant $c>0$ such that when $n$ is large enough the number of such knots is bounded below by $2^{n-5}\/n-n^3> 2^{cn}$. \nNext, we need to estimate how many of these 3 braids are 2-bridge knots using Theorem \\ref{2bridge_theorem}. Using a case by case analysis similar to the method used in the proof of Theorem \\ref{torustwobridge} we can list all vectors of 2-bridge knots with braid index three. If we represent the these 2-bridge knots with an odd length vector using positive integers than we can show that these vector are of the following form:\n$211$, $a12$, $a2b$ and $3a1$  for length three vectors and $a11b1$ and $1a3b1$ for length five vectors,\nwhere $a,b$ are positive integers. No vectors of a length greater than five can have a braid index of three. Thus for a fixed crossing number $n$ the number of different 2-bridge knots with braid index three is linearly bounded above by $c_1 n$  some fixed constant $c_1>0$. So, the number of  3-braids with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)=3$ will still growth exponentially. Roughly a third of these will be one component knots hence the number of one component BB knots also grows exponentially with $n$.\n\\end{proof}\n\nWe end our paper with the following two questions and a conjecture.\n\n\\begin{question} {\\em %\nIs $\\lim_{n\\rightarrow\\infty}\\mathcal{B}_n\/{{\\mathcal{K}_n}}=0$ and does this limit approach zero exponentially fast?\n}\n\\end{question}\n\n\n\n\\begin{question} {\\em Is it true that $\\mathcal{B}_{2n}>\\mathcal{B}_{2n+1}$ for all $n\\ge 4$? And why?}\n\\end{question}\n\n\\begin{conjecture}\\label{conjecture1}{\\em \nThe non alternating Montesinos knots described in Theorem \\ref{braidequalsbridgeMknotnonalt} (or Corollary \\ref{non_un_cor}) are the only non alternating Montesinos knots that are BB knots. \n}\n\\end{conjecture}\n\n\\subsection*{Acknowledgement}\nPh.~R.\\@ was supported by German Research Foundation\ngrant RE~3930\/1--1. We would like to thank S\\\"oren Bartels for fruitful\ndiscussions on the numerical simulation of elastic knots.\n\n\n\\newcommand{\\href}[2]{#2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe problem of noise and decoherence plays the important role both for applications and for fundamental\nphysics. Noise parameters for electron transport provide a valuable information, which is not available from average\ncurrent-voltage dependences. This stimulates development of effective methods of noise investigation. In particular, a lot of\nattention is devoted to shot noise\n\\cite{Blanter}, which is related to discreteness of charge transport and yields direct information on carrier charges. \nEfforts of theorists were invested to studying full counting statistics of shot noise, its non-Gaussian\ncharacter \\cite{Lez} and asymmetry (odd moments)  \\cite{Odd,BKN}. They are also objects of intensive\nexperimental investigations\n\\cite{Reul,Rez}. \n\nRecently it was shown that the zero-bias\nanomaly of the incoherent Cooper-pairs current in a Coulomb blockaded Josephson junction is very\nsensitive to shot noise from an independent source and therefore can be used for noise investigation\n\\cite{SN}. Further experimental and theoretical investigations \\cite{exp,SN-T,Heik} have demonstrated\nthat this method especially useful for studying asymmetry of shot noise, which is connected with \nits non-Gaussian character and is very difficult for detection by other methods of noise spectroscopy\n\\cite{Reul,Rez}. In  addition, other methods to use the Josephson junction as a noise probe were discussed.\n Deblock {\\em et al.} \\cite{Delft} suggested to use the quasiparticle current for noise detection. The\nscheme to use the Josephson junction as a threshold detector of electron counting statistics has also been\ninvestigated theoretically and experimentally \\cite{TN,Pek,Grab}.  The scheme exploited the effect of\nnoise on macroscopical quantum tunneling at currents close to the critical current.\n\nThe goal of this paper is to extend the theoretical analysis of the effect of noise on the low-bias part of\nthe $IV$ curve of a mesoscopic tunnel junction. The previous analysis \\cite{SN,exp,SN-T} addressed the\neffect of shot noise on a Coulomb blockaded Josephson junction in the weak-coupling limit, when \n the Josephson coupling energy $E_J$ was small compared to the Coulomb energy $E_c=e^2\/C$. Here $C$ is the\nrelevant capacitance. Shot noise originated from a low current through a parallel junction, and its effect\non the $IV$ curve of the Josephson junction had asymmetry connected with the non-Gaussian character of shot\nnoise. The effect was  insensitive to counting statistics of the noise current since the latter was so low\nthat the single-electron tunneling events were well separated in time and their correlation was not\nessential. On the basis of the fulfilled analysis one could expect  that a Coulomb blockade normal junction\nis also able to effectively probe noise, and other types of noise different from shot noise can be\ninvestigated, but in order to check these expectations an additional analysis was needed. The present paper\nis to give answers to the following questions:\n\\begin{itemize}\n\\item\nCan the zero-bias anomaly of the probing junction be sensitive to the counting statistics of electrons tunneling through\nthe noise source (noise junction) when the current through the noise junction grows?\n\n\\item Is the zero-bias anomaly of the Coulomb blockaded  {\\em normal} tunnel junction also sensitive to noise similarly to\nthe Coulomb blockaded  Josephson junction?\n\n\\item Is the zero-bias anomaly of the Coulomb blockaded  tunnel junction sensitive to other types of noise different from\nshot noise?\n\n\\item Can the Josephson junction in the strong coupling limit $E_J\\gg E_C$ also probe noise?\n\\end{itemize}\n\nThe paper gives positive answers to all these questions. The content of the paper is the following. Section \\ref{sec2}\nreminds the previous analysis \\cite{SN,exp,SN-T} for the Coulomb blockade Josephson junction in the weak coupling limit $E_J\n\\ll E_c$ and extends it on the case of higher noise currents, when the zero-bias anomaly is expected to be sensitive to the\nelectron counting statistics. The analysis confirms this expectation. At high currents the\neffect is essentially different for the cases of strictly periodical sequence of pulses and of random pulses governed\nby the Poissonian statistics. Section \\ref{sec3} considers\nanother type of noise: phase fluctuations in a monochromatic microwave signal, which result in decoherence of the\nsignal. A similar phase noise  is well known in quantum optics. It is shown that the effect of the microwave on the\n$IV$ curve of the Josephson junction gives direct information on the decoherence time. Section \\ref{sec4} addresses the\neffect of shot noise on the zero-bias anomaly of a normal tunnel junction. The latter is also very sensitive to shot\nnoise and phase fluctuations in an AC input, and therefore can be exploited as a noise detector. Section \\ref{sec5}  considers the\nJosephson junction in the strong coupling limit in low-impedance environment. This case was chosen because it allows to exploit its\nduality to the case of the Josephson junction in the weak coupling limit in high-impedance environment, which was considered in the\nprevious sections. But duality is valid mostly for the equilibrium {Nyquist-Johnson noise. For shot noise\nduality does not work, and the effect of noise on the Josephson junction in low impedance environment \n(superconductive regime) is different from that in high impedance environment (Coulomb blockade regime). The\nimportant feature of the low-impedance case is  the perspective to measure the intrinsic tunneling time,\nwhich is discussed in the end of Sec.\n\\ref{sec5}. \n\n\n\\section{Josephson junction, weak coupling limit, shot noise} \\label{sec2}\n\n\\subsection{Without shot noise}\n\n Let us review well known results concerning the\n$IV$ curve without shot noise. We assume that the  Josephson coupling\nenergy $E_J\\cos \\varphi$ ($\\varphi$ is the Cooper-pair phase difference between two banks of the Josephson\njunction) is weak in comparison with the Coulomb energy $E_c =e^2\/2C$, where\n$C$ is the relevant capacitance (see below), and the  Josephson coupling can be considered as a time-dependent\nperturbation. Then the\nGolden Rule gives for the current  of Cooper pairs \\cite{Aver,SZ,IN}:\n\\begin{equation}\nI={\\pi eE_J^2 \\over \\hbar}[P(2eV)-P(-2eV)]~, \n \\label{IP}\\end{equation}\nwhere the function\n\\begin{equation}\nP(E)={1\\over 2\\pi \\hbar}\\int_0^\\infty dt\\, \\left[e^{iEt\/\\hbar}\\left\\langle\ne^{i\\varphi(t_0)}e^{-i\\varphi (t_0-t)}\\right\\rangle+ e^{-iEt\/\\hbar}\\left\\langle\ne^{i\\varphi (t_0-t)} e^{-i\\varphi(t_0)}\\right\\rangle\\right]\n  \\label{PE} \\end{equation}\ncharacterizes the probability to transfer the energy $E>0$ to environment (or to\nabsorb the energy $|E|$ from environment if $E<0$) at the time $t_0$. In many cases (equilibrium and shot noise included) the\naverage phase correlators do not depend on the time $t_0$ after averaging, and Eq. (\\ref{PE}) is reduced to the Fourier component\nof the average phase correlator \\cite{IN}:\n\\begin{equation}\nP(E)={1\\over 2\\pi \\hbar}\\int_{-\\infty}^\\infty dt\\, e^{iEt\/\\hbar}\\left\\langle\ne^{i\\varphi(t)}e^{-i\\varphi (0)}\\right\\rangle~.\n  \\end{equation}\nHowever studying the linear response to the AC input in Sec. \\ref{AClin} we shall need the more\ngeneral expression Eq.~(\\ref{PE}).\n\n\nSince we use the\nperturbation theory with respect to $E_J$ we can calculate phase fluctuations neglecting $E_J$, i.e.,\ntreating the Josephson junction as a capacitor. The crucial assumption in the phase-fluctuation theory (or\n$P(E)$ theory) is that the phase fluctuations are Gaussian \\cite{IN} and \n\\begin{equation}\n\\langle e^{i\\varphi_0(t_0)}e^{-i\\varphi_0 (t_0-t)}\\rangle \\approx e^{J_0(t)} ~,\n  \\label{Gauss}    \\end{equation}\nwhere the phase--phase correlator\n\\begin{eqnarray}\nJ_0(t) =\\langle[ \\varphi_0(t)\n-\\varphi_0(0)]\\varphi_0(0)\\rangle=J_R(t)+ iJ_I(t) \n= 2\\int_{-\\infty}^\\infty {d\\omega \\over \\omega}\n\\frac{\\mbox{Re} Z(\\omega)}{R_Q}\n \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}\n     \\label{phaseCor} \\end{eqnarray}\nis a complex function of time. The subscript 0 points out that the phase fluctuation $\\varphi_0$ is determined by the\nequilibrium Johnson-Nyquist noise in the environment, i.e., in the electric circuit with the impedance\n$Z(\\omega)$. Here $R_Q=h\/4e^2=\\pi \\hbar\/2e^2$ is the quantum resistance for Cooper pairs and $\\beta =1\/k_BT$ is the inverse\ntemperature. Then\n\\cite{IN}\n\\begin{equation}\nP(E)={1\\over \\pi \\hbar}\\mbox{Re} \\left\\{ \\int_0^\\infty dt \\exp\\left[ J_0(t) +\n\\frac{iEt}{\\hbar}\\right]\\right\\}\\, , \n    \\label{P-E}    \\end{equation}\nand the current is\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n \\sin \\left(2eVt\\over \\hbar\\right)\\right\\}~.\n \\label{WC}  \\end{eqnarray}\nThe zero-bias conductance is given by\n\\begin{equation}\nG_0=\\left.{dI\\over dV}\\right|_{V\\rightarrow 0}=-{4 e^2E_J^2 \\over\n\\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt e^{J_0(t)}\\right\\} \\,.\n          \\label{G-0}\\end{equation}\n\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.5\\linewidth]{ShotL-1.eps}\n   \n    \\caption{Electric circuit. a) The Josephson junction voltage-biased through the shunt resistance $R$. b) Parallel to\nthe Josephson junction the normal junction with the capacitance $C_s$ is switched on, which produces shot noise affecting the\ncurrent through the Josephson junction. The large shunt capacitance $C_{sh}$ transforms at finite frequencies the current bias\ninto the voltage bias.}\n  \\label{fig1}\n  \\end{center}\n  \\end{figure}\n\n\nThe electric circuit for our analysis is shown in Fig. \\ref{fig1}a. The\nJosephson tunnel junction  with capacitance $C_J$ is voltage-biased via the shunt\nresistor $R$. Thus in our case  $Z(\\omega) =(1\/R +i\\omega C)^{-1}$ with $C=C_J$, and \n\\begin{eqnarray}\nJ_0(t) = 2\\rho\\int_{-\\infty}^\\infty {d\\omega \\over \\omega}\n\\frac{1}{1 +\\omega^2\\tau^2}\n \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}~,\n     \\label{phaseCorT} \\end{eqnarray}\nwhere $\\tau=RC$ and $\\rho=R\/R_Q$. At $T \\to 0$\n($t>0$):\n\\begin{eqnarray}\nJ_0(t) =  \\rho \\left[-e^{t\/\\tau}\\mbox{E}_1\\left({t\\over \\tau}\\right)-\ne^{-t\/\\tau}\\mbox{E}_1\\left(-{t\\over \\tau}+i0\\right) \n-2 \\ln{ t \\over \\tau}\n -2\\gamma -i\\pi\\right] \\,,\n     \\label{J-0}     \\end{eqnarray}\n where $\\gamma=0.577$ is the Euler constant, and\n$\\mbox{E}_1(z)=\\int_1^\\infty (e^{-zt}\/t)\\,dt$ is the exponential integral\n\\cite{AS}. The small imaginary correction $+i0$ to the argument of one of the\nexponential integrals is important for analytic continuation of $\\mbox{E}_1(z)$ from real\n$t$ to the complex plane \\cite{AS} (see below). At $T=0$ $P(E)$ vanishes for $E<0$ since it is the probability of the transfer of\nthe energy $|E|$ from the environment to the junction, which is impossible if $T=0$.\n\nFor further analysis we need the expressions for $J_0$ in the limits of short time\n$t \\ll \\tau$:\n\\begin{eqnarray}\nJ_0(t) \\approx  \\rho \\left[{t^2  \\over \\tau^2}\\left(\\ln{t\\over\n\\tau}+\\gamma -{3\\over 2}\\right) -{i\\pi t\\over \\tau}\\right]\\,,\n     \\label{J-0s}     \\end{eqnarray}\nand long time $t \\gg \\tau$:\n\\begin{eqnarray}\nJ_0(t) = - \\rho \\left(2 \\ln{ t \\over \\tau}\n +2\\gamma  +i\\pi\\right) \\, .\n     \\label{J-0l}     \\end{eqnarray}\n\nThe function $P(E)$, as well as the current $I$, which it determines, have been carefully studied and calculated for arbitrary\n$\\rho$ \\cite{Ing}. But for the goals of our analysis it is useful to present a simplified calculation valid only in the high\nimpedance limit  $\\rho \\gg 1$. In  this limit integration in the expression for the conductance $G_0$, Eq. (\\ref{G-0}),\nshould be done over rapidly oscillating functions and it is difficult to see that the integral exactly  vanishes at $T=0$.\nBut it becomes evident  after rotation of the integration path in the plane of complex time, which corresponds to replacing\n$t$ by $-iy$. This rotation transforms the integration over the positive real semiaxis into the integration over the\nnegative imaginary semiaxis. After this transformation, the complex function $J_0(t) \\to J_0(-iy)$ becomes a\npurely real function, and the conductance $G_0$, which is given by the imaginary part of this function, vanishes. This\nis true for any term\n$V^k$ in the expansion of the current $I$ in voltage $V$ as far as the integral over $t$, which determines this\nterm, converges at long time. Using the asymptotic expression Eq. (\\ref{J-0l}) one can see that the integral is\ndivergent if\n$k>2\\rho -2$. So at zero temperature the current $I$ as a function of $V$  does not vanish completely but is given by a\nsmall nonanalytic power-law term, which can be found in the limit of high $\\rho$ from the following simple steepest\ndescent estimation of the integral.\n\nExpecting that the main contribution comes from long times one can rewrite the expression (\\ref{P-E}) for the $P(E)$ function\nusing the asymptotic expression  (\\ref{J-0l}) for $J_0$:\n\\begin{eqnarray}\nP(E)= {1\\over \\pi \\hbar}\\mbox{Re}\\left\\{\\int _0^\\infty dt\\,\\exp \\left[ - \\rho \\left(2 \\ln{ t \\over \\tau}\n +2\\gamma  +i\\pi\\right) +{iE t\\over \\hbar}\\right]\\right\\}~.\n    \\end{eqnarray}\nThe saddle point is determined by\nthe condition of vanishing first derivative of the argument of the exponential function: $\n-{2\\rho \/ t_0}  + {iE \/\\hbar}=0$. This yields $t_0=-2i\\hbar\/E$. Expanding the argument of the exponential\nfunction around the saddle point ($t'=t-t_0$) we receive \n\\begin{eqnarray}\n P(E)= {1\\over \\pi \\hbar}\\mbox{Re}\\left\\{\\exp\\left[\\rho\\left(-2\\gamma -\n2\\ln {2\\rho\\hbar\\over iE\\tau}\n-i\\pi\\right)+2\\rho\\right]\\right\\} \\int_{-\\infty}^\\infty\ndt'\\,\\exp\\left(-{E^2\\over 4\\rho\\hbar^2} t'^2\\right)\\nonumber \\\\\n=\\exp[2\\rho(1-\\gamma)]{\\tau \\over \\sqrt{\\pi\\rho} \\hbar} \\left(E\\tau\n\\over 2\\rho\\hbar\\right)^{2\\rho-1}~.\n              \\end{eqnarray}\nFinally \n\\begin{equation}\nI={\\pi eE_J^2 \\over \\hbar}P(2eV)=\\sqrt{\\pi \\over {\\rho} } \\exp[2\\rho(1-\\gamma)] {eE_J^2 \\tau\\over \\hbar^2}\\left(eV\\tau \\over\n\\rho\\hbar\\right)^{2\\rho-1}~. \n \\label{curr0}\\end{equation}\n\n\nIn summary, if environment provides only the equilibrium noise, the low-bias part of the $IV$ curve is governed by phase\ncorrelations at very long time, which yield in the limit $\\rho \\gg 1$ an extremely small nonanalytic current.\n\n\n\\subsection{With shot noise}\n\nNow we consider the effect of shot noise from an independent source. Parallel to the Josephson junction there is another junction\n(noise junction) with the capacitance $C_s$ connected with an independent\nDC current source (see Fig. \\ref{fig1}b). The resistance of the noise junction is very large compared to the\nshunt resistance $R$. The role of very large capacitance\n$C_{sh}$ is to shortcircuit the large ohmic resistance of the current source for finite frequencies. The\ncurrent\n$I_s$ through the noise junction produces shot noise, which affects the $IV$\ncurve of the Josephson junction. \n\nIn the presence of shot noise the fluctuating phase $\\varphi=\\varphi_0+\\varphi_s$ consists of\ntwo terms: $\\varphi_0$  from Johnson-Nyquist noise, and $\\varphi_s$ from shot noise. \nFor calculation of the shot-noise fluctuations $\\varphi_s$ we assume that\nthe charge transport through the noise junction is a sequence of current\npeaks $\\delta I=\\mbox{sign}(I_s)e\\sum_i\\delta (t-t_i)$, where $t_i$ are random\nmoments of time when an electron crosses the junction \\cite{Blanter}. We neglect \nduration of the tunneling event itself. The positive sign of\n$I_s$ corresponds to the current shown in Fig. \\ref{fig1}b.  Any peak\ngenerates a voltage pulse  at the Josephson junction:\n$V_s(t)=\\mbox{sign}(I_s)(e\/C)\\sum_i \\Theta (t-t_i) e^{-(t-t_i)\/\\tau}$, where\n$\\Theta(t)$ is the step function and $C=C_J+C_s$. The voltage pulses result in phase jumps determined by\nthe Josephson relation $\\hbar \\partial \\varphi_s\/\\partial t=2eV_s$. The sequences of current and voltage\npeaks and phase jumps are shown in Fig. \\ref{fig2}.\n\nCalculating the contribution from shot\nnoise fluctuations $\\varphi_s$ to the phase correlators one should abandon the assumption that noise is Gaussian\n[Eq. (\\ref{Gauss})]. On the\nother hand, we assume that the phase fluctuation\n$\\varphi_s$ is classical and the values of  $\\varphi_s$ at different moments of\ntime commute. Since equilibrium noise and shot noise are uncorrelated,  the\ngeneralization of Eq. (\\ref{WC}) is \n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\left\\langle \\sin \\left({2eVt\\over \\hbar}+\\Delta\n\\varphi_s\\right)\\right\\rangle\\right\\} .\n \\label{curG} \\end{eqnarray}\n\nThe phase difference between two moments $t_0$ and $t_0-t$, $\\Delta \\varphi_s=\n\\varphi_s(t_0)-\\varphi_s(t_0-t)=\\sum _i \n\\delta \\varphi_s(t,t_0-t_i)$, is a sum of contributions from random current peaks, each of them determined\nby\n\\begin{eqnarray}\n\\delta \\varphi_s(t,\\tilde t) \n=\\mbox{sign}(I_s)\\pi \\rho\\left\\{ \\Theta (\\tilde t)\\left [ 1-\ne^{-(\\tilde t)\/\\tau}\\right] \n- \\Theta (\\tilde t-t)\\left [ 1-\ne^{-(\\tilde t-t)\/\\tau}\\right]\\right\\}~, \n     \\end{eqnarray}\nwhere $\\tilde t=t_0-t_i$. Since phase jumps are not  small\nfor $\\rho \\gg 1$, one cannot use the perturbation theory with respect to them. \nThe interpretation of the expression Eq. (\\ref{curG}) for the current is straightforward: As far as the\nrandom phase difference $\\Delta \\varphi_s=2e\\int^{t_0}_{t_0-t}\nV_s(t')dt'\/\\hbar$ is classical,  it should be simply added to the phase difference $\\varphi_V=2eVt\/\\hbar$\ngenerated by the constant voltage bias $V$.\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.9\\linewidth]{ShotL-2.eps}\n   \n    \\caption{Current and voltage pulses, phase jumps.}\n  \\label{fig2}\n  \\end{center}\n  \\end{figure}\n\nLet us consider the case of the Poissonian statistics. If one performs observation during a very long\nperiod of time\n$\\tau_\\infty$, the number of pulses during this period is large and close to $N\\sim |I_s| \\tau_\\infty\/e$. \nKeeping in mind the absence of correlation between pulses, the phase correlator after averaging over\nthe long time $\\tau_\\infty$ is\n\\begin{eqnarray}\n \\left\\langle e^{i\\Delta \\varphi_s}\\right\\rangle =\n \\left\\langle e^{i\\sum _j\\delta \\varphi_s(t, t_0-t_j) }\\right\\rangle \n=  \\left\\langle e^{i\\delta \\varphi_s }\\right\\rangle^N \n\\approx \\left(1+ {\\Phi(t)\\over \\tau_\\infty}\\right)^N ~,\n              \\end{eqnarray}\nwhere $\\Phi(t)=\\Phi_c(t)+i\\Phi_s(t)$\nis determined by the integrals over a single phase jump (corresponding to a single current pulse):\n\\begin{eqnarray}\n\\Phi_s(t)  = \\int_{-\\infty}^\\infty d\\tilde\nt \\sin \\delta \\varphi_s(t,\\tilde t)\n= \\tau {I_s\\over |I_s|} \\left\\{{\\pi \\over 2}+ \\mbox{si} \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right]  +\\sin r \\left[\\mbox{ci} r -\n\\mbox{ci} \\left(r  e^{-t\/\\tau} \\right)\\right] -\\cos r\n\\left[\\mbox{si} r - \\mbox{si} \\left(r \ne^{-t\/\\tau} \\right)\\right] \\right\\}\n        \\label{sin}      \\end{eqnarray}\nand\n\\begin{eqnarray}\n\\Phi_c(t)=\\int_{-\\infty}^\\infty d\\tilde\nt\n\\left[\\cos \\delta \\varphi_s(t, \\tilde t) -1\\right] \n= \\tau \\left\\{ \\mbox{ci} \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right]+\n\\cos r\n\\left[\\mbox{ci} r -\n\\mbox{ci} \\left(r  e^{-t\/\\tau} \\right)\\right]  \\right.\\nonumber \\\\ \\left.\n+ \\sin r\n\\left[\\mbox{si} r - \\mbox{si} \\left(r \ne^{-t\/\\tau} \\right)\\right]     \n-\\gamma-\\ln \\left[r\\left(1-\ne^{-t\/\\tau} \n\\right) \\right] \\right\\}-t~. \n     \\label{cos}         \\end{eqnarray}\nHere $\\mbox{si}(x)=-\\int_x^\\infty\n\\sin t\\, dt\/t$ and $\\mbox{ci}(x)=-\\int_x^\\infty \\cos t\\, dt\/t$ are sine and cosine\nintegral functions \\cite{AS}, and $r=\\pi \\rho$. \nIn the limit $\\tau_\\infty\\to \\infty$ and $N\\to \\infty$ at $N\/\\tau_\\infty=|I_s|\/e$ the phase correlator is \n\\begin{eqnarray}\n \\left\\langle e^{i\\Delta \\varphi_s}\\right\\rangle=  \n\\exp\\left[{N\\Phi(t)\\over \\tau_\\infty}\\right] =\\exp\\left[{|I_s|\\over e}\\Phi(t)\\right]~.\n      \\label{Pois}        \\end{eqnarray}\nFinally the expression for the current, Eq. (\\ref{curG}), can be written as\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\exp \\left({|I_s| \\over e}\\Phi_c \\right) \\sin \\left({2eVt\\over\n\\hbar}+{|I_s| \\over e} \\Phi_s\\right) \\right\\}.\n \\label{curGg} \\end{eqnarray}\n\nIn contrast to the case without shot noise, when the expansion of the current $I$ in a small voltage bias\nstarts from the nonanalytic term $\\propto V^{2\\rho-1}$, in the presence of shot noise\nthe expansion in  $V$ starts with analytic terms:\n\\begin{equation}\nI=I_0 +G_s V+ a V^2+bV^3~,\n    \\label{Vexp}  \\end{equation}\nwhere \n\\begin{eqnarray}\nG_s=-{ 4e^2 E_J^2 \\over \\hbar^3 }\\int_0^\\infty t\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}[\\left\\langle\\cos\n\\Delta\\varphi_s\\right\\rangle-1]\n               \\end{eqnarray}\nis the shot-noise conductance, \n\\begin{eqnarray}\nI_0=-{ 2e E_J^2 \\over \\hbar^2 } \\int_0^\\infty  dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle\n      \\label{ratchet}         \\end{eqnarray}\nis the ratchet current, and the parameters \n\\begin{eqnarray}\na={ 4e^3 E_J^2 \\over \\hbar^4}\\int_0^\\infty t^2\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle\n               \\end{eqnarray}\nand\n\\begin{eqnarray}\nb={ 8 e^4 E_J^2 \\over 3 \\hbar^5} \\int_0^\\infty t^3\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}[\\left\\langle\\cos\n\\Delta\\varphi_s\\right\\rangle-1]\n               \\end{eqnarray}\ndetermine the curvature of the  conductance-voltage plot and the shift of the conductance minimum. Without\nshot noise all these integrals vanish at $\\rho \\gg 1$ [see the paragraph after Eq. (\\ref{J-0l})].\n\nIn Refs. \\onlinecite{exp,SN-T} the parameters of the analytic expansion of $I(V)$ were calculated for low\ncurrents $|I_s|\\ll e\/\\tau$ through the noise junction, when voltage pulses and phase jumps are well\nseparated in time (see Fig.\n\\ref{fig2}). Then  the relevant phase correlators are proportional to the density\n$|I_s|\/e$ of pulses in time \\cite{SN-T}:\n\\begin{equation}\n\\langle e^{i \\Delta \\varphi_s} \\rangle -1= \\langle \\cos \\Delta \\varphi_s \\rangle-1\n+i\\langle \\sin \\Delta \\varphi_s \\rangle = {|I_s| \\over e}\\Phi(t)={|I_s| \\over e}[\\Phi_c(t)+i\\Phi_s(t)]~.\n\\end{equation}\n The parameters \ndetermined by $\\left\\langle\\cos \\Delta\\varphi_s\\right\\rangle$ correspond to ``even'' effects (the conductance\nis symmetric with respect to voltage inversion $V\\to -V$), which are present also without shot noise. Therefore the\ncontribution from shot noise to these parameters  should be\nadded to the values derived from Gaussian equilibrium noise. In contrast, the parameters determined by\n$\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle$  correspond to ``odd'' (asymmetric) effects, which are absent for \nequilibrium noise and are related to the non-Gaussian character of shot noise.\n\nIn the high-impedance limit $\\rho \\gg 1$ it is possible to calculate the parameters\nof the $IV$ curve analytically. As one can see below, the most important\ncontributions to the integrals, which determine $G_s$, $a$, and $b$, come from times $t  \\sim \\tau\n\/\\sqrt{\\rho \\ln \\rho}$ short compared to $\\tau $, and one may use the small-argument expansion for the\nJohnson-Nyquist correlator given in Eq. (\\ref{J-0s}). On the other hand, these times are long enough for using\nasymptotic expansions for the sine and the cosine integrals:\n$\\mbox{si}(x) \\sim -\\cos x\/x -\\sin x\/x^2 $, $\\mbox{ci}(x) \\sim \\sin x\/x -\\cos x\/x^2$. Then one can rewrite Eq. (\\ref{sin})  as \n\\begin{eqnarray}\n\\Phi_s(t) \\approx \\tau \\left\\{{\\pi \\over 2}+ {1\\over r} -{\\cos \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over re^{-t\/\\tau}}-{\\cos \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r(1-e^{-t\/\\tau})} \\right.  \\nonumber \\\\ \\left.\n+{\\sin \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r^2e^{-2t\/\\tau}}-{\\sin \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r^2(1-e^{-t\/\\tau})^2} \\right\\}\n\\approx \\tau \\left[{\\pi \\over 2}-{\\tau\\over rt} \\cos {rt\\over \\tau}  -{\\tau^2\\over r^2t^2} \\sin {rt\\over\n\\tau}\n\\right]  ~.\n      \\label{sinAs}      \\end{eqnarray}\nIn Eq. (\\ref{cos}) it is enough to keep only the main asymptotic terms $\\propto 1\/r$:\n\\begin{eqnarray}\n\\Phi_c(t) \\approx \\tau \\left\\{-{t \\over\\tau} \n+{\\sin \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\over re^{-t\/\\tau}}+{\\sin \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\over\nr(1-e^{-t\/\\tau})}\n -\\gamma-\\ln \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\right\\}\\nonumber \\\\\n\\approx \\tau \\left[-{t \\over\\tau} -\\ln {rt\\over \\tau}  -\\gamma\n+{\\tau \\over rt}\\sin {rt\\over \\tau} \\right]~. \n          \\label{cosAs}  \\end{eqnarray}\n\nLet us consider first the parameters connected with ``even'' effects. For low currents $|I_s| \\ll\ne\/\\tau $:\n\\begin{eqnarray}\nG_s\\approx -{ 4e E_J^2 \\tau \\over \\hbar^3 }|I_s|\\int_0^\\infty t\\, dt \\mbox{Im}\n\\left\\{e^{J_0(t)}\\right\\}{\\tau \\over rt}\\sin {rt\\over \\tau}  \\nonumber \\\\ \n\\approx { 4e E_J^2 \\tau^2\\over \\hbar^3 r}|I_s| \\int_0^\\infty  dt\n\\exp \\left(-\\rho{t^2  \\over \\tau^2}\\ln{\\tau \\over\nt}\\right) \\sin ^2 {rt \\over\\tau} \n\\approx  {\\sqrt{2} \\pi e E_J^2 \\tau^3\\over \\hbar^3 r^{3\/2}\\sqrt{\\ln r}}|I_s|\n\\approx {\\pi^{5\/2}E_J^2 C^3 \\over 4\ne^5 \\sqrt{2\\ln \\rho}}\\rho^{3\/2}|I_s|~,\n    \\label{integr}           \\end{eqnarray}\n\\begin{eqnarray}\nb={ 8 e^3 E_J^2\\tau \\over 3 \\hbar^5}|I_s| \\int_0^\\infty t^3\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}{\\tau \\over rt}\\sin {rt\\over\n\\tau} \n\\nonumber \\\\\n\\approx -{ 8 e^3 E_J^2\\tau^2 \\over 3 \\hbar^5 r}|I_s| \\int_0^\\infty t^2\\, dt \\exp \\left(-\\rho{t^2  \\over \\tau^2}\\ln{\\tau \\over\nt}\\right) \\sin ^2 {rt \\over\\tau} \\approx -{ 2\\pi^2\\sqrt{2} e^3 E_J^2\\tau^5 \\over 3 \\hbar^5 r^{5\/2} (\\ln r)^{3\/2}}|I_s|\n\\approx -{\\pi^2 C^2\\rho\\over 6 e^2\\ln \\rho}G_s ~.\n       \\label{integr-b}        \\end{eqnarray}\nThe main contributions to these integrals come from the last term in the asymptotic expansion Eq.\n(\\ref{cosAs}). The  other terms either vanish (term $\\propto \\gamma$) or are of higher orders (terms $\\propto\nt$ and $\\propto \\ln t$) with respect to $1\/r$. A negative sign of $b$ means that in the center of the\nzero-bias anomaly the curve ``conductance vs. voltage'' has a maximum but not a minimum. However this is\npossible to see only at very low temperatures since finite temperatures give a positive contribution to the\ncurvature parameter $b$.\n\n\nIn a similar way one can calculate the asymmetry integral \n\\begin{eqnarray}\na=-{ 4e^2 E_J^2 \\tau \\over \\hbar^4}I_s\\int_0^\\infty t^2\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n{\\tau^2\\over r^2t^2} \\sin {rt\\over \\tau}={I_s \\over |I_s|}{  C\\over 2e }G_s ~.\n    \\label{integr-a}      \\end{eqnarray}\nHere the main contribution comes from the last term of the asymptotic expansion Eq.  (\\ref{sinAs}). \n\n For the integral in Eq.\n(\\ref{ratchet}), which determines the ratchet current $I_0$, the relevant times are shorter than for the\nother integrals:\n$t\n\\sim \\tau\/\\rho \\sim R_QC$. Therefore one cannot use the asymptotic expansion Eq. (\\ref{sinAs}). Instead one can\nintegrate by parts, and for low noise currents: \n\\begin{eqnarray}\nI_0=-{ 2e E_J^2 \\over \\hbar^2 } {|I_s|\\over e} \\int_0^\\infty  dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\Phi_s(t) \\approx { 2 E_J^2 \\over \\hbar^2 }|I_s| \\int_0^\\infty  dt \\Phi_s(t) \\sin {r t\\over \\tau}\n = { 2 E_J^2\\tau \\over \\hbar^2 r } |I_s| \\int_0^\\infty  dt \\frac {d\\Phi_s(t)}{dt}\\cos {r t\\over \\tau}\n\\nonumber \\\\\n\\approx  { 2E_J^2\\tau^2 \\over \\hbar^2 r }I_s \\int_0^\\infty  {dt\\over t} \\cos {r t\\over \\tau}\n\\sin {r t\\over \\tau}={ \\pi E_J^2\\tau^2 \\over 2 \\hbar^2 r }I_s= {\\pi^2 E_J^2 C^2\\over 8e^4 }\\rho I_s\\, .\n     \\label{integr-r}     \\end{eqnarray}\n\n\\subsection{Shot noise from high currents and electron counting statistics} \\label{sec2c}\n\n\nShot noise is non-Gaussian and asymmetric, but in the case of low noise currents its effect does not\ndepend on statistics of electron transport and is determined only by density of current pulses in time.\nEven if pulses were not random and formed a strictly periodical sequence the effect would be the same. In\norder to obtain information on counting statistics from measurements of the $IV$ curve the noise current\nshould be not small compared with the width of the voltage pulse $\n\\sim \\tau$. One could expect \\cite{SN-T} that in the limit of high noise current $I_s$ discreteness of the\nelectron transport through the junction would be less and less essential and eventually the effect of the\ncurrent\n$I_s$ would be reduced to the constant voltage drop $V_s =I_s  R$ on the shunt resistor $R$ resulting\nin a trivial shift of the voltage applied to the Josephson junction. If it were the case, the long-time asymptote for\nthe phase correlator would be\n\\begin{eqnarray}\n \\langle e^{i\\Delta \\varphi_s }\\rangle \\to \\exp \\left(2ie V_s t\\over \\hbar \\right)\n=\\exp \\left(2ie I_s Rt\\over \\hbar \\right)=\\exp \\left(ir {I_s t\\over e} \\right)~.\n          \\label{asyTr}    \\end{eqnarray} \nThis asymptotic behavior really takes place for a strictly periodic sequence of current pulses with period\n$T_0=e\/|I_s|$. In this case the phase variation for the time interval $nT_0 <t\n< (n+1)T_0$ ($n$ is integer) is:\n\\begin{eqnarray}\n\\varphi(t)=r \\left[n +{1-e^{-(t-nT_0)\/\\tau} \\over1-e^{-T_0\/\\tau}}  \\right]  ~.           \n\\end{eqnarray}\nIn the limit $|I_s| \\to \\infty$ ($T_0 \\to 0$) the maximum deviation of the phase from the linear function\n$\\varphi(t)=rt \/T_0$ is of the order $r T_0^2\/\\tau ^2$, i.e., vanishes in this limit. This yields Eq. (\\ref{asyTr}).\n\nHowever, the expression (\\ref{Pois}) derived from the Poissonian statistics has another asymptotic\nbehavior for $t \\to \\infty$:\n\\begin{eqnarray}\n \\langle e^{i\\Delta \\varphi_s }\\rangle-1\n\\to  \\exp\\left[ \\left(e^{ir}-1\\right){I_s t\\over e}\\right] ~.\n        \\label{asyPoi}       \\end{eqnarray}\nThus the $IV$ curve of the\nJosephson junction essentially depends on whether the current pulses through the parallel (noise) junction are strictly\nperiodical in time, or completely random.  This proves strong sensitivity  to counting statistics.\n\nNote that the right-hand side of Eq. (\\ref{asyPoi}) coincides with the characteristic function\n$\\chi(r)=\\sum_n P_ne^{ir n}$ for the Poissonian distribution $P_n=e^{-\\bar{n}}\\sum_n \\bar{n}^n\/n!$ with\n$\\bar{n}=I_st\/e$. This is a natural result since  the number of phase jumps $n=\\Delta \\varphi_s\/2\\pi$ during a\nlong time $t \\gg \\tau$ coincides with the number of electrons transmitted through the noise junction during\nthe same time interval. Equation (\\ref{asyPoi})  leads to a conclusion that in the case of the Poissonian\nstatistics shot noise affects the $IV$ curve even at high noise currents when one might expect suppression of\nnoise effects. This suppression takes place for usual methods of noise detection probing voltage fluctuations.\nIndeed calculating the voltage variance $\\langle\\Delta V^2 \\rangle$ for the Poissonian statistics,\n\\begin{eqnarray}\n \\langle \\Delta V^2 \\rangle=  \\langle  V^2 \\rangle- \\langle  V \\rangle^2\n= {e\\over 2 C}\\langle  V \\rangle~,\n              \\end{eqnarray} \n one obtains that the relative mean-square-root voltage fluctuation $ \\sqrt{\\langle \\Delta V^2 \\rangle\n}\/\\langle  V \\rangle$ vanishes at $V_s =\\langle  V \\rangle=I_s R \\to \\infty$. However, the\nCoulomb blockaded junction probes not voltage but phase, which is much more sensitive to\nfluctuations. This ``supersensitivity'' of the Coulomb blockaded junction to fluctuations explains why the\neffects of higher moments (or cummulants) are much stronger than in usual methods of noise detection\n\\cite{BKN,Reul,Rez}.\n\n\\section{Josephson junction, weak coupling limit, phase noise in the AC input} \\label{sec3}\n\nNow we want to demonstrate that the Coulomb blockaded junction can effectively probe not only shot noise\nbut other types of noise as well. As an example, phase fluctuations in a monochromatic\nelectromagnetic wave will be considered. This type of noise is very important for lasers \\cite{L}, but the\npresent analysis addresses  microwaves. Suppose that some microwave source provides an input signal described by phase variation at\nthe Josephson junction:\n\\begin{eqnarray}\n \\varphi_i=\\varphi_0 \\sin \\psi(t)=\\varphi_0 \\sin \\left[\\omega t+ \\sum_i \\gamma_i\n\\Theta(t-t_i)\\right]~,\n        \\label{phN}      \\end{eqnarray}\nwhere $\\gamma_i$ are random phase jumps at random moments of time $t_i$. Frequency of phase jumps is characterized by the\naverage time $\\tau_c$ between the jumps. In fact $\\tau_c$ is nothing else as the decoherence time of the microwave signal. We\nshall see that the $IV$ curve of the Josephson junction is very sensitive to the decoherence process. But in order to demonstrate\nit one should first consider the response of the Josephson junction to  a strictly monochromatic noiseless microwave\ninput.\n\n\\subsection{Response to a monochromatic AC input} \\label{AClin}\n\nFor a strictly monochromatic signal the phase  difference between the moments $t_0$ and $t_0-t$ is \n\\begin{eqnarray}\n\\Delta \\varphi_i(t_0,t)=\\varphi_i(t_0)-\\varphi_i(t_0-t)=\\varphi_0 [\\sin \\omega t_0-\\sin \\omega (t_0-t)]\n=\\varphi_0 [\\sin \\omega t_0(1-\\cos \\omega t)+\\cos \\omega t_0\\sin \\omega t]~.\n              \\end{eqnarray}\nIn full analogy with the previous cases, the current is given by\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\sin \\Delta\n\\varphi_i(t_0,t)\\right\\}~, \n       \\end{eqnarray}\nbut now averaging over the moment $t_0$ is not done. The linearization with respect to the input gives\n\\begin{eqnarray}\nI\\approx -{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\Delta\n\\varphi_i(t_0,t)\\right\\} \n= -\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n[\\cos \\omega t_0\\sin \\omega t +\\sin \\omega t_0(1-\\cos \\omega t)]\\right\\} \n \\label{curAC} \\nonumber \\\\\n= I_1 \\cos \\omega t_0 +I_2 \\sin \\omega t_0 ~.\n   \\label{curPh}   \\end{eqnarray}\nThus the tunneling current of Cooper pairs consists of two parts: one is out of phase and another is in phase with the\ninput voltage $ V_i=V_0 \\cos \\omega t$ (the reactive and the active responses)  where $V_0=\\hbar \\omega \\varphi_0\/2e $. Comparing\nthe active response,\n\\begin{eqnarray}\nI_1= -\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\sin \\omega t\\right\\} ~,\n              \\end{eqnarray}\nwith Eq. (\\ref{WC}) one can see that the ohmic current as a function of frequency can be directly received from the \n$IV$ curve $I(V)$  replacing voltage $V$ by $\\hbar \\omega\/2e$. Therefore the  active (ohmic)\nresponse is given by\n\\begin{eqnarray}\nI_1= \\varphi_0 I\\left(\\hbar \\omega \\over 2e\\right)\n= V_0 {2e  \\over \\hbar \\omega} I\\left(\\hbar \\omega \\over 2e\\right)~.\n              \\end{eqnarray}\nThus the ohmic AC linear conductance of the junction is a nonanalytic function of frequency proportional to $\n\\omega^{2\\rho -2}$. \n\nThe reactive response for frequencies $\\omega \\ll \\rho\/\\tau=1\/R_QC$ is \n\\begin{eqnarray}\nI_2=-\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n(1-\\cos \\omega t)\\right\\}\\approx -\\varphi_0{ eE_J^2 \\over\n\\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)} \\omega^2 t^2\\right\\}~ .\n      \\end{eqnarray}\nAfter rotation in the complex plane ($t \\to-iy$):\n\\begin{eqnarray}\nI_2=-\\varphi_0{eE_J^2 \\over \\hbar^2}\\omega^2\\int_0^\\infty y^2 dy e^{-\\pi \\rho y\/\\tau}\n=-\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\omega^2\\left(\\tau\\over \\pi \\rho\\right)^3\n=-V_0{2e\\over \\hbar \\omega}{2 eE_J^2 \\over \\hbar^2}\\omega^2\\left(R_Q C\\over \\pi\n\\right)^3=-V_0\\omega C{E_J^2 C^2\\over 2e^4}~.\n         \\end{eqnarray}\nThe linear dependence on frequency points out that this current corresponds to the capacitance element of the\nelectrical circuit. Indeed, the effective circuit for the Josephson junction has the capacitance $C$ in parallel to the\ntunneling conductance element. But in the presence of the Josephson coupling the geometric capacitance should be\nreplaced by the effective capacitance (like the mass of an electron in a periodic potential should be replaced by the\neffective mass, see, e.g., Ref. \\onlinecite{SZ}). The current component $I_2$ is related with the difference between the\neffective and the geometric capacitance in the weak coupling limit. In order to check it one should calculate the\nsecond-order (with respect to the Josephson coupling) correction to the Coulomb energy $Q^2\/2C$:\n\\begin{eqnarray}\nE={Q^2\\over 2 C}+ \\sum _{Q_G\\neq 0}{V(Q_G)^2 \\over E_0(Q)-E_0(Q+Q_G)}~,\n         \\end{eqnarray}\nwhere summation is over the inverse-lattice ``wave-numbers'' (charges) $Q_G=2e n$ ($n$ is any integer).  For weak\nJosephson coupling only two terms  $ Q_G=\\pm 2e$ of the sum are important, and $V(\\pm 2e)=E_J\/2$. Then \n\\begin{eqnarray}\nE={Q^2\\over 2 C}+ {E_J^2\\over 4}\\left[{2 C \\over Q^2-(Q+2e)^2}+ {2C \\over\nQ^2-(Q-2e)^2}\\right]\n={Q^2\\over 2 C}+{E_J^2\\over 4}{e^2 C\\over e^4 -e^2Q^2}\\approx {Q^2\\over 2 C}+{E_J^2 C\\over 4e^2}-{E_J^2 C\\over\n4e^4}Q^2~.\n         \\end{eqnarray}\nThis yields for the effective capacitance:\n\\begin{eqnarray}\n{1\\over C^*}= {d^2 E\\over dQ^2}={1\\over C}-{E_J^2 C\\over 2e^4}~,\n        \\end{eqnarray}\nand it is evident that the reactive component $I_2$ of the tunneling current is determined by $\\Delta C=C^*-C$. It is\nuseful to rewrite the expression for this correction using  the\nAmbegoakar-Baratoff relation:\n\\begin{eqnarray}\n E_J={\\pi \\hbar \\Delta \\over 4e^2 R_T}={\\Delta_0 \\over 2}{R_Q\\over R_T}~,\n        \\end{eqnarray}\nwhere $\\Delta_0$ is the bulk superconducting gap. Then the capacitance correction is\n\\begin{eqnarray}\n\\Delta C=C {E_J^2 C^2 \\over 2e^4}=C{R_Q^2\\over R_T^2}{\\Delta_0^2 \\over 8e^4\/C^2}~.\n  \\label{Ccor}       \\end{eqnarray}\n\nNow we calculate the nonlinear (quadratic) effect of the AC input on the DC conductance. This requires averaging (over\nthe period of the AC input) of the quadratic term in the expression for the conductance:\n\\begin{eqnarray}\n\\Delta G=-{4 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt e^{J_0(t)}\n\\langle \\cos \\Delta\\varphi_i \\rangle\\right\\} \n\\approx {2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}\\langle  \\Delta \\varphi_i^2 \\rangle\\right\\} \\nonumber \\\\\n=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}(1-\\cos \\omega t)\\right\\}~.\n   \\label{DG}   \\end{eqnarray}\nThe expansion in $\\omega$ contains only the terms even in $t$. The integrals over these terms vanish after rotation in\nthe plane of complex time. This means that $\\Delta G$ is determined by a very small nonanalytic contribution from long\ntimes \\cite{FBS}. The presence of\nnoise essentially changes the situation as we shall see in the next subsection.\n\n\\subsection{Phase noise}\n\nIn the presence of the phase noise, when the phase varies according to Eq. (\\ref{phN}), the squared difference between the phases\nat the moments of time $t_0$ and\n$t_0-t$ averaged over the time $t_0$ is given by\n\\begin{eqnarray}\n\\langle \\Delta \\varphi_i(t_0,t)^2\\rangle =\\varphi_0^2 \\langle[\\sin \\psi(t_0)-\\sin \\psi\n(t_0-t)]\\rangle^2 =\\varphi_0^2 \\{1-\\langle\\cos[\\psi(t_0)-\\psi (t_0-t)]\\rangle\\}\\, .\n    \\end{eqnarray}\nThe average $\\langle\\cos[\\psi(t_0)-\\psi (t_0-t)]\\rangle$ does not vanish only if there is no random phase jumps in the time\ninterval $t$. Assuming the  Poisson distribution the probability of the absence of random phase jumps in the interval $t$\nis\n$e^{-t\/\\tau_c}$, where\n$\\tau_c$ is the average period between random phase jumps. Then one obtains that\n\\begin{eqnarray}\n\\langle \\Delta \\varphi_i(t_0,t)^2\\rangle  =\\varphi_0^2(1- \\cos \\omega t e^{-t\/\\tau_c})~.\n    \\end{eqnarray}\nFinally the correction to the DC conductance [compare with Eq. (\\ref{DG})] is:\n\\begin{eqnarray}\n\\Delta G=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}(1-\\cos \\omega t e^{-t\/\\tau_c})\\right\\}\n\\approx \\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3 \\tau_c}\\mbox{Im}\\left\\{\\int_0^\\infty\nt^2\\,dt e^{J_0(t)}\\right\\}~.\n       \\end{eqnarray}\nAfter rotation in the complex plane the imaginary part of the integral does not vanish, and\n\\begin{eqnarray}\n\\Delta G=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3 \\tau_c}\\int_0^\\infty\ny^2\\,dy e^{-\\pi \\rho y\/\\tau}=\\varphi_0^2{4 e^2E_J^2 \\tau^3\\over\\pi^3\n\\rho^3 \\hbar^3 \\tau_c}=\\varphi_0^2{E_J^2 C^3\\over 2 e^4 \\tau_c}\n=V_0^2{2E_J^2 C^3\\over  e^2 \\hbar ^2 \\omega^2 \\tau_c}~.\n      \\end{eqnarray}\nThis contribution to the zero-bias conductance essentially exceeds the small nonanalytic contribution from the\nstrictly monochromatic microwave and provides direct information on the decoherence time $\\tau_c$. \n\n\\section{Normal junction, shot noise}  \\label{sec4}\n\n\n\\subsection{The $IV$ curve in the time-domain formulation (without shot noise)}\n\nThis subsection reminds the $P(E)$ theory for a normal Coulomb blockaded junction \\cite{IN}. The current is given\nby \n\\begin{equation}\n  I= { e}[\\Gamma^+(V)-\\Gamma^-(V)] ~,\n        \\end{equation}\nwhere the forward tunneling probability is\n\\begin{equation}\n\\Gamma^+(V) = {1\\over e^2 R_T}\\int_{-\\infty}^\\infty dE \\frac{E}{1\n-\\exp\\left(-\\frac{E}{T}\\right)}P(eV-E)~.\n    \\label{ProbE} \\end{equation}\nHere $P(E)$ is given by Eq. (\\ref{P-E}). The backward tunneling rate is $\\Gamma^- (V)=\\Gamma^+(-V)$.\nSince now we consider tunneling of single electrons (instead of Cooper pairs) the correlation function is given by\n\\begin{equation}\nJ_0(t) =\\langle[ \\varphi(t) -\\varphi(0)]\\varphi(0)\\rangle\n= 2\\int_{-\\infty}^\\infty {d\\omega \\over \\omega} \\frac{\\mbox{Re} \nZ(\\omega)}{R_K} \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}~,\n      \\end{equation}\nwhere the single electron quantum resistance $R_K=h\/e^2$ replaces the Cooper-pair quantum resistance $R_Q=h\/2e^2$ in\nEq.~(\\ref{phaseCor}). In the case of purely ohmic environment, $Z(\\omega)=R\/(1+i\\tau \\omega)$, \none can use the time-domain formulation \\cite{JED} fulfilling integration over energy $E$ in Eq. (\\ref{ProbE}):\n\\begin{equation}\n\\Gamma^+(V) = {1\\over \\pi \\hbar e^2 R_T}\\mbox{Re}\\left\\{\\int_0^\\infty dt\n\\gamma(t)\\exp\\left[ J_0(t) + \\frac{ieVt}{\\hbar}\\right] \\right\\}  ~.\n    \\label{ProbT}\\end{equation}\nHere $\\tau=RC$ as before, and\n\\begin{equation}\n\\gamma(t) = \\int_{-\\infty}^\\infty dE \n\\frac{E}{1-\\exp\\left(-\\frac{E}{T}\\right)}\n\\exp\\left( - \\frac{iEt}{\\hbar}\\right) =i\\pi\\hbar ^2 {d\\over dt} \\delta(t)\n- {\\pi^2 \\over \\beta^2} \\frac{1}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)}~.\n   \\end{equation}\nThen after some simple algebra the current is \n\\begin{eqnarray}\n  I= { e}[\\Gamma^+(V)-\\Gamma^-(V)]= {1\\over R_T} \\left[\nV  + {2\\pi \\over e\\hbar \\beta^2} \\mbox{Im}\\left\\{  \\int_0^\\infty \n\\frac{dt}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)} e^{J_0(t)}\\sin \\frac{eVt}{\\hbar}\\right\\} \\right] ~.\n        \\end{eqnarray}\nAt $T=0$:\n\\begin{eqnarray}\n  I= {1\\over R_T} \\left[V  + {2 \\hbar \\over \\pi e } \\mbox{Im}\\left\\{ \\int_0^\\infty \n\\frac{dt}{t ^2 } e^{J_0(t)}\\sin \\frac{eVt}{\\hbar} \\right\\}\\right] ~,\n    \\label{i-v}    \\end{eqnarray}\nand the differential conductance is\n\\begin{equation}\n G= {dI \\over dV}=  {1\\over R_T} \\left[\n1 + {2\\over \\pi } \\mbox{Im}\\left\\{ \\int_0^\\infty \n\\frac{dt}{t} e^{J_0(t)}\\cos \\frac{eVt}{\\hbar}\\right\\} \\right] ~.\n        \\end{equation}\nFor $\\rho=R\/R_K \\gg 1$ in the limit $V \\to 0$ the conductance vanishes:\n\\begin{equation}\n G_0= {1\\over R_T} \\left[1  - {2\\over \\pi }  \\int_0^\\infty \\frac{dt}{ t}\\sin \\left( \\pi \\rho {t\\over\n\\tau}\\right)\n \\right] \\to  0~,\n          \\end{equation}\nbut there remains a small nonanalytic contribution, which originates from long times: $G_0 \\propto V^{2\\rho}$.\n\n\\subsection{Effect of shot noise}\n\nRepeating derivation of the  effect of the random phase from shot noise, which was done above for the Josephson junction,\none receives that the effect on the normal tunnel junction is similar: the  linear phase from the constant voltage,\n$eVt\/\\hbar$, must be replaced  by the phase $eVt\/\\hbar + \\Delta \\varphi_s$, where $\\Delta \\varphi_s=\\varphi_s(t_0)\n-\\varphi_s(t_0-t)$ is the phase difference produced by the current through the noise junction. Note that the Josephson\nrelation between the voltage and the phase from shot noise is different from that for the Josephson junction by the factor\nof 2:\n$V_s(t)=(\\hbar\/e) d\\varphi_s(t)\/dt_s$.  Then the probability of the forward tunneling is [instead of Eq. (\\ref{ProbT})]\n\\begin{equation}\n\\Gamma^+(V) = {1\\over \\pi \\hbar e^2 R_T}\\mbox{Re}\\left\\{\\int_0^\\infty dt\n\\gamma(t)\\left\\langle\\exp\\left( J(t) + \\frac{ieVt}{\\hbar}+i\\Delta \\varphi_s\\right) \\right\\rangle \\right\\} ~,\n    \\end{equation}\nand the total current is \n\\begin{eqnarray}\n I= { e}[\\Gamma^+(V)-\\Gamma^-(V)]= {1\\over R_T} \\left[\nV + V_s  + {2\\pi \\over e\\hbar \\beta^2}\\mbox{Im}\\left\\{  \\int_{0}^\\infty \n\\frac{dt}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)} e^{J(t)}\\left\\langle\\sin\n\\left( \n\\frac{eVt}{\\hbar}+\\Delta \\varphi_s\\right) \\right\\rangle \\right\\}\\right]~.\n        \\end{eqnarray}\nHere $ V_s =I_sR$ is the average voltage generated by the current through the noise junction.  In\nthe\n$T=0$ limit the current is:\n\\begin{eqnarray}\n I= {1\\over R_T} \\left[\nV +V_s + {2\\hbar  \\over \\pi e} \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J(t)}\\left\\langle\\sin\n\\left( \n\\frac{eVt}{\\hbar}+\\Delta \\varphi_s\\right) \\right\\rangle \\right\\}\\right]~.\n        \\end{eqnarray}\n\nFor the normal junction one should replace $r$ by $2r$ in Eqs. (\\ref{sin}) and (\\ref{cos}) if the single-electron\nquantum resistance $R_K$ is used in $r=\\pi \\rho=\\pi R\/R_K$. We need the approximate expressions for the phase\ncorrelator at\n$t\\ll\n\\tau$:\n\\begin{eqnarray}\n\\langle \\sin \\Delta \\varphi_s \\rangle \\approx {I_s \\tau\\over e}\\left(\\mbox{si} {2rt\\over \\tau }  +{\\pi \\over 2} \n\\right)  ={I_s \\tau\\over e}\\mbox{Si} {2rt\\over \\tau }  ~,            \n\\end{eqnarray}\n\\begin{eqnarray}\n(\\langle \\cos \\Delta \\varphi_s \\rangle-1)\\approx {I_s \\tau\\over e}\\left(\\mbox{ci}\n{2rt\\over \\tau } -\\gamma-\\ln {2rt\\over \\tau }\\right)~.\n              \\end{eqnarray}\nWith help  of these expressions one receives the expressions for the ratchet  current,\n\\begin{eqnarray}\n I_0= {1\\over R_T} \\left[V_s + {2\\hbar  \\over \\pi e}  \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J_R(t)}\\sin J_I(t)\\left\\langle \\sin\n\\Delta \\varphi(t_s,t)\\right\\rangle \\right] \\approx\n {1\\over R_T} \\left[V_s - {2\\hbar  \\over \\pi e} {I_s \\tau\\over e}\n\\int_{0}^\\infty \n\\frac{dt}{t^2}  \\sin {rt\\over \\tau }\n\\mbox{Si} {2 rt\\over \\tau }\\right] \\nonumber \\\\\n={1\\over R_T} \\left[V_s - (1+\\ln 2)I_s {\\hbar  \\over e^2} \nr\\right]=0.153 {V_s\\over  R_T}~,\n        \\end{eqnarray}\nand for the shot-noise contribution to the conductance:\n\\begin{eqnarray}\nG_s={d I\\over dV}= {1\\over R_T} \\left[1 + {2 \\over \\pi }  \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_R(t)}\\sin J_I(t)\\left\\langle\\cos\n\\Delta \\varphi(t_s,t)\\right\\rangle \\right] \\nonumber \\\\\n\\approx\n-{1\\over R_T}  {2 \\over \\pi } {|I_s| \\tau\\over e} \\int_{0}^\\infty \n\\frac{dt}{t}\\sin {rt\\over \\tau }\\left(\\mbox{ci}\n{2rt\\over \\tau } -\\gamma-\\ln {2rt\\over \\tau }\n\\right)\n={0.693\\over\nR_T}   {|I_s|\\tau\\over e} ={0.693\\over\nR_T}   {|V_s|\\over e\/C}~.\n        \\end{eqnarray}\n\n\\subsection{Response of the normal junction to an AC input}\n\nThe normal Coulomb blockade junction is sensitive to phase noise in an AC input as well. First let us consider the\nlinear response to a monochromatic input using analogy with the case of the Josephson junction (Sec. \\ref{AClin}):\n\\begin{eqnarray}\n  I= {1\\over R_T} \\left[V_i  + {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}\\sin\\Delta \\varphi_i(t_0,t) \\right]\n\\approx  {1\\over R_T} \\left[V_i  + {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}\\Delta \\varphi_i(t_0,t) \\right]  \\nonumber \\\\\n={1\\over R_T} \\left[V_0 \\cos \\omega t_0  -\\varphi_0 {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}[\\cos \\omega t_0\\sin \\omega t +\\sin \\omega t_0(1-\\cos \\omega\nt)] \\right]\n= I_1 \\cos \\omega t_0 +I_2 \\sin \\omega t_0 ~,\n     \\end{eqnarray}\nwhere $V_i=V_0 \\cos \\omega t_0$ with $V_0=\\hbar \\omega \\varphi_0 \/e$. The ohmic response is\n\\begin{eqnarray}\n I_1= {V_0\\over R_T}  + {2\\hbar  \\over \\pi e R_T}\\varphi_0  \\int_{0}^\\infty \n\\frac{dt}{t^2} \\mbox{Im} \\{e^{J_0(t)}\\}\\sin\\omega t\n= \\varphi_0  I\\left(\\hbar \\omega \\over e\\right)\n=V_0{e\\over \\hbar \\omega}I\\left(\\hbar \\omega \\over\ne\\right)~,\n        \\end{eqnarray}\nwhere the $IV$ curve $I(V)$ of the normal junction without AC input was used [Eq. (\\ref{i-v})] with $V$ replaced by $\\hbar\n\\omega\/e$.\n\nThere is also the reactive response:\n\\begin{eqnarray}\n I_2= {2\\hbar  \\over \\pi e R_T}\\varphi_0 \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J(t)}(1-\\cos\\omega t) \\right\\} \\approx\n{2\\hbar  \\over \\pi e R_T}\\varphi_0 \\omega^2 \\mbox{Im}\\left\\{\\int_{0}^\\infty \ndt  e^{J(t)}\\right\\} \\nonumber \\\\\n=-\\omega V_0{2  \\over \\pi R_T}  \\int_{0}^\\infty \ndy  e^{-\\pi \\rho t\/\\tau}=-\\omega V_0{2 \\tau \\over \\pi^2 \\rho R_T} \n=-\\omega C V_0{2 R_K \\over \\pi^2 R_T} ~.\n        \\end{eqnarray}\nThe reactive response is a result of the Coulomb blockade correction to the geometric capacitance:\n\\begin{eqnarray}\n \\Delta C= C {2 R_K \\over \\pi^2 R_T}\n        \\end{eqnarray}\nsimilar to the correction for the Josephson junction, Eq. (\\ref{Ccor}). \n\n\nQuadratic correction to the DC zero-bias conductance is also sensitive to the decoherence time $\\tau_c$:\n\\begin{eqnarray}\n\\Delta G\\approx -{1\\over R_T} {1 \\over \\pi } \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_0(t)}\\left\\langle\\Delta \\varphi_i^2\\right\\rangle \\right\\}\n=-{1\\over R_T} {\\varphi_0^2 \\over \\pi } \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_0(t)}(1-\\cos \\omega t e^{-t\/\\tau_c})\\right\\} \\nonumber \\\\\n\\approx -{1\\over R_T} {\\varphi_0^2 \\over \\pi \\tau_c} \\mbox{Im}\\left\\{ \\int_{0}^\\infty \ndt e^{J_0(t)}\\right\\} = {1\\over R_T} {\\varphi_0^2 \\over \\pi \\tau_c} \\int_{0}^\\infty \ndy e^{-\\pi \\rho\/\\tau}= {1\\over R_T} {\\varphi_0^2 \\tau\\over \\pi^2 \\rho \\tau_c} \n = {2\\over R_T} {V_0^2 C\\over \\pi\\hbar \\omega^2 \\tau_c}~.\n        \\end{eqnarray}\n\n\\section{Josephson junction, strong coupling limit, shot noise} \\label{sec5}\n\n\\subsection{Without shot noise}\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.8\\linewidth]{Duality.eps}\n    \\bigskip\n    \\caption{Strong coupling limit and environment duality. The dashed squares \nshow the circuit elements, which present the Josephson and the noise junctions with  the capacitance $C=C_J+C_s$. The\ncurrent source $I_\\Sigma=I+I_s$ provides the constant current $I$ and the noisy current $I_s$ in parallel. a)\nThe dual weak-coupling case.  The inductance\n$L_J$ of the Josephson junction is negligible in the weak-coupling limit. b) The dual strong-coupling limit. c) The\nstrong-coupling case taking into account the capacitance\n$C_0$ and the resistance $R_0$ of the leads.}\n  \\label{fig3}\n  \\end{center}\n  \\end{figure}\n\n In the case of equilibrium noise the strong coupling limit can be investigated using the\nduality relations, which connect the parameters of the Josephson junction in the weak coupling and the strong\ncoupling limits \\cite{SZ,Weiss}:\n\\begin{equation}\nE_J \\rightarrow 2 \\Delta~,~~V \\rightarrow {h \\over 4e^2}I = R_Q I~,~~\\varphi\n\\rightarrow {\\pi Q \\over e}~,~~~\\rho={R\\over R_Q} \\rightarrow \\tilde\n\\rho={R_Q\\over R}~,~~~Z \\rightarrow Y~.\n      \\label{du} \\end{equation}\nHere $Y(\\omega)$ is the admittance of the electric circuit, and $\\Delta$ is the matrix element for a transition between adjacent\nstates of the Josephson junction with localized phases $2\\pi n$ and $2\\pi (n+1)$ ($\\hbar\/\\Delta$ is the life time at the state,\nwhich corresponds to some localized phase). Using these relations the\n$IV$ curve in the weak coupling limit, Eq. (\\ref{WC}), transforms to the dual $VI$ curve in the strong-coupling\nlimit:\n\\begin{eqnarray}\nV=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\sin \\left(2eR_Q It\\over \\hbar\\right)\n=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\sin \\left(\\pi It\\over e\\right)~.\n \\end{eqnarray}\nThe charge-charge correlator [see Eq. (149) in Ref. \\onlinecite{IN}] is given by the relation dual to Eq.\n(\\ref{phaseCor}):\n\\begin{eqnarray}\n\\tilde J_0(t)=2R_Q  \\int_0^\\infty\n{d\\omega \\over \\omega} {\\mbox{Re}  Y(\\omega)}\n \\left[\\coth\\left({1\\over 2}\\beta \\hbar \\omega\\right)(\\cos \\omega t -1)\n-i\\sin \\omega t\\right] ~.\n    \\label{Q-IN}  \\end{eqnarray}\n \nIn order to exploit duality completely, i.e., not only for the general expression for the $IV$ curve but also for the\nspecific results for ohmic environment with the effective impedance $Z=(1\/R+i\\omega C )^{-1}$, one should find a\ncorresponding ``dual'' environment for the strong-coupling case. In\nthe weak-coupling case the finite relaxation time $\\tau=RC$ is provided by\nthe imaginary element of the impedance $Z$, which is the capacitance $C=C_J+C_s$ of the two junctions (without the\nnoise source $C_s=0$). In the strong coupling case the both imaginary elements of the Josephson junction (capacitance\nand inductance) do not contribute to the admittance (see Fig. \\ref{fig3}b). In order to get a finite relaxation time\ndual to the time $\\tau=RC$ of the weak coupling limit one should add to the effective electric circuit\nthe inductance $L$ in series with the ohmic shunt resistance $R$ (Fig. \\ref{fig3}b). Then the admittance\ndual to the impedance $Z$ of the weak coupling limit is \n\\begin{equation}\n Y(\\omega)=\\frac{1}{R +i\\omega L}=\\frac{1\/R}{1+i\\omega \\tau_L}~,\n      \\end{equation}\nwhere the circuit relaxation time $\\tau_L =L\/R$ is dual to the relaxation time $\\tau=RC$ in the weak\ncoupling limit. Thus duality relations Eq. (\\ref{du}) should by supplemented by the ``environment\nduality'' relation $C \\rightarrow L$. Note, however, that introduction of finite inductance $L$ is not\nthe only way to achieve a finite relaxation time. For example, Sch\\\"on and Zaikin \\cite{SZ} suggested\nthat the period of Josephson plasma oscillation $\\sim \\hbar\/\\sqrt{E_J E_c}$ should play the role of\nrelaxation time dual to the time $RC$ in the weak coupling limit. \n\n\\subsection{With shot noise}\n\nThe duality concept is valid only for equilibrium noise when symmetry\nbetween current and voltage or impedance and admittance takes place. In the case of shot noise this\nsymmetry is broken since shot noise is a {\\em current} noise, which produces asymmetric effects on the\ncurrent and on the voltage. Though we can still use the duality for description of the equilibrium\ncontribution to the noise (this contribution cannot be ignored even if our goal is the effect of shot\nnoise), the effect of shot noise on the $IV$ curve is essentially different in the weak and the strong\ncoupling limits.\n\nIn the strong-coupling limit the effect of environment is governed by charge fluctuations. Whereas the phase\nfluctuations were determined by the voltage fluctuations via the Josephson relation $\\hbar d\\varphi\/dt=2eV$, the charge\nfluctuations are determined by the current fluctuations via the relation\n$dQ\/dt =I$. Important is that the current $I$ should be the {\\em total current} of the part of the circuit, which describes the\nJosephson junction in the RCSJ model (inductance + capacitance + shunt resistance). Single-electron tunneling through the noise\njunction produces a pulse  of the total current. If the tunneling time is very short, the pulse can be approximated by the\n$\\delta$-function current peak. The effective circuit in  Fig. \\ref{fig3}b cannot broaden the original $\\delta$-function\ncurrent peak. This is in contrast to the voltage pulses, which were relevant for phase fluctuations in the weak coupling\nlimit: the $\\delta$-function current peak produced the voltage pulse of width $\\tau=RC$.\nMeanwhile, the finite width of the current pulse is of principal importance for the shot-noise effect  and therefore cannot be\nignored. Broadening of the current pulse can be determined by additional lump elements of the electric circuit, which are\nconnected with the ohmic resistance $R_0$ and the capacitance $C_0$ of wires connecting the current-noise source and the probe\nJosephson junction (see Fig. \\ref{fig3}c). Bearing in mind this electric circuit the single-electron tunneling at $t-0$ produces\nthe current pulse\n\\begin{equation}\n\\delta I (t)=\\mbox{sign}(I_s){e\\over \\tau_s} e^{-t\/\\tau_s}~,\n      \\end{equation}\nwhich brings the charge\n\\begin{equation}\n\\delta Q_s(t)=\\mbox{sign}(I_s)e(1- e^{-t\/\\tau_s})~.\n      \\end{equation}\nHere $\\tau_s=R_0C_0$. \nThe $VI$ curve with shot noise is given by\n\\begin{eqnarray}\nV=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\left\\langle\\sin \\left({\\pi It\\over e}+{\\pi\\Delta Q_s\\over e}\\right)\\right\\rangle~,\n \\end{eqnarray}\nwhere $\\Delta Q_s=Q_s(t_0)-Q_s(t_0-t)$ and\n\\begin{eqnarray}\nQ_s(t)  =\\mbox{sign}(I_s)\\pi \\sum_i \\Theta (t-t_i)\\left [ 1-\ne^{-(t-t_i)\/\\tau}\\right]  ~.\n     \\end{eqnarray}\nThe shot-noise charge-charge correlators are found by averaging over the the time $t_0$. \n The expressions for $\\langle\\cos (\\pi \\Delta Q_s\/e)\\rangle$ and $\\langle\\sin\n(\\pi \\Delta Q_s\/e)\\rangle$ follow from those for $\\langle\\cos (\\Delta\n\\varphi_s)\\rangle$ and $\\langle\\sin (\\Delta \\varphi_s)\\rangle$ assuming $\\rho=1$ and using the duality\nrelations between the phase and the charge:\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle = \n\\mbox{si} \\pi - \\mbox{si} \\left(\\pi\ne^{-t\/\\tau_s} \\right) +  \\mbox{si} \\left[\\pi\\left(1-\ne^{-t\/\\tau_s} \n\\right) \\right] +{\\pi \\over 2}~,\n       \\label{sinG}       \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s |\\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)= -\n\\mbox{ci} \\pi + \\mbox{ci} \\left(\\pi  e^{-t\/\\tau_s} \\right)-{t\\over\n\\tau_s} +\n\\mbox{ci} \\left[\\pi\\left(1- e^{-t\/\\tau_s} \n\\right) \\right] -\\gamma-\\ln \\left[\\pi\\left(1-\ne^{-t\/\\tau_s} \n\\right) \\right] ~.\n    \\label{cosS}           \\end{eqnarray}\nIn contrast to the weak-coupling case where the single relaxation time $\\tau$ characterized both the\nequilibrium and the shot-noise correlators, in the strong coupling case the equilibrium charge correlator is\ncharacterized by the time $\\tau_L$, whereas the shot-noise charge correlator has the other characteristic time\n$\\tau_s$. We expect that $\\tau_s \\ll \\tau_L$. Therefore we should consider the limit $t \\gg \\tau_s$ in Eqs.\n(\\ref{sinG}) and (\\ref{cosS}):\n\\begin{eqnarray}\n\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle =(2 \\mbox{si} \\pi  +  \\pi) {I_s\\over e} \\tau_s\n =3.704 {I_s\\over e} \\tau_s~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1= -2{|I_s|\\over e} t ~.\n         \\end{eqnarray}\n\nWe expand the shot noise contribution with respect to the current $I$:\n\\begin{eqnarray}\n\\Delta V=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left[ \\sin \\left({\\pi It\\over e}\\right) \\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) +\\cos \\left({\\pi It\\over e}\\right) \\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right]\\nonumber \\\\\n\\approx -{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left[ {\\pi It\\over e} \\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) +\\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right] = V_0+R_s I ~.\n \\end{eqnarray}\nRotating the integration path in the complex plane ($t\\to -iy$)  we obtain the\nratchet voltage $V_0$ and the zero-bias resistance $R_s $:\n\\begin{eqnarray}\nV_0 =-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\n\\nonumber \\\\\n=-{14.8\\pi \\Delta^2 \\over e^2 \\hbar}I_s \\tau_s\\int_0^\\infty dt \\mbox{Im}\n\\left\\{e^{\\tilde J_0(t)}\n\\right\\}={14.8\\pi \\Delta^2 \\over e^2 \\hbar}I_s \\tau_s\\int_0^\\infty dy\ne^{-\\pi\\tilde \\rho y\/\\tau_L}={14.8 \\Delta^2 \\over e^2\n\\hbar}{ \\tau_L\\over \\tilde \\rho }I_s \\tau_s={7.4 \\pi\\Delta^2 \\over e^4}{L\\tau_s\\over R_Q^2 }I_s ~,\n \\end{eqnarray}\n\\begin{eqnarray}\nR_s =-{4\\pi^2 \\Delta^2 \\over e^2 \\hbar}\\int_0^\\infty t\\, dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) \n\\nonumber \\\\\n=| I_s| {8\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\int_0^\\infty t^2\\,dt \\mbox{Im}\n\\left\\{e^{\\tilde J_0(t)}\\right\\}=| I_s| {8\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\int_0^\\infty\ny^2\\,dy e^{-\\pi\\tilde \\rho y\/\\tau_L} =| I_s| {16\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\left(\\tau_L \\over \\pi\n\\tilde\n\\rho\\right)^3 =| I_s| {8 \\Delta^2 \\over  e^5 }{L^3 \\over R_Q^4}~. \n \\end{eqnarray}\nOne can see that the even effect (resistance $R_s$) remains finite in the limit $\\tau_s \\to 0$. In contrast, the\nratchet effect, which is an odd effect, vanishes. More generally, any odd effect should vanish in the limit $\\tau_s\n\\to 0$. This is because in this limit tunneling events lead to instantaneous transfers of charge $\\pm e$. The odd\neffect should depend on the sign of transferred charge, but the current cannot depend on whether the charge $+e$ or\n$-e$ suddenly appears at the Josephson junction due to periodicity of the charge correlator with the period\n$2e$: the instantaneous transfers of the charges $+e$ and $-e$ must produce identical effects. Only finite duration of\nthe current pulse (finite $\\tau_s$) can lead to an odd effect (ratchet voltage $V_0$ as an example).\n\nThis interesting feature of the strong-coupling case becomes even more pronounced if shot noise originates\nfrom the current of Cooper pairs. This can be realized if the noise junction is a SIN junction in the regime of\nthe Andreev reflection or a Josephson junction in the weak coupling limit in the regime of incoherent tunneling\nof Cooper pairs. In the both cases the tunneling event is accompanied by the charge jump\n$Q_s=2e(1-e^{-t\/\\tau_s})\\mbox{sign}(I_s)$ and, in analogy with Eqs. (\\ref{sinG}) and (\\ref{cosS}), the odd and\nthe even charge correlators can be obtained from the phase-phase correlators using duality and assuming\n$\\rho=2$:\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle = \n-\\mbox{si}(2 \\pi) + \\mbox{si} \\left(2\\pi e^{-t\/\\tau_s} \\right) +  \\mbox{si} \\left[2\\pi\\left(1-\ne^{-t\/\\tau_s} \\right) \\right] +{\\pi \\over 2}~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s| \\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)= \\mbox{ci}(2 \\pi) -\n\\mbox{ci} \\left(2\\pi  e^{-t\/\\tau_s} \\right)-{t\\over\n\\tau_s} +\\mbox{ci} \\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right] -\\gamma-\\ln \\left[2\\pi\\left(1-\ne^{-t\/\\tau_s} \\right) \\right] ~.\n               \\end{eqnarray}\nIn the limit $t\\gg \\tau_s$\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle =2\\pi  e^{-t\/\\tau_s}~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s |\\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e}\\right \\rangle-1\\right) \n= 2[\\mbox{ci} (2\\pi) - \\gamma-\\ln\n(2\\pi)] +\\pi^2e^{-2t\/\\tau_s} ~.\n               \\end{eqnarray}\nCalculating the shot-noise contribution to\nthe voltage one cannot expand in $t\/\\tau_s$ but can expand in\n$t\/\\tau_L$ and in the current $I$:\n\\begin{eqnarray}\n\\Delta V =-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\}\\left[ \\left(\\left \\langle \\cos{\\pi\\Delta Q_s\\over e}\\right\\rangle-1\\right))\\sin \\left({\\pi It\\over e}\\right) +\\left\n\\langle\n\\sin {\\pi\\Delta Q_s\\over e}\\right \\rangle\\cos \\left({\\pi It\\over e}\\right) \\right] \\nonumber \\\\\n\\approx {4\\pi^2 \\Delta^2 \\over e \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t\\,dt \\left[\\left (\\left \\langle\n\\cos\n{\\pi\\Delta Q_s\\over e}\\right \\rangle-1\\right){\\pi It\\over e} +\\left \\langle \\sin {\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right]=V_0+R_s I~.\n \\end{eqnarray}\nIntegrating by parts one obtains the ratchet voltage \n\\begin{eqnarray}\nV_0 = {4\\pi^2 \\Delta^2 \\over e \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t\\,dt \\left\\langle \\sin\n{\\pi\\Delta Q_s\\over e} \\right \\rangle={8\\pi^3 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}I_s\n\\int_0^\\infty  e^{-t\/\\tau_s}{t^2\\over 2}\\,dt \\left\\{\\mbox{si}'(2\\pi e^{-t\/\\tau_s}) \\right. \\nonumber \\\\ \\left.\n-\\mbox{si}'\\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right]\\right\\} \n={2\\pi^2 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}I_s\n\\tau_s^3\\int_0^\\infty x^2\\,dx \\left[\\sin(2\\pi e^{-x})-{\\sin\\left[2\\pi\\left(1- e^{-x} \\right) \\right]\\over\ne^x-1}\\right]={7.62\\pi^3 \\Delta^2  \\tau_s^3\\over e^4  L}  I_s ~,\n \\end{eqnarray}\nand the short noise contribution to the zero-bias resistance\n\\begin{eqnarray}\nR_s = {4\\pi^3 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t^2\\,dt  \\left(\\left\\langle\n\\cos{\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)=- {4\\pi^3 \\Delta^2 \\over e^3 \\hbar}{\\tilde \\rho\\over\n\\tau_L}|I_s|\\int_0^\\infty {t^3\\over 3}\\,dt \\left[ \\mbox{ci}'(2\\pi e^{-t\/\\tau_s})2\\pi e^{-t\/\\tau_s}-1\n\\right.\\nonumber \\\\ \\left. +\\mbox{ci}'\\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right]2\\pi\ne^{-t\/\\tau_s} -{e^{-t\/\\tau_s} \\over 1-e^{-t\/\\tau_s}}\\right]\\nonumber \\\\\n= {4\\pi^3 \\Delta^2 \\tau_s^4\\over 3e^3 \\hbar}{\\tilde \\rho\\over \\tau_L}|I_s|\\int_0^\\infty x^3\\,dx\n\\left[1-\\cos(2\\pi e^{-x})  +{1-\\cos\\left[2\\pi\\left(1- e^{-x} \\right) \\right]\\over e^x-1}\\right] \n ={5.08\\pi^4\\Delta^2  \\tau_s^4\\over\ne^5L}|I_s| ~.\n \\end{eqnarray}\nThus shot noise from the current of Cooper pairs produces neither odd nor even effects\nif duration $\\tau_s$ of a single current pulse vanishes.\n\nWhat would happen if experimentalists were able to decrease the time $\\tau_s$ essentially? Eventually this\ntime would be comparable or even less that the intrinsic time, which determines duration of\nsingle-electron tunneling. Then the experimental observation of the zero-bias anomaly would provide an\ninformation on this intrinsic time. Up to now the tunneling time is mostly an issue of theoretical debates\n\\cite{trav,Hauge,Land}: Though some attempts to observe it were undertaken, but their interpretation is ambiguous (see\nRef. \\onlinecite{Land} and references therein). The subject is controversial and the consensus on how this time should\nbe defined is absent. Decreasing of $R_0$ and\n$C_0$ the experimentalists would shift this issue  from the pure theory to the measurement. How realistic could this\nperspective be? Presently it is difficult to achieve\n$R_0$ less than 100 $\\Omega$ and $C_0$ less then 1 fF. This yields $\\tau_s=R_0C_0 >10^{-13}$ s. On the other hand,\nwhatever disagreement on definition of the tunneling time, usually it is estimated as\n$10^{-15}~-~10^{-14}$ s. This yields the hope that further progress in miniaturization of the\nexperimental set up or searching for cases with longer tunneling time could open a new possibility to investigate\nthe tunneling time experimentally.\n\n\\section{Conclusions}\n\nWe have presented the theory of the  effect of shot noise from an independent source on the Coulomb blockaded\nJosephson junction in high-impedance environment. The analysis takes into account asymmetry of \nshot noise characterized by its odd moments. For high impedance environment the effect is so strong that the expansion in\nmoments is not valid and was not used in the analysis. Asymmetry of shot noise results in asymmetry of the $IV$\ncurve: the shift of the conductance minimum from the zero bias and the ratchet effect, which\nhave been observed experimentally \\cite{SN,exp}. At low noise currents (currents responsible for shot noise) the effect is\nproportional to the noise current independently of counting statistics of electron transport. However, at high noise\ncurrents the effect of noise on the $IV$ curve is very sensitive to electron counting statistics. This high\nsensitivity is explained by the fact that in contrast to usual methods of noise detection the Coulomb\nblockaded junction probes phase but not voltage.\n\nThe theory was generalized on  another type of noise (phase noise of a monochromatic AC input) and on a normal\nCoulomb blockaded tunnel junction, In the both cases the effect of noise strongly affects the Coulomb\nzero-bias anomaly of the $IV$ curve. The effect of shot noise on the superconducting Josephson junction in\nlow-impedance environment was also considered. If environment generates only the equilibrium Johnson-Nyquist\nnoise this case can be analyzed using the duality  relations with the case of the Coulomb blockaded Josephson\njunction in high-impedance environment. However, shot noise breaks duality relations between the two cases. An\ninteresting feature of the superconducting Josephson junction in low-impedance environment has been revealed:\nthe effect of shot noise on its $IV$ curve can give information on the time of electron tunneling through the\njunction responsible for shot noise.\n\nAltogether, the analysis demonstrates, that the  low-bias part of the $IV$ curves of tunnel junctions, both Coulomb\nblockaded and superconducting, can be exploited as a\nsensitive detector of various types of noise. \n\n\n\\section*{Acknowledgements}\n\nThe author appreciates collaboration and discussions with Julien Delahaye, Pertti Hakonen,\nTero Heikkil\\\"a, Rene Lindell, Mikko Paalanen, Jukka Pekola, and Mikko Sillanp\\\"a\\\"a. The author also thanks Markus\nB\\\"uttiker, Rosario Fazio, and Bertrand Reulet for interesting discussions of the results of the paper. The work was supported by \nthe Large Scale Installation Program ULTI-3 of the European Union and by the grant of the Israel Academy of Sciences and\nHumanities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe low-energy properties of frustrated quantum spin systems -- loosely speaking,\nsystems of interacting spins in which the local energetic constraints cannot\nall be simultaneously satisfied -- have fascinated researchers for\nmany decades. These systems arise in the description of the spin degrees\nof freedom of Mott insulators, i.e. insulating states where the fluctuations of\nthe charge degrees of freedom have been suppressed by interactions\nwhile the spin degrees of freedom remain free to form non-trivial quantum\nphases. Such states are found in many materials, but can also be\nartificially made in the lab using cold atomic gases~\\cite{greiner2002,jordens2008}.\nIn most situations, the spins collectively order into some pattern\nthat can be described through a local order parameter. A more exciting possibility,\ncoined {\\it spin liquid phase}~\\cite{balents2010}, is that the spins do not order into\nsuch a local pattern; instead, a more exotic state governed by strong quantum\nfluctuations emerges.\n\nIn a famous paper in 1987, Kalmeyer and Laughlin~\\cite{kalmeyer1987}\nhypothesized a scenario where a chiral topological\nspin liquid (CSL) is formed. In this elusive phase of matter, the spins form\na collective state that can only be described in terms of emergent, non-local\ntopological properties. So far, this behavior has experimentally only been observed in fractional\nQuantum Hall systems~\\cite{tsui1982,laughlin1983}.\nSuch topologically ordered liquids~\\cite{wen1990-1} are characterized through a\nnumber of universal properties ranging from topologically protected gapless\nedge states~\\cite{halperin1982,wen1990} and a ground state degeneracies\nthat depend on the topology of the sample~\\cite{wen1990-1} to exotic\nexcitations that carry fractional charge and satisfy neither fermionic nor\nbosonic exchange statistics~\\cite{moore1991}. These \\emph{anyonic particles}~\\cite{wilczek1982}\ncan serve as key ingredients in topological quantum computers~\\cite{nayak2008},\nmaking them relevant also for technological applications.\n\nIn the specific case of the chiral spin liquid as proposed by Kalmeyer and\nLaughlin, the universal properties of the\nground state are captured by the bosonic $\\nu=1\/2$ Laughlin\nstate~\\cite{laughlin1981}. Probably the most striking property of this state\nare its semionic bulk excitations: When exchanging two such semions,\nthe wave function describing the collective state of the system acquires\na complex phase $i$, in stark contrast\nto conventional bosons or fermions, where the factor is 1 and $-1$,\nrespectively. Equally striking is the emergence of a topologically\nprotected chiral edge state with a universal spectrum at the boundary\nof the sample. This leads to unidirectional transport along the\nboundary of the sample, while the bulk remains insulating. The \ncorrespondence between edge and bulk physics has\nbeen used as a powerful experimental probe into the physics of\nfractional quantum Hall systems. Finally, the\nground state degeneracy of this state depends on the topology of the\nunderlying manifold; for example, when placed on a torus, two ground\nstates are found.\n\n\\begin{figure}\n  \\includegraphics[width=1.5in]{kagome.pdf}\n  \\includegraphics[width=1.5in]{hom.pdf}\n  \\caption{\n  {\\it Left panel:} Kagome lattice considered in this manuscript, where grey shading\n  indicates the elementary triangles. Arrows on the bonds indicate the direction\n  induced by the magnetic flux $\\Phi$ enclosed in each triangle. \n  {\\it Right panel:} Visualization of the network-model perspective on the chiral spin liquid phase arising from Hamiltonian~\\eqnref{eqn:ham}. Consistent with a chiral topological phase, a collective edge state encircles the whole systems. In this particular model, additional closed edges encircle each hexagon.\n  \\label{fig:kagome}\n  }\n\\end{figure}\n\nOver the last decades, much research has been devoted to finding\nrealistic spin Hamiltonians that have such a chiral topological phase\nas their ground state, but to this date, the only known examples are\nHamiltonians that are unlikely to be relevant for any material~\\cite{yao2007,schroter2007,tang2011,sun2011,neupert2011,nielsen2013}.\nHere, we study a simple spin model on the Kagome lattice (left panel of Fig.~\\ref{fig:kagome}) that is derived from\nthe Hubbard model, which is the minimal relevant\nmodel for itinerant interacting electrons,\nin the presence of time-reversal symmetry breaking.\nThe Hubbard Hamiltonian reads\n\\begin{equation} \\label{eqn:hubb} \\begin{split}\nH =& - \\sum_{\\langle i,j \\rangle, \\sigma} (t_{ij} c_{i \\sigma}^\\dagger c_{j \\sigma} + t_{ij}^* c_{j \\sigma}^\\dagger c_{i \\sigma}) \\\\\n& + \\frac{h_z}{2} \\sum_i (n_{i \\uparrow} - n_{i \\downarrow}) + U \\sum_{i} n_{i \\uparrow} n_{i \\downarrow}.\n\\end{split} \\end{equation}\nHere, a magnetic field induces both a Zeeman term $h_z$ as well as a flux $\\Phi$ through\neach elementary triangle of the lattice, such that for $i,j,k$ clockwise around a triangle\nwe have $t_{ij} t_{jk} t_{ki} = t^3 \\exp(i \\Phi)$, as indicated by the arrows in the left panel of\nFig.~\\ref{fig:kagome}.\nWhen $\\Phi \\neq 0$ or $h_z \\neq 0$, time-reversal symmetry is broken.\nWhen considered at half-filling, $\\langle n \\rangle = 1$, and in the limit\nof large repulsive interaction strength $U$, a Mott insulating state is formed and an effective\nspin model can be derived from perturbation theory in $t\/U$~\\cite{motrunich2006}.\n\nHere, we will demonstrate conclusively that in a very wide parameter regime where\na large enough magnetic flux $\\Phi$ breaks time-reversal symmetry,\nthe ground state of the effective spin model -- and hence also the Hubbard model in an\nappropriate parameter regime -- is a chiral topological spin liquid with\nemergent anyonic excitations.\n\n\\section{Model} \\label{sec:model}\n\nStarting from from the Hubbard model of Eqn.~\\eqnref{eqn:hubb}, a $t\/U$ expansion\nat half filling gives the following spin Hamiltonian~\\cite{motrunich2006}:\n\\begin{equation} \\label{eqn:ham} \\begin{split}\nH =&~ J_\\text{HB} \\sum_{\\langle i,j \\rangle} \\vec{S}_i \\cdot \\vec{S}_j + h_z \\sum_i S^z_i \\\\\n& +J_\\chi \\sum_{i,j,k \\in \\bigtriangleup} \\vec{S}_i \\cdot (\\vec{S}_j \\times \\vec{S}_k) + \\ldots,\n\\end{split} \\end{equation}\nwhere for the three-spin term, the $i,j,k$ are ordered clockwise around the elementary triangles of the\nKagome lattice. The term $\\chi_{ijk} = \\vec{S}_i \\cdot (\\vec{S}_j \\times \\vec{S}_k)$, referred to\nas the scalar spin chirality~\\cite{wen1989,baskaran1989},\nbreaks time-reversal symmetry and parity, but preserves SU(2) symmetry.\nTo lowest order, the coupling parameters depend on the parameters of\nthe Hubbard model as $J_\\text{HB} \\sim t^2\/U$ and $J_\\chi \\sim \\Phi t^3\/U^2$,\nignoring further subleading terms. We choose\nto parametrize the model using $J_\\text{HB} = J \\cos \\theta$ and $J_\\chi = J \\sin \\theta$\nand set $J=1$.\n\nIn the absence of time-reversal symmetry breaking ($\\theta=0$ and $h_z=0$),\nthis is the Kagome lattice nearest-neighbor Heisenberg antiferromagnet, which has become\na paradigmatic model for frustrated magnetism~\\cite{elser1989,marston1991,sachdev1992} with possible\nspin liquid ground state and relevance to the description of materials such as Herbertsmithite and\nVolborthite~\\cite{lee2007,han2012}.\nRecent numerical work~\\cite{yan2011,jiang2012-1,depenbrock2012} has indicated \nthat this model may realize a time-reversal symmetric $\\mathbb{Z}_2$ topological spin liquid,\nwhile other numerical results give evidence for a gapless spin liquid phase~\\cite{iqbal2011,clark2012}.\n\nHere, we explore the ground state phase diagram of~\\eqnref{eqn:ham} away from\nthe time-reversal invariant Heisenberg point $\\theta=0$. In particular, we find an\nextended chiral spin liquid phase around the point $\\theta=\\pi\/2$ and $h_z=0$,\nwhere the Hamiltonian reduces to the three-spin term,\n\\begin{align} \\label{eqn:csl-ham}\nH_{\\text{CSL}} &= \\sum_{i,j,k \\in \\bigtriangleup} \\chi_{ijk}.\n\\end{align}\nOur numerical results indicate that the CSL is in fact stable almost up to the\nHeisenberg point, namely for all $\\theta \\geq 0.05 \\pi$.\nWe also establish an extended range of stability against other perturbations, including\n(i) the Zeeman field,\n(ii) an easy-axis spin anisotropy in the Heisenberg term,\n(iii) a next-nearest neighbor Heisenberg term, and\n(iv) the Dzyaloshinsky-Moriya (DM) interaction induced by Rashba-type spin-orbit coupling for the fermions.\nWhile the aim of this paper is not to examine the nature of the\ntransitions out of the chiral spin liquid phase, e.g. the expected phase transition\nto the time-reversal symmetric spin liquid of the Heisenberg antiferromagnet,\nthese questions should be addressed in future work.\n\nIn the following, we will use two complementary routes to show that the ground\nstate of Eq.~\\eqnref{eqn:csl-ham} is indeed a chiral spin liquid.\nFirst, we argue for this from a powerful perspective rooted in the physics of network models of\nedge states akin to the Chalker-Coddington network model for the integer\nquantum Hall transition~\\cite{chalker1988}. We will then turn to powerful numerical\ntools to unambiguously identify the universal properties of the chiral spin liquid.\n\n\\begin{figure}\n  \\begin{tabular}{c}\n  \\includegraphics[width=3in]{puddle1.pdf} \\\\ \\hline\n  \\vspace{0.1in} \\includegraphics[width=2.5in]{puddles_2.pdf}  \n  \\end{tabular}\n  \\caption{(color online)\n  {\\it Top:} Sketch of a puddle of topological phase replacing each triangle of three spins.\n  {\\it Bottom:} Behavior of two corner-sharing triangular puddles of the topological phase. \\label{fig:puddle} }\n\\end{figure}\n\nThe key step of our first argument is to view each triangle of spins\nas the seed of a chiral topological phase, a puddle encircled by\nan edge state, as illustrated in the top panel of Fig.~\\ref{fig:puddle}. The natural\ncandidate for the phase filling the puddle is the bosonic $\\nu=1\/2$\nLaughlin state~\\cite{halperin1983}\\nocite{moore1991}, which is the simplest\nbosonic quantum Hall state known to possess the SU(2) symmetry required by our construction.\nIt is also the state envisioned by Kalmeyer and Laughlin~\\cite{kalmeyer1987}.\nForming a lattice out of the elementary triangles, we should then consider\na situation with many individual puddles of this topological phase. To\nsee what collective state is formed, we have to understand how two\ncorner-sharing triangles of the Kagome lattice are joined.\nThis situation of edges meeting at the corner spin shared by two triangles\nis an incarnation of two-channel Kondo physics~\\cite{AffleckLudwig1991,MaldacenaLudwig1997},\nfor which it is well-known that the edges will {\\it heal}~\\cite{eggert1992,kane1992}\nif the coupling\nto the center spin is symmetric, as illustrated in the lower panel\nof Fig.~\\ref{fig:puddle} and discussed in more detail in the appendix. Thus, the corner spin has\nmerged the two triangles to form a larger puddle encircled by\na single edge state, i.e. to form a larger region of the topological phase.\nWe can repeat the above step (Fig.~\\ref{fig:puddle}) for all pairs of\ncorner-sharing triangles of the Kagome lattice. The system then forms\none macroscopic, extended region of a \\emph{single} topological\nphase with one edge state encircling its outer boundary, as illustrated\nin the right panel of Fig.~\\ref{fig:kagome}, and with closed loops encircling the interior\nhexagons of the Kagome lattice. We thus obtain a direct realization of the\nKalmeyer-Laughlin state for a chiral topological spin liquid phase.\n\n\\section{Numerical identification of the CSL} \\label{sec:numerics}\n\nWe now turn to a numerical identification of the CSL \nat the chiral point $\\theta=\\pi\/2$ of Hamiltonian~\\eqnref{eqn:ham} and\nin its vicinity by studying the three\nhallmark properties of a chiral topological phase: the presence of\n(i) a gapped spectrum with a topological degeneracy on the torus,\n(ii) a gapless edge state with a universal spectrum of low-energy excitations,\nand (iii) anyonic bulk excitations.\nOn a technical level, we resort to exact diagonalization and DMRG calculations\nto extract energy spectra, entanglement spectra, and modular matrices for\nvarious system configurations.\nTo label their diameter and boundary condition, we will\nuse the notation introduced in Ref.~\\onlinecite{yan2011}; see also\nthe Methods section.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{spectrum.pdf}\n  \\caption{\n  {\\it (a)} Exact diagonalization excitation energies on small tori of type XT4-0 up to length 5. \n  {\\it (b)} Energy gap on a thin, long strip of width 4 sites; dashed lines indicate the extrapolation to $N \\rightarrow \\infty$. The two different branches denote systems with an even (black) and odd (blue) number of unit cells.\n  {\\it (c)} Entanglement entropy at the center of the same system as (b) on a semi-logarithmic scale for even (black) and odd (blue) number of unit cells; dashed lines indicate a fit. Both fits are consistent with $c=1$.\n  In all panels, $N$ is the total number of lattice sites.\n  \\label{fig:spectrum} }\n\\end{figure}\n\nWe first demonstrate that the system has a finite gap in the thermodynamic limit.\nTo this end, we consider a sequence of XT4-0 tori of length up to 5 unit cells, shown in the left panel\nof Fig.~\\ref{fig:spectrum}.\nFor systems with $N \\geq 18$ sites, there clearly is a low-lying excitation, which can be attributed\nto a two-fold near-degeneracy of the ground state. All excitations above these near-degenerate ground\nstates are separated by a spectral gap of roughly $\\Delta \\approx 0.05$.\nFurther consistent evidence for the gap can be obtained from exact diagonalization of a 36-site\nXT6-3 cluster and on XT4-2 clusters of size up to 30 sites (not shown).\nAs a further consistency check, we can extract the gap for long, thin cylinders using DMRG.\nPerforming this for cylinders of type XC4-0 with up to 100 sites,\nwe confirm that the triplet gap does not depend significantly on the length\nof the system, ruling out the presence of gapless modes propagating along the cylinder.\n\nWe conclude from this that the gap remains finite in the thermodynamic limit. The qualitative\nagreement between the spectral gap extracted for different system sizes and boundary conditions\nis also a strong indication\nthat the correlation length of the system is short compared to the system sizes we are able to\nstudy numerically. To further support this, we have calculated the spin-spin and dimer-dimer\ncorrelation functions as well as an upper bound on the asymptotic correlation\nlength on infinite cylinders. All of these indicate a correlation length on the order\nof {\\it one} unit cell. Taken together, this gives strong evidence that the properties observed on\nsmall tori and quasi-one-dimensional systems are representative of the\ntwo-dimensional phase in the thermodynamic limit.\n\nThe observed two-fold near-degeneracy is consistent with what is\nexpected for the CSL, namely a\n$2^g$-fold ground state degeneracy on a manifold of genus $g$, which\nwill be split by an amount that is exponentially small in the system size.\nWe also find two states $\\ket{\\Psi_a}$ with very similar energy densities for infinite\ncylinders of type XC8-4 and XC12-6. As explained in the\nMethods section, the two states can be identified by\ntheir well-defined topological flux $a$ through the cylinder,\nwhich for the $\\nu=1\/2$ Laughlin state can be the identity\n($a = \\mathds{1}$) or a semion ($a=\\mathrm{s}$).\n\n\\subhead{Edge physics}\nPlacing a chiral topological phase on a cylinder or disk, a gapless chiral edge state emerges with\na universal spectrum governed by a conformal field theory~\\cite{halperin1982,wen1990}.\nTo observe this, we consider the spectral gap of the system\nwhen placed on a thin, long strip with a fixed width of 4 sites.\nIn stark contrast with the case of a thin long cylinder, the spin gap\nvanishes as $a\/L+b\/L^2$, where $a$ and $b$ are\nparameters of the fit (Fig.~\\ref{fig:spectrum}(b)).\nThis is a hallmark signature of a gapless edge mode. We can further\npinpoint the universality class of the edge theory by extracting its central charge $c$\nfrom the entanglement entropy. As shown in the Fig.~\\ref{fig:spectrum}(c), we find good agreement\nwith a value of $c=1$, which is precisely that expected for the chiral SU(2)$_1$ Wess-Zumino-Witten\nconformal field theory describing the edge of a $\\nu=1\/2$ Laughlin state.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{cft.pdf}\n  \\caption{Entanglement spectrum of the reduced density matrix $\\rho_a$ for one half of an infinite cylinder obtained for both ground states $\\ket{\\Psi_\\mathds{1}}$ (left) and $\\ket{\\Psi_\\mathrm{s}}$ (right panel). The entanglement energies shown on the vertical axis, up to the global shift and rescaling, are given by $E_{a,\\sigma} = -\\log (p_{a,\\sigma})$, where $p_{a,\\sigma}$ are the eigenvalues of $\\rho_a$. The horizontal axis shows the momentum in the transverse direction of the corresponding eigenvectors of $\\rho_a$. Each tower is identified by its $S^z$ quantum number as indicated by the blue label; we have offset the momentum of different towers by $2\\pi$ to improve clarity. The cylinder used here is XC12-6.\n }\n \\label{fig:cft}\n\\end{figure}\n\nAs an even more refined probe, we use the entanglement spectrum, which\nreflects the same universal properties as the physical edge spectrum~\\cite{li2008,qi2012,chandran2011,dubail2012,swingle2012}.\nFor each of the two ground states $\\ket{\\Psi_a}$ obtained for an infinite cylinder,\nthe entanglement spectrum, see Fig.~\\ref{fig:cft}, is consistent with\nthe corresponding sector of the\nchiral SU$(2)_1$ Wess-Zumino-Witten conformal field theory:\nThe entanglement spectrum of $\\ket{\\Psi_\\mathds{1}}$ displays precisely the sequence\nof degeneracies of the tower of Kac-Moody descendants\nof the identity primary field (1-1-2-3-5-...). These are reproduced by counting\nthe number of low-lying close-by states in each tower grouped by momentum and spin quantum numbers.\nSimilarly, the entanglement spectrum of $\\ket{\\Psi_\\mathrm{s}}$ displays the degeneracies of\nthe spin-1\/2 primary field and its descendants (also 1-1-2-3-5-...).\nWe note that in the identity sector, all towers carry integer representations of the spin quantum number, whereas in the semion\nsector they carry half-integer representations. In both ground states, the levels can be grouped into\nSU(2) multipletts.\n\n\\subhead{Emergent anyons}\nThe bulk of the chiral spin liquid phase has anyonic excitations, referred to as semions.\nThe topological properties of these quasiparticles can be characterized through their modular\n$T$ and $S$ matrices~\\cite{nayak2008}. The $T$ matrix contains the\ncentral charge $c$ and the self-statistics of the anyonic particles, i.e. the phase\nthat is obtained when two particles of the same kind are exchanged.\nThe $S$ matrix contains the mutual statistics of the anyonic quasiparticles, their\nquantum dimensions (counting the internal degrees of freedom\nof each particle), and the total quantum dimension of the phase. More detailed\ndefinitions of these quantities are given in the Methods summary.\n\nFor a fixed number of quasiparticles, only a finite number of\npossible $S$ and $T$ matrices exist~\\cite{rowell2009,bruillard2013}. For two types of\nquasiparticles (as in the case of the $\\nu=1\/2$ bosonic Laughlin state) only two choices are possible~\\cite{rowell2009}.\nTherefore, by numerically calculating the $S$ and $T$ matrices and\ncomparing them against the two possibilities, we have fully identified\nthe universal properties of the topological phase.\nFor the $\\nu=1\/2$ Laughlin state, the modular matrices are\n\\begin{align} \\label{TS}\nT&=e^{-i \\frac{2\\pi}{24}} \\left[\\begin{matrix} 1 & 0 \\\\ 0 & i \\end{matrix} \\right]\n&S&=\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix} 1 & 1 \\\\ 1 & -1 \\end{matrix}\\right].\n\\end{align}\n\nFor an XT8-4 torus of $48$ sites at $\\theta=0.05 \\pi$, where the\nfinite-size corrections to this quantity are minimal, we obtain\n\\begin{eqnarray}\nT &=& e^{-i \\frac{2\\pi}{24} 0.988} \\left[\n\\begin{matrix}\n1 & 0 \\\\\n0 & i \\cdot e^{-i0.0021 \\pi}\n\\end{matrix}\n\\right] \\ , \\label{numT} \\label{numS} \\\\\nS &=& \\frac{1}{\\sqrt{2}} \\left[\n\\begin{matrix}\n0.996 & 0.995 \\\\\n0.996 & -0.994 e^{-i 0.0019 \\pi}\n\\end{matrix} \\right] \\ . \\nonumber\n\\end{eqnarray}\nThis is in very good agreement with the $T$ and $S$ matrices for the $\\nu=1\/2$ Laughlin state\ngiven in Eqn.~\\eqnref{TS} and provides the strongest confirmation of the nature of the\nbulk topological phase.\nThe correct normalization of the first row or column of the $S$ matrix\nindicates that we have indeed obtained a full set of ground states.\nWe can also read off the total quantum dimension $\\mathcal{D}=1\/S_{\\mathds{1} \\mathds{1}}=\\sqrt{2}\/0.996$ of the phase,\nwhich determines the topological entanglement entropy~\\cite{kitaev2006-tee,levin2006}\nthat has been widely used to identify topological phases.\nFurthermore, the central charge $c=0.988$ is in excellent agreement with the prediction\nand the value extracted from the edge above.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{pd.pdf}\n  \\caption{\n  Singlet and triplet gaps as a function of $\\theta$ for an infinite XC8-4 cylinder.\n  The triplet gap is a lower bound on the critical magnetic field $h_c$; hence, the shaded region in\n  the middle panel indicates the minimal stability of the phase in the $\\theta$-$h_z$ phase diagram.\n  \\label{fig:pd} }\n\\end{figure}\n\n\\subhead{Stability of the chiral spin liquid}\nTo establish the region in which the phase persists as $\\theta$ is tuned in the range $\\theta \\in [0,\\pi\/2]$,\nwe first consider the fidelity~\\cite{zanardi2006} $F(\\theta) = \\langle \\Psi_a(\\theta-\\epsilon) | \\Psi_a(\\theta+\\epsilon) \\rangle$\nshown in the bottom panel of Fig.~\\ref{fig:pd} (for the precise definition of this quantity for infinite\nsystems, see the appendix). The fidelity remains near unity as $\\theta$ is tuned away from\nthe chiral point $\\theta=\\pi\/2$, until it suddenly drops for $\\theta < 0.05 \\pi$, indicating a transition.\nFurther support for the extended stability of the CSL is found in various other characteristics,\nincluding the spectral gap, the modular matrices and the entanglement spectrum.\nThis is remarkable as it indicates that tuning away from the Heisenberg model ($\\theta=0$)\nwith a small critical chiral coupling of $(J_\\chi\/J_\\text{HB})_\\text{crit} \\leq \\tan(0.05 \\pi) \\approx 0.16$\nis sufficient to drive the system into the chiral phase. \n\nIn experimental scenarios, a Zeeman magnetic field $h_z$ is likely to be generated along with\nthe orbital magnetic field that induces the three-spin chiral term. The relative strength of the orbital\nmagnetic field and the Zeeman field is determined by the $g$-factor and the ratio $t\/U$.\nThe energy gap in the triplet sector gives a lower bound on the critical field strength $h_c$ up \nto which the CSL phase is stable. The values for the triplet gap shown in the middle panel of Fig.~\\ref{fig:pd}\nremain large all the way from the fully chiral point ($\\theta=\\pi\/2$) to the transition out of the\nCSL towards the Heisenberg point ($\\theta=0$).\n\nIn the top panel of Fig.~\\ref{fig:pd}, we also show the singlet gap, i.e. the gap to the lowest excitation\nin the $S_z=0$ sector. As opposed to the triplet gap, which appears to remain large across the\ntransition, the singlet gap decreases as the transition out of the CSL is approached. Note that the gaps may\nbe rounded off at the transition by effects due to either the finite diameter or the finite bond dimension\nof the matrix-product state ansatz.\nWe point out, however, that a closing only of the singlet and not the triplet\ngap is consistent with the scenario for the transition from the chiral spin liquid into a\ndoubled semion phase (twisted $\\mathbb{Z}_2$ topological phase) studied in Ref.~\\onlinecite{barkeshli2013}.\n\n\\section{Outlook}\n\nWe have taken an important step towards finding realistic models for a chiral spin\nliquid in a frustrated spin system. \nWe believe that this will nucleate new research efforts both by theorists and\nexperimentalists. From a theoretical point of view, studying the transition from\nthe chiral spin liquid to the putative time-reversal symmetric spin liquid in the\nHeisenberg model will provide the unique opportunity to study a topological\nphase transition in a realistic model, and may provide invaluable insights into\nthe physics of frustrated spins on the Kagome lattice. For experimentalists, our\nwork will provide a guide in searching for realizations of bosonic fractional\nQuantum Hall physics in the lab, be it in materials that have Kagome lattice\nstructure and form Mott insulators, or by engineering such systems in cold atomic gases.\n\n \\vspace{0.15in}\n{\\small During completion of this work, we became aware of related work in Refs.~\\onlinecite{he2013,gong2013}.}\n\n\\acknowledgements\nThe DMRG code was developed with support from the Swiss Platform for High-Performance\nand High-Productivity Computing (HP2C) and based on the ALPS libraries~\\cite{bauer2011-alps}.\nS.T. was supported, in part, by  SFB TR 12 of the DFG. A.W.W.L was supported, in part, by\nNSF DMR-1309667. We acknowledge illuminating discussions with participants of the KITP\nworkshop \\emph{Frustrated Magnetism and Quantum Spin Liquids} (Fall 2012) as well as\nM. Barkeshli and P. Bonderson. This research was supported in part by Perimeter Institute for\nTheoretical Physics. Research at Perimeter Institute is supported by the Government of Canada\nthrough Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nMrk~766 is a bright ($F_{(2-10)}\\sim 2 \\times 10^{-11} \\rm erg\\,\ncm^{-2} s^{-1}$), soft ($\\Gamma_{0.1-2.4} \\sim 2.7$) X-ray source at\nredshift $z=0.012$.  The spectrum measured with the {\\it Einstein} IPC\nand MPC was complex and ultra-soft ($\\Gamma=1.77$; $kT=18.6 \\rm eV$;\nUrry {\\it et al.}\n\\markcite{34} 1990).  A shortest time scale of variability of 1000\nseconds and a steep and variable power law index was found in a long\nobservation using {\\it EXOSAT} (Molendi, Maccacaro \\& Schaeidt\n\\markcite{20} 1993).  During the {\\it ROSAT} All Sky Survey, \nMrk~766 was bright ($F_{0.1-2.4} \\sim 1.5\n\\times 10^{-10} \\rm erg\\, cm^{-2} s^{-1}$ (unabsorbed)) and\nvariability by a factor of three with no accompanying spectral\nvariability was observed in 10--12 hours (Molendi, Maccacaro \\&\nSchaeidt \\markcite{20} 1993).  Pointed {\\it ROSAT} observations\nrevealed spectral variability that Netzer, Turner\n\\& George (1994) \\markcite{24} showed could not be explained by a\nchange in ionization of a warm absorber, and Molendi \\& Maccacaro\n(1994) \\markcite{19} attributed to a change in the accretion rate.\n\nMrk~766 is a member of the X-ray narrow line Seyfert 1 (NLS1) galaxy\nclass (Osterbrock \\& Pogge \\markcite{25} 1985; Goodrich \\markcite{13}\n1989).  {\\it ROSAT} observations of NLS1s find soft 0.1--2.4 keV X-ray\nspectra and rapid, large amplitude soft X-ray variability.  The soft\nX-ray spectra of NLS1s are systematically steeper than the spectra of\nbroad-line Seyfert 1 galaxies (Boller, Brandt \\& Fink \\markcite{3}\n1996 and references therein). A harder high energy power law component\ngenerally was not observed in the relatively soft {\\it ROSAT} band.\nOnly a few observations at higher energies have been reported.  The\n{\\it ASCA} spectrum of the NLS1 object IRAS~13224-3809 is dominated\nbelow $\\sim 2$ keV by a soft excess and from 2 to 10 keV by a hard\n($\\Gamma \\sim 1.3$) power law (Otani \\markcite{26} 1995).  In\ncontrast, a very steep spectrum with $\\Gamma_{(2-10keV)} \\sim 2.6$ was\nfound from NLS1 object RE~1034+39 (Pounds, Done \\& Osborne\n\\markcite{39} 1995).\n\nWe report the results from December 1993 {\\it ROSAT} and {\\it ASCA}\nobservations of Mrk~766. Timing analyses of two {\\it ROSAT} archival\nobservations are also presented. In section 2 the data reduction is\ndiscussed briefly.  In section 3 timing analyses using normalized\nvariability amplitudes and hardness ratios are presented.  In section\n4 the spectral analysis of the {\\it ASCA} data is described.  The\nresults are discussed in terms of standard models in Section 5 and\ncompared with reported results from other NLS1s.  A summary and\nconclusions are given in Section 6.\n\n\\section{{\\it ASCA} and {\\it ROSAT} Observations of Mrk~766}\n\nWe observed Mrk~766 with {\\it ASCA} and the {\\it ROSAT} PSPC during\nDecember 1993.  It had been previously observed with the {\\it ROSAT}\nPSPC several times. Two longer observations were made 1991 June 15 and\n1992 December 21 and these data were extracted from the {\\it ROSAT}\narchive. The observation log is given in Table 1.\n\nThe data were reduced using Xselect. To ensure all soft photons were\ncollected, {\\it ROSAT} extraction regions of $3 ^\\prime$ for the\non-axis 1993 observation and $4^\\prime$ for the off-axis 1991 and 1992\nobservations were used. Extraction of background subtracted light\ncurves from the events files was done using IDL software.  The light\ncurves from the off-axis 1991 and 1992 observations were corrected for\nvignetting. In the 1991 observation, Mrk~766 was periodically occulted\nby the detector rib so time periods in which the flux dropped to zero\nbecause of occultation were excluded (e.g. Brandt et al.\n\\markcite{2} 1993).  A region of the same radius located diametrically\nacross the detector and subjected to the same good time interval\nselection provided an approximately correctly normalized background.\n\n{\\it ASCA} data were reduced using standard selection and cleaning\ncriteria.  Background for the GIS was obtained from a source free\nregion in the GIS field of view approximately the same distance from\nthe optical axis as the source.  Background SIS spectra were obtained\nfrom blank sky fields, while background count rates for the light\ncurves were determined from the edges of the SIS chips. The time\ndependent gain shift in the SIS was accounted for by filling and using\nPI columns.  Spectra were extracted using Xselect, and background\nsubtracted light curves were obtained from the cleaned events files\nusing IDL.\n\nFigure~\\ref{fig1}a shows the {\\it ROSAT} PSPC and {\\it ASCA} SIS0\nlight curves from the most recent, quasi-simultaneous 1993\nobservations.  Error bars represent $1\\sigma$ statistical error. The\n{\\it ROSAT} observation started $\\sim 20,000$ seconds before the {\\it\nASCA} observation but was unfortunately cut short as the satellite\nwent into safe hold mode.  The {\\it ROSAT} PSPC light curves from the\n1991 and 1992 observations are shown in Figure~\\ref{fig1}b.  Mrk~766\nwas brightest in the {\\it ROSAT} band in 1992 at $<5 \\rm\\,\ncnts\\,s^{-1}$.  This is comparable to the flux level observed during\nthe {\\it ROSAT} All Sky Survey observation (Molendi, Maccacaro \\&\nSchaeidt \\markcite{20} 1993).  In order to make a direct comparison\nwith the other data, only the first $\\sim 80$ks of the 1992\nobservation was considered.  The remainder consists of data from only\ntwo orbits, is separated from the main part of the observation by\nnearly one day and the full light curve is shown in Netzer, George \\&\nTurner \\markcite{24} 1994.\n\n\\section{Timing  Analysis}\n\n\\subsection{Fastest Observed Variability}\n\nThe {\\it ASCA} light curves were examined in order to find instances\nof rapid variability that could be clearly identified in all four\ndetectors.  A dip in flux occurred at $\\sim 50,000\\rm \\, s$ from the\nstart of the observation; the light curve from the SIS0 detector is\nshown embedded in Figure~\\ref{fig1}a.  At the end of the dip, the flux\nincreased by nearly a factor of two in $\\sim 1000$ s. This timescale\nis the same order as those observed in {\\it ASCA} data from other\nSeyfert 1 nuclei, including NGC 4051 ($\\Delta t \\sim 200 \\rm \\,s$;\nMihara et al. \\markcite{18} 1994).\n\nAssuming that the X-ray emission originates from a single region, the\nminimum flux doubling time scale $\\Delta t$ gives an upper limit on\nthe source size as $R \\sim c \\Delta t$ of $3 \\times 10^{13}\\, \\rm cm$,\nwhere $c$ is the speed of light.  This estimate breaks down if the\nX-rays are emitted from many small regions.\n\nBecause of the telescope wobble, variability on timescales less than\n$400\\rm \\, s$ generally cannot be detected in {\\it ROSAT} data.\nSignificant variability was observed consistently between adjacent\norbits ($\\Delta t \\sim 6000$ s).  The fastest variability of the three\nobservations was found when it was brightest during the 1992\nobservation.  A 30\\% increase in 2400 s occurred 3400 s from the\nbeginning of the observation (Figure~\\ref{fig1}b).\n\n\\subsection{Variance analysis}\n\nThe normalized variability amplitude (NVA), defined to be the standard\ndeviation divided by the mean intensity, provides a simple way to\nquantify the variability in different energy bands (e.g. Edelson\n\\markcite{5} 1992).  If the width of the energy band is chosen so that the\nnumber of photons in each light curve is the same, the NVA collapses\nto the square root of the variance.\n\nThe {\\it ASCA} light curves from each detector were accumulated with\n100 s binning in the 4--10 keV band where the power law component\nshould dominate the spectrum.  The light curves were not background\nsubtracted.  However, the dilution of the variability by the constant\nbackground was small since the background rate in the highest energy\nband where contribution is largest was only e.g. $\\sim 8$\\% of the\nSIS0 count rate.  The data below 4 keV were divided into energy bands\nwith bounds chosen so that the light curves had the same mean square\nmeasurement error $\\sigma^2_{err}$ as in the 4--10 keV band.  Note\nthat each resulting band was wider than the energy resolution at that\nband.  The true variance of the data given by\n$\\sigma^2_{int}=\\sigma^2_{obs}-\\sigma^2_{err}$ is plotted as a\nfunction of energy in Figure~\\ref{fig2}.  The $1\n\\sigma$ uncertainties in the variance are less than 10\\% of the values\n(Equation 2 of Done et al.  \\markcite{4} 1992).  These variance plots\ndemonstrate that the variability amplitude over the whole observation\nis smallest at hard energies, peaks near 1 keV, and decreases towards\nlower energies in the SISs.\n\nFor the {\\it ROSAT} light curves, three standard energy bands were\nconsidered: the `{\\it a} band' (channels 11--41, energy 0.1--0.4 keV),\nthe `{\\it c} band' (channels 52--90, energy 0.5--0.9 keV) and the\n`{\\it d} band' (channels 91--201, energy 0.9--2 keV). These bands are\nindependent and roughly correspond to regions where different spectral\nfeatures will dominate: the soft excess in the {\\it a} band, the warm\nabsorber in the {\\it c} band, and the power law in the {\\it d} band.\nThe binning of the background subtracted light curves was chosen to be\n400~s to account for the telescope wobble.  \n\nBecause of the rib occultations in the 1991 data, it was necessary to\ninclude data with net exposure per bin of 200--400~s, even though the\nshorter exposure bins add noise to the light curve. The light curves\nin these three bands are shown on the left in Figures~\\ref{fig3}a, b\nand c.  Mrk~766 is bright enough and the bin size is long enough that\nthe signal to noise is better than 6 in all bins. Variability was\ndetected in each energy band and the $\\chi^2$  values for a constant\nhypothesis model fit and the computed NVAs are listed in Table 2. The\nNVAs show that in all observations the source is significantly less\nvariable in the lowest energy band compared with the higher energy\nbands.\n\n\\subsection{Flux Ratios}\n\nThe hardness ratio (4.0--10.5 keV\/1.0--1.35 keV) and softness ratio\n(0.4--0.7 keV\/1.0--1.35 keV) light curves from the {\\it ASCA} SIS0\ndetector are shown in Figures~\\ref{fig4}a and 4b, respectively.\nFollowing Ptak et al.  \\markcite{27} (1994) the values were computed\nusing variable bin sizes corresponding to good time intervals longer\nthan 300 seconds.  A large decrease in hardness corresponding to a\nsoftening of the spectrum was observed in the hardness ratio light\ncurve at about 20,000~s from the beginning of the observation\n(Figure~\\ref{fig4}a).  The light curves show that the spectral change\nis due to a large increase in 1.0--1.35 keV flux while the 4--10 keV\nflux remained nearly constant.  Significant variability in the\nsoftness ratio was also observed at the same time, such that the\nspectrum below 1 keV hardens when the flux increases\n(Figure~\\ref{fig4}b).  The hardening of the spectrum may be more\ngradual and it is not clearly completed until after $\\sim 25,000$~s.\nVariability was observed in both energy bands but the amplitude is\nlarger in the 1.0--1.35 keV band.  The spectral variability is\nconfined to the region around 20,000~s elapsed time.  Other instances\nof large amplitude flux variability occurred (e.g. at $\\sim 35,000$ s)\nwith no corresponding spectral change.  Thus, the spectral variability\nis not strictly correlated with the flux.\n\n{\\it ROSAT} softness ratio light curves (0.1--0.4 keV\/0.9--2.0 keV and\n0.5--0.9 keV\/0.9--2.0 keV) are shown on the right sides of\nFigures~\\ref{fig3}a, b and c.  Spectral variability is most pronounced\nbetween the hardest and softest bands, with the spectrum generally\nhardening as the flux increased.  This result is also true from\nobservation to observation; the spectrum was hardest when the source\nwas brightest in 1992, and softest when it was dimmest in 1991.  On\nlong time scales the softest band varies proportionately less than the\nharder bands.  On short time scales the spectral variability is not\ncorrelated with the flux.  At high flux, when the source was most\nrapidly variable, correlated variability occurred with no change in the\nhardness ratio (1992: at the beginning of the observation).  Similar\ncorrelated variability was observed during the {\\it ROSAT} All Sky\nSurvey when the source was also quite bright (Molendi, Maccacaro and\nSchaeidt \\markcite{20} 1993).  In contrast, at low flux in the 1991\nand 1992 observations, orbit to orbit deviations in the hardest band\noccurred which were not followed in the softest band (1991: at $\\sim\n23000$ and $\\sim 58000$~s; 1992: at $\\sim 66000$~s).  These short time\nscale hard band excursions result in dips in the softness ratio.\n\n\\section{Spectral Analysis}\n\nBased on the softness and hardness ratios, the {\\it ASCA} data were\ndivided in two as shown in Figures~\\ref{fig4}a and 4b, and spectra were\naccumulated avoiding the transition region between $\\sim 15,000$ and\n$\\sim 25,0000$~s from the beginning of the observation.  These spectra\nare referred to as the low state and high state spectra and represent\nexposures of $\\sim 8,000$ and $20,000$~s respectively.\nFigure~\\ref{fig5} shows the pha ratios from the low state divided by\nthe high state of the summed SIS0 and SIS1 spectra in the top panel\nand of the summed GIS2 and GIS3 spectra in the bottom panel.  These\nconfirm the results of the variance analysis which were that the\nspectrum is most variable at around 1 keV, the variability amplitude\ndecreases to lower and higher energies, and the flux is essentially\nconstant at about 10 keV.\n\n\\subsection{The Hard X-ray Spectrum}\n\nAn estimation of the power law index, assuming that any warm absorber\ndoes not have a large column density, was obtained by fitting the four\nspectra above 2 keV with a power law plus iron $K\\alpha$ line model.\nThe low and high state spectral indices were $1.57\\pm 0.07$ and\n$2.00^{+0.03}_{-0.04}$ respectively, and the results from fitting each\ndetector separately were consistent.  Throughout this paper, quoted\nuncertainties are determined assuming 90\\% confidence and 1 parameter\nof interest ($\\Delta \\chi^2=2.71$).\n\n\\subsection{The Soft X-ray Spectrum}\n\nThe shape of the soft X-ray spectrum can be seen by fixing the power\nlaw indices to the values found above 2 keV, including the Galactic\nabsorption ($1.77 \\times 10^{20} \\rm \\, cm^{-2}$; Elvis, Wilkes \\&\nLockman\n\\markcite{6} 1989), and plotting the ratio of the data to the model.\nThese plots are shown in Figure~\\ref{fig6}a and 6b for the SIS0 and\nGIS2 low state and high state spectra respectively.  In the low state\nthere is excess emission below 0.8 keV, suggesting the presence of a\nsoft excess component (e.g.  Mihara et al. \\markcite{18} 1994).  In\nthe high state there is a deficit between 0.8 and 1.3 keV, suggesting\nthe presence of a partially ionized absorber (e.g. Fabian et al.\n\\markcite{8} 1994).  It is clear that an absorbed power law plus\nnarrow iron line cannot model the spectra.  Spectral fitting results\nof the SIS0:GIS2 and SIS1:GIS3 pairs are listed in Table 3 for low and\nhigh states separately.  The iron line is discussed separately in\nSection 4.5 and fit results are given in Table 4.\n\nThe soft excess component and warm absorber can be simply\nparameterized using a black body model and an absorption edge,\nrespectively.  The results from adding each of these components\nseparately are given in the second and third panels of Table 3.\nAddition of either model component significantly improved the fits of\nboth the low and high flux spectra.  However, the addition of the\nblack body improved the fit more than the edge in the low flux case,\nwhile the reverse was true for the high flux case, as expected from\nthe residuals shown in Figure~\\ref{fig6}.\n\nThe fourth panel of Table 3 lists results from fitting with a model\nincluding both a blackbody and an edge.  These fit results confirm the\nimportance of the black body component in the low state spectra and\nthe edge in the high state spectra. The black body temperatures and\nedge energies of the low and high state spectra are consistent. The\nedge energy near $0.74\\rm \\, keV$ is consistent with an origin of\ntransmission through a partially ionized absorber dominated by\n\\ion{O}{7}.  The power law slope of the low flux spectra remains\nsignificantly flatter than that of the high flux spectra ($\\Delta\n\\Gamma \\sim 0.35$).\n\nNext, to better simulate the warm absorber, a two edge plus power law\nmodel was tried and the results are listed in the fifth panel of Table\n3.  The power law indices are steep in both the low and high state,\nand the implied change in index is much smaller ($\\Delta\\chi^2\n\\sim 0.1$).  This model has the same number of degrees of freedom as\nthe power law plus black body and edge model, but the fit is much\npoorer for the low flux spectra ($\\Delta \\chi^2$ of 24 and 50 for the\nSIS0:GIS2 and SIS1:GIS3 pairs, respectively).  In contrast, the fits\nof the high flux spectra are improved somewhat by the additional edge\n($\\Delta \\chi^2$ of 9 and 11).  Addition of a black body component,\nwhile not necessary for the high flux spectra, greatly improves the\nfit of the low flux spectra ($\\Delta \\chi^2$ of 28 and 52), and\nresults in a decrease in the low flux photon index and an increase in\nthe implied index change ($\\Delta\\Gamma \\sim 0.35$).  The temperature\nof the black body component in the low and high states is consistent\nat $kT \\sim 120\\rm eV$, although the limits on the temperature in the\nhigh state cannot be determined well for the SIS1:GIS3 pair.  The edge\nenergies near 0.74 and $0.87\\rm \\, keV$ found in the high state are\napproximately consistent with absorption by \\ion{O}{7} and \\ion{O}{8}.\nIn the low state, the \\ion{O}{7} edge is clearly detected but the\nsecond edge is not necessary.  This suggests that the ionization of\nthe gas in the high state is higher than in the low state.\n\nWe also modeled the warm absorber using a table model (e.g.  Yaqoob,\nWarwick \\& Pounds \\markcite{37} 1989).  The model used here assumes\nthat the power law is the sole source of ionizing photons. The density\nof the gas was assumed to be $10^{9.5} \\rm cm^{-3}$.  An analytic\napproximation based on a large number of CLOUDY runs was used to model\nthe temperature as a function of ionization parameter.  It was found\nto range between $\\sim 5 \\times 10^4$K and $\\sim 10 \\times 10^4$K. The\nadvantages of using the warm absorber table model compared with the\ntwo edge model are that there are two fewer parameters, and the\nparameters ($log(U)$ and $log(N_w)$) can be directly interpreted.  The\ndisadvantage is that we must assume a particular model for the\nionizing spectrum and gas.  Further, in this model, only absorption is\nconsidered, and the emission lines expected if the warm absorber has a\nlarge covering factor are ignored (e.g. Netzer \\markcite{23} 1993).\nHowever, this model is sufficient for a general discussion given the\nstatistical quality of these data (but see Section 4.3.1).\n\nThe four groups of spectra were first fit with a power law plus warm\nabsorber model (Table 3).  As found using the two edge model, the fits\nof the high flux spectra are acceptable, but the fits of the low flux\nspectra are unacceptable.  Addition of a black body component\nsubstantially improves the fit of the low flux spectra, as shown in\nthe eighth panel of Table 3, but it is not necessary to model the high\nflux spectra.  The ionized column density is consistent between the\nlow and high flux states at $log(N_w) \\sim 21.8$, while the ionization\nstate $\\log(U)$ is lower for the low flux spectra ($-0.78$) compared\nto the high flux spectra ($-0.4$).  In this model, ionization states\n$log(U)$ below and above $\\sim -0.4$ are dominated by \\ion{O}{7} and\n\\ion{O}{8} absorption, respectively. In contrast with the two edge\ndescription of the warm absorber, the use of the warm absorber table\nmodel resulted in a higher black body temperature in the high flux\nstate; however, the black body was barely detected in the high flux\nspectra ($\\Delta \\chi^2$ of 4 and 5).  Thus the temperature of the\nsoft component is model dependent, but the necessity of this component\nto model the low flux spectra is not. The table model description of\nthe warm absorber results in a slightly steeper index compared with\nthe two edge description.  This is because the table model properly\ntreats the absorption by gas with cosmic abundances (i.e. not only\noxygen) producing curvature in the model between 1.5 and 2.5 keV.\nThus the value of the indices is slightly model dependent, but the\nchange in index between the low and high states is not.\n\nIn summary, these results show that to model the high and low flux\nspectra consistently, both a soft excess component and a warm absorber\nare required.  When both of these components are included in the\nspectrum, the low flux spectral index is significantly flatter than\nthe high flux index ($\\Delta\\Gamma \\sim 0.35$).\n\n\\subsection{Other Models}\n\n\\subsubsection{Other Soft Excess Models}\n\nThe soft excess component in the low flux spectra was modeled\nadequately with a black body (with a single edge, $\\chi^2 = 230$\/266\nd.o.f. for the SIS0:GIS2 pair).  For comparison, other soft excess\nmodels including Raymond--Smith (cosmic abundances), bremsstrahlung,\ndisk black body, and power law were tried.  The power law plus soft\ncomponent model alone did not fit the low flux spectra well.  The disk\nblack body model fit the best, with $\\chi^2$  of 247\/268 d.o.f.  Including\nan edge generally improved the fits, but the Raymond--Smith and the\npower law models could not describe the low flux spectra well ($\\chi^2$  of\n277 and 259\/266 d.o.f. respectively), while the bremsstrahlung and the\ndisk black body models could ($\\chi^2$  of 234 and 231\/266 d.o.f.\nrespectively).  A slightly higher temperature was found using these\nmodels ($kT=200$ eV and $kT=145$ eV) compared with the black body\nmodel ($kT=117$ eV) but nearly consistent edge energy, edge depth and\nphoton index were found.  The indices obtained were flat ($\\Gamma =\n1.57$ in both cases).\n\nThe warm absorber table model used thus far models only the absorption\nedges of various ionized species.  Since the line emission expected\nfrom a physical warm absorber might appear as an excess emission\ncomponent, we also tested models calculated using XSTAR which include\nemission lines from the warm absorber (Kallman \\& Krolik\n\\markcite{16} 1993).  Fitting the SIS0:GIS2 pair with a power law\nplus emission only from a physical warm absorber in the line of sight\nresulted in a better fit with $\\chi^2$  of 292\/277 d.o.f.  Including an\nedge, to simulate the absorption by a warm absorber, resulted in a $\\chi^2$ \nof 261\/268 d.o.f.  However, a significantly better fit was found when\na blackbody was also included ($\\chi^2$  = 228\/264 d.o.f.) and the resulting\npower law index was again flat ($\\Gamma = 1.55$).  Similarly, emission\nfrom reflection by a warm absorber could not alone describe the\nspectrum (alone: $\\chi^2=299\/268$ d.o.f.; with an edge:\n$\\chi^2=268\/266$ d.o.f.; with an edge and a black body:\n$\\chi^2=227\/264$ d.o.f.).  In the final case, the index was flat\n($\\Gamma = 1.55$).\n\nSoft excess emission can also be produced by reflection from an\nionized disk (Ross \\& Fabian \\markcite{28} 1993; Zycki et al.\n\\markcite{38} 1994).  We fit the low flux spectrum with an ionized\ndisk table model computed according to Zycki et al. \\markcite{38}\n1994.  For power law plus disk emission only the $\\chi^2$  was 262\/269\nd.o.f., but the ratio of the reflected flux to direct emission was\n5.8.  Since the ratio should be near 1 for the isotropic static case.\nwe considered this result to be unphysical.  Addition of an edge gave\n$\\chi^2$  of 248\/267 d.o.f., but again the ratio was too large at 5.5.\nAddition of a black body gave $\\chi^2=228\/265$ d.o.f., with the ratio\nreduced to a physical value of $R=1.23$ and a flat index $\\Gamma =\n1.61$.\n\nThese results show that soft excess models without line emission fit\nthe spectra well, but we cannot distinguish among them, possibly\nbecause of the poor statistics due to the relatively short exposure in\nthe low state and the decrease in sensitivity toward low energies of\nthe {\\it ASCA} SIS.  Further, the flat spectral index obtained in the\nlow state is robust against changes in the soft excess model.\n\n\\subsubsection{Partial Covering Models}\n\nAnother possible origin of soft emission is leakage through a\npartially covering absorber (e.g. NGC 4151: Weaver et al.\n\\markcite{35} 1994).  The spectral variability would then result from\na change in the covering fraction.  However, the partial covering\nmodel does not fit the low flux SIS0 and GIS2 spectra well\n($\\chi^2=298$ for 268 d.o.f.), the resulting power law is forced to be\nsteep ($\\Gamma=2.42$), and both low and high energy residuals are\nseen.  These can be modeled by reflection, in which case the fit is\ngood ($\\chi^2=225$\/263 d.o.f.), but the power law is very steep\n($\\Gamma=3.0$) and the ratio of the reflected emission to primary\nemission is required to be 20.  This model is unphysical so we\nconclude that partial covering cannot adequately model the soft excess\ncomponent. Finally, a decrease in the fraction of the source covered can\nonly produce a steepening of the spectrum with an increase in flux and\ncannot explain the hardening of the spectrum below 1 keV indicated by\nthe softness ratio (Figure~\\ref{fig4}b).\n\nA scattering and dual absorber model was used to describe the complex\nX-ray spectrum of NGC 4151 (Weaver et al.  \\markcite{35} 1994).  For\nMrk~766 this model does not give a good fit ($\\chi^2=244$\/266 d.o.f.)\nand the photon index is steep ($\\Gamma=2.84$). Low energy residuals\nsuggest an unmodeled absorption edge.  When an edge is included the\nfit is good ($\\chi^2$ =222\/264 d.o.f.), but the power law index is very\nsteep ($\\Gamma=2.94$).  Again, this steep index seems unphysical, so\nthe dual absorber model is rejected.\n\n\\section{Reflection}\n\nThe spectral index change is robust against changes in models of the\nsoft excess component and warm absorber because the low flux spectrum\nis flat at high energies.  The reflection spectrum is also flat (e.g.\nGeorge \\& Fabian  \\markcite{9} 1991), and could produce a hard tail if\nthe reflection component normalization is high compared with the power\nlaw normalization.  This could be observed if the response of the\nreflection component lags variability of the incident X-rays and would\nbe expected if the light crossing timescale of the reflection region\nis long compared with the source variability timescale, or if the\nreflection region is located far from the X-ray source (e.g. in the\nmolecular torus; Ghisellini, Haardt\n\\& Matt \\markcite{12} 1994; Krolik, Madau \\& Zycki\n\\markcite{17} 1994; Leighly et al. \\markcite{42} 1996).\n\nWe define the reflection ratio to be 1 under the conditions that\nnonvarying primary power law emission from an isotropic point source\nilluminates an infinite optically thick disk.  In this case, the\nreflection spectral component is not important in the spectrum below 5\nkeV.  Thus, we can estimate the contribution of the reflection by\nfitting the spectra below 5 keV and comparing with the fit results\nover the full range.  A power law plus two edge fit of the SIS0:GIS2\nspectra results in a steep photon index (low state: $\\Gamma=1.95$ and\n$\\chi^2$  = 214.4\/230 d.o.f., high state:  $\\Gamma=2.02$ and $\\chi^2$ =524\/519\nd.o.f.).  Addition of a black body component to the model improved the\nfit of the low flux spectra and flattened the photon index\n($\\Gamma=1.67$ and $\\chi^2=197\/228$ d.o.f.), but had no effect on the\nhigh flux spectral fit.  If the table model is used to model the warm\nabsorber, similar results are obtained although both indices are found\nto be slightly steeper.  Thus, fitting below 5 keV shows that the\nindex variability is still required and also that the measured low\nenergy index is consistent with the data above $\\sim 5 \\rm keV$.\n\nA reflection ratio much larger than 1 results in a flat spectrum\ntowards high energies.  Fitting the spectra with a power law plus 2\nedges and reflection allowing the ratio to be free produces a good fit\nwith a steep index which is consistent between the low and high flux\nspectra (Table 3).  However, to explain the low flux spectrum, the\nreflection ratio must be 5--8, while the high flux spectra require a\nreflection ratio of only 0.5.  Most importantly, the reflection\ncomponent normalization is required to be significantly {\\it higher}\nin the low flux state compared with the high flux state, so a\nreflection lag cannot explain the spectral variability.  Further, such\na large reflection ratio should be accompanied by a large equivalent\nwidth narrow iron line in the low state spectrum and such a line is\nnot observed (Section 4.5).\n\nThe timing analysis results also rule out a lag in neutral reflection as\nthe origin of the spectral variability.  Since the neutral reflection\nspectrum flux decreases towards low energies, only spectral softening\nwith an increase in power law flux is predicted, whereas a hardening\nof the spectrum below 1 keV was observed. If the surface of the\nreflector is ionized, the opacity is reduced at low energies and a\nsoft X-ray reflection component plus emission lines should be observed\n(Ross \\& Fabian \\markcite{28} 1993; Zycki et al. \\markcite{38} 1994),\nwhich results in hardening of the low flux spectrum with an increase\nin power law flux.  It was shown in Section 4.3.1 that the ionized\ndisk model does not produce a good fit alone, mostly because the soft\nexcess does not show evidence for line emission.\n\n\\subsection{The Iron Line}\n\nThe presence of the iron $K\\alpha$ line in Seyfert 1 nuclei spectra\nis well established (e.g. Nandra \\& Pounds \\markcite{22} 1994).  Mrk\n766 was not observed using Ginga, and a iron $K\\alpha$ line had not\nyet been observed in its spectrum.  To look for the iron line, we fit\nthe spectra from all four detectors simultaneously above 2 keV.\nBecause of the continuum spectral variability the low and high state\nspectra were fit separately.  A power law model resulted in\n$\\chi^2$\/d.o.f. of 206\/253 and 806\/759 for the low and high states\nrespectively.  Addition of a narrow ($\\sigma=0$ keV) redshifted line\nwith the energy fixed at 6.4 keV gave the results presented in Table\n4.  There is no strong evidence for a narrow line with rest energy 6.4\nkeV in the low flux spectra ($\\Delta\\chi^2 \\sim 2$). In the high flux\nspectra $\\Delta\\chi^2$ is 13 corresponding to an F statistic value of\n12.4, indicating the presence of a line with confidence greater than\n99.9\\%. Thus the presence of an iron emission line is confirmed in\nthis source.  The ratio of data to power law model for the summed SIS0\nand SIS1 spectra shows the narrow line and ionized iron edge\n(Figure~\\ref{fig7}).  The non-detection in the low flux spectra can be\nexplained by the poor statistics resulting from lower flux and shorter\nexposure.  If the high flux data are divided into several spectra\ncharacterized by shorter exposures, the presence of a line in the\nspectra from all detectors separately cannot be confirmed.  Next, the\nline energy was allowed to be free.  The high state line energy is\nconsistent with 6.4 keV and a lower energy line is marginally detected\nin the low state spectra.  The line energy in both states is\nconsistent with an origin in primarily neutral material, and emission\nfrom highly ionized material is excluded. The line equivalent width is\n$\\sim 100 eV$ consistent with that expected from emission from an\naccretion disk (e.g. George\n\\& Fabian  \\markcite{9} 1991).  Broad iron lines have been discovered in the\n{\\it ASCA} spectra of several AGN (e.g.  Mushotzky et al.\n\\markcite{21} 1995).  A broad line is preferred in the low flux\nspectra, although the addition of another parameter reduces the $\\chi^2$  by\nonly 5, so again the detection is marginal. The line is narrow in the\nhigh flux spectra, constrained with $\\sigma < 0.2\\,\\rm keV$.  The\nshape of the high energy continuum changes the measured line\nparameters.  When reflection with ratio fixed to 1 is included,\nsimilar results were obtained as before, but the measured equivalent\nwidths were smaller (Table~4).\n\nThese results also support the hypothesis that a lag in the reflection\ncomponent cannot be the origin of the spectral variability.  A lag\nimplies that the bulk of the reflected emission should come from a\nsubstantial distance from the source, and so the line would be\nexpected to be narrow.  However, the large reflection ratio required\nto fit the low flux spectra predicts a very large equivalent width ($>\n\\sim 500\\rm eV$) narrow line, which would be easily detected if\npresent.\n\n\\subsection{Quantifying the Spectral Variability}\n\nThe results of the previous sections indicate that the power law with\nblack body and warm absorber model provides the best fit to both the\nlow and high flux spectra.  A change in spectral index seems to be\nindicated.  However, since the model is complex, spectra from the two\nstates must be fit simultaneously to determine which parameters\nnecessarily change.  Neutral absorption, originating in our Galaxy and\nthe host galaxy, was assumed not to change, and the iron line was\nmodeled as narrow with fixed energy.  Combined fits were done using both\ndescriptions of the warm absorber.\n\nWhen two edges were used to model the warm absorber, the edge energies\nwere equated in the high and low state models.  This was done because\nthe low state spectra could not constrain the higher edge energy and\nsince the edges are identified as \\ion{O}{7} and\n\\ion{O}{8} edges, no change is expected in the energies.  This model fit the\nlow and high spectra well ($\\chi^2$  of 860.1\/859 d.o.f. and 789.8\/844\nd.o.f. for the SIS0:GIS2 and SIS1:GIS3 spectra respectively).  The\ntemperatures of the black body were consistent between the low and\nhigh states so these were equated resulting in a negligible increase\nin $\\chi^2$.  No other parameters could be equated without resulting\nin a large change in $\\chi^2$.  The results are listed in the top\npanel of Table~5, and they indicate that a significant change in index\nand black body normalization occurred.  The edge energies are roughly\nconsistent with absorption by \\ion{O}{7} and \\ion{O}{8}.  The optical\ndepths of these edges are consistent between the low and high state,\nso no change in the ionization of the warm absorber can be determined\nfrom these fits.  However, the best fit value of $\\tau_{OVII}$ is\nlarger in the low state than in the high state, and the reverse is\napproximately true for $\\tau_{OVIII}$.  This suggests that an increase\nin the ionization occurred, but the statistics are too poor to require\nthis conclusion.\n\nThe results of the combined fits using the warm absorber table model\nare listed in the second panel of Table 5.  The ionized column\ndensities were consistent so they were equated in the spectral\nfitting.  As noted previously, when the warm absorber was described\nusing the table model, the black body temperature was found to be\nhigher in the high state than in the low state (Table 3).  When the\ntemperatures are equated in the combined fits, the increase in\n$\\chi^2$ is 2.7 and 5.7 for the SIS0:GIS2 and SIS1:GIS3 respectively,\nsignificant with 90\\% and 97.5\\% confidence. However, this effect is\nclearly model-dependent and may be due to the shape of the warm\nabsorber model and possibly the absence of emission lines in the\nmodel, and thus the implied black body temperature change is unlikely\nto be physical.  Three parameters changed between the low and high\nstates: the power law index, the black body normalization, and the\nionization parameter.  As these parameters are coupled in the spectral\nfitting, to evaluate the significance of the change, the $\\chi^2$  contours\nwere plotted for each pair of parameters (Figure~\\ref{fig8}a, b, and\nc).  These show that the results are consistent between the SIS0:GIS2\nand SIS1:GIS3 spectral pairs, and that the index change, the\nionization state change and the black body normalization change are\nsignificant with $>99$\\%, 90\\% and 68\\% confidence respectively.  We\nnote that the change in the ionization state is a model dependent\nresult, as we cannot demonstrate a change in the optical depths of the\ntwo oxygen edges.  Figure~\\ref{fig9}a and b show the best fitting\nmodels, spectra, and ratios between spectra and model for the low and\nhigh state SIS0:GIS2 spectra.  Figure~\\ref{fig10} shows the best fit\nmodels for the low and high state SIS0 spectra.  Note that the pivot\npoint for the power law change is $\\sim 9$ keV.\n\n\\subsection{{\\it ROSAT} Spectral Fitting}\n\nThe {\\it ASCA} data show that the soft X-ray spectrum is complex,\ncomprised of a power law with variable index, a warm absorber with\nvariable ionization state and a black body with marginally variable\nnormalization and model dependent temperature.  Seven parameters are\nrequired to describe the spectrum in the {\\it ROSAT} band.  Because of\nthe poor spectral resolution, the PSPC spectra have 5 independent\nchannels.  Thus, detailed fitting of the PSPC spectra is of limited\nvalue, as multiple models are acceptable.  Qualitatively, the spectra\nfrom the 1992 and 1993 observations are very soft and cannot be\nadequately modeled using a single power law plus absorption.  A soft\ncomponent like a black body gives a good fit, with $kT\n\\sim$70--90 eV, and a power law with index $\\sim $1.9--2.0.  An edge\ncan also model the spectrum, but the spectral index is steeper at\n$\\sim 2.5$.\n\n\\section{Discussion}\n\n\\subsection{The Hard Spectral Variability}\n\nThe most significant result of this study was the observation of\ndramatic photon index variability from $\\sim 1.6$ to $\\sim 2.0$, over\nseveral thousand seconds and confined to a single event.  We discuss this\nresult in light of current models of the X-ray power law emission in AGN.\n\n\\subsubsection{General Considerations}\n\nThe high energy power law observed from AGN can be successfully and\nplausibly explained by inverse Comptonization by high energy electrons\nof soft UV photons likely originating in the accretion disk. The rapid\nX-ray variability of some AGN implies a high radiation density in the\nnucleus which results in production of electron-positron pairs. The\nimportance of pair production is determined by the compactness\nparameter, $$l= L\\sigma_T\/R m_e c^3,\\eqno(1)$$ where $L$ is the\nluminosity, $R$ is the source size, $\\sigma_T$ is the Thompson\nscattering cross section, $m_e$ is the mass of the electron, and $c$\nis the speed of light.  If the compactness is high, the optical depth\nto pair production will exceed unity and pairs will be produced which\ncan substantially modify the emerging spectrum.  Generally speaking,\nthese models can be differentiated by whether the high energy\nelectrons are thermal or accelerated by non-thermal processes, since\npair production limits the highest energy attainable in the thermal\nplasma.  In rapidly variable AGN, however, both thermal and\nnon-thermal processes may be present (Ghisellini, Haardt \\& Fabian\n\\markcite{10} 1993).\n\nThe results presented here suggest that Mrk~766 is compact enough that\npairs should be produced in the nucleus.  The X-ray flux was observed\nto change by a factor of two in $\\sim 1000$ seconds. This corresponds\nto a source size upper limit of $R < c\\Delta t \\sim 3 \\times 10^{13}\n\\rm \\, cm$.  The 2--10 keV luminosity is $1.3 \\times 10^{43}\\, \\rm ergs \\,\ns^{-1}$ in the high state, implying an X-ray compactness parameter of\n$l_x \\sim 12$.  The hard compactness parameter, proportional to the\ntotal luminosity in the hard component, could be substantially larger.\nIf the compactness parameters are larger than 10, pair production\nshould be important if there are an adequate number of $\\gamma$-ray\nphotons present.  In non-thermal models, the $\\gamma$-rays are\nproduced through upscattering of soft photons by extremely\nrelativistic electrons.  In thermal models, the origin is primarily the\nhigh energy tail of the thermal spectrum and thus the number of\n$\\gamma$-rays depends on the temperature of the plasma. OSSE\nobservations of a few AGN find that the temperature is large enough\nthat electron positron pairs should be produced (e.g. see Figure 1 of\nFabian  \\markcite{7} 1994); however, there have been no high energy\nspectra obtained from Mrk~766.\n\n\\subsubsection{Simple Thermal Comptonization Models}\n\nIf the source does not contain many pairs, and if the power law\nresults from unsaturated Comptonization of soft photons, the spectral\nparameters are very simply related to one another (e.g. Rybicki \\&\nLightman \\markcite{29} 1979).  The slope of the power law depends\ninversely on the Compton $y$ parameter which is proportional to a\npower of the temperature, so an increase in temperature implys a\ndecrease in index.  However, if the soft photon input is constant, the\npivot point energy of the photon index change should be the energy of\nthe soft input photons.  In contrast, we observe the pivot point to be\nmuch higher, at $\\sim 9$ keV.\n\nIf the thermal plasma is pair dominated, Ghisellini \\& Haardt (1994)\n\\markcite{11} show that there is a one--to--one mapping of the\nobservables ($kT$ and $\\alpha$, where $\\alpha=\\Gamma-1$ is the energy\nindex) to the plasma parameters ($l_H$ and $l_H\/l_S$, where $l_H$ and\n$l_S$ are the hard and soft compactnesses, characteristic of the\nrelativistic electrons and the soft (UV) seed photons, respectively).\nOur observed increase in photon index by $\\Delta \\alpha \\sim 0.4$\nimplies a decrease in $l_H\/l_S$ by a factor of 10 (Figure 2 of\nGhisellini \\& Haardt \\markcite{11} 1994).  We observed a 2--10 keV\nflux increase by a factor of 1.3, but since the power law pivot point\nis $\\sim 9\\rm keV$, integration to high energies may show that the low\nflux state luminosity is actually larger than the high flux state\nluminosity.  OSSE observations of several Seyfert 1 galaxies have\nfound that the power spectrum is cut off above several hundred keV\n(e.g. Fabian \\markcite{7} 1994).  Integration of the power law from\n2~keV to the generous upper limit of 500~keV shows that the low state\n$l_H$ is at most a factor of 2 larger than the high state $l_H$,\npredicting an increase in index by only $\\sim 0.1$.  Further, the time\nscale of the spectral variability, less than 10,000 seconds, precludes\na large increase in $l_S$, since this short time scale would be the\norder of the orbital period at the innermost stable orbit for a $< 5\n\\times 10^{7} M_\\odot$ black hole.  A change in the accretion rate\nshould be characterized by the viscous or radial drift time scale,\nestimated by Molendi \\& Maccacaro (1994) \\markcite{19} to be 2.6 days.\nFurther, the {\\it ROSAT} spectral variability can be most naturally\nexplained by a constant (on time scales of one day) soft component\ndominating the softest X-ray band (see Section 5.2).  Finally, as\nGhisellini \\& Haardt (1994) \\markcite{11} note, if reprocessing in the\ndisk is important, $l_H\/l_S$ would be expected to remain approximately\nconstant, and little intrinsic index variability should be observed.\nHowever, this model is very simple, and predictions may change\nsubstantially if a realistic geometry or self-consistent treatment of\nthe two phases is considered.\n\n\\subsubsection{Nonthermal Comptonization Models}\n\nStatic nonthermal models of X-ray emission have been investigated by\nseveral authors (e.g. Svensson \\markcite{31} 1994 and references\ntherein) and 2--10 keV spectral index variability has been studied by\nYaqoob \\markcite{36} (1992). Most simply and generally, soft photons\nwith dimensionless frequency $x_S$ and compactness $l_S$ are scattered\nby relativistic electrons with Lorentz factor $\\gamma_0$ and\ncompactness $l_H$.  First order scattering produces a flat photon\nspectrum with $\\Gamma \\sim 1.5$ extending to\n$x_{max,1}=max[4\/3\\gamma_0^2 x_s,\\gamma_0]$ (Svensson \\markcite{30}\n1987). Pairs are produced if the photon spectrum extends to\nsufficiently high energies and if the optical depth to pair production\nis greater than unity.  Soft photons reprocessed by pairs have a\nsteeper spectrum with $\\Gamma=1.75$ breaking sharply at $x_B=2\n\\gamma_0^4 (2\/3 x_s)^3$. If the energy of pair reprocessed photons is\nrelatively low, they could be observed as an X-ray soft excess.  If\nthe energy of reprocessed photons is high enough, additional pair\ngenerations will be produced resulting in a pair cascade.  In that\ncase, the photon spectrum is steep with $\\Gamma$ approaching 2 (e.g.\nSvensson \\markcite{30} 1987).\n\nA nonthermal model can naturally explain the observed change in index,\nthe disappearance of the soft excess component and the confinement of\nthe spectral variability to a single event. The photon index\nvariability could result from a transition from a first order pair\nspectrum to a cascade caused by a sudden increase in the Lorentz\nfactor of the relativistic electrons.  Thus, the low flux spectrum is\ncomprised of the inverse Compton cooling spectrum, characterized by\nthe hard X-ray power law with photon index near $1.5$, and the first\norder optically thin pair reprocessed spectrum, observed as the soft\nexcess component.  The high flux spectrum is comprised of a pair\ncascade spectrum, characterized by the hard X-ray power law with\nphoton index near 2.  The soft excess component disappears in the high\nflux spectrum as the maximum energy of the pair reprocessed spectrum\nincreases far beyond the observed X-ray band, resulting in the\ncascade.  The change in flux in the X-ray band depends on the change\nof hard compactness $l_H$ which can be expected to accompany the\nchange in Lorentz factor, and flux variability uncorrelated with\nspectral variability would occur through variation in $l_H$ alone.\n\nA possible difficulty with non-thermal models which produce flat\nspectra is that, in general, they overpredict the gamma ray\nbackground.  However, recent work shows that if the non-thermal plasma\nis in a corona above a disk, $\\gamma$-ray photons are more efficiently\ndepleted and, depending on the compactness, a strong spectral cut off\nbelow 200 keV is predicted (Tritz \\markcite{32} 1990; Tsuruta \\&\nKellen \\markcite{40} 1995).  \n\n\\subsection{The Soft Spectral Variability}\n\nThe {\\it ROSAT} spectrum was observed to become harder as the flux\nincreased, consistent with the lower amplitude variability observed at\nsoftest energies. A most natural explanation for this behavior is\nvariability between the relative normalizations of the power law and\nsoft excess component.  This would be observed if the flux of the soft\ncomponent were nearly constant on the time scale of an observation.\nIn terms of current physical models it could imply that the soft\ncomponent is dominated by primary emission from an accretion disk and\nreprocessed hard emission is relatively less important.  The fraction\nof black body flux in the softest band can be estimated by comparing\nthe relative variability of the hardest band where the power law\ndominates with the relative variability of the softest band where the\nsoft component dominates.  This scenario predicts that if a static\nsoft excess component comprises a large fraction of the flux in the\nsoftest band, any flux change must be accompanied by a hardness ratio\nchange, while if the black body comprises only a small fraction of the\nflux, variability in hard and soft bands should be correlated.\n\nOverall, the {\\it ROSAT} PSPC data are qualitatively consistent with\nthis scenario. In the 1992 observation, the spectrum was harder at\nhigh flux and correlated variability was observed, while at low flux\nuncorrelated variability was found.  In the 1991 observation, when the\nflux was lower and the spectrum generally softer, only uncorrelated\nvariability was observed.  This scenario can also explain the lack of\nspectral variability seen in the {\\it ROSAT} All Sky Survey data\n(Molendi, Maccacaro and Schaeidt \\markcite{20} 1993) since at that\ntime the source was bright and power law emission may have dominated\nthe soft component emission.  Qualitatively and on short time scales\nthere are difficulties with this scenario.  In 1992, the hardest and\nsoftest bands decrease by 50\\% and 25\\% respectively overall, implying\nthe soft component must contribute the same percentages of the soft\nband flux in the high and low states respectively.  However, at high\nflux there is a dip in flux by 30\\% just after the start of the\nobservation, and no change in the softness ratio was observed.\nSimilarly, at low flux, there is an increase by 30\\% in the hard band\napproximately $6\\times 10^4\\,\\rm s$ after the start of the observation,\nbut no change in the soft band emission was observed.  In combination,\nthese two results cannot be explained if the black body flux is constant\nand no other parameters change.  Reprocessing, neglected so far, may\nbe able to explain large amplitude correlated variability at high\nflux.\n\nOther models cannot explain the observed spectral variability.\nNetzer, Turner \\& George (1994) \\markcite{24} showed that variability\nof the warm absorber in response to ionizing flux changes could not\nexplain the spectral variability found during the 1992 observation.\nIn general, if the soft component were dominated by reprocessing, the\nsoft X-ray variability would be expected to track the hard component\nvariability.  However, substantial spectral variability of the\nincident continuum could result in some spectral variability of the\nreprocessed component.  Changes in the accretion rate (Molendi \\&\nMaccacaro \\markcite{19} 1994) cannot explain the orbit-to-orbit\nspectral variability as the predicted time scales are much longer.\n\n\\subsection{The Change in the Warm Absorber}\n\nIn the {\\it ASCA} spectra, we found that there was no evidence that\nthe ionized absorption column density changed between the high state\nand the low state; however, we found that the ionization parameter\nchanged with 90\\% confidence.  This result is model dependent, as we\ncould not demonstrate that the optical depth of the oxygen edges\nchanged significantly.\n\nThe best fit ionization parameter $\\log(U)$ changed from $\\sim -0.85$\nin the low state to $\\sim -0.42$ in the high state, implying an\nincrease in flux of ionizing photons by a factor of 2.7.  It is\ninteresting that this is quite close to the implied change in flux by\na factor of 3.1 of the intrinsic power law at 0.7 keV. The observed\nphoton index variability implies a larger change in the flux of\nionizing photons, assuming the power law extends to low energies.  The\nspectral changes occurred over a time scale of several thousand\nseconds.  The recombination time scale for \\ion{O}{8} is about $2\n\\times 10^{11}T_e^{0.5} n^{-1}$ seconds (e.g.  Turner et al.\n\\markcite{33} 1993) or about 5.5 hours using the parameters assumed in\nthis model (or longer if the gas is rarer).  Thus the gas may not be\nin photoionization equilibrium in the high flux state, and the larger\npopulation of \\ion{O}{8} implied by the increase in the ionization\nparameter may result from ions which are directly stripped of an\nelectron by the increased number of photons with energy near 0.7 keV.\nFurther observations are necessary to determine the response of the\nionized material to a decrease in flux since a change in ionization\nwould be observed only if recombination had occurred.\n\nA change in the ionization correlated with an increase in flux has not\nbeen previously reported from {\\it ASCA} data.  An increase in column\ndensity and no change in ionization accompanied an increase in flux in\nMCG--6-30-15 (Fabian et al. \\markcite{8} 1994).  Explaining the\nspectral variability during an increase in flux by a change in warm\nabsorber properties in NGC~3227 required a decrease in ionization and\nan increase in column (Ptak et al. \\markcite{27} 1994). In another\nobservation of MCG--6-30-15, an increase in the optical depth of the\n\\ion{O}{8} edge during a flux decrease was interpreted as evidence for\nrecombination of \\ion{O}{9} (Otani \\markcite{26} 1995).\n\n\\subsection{Hard X-ray Emission from Narrow Line Seyfert 1s}\n\\vskip 1pc\n\nThe soft X-ray properties of narrow line Seyfert 1s are well studied\n(Boller, Brandt \\& Fink \\markcite{1} 1996) but few hard X-ray\nobservations have been reported. The {\\it ASCA} observation of Mrk~766\nrepresents one of the first observations of the hard emission from\nthese objects.\n\nWe found a hard power law with variable photon index in Mrk~766, and\nthe spectral variability which was not strictly flux correlated.\nSimilar photon index variability which was flux correlated has been\ndiscovered from NGC~4051 (Guainazzi et al. \\markcite{15} 1996), a\nSeyfert 1 galaxy which shares many properties with NLS1s.  In\ncontrast, a very steep spectrum with photon index $\\sim 2.6$, a\ndominate soft excess and no variability was observed from narrow-line\nSeyfert 1 RE~1034+39 (Pounds, Done \\& Osborne \\markcite{39} 1995).\nBecause these spectral and variability properties are similar to those\ncharacteristic of black hole candidates in the high state (e.g. Nowak\n\\markcite{41} 1991) it was postulated that RE~1034+39 represents a\nSeyfert 1 galaxy in the high state (Pounds, Done \\& Osborne\n\\markcite{39} 1995).\n\nThe marked differences between the hard X-ray properties of Mrk~766\nand RE~1034+39 are interesting.  Black hole candidates in the low\nstate are characterized by a flat hard X-ray power law and more rapid,\nlarger amplitude hard X-ray variability (e.g. Nowak \\markcite {41}\n1995).  These properties more closely resemble the observational\nresults from Mrk~766 and NGC~4051 than do the properties of black hole\ncandidates in the high state.  Further observations of NLS1s may find\nthat the spectral and variability properties fall into two classes:\nthose with steep hard X-ray spectra, dominant soft X-ray emission and\nlower amplitude short term hard X-ray variability, and those with flat\nhard X-ray spectra, less soft X-ray emission and rapid hard X-ray\nvariability.  There is perhaps already some evidence for such a\ndivision. While many NLS1s are very bright soft X-ray objects commonly\nfound in soft X-ray samples, relatively few have hard X-ray detections\nby HEAO-1 A2.\n\nFurther support for this scenario may come if repeated X-ray\nobservations discover that some objects have made the transition\nbetween two states.  This could have been what had occurred in objects\nobserved to have varied by factors of 10 or more between two {\\it\nROSAT} observations (e.g. Zwicky 159.034, Brandt, Pounds \\& Fink\n\\markcite{3} 1995; WPVS007, Grupe et al. \\markcite{14} 1995).\nWell-studied broad-line Seyfert 1 galaxies are not observed to undergo\nthis kind of transition.  The fact that many well-studied AGN are hard\nX-ray selected while NLS1s are clearly soft X-ray selected objects\nfurther supports this hypothesis.\n\nIn black hole candidates, it is widely believed that the high state\nis characterized by a relatively larger accretion rate compared with\nthe low state.  Thus the behavior of NLS1s may result from a\nrelatively larger accretion rate.  If the processes fueling AGNs are\ncommon for Seyferts, a relatively larger accretion rate would be\nobserved in systems with relatively small black hole masses.\n\n\\section{Conclusions}\n\nWe report analysis of {\\it ASCA} and {\\it ROSAT} observations of the\nnarrow-line Seyfert 1 galaxy Mrk~766.  In the {\\it ASCA} observation\nrapid variability with doubling time scale of order $\\sim 1000$\nseconds was observed, and dramatic spectral variability over as time\nperiod of less than $\\sim 10,000\\,\\rm s$ was discovered. Confined to a\nsingle event, during a 2--10 keV flux increase the spectrum above and\nbelow $\\sim 1$ keV softened and hardened respectively. The low and\nhigh flux spectra could be described with a model consisting of a\npower law, iron line, warm absorber and soft excess modeled as a black\nbody.  The spectral variability was a result of a highly significant\nincrease in the intrinsic power law index from $\\sim 1.6$ to $\\sim\n2.0$ with the pivot point at $\\sim 9$ keV, a model dependent increase\nin the ionization of the warm absorber, and a marginal decrease in the\nsoft excess component.  A $100 \\rm \\, eV$ equivalent width narrow iron\nline was detected in the high flux spectrum but not in the low flux\nspectrum, most likely because of poor statistics.  Variability on time\nscales as short as $\\sim 2400$ seconds was found in the {\\it ROSAT}\ndata.  Because the variability in the softest {\\it ROSAT} band, below\n0.4 keV, had relatively lower amplitude than the harder bands,\nspectral hardening during flux increases was detected on time scales\nas short as the orbital period of $ \\sim 6000 \\,\\rm s$.\n\nThe spectral index change, the disappearance of the soft component in\nthe {\\it ASCA} band and the confinement of the spectral variability to\na single event could be naturally explained in terms of non-thermal\nComptonization models.  We postulate that the index change occurred\nthrough a transition from a first order pair reprocessed spectrum to a\npair cascade spectrum brought about by a sudden increase in the\nLorentz factor of the injected relativistic electrons.  The first\norder pair reprocessed spectrum observed in the low state as a soft\nexcess disappeared in the high state cascade spectrum. Variations in\nthe hard compactness resulted in pure flux variability.  The measured\nincrease in the warm absorber ionization corresponds to the increase\nin flux near the oxygen edges resulting from the power law index\nchange.  The spectral variability in the {\\it ROSAT} data was most\nnaturally explained by a variable hard component and a nonvariable\nsoft component which dominated the softest band and may be primary\nemission from an accretion disk perhaps implying that reprocessing is\nrelatively less important in this object.\n\nThe flat and variable hard power law index observed in Mrk~766 is\nsimilar to that observed in NGC 4051 (Guainazzi et al. \\markcite{15}\n1996), a Seyfert 1 with many properties common to NLS1s, but contrasts\nmarkedly with the very steep hard X-ray index $\\Gamma \\sim 2.6$ found\nin NLS1 object RE~1034+39 (Pounds, Done \\& Osborne \\markcite{39} 1995).\nFurther hard X-ray observations of NLS1s using {\\it ASCA} are\nnecessary to clearly understand the hard X-ray properties of these\nsources.\n\n\\acknowledgements\n\nThis research has made use of data obtained through the High Energy\nAstrophysics Science Archive Research Center Online Service, provided\nby the NASA-Goddard Space Flight Center. The authors thank T. Kallman\nand P. Zycki for the use of their spectral table models.  KML\nacknowledges receipt of a National Research Council fellowship at\nNASA\/GSFC and a Japanese Science and Technology Agency fellowship at\nRIKEN and useful conversations with M. Cappi.  KML acknowledges\nsupport by a {\\it ROSAT} AO4 guest observer grant and an {\\it ASCA}\nAO1 guest observer grant.\n\n\\clearpage\n\n\\begin{deluxetable}{lrrrrr}\n\\tablewidth{0pc}\n\\tablenum{1}\n\\tablecaption{Observing Log}\n\\tablehead{\n\\colhead{Instrument}   & \\colhead{Observation} & \n\\colhead{Exposure}     & \\colhead{Total Counts} &\n\\colhead{Background}   \\\\\n\\colhead{}             & \\colhead{Date} &\n\\colhead{(s)}          & \\colhead{} &\n\\colhead{Count Rate}  }\n\n\\startdata\n{\\it ROSAT} PSPC & 15\/06\/91 & 15179\\tablenotemark{a} & 21804 & 0.081   \\nl\n{\\it ROSAT} PSPC & 21\/12\/92 & 16303\\tablenotemark{b} & 53865 & 0.056  \\nl\n{\\it ROSAT} PSPC & 17\/12\/93 & 3146 & 7309 & 0.033  \\nl\n{\\it ASCA} S0 & 18\/12\/93 & 32947 & 32823 & 0.030  \\nl\n\\hphantom{{\\it ASCA}} S1 & & 32735 & 26853 & 0.020 \\nl\n\\hphantom{{\\it ASCA}} G2 & & 35805 & 16991 & 0.033 \\nl\n\\hphantom{{\\it ASCA}} G3 & & 35808 & 19532 & 0.032 \\nl\n\\tablecomments{Column 4 is the total (not background subtracted) counts in the\nsource extraction region.  \nColumn 5 is the background rate scaled to the source\nextraction region.  }\n\\tablenotetext{a} {Exposure time after data selection to remove periods\nwhere the source was occulted by a rib.}\n\\tablenotetext{b} {Exposure time in the first 85,000 seconds of\nobservation.  The total observation time was 19870 seconds.}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\\begin{deluxetable}{lrrr}\n\\tablewidth{0pc}\n\\tablenum{2}\n\\tablecaption{PSPC Variability}\n\\tablehead{\n\\colhead{Band}   & \\colhead{Mean} & \n\\colhead{$\\chi^2_\\nu$}     & \\colhead{NVA}}\n\n\\startdata\n\n\\multicolumn{4}{l}{1991 Data (44 points):} \\nl\nTotal & 1.35 & 17.6 & 0.22 \\nl\n0.2--0.5 & 0.84 & 6.0 & 0.16 \\nl\n0.5--0.9 & 0.20 & 7.05 & 0.36 \\nl\n0.9--2.0 & 0.22 & 8.54 & 0.37 \\nl\n\\tableline\n\\multicolumn{4}{l}{1992 Data (41 points):} \\nl\nTotal & 3.06 & 77.5 & 0.26 \\nl\n0.2--0.5 & 1.63 & 22.7 & 0.19 \\nl\n0.5--0.9 & 0.58 & 24.9 & 0.34 \\nl\n0.9--2.0 & 0.67 & 35.1 & 0.38 \\nl\n\\tableline\n\\multicolumn{4}{l}{Quasi-simultaneous} \\nl\n\\multicolumn{4}{l}{Data (8 points):} \\nl\nTotal & 2.24 & 29.9 & 0.19 \\nl\n0.2--0.5 & 1.33 & 8.2 & 0.13 \\nl\n0.5--0.9 & 0.39 & 13.9 & 0.31 \\nl\n0.9--2.0 & 0.39 & 13.6 & 0.31 \\nl\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\\begin{deluxetable}{lcccc}\n\\footnotesize\n\\tablewidth{0pc}\n\\tablenum{3}\n\\tablecaption{{\\it ASCA} Spectral Fitting Results}\n\\tablehead{\n\\colhead{Parameter}   & \\multicolumn{2}{c}{Low State} &\n\\multicolumn{2}{c}{High State} \\\\\n\\colhead{} & \\colhead{SIS0:GIS2} & \n\\colhead{SIS1:GIS3} & \\colhead{SIS0:GIS2} &\n\\colhead{SIS1:GIS3}}\n\n\\startdata\n\\multicolumn{5}{l}{Power law model:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ &\n $\\rm Gal < 0.19$ & $\\rm Gal<0.20$ & $\\rm Gal<0.19$ & $\\rm Gal<0.192$ \\nl\nIndex & $1.65 \\pm 0.04$ & $1.64^{+0.04}_{-0.05}$ & $1.92 \\pm 0.02$ & \n$1.91^{+0.01}_{-0.02}$ \\nl\n$\\chi^2$\/d.o.f. & 422\/270 & 341\/254 &\n 819\/598 & 765\/597 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law plus black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $0.84^{+0.51}_{-0.48}$ & \n$0.75^{+0.53}_{-0.49}$\n& $0.95^{+0.22}_{-0.20}$ & $1.02^{+0.22}_{-0.20}$ \\nl\nIndex & $1.54^{+0.06}_{-0.07}$ & $1.54^{+0.08}_{-0.07}$ & $1.96 \\pm 0.03$ &\n$1.98^{+0.04}_{-0.03}$ \\nl\nSIS PL norm\\tablenotemark{a}  & $3.0^{+0.3}_{-0.2}$ & $3.0^{+0.3}_{-0.2}$ &\n$7.7 \\pm 0.03$ & $7.9 \\pm 0.03$ \\nl\nkT (eV)  & $88^{+8}_{-6}$ & $87^{+9}_{-7}$ & $77^{+5}_{-4}$ & $72 \\pm 5$ \\cr\nSIS bb norm\\tablenotemark{b} & $2.5^{+2.1}_{-1.2}$ & $2.3^{+2.1}_{-1.2}$ &\n$4.2^{+1.5}_{-1.3}$ & $5.4^{+2.0}_{-1.7}$ \\nl\n$\\chi^2$\/d.o.f. & 245\/268 & 208\/254 & 681\/596 & 647\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law plus edge:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal < 0.18$ & $\\rm Gal<0.20$ & $\\rm Gal<0.26$ &\n$\\rm Gal< 0.23$ \\nl\nIndex & $1.87 \\pm 0.04$ & $1.83^{+0.06}_{-0.05}$ & $2.01 \\pm 0.02$ &\n$2.01^{+0.03}_{-0.02}$ \\nl\nSIS PL norm\\tablenotemark{a}\n & $4.38 \\pm 0.18$ & $4.23^{+0.24}_{-0.23}$ & $8.06^{+0.17}_{-0.15}$\n& $8.06^{+0.23}_{-0.17}$ \\nl\nEdge Energy (keV) & $0.81 \\pm 0.02$ & $0.82^{+0.02}_{-0.03}$ &\n$0.78^{+0.01}_{-0.03}$ & $0.75 \\pm 0.02$ \\nl\n$\\tau$ & $1.01 \\pm 0.15$ & $0.89^{+0.21}_{-0.19}$ & $0.49^{+0.07}_{-0.06}$ &\n$0.53 \\pm 0.70$ \\nl\n$\\chi^2$\/d.o.f. & 344\/268 & 279\/254 & 651\/596 & 608\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law, black body and edge:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.28<0.91$ & $\\rm Gal<0.42<0.88$ &\n$0.43^{+0.18}_{-0.20}$ & $\\rm Gal<0.35<0.63$ \\nl\nIndex & $1.55^{+0.09}_{-0.05}$ & $1.57 \\pm 0.07$ & $1.99^{+0.02}_{-0.04}$ &\n$2.00^{+0.04}_{-0.03}$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.1^{+0.3}_{-0.2}$ & $3.2 \\pm 0.30$ &\n $7.9^{+0.2}_{-0.3}$ & $8.0^{+0.4}_{-0.3}$ \\nl\nEdge energy (keV) & $0.76^{+0.03}_{-0.04}$ & $0.74 \\pm 0.02$ &\n$0.74^{+0.03}_{-0.02}$ & $0.75^{+0.02}_{-0.03}$ \\nl\n$\\tau$ & $0.60^{+0.28}_{-0.25}$ & $0.77^{+0.28}_{-0.30}$ & \n$0.43^{+0.09}_{-0.10}$ & $0.46^{+0.12}_{-0.11}$ \\nl \n$kT$ (eV) & $117 \\pm 18$ & $121^{+21}_{-16}$ & $101^{+19}_{-13}$\n & $91^{+48}_{-35}$ \\nl\nSIS bb norm\\tablenotemark{b}\n & $0.95^{+1.19}_{-0.26}$ & $1.12^{+0.82}_{-0.44}$ & $0.81^{+0.56}_{-0.50}$\n& $0.54^{+1.04}_{-0.52}$ \\nl\n$\\chi^2$\/d.o.f. & 230\/266 & 191\/253 & 640\/594 & 605\/593 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power law and 2 edges:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.22$ & $\\rm Gal<0.22$ & $1.1<2.8$ &\n$2.5<3.4$ \\nl\nIndex & $1.91^{+0.05}_{-0.04}$ & $1.90 \\pm 0.05$ & $2.02^{+0.05}_{-0.01}$ &\n$2.05 \\pm 0.04$ \\nl\nSIS PL norm\\tablenotemark{a} & $5.0 \\pm 0.20$ & $4.9 \\pm 0.30$ & \n$8.2^{+0.5}_{-0.2}$ & $8.5^{+0.5}_{-0.4}$ \\nl\nEdge energy (keV) & $0.78^{+0.01}_{-0.02}$ & $0.77 \\pm 0.03$ &\n$0.74^{+0.02}_{-0.04}$ & $0.73 \\pm 0.02$ \\nl\n$\\tau$ & $1.00^{+0.18}_{-0.15}$ & $1.01^{+0.23}_{-0.18}$ &\n $0.43^{+0.08}_{-0.19}$ & $0.51^{+0.09}_{-0.07}$ \\nl\nEdge energy (keV) & $1.19^{+0.05}_{-0.03}$ & $1.24^{+0.07}_{-0.14}$ &\n$0.94^{+0.06}_{-0.24}$ & $0.98^{+0.05}_{-0.04}$ \\nl\n$\\tau$ & $0.55 \\pm 0.11$ & $0.49^{+0.15}_{-0.11}$ & $0.16^{+0.15}_{-0.06}$ &\n$0.14^{+0.07}_{-0.06}$ \\nl\n$\\chi^2$\/d.o.f. & 254\/266 & 241\/252 & 631\/594 & 594\/593 \\nl\n\\tableline\n\\tablebreak\n\\multicolumn{5}{l}{Power law, 2 edges and black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $Gal<0.32<0.81$ & $\\rm Gal<0.70$ &\n$\\rm Gal<0.29<0.44$ & $\\rm Gal<0.49<0.52$ \\nl\nIndex & $1.64^{+0.10}_{-0.09}$ & $1.62 \\pm 0.09$ & $2.01 \\pm 0.04$ &\n$2.05 \\pm 0.04$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.5^{+0.5}_{-0.4}$ & $3.4^{+0.4}_{-0.3}$ &\n$8.2 \\pm 0.4$ & $8.5^{+0.5}_{-0.4}$ \\nl\nEdge energy (keV) & $0.76 \\pm 0.03$ & $0.75^{+0.02}_{-0.03}$ &\n$0.73^{+0.02}_{-0.04}$ & $0.73 \\pm 0.02$ \\nl\n$\\tau$ & $0.67^{+0.26}_{-0.27}$ & $0.84^{+0.37}_{-0.30}$ & \n$0.40^{+0.15}_{-0.16}$ & $0.51^{+0.08}_{-0.07}$ \\nl\n$kT$ (eV) & $115 \\pm 21$ & $134^{+31}_{-33}$ & $126^{+34}_{-29}$ & \n84\\tablenotemark{c} \\nl\nSIS bb norm\\tablenotemark{b} & $0.88^{+1.00}_{-0.37}$ &\n$0.69^{+1.04}_{-0.13}$ & $0.29^{+0.31}_{-0.26}$ & $0<0.08$ \\nl\nEdge energy (keV) & $1.20^{+0.07}_{-0.17}$ & 1.104\\tablenotemark{c}\n & $0.87^{+0.12}_{-0.06}$ & $0.98^{+0.05}_{-0.04}$ \\nl\n$\\tau$ & $0.18^{+0.16}_{-0.14}$ & $0<0.16<0.27$ & $0.18^{+0.14}_{-0.09}$ &\n$0.14 \\pm 0.07$ \\nl\n$\\chi^2$\/d.o.f.  & 226\/264 & 189\/250 & 629\/592 & 594\/591 \\nl\n\\tableline\n\\multicolumn{5}{l}{Warm Absorber:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal<0.18$ & $\\rm Gal < 0.18$ &\n$\\rm Gal<0.23<0.30$ & $0.26^{+0.09}_{-0.08}$ \\nl\nIndex & $2.05^{+0.04}_{-0.07}$ & $1.98 \\pm 0.08$ & $2.12^{+0.05}_{-0.04}$ &\n$2.13^{+0.05}_{-0.04}$ \\nl\nSIS pl norm\\tablenotemark{a} & $6.4^{+0.5}_{-0.7}$ & $5.7^{+0.5}_{-0.3}$ &\n9$.7 \\pm 0.5$ & $9.9 \\pm 0.6$ \\nl\n$log(U)$ & $-0.17^{+0.08}_{-0.19}$ & $-0.28^{+0.06}_{-0.09}$ &\n$-0.38^{+0.05}_{-0.06}$ & $-0.43^{+0.08}_{-0.06}$ \\nl\n$log(N_w)$\\tablenotemark{c} & $22.27^{+0.09}_{-0.17}$ & $22.13 \\pm 0.09$ &\n$21.71 \\pm 0.07$ & $21.68^{+0.08}_{-0.06}$ \\nl\n$\\chi^2$\/d.o.f. & 294\/268 & 269\/254 & 634\/596 & 601\/595 \\nl\n\\tableline\n\\multicolumn{5}{l}{Warm Absorber plus black body:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & Gal<0.48<0.94 & $\\rm Gal<0.36<0.79$ &\n$\\rm Gal<0.29<0.42$ & $\\rm Gal<0.23<0.40$ \\nl\nIndex & $1.70^{+0.13}_{-0.11}$ & $1.70^{+0.14}_{-0.10}$ &\n $2.12^{+0.05}_{-0.07}$ & $2.10^{+0.07}_{-0.06}$ \\nl\nSIS PL norm\\tablenotemark{a} & $3.9^{+1.0}_{-0.6}$ & $4.0^{+1.1}_{-0.6}$ &\n$9.8^{+0.9}_{-0.8}$ & $9.5^{+1.0}_{-0.6}$ \\nl\n$log(U)$ & $-0.78^{+0.21}_{-0.32}$ & $-0.79^{+0.25}_{-0.26}$ & \n$-0.41^{+0.08}_{-0.07}$ & $-0.42^{+0.06}_{-0.05}$ \\nl\n$log(N_w)^{d}$ & $21.78^{+0.36}_{-0.39}$ & $21.84^{+0.40}_{-0.36}$ & \n$21.82^{+0.13}_{-0.20}$ & $21.83 \\pm 0.11$ \\nl\n$kT$ (eV) & $119^{+47}_{-18}$ & $131^{+55}_{-24}$ & $184^{+76}_{-103}$ &\n$230^{+105}_{-57}$ \\nl\nSIS bb norm\\tablenotemark{b}\n & $1.52^{+1.34}_{-0.66}$ & $1.37^{+1.70}_{-0.48}$ & $0.49^{+0.71}_{-0.46}$\n& $0.49^{+0.65}_{-0.44}$ \\nl\n$\\chi^2$\/d.o.f. & 234\/266 & 194\/253 & 630\/594 & 596\/593 \\nl\n\\tableline\n\\multicolumn{5}{l}{Power Law, two edges and Reflection:} \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & $\\rm Gal < 0.39$ & $\\rm Gal < 0.50$ &\n$\\rm Gal<0.23<0.37$ & $\\rm Gal<0.28<0.44$ \\nl\nIndex & $2.05^{+0.12}_{-0.06}$ & $2.04^{+0.17}_{-0.06}$ & \n$2.06^{+0.08}_{-0.05}$ & $2.07^{+0.09}_{-0.06}$ \\nl\nSIS PL norm\\tablenotemark{a} & $4.87^{+0.52}_{-0.22}$ & $4.75^{+0.83}_{-0.26}$ &\n$8.40^{+0.61}_{-0.36}$ & $8.65^{+0.58}_{-0.51}$ \\nl\nSIS Refl norm\\tablenotemark{a}\n & $2.73^{+1.83}_{-1.10}$ & $2.56^{+2.42}_{-0.94}$ & $0.21<1.24$ &\n$0.26<1.35$ \\nl\nRefl. Ratio & $5.5^{+2.5}_{-2.1}$ & $8.3^{+6.1}_{-2.6}$ & $0.4<1.5$ &\n$0.3<1.5$ \\nl\nEdge Energy (keV) & $1.18^{+0.05}_{-0.1}$ & $1.11^{+0.05}_{-0.06}$ &\n$0.94^{+0.07}_{-0.11}$ & $0.99 \\pm 0.04$  \\nl\n$\\tau$ & $0.44^{+0.08}_{-0.12}$ & $0.38^{+0.13}_{-0.12}$ & \n$0.18^{+0.06}_{-0.08}$ & $0.15 \\pm 0.07$ \\nl\nEdge Energy (keV) & $0.77 \\pm 0.02$ & $0.75 \\pm 0.02$ &\n$0.73 \\pm 0.02$ & $0.73^{+0.02}_{-0.01}$ \\nl\n$\\tau$ & $0.98 \\pm 0.17$ & $0.90^{+0.18}_{-0.17}$ & $0.44^{+0.10}_{-0.20}$ &\n$0.53\\pm 0.09$ \\nl\n$\\chi^2$\/d.o.f. & 230\/263 & 201\/250 & 631\/593 & 594\/590 \\nl\n\\tablecomments{The value ``Gal'' for the absorption refers to the spectral\nfit lower limit set to the Galactic value, $1.77 \\times 10^{20} \\rm \\,\ncm^{-2}$ (Elvis, Wilkes \\& Lockman \\markcite{6} 1989).}\n\\tablenotetext{a}{$\\times\n10^{-3} \\rm photons\\,keV^{-1}cm^{-2}s^{-1}$}\n\\tablenotetext{b}{$\\times 10^{-4} L_{39}\/{D_{10}^2}$, where \n$L_{39}$ is the source\nluminosity in $10^{39}\\rm erg\\,s^{-1}$ and $D_{10}$ is the distance to\nthe source in $10^{10} \\rm kpc$}\n\\tablenotetext{c}{Unconstrained}\n\\tablenotetext{d}{log of the ionized column in units of $\\rm cm^{-2}$.}\n\\enddata\n\n\\end{deluxetable}\n\\clearpage\n\n\n\\begin{deluxetable}{lcc}\n\\tablewidth{0pc}\n\\tablenum{4}\n\\tablecaption{Iron Line Fitting Results}\n\\tablehead{\n\\colhead{Parameter}   & \\colhead{Low State} & \n\\colhead{High State}}\n\n\\startdata\n\\multicolumn{3}{l}{Power law plus narrow line:} \\nl\nIndex & $1.57 \\pm 0.07$ & $2.00^{+0.03}_{-0.04}$ \\nl\nLine Flux\\tablenotemark{a} & $1.8<3.0$ & $2.3^{+1.0}_{-1.2}$ \\nl\nLine Eq. Width (eV) & $100<170$ & $110^{+40}_{-55}$ \\nl\n$\\chi^2$\/d.o.f. & 204\/252 & 793\/758 \\nl\n\\tableline\n\\multicolumn{3}{l}{Power law, narrow line and reflection:} \\nl\nIndex & $1.66 \\pm 0.07$ & $2.08^{+0.04}_{-0.03}$ \\nl\nLine Flux\\tablenotemark{a}& $1.1<2.6$ & $1.8^{+0.9}_{-1.0}$ \\nl\nLine Eq. Width (eV) & $35<84$ & $47 \\pm 25$ \\nl\n$\\chi^2$\/d.o.f. & 204\/252 & 793\/758 \\nl\n\\tablenotetext{a}{$\\times 10^{-5} \\rm photons\\,cm^{-2}s^{-1}$ in the\nline.}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\n\\begin{deluxetable}{lcccc}\n\\footnotesize\n\\tablewidth{0pc}\n\\tablenum{5}\n\\tablecaption{Combined Spectral Fits}\n\\tablehead{\n\\colhead{Parameter}   & \\multicolumn{2}{c}{SIS0:GIS2}\n   & \\multicolumn{2}{c}{SIS1:GIS3} \\\\\n\\colhead{} & \\colhead{Low State} &\n\\colhead{High State} & \\colhead{Low State} &\n\\colhead{High State}}\n\n\\startdata\n\\multicolumn{5}{l}{Two Edge Model: } \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & \\multicolumn{2}{c}{$\\rm Gal<0.27<0.45$} &\n\\multicolumn{2}{c}{$\\rm Gal<0.25<0.37$} \\nl\nIndex & $1.58 \\pm 0.08$ & $2.02 \\pm 0.04$ & $1.60^{+0.08}_{-0.07}$ &\n$2.05 \\pm 0.04$ \\nl \nSIS PL norm\\tablenotemark{a} & $3.2 \\pm 0.30$ & $8.3^{+0.4}_{-0.3}$ \n& $3.2 \\pm 0.3$ & $8.6 \\pm 0.4$ \\nl\nkT (eV) & \\multicolumn{2}{c}{$123^{+22}_{-17}$} &\n \\multicolumn{2}{c}{$133^{+26}_{-21}$} \\nl\nSIS BB norm\\tablenotemark{b} & 0$.89^{+0.34}_{-0.22}$ & $0.19<0.61$ & \n$0.84^{+0.24}_{-0.16}$ & $0<0.23$ \\nl\nEdge Energy (keV) & \\multicolumn{2}{c}{$0.73^{+0.03}_{-0.01}$} &\n\\multicolumn{2}{c}{$0.74^{+0.01}_{-0.02}$} \\nl\n$\\tau$ & $0.66^{+0.28}_{-0.25}$ & $0.47^{+0.08}_{-0.09}$ & \n$0.83^{+0.19}_{-0.17}$ & $0.52^{+0.07}_{-0.08}$ \\nl\nEdge Energy (keV) & \\multicolumn{2}{c}{$0.97^{+0.05}_{-0.16}$} &\n\\multicolumn{2}{c}{ $0.99\\pm 0.04$ } \\nl\n$\\tau$ & $0.11<0.30$ & $0.15^{+0.06}_{-0.08}$ & $0.13<0.30$ & \n$0.14^{+0.07}_{-0.06}$\\nl\n$\\chi^2$\/d.o.f & \\multicolumn{2}{c}{860.1\/860} &\n\\multicolumn{2}{c}{790.7\/845 } \\nl \n\\tableline \n\\multicolumn{5}{l}{Warm Absorber Model: } \\nl\n$N_H (\\times 10^{21}\\rm cm^{-2})$ & \\multicolumn{2}{c}{$0.33^{+0.16}_{-0.15}$} &\n\\multicolumn{2}{c}{$0.30^{+0.17}_{-0.10}$} \\nl\nIndex & $1.66^{+0.07}_{-0.06}$ & $2.12^{+0.05}_{-0.04}$ &\n$1.67^{+0.07}_{-0.06}$ & $2.14^{+0.05}_{-0.04}$ \\nl\nSIS PL norm\\tablenotemark{a}\n & $3.7^{+0.2}_{-0.3}$ & $9.8 \\pm 0.6$ & $3.6^{+0.3}_{-0.2}$ &\n$10.0 \\pm 0.6$ \\nl \nkT (eV)  & \\multicolumn{2}{c}{$117^{+13}_{-14}$} & \n\\multicolumn{2}{c}{$121^{+12}_{-14}$} \\nl\nSIS BB norm\\tablenotemark{b}\n & $1.34^{+0.43}_{-0.38}$ & $0.42<0.97$ & $1.20^{+0.49}_{-0.28}$ &\n$0.02<0.52$ \\nl\n$log(N_w)$\\tablenotemark{c} & \\multicolumn{2}{c}{$21.68^{+0.04}_{-0.08}$} \n& \\multicolumn{2}{c}{$21.69 \\pm 0.08$}\n\\nl\n$log(U)$ & $-0.85^{+0.16}_{-0.15}$ & $-0.42 \\pm 0.08$  &\n$-0.87^{+0.16}_{-0.15}$ & $-0.42^{+0.07}_{-0.09}$ \\nl\n$\\chi^2$\/d.o.f & \\multicolumn{2}{c}{867.1\/863} & \\multicolumn{2}{c}{801.8\/848} \\nl\n\\tablecomments{The value ``Gal'' for the absorption refers to the spectral\nfit lower limit set to the Galactic value, $1.77 \\times 10^{20} \\rm \\,\ncm^{-2}$ (Elvis, Wilkes \\& Lockman \\markcite{6} 1989).}\n\\tablenotetext{a} {$\\times 10^{-3} \\rm\nphotons\\,keV^{-1}cm^{-2}s^{-1}$}\n\\tablenotetext{b} {$\\times 10^{-4} L_{39}\/{D_{10}^2}$, where $L_{39}$\n is the source\nluminosity in $10^{39}\\rm erg\\,s^{-1}$ and $D_{10}$ is the distance to\nthe source in $10^{10} \\rm kpc$.}\n\\tablenotetext{c}{log of the ionized column in units of $ \\rm\ncm^{-2}$}\n\\enddata\n\\end{deluxetable}\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\nThe coupling between density fluctuations of different wavelengths is one of the\nmost important topics in the study of large-scale structure \\cite{Bernardeau:2001qr}.\nThese couplings can be imprinted in the inflationary initial conditions or develop \nthrough  gravitational evolution.  In the latter class, a long-wavelength density mode\naffects the evolution of all short-wavelength modes that are embedded in it leading\nto changes in the power spectrum \\cite{Takada:2013bfn,Li:2014jra,Chiang:2014oga,Chiang:2015eza}\nand the dark matter halo abundance which gives rise to halo bias \\cite{Mo:1995cs,Seljak:2012tp}.\nIn the limit of a large separation in these scales, one can use the ``separate universe''\n(SU) approach to describe these and other effects through a change in the background\ndensity within which small scale structure evolves \n\\cite{1993MNRAS.262..717B,Baldauf:2011bh,Sherwin:2012nh}.\n\n\nThe SU approach is not only conceptually straightforward to understand but can also\nbe readily implemented in cosmological simulations, arbitrarily deep into the nonlinear\nregime where perturbation theory breaks down \\cite{Sirko:2005uz,Gnedin:2011kj,Li:2014sga,Wagner:2014aka}.\nIn particular, SU simulations have enabled studies of the squeezed-limit $n$-point\ncorrelation functions \\cite{Wagner:2015gva} and their impact on the power spectrum\ncovariance \\cite{Li:2014sga}, the halo bias \\cite{Li:2015jsz,Lazeyras:2015lgp,Baldauf:2015vio},\nand the Lyman-$\\alpha$ forest \\cite{McDonald:2001fe,Cieplak:2015kra}. Since the whole\ntime evolution of the long-wavelength mode is properly captured, as opposed to just\na single epoch such as the time of observation, temporally nonlocal effects on small-scale\nobservables such as the nonlinear power spectrum \\cite{Ma:2006zk} and halo bias\n\\cite{Senatore:2014eva,LoVerde:2014pxa} are correctly modeled.\n\n\nPrevious studies have focused on SU simulations in the $\\Lambda$CDM cosmology,\nwhere only matter clusters at low redshift. If the system contains additional\nclustering components such as dynamic dark energy or massive neutrinos, then\none has to be careful when applying the separate universe principle. Specifically,\nthe separate universe construction is only strictly true if long-wavelength\nperturbations evolve under gravitational forces alone and not internal stress\ngradients \\cite{Dai:2015jaa,Hu:2016ssz}. This means that a SU description would\nseem to require that the long-wavelength mode be larger than the Jeans or free\nstreaming scale of the system. On the other hand, if the impact on small-scale\nstructure of these extra components is only gravitational, then it can be correctly\nmodeled by matching the local expansion rate to an SU Hubble rate in a ``fake''\nSU approach which implicitly requires fictitious energy density components \\cite{Hu:2016ssz}.\n\n\nIn this work, we implement and test this multi-component SU method in simulations\nwith quintessence dark energy. In particular, the growth of long-wavelength matter\nfluctuations above or below the Jeans scale of quintessence differs due to clustering\nof the dark energy. As a result, the SU expansion history depends on the scale of\nthe long-wavelength matter fluctuation as does the response of small-scale observables\nsuch as the power spectrum and halo mass function. The latter implies that halo bias\nitself will become scale dependent.\n\n\nThe rest of the paper is organized as follows.\nIn \\refsec{theory}, we describe the mapping of perturbations in the quintessence model\nonto the SU background above and below the Jeans scale.\nIn \\refsec{sims}, we implement  the SU approach in quintessence simulations.\nWe present the results of SU simulations in \\refsec{pk} and \\refsec{bias} for \nthe power spectrum response and the  halo bias respectively.\nWe discuss these results in \\refsec{discussion}.\nIn \\refapp{bias_model}, we compare our results to the predictions of scale-dependent halo bias models\nin the recent literature.\n\n\nThroughout the paper, we  adopt a spatially flat cosmology with a Hubble constant\n$h=0.7$, matter density $\\Omega_m=0.3$, quintessence energy density $\\Omega_Q=0.7$,\nquintessence equation of state $w_Q=-0.5$, and an initial curvature power spectrum\nwith scale-invariant tilt $n_s=1$ and amplitude which sets $\\sigma_8=1$ today.\nThese parameters are chosen to highlight the scale dependence of quintessence\nrather than for observational viability.\n\n\n\\section{Quintessential Separate Universe}\n\\label{sec:theory}\n\n\nFollowing Ref.~\\cite{Hu:2016ssz}, we review here the construction of the separate universe\nfor the case where components other than the cold dark matter possess Jeans scales. In\n\\refsec{expansion}, we show that the influence of these components is captured by a modified\nexpansion history that is defined by the growth history of the large-scale matter density\nfluctuation. We apply this construction to quintessence dark energy models in \\refsec{quintessence}.\n\n\\subsection{Expansion History}\n\\label{sec:expansion}\n\nA observer sitting within a long-wavelength matter fluctuation $\\delta_m$\nwould measure the {\\it local} mean matter density as\n\\begin{equation}\n \\bar{\\rho}_{mW}(a)=\\bar{\\rho}_m(a)[1+\\delta_m(a)] \\,,\n \\label{eq:rhoW}\n\\end{equation}\nwhere $W$ denotes a windowed average across a scale much smaller than the\nlong-wavelength mode. In the SU picture, the local mean evolves as if the\nobserver were in a SU whose scale factor \n\\begin{equation}\n a_W=\\frac{a}{(1+\\delta_m)^{1\/3}}\\approx a\\left(1-\\frac{\\delta_m}{3}\\right) \\,,\n \\label{eq:aW}\n\\end{equation}\nso that $\\bar{\\rho}_{mW} \\propto a_W^{-3}$.\nNote that at early times\n\\begin{equation}\n \\lim_{t\\to 0}\\delta_m\\to 0 \\,, ~~ \\lim_{t\\to 0}a_W\\to a \\,,\n\\end{equation}\nand the physical conditions of the local and global cosmology coincide. We\nhave implicitly assumed that there is a universal time coordinate between\nthe two and so in the relativistic limit $\\delta_m$ is specifically the\nsynchronous gauge density perturbation \\cite{Hu:2016ssz}.\n\n\nNotice that the SU construction requires only $\\delta_m(a)$ itself, not the\nevolution of any other density component in the universe. The other components \ndetermine the evolution of $\\delta_m(a)$, but they do not enter into $a_W$\nexplicitly. If these components only influence small-scale observables through\ntheir impact on $\\delta_m(a)$, their effects can be characterized by $a_W$\nand the {\\it local} Hubble expansion\n\\begin{equation}\n H_W=\\frac{\\dot{a}_W}{a_W}=H-\\frac13\\dot{\\delta}_m=H\\left(1-\\frac13\\delta'_m\\right) \\,,\n\\label{eq:H_W}\n\\end{equation}\nwhere $'\\equiv d\/d\\ln a$. This expansion history does not even need to be\ngiven by a SU Friedmann equation involving the local energy densities and\ncurvature \\cite{Gnedin:2011kj,Hu:2016ssz}. With $a_W$ and $H_W$ alone, we\ncan model the small-scale observables using $N$-body simulations with this\nSU expansion rate.\n\n\nThis construction includes cases where the other components experience\nnon-gravitational forces which define their Jeans scales. In these cases,\nthe growth history of $\\delta_m(a)$ depends on scale. Since the SU expansion\nhistory depends on the whole growth history, regions of different sizes that\nshare a common $\\delta_m$ at a fixed $a$ will produce different responses\nin the small-scale observables. In other words, these observables cannot be\ndescribed solely by the change in the local density at the time of observation\nalone. For example, as we shall see in \\refsec{bias}, the response of the\ndark matter halo abundance to $\\delta_m(a)$ leads to a halo bias that violates\nthe local bias expectation of scale independence in the linear regime.\n\n\n\\subsection{Quintessence}\n\\label{sec:quintessence}\n\n\nQuintessence or scalar field dark energy models provide a simple arena to\nexplore the response of small-scale observables to long-wavelength fluctuations,\nin particular their amplitude, scale, and growth history. The construction\nof the SU with quintessence perturbations has been extensively discussed in\nRef.~\\cite{Hu:2016ssz}. Here we only summarize the results that are related\nto simulating observable responses above and below the quintessence Jeans scale.\n\n\nThe sound speed of quintessence $c_Q$ sets the sound horizon or Jeans scale\n$r_J \\sim c_Q \/a H$ across which its influence on the evolution of $\\delta_m$\ndiffers. If $\\delta_m$ has a wavelength smaller than $r_J$, the quintessence\nperturbation is Jeans stable and becomes negligible in comparison. Thus the\nmatter fluctuations evolve under\n\\begin{equation}\n \\del{\\downarrow}''+\\left(2+\\frac{H'}{H}\\right)\\del{\\downarrow}'\n =\\frac32\\frac{H_0^2}{H^2}\\frac{\\Omega_m}{a^3}\\del{\\downarrow}\\,,\n\\label{eq:dm_subJ}\n\\end{equation}\nwhere $\\delta_m=\\del{\\downarrow}$ and the down arrow in the subscript denotes\nthe sub-Jeans case. On the other hand, if $\\delta_m$ has a wavelength larger\nthan $r_J$, then the quintessence perturbation $\\delta_Q$ has an impact on\n$\\delta_m$. Assuming that all fluctuations arise from initial curvature\nfluctuations and the sound speed of quintessence is much smaller than speed\nof light, we have for the two-component system \\cite{Hu:2016ssz}\n\\ba\n \\delta'_Q-3w_Q\\delta_Q\\:&=(1+w_Q)\\del{\\uparrow}' \\,, \\nonumber\\\\\n \\del{\\uparrow}'' +\\left(2+\\frac{H'}{H}\\right)\\del{\\uparrow}'\\:&=\n \\frac32\\frac{H_0^2}{H^2}\\left[\\frac{\\Omega_m \\del{\\uparrow}}{a^3}\n +\\frac{\\Omega_Q \\delta_Q}{a^{3(1+w_Q)}}\\right] \\,,\n\\label{eq:dm_supJ}\n\\ea\nwhere $\\delta_m =\\del{\\uparrow}$ and the up arrow  in the subscript denotes\nthe super-Jeans case. For simplicity we have also taken the quintessence\nequation of state parameter $w_Q=\\bar p_Q\/\\bar \\rho_Q$ to be a constant.\nWith the assumed curvature initial conditions, the initial conditions for\nthe fields are set by taking $\\delta_m=\\del{\\uparrow}=\\del{\\downarrow}$\nand $\\delta_Q$ are all proportional to $a$ in the matter dominated limit.\n\n\nIn \\refFig{dm} we plot $\\delta_m\/\\delta_{m0}$ as a function of the global\nscale factor, where $\\delta_{m0}=\\delta_m(a=1)$ is the present-day overdensity.\nThe red solid and blue dashed lines show the sub-Jeans and super-Jeans SUs.\nNormalized to the same $\\delta_{m0}$, the super-Jeans SU is always closer\nto the global universe ($\\delta_m=0$) in the past than the sub-Jeans SU in\nits expansion history. This implies that the response of the small-scale\nobservables such as the power spectrum and halo abundance should be smaller\nin the super-Jeans than the sub-Jeans SU.\n \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dm.pdf}\n\\caption{(Top) Scale-dependent growth in $\\delta_m$ as a function of the scale\nfactor for super-Jeans (red solid) and sub-Jeans (blue dashed) long-wavelength\nmodes. (Bottom) The ratio of $\\delta_m$ in super-Jeans to sub-Jeans cases. When\ncharacterized as a separate universe, the former is closer to the global universe\nthan the latter in the past for the same density fluctuation today.}\n\\label{fig:dm}\n\\end{figure}\n\n\nFinally for setting up simulations in the next section it is useful to define\nthe linear growth of short-wavelength structure in the SU. If the small-scale\nmatter fluctuations of interest are well within $r_J$, the growth function $D_W$\nis simply \\refeq{dm_subJ} with the SU expansion history\n\\begin{equation}\n \\frac{d^2D_W}{d\\ln a_W^2}+\\left(2+\\frac{d\\ln H_W}{d\\ln a_W}\\right)\\frac{dD_W}{d\\ln a_W}\n =\\frac32\\frac{H_{0W}^2}{H_W^2}\\frac{\\Omega_{mW}}{a_W^3}D_W \\,.\n\\label{eq:DW}\n\\end{equation}\nNote that \n\\begin{equation}\n \\Omega_{mW} H_{0W}^2 = \\Omega_m H_0^2 \\,,\n\\end{equation}\nand so only the SU expansion rate $H_W$ from \\refeq{H_W} is required to solve\nfor $D_W(a_W)$. As we shall see next, we can generalize this statement to nonlinear\nobservables with SU $N$-body simulations.\n\n\n\\section{Separate Universe Simulations}\n\\label{sec:sims}\n\n\nThe growth history of the long-wavelength matter fluctuation $\\delta_m(a)$ in\nthe global universe alone sets the expansion history in the separate universe.\nAll effects from long-wavelength fluctuations in other species are incorporated\ninto its growth history. In the quintessence model, within the Jeans scale dark\nenergy perturbations can be ignored and so the response of small scale observables\ncan be calibrated using $N$-body simulations with just this change in the expansion\nhistory.\n   \n\nUnlike the SU technique in $\\Lambda$CDM, the change of cosmological parameters \nand their correspondence with real energy densities and curvature becomes non-trivial\nfor the quintessence model \\cite{Hu:2016ssz} whereas the direct change in the\nexpansion rate $H_W$ remains simply determined by $\\delta_m(a)$. Thus, while\nsome steps are similar to the $\\Lambda$CDM SU techniques for running and analyzing\nsimulations (see e.g. \\cite{Li:2014sga,Wagner:2014aka,Li:2015jsz,Lazeyras:2015lgp}),\nthere are some major differences in performing the SU simulations with quintessence\nwhich we now describe.\n\n\nLet us start with setting the initial conditions for the simulations. Recall that \nat high redshift the separate and global universes are identical in their physical\ndescription. To achieve this, we first compute the linear power spectrum with the\nglobal cosmology at $z=0$ using CAMB \\cite{Lewis:1999bs,Howlett:2012mh}. We then\nrescale this power spectrum to the initial redshift of the simulations $a_{Wi}=0.02$ as\n\\begin{equation}\n P_W(k,a_{Wi})=P(k,a_0)\\left[\\frac{D_W(a_{Wi})}{D(a_0)}\\right]^2 \\,,\n\\end{equation}\nwhere $D$ is the linear growth in the global universe and $D_W$ is the linear\ngrowth of the SU following \\refEq{DW}. The growth functions are normalized in\nthe matter dominated epoch as\n\\begin{equation}\n \\lim_{a \\rightarrow 0} D(a)=a, \\quad \\lim_{a_W \\rightarrow 0} D(a_W) = a_W \\,.\n\\label{eq:D_ic}\n\\end{equation}\nNote that $D_W$ in sub-Jeans and super-Jeans SUs are different, as they have different\nexpansion histories. Another subtlety is that the change in the expansion rate of the\nSU makes the traditional unit of comoving $[h\\,{\\rm Mpc}^{-1}]$ inconvenient. Throughout this paper\nwe avoid this confusion by using units of comoving $[{\\rm Mpc}]$ and convert for code\npurposes as necessary. Given the different scale factors $a$ and $a_W$, the correspondence\nbetween comoving wavenumber and physical wavenumber in the global universe differ. Since\nthis represents a simple dilation of scales, we can account for it in the interpretation\nof observable responses rather than in the simulations directly \\cite{Li:2014sga}.\n \n\nThe initial conditions are then set up using realizations of Gaussian random\nfields for the primordial fluctuations and evolved to $a_{Wi}$ using second-order\nLagrangian perturbation theory (2LPT) \\cite{Crocce:2006ve}. Usual 2LPT codes,\nsuch as the publicly available 2LPTIC \\cite{2lptic}, compute the linear growth and\ngrowth rate $f_W=d\\ln D_W\/d\\ln a_W$ at $a_{Wi}$ from the cosmological parameters.\nWe modify the pipeline such that $D_W$ and $f_W$ from the numerical solution of\n\\refEq{DW} determine the initial positions and velocities of the particles.\n\n\nWe use Gadget-2 \\cite{Springel:2005mi} to carry out the simulations. Standard\nGadget-2 computes the Hubble expansion as a function of the scale factor using\nthe input cosmological parameters. Instead of finding the corresponding cosmological\nparameters, we first compute $H_W$ as a function of $a_W$ with \\refEq{H_W} and\n\\refEqs{dm_subJ}{dm_supJ}, pass the table $(a_W,H_W)$ to the code, and then\ninterpolate the value of $H_W(a_W)$ when necessary\\footnote{Specifically, we\nonly need to modify \\texttt{driftfac.c} and \\texttt{timestep.c}. Also since\nGadget-2 checks the consistency of the input parameters, we provide the SU\n$\\Omega_{mW}$ as well as $h_W$, and $L_W$ where $L_W$ is the box size of the\nsimulations}. We have verified that the SU results are in excellent agreement\nwith those of the standard 2LPTIC Gadget-2 pipeline in $\\Lambda$CDM where the\nSU is implemented by varying cosmological parameters.\n\n\nFollowing the procedures in Ref.~\\cite{Lazeyras:2015lgp}, we identify halos with\nthe Amiga Halo Finder \\cite{Knollmann:2009pb,Gill:2004km}, which is based on the\nspherical overdensity algorithm. The key quantity of the spherical overdensity\nalgorithm is the density threshold, and we set it to be $\\Delta=200$ in the global\nuniverse. To match halos identified in the global cosmology, the threshold relative\nto the mean in the SU needs to be rescaled as \\cite{Li:2015jsz,Lazeyras:2015lgp}\n\\begin{equation}\n \\Delta_W=\\frac{\\Delta}{1+\\delta_m(t)}\\approx\\Delta[1-\\delta_m(t)] \\,.\n\\end{equation}\nIn other words, in the overdense (underdense) universe the threshold becomes smaller\n(larger) due to the background fluctuations. From each simulation we obtain one halo\ncatalog, and we consider only halos with more than 400 particles. We also neglect\nsub-halos for simplicity.\n\n\nIn this paper, we perform both the sub-Jeans and super-Jeans SU simulations with\n$\\del{\\uparrow\\downarrow 0}=\\del{\\uparrow\\downarrow}(a=1)=\\pm0.01$, totaling 4\nsimulations per set. For each of the 20 sets, we fix their initial phases so that\nwhen we take the difference of the observables between overdense and underdense\nSU simulations a large amount of noise due to sample variance is removed. We also\nrun 40 simulations of the global $\\delta_m=0$ universe in order to characterize\nthe clustering bias for comparison in \\refsec{clusteringbias}. The first 20 have\nthe same initial phases as their SU counterparts. For each of these sets we take\na comoving box size $L=1000\\,$Mpc and number of particles $N_p=1024^3$, denoted\nas small-box.\n\n\nWe also run 20 simulations with $L=2800$ Mpc and $N_p=1024^3$ particles in the global\nuniverse, denoted as big-box simulations. These big-box simulations are used to measure\nthe position-dependent power spectrum \\cite{Chiang:2014oga} for comparison with the\npower spectrum response of the sub-Jeans simulations. The details of the simulations\nare summarized in \\reftab{sims}.\n\n\n\\begin{table}[h]\n \\begin{tabular}{c c c c c c c}\n  \\hline\n  type & SU & $L$ [Mpc] & $N_p$ & $\\del{m0}$ & $N_{\\rm sets}$\\\\\n  \\hline\n  small-box & $\\uparrow\\downarrow$  &1000 & $1024^3$ & $\\pm0.01$ & 20 \\\\\n  small-box &  no &1000 & $1024^3$ & 0 & 40 \\\\\n  big-box &  no&  2800 & $1024^3$ & 0 & 20 \\\\\n  \\hline\n \\end{tabular}\n \\caption{Summary of the simulations.}\n\\label{tab:sims}\n\\end{table}\n\n\n\\section{Power Spectrum Response}\n\\label{sec:pk}\n\n\nIn this section, we calibrate the responses in the locally measured power spectrum\nto a long-wavelength mode above and below the Jeans scale. In \\refsec{resp} we extract\nthese responses from the SU simulations and show that they are scale dependent and\nsmaller for modes above the Jeans scale than below. We test these responses against\npredictions from perturbation theory in \\refsec{pt} and the local, position-dependent,\npower spectrum from the big-box simulations with long-wavelength sub-Jeans scale modes\nin \\refsec{ibn}. The good agreement implies that the SU simulation technique provides\naccurate predictions for these small scale observables without the need for direct\nsimulations of quintessence clustering.\n\n\n\\subsection{Separate Universe Calibration}\n\\label{sec:resp}\n\n\nIn the presence of a long-wavelength density fluctuation $\\delta_m$, the power spectrum\nobserved locally will differ from the global average. We can characterize the fractional\nchange in the local power spectrum  as a ``response'' $R_{\\rm tot}$ to  $\\delta_m$\n\\begin{equation}\n \\frac{\\Delta P}{P} \\approx \\frac{d \\ln P}{d\\delta_m} \\delta_m \\equiv R_{\\rm tot} \\delta_m \\,.\n\\end{equation}\nSince to the leading order $R_{\\rm tot}$ is independent of $\\delta_m$, it can be calibrated\nusing the SU simulations once and for all rather than with simulations that follow the\ndynamics of individual long-wavelength modes. This is especially advantageous for quintessence,\nwhere super-Jeans modes require simulations with quintessence clustering.\n\n\nThis effect can be observed in a local sample of our universe by dividing it into subvolumes\nand measuring the correlation between the local power spectra and the subvolume mean overdensities,\nwhich is known as the position-dependent power spectrum \\cite{Chiang:2014oga,Chiang:2015eza}.\nEven if only the undivided volume is employed, the coherent change in the local power spectrum\n$\\Delta P(k)$ due to wavelengths larger than the sample induces a ``super-sample'' covariance\nbetween measurements of different $k$ modes \\cite{Takada:2013bfn,Li:2014sga,Li:2014jra}.\n\n\nIn practice, the calibration of the total response with SU simulations involves three pieces:\ngrowth, dilation, and reference-density \\cite{Li:2014sga}\n\\begin{equation}\n R_{\\rm tot}  = R_{\\rm growth} + R_{\\rm dilation} + R_{\\bar \\rho} \\,.\n\\end{equation}\n$R_{\\rm growth}$ describes the change in the growth of a small-scale density fluctuation\nat a fixed comoving $k$ in the separate and global universe relative to their own scale\nfactors. $R_{\\rm dilation}$ changes the scale to a fixed wavenumber in the global universe\nor physical wavenumber in each. Finally $R_{\\bar \\rho}$ accounts for the different mean\ndensity of the two universes in the definition of the density fluctuation.\n\n\nTo measure the growth response from SU simulations, we first distribute the dark matter\nparticles onto a $1024^3$ grid by the cloud-in-cell (CIC) density assignment scheme to construct\nthe density fluctuation, and Fourier transform the density fluctuations with FFTW \\cite{fftw}\nto form the power spectrum. For each set of super $\\uparrow$ or sub $\\downarrow$ Jeans\nscale SU simulations with the same initial phases, we estimate the growth response,\n\\begin{equation}\n R_{\\rm growth,\\uparrow\\downarrow} \\equiv \n R_{\\uparrow\\downarrow}\n \\equiv \\frac{d\\ln P_{\\uparrow\\downarrow}}{d\\del{\\uparrow\\downarrow}}\n\\end{equation}\nas\n\\begin{equation}\n \\hat{R}_{\\uparrow\\downarrow}(k,a)=\n \\frac{\\hat{P}_{\\uparrow\\downarrow}(k,a|{\\scriptstyle +}\\del{\\uparrow\\downarrow,0})\n -\\hat{P}_{\\uparrow\\downarrow}(k,a|{\\scriptstyle -}\\del{\\uparrow\\downarrow,0})}\n {2\\hat{P}(k,a) \\del{\\uparrow\\downarrow}(a)} \\,,\n\\end{equation}\nwhere we difference the overdense and underdense pairs for each $|\\del{\\uparrow\\downarrow,0}|$.\nWe then compute the variance of $\\hat{R}_{\\uparrow\\downarrow}$ from the 20 small-box realizations.\n\n\nThe dilation response accounts for the fact that the same comoving $k$ in the SU corresponds\nto a different physical $k$ in the global universe. Given the change in the scale factor from\n\\refeq{aW}, it is analytically related to the local slope in power spectrum as \\cite{Li:2014sga}\n\\begin{equation}\n R_{\\rm dilation}(k,a)=-\\frac{1}{3}\\frac{d\\ln k^3P(k,a)}{d\\ln k} \\,.\n\\label{eq:Rdilation}\n\\end{equation}\nTo compute the dilation response from simulations, we take the log-derivative of the\nmean power spectrum measured from 40 small-box simulations with the global cosmology.\nAs a result, the dilation response is the same in both sub-Jeans and super-Jeans SUs.\nFinally the $\\bar\\rho$ response is due to the change in the definition of a density\nfluctuation\n\\begin{equation}\n \\delta_m \\equiv \\frac{\\delta\\rho_m}{\\bar\\rho_m} = \\delta_{mW} \\frac{ \\bar\\rho_{mW}}{ \\bar\\rho_m} \\,,\n\\end{equation}\nso that to the leading order \\refeq{rhoW} implies\n\\begin{equation}\n R_{\\bar \\rho}(k,a)=2 \\,.\n\\label{eq:Rreference}\n\\end{equation}\nNote that the last two responses, dilation and reference-density, do not involve\nthe SU simulations.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.46\\textwidth]{pk_response_component_z0.pdf}\n\\caption{(Top) Different components of the separate universe power spectrum responses\nat $z=0$: growth response (sub-Jeans, red solid; super-Jeans blue dashed), absolute\nvalue or negative of the dilation response (green dot-dashed), and the reference-density\nresponse (black dotted). (Bottom) Ratios of the mean of the total response to that of\nthe sub-Jeans separate universe. Shaded bands reflect the error on the mean response\nof the simulations. The clear distinction between the sub-Jeans and super-Jeans power\nspectrum responses is the first important result of our separate universe simulations.}\n\\label{fig:resp_tot}\n\\end{figure}\n\n\nThe top panel of \\reffig{resp_tot} shows the various power spectrum responses at $z=0$,\nand the bottom panel shows the ratios of the total power spectrum response, $R_{\\rm tot}$,\nto that of the sub-Jeans SU, $R_{{\\rm tot},\\downarrow}$. We find that the response is\nroughly 2\\% smaller in super-Jeans than in sub-Jeans SUs, and the distinction is statistically\nsignificant. Note also the small errors ($\\sim0.1\\%$ at low-$k$ and $\\sim 0.3\\%$ at high-$k$)\nestimated from small-box SU simulations, demonstrating the power of the SU technique to\nprecisely characterize the response down to arbitrarily small scales.\n\n\nThe fact that the growth response is smaller in super-Jeans than in sub-Jeans SUs can be\nunderstood qualitatively from \\reffig{dm}. Normalized to a given observation redshift,\n$\\delta_m$ in the super-Jeans limit is always smaller in the past than that in the sub-Jeans\nlimit. Consequently, the super-Jeans SU is closer to the global universe along the growth\nhistory, and so the response is smaller.\n\n\nThis difference between super-Jeans and sub-Jeans scales produces an observable change\nin the local power spectrum, and so can in principle be used as a new probe of the sound\nspeed of quintessence. In the real universe, the small-scale power spectrum responds\nto long modes of all scales, the difference of the responses in sub-Jeans and super-Jeans\nlimit would thus appear as the scale-dependent squeezed-limit bispectrum for a fixed\nsmall-scale mode. However in quintessence models with initial curvature perturbations,\nthe predicted amplitude for adiabatic quintessence fluctuations is proportional to\n$(1+w_Q)$ (see Eq.~(95) of Ref. \\cite{Hu:2016ssz}) and so goes to zero as $w_Q \\rightarrow -1$.\nMore generally, this growth history dependence demonstrates that the nonlinear matter power\nspectrum cannot simply be a functional of the linear power spectrum at the same epoch as\nis commonly assumed in simple halo model and nonlinear fitting procedures (see also \\cite{Ma:2006zk}).\n\n\n\\subsection{Perturbation Theory}\n\\label{sec:pt}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnDW.pdf}\n\\caption{The response of the separate universe growth function as a function of the global\nscale factor $a$ in the global universe for super-Jeans (red solid) and sub-Jeans (blue\ndashed) cases. Above the Jeans scale, the response to the same $\\delta_m$ at $a$ is smaller\nthan below.}\n\\label{fig:dlnDW}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnpk_linear.pdf}\n\\caption{Linear perturbation theory predictions for the growth responses compared with\nthe measurements from the 20 small-box separate universe simulations at $z=3$ (top)\nand 0 (bottom). The red solid and blue dashed lines with shaded areas show the sub-Jeans\nand super-Jeans separate universes measurements as in Fig.~\\ref{fig:resp_tot}, whereas\nthe red dot-dashed and blue long-dashed lines show the corresponding linear perturbation\ntheory predictions, i.e.~\\refeqs{Rgrowth}{linear}. Note that the range of $y$-axes is\nsmaller in the top than in the bottom panel. Also the cusp feature at $k\\sim0.078~{\\rm Mpc}^{-1}$\nis a visual artifact due to binning, which we choose to be $\\Delta k= 2\\pi\/L$.}\n\\label{fig:dlnpk_linear}\n\\end{figure}\n\n\nTo better understand the growth responses quantitatively, we compute them in perturbation\ntheory and check their agreement with the SU simulations at various redshifts. In perturbation\ntheory, the effect can be modeled through the SU linear growth function $D_W$ as\n\\begin{equation}\n R_{\\rm growth}(k,a)=\\frac{d\\ln P(k,a)}{d\\ln D_W(a)}\\frac{d\\ln D_W(a)}{d\\delta_m(a)} \\,.\n\\label{eq:Rgrowth}\n\\end{equation}\nIn the linear regime $P(k,a) \\approx P_{\\rm lin}(k,a) \\propto D_W^2(a)$ and so \n\\begin{equation}\n \\frac{d\\ln P(k,a)}{d\\ln D_W(a)}\\approx 2 \\,.\n \\label{eq:linear}\n\\end{equation}\nTo determine the response of $D_W$ in \\refEq{Rgrowth}, we solve \\refEq{DW} with the initial\ncondition \\refEq{D_ic}, and the result is shown in \\reffig{dlnDW}. It approaches the matter\ndominated ($\\Omega_m=1$) limit $13\/21$ at high redshift for both super-Jeans and sub-Jeans\nscale responses, and at low redshift is smaller in the super-Jeans case as expected from\n\\reffig{dm}. In \\reffig{dlnpk_linear}, we compare the measured power spectrum response in\nsub-Jeans and super-Jeans SUs to the corresponding linear perturbation theory predictions,\ni.e.~\\refeqs{Rgrowth}{linear}. We find that in both cases the linear perturbation theory\nagrees with the measured responses in the linear regime, i.e., at sufficiently low $k$ or\nhigh $z$.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dlnpk_1loop.pdf}\n\\caption{Same as \\reffig{dlnpk_linear}, but for the 1-loop predictions,\ni.e. \\refeq{Rgrowth} and \\refeq{1loop}.}\n\\label{fig:dlnpk_1loop}\n\\end{figure}\n\n\nHowever, as we move to lower redshift as well as higher $k$, the measured responses\nbecome nonlinear and perturbation theory predictions deviate from SU simulation\nmeasurements. Unlike perturbation theory, the SU response calibration is not limited\nto large scales. On the other hand, we can understand the onset of nonlinearity in\nthe simulations through higher order perturbation theory. The 1-loop power spectrum\nfrom standard perturbation theory is given by (see e.g. Ref.~\\cite{Jeong:2006xd})\n\\begin{equation}\n P_{\\rm 1-loop}(k,a)=P_{\\rm lin}(k,a)+P_{22}(k,a)+2P_{13}(k,a) \\,,\n\\end{equation}\nwhere the nonlinear corrections $P_{22}$ and $P_{13}$ are proportional to $D_W^4$\nif $\\Omega_{mW}(a_W)\/f_{W}^2(a_W)\\approx1$.\nTherefore,\n\\begin{equation}\n \\frac{d\\ln P_{\\rm 1-loop}(k,a)}{d\\ln D_W(a)}\n =2\\left[1+\\frac{P_{22}(k,a)+2P_{13}(k,a)}{P_{\\rm 1-loop}(k,a)}\\right] \\,,\n \\label{eq:1loop}\n\\end{equation}\nwhich now is a function of $k$. Note that in the global cosmology $\\Omega_m\/f^2=1.034$\nat $z=3$ and 1.203 at $z=0$, and the standard perturbation theory should work better\nat $z=3$ than at $z=0$. Since the long-wavelength perturbation we consider is small\n($|\\delta_{\\uparrow\\downarrow0}|=0.01$), the standard perturbation theory should work\nas well in both the sub-Jeans and super-Jeans SUs as the global universe.\n\nIn \\reffig{dlnpk_1loop} we plot the 1-loop predictions in sub-Jeans (red dot-dashed)\nand super-Jeans (blue long-dashed). We find that the 1-loop predictions extend the\nagreement with the $N$-body measurement to smaller scales compared to the linear\npredictions. More precisely, the difference between the 1-loop model and the measurement\nat $z=3$ ($z=0$) is 1\\% (3\\%) at $k\\sim0.1~{\\rm Mpc}^{-1}$ and 4\\% (6\\%) at $k\\sim0.2~{\\rm Mpc}^{-1}$.\nAt even smaller scale or lower redshift, the nonlinearity is too large to be modeled\nby the 1-loop perturbation theory. The SU simulation calibration technique itself is\nnot limited in wavenumber and our $N$-body implementation is instead only limited by\nresolution as well as the lack of baryonic and astrophysical modeling in the deeply\nnonlinear regime.\n\n\n\\subsection{Position-Dependent Power Spectrum}\n\\label{sec:ibn}\nThe power spectrum response can also be tested in simulations and observed in surveys \nthrough the position-dependent power spectrum. As a simulation based test, it also serves\nto check the SU calibration of the power spectrum response deep into the nonlinear regime.\n\n\nSpecifically we compare the response measured from the small-box SU simulations to the\nsqueezed-limit position-dependent power spectrum measured from the big-box simulations\nwith the global cosmology (without a uniform long-wavelength density fluctuation). In\nthe latter, we assume that the dark energy does not cluster with matter and so its\nposition-dependent power spectrum should match the sub-Jeans SU prediction.  \n\n\nThe procedure of measuring the position-dependent power spectrum is explained in detail\nin \\cite{Chiang:2014oga}. In short, we first distribute the dark matter particles onto\na $2048^3$ grid by the CIC density assignment scheme to construct the density fluctuation.\nWe next divide the big-box simulations in each dimension by 8, so there are  $N_s=512$\nsubvolumes in total  with  comoving side length of $L=350$ Mpc. In each subvolume centered at\n${\\mathbf r}_L$, we measure the local power spectrum as $\\hat{P}(k,a|{\\mathbf r}_L)$\nand the mean overdensity (with respect to the entire box) as $\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)$\nand construct \n\\begin{equation}\n \\frac{\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{P}(k,a|{\\mathbf r}_L)\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)}\n {\\left[\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{P}(k,a|{\\mathbf r}_L)\\right]\n \\left[\\frac{1}{N_s}\\sum_{{\\mathbf r}_L}\\hat{\\bar{\\delta}}_m^2({\\mathbf r}_L)\\right]} \\,,\n\\end{equation}\nwhere the summation is over the 512 subvolumes in one big-box realization. The correlation\nbetween $\\hat{P}(k,a|{\\mathbf r}_L)$ and $\\hat{\\bar{\\delta}}_m({\\mathbf r}_L)$ quantifies\nthe integrated bispectrum, and in the squeezed limit where $k\\ll1\/L$ the integrated bispectrum\ncan be understood as the {\\it total} response of the power spectrum to the long-wavelength\noverdensity.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{ibn_z0.pdf}\n\\caption{Comparison of the squeezed-limit position-dependent power spectrum of big-box,\nglobal simulations (green dot-dashed), and the total power spectrum response of sub-Jeans\n(red solid) as well as super-Jeans (blue dashed) separate universe simulations at $z=0$.\nThe shaded areas show the error on the mean. This figure summarizes our main power spectrum\nresponse results: the position-dependent power spectrum and the sub-Jeans power spectrum\nresponse agree significantly better than the difference in response across the Jeans scale\nconfirming the scale dependence in this observable deep into the nonlinear regime.}\n\\label{fig:ibn}\n\\end{figure}\n\n\\refFig{ibn} shows the comparison at $z=0$ between the total power spectrum response from\nthe previous section (red solid for sub-Jeans and blue dashed for super-Jeans cases) and\nthe position-dependent power spectrum  (green dot-dashed) averaged over the 20 realizations\nwith its error. To reach the squeezed limit, we require $k\\gtrsim100\/L\\sim0.3~{\\rm Mpc}^{-1}$,\nand in this regime the agreement with the SU response is better than a percent. This agreement\nis significantly better than the difference between the super-Jeans and sub-Jeans power spectrum\nresponses and thus verifies the SU calibration technique. With the SU technique tested into\nthe nonlinear regime, we can apply these results to the super-Jeans case of the position-dependent\npower spectrum without the need for costly simulations that include dark energy clustering.\n\n\n\\section{Scale-Dependent Halo Bias}\n\\label{sec:bias}\nThe SU simulations also calibrate the response of the halo mass function to a long-wavelength \nmode and hence the bias of the halo number density due to that mode. In \\refsec{responsebias},\nwe review the technique for measuring halo bias from SU simulations and show that in the\nquintessence model it acquires a scale dependence at the Jeans scale. In \\refsec{clusteringbias},\nwe test this response bias in the SU simulations against the clustering bias extracted from\n40 small-box global simulations. We discuss the implications of scale dependence for the\ntemporal nonlocality of halo bias and the observability of features in the halo power\nspectrum in \\refsec{interpretation}.\n\n\n\\subsection{Response Bias}\n\\label{sec:responsebias}\n\nThe linear density bias $b_1(M)$ of halos of mass $M$ can be defined as the response\nof the differential halo abundance $n_{\\ln M}=dn\/d\\ln M$ to the long-wavelength mode\n\\begin{equation}\n b_1(M) \\equiv \\frac{d\\delta_h}{d\\delta_m} = \\frac{d\\ln n_\\lnM}{d\\delta_m} \\,,\n \\label{eq:biasasresponse}\n\\end{equation}\nwhich we call ``response bias''. Thus by measuring the response of the halo mass\nfunction in the SU simulations we have a direct calibration of response bias\n\\cite{Li:2015jsz,Lazeyras:2015lgp,Baldauf:2015vio}. Note that the derivative in\n\\refeq{biasasresponse} is evaluated at a fixed time, but will depend on the whole\ngrowth history of $\\delta_m(a)$. This temporal nonlocality implies that response\nbias can be scale dependent if that growth history is also scale dependent. For\nquintessence, the SU simulations allow us to calibrate the bias above and below\nthe Jeans length of quintessence without direct simulations of its clustering properties. \n\n\nAs discussed in Ref.~\\cite{Li:2015jsz}, response bias largely reflects the change\nin the masses of halos due to the same local change in growth that affects the power\nspectrum. The enhanced growth in $\\delta_m>0$ regions makes halos more massive locally\nthan in their $\\delta_m<0$ counterparts. Hence halos of a fixed mass are associated\nwith the more abundant lower peaks in the initial density field in the former and\nthe less abundant higher peaks in the latter. This also means that measuring the\nchange in abundance of halos in fixed mass bins between the SU simulation pairs is\nan inefficient way to quantify response bias. Halos with small changes in mass across\nwide mass bins would register their response only when individual halos move across\nmass bins.\n\n\nWe instead adopt abundance matching as introduced in Ref.~\\cite{Li:2015jsz}, which\nwe now summarize. By finding the mass threshold above which the cumulative abundance\nis fixed, we largely eliminate the sampling noise from the discrete nature of halos.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{s_z0.pdf}\n\\caption{Threshold mass shift as a response of varying $\\delta_m$ at fixed cumulative abundance\nat $z=0$. The solid line and shaded region show the smoothed estimate and the bootstrap error.}\n\\label{fig:s_z0}\n\\end{figure}\n\n\nSpecifically for either the sub-Jeans or the super-Jeans case, we first combine halo catalogs\nof all realizations of the same $\\delta_m$ in the small-box suite. The masses of the $i^{\\rm th}$\nmost massive halo $M_i^\\pm$ from the $\\delta_m = \\pm 0.01$ SU simulations determine the\ndiscrete threshold mass shift\n\\begin{equation}\n s_i(\\lnM_i) = \\frac{\\lnM_i^+ - \\lnM_i^-}{2|\\delta_m|} \\,,\n\\end{equation}\nwhere $M_i$ is the geometric mean of $M_i^+$ and $M_i^-$. We then use the smoothing spline\ntechnique to estimate the ensemble average threshold mass shift $\\hat s(\\ln M)$ as well as\nthe cumulative halo abundance above threshold mass $\\hat n(\\ln M)$. \\refFig{s_z0} shows the\nmass shift measured  from 20 sub-Jeans and super-Jeans SU simulations at $z=0$ as a function\nof halo mass. We find that the mass shift due to varying $\\delta_m$ is smaller above the\nJeans scale, which reflects the fact that its growth history makes it closer to the global\nuniverse than below the Jeans scale.\n\n\nThe halo mass function follows as the derivative of the cumulative mass function\n$\\hat n_\\lnM=-d\\hat n\/d\\lnM$. We can then estimate the Lagrangian halo bias above\nthreshold mass $M$ as\n\\begin{equation}\n \\hat{\\bar{b}}_1^\\Lr(M) = \\frac{\\hat n_\\lnM(\\ln M)\\,\\hat s(\\ln M)}{\\hat n(\\ln M)} \\,.\n\\label{eq:am}\n\\end{equation}\nThis quantity is the Lagrangian bias since the SU simulations are performed with the\nsame comoving rather than physical volume. The dilation of the volume from the change\nin scale factors brings the cumulative Eulerian bias to\n\\begin{equation}\n \\hat{\\bar{b}}_1(M) =1+ \\hat{\\bar{b}}_1^\\Lr(M)\\,.\n \\label{eq:eulerianb}\n\\end{equation}\nIn \\reffig{b_resp} we compare the response bias on super-Jeans (blue dashed) and sub-Jeans \n(red solid) scales as a function of halo mass at $z=0$. The bias at a fixed mass is smaller\nin the super-Jeans case. Just like for the power spectrum response, above the Jeans scale\nfor the same final $\\delta_m$, the SU is closer to global at high redshift. Thus the change\nin growth and the consequent change in halo masses and abundances is smaller. We find that\nthe mild mass dependence of the fractional difference between the super-Jeans and sub-Jeans\nresponse biases is due mainly to the dilation effect in \\refeq{eulerianb}, since that of\nthe Lagrangian bias is fairly mass independent due to the similar shapes of the mass shift\ndisplayed in \\reffig{s_z0} (see also \\reffig{bLversusmodels}).\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.46\\textwidth]{b_response_z0.pdf}\n\\caption{(Top) The $z=0$ response biases measured from 20 sub-Jeans (red solid) and\nsuper-Jeans (blue dashed) separate universe simulations. The lines and shaded areas\nshow the smoothed estimate and the bootstrap error. (Bottom) The ratios of the response\nbiases to that of the sub-Jeans response bias. The difference between the sub-Jeans\nand super-Jeans response biases, which indicates that the linear halo bias is scale\ndependent in the presence of the scale-dependent growth, is the second central result\nof our separate universe simulations.}\n\\label{fig:b_resp}\n\\end{figure}\n\n\n\\subsection{Clustering Bias}\n\\label{sec:clusteringbias}\n\nTo verify the SU calibration of halo bias through the mass function response, we can\ncompare it to how linear halo bias is commonly measured from the two-point statistics,\nwhich we call clustering bias\n\\begin{equation}\n\\bar b_1(M) = \\lim_{k\\to0}\\frac{P_{hm}(k;M)}{P_{mm}(k)} \\,,\n \\label{eq:biasascrosspower}\n\\end{equation}\nwhere $P_{hm}$ is the cumulative halo number density cross power spectrum with the\nmatter density. Where no confusion should arise, we omit the $M$ argument of the\ncumulative bias. Above the Jeans scale of quintessence, this approach would require\nsimulations of quintessence clustering even for linear halo bias. Below the Jeans\nscale, we can test the equivalence of response and clustering bias with global\nsimulations where quintessence enters only at the background level.  \n\n\nIn order to extract the $k\\rightarrow 0$ limit, we first compute\n\\begin{equation}\n \\bar{q}(k)=\\frac{{P}_{hm}(k)}{P_{mm}(k)} \\,,\n\\end{equation}\nfor each of the 40 simulations of the global cosmology for a set of mass thresholds.    \nMotivated by Ref.~\\cite{Assassi:2014fva}, we fit $\\bar{q}(k)$ to the model\n\\begin{equation}\n \\bar{b}(k)=\\bar{b}_1+\\sum_{i=1}^n \\bar{b}_{k^{2n}}k^{2n} \\,,\n\\end{equation}\nwhere we treat $\\bar{b}_{k^{2n}}$ as nuisance parameters that absorb the loop\ncorrections in the large-scale limit. We then get the best-fit bias parameters\nby minimizing\n\\begin{equation}\n \\chi^2=\\sum_{k}^{k_{\\rm max}}\\frac{[\\bar{q}(k)-\\bar{b}(k)]^2}{\\sigma^2[\\bar{q}(k)]} \\,,\n\\label{eq:chi2}\n\\end{equation}\nwhere $\\sigma^2[\\bar{q}(k)]$ is the variance of $\\bar{q}(k)$ measured from 40 global\nsmall-box simulations.\n\n\nTo ensure the robustness of the fitted clustering bias, especially as compared with the\nsmall predicted difference between sub-Jeans and super-Jeans response biases, we examine\nthe bias models with $n=0$, 1, and 2 for various $k_{\\rm max}$. We seek consistent result for\ndifferent bias models (different $n$) and $k_{\\rm max}$. The general principle is that the\nlarger the $k_{\\rm max}$, the larger the $n$ required to account for the nonlinearity and to\navoid underfitting. Conversely, for models with $n>0$ $k_{\\rm max}$ cannot be too small or the\nfit would suffer from overfitting.\n\n\nWith each bias model and  $k_{\\rm max}$, we visually inspect its goodness of fit to $\\bar{q}(k)$\nfor various threshold halo masses. We find that across two decades in halo mass\n($2\\times10^{13}-2\\times10^{15}\\,M_\\odot$), the bias models of $n=0$, 1, and 2 with the\nbiases fitted to $k_{\\rm max}=0.014-0.028\\,{\\rm Mpc}^{-1}$, $0.042-0.049\\,{\\rm Mpc}^{-1}$,\nand $0.056-0.07\\,{\\rm Mpc}^{-1}$ are in agreement with the mean $\\bar{q}(k)$, and the\nagreement even extends to $k>k_{\\rm max}$. This shows that the fit is free from overfitting\nand underfitting problems. For a given halo mass, the best-fit clustering bias varies\nup to 0.2\\%, 0.5\\%, and 2\\% among different $n$ and $k_{\\rm max}$ at $2\\times10^{13}\\,M_\\odot$,\n$2\\times10^{14}\\,M_\\odot$, and $2\\times10^{15}\\,M_\\odot$, respectively. Given the fact\nthat the clustering bias is stable for various bias models and fitting range, we conclude\nthat systematic error due to $n$ and $k_{\\rm max}$ is at most comparable to our statistical\nerror, and is the largest at the high-mass end at which the statistical error is also large.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{b_clustering_ratioonly_z0.pdf}\n\\caption{The ratios of the linear biases to that of the sub-Jeans response bias at $z=0$.\nThe red solid and blue dashed lines show the sub-Jeans and super-Jeans response biases\nmeasured from 20 separate universe simulations, whereas the green dot-dashed line shows\nthe clustering bias measured from 40 global simulations. The error of the clustering bias\nis measured from the scatter of the 40 simulations. This figure summarizes the main results\non halo bias: the agreement between the clustering bias measured from global simulations\nand the sub-Jeans response bias verifies the observable difference in halo bias across the\nJeans scale inferred from the separate universe simulations.}\n\\label{fig:b_clus}\n\\end{figure}\n\n\nIn \\reffig{b_clus} we shows our fiducial results of the clustering bias measurement for\nthe quadratic model ($n=1$) with $k_{\\rm max}=0.49\\,{\\rm Mpc}^{-1}$, which gives the smallest\nstatistical errors. We find that the clustering bias is in good agreement with the sub-Jeans\nresponse bias across two decades in halo mass, confirming the validity of the SU technique.\nThis agreement is substantially better than the difference between the super-Jeans and sub-Jeans\nresponse bias at low- and mid-mass regime, even after including the systematic differences\nbetween the fitting techniques.\n\n\nTo further test robustness of the scale-dependent bias result, we also try the halo\nfinding algorithm provided in Ref.~\\cite{Li:2015jsz}, another spherical overdensity finder\nsimilar to that in Ref.~\\cite{Tinker:2008ff}. We find that the clustering bias is statistically\nin equally good agreement with the sub-Jeans response as well.\n\n\nWith this verification of the SU calibration of halo bias, our results represent the first\nsimulation confirmation of scale-dependent halo bias from scale-dependent growth. A related\neffect on the void bias has been measured in the simulations with cold dark matter and massive\nneutrinos \\cite{Banerjee:2016zaa}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{phh_sup_to_sub.pdf}\n\\caption{Fractional difference or ``step'' across the Jeans scale in the matter (red solid,\n\\refeq{Smm}) versus the halo (blue dashed, \\refeq{Shh}) power spectra. The step in the halo\npower spectrum is reduced by approximately a factor of 2 compared with the matter power\nspectrum at high mass where the Lagrangian bias contribution dominates. Bias prescriptions\nwhich assume locality at either the observed or initial redshift predict the same step or\nzero step respectively at high masses.}\n\\label{fig:phh_ratio}\n\\end{figure}\n\n\\subsection{Scale-Dependent Bias and Power Spectra}\n\\label{sec:interpretation}\n\n\nSince the halo bias is smaller above versus below the Jeans scale of quintessence, its\nscale dependence counters the growth rate effects in the matter power spectrum. This\nis especially true at high masses where the Lagrangian bias dominates. The change in\nthe linear growth function above $(D^\\uparrow)$ versus below $(D^\\downarrow)$ the Jeans\nscale leads to a step in the linear matter power spectrum of approximately\n\\begin{equation}\n\\label{eq:Smm}\nS_{mm} \\equiv \n2 \\frac{D^\\uparrow - D^\\downarrow}{D^\\downarrow}\\,,\n\\end{equation}\nwhereas the step in the cumulative halo power spectrum is\n\\begin{equation}\nS_{hh}\n \\equiv\n  2 \\frac{D^\\uparrow \\bar b_1^\\uparrow - D^\\downarrow \\bar b_1^\\downarrow}{D^\\downarrow \\bar b_1^\\downarrow} \\,.\n  \\label{eq:Shh}\n\\end{equation}\nIn \\reffig{phh_ratio}, we show the amplitude of these steps as a function of mass.\nAt the high mass end the halo power spectrum has half the step amplitude of the\nmatter power spectrum.\n\n\nThis result not only confirms that scale-dependent linear growth leads to scale-dependent\nbias, but it does so in a way that both reduces the observability of features in the halo\npower spectrum \\cite{LoVerde:2014pxa} and violates principles that underlie simple models\nfor bias. It is commonly assumed that the statistics of halos at any observation epoch is\ndetermined solely by the statistics of the linear density field at a single epoch, and\nhence bias is scale-free with respect to the matter power spectrum at that epoch. For\nmodels with scale-dependent growth, this epoch is commonly taken to be  the initial epoch\nfor models where the growth becomes scale-free during matter domination \\cite{Parfrey:2010uy}.\n\n\nFor example in the excursion set, Lagrangian bias is given by the conditional probability\nthat the initial density field crosses some barrier at a smoothing scale $R_S$ corresponding\nto the mass $M$ at the background density given that it takes the value $\\delta_m$ at some\nlarger scale $R_L$ via a random walk between the two. For our quintessence case where these\nscales are arbitrarily well separated by our SU assumption, the lack of correlation in the\nGaussian random initial conditions between these scales means that halo bias is local in the\ninitial density field. Specifically, the conditional probability cannot depend on steps in\nthe random walk with $R>R_L$, and hence whether $\\delta_m$ was achieved from super-Jeans or\nsub-Jeans scale fluctuations. This holds regardless of the shape of the barrier, its dependence\non the redshift of observation, or some putative intermediate epoch of halo formation. As\na result, Lagrangian halo bias should be scale-free with respect to the matter power spectrum\nat the initial epoch.\n\n\nAs emphasized in Ref.~\\cite{Parfrey:2010uy}, even if the Lagrangian bias is scale-free with\nrespect to the initial power spectrum, it becomes scale dependent with respect to the matter\npower spectrum at the observation epoch. This effect is solely due to the scale-dependent\ngrowth in the latter, and hence the scale dependence takes a simple and specific form\n$b_1^{L\\uparrow} = (D^\\downarrow\/D^\\uparrow) b_1^{L\\downarrow}$. In \\refapp{bias_model},\nwe review this construction in more detail. In the quintessence model this means that the\nstep in the halo power spectrum should be absent when the Lagrangian bias dominates\n$\\lim_{M\\rightarrow \\infty}S_{hh}=0$ (see \\refeq{Shh}). Our results significantly violate\nthis prediction.\n\n\nSimilarly models of halo bias that rely on a universal mass function ansatz, characterize\nthe bias as its derivative with respect to $\\delta_m$ at the observation epoch. If this\nderivative depends {\\it only} on the local density field $\\delta_m$ at the observation epoch,\nfor example by assuming a change in the spherical collapse threshold $d\\delta_c\/d\\delta_m=-1$\n(see \\refapp{bias_model}), then the Lagrangian bias would be local and hence scale-free with\nrespect to the matter power spectrum at the observation epoch. Our results for scale-dependent\nbias directly violate this prediction and are essentially half-way between these two extreme\nmodels. We find that halo bias is nonlocal in time and cannot be characterized by the statistics\nof the density field at a single epoch, initial or observed when there is scale-dependent\nlinear growth.\n\n\nIn \\refapp{bias_model}, we show that encapsulating the dependence on the growth history of\n$\\delta_m(a)$ through its impact on the spherical collapse threshold at the observation epoch\nand assuming a universal mass function characterizes the quintessence SU simulation results\nbetter than either of these simplistic models. However given the assumptions underlying this\ntype of modeling, its validity in other contexts should be tested directly in simulations.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\nQuintessence dark energy provides an arena to explore the response of small-scale\nobservables to the amplitude, scale, and growth history of long-wavelength fluctuations\nwith the separate universe technique. In the presence of quintessence fluctuations,\nthe growth of long-wavelength fluctuations differ above and below its Jeans scale.   \nWe verify that even below the Jeans scale, where a naive separate universe picture\ndoes not strictly apply because the local curvature evolves due to non-gravitational\nforces which keep the quintessence smooth, the response of small-scale observables\ncan still be accurately modeled by a modified expansion history alone. One implication\nof this finding is that halo bias is not directly a response of halo number density\nto the local curvature, but rather to the local expansion history.\n\n\nUsing this technique, we show that in the presence of the scale-dependent growth,\nthe local power spectrum and halo mass function acquire a dependence on the scale\nof the long-wavelength mode. Equivalently, the squeezed bispectrum and halo bias\nbecome scale dependent. To our knowledge, our results are the first verification\nof scale-dependent bias from scale-dependent growth using simulations. Moreover\nthey violate predictions of models where bias is effectively local in the density\nfield at a single epoch, initial or observed, and show that halo bias is temporally\nnonlocal. Likewise the nonlinear matter power spectrum cannot simply be a function\nof the linear power spectrum at the same epoch.\n\n\nSpecifically, we use the separate universe (SU) technique to perform $N$-body\nsimulations in the sub-Jeans and super-Jeans SUs. By differencing pairs of\noverdense and underdense SU simulations with the same Gaussian realizations\nof initial phases, much of the sample variance is canceled, and so we can\nprecisely characterize the responses of the power spectrum (which is equivalent\nto the squeezed-limit bispectrum) and the halo mass function (which gives the\nlinear halo bias) to the long-wavelength matter fluctuation.\n\n\nWe validate the SU approach by comparing to perturbation theory predictions\nfor the power spectrum response in both the super-Jeans and sub-Jeans limits\n(see \\reffigs{dlnpk_linear}{dlnpk_1loop}). Since it is the sub-Jeans limit\nwhere the SU technique might naively fail, we further test it with direct\nsimulations that possess long-wavelength matter modes in big-box simulations\nwith smooth dark energy. We find that the squeezed-limit position-dependent\npower spectrum measured from the big-box simulations agrees with the power\nspectrum response to the resolution limit $k\\sim1\\,{\\rm Mpc}^{-1}$. Similarly,\nthe clustering bias is statistically consistent with the response bias across\ntwo decades in halo mass ($\\sim\\!10^{13}\\!-\\!10^{15}\\,M_\\odot$). Thus, with\nthe SU technique verified into the nonlinear regime, we can robustly assess\nthe scale-dependence of the power spectrum and halo density responses across\nthe Jeans scale without costly simulations that include quintessence clustering.\n\n\nWe show that for both responses there is a statistically significant distinction\nbetween sub-Jeans and super-Jeans SUs at $z=0$. More precisely, the power spectrum\nresponse in the super-Jeans SU is roughly 2\\% smaller than that in the sub-Jeans\nSU for $k\\lesssim1\\,{\\rm Mpc}^{-1}$; the halo bias in the super-Jeans SU is roughly\n1\\% and 3\\% smaller than that in the sub-Jeans SU for halo mass of $2 \\times 10^{13}$\nand $2\\times 10^{15}~M_\\odot$ respectively. The fact that the response is smaller\nin the super-Jeans SU is because quintessence enhances the growth of matter\nfluctuations there, and so the super-Jeans overdensity was smaller in the past.\nThese key SU results, along with the comparison to the global simulations,\nare summarized in \\reffig{ibn} and \\reffig{b_clus}.\n\n\nMore generally, this dependence on the growth history of the long wavelength fluctuation\nindicates that the response of small scale observables is nonlocal in time. In particular,\nthe statistically significant difference between sub-Jeans and super-Jeans response biases\nmeasured in our SU simulations falsifies the standard Lagrangian picture where the statistics\nof halos at any observation epoch is determined solely by the statistics of the linear density\nfield at a single epoch.\n\n\nThese effects are in principle important for interpreting observational tests of quintessence\nclustering from galaxy surveys and their cross correlation with the CMB (e.g. \\cite{Bean:2003fb,Hu:2004yd}).\nIn particular, the step feature in the halo power spectrum is smaller by up to a factor of 2\ncompared with the matter. However these corrections, while significant relative to the clustering\neffects on the matter power spectrum itself, are small in an absolute sense for observationally\nviable dark energy equations of state (i.e. $w_Q\\approx-1$) in the absence of quintessence\nisocurvature fluctuations \\cite{Gordon:2004ez,Hu:2016ssz}. \n\n\nOn the other hand, the same technique which has been validated here using quintessence,\ncan be applied to more observationally viable cosmological models, such as those with\nmassive neutrinos. Massive neutrinos cluster with dark matter on large scales, but their\nfree streaming sets an effective Jeans scale. This would generate not only a feature in\nthe two-point function of the total matter \\cite{Lesgourgues:2006nd}, but also influence\nthe high-order statistics \\cite{Shoji:2009gg,Blas:2014hya,Fuhrer:2014zka,Levi:2016tlf}\nas well as the halo bias \\cite{LoVerde:2014pxa}. We intend to apply the SU technique\nto study how massive neutrinos affect the small-scale structure formation in a\nfuture work.\n\n\n\\acknowledgements\nWe thank Eiichiro Komatsu and Fabian Schmidt for useful discussions.\nWe would also like to thank Alexander Knebe for guiding us to implement the dark energy model into Amiga Halo Finder. \nWH thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, \nwhere part of this work was completed.\nResults in this paper were obtained using the high-performance computing system\nat the Institute for Advanced Computational Science at Stony Brook University\nand with the computation and storage resources\nprovided by the University of Chicago Research Computing Center.\nCC and ML are supported by grant NSF PHY-1316617.\nWH was supported by U.S.~Dept.\\ of Energy contract DE-FG02-13ER41958,\nNASA ATP NNX15AK22G, and  the Kavli Institute for Cosmological Physics\nat the University of Chicago through grants NSF PHY-0114422\nand NSF PHY-0551142.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\newlength{\\szerkol}\n\n\\setlength{\\szerkol}{0.5\\textwidth}\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hspace{-1.em}\\includegraphics[width=1\\szerkol,clip,viewport=2 150 575 690]{t1png_withsn.ps} &\n\\hspace{-1.5em}\\includegraphics[width=1\\szerkol,clip,viewport=2 150 575 690]{t2png_withsn.ps}\\\\\n\n\\end{tabular}\n\\end{center}\n\\caption{\nLow resolution  {\\sc Hi} map\n(the beam size of $72\\arcsec\\times62\\arcsec$)\nsuperimposed on an optical image of M74.\nThe position of {SN\\,2002ap} is marked as the red dot\n(credit: Kamphuis \\& Briggs, 1992, reproduced with permission $\\copyright$ ESO).\nLeft: Zeroth moment map (integrated emission).\nRight:\nFirst moment map (velocity fields).\nThe asymmetric tail with an irregular velocity field is visible at the south-western outskirt of the atomic disc (around the position of $\\mbox{R.A.}=1^h 33^m20^s$, $\\mbox{Dec.}=15^\\circ 25^m$).\nNorth is up and east is to the left.\n}\n\\label{fig:lowres}\n\\end{figure*}\n\n\n\\setlength{\\szerkol}{0.3\\textwidth}\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom1.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_NA_mom2.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom1.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_HI_RO_mom2.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_CO21_mom0.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_CO21_mom1.eps} & \n\\end{tabular}\n\\end{center}\n\\caption{Gas distribution in M74. \nTop and middle: \n{\\sc Hi} data with a resolution of \n$11.9\\arcsec\\times9.3\\arcsec$ and\n$6.9\\arcsec\\times5.6\\arcsec$, respectively \\citep{walter08}. Bottom: CO(2-1) data with a resolution of $13.4\\arcsec$ \\citep{leroy09}. \nLeft: Zeroth moment maps (integrated emission).\nMiddle:\nFirst moment maps (velocity fields) relative to $z=0.00219$ (656.545\\,{\\mbox{km\\,s$^{-1}$}}).\nRight: Second moment maps (velocity dispersion). The positions of SNe are marked by grey circles. The lines outline the main spiral arm. \nEach panel is 10{\\arcmin} per side. North is up and east is to the left.\n}\n\\label{fig:image}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=\\szerkol,clip]{M74_UVW2_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_Ha_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_36_cont.eps} \\\\\n\\includegraphics[width=\\szerkol,clip]{M74_80_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_160_cont.eps} & \n\\includegraphics[width=\\szerkol,clip]{M74_31_cont.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Ultraviolet, H$\\alpha$, mid-IR, far-IR, and radio images of M74.\nThe positions of SNe are marked by grey circles. The lines outline the main spiral arm.\nEach panel is 10{\\arcmin} per side. North is up and east is to the left.\n}\n\\label{fig:image2}\n\\end{figure*}\n\nThe nature of various types of supernovae (SNe) carry crucial information about stellar evolution.\nA subclass of SNe with no hydrogen, helium, or silicon lines in the spectrum (known as type Ic) are believed to be explosions of stars born with very high masses.\nThose exhibiting broad emission lines, indicating high velocities of the ejected material (up to a few $10^4\\,\\mbox{km\\,s$^{-1}$}$), are called `hypernovae' or SNe type Ic-BL (broad lined). Some of these SNe also show relativistic ejecta, for example SN\\,1998bw \\citep[GRB\\,980425][]{galamanature} and 2009bb \\citep{soderberg10}. These relativistic features are interpreted as a jet, and indeed some Ic-BL SNe have been associated with gamma-ray bursts (GRBs; \\citealt{hjorthsn}).\n\nObservations of atomic and molecular gas (through 21\\,cm {\\sc Hi} and carbon monoxide [CO] lines, respectively)  in  host galaxies of GRBs and SNe have recently been used to learn about the nature of the explosions themselves, as well as the star formation event during which their progenitors were born.\n\\citet{michalowski15hi,michalowski16,michalowski18} and \\citet{arabsalmani15b,arabsalmani19} showed that GRBs and a relativistic SN type Ic-BL exploded close to the most {\\sc Hi}-rich region of their hosts, which was interpreted as being the result of a recent gas accretion or a galaxy merger.\nWhile this conclusion was based on very small samples, if substantiated it would have important consequences for our understanding of the conditions necessary for such explosions, as well as for triggering star formation in general. This motivates us to study the gas properties of another Ic-BL SN (to date, amongst the hosts of Ic-BL SNe, atomic gas was studied in only one case;\n\\citealt{michalowski18}).\n\n{SN\\,2002ap}  \\citep{nakano12circ} \nexploded $\\sim4.7\\arcmin$ ($\\sim12.7$\\,kpc)\nfrom the centre of \\object{Messier 74} (\\object{M74} or \\object{NGC\\,628}) \nand was classified as a type Ic-BL \\citep{mazzali02,kinugasa02,\ngalyam02,foley03%\n}. Its estimated progenitor mass is $20$--$25\\,\\mbox{$M_\\odot$}$, lower than other hypernovae, including SN\\,1998bw \\citep{mazzali02}. {SN\\,2002ap} was also shown to have only modest relativistic ejecta, and hence no detectable jet \\citep{berger02}. The progenitor was proposed to be a Wolf-Rayet (WR) star or a massive star in an interacting binary \\citep{smartt02,wang03%\n}.\n\nM74 has hosted three other known SNe: 2003gd (type IIP; $\\sim2.7\\arcmin$ or $\\sim7.3$\\,kpc from the galaxy centre; \\citealt{hendry05}), 2013ej (IIP; $\\sim2.2\\arcmin$ or $\\sim5.9$\\,kpc;  \\citealt{valenti14}), and 2019krl (IIn; $1.9\\arcmin$ or $\\sim5.2$\\,kpc; \\citealt{ho19rep,\nandrews19atel}). The progenitor of SN\\,2003gd was confirmed to be an M-type supergiant with a mass of $\\sim8\\,\\mbox{$M_\\odot$}$,\nby examining the SN position in pre- and post-explosion images, which revealed that this star was missing in the latter\n\\citep{maund09,vandyk03,smartt04}. In a similar way, the mass of the progenitor of SN\\,2013ej was estimated to be $8.0$--$15.5\\,\\mbox{$M_\\odot$}$\\citep{fraser14,mauerhan17}.\n \nThe objectives of this paper are to {\\it i}) test whether {SN\\,2002ap} and other SNe in M74 were born in concentrations of gas indicating a recent gas accretion, and {\\it ii}) investigate what this tells us about the formation of SN progenitors.\nWe adopt a redshift of M74 of $z=0.00219$ \\citep{lu93}, a distance of 9.4\\,Mpc, and a corresponding scale of 2.7 kpc arcmin$^{-1}$.\nThis  assumes a cosmological model with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$,  $\\Omega_\\Lambda=0.7$, and $\\Omega_{\\rm m}=0.3$.\n\n\n\n\n\n\\section{Selection and data}\n\\label{sec:data}\n\n\n\n{The supernova SN\\,2002ap} and its host galaxy M74 were selected as part of a larger study of gas in SN hosts (Gotkiewicz \\& Micha\\l owski, in prep.). We investigated all known SNe up to Aug 2018 with redshifts $z<0.1$ from the Open Supernova Catalog\\footnote{{\\tt https:\/\/sne.space}} \\citep{snespace} and searched the NASA\/IPAC Extragalactic Database (NED) for {\\sc Hi} data for their hosts.  {SN\\,2002ap} was the only SN type Ic-BL that satisfied these criteria.\n\nThe archival {\\sc Hi} and CO data for M74 are shown in Figs.~\\ref{fig:lowres} and \\ref{fig:image}, while the rest of the data are shown in Fig.~\\ref{fig:image2}.\n{\\sc Hi} data are from \\citet[][a resolution of $72\\arcsec\\times62\\arcsec$]{kamphuis92}  and The {\\sc Hi} Nearby Galaxy Survey \\citep[THINGS;][a resolution of $11.9\\arcsec\\times9.3\\arcsec$ and\n$6.9\\arcsec\\times5.6\\arcsec$]\n{walter08}. The CO(2-1) data are from the HERA CO Line Extragalactic Survey \\citep[HERACLES;][a resolution of $13.4\\arcsec$]{leroy09}. The H$\\alpha$ image is from \\citet{marcum01}. In addition, we use the following continuum data: \nthe {\\it Swift} \\citep{swift,uvot} \nUVW2 0.2\\,{\\mbox{$\\mu$m}} image \\citep{brown14};\n{\\it Spitzer} \\citep{spitzer,irac} \n3.6 and 8.0\\,{\\mbox{$\\mu$m}} images \\citep{dale09},\n {\\it Herschel}\n\n\\citep{herschel,pacs} \n160\\,{\\mbox{$\\mu$m}} image \\citep{kingfish}, and\nNational Science Foundation's (NSF's) Karl G. Jansky Very Large Array (VLA) 3.1\\,GHz image \\citep{mulcahy17}.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nAccording to \\citet{kamphuis92}, M74 harbours an off-centre asymmetric {\\sc Hi} tail, located \non the south-western outskirts of the galaxy, outside the optical disc\n(Fig.~\\ref{fig:lowres}). The feature is detected over 20{\\arcmin} (55\\,kpc) and contains \n$\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=8.95$, or 7.5\\% of the total atomic gas of M74. {SN\\,2002ap} is located \nwhere this feature connects with the symmetric disc of M74.\nThe velocity pattern of the feature is irregular \n(Fig.~\\ref{fig:lowres}).\nThis gas does not follow the overall rotation of the gas disc, as evidenced by both negative and positive velocity residuals from the disc models at this location presented in Figs.~8 and 9 of \\citet{kamphuis92}. \n\nThe higher resolution THINGS data are not sensitive to such large scales, but allow detailed investigation of the local environments of SNe.\nAt this resolution, the position of {SN\\,2002ap} is not associated with any strong concentration of atomic or molecular gas (Fig.~\\ref{fig:image}).\n{SN\\,2002ap} is located $\\sim80\\arcsec$ (3.6\\,kpc) south-west of the main spiral arm running from the south to the west of the galaxy (marked as a curved region on Figs.~\\ref{fig:image} and \\ref{fig:image2}, traced clearly on all images up to the southernmost point, and by the {\\sc Hi} and 3.1\\,GHz images to the west) \nand $\\sim20\\arcsec$ (0.9\\,kpc) from \na bright {\\sc Hi} knot to the north. The main spiral arm in the south is also visible in the {\\sc Hi} velocity map in which \nthe integrated mean velocities show a larger deviation from the systemic velocity in the interarm regions.\n\nMoreover, {SN\\,2002ap} exploded away from regions of significant star formation activity and very little emission is present at its position at any wavelength (Fig.~\\ref{fig:image2}). There is, however, a faint star-forming region visible in the UV $\\sim4\\arcsec$ (180\\,pc) away from the SN position (this is not the object $10\\arcsec$ away mentioned by \\citealt{crowther13}).\n{SN\\,2002ap} is also outside the CO disc.\n\nThe other three type II SNe in M74 exploded along the most prominent spiral arm (running from the east to south of M74; curved regions on Figs.~\\ref{fig:image} and \\ref{fig:image2}), but are displaced from the arm towards the outside \nby $\\sim25\\arcsec$ ($\\sim1$\\,kpc). This is especially evident in the H$\\alpha$, $3.6\\,\\mbox{$\\mu$m},$ and CO images. SN\\,2013ej and 2019krl exploded in interarm regions with very little {\\sc Hi}, 8, 160\\,{\\mbox{$\\mu$m},} and 3.1\\,GHz emission.  All three type II SNe exploded in regions with undetectable CO emission.\n\nWe have investigated the large-scale environment of M74. Within 150\\,kpc (55{\\arcmin}) and $\\pm500\\,\\mbox{km\\,s$^{-1}$}$ from M74 (velocity of $627\\,\\mbox{km\\,s$^{-1}$}$), NED lists three galaxies:\nUGC\\,1171,\nUGC\\,1176,\nboth to the east,\nand the much fainter SDSS J013800.30+145858.1\nto the south. \nAll of them are detected in {\\sc Hi} by the Arecibo Legacy Fast ALFA Survey (ALFALFA; \\citealt{haynes18}). In Table~\\ref{tab:other} we list their properties.\n\nIn the ALFALFA catalogue within 200{\\arcsec} (540\\,kpc) of M74 there are in total 13 galaxies. All but one have atomic gas masses $7<\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})<9$. Only NGC\\,660, 426\\,kpc away, with $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=9.59$ has a mass comparable to that of M74.  \nThe positions of these galaxies are shown on Fig.~\\ref{fig:comp}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,clip]{companion.eps} \n\\end{center}\n\\caption{Large-scale environment around M74 within $\\pm500\\,\\mbox{km\\,s$^{-1}$}$. The width of the panel is 400{\\arcmin} ($\\sim1$\\,Mpc). Circles show the sizes of larger galaxies: 20{\\arcmin} for M74, 10{\\arcmin} for NGC\\,660, 5{\\arcmin} for UGC\\,1176, and 1.4{\\arcmin} for UGC\\,1171 (a small circle next to UGC\\,1176). Crosses show the positions of additional galaxies detected at the {\\sc Hi} line by ALFALFA \\citep{haynes18}. \n}\n\\label{fig:comp}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Properties of galaxies in the vicinity of M74 from the ALFALFA survey  \\citep{haynes18}.}\n    \\centering\n    \\begin{tabular}{lrccrccccc}\n    \\hline\\hline\n    Galaxy & ID & RA & Dec & \\multicolumn{2}{c}{Dist$_{\\rm M74}$} & $z$ & $V_{\\rm helio}$ & $f_{HI}$ & $\\log(M_{\\rm HI})$ \\\\ \n     & & (deg) & (deg) & ($'$) & (kpc) & & ($\\mbox{km\\,s$^{-1}$}$) & (Jy\\,$\\mbox{km\\,s$^{-1}$}$) & ($\\mbox{$M_\\odot$}$) \\\\\n     \\hline\n    M74 & 1149 & 24.18500 & 15.79222 & $\\cdots$ & $\\cdots$ & 0.002190 & 657 & $424.30\\pm0.18$ & $9.73\\pm0.1\\phantom{0}$ \\\\\n    UGC\\,1171 & 1171 & 24.93708 & 15.89611 & 44.6 & 120 & 0.002463 & 738 & $\\phantom{10}2.07\\pm0.05$ & $7.42\\pm0.09$ \\\\\n    UGC\\,1176 & 1176 &  25.03167 & 15.90167 &  50.6 & 137 & 0.002103 & 630 & $\\phantom{1}31.22\\pm0.06$ & $8.78\\pm0.1\\phantom{0}$ \\\\\n    SDSSJ013\n    & 112503 & 24.50583 & 14.99333 & 51.6 & 139 & 0.002478 & 743 & $\\phantom{10}0.56\\pm0.05$ & $7.14\\pm0.2\\phantom{0}$ \\\\\n    NGC\\,660 & 1201 & 25.76167 & 13.64000 &  157.9 & 426 & 0.002830 & 848 & $148.39\\pm0.14$ & $9.59\\pm0.34$ \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:other}\n    \\tablecomments{The columns show the galaxy name (that of SDSSJ013800.30+145858.1 has been abbreviated), ALFALFA ID, position, projected distance to M74, redshift, heliocentric velocity, {\\sc Hi} flux, and the atomic gas mass.}\n\\end{table*}\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe estimated mass of the progenitor of  {SN\\,2002ap} implies that it was formed 3.2--5.6\\,Myr before the explosion, whereas the progenitors of \nSN\\,2003gd and 2013ej\nwere formed 10--55\\,Myr before the explosions \n(assuming a main-sequence lifetime of $10^{10}\\,\\mbox{yr} \\times [M\/\\mbox{$M_\\odot$}\\mbox{$]$} ^{-2.5}$; \\citealt{kippenhahn90}).\nThis lifetime for {SN\\,2002ap} agrees with the single-progenitor estimate of \\citet[][5\\,Myr]{zapartas17}, but is lower than the binary-progenitor estimate (20\\,Myr). Similarly, \\citet{maund18} obtained an age of 15\\,Myr based on the analysis of stars in the vicinity of {SN\\,2002ap}.\n\n\\subsection{Type Ic-BL {SN\\,2002ap}}\n\n{The supernova SN\\,2002ap} is the fourth known explosion of a presumably massive star located close to a concentration of atomic gas. Similarly to {SN\\,2002ap},  GRB\\,060505 and SN\\,2009bb were both located close to {\\sc Hi} extensions, whereas GRB\\,980425 was located close to the most significant {\\sc Hi} concentration \\citep{michalowski15hi,michalowski18,arabsalmani15b,arabsalmani19}.\n\nWe can assess the statistical significance of the associations of the {\\sc Hi} off-centre concentrations with the GRB and SN positions by investigating the probability of four GRBs and SNe exploding by chance in the quadrants of their hosts at which these {\\sc Hi} concentrations are located. For a given explosion this probability is $0.25$, so for four out of four analysed cases this is $(0.25)^4\\sim0.004$. This corresponds to $\\sim3\\sigma$, so the associations are unlikely to be random, but a larger sample is needed to confirm this result.\n\nThe case of {SN\\,2002ap} adds to \nthe hypothesis put forward in \\citet{michalowski15hi} that the progenitors of these explosions are born when atomic gas is accreted from the intergalactic medium. Indeed, \\citet{kamphuis92} concluded that the south-western {\\sc Hi} extension in M74 is a result of an accretion event \nbecause it has not settled yet, and is inconsistent with the rotation of the gas disc. This extension may be the gas flowing in and feeding star formation directly, or distorting gas on the outskirts of the optical disc of the galaxy leading to star formation at the position of {SN\\,2002ap}.\n\nThe asymmetric tail in M74 has a mass of $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=8.95$. This is comparable to the mass of the most massive galaxy within 150\\,kpc  (UGC\\,1176) and only a factor of four less than the atomic gas mass of NGC\\,660.\nThe sum of atomic masses of all galaxies within 200{\\arcmin} (540\\,kpc) excluding NGC\\,660 is $\\log(M_{\\rm HI}\/\\mbox{$M_\\odot$})=9.36$.\nHence, the tail in M74 might have come from UGC\\,1176 if it was significantly distorted by the interaction and lost half of its gas, or from NGC\\,660.\nIt could also be a remnant of a galaxy similar to UGC\\,1176 that has been accreted entirely (as postulated by \\citealt{kamphuis92}), or a result of the accretion of intragroup medium.\n\nIn principle this tail could be a remnant of a  tidal feature created by interaction with these galaxies, but we found this interpretation unlikely. First, the tail does not resemble recent tidal features, whereas an older feature would  wind almost symmetrically around the galaxy. Second, simulations shows that tidal tails are created on both sides of interacting galaxies \\citep{hopkins06b,hayward12,hayward14,pettitt16,oh16}, whereas M74 does not have such feature on the other side.\nWe also note that the asymmetric nature of M74, with the southern arm being stronger than the northern one (Fig.~\\ref{fig:image}), could be a result of interaction with the UGC\\,1176\/1171 pair.\n\n\nThe only possible counter-example of a potential  explosion of a massive star without an associated gas concentration is the enigmatic transient AT\\,2018cow, whose host galaxy does not show such off-centre asymmetric {\\sc Hi} features  \\citep{michalowski19} and possibly only a gas ring \\citep{roychowdhury19}. \nSuch an {\\sc Hi} ring would also be apparent for M74 if it was further away so the sensitivity and  resolution were poorer, because the central part is devoid of atomic gas, likely due to conversion to the molecular phase (Fig.~\\ref{fig:image}).\nTo demonstrate this we smoothed the {\\sc Hi} VLA image with a Gaussian with a full width half maximum of 100{\\arcsec} (4.5\\,kpc;   Fig.~\\ref{fig:smooth} in the appendix). At this resolution the spiral structure of M74 resembles an irregular ring, similar to that detected for the AT\\,2018cow host (for which the resolution was around 2\\,kpc).\nHowever, the nature of AT\\,2018cow is not clear, so it may not be connected with the explosion of a massive star\n(\\citealt{prentice18,liu18,kuin19,perley19,soker19,lyutikov19,bietenholz20}, but see \\citealt{prentice18,riverasandoval18,margutti19,fox19,huang19}).\n\nIt is unlikely that the lack of molecular gas or star formation at the position of {SN\\,2002ap} is due to the progenitor being kicked out of a star-forming region. The velocities of runaway stars are up to 200\\,{\\mbox{km\\,s$^{-1}$}} \\citep{blaauw93,%\nhoogerwerf01,%\neldridge11%\n}, which corresponds to 1\\,kpc per 5\\,Myr. This is only $\\sim20\\arcsec$ at the distance of M74, so not sufficient to move the birth place of the {SN\\,2002ap} progenitor to any place of significant star formation or CO concentration.\nThis is true even if the lifetime of the progenitor is three to four times longer (15--20\\,My; \\citealt{zapartas17}, \\citealt{maund18}).\nCO deficiency at  GRB positions was also claimed by \\citet{hatsukade14}, \\citet{stanway15}, and \\citet{michalowski16}, but \\citet{perley17}, \\citet{michalowski18co}, and \\citet{arabsalmani18} suggest an alternative. If the lack of molecular gas is confirmed for a larger sample of type Ic-BL SNe, this would support the hypothesis of {\\sc Hi}-fuelled star formation giving rise to the birth of their progenitors \\citep{michalowski15hi}.\n\n\n\nIt is unlikely that the {SN\\,2002ap} progenitor moved to its explosion position due to a random kick.\nAssuming a lifetime of 5\\,Myr, the {SN\\,2002ap} progenitor could not be born in the main arm, as the required velocity is 700\\,{\\mbox{km\\,s$^{-1}$}} to cross 3.6\\,kpc. Even the closest bright {\\sc Hi} knot \nto the north\nis likely too far to be the birthplace, as this would require a velocity of 175\\,{\\mbox{km\\,s$^{-1}$}}. \nSuch  velocity kicks are at the high end for runaway stars \\citep{hoogerwerf01}. For the longer lifetime estimates of 15--20\\,Myr, the required velocities from the spiral arm would be 235--175\\,\\mbox{km\\,s$^{-1}$}, so still too high for the {SN\\,2002ap} progenitor to have been born there. However, in such a case it is feasible that it was born in the closest bright {\\sc Hi} knot to the north,\nas this would require velocities of 60--40\\,\\mbox{km\\,s$^{-1}$}.\n\n\\subsection{Type II SN\\,2003gd, 2013ej, and 2019krl}\n      \nAll type II SNe in M74 \nare not located close to the {\\sc Hi} extension, so are unlikely connected to gas accretion.\nThey are located at the outside edge of a spiral arm. This can be explained by either of two scenarios: \nby a gas density build-up and shock scenario at the edge of the arm, or by SN progenitors moving away from the arm during their lifetimes.\n\nThe first possibility is that the SN progenitors are born when gas is piling up and shocked at the edge of the arm when gas clouds are being swept up by the arm, as explained by the spiral density wave theory (\\citealt{shu16}). This is similar to the hypothesis presented in \\citet{michalowski14} that GRB progenitors are preferentially born in high-density gas.\nWe note that the amount of gas piling up at the edge of the arm giving rise to the birth of SN progenitors cannot be large, because the concentration is not detected with {\\sc Hi} or CO observations. \n\nThe SNe in M74 are on the outside of the spiral arm, so this scenario is only valid if they are outside the corotation radius (where the orbital velocity is equal to the spiral pattern speed), so the arm is catching up with gas that is moving slower \\citep{shu16}. \n\\citet{aramyan16} found that core-collapse SNe are indeed shifted towards the outside edge of spiral arms as long as they are  outside the corotation radius.\nUnfortunately the accuracy of the estimate of the corotation radius for M74 is not sufficient to test this. The corotation radius given by \\citet{scarano13}\\footnote{$4.6\\pm1.2$\\,kpc and their adopted scale of 1.95 kpc arcmin$^{-1}$.} is $(2.4\\pm0.6)\\arcmin$, corresponding to $(6.4\\pm1.7)\\,$kpc with our adopted distance, so it cannot be established whether type II SNe are inside or outside this radius. \n\\citet{karapetyan18} quoted a slightly lower (but consistent within errors) value of the corotation radius\\footnote{The ratio of the corotation and isophotal  ($R_{25}=5.52\\arcmin$) radii of  $0.34\\pm0.09$.} of $(1.9\\pm0.5)\\arcmin$, concluding that SNe\\,2003gd and 2013ej are  indeed outside the corotation radius, supporting the spiral density wave scenario for their formation.\nThis scenario cannot explain the birth of the {SN\\,2002ap} progenitor, because gas should not pile up so far away (3.6\\,kpc; Fig.~\\ref{fig:image}) from the spiral arm.\n\n\nHowever, the region of increased gas density and shocks, predicted by the spiral density wave theory, does manifest itself with increased star formation, and therefore more intense UV and H$\\alpha$ emission. These main sites of star formation are where SN progenitors should be born, not 1\\,kpc away. This scenario does not explain this discrepancy.\n\nThe second possibility is that SN progenitors are in fact born in the spiral arm, but, due to their orbital motions, move away before they explode. \nInside the corotation radius stars (and gas) are moving faster than the spiral pattern, so are constantly drifting towards the outside of the arm. \nThis means that SNe with progenitors with long enough lifetimes should be happening preferentially outside the arm. On the other hand, H$\\alpha$ emission is dominated by stars born very recently (i.e. after the SN progenitors), which have therefore not yet moved away from the arm.\n\nThe lifetimes of type II progenitors imply that they would need to have (reasonable) velocities of $18$--$100\\,\\mbox{km\\,s$^{-1}$}$ with respect to the arm to cross 1\\,kpc from the arm to their current position.\nWe note that this velocity is not due to any random-direction kick, but is the orbital velocity minus the spiral pattern speed.\n\n\n\n\n\nThe orbital migration scenario requires that type II SNe progenitors are inside the corotation radius, so their orbital motion is faster than the spiral pattern and so they can leave it on the outside edge \\citep[e.g.][]{aramyan16}.\nThis effect is also visible in M51 where younger stellar clusters have a distribution that has the strongest correlation with the distribution of star formation, and they are shifted towards the outside of the arm inside the corotation radius \\citep{scheepmaker09}.\n\nThis scenario cannot explain the position of {SN\\,2002ap}, because it is located securely outside the corotation radius, so cannot leave the spiral arm at the outside edge. Instead, as we discuss above, it may\nbe connected with the gas accretion visible in the {\\sc Hi} map of \\citet{kamphuis92}.\n\nThe position of SNe away from the main sites of star formation (spiral arms) is not unique to M74. We investigated galaxies with four or more core-collapse SNe and with well-separated spiral structures from the list of \\citet{thone09}. NGC\\,6946 and 4303 hosted nine and six type II SNe, respectively, and only one (1948B) and two (1999gn and 2006ov) exploded in spiral arms; the rest exploded in interarm regions or outside the detectable stellar disk (Figs.~9 and 11 of \\citealt{anderson13}). \nThis is consistent with the spiral density wave theory and the migration scenario described above.\nIndeed, larger samples of type II SNe show that they do not concentrate in the brightest regions of their hosts \\citep{fruchter06,anderson08,\nleloudas11}.\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nWe have used archival {\\sc Hi} and CO data for M74 (not previously investigated in the context of SN positions), together with H$\\alpha$ and continuum images.\n{SN\\,2002ap} is located  at  the end of  an  off-centre asymmetric 55\\,kpc-long {\\sc Hi} extension containing 7.5\\% of the total atomic gas of M74. \nIt is the fourth known explosion of a presumably massive star that is located close to the concentration of atomic gas (after GRB\\,980425, 060505, and SN\\,2009bb). \nIt is unlikely that all these associations are random (at a $3\\sigma$ significance), so\nthe case of {SN\\,2002ap} adds to\n the evidence that the birth of the progenitors of type Ic-BL SNe and GRBs is connected with the accretion of atomic gas from the intergalactic medium.\nThe {\\sc Hi} extension could come from tidally disrupted companions of M74, or be a remnant of a galaxy or a gas cloud in the intragroup medium accreted entirely. \n\nThe type II SNe in M74 do not seem to be related to gas accretion.\nThe fact that \nthey\nare located at the outside edge of a spiral arm suggests either that their progenitors are born when gas is piling up there, reaching high density, or that SN progenitors move away from the arm during their lifetimes, due to their orbital motions.\nThis is also similar for NGC\\,6946 and 4303, with eight out of nine and four out of six type II SNe, respectively, located in interarm regions or outside the detectable stellar discs, not in the spiral arms.\n\n\\begin{acknowledgements}\n\n\nWe wish to thank the referee for careful and important suggestions, Joanna Baradziej and Phillip Hopkins  for discussion and comments, and Frank Briggs for permission to use the figure from \\citet{kamphuis92}.\n\nM.J.M.~acknowledges the support of \nthe National Science Centre, Poland through the SONATA BIS grant 2018\/30\/E\/ST9\/00208, and of the Polish-U.S. Fulbright Commission.\nJ.H.~was supported by a VILLUM FONDEN Investigator grant (project number 16599).\nP.K.~is partially supported by the BMBF project 05A17PC2 for D-MeerKAT.\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThis work made use of HERACLES, `The HERA CO-Line Extragalactic Survey'.\nThis research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.\nThis work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.\n{\\it Herschel} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.\nThis research has made use of \nthe Open Supernova Catalog (\\url{https:\/\/sne.space});\nNASA\/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;\nSAOImage DS9, developed by the Smithsonian Astrophysical Observatory \\citep{ds9};\nEdward Wright cosmology calculator \\citep{wrightcalc};\nthe WebPlotDigitizer of Ankit Rohatgi ({\\tt arohatgi.info\/WebPlotDigitizer});\nand NASA's Astrophysics Data System Bibliographic Services.\n\\end{acknowledgements}\n\n\n\\input{ms.bbl}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn \\emph{alternating sign matrix} (ASM) is a square matrix with entries in $\\{0,1,-1\\}$ such that in each row and each column the non-zero entries alternate and sum up to $1$. Robbins and Rumsey introduced alternating sign matrices in the 1980s \\cite{lambda}  when studying their \\emph{$\\lambda$-determinant} (a generalization of the classical determinant) and showing that the $\\lambda$-deter\\-mi\\-nant can be expressed as a sum over all alternating sign matrices of fixed size. The classical determinant is obtained from this by setting $\\lambda=-1$, in which case the sum reduces so that it extends only over all ASMs \\emph{without} $-1$'s, i.e., permutation matrices, and the well-known formula of Leibniz is recovered.\nNumerical experiments led Robbins and Rumsey to conjecture that the number of $n \\times n$ alternating sign matrices is given by the surprisingly simple product formula\n\\begin{equation}\n\\label{asm}\n\\prod_{i=0}^{n-1} \\frac{(3i+1)!}{(n+i)!}.\n\\end{equation}\n\n\\medskip\n\nBack then the surprise was even bigger when they learned from Stanley (see \\cite{BrePro99,Bre99}) that this product formula had recently also appeared in Andrews' paper \\cite{And79} on his proof of the weak Macdonald conjecture, which in turn provides a formula for the number of \\emph{cyclically symmetric plane partitions}. As a byproduct, Andrews had introduced \\emph{descending plane partitions}\u00a0and had proven that the number of descending plane partitions (DPPs) with parts at most $n$ is also equal to \\eqref{asm}. Since then the problem of finding an explicit bijection between alternating sign matrices and descending plane partitions has attracted considerable attention from combinatorialists and to many of them it is a miracle that such a bijection has not been found so far. All the more so because Mills, Robbins and Rumsey had also introduced several ``statistics'' on alternating sign matrices and on descending plane partitions for which they had strong numerical evidence that the joint distributions coincide as well, see \\cite{MilRobRum83}.\n\n\\medskip\n\nThere were a few further surprises yet to come. Robbins introduced a new operation on plane partitions, \\emph{complementation}, and had strong numerical evidence that totally symmetric self-complementary plane partitions (TSSCPPs) in a $2n \\times 2n \\times 2n$-box are also counted by \\eqref{asm}. Again this was further supported by statistics that have the same joint distribution as well as certain refinements, see \\cite{MilRobRum86,Kra96,krattsurvey,bianecheballah}. We still lack an explicit bijection between TSSCPPs and ASMs, as well as between TSSCPPs and DPPs.\n\n\\medskip\n\nIn his collection of bijective proof problems (which is available from his webpage) Stanley says the following about the problem of finding all these bijections: ``\\emph{This is one of the most intriguing open problems in the area of bijective proofs.}'' In Krattenthaler's survey on plane partitions \\cite{krattsurvey} he expresses his opinion by saying: ``\\emph{The greatest, still unsolved, mystery concerns the question of what plane partitions have to do with alternating sign matrices.}''\n\n\\medskip\n\nMany of the above mentioned conjectures have since been proved by non-bijective means: Zeilberger \\cite{Zei96a} was the first who proved that $n \\times n$ ASMs are counted by \\eqref{asm}. Kuperberg gave another shorter proof \\cite{Kup96}\u00a0based on the remarkable observation that the \\emph{six-vertex model} (which had been introduced by physicists several decades earlier) with domain wall boundary conditions is equivalent to ASMs, see \\cite{ElkKupLarPro92a,ElkKupLarPro92b}, and he used the techniques that had been developed by physicists to study this model. Andrews enumerated TSSCPPs in \\cite{And94}. The equivalence of certain statistics for ASMs and of certain statistics for DPPs has been proved in \\cite{BehDifZin12,BehDifZin13}, while for ASMs and TSSCPPs see \\cite{Zei96b,FonZin08}, and note in particular that already in Zeilberger's first ASM paper \\cite{Zei96a} he could deal with an important refinement.\nFurther work including the study of \\emph{symmetry classes} has been accomplished; for a more detailed description of this we defer to \\cite{BehFisKon17}. Then, in very recent work, alternating sign triangles (ASTs) were introduced in \\cite{AyyBehFis16}, which establishes a fourth class of objects that are equinumerous with ASMs, and also in this case nobody has so far been able to construct a bijection.\n\n\\medskip\n\nAnother aspect that should be mentioned here is Okada's work \\cite{Oka06}\u00a0(see also \\cite{stro}), which hints at a connection between ASMs and representation theory that has not yet been well understood. He observed that a certain multivariate generating function (a specialization at a root of unity of the partition function that had been introduced by physicists in their study of the six-vertex model) can be expressed---up to a power of $3$---by a single \\emph{Schur polynomial}. Since Schur polynomials are generating functions of semistandard tableaux, this establishes yet another challenging open problem for combinatorialists inclined to find bijections.\n\n\\medskip\n\nThe proofs of the results briefly reviewed above contain rather long and complicated computations, and include hardly any arguments of a combinatorial flavor. In fact it seems that all ASM-related identities for which there exists a bijective proofs are trivial, with the exception of the rotational invariance of fully packed loop configurations. This was proved by Wieland  \\cite{wieland} bijectively and is also used in the celebrated proof of the Razumov-Stroganov (ex-)conjecture \\cite{razstrog}.\n\n\n\\medskip\n\nWe come now to the purpose of the current paper. This is the first paper in a planned series that seeks to give the first bijective proofs of several results described so far. The seed of the idea to do so came from a brief discussion of the first author with Zeilberger on the problem of finding such bijections at the AMS-MAA Joint Mathematics Meetings 2019. Zeilberger mentioned that such bijections can be constructed from existing ``computational'' proofs, however, most likely these bijections are complicated. The authors of the current paper agree, in fact the first author gave her ``own'' proof of the ASM theorem in \\cite{Fis06,Fis07,Fis16} and expressed some speculations in this direction in the final section of the last paper. It is also not implausible that a simple satisfactory bijective proof of the ASM theorem does not exist at all. Combinatorialists have failed to find such bijections for decades now, and we may start to ask ourselves why we are not rewarded for these efforts.\n\n\\medskip\n\nThis is how the authors of the current paper decided to work on converting the proof in \\cite{Fis16} into a bijective proof. After having figured out how to actually convert computations and also having shaped certain useful fundamental concepts related to  \\emph{signed sets} (see Section~\\ref{sec:ss}), the translation of several steps became quite straightforward; some steps were quite challenging. Then a certain type of (exciting) dynamics evolved, where the combina\\-to\\-rial point of view led to simplifications and other modifications, and after this process the original ``computational'' proof is in fact rather difficult to recognize. For several obvious reasons, we find it essential to check all our constructions with computer code; to name one it can possibly be used to identify new equivalent statistics.\n\n\\medskip\n\nAfter the above mentioned simplifications, it seems that \\emph{signs} seem to be unavoidable. After all, if there would be a simple bijective proof that avoided signs, would it not also be plausible that such a proof could be converted into a simple ``computational'' proof that avoids signs? Such a proof has also not been found so far.\n\n\\medskip\n\nIn the remainder of the introduction we discuss the result that is proved bijectively in this paper, in particular we discuss why signed enumerations seem to be unavoidable from this point of view. We also sketch a few ideas informally before giving rigorous definitions and proofs later on.\n\n\\subsection*{The operator formula}\u00a0\nWe use the well-known correspondence between order $n \\times n$ ASMs and \\emph{monotone triangles} with bottom row $1,2,\\ldots,n$. A \\emph{monotone triangle} is a triangular array $(a_{i,j})_{1 \\le j \\le i \\le n}$ of integers, where the elements are usually arranged as follows\n\\begin{equation}\n\\label{triangle}\n\\begin{array}{ccccccccccccccccc}\n  &   &   &   &   &   &   &   & a_{1,1} &   &   &   &   &   &   &   & \\\\\n  &   &   &   &   &   &   & a_{2,1} &   & a_{2,2} &   &   &   &   &   &   & \\\\\n  &   &   &   &   &   & \\dots &   & \\dots &   & \\dots &   &   &   &   &   & \\\\\n  &   &   &   &   & a_{n-2,1} &   & \\dots &   & \\dots &   & a_{n-2,n-2} &   &   &   &   & \\\\\n  &   &   &   & a_{n-1,1} &   & a_{n-1,2} &  &   \\dots &   & \\dots   &  & a_{n-1,n-1}  &   &   &   & \\\\\n  &   &   & a_{n,1} &   & a_{n,2} &   & a_{n,3} &   & \\dots &   & \\dots &   & a_{n,n} &   &   &\n\\end{array},\n\\end{equation}\nsuch that the integers increase weakly along $\\nearrow$-diagonals and $\\searrow$-diagonals, and increase strictly along rows, i.e.,\n$a_{i,j} \\le a_{i-1,j} \\le a_{i,j+1}$ and $a_{i,j} < a_{i,j+1}$  for all $i,j$ with $1 \\le j < i \\le n$. In order to convert an ASM into the corresponding monotone triangle, add to each entry all the entries that are in the same column above it, and record then row by row the positions of the $1$'s, see Figure~\\ref{ASM-MT} for an example.\n\n\n\\begin{figure}\n$$\n\\left(\n\\begin{matrix}\n0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 1 & 0 & -1 & 1 & 0 \\\\\n1 & -1 & 0 & 1 & -1 & 1 \\\\\n0 & 1  & 0 & -1 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0\n\\end{matrix} \\right) \\rightarrow\n\\left(\n\\begin{matrix}\n0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 1 & 0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 1 & 0 & 1 \\\\\n1 & 1  & 0 &0 & 1 & 1 \\\\\n1 & 1 & 0 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1\n\\end{matrix} \\right) \\rightarrow\n\\begin{array}{ccccccccccc}\n  & & & & & 4  & & & & &  \\\\\n  & & & & 2 &  & 5  & & & &  \\\\\n  & & & 1 &  & 4 &   & 6 & & &  \\\\\n & & 1 &  & 2  &  & 5  &  & 6 & &  \\\\\n &1  &  & 2 &   & 4  &   & 5  &  & 6 &  \\\\\n1  &  & 2 &  & 3  &   & 4   &   & 5  &  & 6\n\\end{array}\n$$\n\\caption{\\label{ASM-MT} ASM $\\rightarrow$ partial columnsums $\\rightarrow$ monotone triangle}\n\\end{figure}\n\n\\medskip\n\nThe following \\emph{operator formula}  for the number of monotone triangles with prescribed bottom row was first proved in \\cite{Fis06} (see \\cite{Fis10,Fis16} for simplifications and generali\\-za\\-tions). Note that we allow arbitrary strictly increasing bottom rows.\n\n\\begin{theorem}\n\\label{operator}\nLet $k_1 < k_2 < \\ldots < k_n$ be a sequence of strictly increasing integers. The number of monotone triangles with bottom row $k_1,\\ldots,k_n$ is\n\\begin{equation}\n\\label{operatorexpr}\n\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right) \\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i},\n\\end{equation}\nwhere ${\\operatorname{E}}_x$ denotes the shift operator, i.e., ${\\operatorname{E}}_x p(x) = p(x+1)$.\\footnote{The formula has to be understood as follows: Take $\\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i}$ and treat the $k_i$'s as indeterminates. Apply $\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right)$ to this polynomial to obtain another polynomial. Only then the $k_i$'s can be specialized to the actual values.}\n\\end{theorem}\n\n\n\nThe purpose of this paper is to provide a bijective proof of Theorem~\\ref{operator}. While the operator formula is an interesting result in its own right, it has also been the main tool for proofs of several results mentioned above. This will be reviewed in the final section of this paper along with indications for future projects on converting also these proofs into bijective proofs.\n\n\\medskip\n\nIn order to be able to construct a bijective proof of Theorem~\\ref{operator}, we need to interpret \\eqref{operatorexpr} combinatorially. Recall that \\emph{Gelfand-Tsetlin patterns} are defined as monotone triangles with the condition on the strict increase along rows being dropped, see  \\cite[p.\\ 313]{Sta99} or \\cite[(3)]{gelfand} for the original reference\\footnote{Gelfand-Tsetlin patterns with bottom row $0 \\le k_1 \\le k_2 \\le \\ldots \\le k_n$ are in an easy bijective correspondence with seminstandard tableaux of shape $(k_n,k_{n-1},\\ldots,k_1)$ and entries in $\\{1,2,\\ldots,n\\}$.}. It is well known that the number of Gelfand-Tsetlin patterns with bottom row $k_1 \\le k_2 \\le \\ldots \\le k_n$ is\n\\begin{equation}\n\\label{gelfandtsetlin}\n\\prod_{1 \\le i < j \\le n} \\frac{k_j-k_i+j-i}{j-i},\n\\end{equation}\nwhich is the operand in the operator formula \\eqref{operatorexpr}. Expanding $\\prod_{1 \\le p < q \\le n}\u00a0\\left( {\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1} \\right)$ into $3^{\\binom{n}{2}}$ monomials in ${\\operatorname{E}}^{\\pm 1}_{k_1}, {\\operatorname{E}}^{\\pm 1}_{k_2}, \\ldots, {\\operatorname{E}}^{\\pm 1}_{k_n}$ (keeping a copy for each multiplicity), \\eqref{operatorexpr}\u00a0is a signed enumeration of certain Gelfand-Tsetlin patterns, where each monomial causes a deformation of the bottom row $k_1,\\ldots,k_n$. It is useful to encode these deformations by \\emph{arrow patterns} as defined in Section~\\ref{sec:recursion}, where we choose $\\swarrow$ if we pick ${\\operatorname{E}}_{k_p}$ from ${\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$, we choose $\\searrow$ if we pick ${\\operatorname{E}}_{k_q}^{-1}$, while we choose $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if we pick $-{\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$. Arranging the $\\binom{n}{2}$ arrows in a triangular manner so that the arrows coming from ${\\operatorname{E}}_{k_p} + {\\operatorname{E}}_{k_q}^{-1} - {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$ are situated in the $p$-th $\\swarrow$-diagonal and the $q$-th $\\searrow$-diagonal, and placing $k_1,\\ldots,k_n$ in the bottom row will allow us to describe the deformation coming from a particular monomial in a convenient way. The combinatorial objects associated with \\eqref{operatorexpr} then consist of a pair of such an arrow pattern and a Gelfand-Tsetlin pattern where the bottom row is a deformation of $k_1,\\ldots,k_n$ as prescribed by the arrow pattern. This will lead directly to the definition of \\emph{shifted Gelfand-Tsetlin patterns}.\n\n\\medskip\n\nA sign comes from picking $- {\\operatorname{E}}_{k_p} {\\operatorname{E}}_{k_q}^{-1}$, but there is also a more subtle appearance. The deformation induced by the arrow pattern can cause a deformation of the increasing bottom row $k_1,k_2,\\ldots,k_n$  into a sequence that is not increasing. Therefore we are in need of an extension of the combinatorial interpretation of \\eqref{gelfandtsetlin} to any sequence $k_1,\\ldots,k_n$ of integers. Such an interpretation was given in \\cite{Fis05} and is repeated below in Section~\\ref{sec:gt}.\n\n\\subsection*{Outline of the bijective proof} Given a sequence $k_1 < \\ldots < k_n$, it suffices to find an injective map from the set of monotone triangles with bottom row $k_1,\\ldots,k_n$ to our shifted Gelfand-Tsetlin patterns associated with $k_1,\\ldots,k_n$ so that the images under this map have positive signs, along with a sign-reversing involution on the set of shifted Gelfand-Tsetlin patterns that are not the image of a monotone triangle.\n\n\\medskip\n\nWe will accomplish something more general, as we will also consider an extension of monotone triangles to all integer sequences $k_1,\\ldots,k_n$, see Section~\\ref{sec:recursion}, along with a sign function on these objects, and prove that the operator formula also holds in this instance. To do that, we will construct a sign-reversing involution on a subset of monotone triangles, another sign-reversing involution on a subset of shifted Gelfand-Tsetlin patterns, and a sign-\\emph{preser\\-ving} bijection between the remaining monotone triangles and the remaining shifted Gelfand-Tsetlin patterns. Note that this is actually equivalent to the construction of a bijection between the (disjoint) union of the ``positive'' monotone triangles and the ``negative'' shifted Gelfand-Tsetlin patterns, and the (disjoint) union of the ``negative'' monotone triangles and the ``positive'' shifted Gelfand-Tsetlin patterns. We call such maps \\emph{sijections} for general signed sets.\n\n\\medskip\n\nThe actual construction here will make use of the recursion underlying monotone triangles. For a monotone triangle with bottom row $k_1,\\ldots,k_n$, the eligible penultimate rows $l_1,\\ldots,l_{n-1}$ are those with\n$$\nk_1 \\le l_1 \\le k_2 \\le l_2 \\le \\ldots \\le l_{n-1} \\le k_n,\n$$\nand $l_1 < l_2 < \\ldots < l_{n-1}$. This establishes a recursion that can be used to construct all monotone triangles. Phrased differently, ``at'' each $k_i$ we need to sum over all $l_{i-1},l_i$ such that $l_{i-1}\u00a0\\le k_i \\le l_i$ and $l_{i-1} < l_{i}$.\\footnote{The degenerate cases $k_1$ and $k_n$ are slightly different.} However, we can split this into the following three cases:\n\\begin{enumerate}\n\\item Consider all $l_{i-1},l_i$ with $l_{i-1} < k_i \\le  l_i$.\n\\item Consider all $l_{i-1},l_i$ with $l_{i-1} \\le  k_i < l_i$.\n\\item Combining (1) and (2), we have done some double counting, thus we need to subtract the intersection, i.e., all $l_{i-1},l_i$ with $l_{i-1} < k_i < l_i$.\n\\end{enumerate}\nThis can be written as a recursion. The \\emph{arrow rows} in Section~\\ref{sec:recursion} are used to describe this recursion: we choose $\\nwarrow$ ``at'' $k_i$ if we are in Case (1), $\\nearrow$ in Case (2), and $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in Case (3). Our main effort will be to show ``sijectively'' that shifted Gelfand-Tsetlin patterns also fulfill this recursion.\n\n\\subsection*{Outline of the paper} The remainder of this paper is devoted to the bijective proof of Theorem~\\ref{operator} (or rather, the more general version with the increasing condition on $k_1,\\ldots,k_n$ dropped). In Section~\\ref{sec:ss} we lay the groundwork by defining concepts like signed sets and sijections, and we extend known concepts such as disjoint union, Cartesian product and composition for ordinary sets and bijections to signed sets and sijections. The composition of sijections will use a variation of the well-known Garsia-Milne involution principle \\cite{GarsiaMilne2,GarsiaMilne1}. Many of the signed sets we will be considering are signed boxes (Cartesian products of signed intervals) or at least involve them, and we define some sijections on them in Section~\\ref{sec:sb}. These sijections will be the building blocks of our bijective proof later on. In\nSection~\\ref{sec:gt} we introduce the extended Gelfand-Tsetlin patterns and construct some related sijections. In Section~\\ref{sec:recursion}, we finally define the extended monotone triangles as well as the shifted Gelfand-Tsetlin patterns (i.e., the combinatorial interpretation of \\eqref{operatorexpr}), and use all the preparation to construct the sijection between monotone triangles and shifted Gelfand-Tsetlin patterns. In the final section, we discuss further projects.\n\n\\medskip\n\nTo emphasize that we are not merely interested in the fact that two signed sets have the same size, but want to use the constructed signed bijection later on, we will be using a convention that is slightly unorthodox in our field. Instead of listing out results as lemmas and theorems with their corresponding proofs, we will be using the Problem--Construction terminology. See for instance \\cite{Voevodsky} and \\cite{Bauer}.\n\n\\section{Signed sets and sijections} \\label{sec:ss}\n\n\\subsection*{Signed sets}\n\nA \\emph{signed set} is a pair of disjoint finite sets: $\\u S = (S^+,S^-)$ with $S^+ \\cap S^- = \\emptyset$. Equivalently, a signed set is a finite set $S$ together with a sign function  $\\sign \\colon S \\to \\{1,-1\\}$. While we will mostly avoid the use of the sign function altogether (with the exception of monotone triangles defined in Section \\ref{sec:recursion}), it is useful to keep this description at the back of one's mind. Note that throughout the paper, signed sets are underlined. We will write $i \\in \\u S$ to mean $i \\in S^+ \\cup S^-$.\n\n\\medskip\n\nThe \\emph{size} of a signed set $\\u S$ is $|\\u S| = |S^+| - |S^-|$. The \\emph{opposite} signed set of $\\u S$ is $- \\u S = (S^-,S^+)$. We have $|{- \\u S}  | = -|\\u S|$. The \\emph{Cartesian product} of signed sets $\\u S$ and $\\u T$ is\n$$\\u S \\times \\u T = (S^+ \\times T^+ \\cup S^- \\times T^-,S^+ \\times T^- \\cup S^- \\times T^+),$$\nand we can similarly (or recursively) define the Cartesian product of a finite number of signed sets. We have\n$$|\\u S \\times \\u T| = |S^+| \\cdot |T^+| + |S^-| \\cdot |T^-| - |S^+| \\cdot |T^-| - |S^-| \\cdot |T^+| = |\\u S| \\cdot |\\u T|.$$\n\nThe intersection of signed sets $\\u S$ and $\\u T$ is defined as $\\u S \\cap \\u T = (S^+ \\cap T^+, S^- \\cap T^-)$, while the union $\\u S \\cup \\u T = (S^+ \\cup T^+, S^- \\cup T^-)$ is only defined when $S^+ \\cap T^- = S^- \\cap T^+ = \\emptyset$. Again, we can extend these definitions to a finite family of signed sets.\n\n\\begin{example}\n One of the crucial signed sets is the \\emph{signed interval}\n $$\\u{[a,b]} = \\begin{cases} ([a,b],\\emptyset) & \\mbox{if } a \\leq b \\\\ (\\emptyset,[b+1,a-1]) & \\mbox{if } a > b \\end{cases}$$\n for $a,b \\in {\\mathbb Z}$, {where $[a,b]$ stands for an interval in $\\mathbb{Z}$ in the usual sense}. We have $\\si{b+1}{a-1} = -\\si a b$ and $|\\u{[a,b]}| = b - a + 1$.\\\\\n We will also see many \\emph{signed boxes}, Cartesian products of signed intervals. Note that $S^+ = \\emptyset$ or $S^- = \\emptyset$ for every signed box $\\u S$.\n\\end{example}\n\n{Signed subsets $\\u T \\subseteq \\u S$ are defined in an obvious manner, in particular, for $s \\in \\u S$, we have}\n$$\\u{\\{s\\}} = \\begin{cases} (\\{s\\},\\emptyset) & \\mbox{if } s \\in S^+ \\\\ (\\emptyset,\\{s\\}) & \\mbox{if } s \\in S^- \\end{cases}.$$\n\nThe \\emph{disjoint union} of signed sets $\\u S$ and $\\u T$ is the signed set\n$$\\u S \\sqcup \\u T = (\\u S \\times (\\{0\\},\\emptyset)) \\cup (\\u T \\times (\\{1\\},\\emptyset))$$\nwith elements $(s,0)$ for $s \\in \\u S$ and $(t,1)$ for $t \\in \\u T$. If $\\u S$ and $\\u T$ are signed sets with $(S^+ \\cup S^-) \\cap (T^+ \\cup T^-) = \\emptyset$, we can identify $\\u S \\cup \\u T$ and $\\u S \\sqcup \\u T$.\n\n\\medskip\n\nMore generally, we can define the disjoint union of a family of signed sets $\\u S_t$, where the family is indexed with a signed set $\\u T$:\n$$\\bigsqcup_{t \\in \\u T} \\u S_t = \\bigcup_{t \\in \\u T} (\\u S_t \\times \\u{\\{t\\}}).$$\n{We get $\\bigsqcup_{t \\in \\u{[0,1]}} \\u S_t = S_0 \\sqcup S_1$. For $a,b \\in \\mathbb{Z}$, we may also write $\\bigsqcup_{i=a}^b \\u S_i$ instead of $\\bigsqcup_{i \\in \\si a b} \\u S_i$.}\n{As for the size, we have $$\\left| \\bigsqcup_{t \\in \\u T} \\u S_t  \\right|= \\sum_{t \\in T} | \\u{S_t} | \\cdot\n| \\u { \\{ t \\} } |.$$}\n\n\\medskip\n\nThe usual properties such as {associativity $(\\u S \\sqcup \\u T) \\sqcup \\u U = \\u S \\sqcup (\\u T \\sqcup \\u U)$ and} {distributivity}\n$(\\u S \\sqcup \\u T) \\times \\u U = \\u S \\times \\u U \\sqcup \\u T \\times \\u U$ also hold. {Strictly speaking, the $=$ sign here and sometimes later on indicates that there is an obvious and natural sign-preserving bijection between the two signed sets.} We summarize a few more basic properties that will be needed in the following and that are easy to prove.\n\\begin{enumerate}\n\\item $$\\bigsqcup\\limits_{\\q l \\in \\si{a_1}{b_1} \\times \\ldots  \\times \\si{a_{n}}{b_{n}}} \\u S_{l_1+c_1,\\ldots,l_n+c_n}  =\n\\bigsqcup\\limits_{\\q l \\in \\si{a_1+c_1}{b_1+c_1} \\times \\ldots  \\times \\si{a_{n}+c_{n}}{b_{n}+c_{n}}} \\u S_{l_1,\\ldots,l_n}$$\n\\item $$\n\\bigsqcup_{t \\in \\u T} \\bigsqcup_{u \\in \\u U} \\u S_{t,u} = \\bigsqcup_{(u,t) \\in \\u U \\times \\u T}\u00a0\n\\u S_{t,u} =    \\bigsqcup_{(t,u) \\in \\u T \\times \\u U}\u00a0\\u S_{t,u} =\n\\bigsqcup_{u \\in \\u U} \\bigsqcup_{t \\in \\u T} \\u S_{t,u}\n$$\n\\item\n$$\n\\bigsqcup_{t \\in \\bigsqcup_{u \\in \\u U} \\u T_u} \\u S_t = \\bigsqcup_{u \\in \\u U} \\bigsqcup_{t \\in \\u T_u} \\u S_t\n$$\n\\item $$\n- \\bigsqcup_{t \\in \\u T} \\u S_t  = \\bigsqcup_{t \\in \\u T} - \\u S_t= \\bigsqcup_{t \\in -\\u T} \\u S_t.\n$$\n\\end{enumerate}\n\n\\medskip\n\n\\subsection*{Sijections}\n\nThe role of bijections for signed sets is played by ``signed bijections'', which we call \\emph{sijections}. A sijection $\\varphi$ from $\\u S$ to $\\u T$,\n$$\\varphi \\colon \\u S \\Rightarrow \\u T,$$\nis an involution on the set $(S^+ \\cup S^-) \\sqcup (T^+ \\cup T^-)$ with the property $\\varphi(S^+ \\sqcup T^-) = S^- \\sqcup T^+$, {where $\\sqcup$ refers to the disjoint union for ordinary (``unsigned'') sets.}\n It follows that also $\\varphi(S^- \\sqcup T^+) = S^+ \\sqcup T^-$. {There is an obvious sijection $\\id_{\\u S} \\colon \\u S \\Rightarrow \\u S$.}\n\\medskip\n\nWe can think of a sijection as a collection of a sign-reversing involution on a subset of $\\u S$, a sign-reversing involution on a subset of $\\u T$, and a {sign-preserving} matching between the remaining elements of $\\u S$ with the remaining elements of $\\u T$. When $S^- = T^- = \\emptyset$, the signed sets can be identified with ordinary sets, and a sijection in this case is simply a bijection.\n\n\\medskip\n\n{A sijection is a  manifestation of the fact that two signed sets have the same size. Indeed,}\nif there exists a sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$, we have $|S^+| + |T^-| = |S^+ \\sqcup T^-| = |S^- \\sqcup T^+| = |S^-| + |T^+|$ and therefore $|\\u S| = |S^+| - |S^-| = |T^+| - |T^-| = |\\u T|$. A sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$ has an inverse $\\varphi^{-1} \\colon \\u T \\Rightarrow \\u S$ that we obtain by identifying $(T^+ \\cup T^-) \\sqcup (S^+ \\cup S^-)$ with $(S^+ \\cup S^-) \\sqcup (T^+ \\cup T^-)$.\n\n\\medskip\n\n\\comment{A sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$ is \\emph{simple} if  $\\varphi(S^+) = T^+$ and  $\\varphi(S^-) = T^-$.}\n\n\\medskip\n\nFor a signed set $\\u S$, there is a natural sijection $\\varphi$ from $\\u S \\sqcup (- \\u S)\n$\nto the empty signed set $\\u \\emptyset = (\\emptyset,\\emptyset)$. Indeed, the {involution} should be defined on {$(S^+ \\times \\{0\\} \\cup S^- \\times\\{1\\}) \\cup (S^- \\times \\{0\\} \\cup S^+ \\times \\{1\\})$ and map $S^+ \\times \\{0\\} \\cup S^- \\times\\{1\\}$ to $S^+ \\times \\{1\\} \\cup S^- \\times \\{0\\}$}, and {so} we can take $\\varphi((s,0),0) = ((s,1),0)$, $\\varphi((s,1),0) = ((s,0),0)$. {Note that in general, a sijection from a signed set $\\u S$ to $\\u \\emptyset$ is simply a sign-reversing involution on $\\u S$, in other words, a bijection between $S^+$ and $S^-$.} \\comment{In other words, a sijection $\\varphi \\colon \\u S \\Rightarrow \\u \\emptyset$ is equivalent to a simple sijection $\\varphi \\colon \\u S \\Rightarrow - \\u S$.}\n\n\\medskip\n\n{If we have a sijection $\\varphi \\colon \\u S \\Rightarrow \\u T$, there is a natural sijection $-\\varphi \\colon -\\u S \\Rightarrow -\\u T$ (as a map, it is actually precisely the same).}\n\n\\medskip\n\n{If we have sijections $\\varphi_i \\colon \\u S_i \\Rightarrow \\u T_i$ for $i=0,1$, then there is a natural sijection $\\varphi \\colon \\u S_0 \\sqcup \\u S_1 \\Rightarrow \\u T_0 \\sqcup \\u T_1$.}\nMore interesting ways to create new sijections are described below in Proposition \\ref{prop:sijections}, but we will need this in our first construction {for the special case $\\u S_0 = \\u T_0$ and $\\varphi_0 = \\id_{\\u S_0}$.}\n\n\\medskip\n\nTo motivate our first result, note that if $a \\leq b \\leq c$ or $c < b < a$, then $\\si a c = \\si a b \\cup \\si{b+1}c= \\si a b \\sqcup \\si{b+1}c$. Of course, this does not hold in general; for $a = 1$, $b = 8$, $c = 5$, we have $\\si 1 5 = (\\{1,2,3,4,5\\},\\emptyset)$, $(\\si 1 8 \\sqcup \\si 9 5)^+ = (\\{(1,0),(2,0),(3,0),(4,0),(5,0),(6,0),(7,0),(8,0)\\}$ and $(\\si 1 8 \\sqcup \\si 9 5)^- = \\{(6,1),(7,1),(8,1)\\})$. The following, however, tells us that there is in general a sijection between {$\\si a c$ and\n$\\si a b \\sqcup \\si{b+1}c$.}\nThis map will be the crucial building block for more complicated sijections.\n\n\\begin{problem} \\label{prob:alpha}\n Given $a,b,c \\in {\\mathbb Z}$, construct a sijection\n $$\\alpha = \\alpha_{a,b,c} \\colon \\si a c \\Rightarrow \\si a b \\sqcup \\si{b+1}c.$$\n\\end{problem}\n\\begin{proof}[Construction]\n For $a \\leq b \\leq c$ and $c < b < a$, there is nothing to prove. For, say, $a \\leq c < b$, we have\n $$\\si a b \\sqcup \\si{b+1}c = (\\si a c \\sqcup \\si{c+1} b) \\sqcup \\si{b+1}c = \\si a c \\sqcup (\\si{c+1} b \\sqcup (-\\si {c+1}{b}))$$\n and since there is a sijection $\\si{c+1} b \\sqcup (-\\si {c+1}{b}) \\Rightarrow \\u \\emptyset$, we get a sijection $\\si a b \\sqcup \\si{b+1}c \\Rightarrow \\si a c$. The cases $b < a \\leq c$, $b \\leq c < a$, and $c < a \\leq b$ are analogous. \\comment{Note that in all cases, $\\si a b$ and  $\\si{b+1}c$ are either disjoint or one set is contained in the other.}\n\\end{proof}\n\n\\medskip\n\nThe following proposition describes composition, Cartesian product, and disjoint union of sijections. The composition is a variant of the well-known Garsia-Milne involution principle. All the statements are easy to prove{, and the proofs are left to the reader.}\n\n\\begin{prop} \\leavevmode \\label{prop:sijections}\n \\begin{enumerate}\n  \\item (Composition) Suppose that we have sijections $\\varphi \\colon \\u S \\Rightarrow \\u T$ and $\\psi \\colon \\u T \\Rightarrow \\u U$. For $s \\in \\u S$ (resp.\\ $u \\in \\u U$), define $\\psi \\circ \\varphi(s)$ {(resp.\\ $\\psi \\circ \\varphi(u)$)} as the last well-defined element in the sequence $s, \\varphi(s), \\psi(\\varphi(s)), \\varphi(\\psi(\\varphi(s))),\\ldots$ (resp.\\ $u, \\psi(u), \\varphi(\\psi(u)), \\psi(\\varphi(\\psi(u))),\\ldots$). Then $\\psi \\circ \\varphi$ is a well-defined sijection from $\\u S$ to $\\u U$. \\comment{If $\\varphi$ and $\\psi$ are simple, so is $\\psi \\circ \\varphi$.}\n  \\item (Cartesian product) Suppose we have sijections $\\varphi_i \\colon \\u S_i \\Rightarrow \\u T_i$, $i = 1,\\ldots,k$. Then $\\varphi = \\varphi_1 \\times \\cdots \\times \\varphi_k$, defined by\n$$\\varphi(s_1,\\ldots,s_k) = \\begin{cases} (\\varphi_1(s_1),\\ldots,\\varphi_k(s_k)) & \\mbox{if } \\varphi_i(s_i) \\in \\u T_i \\mbox{ for } i=1,\\ldots,k \\\\ (s_1,\\ldots,s_{j-1},\\varphi_j(s_j),s_{j+1},\\ldots,s_k) & \\mbox{if } \\varphi_j(s_j) \\in \\u S_j, \\varphi_i(s_i) \\in \\u T_i \\mbox{ for } i < j \\end{cases}$$\n{if $(s_1,\\ldots,s_k) \\in \\u S_1 \\times  \\dots \\times \\u S_k$ and\n$$\\varphi(t_1,\\ldots,t_k) = \\begin{cases} (\\varphi_1(t_1),\\ldots,\\varphi_k(t_k)) & \\mbox{if } \\varphi_i(t_i) \\in \\u S_i \\mbox{ for } i=1,\\ldots,k \\\\ (t_1,\\ldots,t_{j-1},\\varphi_j(t_j),t_{j+1},\\ldots,t_k) & \\mbox{if } \\varphi_j(t_j) \\in \\u T_j, \\varphi_i(t_i) \\in \\u S_i \\mbox{ for } i < j \\end{cases}$$\nif $(t_1,\\ldots,t_k) \\in \\u T_1  \\times \\dots \\times \\u T_k$,}\n  is a well-defined sijection from $\\u S_1 \\times \\cdots \\times \\u S_k$ to $\\u T_1 \\times \\cdots \\times \\u T_k$. \\comment{If $\\varphi_i$ are all simple sijections, so is $\\varphi$.}\n  \\item (Disjoint union) {Suppose we have signed sets $\\u T, \\u{\\widetilde T}$ and a sijection $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$. Further\\-more, suppose that for every $t \\in \\u T \\sqcup \\u{\\widetilde T}$, we have a signed set $\\u S_t$ and a sijection $\\varphi_t \\colon \\u S_t \\Rightarrow \\u S_{\\psi(t)}$ satisfying $\\varphi_{\\psi(t)} = \\varphi_t^{-1}$.  Then $\\varphi = \\bigsqcup_{t \\in \\u T \\sqcup \\u{\\widetilde T}} \\varphi_t$, defined by\n$$\\varphi(s_t,t) = \\begin{cases}\n(\\varphi_t(s_t),t) & \\mbox{if } t \\in \\u T \\sqcup \\u {\\widetilde T}, s_t \\in \\u S_t, \\varphi_t(s_t) \\in \\u S_t \\\\\n(\\varphi_t(s_t),\\psi(t)) & \\mbox{if } t \\in \\u T \\sqcup \\u {\\widetilde T}, s_t \\in \\u S_t, \\varphi_t(s_t) \\in \\u {S}_{\\psi(t)} \\\\\n\\end{cases}\n$$\nis a sijection $\\bigsqcup_{t \\in \\u T} \\u S_t \\Rightarrow \\bigsqcup_{t \\in \\u {\\widetilde T}} \\u {\\widetilde S}_t$.} \\comment{If $\\psi$ and $\\varphi_t$ are all simple sijections, so is $\\varphi$.}\n\\end{enumerate}\n\\end{prop}\n\n\n{One} important special case of Proposition \\ref{prop:sijections} (3) is $\\u T =  \\u {\\widetilde T}$ and $\\psi = \\id$. We have two sets of signed sets indexed by $\\u T$, $\\u S_{(t,0)} =: \\u{S}^{{0}}_t$ and $\\u S_{(t,1)} =: \\u{S}^{{1}}_t$, and sijections $\\varphi_t \\colon \\u{S}^{{0}}_t \\Rightarrow \\u{S}^{{1}}_t$. By the proposition, these sijections have a disjoint union that is a sijection $\\bigsqcup_{t \\in \\u T} \\u S^{{0}}_t \\Rightarrow \\bigsqcup_{t \\in \\u T} \\u S^{{1}}_t$.\n\n\\medskip\n\nBy the proposition, the relation\n$$\\u S \\approx \\u T \\iff \\mbox{there exists a sijection from } \\u S \\mbox{ to } \\u T$$\nis an equivalence {relation}.\n\n\\subsection*{Elementary signed sets and normal sijections}\n\nOften, we will be interested in disjoint unions of Cartesian products of signed intervals. An element of such a signed set is a pair, consisting of a tuple of integers and an element of the indexing signed set. Intuitively, the first one is ``more important'', as the second one serves just as an index. We formalize this notion in the following definition.\n\n\\begin{definition}\n A signed set $\\u A$ is \\emph{elementary of dimension $n$ and depth $0$} if its elements are in ${\\mathbb Z}^n$. A signed set $\\u A$ is \\emph{elementary of dimension $n$ and depth $d$}, $d \\geq 1$, if it is of the form\n $$\\bigsqcup_{t \\in \\u T} \\u S_t,$$\n where $\\u T$ is a signed set, and $\\u S_t$ are all signed sets of dimension $n$ and depth at most $d-1$, with the depth of at least one of them equal to $d-1$. A signed set $\\u A$ is \\emph{elementary of dimension $n$} if it is an elementary signed set of dimension $n$ and depth $d$ for some $d \\in {\\mathbb N}$.\\\\\n The \\emph{projection map} on an elementary set of dimension $n$ is the map\n $$\\xi \\colon \\u A \\to {\\mathbb Z}^n$$\n defined as follows. If the depth of $\\u A$ is $0$, then $\\xi$ is simply the inclusion map. Once $\\xi$ is defined on elementary signed sets of depth $< d$, and the depth of $\\u A$ is $d$, then $\\u A = \\bigsqcup_{t \\in \\u T} \\u S_t$, where $\\xi$ is defined on all $\\u S_t$. Then define $\\xi(s,t) = \\xi(s)$ for $(s,t) \\in \\u A$.\\\\\n A sijection $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$ between elementary signed sets $\\u T$ and $\\u{\\widetilde T}$ of the same dimension is \\emph{normal} if $\\xi(\\psi(t)) = \\xi(t)$ for all $t \\in \\u T \\sqcup \\u{\\widetilde T}$.\n\\end{definition}\n\nSimple examples of elementary signed sets are $\\si a c$, $\\si a b \\sqcup \\si {b+1} c$ and $\\si a c \\sqcup (\\si a b \\sqcup \\si {b+1} c)$. They are all of dimension $1$ and depth $0$, $1$ and $2$, respectively.\\footnote{{To avoid ambiguity, we should consider signed intervals in this case to be subsets of ${\\mathbb Z}^1$ ($1$-tuples of integers), not ${\\mathbb Z}$. Otherwise, $\\si{0}{1} \\sqcup \\si{2}{3} = (\\{(0,0),(1,0),(2,1),(3,1)\\},\\emptyset)$, and this can be seen either as an elementary set of dimension $1$ and depth $1$, or as an elementary signed set of dimension $2$ and depth $0$. So the interpretation depends on the ``representation'' of the set as disjoint union. Instead, we should understand $\\si{0}{1} \\sqcup \\si{2}{3}$ to mean $ (\\{((0),0),((1),0),((2),1),((3),1)\\},\\emptyset)$, with dimension $1$ and depth $1$. For coding, the distinction is important, but in the paper we nevertheless think of elements of signed intervals as integers.}} It is easy to see that the sijection $\\alpha_{a,b,c}$ from Problem \\ref{prob:alpha} is normal.\n\nLet us illustrate this with the example $a = 1$, $b = 5$, $c = 3$. We have $\\si a c = (\\{1,2,3\\},\\emptyset)$ and $\\si a b \\sqcup \\si {b+1} c = (\\{(1,0),(2,0),(3,0),(4,0),(5,0)\\},\\{(4,1),(5,1)\\})$. The sijection $\\alpha_{1,5,3}$ is the involution on $\\si 1 3 \\sqcup (\\si 1 5 \\sqcup \\si {6} 3)$ defined by\n$$(1,0) \\leftrightarrow ((1,0),1), \\quad (2,0) \\leftrightarrow ((2,0),1), \\quad (3,0) \\leftrightarrow ((3,0),1),$$\n$$((4,0),1) \\leftrightarrow ((4,1),1), \\quad ((5,0),1) \\leftrightarrow ((5,1),1).$$\nSince $\\xi(i,0) = i$ for $i = 1,2,3$, $\\xi((i,0),1) = i$ for $i = 1,2,3,4,5$ and $\\xi((i,1),1) = i$ for $i=4,5$, $\\alpha_{1,5,3}$ is indeed normal.\n\n\\medskip\n\nOther examples of elementary signed sets appear in the statements of Problems \\ref{prob:beta} and \\ref{prob:gamma} (in both cases, they are of dimension $n-1$).\n\n\\medskip\n\nNormality is preserved under Cartesian product, disjoint union etc. For example, the sijection\n\\begin{multline*}\n  \\si{a_1}{c_1} \\times \\si{a_2}{c_2} \\Rightarrow \\\\\n  \\si{a_1}{b_1} \\times \\si{a_2}{b_2} \\sqcup \\si{a_1}{b_1} \\times \\si{b_2+1}{c_2} \\sqcup \\si{b_1+1}{c_1} \\times \\si{a_2}{b_2} \\sqcup \\si{b_1+1}{c_1} \\times \\si{b_2+1}{c_2},\n\\end{multline*}\nobtained by using $\\alpha_{a_1,b_1,c_1} \\times \\alpha_{a_2,b_2,c_2}$ and distributivity on disjoint unions, is normal.\n\n\\medskip\n\nThe main reason normal sijections are important is that they give a very natural special case of Proposition \\ref{prop:sijections} (3). Suppose that $\\u T$ and $\\u{\\widetilde T}$ are elementary signed sets of dimension $n$, and that $\\psi \\colon \\u T \\Rightarrow \\u{\\widetilde T}$ is a normal sijection. Furthermore, suppose that we have a signed set $\\u S_{\\q k}$ for every $\\q k \\in {\\mathbb Z}^n$. Then we have a sijection\n$$\\bigsqcup_{t \\in \\u T} \\u S_{\\xi(t)} \\Rightarrow \\bigsqcup_{t \\in \\u{\\widetilde T}} \\u S_{\\xi(t)}.$$\nIndeed, Proposition \\ref{prop:sijections} gives us a sijection provided that we have a sijection $\\varphi_t \\colon \\u S_{\\xi(t)} \\Rightarrow \\u S_{\\xi(\\psi(t))}$ satisfying $\\varphi_{\\psi(t)} = \\varphi_t^{-1}$ for every $t \\in \\u T \\sqcup \\u{\\widetilde T}$. But since $\\xi(\\psi(t)) = \\xi(t)$, we can take $\\varphi_t$ to be the identity.\n\n\\comment{The following observation is crucial: Suppose $\\u T \\approx \\u {\\widetilde T}$, then in general we do not have\n\\begin{equation}\n\\label{indexset_equivalence}\n\\bigsqcup_{t \\in \\u T} S_t \\approx \\bigsqcup_{t \\in \\u {\\widetilde T}} S_t.\n\\end{equation}\nHowever, if $\\u T=\\si{a}{c}$ and $\\u {\\widetilde T}=\\si{a}{b} \\sqcup \\si{b+1}{c}$ (see Problem~\\ref{prob:alpha}), then the identity is true because either $\\si{a}{b}$ and\n$\\si{b+1}{c}$ are disjoint (in which case the two boxes have the same sign) or one set is contained in the other (in which case the two boxes have opposite sign).}\n\n\\section{Some sijections on signed boxes} \\label{sec:sb}\n\nThe first sijection in this section will serve as the base of induction for Problem \\ref{prob:pi}.\n\n\\comment{\n\n\\begin{example} \\label{ex:toempty}\n  For $a,b \\in {\\mathbb Z}$, define $\\u T = \\si{a+1}{b+1} \\times \\si a b$. Then $\\psi: \\u T \\Rightarrow \\u T$, defined by $\\psi((l_1,l_2),i) = ((l_2+1,l_1-1),1-i)$ for $l_1 \\in \\si{a+1}{b+1}, l_2 \\in \\si a b, i \\in \\{0,1\\}$, is a (non-normal) sijection. Define $\\u S_{((l_1,l_2),i)} = (-1)^i \\si{l_1}{l_2}$. Since $\\u S_{((l_2+1,l_1-1),i)} = - \\u S_{((l_1,l_2),i)}$ and $\\u S_{((l_1,l_2),1-i)} = - \\u S_{((l_1,l_2),i)}$, we have $\\u S_{\\psi((l_1,l_2),i)} = \\u S_{((l_1,l_2),i)}$. Therefore we can use Proposition \\ref{prop:sijections} with $\\varphi_t = \\id$, and we obtain a sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} -\\si{l_1}{l_2},$$\n  which is equivalent to a sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\u \\emptyset.$$\n\\end{example}}\n\n\\begin{example} \\label{ex:toempty}\n  For $a,b \\in {\\mathbb Z}$, we have a normal sijection\n  $$\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2} \\Rightarrow \\u \\emptyset$$\n  defined by $\\varphi((x,(l_1,l_2)),0) = ((x,(l_2+1,l_1-1)),0)$. It is well defined because $(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b$ if and only if $(l_2+1,l_1-1) \\in \\si{a+1}{b+1} \\times \\si a b$, and because $x \\in \\si{l_1}{l_2}$ if and only if $x \\in \\si{l_2+1}{l_1-1}$.\n\\end{example}\n\nNote that the $0$ as the second coordinate in the example comes from the fact that a sijection in question is an involution on the disjoint union\n$$\\left(\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2}\\right) \\sqcup \\u \\emptyset = \\left(\\bigsqcup_{(l_1,l_2) \\in \\si{a+1}{b+1} \\times \\si a b} \\si{l_1}{l_2}\\right) \\times \\u{\\{0\\}} \\cup \\u \\emptyset \\times \\u{\\{1\\}}.$$\nWe could be a little less precise and write $\\varphi(x,(l_1,l_2)) = (x,(l_2+1,l_1-1))$ without causing confusion.\n\n\\medskip\n\nThe following generalizes the construction of Problem \\ref{prob:alpha}; indeed, for $n = 2$ the construction gives a sijection from $[a_1,b_1]$ to $[a_1,x] \\sqcup (-[b_1+1,x])$.\n\n\\begin{problem} \\label{prob:beta}\n Given $\\q a =(a_1,\\ldots,a_{n-1}) \\in {\\mathbb Z}^{n-1}$, $\\q b=(b_1,\\ldots,b_{n-1}) \\in {\\mathbb Z}^{n-1}$, $x \\in {\\mathbb Z}$, construct a normal sijection\n $$ \\beta = \\beta_{\\q a,\\q b,x} \\colon \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x,$$\n where $\\u S_i = (\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$\n\\end{problem}\n\n{Note that $(\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$ can be identified with\n$(\\{a_i\\},\\{b_i+1\\})$ if $a_i \\not= b_i+1$.}\n\n\\begin{proof}[Construction]\n The proof is by induction, with the case $n = 1$ being trivial and the case $n=2$ was constructed in Problem~\\ref{prob:alpha}. Now, for $n \\ge 3$,\n \\begin{multline*}\n   \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\approx \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_2,\\ldots,l_{n-1}) \\in \\u S_2 \\times \\cdots \\times \\u S_{n-1}} \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}}x \\\\\n   \\approx \\left( \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} \\si{a_2}{l_3} \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\ \\sqcup \\left( \\si{a_1}{b_1} \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} (-\\si{b_2+1}{l_3}) \\times \\cdots \\times \\si{l_{n-1}}x \\right),\n\\end{multline*}\nwhere we used induction for the first equivalence, and distributivity and the fact that $S_2 = (\\{a_2\\},\\emptyset) \\sqcup (\\emptyset,\\{b_2+1\\})$ for the second equivalence. By Problem~\\ref{prob:alpha} and Proposition~\\ref{prop:sijections} (2), there exists a sijection from the last expression to\n\\begin{multline*}\n   \\left( \\left(\\si{a_1}{a_2} \\sqcup (-\\si{b_1+1}{a_2}) \\right) \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} \\si{a_2}{l_3} \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\\n   \\sqcup \\left( \\left( \\si{a_1}{b_2+1} \\sqcup (-\\si{b_1+1}{b_2+1}) \\right) \\times \\bigsqcup_{\\q (l_3,\\ldots,l_{n-1}) \\in \\u S_3 \\times \\cdots \\times \\u S_{n-1}} (-\\si{b_2+1}{l_3}) \\times \\cdots \\times \\si{l_{n-1}}x \\right) \\\\\n   \\approx \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\cdots \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x,\n\\end{multline*}\nwhere for the last equivalence we have again used distributivity. Normality follows from the normality of all the sijections involved in the construction.\n\\end{proof}\n\n\\begin{problem} \\label{prob:gamma}\n Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a normal sijection\n \\begin{multline*}\n   \\gamma = \\gamma_{\\q k,x} \\colon \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n   \\Rightarrow \\bigsqcup_{i = 1}^n \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n   \\sqcup \\bigsqcup_{i = 1}^{n-2} \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots .\n   \\end{multline*}\n \\end{problem}\n\n\\begin{proof}[Construction] The proof is by induction with respect to $n$. The case $n=1$ is trivial, and $n=2$ is Problem~\\ref{prob:alpha}. Now take $n > 2$. By the induction hypothesis (for $(k_1,\\ldots,k_{n-1})$ and $x+1$), we have\n\\begin{multline*}\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\approx \\bigg( \\bigsqcup_{i = 1}^{n-1} \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-2}}{k_{n-1}}\\\\\n  \\sqcup \\bigsqcup_{i = 1}^{n-3} \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\cdots \\times \\si{k_{n-2}}{k_{n-1}} \\bigg) \\times \\si{k_{n-1}}{k_n}.\n  \\end{multline*}\nWe use distributivity. We keep all terms except the one corresponding to $i = n-1$ in the first part. Because\n\\begin{multline*}\n\\si{k_{n-2}}{x+1} \\times \\si{k_{n-1}}{k_n} \\approx \\si{k_{n-2}}{x+1} \\times (\\si{k_{n-1}}x \\sqcup \\si{x+1}{k_n}) \\\\\n\\approx (\\si{k_{n-2}}{k_{n-1}} \\sqcup \\si{k_{n-1}+1}{x+1}) \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-2}}{x+1} \\times \\si{x+1}{k_n} \\\\\n{\\approx \\si{k_{n-2}}{k_{n-1}} \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-1}+1}{x+1} \\times \\si{k_{n-1}}x \\sqcup \\si{k_{n-2}}{x+1} \\times \\si{x+1}{k_n}},\n\\end{multline*}\nwe obtain the required Cartesian products for the first term on the right-hand side at $i = n$, the second term at $i = n-2$, and the first term at $i = n-1$. Again, normality follows from the fact that $\\alpha$ is normal.\n\\end{proof}\n\n\n\\section{Gelfand-Tsetlin patterns} \\label{sec:gt}\n\nUsing our definition of a disjoint union of {signed sets}, it is easy to define generalized Gelfand-Tsetlin patterns, or GT patterns for short (compare with \\cite{Fis05}).\n\n\\begin{definition}\n For $k \\in {\\mathbb Z}$, define ${\\GT(k) = (\\{\\cdot\\},\\emptyset)}$, \n and for $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$, define {recursively}\n $$\\GT(\\q k) = \\GT(k_1,\\ldots,k_n) = \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(l_1,\\ldots,l_{n-1}).$$\n\\end{definition}\n\nIn particular, $\\GT(a,b) \\approx \\si a b$.\n\n\\medskip\n\nOf course, one can think of an element of $\\GT(\\q k)$ in the usual way, as a triangular array $A=(A_{i,j})_{1 \\leq j \\leq i \\le n}$ of $\\binom {n+1}2$ numbers,\narranged as\n$$\n\\begin{array}{ccccccccc}\n &&&& A_{1,1} &&&& \\\\\n &&& A_{2,1} && A_{2,2} &&& \\\\\n && A_{3,1} && A_{3,2} && A_{3,3} && \\\\\n & \\iddots & \\vdots & \\ddots & \\vdots & \\iddots & \\vdots & \\ddots &  \\\\\nA_{n,1} && A_{n,2} && \\ldots && \\ldots && A_{n,n},\n \\end{array}\n$$\nso that $A_{i+1,j} \\leq A_{i,j} \\leq A_{i+1,j+1}$ or $A_{i+1,j} > A_{i,j} > A_{i+1,j+1}$ for $1 \\leq j \\leq i < n$, and $A_{n,i}=k_i$. The sign of such an array is $(-1)^m$, where $m$ is the number of $(i,j)$ with $a_{i,j} > a_{i,j+1}$.\n\n\\medskip\n\nSome crucial sijections for GT patterns are given by the following constructions.\n\n\\begin{problem} \\label{prob:rho}\n  Given $\\q a =(a_1,\\ldots,a_{n-1}) \\in {\\mathbb Z}^{n-1}$, $\\q b=(b_1,\\ldots,b_{n-1}) \\in {\\mathbb Z}^{n-1}$, $x \\in {\\mathbb Z}$, construct a sijection\n  $$ \\rho = \\rho_{\\q a,\\q b,x} \\colon \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(l_1,\\ldots,l_{n-1},x),$$\n  where $\\u S_i = (\\{a_i\\},\\emptyset) \\sqcup (\\emptyset,\\{b_i+1\\})$.\n\\end{problem}\n\\begin{proof}[Construction]\n In Problem \\ref{prob:beta}, we constructed a normal sijection\n $$ \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}} \\Rightarrow \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x.$$\n By Proposition~\\ref{prop:sijections} (3) (see the comment at the end of Section \\ref{sec:ss}), this gives a sijection\n $$ \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_{n-1}}{b_{n-1}}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{\\q m \\in \\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x} \\GT(\\q m).$$\n By basic sijection constructions, we get that this is equivalent to\n $$\\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\bigsqcup_{\\q m \\in \\si{l_1}{l_2} \\times \\si{l_2}{l_3} \\times \\cdots \\times \\si{l_{n-2}}{l_{n-1}} \\times \\si{l_{n-1}} x} \\GT(\\q m),$$\n and by definition of $\\GT$, this is equal to $\\bigsqcup_{\\q (l_1,\\ldots,l_{n-1}) \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(l_1,\\ldots,l_{n-1},x)$.\n\\end{proof}\n\nThe result is important because while it adds a dimension to GT patterns, it (typically) greatly reduces the size of the indexing signed set. {In fact, there is an analogy to the fundamental theorem of calculus: instead of extending the disjoint union over the entire signed box, it suffices to consider the boundary; $x$ corresponds in a sense to the constant of integration.}\n\n\\begin{problem} \\label{prob:pi}\n  Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $i$, $1 \\leq i \\leq n-1$, construct a sijection\n $$\\pi = \\pi_{\\q k,i} \\colon \\GT(k_1,\\ldots,k_n) \\Rightarrow -\\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,k_i-1,k_{i+2},\\ldots,k_n).$$\n Given $\\q a =(a_1,\\ldots,a_n) \\in {\\mathbb Z}^n$,  $\\q b =(b_1,\\ldots,b_n) \\in {\\mathbb Z}^n$ such that for some $i$, $1 \\leq i \\leq n-1$, we have $a_{i+1} = a_i - 1$ and $b_{i+1} = b_i - 1$, construct a sijection\n $$\\sigma = \\sigma_{\\q a,\\q b,i} \\colon \\bigsqcup_{\\q l \\in \\si{a_1}{b_1} \\times \\cdots \\times \\si{a_n}{b_n}} \\GT(\\q l) \\Rightarrow \\u \\emptyset.$$\n\\end{problem}\n\\begin{proof}[Construction]\n The proof is by induction, with the induction step for $\\pi$ using $\\sigma$ and vice versa. For $n = 1$, there is nothing to prove. For $n=2$ and $i=1$, the existence of $\\pi$ follows from the statement $\\si{k_1}{k_2} = -\\si{k_2+1}{k_1-1}$, and $\\sigma$ was constructed in Example \\ref{ex:toempty}. Assume that $n > 2$ and $1 < i < n-1$. We have\n$$\\GT(k_1,\\ldots,k_n) =  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}).$$\nBy using $\\id \\times \\cdots \\times \\id \\times \\alpha_{k_{i-1},k_{i+1}+1,k_i} \\times \\id \\times \\alpha_{k_{i+1},k_i-2,k_{i+2}} \\times \\id \\times \\cdots \\times \\id$ and distributivity, we get a normal sijection\n\\begin{multline*}\n\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\Rightarrow \\\\\n\\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\\\\n\\sqcup \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}\n\\end{multline*}\n By Proposition~\\ref{prop:sijections} (3), this gives a sijection\n \\begin{multline*}\n  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}) \\Rightarrow \\\\\n   \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} - \\GT(l_1,\\ldots,l_{n-1}) \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_{i+1}+1} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1})  \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i}-1}{k_{i+2}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1})\n \\\\ \\sqcup  \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} -\\GT(l_1,\\ldots,l_{n-1}).\n \\end{multline*}\n By definition, the first signed set on the right-hand side is $-\\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,k_i-1,k_{i+2},\\ldots,k_n)$. The other three disjoint unions all satisfy the condition needed for the existence of $\\sigma$ {(for $i$, for $i-1$ and for both, $i-1$ and $i$, respectively)}, and hence we can siject them to $\\u \\emptyset$.\\\\\n If $i = 1$ or $i = n-1$, the proof is similar but easier (as we only have to use $\\alpha$ once, and we get only two factors after using distributivity). Details are left to the reader.\\\\\n Now take $\\q l = (l_1,\\ldots,l_n)$ and $\\q{l'}=(l_1,\\ldots,l_{i-1},l_{i+1}+1,l_i-1,l_{i+2},\\ldots,l_n)$. The sijection $\\sigma$ can then be defined as\n $$\\sigma_{\\q a,\\q b,i}(A,\\q l) = \\begin{cases} (\\pi_{\\q l,i}(A),\\q l) & \\mbox{if }\\pi_{\\q l,i}(A) \\in \\GT(\\q l) \\\\ (\\pi_{\\q l,i}(A),\\q {l'}) & \\mbox{if } \\pi_{\\q {l},i}(A) \\in \\GT(\\q {l'})  \\end{cases}.$$\n It is easy to check that this is a sijection. Compare with Example \\ref{ex:toempty}.\n \\comment{These follow immediately from such sijections on each of the following two signed sets,\n $$\n \\bigsqcup_{(l_{i-1},l_i) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}}  \\GT(l_1,\\ldots,l_{n-1})\n$$\nand\n$$\n \\bigsqcup_{ (l_i,l_{i+1}) \\in  \\si{k_{i+1}+1}{k_i-1} \\times \\si{k_{i+1}}{k_{i}-2}}  \\GT(l_1,\\ldots,l_{n-1}),\n$$\nfixing arbitrary integers $l_1,\\ldots,l_{i-2},l_{i+1},\\ldots,l_{n-1}$ in the first case and $l_1,\\ldots,l_{i-1},l_{i+2},\\ldots,l_{n-1}$ in the second case. As for the first case, we let\n$$\nA \\times \\u {\\{ (l_{i-1},l_{i}) \\}}  \\in  \\bigsqcup_{(l_{i-1},l_i) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}}  \\GT(l_1,\\ldots,l_{n-1})\n$$\nand map it to\n$$\n\\begin{cases}\n\\,\\, \\pi_{\\q l,i-1} (A) \\times \\u {\\{ (l_{i-1},l_{i}) \\}} & \\mbox{if } \\pi_{\\q l,i-1}(A) \\in  \\GT(l_1,\\ldots,l_{n-1}) \\\\\u00a0\n-\\pi_{\\q l,i-1} (A) \\times \\u {\\{ (l_{i}+1,l_{i-1}-1) \\}} & \\mbox{if } \\pi_{\\q l,i-1}(A) \\in  -\\GT(l_1,\\ldots,l_{i-2},l_i+1,l_{i-1}-1,l_{i+1},\\ldots,l_{n-1})\n\\end{cases}\n$$\nwhich is feasible since $(l_{i}+1,l_{i-1}-1) \\in  \\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}$, and $\\u {\\{ (l_{i-1},l_{i}) \\}}$ and\n$\\u {\\{ (l_{i}+1,l_{i-1}-1) \\}}$ have the same sign in $\\si{k_{i+1}+2}{k_i} \\times \\si{k_{i+1}+1}{k_i-1}$. The second case is}\n\\end{proof}\n\n\\begin{problem} \\label{prob:tau}\n Given $\\q k =(k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n $$\\tau = \\tau_{\\q k,x} \\colon \\GT(k_1,\\ldots,k_n) \\Rightarrow \\bigsqcup_{i = 1}^n \\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n).$$\n\\end{problem}\n\\begin{proof}[Construction]\n In Problem \\ref{prob:gamma}, we constructed a normal sijection\n \\begin{multline*}\n   \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n} \\Rightarrow \\bigsqcup_{i = 1}^n \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}\\\\\n   \\sqcup \\bigsqcup_{i = 1}^{n-2} \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots ,\n   \\end{multline*}\nwhich gives a sijection\n\\begin{multline*}\\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{i = 1}^n \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}}\\GT(\\q l) \\\\\n\\sqcup \\bigsqcup_{i = 1}^{n-2} \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{k_i} \\times \\si{k_{i+1}+1}{x+n-i-1} \\times \\si{k_{i+1}}{x+n-i-2} \\times \\si{k_{i+2}}{k_{i+3}} \\times\\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l).\n\\end{multline*}\nAll disjoint unions in the second term satisfy the conditions for the existence of $\\sigma$ from Problem \\ref{prob:pi}, so we can siject them to $\\u \\emptyset$. This gives a sijection\n $$\\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{n-1}}{k_n}} \\GT(\\q l) \\Rightarrow \\bigsqcup_{i = 1}^n \\bigsqcup_{\\q l \\in \\si{k_1}{k_2} \\times \\cdots \\times \\si{k_{i-1}}{x+n-i} \\times \\si{x+n-i}{k_{i+1}} \\times \\cdots \\times \\si{k_{n-1}}{k_n}}\\GT(\\q l),$$\n which is, by the definition of $\\GT$, a sijection $\\GT(k_1,\\ldots,k_n) \\Rightarrow \\bigsqcup_{i = 1}^n \\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n)$.\n \\comment{Finally, for each $i$, apply the sijection\n $$\\pi_{(k_1,\\ldots,k_{i-1},k_{i+1}+1,\\ldots,k_{n-1}+1,x+1,k_n),n-1} \\circ \\cdots \\circ \\pi_{(k_1,\\ldots,k_{i-1},k_{i+1}+1,x+n-i-1,k_{i+2},\\ldots,k_n),i+1} \\circ \\pi_{(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n),i}$$\n from $\\GT(k_1,\\ldots,k_{i-1},x+n-i,k_{i+1},\\ldots,k_n)$ to $(-1)^{n-i} \\GT(k_1,\\ldots,k_{i-1},k_{i+1}+1,\\ldots,k_n+1,x)$.}\n\\end{proof}\n\n\n\\section{Combinatorics of the monotone triangle recursion} \\label{sec:recursion}\n\n\\subsection*{Monotone triangles}\n\nSuppose that $\\q k = (k_1,\\ldots,k_n)$ and $\\q l = (l_1,\\ldots,l_{n-1})$ are two sequences of integers. We say that $\\q l$ \\emph{interlaces} $\\q k$, $\\q l \\prec \\q k$, if the following holds:\n\\begin{enumerate}\n  \\item for every $i$, $1 \\leq i \\leq n-1$, $l_i$ is in the closed interval between $k_i$ and $k_{i+1}$;\n  \\item if $k_{i-1} \\leq k_i \\leq k_{i+1}$ for some $i$, $2 \\leq i \\leq n-1$, then $l_{i-1}$ and $l_i$ cannot both be $k_i$;\n  \\item if $k_i > l_i = k_{i+1}$, then $i \\leq n-2$ and $l_{i+1} = l_i = k_{i+1}$;\n  \\item if $k_i = l_i > k_{i+1}$, then $i \\geq 2$ and $l_{i-1} = l_i = k_i$.\n\\end{enumerate}\n\nFor example, if $k_1 < k_2 < \\ldots < k_n$, then $l_i \\in [k_i,k_{i+1}]$ and $l_1 < l_2 < \\ldots < l_{n-1}$.\n\n\\medskip\n\nA \\emph{monotone triangle of size $n$} is a map $T \\colon \\{(i,j) \\colon 1 \\leq j \\leq i \\leq n \\} \\to {\\mathbb Z}$ so that line $i-1$ (i.e.~the sequence $T_{i-1,1},\\ldots,T_{i-1,i-1}$) interlaces line $i$ (i.e.~the sequence $T_{i,1},\\ldots,T_{i,i}$).\n\\begin{example} {The following is a monotone triangles of size $5$:}\n$$\\begin{array}{ccccccccc}\n&&&& 4 &&&& \\\\\n&&& 3 && 5 &&& \\\\\n&& 3 && 4 && 5 && \\\\\n& 3 & & 3 & & 4 & & 5 & \\\\\n5 & & 3 & & 1 & & 4 & & 6\n\\end{array}\n$$\n\\end{example}\n{This notion of (generalized) monotone triangle was introduced in \\cite{Rie13}. Other notions appeared in \\cite{Fis12}.}\u00a0\n\n\n\\medskip\n\nThe \\emph{sign} of a monotone triangle $T$ is $(-1)^r$, where $r$ is the sum of:\n\\begin{itemize}\n  \\item the number of strict descents in the rows of $T$, i.e.~the number of pairs $(i,j)$ so that $1 \\leq j < i \\leq n$ and $T_{i,j} > T_{i,j+1}$, and\n  \\item the number of $(i,j)$ so that $1 \\leq j \\leq i - 2$, $i \\leq n$ and $T_{i,j} > T_{i-1,j} = T_{i,j+1} = T_{i-1,j+1} > T_{i,j+2}$.\n\\end{itemize}\n{The sign of our example is $-1$.}\n\n\\medskip\n\nWe denote the signed set of all monotone triangles with bottom row $\\q k$ by ${\\MT(\\q k)}$.\n\n\\medskip\n\nIt turns out that $\\MT(\\q k)$ satisfies a recursive ``identity''. Let us define the signed set of \\emph{arrow rows of order $n$} as\n$$\\AR_n = (\\{\\nearrow,\\nwarrow\\},\\{\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\})^{n}.$$\nAlternatively, we can think of them as rows of length $n$ with elements $\\nwarrow, \\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$, where the positive elements are precisely those with an even number of $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$'s.\n\n\\medskip\n\nThe role of an arrow row $\\mu$ of order $n$ is that it induces a deformation of $\\si{k_1}{k_2} \\times \\si{k_2}{k_3} \\times \\cdots \\times \\si{k_{n-1}}{k_n}$ as follows. Consider\n$$\n\\begin{array}{ccccccccccc}\n & \\si{k_1}{k_2} &   & \\si{k_2}{k_3} &  &  \\ldots &  & \\si{k_{n-2}}{k_n-1} &  & \\si{k_{n-1}}{k_n} & \\\\\n \\mu_1 & & \\mu_2  & & \\mu_3 &  &  \\ldots &   & \\mu_{n-1} & & \\mu_{n} ,\n \\end{array}\n $$\nand if $\\mu_i \\in \\{\\nwarrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$ (that is we have an arrow pointing towards $\\si{k_{i-1}}{k_i}$) then\n$k_i$ is decreased by $1$ in $\\si{k_{i-1}}{k_i}$, while there is no change for this $k_i$ if $\\mu_i=\\nearrow$. If $\\mu_i \\in \\{\\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$ (that is we have an arrow pointing towards $[k_i,k_{i+1}]$) then $k_i$ is increased by $1$ in $[k_i,k_{i+1}]$, while there is no change for this $k_i$ if $\\mu_i=\\nwarrow$.\n\n\\medskip\n\nFor a more formal description, we let $\\delta_{\\nwarrow}(\\nwarrow) = \\delta_{\\nwarrow}(\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow) = \\delta_{\\nearrow}(\\nearrow) = \\delta_{\\nearrow}(\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow) = 1$ and $\\delta_{\\nwarrow}(\\nearrow) = \\delta_{\\nearrow}(\\nwarrow) = 0$, and we define\n$$e(\\q k,\\mu)  = \\si{k_1+\\delta_{\\nearrow}(\\mu_1)}{k_2-\\delta_{\\nwarrow}(\\mu_2)} \\times \\ldots \\times \\si{k_{n-1}+\\delta_{\\nearrow}(\\mu_{n-1})}{k_n-\\delta_{\\nwarrow}(\\mu_n)}.$$\nfor $\\q k = (k_1,\\ldots,k_n)$ and $\\mu \\in \\AR_n$.\n\n\\begin{problem} \\label{prob:Xi}\n  Given $\\q k = (k_1,\\ldots,k_n)$, construct a sijection\n  $$\\Xi = \\Xi_{\\q k} \\colon \\MT(\\q k) \\Rightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\MT(\\q l).$$\n\\end{problem}\n\\begin{proof}[Construction]\n All elements on the left are mapped to the right with $\\Xi$, while there are quite a few cancellations on the right-hand side. More specifically, take a monotone triangle $T$ with bottom row $\\q k$. Then $\\Xi(T) = ((T',\\q l),\\mu)$, where $T'$ is the monotone triangle we obtain from $T$ by deleting the last row, $\\q l$ is the bottom row of $T'$, and $\\mu = (\\mu_1,\\ldots,\\mu_n)$ is the arrow row defined as follows:\n \\begin{itemize}\n  \\item $\\mu_1 = \\nwarrow$;\n  \\item $\\mu_n = \\nearrow$;\n  \\item for $ 1 < i < n$, $\\mu_i$ is determined as follows:\n  \\begin{enumerate} \\item if $k_{i-1} \\leq l_{i-1} = k_i$, take $\\mu_i = \\nearrow$;  \\item if $k_{i-1} > l_{i-1} = k_i = l_{i} > k_{i+1}$, take $\\mu_i = \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$; \\item otherwise, take $\\mu_i = \\nwarrow$. \\end{enumerate}\n \\end{itemize}\n It is easy to check that $\\q l$ is indeed in $e(\\q k,\\mu)$.\n Note that in (1) and (2) of the third bullet point, $\\mu_i$ is forced if we require $\\q l \\in e(\\q k,\\mu)$. In (3), $\\mu_i=\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ would also be possible if and only if  $\\mu_i=\\nearrow$ would also be possible.\\\\\n On the other hand, for $((T',\\q l),\\mu)$, define $\\Xi((T',\\q l),\\mu)$ as follows. For the construction it is useful to keep in mind that $\\q l \\in e(\\q k, \\mu)$ implies that conditions (1) and (2) for $\\q l \\prec \\q k$ are satisfied.\n \\begin{itemize}\n  \\item if $\\mu_1 \\neq \\nwarrow$, take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $1$ by $\\nearrow$ and vice versa;\n  \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n \\neq \\nearrow$, take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $n$ by $\\nwarrow$ and vice versa;\n  \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n = \\nearrow$, and $\\q l \\not\\prec \\q k$, find the smallest $i$ between $2$ and $n-1$ such that:\n \\begin{itemize}\n   \\item condition (3) of $\\q l \\prec \\q k$ is not satisfied at $i$, i.e.\n   $k_{i-1} > l_{i-1} = k_{i} \\not= l_{i}$ (which implies $\\mu_i \\in \\{\\nwarrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$), or\n   \\item condition (4) of $\\q l \\prec \\q k$ is not satisfied at $i$, i.e.\n   $l_{i-1} \\not= k_i = l_i > k_{i+1}$ (which implies  $\\mu_i \\in \\{\\nearrow, \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow\\}$).\n\\end{itemize}\nThen take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nwarrow$ and vice versa in the first case, and replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nearrow$ and vice versa in the second case;\n \\item if $\\mu_1 = \\nwarrow$ and $\\mu_n = \\nearrow$, and $\\q l \\prec \\q k$, find the smallest $i$ for an instance of (3) of the third bullet point in the first paragraph of the proof with $\\mu_i \\not= \\nwarrow$ (if such an $i$ exists). Then take $\\Xi((T',\\q l),\\mu) = ((T',\\q l),\\mu')$, where we obtain $\\mu'$ from $\\mu$ by replacing $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ in position $i$ by $\\nwarrow$ and vice versa.\n \\end{itemize}\nIf no such $i$ exists, we take $\\Xi((T',\\q l),\\mu) = T$, where we obtain $T$ from $T'$ by adding $\\q k$ as the last row. It is easy to see that this is a well-defined sijection.\n\\end{proof}\n\n\n\\begin{remark} The previous construction could have been avoided by using alternative extensions of monotone triangles provided in \\cite{Fis12}. However, the advantage of the definition used in this paper is that it is more reduced than the others in the sense that it can obtained from these by cancelling elements using certain sign-reversing involutions.\n\\end{remark}\n\n\\subsection*{Arrow patterns and shifted GT patterns}\n\nDefine the signed set of \\emph{arrow patterns of order $n$} as\n$$\\AP_n = (\\{\\swarrow, \\searrow\\},\\{\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\})^{\\binom n 2}.$$\n\n\\medskip\n\nAlternatively, we can think of an arrow pattern of order $n$ as a triangular array $T=(t_{p,q})_{1 \\le p < q \\le n}$ arranged as\n$$T = \\begin{smallmatrix} & & & & t_{1,n} & & & & \\\\ & & & t_{1,n-1} & & t_{2,n} & & & \\\\ & & t_{1,n-2} & & t_{2,n-1}& & t_{3,n} & & \\\\ & \\vstretch{0.35}{\\udots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\ddots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\udots} & \\vstretch{0.35}{\\vdots} & \\vstretch{0.35}{\\ddots} & \\\\ t_{1,2} & & t_{2,3}  & & \\ldots  & & \\ldots  & & t_{n-1,n}  \\end{smallmatrix},$$\nwith $t_{p,q} \\in \\{\\swarrow, \\searrow, \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\}$, and the sign of an arrow pattern is $1$ if the number of $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$'s is even and $-1$ otherwise.\n\n\\medskip\n\nThe role of an arrow pattern of order $n$ is that it induces a deformation of $(k_1,\\ldots,k_n)$, which can be thought of as follows. Add $k_1,\\ldots,k_n$ as bottom row of $T$ (i.e., $t_{i,i}=k_i$), and for each $\\swarrow$ or$ \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ which is in the same $\\swarrow$-diagonal as $k_i$ add $1$ to $k_i$, while for each $\\searrow$ or $\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ which is in the same $\\searrow$-diagonal as $k_i$ subtract $1$ from $k_i$.\nMore formally, letting $\\delta_{\\swarrow}(\\swarrow) = \\delta_{\\swarrow}(\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow) = \\delta_{\\searrow}(\\searrow) = \\delta_{\\searrow}(\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow) = 1$ and $\\delta_{\\swarrow}(\\searrow) = \\delta_{\\searrow}(\\swarrow) = 0$, we set\n$$c_i(T) = \\sum_{j=i+1}^{n} \\delta_{\\swarrow}(t_{i,j}) - \\sum_{j=1}^{i-1} \\delta_{\\searrow}(t_{j,i}) \\mbox{ and } d(\\q k,T) = (k_1+c_1(T),k_2+c_2(T),\\ldots,k_n+c_n(T))$$\nfor $\\q k = (k_1,\\ldots,k_n)$ and $T \\in \\AP_n$.\n\n\\medskip\n\nFor $\\q k =(k_1,\\ldots,k_n)$ define \\emph{shifted Gelfand-Tsetlin patterns}, or SGT patterns for short, as the following disjoint union of GT patterns over arrow patterns of order $n$:\n$$\n\\SGT(\\q k) = \\bigsqcup_{T \\in \\AP_n} \\GT(d(\\q k,T))\n$$\n\n\n\\medskip\n\nConsidering that $|(\\{\\swarrow, \\searrow\\},\\{\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\})| = 1$ and therefore $|\\AP_n| = 1$, the following is not surprising.\n\n\\begin{problem} \\label{prob:Psi}\n Given $n$ and $i$, $1 \\leq i \\leq n$, construct a sijection\n $$\\Psi = \\Psi_{n,i} \\colon  \\AP_{n-1} \\Rightarrow \\AP_n.$$\n\\end{problem}\n\\begin{proof}[Construction]\n For $T \\in \\AP_{n-1}$, take $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$ to be the arrow pattern defined by\n $$t'_{p,q} = \\begin{cases} t_{p,q} & \\mbox{if } p < q < i \\\\ t_{p,q-1} & \\mbox{if } p < i < q \\\\ t_{p-1,q-1} & \\mbox{if } i < p < q \\\\ \\searrow & \\mbox{if } p < q = i \\\\ \\swarrow & \\mbox{if } i = p < q \\end{cases}.$$\n{An example for $n=6$ and $i=4$ is\n$$\n\\begin{array}{ccccccc}\u00a0\n&&& \\searrow &&& \\\\\n&& \\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\\\\n& \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\swarrow && \\swarrow & \\\\\n\\searrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && \\searrow && \\swarrow\n\\end{array} \\quad \\stackrel{\\Psi}{\\Rightarrow} \\quad\n\\begin{array}{ccccccccc}\u00a0\n&&&& \\searrow &&&& \\\\\n&&& \\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow &&& \\\\\n&& {\\color{red} \\searrow} && \\swarrow &&  \\swarrow && \\\\\n& \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow &&  {\\color{red} \\searrow} && \\searrow && {\\color{red} \\swarrow}  & \\\\\n\\swarrow && \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow && {\\color{red} \\searrow}\u00a0&& {\\color{red} \\swarrow} && \\swarrow\n\\end{array},\n$$\nwhere the new arrows are indicated in red.}\n If $T \\in \\AP_{n}$, $t_{p,i} = \\searrow$ for $p = 1,\\ldots,i-1$, $t_{i,q} = \\swarrow$ for $q = i+1,\\ldots,n$, take $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n-1}$, where\n $$t'_{p,q} = \\begin{cases} t_{p,q} & \\mbox{if } p < q < i \\\\ t_{p,q+1} & \\mbox{if } p < i \\leq q \\\\ t_{p+1,q+1} & \\mbox{if } i \\leq p < q \\end{cases}.$$\nOtherwise, there either exists $p$ so that $t_{p,i} \\neq \\searrow$, or there exists $q$ so that $t_{i,q} \\neq \\swarrow$. In the first case, define $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$, where $t'_{p,i}=\\swarrow$ if $t_{p,i} = \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ and $t'_{p,i}=\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if $t_{p,i} = \\swarrow$, and all other array elements are equal. In the second case, define $\\Psi(T) = (t'_{p,q})_{1 \\leq p < q \\leq n}$, where $t'_{i,q}=\\searrow$ if $t_{i,q} = \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ and $t'_{i,q}=\\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow$ if $t_{i,q} = \\searrow$, and all other array elements are equal. It is easy to see that this is a sijection.\n\\end{proof}\n\nThe difficult part of this paper is to prove that $\\SGT$ satisfies the same ``recursion'' as $\\MT$. While the proof of the recursion was easy for monotone triangles, it is very involved for shifted GT patterns, and needs almost {all} the sijections we have constructed in this and previous sections.\n\n\\begin{problem}\nGiven $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n$$\\Phi = \\Phi_{\\q k,x} \\colon \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\Rightarrow \\SGT(\\q k).$$\n\\end{problem}\n\\begin{proof}[Construction]\nTo make the construction of $\\Phi$ a little easier, we will define it as the composition of several sijections. The first one will reduce the indexing sets (from a signed box to its ``corners'') using Problem \\ref{prob:rho}. The second one increases the order of the arrow patterns using the sijection from Problem \\ref{prob:Psi}. The third one further reduces the indexing set (from a signed set with $2^{n-1}$ elements to $\\si{1}{n}$). The last one gets rid of the arrow row and then uses Problem \\ref{prob:tau}.\\\\\nFor $\\mu \\in \\AR_n$, define $\\u S_i = (\\{k_i+\\delta_{\\nearrow}(\\mu_i)\\},\\emptyset) \\sqcup (\\emptyset,\\{k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1\\})$. Then $\\Phi$ is the composition of sijections\n\\begin{multline*}\n  \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\\\\\stackrel{\\Phi_1}\\Longrightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\\\\n\\stackrel{\\Phi_2}\\Longrightarrow \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{\\q m \\in \\u S_1 \\times \\cdots \\times \\u S_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\\\ \\stackrel{\\Phi_3} \\Longrightarrow\n\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{i=1}^n \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots) \\\\\n\\stackrel{\\Phi_4}\\Longrightarrow \\SGT(\\q k), \\hspace*{12cm}\n\\end{multline*}\nwhere $\\Phi_1$, $\\Phi_2$, $\\Phi_3$, and $\\Phi_4$ are constructed as follows.\\\\\n\\emph{Construction of $\\Phi_1$.} By definition of $\\SGT$, we have\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) = \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\bigsqcup_{T \\in \\AP_{n-1}} \\GT(d(\\q l,T)).$$\nBy switching the inner disjoint unions, we get a sijection to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)}  \\GT(d(\\q l,T)).$$\nThere is an obvious sijection from this signed set to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q l \\in d(e(\\q k,\\mu),T)}  \\GT(\\q l),$$\n{by abuse of notation setting}\n$${ d(\\si{x_1}{y_1} \\times \\ldots \\si{x_{n-1}}{y_{n-1}},T)=\\si{x_1+c_1(T)}{y_1+c_1(T)} \\times \\cdots \\times \\si{x_{n-1}+c_{n-1}(T)}{y_{n-1}+c_{n-1}(T)}.}$$\nNow for each $\\mu$ and $T$, use the map $\\rho$ from Problem \\ref{prob:rho} for $a_i = k_i+\\delta_{\\nearrow}(\\mu_i)+c_i(T)$, $b_i = k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_i(T)$, and $x$. We get a sijection to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u {S}'_1 \\times \\cdots \\u{S}'_{n-1}} \\GT(m_1,\\ldots,m_{n-1},x),$$\nwhere $\\u{S}'_i = (\\{k_i+\\delta_{\\nearrow}(\\mu_i)+c_i(T)\\},\\emptyset) \\sqcup (\\emptyset,\\{k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_i(T)+1\\})$. Finally, there is an obvious sijection from this signed set to\n$$\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n-1}} \\bigsqcup_{\\q m \\in \\u {S}_1 \\times \\cdots \\u{S}_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x).$$\n\\emph{Construction of $\\Phi_2$.} In Problem \\ref{prob:Psi}, we constructed sijections $\\Psi_{n,i} \\colon \\AP_{n-1} \\Rightarrow \\AP_n$. We construct $\\Phi_2$ by using Proposition \\ref{prop:sijections}~ (3) for $\\psi = \\Psi_{n,n}$, $\\u T = \\AP_{n-1}$, $\\u{\\widetilde T} = \\AP_n$,\n$$\\u S_T =  \\bigsqcup_{\\q m \\in \\u {S}_1 \\times \\cdots \\u{S}_{n-1}} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\quad \\mbox{for } T \\in \\AP_{n-1} \\sqcup \\AP_n$$\nand $\\varphi_T = \\u\\id$. This is well defined because $c_i(T) = c_i(\\Psi_{n,n}(T))$ for $T \\in \\AP_{n-1} \\sqcup \\AP_n$ and $i=1,\\ldots,n-1$.\\\\%Since $x(\\Phi(T)) = x(T)$ for all $T \\in \\AP_{n-1} \\sqcup \\AP_n$, this is well defined. \\\\\n\\emph{Construction of $\\Phi_3$.} Let $\\eta$ be the involution that maps $\\swarrow \\leftrightarrow \\nearrow, \\searrow \\leftrightarrow \\nwarrow, \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\leftrightarrow \\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$. The elements of the signed set $\\u S = \\u {S}_1 \\times \\cdots  \\times \\u{S}_{n-1}$ are $(n-1)$-tuples of elements that are either $(k_i+\\delta_{\\nearrow}(\\mu_i),0)$ or $(k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1,1)$. Define $\\u S'$ as the subset of $\\u S$ containing tuples of the form $(\\ldots,(m_i,1),(m_{i+1},0),\\ldots)$, i.e.~the ones where we choose $k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1$ in position $i$ and $k_{i+1}+\\delta_{\\nearrow}(\\mu_{i+1})$ in position $i+1$ for some $i$. Then we can define a sijection\n$$ \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{\\q m \\in \\u S'} \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x) \\Rightarrow \\u \\emptyset$$\nas follows: given $\\mu \\in \\AR_n$, $T \\in \\AP_n$, $\\q m = (\\ldots,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+1,k_{i+1}+\\delta_{\\nearrow}(\\mu_{i+1}),\\ldots)$ (and $i$ is the smallest index where this happens), $A \\in \\GT(m_1+c_1(T),\\ldots,m_{n-1}+c_{n-1}(T),x)$, map $(((A,\\q m),T),\\mu)$ to $(((A',\\q m'),T'),\\mu')$, where:\n\\begin{itemize}\n \\item $A' = \\pi_{i,n}(A)$;\n \\item $T'$ is $T$ if $A'$ has the same bottom row as $A$; otherwise, $T'$ is obtained from $T$ by interchanging {$t_{i,j}$ and $t_{i+1,j}$ for $j>i+1$ as well as $t_{j,i}$ and $t_{j,i+1}$ for $j<i$, and setting $t'_{i,i+1} = \\eta(\\mu_{i+1})$;}\n \\item $\\mu'$ is $\\mu$ if $A'$ has the same bottom row as $A$; otherwise, $\\mu'$ is obtained from $\\mu$ by replacing {$\\mu_{i+1}$ with $\\eta(t_{i,i+1})$};\n \\item $\\q m' = (\\ldots,k_{i+1}-\\delta_{\\nwarrow}(\\mu'_{i+1})+1,k_{i+1}+\\delta_{\\nearrow}(\\mu'_{i+1}),\\ldots)$.\n\\end{itemize}\n{What remains is\n$$\n\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{T \\in \\AP_{n}} \\bigsqcup_{i=1}^n (-1)^{n-i} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T),k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T)+1,\\ldots,x),\n$$\nand we can now apply $\\pi_{i,n} \\circ \\pi_{i,n} \\circ \\dots \\circ \\pi_{n-1,n}$ to obtain what is claimed.}\n\n\\emph{Construction of $\\Phi_4$.} By switching the order in which we do disjoint unions, we arrive at\n$$\\bigsqcup_{T \\in \\AP_{n}}  \\bigsqcup_{i=1}^n \\bigsqcup_{\\mu \\in \\AR_n} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots).$$\nLet us define a sijection $\\Lambda_{n,i} \\colon \\AR_n \\Rightarrow (\\{\\cdot\\},\\{\\})$. For $\\mu' = (\\nwarrow,\\ldots,\\nwarrow,\\nearrow,\\ldots,\\nearrow)$ (with $i$ $\\nwarrow$'s), take $\\Lambda_{n,i}(\\mu') = \\cdot$ and $\\Lambda_{n,i}(\\cdot) = \\mu'$. For every other $\\mu$, take the smallest $p$ so that $\\mu_p \\neq \\mu'_p$. If $p \\leq i$, replace $\\nearrow$ with $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ and vice versa in position $p$ to get $\\Lambda_{n,i}(\\mu)$ from $\\mu$, and if $p > i$, replace $\\nwarrow$ with $\\nwarrow \\!\\!\\!\\!\\!\\;\\!\\! \\nearrow$ and vice versa in position $p$ to get $\\Lambda_{n,i}(\\mu)$ from $\\mu$. If $\\mu \\neq \\mu'$, $(\\delta_{\\nearrow}(\\mu_1),\\ldots,\\delta_{\\nearrow}(\\mu_{i-1}),\\delta_{\\nwarrow}(\\mu_{i+1}),\\ldots,\\delta_{\\nwarrow}(\\mu_n))$ are unaffected by this sijection, so it induces a sijection\n\\begin{multline*}\n  \\bigsqcup_{\\mu \\in \\AR_n} \\GT(\\ldots,k_{i-1}+\\delta_{\\nearrow}(\\mu_{i-1})+c_{i-1}(T), x + n-i,k_{i+1}-\\delta_{\\nwarrow}(\\mu_{i+1})+c_{i}(T),\\ldots) \\\\\n  \\Rightarrow \\GT(\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i}(T),\\ldots).\n\\end{multline*}\nWe switch disjoint unions again, and we get\n$$\\bigsqcup_{i=1}^n \\bigsqcup_{T \\in \\AP_{n}}  \\GT(k_1+c_1(T),\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i}(T),\\ldots,k_{n} + c_{n-1}(T)).$$\nFor chosen $i$, use Proposition \\ref{prop:sijections} (3) for $\\psi = \\Psi_{n,i} \\circ \\Psi_{n,n}^{-1}$ and $\\varphi_t = \\id$. We get a sijection to\n$$\\bigsqcup_{i=1}^n \\bigsqcup_{T \\in \\AP_{n}}  \\GT(k_1+c_1(T),\\ldots,k_{i-1}+c_{i-1}(T), x + n-i,k_{i+1}+c_{i+1}(T),\\ldots,k_{n} + c_{n}(T)).$$\nIf we switch disjoint unions one last time, we can use the sijection $\\tau^{-1}$ (see Problem \\ref{prob:tau}), and we get\n$$\\bigsqcup_{T \\in \\AP_n} \\GT(d(\\q k,T)) = \\SGT(\\q k).$$\nThis completes the construction of $\\Phi_4$ and therefore of $\\Phi$.\n\\end{proof}\n\n\\begin{problem}\n Given $\\q k = (k_1,\\ldots,k_n) \\in {\\mathbb Z}^n$ and $x \\in {\\mathbb Z}$, construct a sijection\n $$\\Gamma = \\Gamma_{\\q k,x} \\colon \\MT(\\q k) \\Rightarrow \\SGT(\\q k).$$\n\\end{problem}\n\\begin{proof}[Construction]\n The proof is by induction on $n$. For $n = 1$, both sides consist of one (positive) element, and the sijection is obvious. Once we have constructed $\\Gamma$ for all lists of length less than $n$, we can construct $\\Gamma_{\\q k,x}$ as the composition of sijections\n $$\\MT(\\q k) \\stackrel{\\Xi_{\\q k}}{\\Longrightarrow} \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\MT(\\q l) \\stackrel{\\sqcup \\sqcup \\Gamma}{\\Longrightarrow} \\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\SGT(\\q l) \\stackrel{\\Phi_{\\q k,x}}{\\Longrightarrow} \\SGT(\\q k),$$\n where $\\sqcup \\sqcup \\Gamma$ means $\\bigsqcup_{\\mu \\in \\AR_n} \\bigsqcup_{\\q l \\in e(\\q k,\\mu)} \\Gamma_{\\q l,x}.$\n\\end{proof}\n\nRunning the code shows that the main sijection $\\Gamma$ indeed depends on the choice $x$. As an example, take $\\q k = (1,2,3)$. In this case, $\\MT(\\q k)$ has $7$ positive elements, and $\\SGT(\\q k)$ has $10$ positive and $3$ negative elements. For $x = 0$, the sijection is given by\n$$\\mtthree 1 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 1 1 1 1,\\mttwo  \\searrow \\searrow \\searrow\\right) \\qquad \\mtthree 2 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 2 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right) \\qquad \\mtthree 1 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 2 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 3 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 2 3 2 2 3, \\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 2 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 3 1 2, \\mttwo  \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow\\right)$$\n$$\\mtthree 3 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 3 3 3 3 3,\\mttwo  \\swarrow \\swarrow \\swarrow\\right) \\qquad \\left(\\mtthree 2 2 2 2 2 3, \\mttwo \\swarrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\swarrow\\right)$$\n$$\\left(\\mtthree 2 2 2 2 3 1,\\mttwo \\searrow \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow\\right) \\qquad \\left(\\mtthree 2 2 2 1 2 2,\\mttwo \\searrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\swarrow\\right)$$\nwhile for $x = 1$, it is given by\n$$\\mtthree 1 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 1 1 1 1,\\mttwo  \\searrow \\searrow \\searrow\\right) \\qquad \\mtthree 2 1 2 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 3,\\mttwo \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 1 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 1 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 2 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 3 1 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 2 3 2 2 3,\\mttwo  \\swarrow \\searrow \\swarrow\\right) \\qquad \\mtthree 2 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 2 2 2 1 2 2,\\mttwo \\searrow \\searrow \\swarrow\\right)$$\n$$\\mtthree 3 2 3 1 2 3 \\leftrightarrow \\left(\\mtthree 3 3 3 3 3 3,\\mttwo  \\swarrow \\swarrow \\swarrow\\right) \\qquad \\left(\\mtthree 2 1 2 1 2 2,\\mttwo  \\searrow \\searrow \\swarrow\\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2,\\mttwo  \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\swarrow\\right)$$\n$$\\left(\\mtthree 2 2 2 3 1 2, \\mttwo  \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\searrow \\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2, \\mttwo  \\swarrow \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\right) \\qquad \\left(\\mtthree 2 2 2 2 3 1,\\mttwo \\searrow \\swarrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\right) \\leftrightarrow \\left(\\mtthree 2 2 2 2 2 2, \\mttwo  \\searrow \\swarrow \\!\\!\\!\\!\\!\\;\\!\\! \\searrow \\swarrow\\right)$$\n\n\\section{Concluding remarks}\n\n\\subsection*{Future work}\n\nIn this article, we have presented the first bijective proof of the operator formula. The operator formula is the main tool for non-combinatorial proofs of several results where alternating sign matrix objects are related to plane partition objects, or simply for showing that $n \\times n$ ASMs are enumerated by \\eqref{asm}.\n\n\\begin{itemize}\n\\item The operator formula was used in \\cite{Fis07} to show that $n \\times n$ ASMs are counted by \\eqref{asm} and, more generally, to count ASMs with respect to the position of the unique $1$ in the top row.\n\\item While working on this project, we actually realized that the final calculation in \\cite{Fis07} also implies that ASMs are equinumerous with DPPs without having to use Andrews' result \\cite{And79} on the number of DPPs; more generally, we can even obtain the equivalence of the refined count of $n \\times n$ ASMs with respect to the position of the unique $1$ in the top row and the refined count of DPPs with parts no greater than $n$ with respect to the number of parts equal to $n$. This was conjectured in  \\cite{MilRobRum83} and first proved in \\cite{BehDifZin12}.\n\\item In \\cite{Fis19a}, the operator formula was used to show that ASTs with $n$ rows are equinumerous with TSSCPPs in a $2n \\times 2n \\times 2n$-box. Again we do not rely on Andrews' result \\cite{And94} on the number of TSSCPPs and we were actually able to deal with a refined count again (which has also the same distribution as the position of the unique $1$ in the top row of an ASM).\n\\item In \\cite{Fis19b}, we have considered alternating sign trapezoids (which generalize ASTs) and, using the operator formula, we have shown that they are equinumerous with objects generalizing DPPs. These objects were already known to Andrews and he actually enumerated them in \\cite{And79}. Later Krattenthaler \\cite{Kra06} realized that these more general objects are (almost trivially) equivalent to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. Again we do not rely on Andrews' enumeration of these generalized DPPs, and in this case we were able to include three statistics.\n\\end{itemize}\n\nWe plan to work on converting the proofs just mentioned into bijective proofs. For those mentioned in the first and second bullet point, this has already been worked out.\nThe attentive reader will have noticed that working out all of them will link all four known classes of objects that are enumerated by \\eqref{asm}.\n\n\\subsection*{Computer code}\n\nAs mentioned before, we consider computer code for the constructed sijections an essential part of this project. The code (in python) is available at \n\\begin{center}\n\\url{https:\/\/www.fmf.uni-lj.si\/~konvalinka\/asmcode.html}.\n\\end{center}\n All the constructed sijections are quite efficient.  If run with pypy, checking that $\\Gamma_{(1,2,3,4,5),0}$ is a sijection between $\\MT(1,2,3,4,5)$ (with $429$ positive elements and no negative elements) and $\\SGT(1,2,3,4,5)$ (with $18913$ positive elements and $18484$ negative elements) takes less than a minute. Of course, the sets involved can be huge, so checking that  $\\Gamma_{(1,2,3,4,5,6),0}$ is a sijection between $\\MT(1,2,3,4,5,6)$ (with $7436$ positive elements and no negative elements) and $\\SGT(1,2,3,4,5,6)$ (with $11167588$ positive elements and $11160152$ negative elements) took almost 20 hours.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nGiven a complex non-K\\\"ahler $n$-dimensional manifold $(\\rm{M},\\rm{J})$ it is a natural and meaningful problem to find special Hermitian metrics which might help in understanding the geometry of $\\rm{M}$. Great effort has been spent in the last decades in this research topic and among special metrics the pluriclosed and the balanced conditions have shown to be highly significant.\\par \nThe balanced condition can be defined saying that the fundamental form $\\o=h(\\cdot,\\rm{J}\\cdot)$ of a Hermitian metric $h$ satisfies the non-linear condition $d\\o^{n-1}=0$ or equivalently, $-\\rm{J} \\theta = \\d\\o=0$, where $\\d$ denotes the codifferential and $\\theta$ the torsion $1$-form (see e.g. \\cite{Ga2}).\nWhile this concept appears in \\cite{Ga1} under the name of semi-K\\\"ahler (see also \\cite{Gra}), in \\cite{Mi} the balanced condition was started to be thoroughly investigated, highlighting also the duality with the K\\\"ahler condition and establishing necessary and sufficient conditions for the existence of these metrics in terms of currents. While K\\\"ahler metrics are obviously balanced and share with these the important relation among Laplacians $\\Delta_{\\partial}=\\Delta_{\\overline\\partial}=\\frac 12 \\Delta$ (see \\cite{Ga1}), there are many examples of non-K\\\"ahler manifolds carrying balanced metrics. Basic examples are given by compact complex parallelizable manifolds, which are covered by complex unimodular Lie groups $\\rm{G}$ and every left invariant Hermitian metric turns out to be balanced (see \\cite{AG}\\cite{Ga1}\\cite{Gr}). Further examples of balanced metrics are provided by any Hermitian invariant metric on a compact homogeneous flag manifold (see also \\cite{FGV} for a characterization of compact homogeneous complex manifolds carrying balanced metrics) as well as by twistor spaces of certain self-dual $4$-manifolds (\\cite{Mi}) and more generally (\\cite{To}) by twistor spaces of compact hypercomplex manifolds (see also \\cite{Fo} for other examples on toric bundles over hyperk\\\"ahler manifolds). Contrary to the K\\\"ahlerness condition, being balanced is a birational invariant (see \\cite{AB1}, so that e.g. Moishezon manifolds are balanced) and compact complex manifolds $X$ which can be realized as the base of a holomorphic proper submersion $f:Y\\to X$ inherit the balanced condition whenever $Y$ has it (\\cite{Mi}), while the balanced property is not stable under small deformations of the complex structure (see \\cite{AB2},\\cite{FuY}, \\cite{AU}). On the other hand, the balanced condition is obstructed, as on compact manifolds with balanced metrics no compact complex hypersurface is homologically trivial, so that for instance Calabi-Eckmann manifolds do not carry balanced metrics. This is in contrast with the fact that Gauduchon metrics, which statisfy the weaker condition  $\\partial\\bar\\partial\\o^{n-1}=0$, always exist on a compact complex manifold.  \\par \nIn more recent years, the rising interest in the Strominger System (see \\cite{GF} and \\cite{FeY} for the case of invariant solutions on complex Lie groups) has given balanced metrics a really central role in non-K\\\"ahler geometry, as the equivalence between the dilatino equation (i.e. one of the equations of the system) and the conformally balanced equation requires the solutions of the system to be necessarily balanced. We refer also to the work \\cite{FLY}, where new examples of balanced metrics are constructed on some Calabi Yau non-K\\\"ahler threefolds, as well as to the results in \\cite{BV}, where a new balanced flow is introduced and investigated.  \\par \nThe main goal of this paper is to search for invariant special Hermitian, in particular balanced, metrics in the class of semisimple real non-compact Lie groups and on their compact (non-K\\\"ahler) quotients by a cocompact lattice; actually it appears that, despite invariant complex structures on semisimple (reductive) Lie algebras being fully classified in \\cite{Sn} (after the special case of compact Lie algebras had been considered by Samelson (\\cite{Sam}) and later in \\cite{Pi}), they have never been deeply investigated from this point of view. In contrast, the case of $\\K$ compact is fully understood, as in such a case it is very well known that every invariant complex structure can be deformed to an invariant one for which the opposite of the Cartan-Killing form is a pluriclosed Hermitian metric $h$, i.e. it satisfies $dd^c\\o_h=0$. Moreover it has been proved in \\cite{FGV} that $\\K$ does not carry {\\it any} balanced metric at all, fueling the conjecture (\\cite{FV}) that a compact complex manifold carrying two Hermitian metrics, one balanced and the other pluriclosed, must be actually K\\\"ahler.\\par\nMore specifically, in this work we focus on a large class of simple non-compact real Lie algebras $\\gg_o$ of even dimension, namely those which are of inner type, i.e. when the maximal compactly embedded subalgebra $\\mathfrak{k}$ in a Cartan decomposition of $\\gg_o$ contains a Cartan subalgebra. In these algebras we construct standard invariant complex structures (regular in \\cite{Sn}) and write down the balanced condition for invariant Hermitian metrics. A careful analysis of the resulting equation together with some general argument on root systems allows us to show the existence of a suitable invariant complex structure and a corresponding Hermitian metric satisfying the balanced equation. By Borel's Theorem, every semisimple Lie group $\\rm{G}_o$ admits a cocompact lattice $\\Gamma$ so that the compact quotient $\\rm{G}_o\/\\Gamma$ inherits the invariant balanced structure from $\\rm{G}_o$. We note here that the resulting metrics come in families and moreover the same kind of arguments can be applied to show the existence of balanced structures on quotients $\\rm{G}_o\/\\S$, where $\\rm{G}_o$ is any simple non-compact Lie group of inner type of any dimension and $\\S$ is a suitable abelian closed subgroup.\\par \nWe are also able to prove that the compact quotients $\\rm{M}=\\rm{G}_o\/\\Gamma$, endowed with the invariant complex structure that allows the existence of balanced metrics, do not carry any pluriclosed metric. This result is in accordance with the conjecture by Fino and Vezzoni and in some sense reflects a kind of duality between the compact and non-compact case, switching the existence of balanced\/pluriclosed Hermitian metrics. In the last section, we prove that these balanced manifolds $\\rm{M}$, despite having vanishing first Chern class, carry no non trivial holomorphic $(n,0)$-forms; furthermore  we prove that they have vanishing Chern scalar curvature. This last property may allow to better understand the geometry of these manifolds, according to some more recent results concerning the implications of vanishing Chern-scalar curvature on some geometric features (see \\cite{Y}). \\par\nThe paper is structured as follows. In Section 2, we review basic facts on simple real non-compact Lie algebras with invariant complex structures and we consider a class of invariant Hermitian metrics for which we write down the balanced condition in terms of roots. In section 3 we state our main result, namely Theorem \\ref{main}, and we prove it by means of several steps. We first rewrite the balanced equation in terms of simple roots and then the key Lemma \\ref{L1} allows us to select an invariant complex structure so that the relative balanced equation admits solutions. In section 4 we prove that the complex manifolds that we constructed in the previous section and that admit balanced metrics, do not carry {\\it any} pluriclosed metric. In the last section, we show in Theorem 5.1 that these complex compact manifolds $(\\rm{M},\\rm{J})$ have trivial first Chern class and that the balanced metrics we have constructed have vanishing Chern scalar curvature; as a consequence we show that the Kodaira dimension $\\kappa(\\rm{M})=-\\infty$.\n\n\n\\par \n\\vspace{0.5cm}\n{\\bf Aknowledgements.} The second author was supported by GNSAGA of INdAM and by the project PRIN 2017 ``Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics'', n. 2017JZ2SW5.\\par \nThe authors would like to thank Daniele Angella for valuable conversations.  \n\n\\section{Preliminaries}\n\nLet $\\gg_o$ be a real simple $2n$-dimensional Lie algebra. It is well known that either the complexification $\\gg_o^c$ is a complex simple Lie algebra (and in this case $\\gg_o$ is called absolutely simple) or $\\gg_o$ is the realification $\\gg_\\mathbb R$ of a complex simple Lie algebra $\\gg$ (see e.g. \\cite{He}). \\par \nWhen $\\gg_o$ is even dimensional, it is known (\\cite{Mo}, see also \\cite{Sas}) that $\\gg_o$ admits an invariant complex structure, namely an endomorphism $\\rm{J}\\in \\End(\\gg_o)$ with $\\rm{J}^2=-\\rm{Id}$ and vanishing Nijenhuis tensor or, equivalently, such that \n$$\\gg_o^c = \\gg_o^{10}\\oplus \\gg_o^{01},\\qquad [\\gg_o^{10},\\gg_o^{10}]\\subseteq \\gg_o^{10}.$$\nIf $\\rm{G}_o$ is any Lie group with Lie algebra $\\gg_o$, then the endomorphism $\\rm{J}$ defines a (left)-invariant complex structure on $\\rm{G}_o$. Moreover, thanks to a result due to Borel (\\cite{Bo}), there exists a discrete, torsionfree cocompact lattice $\\Gamma$ so that $\\rm{M}:= \\rm{G}_o\/\\Gamma$ is compact and the left-invariant complex structure $\\rm{J}$ on \n$\\rm{G}_o$ descends to a complex structure $\\rm{J}$ on $\\rm{M}$.\\par \nWe recall that when $\\rm{G}_o$ is compact and even-dimensional, i.e. $\\gg_o$ is of compact type, the existence of an invariant complex structure was already established by Samelson (\\cite{Sam}), while in \\cite{Pi} it was shown that every invariant complex structure on $\\rm{G}_o$ is obtained by means of Samelson's construction. \\par\nIf we now consider an even-dimensional $\\rm{G}_o$ and a compact quotient $\\rm{M}$ endowed with an invariant complex structure $\\rm{J}$, we are interested in the existence of special Hermitian metrics $h$. The following proposition states a known fact, namely the non-existence of (invariant) K\\\"ahler structures.\n\\begin{prop}\\label{invK} The group $\\rm{G}_o$ does not admit any invariant K\\\"ahler metric and the compact quotient $\\rm{M}=\\rm{G}_o\/\\Gamma$ is not K\\\"ahler.\n\\end{prop} \n\\begin{proof} The first assertion is contained in \\cite{Ch}, but we give here an elementary proof. If $\\o$ is an invariant symplectic form \non $\\gg_o$, then the closedness condition $d\\o=0$ can be written as follows for $x,y,z\\in\\gg_o$  \n\\begin{equation}\\label{closed}\\o([x,y],z)+\\o([z,x],y) + \\o([y,z],x)=0.\\end{equation}\nIf $B$ denotes the non-degenerate Cartan-Killing form of $\\gg_o$, then we can define the endomorphism $F\\in\\End(\\gg_o)$ by $B(Fx,y)=\\o(x,y)$ ($x,y\\in\\gg_o$) so that $F$ turns out to be a derivation by \\eqref{closed}. As $\\gg_o$ is semisimple, there exists a unique $z\\in \\gg_o$ with $F=\\ad(z)$, so that $z\\in \\ker\\o$, a  contradiction.  \\par\nWe now suppose that the compact manifold $\\rm{M}$ has a K\\\"ahler metric with K\\\"ahler form $\\o$. Using $\\o$ and a symmetrization procedure that goes back to \\cite{Be}, we now construct an {\\it invariant} K\\\"ahler form on $\\rm{G}_o$, obtaining a contradiction. We fix a basis $x_1,...,x_{2n}$ of $\\gg_o$ and we extend each vector as a left invariant vector fields on $G_o$; these vector fields can be  projected down to $M$ as vector fields $x_1^*,\\ldots,x_{2n}^*$ that span the tangent space $TM$ at each point. As $\\rm{G}_o$ is semisimple, we can find a biinvariant volume form $d\\mu$, that also descends to a volume form on $\\rm{M}$. We now define a left-invariant non-degenerate $2$-form $\\phi$ on $G_o$ by setting\n$$\\phi_e(x_{i},x_j) := \n\\int_M \\o(x_{i}^*,x_{j}^*)\\ d\\mu.$$\nAs $\\mathcal L_{x_k^*}d\\mu = 0$ for every $k$, we have for every $i,j,k=1,\\ldots,2n$\n$$\\int_M x_k^*\\o(x_{i}^*,x_{j}^*)\\ d\\mu = \\int_M \\mathcal L_{x_k^*}( \\o(x_{i}^*,x_{j}^*)\\ d\\mu) = 0$$ by Stokes' theorem and therefore we obtain that \n$$d\\phi(x_{i},x_j,x_k) = \n\\int_M d\\o(x_{i}^*,x_j^*,x_{k}^*)\\ d\\mu = 0.$$\nThis implies that $\\phi$ is a symplectic form and the proof is concluded.\\end{proof}\nTherefore we are interested in the existence of special Hermitian metrics on the complex manifold ($\\rm{M},\\rm{J}$), in particular balanced and pluriclosed metrics, when the group $\\rm{G}_o$ is of non-compact type. \\par \nThe case of a simple Lie algebra $\\gg_o$ which is the realification of a complex simple Lie algebra $\\gg$ can be easily treated and will be dealt with in subsection 2.3.\\par\nWe will now focus on some subclasses of simple real algebras, namely those which are absolutely simple and of inner type. \\par\n\\subsection{Simple Lie algebras of inner type}\\label{simple} Let $\\gg_o$ be an absolutely simple real algebra of non-compact type. It is well-known that $\\gg_o$ admits a Cartan decompositon \n$$\\gg_o = \\mathfrak{k} + \\mathfrak{p},$$\nwhere $\\mathfrak{k}$ is a maximal compactly embedded subalgebra and \n$$[\\mathfrak{k},\\mathfrak{p}]\\subseteq \\mathfrak{p},\\quad [\\mathfrak{p},\\mathfrak{p}]\\subseteq \\mathfrak{k},$$\nso that $(\\gg_o,\\mathfrak{k})$ is a symmetric pair. Moreover the algebra $\\gg_o$ is said to be {\\it of inner type} when the symmetric pair $(\\gg_o,\\mathfrak{k})$ is of inner type, i.e. when a Cartan subalgebra $\\mathfrak{t}$ of $\\mathfrak{k}$ is a Cartan subalgebra of $\\gg_o$, i.e. its complexification $\\mathfrak{t}^c$ is a Cartan subalgebra of $\\gg_o^c$. Using the notation as in \\cite{He}, p.~126, we obtain the list of all inner symmetric pairs $(\\gg_o,\\mathfrak{k})$ of non-compact type with $\\gg_o$ simple and even dimensional (Table 1).  \\par\n\\begin{table}[ht]\\label{T1}\n\t\\centering\n\t\\renewcommand\\arraystretch{1.1}\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t{\\mbox{Type}}\t\t\t\t& \t$\\gg$\t\t\t\t\t&\t$\\mathfrak{k}$\t& \t{\\mbox{conditions}}\t\t\t \t\\\\ \\hline \\hline\n\t$A$ &\t$\\mathfrak{su}(p,q)$ & $\\mathfrak{su}(p) + \\mathfrak{su}(q) +\\mathbb R$ & $p\\geq q\\geq 1$, \\ $p+q\\ {\\rm{odd}}$\t\t\t\t\\\\ \\hline\n\t$B$ & $\\mathfrak{so}(2p+1,2q)$ & $\\mathfrak{so}(2p+1) + \\mathfrak{so}(2q)$ & $p\\geq 0,q\\geq 1$, \\ $p+q\\ {\\rm{even}}$\t\t\t\\\\ \\hline\n\t$C$ & $\\mathfrak{sp}(2n,\\mathbb R)$ & $\\mathfrak{su}(2n)+\\mathbb R$ & $n\\geq 1$\t\t\t\\\\ \\hline\n\t$C$ & $\\mathfrak{sp}(p,q)$ & $\\mathfrak{sp}(p) + \\mathfrak{sp}(q)$ & $p,q\\geq 1$, \\ $p+q\\ {\\rm{even}}$\t\t\t\\\\ \\hline\n\t$D$ & $\\mathfrak{so}(4n)^*$ & $\\mathfrak{su}(2n)+\\mathbb R$ & $n\\geq 2$\t\t\t\\\\ \\hline\n\t$D$ & $\\mathfrak{so}(2p,2q)$ & $\\mathfrak{so}(2p) + \\mathfrak{so}(2q)$ & $p,q\\geq 1,p+q\\ \\rm{even}\\ \\geq 4$\t\t\t\\\\ \\hline\n\t$G$ & $\\gg_{2(2)}$ & $\\mathfrak{su}(2)+\\mathfrak{su}(2)$ & \t\t\t\\\\ \\hline\n\t$F$ & $\\mathfrak{f}_{4(-20)}$ & $\\mathfrak{so}(9)$ & \t\t\t\\\\ \\hline\n\t$F$ & $\\mathfrak{f}_{4(4)}$ & $\\mathfrak{su}(2)+\\mathfrak{sp}(3)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{6(2)}$ & $\\mathfrak{su}(2)+\\mathfrak{su}(6)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{6(-14)}$ & $\\mathfrak{so}(10)+\\mathbb R$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{8(8)}$ & $\\mathfrak{so}(16)$ & \t\t\t\\\\ \\hline\n\t$E$ & $\\mathfrak{e}_{8(-24)}$ & $\\mathfrak{su}(2)+\\mathfrak{e}_7$ & \t\t\t\\\\ \\hline\n\t\\end{tabular}\n\t\\vspace{0.1cm}\n\t\\caption{Inner symmetric pairs $(\\gg,\\mathfrak{k})$ of non-compact type with $\\gg$ simple and even dimensional.}\\label{table}\n\\end{table}\n\n\\subsection{Invariant complex structures} In this section we will describe how to construct invariant complex structures on even-dimensional absolutely simple non-compact Lie algebras $\\gg_o$. \\par\nWe fix a maximal abelian subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$, so that $\\mathfrak{h}:= \\mathfrak{t}^c$ is a Cartan subalgebra of $\\gg:= \\gg_o^c$. Note that if $\\gg_o$ is even dimensional , the same holds for $\\mathfrak{t}$. The corresponding root system is denoted by $R$ and we have the following decompositions \n$$\\mathfrak{k}^c = \\mathfrak{t}^c \\oplus \\bigoplus_{\\a\\in R_\\mathfrak{k}}\\gg_\\a,\\quad \\mathfrak{p}^c = \\bigoplus_{\\a\\in R_\\mathfrak{p}}\\gg_\\a,$$\nwhere a root $\\a$ will be called {\\it compact} (resp. {\\it non-compact}), when $\\gg_\\a\\subseteq \\mathfrak{k}^c$ (resp. $\\gg_\\a\\subseteq \\mathfrak{p}^c$) and the set of all compact (resp. non-compact) roots is denoted by $R_\\mathfrak{k}$ (resp. $R_\\mathfrak{p}$). It is a standard fact that $\\mathfrak{u}:= \\mathfrak{k} + i\\mathfrak{p}\\subseteq \\gg$ is a compact real form of $\\gg$ and that we can choose a basis $\\{E_\\a\\}_{\\a\\in R}$ of root spaces so that \n$$\\tau(E_\\a) = -E_{-\\a}, \\qquad B(E_\\a,E_{-\\a}) = 1,\\qquad [E_\\a,E_{-\\a}] = H_\\a$$\nwhere $\\tau$ denotes the anticomplex involution defining $\\mathfrak{u}$, $B$ is the Cartan Killing form of $\\gg$ and $H_\\a$ is the $B$-dual of $\\a$ (see e.g. \\cite{He}). If $\\sigma$ is the involutive anticomplex map defining $\\gg_o$, we then have that \n$$\\sigma(E_\\a) = -E_{-\\a},\\quad \\a\\in R_\\mathfrak{k},$$\n$$\\sigma(E_\\a) = E_{-\\a},\\quad \\a\\in R_\\mathfrak{p}.$$\nIf we fix an ordering , namely a splitting $R = R^+\\cup R^-$ with $R^-=-R^+$ and $(R^++R^+)\\cap R \\subseteq R^+$, we can define a subalgebra \n$$\\mathfrak{q} := \\mathfrak{h}_1 \\oplus \\bigoplus_{\\a\\in R^+}\\gg_\\a,$$\nwhere $\\mathfrak{h}_1\\subset \\mathfrak{h}$ is a subspace so that $\\mathfrak{h}_1\\oplus \\sigma(\\mathfrak{h}_1)= \\mathfrak{h}$. The so defined subalgebra $\\mathfrak{q}\\subset \\gg$ satisfies \n$$\\gg = \\mathfrak{q} \\oplus \\sigma(\\mathfrak{q})$$\nand therefore it defines a complex structure $\\rm{J}$ on $\\gg_o$ with the property that $\\mathfrak{q} = \\gg_o^{10}$. This complex structure depends on the arbitrary choice of $\\mathfrak{h}_1$, i.e. on the arbitrary choice of a complex structure on $\\mathfrak{t}$. \\par \nWe remark that the complex structure $\\rm{J}$ enjoys the further property of being $\\ad(\\mathfrak{t})$-invariant, namely \n$$[\\ad(x),\\rm{J}] = 0,\\quad x\\in\\mathfrak{t}.$$\nTherefore if $\\rm{G}_o$ is a Lie group with Lie algebra $\\gg_o$, then $\\rm{J}$ extends to a left-invariant complex structure on $\\rm{G}_o$ and it will be also right-invariant with respect to right translations by elements $h\\in {\\rm{T}}:= \\exp(\\mathfrak{t})$ (note that ${\\rm{T}}$ might be non-compact, unless $\\rm{G}_o$ has finite center). \\par\nWe will call such an invariant complex structure {\\it standard}.\n\\begin{remark} In \\cite{Sn} the class of (simple) real Lie algebras of inner type is called ``Class I'' and it is then proved that {\\it every} invariant complex structure in these algebras are standard, with respect to a suitable choice of a Cartan subalgebra (such complex structures are called regular in \\cite{Sn}).\n\\end{remark}\n\n\\subsection{Invariant metrics and the balanced condition}\\label{inv}\nLet $\\rm{M}$ be a compact complex manifold of the form $\\rm{G}_o\/\\Gamma$, endowed with a complex structure $\\rm{J}$ which is induced by a standard invariant complex structure $\\rm{J}$ on $\\rm{G}_o$, as in the previous section. It is clear that any left invariant $\\rm{J}$-Hermitian metric $h$ on $\\rm{G}_o$ induces an Hermitian metric $\\bar h$ on $\\rm{M}$ and $\\bar h$ is balanced or pluriclosed if and only if $h$ is so. For the converse, we prove the following\n\\begin{prop} If $(\\rm{M},\\rm{J})$ admits  a balanced (pluriclosed) Hermitian metric, there exists a left invariant and right $\\T$-invariant Hermitian metric on $\\rm{G}_o$ which is balanced (pluriclosed resp.).  \\end{prop}\n\\begin{proof} Suppose we have a balanced metric $h$ on $M$ with associated fundamental form $\\o$. Then using the same notation and arguments as in the proof of Prop.\\ref{invK}, we define a left-invariant positive $(n-1,n-1)$-form $\\phi$ on $G_o$ as follows\n$$\\phi_e(x_{i_1},\\ldots,x_{i_{2n-2}}) := \n\\int_M \\o^{n-1}(x_{i_1}^*,\\ldots,x_{i_{2n-2}}^*)\\ d\\mu.$$\nAs $d\\o^{n-1}=0$, we obtain that also $d\\phi=0$. Therefore, we\ncan find an unique $(1,1)$-form $\\hat \\o$ so that $\\hat\\o^{n-1} = \\phi$ (see \\cite{Mi}) and the metric given by $\\hat \\o$ is balanced. As $\\phi$ is left invariant, so is $\\hat \\omega$ by uniqueness. Now, the group $\\Ad(\\T)$ is compact and using a standard avaraging process we can make $\\phi_e$ also $\\Ad(\\T)$-invariant. This means that $\\phi$ is also invariant under right $\\T$-translations. Again, by the uniqueness, the same will hold true for $\\hat\\omega$.\\par As for the pluriclosed condition, the lifted metric from $\\rm{M}$ to $\\rm{G}_o$ is clearly pluriclosed and can be made $T$-invariant by a standard averaging. \\end{proof}\n\\begin{remark} We can now deal with the case when $\\gg_o$ is the realification of a simple Lie algebra $\\gg$. In this case the complex structure $\\rm{J}$ commutes with $\\ad(\\gg_o)$ and $\\gg_o = \\mathfrak{u} + i\\mathfrak{u}$ is a Cartan decomposition, where $\\mathfrak{u}$ is a compact real form of $\\gg$. Let $\\rm{G}_o$ be a real group with algebra $\\gg_o$ and let $\\U$ be the compact subgroup with algebra $\\mathfrak{u}$. Then the metric $h$ which coincides with $-B$ on $\\mathfrak{u}$, with $B$ on $i\\mathfrak{u}$ and such that $h(\\mathfrak{u},i\\mathfrak{u})=0$ is a Hermitian metric which is balanced. Indeed, $h$ is $\\Ad(\\U)$-invariant and therefore the corresponding $\\d\\o$ is  $\\Ad(\\U)$-invariant $1$-form, hence it vanishes identically. This is consistent with the fact that complex parallelizable manifolds carry balanced metrics as they carry Chern-flat metrics, as noted in  \\cite{Ga1}, p. 121 (see also \\cite{AG},\\cite{Gr}). \\par \nOn the other hand, $\\rm{G}_o$ admits no invariant pluriclosed metric. Indeed, any such metric $h$ can be avaraged to produce an $\\Ad(\\U)$-invariant pluriclosed metric, which would be balanced by the previous argument. This is not possible, as a metric which is balanced and pluriclosed at the same time has to be K\\\"ahler (see e.g. \\cite{AI}), contrary to Prop \\ref{invK}.\\end{remark}\n\nWe now focus on the case where $\\gg_o$ is absolutely simple of inner type, endowed with an invariant complex structure.  \nWe fix a Cartan subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$ with corresponding root system $R=R_\\mathfrak{k} \\cup R_\\mathfrak{p}$ as in section \\ref{simple} and we consider an ordering \n$R = R^+\\cup R^-$ giving an invariant complex structure $\\rm{J}_o$ on $\\gg_o\/\\mathfrak{t}$. We extend $\\rm{J}_o$ to an invariant complex structure $\\rm{J}$ on $\\gg_o$. \\par \nWe also fix a basis of a complement of $\\mathfrak{t}$ in $\\gg_o$\n$$v_\\a := \\frac 1{\\sqrt 2}(E_\\a-E_{-\\a}),\\ \nw_\\a := \\frac i{\\sqrt 2}(E_\\a+E_{-\\a}),\\ \\a\\in R_\\mathfrak{k}^+,$$\n$$v_\\a := \\frac 1{\\sqrt 2}(E_\\a+E_{-\\a}),\\ \nw_\\a := \\frac i{\\sqrt 2}(E_\\a-E_{-\\a}),\\ \\a\\in R_\\mathfrak{p}^+,$$\nso that $v_\\a,w_\\a\\in \\gg_o$ for every $\\a\\in R^+$ and moreover \n$$Jv_\\a = w_\\a, \\ Jw_\\a = -v_\\a,$$\n$$[H,v_\\a] = -i\\a(H)w_\\a,\\ H\\in\\mathfrak{h},$$ \n$$[v_\\a,w_\\a] = iH_\\a,\\ \\a\\in R_\\mathfrak{k}^+,$$\n$$[v_\\a,w_\\a] = -iH_\\a,\\ \\a\\in R_\\mathfrak{p}^+.$$\n\n\nWe now construct invariant Hermitian metrics $h$ on $\\gg_o$. First, we define $h$ on $\\mathfrak{t}$ by choosing a $J$-Hermitian metric $h_\\mathfrak{t}$ on $\\mathfrak{t}$. If we set $\\mathfrak{m}_\\a := {\\rm{Span}}\\{v_\\a,w_\\a\\}_{\\a\\in R^+}$, we define for $\\a\\neq \\b\\in R^+$\n$$h(\\mathfrak{t},\\mathfrak{m}_\\a)= 0,\\quad h(\\mathfrak{m}_\\a,\\mathfrak{m}_\\b)= 0,$$\n$$h(v_\\a,v_\\a) = h(w_\\a,w_\\a) = h_\\a^2,\\quad h(v_\\a,w_\\a) = 0$$\nfor $h_a\\in \\mathbb R^+$.\\par\nIn particular we are interested in constructing balanced Hermitian metrics, namely Hermitian metrics whose associated $(1,1)$-form $\\o=h(\\cdot,\\rm{J}\\cdot)$ satisfies $d\\o^{n-1}=0$ or equivalently $\\d\\o=0$, where $\\d$ denotes the codifferential. \\par \nWe use the expression \n$$\\d\\o (x) = -{\\rm{Tr}}\\nabla_{\\cdot}\\o(\\cdot,x) = - \\sum_{i}^{2n}\\nabla_{e_i}\\o(e_i,x) = $$\n$$= \\sum_i \\o(\\nabla_{e_i}e_i,x) + \\o(e_i,\\nabla_{e_{i}}x),$$\nwhere $\\nabla$ denotes the Levi Civita connection of $h$ and $\\{e_i\\}$ is an orthonormal basis of $\\gg_o$ w.r.t. $h$. Note that both $h$ and $\\rm{J}$ are $\\ad(\\mathfrak{t})$-invariant and therefore $\\d\\o$, which is also $\\ad(\\mathfrak{t})$-invariant, does not vanish only when evaluated on elements $x\\in \\mathfrak{t}$.\\par We have the following expression for the Levi Civita connection, namely for $x,y,z\\in \\gg_o$ \n$$2h(\\nabla_xy,z) = h([x,y],z) + h([z,x],y) + h([z,y],x).$$\nThen for every $x\\in\\mathfrak{t}$, $y\\in \\gg_o$ \n$$h(\\nabla_{y}y,x) = h([x,y],y) = 0.$$\nTherefore for $x\\in\\mathfrak{t}$ we have\n\\begin{equation}\\label{comp}\\d\\o(x) = \\sum_i\\o(e_i,\\nabla_{e_i}x) = -\\sum_ih(Je_i,\\nabla_{e_i}x) = \\end{equation}\n$$=-\\frac 12\\left( h([e_i,x],Je_i)+ \n           h([Je_i,e_i],x) + h([Je_i,x],e_i)\\right).$$\nWe now observe that $J$ is $\\ad(\\mathfrak{t})$-invariant and therefore \n$h([Je_i,x],e_i) = -h([e_i,x],Je_i)$ for every $i=1,\\ldots,2n$, \nso that \\eqref{comp} can be written as \n$$-\\d\\o(x) = \\frac 12 \\sum_ih([Je_i,e_i],x)= $$\n$$= \\frac 12 \\cdot 2 \\left(\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2} h([w_\\a,v_\\a],x)+ \n\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}h([w_\\a,v_\\a],x)\\right)=$$\n$$= \\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}h(-iH_\\a,x) + \n\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}h(iH_\\a,x),$$\nso that \n$\\d\\o|_{\\mathfrak{t}} =0$ if and only if \n$$-\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}H_\\a + \\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}H_\\a=0.$$\nSumming up, the metric $h$ is balanced when the following equation is satisfied\n\\begin{equation}\\label{eq}\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac 1{h_\\a^2}\\a = \\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac 1{h_\\a^2}\\a.\\end{equation}\\par\\medskip\nNote that this does {\\it not} depend on the choice of the metric along the toral part $\\mathfrak{t}$.\n\n\\section{Main result}\\label{proof}\nIn this section we will prove our main result\n\\begin{theorem}\\label{main} Every non-compact simple Lie group $\\rm{G}_o$ of even dimension and of inner type admits an invariant complex structure $\\rm{J}$ and an invariant balanced $\\rm{J}$-Hermitian metric.\\end{theorem}\nNote that by Borel's Theorem, we can use a cocompact latice $\\Gamma\\subset\\rm{G}_o$ to obtain compact quotients $\\rm{M}=\\rm{G}_o\/\\Gamma$, which will inherit the same balanced structure. \\par \n\n\nWe start noting that equation \\eqref{eq} involves the unknowns $\\{h_\\a\\}_{\\a\\in R^+}$ and also a choice of positive roots, i.e. an ordering or equivalenty a complex structure on $\\gg_o$. We will always fix a complex structure on $\\mathfrak{t}$ once for all. It is known that giving an ordering on the root system $R$ is equivalent to the choice of a system of simple roots $\\Pi$ and that two systems of simple roots are conjugate under the action of the Weyl group $W$. We may fix a system of simple roots $\\Pi = \\{\\a_1,\\ldots,\\a_r\\}$ and put \n$\\Pi = \\Pi_c \\cup \\Pi_{nc}$, where $\\Pi_{c\/nc}$ denotes the set of simple roots which are compact or noncompact. We set $\\Pi_c=\\{\\phi_1,\\ldots,\\phi_k\\}$, $\\Pi_{nc}=\\{\\psi_1,\\ldots,\\psi_l\\}$, $k+l=r = {\\rm{rank}}(\\gg_o)$.\nEach root $\\a\\in R^+$ can be written as \n$$\\a = \\sum_{i=1}^k n_i(\\a)\\phi_i + \\sum_{j=1}^l m_j(\\a)\\psi_j$$\nfor $n_i(\\a),m_j(\\a)\\in\\mathbb N$ nonnegative integers. If we set $g_\\a:= \\frac 1{h_\\a^2}$ and $g_j:= g_{\\phi_j}, h_j := g_{\\psi_j}$, equation \\eqref{eq} can be written as \n$$\\sum_{\\a\\in R_\\mathfrak{k}^+,\\a\\not\\in\\Pi} g_\\a\\left(\\sum n_j(\\a)\\phi_j + \\sum_j m_j(\\a)\\psi_j\\right) + \\sum_j g_j \\phi_j = $$\n$$= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\a\\not\\in\\Pi} g_\\a\\left(\\sum n_j(\\a)\\phi_j + \\sum_j m_j(\\a)\\psi_j\\right) + \\sum_j h_j \\psi_j,$$\nand therefore \n\\begin{equation}\\label{sys1}\n\\left\\{\\begin{aligned}\ng_j &= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(\\a) &- \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(a),{}\\ \\  &j=1,\\ldots,k,\\\\\nh_j &= \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(\\a) &- \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),{}\\ \\  &j=1,\\ldots,l.\n\\end{aligned}\\right.\\end{equation}\n\\begin{remark} If we consider for instance the case $\\gg_o=\\mathfrak{su}(p,q)$ ($p+q$ even, $p,q\\geq 2$) and the standard system of simple roots $\\Pi=\\{\\epsilon_1-\\epsilon_2,\\epsilon_2-\\epsilon_3,\\ldots,\\epsilon_{p-1}-\\epsilon_p,\\epsilon_p-\\epsilon_{p+1},\\ldots,\\epsilon_{p+q-1}-\\epsilon_{p+q}\\}$ of $\\sl(p+q,\\mathbb C)$, then $\\Pi_{nc}=\\{\\epsilon_p-\\epsilon_{p+1}\\}$ and $\\Pi_c$ gives a system of simple roots for the semisimple part $\\mathfrak{k}_{ss}$ of $\\mathfrak{k}$. This means that every root $\\a\\in R_\\mathfrak{k}^+,\\a\\not\\in \\Pi$\nis a linear combination of roots in $\\Pi_c$ and therefore the righthandside of the last equation in \\eqref{sys1} is non-positive, so that \\eqref{sys1} has no solution. This shows that the choice of the invariant complex structure might not be straightforward.   \n\\end{remark}\n\nThe following lemmata provide key tools in our argument.\n\\begin{lemma}\\label{L1} For each symmetric pair $(\\gg_o,\\mathfrak{k})$ as in Table 1, $(\\gg_o,\\mathfrak{k})\\not\\cong (\\mathfrak{so}(1,2n),\\mathfrak{so}(2n))$ and given a Cartan subalgebra $\\mathfrak{t}\\subseteq \\mathfrak{k}$ with corresponding root system $R$, there exists an ordering of the roots, hence a system of simple roots $\\Pi$, such that \n\\begin{equation}\\label{prop} \\forall \\psi \\in \\Pi_{nc}\\ \\exists \\psi'\\in \\Pi_{nc}\\ {\\rm{with}}\\ \\psi+\\psi'\\in R.\\end{equation} \n\\end{lemma}\nThis implies that, if $\\Pi_{nc}=\\{\\psi_1,\\ldots,\\psi_l\\}$, then for every $\\psi_j\\in \\Pi_{nc}$ there exists $\\a\\in R_\\mathfrak{k}^+$ with $m_j(\\a)\\neq 0$ and $\\a\\in {\\rm{Span}}\\{\\Pi_{nc}\\}$.\\par\\medskip\n\\noindent {\\bf Remark} Note that $\\sp(1,1)\\cong\\mathfrak{so}(1,4)$ is also not admissible in the above Lemma. In general, for $\\gg_o = \\mathfrak{so}(1,2n)$ we have the standard system $\\Pi=\\{\\epsilon_i-\\epsilon_{i+1},\\epsilon_n,\\ i=1,\\ldots,n-1\\}$ with \n$\\Pi_{nc}= \\{\\epsilon_n\\}$. As $R_c$ consists precisely of all the short roots, it is clear that for any element $\\sigma$ of the Weyl group $W\\cong \\mathbb Z_2^n\\ltimes \\mathcal S_n$ we have that $\\sigma(\\Pi)_{nc}$ consists of one element. We will deal with this case later on.\n\\begin{proof} We first deal with the classical case. We start with the standard system of simple roots $\\Pi$, following the notation as in \\cite{He}. It is immediate to check that in this case $\\Pi_{nc}$ consists of a single root $\\psi$. \\par \nWe first deal with the case where $\\psi$ is a short root. Let $\\Lambda$ be the set of all simple roots which are connected to $\\psi$ in the Dynkin diagram relative to $\\Pi$. If $s\\in W$ denotes the reflection around $\\psi$, then $s$ leaves every element $\\Pi\\setminus \\Lambda$ pointwise fixed. We observe that $\\Lambda$ consists of either at most three short roots or it contains a long root.  In the first case, $s(\\Lambda) = \\{\\psi + \\lambda|\\ \\l\\in\\Lambda\\}\\subseteq R_\\mathfrak{p}$ so that $s(\\Pi)_{nc} = \\{-\\psi,s(\\Lambda)\\}$ and therefore the system of simple roots $s(\\Pi)$ satisfies \\eqref{prop}. If $\\Lambda$ contains a long root, then it also contains a short root, unless $(\\gg_o,\\mathfrak{k})=(\\mathfrak{so}(2,3),\\mathbb R + \\mathfrak{so}(3))$, that is isomorphic to $(\\sp(2),\\mathfrak{u}(2))$; this case will be dealt with in the second part of the proof.  Therefore $\\Lambda = \\{\\phi_1,\\phi_2\\}$ with $\\phi_1$ short and $\\phi_2$ long. Again the reflection $s$ around $\\psi$ gives $s(\\phi_1) = \\psi+\\phi_1$ and $s(\\phi_2) = \\phi_2+2\\psi\\in R_\\mathfrak{k}$ or $s(\\phi_2) = \\psi+\\phi_2\\in R_\\mathfrak{p}$. This implies that the system of simple roots $s(\\Pi)$ has $s(\\Pi)_{nc} = \\{-\\psi,\\psi+\\phi_1\\}$ or $\\{-\\psi,\\psi+\\phi_1,\\psi+\\phi_2\\}$ and in both cases it satisfies \\eqref{prop}.\\par \nWe are left with the case where $\\psi$ is a long root, namely the case where $\\gg_o=\\sp(2n,\\mathbb R)$ and $\\mathfrak{k} = \\mathfrak{u}(2n)$. A standard system of simple roots is given by $\\Pi=\\{\\epsilon_1-\\epsilon_2,\\epsilon_2-\\epsilon_3,\\ldots,\\epsilon_{2n-1}-\\epsilon_{2n},2\\epsilon_{2n}\\}$ and $\\Pi_{nc}= \\{\\psi=2\\epsilon_{2n}\\}$. Again using $s_{\\b}$, we see that $s_\\b(\\Pi)_{nc}=\\{-2\\epsilon_{2n},\\epsilon_{2n-1}+\\epsilon_{2n}\\}$ so that condition \\eqref{prop} is satisfied.\\par\nWe may now deal with the exceptional cases. Starting with the standard system of simple roots $\\Pi$, we list the set $\\Pi_{nc}$, that turns out to consist of a single root $\\b$.  For each case, using the symmetry $s_\\b$ we obtain  the system of simple roots $\\Pi':=s_\\b(\\Pi)$ that satisfies condition \\eqref{prop}.\\par \n\\noindent (1)\\ $(\\gg_o,\\mathfrak{k}) = (\\gg_2,\\mathfrak{su}(2)+\\mathfrak{su}(2))$. Here $\\Pi=\\{\\a,\\b\\}$, with $\\b$ long. We have $\\Pi_{nc}=\\{\\b\\}$ and $\\Pi'=\\{-\\b,\\a+\\b\\}$.\\par\n\\noindent (2)\\ $(\\gg_o,\\mathfrak{k}) = (\\mathfrak{f}_{4(-20)},\\mathfrak{so}(9))$. According to \\cite{He}, the standard system of simple roots is $\\Pi=\\{\\a_1=\\epsilon_2-\\epsilon_3,\\a_2=\\epsilon_3-\\epsilon_4,\\a_3=\\epsilon_4,\\a_4=\\frac 12(e_1-\\epsilon_2-\\epsilon_3-\\epsilon_4)\\}$ so that \n$\\Pi_{nc}=\\{\\a_4\\}$ and therefore  $\\Pi'_{nc}=\\{-\\a_4,\\a_4+\\a_3\\}$.\\par \n\\noindent (3)\\ $(\\gg_o,\\mathfrak{k}) = (\\mathfrak{f}_{4(4)},\\mathfrak{su}(2)+\\mathfrak{sp}(3))$. In this case $\\Pi_{nc}=\\{\\a_1\\}$ and therefore $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_2\\}$. \\par\n\\noindent\\ (4)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{8(8)},\\mathfrak{so}(16))$. For $\\mathfrak{e}_8$ we have the standard system of simple roots \n$$\\a_1=\\frac 12(\\epsilon_1+\\epsilon_8)-\\frac 12(\\epsilon_2+\\epsilon_3+\\epsilon_4+\\epsilon_5+\\epsilon_6+\\epsilon_7), \\a_2=\\epsilon_1+\\epsilon_2, $$ \n$$\\a_j=\\epsilon_{j-1}-\\epsilon_{j-2},\\ j=3,\\ldots,8.$$\nThen $\\Pi_{nc}=\\{\\a_1\\}$ and $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_3\\}$.\\par \n\\noindent\\ (5)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{8(-24)},\\mathfrak{su}(2)+\\mathfrak{e}_7)$. Keeping the same notation for simple roots as above, we have $\\Pi_{nc} = \\{\\a_8\\}$ and $\\Pi'_{nc}=\\{-\\a_8,\\a_8+\\a_7\\}$.\\par\n\\noindent\\ (6)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{6(2)},\\mathfrak{su}(2)+\\mathfrak{su}(6))$. As the system root system $\\Pi$ can be taken to be composed of the simple roots $\\{\\a_1,\\ldots,\\a_6\\}$ of $\\mathfrak{e}_8$, we have \n$\\Pi_{nc}=\\{\\a_2\\}$ and $\\Pi'_{nc}=\\{-\\a_2,\\a_2+\\a_4\\}$.\\par \n\\noindent\\ (7)\\ $(\\gg_o,\\mathfrak{k})=(\\mathfrak{e}_{6(-14)},\\mathbb R+\\mathfrak{so}(10))$. We have $\\Pi_{nc}=\\{\\a_1\\}$ and $\\Pi'_{nc}=\\{-\\a_1,\\a_1+\\a_3\\}$.\\end{proof}\n\n\n\\begin{lemma}\\label{L2} For every system of simple roots $\\Pi = \\Pi_{c}\\cup \\Pi_{nc}$ \n with $\\Pi_{c}=\\{\\phi_1,\\ldots,\\phi_k\\}$ we have \n$$\\forall\\ j=1,\\ldots,k,\\ \\exists\\ \\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in \\Pi : \\ n_j(\\a)\\neq 0,$$\n where $n_j(\\a)$ denotes the coordinate of $\\a$ along the root $\\phi_j$.\n\\end{lemma}\n\\begin{proof} We start noting that the centralizer $C_{\\mathfrak{k}^c}(\\mathfrak{p}^c) = C_{\\mathfrak{k}}(\\mathfrak{p})^c = \\{0\\}$. It then follows that $[E_{\\phi_j},\\mathfrak{p}^c]\\neq \\{0\\}$, hence there exists $\\gamma\\in R_\\mathfrak{p}$ with $[E_{\\phi_j},E_\\gamma]\\neq 0$, i.e. $\\phi_j+\\gamma \\in R_\\mathfrak{p}$. Now, if $\\gamma>0$, then $\\a:= \\phi_j+\\gamma\\in R_\\mathfrak{p}^+\\setminus\\Pi$ and $n_j(\\a)\\geq 1$. Suppose now $\\gamma<0$. We write $\\gamma=c_j\\phi_j + \\sum_{\\theta\\in \\Pi\\setminus{\\phi_j}} c_\\theta\\theta$ for some nonpositive integers $c_j,c_\\theta$. As $\\gamma\\neq -\\phi_j$, there exists at least one negative coefficient $c_\\theta<0$, for some $\\theta\\in \\Pi,\\theta\\neq\\phi_j$. Therefore the root $\\gamma+\\phi_j$ must be negative and $1+c_j\\leq 0$, i.e. $\\a:=-\\gamma\\in R_\\mathfrak{p}^+\\setminus\\Pi$ and $n_j(\\a)=-c_j\\geq 1$.\n\\end{proof}\n\n\nWe now fix a system of simple roots $\\Pi$ as in Lemma \\ref{L1}. In order to solve the corresponding system of equations \\eqref{sys1} for the positive unknowns $\\{g_i,h_j,g_\\a\\}$, we will show how to choose the positive values $\\{g_\\a\\}_{\\a\\in R^+\\setminus\\Pi}$ in such a way to guarantee that the constants $\\{g_i,h_j\\}$,\ndefined to satisfy \\eqref{sys1}, are positive.\\par \nWe set \n$$\\Sigma_\\mathfrak{k} := \\{\\a\\in R_\\mathfrak{k}^+|\\ \\a\\not\\in \\Pi, \\a\\in {\\rm{Span}}\\{\\Pi_{nc}\\}\\},\\qquad A_\\mathfrak{k} = (R_\\mathfrak{k}\\setminus \\Pi_c) \\setminus \\Sigma_\\mathfrak{k}.$$\nThen the system of equations \\eqref{sys1} can be written as \n\\begin{equation}\\label{sys2}\n\\left\\{\\begin{aligned}\ng_j &= \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a n_j(\\a) - \\sum_{\\a\\in A_\\mathfrak{k}} g_\\a n_j(a),{}\\ \\  &j=1,\\ldots,k,\\ &(1)\\\\\nh_j &= \\sum_{\\a\\in R_\\mathfrak{k}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(\\a) - \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),{}\\ \\  &j=1,\\ldots,l.\\ &(2)\n\\end{aligned}\\right.\\end{equation}\nWe start assigning $g_\\a=1$ for every $\\a\\in A_\\mathfrak{k}$. \\par \nThen, for every $j=1,\\ldots,k$, we use Lemma \\ref{L2} selecting a root $\\a\\in R_\\mathfrak{p}^+$ with $n_j(\\a)\\neq 0$, $\\a\\not\\in \\Pi$. This root $\\a$, which depends on $j$, contributes to the first sum in the righthandside of equation (1) in \\eqref{sys2} and the value $g_\\a$ can be chosen big enough so that $g_j$ is strictly positive. Summing up, we can assign values \n$\\{g_\\a\\}_{\\a\\in R_\\mathfrak{p}^+\\setminus \\Pi_{nc}}$ so that all $g_j$, $j=1,\\ldots,k$ can be defined as in \\eqref{sys2}, (1), and are strictly positive. \\par \nWe now turn to equation \\eqref{sys2}-(2), which can now be written as \n\\begin{equation}\\label{2}h_j= \\sum_{\\a\\in \\Sigma_\\mathfrak{k}} g_\\a m_j(\\a) +\\sum_{\\a\\in A_\\mathfrak{k}} m_j(\\a)  - \\sum_{\\a\\in R_\\mathfrak{p}^+,\\ \\a\\not\\in\\Pi} g_\\a m_j(a),\\end{equation}\nwhere in the righthandside the last two sums have a fixed value. Now, by Lemma \\ref{L1}, we know that for every $j=1,\\ldots,l$, we can find $\\a\\in \\Sigma_\\mathfrak{k}$ with $m_j(\\a)\\neq 0$. These roots can be used to choose the coefficients $g_\\a$ big enough to guarantee that $h_j>0$, when defined to satisfy \\eqref{2}, is strictly positive.\\par \nIn order to complete the proof of our main result Theorem \\eqref{main}, we are left with the case $(\\gg_o,\\mathfrak{k})= (\\mathfrak{so}(1,2n),\\mathfrak{so}(2n))$ with standard system of simple roots $\\Pi=\\{\\epsilon_i-\\epsilon_{i+1},\\epsilon_n,\\ i=1,\\ldots,n-1\\}$, $\\Pi_{nc}=\\{\\epsilon_n\\}$. We see that \n$$R_\\mathfrak{k}^+=\\{\\epsilon_i\\pm\\epsilon_j,\\ i< j\\},\\quad R_\\mathfrak{p} =\\{\\epsilon_1,\\ldots,\\epsilon_n\\}.$$\nNow, we use equation \\eqref{eq} and search for positive real numbers $\\{x,y,z_i,\\ i=1,\\ldots,n\\}$ so that \n$$x\\cdot\\sum_{i<y}\\epsilon_i-\\epsilon_j + y\\cdot \\sum_{i< j} \\epsilon_i+e_j = \\sum_{i=1}^n z_i\\epsilon_i,$$\ni.e.\n$$\\sum_{i=1}^n [(x+y)(n-i)+(x-y)(i-1)]\\epsilon_i = \\sum_{i=1}^n z_i\\epsilon_i.$$\nIt is clear that the above equation has positive solutions by simply choosing $x>y>0$.\n\\begin{remark} We can consider the metric $h_o$ which coincides with $-B$ on the compact part $\\mathfrak{k}$, with $B$ on $\\mathfrak{p}$ and such that $h_o(\\mathfrak{k},\\mathfrak{p})=0$. This metric is easily seen to depend only on $\\gg_o$ and {\\it not} on the Cartan decomposition $\\gg_o=\\mathfrak{k}+\\mathfrak{p}$. We could then ask whether there exists a suitable complex structure such that the metric $h_o$ turns out to be balanced. The resulting equation has been already treated in \\cite{AP} and has a solution if and only if $\\gg_o = \\mathfrak{su}(p,p+1)\\cong \\mathfrak{su}(p+1,p)$ for $p\\geq 1$.   \n \\end{remark}\n\\section{Non-existence of pluriclosed metrics}\nIn this section we prove the following non-existence result\n\\begin{proposition} The compact complex manifolds ($\\rm{M},\\rm{J}$), wheer $\\rm{M}=\\rm{G}_o\/\\Gamma$, do not admit any pluriclosed metric.\n \\end{proposition}\nNote that in the above statement $\\rm{J}$ is the complex structure we have exhibited in section \\ref{proof}. \\par \nNow, if $h$ is any such metric, we can obtain a pluriclosed invariant metric $h$ on $\\rm{G}_o$ which is also invariant under right $\\T$-translations. It follows that on $\\gg$ we have \n$$h(\\gg_\\a,\\gg_\\b) = 0 \\quad {\\rm{if}}\\quad \\b\\neq-\\a.$$\nIn order to write down the condition $dd^c\\o=0$, where $\\o$ is the fundamental form of $h$, we recall the Koszul's formula for the differential of invariant forms. If $\\phi$ is any invariant $k$-form on $\\rm{G}_o$ or equivalently on $\\gg_o$, then for every $v_o,\\ldots,v_{k}$ in $\\gg_o$ \n$$d\\phi(v_o,\\ldots,v_{k}) = \\sum_{i<j}(-1)^{i+j}\\phi([x_i,x_j],\nv_1,\\ldots,\\widehat{v_i},\\ldots,\\widehat{v_j}\\ldots,v_{k}).$$\nWe set $\\phi:=d^c\\o$ and compute $d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b})$ for $\\a,\\b\\in R^+$. We have \n$$d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -\\phi(H_\\a,E_\\b,E_{-\\b}) +\n\\phi(N_{\\a\\b}E_{\\a+\\b},E_{-\\a},E_{-\\b})$$\n$$-\\phi(N_{\\a,-\\b}E_{\\a-\\b},E_{-\\a},E_\\b)-\\phi(N_{-\\a,\\b}E_{\\b-\\a},E_\\a,E_{-\\b})+\\phi(N_{-\\a,-\\b}E_{-\\a-\\b}E_\\a,E_\\b)$$\n$$- \\phi(H_\\b,E_\\a,E_{-\\a}), $$\nwhere we use the standard notation $[E_{\\gamma},E_{\\epsilon}]=N_{\\gamma,\\epsilon}E_{\\gamma+\\epsilon}$ for every $\\gamma,\\epsilon\\in R$. Using the known identities for the Weyl basis (see \\cite{He}, p. 172,176), we can write that \n$$d\\phi(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -\\phi(H_\\a,E_\\b,E_{-\\b}) \n- \\phi(H_\\b,E_\\a,E_{-\\a})$$\n$$+2\n\\phi(N_{\\a\\b}E_{\\a+\\b},E_{-\\a},E_{-\\b}) - 2\\phi(N_{\\a,-\\b}E_{\\a-\\b},E_{-\\a},E_\\b).$$\nWe also introduce the notation $JE_\\gamma= i\\epsilon_\\gamma E_\\gamma$ for every $\\gamma\\in R$, where $\\epsilon_\\gamma=\\pm 1$ according to $\\gamma\\in R^{\\pm}$. Then\n$$dd^c\\o(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) =  -d\\o(JH_\\a,E_\\b,E_{-\\b}) -  d\\o(JH_\\b,E_\\a,E_{-\\a}) $$\n$$- 2iN_{\\a,\\b} d\\o(E_{\\a+\\b},E_{-\\a},E_{-\\b}) - 2iN_{\\a,-\\b}\\epsilon_{\\a-\\b}d\\o(E_{\\a-\\b},E_{-\\a},E_\\b).$$ \nNow we easily compute \n$$d\\o(JH_\\a,E_\\b,E_{-\\b}) = -\\o(H_\\b,JH_\\a)$$\nand \n$$d\\o(E_{\\a+\\b},E_{-\\a},E_{-\\b}) = N_{\\a,\\b}(\\o(E_\\a,E_{-\\a})+\\o(E_\\b,E_{-\\b}) - \\o(E_{\\a+\\b},E_{-\\a-\\b})),$$\nwhere we have used the fact that $N_{\\a,\\b}= N_{\\a+\\b,-\\b}=-N_{\\a+\\b,-\\a}$ (see \\cite{He}, p. 172). Similarly,\n$$d\\o(E_{\\a-\\b},E_{-\\a},E_\\b) = N_{\\a,-\\b}( - \\o(E_\\b,E_{-\\b}) + \\o(E_\\a,E_{-\\a})-\\o(E_{\\a-\\b},E_{\\b-\\a})).$$\nSumming up we have \n$$dd^c\\o(E_\\a,E_{-\\a},E_\\b,E_{-\\b}) = -2\\o(JH_\\a,H_\\b)$$\n$$-2iN_{\\a,\\b}^2(\\o(E_\\a,E_{-\\a})+\\o(E_\\b,E_{-\\b}) - \\o(E_{\\a+\\b},E_{-\\a-\\b}))$$\n$$-2iN_{\\a,-\\b}^2\\epsilon_{\\a-\\b}(- \\o(E_\\b,E_{-\\b}) + \\o(E_\\a,E_{-\\a})-\\o(E_{\\a-\\b},E_{\\b-\\a})).$$\nWe now set $a_\\a : h(E_\\a,E_{-\\a})$. The pluriclosed condition implies that \n$$0=-h(H_\\a,H_\\b) -iN_{\\a,\\b}^2(-i a_\\a -i a_\\b + ia_{\\a+\\b})$$\n$$- iN_{\\a,-\\b}^2\\epsilon_{\\a-\\b}( -i\\epsilon_{\\b-\\a}a_{\\a-\\b}  + i a_\\b- ia_\\a)$$\nhence\n\\begin{equation}\\label{eq1} h(H_\\a,H_\\b) = N_{\\a,\\b}^2(a_{\\a+\\b}- a_\\a - a_\\b)\n+ N_{\\a,-\\b}^2\\epsilon_{\\a-\\b}( \\epsilon_{\\a-\\b}a_{\\a-\\b}  +  a_\\b-  a_\\a)\\end{equation}\nWe recall that \n$$a_\\a = h(E_\\a,E_{-\\a}) = - h(v_\\a,v_\\a) < 0,\\qquad \\a\\in R_\\mathfrak{k}^+,$$\n$$a_\\a = h(E_\\a,E_{-\\a}) = h(v_\\a,v_\\a) > 0, \\qquad \\a\\in R_\\mathfrak{p}^+,$$\n$$h(H_\\a,H_\\b) = -h(iH_\\a,iH_\\b)\\in \\mathbb R,\\qquad \nh(H_\\a,H_\\a) < 0.$$\n\nNow, we recall that the existence of the complex structure $\\rm{J}$, which we constructed in section \\ref{proof}, relies on Lemma \\ref{L1}. In particular, when $\\gg_o\\neq \\mathfrak{so}(1,2n)$, we have the existence of two simple roots $\\psi_1,\\psi_2\\in \\Pi_{nc}$ with $\\psi_1+\\psi_2=\\phi\\in R_\\mathfrak{k}$. The following lemma is elementary.\n\\begin{lemma} Either $\\psi_1+2\\psi_2\\not\\in R$ or $\\psi_2+2\\psi_1\\not\\in R$.\\end{lemma}\n\\begin{proof} As $\\psi_1,\\psi_2$ are simple, we have $\\pm(\\psi_1-\\psi_2)\\not\\in R$. Now, $\\psi_i+n\\psi_j\\in R$ if and only if $0\\leq n\\leq q_j$ with \n$q_j = -2\\frac{\\langle \\psi_1,\\psi_2\\rangle}{||\\psi_j||^2}\\in\\mathbb N$ for $i\\neq j$. It is then clear that $q_1,q_2\\geq 2$ is impossible, as $\\psi_1\\neq \\psi_2$ implies $q_1\\cdot q_2 < 4$.\\end{proof}\nSuppone then that $\\phi+\\psi_1 = \\psi_2+2\\psi_1\\not \\in R$. We now apply \\eqref{eq1} with two possible choices for $\\a,\\b$, namely:\\par \\medskip\n\\noindent (1)\\ $\\a=\\psi_1,\\b=\\psi_2$. Then \n$$h(H_{\\psi_1},H_{\\psi_2}) = N_{\\psi_1,\\psi_2}^2(a_\\phi-a_{\\psi_1}-a_{\\psi_2}).$$\n(2)\\ $\\a=\\phi,\\b=\\psi_1$. Then \n$$h(H_{\\phi},H_{\\psi_2}) = N_{\\phi,-\\psi_1}^2(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi}).$$\nSubtracting (1) from (2) we get \n$$h(H_{\\psi_2},H_{\\psi_2}) = \\left(N_{\\phi,-\\psi_1}^2+N_{\\psi_1,\\psi_2}^2\\right)(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi}).$$\nThis is a contradiction, as $h(H_{\\psi_2},H_{\\psi_2})<0$, while \n$a_{\\psi_i}>0$ for $i=1,2$ and $a_\\phi<0$.\\par \nWe are left with the case $\\gg_o=\\mathfrak{so}(1,2n)$, that we have dealt with separately in section \\ref{proof}. In this case the complex structure $\\rm{J}$ is defined by the standard system of positive roots, namely $R^+=\\{\\epsilon_i\\pm\\epsilon_j, \\epsilon_i,\\ 1\\leq i\\neq j\\leq n\\}$. In particular $R_\\mathfrak{k}^+ =\\{\\epsilon_i\\pm\\epsilon_j\\}_{i\\neq j}$ and $R_\\mathfrak{p}^+=\\{\\epsilon_i\\}_{i=1,\\ldots,n}$. We now consider $\\psi_i=\\epsilon_i$, $i=1,2$, $\\phi_1=\\psi_1+\\psi_2\\in R_\\mathfrak{k}^+$ and $\\phi_2=\\psi_1-\\psi_2\\in R_\\mathfrak{k}^+$.   We apply \\eqref{eq1} in two different ways: \\par\n\\noindent (1)\\ $\\a=\\psi_1,\\b=\\psi_2$. Then \n$$h(H_{\\psi_1},H_{\\psi_2}) = N_{\\psi_1,\\psi_2}^2(a_\\phi-a_{\\psi_1}-a_{\\psi_2}) + N_{\\psi_1,-\\psi_2}^2(a_{\\phi_2}+a_{\\psi_2}-a_{\\psi_1}).$$\n(2)\\ $\\a=\\phi_1,\\b=\\psi_2$. Note that $\\phi_1+\\psi_1\\not\\in R$. Then \n$$h(H_{\\phi_1},H_{\\psi_1}) =  N_{\\phi_1,-\\psi_1}^2(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi_1}).$$\nTherefore \n$$h(H_{\\psi_1},H_{\\psi_1}) = (N_{\\phi_1,-\\psi_1}^2+N_{\\psi_1,\\psi_2}^2)(a_{\\psi_2}+a_{\\psi_1}-a_{\\phi_1})+ N_{\\psi_1,-\\psi_2}^2(a_{\\psi_1} -a_{\\phi_2}-a_{\\psi_2})$$\nWe now recall that, if $\\gamma,\\d\\in R$, then $N_{\\gamma,\\d}^2= \\frac{q(1-p)}{2}||\\gamma||^2$, where $\\d+n\\gamma$, $p\\leq n\\leq q$, is the $\\gamma$-series containing $\\d$ (see \\cite{He}, p.176). We then immediately see that $N_{\\psi_1,\\psi_2}^2 = N_{\\psi_1,-\\psi_2}^2$ and noting furthermore that $N_{\\phi_1,-\\psi_1}^2 = N_{\\psi_1,\\psi_2}^2$,  we can write \n$$h(H_{\\psi_1},H_{\\psi_1}) = N_{\\psi_1,\\psi_2}^2(a_{\\psi_2}+3 a_{\\psi_1}-2a_{\\phi_1}- a_{\\phi_2}),    $$\ngiving the contradiction $h(H_{\\psi_1},H_{\\psi_1}) >0$.\n\n\n\n\\section{Geometric properties}\nIn this section, we prove the following result, which may contribute to shed to some light on the geometry of the complex balanced manifolds we have constructed in the previous sections. \n\\begin{theorem} If $(\\rm{M},\\rm{J},h)$ is a balanced $n$-dimensional manifold, where $\\rm{M}=\\rm{G}_o\/\\Gamma$, $\\rm{J}$ is a standard invariant complex structure and $h$ is a balanced Hermitian metric, then the metric $h$ has vanishing Chern scalar curvature.\\par \nMoreover $c_1(\\rm{M})=0$ and the Kodaira dimension $\\kappa(M) = -\\infty$.\n \\end{theorem}\n\nWe consider a standard complex structure $\\rm{J}$ on a manifold $\\rm{M} = \\rm{G}_o\/\\Gamma$. We denote by $D$ the Chern connection relative to a Hermitian metric $h$ which is induced by an invariant metric on $\\rm{G}_o$, again denoted by $h$. We can moreover suppose that $h$ is invariant by the right $\\T$-translations. \\par \nIf $x\\in\\gg_o$ and if we still denote by $x$ the induced left-invariant vector field on $\\rm{G}_o$, we consider $D_x\\in\\End(\\gg_o)$ the endomorphism of $\\gg_o$ which assigns to every $y\\in\\gg_o$ the element $D_xy$ corresponding to the left invariant vector field $D_xy$. Clearly $D_x\\in \\mathfrak{so}(\\gg_o,h)$ and $[D_x,\\rm{J}]=0$. Moreover \n\\begin{equation}\\label{chern}D_xy = [x,y]^{10},\\qquad \\forall x\\in \\gg_o^{01},\\ y\\in \\gg_o^{10},\\end{equation}\nthat follows from the fact that $T^{1,1}=0$, where $T$ is torsion of $D$.\\par \nIf $R$ denote the curvature, where $R_{xy} = [D_x,D_y]- D_{[x,y]}$, we are interested in the first Ricci tensor $\\rho$ given by\n$$\\rho(x,y) = -\\frac 12{\\rm{Tr}}(J\\circ R_{xy}).$$\nAs the complex structure and the metric are both invariant under the adjoint action of the group $T = \\exp(\\mathfrak{t})$, we see that \n$$\\rho(\\mathfrak{t},E_\\a) = 0,\\ \\forall \\a\\in R,$$\n$$\\rho(E_\\a,E_\\b)\\neq 0\\ {\\rm{implies}}\\ \\b=-\\a,\\ \\a,\\b\\in R.$$\nTherefore we can compute \n$$\\rho(E_\\a,E_{-\\a}) = \\frac 12 {\\rm{Tr}}(JD_{H_\\a}).$$\n\\begin{lemma} For every $x\\in \\mathfrak{h}$\n$$D_x = \\ad(x).$$\n\\end{lemma}\n\\begin{proof} We use similar arguments as in \\cite{Po}. It will suffice to consider the case where $x\\in \\mathfrak{h}^{10}$; then for every $\\a\\in R^+$ we have \n$$D_xE_{-\\a} = [x,E_{-\\a}]^{01} = [x,E_{-\\a}],\\quad D_x\\mathfrak{h}^{01} = 0$$\nby \\eqref{chern}. Then if $\\b\\in R^+$ we have \n$$h(D_xE_\\a,E_{-\\b}) = - h(E_\\a,D_xE_{-\\b}) = -\\b(x)h(E_\\a,E_{-\\b}) = 0 \\qquad {\\rm{if}}\\ \\a\\neq\\b,$$\nso that $D_xE_\\a=\\a(x)E_\\a = [x,E_\\a]\\ ({\\rm{mod}}\\ \\mathfrak{h})$. As $h(D_xE_\\a,\\mathfrak{h}^{01}) = -h(E_\\a,D_x\\mathfrak{h}^{01}) = 0$, we conclude that \n$$D_xE_\\a = [x,E_\\a].$$\nFinally, $h(D_x\\mathfrak{h}^{10},\\mathfrak{h}^{01})=0$ and $h(D_x\\mathfrak{h}^{10},E_{-\\a}) = \n-h(\\mathfrak{h}^{10},[x,E_{-\\a}]) = 0$, so that $D_x\\mathfrak{h} = 0 = [x,\\mathfrak{h}]$.\\end{proof}\n\nIt follows that \n$$\\rho(\\mathfrak{h},\\mathfrak{h})=0$$ \nand \n$$\\rho(E_\\a,E_{-\\a}) = \\frac 12\\left(2\\sum_{\\beta\\in R^+}i\\beta(H_\\a)\\right) = B(H_\\a,\\delta),$$\nwhere \n$$\\delta = \\sum_{\\beta\\in R^+} iH_\\b\\in \\mathfrak{t} .$$\nThis means that for every $\\a,\\b\\in R$ we have\n$$\\rho(E_\\a,E_\\b) = - B(E_\\a,[\\d,E_\\b]) = B([E_\\a,E_\\b],\\d)$$\nand therefore for every $x,y\\in \\gg_o$\n\\begin{equation} \\label{rho}\\rho(x,y) = B([x,y],\\delta).\\end{equation}\nThis means that $\\rho = d\\phi$, where $\\phi$ is the left-invariant $1$-form that is given by $\\phi(v) = B(v,\\delta)$. Then clearly $c_1(M)=0$.\\par\nWe now show that the tensor powers $K_{\\rm{M}}^{\\otimes k}$ are holomorphically non trivial for every $m\\geq 1$. Indeed, the metric $h$ induces a Hermitian metric on the line bundles $K_{\\rm{M}}^{\\otimes m}$ with curvature form $m\\rho$. If $\\Omega$ is a nowhere vanishing holomorphic section of $K_{\\rm{M}}^{\\otimes k}$ , then $m\\rho =  \n-i\\partial\\overline\\partial \\ln(||\\Omega||^2)$. \nIf we denote by\\, $\\widehat{}$\\,  the result of the symmetrization process, which commutes with the operators $\\partial$ and $\\overline\\partial$, we obtain on $\\rm{G}_o$ that $\\widehat{\\rho} =-i \\partial\\overline\\partial\\widehat{\\ln(||\\Omega||^2)} = 0$. As $\\rho$ is invariant, $\\widehat\\rho=\\rho=0$ and we get a contradiction as $\\d\\ne 0$.\\par \nWe now compute the Chern scalar curvature $s^{Ch}$ of the metric $h$ using formula \\eqref{rho}. We use the orthonormal frame $e_1,\\ldots,e_{2n}$. Then \n$$s^{Ch} = \\sum_i \\rho(Je_i,e_i) = \n-2\\sum_{\\a\\in R^+} \\frac{1}{h_\\a^2} \\rho(v_\\a,w_\\a) = $$\n$$= 2iB\\left(\\sum_{\\a\\in R_\\mathfrak{p}^+}\\frac{1}{h_\\a^2} H_\\a-\\sum_{\\a\\in R_\\mathfrak{k}^+}\\frac{1}{h_\\a^2} H_\\a,\\d\\right) = 0$$\nif we consider the system of positive roots satisfying equation \\eqref{eq}.\nThe claim $\\kappa(\\rm{M})=-\\infty$ now follows from \nThm. 1.4 in \\cite{Y}.\\par \n\n\n\n\n\n\\begin{remark} Note that also for a compact group $\\K$ endowed with an invariant complex structure we have $h^{n,0}(\\K)=0$, see \\cite{Pi}, Prop. 3.7.\\par\nWe finally remark here that the balanced condition implies that the two scalar curvatures that one can obtain tracing the Chern curvature tensor coincide (see \\cite{Ga3}, p. 501).\\end{remark}\n\n\n\n\n\n\n\n\n\n\\vspace{2cm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn mechanical and structural systems the knowledge of all possible\nsolutions is crucial for safety and reliability. In devices modelled\nby linear ordinary differential equations we can predict the existing\nsolutions using analytical methods \\cite{rao1995mechanical,nayfeh2008linear}.\nHowever, in case of complex, nonlinear systems analytical methods\ndo not give the full view of system's dynamics \\cite{warminski2003approximate,he2004homotopy,belendez2006analytical,nayfeh2011introduction}.\nDue to nonlinearity, for the same set of parameters more then one\nstable solution may exist \\cite{feudel1998dynamical,ChudzikPSK11,Yanchuk2011,gerson2012design,brzeski2012dynamics,menck2013basin,Stability_threshold}.\nThis phenomenon is called multistability and has been widely investigated\nin all types of dynamical systems (mechanical, electrical, biological,\nneurobiological, climate and many more). The number of coexisting\nsolutions strongly depends on the type of nonlinearity, the number of degrees\nof freedom and the type of coupling between the subsystems. Hence, usually\nthe number of solutions vary strongly when values of system's parameters changes. \n\nAs an example, we point out the classical tuned mass absorber\n\\cite{arnoldfr19955,Cartmell1994173,Alsuwaiyan2002791,fischer2007wind,BVG2008,Ikeda2010,Chung2012,Brzeski2014298}.\nThis device is well known and widely used to absorb energy and mitigate\nunwanted vibrations. However, the best damping ability is achieved\nin the neighbourhood of the multistability zone \\cite{brzeski2012dynamics}.\nAmong all coexisting solutions only one mitigates oscillations effectively.\nOther solutions may even amplify an amplitude of the base system.\nSo, it is clear that only by analyzing all possible solutions we can\nmake the device robust. \n\nSimilarly, in systems with impacts one solution can ensure correct\noperation of a machine, while others may lead to damage or destruction\n\\cite{Brzeski_bells,blazejczyk1998co,de2001basins,qun2003coexisting,de2004controlling}.\nThe same phenomena is present in multi-degree of freedom systems where\ninteractions between modes and internal resonances play an important\nrole \\cite{bux1986non,haquang1987non,cartmell1988simultaneous,orlando2013influence}. \n\nPractically, in nonlinear dynamical systems with more then one degree\nof freedom it is impossible to find all existing solutions without\nhuge effort and using classical methods of analytical and numerical\ninvestigation (path-following, numerical integration, basins of attractions),\nespecially in cases when we analyse a wider range of system's parameters and we\ncannot precisely predict the initial conditions. Moreover, solutions\nobtained by integration may have meager basins of attraction and it\ncould be hard or even impossible to achieve them in reality. That is why\nwe propose here a new method basing on the idea of basin stability \\cite{menck2013basin}.\nThe classical basin stability method is based on the idea of Bernoulli\ntrials, i.e., equations of system's motion are integrated $N$ times\nfor randomly chosen initial conditions (in each trial they are different).\nAnalyzing the results we asses the stability of each solution. If\nthere exist only one solution the result of all trials is the same.\nBut, if more attractors coexist we can estimate the probability of their\noccurrence for a chosen set of initial conditions. In mechanical and\nstructural systems we want to be sure that a presumed solution is stable\nand has the dominant basin of attraction in a given range of system's parameters.\nTherefore, we build up a basin stability method by drawing values of\nsystem's parameters. We take into account the fact that values of\nparameters are measured or estimated with some finite precision and\nalso that they can slightly vary during normal operation. \n\nThe paper is organized as follows. In Section 2 we introduce simple\nmodels which we use to demonstrate the main idea of our approach.\nIn the next section we present and describe the proposed method. Section\n4 includes numerical examples for systems described in Section 2.\nFinally, in Section 5 our conclusions are given.\n\n\n\\section{Model of systems\\label{sec:Model-of-systems}}\n\nIn this section we present systems that we use to present our method.\nTwo models are taken from our previous papers \\cite{brzeski2012dynamics,czolczynski2012synchronization}\nand the third one was described by Pavlovskaia et. al.\n\\cite{pavlovskaia2010complex}. We deliberately picked models whose\ndynamics is well described because we can easily evaluate the correctness\nand efficiency of the method we propose. \n\n\n\\subsection{Tuned mass absorber coupled to a Duffing oscillator}\n\nThe first example is a system with a Duffing oscillator and a tuned\nmass absorber. It was investigated in \\cite{brzeski2012dynamics}\nand is shown in Figure \\ref{fig:Duffing1}. The main body consists\nof mass $M$ fixed to the ground with nonlinear spring (hardening\ncharacteristic $k_{1}+k_{2}y^{2}$) and a viscous damper (damping coefficient\n$c_{1}$). The main mass is forced externally by a harmonic excitation\nwith amplitude $F$ and frequency $\\omega$. The absorber is modelled\nas a mathematical pendulum with length $l$ and mass $m$. A small\nviscous damping is present in the pivot of the pendulum. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure1}\n\\par\\end{centering}\n\n\\caption{The model of the first considered system. Externally forced Duffing\noscillator with attached pendulum (tuned mass absorber). \\label{fig:Duffing1}}\n\\end{figure}\n\n\nThe equations of the system's motion are derived in \\cite{brzeski2012dynamics},\nhence we do not present their dimension form. Based on the following\ntransformation of coordinates and parameters we reach the dimensionless\nform: \\foreignlanguage{english}{$\\omega_{1}^{2}=\\frac{k_{1}}{M+m}$,\n$\\omega_{2}^{2}=\\frac{g}{l}$, $a=\\frac{m}{M+m}$, $b=\\left(\\frac{\\omega_{2}}{\\omega_{1}}\\right)^{2}$,\n$\\alpha=\\frac{k_{2}l^{2}}{(M+m)\\omega_{1}^{2}}$, $f=\\frac{F}{(M+m)l\\omega_{1}^{2}},$\n$d_{1}=\\frac{c_{x}}{(M+m)\\omega_{1}}$, $d_{2}=\\frac{c_{\\varphi}}{ml^{2}\\omega_{2}}$,\n$\\mu=\\frac{\\omega}{\\omega_{1}}$, $\\tau=t\\omega_{1}$, $x=\\frac{y}{l}$,\n$\\dot{x}=\\frac{\\dot{y}}{\\omega_{1}l}$, $\\ddot{x}=\\frac{\\ddot{y}}{\\omega_{1}^{2}l}$,\n$\\gamma=\\varphi,$ $\\dot{\\gamma}=\\frac{\\dot{\\varphi}}{\\omega_{2}},$\n$\\gamma=\\frac{\\ddot{\\varphi}}{\\omega_{2}^{2}}$.} \n\nThe dimensionless equations are as follows:\n\n\\begin{equation}\n\\begin{array}{c}\n\\ddot{x}-ab\\ddot{\\gamma}\\sin\\gamma-ab\\dot{\\gamma}^{2}\\cos\\gamma+x+\\alpha x^{3}+d_{1}\\dot{x}=f\\cos\\mu\\tau,\\\\\n\\\\\n\\ddot{\\gamma}-\\frac{1}{b}\\ddot{x}\\sin\\gamma+\\sin\\gamma+d_{2}\\dot{\\gamma}=0,\n\\end{array}\\label{eq:row bez}\n\\end{equation}\n where $\\mu$ is the frequency of the external forcing and we consider it\nas controlling parameter. The dimensionless parameters have the following\nvalues: $f=0.5$, $a=0.091$, $b=3.33$, $\\alpha=0.031$, $d_{1}=0.132$\nand $d_{2}=0.02$. Both subsystems (Duffing oscillator and the pendulum)\nhave a linear resonance for $\\mu=1.0$. \n\n\n\\subsection{System with impacts}\n\nAs the next example we analyse a system with impacts \\cite{pavlovskaia2010complex}.\nIt is shown in Figure \\ref{fig:Impact} and consists of mass\n$M$ suspended by a linear spring with stiffness $k_{1}$ and a viscous\ndamper with the damping coefficient $c$ to harmonically moving frame.\nThe frame oscillates with amplitude $A$ and frequency $\\Omega$.\nWhen amplitude of mass $M$ motion reaches the value $g$, we observe soft\nimpacts (spring $k_{2}$ is much stiffer than spring $k_{1}$). \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure2}\n\\par\\end{centering}\n\n\\caption{The model of the second considered system. Externally forced oscillator with\nimpacts. \\label{fig:Impact}}\n\\end{figure}\n\n\nThe dimensionless equation of motion is as follow (for derivation\nsee \\cite{pavlovskaia2010complex}) :\n\n\\[\n\\ddot{x}+2\\xi\\dot{x}+x+\\beta\\left(x-e\\right)\\mathrm{H}\\left(x-e\\right)=a\\omega^{2}\\sin\\left(\\omega\\tau\\right)\n\\]\n\n\nwhere $x=\\frac{y}{y_{0}}$ is the dimensionless vertical displacement\nof mass $M$, $\\tau=\\omega_{n}t$ is the dimensionless time, $\\omega_{n}=\\frac{k_{1}}{M}$,\n$\\beta=\\frac{k_{2}}{k_{1}}$ the stiffness ratio, $e=\\frac{g}{y_{0}}$\n the dimensionless gap between equilibrium of mass $M$ and the\nstop suspended on the spring $k_{2}$, $a=\\frac{A}{y_{0}}$ and $\\omega=\\frac{\\Omega}{\\omega_{n}}$\nare dimensionless amplitude and frequency of excitation, $\\xi=\\frac{c}{2m\\omega_{n}}$\nis the damping ratio, $y_{0}=1.0\\:[\\mathrm{mm}]$ and $\\mathrm{H}(\\cdot)$\n the Heaviside function. In our calculations we take the following\nvalues of system's parameters: $a=0.7$, $\\xi=0.01$, $\\beta=29$,\n$e=1.26$. As a controlling parameter we use the frequency of excitation\n$\\omega$. \n\n\n\\subsection{Beam with suspended rotating pendula}\n\nThe last considered system consists of a beam which can move in\nthe horizontal direction and $n$ rotating pendula. The beam has the mass\n$M$ and supports $n$ rotating, excited pendula. Each pendulum has\nthe same length $l$ and masses $m_{i}$ $(i=1,\\:2,\\ldots,\\:n)$.\nWe show the system in Figure \\ref{fig:Beam_model} \\cite{czolczynski2012synchronization}.\nThe rotation of the $i$-th pendulum is given by the variable $\\varphi_{i}$\nand its motion is damped by the viscous friction described by the damping\ncoefficient $c_{\\varphi}$. The forces of inertia of each pendulum\nacts on the beam causing its motion in the horizontal direction (described\nby the coordinate $x$). The beam is considered as a rigid body, so\nwe do not consider the elastic waves along it. We describe the phenomena\nwhich take place far below the resonances for longitudinal oscillations\nof the beam. The beam is connected to a stationary base by a light\nspring with the stiffness coefficient $k_{x}$ and viscous damper with\na damping coefficient $c_{x}$. The pendula are excited by external\ntorques proportional to their velocities: $N_{0}-\\dot{\\varphi}_{i}N_{1}$,\nwhere $N_{0}$ and $N_{1}$ are constants. If no other external forces\nact on the pendulum, it rotates with the constant velocity $\\omega=N_{0}\/N_{1}$.\nIf the system is in a gravitational field (where $g=9.81\\:[\\mathrm{m\/s^{2}}]$\nis the acceleration due to gravity), the weight of the pendulum causes\nthe unevenness of its rotation velocity, i.e., the pendulum slows\ndown when the centre of its mass goes up and accelerates when the\ncentre of its mass goes down. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure3}\n\\par\\end{centering}\n\n\\caption{The model of the third considered system. Horizontally moving beam\nwith attached pendulums. \\label{fig:Beam_model}}\n\\end{figure}\n\n\nThe system is described by the following set of dimensionless equations:\n\n\\begin{equation}\nm_{i}l^{2}\\ddot{\\varphi}_{i}+m_{i}\\ddot{x}l\\cos{\\varphi_{i}}+c_{\\varphi}\\dot{\\varphi}_{i}+m_{i}gl\\sin{\\varphi_{i}}=N_{0}-\\dot{\\varphi}_{i}N_{1}\\label{eq:Beam1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M+\\sum\\limits _{i=1}^{n}{m_{i}}}\\right)\\ddot{x}+c_{x}\\dot{x}+k_{x}x=\\sum\\limits _{i=1}^{n}{m_{i}l\\left({-\\ddot{\\varphi}_{i}\\cos\\varphi_{i}+\\dot{\\varphi}_{i}^{2}\\sin\\varphi_{i}}\\right)}\\label{eq:Beam2}\n\\end{equation}\nIn our investigation we analyze two cases: a system with two pendula\n(where $n=2$ and $i=1,\\:2$) and with $20$ pendula ($n=20$ $i=1,\\:2,\\:...,n$)..\nThe values of the parameters are as follows: $m_{i}=\\frac{2.00}{n}$,\n$l=0.25$, $c_{\\varphi}=\\frac{0.02}{n}$, $N_{0}=5.0$0, $N_{1}=0.50$,\n$M=6.00$, $g=9.81$, $c_{x}=\\frac{\\ln\\left(1.5\\right)}{\\pi}\\sqrt{k_{x}\\left(M+\\sum\\limits _{i=1}^{n}{m_{i}}\\right)}$and\n$k_{x}$ is a controlling parameter. The derivation of the system's equations\ncan be found in \\cite{czolczynski2012synchronization}. We present the\ntransformation to a dimensionless form in Appendix A. \n\n\n\\section{Methodology}\n\nIn \\cite{menck2013basin} Authors present a ``basin stability''\nmethod which let us estimate the stability and number of solutions\nfor given values of system parameters. The idea behind basin stability\nis simple, but it is a powerful tool to assess the size of complex\nbasins of attraction in multidimensional systems. For fixed values\nof system's parameters, $N$ sets of random initial conditions are\ntaken. For each set we check the type of final attractor. Based on\nthis we calculate the chance to reach a given solution and determine\nthe distribution of the probability for all coexisting solutions. This\ngives us information about the number of stable solutions and the\nsizes of their basins of attraction.\n\nWe consider the dynamical system $\\dot{\\mathbf{x}}=f\\,(\\mathbf{x},\\,\\omega)$,\nwhere $\\mathbf{x}\\in\\mathtt{\\mathbb{R}^{n}}$ and $\\omega\\in\\mathbb{R}$\nis the system's parameter. Let ${\\cal B\\subset\\mathbb{\\mathtt{\\mathbb{R}^{n}}}}$\nbe a set of all possible initial conditions and ${\\cal C}\\subset\\mathbb{\\mathtt{\\mathbb{R}}}$\na set of accessible values of system's parameter. Let us assume that an\nattractor ${\\cal A}$ exists for $\\omega\\in{\\cal C_{A}}\\subset{\\cal C}$\nand has a basin of attraction $\\beta({\\cal A})$. Assuming random\ninitial conditions the probability that the system will reach attractor\n${\\cal A}$ is given by $p\\left({\\cal A}\\right)$. If this probability\nis equal to $p\\left({\\cal A}\\right)=1.0$ this means that the considered\nsolution is the only one in the taken range of initial conditions\nand given values of parameters. Otherwise other attractors coexist.\nThe initial conditions of the system are random from set ${\\cal B_{A}}\\subset{\\cal B}$.\nWe can consider two possible ways to select this set. \n\n\\begin{description}\n  \\item[I] The first ensures that set the ${\\cal B_{A}}$ includes values of initial \nconditions leading to all possible solutions. This approach is appropriate if \nwe want to get a general overview of the system's dynamics. \n  \\item[II] In the second approach we use a narrowed set of initial conditions that \ncorresponds to practically accessible initial states.\\ldots\n\\end{description}\n\nIn our method we chose the second approach because it let us take into \naccount constrains imposed on the system and because in engineering we \nusually know or expect the initial state of the system with some finite \nprecision. \n\nIn the classical approach of Menck et. al. \\cite{menck2013basin} the\nvalues of system's parameters are fixed and do not change during calculations.\nThe novelty of our method is that we not only draw initial conditions\nbut also values of some selected parameters of the system. We assume\nthat the initial conditions and some of the system's parameters are chosen\nrandomly. Then using $N$ trials of numerical simulations we estimate\nthe probability that the system will reach a given attractor ${\\cal A}$\n($p\\left({\\cal A}\\right)$). The idea is to take into consideration\nthe fact that the values of system's parameters are measured or estimated\nwith some finite accuracy which is often hard to determine. Moreover\nvalues of parameters can vary during normal operation. Therefore drawing\nvalus of parameters we can describe how a mismatch in their values influences\nthe dynamics of the system and estimate the risk of failure. In many\npractical applications one is interested in reaching only one presumed\nsolution ${\\cal A}$, and the precise description of other coexisting\nattractors is not necessary. We usually want to know the probability\nof reaching the expected solution $p\\left({\\cal A}\\right)$ and the chance\nthat the system behave differently. If $p\\left({\\cal A}\\right)$ is\nsufficiently large, we can treat the other attractors as an element\nof failure risk. \n\nIn our approach we perform the following steps:\n\\begin{description}\n\t\\item[I] We pick values of system's parameters from the set\n\t\t${\\cal C_{A}}\\subset{\\cal C}$. \n\t\\item[II]We select the set ${\\cal C_{A}}$ so that\n\t\tit consists of all practically accessible values of system's parameters\n\t\t$\\omega$ . This let us ensure that a given solution indeed exists in a practically\n\t\taccessible range (taking into account the mismatch in parameters).\n\t\\item[III] We subdivide the set $C_{A}$ in to $m=1,2,\\dots M$ equally spaced\n\t\tsubsets. The subsets ${\\cal C}_{A}^{m}$ do not overlap and the relation\n\t\t$\\bigcup_{m=1\\dots M}{\\cal C}_{A}^{m}=C_{A}$ is always fulfilled.\n\t\\item[IV] Then for each subset ${\\cal C}_{A}^{m}$ we randomly pick $N$ sets\n\t\tof initial conditions and value of the considered parameter. For each\n\t\tset we check the final attractor of the system. \n\t\\item[V] After a suficient number\n\t\tof trials we calculate the probability of reaching a presumed solution\n\t\tor solutions. \n\t\\item[VI]Finally we describe the relation between the\n\t\tvalue of the system's parameter and the ``basin stability'' of reachable\n\t\tsolutions.\\ldots\n\\end{description}\n\nIn our calculations for each range of parameter values\n(subset ${\\cal C}_{A}^{m}$) we draw from $N=100$ up to $N=1000$\nsets of initial conditions and parameter. The value of $N$ strongly depends\non the complexity of the analysed system. Also the computation time for a\nsingle trial should be adjusted for each system independently\nsuch that it can reach the final attractor. In general, we recommend that\nin most cases $N$ should be at least 100.\n\n\n\\section{Numerical results}\n\n\n\\subsection{Tuned mass absorber coupled to a Duffing oscillator}\n\nAt the beginning we want to recall the results we present in our previous\npaper \\cite{brzeski2012dynamics}. As a a summary we show Figure \\ref{fig:Two-paramters_colour}\nwith a two dimensional bifurcation diagram obtained by the path-following\nmethod. It gives bifurcations for varying amplitude $f$ and frequency\n$\\mu$ of the external excitation (see Eq. \\ref{eq:row bez}). Lines shown\nin the plot correspond to different types of bifurcations (period\ndoubling, symmetry breaking, Neimark-Sacker and resonance tongues).\nWe present these lines in one style because the structure is too complex\nto follow bifurcation scenarios and we do not need that data (details\nare shown in \\cite{brzeski2012dynamics}). We mark areas where we\nobserve the existence of one solution (black colour), or the coexistence of two (grey) and three\n(hatched area) stable solutions. The remaining part of the diagram\n(white area) corresponds to situations where there are four or more\nsolutions. Additionally, by white colour we also mark areas where\nonly the Duffing system is oscillating in 1:1 resonance with the frequency\nof excitation and the pendulum is in a stable equilibrium position,\ni.e., HDP (hanging down pendulum) state. In this case the dynamics\nof the system is reduced to the oscillations of summary mass ($M+m$).\n\nThe detailed analysis of system \\ref{eq:row bez} is time consuming\nand creation of Figure \\ref{fig:Two-paramters_colour} was preceded\nby complex analysis done with large computational effort. Additionally,\nthe obtained results give us no information about the size of the basins\nof attraction of each solution - which practically means that some\nof the solutions may occur only very rarely in the real system (i.e. due to not accessible\ninitial conditions). Nevertheless, such analysis gives us an in-depth\nknowledge about the bifurcation structure of the system. As we can see, the\nrange where less then three solutions exist is rather small, especially\nfor $\\mu<2.0$. To illustrate our method of analysis, we focus on three\nsolutions: $2:1$ oscillating resonance, HDP and $1:1$ rotating resonance\nassuming that only they have some practical meaning.\n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure4}\n\\par\\end{centering}\n\n\\caption{Two-parameter bifurcations diagram of the system (1) in the plane $\\left(f,\\,\\mu\\right)$\nshowing periodic oscillations and rotations of the pendulum. Black colour\nindicates one attractor, grey colour shows two coexisting attractors\n(the same as for black but with a coexisting stable steady state of\nthe pendulum). In the hatched area we observe the coexistence of stable\nrotations and a stable steady state of the pendulum. A detailed analysis\nis presented in \\cite{brzeski2012dynamics}. \\label{fig:Two-paramters_colour}}\n\\end{figure}\n\n\nTo show our results obtained with integration, we compute bifurcation diagrams\nfor $f=0.5$ in the range $\\mu\\in[0.1,\\:3.0]$ (see Figure \\ref{fig:tma_bif}).\nIn Figure \\ref{fig:tma_bif}(a) we increase $\\mu$ from $0.1$ to\n$3.0$ and in Figure \\ref{fig:tma_bif}(b) we decrease $\\mu$ from $3.0$\nto $0.1$. As the initial conditions we take the equilibrium position\n($x_{0}=\\dot{x}_{0}=0.0$ and $\\gamma_{0}=\\dot{\\gamma}_{0}=0.0$).\nIn both panels we plot the amplitude of the pendulum $\\gamma$. Ranges where the \ndiagrams differ we mark by grey rectangles. It is easy to see that there\nare two dominating solutions: HDP and $2:1$ internal resonance. Near\n$\\mu=1.0$ we observe a narrow range of $1:1$ and $9:9$ resonances\nand chaotic motion (for details see Figure 6 in \\cite{brzeski2012dynamics}).\nBased on previous results we know that we detected all solutions existing\nin the considered range, however we do not have information about the size of\ntheir basins of attraction and coexistence. Hence the analysis with the \nproposed method should give us new important information about the system's\ndynamics. Contrary to the bifurcation diagram obtained by path-following\nin Figure \\ref{fig:tma_bif}, we do not observe rotating solutions\n(the other set of initial conditions should be taken).\n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure5}\n\\par\\end{centering}\n\n\\caption{Bifurcation diagram showing the behaviour of the pendulum suspended\non the Duffing oscillator. For subplot (a) the value of the bifurcation parameter\n$\\mu$ was increased, while for subplot (b) we decreased the value of $\\mu$.\nGray rectangles mark the range of the bifurcation parameter $\\mu$ for\nwhich different attractors coexist. A detailed analysis is presented\nin \\cite{brzeski2012dynamics}. \\label{fig:tma_bif}}\n\\end{figure}\n\n\nIn Figure \\ref{fig:Duff_prob} we show the probability of reaching the\nthree aforementioned solutions obtained using the proposed method.\nThe initial conditions are random numbers drawn from the following\nranges: $x_{0}\\in[-2,\\:2]$, $\\dot{x}_{0}\\in[-2,\\:2]$, $\\gamma_{0}\\in[-\\pi,\\:\\pi]$\nand $\\dot{\\gamma}_{0}\\in[-2.0,\\:2.0]$ (ranges there selected basing\non the results from \\cite{brzeski2012dynamics}). The frequency of excitation\nis within a range $\\mu\\in[0,\\:3.0]$ (Figure \\ref{fig:Duff_prob}(a,c)\n), then we refine it to $\\mu\\in[1.25,\\:2.75]$ (Figure \\ref{fig:Duff_prob}(b,d)\n). In both cases we take $15$ equally spaced subsets of $\\mu$ and\nin each subset we calculate the probability of reaching a given solution.\nFor each subset we calculate $100$ trials each time drawing initial\nconditions of the system and a value of $\\mu$ from the appropriate\nrange. Then we plot the dot in the middle of the subset which indicate\nthe probability of reaching a given solution in each considered range.\nLines that connect the dots are shown just to ephasize the tendency. For\neach range we take $N=1000$ because we want to estimate the probability\nof a solution with small a basin of stability (1:1 rotating periodic solution).\n\nAs we can see in Figure \\ref{fig:Two-paramters_colour}, the $2:1$\nresonance solution exists in the area marked by black colour around\n$\\mu=2.0$ and coexists with HDP in the neighbouring grey zone. In Figure\n\\ref{fig:Duff_prob} we mark the probability of reaching the $2:1$ resonance\nusing blue dots. As we expected, for $\\mu<1.4$ and $\\mu>2.2$ the solution\ndoes not exist. In the range $\\mu\\in[1.4,\\:2.2]$ the maximum value of\nprobability $p(2:1)=0.971$ is reached in the subset $\\mu\\in[1.8,\\:2.0]$\nand outside that range the probability decreases. To check if we can\nreach $p(2:1)=1.0$, we decrease the range of parameter's values to\n$\\mu\\in[1.25,\\:2.75]$ and the size of subset to $\\Delta\\mu=0.1$\n(we still have 15 equally spaced subsets). The results are shown in\nFigure \\ref{fig:Duff_prob}(b) similarly in blue colour. In the range\n$\\mu\\in[1.95,\\:2.05]$ the probability $p(2:1)$ is equal to unity\nand in the range $\\mu\\in[1.85,\\:1.95]$ it is slightly smaller $p(2:1)=0.992$.\nHence, for both subsets we can be nearly sure that the system reaches the $2:1$\nsolution. This gives us indication of how precise we have to set the\nvalue of $\\mu$ to be sure that the system will behave in a presumed\nway.\n\nA similar analysis is performed for HDP. The values of probability\nis indicated by the red dots. As one can see for $\\mu<0.8$, $\\mu\\in[1.2,\\:1.4]$\nand $\\mu\\in[2.6,\\:2.8]$, the HDP is the only existing solution. The\nrapid decrease close to $\\mu\\approx1.0$ indicates the $1:1$ resonance\nand the presence of other coexisting solutions in this range (see\n\\cite{brzeski2012dynamics}). In the range $\\mu\\in[1.2,\\:1.4]$ the probability\n$p(\\mathrm{HDP})=1.0$ which corresponds to a border between solutions\nborn from $1:1$ and $2:1$ resonance. Hence, up to $\\mu=2.0$ the\nprobability of the HDP solution is a mirror refection of $p(2:1)$. The\nsame tendency is observed in the narrowed range as presented in Figure\n\\ref{fig:Duff_prob}(b). Finally, for $\\mu>2.0$ the third considered\nsolution comes in and we start to observe an increase of probability\nof the rotating solution $S(\\mu,\\:\\mathrm{HDP})$ as shown in Figure \\ref{fig:Duff_prob}(c).\nHowever, the chance of reaching the rotating solution remains small and never\nexceeds $p(\\mathrm{1:1})=8\\times10^{-3}$. We also plot the probability\nof reaching the rotating solution in the narrower range of $\\mu$ in Figure\n\\ref{fig:Duff_prob}(d). The probability is similar to the one presented\nin Figure \\ref{fig:Duff_prob}(c) - it is low and does not exceed $p(\\mathrm{1:1})=8\\times10^{-3}$.\nNote that the results presented in Figure \\ref{fig:Duff_prob}(a,b) and\nFigure \\ref{fig:Duff_prob}(c,d) are computed for different sets of\nrandom initial conditions and parameter values; hence the obtained\nprobability can be slightly different. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure6}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in (1) system with tuned mass\nabsorber. Subplots (a,b) present solutions with $2:1$ periodic oscillations\n(blue) and without motion of the pendulum (red). Subplots (c,d) present the\nprobability of reaching $1:1$ rotations (black). (Please note that\nin both cases (a,b) and (c,d) the initial conditions and parameter\nare somehow random, hence the results may slightly differ). \\label{fig:Duff_prob}}\n\\end{figure}\n\n\n\n\\subsection{System with impacts}\n\nIn this subsection we present our analysis of different periodic solutions\nin the system with impacts. A discontinuity usually increases the number\nof coexisting solutions. Hence, in the considered system we observe\na large number of different stable orbits and their classification\nis necessary. In Figure \\ref{fig:ImpactBif} we show two bifurcation\ndiagrams with $\\omega$ as controlling parameter. Both of them start\nwith initial conditions $x_{0}=0.0$ and $\\dot{x}_{0}=0.0$. In panel\n(a) we increase $\\omega$ from $0.801$ to $0.8075$; while in panel\n(b) we decrease $\\omega$ in the same range. We select the range of\n$\\omega$ basing on the results presented in \\cite{pavlovskaia2010complex}.\nAs one can see, both diagrams differ in two zones marked by grey colour.\nHence, we observe a coexistence of different solutions, i.e., in the range\n$\\omega\\in[0.8033,\\:0.8044]$ solutions with period-3 and -2 are present,\nwhile in the range $\\omega\\in[0.8068,\\:0.8075]$ we detected solutions\nwith period-2 and -5. As presented in \\cite{pavlovskaia2010complex}\nsome solutions appear from a saddle-node bifurcation and we are not\nable to detect them with the classical bifurcation diagram. The proposed\nmethod solves this problem and shows all existing solutions in\nthe considered range of excitation frequency. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure7}\n\\par\\end{centering}\n\n\\caption{Bifurcation diagram showing the behaviour of impacting oscillator (2).\nFor subplot (a) the value of the bifurcation parameter $\\omega$ was increased\nwhile for subplot (b) we decreased the value of $\\omega$. Grey rectangles\nmark the range of the bifurcation parameter $\\omega$ for which different\nattractors coexist. Further analysis can be found in \\cite{pavlovskaia2010complex}.\n\\label{fig:ImpactBif}}\n\\end{figure}\n\n\nWe focus on periodic solutions with periods that are not longer than\neight periods of excitation. We observe periodic solutions with higher\nperiods in the narrow range of $\\omega$ but the probability that\nthey will occur is very small and we can neglect them. All non-periodic\nsolutions are chaotic (quasiperiodic solutions are not present in\nthis system). The results of our calculations are shown in Figure\n\\ref{fig:Impact_prob}(a,b). We take initial conditions from the following\nranges $x_{0}\\in[-2,\\:2]$, $\\dot{x}_{0}\\in[-2,\\:2]$. The controlling\nparameter $\\omega$ is changed from $0.801$ to $0.8075$ with step\n$\\Delta\\omega=0.0005$ in Figure \\ref{fig:Impact_prob}(a) and from\n$0.806$ to $0.8075$ with the step $\\Delta\\omega=0.0001$ in Figure \\ref{fig:Impact_prob}(b)\n(in each subrange of excitation's frequency we pick the exact value of\n$\\omega$ randomly from this subset). The probability of periodic\nsolutions is plotted by lines with different colours and markers.\nWe detect the following solutions: period-1, -2, -3, -5 (two different\nattractors with large and small amplitude), -6 and -8. The dot lines\nindicate the sum of all periodic solutions' probability (also with\nperiod higher then eight). Hence, when its value is below $1$, chaotic solution exist. Dots are drawn for mean value i.e, middle\nof the subset. For each range we take $N=200$ and we increase the calculation\ntime because the transient time is sufficiently larger than in the\nprevious example due to the piecewise smooth characteristic of spring's\nstiffness.\n\nAs we can see, the chance of reaching a given solution strongly depends\non $\\omega$. Hence, in the sense of basin stability we can say that\nstability of solutions rely upon the $\\omega$ value. In Figure \\ref{fig:Impact_prob}(a)\nthe probability of a single solution is always smaller than one. Nevertheless,\nwe observe two dominant solutions: period-5 with large amplitude in\nthe first half of the considered $\\omega$ range and period-2 in the second\nhalf of the range. The maximum registered value of probability is $p(\\mathrm{period-2})=0.92$\nand it refers to the period-2 solution for $\\omega\\approx0.80675$. To\ncheck if we can achieve even higher probability we analyse a narrower\nrange of $\\omega$ and decrease the step (from $\\Delta\\omega=0.0005$\nto $\\Delta\\omega=0.0001$). In Figure \\ref{fig:Impact_prob}(b) we\nsee that in range $\\omega\\in[0.8069,\\:0.807]$ the probability of reaching\nthe period-2 solution is equal to $1$. Hence, in the sense of basin stability\nit is the only stable solution. Also in the range $\\omega\\in[0.8065,\\:0.8072]$\nthe probability of reaching this solution is higher then $0.9$ and\nwe can say that its basin of attraction is strongly dominant. \n\nOther periodic solutions presented in Figure \\ref{fig:Impact_prob}(a)\nare: period-1 is present in the range $\\omega\\in[0.801,\\:0.8025]$ with the\nhighest probability $p(\\mathrm{period-1})=0.4$, period-3 exists in the\nrange $\\omega\\in[0.803,\\:0.805]$ with the maximum probability $p(\\mathrm{period-3})=0.36$,\nperiod-2 is observed in two ranges $\\omega\\in[0.8025,\\:0.8035]$ and\n$\\omega\\in[0.804,\\:0.8045]$ with the highest probability equal to $0.18$\nand $0.12$ respectively. Solution with period-5 (small amplitude's\nattractor) exists also in two ranges $\\omega\\in[0.8055,\\:0.8065]$\nand $\\omega\\in[0.807,\\:0.8075]$ with the highest probability equal to\n$0.14$ and $0.4$3 respectively.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure8}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in the impacting system. Subplots\n(a,b) present different periodic solutions and the summary probability\nof reaching any periodic solution. In Subplot (a) we analyze $\\omega\\in[0.801,\\:0.8075]$\nwith the step $\\Delta\\omega=0.0005$, and in subplot (b) we narrow\nthe range $\\omega\\in[0.806,\\:0.8075]$ and decrease the step size\n$\\Delta\\omega=0.0001$. (Please note that in cases (a) and (b) the\ninitial conditions and parameter are somehow random, hence the results\nmay slightly differ).\\label{fig:Impact_prob}}\n\\end{figure}\n\n\n\n\\subsection{Beam with suspended rotating pendula}\n\nThe third considered system consists of a beam that can move horizontally\nwith two ($n=2$) or twenty ($n=20$) pendula suspended on it. As\na control parameter we use $k_{x}$ which describes the stiffness\nof the beam's support. For the considered range of $k_{x}\\in[100,\\,5000]$\ntwo stable periodic attractors exist in that system. One corresponds\nto complete synchronization of the rotating pendula. The second one is called anti-phase synchronization and refers to the state when\nthe pendula rotate in the same direction but are shifted in phase by $\\pi$.\n\nIn Figure \\ref{fig:pendulaBif} we show four bifurcation diagrams\nwith $k_{x}$ as the controlling parameter and a Pioncare map of rotational\nspeed of the pendula. The subplots (a,b) refer to the system with two\npendula ($n=2$). We start with zero initial conditions: $x_{0}=0.0$,\n$\\dot{x}_{0}=0.0$, $\\varphi_{10}=0.0$, $\\dot{\\varphi}_{10}=0.0$,\n$\\varphi_{20}=0.0$, $\\dot{\\varphi}_{20}=0.0$ and take $k_{x}\\in[100,\\,5000]$.\nThe parameter $k_{x}$ is increasing in subplot (a) and decreasing in\n(b). We see that in the range marked by grey rectangle both complete\nand anti-phase synchronization coexist. In subplots (c,d) we present\nresults for twenty pendula ($n=20$). We start the integration from initial\nconditions that refer to anti-phase synchronization (two clusters\nof 10 pendula shifted by $\\pi$) i.e. $x_{0}=0.1$, $\\dot{x}_{0}=0.00057$,\n$\\varphi_{k0}=0.0$, $\\dot{\\varphi}_{k0}=9.81$, $\\varphi_{j0}=3.09$,\n$\\dot{\\varphi}_{j0}=9.784$ where: $k=1,2,\\ldots10$ and $j=11,12,\\ldots20$.\nThe value of $k_{x}$ is increasing in subplot (c) and decreasing in (d).\nSimilarly as in the two pendula case, we observe the region ($k_{x}\\in[100,\\,750]$)\nwhere two solutions coexist: anti-phase synchronization and non-synchronous\nstate. To further analyse multistability in that system we use \nproposed method.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure9}\n\\par\\end{centering}\n\n\\caption{Bifurctaion diagram showing the behaviour of two (a,b) and twenty\n(c,d) pendula suspended on the moving beam. For subplots (a,c) the value\nof the bifurcation parameter $k_{x}$ was increased, while for subplots\n(b,d) we decreased the value of $k_{x}$. Grey rectangles mark the\nranges of the bifurcation parameter $k_{x}$ for which different attractors\ncoexist. Further analysis of number of solutions can be found in \\cite{czolczynski2012synchronization}.\n\\label{fig:pendulaBif}}\n\\end{figure}\n\n\nIn Figure \\ref{fig:Pendula_prob} we present how the probability of\nreaching a given solution depends on the parameter $k_{x}$ . In subplot\n(a) we show the results for the system with 2 pendula, while in subplot\n(b) results obtained for the system with 20 pendula suspended\non the beam are given. In both cases we consider $k_{x}\\in[0,\\:5000]$ and assume\nthe following ranges of initial conditions: $x_{0}\\in[-0.15,\\:0.15]$,\n$\\dot{x}_{0}\\in[-0.1,\\:0.1]$, $\\varphi_{i0}\\in[-\\pi,\\:\\pi]$, $\\varphi_{20}\\in[-\\pi,\\:\\pi]$,\n$\\dot{\\varphi}_{10}\\in[-3.0,\\:3.0]$ and $\\dot{\\varphi}_{20}\\in[-3.0,\\:3.0]$\nin Figure \\ref{fig:Pendula_prob}(a) and $x_{0}\\in[-0.15,\\:0.15]$,\n$\\dot{x}_{0}\\in[-0.1,\\:0.1]$, $\\varphi_{i0}\\in[-\\pi,\\:\\pi]$, $\\dot{\\varphi}_{i0}\\in[-\\pi,\\:\\pi]$\nwhere $i=1\\dots\\:20$ in Figure \\ref{fig:Pendula_prob}(b). We take\n$20$ subsets of parameter $k_{x}$ values with the step equal to\n$\\Delta k_{x}=250$ and mark their borders with vertical lines. For\neach set we run $N=100$ simulations; each one with random initial\nconditions and $k_{x}$value drawn from the respective subset. Then,\nwe estimate the probability of reaching given solution. The dots in\nFigure \\ref{fig:Pendula_prob} indicate the probability of reaching a\ngiven solution in the considered range (dots are drawn for mean value,\ni.e, middle of subset). Contrary to both already presented systems,\nthis one has a much larger dimension of phase space (six and forty two),\nhence we decide to decrease number of the trials to $N=100$ in order\nto minimise the time of calculations. \n\nIn Figure \\ref{fig:Pendula_prob}(a) we show the results for 2 pendula.\nWhen $k_{x}\\in[0,\\:250]$ only anti-phase synchronization is possible.\nThen, with the increase of $k_{x}$ we observe a sudden change\nin the probability and for $k_{x}\\in[750,\\:1750]$ only complete synchronization\nexists. For $k_{x}>2000$ a probability of reaching both solutions fluctuates\naround $p(\\mathrm{complete})=0.7$ for complete and $p(\\mathrm{\\mathrm{anti-phas}e})=0.3$\nfor anti-phase synchronization. Further increase of $k_{x}$\ndoes not introduce any significant changes. \n\nIn Figure \\ref{fig:Pendula_prob}(b) we show the results for twenty\npendula. For $k_{x}\\in[0,\\:250]$ the system reaches solutions different\nfrom the two analysed (usually chaotic). Then, the probability of reaching\ncomplete synchronization drastically increases and for $k_{x}\\in[750,\\:5000]$\nit is equal to $p(complete)=1.0$ which means that the pendula always\nsynchronize completely. We also present the magnification of the plot\nwhere we see that in fact for $k_{x}\\in[715,\\:5000]$ we will always\nobserve complete synchronization of the pendula. Please note that\nfor calculating both plots we use random initial conditions and $k_{x}$\nvalue hence, the results for a narrower range may differ. Anti-phase\nsynchronization was never achieved with randomly chosen initial conditions.\nThis means that even though this solution is stable for $k_{x}\\in[100,\\:750]$\n(see Figure \\ref{fig:pendulaBif}(c)) it has a much smaller basin of attraction\nand is extremely hard to obtain in reality. The results presented in Figure\\ref{fig:Pendula_prob}\nprove that by proper tuning of the parameter $k_{x}$ we can control the\nsystems behaviour even if we can only fix the $k_{x}$ value with finite\nprecision. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure10}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in the system with rotating\npendula. Subplot (a) refers to the case with two pendula and (b) with\ntwenty pendula. (Please note that on plot (b) and its magnification\nthe initial conditions and parameter are somehow random, hence the\nresults may slightly differ). \\label{fig:Pendula_prob}}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nIn this paper we propose a new method of detection of solutions' in \nnon-linear mechanical or structural systems. The method allows\nto get a general view of the system's dynamics and estimate the risk that\nthe system will behave behave differently than assumed. To achieve\nthis goal we extend the method of basin stability \\cite{menck2013basin}.\nWe build up the classical algorithm and draw not only initial conditions\nbut also values of system's parameters. We take this into account\nbecause the identification of parameters' values is quite often not\nvery precise. Moreover values of parameters often slowly vary during\noperation. Whereas in practical applications we usually need certainty\nthat the presumed solution is stable and its basin of stability is large\nenough to ensure its robustness. Hence, there is a need to describe\nhow small changes of parameters' values influence the behaviour of\nthe system. Our method provides such a description and allows us to estimate\nthe required accuracy of parameters values and the risk of unwanted\nphenomena. Moreover it is relatively time efficient and does not require\nhigh computational power.\n\nWe show three examples, each for a different class of systems: a tuned\nmass absorber, a piecewise smooth oscillator and a multi-degree of freedom\nsystem. Using the proposed method we can estimate the number of existing\nsolutions, classify them and predict their probability of appearance.\nNevertheless, in many cases it is not necessary to distinct all solutions\nexisting in a system but it is enough to focus on an expected solution,\nwhile usually other periodic, quasi-periodic and chaotic solutions\nare classified as undesirable. Such a strategy simplifies the analysis and\nreduces the computational effort. We can focus only on probable solutions\nand reduce the number of trials omitting a precise description of solutions\nwith low probability. \n\nThe proposed method is robust and can be used not only for mechanical\nand structural systems but also for any system given by differential\nequations where the knowledge about existing solutions is crucial. \n\n\n\\section*{Acknowledgement}\n\n\nThis work is funded by the National Science Center Poland based on\nthe decision number DEC-2015\/16\/T\/ST8\/00516. PB\nis supported by the Foundation for Polish Science (FNP).\n\n\n\\section*{Appendix A}\n\nThe motion of the system presented in Figure \\ref{fig:Beam_model}\nis described by the following set of two second order ODEs:\n\n\\begin{equation}\nm_{iD}l_{D}^{2}\\ddot{\\varphi'}_{i}+m_{iD}\\ddot{x'}l_{D}\\cos{\\varphi'_{i}}+c_{\\varphi D}\\dot{\\varphi'}_{i}+m_{iD}g_{D}l_{D}\\sin{\\varphi'_{i}}=N_{0D}-\\dot{\\varphi'}_{i}N_{1D}\\label{eq:Beam1-1-1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M_{D}+\\sum\\limits _{i=1}^{n}{m_{iD}}}\\right)\\ddot{x'}+c_{xD}\\dot{x'}+k_{xD}x'=\\sum\\limits _{i=1}^{n}{m_{iD}l_{D}\\left({-\\ddot{\\varphi'}_{i}\\cos\\varphi'_{i}+\\dot{\\varphi'}_{i}^{2}\\sin\\varphi'_{i}}\\right)}\\label{eq:Beam2-1-1}\n\\end{equation}\n\n\nThe values of parameters and their dimensions are as follow: $m_{iD}=\\frac{2.00}{n}\\,[kg]$,\n$l_{D}=0.25\\,[m]$, $c_{\\varphi D}=\\frac{0.02}{n}\\,[Nms]$, $N_{0D}=5.00\\,[Nm]$,\n$N_{1D}=0.50\\,[Nms]$, $M_{D}=6.00\\,[kg]$, $g_{D}=9.81\\,[\\frac{m}{s^{2}}]$,\n$c_{x_{D}}=\\frac{\\ln\\left(1.5\\right)}{\\pi}\\sqrt{k_{x}\\left(M+\\sum\\limits _{i=1}^{n}{m_{i}}\\right)}\\,[\\frac{Ns}{m}]$\nand $k_{xD}\\,[\\frac{N}{m}]$ is controlling parameter. The derivation\nof the above equations can be found in \\cite{czolczynski2012synchronization}.\n\\foreignlanguage{english}{We perform a transformation to a dimensionless\nform in a way that enables us to hold parameters' values. It is because\nwe want to present new results in a way that thay can be easily compared\nto results of the investigation presented in }\\cite{czolczynski2012synchronization}\\foreignlanguage{english}{.\nWe introduce dimensionless time $\\tau=t\\omega_{0}$, where $\\omega_{0}=1\\,\\mathrm{[Hz]}$,\nand unit parameters $m_{0}=1.0\\,[kg]$, }$l_{0}=1.0\\,[m]$\\foreignlanguage{english}{\nand reach the dimensionless equations:}\n\n\\begin{equation}\nm_{i}l^{2}\\ddot{\\varphi}_{i}+m_{i}\\ddot{x}l\\cos{\\varphi_{i}}+c_{\\varphi}\\dot{\\varphi}_{i}+m_{i}gl\\sin{\\varphi_{i}}=N_{0}-\\dot{\\varphi}_{i}N_{1}\\label{eq:Beam1-1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M+\\sum\\limits _{i=1}^{n}{m_{i}}}\\right)\\ddot{x}+c_{x}\\dot{x}+k_{x}x=\\sum\\limits _{i=1}^{n}{m_{i}l\\left({-\\ddot{\\varphi}_{i}\\cos\\varphi_{i}+\\dot{\\varphi}_{i}^{2}\\sin\\varphi_{i}}\\right)}\\label{eq:Beam2-1}\n\\end{equation}\n\n\n\\selectlanguage{english}%\nwhere: $x=\\frac{x'}{l_{0}}$, $\\dot{x}=\\frac{\\dot{x'}}{l_{0}\\omega_{0}}$,\n$\\ddot{x}=\\frac{\\ddot{x'}}{l_{0}\\omega_{0}^{2}}$, $\\varphi_{i}=\\varphi'_{i}$,\n$\\dot{\\varphi}_{i}=\\frac{\\dot{\\varphi'}_{i}}{\\omega_{0}}$, $\\ddot{\\varphi}_{i}=\\frac{\\ddot{\\varphi'}_{i}}{\\omega_{0}^{2}}$,\n$m_{i}=\\frac{m_{iD}}{m_{0}}$, $l=\\frac{l_{D}}{l_{0}}$, $c_{\\varphi}=\\frac{c_{\\varphi D}}{m_{0}l_{o}^{2}\\omega_{0}}$,\n$N_{0}=\\frac{N_{0D}}{m_{0}l_{o}^{2}\\omega_{0}^{2}}$, $N_{1}=\\frac{N_{1D}}{m_{0}l_{o}^{2}\\omega_{0}}$,\n$M=\\frac{M_{D}}{m_{0}}$, $g=\\frac{g_{D}}{l_{o}\\omega_{0}^{2}}$,\n$c_{x}=\\frac{c_{xD}}{m_{0}\\omega_{0}}$ and dimensionless control\nparameter $k_{x}=\\frac{k_{xD}}{m_{0}\\omega_{0}^{2}}$. Dimensionless\nparameters have the following values: {$m_{i}=\\frac{2.0}{n}$,\n$l=0.25$, $c_{\\varphi}=\\frac{0.02}{n}$, $N_{0}=5.0$, $N_{1}=0.5$,\n$M=6.0$, $g=9.81$.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\nThough up to now there is no experimental indication why quantum evolution may be nonlinear, it has been traditionally considered both as a possible way out to the measurement problem\\cite{Wig62a} or as matter of theoretical considerations to be contrasted with high-finesse experiments\\cite{Wei89a}. One of the most remarkable consequences of these considerations\\cite{Gis90a} was the possibility, under the nonlinearity assumption, of supraluminal communication between two spatially separated parties. This soon led some authors to conclude that any nonlinear quantum evolution would necessarily entail the possibility of such a communication\\cite{Gis89a,GisRig95a} and even to consider the relativistic postulate of 'no-faster-than-light' phenomena as the theoretical basis for the quantum evolution to be linear\\cite{SimBuzGis01a}. Recently\\cite{FerSalSan03b} we have proven that this implicaction is not strict, i.e.\\ that there exist possible nonlinear quantum evolutions not implying this fatal supraluminal communication.\\\\\n\nHere we extend our previous result to finite quantum systems of arbitrary dimensions. We formulate the 'no-signaling' condition for these systems and show a full-flegded infinity of examples fulfilling this condition. Everything is expressed in Bloch space language\\cite{Kim03a,ByrKha03a}, i.e.\\ the states of quantum systems are expressed as \n\n\\begin{equation}\n\\rho(t)=\\frac{1}{N}\\left(\\mathbb{I}_{N}+\\mathbf{r}(t)\\cdot\\mathbf{\\sigma}\\right)\n\\end{equation}\n\n\\noindent and orthogonal projectors as\n\n\\begin{equation}\nP=P_{0}\\mathbb{I}_{N}+\\mathbf{P}\\cdot\\mathbf{\\sigma}\n\\end{equation}\n\\noindent where $\\mathbf{r}(t)$ is a time-dependent so-called Bloch vector belonging to a particular convex subset of $\\mathbb{R}^{N^{2}-1}$, $\\mathbf{\\sigma}\\equiv\\left(\\sigma_{1},\\dots,\\sigma_{N^{2}-1}\\right)$ are the traceless orthogonal generators of $SU(N)$ and $(P_{0},\\mathbf{P})\\equiv(P_{0},P_{1},\\dots,P_{N^{2}-1})$ are real numbers subjected to certain restrictions (cf.\\ \\cite{Kim03a} for the details).\n\n\n\\section{The 'no-signaling' condition}\n\n\nAs remarked in \\cite{FerSalSan03b}, the impossibility of communication through the projection postulate, i.e.\\ at a speed faster than that of light, is obtained only after imposing that \\emph{the \\textbf{probability distribution} of any observable of one subsystem \\textbf{only} depends on its own reduced state}. The mathematical translation of this criterion is straightforward provided one is familiar with the preceding language. Let us consider a two-partite system of subsystems $1$ and $2$, which have dimensions $N_{1}$ and $N_2$, respectively. Their common density matrix, using a tensor product basis, will be given by\n\n\\begin{equation}\n\\rho_{12}=\\frac{1}{N_{1}N_{2}}\\left(\\mathbb{I}_{N_{1}N_{2}}+\\mathbf{r}^{(1)}\\cdot\\mathbf{\\sigma}\\otimes\\mathbb{I}_{N_{2}}+\\mathbb{I}_{N_{1}}\\otimes\\mathbf{r}^{(2)}\\cdot\\lambda+\\sum_{ij}r_{ij}^{(12)}\\sigma_{i}\\otimes\\lambda_{j}\\right)\n\\end{equation}\n\n\\noindent and an orthogonal projector for each of them by\n\n\\begin{equation}\nP^{(1)}=P_{0}^{(1)}\\mathbb{I}_{N_{1}}+\\mathbf{P}^{(1)}\\cdot\\sigma\\quad P^{(2)}=P_{0}^{(2)}\\mathbb{I}_{N_{2}}+\\mathbf{P}^{(2)}\\cdot\\lambda\n\\end{equation}\n\n\\noindent respectively, where $\\sigma$ ($\\lambda$) stands for the traceless orthogonal generators of $SU(N_{1})$ ($SU(N_{2})$) and $\\mathbf{P}^{(1)}$ ($\\mathbf{P}^{(2)})$ is a $(N^{2}_{1}-1)$($(N^{2}_{2}-1)$)-dimensional vector restricted to some given subset\\footnote{Namely, $P_{0}=P_{0}^{2}+\\mathbf{P}\\cdot\\mathbf{P}$ and $2P_{0}P_{n}+z_{ijn}P_{i}P_{j}=P_{n}$, where $z_{ijk}\\equiv g_{ijk}+if_{ijk}$, the latter denoting the completely symmetric and antisymmetric tensors of the Lie algebra $\\mathfrak{su}(N_{j})$, respectively.}.\\\\\nSuppose now that an orthogonal projector $(u_{0},\\mathbf{u})$ is measured upon subsystem $2$. Then $N_{2}$ possible outcomes $(u_{0}^{(k)},\\mathbf{u}^{(k)})$ will result with probabilites $p_{k}=u_{0}^{(k)}+\\mathbf{u}^{(k)}\\cdot\\mathbf{r}^{(2)}$ given by the trace rule. Also, the projection postulate allows us to conclude that after such a measurement, the reduced density operator for its partner, subsystem $1$ will be given by\n\n\\begin{equation}\n\\rho_{k}^{(1)}(0)=\\frac{1}{N_{1}}\\left(\\mathbb{I}_{N_{1}}+\\mathbf{r}^{(1;k)}\\cdot\\sigma\\right)\n\\end{equation} \n\n\\noindent where $\\mathbf{r}^{(1;k)}$ is an $(N^{2}_{1}-1)$-dimensional vector ($k=1,\\dots,N_{2}$ possible outcomes) dependent on the joint state $\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r_{ij}^{(12)}$ and on the measured observable $(u_{0},\\mathbf{u})$:\n\n\\begin{equation}\n\\mathbf{r}^{(1;k)}_{j}=\\frac{u_{0}^{(k)}r_{j}^{(1)}+\\sum_{n=1}^{N_{2}^{2}-1}r_{jn}^{(12)}u_{n}^{(k)}}{u_{0}^{(k)}+\\mathbf{u}^{(k)}\\cdot\\mathbf{r}^{(2)}}\\equiv r^{(1;k)}_{j}(0)\n\\end{equation}\n\nIn these conditions, the probability distribution $\\mathbb{P}$ of an arbitrary orthogonal projector $(v_{0},\\mathbf{v})$ with $p=1,\\dots,N_{1}$ possible outcomes $(v_{0}^{(p)},\\mathbf{v}^{(p)})$ at time $t$ of subsystem $1$ will be given by \n\n\\begin{equation}\n\\mathbb{P}^{(1)}(t;v^{(p)})=\\sum_{k=1}^{N_{2}}(u_{0}^{(k)}+\\mathbf{r}^{(2)}\\cdot\\mathbf{u}^{(k)})(v_{0}^{(p)}+\\mathbf{v}^{(p)}\\cdot\\mathbf{r}^{(1)}(t;\\mathbf{r}^{(1;k)}(0))\n\\end{equation}\n\n\\noindent where $\\mathbf{r}^{(1)}(t;\\mathbf{r}^{(1;k)}(0))$ denotes the Bloch vector of subsystem $1$ at time $t$ with initial condition $\\mathbf{r}^{(1;k)}(0)$.\\\\\n \nThe 'no-signaling' condition can then be easily formulated. The independece with respect to other partners' reduced state and their mutual correlations will be expressed as\n\n\\begin{eqnarray}\\label{NoSig1}\n\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial r_{k}^{(2)}}&=&0\\\\\n\\label{NoSig2}\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial r_{ij}^{(12)}}&=&0\n\\end{eqnarray}\n\n\\noindent Finally, the independence with respect to observables to be measured in spatially separated subsystems will be expressed as\n\n\\begin{equation}\\label{NoSig3}\n\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial u^{(k)}_{\\mu}}=0\\quad\\mu=0,1,\\dots,N_{1}^{2}-1\n\\end{equation}\n\nThese three conditions are the mathematical translation of the previously formulated 'no-signaling' condition. The reader may check for himself that, as expected, the usual linear quantum evolution fulfills each of them (see also below).\n\n\\section{Consequences}\n\nOne of the main consequences of eqs.\\ (\\ref{NoSig1}), (\\ref{NoSig2}) and (\\ref{NoSig3}) arises after noticing that they must be valid for any particular value of the parameters involved, which implies $\\mathbf{r}^{(i)}(t;\\mathbf{r}_{k})=A^{(i)}(t)\\mathbf{r}_{k}$, where $A^{(i)}(t)$ is a time-dependent matrix. In other words, the reduced dynamics in absence of interactions (spatial separation) must be linear. Note that this does not exhaust the possibility of having nonlinear joint evolution. Indeed reduced linearity in absence of interactions entails neither joint linearity nor even reduced unitarity. Expressing this in Bloch vector language, if $(\\mathbf{r}^{(1)}(t),\\mathbf{r}^{(2)}(t),r_{ij}^{(12)}(t))$ denotes the Bloch vector of a two-partite system and if $H=H_{0}\\mathbb{I}_{N_{1}N_{2}}+\\mathbf{H}\\cdot\\sigma_{12}$ ($\\mathbf{H}=(\\mathbf{H}^{(1)},\\mathbf{H}^{(2)},H^{(12)})$ and $\\sigma_{12}=(\\sigma\\otimes\\mathbb{I}_{N_{2}},\\mathbb{I}_{N_{1}}\\otimes\\lambda,\\sigma\\otimes\\lambda)$) denotes its joint Hamiltonian, then any evolution given by\n\n \\begin{eqnarray}\n\\mathbf{r}^{(1)}(t)&=&\\mathbf{F}_{1}(t;H,\\mathbf{r}(0))\\\\\n\\mathbf{r}^{(2)}(t)&=&\\mathbf{F}_{2}(t;H,\\mathbf{r}(0))\\\\\nr^{(12)}(t)&=&F_{12}(t;H,\\mathbf{r}(0))\n\\end{eqnarray}\n\n\\noindent such that in absence of interactions ($H^{(12)}=0$) satisfies\n\n\\begin{eqnarray}\n\\label{RedLin1}\\mathbf{r}^{(1)}(t)&=&M^{(1)}(t;\\mathbf{H}^{(1)})\\mathbf{r}^{(1)}(0)\\\\\n\\label{RedLin2}\\mathbf{r}^{(2)}(t)&=&M^{(2)}(t;\\mathbf{H}^{(2)})\\mathbf{r}^{(2)}(0)\n\\end{eqnarray}\n\n\n\\noindent where $M^{(k)}(t;\\mathbf{H}^{(k)})$ denotes a time-dependent matrix depending only on the Hamiltonian of the $k$th subsystem, is free of supraluminal communication.\\\\\n\nIt should be clear that this nonlinearity only affects the evolution and never the static structure of the theory, i.e.\\ the principle of superposition of quantum states at a given instant of time is still valid, only the evolution of these states is affected.\\\\\n\nAlternatively, one can express these nonlinearities through the evolution equations:\n\n\\begin{eqnarray}\n\\frac{dr^{(1)}_{i}}{dt}&=&\\left(\\sum_{m,n=1}^{N_{1}^{2}-1}f_{imn}^{(1)}H^{(1)}_{m}r^{(1)}_{n}+\\sum_{j,m,n=1}^{N_{1}^{2}-1}f_{ijm}^{(1)}H^{(12)}_{jn}r^{(12)}_{mn}\\xi_{i;jn}^{(1)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\nonumber\\\\\n&&\\\\\n\\frac{dr^{(2)}_{i}}{dt}&=&\\left(\\sum_{m,n=1}^{N_{2}^{2}-1}f^{(2)}_{imn}H^{(2)}_{m}r^{(2)}_{n}+\\sum_{j,m,n=1}^{N_{2}^{2}-1}f^{(2)}_{ijm}H^{(12)}_{jn}r^{(12)}_{nm}\\xi_{i;jn}^{(2)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\\\\n\\frac{dr^{(12)}_{pq}}{dt}&=&2\\left(\\sum_{i,j=1}^{N_{1}^{2}-1}f^{(1)}_{jip}H^{(1)}_{j}r^{(12)}_{iq}+\\sum_{i,j=1}^{N_{2}^{2}-1}f^{(2)}_{jip}H^{(1)}_{j}r^{(12)}_{qi}+\\right.\\nonumber\\\\\n&+&\\sum_{i,j=1}^{N_{1}^{2}-1}\\sum_{m,n=1}^{N_{2}^{2}-1}\\textrm{Im}\\left[z_{ijp}^{(1)}z_{mnq}^{(2)}\\right]H_{im}^{(12)}r_{jn}^{(12)}\\xi_{pq;im}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})+\\nonumber\\\\\n&+&\\left.\\sum_{i,j=1}^{N_{1}^{2}-1}f_{ijp}^{(1)}H^{(12)}_{iq}r^{(1)}_{j}\\xi_{pq;iq}^{(12)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})+\\sum_{i,j=1}^{N_{2}^{2}-1}f_{ijp}^{(1)}H^{(12)}_{qi}r^{(2)}_{j}\\xi_{pq;iq}^{(12)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\nonumber\\\\\n\\end{eqnarray}\n\n\\noindent where the functions $\\xi$ are completely arbitrary. Notice that in absence of interactions ($H^{(12)}=0$), one recovers the usual well-known quantum evolution.\n\n\\section{Conclusions\n\nThe main two conclusions to be drawn are that (i) nonlinear evolution does not necessarily imply the possibility of supraluminal communication between two arbitrary finite quantum systems, and (ii) non linear terms, in order to fulfill the no-signaling condition, must be necessarily associated to interactions.\\\\\n\nThis reopens a door, originally suggested by Wigner, to explore possible solutions to the measurement problem without contradicting other well contrasted theories.\\\\\n\nA third generalization of this approach can be undertaken by focusing on non-projective measurements, but on generalized measurements, i.e.\\ on POVM's \\cite{Per93a}.\n\n\n\\section*{Acknowledgements}\nWe acknowlegde financial support from Spanish Ministry of Science and Techmology through project no.\\ FIS2004-01576. M.F.\\ also acknowledges financial support from Oviedo University (ref.\\ no.\\ MB-04-514).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nWe propose a novel FL method (CoFED) that is simultaneously compatible with heterogeneous tasks, heterogeneous models, and heterogeneous training processes. Compared with the traditional method, CoFED is more suitable for CS-FL settings with fewer participants but higher heterogeneity. CoFED decouples the models and training processes of different participants, thereby enabling each participant to train its independently designed model for its unique task via its optimal training methods. In addition, CoFED protects private data, models and training methods of all participants under FL settings. CoFED enables participants to share multiparty knowledge to increase their local model performance. The method produces promising results under non-IID data settings for models with heterogeneous architectures, which is more practical but is usually difficult to handle in existing FL methods. Moreover, the CoFED method is efficient since training can be performed in only one communication round.\n\nThe CoFED method may be limited by the availability of public unlabeled datasets. Although we conduct numerous experiments to demonstrate that CoFED has low requirements for public datasets and that the use of irrelevant or randomly generated datasets is still effective, some failure scenarios may still occur; this is a problem that we hope to address in the future.\n\\section{Experiments}\n\nIn this section, we execute the CoFED method under different FL settings to explore the impacts of different conditions and compare it with existing FL methods. The source code can be found at https:\/\/github.com\/flcao\/CoFED.\n\n\n\n\n\n\n\\subsection{Model Settings}\nThe CoFED method enables participants to independently design different models. For example, some participants use CNNs as classifiers, while others use support vector machine (SVM) classifiers. We use randomly generated 2-layer or 3-layer CNNs with different architectures as the models for the different participants in image classification tasks to illustrate that CoFED is compatible with heterogeneous models. Ten of the 100 employed model architectures are shown in Table \\ref{netStruct}.\n\n\\begin{table}[ht]\n\\caption{Network Architectures}\n\\label{netStruct}\n\\centering\n\\begin{tabular}{cccc}\n\\toprule\nModel & \\begin{tabular}[c]{@{}c@{}}1st \\\\ conv layer\\end{tabular} & \\ \\begin{tabular}[c]{@{}c@{}}2nd \\\\ conv layer\\end{tabular} & \\ \\begin{tabular}[c]{@{}c@{}}3rd \\\\ conv layer\\end{tabular} \\\\ \\midrule\n1 & 24 3x3 & 40 3x3 & none \\\\\n2 & 24 3x3 & 32 3x3 & 56 3x3 \\\\ \n3 & 20 3x3 & 32 3x3 & none \\\\ \n4 & 24 3x3 & 40 3x3 & 56 3x3 \\\\ \n5 & 20 3x3 & 32 3x3 & 80 3x3 \\\\ \n6 & 24 3x3 & 32 3x3 & 80 3x3 \\\\ \n7 & 32 3x3 & 32 3x3 & none \\\\ \n8 & 40 3x3 & 56 3x3 & none \\\\ \n9 & 32 3x3 & 48 3x3 & none \\\\ \n10 & 48 3x3 & 56 3x3 & 96 3x3 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo demonstrate that CoFED is applicable to broader ranges of models and task types than other approaches, we choose four types of classifiers as the participant models in the Adult dataset experiment: decision trees, SVMs, generalized additive models, and shallow neural networks.\n\n\\subsection{CIFAR Dataset Experiment}\nWe set the number of participants to 100 in our CIFAR experiments. For each participant, we randomly select a few of the 20 superclasses of CIFAR100 \\cite{krizhevsky2009learning} as its label space. Since we try to study the effect of the proposed method on heterogeneous tasks, the label spaces of different participants are generally different, but some overlap may occur. The local dataset of each participant consists of the samples in its label space.\n\nThe distributions of each superclass possessed by the different participants who own this superclass sample may encounter two situations. In the first case, we assume that the samples of each participant with the superclass are uniformly and randomly sampled from all samples of the superclass in CIFAR100 (that is, the IID setting). In this case, each participant usually has samples belonging to all 5 subclasses of each superclass in its label space. In the second case, we assume that each participant who owns the superclass only has samples belonging to some of the subclasses of its superclass (in our experiment, 1 or 2 subclasses), that is, the non-IID setting. The details of the experimental data settings are as follows.\n\n\\textbf{IID Data Setting}: Each participant is randomly assigned 6 to 8 superclasses in CIFAR100 as its label space. For the local training sets, each participant has 50 instances from each superclass in the CIFAR100 training set, and these 50 samples are evenly sampled from the samples of this superclass in the training set of CIFAR100. No overlap occurs between the training sets of any participants. For the test sets, all instances in the test set of CIFAR100 are used, whereby each participant's test set has 500 instances for each superclass.\n\n\\textbf{Non-IID Data Setting}: This setting is almost the same as the IID setting; the difference is that the sample of a superclass of each participant is only randomly acquired from 1 to 2 subclasses of the superclass in the CIFAR100 training set. The configuration of the test set is exactly the same as that used with the IID setting. The non-IID data setting is often regarded as more difficult than the IID one since a model that learns only 1 or 2 subclasses of a superclass during the training process is required to recognize all 5 subclasses included in the test set.\n\nIn this experiment, the public unlabeled dataset used in CoFED is the training set of CIFAR10 \\cite{krizhevsky2009learning}. Since CoFED is compatible with heterogeneous training processes, we conduct a grid search to determine the optimal training parameter settings for each participant task. The training configuration optimized for each participant (including the trainer, learning rate, and learning rate decay) is used in the initial local training phase, and the update training settings in the final step are adjusted based on these parameters. We always use a minibatch size of 50 for the local training process and 1000 for the update training process.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/iidcifar_cifar10.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/noniidcifar_cifar10.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Results of the CIFAR experiment. The X-axis value is the relative test accuracy, which is the ratio of the CoFED test accuracy to the local test accuracy.}\n  \\label{cifar_cifar10}\n\\end{figure}\n\n\nWe test our proposed CoFED method separately under the IID setting and non-IID setting, and the hyperparameter $\\alpha$ is set to 0.3 for both settings. We compare the test classification accuracies of the models trained by CoFED with those of the models utilizing local training, and the results are shown in Fig. \\ref{cifar_cifar10}(a) and Fig. \\ref{cifar_cifar10}(b). Under the IID setting, CoFED method improves the relative test accuracy of each participant model by 10\\%-32\\% with an average of 17.3\\%. This demonstrates that CoFED can increase participant model performance even when the divergences of models are not large (such as those under the IID data setting). For the non-IID setting, CoFED can lead to greater model performance gains due to the greater divergence of data distributions that make locally trained models more divergent. In this experiment, CoFED achieves a relative average test accuracy improvement of 35.6\\%, and for each participant, the improvement ranges from 14\\% to 67\\%. The performance boost under the Non-IID setting is better than the IID setting, which indicates that CoFED suffers less from statistical heterogeneity.\n\n\n\\subsection{FEMNIST Dataset Experiment}\nFEMNIST dataset is a handwritten character dataset of LEAF \\cite{caldas2018leaf} which is a benchmark framework for FL. It consists of samples of handwritten characters from different users, and we select the 100 users with the most samples of FEMNIST as participants. Forty percent of selected samples are used as training set, and the rest are test set. \n\nThe architectures of participant models are the same as those in the CIFAR experiment, and the local training hyperparameters are tuned in similarly to those in the CIFAR experiment. In this experiment, we use random crops of the images in the Chars74k-Kannada dataset \\cite{de2009character} to construct an unlabeled public dataset with nearly 50,000 items. Chars74k-Kannada contains handwritten character images of English and Kannada, and only the handwritten Kannada characters are used as unlabeled public dataset in our experiment. The hyperparameter $\\alpha$ is set to 0.01, and the results are shown in Fig. \\ref{fem_kan}. CoFED improves the relative test accuracies of the models for almost all participants, with an average improvement of 15.6\\%.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Figures\/fem_kan.eps}\n\\caption{Results of the FEMNIST experiment.}\n\\label{fem_kan}\n\\end{figure}\n\n\n\\subsection{Public Unlabeled Dataset Availability}\nOne of the major concerns regarding the CoFED method concerns whether a public dataset is available. The role of the public dataset is to express and share the knowledge that the participant models need to learn. Generally, the participants in FL cannot learn knowledge from samples generated entirely by random values, but this does not mean that we must use samples that are highly related to the participants' local data to construct a public dataset.\n\nFor example, in the CIFAR experiment, we use the samples of CIFAR100 to construct the local data of the participants, but we use CIFAR10 as the public dataset. The categories of the data samples contained in CIFAR10 and CIFAR100 only overlap slightly. Therefore, we can regard CIFAR10 as a dataset composed of pictures randomly obtained from the Internet without considering the similarity between its contents and the samples of participants (from CIFAR100). A large morphology difference between English characters and Kannada characters is also observed in the FEMNIST experiment. However, CoFED can effectively improve the performance of almost all participant models in both experiments, which makes us want to know how different participant models use public datasets that are not relevant to them to share knowledge. Therefore, we review the results of pseudolabel aggregation and check how these models classify the irrelevant images. Fig. \\ref{pseudo} shows a partial example of the pseudolabel aggregation results of the 10 participant models (out of 100 participants).\n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\linewidth]{Figures\/aggregating.eps}\n\\caption{The results of pseudolabel aggregation. The images obtained from the training set of CIFAR10 are scattered across the superclasses of CIFAR100. Ten images per superclass are randomly selected.}\n\\label{pseudo}\n\\end{figure*}\n\nFirst, we notice that some trucks and automobiles are correctly classified into the vehicles\\_1 category. This indicates that the unlabeled instances whose categories are contained in the label spaces of the participant models are more likely to be assigned to the correct category. However, exceptions occur; some automobiles are assigned to the {\\it{flowers}} and {\\it{fruit and vegetables}} categories. A common feature possessed by these automobile images is that they contain many red parts, which may be regarded as a distinctive feature of {\\it{flowers}} or {\\it{fruit and vegetables}} by the classifiers. In addition, the unlabeled samples that are not included in the label spaces of any classifier are also classified into the groups that match their visual characteristics. For example, the corresponding instances of aquatic\\_mammals and fish usually have blue backgrounds, which resemble water. Another interesting example is the people category. Although almost no human instances are contained in the training set of CIFAR10, some of the closest instances are still given, including the person on a horse.\n\nMoreover, we also try to replace the public dataset in the CIFAR experiment with the ImageNet dataset \\cite{van2016pixel} with almost the same size as that of CIFAR10. The CoFED method can still achieve considerable performance improvements, as shown in Fig. \\ref{cifar_imagenet}(a) and Fig. \\ref{cifar_imagenet}(b).\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/iidcifar_imagenet.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/noniidcifar_imagenet.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Results of the CIFAR experiment obtained by using the ImageNet dataset as an unlabeled public dataset.}\n  \\label{cifar_imagenet}\n\\end{figure}\n\n\n\\subsection{Unlabeled Dataset Size}\nAccording to (\\ref{13}), (\\ref{14}) and (\\ref{15}), when an existing classifier has a higher generalization accuracy and a larger labeled training set, a larger unlabeled dataset is needed to improve its accuracy. This suggests that in the CoFED method, a larger unlabeled dataset can produce a more obvious performance improvement for a given group of participant models. We redo the CIFAR experiment with 10 participants and vary the size of the unlabeled dataset between 500 and 50,000 samples in the training set of CIFAR10. The experimental results are shown in Fig. \\ref{pubsize}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/pubsize.eps}\n\\caption{Size of the unlabeled public dataset vs. the mean relative test accuracy.}\n\\label{pubsize}\n\\end{figure}\n\nThe experimental results are consistent with the theoretical analysis. This shows that the strategy of increasing the size of the unlabeled dataset can be used to boost the performance improvements exhibited by all participant models. Considering that the difficulty of collecting unlabeled data is much lower than that of collecting labeled data in general, this strategy is feasible in many practical application scenarios.\n\n\n\\subsection{Hyperparameter $\\alpha$}\nFrom the theoretical analysis, increasing the reliability of the pseudolabeling process is likely to bring more significant performance improvements, which is also very intuitive. Therefore, we use a hyperparameter $\\alpha$ to improve the reliability of pseudolabel aggregation. A larger $\\alpha$ requires a higher percentage of participants to agree to increase the reliability of the pseudolabels, but this may also reduce the number of available pseudolabel instances. In particular, when the participants disagree greatly, requiring excessive consistency across the results of different participant models may stop the spread of multiparty knowledge.\n\nIn this section, we repeat the CIFAR experiment for 10 participants with different $\\alpha$ and record the changes in the total number of samples generated by pseudolabel aggregation when different $\\alpha$ values are taken in Fig. \\ref{idxSiz}. The changes in the test accuracies of all participant models in the CoFED method are shown in Fig. \\ref{alphasize}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/idxSize.eps}\n\\caption{$\\alpha$ vs. the size of the pseudolabel aggregation results. For $\\alpha=1$, the total numbers of IID and non-IID data are both 0, which cannot be shown with the log scale.}\n\\label{idxSiz}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/alphasize.eps}\n\\caption{$\\alpha$ vs. the mean relative test accuracy.}\n\\label{alphasize}\n\\end{figure}\n\nThe results show that when the value of $\\alpha$ changes, its impact on CoFED is not monotonic. Although an excessively large value of $\\alpha$ can increase the reliability of the generated pseudolabels, this also greatly reduces the number of samples in the pseudolabel aggregation results, which may degrade the training results. In the FEMNIST experiment, we find that a larger $\\alpha$ value may greatly reduce the number of samples in the pseudolabel aggregation results, so we set the $\\alpha$ value to 0.01.\n\nAt the same time, an excessively large $\\alpha$ value makes the pseudolabel aggregation process more inclined to choose the samples that most participants agree on. Since these sample are approved by most participants, it is almost impossible to bring new knowledge to these participants. In addition, we find that a larger $\\alpha$ value has a more severe impact on the non-IID data setting, where performance degradation is more significant than in the IID cases. This is because the differences between the models trained on the non-IID data are greater, and the number of samples that most participants agree on is smaller. Therefore, when $\\alpha$ increases, the number of available samples decreases faster than in the IID case, as shown in Fig. \\ref{idxSiz}, which causes greater performance degradation under the non-IID setting.\n\nOn the other hand, an $\\alpha$ that is too small decreases the reliability of the pseudolabel aggregation results, which may introduce more mislabeled samples, making the CoFED method less effective. In addition, an excessively small $\\alpha$ value may cause a large increase in the number pseudolabel aggregation samples, resulting in increased computation and communication overheads.\n\nIn summary, the effectiveness of the CoFED method can be affected by the value of the hyperparameter $\\alpha$, and adopting an appropriate $\\alpha$ value can yield greater performance improvements and avoid unnecessary computation and communication costs.\n\n\\subsection{Adult Dataset Experiment}\nAdult dataset\\cite{kohavi1996scaling} is a census income dataset to be used to classify a person's yearly salary based on their demographic data. We set the number of participants to 100 in our Adult experiments. For each participant, we randomly select 200 samples from the training set of the Adult dataset without replacement. Since we try to study the effect of the proposed method on heterogeneous models, 4 types of classifiers are chosen: decision trees, SVMs, generalized additive models, and shallow neural networks. The number of utilized classifiers of each type is 25. The size of unlabeled public dataset in this experiment is 5000, and each sample is randomly generated with valid values for the input properties used by the classifiers. The Adult test set is used to measure the performance of each participant's classifier.\n\nThe results are shown in Fig. \\ref{4c}. The proposed method improves the test accuracies of most classifiers, with an average of 8.5\\%, and the performance of the classifier that benefits the most increases by more than 25\\%. This demonstrates that the proposed method is applicable not only to image classification tasks involving CNN models but also to nonimage classification tasks with traditional machine learning models.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Figures\/4classifiers.eps}\n\\caption{Results of the Adult experiment.}\n\\label{4c}\n\\end{figure}\n\n\\subsection{Comparison with Other FL Methods}\nTo the best of our knowledge, CoFED is the first FL method that tries to be compatible with heterogeneous models, tasks, and training processes simultaneously. Therefore, FL methods are available for comparison under HFMTL settings. We use two well-known comparison methods. One is the personalized FedAvg method, which represents the classic parameter aggregation-based FL strategy that is not compatible with heterogeneous models with different architectures; the other is the FedMD method \\cite{li2019fedmd}, which supports different neural network architectures and is used to evaluate the performance of CoFED under heterogeneous models.\n\nTo enable the personalized FedAvg and FedMD methods to handle heterogeneous classification tasks, we treat each participant's local label space $\\mathcal{Y}_i$ as the union of the spaces $\\mathcal{Y}$ in (\\ref{union}). In this way, we can handle heterogeneous tasks with the idea of personalized FedAvg and FedMD. The main steps are as follows.\n\\begin{enumerate}\n    \\item Use the FedAvg or FedMD algorithm to train a global model whose label space is the union of the label spaces of all participants.\n    \\item The global model is fine-tuned on each participant's local dataset for a personalized model for their local task..\n\\end{enumerate}\n\nIn this comparison experiment, we use the data configuration presented in Section 4.3. Considering that the personalized FedAvg method can not be used for models with different architectures, 100 participants share the same architecture neural network model. In the FedMD comparison experiment, we use the same setup as that in Section 4.3; that is, we select 100 participants with different neural network architectures.\n\nWe try a variety of different hyperparameter settings to achieve better performance in the personalized FedAvg experiment. We use $E=20$ as the number of local training rounds in each communication iteration, $B=50$ is set as the local minibatch size used for the local updates, and participants also perform local transfer learning to train their personalized models in each communication iteration. In the FedMD experiment, We use the similar settings of \\cite{li2019fedmd}, and it should be noted that FedMD uses the labeled data of CIFAR10 as the public dataset, which is different from the unlabeled data used in CoFED.\n\n\\begin{table}[]\n\\caption{Comparison Results}\n\\label{iidtb}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{@{}c@{}c@{}ccc}\n\\hline\n                         &          & \\begin{tabular}[c]{@{}c@{}}Personalized\\\\ FedAvg\\end{tabular} & FedMD & CoFED \\\\ \\hline\n\\multirow{2}{*}{IID}     & Accuracy & \\textbf{1.06}                                                          & 0.87  & 1.00\\tnote{*}  \\\\\n                         & Rounds   & 100                                                           & 8\\tnote{\\dag}     & \\textbf{1}     \\\\ \\hline\n\\multirow{2}{*}{Non-IID} & Accuracy & 0.94                                                          & 0.94  & \\textbf{1.00}\\tnote{*}  \\\\\n                         & Rounds   & 150                                                           & 17\\tnote{\\dag}    & \\textbf{1}     \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item[*] Reference value.\n\\item[\\dag] The round with the best performance.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedavg_iid.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedavg_noiid.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Personalized FedAvg method vs. CoFED. To facilitate the comparison, the test accuracy of the CoFED model is used as the reference accuracy, and the value of the Y-axis is the ratio of the comparison algorithm's test accuracy to the reference accuracy. Since each participant has a corresponding ratio, the blue line represents the average of the ratios of all participants corresponding to the given number of iterations.}\n  \\label{fedavg}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedmd_iid.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedmd_noiid.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{FedMD vs. CoFED.}\n  \\label{fedmd}\n\\end{figure}\n\nThe results of the comparison between FedAvg and CoFED are shown in Fig. \\ref{fedavg}. The relative test accuracy is calculated as the average of the ratios of all participants to the CoFED test accuracy. Under the IID setting, FedAvg reaches the performance level of CoFED after 42 communication rounds and is finally 6\\% ahead of COFED in Fig. \\ref{fedavg}(a). Under the non-IID setting, FedAvg stabilizes after 150 communication rounds. At this time, CoFED still leads FedAvg by 6\\% in Fig. \\ref{fedavg}(b). The comparing results of personalized FedAvg and CoFED are shown in Fig. \\ref{fedmd}. For both data settings, CoFED outperforms FedMD, and CoFED leads by 14\\% under the IID setting and 35\\% under the non-IID one.\n\nIn terms of communication cost, if we do not consider the communication overhead required to initially construct the public dataset, CoFED achieves better performance with lower communication overheads in all cases because CoFED only needs to pass through the label data (not the sample itself) during the training process, and iterating for multiple rounds is not required. This assumption is not unrealistic because the construction of public datasets may not require the central server to distribute data to the participants; the participants can instead obtain data from a third party, which does not incur communication costs between the participants and the central server. Even if we include that paradigm, CoFED still achieves better performance with lower communication costs except under the IID and identical architecture model settings. In fact, the IID setting used in the FedAvg comparison experiment is not the scenario that is considered most by CoFED because the model architectures are the same and can be shared under that setting.\n\\section{Introduction}\nFederated learning (FL) allows different participants to collaborate in solving machine learning problems under the supervision of a center without disclosing participants' private data \\cite{kairouz2021advances}. The main purpose of FL is to achieve improved model quality by leveraging the multiparty knowledge from participants' private data without the disclosure of data themselves.\n\nFL was originally proposed for training a machine learning model across a large number of users' mobile devices without logging their private data to a data center \\cite{konevcny2015federated}. In this scenario, a center orchestrates edge devices to train a global model that serves a global task. However, some new FL settings have emerged in many fields, including medicine \\cite{rieke2020future}, \\cite{xiao2021federated}, \\cite{raza2022designing}, \\cite{courtiol2019deep}, \\cite{adnan2022federated}, finance \\cite{li2020preserving}, \\cite{gu2021privacy}, and network \\cite{zhang2021survey}, \\cite{nguyen2021federated}, \\cite{ghimire2022recent}, \\cite{regan2022federated}, where the participants are likely companies or organizations. Generally, the terms \\textit{cross-device federated learning} (CD-FL) and \\textit{cross-silo federated learning} (CS-FL) can refer to the above two FL settings \\cite{kairouz2021advances}. However, the majority of these studies' contributions concern training reward mechanisms \\cite{tang2021incentive}, topology designs \\cite{marfoq2020throughput}, data protection optimization approaches \\cite{zhang2020batchcrypt}, etc., and for their core CS-FL algorithms, they simply follow the idea of gradient aggregation used in CD-FL. Such studies ignore the fact that organizations or companies, as participants, may be more heterogeneous than device participants.\n\nOne of the most important heterogeneities to address under the CS-FL setting is model heterogeneity; that is, models may have different architecture designs. Different devices in CD-FL typically share the same model architecture, which is given by a center. As a result, the models obtained through local data training on different devices differ only in terms of their model parameters, allowing the center to directly aggregate the gradients or model parameters uploaded by the participating devices. However, participants in a CS-FL scenario are usually independent companies or organizations, and they are capable of designing unique models. They prefer to use their own independently designed model architectures rather than sharing the same model architecture with others. At this time, the strategy used in CD-FL cannot be applied to CS-FL models with different architectures. Furthermore, model architectures may also be intellectual properties that need to be protected, and companies or organizations that own these properties do not want them to be exposed to anyone else, which makes model architecture sharing hard for CS-FL participants to accept. An ideal CS-FL method should treat each participant's model as a black box, without the need for its parameters or architecture.\n\nThe heterogeneity among models results in the need for training heterogeneity. Under the CD-FL setting, participants usually train their models according to the configuration of a center, which may include training algorithms (such as stochastic gradient descent) and parameter settings (such as the learning rate and minibatch size). However, when participants like companies or organizations use their independently designed model architectures in CS-FL, they need to choose different local training algorithms that are suitable for their models and exert full control over their local training processes. Decoupling the local training processes of different participants not only enables them to choose suitable algorithms for their models but also prevents the leakage of their training strategies, which may be their intellectual properties.\n\nIn addition, CS-FL is more likely to face task heterogeneity than CD-FL. In CD-FL scenarios, all devices usually share the same target task. In terms of classification tasks, the task output categories of all devices are exactly the same. Under the CS-FL setting, because the participating companies or organizations are independent of each other and have different business needs, their tasks may be different. Of course, we must assume that there are still similarities between these tasks. In terms of classification tasks, the task output categories of different participants in CS-FL may be different. For example, an autonomous driving technology company and a home video surveillance system company both need to complete their individual machine learning classification tasks. Although both of them need to recognize pedestrians, the former must also recognize vehicles, whereas the latter must recognize indoor fires. Therefore, the task output categories of the autonomous driving technology company include pedestrians and vehicles without indoor fires, whereas the task output categories of the home video surveillance system company include pedestrians and indoor fires without vehicles. It is easy to see that, in contrast to the complete task consistency of CD-FL, CS-FL participation by independent companies or organizations is more likely to encounter situations in which the different participants possess heterogeneous tasks.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fig0.eps}\n    \\caption{CS-FL setting without heterogeneity}\n  \\end{subfigure}%\n  \\par\\bigskip\n  \\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fig1.eps}\n    \\caption{CS-FL setting with heterogeneity (HFMTL)}\n  \\end{subfigure}%\n  \\caption{(b) is an example of an HFMTL setting with 3 participants. A central server needs to coordinate 3 participants to solve their classification tasks via federated training. Compared with (a), which has no heterogeneity, (b) also contains different machine learning model architectures and training optimization algorithms, and the tasks of participants are distinct from one another(i.e., different label spaces).}\n  \\label{fig1}\n\\end{figure}\n\nAlthough the number of participants in CS-FL is much smaller than that in CD-FL in general, the heterogeneity of the participants, including model heterogeneity, training heterogeneity and task heterogeneity, may bring more challenges. Overall, we use the term \\textit{heterogeneous federated multitask learning} (HFMTL) to refer to the FL settings that contain the above three heterogeneity requirements, and Fig. \\ref{fig1}(b) shows an example of the HFMTL setting with three participants. Different participants may have different label spaces. For example, Participant $B$ has a label space $\\left\\{0, 1, 2, 3\\right\\}$. This space can be different from those of other participants, e.g., Participant $C$ has $\\left\\{1, 4, 5\\right\\}$. The classification task of Participant $B$ is to classify inputs with the labels in $\\left\\{0, 1, 2, 3\\right\\}$, while Participant $C$ aims to classify inputs with the labels in $\\left\\{1, 4, 5\\right\\}$. Therefore, they have different tasks and usually have different categories of local training data.\n\nOur main motivation in this paper is to address the needs of \\textit{model heterogeneity}, \\textit{training heterogeneity}, and \\textit{task heterogeneity} in CS-FL settings. Aiming at the HFMTL scenario with these heterogeneities, we propose a novel FL method. The main contributions of this paper are as follows.\n\n\\begin{itemize}\n\\item We propose an FL method that is simultaneously compatible with heterogeneous tasks, models, and training processes, and it boost the performance of each participant's model.\n\\item Compared with the existing FL methods, the proposed method not only protects private local data but also protects the architectures of private models and private local training methods.\n\\item Compared with these other methods, the proposed method achieves more performance improvements for the non-independent and identically distributed (non-IID) data settings in FL and can be completed in one round.\n\\item We conduct comprehensive experiments to corroborate the theoretical analysis conclusions and the impacts of different settings on the suggested method.\n\\end{itemize}\n\nThe rest of this paper is structured as follows. Section 2 provides a brief overview of relevant works. Section 3 contains the preliminaries of our method. Section 4 describes the proposed method in detail. Section 5 offers the experimental results and analysis. Section 6 summarizes the paper.\n\\section*{Acknowledgment}\nThis research was partially supported by the National Key Research and Development Program of China (2019YFB1802800), PCL Future Greater-Bay Area Network Facilities for Large-scale Experiments and Applications (PCL2018KP001).\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Methodology}\n\\subsection{The Overall Steps of CoFED}\nThe main incentive of FL is that it improves the performance of participant models. The poor performance of locally trained models is mainly due to insufficient local training data, so these models fail to learn enough task expertise. Hence, to increase the participant model performance, it is vital to allow them to learn from other participants. The most popular and straightforward methods of sharing knowledge are shared models or data, however both are forbidden by FL settings, thus we must devise alternative methods.\n\nA research \\cite{wang2007analyzing} discovered that a sufficiently enough amount of variety across the models is necessary to increase classification model performance. Generally, it is difficult to generate models with large divergences under single-view settings. However, under FL settings, The architectures of the participant models may be highly varied, and they may have been trained on distinct local datasets that are very likely to be distributed differently. All of these may increase the diversity of different participants' models. Therefore, if enough unlabeled data are obtained, we can regard the federated classification problem as a semisupervised learning problem, and it is very suitable to adopt cotraining-like techniques due to the high diversity between different participant models. We provide the overall steps of CoFED as follows.\n\n\\begin{enumerate}\n    \\item \\textbf{Local training}: Each participant independently trains a local model on its private dataset.\n    \\item \\textbf{Pseudolabeling}: Each participant pseudolabels an unlabeled dataset, which is public to all participants, with its locally trained model.\n    \\item \\textbf{Pseudolabel aggregation}: Each participant uploads its pseudolabeling results to a central server, and the center votes for the pseudolabeled dataset with high confidence for each category based on the category overlap statuses of different participants and the pseudolabeling results. After that, the center sends all pseudolabels to each participant.\n    \\item \\textbf{Update training}: Each participant trains its local model on the new dataset created by combining the local dataset with the received pseudolabeled dataset.\n\\end{enumerate}\n\nIt can be seen that there are some differences between CoFED and the cotraining process under single-view settings. First, in cotraining, the training sets used by different classifiers are the same, while the training sets of the different classifiers in CoFED come from different participants, so they are usually different. Second, the target tasks of different classifiers in cotraining are the same; that is, they have the same label space. However, in CoFED, the label spaces of different classifiers are different, and the pseudolabeling of unlabeled samples need to be performed according to the pseudolabel aggregation results of the overlapping classification process. Furthermore, cotraining completes training by repeating the above process, while the process is performed only once in CoFED.\n\n\\subsection{Analysis}\n\nSuppose that we are given two binary classification models $f$ and $g$ from a hypothesis space $\\mathcal{H}: \\mathcal{X} \\rightarrow \\mathcal{Y}, |\\mathcal{H}| < \\infty$, and an oracle model $h \\in \\mathcal{H}$ whose generalization error is zero. We can define the generalization disagreement between $h_1 \\in \\mathcal{H}$ and $h_2 \\in \\mathcal{H}$ as:\n\\begin{equation}\n    \\begin{aligned}\n     d(h_1,h_2)&=d(h_1,h_2|\\mathcal{X}) \\\\\n               &=\\mathbf{Pr}(h_1(x) \\ne h_2(x) | x \\in \\mathcal{X})\n    \\end{aligned}\n\\end{equation}\n\nTherefore, the generalization errors of $f$ and $g$ can be computed as $d(f,h)$ and $d(g,h)$, respectively. Let $\\varepsilon$ bound the generalization error of a model, and let $\\delta>0$; a learning process generates an approximate model $h'$ for $h$ with respect to $\\varepsilon$ and $\\delta$ if and only if:\n\\begin{equation}\n    \\mathbf{Pr}(d(h', h) \\ge \\varepsilon) \\le \\delta\n\\end{equation}\n\nSince we usually have only a training dataset $X \\subset \\mathcal{X}$ containing finite samples, the training process minimizes the disagreement over $X$:\n\\begin{equation}\n    \\mathbf{Pr}(d(f,h|X) \\ge \\varepsilon) \\le \\delta\n\\end{equation}\n\n\\textbf{Theorem 1.} \\textit{\nGiven that $f$ is a probably approximately correct (PAC) learnable model trained on $L \\subset \\mathcal{D}$, $g$ is a PAC model trained on $L_{g} \\subset \\mathcal{D}$, and $\\varepsilon_f < \\frac{1}{2}$, $\\varepsilon_g < \\frac{1}{2}$. $f$ and $g$ satisfy that the following:\n\\begin{align}\n    \\label{9}\n    & \\mathbf{Pr}(d(f, h) \\ge \\varepsilon_f) \\le \\delta \\\\\n    \\label{10}\n    & \\mathbf{Pr}(d(g, h) \\ge \\varepsilon_g) \\le \\delta\n\\end{align}\nIf we use $g$ to pseudolabel an unlabeled dataset $X_u \\subset \\mathcal{X}$, we generate a pseudolabeled dataset\n\\begin{equation}\n    P=\\left \\{ (x,y)|x \\in X_u, y=g(x) \\right \\}\n\\end{equation}\nand combine $P$ and $L$ into a new training dataset $C$. After that, $f'$ is trained on $C$ by minimizing the empirical risk. Moreover,}\n\\begin{align}\n    \\label{13}\n    & |L| \\varepsilon_f < \\sqrt[|P|\\varepsilon_g]{(|P|\\varepsilon_g)!} \\hspace{1mm} e-|P|\\varepsilon_g \\\\\n    \\label{14}\n    & \\varepsilon_{f'} = \\max \\left \\{\\varepsilon_f + \\frac{|P|}{|L|}(\\varepsilon_g-d(g,f')), 0 \\right \\}\n\\end{align}\n\\textit{where $e$ is the base for natural logarithms; then,}\n\\begin{equation}\n    \\label{15}\n    \\mathbf{Pr}(d(f',h) \\ge \\varepsilon_{f'}) \\le \\delta\n\\end{equation}\n\nTheorem 1 has been proven in \\cite{wang2007analyzing}. Assume that $f$ and $g$ are 2 models from different participants that satisfy (\\ref{9}) and (\\ref{10}). The right side of (\\ref{13}) monotonically increases as $|P|\\varepsilon_g \\in (0, \\infty)$, which indicates that a larger pseudolabeled dataset $P$ enables a larger upper bound of $|L| \\varepsilon_f$. That is, if $f$ is a model trained on a larger training dataset with higher generalization accuracy, a larger unlabeled dataset is required to further improve the generalization accuracy of $f$.It can be seen from (\\ref{14}) and (\\ref{15}) that when $d(g,f')$ is larger, the lower bound of the generalization error of $f'$ under the same confidence is smaller, which is because that $f'$ is trained on $L$ and $P$, and $P$ is generated by $g$, $d(g,f')$ mainly depends on  the degree of divergence between $f$ and $g$, i.e., the difference between the training dataset of $f$ and $g$. In FL settings, substantial diversity across different local training datasets is fairly prevalent, hence the performance improvement requirement is generally satisfied. The same conclusion applies to the boost version $g'$ obtained by switching $f$ and $g$ due to symmetry.\n\nOn the other hand, if $|P|$ is sufficiently large since $P$ is generated by $g$, the $f'$ trained on $C$ can be treated as proximal to $g$. That is, if we repeat the above process on $f'$ and $g$, $f'$ and $g$ may be too similar to improve them. Therefore, we utilize a large unlabeled dataset and only perform the above process once instead of iterating for multiple rounds; this technique can also avoid the computational cost increase caused by multiple training iterations.\n\nAn intuitive explanation for the CoFED method is that when the different participant models have great diversity, the differences in the knowledge possessed by the models are greater. As a result, the knowledge acquired by each participant model from others contains more knowledge that is unknown to itself. Therefore, more distinctive knowledge leads to greater performance gains. In addition, when mutual learning between different models is sufficient, the knowledge difference between them will almost disappear, and it is difficult for mutual learning to provide any participant's model with distinctive knowledge.\n\n\\subsection{Pseudolabel Aggregation}\n\nAfter each participant uploads its pseudolabeling results to the public unlabeled dataset, the center exploits their outputs to pseudolabel the unlabeled dataset. In this subsection, we explain the implementation details of this step.\n\nAssume that the union of the label spaces of all participants' tasks is\n\\begin{equation}\n\\label{union}\n\\mathcal{Y}=\\bigcup_{i=1}^N{\\mathcal{Y}_i=\\left\\{ c_k \\right\\} , k=1,2,\\cdots,n_c}\n\\end{equation}\nwhere $n_c$ is the number of elements in the whole label space $\\mathcal{Y}$, and each category $c_k$ exists in the label space of one or multiple participants' label spaces:\n\\begin{equation}\n    c_k\\in \\bigcap_{j=1}^{m_k}{\\mathcal{Y}_{i_j}, 1\\leq i_1 < i_2 < \\cdots< i_{m_k}\\leq N}\n\\end{equation}\n$m_k$ is the number of participants who possess category $c_k$. At the same time, we define the pseudolabeled dataset $P_k$ for category $c_k$, and $P_k$ is used to store the indices of the instances in the public dataset corresponding to each category $c_k$ after pseudolabel aggregation.\n\nFor each category $c_k$ existing in $\\mathcal{Y}_{i_j}$, the model $f_{i_j}$ classifies the instances in the public unlabeled dataset $D^{pub}$ as belonging to category $c_k$, and the set of these instances can be defined as\n\\begin{equation}\n    S_j=\\left\\{ x|f_{i_j}\\left( x \\right) \\equiv c_k,x\\in D^{pub} \\right\\} , j=1,2,\\cdots,m_k\n\\end{equation}\n\nFor an instance $x\\in D^{pub}$, if we regard the outputs of different participant models on $x$  for category $c_k$ as the outputs of a two-class (belonging to category $c_k$ or not) ensemble classifier $g$, since (\\ref{14}) suggests that a lower generalization error bound $\\varepsilon_g$ is helpful, we can set a hyperparameter $\\alpha$ to make the results more reliable. That is, if\n\\begin{equation}\n    \\frac{|\\left\\{ S_j|x\\in S_j \\right\\} |}{m_k} > \\alpha, 0 \\le \\alpha \\le 1,\n\\end{equation}\nan $\\alpha$ value of 0 means that whether $x$ is marked as belonging to category $c_k$ requires only one participant to agree, while an $\\alpha$ of 1 means that the consent of all participants is required. After that, we can put the index of $x$ into $P_k$. After all $P_k$ are generated, the central server sends the corresponding pseudolabeled dataset to each participant's task based on its label space $\\mathcal{Y}_i$. For the participants whose label space is $\\mathcal{Y}_i$, the corresponding pseudolabeled dataset received from the center is:\n\\begin{equation}\n    R_i = \\{(P_k, c_k)|c_k \\in \\mathcal{Y}_i\\}\n\\end{equation}\nAssuming that $C_i[index]$ stores the results of $f_i(D^{pub}[index])$, $\\mathcal{P}=\\{P_k|c_k \\in \\mathcal{Y}\\}$ and $M=|D^{pub}|$, Algorithm 1 describes the above process.\n\\begin{algorithm}[!t]\n\\DontPrintSemicolon\n \\caption{Pseudolabel aggregation}\n \\KwIn{$\\{C_i\\}$, $\\mathcal{Y}_i$, $\\mathcal{Y}$, $\\alpha$, $M$}\n \\KwOut{$\\{P_k\\}$}\n \\SetKwBlock{Begin}{function}{end function}\n \\Begin(\\text{Aggregation} {$(\\{C_i\\}$, $\\mathcal{Y}_i$, $\\mathcal{Y}$, $\\alpha$, $M)$})\n {\n  \\tcp*[l]{Initialization}\n  $TOTAL$ = empty dict\\;\n  $\\mathcal{P}$ = empty set\\;\n  \n  \\tcp*[l]{Counting $c_k$}\n  \\ForAll {$c_k \\in \\mathcal{Y}$}\n  {\n    $TOTAL[c_k] = |\\{i|c_k \\in \\mathcal{Y}_i\\}|$\\;\n    $\\mathcal{P}[c_k]$ = empty set\\;\n  }\n  \n  \\tcp*[l]{Label aggregating}\n  \\ForAll {$index = 1$ \\textbf{to} $M$}\n  {\n    $COUNT$ = empty dict with default value 0\\;\n    \\ForAll {$i = 1$ \\textbf{to} $N$}\n    {\n        $COUNT[C_i[index]] = COUNT[C_i[index]] + 1$\\;\n    }\n    \\ForAll {$c_k \\in keys(COUNT)$}\n    {\n        \\uIf {$\\frac{COUNT(c_k)}{TOTAL(c_k)}>\\alpha$}\n        {\n            add $index$ to $P_k$\\;\n        }\n        \\Else\n        {\n           continue\\;\n        }\n        \n    }\n  }\n\n \\Return{$\\{P_k\\}$}\n }\n \\end{algorithm}\n\nIt should be pointed out that some indices of $x \\in D^{pub}$ may exist in multiple $P_k$ because all $\\mathcal{Y}_i$ are different from each other. Therefore, the different $P_k$ in $R_i$ may overlap, resulting in contradictory labels. To build a compatible pseudolabeled dataset $R_i$, the indices contained by different $P_k$ should be removed from $R_i$.\n\n\\subsection{Unlabeled Dataset}\nTo perform the CoFED method, we need to build a public unlabeled dataset for all participants. Although an unlabeled dataset that is highly relevant to the classification tasks is preferred, we find that even less relevant datasets can yield sufficient results in our experiments. For the image classification tasks that we focus on, the almost unlimited image resources that are publicly accessible on the Internet can be built into an unlabeled dataset. Another benefit of utilizing resources that each participant can independently obtain is that this strategy can prevent the distribution of unlabeled datasets by a central server, thereby saving the limited communication resources in an FL scenario.\n\nThe reason why less relevant datasets work is that even though different objects may have commonality, we can use this commonality as the target of the tasks involving different objects. For example, if someone asks you what an apple looks like when you have nothing else but a pear, you might tell the person that it looks similar to a pear. This may give him or her some incorrect perceptions about apples, but it is better than nothing, and this person will at least be less likely to recognize a banana as an apple. That is, although you have not been able to tell him or her exactly what an apple looks like, the process still improves his or her ability to recognize apples. For a similar reason, even if a less relevant unlabeled dataset is used to transfer knowledge, it can also yield improved model performance in FL scenarios.\n\n\\subsection{Training Process}\nIn the CoFED method, each participant needs to train its model twice, i.e., local training and update training. Both training processes are performed locally, where no exchange of any data with other participants or the central server is necessary. The benefits of this approach are as follows.\n\\begin{enumerate}\n    \\item It prevents the leakage of participant data, including the local data and models.\n    \\item It avoids the loss of stability and performance through communication.\n    \\item It decouples the training processes of different participants so that they can independently choose the training algorithms and training configurations that are most suitable for their models.\n\\end{enumerate}\n\n\\subsection{Different Participant Credibility Levels}\nIn practical applications, the problem of different participant credibility levels may be encountered, resulting in uneven pseudolabel quality for different participants. Credibility weights can be assigned to the pseudolabels provided by different participants. Accordingly, Algorithm 1 can be modified to calculate $TOTAL(c_k)$ and $COUNT(c_k)$ by adding the weights of different participant, and the unmodified version of Algorithm 1 is equivalent to the case in which the weight of each participant is 1.\n\nDifferent bases can be used for setting the weights. For example, since the quality of the model trained on a larger training set is generally higher, weights based on the size of the utilized local dataset may be helpful for the unbalanced data problem. The test accuracy can also be used as a basis for the weights, but the participants may be reluctant to provide this information. Therefore, we can make decisions according to the actual scenario.\n\n\\subsection{Non-IID Data Settings for Heterogeneous Tasks}\nData can be non-IID in different ways. We have pointed out that the heterogeneous task setting itself is an extreme non-IID case of the personalized task setting. However, under the heterogeneous task setting, the instance distributions of a single category in the local datasets of different participants can still be IID or non-IID. The IID case means that the instances of this category owned by different participants have the same probability distribution, while in the extreme non-IID case, each participant may only have the instances of one subclass of this category, and different participants have different subclasses. A non-IID example is a case in which pets are contained in the label spaces of two participants, but all pet instances in the local training set of one participant are dogs, while the other participant only has cat images in its local dataset.\n\nThe existing FL methods based on parameter aggregation usually work well for IID data but suffer when encountering non-IID data. Zhao \\textit{et al.} \\cite{zhao2018federated} showed that the accuracy of convolutional neural networks (CNNs) trained with the FedAvg algorithm could be significantly reduced, by up to 55\\%, with highly skewed non-IID data. Since non-IID data cannot be prevented in practice, addressing them has always been regarded as an open challenge in FL \\cite{kairouz2021advances}, \\cite{li2020federated}. Fortunately, in the CoFED method, where model diversity is helpful for improving performance, a non-IID data setting is usually beneficial. This is because models trained on non-IID data generally have more divergences than models trained on IID data.\n\\section{Related Work}\nThe federated average (FedAvg) algorithm was originally proposed by McMahan \\textit{et al.} \\cite{mcmahan2017communication} to solve machine learning federated optimization problems on mobile devices. The core idea of FedAvg method is to pass model parameters instead of private data from different data sources, and use a weighted average of model parameters from different data sources as a global model. In each round of communication in the FedAvg method, the central server broadcasts the global model to participants, and then the participants who received the global model continue to train the global model using local private data. Following the local training phase, participants submit trained models to the center which utilizes the weighted average of different participant models as the global model for the next round of communication.\n\n\nInspired by the FedAvg algorithm, many FL methods \\cite{wang2019adaptive}, \\cite{li2020federatedhn} based on the aggregation of model parameters have been proposed. These methods are mainly suitable for federated training under the CD-FL setting where the tasks and model architectures are usually published by a central server, and all participants (i.e. devices) sharing the same model architecture makes aggregation of model parameters an effective knowledge sharing strategy. In addition, under the CD-FL setting, it is also practical for the central server to control the local training process and parameters (such as learning rate, epoch number, and mini-batch size, etc.). However, under the CS-FL setting where the model architectures and tasks of different participants may be different, and the training process and parameters are reluctant to be controlled by the center, these FL methods based on aggregation of model parameters are generally unable to cope with these heterogeneity challenges. FedProx proposed by Li \\textit{et al.} \\cite{li2020federatedhn} introduced a proximal term to overcome statistical heterogeneity and systematic heterogeneity (i.e., stragglers), but FedProx is unable to cope with heterogeneous model architectures due to model parameter aggregation.\n\nIn recent years, personalized federated learning has been proposed to address the personalized needs of participants. Smith \\textit{et al.} \\cite{smith2017federated} proposes a multitask FL framework MOCHA which clusters different tasks according to their relevance by an estimated matrix. Khodak \\textit{et al.} \\cite{khodak2019adaptive} proposed an adaptive meta-learning framework utilizing online learning ARUBA, which has the advantage of eliminating the need for hyperparameter optimization during personalized FL. Lin \\textit{et al.} \\cite{lin2020meta} and Fallah \\textit{et al.} \\cite{fallah2020personalized} proposed a model-agnostic meta-learning method and their variants to achieve personalized federated learning. Dinh \\textit{et al.} \\cite{dinh2020personalized} proposed a personalized FL method based on meta-learning by Moreau envelope, which leverages the $l2$-norm regularization loss to balance the personalization performance and generalization performance.\n\n\nIt should be pointed out that personalized tasks are different from heterogeneous tasks. Tasks with personalized settings always have the same label spaces, while tasks with heterogeneous settings have different label spaces. A typical example of personalized tasks that can be presented from \\cite{smith2017federated} is to classify users' activities by using their mobile phone accelerometer and gyroscope data. For each user, the model needs to provide outputs from the same range of activity categories, but the classification for each user is regarded as a personalized task due to the differences in their heights, weights, and personal habits. Despite these differences, from the data distribution perspective, the data of heterogeneous classification tasks can be regarded as non-IID sampling results from an input space $\\mathcal{X}$, which contains all instances of all labels in a label space $\\mathcal{Y}$, and $\\mathcal{Y}$ is the union of the label spaces of all participants. Each participant only has instances of some categories in $\\mathcal{Y}$ since its label space is a subset of $\\mathcal{Y}$. Therefore, an FL method that is compatible with personalized tasks can also be used for heterogeneous tasks.\n\nHowever, the core idea of all these methods is model parameter aggregation, which requires the model architectures of all participants to be consistent or only partially different. However, under CS-FL settings, participants as companies or organizations usually need to use their own independently designed model architectures. Li \\textit{et al.} \\cite{li2019fedmd} proposed a FL method FedMD leveraging model distillation to transfer the knowledge of different participant's model by aligning the output of neural networks models on a public dataset. Although FedMD participants are allowed to use neural network models of different architectures, they must be consistent in the logit output layer, and FedMD is not compatible with participants using non-neural network models. Also, FedMD needs a large number of labeled data for participant model alignment, which raises the bar for its application.\n\nModel parameter aggregation also leads to the leakage of participant models. Many studies try to overcome model leakage using various secure computing methods including differential privacy \\cite{wei2020federated}, \\cite{hu2020personalized}, \\cite{adnan2022federated}, secure multiparty computing \\cite{yin2020fdc}, \\cite{liu2020secure}, homomorphic encryption \\cite{jia2021blockchain}, \\cite{fang2021privacy}, \\cite{zhang2020batchcrypt}, blockchain \\cite{li2022blockchain}, \\cite{otoum2022federated}, and trusted execution environments \\cite{mo2021ppfl}, \\cite{chen2020training}, However, these technologies still have certain drawbacks, such as high computational costs or hardware specific.\n\\section{Preliminaries}\nWe first formulate the HFMTL problem and introduce the cotraining method that inspires us to propose the communication-efficient FL (CoFED) method.\n\\subsection{HFMTL}\nAn FL setting contains $N$ participants, and each of them has its own classification task $T_i$, input space $\\mathcal{X}_i$, output space $\\mathcal{Y}_i$, and $\\mathcal{D}_i$ consisting of all valid pairs $(\\boldsymbol{x}, y)$, where $\\boldsymbol{x} \\in \\mathcal{X}_i, y \\in \\mathcal{Y}_i$. Since we indicate that $T_i$ is a classification task, $\\mathcal{Y}_i$ is the label space of $T_i$. Each participant trains its machine learning model with supervised learning to perform a classification task, so each participant has its own local data $D_i \\subset \\mathcal{D}_i$.\n\nSince the HFMTL settings contain heterogeneous tasks and heterogeneous models $f$, we can assume that for the general $i \\ne j$:\n\\begin{equation}\n\\mathcal{Y}_i\\ne \\mathcal{Y}_j, f_i\\ne f_j\n\\end{equation}\nOn the other hand, the classification tasks of each participant should have commonality with the tasks of other participants. In our setting, this commonality manifests as the overlap between the label spaces. This means that:\n\\begin{equation}\n    \\forall \\mathcal{Y}_i\\rightarrow \\left\\{ j|\\mathcal{Y}_i\\cap \\mathcal{Y}_j\\ne \\oslash , i\\ne j \\right\\} \\ne \\oslash \n\\end{equation}\n\nAssuming that a model $f_i$ has been trained for $T_i$, its generalization accuracy can be defined as:\n\\begin{equation}\n    GA(f_i) = \\mathbf{E}_{(\\boldsymbol{x}, y) \\sim \\mathcal{D}_i}[\\mathbf{I}\\left( f_i\\left( \\boldsymbol{x} \\right) \\equiv y \\right)]\n\\end{equation}\nwhere $\\mathbf{E}[\\cdot]$ is the expectation and $\\mathbf{I}(\\cdot)$ is an indicator function: $\\mathbf{I}(\\text{TRUE}) \\equiv 1 $ and $\\mathbf{I}(\\text{FALSE}) \\equiv 0 $.\n\nThe goal of this paper is to propose an HFMTL method that is compatible with heterogeneous tasks, models, and training processes. This method should help to improve the performance of each participant model without sharing the local private datasets $D_i$ and private models $f_i$ of these participants:\n\\begin{equation}\n    f_i^{fed} = \\underset{f_i}{arg\\max}\\,\\,GA\\left( f_i \\right) \n\\end{equation}\nIn addition, assuming that the model locally trained by the participant is $f_{i}^{loc}$, we expect that the model $f_{i}^{fed}$ trained with our FL method yields better performance for each participant:\n\\begin{equation}\n    GA(f_{i}^{fed}) > GA(f_{i}^{loc}), i=1,2,\\cdots,N\n\\end{equation}\n\n\\subsection{Cotraining}\nIn HFMTL settings, many tricky problems come from the differences between participant models and tasks. However, cotraining is an effective method for training different models. Therefore, ideas that are similar to cotraining can be used to solve problems under HFMTL settings.\n\nCotraining is a semi-supervised learning technique proposed by Blum \\textit{et al.} \\cite{blum1998combining} that utilizes unlabeled data from different views. The \\textit{view} means a subset of instance attributes, and two classifiers that can be obtained by training on different view data. Cotraining lets the two classifiers jointly give some pseudo-labels for unlabeled instances, and then retraining the two classifiers on the original training set and the pseudo-labeled dataset can improve the classifier performance. Wang \\textit{et al.} \\cite{wang2007analyzing} pointed out that cotraining is still effective for a single view, and applying cotraining can bring greater classifier performance improvement when  the divergence of different classifiers is larger. In FL, local models of different participants are trained from their local data. Under the HFMTL setting, due to differences in model architecture and data distribution, models of different parties are likely to have large divergences, which is beneficial for applying Cotraining.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\n\n\tWe studied the properties of the MSSM Higgs bosons, h and H through the decay into b-quarks in associated production with a top-quark pair. There was used the tree-level Higgs sector described by two parameters M$_A$ and tan$\\beta$  and found their optimal values according to experimental data of ATLAS detector. Using the restricted parameter space we calculated cross sections of associated $t\\bar{t}h(H)$ production at 13 and 14 TeV, the corresponding kinematical cuts, mass distributions and Branching Ratios of h and H decays into $b\\bar{b}$ quark pair. \n\n\\section{Introduction}\n\n    Supersymmetry searches are the most attractive in the aspect of searching for new physics beyond the standard model. The search for an extended sector of Higgs bosons is especially urgent, since they are the lightest candidates for supersymmetric particles and information on their production cross sections and decay widths provides additional knowledge about the Yukawa coupling constants. The dependences on these couplings of the cross section and Higgs branching ratios have been studied intensively and it was found, that there are indirect constraints from experimental data on the scalar and \npseudoscalar H-top couplings k$_t$ and $\\tilde{k}$, and these constraints are relatively weak, \\cite {1.}  \n\t\n\tOne of the important channels of such searches is the $t\\bar{t}H$ Higgs boson production channel. The search strategy for the $t\\bar{t}H$ process has been studied in various Higgs decay modes: $b\\bar{b}$, \\cite{2.}, $\\tau\\bar{\\tau}$, \\cite{3.} and WW$^*$, \\cite{4.}.  Furthermore, from experimental point of view the $H\\rightarrow b\\bar{b}$ decay mode is more prefferable due to the possibility of the reconstruction of the Higgs boson kinematics, which allows to extract the information about the top\u2013Higgs interaction. \n    As the decay of Higgs boson into two b-quarks ($H\\rightarrow b\\bar{b}$) is the most probable, \\cite{5.} our paper is devoted to the consideration of this decay channel. Higgs boson decay into b-quarks in associated production with a top-quark pair is connected with\ntesting the predictions of the Standard Model (SM) and\nvery sensitive to effects of physics beyond the SM (BSM). So,\nwe used Minimal Supersymmetric Standard Model (MSSM) as base theory for futher calculations. Our purpose was to calculate production cross sections $\\sigma(t\\bar{t}h),\\ \\sigma(t\\bar{t}H)$ at the centre-of-mass energy of $\\sqrt{s}$ = 13 TeV and to compare obtained data with experimental data. We also found p$_T$ and rapidity distributions, parameter space and mass distributions, which corresponds to the best fit with experimental data.\n\n\\section{Search channels and parameter cuts of Higgs boson production}\n\tATLAS \\cite{6.} and CMS \\cite{7.} have searched for the process of  Higgs boson production ($t\\bar{t}H(b\\bar{b})$)  intensively using the 8 TeV data set. Later new data collected in proton\u2013proton collisions at the LHC between 2015 and 2018 at a centre-of-mass energy of $\\sqrt{s}$ = 13 TeV were analysed, corresponding to an integrated luminosity of 139 fb$^{-1}$, \\cite{8.}. The measured signal strength, defined as the ratio of the measured signal yield to that predicted by the SM, \n\\[ \\mu = 0.35\\pm0.20 (stat.)^{+0.30}_{-0.28}(syst.)=0.35^{+0.36}_{-0.34} \\ ,\\]\ncorresponds to an observed (expected) significance of 1.0 (2.7) standard deviations. The measured 95$\\%$ confidence level (CL) cross-section upper limits in each bin for simplified template cross-sections (STXS) formalism are shown in Fig.1\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{1.eps}\\\\\n\\emph{\\textbf{Fig.1}} {\\emph{The measured 95$\\%$ CL cross-section upper limits with the theoretical uncertainty \nconnected with signal scale and PDF uncertainties.}}\n\\end{center}\n\n        \n    The Higgs sector of the Minimal Supersymmetric extension of the SM \\cite{9.} consists of five physical Higgs bosons, two neutral CP-even bosons, h, H, one neutral CP-odd boson, A, and a charged Higgs pair, H$^{\\pm}$. \nAs there is the experimental deviation from the SM, we used BSM model - MSSM and studied the properties of two Higgs bosons: the lightest Higgs boson, h and CP-even Higgs boson, H. Our analysis of  Higgs boson production in association with a pair of top quarks and decaying into a pair of b-quarks ($t\\bar{t}H(b\\bar{b})$) is presented by Feynman diagrams in Fig.2.\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{2.eps}\\\\\n\\emph{\\textbf{Fig.2}} {\\emph{Representative tree-level Feynman diagrams for the production of a Higgs boson in association with a\ntop-quark pair ($t\\bar{t}H$) in (a) the t-channel and (b) the s-channel and the subsequent decay of the Higgs boson into $b\\bar{b}$, from \\cite{8.}.\n}}\n\\end{center}\nThe tree-level Higgs sector, can be described by two parameters, the mass of the CP-odd Higgs boson, M$_A$, and the ratio of the two vacuum expectation values of the two Higgs doublets, tan$\\beta = v_2\/v_1$. So, our purpose was to chooose the optimal value of the correspomding parameters M$_A$ and tan$\\beta$. We calculated Branching ratio (BR) of h and H using FeynHiggs program \\cite{10.} as the function of M$_A$ at 13 TeV (Fig. 3) as well as production cross section as the function of tan$\\beta$ at M$_A$=200 GeV (Fig.4).\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{3.eps}\\\\\n\\emph{\\textbf{Fig.3}} {\\emph{Branching ratio (BR) of h and H decays using FeynHiggs program as the function of M$_A$ at 13 TeV.}}\n\\end{center}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{4.eps}\\\\\n\\emph{\\textbf{Fig.4}} {\\emph{Production cross section of Higgs bosons h and H for two search channels as the function of tan$\\beta$ at M$_A$=200 GeV at 13 TeV.}}\n\\end{center}\n    \nFrom the obtained data we came to the conclusion about the optimum parameters of M$_A$=200 GeV and tan$\\beta$ =2.\n\n\\section{Results of calculations}\n    As gluon-gluon and quark-antiquark processes are most preferable for the Higgs boson production we have considered the following search channels using Pythia program \\cite{11.}, presented in Table 1 and Table 2\n\n\\begin{center}\n{\\it\\normalsize Table 1. Search channels and production cross sections of h and H bosons at the centre-of-mass energy of $\\sqrt{s}$ = 13 TeV at M$_A$ = 200 GeV and tan$\\beta$ =2}\\\\\n\\vspace*{3mm}\n\\begin{tabular}{|c|c|} \\hline \nSearch channels& Production cross sections (pb) \\\\ \n& (with stat. err.) \\\\ \\hline \\hline\ngg $\\rightarrow$  $ht\\bar{t}$ &   2.609e-01 +\/- 5.084e-03   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $ht\\bar{t}$   &   1.034e-01 +\/- 1.287e-03\n   \\\\ \\hline\ngg $\\rightarrow$  $Ht\\bar{t}$\u00a0\u00a0\u00a0  &   2.017e-02 +\/- 5.618e-04   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$ $Ht\\bar{t}$   &    7.524e-03 +\/- 3.862e-04\n   \\\\ \\hline\n     gg $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0  &    2.530e-1\u00a0+\/- 2.704e-3  \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0   &    1.041e-1 +\/-\u00a0 2.572e-3\n   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n{\\it\\normalsize Table 2. Search channels and production cross sections of h and H bosons at the centre-of-mass energy of $\\sqrt{s}$ = 14 TeV at M$_A$ = 200 GeV and tan$\\beta$ =2}\\\\\n\\vspace*{3mm}\n\\begin{tabular}{|c|c|} \\hline \nSearch channels& Production cross sections (pb) \\\\ \n& (with stat. err.)\\\\ \\hline \\hline\ngg $\\rightarrow$ $ht\\bar{t}$\u00a0     &  3.134e-01 +\/- 4.095e-04   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $ht\\bar{t}$   &  1.171e-01 +\/- 1.660e-04  \\\\ \\hline\ngg $\\rightarrow$ $Ht\\bar{t}$\u00a0\u00a0\u00a0   &  2.514e-02 +\/- 6.099e-05   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ &  9.221e-03 +\/- 5.913e-05 \\\\ \\hline\n     gg $\\rightarrow$ $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0   &  2.530e-1\u00a0+\/- 2.704e-3   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0   &  1.041e-1\u00a0+\/- 2.572e-3  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\nThe  kinematical cuts on h boson corresponding to the production cross section are presented in Fig. 5.\n\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{5.eps}\\\\\n\\emph{\\textbf{Fig.5}} {\\emph{Transverse momentum (up) and rapidity distributions (down) of h boson at the energy of 13 TeV.}}\n\\end{center}\nThe  kinematical cuts on H boson corresponding to the production cross section are presented in Fig. 6.\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{6.eps}\\\\\n\\emph{\\textbf{Fig.6}} {\\emph{Transverse momentum (up) and rapidity distributions (down) of H boson at the energy of 13 TeV.}}\n\\end{center}\nFrom the obtained data we came to the conclusion about the most suitable range of transverse momentum variation of h boson (50; 150) GeV and (100; 300) GeV of H boson. As for rapidity distributions of both Higgs bosons, their maximum region is  (-1,1), which signals about the angle range along the longitudinal (beam) direction ~ 45-90$^{\\circ}$ of the Higgs bosons. \n\t\n    As for the mass detemination of both Higgs bosons, we received the mass distributions, presented in Fig. 7\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{7.eps}\\\\\n\\emph{\\textbf{Fig.7}} {\\emph{Mass distributions of h (up) and H (down) Higgs bosons obtained at 13 TeV.}}\n\\end{center}\nFrom Fig.7 we see, that the mass of h boson is about 126 GeV, which coincides with the mass of the SM Higgs boson. As for the H boson, it's mass is approximately 330 GeV. We also calculated kinematical and mass distributions of both Higgs bosons at 14 TeV and didn't find a significant difference with previous calculations at 13 TeV.\n \\section{Conclusions}\n\nThe study of the properties of the Higgs boson is an urgent task, as evidenced by the experimental data of the ATLAS and CMS collaborations. We have presented the actual experimental data of the cross sections of Higgs boson decay into b-quarks  in associated production with a top-quark pair in  pp collisions at $\\sqrt{s}$ = 13 TeV and an integrated luminosity of 139 fb$^{-1}$ with the ATLAS detector. This result corresponds to an observed (expected) significance of 1.0 (2.7) standard deviations. To clarify the ambiguities, we decided to consider the minimal extension of SM \u2013 MSSM model. We used the tree-level Higgs sector which can be described by two parameters M$_A$ and tan$\\beta$. The calculation of BR($h\\rightarrow b\\bar{b}$) as the function of M$_A$ and production cross section of both Higgs bosons as the function of tan$\\beta$ gives us the possibility to choose the optimal parameter space corresponding to the maximum values of BR and the production cross section. Using received parameters (M$_A$ = 200, tan$\\beta$=2) we calculated cross sections of associated $t\\bar{t}h(H)$ production at 13 and 14 TeV and the corresponding kinematical cuts on transverse momentum and rapidity. We found out the value of the mass of h (126 GeV) and H (330 GeV) at the chosen parameters from the constructed mass distribution. The values of BR($h\\rightarrow b\\bar{b}$) and BR($H\\rightarrow b\\bar{b}$)  are equal to 0.85 and 0.05 correspondingly.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIt is well known that pulsars have considerably steeper spectral indices than the background population of radio sources. Their flux density (S$_\\nu$) can be described by a single power-law with slope $\\alpha$ (i.e. S$_\\nu\\propto{\\nu}^\\alpha$).  Observationally-derived spectral indices have been determined variously to be in the range of $-1.6\\pm{0.3}$ \\citep{lylg95} to  $-1.8\\pm{0.2}$ \\citep{mkkw00}. \\citet{blv13} attempted to remove pulsar survey biases to derive an intrinsic spectral index of $-1.4\\pm{1.0}$. There have been claims that millisecond pulsars have more shallow spectral indices on average than ``normal\" (i.e. non recycled)  pulsars \\citep{kll+99}, but this may be due to an observational bias \\citep{blv13}. \n\nDeviations from this pure power-law behavior have been seen at both high and low frequencies, with some fraction (10\\%) having evidence for flat spectra and spectral steepening above several GHz \\citep{mkkw00}. Low frequency turnovers first seen by \\cite{sie73}, have now been measured for both normal and millisecond (MSP) pulsars, typically below 100 MHz \\citep[e.g.][]{drt+13,kvl+15}. External free-free absorption, either in the immediate environment of the pulsar or along the line of sight, gives a good explanation for the origin of the high frequency turnover pulsars \\citep{lrkm15,rla16}.  The origin of the low frequency turnovers, is not so clear. They could be telling us something fundamental about the energy distribution of the coherent emitting electrons, or the turnover could be due to absorption, occurring either within the pulsar magnetosphere or along the line of sight.\n\nProgress in understanding these low frequency behaviors and their dependence (if any) on known pulsar parameters has been slow, owing to a shortage of flux density measurements below 1 GHz \\citep{mgj+94}. The best efforts to date are those of \\citet{mms00} who made measurements of 235 pulsars at 102.5 MHz, while 30 MSPs were observed at 102 and  110 MHz  \\citep{kl01}.  Fortunately, the situation is changing with new instruments such as the LOw-Frequency ARray (LOFAR)  and the Long Wavelength Array (LWA). There are more recent LWA measurements of 44 pulsars from 10-88 MHz \\citep{srb+15} and LOFAR has now observed large samples of normal and MSPs at 110-188 MHz and \\citep{bkk+15, kvh+16}. \n \nIn this paper we use the recently completed GMRT Sky Survey (TGSS ADR) to study known radio and gamma-ray pulsar populations at 150 MHz. This paper is arranged as follows. In \\S\\ref{survey} we briefly describe the TGSS ADR survey while in \\S\\ref{method} we outline our search methods for both the radio-loud and radio-quiet samples.  The results are discussed in \\S\\ref{results} where we derive estimates of the spectral index distribution of the TGSS ADR pulsars, and we compare the derived flux densities and the detection statistics with previously published samples. Our conclusions and suggestions for future work are given in \\S\\ref{sec:conclude}.\n\n\\section{The TGSS ADR Survey}\\label{survey}\n\nThe Giant Metrewave Radio Telescope (GMRT) was used to carry out a radio continuum survey at a frequency of 150 MHz using a total of 2000 hrs of observations. The entire sky was surveyed in over 5000 partially overlapping pointings from $-55$\\degr\\, declination to the northern polar cap covering 37,000 deg${^2}$.\n\nThe entirety of these data have recently been re-processed \\citep[TGSS ADR;][]{int16} creating high-quality images of approximately 90\\% of the entire sky. The TGSS ADR achieves a median rms noise level of \\mjybeam{3.5} and an angular resolution of \\asec{25} for Dec.$>19^{\\circ}$, and \n\\asec{25}$\\times$\\asec{25}\/cos(DEC-19$^\\circ$) for more southern declinations. In the final catalog there are some 0.62 million radio sources down to the 7$\\sigma$ level. Compared to existing meter-wavelength surveys \\citep{lcv+14,hpo+15,wlb+15},  the TGSS ADR represents a significant improvement in terms of number of radio sources, sensitivity and angular resolution. The improved angular resolution in particular allows accurate matching of radio sources with counterparts at other wavelengths. The capabilities of the TGSS ADR are well-matched to existing surveys \\citep{bwh95,ccg+98,bls+99} and provide a large frequency leverage arm for spectral index measurements. For more details on this survey and how to obtain the publicly available mosaic images and source catalog, see \\citet{int16}.\n\nThe observing bandwidth and integration time of the survey are especially relevant to the detection of the phase-averaged emission from pulsars. The original data were recorded with 256 frequency channels across 16.7 MHz of total bandwidth centered on 147.5 MHz. \nThe GMRT visibility data from the archive were saved as 16.1 s averages. Typically each (pointing) direction on the sky was observed as a series of short snapshots 3 to 5 times over the course of a single night's observing. The total integration time was 15 minutes per pointing on average.\nDuring imaging of the pointings, these data were combined in time and frequency to create a single Stokes-I image.\nThe full duration spent on any given point on the sky is more difficult to quantify. The final data products used for the analysis are 25 deg$^2$ image mosaics, formed by combining overlapping 7.6 deg$^2$ pointing images.\nNote that some pointings were observed repeatedly during multiple observing sessions, imaged separately, and combined in creating the mosaics.\nAs a result, many sources have been observed more than once, sometimes separated by days or even months.\nAs the survey's pointing centers followed the FIRST survey hexagonal grid strategy \\citep{bwh95}, the TGSS ADR will have similar duration statistics \\citep[see][]{thw+11}.\n\n\\section{Methods\\label{method}}\n\n\\subsection{Radio Loud Pulsars}\\label{sec:loud}\n\nWhile pulsars are typically detected by their pulsed, periodic emissions, they can also be identified in interferometric images as phase-averaged continuum point sources. \\citet{kcac98} were the first to employ a wide-field radio survey for this purpose. They used the 1.4 GHz NRAO VLA Sky Survey \\citep[NVSS;][]{ccg+98} to identify 79 known pulsars from the total intensity alone, while \\citet{ht99} added in the polarized intensity to identify 97 pulsars from the same survey. The 325 MHz Westerbork Northern Sky Survey  \\citep[WENSS;][]{rtb+97} was used by \\citet{kou00} to find radio emission toward 25 known pulsars. For this project we employ the TGSS ADR at 150 MHz, using a version of the source catalog that was formed by running the source extraction algorithm PyBDSM \\citep{mr15}\\footnote{http:\/\/www.astron.nl\/citt\/pybdsm\/} with its default parameters searching the mosaicked images down to a 5$\\sigma$ detection threshold. For our list of known pulsars we used the HEASARC 27 December 2015 version of the ATNF Pulsar Catalog \\citep{mhth05}. A total of 1238 pulsars were selected with dec.$\\geq-52^\\circ$ and having known positions with $\\Delta$dec.$<\\pm{3^{\\prime\\prime}}$ or $\\Delta$R.A.$<0.35^s$. {A history file listing the contents and any changes to the ATNF database as of December 2015 is at their web site\\footnote{http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat\/catalogueHistory.html}. Up to approximately November 2015, our sample included all well-localized normal pulsars from the \\emph{Fermi}  sample at Stanford University as well as the MSP sample at the University of West Virginia.} The pulsar positions were corrected for proper motion using the mean epoch of the TGSS ADR of 11 January 2011 (MJD\\,55579.0). \n\nFollowing \\citet{hwb15}, we searched for matches between the PSR and TGSS ADR catalogs out to a radius of 30\\% of the FWHM of the 25$^{\\prime\\prime}$ beam, or 7.5$^{\\prime\\prime}$. We find 200 known pulsars, or 16\\% of the sample, are associated with TGSS ADR sources.  We list these detections in Table \\ref{tab:bright} along with some basic pulsar parameters (period and dispersion measure) along with the total flux density and peak flux from the TGSS ADR.  {The ratio of the total flux density and the peak flux can be a useful proxy in helping decide whether the radio emission is extended or unresolved, and therefore likely phase-averaged pulsar emission. We return to this point, as well as discussing the rates of false positives in \\S\\ref{ids}.} For all matches, we derived two-point spectral indices between the TGSS ADR total flux densities at 150 MHz and the 1400 MHz values from the pulsar catalog. The 400 MHz flux density was used in those few cases where the 1400 MHz values were missing. When no values were provided in the pulsar catalog database, we obtained flux densities from the original literature.\n\nThe search method above was supplemented with an image-based approach for pulsars below the 5$\\sigma$ limit of the catalog. For all  of the original 1238 well-localized pulsars, we measured the peak flux in the TGSS ADR images at the pixel corresponding to the pulsar position, along with an estimate of the rms noise in immediate vicinity. We did not attempt to search over some radius and fit a Gaussian since below 5$\\sigma$ such a approach would likely lead to many more false positives. The positive identifications are defined as having S$_p$\/$\\sigma_{rms}\\geq$2.5. Our justification for this choice of threshold is shown in  Fig. \\ref{fig:noise}. We make an estimate of the shape of the signal-to-noise distribution as an estimate of the blank sky in the vicinity of these pulsars (solid line). The positive S\/N peaks are strongly skewed above that expected from Gaussian noise. From the ratio of the levels of the positive and negative 2.5$\\sigma$ bins we estimate that 4\\% of the detections at this level will be false positives, or about 2 sources.\n\nWith this image-based approach we find significant emission towards another 88 pulsars. For each of these we inspected the mosaic images to verify that the emission was coming from a point centered on the pulsar position and was not due to a nearby extended source or an image artifact. Table \\ref{tab:faint} lists the peak flux and rms toward all 88 pulsars, along with some basic pulsar parameters identical to those in Table \\ref{tab:bright}.  Given the lower significance, we are not as confident in these identifications as we are with those in Table \\ref{tab:bright}.\n\n\n\n\\subsection{Radio Quiet Pulsars}\\label{quiet}\n\nMotivated by recent claims of the detection of pulsed radio emission from the ``radio-quiet'' PSR\\,J1732$-$3131 \\citep{mad12}, we carried out a search for emission at 150 MHz. In the Second \\emph{Fermi}  Large Area Telescope (LAT) Pulsar Catalog \\citep[2PC;][]{aaa+13} there are 35 PSRs that are radio quiet, defined as having a phase-averaged flux density S(1.4 GHz)$\\leq 30$ $\\mu$Jy. In the meantime the sample has grown; an updated list of radio quiet pulsars is available on the LAT team web site\\footnote{\\url{https:\/\/confluence.slac.stanford.edu\/display\/GLAMCOG\/Public+List+of+LAT-Detected+Gamma-Ray+Pulsars}}. For accurate pulsar positions we began with the recent compilation of \\citet{krj+15}, which has a table of positions for normal pulsars and MSPs obtained from both timing and multi-wavelength observations. All positions from \\citet{krj+15} are computed at the epoch MJD 55555 (25 Dec 2010). This is useful when comparing to the TGSS ADR which was observed around the same epoch. Additional X-ray positions are taken from \\citet{mmd+15}. \n\nThe final list consisted of 30 radio quiet pulsars in the declination range of TGSS ADR with localizations of an arcsecond or better (Table \\ref{tab:unknown}). For all pulsars we extracted image cutouts and looked for faint point sources at the pulsar position. We then made a final stacked image of all 30 pointings weighted by the inverse square of the local rms noise for each image. As the image pixel size is the same \\asec{6.2} in all images, this ensures accurate image stacking.  No source is detected. The rms noise is \\mjybeam{0.7} and the max\/min on the image is  approximately $\\pm$\\mjybeam{2.3}. \n\n\\section{Results and Discussion\\label{results}}\n\n\\subsection{Radio Loud Pulsars}\n\n\\subsubsection{Identifications}\\label{ids}\n\nThe majority of the emission that was detected at 150 MHz is likely due to phase-averaged pulsar emission. In support of this we note that the distribution of PSR-TGSS ADR offsets follows the expected Rayleigh distribution, with 95\\% of the identifications matched within a \\asec{4.5} radius. This is consistent with the astrometric accuracy (68\\% confidence) derived for the TGSS ADR of \\asec{1.55} \\citep{int16}. Given the source density of the TGSS ADR at the completeness limit of 17.6 source\/deg$^2$, and a search of 1238 positions each of radius \\asec{7.5} \n(see \\S\\ref{sec:loud}), we expect less than one false positive (i.e. a background radio source not associated with a pulsar).\n\nFurther support for pulsar identifications comes from Fig.\\ref{fig:ff}, in which we the show all pulsars (crosses) with published 400 MHz and 1400 MHz flux densities in the ATNF catalog. Those pulsars with TGSS ADR detections are indicated by circles. This figure shows that the TGSS ADR associations are well-correlated with the brightest pulsars, and thus the number of false positives are likely to be low. Furthermore, the number of associations drop off sharply for S$_{1400}<0.6$ mJy and S$_{400}<5$ mJy, as would be expected for steep spectrum pulsars given that the median noise of the TGSS ADR is \\mjybeam{3.5}. The two outliers in the bottom left corner of Fig.\\ref{fig:ff} are PSR\\, J2229+6114 and the LOFAR-identified PSR\\, J0613+3731. Their flux densities at 150 MHz appear to be dominated a pulsar wind nebula called `The Boomerang\" \\citep{kru06} in the first case and some unidentified extended emission in the second case. \n\nAs the above example illustrates, not all the matches in the TGSS ADR are from phase-averaged pulsar emission. Some radio emission is due an associated nebula (e.g. Crab), or is an from an ensemble of pulsars in a globular cluster.  {In \\citet{int16} we derive an empirical formula to help decide when a radio source is unresolved. In the high signal--to-noise case this reduces to S$_t\/$S$_p\\leq 1.13$. However, some caution is needed in applying this criteria to pulsars since they can show strong time-variability during an integration time, violating one of the central assumptions of the van Cittert-Zernike theorem upon which radio interferometric imaging is based. This can lead to deviations in the Gaussian fitted beam, or in especially strong cases, diffraction spikes around the pulsar. A visual inspection of the images is required to be sure since this same condition is likely met by strongly scintillating pulsars like PSR\\,B1937+21. We examined the images of {\\it all} of the detections in Table \\ref{tab:bright} and find that likely non-pulsar candidates are those entries for which the total flux density exceeds the peak flux (i.e. S$_t>$S$_p$) by more than 50\\%.}  For those small number of TGSS ADR detections (11) that we suspect are contaminated in this way, we add a comment in Table \\ref{tab:bright} and we do not derive a spectral index.\n\n\\subsubsection{Spectral Index Distribution}\n\nThe distribution of the two-point spectral indices of the TGSS ADR sample from Table \\ref{tab:bright}  is shown in Fig. \\ref{fig:spec_histo}. For comparison we have plotted the more comprehensive sample of 329 pulsars from the ATNF Pulsar catalog with non-zero spectral indices. As expected, the two histograms are in reasonable agreement with each other, both in terms of the width and the median of the two distributions. \n\nIf we order the spectral index values by pulsar period (Fig. \\ref{fig:alpha}) an unusual feature of our 150 MHz sample appears. The steep spectrum tail of the $\\alpha$ distribution measured at low frequencies is dominated by short period pulsars. This effect is not seen in the ATNF pulsar catalog.  We have detected many of the fastest rotating MSPs at 150 MHz, and these pulsars show a marked preference for steeper spectral index values. Of the 16 pulsars with $\\alpha<-2.5$, all but four are MSPs. Of these MSPs, all except one has been detected by the \\emph{Fermi}  gamma-rays mission including several eclipsing MSPs such as PSR\\,J1816+4510, with the steepest spectral index in Fig. \\ref{fig:alpha}. The 18 MSPs in Table \\ref{tab:faint} do not have ultra-steep spectra. \\citet{kvl+15} were the first to note a tendency for the gamma-ray MSPs to be steep-spectrum outliers based on a smaller sample. \n\nSince the values in Table \\ref{tab:bright} and Fig. \\ref{fig:alpha} are two-point values, we suspected measurement error as the source of these large values. As a first step we re-calculated the spectral index of all pulsars with $\\alpha<-2.5$ using the flux density and observing frequency taken from the original references. If no rms noise was given we assumed a fractional error of 50\\% for the flux density when estimating the uncertainty on $\\alpha$.\n\nThere are several useful compilations of flux density measurements and spectral indices we can use to cross check our measurements \\citep{tbm+98,kxl+98,kvl+15}. We find reasonable agreement in the $\\alpha$ values for all of the MSPs within the errors. For PSR\\,J1816+4510, the pulsar with the steepest 2-point spectral index in our sample, we re-fit our 150 MHz measurement along with a value at 74 MHz \\citep{kvl+15} and flux densities at 350 and 820 MHz \\citep{slr+14}. The latter two measurements were estimated from the radiometer equation so we have taken typical errors of $\\pm$50\\% on these two values. The mean spectral index is $-3.46\\pm0.10$ in agreement with a preliminary value from \\citet{kvl+15}.\n\nSince there is no evidence that the distribution of MSP spectral indices is steeper than the general population \\citep{tbm+98,kll+99}, we suspect this trend is the result of some low frequency bias.  Most of the steep spectrum MSPs in Table \\ref{tab:bright}, were discovered in low frequency searches \\citep[e.g.][]{fst+88,bhl+94,hrm+11,slr+14}. Two pulsars (B1937+21 and J0218+4232) had such steep spectral indices that they were initially identified in imaging data \\cite[e.g.][]{nbf+95}. Thus it is reasonable to expect that the TGSS ADR survey at 150 MHz would be sensitive to steep-spectrum radio sources, with a similar bias as low-frequency searches for pulsations \\citep{blv13}. This explanation, however, does not account for the preponderance of gamma-ray pulsars among our sample, nor for the unusually large fraction of (eclipsing) binaries. We know of no intrinsic property of the MSP population that would produce such an effect. {\\citet{ckr+15} noted that the nearby MSPs were susceptible to deep flux density variations at decimeter wavelengths, with strong expoential statistics such that the measured median flux density is less than the mean, skewing the spectral index to steeper values.} Since many of these systems have been found within the error ellipses of \\emph{Fermi}  unassociated sources \\citep[e.g.][]{krj+15}, a more prosaic explanation may be that the \\emph{Fermi}  mission has been such a prolific source of MSPs that they are over-represented in any sample. \n\n\\subsubsection{Comparison with the LOFAR Sample}\n\nIt is illustrative to compare the TGSS ADR and LOFAR samples. While both surveys were undertaken at the same frequency, they were observed in very different ways.  Thus a comparison could give us some insight into the different biases of each survey. LOFAR has carried out a search for pulsed emission from all northern radio pulsars \\citep{bkk+15,kvh+16}. This census was primarily  conducted with the LOFAR high-band antennas (HBA) between 110 and 188 MHz, with 400 channels each of 0.195 MHz in width, or a bandwidth of 78 MHz. Each pulsar was observed once for at least 20 minutes, although long period (P$>3$ s) normal pulsars and faint MSPs were observed up to 60 minutes in duration. Pulsed emission from a  total of 158 normal pulsars and 48 MSPs were detected. The GMRT observing method is summarized in \\S\\ref{survey} and the pulsar yield is given in \\S\\ref{ids}. We find 92 pulsars commonly detected in both the  LOFAR and TGSS ADR surveys (Tables \\ref{tab:bright} and \\ref{tab:faint}).\n\nFigure \\ref{fig:compare} (left) is a flux-flux plot of LOFAR and TGSS measured flux densities, while the same figure (right) shows a flux ratio plot of the same sample. The flux densities of the LOFAR and TGSS ADR pulsars do not agree. On average, the LOFAR pulsars are about two times brighter than the the TGSS ADR. The result persists even if we use only the bright pulsars in common (i.e. Table \\ref{tab:bright}). There are some significant outliers, dominated by bright, scintillating MSPs such as PSR\\,B1937+21 and PSR\\,J0218+4232, but the overall trend is clear. \n\nWe can immediately rule out frequency-dependent effects for this difference in the flux density scales since the surveys were performed at similar frequencies. Spectral curvature was the most likely explanation offered by \\citet{kvh+16} for why the LOFAR flux densities for one third of their MSPs are {\\it lower} than the predicted values based on an extrapolation from higher frequencies. Diffractive scintillation, while clearly important for the outliers, is not the likely origin for the systematic difference. The large observing bandwidths and the long integration times relative to the scintillation values for both the LOFAR and GMRT observations (\\S\\ref{survey}) suggest modulation of the flux density is not widespread; see \\S\\ref{sec:missing} and Appendix A of \\citet{bkk+15}. There is one important difference: the typical 20-min LOFAR integration time is a single integration, while the 15-min GMRT observations are typically subdivided into 3--5 short observations taken over a night of observing. The later is a more optimal detection strategy when there are intensity variations caused by the phase fluctuations in the interstellar medium \\citep{cl91}. If this effect is important, however, it would result in the LOFAR flux densities being {\\it lower} on average that the GMRT values, the opposite of what is seen. Temporal scattering can also reduce the measured flux density for pulsed surveys but as an imaging survey, the TGSS ADR is not sensitive to pulse smearing caused by interstellar scattering. While the LOFAR surveys are sensitive to such effects, they would also act to {\\it lower} the measured flux density. \n\nWe are left with instrumental effects associated with gain calibration. The TGSS ADR flux density scale is good to about 10\\% over the full sky.\nTaken in interferometric imaging mode, the data each day were calibrated back to several low frequency primary flux density calibrators (3C\\,48, 3C\\,147, 3C\\,286 and 3C\\,468.1). After calibration of the full survey, the accuracy of the flux density scale was cross-checked against other sky surveys such as 7C \\citep{hrw+07} and the LOFAR Multi-frequency Snapshot Survey \\citep[MSSS;][]{hpo+15} and they were found to agree at the $\\sim$5\\% level. On the other hand, the flux density calibration for the LOFAR pulsar survey was done directly using the radiometer equation for direction-dependent estimates of the antenna gain and the sky system temperature. The calibration was cross-checked with regular observations of a sample of normal pulsars and MSPs with well-determined spectra. Variations at a level 2--4 times larger than expected from scintillation alone were seen to occur and thus the resulting flux density scale was quoted with errors of $\\pm$50\\%. \n\nWe tentatively suggest that our TGSS ADR pulsar sample shows that there remains an unaccounted gain error in the LOFAR pulsar observing system that results in an overestimate of the flux density scale by about a factor of two.\n\n\\subsubsection{The Missing Pulsars}\\label{sec:missing}\n\nDespite the high yield, there are also a number pulsars in Fig. \\ref{fig:ff}  with large {decimeter} flux densities but with no TGSS ADR counterpart in Table \\ref{tab:bright}. Likewise, we failed to detect several bright pulsars which had been found in previous low frequency pulsation surveys \\citep[e.g.][]{kl01,bkk+15}. To investigate the origin of these missing pulsars, we defined a radio-bright sample from the original 1238 well-localized pulsars in \\S\\ref{sec:loud} as having 400 and 1400 MHz flux densities greater than 21 mJy and 1.8 mJy, respectively.  For a canonical pulsar spectral index these flux densities extrapolate at 150 MHz to the completeness limit of the TGSS ADR \\citep{int16}.  There are 232 such pulsars. Of this sample, 70\\% are detected and are listed in Tables \\ref{tab:bright} and \\ref{tab:faint}.\n\nWe can identify three possible reasons that about one third of this radio-bright sample of pulsars would not be detected in the TGSS ADR. The local rms noise may be too high, the pulsar spectrum may be flat or turn over at 150 MHz, or the signal may be reduced due to interstellar scintillation. It may be possible that one of these effects dominant or they are working in tandem. We will look at each of these in turn.\n\nAt low radio frequencies the synchrotron and thermal emission from the Galactic plane makes a non-negligible contribution to the system temperature of the receivers. The frequency dependence of the brightness temperature goes approximately at T$_b\\propto\\nu^{-2.6}$ \\citep{hks+81} so unless the pulsar spectrum is steeper than this value, they become increasingly more difficult to detect. While the increased brightness temperature affects pulsed and imaging searches equally, the later also suffers from increased rms due to confusion and reduced image fidelity in the presence of bright Galactic HII regions or supernova remnants.  The pulsar B\\,2319+60 is a good example of a bright pulsar confused by nearby bright, extended emission. We looked at the rms noise statistics of the  detected and non-detected samples, following up the large rms cases with a visual inspection of the TGSS ADR image data at the PSR positions. We find evidence that the rms noise of the images has some influence on the detectability of the pulsars. The median rms noise for the detections is \\mjybeam{3.5}. while for the non-detections it is nearby twice this value (\\mjybeam{6.9}).\n\nThe intrinsic spectral shape of the pulsar emission will also affect the detectability at low frequencies. The mean pulsar spectral index, while steep, has a wide scatter (\\S\\ref{sec:intro}). Likewise, for approximately 10\\% of known pulsars there is evidence of a low-frequency spectral turnover, typically around 100 MHz (\\S\\ref{sec:intro}). Our 150 MHz sample has a number of pulsars with known spectral turnovers including PSR\\,J2145-0750 \\citep{drt+13}. We lack a large public database of accurate pulsar flux densities that would be sufficient to look for a turnover frequency for our non-detections, but fortunately most of them have single power-law measurements in the ATNF pulsar catalog. The median spectral index for the detections is $\\alpha=-1.9$ and there are no pulsars in this sample as shallow as $\\alpha\\leq-0.5$. The non-detections have a much flatter median spectral index of $\\alpha=-1.3$. At least one third of our non-detections have spectral indices that are so flat that we do not expect to detect them at 150 MHz based on an extrapolation of their 400 or 1400 MHz catalog flux densities. \n\nDensity fluctuations in the ionized interstellar medium of our Galaxy can induce intensity fluctuations that may depress the flux density of a pulsar during an integration time. The characteristic time and frequency scale depends on many factors including the distance of the pulsar, the turbulent properties of the gas along the line of sight, and the relative velocities of the pulsar and the ionized gas \\citep{cwf+91}. To estimate the magnitude of strong scattering on the phase-averaged pulsar flux densities we followed the method of \\citet{kcac98}. We first estimated the scattering bandwidth and scattering time at 150 MHz for each pulsar using the NE2001 model of \\citet{cl02}. Typical scintillation timescales and bandwidths at these frequencies are small, of order 1 minute and below 1 MHz, respectively. Our values are similar to the values estimated at the same frequency by \\citet{bkk+15}. We then estimate the number of ``scintles'' that are averaged over the observed bandwidth and the duration of the observation. The intensity modulation is equal to the square root of this value. The observed bandwidth is given in \\S\\ref{survey} as 16.7 MHz. The duration of the GMRT observations are more difficult to estimate. The total integration on source is 15 minutes but it is split into 3--5 short snapshots spaced over a full night's observing. As an added complication, the image mosaics are additions of many overlapping fields and so it is possible that a single pixel may contain observations from more than one night. This sampling has the effect of smoothing out any large intensity modulations, so as a (pessimistic) estimate we take the duration as 15 min but we recognize that there may be additional temporal smoothing. Our results by and large suggest that the TGSS ADR pulsar flux densities are only being weakly modulated by scintillation in most cases. There are pulsars that are predicted to be undergoing strong diffractive scintillation at this frequency (e.g. PSR\\,B0950+08 and PSR\\,1929+10) and there are diffraction spikes centered on the MSP PSR\\,B1937+12, likely caused by intensity variations on timescales comparable to the dump time. However, we can find no systematic trend for the non-detected pulsars to have greater predicted modulations from scattering.\n\nSummarizing, we find that the bright cataloged pulsars with no TGSS ADR counterpart may be due to a combination of effects. There is evidence that the non-detections at 150 MHz have more shallow spectral indices than average, and that some of the non-detections are caused by high rms and confusion in the image plane. Strong intensity variations by interstellar scintillation is undoubtedly occurring for some pulsars but we cannot show that the non-detections differ from the detections in their scattering properties. The difficulty in estimating the true GMRT integration time for each pulsar may be masking this effect. \n\n\n\n\\subsection{Radio Quiet Pulsars}\n\nOur search did not find any significant radio emission at 150 MHz toward individual radio quiet pulsars, nor in a weighted stack of all 30 pulsars. The peak of the stacked image in Fig. \\ref{fig:stack} is 0.1$\\pm$0.7 mJy beam$^{-1}$, with upper limit (peak + 2$\\sigma$) of $<$1.5 mJy beam$^{-1}$. Recall from \\S\\ref{quiet} that ``radio-quiet\" pulsars are observationally defined as having a phase-averaged flux density at 1.4 GHz S$_\\nu<$30 $\\mu$Jy. The simplest hypothesis is that radio quiet pulsars are beamed away from the line of sight \\citep{ckr+12}. \n\nRadio quiet gamma-ray pulsars, with Geminga as the prototype, are expected, given what we know about the structure of neutron star magnetospheres \\citep{car14}. The radio emission is thought to originate further down the poles than the gamma-rays, and thus the radio will be beamed into a narrowing opening angle, increasing the probability that the beam sweeps out away from the observer's line of sight. However, it is well-known that both the radio pulse width and the component separation are frequency dependent \\citep{thor91,mr02}. As noted by \\citet{mad12}, this widening of radio beams at low frequencies might be used to detect radio quiet, gamma-ray pulsars. Such pulsars would be recognized in the image plane as having a spectral index that is much steeper than the canonical value. PSR\\,B1943+10 may be thought of the prototype of such systems, bright at 400 MHz and below but weak at 1.4 GHz, with a  spectral index $\\alpha$ steeper than $-3.0$ \\citep[see Table \\ref{tab:bright};][]{wcl+99}. The lower limit estimate on spectral index that we derive from the weighted stack at 150 MHz, assuming the defining radio-quiet 1.4 GHz flux density of 30 $\\mu$Jy, gives $\\alpha>-1.75\\pm0.20$.  This is a spectral index limit that is well within the canonical value for normal pulsars. Thus we find no evidence that these gamma-ray pulsars have radio beams that sweep close to our line of sight.\n\n\n\n\n\n\\section{Conclusion}\\label{sec:conclude}\n\nWe have identified nearly 300 pulsars at 150 MHz based in their phase-averaged emission on all-sky images. This imaging approach is complementary to pulsation studies since it is not affected by {pulse scatter} broadening or dispersion, making it sensitive to both normal and millisecond pulsars equally. Our sample includes many southern pulsars which are being detected at low radio frequencies for the first time. We anticipate that these 150 MHz flux densities will be used to study large numbers of pulsar over a wider frequency range than has hitherto been possible, and to addresses questions about the incidence and origins of low-frequency spectral turnovers. Accurate calibration between telescopes remains an important issue and we have identified a discrepancy between the flux densities of pulsars in common between GMRT and LOFAR. We suggest that the LOFAR sample may be overestimating the flux density scale by about a factor of two. It should be straightforward to test this hypothesis by observing a sample of pulsars with LOFAR in both imaging and phase-binning modes, calibrating the interferometric data in the standard way to allow proper comparison with each other and with the GMRT.\n\nWe have carried out a preliminary spectral index study of our sample. Generally there is good agreement with past work, except that we find a curious preponderance of gamma-ray MSPs with unusually steep spectral indices ($\\alpha\\leq-2.5$). Regardless of its origins, this suggests a possible way to identify new MSP candidates in \\emph{Fermi}  unassociated sources on the basis of their unusually steep spectrum at low radio frequencies. Such pulsars may have been missed in radio pulsation searches due to propagation effects caused by the interstellar medium or they may be in binary systems and thus more difficult to discover. In such cases, imaging \\emph{Fermi}  error regions with LOFAR and the GMRT could provide accurate enough positions to enable blind gamma-ray searches for pulsations.\n\n\\acknowledgments\n\nThis research has made use of data and\/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA\/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. DAF thanks T. Readhead and S. Kulkarni  for their hospitality at Caltech while this work was being written up.\nHTI acknowledges financial support through the NL-SKA roadmap project funded by the NWO. We thank A. Bilous and J. Hessels for sharing their knowledge of LOFAR pulsar flux density calibration.\n\n{\\it Facilities:} \\facility{GMRT}, \\facility{Fermi (LAT)}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn the composite fermion (CF) picture of the fractional quantum Hall\n(FQH) effect, fundamental interactions are taken into account at the\nmean field level by mapping the system of strongly interacting 2D\nelectrons in magnetic field into a system of weakly interacting\nComposite Fermions (CF) moving in a reduced effective magnetic field\n\\cite{Jain}. The reduction in magnetic field follows from the\nbinding of $\\phi$ flux quanta to electrons, so that effective\nmagnetic field experienced by CF quasiparticles is $B^*=\\pm B\/(\\phi\np\\pm 1)$, where $p$ is an integer that enumerates members of a\nparticular sequence and $\\phi$ is a even integer that labels\ndifferent sequences. In this picture, the FQH effect can be\nunderstood by the emergence of CF Landau levels  with cyclotron\nfrequency: $\\omega_{CF}=\\frac{eB^*}{cm^{*}}$, where $m^{*}$ is an\neffective CF mass. Evidence for a spin split Landau level structure\nof CF for the $\\phi=2$ sequence has been provided by\nmagnetotransport experiments in tilted magnetic fields at filling\nfactors near $\\nu=3\/2$ \\cite{Du} and by inelastic light scattering\nstudies of spin excitations in the range $1\/3<\\nu<2\/5$ \\cite{Irene}.\nFor the $\\phi=4$ sequence  (i.e. $\\nu\\lesssim1\/3$), however, direct\nevidence for such CF Landau level structure is lacking. Studies of\nthe $\\phi=4$ sequence are more difficult because of the higher\nmagnetic fields that are required and, compared to the $\\phi=2$\nsequence, the smaller energy scales in the excitations. Insight on\nthe energy scales for excitations were revealed by activated\ntransport measurements \\cite{Pan} and by the recent observations of\n$\\phi=4$ quasiparticle excitations in light scattering experiments\n\\cite{Cyrus}.\n\nIn this work, we present a resonant inelastic light scattering study\nof spin excitations for $\\nu\\lesssim1\/3$. The excitations are spin\nwaves (SW) and spin-flip (SF) modes. The SF excitations involve a\nchange in both the spin orientation and CF Landau level quantum\nnumber. We monitor the evolution of these spin excitations below and\naway from $\\nu=1\/3$ when the population of the excited CF Landau\nlevel increases. Our results reveal the existence of spin split CF\nLandau levels in the $\\phi=4$ sequence. The SW-SF splitting is\nlinear in magnetic field. This determination suggests an effective\nmass significantly larger than the activation mass of CF with\n$\\phi=4$.\n\\section{Sample and Experiment}\nThe 2D electron (2DES) system studied here is a GaAs single quantum\nwell of width $w=330~\\AA$. The electron density at small magnetic\nfields is n=5.5$\\times$10$^{10}$~cm$^{-2}$ and the low temperature\nmobility is $\\mu$=7.2$\\times$10$^6$\/Vs. The sample is mounted in a\nbackscattering geometry, making an angle $\\theta$ between the\nincident photons and the normal to the sample surface. The magnetic\nfield perpendicular to the sample is B=B$_T$cos$\\theta$, where B$_T$ is\nthe total applied field. The results reported here have been\nobtained at $\\theta$=50$\\pm$2 degrees. Similar results have been seen at 30 degrees \\cite{Cyrus-thesis}.The ingoing and outgoing\npolarizations were chosen to be orthogonal (depolarized spectra)\nsince excitations which involve a change in the spin degrees of\nfreedom are stronger in this configuration. The sample was cooled in\na dilution refrigerator with windows for optical access. All the\nmeasurements were performed at the base temperature T=23~mK and the\npower density was kept below 10$^{-5}$W\/cm$^2$ to avoid heating the\nelectron system. The energy of the incident photons was tuned to be\nin resonance with the excitonic optical transitions of the 2DES in\nthe FQH regime \\cite{Goldberg,Bar-joseph,Cyrus2}.\n\\section{Results and discussion}\n\\begin{figure}\n\\centering \\epsfig{figure=spectres.eps, width=0.79\\linewidth, clip=}\n\\caption{Low energy spectra of spin excitations in the filling\nfactor range 0.31$<\\nu<$0.33. The most intense peak is the long\nwavelength spin-wave at the Zeeman energy E$_z$ while the peak its\nlow energy side is assigned to a spin-flip transition (see text and\nfigure \\ref{levels}). The left inset shows the result of a\ntwo-gaussian fitting procedure for the two peaks to extract their\nrespective energies. The right inset shows the backscattering\nconfiguration} \\label{spectres}\n\\end{figure}\nFigure \\ref{spectres} shows the evolution of the low energy spectrum\nfor $\\nu\\lesssim1\/3$. $\\nu=1\/3$ corresponds to a perpendicular field\nof 6.5T and the filling factor range studied corresponds to the\nrange 0.31$<\\nu<$0.33. Close to $\\nu=1\/3$, the spectra are dominated\nby the long wavelength SW at the 'bare' Zeeman energy\nE$_z=g\\mu_BB_T$, where g=0.44 is the Lande factor for electrons in\nGaAs and $\\mu_B$ is the Bohr magneton. For filling factors away from\n$\\nu=1\/3$ an excitation emerges on the low energy side of E$_z$. We\nassign this excitation to a SF mode linked to transitions in the CF\nframework that involve the first excited CF Landau level as depicted\nin figure \\ref{levels} . At $\\nu=1\/3$ the first CF Landau\n($\\ket{0,\\uparrow}$) level is fully occupied while for $\\nu=2\/7$ the\nfirst two CF Landau levels ($\\ket{0,\\uparrow}$ and\n$\\ket{1,\\uparrow}$) are occupied. In between the two incompressible\nstates, the first excited Landau level is partially populated and SF\ntransitions between $\\ket{1,\\uparrow}$ and $\\ket{0,\\downarrow}$\nstarting from the partially filled level become possible. Thus the\nstudy of the SF excitations in the filling factor range\n2\/7$<\\nu<$1\/3 probe directly the CF level structure for $\\phi=4$.\n\\begin{figure}\n\\centering \\epsfig{figure=levels.eps, width=0.7\\linewidth, clip=}\n\\caption{Structure of spin split CF Landau levels for 2\/7$<\\nu<$1\/3 ($\\phi=4$).\nTwo $^4$CF spin transitions are possible. The large $q$ spin wave is\nat E$^*_z$=E$_z$+E$^{\\uparrow\\downarrow}$ where\nE$^{\\uparrow\\downarrow}$ is the spin reversal energy. The spin-flip\nexcitation at E$_{SF}$. The spin-flip excitation emerges when the\n$\\ket{1,\\uparrow}$ level is populated.} \\label{levels}\n\\end{figure}\nFor small occupation of the $\\ket{1,\\uparrow}$ excited level and\nwhen the coupling between the excited quasiparticle and its\nquasihole is negligible, the SF transition energy can be written as\nin the $\\phi=2$ case:\n\\begin{equation}\nE_{SF}=E_z+E^{\\uparrow\\downarrow}-\\hbar\\omega_c\n\\label{ESF}\n\\end{equation}\nwhere E$^{\\uparrow\\downarrow}$ is the spin reversal energy which is\na measure of the residual interactions between $\\phi=4$ CF\nquasiparticles \\cite{Pinczuk,Longo,Aoki,Mandal,Irene}.\n\\begin{figure}\n\\centering \\epsfig{figure=energies.eps, width=0.65\\linewidth, clip=}\n\\caption{Magnetic field dependence of the Zeeman (E$_z$) and\nspin-flip excitation energies for 0.31$<\\nu<$0.33. Also shown is the\nevolution of the splitting E$_z$-E$_{SF}$.} \\label{energies}\n\\end{figure}\nThe energy E$_{SF}$ was extracted for each filling factor by\nperforming a simple analysis of the low energy spectra using a\ntwo-gaussian fitting procedure as shown in the inset of figure\n\\ref{spectres}. Figure \\ref{energies} displays the corresponding\nenergies, E$_z$ and E$_{SF}$ as a function of filling factor. The\nstrong dependence of E$_{SF}$ confirms our assignment of the peak as\nexcitation involving spin degrees of freedom. More importantly, the\nspacing between E$_z$ and E$_{SF}$ is not constant and decreases\nwith the filling factor. From equation \\ref{ESF}, we easily see that\nthis spacing is directly related to the CF cyclotron energy so that\nthe splitting between the two spin excitations is\nE$_z$-E$_{SF}$=$\\hbar\\omega_c$-E$^{\\uparrow\\downarrow}$.\nThe magnetic field dependence of the E$_z$-E$_{SF}$ spacing is set\nby the effective field B$^*$. For the $\\phi=2$ sequence, $^2$CF\nemanate from the $\\nu=1\/2$ state and the effective field has its\norigin at B$_{1\/2}$. For the $\\phi=4$ sequence however, $^4$CF\nemanate from the $\\nu$=1\/4 state and the origin is at B$_{1\/4}$. and\nthe effective magnetic field should then decrease when going from\n0.33 to 0.31. This is indeed consistent with our data and to the\nexistence of $^4$CF or $\\phi=4$ spin-flip excitations below\n$\\nu$=1\/3. Our results support the CF Landau level picture shown figure\n\\ref{levels} for the $\\phi=4$ sequence.\n\nThe linear decrease of E$_z$-E$_{SF}$ with the effective magnetic\nfield makes very tempting the evaluation of an effective mass by using\na slope that is simply given by $\\frac{\\hbar e}{m^{*}c}$ in the\nframework of equation \\ref{ESF} for E$_{SF}$. Our data between\nfilling factors 0.33 and 0.31 give m$^{*}$=1.5($\\pm 0.1$)~m$_e$\nwhere m$_e$ is the bare electron mass. Previous determinations using\nthe activation gap values at $\\nu$=2\/7, 3\/11 and 4\/15 on samples\nwith similar densities yield values around 0.9~m$_e$ \\cite{Pan}. We\nnote that while our determination of m$^{*}$ is performed between\n$\\nu$=0.33 and $\\nu$=0.31, i.e. in between the incompressible states\nat 1\/3 and 2\/7, the effective mass determined via transport\nmeasurements comes from a linear scaling of the activation gap\nvalues at the incompressible states. The high\neffective mass obtained in our analysis may be linked to the onset\nof significant CF interactions in the partially populated CF Landau\nlevel.\n\\begin{figure}\n\\centering\n\\epsfig{figure=intensities.eps, width=0.65\\linewidth, clip=}\n\\caption{Evolution of the SF integrated intensity for 0.31$<\\nu<$0.33.}\n\\label{intensities}\n\\end{figure}\nAdditional insights can be obtained by tracking the evolution of the\nintensity of the SF excitation when the population of the first\nexcited CF level increases. This is done in Fig. \\ref{intensities}\nwhere the integrated spectral weight of the SF excitation is plotted\nas a function of filling factor. As expected, the SF intensity\nincreases when the population of $\\ket{1,\\uparrow}$ increases but\ndisplays an intriguing saturation around below $\\nu$=0.32. As\nalready mentioned for the effective mass, the saturation may result\nfrom increasing impact of CF residual interactions. These\ninteractions could possibly lead to further condensation into higher\norder CF in the partially populated level. Recent transport\nmeasurements have indeed shown the possible existence of such higher\norder states even for the $\\phi$=4 sequence \\cite{Pan2}.\n\\section{Conclusion}\nIn this study, we have shown the existence of spin-flip excitations\nof $^4$CF quasiparticle below $\\nu$=1\/3. The results indicate the\nexistence of a spin-split CF Landau level structure for the $\\phi=4$\nsequence of the fractional quantum Hall effect that is similar to\nthe one found for the $\\phi=2$ sequence. The evolution of the SF\nenergy with effective magnetic field yields an effective mass\nof 1.5m$_e$. The evolution of the SF intensity with filling factor\nmight signal the onset of significant CF-CF interactions that could\npossibly lead to further CF condensation.\n\nThis work is supported by the National Science Foundation under Grant No. NMR-0352738, by the Department of Energy under Grant No. DE-AIO2--04ER46133, and by a research grant from the W.M. Keck Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRare flavour-changing neutral-current (FCNC) $B^0_{(s)} \\rightarrow \\mu^+ \\mu^-$  and $b \\rightarrow s \\mu^+ \\mu^-$ decays\nare considered among the most promising  probes\nof the standard model (SM) and its extensions.\nThe precise measurement of several observables (total and differential\nbranching ratios, angular distributions of the decay products) of these decays\nmight provide interesting clues for new physics (NP) phenomena,\nif any sizable deviation  from the SM predictions is observed.\nIn this paper we review the current status of these indirect searches at the Collider \nDetector at Fermilab (CDF II), which\nreached a sensitivity very close to SM predictions after a \ndecade of Tevatron leadership in the exploration of $B^0_s$ dynamics.\nThe Tevatron $p \\bar{p}$ collider, whose operations ended in October 2011 after 20 years of operation, \nprovided excellent opportunities to study $B$ physics.\nCDF II is a multipurpose detector, consisting of a central charged particle tracking system, \nsurrounded by calorimeters and muon chambers. It collected a final data set corresponding\nto about 10 fb$^{-1}$ of integrated luminosity. \n\nIn the search for rare $B$ decays,\nthe  experimental challenge is to reject a huge background while keeping the signal efficiency high.\nA dedicated dimuon trigger has been used to select \nevents with a pair of muons in the final state in the pseudorapidity region $|\\eta|<$1.1.\nIn the reconstruction and analysis of $B$ hadron decays, CDF takes advantage of\nan excellent transverse momentum resolution, \n$\\frac{\\sigma_{p_T}}{p_T} = 0.07\\%\\, p_T$ (GeV\/c),\nwhich implies a resolution on the invariant\ndimuon mass of 24 MeV in the $B^0_{(s)} \\rightarrow \\mu^+ \\mu^-$ decay, \na vertex resolution  of about 30 $\\mu$m in the transverse plane, and \nparticle identification (PID) capability, \nbased on multiple measurements of the ionization per unit of path length ($dE\/dx$) in the drift chambers.  \n\\section{Search for $B^0_{(s)} \\rightarrow \\mu^{+} \\mu^{-}$}\n$B_{(s)}^0 \\rightarrow \\mu^{+} \\mu^{-}$ decays are mediated by FCNC and thus forbidden at first order in the SM. \nMoreover they are further suppressed by helicity factors $(m_{\\mu}\/m_B)^2$ in the final dimuon state.\nThey can only occur at second order through penguin and box diagrams. \nThe SM predicts very low rates for these processes: $\\ensuremath{\\mathcal B}(B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(3.2 \\pm 0.2\\right)\\times 10^{-9}$ \nand  $\\ensuremath{\\mathcal B}(B^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(1.0 \\pm 0.1\\right)\\times 10^{-10}$ \\cite{SM1,SM2}.\nHowever, a wide variety of beyond the standard model (BSM) theories predict enhancement of their branching ratios\n by several order of magnitudes, \nmaking these decays one of the most sensitive probes in indirect searches for NP. \n\nIn 2011, CDF observed an intriguing $\\sim$2.5$\\sigma$ excess over background in\n$B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}$ using  7 fb$^{-1}$ of data \\cite{CDF}.\nThough it was compatible with other experimental results (LHCb \\cite{LHCb}, CMS \\cite{CMS})\nand the SM prediction, it could be interpreted as the first indication of a signal\nand allowed CDF to set a two-sided bound on the rate \n$\\ensuremath{\\mathcal B}(B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(1.8^{+1.1}_{-0.9}\\right)\\times 10^{-8}$ .\nTo further investigate the nature of the excess, we repeated the analysis unchanged using the whole\n Run II data set, corresponding to 9.7 fb$^{-1}$ of integrated\nluminosity, about 30\\% more data with respect to the 2011 analysis. Here we report on \nthe final results of this search \\cite{CDF2}.\n\\begin{figure}\n  \\centering\n\\subfigure[]{\\label{fig:bmumu1}\n  \\includegraphics[width=.85\\textwidth]{bd-8NNbins_poiE_bgE_v12.eps}\n}\n\\subfigure[]{\\label{fig:bmumu2}\n  \\includegraphics[width=.85\\textwidth]{bs-8NNbins_poiE_bgE_v12.eps}\n}\n\\caption{\nDimuon mass distributions for the (a) $B^0 \\rightarrow \\mu^+ \\mu^-$ and (b) \n$B_s^0 \\rightarrow \\mu^+ \\mu^-$ signal region in the eight NN bins. \nThe observed data (points) are compared to the total background expectation (light gray\nhistogram). The hatched region is the total uncertainty on the background expectation.\nIn (b) the dark gray histogram represents the SM signal expectation enhanced by a factor 4.1. \n}\n\\label{fig:bmumu}\n\\end{figure}\n\nThe baseline selection  requires high quality muon candidates with opposite charge,\n transverse momentum $p_T >$ 2 GeV\/c, and a dimuon invariant mass $m_{\\mu\\mu}$ in the range 4.669-5.969 GeV\/c$^2$. \nThe muon pairs are constrained to originate from a common, well-measured reconstructed decay point. \nA likelihood-based muon identification method, \nis used to suppress contributions \nfrom hadrons misidentified as muons. \nThe branching ratios of $B_{(s)}^0 \\rightarrow \\mu^{+} \\mu^{-}$ are measured\nby normalizing to a sample of 40225$\\pm$ 267\n $B^+ \\rightarrow J\/\\psi (\\rightarrow \\mu^+ \\mu^-)\\,K^+$ candidates, \nselected with the same baseline requirements.\nA Neural Network (NN) classifier is used to improve signal over background  separation. \nFourteen variables are used to construct the NN discriminant that ranges between 0 and 1. \nThe six most discriminating variables \ninclude the 3D opening angle between the dimuon momentum and the displacement vector between the\nprimary and secondary vertex; \nthe isolation $I$ \\footnote{$I = |\\vec{p}_T^{\\mu\\mu}|\/(\\sum_i p_T^i + |\\vec{p}_T^{\\mu\\mu}|)$, where $\\vec{p}^{\\mu\\mu}$ is the momentum\nof the dimuon pair; the sum is over all tracks with $\\sqrt{ (\\Delta\\phi)^2 + (\\Delta\\eta)^2} \\le 1$; $\\Delta\\eta$ and $\\Delta\\phi$ are the relative azimuthal angle and pseudorapidity of track $i$ with respect to $\\vec{p}^{\\mu\\mu}$.}\nof the candidate $B^0_{s}$ ;\nthe muon and $B^0_{(s)}$ impact parameters; the $B^0_{(s)}$ decay length significance; the vertex-fit $\\chi^2$.\nThe NN from the 2011 analysis was used with the same training.\nThe final search region in the dimuon invariant mass has a half width of about 60 MeV corresponding to 2.5 times the dimuon \nmass resolution. \nThe NN was validated with signal and background events.\nCareful checks for possible mass-biases  of the NN output and overtraining show no anomalies. \n\nExtensive and detailed background estimates and checks have been performed.\nBackground is due to  both combinatorial and peaking contributions in the signal region. \nCombinatorial background is estimated by fitting the sidebands  to linear functions,\nafter blinding the signal region in the dimuon mass distribution.\nThe peaking background is due to $B \\rightarrow h^+ h^{'-}$ decays where the hadrons \n($h,h'$ stand for $\\pi$ or $K$) are misidentified as  muons. \nThis has been estimated from both MC and data.  \nThe misidentification probability is parametrized as a function of the track transverse momentum\n using $D^*$-tagged $D^0 \\rightarrow \\pi^+ K^-$ events. \nIt turned out that the peaking background is  about 10\\% of the combinatorial background \nin $B_s^0 \\rightarrow \\mu^+ \\mu^-$ and about 50\\% of the total background in \nthe $B^0 \\rightarrow \\mu^+ \\mu^-$ channel.\nBackground estimates have been cross-checked using independent background-dominated control samples, \nin which the muons have the same measured charge or the reconstructed\ndimuon candidate lifetime is negative. \nNo significant discrepancies between the expected and observed number of events in the control samples have been found.\\\\\nThe data are divided into 8 bins of the NN discriminant to exploit the improved background suppression at high NN values, \nand five bins of mass in the search region.\nIn the $B^0$ search region  data are consistent with the background \nprediction (Fig.~\\ref{fig:bmumu1}) and yield the limit of\n $\\mathcal{B}(B^0 \\rightarrow \\mu^+ \\mu^-) < 3.8 (4.6)\\times 10^{-9}$ at 90\\% (95\\%) C.L.. \nThe significance of the background-only hypothesis expressed as a p-value, estimated from an \nensemble of background-only pseudo-experiments, is 41\\%.\n\nIn the $B^0_s$ search region, a moderate excess in the highest NN bins ($>$0.97)\nis observed (Fig.~\\ref{fig:bmumu2}).\nThe p-value for the\nbackground-only hypothesis is 0.94\\%. We also\nproduce an ensemble of simulated experiments that includes\na  $B^0_s \\rightarrow \\mu^+ \\mu^-$ contribution at the expected SM\nbranching fraction which yields a p-value of 7.1\\%.\nWith respect to the previous CDF result with 7 fb$^{-1}$ of data \\cite{CDF}, the excess in the third significant NN bin\n(0.97-0.987)  softened, as expected for a statistical fluctuation.\nThough the 2011 hint of signal is not reinforced by the new data,\nit is still present and  remains $>$2$\\sigma$ significant over background.\nAssuming the observed\nexcess in the $B^0_s$ region is due to signal, CDF finds \n$\\mathcal{B}(B_s^0 \\rightarrow \\mu^+ \\mu^-) = \\left(1.3^{+0.9}_{-0.7}\\right)\\times 10^{-8}$, \nwhich is still compatible with both\n the SM expectation and the latest  constraints from LHC experiments \\cite{LHCbmumu, LHCb2}.\n\n\\begin{figure}[!h]\n  \\centering\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kstmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_phimm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kstmm_kspi_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_ksmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_lmmm_data_prl.eps}\n}\n\\caption{Invariant mass of\n(a) $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $, \n(b) $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$,\n(c) $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$,\n(d) $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$,\n(e) $B^0 \\rightarrow K_S \\mu^+ \\mu^-$, \n(f) $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ with fit results overlaid. }\n\\label{fig:bsmumuyield}\n\\end{figure}\n\\section{$b \\rightarrow  s \\mu^{+} \\mu^{-}$  decays}\nRare decays of bottom hadrons mediated by the FCNC process $b \\rightarrow s \\mu^+ \\mu^-$ \nare suppressed at tree level in the SM and must\noccur through higher-order loop amplitudes. Their expected branching ratios are of the order of 10$^{-6}$.\nBecause of their clean experimental signature and the reliable theoretical predictions for their rates, \nthese are excellent channels for NP searches.\n\n\\begin{table}[h]\n\\begin{center} \n  \\begin{tabular}{l l}\n    \\hline\\hline Decay mode & $\\mathcal{B}(10^{-6})$  \\\\\n    \\hline\n    $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $ & $0.46 \\pm 0.04 \\pm 0.02 $\\\\\n    $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$ &  $1.02 \\pm 0.10 \\pm 0.06$\\\\\n    $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$ &  $1.47 \\pm 0.24 \\pm 0.46$\\\\\n    $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ & $0.95 \\pm 0.32 \\pm 0.08$\\\\\n    $B^0 \\rightarrow K_S \\mu^+ \\mu^-$ & $0.32 \\pm 0.10 \\pm 0.02$\\\\\n    $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ & $1.73 \\pm 0.42 \\pm 0.55$\\\\\n    \\hline\\hline\n  \\end{tabular}\n\\caption{Branching ratio of the $H_b\\rightarrow h \\mu^+ \\mu^-$ decays measured by CDF.\nThe first quoted uncertainty is statistical, the second is systematic.}\n  \\label{tabBR}\n\\end{center}\n\\end{table}\nCDF has studied the  FCNC $H_b\\rightarrow h \\mu^+ \\mu^-$ decays (where $H_b$ and $h$ indicate hadrons \ncontaining a $b$ and $s$ quark, respectively) listed in Table~\\ref{tabBR}, \nusing 6.8 fb$^{-1}$ of data collected with the dimuon trigger \\cite{CDFbsmumuBR}. \nCandidates for each decay have been selected by standard kinematics cuts and a NN optimized for best sensitivity.\nSignal yields are obtained by an unbinned maximum log-likelihood fit to the $b$-hadron mass distributions \n(Fig.~\\ref{fig:bsmumuyield}), modelling \nthe signal peak with a Gaussian, and the combinatorial background with a linear function.\nTo cancel dominant systematic uncertainties, the branching ratio of each rare decay  $H_b\\rightarrow h \\mu^+ \\mu^-$ \nis measured relative to the corresponding resonant channel $H_b\\rightarrow J\/\\psi\\, h$,\nused as a normalization and a cross-check of the whole analysis.\nThe results of the total branching ratios are reported in Table~\\ref{tabBR} and include\nthe first observation of the baryonic FCNC decay $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$, \nand the first measurement  of the $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ and  $B^0 \\rightarrow K_S\\,\\mu^+ \\mu^-$ decays\nat a hadron collider.\n\\begin{figure}[h] \n  \\centering\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kstmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_phimm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kstmm_kspi.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_ksmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_lmmm.eps}\n}\n\\caption{\nDifferential branching ratios of \n(a) $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $, \n(b) $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$,\n(c) $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$,\n(d) $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$,\n(e) $B^0 \\rightarrow K_S\\,\\mu^+ \\mu^-$, \n(f) $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$.\nThe points are the fit result from data. The solid curves are the SM\nexpectation \\cite{SMexp, SMexp2, SMexp3, SMexp4}. The dashed line in (f) is the SM prediction normalized to our total branching\nratio measurement. The hatched regions are the charmonium veto regions.}\n\\label{fig:bsmumudiff}\n\\end{figure}\n\nRich information about the $b \\rightarrow s \\mu^+ \\mu^-$\ndynamics can be obtained by precise measurements of the differential branching ratio as a function of $q^2 = m_{\\mu\\mu}^2 c^2$\nand the angular distributions of the decay products.\n\\begin{figure}[h] \n  \\centering\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_afb_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_fl_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_at2_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_aim_6bin_all_prl.eps}\n}\n\n\\caption{Measurements of angular observables (a) $A_{FB}$, (b) $F_L$, (c) $A_T^{(2)}$, and (d) $A_{im}$ \n as a function of dimuon mass squared $q^2$\nin the combined decay mode $B \\rightarrow K^{*} \\mu^+ \\mu^-$. \nThe points are the fit results from data.\nThe solid and dotted curves represent expectations from the SM \nand a particular BSM scenario, respectively.}\n\\label{fig:angular}\n\\end{figure}\nThe differential branching ratios with respect to $q^2$ have been measured by dividing \nthe signal region  into six bins in $q^2$ \nand fitting the signal yield in each bin.\nIn each fit, the mean of the $H_b$ mass and the background slope were fixed to the\nvalue from the global fit, so that only the signal fraction\nwas allowed to vary in the fit.\nThe results are shown in Fig.~\\ref{fig:bsmumudiff}. \nFor $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$ and $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ \nthese are the first such measurements. At present no significant discrepancy from SM prediction is found.\n\nThe angular distributions \nof the combined  $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$\nand $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ decays have been measured and  \nparametrized to four angular observables:\nthe muon forward-backward asymmetry $A_{FB}$, \nthe $K^*$ longitudinal polarization fraction $F_L$,  \nthe transverse polarization asymmetry $A_T^{(2)}$,\nthe time-reversal-odd charge-and-parity asymmetry $A_{im}$, defined in \\cite{ANG,ANG2}. \n$A_T^{(2)}$ and $A_{im}$  have been measured for the first time by CDF \\cite{CDFbsmumuAfb}.\nThe results for these observables, shown in Fig.~\\ref{fig:angular}, are among the most precise to date and consistent \nwith SM predictions and other experiments, but\nstill statistically limited in providing stringent tests on\nvarious BSM models.\n\\section{Conclusion}\nWe have summarized the recent updates on the searches of rare $b$-hadron decays at CDF.\nThe intriguing excess in $B_s^0 \\rightarrow \\mu^+ \\mu^-$ reported in 2011\nis confirmed with the full data set, though its significance is softened to the level of 2$\\sigma$ over background. \nThe measured $B_s^0 \\rightarrow \\mu^+ \\mu^-$ branching ratio   is still compatible with \nthe SM expectation and recent combined results from LHC experiments.\n\nThe dynamics of several rare decays mediated by the FCNC process $b \\rightarrow s \\mu^- \\mu^-$ \nhas been studied in detail extending the reach to new angular observables. \nThe $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ decay has been observed for the first time. \nAnalyses of the  $b \\rightarrow s \\mu^- \\mu^-$  decays \nare still in progress and may yield interesting results in the\nnear future. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background on \\texorpdfstring{$p$}{p}-local compact groups}\\label{Background}\n\nLet $p$ a prime, to remain fixed for the rest of the paper unless otherwise stated. In this section we review all the definitions and results about $p$-local compact groups that we need in this paper. The main references for this section are the works of C. Broto, R. Levi and B. Oliver \\cite{BLO2, BLO3, BLO6}.\n\nRoughly speaking, $p$-local compact groups are abstractions of the fusion data obtained from finite and compact Lie groups. This idea already implies the existence of some sort of \\emph{Sylow $p$-subgroup}. In the finite case this role was played by finite $p$-groups, while in this more general setup we use discrete $p$-toral groups. Let $\\Z\/p^{\\infty} = \\bigcup_{n \\geq 1} \\Z\/p^n$ be the union of the cyclic $p$-groups $\\Z\/p^n$ under the obvious inclusions.\n\n\\begin{defi}\\label{defidiscptor}\n\nA \\emph{discrete $p$-toral group} $P$ is a group that contains a normal subgroup $P_0$ of finite index and which is isomorphic to $(\\Z\/p^{\\infty})^{\\times r}$ for some finite $r \\geq 0$.\n\n\\end{defi}\n\nIn other words, a discrete $p$-toral group is a group $P$ fitting in an exact sequence\n$$\n\\{1\\} \\to P_0 \\Right4{} P \\Right4{} \\pi \\to \\{1\\},\n$$\nwhere $\\pi$ is a finite $p$-group and $P_0 \\cong (\\Z\/p^{\\infty})^{\\times r}$. The \\emph{rank} of $P$, denoted by $\\mathrm{rk}(P)$, is $r$, and the \\emph{order} of $P$ is then defined as the pair $|P| \\stackrel{def} = (\\mathrm{rk}(P), |\\pi(P)|)$, considered as an element of $\\mathbb{N}^2$. This way we can compare the order of two discrete $p$-toral groups, by writing $|Q|\\leq |P|$ if either $\\mathrm{rk}(Q) < \\mathrm{rk}(P)$, or $\\mathrm{rk}(Q) = \\mathrm{rk}(P)$ and $|\\pi(Q)|\\leq |\\pi(P)|$. Given a discrete $p$-toral group $S$ and subgroups $P, Q\\leq S$, define\n$$\n\\mathrm{Hom}_S(P, Q) = \\{f = c_x \\in \\mathrm{Hom}(P, Q) \\,\\, | \\,\\, \\exists x \\in S \\mbox{ such that } x P x^{-1}\\leq Q\\}.\n$$\n\n\\begin{defi}\n\nGiven a discrete $p$-toral group $S$, a \\emph{fusion system} over $S$ is a category $\\mathcal{F}$ with $\\mathrm{Ob}(\\mathcal{F}) = \\{P\\leq S\\}$, and whose morphisms are actual homomorphisms satisfying the following:\n\\begin{enumerate}[(i)]\n\n\\item $\\mathrm{Hom}_S(P,Q) \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\subseteq \\operatorname{Inj}\\nolimits(P,Q)$  for all $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$; and\n\n\\item every morphism in $\\mathcal{F}$ is the composition of an isomorphism in $\\mathcal{F}$, followed by an inclusion.\n\n\\end{enumerate}\nThe \\emph{rank of $\\mathcal{F}$} is the rank of $S$.\n\n\\end{defi}\n\nThe following notation will be used tacitly throughout the rest of the paper. Let $\\mathcal{F}$ be a fusion system over $S$, and let $P, Q, X\\leq S$. As objects in $\\mathcal{F}$, we say that $P$ and $Q$ are \\emph{$\\mathcal{F}$-conjugate} if they are isomorphic as objects in $\\mathcal{F}$. The notation $P^X$ and $P^{\\mathcal{F}}$, respectively, stands for the $X$-conjugacy and $\\mathcal{F}$-conjugacy classes of $P$. Note also that $\\mathrm{Aut}_{\\mathcal{F}}(P)$ is a group, by definition of fusion system, and that $\\mathrm{Inn}(P)\\leq \\mathrm{Aut}_{\\mathcal{F}}(P)$. Thus, it is reasonable to define\n$$\n\\mathrm{Out}_S(P) \\stackrel{def} = \\mathrm{Aut}_S(P)\/\\mathrm{Inn}(P) \\qquad \\mbox{and} \\qquad \\mathrm{Out}_{\\mathcal{F}}(P) \\stackrel{def} = \\mathrm{Aut}_{\\mathcal{F}}(P)\/\\mathrm{Inn}(P).\n$$\nFinally, we say that $P$ is \\emph{fully $\\mathcal{F}$-centralized}, respectively \\emph{fully $\\mathcal{F}$-normalized}, if $|C_S(P)| \\geq |C_S(Q)|$ for all $Q \\in P^{\\mathcal{F}}$, respectively  if $|N_S(P)| \\geq |N_S(Q)|$ for all $Q \\in P^{\\mathcal{F}}$.\n\n\\begin{defi}\\label{defisat}\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a fusion system over $S$. We say that $\\mathcal{F}$ is a \\emph{saturated fusion system} if the following conditions are satisfied.\n\\begin{enumerate}[(I)]\n\n\\item If $P\\leq S$ is a fully $\\mathcal{F}$-normalized subgroup, then it is also fully $\\mathcal{F}$-centralized. Moreover, in this case $\\mathrm{Out}_{\\mathcal{F}}(P)$ is a finite group, and $\\mathrm{Out}_S(P) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Out}_{\\mathcal{F}}(P))$.\n\n\\item Suppose $P\\leq S$ and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$ are such that $f(P)$ is fully $\\mathcal{F}$-centralized, and set\n$$\nN_f = \\{g \\in N_S(P) \\,\\, | \\,\\, f \\circ c_g \\circ f^{-1} \\in \\mathrm{Aut}_S(f(P))\\}.\n$$\nThen, there exists $\\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(N_f, S)$ such that $\\4{f}|_P = f$.\n\n\\item Let $P_1\\leq P_2\\leq P_3\\leq \\ldots$ be a sequence of subgroups of $S$, and set $P = \\bigcup_{n = 1}^{\\infty} P_n$. If $f \\in \\mathrm{Hom}(P,S)$ is a homomorphism such that $f|_{P_n} \\in \\mathrm{Hom}_{\\mathcal{F}}(P_n,S)$ for all $n$, then $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nWe also recall the definition of centric and radical subgroups, which are crucial concepts in the $p$-local group theory.\n\n\\begin{defi}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$.\n\\begin{itemize}\n\n\\item A subgroup $P\\leq S$ is \\emph{$\\mathcal{F}$-centric} if $C_S(Q) = Z(Q)$ for all $Q \\in P^{\\mathcal{F}}$.\n\n\\item A subgroup $P\\leq S$ is \\emph{$\\mathcal{F}$-radical} if $\\mathrm{Out}_{\\mathcal{F}}(P)$ contains no nontrivial normal $p$-subgroup.\n\n\\end{itemize}\n\n\\end{defi}\n\nGiven a saturated fusion system $\\mathcal{F}$ over a discrete $p$-toral group $S$, we denote by $\\mathcal{F}^c$ and $\\mathcal{F}^r$ the full subcategories of $\\mathcal{F}$ with object sets the collections of $\\mathcal{F}$-centric and $\\mathcal{F}$-radical subgroups, respectively. We also write $\\mathcal{F}^{cr} \\subseteq \\mathcal{F}$ for the full subcategory of $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups.\n\nProving that a given fusion system is saturated is a rather difficult task, even when the fusion system is finite, but there are some techniques that may be helpful. One of these techniques, which we will use in later sections, is \\cite[Theorem A]{BCGLO1}, restated as Theorem \\ref{5A} below.\n\n\\begin{defi}\n\nLet $\\mathcal{F}$ be a fusion system over a finite $p$-group $S$, and let $\\mathcal{H} \\subseteq \\mathrm{Ob}(\\mathcal{F})$ be a subset of objects.\n\\begin{itemize}\n\n\\item $\\mathcal{F}$ is \\emph{$\\mathcal{H}$-generated} if every morphism in $\\mathcal{F}$ can be described as a composite of restrictions of morphisms in $\\mathcal{F}$ between subgroups in $\\mathcal{H}$.\n\n\\item $\\mathcal{F}$ is \\emph{$\\mathcal{H}$-saturated} if the saturation axioms hold for all subgroups in the set $\\mathcal{H}$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{thm}\\label{5A}\n\nLet $\\mathcal{F}$ be a fusion system over a finite $p$-group $S$, and let $\\mathcal{H}$ be a subset of objects of $\\mathcal{F}$ closed under $\\mathcal{F}$-conjugacy and such that $\\mathcal{F}$ is $\\mathcal{H}$-generated and $\\mathcal{H}$-saturated. Suppose further that, for each $\\mathcal{F}$-centric subgroup $P \\notin \\mathcal{H}$, $P$ is $\\mathcal{F}$-conjugate to some $Q$ such that\n$$\n\\mathrm{Out}_S(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}}(Q)) \\neq \\{1\\}.\n$$\nThen $\\mathcal{F}$ is saturated.\n\n\\end{thm}\n\nThe following result will be useful in later sections when checking the condition displayed in the previous theorem. We state it in full generality since in fact it will apply in different situations throughout this paper.\n\n\\begin{lmm}\\label{Kpgp}\n\nLet $\\mathcal{F}$ be a fusion system over a discrete $p$-toral group $S$. Let also $P\\leq S$ be a subgroup, and let $P_0 \\lhd P$ be a normal subgroup such that $f|_{P_0} \\in \\mathrm{Aut}_{\\mathcal{F}}(P_0)$ for all $f \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$. Set\n$$\nK_P \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(P_0) \\times \\mathrm{Aut}(P\/P_0)).\n$$\nThen, $K_P\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}}(P))$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nBy definition $K_P$ is normal in $\\mathrm{Aut}_{\\mathcal{F}}(P)$, and thus we only have to show that $K_P$ is a discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ (that is, every element of $K_P$ has order a power of $p$). This in turn follows by \\cite[Theorem 3.2]{Gor}: although the result in \\cite{Gor} is stated for finite groups, the arguments in its proof apply here without modification, since $\\mathrm{Aut}_{\\mathcal{F}}(P)$ is a locally finite group.\n\\end{proof}\n\nThe concept of transporter system associated to a fusion system was first introduced in \\cite{OV} for fusion systems over finite $p$-groups, and then extended to discrete $p$-toral groups in \\cite{BLO6}, with centric linking systems as a particular case. We refer the reader to the aforementioned sources for further details.\n\nLet $G$ be a group and let $\\mathcal{H}$ be a set of subgroups of $G$ that is closed by overgroups, i.e. if $H \\in \\mathcal{H}$ and $K \\geq H$, then $K \\in \\mathcal{H}$, and closed by conjugation in $G$, i.e. if $H \\in \\mathcal{H}$ and $g \\in G$, then $gHg^{-1} \\in \\mathcal{H}$. The transporter category of $G$ with respect to $\\mathcal{H}$ is the category $\\mathcal{T}_{\\mathcal{H}}(G)$ whose object set is $\\mathcal{H}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{T}_{\\mathcal{H}}(G)}(P,Q) = \\{x \\in G \\,\\, | \\,\\, x \\cdot P \\cdot x^{-1}\\leq Q\\}\n$$\nfor each $P,Q \\in \\mathcal{H}$.\n\n\\begin{defi}\\label{defitransporter}\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a fusion system over $S$. A \\emph{transporter system} associated to $\\mathcal{F}$ is a nonempty category $\\mathcal{T}$ whose object set $\\mathrm{Ob}(\\mathcal{T})$ is a subset of $\\mathrm{Ob}(\\mathcal{F})$ that is closed by overgroups and conjugation in $\\mathcal{F}$, together with functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{T})}(S) \\Right4{\\varepsilon} \\mathcal{T} \\qquad \\mbox{and} \\qquad \\mathcal{T} \\Right4{\\rho} \\mathcal{F}\n$$\nsatisfying the following conditions.\n\\begin{itemize}\n\n\\item[(A1)] The functor $\\varepsilon$ is the identity on objects and an inclusion on morphism sets, and the functor $\\rho$ is the inclusion on objects and a surjection on morphism sets.\n\n\\item[(A2)] For each $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, the set $\\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ has a free action of\n$$\nE(P) \\stackrel{def} = \\mathrm{Ker} \\big[\\rho_P \\colon \\mathrm{Aut}_{\\mathcal{T}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(P) \\big]\n$$\nby right composition, and $\\rho_{P,Q}$ is the orbit map of this action. Also, $E(Q)$ acts freely on $\\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ by left composition.\n\n\\item[(B)] Let $P,Q \\in \\mathrm{Ob}(\\mathcal{T})$. Then, the map $\\varepsilon_{P,Q} \\colon N_S(P,Q) \\to \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ is injective, and\n$$\n(\\rho_{P,Q} \\circ \\varepsilon_{P,Q})(g) = c_g \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)\n$$\nfor all $g \\in \\mathrm{Mor}_{\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{T})}(S)}(P, Q) = N_S(P, Q)$.\n\n\\item[(C)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, for all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$, and for all $g \\in P$, the following is a commutative diagram in $\\mathcal{T}$.\n$$\n\\xymatrix{\nP \\ar[r]^{\\varphi} \\ar[d]_{\\varepsilon_P(g)} & Q \\ar[d]^{\\varepsilon_Q(\\rho(\\varphi)(g))} \\\\\nP \\ar[r]_{\\varphi} & Q\n}\n$$\n\n\\item[(I)] Each isomorphism class of objects in $\\mathrm{Ob}(\\mathcal{T})$ contains an element $P$ such that\n$$\n\\varepsilon_P(N_S(P)) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(P));\n$$\nor, in other words, such that $\\varepsilon(N_S(P))$ has finite index prime to $p$ in $\\mathrm{Aut}_{\\mathcal{T}}(P)$.\n\n\\item[(II)] Let $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ be isomorphic objects, and let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{T}}(P,Q)$. Let also $\\4{P}\\leq N_S(P)$ and $\\4{Q}\\leq N_S(Q)$ be such that $\\varphi \\circ \\varepsilon_P(\\4{P}) \\circ \\varphi^{-1}\\leq \\varepsilon_Q(\\4{Q})$. Then there is some morphism $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{T}}(\\4{P}, \\4{Q})$ such that\n$$\n\\4{\\varphi} \\circ \\varepsilon_{P, \\4{P}}(1) = \\varepsilon_{Q, \\4{Q}}(1) \\circ \\varphi.\n$$\n\n\\item[(III)] Let $P_1\\leq P_2\\leq P_3\\leq \\ldots$ be a sequence in $\\mathrm{Ob}(\\mathcal{T})$, and let $\\varphi_n \\in \\mathrm{Mor}_{\\mathcal{T}}(P_n,S)$ be such that $\\varphi_n = \\varphi_{n+1} \\circ \\varepsilon_{P_n, P_{n+1}}(1)$ for all $n \\geq 1$. Then, upon setting $P = \\bigcup_{n \\geq 1} P_n$, there is a morphism $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,S)$ such that $\\varphi_n = \\varphi \\circ \\varepsilon_{P_n,P}(1)$ for all $n \\geq 1$.\n\n\\end{itemize}\nThe \\emph{rank of $\\mathcal{T}$} is the rank of $S$. A \\emph{centric linking system} associated to a saturated fusion system $\\mathcal{F}$ is a transporter system $\\mathcal{L}$ such that $\\mathrm{Ob}(\\mathcal{L})$ is the collection of all $\\mathcal{F}$-centric subgroups of $S$ and $E(P) = \\varepsilon(Z(P))$ for all $P \\in \\mathrm{Ob}(\\mathcal{L})$.\n\n\\end{defi}\n\n\\begin{rmk}\\label{rmktransp}\n\nThe above definition of centric linking system is taken from \\cite{BLO6}, and it is seen in \\cite[Corollary A.5]{BLO6} to coincide with the original \\cite[Definition 4.1]{BLO3}. Notice also that axiom (I) above differs from the corresponding axiom for the finite case, see \\cite[Definition 3.1]{OV}, in that condition (I) above seems to be more restrictive than the corresponding condition in \\cite{OV}:\n\\begin{itemize}\n\n\\item[(I')] $\\varepsilon_{S,S}(S) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(S))$.\n\n\\end{itemize}\nHowever, \\cite[Proposition 3.4]{OV} implies that both definitions, \\cite[Definition 3.1]{OV} and the above, agree in the finite case.\n\n\\end{rmk}\n\n\\begin{lmm}\\label{epimono}\n\nIn a transporter system, all morphisms are monomorphisms and epimorphisms in the categorical sense.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is \\cite[Proposition A.2 (d)]{BLO6}.\n\\end{proof}\n\n\\begin{defi}\n\nA \\emph{$p$-local compact group} is a triple $\\mathcal{G} = (S, \\FF, \\LL)$ formed by a discrete $p$-toral group $S$, a saturated fusion system $\\mathcal{F}$ over $S$, and a centric linking system $\\mathcal{L}$ associated to $\\mathcal{F}$. The \\emph{classifying space} of a $p$-local compact group $\\mathcal{G}$ is the $p$-completed nerve of $\\mathcal{L}$, denoted by $B\\mathcal{G} = |\\mathcal{L}|^{\\wedge}_p$. The \\emph{rank of $\\mathcal{G}$} is the rank of $S$.\n\n\\end{defi}\n\nGeneralizing work of \\cite{Chermak} and \\cite{Oliver}, it is proved in \\cite{Levi-Libman} that every saturated fusion system over a discrete $p$-toral group has an associated centric linking system which is unique up to isomorphism. Thus, from now on we speak of \\emph{the} associated centric linking system for a given saturated fusion system. \n\nFinally, we recall the ``bullet construction'' on a $p$-local compact group.\n\n\\begin{defi}\\label{defibullet}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$. Let also $T\\leq S$ be the maximal torus, and let $W = \\mathrm{Aut}_{\\mathcal{F}}(T)$. Set the following\n\n\\begin{enumerate}[(i)]\n\n\\item The exponent of $S\/T$ is $e = \\exp(S\/T) = \\min\\{k \\in \\mathbb{N} \\, | \\, x^{p^k} \\in T \\mbox{ for all } x \\in S\\}$.\n\n\\item For each $P\\leq T$, let $I(P) = \\{t \\in T \\, | \\, \\omega(t) = t \\mbox{ for all } \\omega \\in W \\mbox{ such that } \\omega|_P = \\mathrm{Id}_P\\}$, and let $I(P)_0$ denote its maximal torus.\n\n\\item For each $P\\leq S$, set $P^{[e]} = \\{x^{p^e} \\, | \\, x \\in P\\}\\leq T$, and set\n$$\nP^{\\bullet} = P \\cdot I(P^{[e]})_0 = \\{xt \\, | \\, x \\in P, \\, t \\in I(P^{[e]})_0\\}.\n$$\n\n\\item Let $\\mathcal{F}^{\\bullet}$ be the full subcategory of $\\mathcal{F}$ with object set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) = \\{P^{\\bullet} \\, | \\, P\\leq S\\}$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nThe following summarizes the main properties of the ``bullet construction''.\n\n\\begin{prop}\\label{3.2BLO3}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for each $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$ and each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ there is a unique $f^{\\bullet} \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\bullet}, Q^{\\bullet})$ whose restriction to $P$ is $f$. This way, the ``bullet construction'' makes $P \\mapsto P^{\\bullet}$ into a functor $(-)^{\\bullet} \\colon \\mathcal{F} \\to \\mathcal{F}$ that satisfies the following properties.\n\\begin{enumerate}[(i)]\n\n\\item The set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) = \\{P^{\\bullet} \\,  | \\, P\\leq S\\}$ contains finitely many $S$-conjugacy classes of subgroups of $S$.\n\n\\item For all $P\\leq S$, $(P^{\\bullet})^{\\bullet} = P^{\\bullet}$.\n\n\\item If $P\\leq Q\\leq S$, then $P^{\\bullet}\\leq Q^{\\bullet}$.\n\n\\item For all $P, Q\\leq S$, $N_S(P, Q) \\subseteq N_S(P^{\\bullet}, Q^{\\bullet})$.\n\n\\item For all $P\\leq S$, $C_S(P) = C_S(P^{\\bullet})$.\n\n\\item The functor $(-)^{\\bullet}$ is a left adjoint to the inclusion of $\\mathcal{F}^{\\bullet}$ as a full subcategory of $\\mathcal{F}$.\n\n\\item All $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ are in $\\mathcal{F}^{\\bullet}$. In particular, there are only finitely many $\\mathcal{F}$-conjugacy classes of such subgroups.\n\n\\end{enumerate}\nMoreover, if we denote by $\\mathcal{L}^{\\bullet} \\subseteq \\mathcal{L}$ the full subcategory with $\\mathrm{Ob}(\\mathcal{L}^{\\bullet}) = \\{P^{\\bullet} \\, | \\, P \\in \\mathrm{Ob}(\\mathcal{L})\\}$, then there is a unique functor $(-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{L}^{\\bullet}$ such that the following holds.\n\\begin{enumerate}[(a)]\n\n\\item $(-)^{\\bullet} \\circ \\rho = \\rho \\circ (-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{F}$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\mathcal{L})$ and all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(P,Q)$, we have $\\varepsilon_{Q, Q^{\\bullet}}(1) \\circ \\varphi = \\varphi^{\\bullet} \\circ \\varepsilon_{P, P^{\\bullet}}(1)$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\mathcal{L})$ and all $g \\in N_S(P,Q)$, we have $\\varepsilon_{P,Q}(g)^{\\bullet} = \\varepsilon_{P^{\\bullet}, Q^{\\bullet}}(g)$.\n\n\\item The functor $(-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{L}$ is left adjoint to the inclusion of $\\mathcal{L}^{\\bullet}$ as a full subcategory of $\\mathcal{L}$. In particular, the inclusion $\\mathcal{L}^{\\bullet} \\subseteq \\mathcal{L}$ induces an equivalence $|\\mathcal{L}^{\\bullet}|\\simeq |\\mathcal{L}|$.\n\n\\end{enumerate}\n\n\\end{prop}\n\n\\begin{proof}\n\nThe first part of the statement corresponds to \\cite[Proposition 3.3]{BLO3}. Parts (i), (ii) and (iii)  correspond to \\cite[Lemma 3.2 (a), (b) and (c)]{BLO3} respectively. Part (iv) is an easy variation of \\cite[Lemma 3.2 (b)]{BLO3} (details are left to the reader). For part (v), let $P\\leq S$. Since $P\\leq P^{\\bullet}$, we have $C_S(P) \\geq C_S(P^{\\bullet})$. Let $x \\in C_S(P)$. By (iv), we have $x \\in N_S(P^{\\bullet})$. Since $c_x = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$ extends uniquely to $c_x = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P^{\\bullet})$, it follows that $x \\in C_S(P^{\\bullet})$. Part (vi) corresponds to \\cite[Corollary 3.4]{BLO3}, and part (vii) corresponds to \\cite[Corollary 3.5]{BLO3}. The last part of the statement, including parts (a), (b) and (c) corresponds to \\cite[Proposition 1.12]{JLL}. Part (d) corresponds to \\cite[Proposition 4.5 (a)]{BLO3}.\n\\end{proof}\n\n\n\\subsection{Isotypical equivalences and unstable Adams operations}\\label{Ssisotyp}\n\nIn this subsection we review the concept of isotypical equivalence, with particular interest on the unstable Adams operations for $p$-local compact groups originally introduced in \\cite{JLL}.\n\n\\begin{defi}\n\nLet $(\\mathcal{T}, \\varepsilon, \\rho)$ be a transporter system associated to a fusion system $\\mathcal{F}$. An automorphism $\\Psi \\colon \\mathcal{T} \\to \\mathcal{T}$ is \\emph{isotypical} if $\\Psi(\\varepsilon_P(P)) = \\varepsilon_{\\Psi(P)}(\\Psi(P))$ for each $P \\in \\mathrm{Ob}(\\mathcal{T})$.\n\n\\end{defi}\n\nWe denote by $\\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ the group of isotypical automorphisms $\\Psi$ of $\\mathcal{T}$ which in addition satisfy $\\Psi(\\varepsilon_{P,Q}(1)) = \\varepsilon_{\\Psi(P), \\Psi(Q)}(1)$ whenever $P\\leq Q$. Notice that if $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{T})$, then $\\Psi$ induces an automorphism of $S$ by restricting to the object $S \\in \\mathrm{Ob}(\\mathcal{T})$. By abuse of notation, we will denote the induced automorphism by $\\Psi \\in \\mathrm{Aut}(S)$.\n\nNext we review the concept of unstable Adams operations for $p$-local compact groups. Our definition corresponds to the definition of \\emph{normal Adams operation} in \\cite[Definition 3.3]{JLL}, conveniently adapted to our notation. By $(\\Z^\\wedge_p)^{\\times}$ we denote the subgroup of multiplicative units in the ring of $p$-adic integers $\\Z^\\wedge_p$.\n\n\\begin{defi}\\label{uAo}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. An \\emph{unstable Adams operation of degree $\\zeta \\in (\\Z^\\wedge_p)^{\\times}$} on $\\mathcal{G}$ is an isotypical equivalence $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L})$ such that the induced automorphism $\\Psi \\in \\mathrm{Aut}(S)$ satisfies\n\\begin{enumerate}[(i)]\n\n\\item the restriction of $\\Psi$ to the maximal torus $T\\leq S$ is the $\\zeta$-power automorphism; and\n\n\\item $\\Psi$ induces the identity on $S\/T$.\n\n\\end{enumerate}\nAn unstable Adams operation is \\emph{fine} if its degree is $\\zeta \\neq 1$, with $\\zeta$ congruent to $1$ modulo $p$.\n\n\\end{defi}\n\nAs proved in \\cite[Theorem 4.1]{JLL}, unstable Adams operations exist for all $p$-local compact groups, and in particular this applies to the existence of fine unstable Adams operations.\n\n\\begin{thm}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for some large enough $m \\in \\mathbb{N}$, $\\mathcal{G}$ has unstable Adams operations of degree $\\zeta$, for each $\\zeta \\in 1 + p^m\\Z^\\wedge_p$.\n\n\\end{thm}\n\n\\begin{rmk}\\label{uAo1}\n\nRoughly speaking, the construction of unstable Adams operations in \\cite{JLL} is done by defining $\\Psi$ to fix enough objects and morphisms in $\\mathcal{L}$. More specifically, $\\Psi$ fixes\n\\begin{enumerate}[(a)]\n\n\\item a set $\\mathcal{H}$ of representatives of the $S$-conjugacy classes $\\mathrm{Ob}(\\mathcal{L}^{\\bullet})$; and\n\n\\item for each $P \\in \\mathcal{H}$, a set of representatives $\\mathcal{M}_P \\subseteq \\mathrm{Aut}_{\\mathcal{L}}(P)$ of the classes in $\\mathrm{Aut}_{\\mathcal{L}}(P)\/P \\cong \\mathrm{Out}_{\\mathcal{F}}(P)$.\n\n\\end{enumerate}\nThis properties will be crucial in our constructions in Section \\ref{Sfam}.\n\n\\end{rmk}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$ (not necessarily saturated), and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. Let also $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{T})$ be an isotypical automorphism. Set also\n\\begin{equation}\\label{fixS}\nC_S(\\Psi) = \\{g \\in S \\, | \\, \\Psi(\\varepsilon_S(g)) = \\varepsilon_S(g)\\}\\leq S,\n\\end{equation}\nthe subgroup of fixed points of $S$ by $\\Psi$. The following result is the main tool in detecting objects and morphisms in $\\mathcal{T}$ that are invariant by $\\Psi$.\n\n\\begin{lmm}\\label{invar1}\n\nThe following holds.\n\\begin{enumerate}[(i)]\n\n\\item Let $P\\leq C_S(\\Psi)$, and let $Q \\in P^S$. Then, $Q\\leq C_S(\\Psi)$ if and only if, for some $x \\in N_S(Q,P)$,\n$$\nx^{-1} \\cdot \\Psi(x) \\in C_S(Q).\n$$\n\n\\item Let $P, P' ,Q, Q'\\leq C_S(\\Psi)$ be such that $P' \\in P^S$ and $Q' \\in Q^S$, and suppose $P, P', Q, Q' \\in \\mathrm{Ob}(\\mathcal{L})$. Let also $x \\in N_S(P',P)$ and $y \\in N_S(Q', Q)$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ be such that $\\Psi(\\varphi) = \\varphi$. Set $\\varphi' = \\varepsilon(y^{-1}) \\circ \\varphi \\circ \\varepsilon(x) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P', Q')$. Then, $\\Psi(\\varphi') = \\varphi'$ if and only if\n$$\n\\varepsilon(y^{-1} \\cdot \\Psi(y)) \\circ \\varphi' = \\varphi' \\circ \\varepsilon(x^{-1} \\cdot \\Psi(x)).\n$$\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nFor part (i), let $P\\leq C_S(\\Psi)$, and let $Q \\in P^S$. Let also $g \\in Q$ and $x \\in N_S(Q,P)$, and set $h = x \\cdot g \\cdot x^{-1} \\in P$. Since $P\\leq C_S(\\Psi)$, we get\n$$\nx \\cdot g \\cdot x^{-1} = h = \\Psi(h) = \\Psi(x \\cdot g \\cdot x^{-1}) = \\Psi(x) \\cdot \\Psi(g) \\cdot \\Psi(x)^{-1}.\n$$\nThus, if $Q\\leq C_S(\\Psi)$ then clearly $x^{-1} \\cdot \\Psi(x) \\in C_S(Q)$, and conversely if $x^{-1} \\cdot \\Psi(x) \\in C_S(Q)$ then $h \\in C_S(\\Psi)$. Since the argument works for any $g \\in Q$ and any $x \\in N_S(Q, P)$, part (i) follows.\n\nFor part (ii), let $P, P' ,Q, Q'\\leq C_S(\\Psi)$, with $P' \\in P^S$ and $Q' \\in Q^S$. Let also $x \\in N_S(P',P)$ and $y \\in N_S(Q', Q)$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ be such that $\\Psi(\\varphi) = \\varphi$, with $\\varphi' = \\varepsilon(y^{-1}) \\circ \\varphi \\circ \\varepsilon(x) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P', Q')$. We have\n$$\n\\varepsilon(y) \\circ \\varphi' \\circ \\varepsilon(x^{-1}) = \\varphi = \\Psi(\\varphi) = \\Psi(\\varepsilon(y) \\circ \\varphi' \\circ \\varepsilon(x^{-1})) = \\varepsilon(\\Psi(y)) \\circ \\Psi(\\varphi') \\circ \\varepsilon(\\Psi(x)^{-1}),\n$$\nand (ii) follows easily.\n\\end{proof}\n\n\n\\subsection{Normalizers, centralizers, and related constructions}\\label{Squotient}\n\nIn this subsection we review the construction of the centralizer and normalizer $p$-local compact subgroups for a given $p$-local compact group. The main references here are \\cite[Appendix A]{BLO2} and \\cite[Section 2]{BLO6}. For the rest of this subsection, fix a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$, a subgroup $A\\leq S$, and a subgroup $K\\leq \\mathrm{Aut}(A)$, and define the following:\n\\begin{itemize}\n\n\\item $\\mathrm{Aut}_{\\mathcal{F}}^K(A) = K \\cap \\mathrm{Aut}_{\\mathcal{F}}(A)$;\n\n\\item $\\mathrm{Aut}_S^K(A) = K \\cap \\mathrm{Aut}_S(A)$; and\n\n\\item $N_S^K(A) = \\{x \\in N_S(A) \\, | \\, c_x \\in K\\}$.\n\n\\end{itemize}\nThe subgroup $A$ is \\emph{fully $K$-normalized in $\\mathcal{F}$} if we have $|N_S^K(A)| \\geq |N_S^{^{f}K}(f(A))|$ for each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(A, S)$, where $^{f}K = \\{f \\gamma f^{-1} \\, | \\, \\gamma \\in K\\}\\leq \\mathrm{Aut}(f(A))$.\n\n\\begin{defi}\\label{definorm}\n\nThe \\emph{$K$-normalizer fusion system of $A$ in $\\mathcal{F}$}, is the fusion system $N_{\\mathcal{F}}^K(A)$ over $N_S^K(A)$ with morphism sets\n$$\n\\begin{aligned}\n\\mathrm{Hom}_{N_{\\mathcal{F}}^K(A)}&(P,Q) = \\\\\n & = \\{f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\,\\, | \\,\\, \\exists \\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(PA, QA) \\mbox{ with } \\4{f}|_P = f \\mbox{ and } \\4{f}|_A \\in K\\}\n\\end{aligned}\n$$\nfor each $P, Q\\leq N_S^K(A)$.\n\n\\end{defi}\n\nBy \\cite[Theorem 2.3]{BLO6} we know that $N_{\\mathcal{F}}^K(A)$ is a saturated fusion system whenever $A$ is fully $K$-normalized in $\\mathcal{F}$. For this reason, for the rest of this subsection we assume that $A$ satisfies this property.\n\n\\begin{lmm}\\label{centricNFKA}\n\nIf $P\\leq N_S^K(A)$ is $N_{\\mathcal{F}}^K(A)$-centric, then $P \\cdot A$ is $\\mathcal{F}$-centric.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $P\\leq N_S^K(A)$ be $N_{\\mathcal{F}}^K(A)$-centric. We have to check that, for each $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(P \\cdot A, S)$, there is an inclusion $C_S(\\gamma(P \\cdot A))\\leq \\gamma(P \\cdot A)$. We can apply \\cite[Proposition A.2]{BLO2}, since the proof in \\cite{BLO2} works without modifications in the compact setup, and it follows that the subgroup $A$ is fully centralized in $\\mathcal{F}$, and there is some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S^{^{\\gamma}K}(\\gamma(A)) \\cdot \\gamma(A),S)$ such that $(f \\circ \\gamma)|_A \\in K$. Thus $f \\circ \\gamma$ is a morphism in $N_{\\mathcal{F}}^K(A)$.\n\nNote that $C_S(\\gamma(P \\cdot A))\\leq C_S(\\gamma(A))\\leq N_S^{^{\\gamma} K}(\\gamma(A))$, and we have inclusions\n$$\n\\begin{aligned}\nf(C_S(\\gamma(P \\cdot A))) &\\leq C_S((f \\circ \\gamma)(P \\cdot A)) = C_S((f \\circ \\gamma)(P) \\cdot A)\\leq \\\\\n &\\leq C_S((f \\circ \\gamma)(P)) \\cap C_S(A)\\leq C_S((f \\circ \\gamma)(P)) \\cap N_S^K(A)\\leq (f \\circ \\gamma)(P),\n\\end{aligned}\n$$\nwhere the last inequality holds since $P \\in N_{\\mathcal{F}}^K(A)^c$. Thus,\n$$\nC_S(\\gamma(P \\cdot A))\\leq \\gamma(P)\\leq \\gamma(P) \\cdot \\gamma(A) = \\gamma(P \\cdot A),\n$$\nand this proves that $P \\cdot A \\in \\mathcal{F}^c$.\n\\end{proof}\n\nIn view of the above, we can now define $N_{\\mathcal{L}}^K(A)$ as the category with objects the set of $N_{\\mathcal{F}}^K(A)$-centric subgroups of $N_S^K(A)$ and with morphism sets\n$$\n\\mathrm{Mor}_{N_{\\mathcal{L}}^K(A)}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(PA, QA) \\,\\, | \\,\\, \\rho(\\varphi)|_P \\in \\mathrm{Hom}_{N_{\\mathcal{F}}^K(A)}(P,Q) \\mbox{ and } \\rho(\\varphi)|_A \\in K\\}.\n$$\nIn general, $N_{\\mathcal{L}}^K(A)$ need not be a transporter system associated to $N_{\\mathcal{F}}^K(A)$, but there are two particular situations where this is indeed the case.\n\n\\begin{lmm}\n\nIf either $K = \\{\\mathrm{Id}\\}$ or $K = \\mathrm{Aut}(A)$, then the category $N_{\\mathcal{L}}^K(A)$ is a centric linking system associated to $N_{\\mathcal{F}}^K(A)$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThe case $K = \\{\\mathrm{Id}\\}$ corresponds to \\cite[Proposition 2.5]{BLO2} in the finite case, while the case $K = \\mathrm{Aut}(A)$ corresponds to \\cite[Lemma 6.2]{BLO2} for $p$-local finite groups. In both situations, the proof for $p$-local finite groups applies here without modification to show that $N_{\\mathcal{L}}^{\\{\\mathrm{Id}\\}}(A)$ satisfies all the condition of a centric linking system, except perhaps axiom (III), which is easily checked.\n\\end{proof}\n\n\\begin{defi}\\label{rmknorm}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $A\\leq S$.\n\\begin{enumerate}[(a)]\n\n\\item If $A$ is fully $\\mathcal{F}$-centralized, the \\emph{centralizer $p$-local compact group of $A$ in $\\mathcal{G}$} is the triple\n$$\nC_{\\mathcal{G}}(A) = (C_S(A), C_{\\mathcal{F}}(A), C_{\\mathcal{L}}(A)) \\stackrel{def} = (N_S^{\\{\\mathrm{Id}\\}}(A), N_{\\mathcal{F}}^{\\{\\mathrm{Id}\\}}(A), N_{\\mathcal{L}}^{\\{\\mathrm{Id}\\}}(A)).\n$$\n\n\\item If $A$ is fully $\\mathcal{F}$-normalized, the \\emph{normalizer $p$-local compact group of $A$ in $\\mathcal{G}$} is the triple\n$$\nN_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A)) \\stackrel{def} = (N_S^{\\mathrm{Aut}(A)}(A), N_{\\mathcal{F}}^{\\mathrm{Aut}(A)}(A), N_{\\mathcal{L}}^{\\mathrm{Aut}(A)}(A)).\n$$\n\n\\end{enumerate}\nA subgroup $A\\leq S$ is called \\emph{central in $\\mathcal{F}$} if $C_{\\mathcal{G}}(A) = \\mathcal{G}$. Similarly, $A\\leq S$ is called \\emph{normal in $\\mathcal{F}$} if $N_{\\mathcal{G}}(A) = \\mathcal{G}$. Clearly, if $A\\leq S$ is central in $\\mathcal{F}$ then in particular it is normal in $\\mathcal{F}$.\n\n\\end{defi}\n\n\\begin{lmm}\\label{central1}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $P\\leq S$. Then $P$ is fully $\\mathcal{F}$-centralized if and only if $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized. Furthermore, if this is the case then $C_{\\mathcal{G}}(P) = C_{\\mathcal{G}}(P^{\\bullet})$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nSuppose first that $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized. By Proposition \\ref{3.2BLO3} (v), we have $C_S(P) = C_S(P^{\\bullet})$. If $Q \\in P^{\\mathcal{F}}$, then $Q^{\\bullet} \\in (P^{\\bullet})^{\\mathcal{F}}$ by Proposition \\ref{3.2BLO3}, and we have\n$$\n|C_S(Q)| = |C_S(Q^{\\bullet})|\\leq |C_S(P^{\\bullet})| = |C_S(P)|,\n$$\nwhich implies that $P$ is fully $\\mathcal{F}$-centralized.\n\nConversely, suppose that $P$ is fully $\\mathcal{F}$-centralized, and let $R \\in (P^{\\bullet})^{\\mathcal{F}}$ be fully $\\mathcal{F}$-centralized. Choose some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\bullet} C_S(P^{\\bullet}), S)$ such that $\\gamma(P^{\\bullet}) = R$, and set $Q = \\gamma(P)$, with $Q^{\\bullet} = \\gamma(P^{\\bullet}) = R$. By Proposition \\ref{3.2BLO3} (v), we have $C_S(P) = C_S(P^{\\bullet})$, and thus\n$$\n\\gamma(C_S(P^{\\bullet})) = \\gamma(C_S(P)) = C_S(Q) = C_S(Q^{\\bullet}) = C_S(R),\n$$\nwhere the leftmost and rightmost equalities hold by Proposition \\ref{3.2BLO3} (v), and the equality in the middle holds since $P$ is fully $\\mathcal{F}$-centralized. It follows that $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized.\n\nTo finish the proof, suppose that $P$ and $P^{\\bullet}$ are fully $\\mathcal{F}$-centralized, and consider $C_{\\mathcal{G}}(P)$ and $C_{\\mathcal{G}}(P^{\\bullet})$, which are $p$-local compact groups with Sylow $C_S(P) = C_S(P^{\\bullet})$. By definition, it is enough to show that $C_{\\mathcal{F}}(P) = C_{\\mathcal{F}}(P^{\\bullet})$. Notice that there is an obvious inclusion $C_{\\mathcal{F}}(P^{\\bullet}) \\subseteq C_{\\mathcal{F}}(P)$. Let $Q, R\\leq C_S(P)$, and let $f \\in \\mathrm{Hom}_{C_{\\mathcal{F}}(P)}(Q,R)$. By definition of $C_{\\mathcal{F}}(P)$, there is some $\\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(QP, RP)$ such that $\\4{f}|_Q = f$ and $\\4{f}|_P = \\mathrm{Id}$. Let $\\gamma = \\4{f}$, and consider $\\gamma^{\\bullet} \\in \\mathrm{Hom}_{\\mathcal{F}}((QP)^{\\bullet}, (RP)^{\\bullet})$. Then $\\gamma^{\\bullet}$ restricts to a morphism $\\omega \\in \\mathrm{Hom}_{\\mathcal{F}}(QP^{\\bullet}, RP^{\\bullet})$. Furthermore, by definition of $\\omega$, we have\n$$\n\\omega|_Q = \\gamma^{\\bullet}|Q = \\gamma|Q = f \\qquad \\mbox{and} \\qquad \\omega|_{P^{\\bullet}} = \\gamma^{\\bullet}|_{P^{\\bullet}} = (f|_P)^{\\bullet} = \\mathrm{Id}.\n$$\nThus $f$ is a morphism in $C_{\\mathcal{F}}(P^{\\bullet})$, and $C_{\\mathcal{F}}(P) = C_{\\mathcal{F}}(P^{\\bullet})$.\n\\end{proof}\n\n\\begin{cor}\\label{central2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for each $P\\leq S$ which is fully $\\mathcal{F}$-centralized, there is a sequence of finite subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$ and such that the following conditions hold for all $n \\geq 0$.\n\\begin{enumerate}[(i)]\n\n\\item $P_n$ is fully $\\mathcal{F}$-centralized and $P_n^{\\bullet} = P^{\\bullet}$.\n\n\\item $C_{\\mathcal{G}}(P_n) = C_{\\mathcal{G}}(P)$.\n\n\\end{enumerate}\n\n\\end{cor}\n\n\\begin{proof}\n\nLet $P\\leq S$. Since $S$ is locally finite, so is $P$, and we can find some sequence of finite subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$. Furthermore, there is some $M \\in \\mathbb{N}$ such that $P_n^{\\bullet} = P^{\\bullet}$ for all $n \\geq M$, and we may assume for simplicity that $M = 0$. By Lemma \\ref{central1}, $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized, and then so is $P_n$, for all $n \\geq 0$. Furthermore,\n$$\nC_{\\mathcal{F}}(P_n) = C_{\\mathcal{F}}(P_n^{\\bullet}) = C_{\\mathcal{F}}(P^{\\bullet}) = C_{\\mathcal{F}}(P),\n$$\nand this finishes the proof.\n\\end{proof}\n\nTo finish this section, we recall the construction of the quotient of a transporter system by a $p$-group. This quotient was already explored in \\cite[Appendix A]{Gonza2}, and here we only recall the necessary definitions. Let $(\\mathcal{T}, \\varepsilon, \\rho)$ be a transporter system associated to a fusion system $\\mathcal{F}$, and let $A\\leq S$ be a normal subgroup in $\\mathcal{F}$. If $P, Q\\leq S$ are such that $A\\leq P,Q$, then each morphism $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ restricts to an automorphism of $A$, and hence it also induces a homomorphism $\\mathrm{ind}(f) \\colon P\/A \\to Q\/A$. For a subgroup $P\/A\\leq S\/A$, we will denote by $P\\leq S$ the unique subgroup of $S$ that contains $A$ with image $P\/A$ through the projection $S \\to S\/A$.\n\n\\begin{defi}\\label{quotient1}\n\nLet $A$ is a normal subgroup in $\\mathcal{F}$. The \\emph{quotient} of $\\mathcal{T}$ by $A$ is the transporter system $(\\mathcal{T}\/A, \\3{\\varepsilon}, \\3{\\rho})$ associated to the fusion system $\\mathcal{F}\/A$, where\n\\begin{itemize}\n\n\\item $\\mathcal{F}\/A$ is the fusion system over $S\/A$ with morphism sets\n$$\n\\begin{aligned}\n\\mathrm{Hom}_{\\mathcal{F}\/A}&(P\/A, Q\/A) = \\\\\n & = \\{\\overline{f} \\in \\mathrm{Hom}(P\/A,Q\/A) \\,\\, | \\,\\, \\exists f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\mbox{ such that } \\overline{f} = \\mathrm{ind}(f)\\}.\n\\end{aligned}\n$$\n\n\\item $\\mathcal{T}\/A$ is the category with object set $\\{P\/A\\leq S\/A \\, | \\, A\\leq P \\in \\mathrm{Ob}(\\mathcal{T})\\}$ and morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{T}\/A}(P\/A,Q\/A) = \\mathrm{Mor}_{\\mathcal{T}}(P,Q)\/\\varepsilon_P(A).\n$$\nThe structural functors $\\3{\\varepsilon}$ and $\\3{\\rho}$ are induced, respectively, by the structural functors $\\varepsilon$ and $\\rho$ of $\\mathcal{T}$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{rmk}\\label{quotient21}\n\nBy \\cite[Proposition A.2]{Gonza2}, $(\\mathcal{T}\/A, \\3{\\varepsilon}, \\3{\\rho})$ is a transporter system associated to $\\mathcal{F}\/A$. \n\n\\end{rmk}\n\n\\begin{lmm}\\label{quotient22}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a saturated fusion system over $S$, and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$, such that $\\mathrm{Ob}(\\mathcal{T})$ contains all the centric subgroups of $\\mathcal{F}$. Let also $A\\leq S$ be normal in $\\mathcal{F}$. Then the following holds:\n\\begin{enumerate}[(i)]\n\n\\item the fusion system $\\mathcal{F}\/A$ is saturated; and\n\n\\item the transporter system $\\mathcal{T}\/A$ contains all the $\\mathcal{F}\/A$-centric subgroups of $S\/A$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nPart (i) corresponds to \\cite[Proposition A.3]{Gonza2}, and part (ii) is easily checked: let $P\/A\\leq S\/A$ be $\\mathcal{F}\/A$-centric, and let $Q\/A$ be $\\mathcal{F}\/A$-conjugate to $P\/A$. If $P, Q\\leq S$ denote the preimages of $P\/A$ and $Q\/A$ in $S$ respectively, then $Q$ is $\\mathcal{F}$-conjugate to $P$ by definition of $\\mathcal{F}\/A$. Moreover,\n$$\nC_S(Q)A\/A\\leq C_{S\/A}(Q\/A)\\leq Q\/A,\n$$\nand thus $C_S(Q)\\leq Q$ (since $A\\leq Q$). It follows that $P$ is $\\mathcal{F}$-centric, and hence an object in $\\mathcal{T}$. This implies that $P\/A \\in \\mathrm{Ob}(\\mathcal{T}\/A)$.\n\\end{proof}\n\n\n\\section{Telescopic transporter systems}\\label{Sbig}\n\nIn this section we describe a general procedure to add new objects to a given transporter system. The constructions in this section play a crucial role in the next section.\n\n\\begin{defi}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$ (not necessarily saturated), and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. The transporter system $\\mathcal{T}$ is \\emph{telescopic} if it satisfies the following condition.\n\\begin{itemize}\n\n\\item[(T)] For each $P \\in \\mathrm{Ob}(\\mathcal{T})$ there is a sequence $P_0\\leq P_1\\leq \\ldots$ of objects in $\\mathcal{T}$ such that $P = \\bigcup_{i \\geq 0} P_i$, and such that $P_i$ is a finite subgroup of $S$ for all $i \\geq 0$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{lmm}\\label{equinerv}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be a telescopic transporter system associated to $\\mathcal{F}$. Suppose in addition that $\\widetilde{\\LL}$ contains $\\mathcal{L}$ as a full subcategory. Then, the inclusion $\\mathcal{L} \\subseteq \\widetilde{\\LL}$ induces an equivalence between the corresponding nerves.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is an immediate consequence of \\cite[Proposition A.9]{BLO6}.\n\\end{proof}\n\nIn terms of the above definition, in this section we study some situations where we can add objects to a given transporter system to produce a telescopic transporter system, without changing the homotopy type of the nerve of the original transporter system.\n\n\\begin{defi}\\label{compsyst}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a saturated fusion system over $S$, and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. Let also $(-)^{\\star}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}$ and $(-)^{\\star}_{\\mathcal{T}} \\colon \\mathcal{T} \\to \\mathcal{T}$ be a pair of idempotent functors, and let $\\mathcal{C}^{\\star} \\subseteq \\mathcal{C}$, with $\\mathcal{C} = \\mathcal{F}, \\mathcal{T}$, be the full subcategory with $\\mathrm{Ob}(\\mathcal{C}^{\\star}) = \\{P^{\\star} \\, | \\, P \\in \\mathrm{Ob}(\\mathcal{C})\\}$. The pair $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ is a \\emph{finite retraction pair} if the following conditions are satisfied.\n\\begin{itemize}\n\n\\item[(1)] For each $P\\leq S$, $P\\leq (P)^{\\star}_{\\mathcal{F}}$. Moreover, if $P \\in \\mathrm{Ob}(\\mathcal{T})$, then $(P)^{\\star}_{\\mathcal{F}} = (P)^{\\star}_{\\mathcal{T}} = P^{\\star}$.\n\n\\item[(2)] For each $P, Q\\leq S$ and each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$, $(f)^{\\star}_{\\mathcal{F}} \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star})$ extends $f$, and it is the unique extension.\n\n\\item[(i)] $\\mathrm{Ob}(\\mathcal{F}^{\\star})$ contains finitely many $S$-conjugacy classes of subgroups of $S$.\n\n\\item[(ii)] For all $P\\leq S$, $P^{\\star} = (P^{\\star})^{\\star}$.\n\n\\item[(iii)] If $P\\leq Q\\leq S$, then $P^{\\star}\\leq Q^{\\star}$.\n\n\\item[(iv)] For all $P, Q\\leq S$, $N_S(P,Q) \\subseteq N_S(P^{\\star}, Q^{\\star})$.\n\n\\item[(v)] For all $P\\leq S$, $C_S(P) = C_S(P^{\\star})$.\n\n\\item[(a)] $(-)^{\\star}_{\\mathcal{F}} \\circ \\rho = \\rho \\circ (-)^{\\star}_{\\mathcal{T}}$.\n\n\\item[(b)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ and all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$, we have $\\varepsilon_{Q, Q^{\\star}}(1) \\circ \\varphi = (\\varphi)^{\\star}_{\\mathcal{T}} \\circ \\varepsilon_{P, P^{\\star}}(1)$.\n\n\\end{itemize}\n\n\\end{defi}\n\nThe above definition is inspired in the pair of ``bullet'' functors described in Proposition \\ref{3.2BLO3}. In particular, conditions (i)-(v) and (a)-(b) are labelled to emphasize the relation with the motivating example. Properties (vi), (vii), (c) and (d) in \\ref{3.2BLO3} are actually consequences of the definition, as we prove below.\n\n\\begin{lmm}\\label{extraprop}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$, let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$, and let $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ be a finite retraction pair. Then the following properties hold (where we label the properties according to \\ref{3.2BLO3} to emphasize the correspondence).\n\\begin{itemize}\n\n\\item[(vi)] The functor $(-)^{\\star}_{\\mathcal{F}}$ is left adjoint to the inclusion of $\\mathcal{F}^{\\star}$ as a full subcategory of $\\mathcal{F}$.\n\n\\item[(vii)] All $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ are in $\\mathcal{F}^{\\star}$.\n\n\\item[(c)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ and all $g \\in N_S(P,Q)$, we have $(\\varepsilon_{P,Q}(g))^{\\star}_{\\mathcal{T}} = \\varepsilon_{P^{\\star}, Q^{\\star}}(g)$.\n\n\\item[(d)] The functor $(-)^{\\star}_{\\mathcal{T}}$ is left adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\mathcal{T}$. In particular, the inclusion $\\mathcal{T}^{\\star} \\subseteq \\mathcal{T}$ induces an equivalence $|\\mathcal{T}^{\\star}| \\simeq |\\mathcal{T}|$.\n\n\\end{itemize}\n\n\\end{lmm}\n\n\\begin{proof}\n\nProperty (vi) follows from condition (2) \\ref{compsyst}, since it implies that, for all $P, Q\\leq S$, the restriction map $\\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star}) \\to \\mathrm{Hom}_{\\mathcal{F}}(P, Q^{\\star})$ is a bijection. To prove property (vii), let $P\\leq S$ be $\\mathcal{F}$-centric, and suppose that $P \\notin \\mathrm{Ob}(\\mathcal{F}^{\\star})$. Then, $P \\lneqq P^{\\star}$, which implies that $P \\lneqq N_{P^{\\star}}(P)$. Since every element of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ extends uniquely to an element of $\\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})$, it is not hard to see that $1 \\neq N_{P^{\\star}}(P)\/\\mathrm{Inn}(P)$ is normalized by $\\mathrm{Aut}_{\\mathcal{F}}(P)$, and hence $P$ cannot be $\\mathcal{F}$-radical. Property (c) is an immediate consequence of property (b) in \\ref{compsyst}, applied to $\\varphi = \\varepsilon_{P,Q}(g)$, together with Lemma \\ref{epimono}.\n\nFinally, we prove property (d). For each $P \\in \\mathrm{Ob}(\\mathcal{T})$, set as usual\n$$\nE(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{T}}(P) \\Right2{}\\mathrm{Aut}_{\\mathcal{F}}(P)).\n$$\nWe claim that $E(P) = E(P^{\\star})$ for all $P \\in \\mathrm{Ob}(\\mathcal{T})$. Clearly, restriction from $P^{\\star}$ to $P$ maps $E(P^{\\star})$ to $E(P)$, and this restriction is injective by \\ref{epimono}. Let now $\\varphi \\in E(P)$, and consider $(\\varphi)^{\\star}_{\\mathcal{T}} \\in \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star})$. By assumption, $f = \\rho(\\varphi) = \\mathrm{Id}$, and thus $(f)^{\\star}_{\\mathcal{F}} = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})$ by property (2) in \\ref{compsyst}. By property (a) in \\ref{compsyst} we get\n$$\n\\rho((\\varphi)^{\\star}_{\\mathcal{T}}) = (\\rho(\\varphi))^{\\star}_{\\mathcal{F}} = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id},\n$$\nand $(\\varphi)^{\\star}_{\\mathcal{T}} \\in E(P^{\\star})$. Using axiom (A2) of transporter systems, together with property (vi) above, it is easy to deduce now that the functor $(-)^{\\star}_{\\mathcal{T}}$ is left adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\mathcal{T}$, and property (d) follows.\n\\end{proof}\n\n\\begin{defi}\\label{wT}\n\nLet $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ be a finite retraction pair. Define $\\4{\\mathcal{T}}$ to be the category with object set $\\mathrm{Ob}(\\4{\\mathcal{T}}) = \\{P\\leq S \\, | \\, P^{\\star} \\in \\mathrm{Ob}(\\mathcal{T})\\}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star}) \\, | \\, \\varphi \\circ \\varepsilon_{P^{\\star}}(g) \\circ \\varphi^{-1} \\in \\varepsilon_{Q^{\\star}}(Q) \\mbox{, for all } g \\in P\\},\n$$\nfor all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. Composition in $\\4{\\mathcal{T}}$ is given by composition in $\\mathcal{T}$. Define also functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\4{\\mathcal{T}})}(S) \\Right3{\\4{\\varepsilon}} \\4{\\mathcal{T}} \\qquad \\mbox{and} \\qquad \\4{\\mathcal{T}} \\Right3{\\4{\\rho}} \\mathcal{F}\n$$\nas follows. The functor $\\4{\\varepsilon}$ is the identity on objects, and the functor $\\4{\\rho}$ is injective on objects. For all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, all $g \\in N_S(P,Q)$, and all $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$, define\n$$\n\\4{\\varepsilon}_{P,Q}(g) = \\varepsilon_{P^{\\bullet}, Q^{\\bullet}}(g) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) \\qquad \\mbox{and} \\qquad \\4{\\rho}(\\varphi \\colon P \\to Q) = \\rho(\\varphi \\colon P^{\\bullet} \\to Q^{\\bullet})|_P.\n$$\nThe properties of the functors $(-)^{\\star}_{\\mathcal{F}}$ and $(-)^{\\star}_{\\mathcal{T}}$ imply that both $\\4{\\mathcal{T}}$ and the above functors are well-defined, and that $\\mathcal{T}$ is a full subcategory of $\\4{\\mathcal{T}}$.\n\n\\end{defi}\n\n\\begin{prop}\\label{extendL}\n\nFor each finite retraction pair $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$, the category $\\4{\\mathcal{T}}$, with the functors $\\4{\\varepsilon}$ and $\\4{\\rho}$ defined above, is a telescopic transporter system associated to $\\mathcal{F}$. Furthermore, the functor $(-)^{\\star}_{\\mathcal{T}} \\colon \\mathcal{T} \\to \\mathcal{T}$ extends to a functor $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}$, which is unique satisfying the following properties\n\\begin{enumerate}[(a)]\n\n\\item There is an equality $(-)^{\\star}_{\\mathcal{F}} \\circ \\4{\\rho} = \\rho \\circ (-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{F}$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ and all $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$, we have $\\4{\\varepsilon}_{Q,Q^{\\star}}(1) \\circ \\varphi = (\\varphi)^{\\star}_{\\4{\\mathcal{T}}} \\circ \\4{\\varepsilon}_{P,P^{\\star}}(1)$.\n\n\\end{enumerate}\nIn particular, the inclusion of $\\mathcal{T}$ in $\\4{\\mathcal{T}}$ as a full subcategory induces an equivalence $|\\mathcal{T}| \\simeq |\\4{\\mathcal{T}}|$.\n\n\\end{prop}\n\n\\begin{proof}\n\nBy definition, $\\4{\\mathcal{T}}$ contains $\\mathcal{T}$ as a full subcategory, and the functor $(-)_{\\mathcal{T}}^{\\star} \\colon \\mathcal{T} \\to \\mathcal{T}^{\\star}$ can be extended to a functor $(-)_{\\4{\\mathcal{T}}}^{\\star} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$ as follows. On objects, $(P)^{\\star}_{\\4{\\mathcal{T}}} = (P)^{\\star}_{\\mathcal{F}} = P^{\\star}$. On morphisms, $(-)^{\\star}_{\\4{\\mathcal{T}}}$ is defined by the inclusion\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q) \\subseteq \\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})\n$$\ngiven by definition of $\\4{\\mathcal{T}}$. The proof of (a) and (b), as well as the uniqueness of $(-)^{\\star}_{\\4{\\mathcal{T}}}$ satisfying these conditions, is left to the reader as an easy exercise.\n\nNext we show that $\\4{\\mathcal{T}}$ is indeed a transporter system. Conditions (A1), (B) and (C) are clear. Condition (A2) follows from the properties of the functor $(-)^{\\star}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}^{\\star}$. Indeed, for each $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ set\n$$\n\\begin{array}{c}\n\\4{E}(P) = \\mathrm{Ker}(\\4{\\rho}_P \\colon \\mathrm{Aut}_{\\4{\\mathcal{T}}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) \\\\\nE(P^{\\star}) = \\mathrm{Ker}(\\rho_{P^{\\star}} \\colon \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star}) \\to \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})).\n\\end{array}\n$$\nIf $\\varphi \\in E(P^{\\star})$, then, by definition of $\\4{\\mathcal{T}}$, together with axiom (C) on $\\mathcal{T}$, it follows that $\\varphi \\in \\4{E}(P)$. Conversely, if $\\varphi \\in \\4{E}(P)$, then by definition $\\varphi \\in \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star})$ is such that $\\rho(\\varphi)|_P = \\mathrm{Id}$. By property (a) on $(-)^{\\star}_{\\4{\\mathcal{T}}}$, we have\n$$\n\\rho(\\varphi) = \\rho((\\varphi)^{\\star}_{\\4{\\mathcal{T}}}) = (\\rho(\\varphi)|_P)^{\\star}_{\\mathcal{F}} = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id},\n$$\nand thus $\\varphi \\in E(P^{\\star})$. Hence $\\4{E}(P) = E(P^{\\star})$, and the freeness of the action of $\\4{E}(P)$ on $\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$ follows from property (A2) in $\\mathcal{T}$. That $\\4{\\rho}_{P,Q}$ is the orbit map of this action follows easily.\n\nNext we check condition (I) for $\\4{\\mathcal{T}}$. Fix $Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. If $Q \\in \\mathrm{Ob}(\\mathcal{T})$ then there is nothing to show, since $\\mathcal{T}$ is a full subcategory of $\\4{\\mathcal{T}}$. Thus, assume that $Q \\notin \\mathrm{Ob}(\\mathcal{T})$. We can choose $Q$ such that $Q^{\\star}$ is fully $\\mathcal{F}$-normalized, so that\n$$\n\\varepsilon_{Q^{\\star}}(N_S(Q^{\\star})) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(Q^{\\star})).\n$$\nSet for short $G = \\mathrm{Aut}_{\\mathcal{T}}(Q^{\\star})$ and $K = \\varepsilon_{Q^{\\star}}(N_S(Q^{\\star}))$.\n\nWe claim first that every subgroup $H$ of $G$ has Sylow $p$-subgroups. Notice that $G\/Q^{\\star} \\cong \\mathrm{Out}_{\\mathcal{F}}(Q^{\\star})$, which is a finite group. Thus, $H\/(H \\cap Q^{\\star}) \\cong HQ^{\\star}\/Q^{\\star}\\leq G\/Q^{\\star}$, and thus $H \\cap Q^{\\star}$ is a discrete $p$-toral normal subgroup of $H$ with finite index. The claim follows by \\cite[Lemma 8.1]{BLO3}.\n\nNow, by definition we can consider $H = \\mathrm{Aut}_{\\4{\\mathcal{T}}}(Q)$ as a subgroup of $G$, and in particular the above discussion implies that $H$ has Sylow $p$-subgroups. Fix $R \\in \\operatorname{Syl}\\nolimits_p(H)$ such that $\\varepsilon_Q(N_S(Q))\\leq R$. Since $K \\in \\operatorname{Syl}\\nolimits_p(G)$, there is some $\\varphi \\in G$ such that $\\varphi \\circ R \\circ \\varphi^{-1}\\leq K$. Set $P = \\rho(\\varphi)(Q)\\leq Q^{\\star}$. Note that $P^{\\star}\\leq Q^{\\star}$ by definition of $P$, and $P$ is $\\mathcal{F}$-conjugate to $Q$, which implies that $P^{\\star}$ is $\\mathcal{F}$-conjugate to $Q^{\\star}$ This implies that $P^{\\star} = Q^{\\star}$. Thus,\n$$\n\\mathrm{Aut}_{\\4{\\mathcal{T}}}(P) = \\varphi \\circ H \\circ \\varphi^{-1} = \\varphi \\circ \\mathrm{Aut}_{\\4{\\mathcal{T}}}(Q) \\circ \\varphi^{-1}\\leq G,\n$$\nand $\\varepsilon_P(N_S(P)) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\4{\\mathcal{T}}}(P))$. Condition (I) follows.\n\nCondition (II) for $\\4{\\mathcal{T}}$ follows easily from condition (II) for $\\mathcal{T}$. Indeed, let $\\varphi \\in \\mathrm{Iso}_{\\4{\\mathcal{T}}}(P,Q)$, $P \\lhd \\4{P}\\leq S$ and $Q \\lhd \\4{Q}\\leq S$ be such that $\\varphi \\circ \\4{\\varepsilon}_P(\\4{P}) \\circ \\varphi^{-1}\\leq \\4{\\varepsilon}_Q(\\4{Q})$. By applying the functor $(-)_{\\4{\\mathcal{T}}}^{\\star} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$, we get $\\varphi^{\\star} \\in \\mathrm{Iso}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})$, and\n$$\nP^{\\star} \\lhd \\widehat{P} \\stackrel{def} = N_{(\\4{P})^{\\star}}(P^{\\star})\\leq S \\qquad \\mbox{and} \\qquad Q^{\\star} \\lhd \\widehat{Q} \\stackrel{def} = N_{(\\4{Q})^{\\star}}(Q^{\\star})\\leq S,\n$$\nsuch that $\\varphi^{\\star} \\circ \\varepsilon_{P^{\\star}}(\\widehat{P}) \\circ (\\varphi^{\\star})^{-1}\\leq \\varepsilon_{Q^{\\star}}(\\widehat{Q})$. Axiom (II) in $\\mathcal{T}$ implies that there exists some $\\widehat{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{T}}(\\widehat{P}, \\widehat{Q})$ such that\n$$\n\\widehat{\\varphi} \\circ \\varepsilon_{P^{\\star}, \\widehat{P}}(1) = \\varepsilon_{Q^{\\star}, \\widehat{Q}}(1) \\circ \\varphi^{\\star}.\n$$\nNote that $\\4{P}\\leq \\widehat{P}$ and $\\4{Q}\\leq \\widehat{Q}$ by property (iv) in \\ref{compsyst}. Thus, we may restrict the morphism $\\widehat{\\varphi}$ to $\\4{P}$, and condition (II) follows.\n\nCondition (III) for $\\4{\\mathcal{T}}$ follows easily by condition (III) for $\\mathcal{T}$, together with the properties of the functor $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$, since $\\mathcal{T}^{\\star}$ contains finitely many isomorphism classes of objects by property (i) in \\ref{compsyst}.\n\nLet us now prove that that $\\4{\\mathcal{T}}$ is a telescopic transporter system. Let $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. By \\cite[Lemma 1.9]{BLO3}, there is a sequence $P_0\\leq P_1\\leq \\ldots$ of finite subgroups of $P$ such that $P = \\bigcup_{i \\geq 0} P_i$. Since $\\mathcal{F}^{\\star}$ contains finitely many $S$-conjugacy classes of subgroups by property (i) in \\ref{compsyst}, it follows that there is some $M \\in \\mathbb{N}$ such that $(P_i)^{\\star} = P^{\\star}$ for all $i \\geq M$, and we may assume that $M = 0$ for simplicity. This way, since $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, it follows that $P_i \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ for all $i \\geq 0$. Thus $\\4{\\mathcal{T}}$ is a telescopic transporter system.\n\nFinally, we check that the inclusion of $\\mathcal{T}$ in $\\4{\\mathcal{T}}$ as a full subcategory induces an equivalence between the corresponding nerves. Recall from property (d) in \\ref{extraprop} that the inclusion $\\mathcal{T}^{\\star} \\subseteq \\mathcal{T}$ induces an equivalence $|\\mathcal{T}^{\\star}| \\simeq |\\mathcal{T}|$. Thus, we only need to show that $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$ is (left) adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\4{\\mathcal{T}}$. That is, given $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, we have to show that the restriction map\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P^{\\star}, Q^{\\star}) \\Right2{} \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P, Q^{\\star})\n$$\nis a bijection. Let $\\4{E}(P) = E(P^{\\star})$ as above, and recall that there is a bijection between the sets $\\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star})$ and $\\mathrm{Hom}_{\\mathcal{F}}(P, Q^{\\star})$, given by the restriction map, by (vi) in \\ref{extraprop}. Thus, by axiom (A2) of transporter systems,\n$$\n\\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})\/E(P^{\\star}) = \\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star}) \\cong \\mathrm{Hom}_{\\mathcal{F}}(P,Q) = \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P, Q^{\\star})\/\\4{E}(P),\n$$\nand the claim follows.\n\\end{proof}\n\nWe call $\\4{\\mathcal{T}}$ the \\emph{telescopic transporter system associated to $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$}, or simply the telescopic transporter system associated to $\\mathcal{T}$ if there is no need to specify $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$.\n\n\\begin{prop}\\label{extendL2}\n\nEach $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ extends uniquely to some $\\4{\\Psi} \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\4{\\mathcal{T}})$.\n\n\\end{prop}\n\n\\begin{proof}\n\nLet $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ and let $\\psi \\in \\mathrm{Aut}(S)$ be the automorphism induced by $\\Psi$. Then $\\Psi$ extends to $\\4{\\mathcal{T}}$ by the formulas\n$$\n\\4{\\Psi}(P) = \\psi(P) \\qquad \\mbox{and} \\qquad \\4{\\Psi}(\\varphi \\colon P \\to Q) = \\Psi(\\varphi^{\\star}_{\\4{\\mathcal{T}}} \\colon P^{\\star} \\to Q^{\\star}).\n$$\nClearly, this determines an isotypical equivalence $\\4{\\Psi}$ of $\\4{\\mathcal{T}}$. Moreover, since $\\4{\\Psi}$  is isotypical, that is $\\4{\\Psi}(\\4{\\varepsilon}_{P,Q}(1)) = \\4{\\varepsilon}_{\\4{\\Psi}(P), \\4{\\Psi}(Q)}(1)$ for all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ with $P\\leq Q$, and since morphisms in $\\4{\\mathcal{T}}$ are monomorphisms and epimorphisms in the categorical sense by Lemma \\ref{epimono}, it follows that $\\4{\\Psi}$ is the unique extension of $\\Psi$ to $\\4{\\mathcal{T}}$.\n\\end{proof}\n\nBelow we analyze some examples which will be of interest in later sections.\n\n\\begin{expl}\\label{expl1}\n\nWe start with the most obvious example. Let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $(-)^{\\bullet}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}$ and $(-)^{\\bullet}_{\\mathcal{L}} \\colon \\mathcal{L} \\to \\mathcal{L}$ be the usual ``bullet'' functors. Then, clearly $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ is a finite retraction pair (by \\ref{3.2BLO3}), and this way we obtain a telescopic transporter system $\\widetilde{\\LL}$ which in addition satisfies the following properties.\n\\begin{itemize}\n\n\\item[(1)] For each $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, we have \n$$\n\\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) = \\varepsilon_P(C_S(P)) = \\varepsilon_P(Z(P^{\\bullet})).\n$$\n\n\\end{itemize}\nLet $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, and set $E(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P))$. If $P \\in \\mathrm{Ob}(\\mathcal{L})$ then $\\mathrm{Aut}_{\\widetilde{\\LL}}(P) = \\mathrm{Aut}_{\\mathcal{L}}(P)$, and there is nothing to prove. Suppose that $P \\notin \\mathrm{Ob}(\\mathcal{L})$. By definition $P^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L})$, and there is a commutative diagram of group extensions\n$$\n\\xymatrix{\nE(P^{\\bullet}) \\ar[r] & \\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\bullet}) \\ar[r] & \\mathrm{Aut}_{\\mathcal{F}}(P^{\\bullet}) \\\\\nE(P) \\ar[r] \\ar[u] & \\mathrm{Aut}_{\\mathcal{L}}(P) \\ar[r] \\ar[u] & \\mathrm{Aut}_{\\mathcal{F}}(P) \\ar[u]\n}\n$$\nwhere all the vertical arrows are inclusions. Thus, we have\n$$\nE(P)\\leq E(P^{\\bullet}) = \\varepsilon_{P^{\\bullet}}(Z(P^{\\bullet})) = \\varepsilon_{P^{\\bullet}}(C_S(P^{\\bullet})) = \\varepsilon_P(C_S(P)),\n$$\nwhere $C_S(P^{\\bullet}) = Z(P^{\\bullet})$ since $P^{\\bullet}$ is $\\mathcal{F}$-centric, and $C_S(P^{\\bullet}) = C_S(P)$ by property (v) in \\ref{compsyst}. The inclusion $\\varepsilon_P(C_S(P))\\leq E(P)$ is clear. This proves (1). In particular, every object in $\\widetilde{\\LL}$ is \\emph{quasicentric} (that is $C_{\\mathcal{F}}(P)$ is the fusion system of $C_S(P)$ for all $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$), and in this sense $\\widetilde{\\LL}$ is a \\emph{quasicentric linking system}.\n\\begin{itemize}\n \n\\item[(2)] There is an isomorphism $\\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L}) \\cong \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\widetilde{\\LL})$.\n\n\\end{itemize}\nThis follows by Proposition \\ref{extendL2}, together with the observation that every isotypical automorphism of $\\widetilde{\\LL}$ must restrict to an isotypical automorphism of $\\mathcal{L}$, since $\\mathrm{Ob}(\\mathcal{L})$ is the set of all $\\mathcal{F}$-centric subgroups of $S$. Moreover, this restriction is injective as a consequence of Lemma \\ref{epimono},  and since every morphism in $\\widetilde{\\LL}$ is the restriction of some morphism in $\\mathcal{L}$.\n\n\\end{expl}\n\n\\begin{expl}\\label{expl3}\n\nThe following is a less obvious example. Again, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair in \\ref{expl1}, and let $\\widetilde{\\LL}$ be the telescopic transporter system associated to $\\mathcal{L}$. Let also $A\\leq S$ be a fully $\\mathcal{F}$-normalized subgroup such that $N_S(A)$ has finite index in $S$ (for example $A\\leq T$ a subgroup of the maximal torus of $S$).\n\nLet $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ be the normalizer $p$-local compact group of $A$, as defined in \\ref{rmknorm}. Again, in general $N_{\\mathcal{L}}(A)$ is not a telescopic linking system. In this case, since $N_{\\mathcal{G}}(A)$ is a $p$-local compact group, one could apply Example \\ref{expl1} to produce a telescopic transporter system associated to $N_{\\mathcal{L}}(A)$. However, this way there is no obvious correspondence between the telescopic transporter systems for $N_{\\mathcal{L}}(A)$ and $\\mathcal{L}$, mainly because usually the ``bullet'' functors in $N_{\\mathcal{F}}(A)$ and in $\\mathcal{F}$ do not agree, that is $(P)^{\\bullet}_{\\mathcal{F}} \\neq (P)^{\\bullet}_{N_{\\mathcal{F}}(A)}$ in general.\n\nInstead, we propose a different construction. Set for short $N = N_S(A)$, $\\mathcal{E} = N_{\\mathcal{F}}(A)$ and $\\mathcal{T} = N_{\\mathcal{L}}(A)$. We define a finite retraction pair $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$ as follows. For each $P\\leq N$, notice that $(P)^{\\bullet}_{\\mathcal{F}}\\leq N_S(A)$, since $N_S(A)$ already contains the maximal torus of $S$. Thus, we can define\n$$\nP^{\\star} = P^{\\bullet}.\n$$\nOn morphisms, let $P, Q\\leq N$, and let $f \\in \\mathrm{Hom}_{\\mathcal{E}}(P,Q)$. By definition of $\\mathcal{E}$, the morphism $f$ extends to some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(PA, QA)$ such that $\\gamma|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Applying the functor $(-)^{\\bullet}_{\\mathcal{F}}$ to the commutative square\n$$\n\\xymatrix{\nPA \\ar[r]^{\\gamma} & QA\\\\\nP \\ar[u]^{\\mathrm{incl}} \\ar[r]_{f} & Q \\ar[u]_{\\mathrm{incl}}\n}\n$$\nwe see that $(f)^{\\bullet}_{\\mathcal{F}}$ extends to $(\\gamma)^{\\bullet}_{\\mathcal{F}}$, and the latter restricts in turn to a morphism $\\5{\\gamma} \\in \\mathrm{Hom}_{\\mathcal{F}}((P)^{\\bullet}_{\\mathcal{F}}A, (Q)^{\\bullet}_{\\mathcal{F}}A)$ such that $\\5{\\gamma}|_A = \\gamma|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. We define\n$$\n(f)^{\\star}_{\\mathcal{E}} = (f)^{\\bullet}_{\\mathcal{F}}.\n$$\nProperties (1)-(2) and (i)-(v) in \\ref{compsyst} for $(-)^{\\bullet}_{\\mathcal{F}}$ imply that $(-)^{\\star}_{\\mathcal{E}}$ also satisfies these conditions.\n\nOn $\\mathcal{T}$, define $(-)^{\\star}_{\\mathcal{T}}$ as follows. Let $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$. By definition, $\\varphi$ is a morphism in $\\mathrm{Mor}_{\\mathcal{L}}(PA, QA)$ such that $\\rho(\\varphi)|_P \\in \\mathrm{Mor}_{\\mathcal{E}}(P,Q)$, and $\\rho(\\varphi)|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Clearly,\n$$\nP^{\\bullet}A\\leq (PA)^{\\bullet} \\qquad \\mbox{and} \\qquad Q^{\\bullet}A\\leq (QA)^{\\bullet},\n$$\nand thus $(\\varphi)^{\\bullet}_{\\mathcal{L}}$ restricts to a morphism $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{L}}(P^{\\bullet}A, Q^{\\bullet}A)$ such that $\\rho(\\4{\\varphi})|_{P^{\\bullet}} \\in \\mathrm{Hom}_{\\mathcal{E}}(P^{\\bullet}, Q^{\\bullet})$ and $\\rho(\\4{\\varphi})|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Define $(\\varphi)^{\\star}_{\\mathcal{T}} = \\4{\\varphi}$. It is not difficult to check that $(-)^{\\bullet}_{\\mathcal{T}}$ satisfies properties (a)-(b) in \\ref{compsyst}.\n\nLet $\\widetilde{\\LL}$ and $\\4{\\mathcal{T}}$ be the associated telescopic transporter systems for $\\mathcal{L}$ and $\\mathcal{T}$, respectively. In general, $\\mathcal{T}$ is not a subcategory of $\\mathcal{L}$, and neither is $\\4{\\mathcal{T}}$ a subcategory of $\\widetilde{\\LL}$. Let $\\mathcal{T}_{\\geq A} \\subseteq \\mathcal{T}$ be the full subcategory of subgroups that contain $A$, and let $\\4{\\mathcal{T}}_{\\geq A} \\subseteq \\4{\\mathcal{T}}$ be the full subcategory of subgroups $P$ such that $A\\leq P^{\\bullet}$. Then the following is easily checked.\n\\begin{enumerate}[(a)]\n\n\\item $\\mathcal{T}_{\\geq A}$ is a subcategory of $\\mathcal{L}$, and $\\4{\\mathcal{T}}_{\\geq A}$ is a subcategory of $\\widetilde{\\LL}$.\n\n\\item $\\mathcal{T}_{\\geq A}$ contains all the centric radical subgroups of $\\mathcal{E}$.\n\n\\item The functor $(-)^{\\star}_{\\4{\\mathcal{T}}}$ coincides with the functor $(-)^{\\bullet}_{\\widetilde{\\LL}}$ on the subcategory $\\4{\\mathcal{T}}_{\\geq A}$.\n\n\\end{enumerate}\nThis example, including the above remarks, will be very useful in the next section, when we have to compare certain constructions on a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ and on the normalizer $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ of a certain subtorus $A\\leq S$.\n\n\\end{expl}\n\n\n\\section{Families of unstable Adams operations}\\label{Sfam}\n\nIn this section we prove Theorem \\ref{thmA}, restated as Theorem \\ref{fix6} below: every $p$-local compact group can be approximated by $p$-local finite groups. Roughly speaking, given a $p$-local compact group we produce an \\emph{approximation of $\\mathcal{G}$ by $p$-local finite groups} (defined below) by considering \\emph{fixed point subcategories} of a telescopic transporter system associated to $\\mathcal{L}$ by iterations of a given unstable Adams operation on $\\mathcal{G}$. \n\nEssentially, we follow the same lines as \\cite{Gonza1}. However, the introduction of telescopic transporter systems means a great deal of simplification, and it is actually thanks to this that we are finally able to prove Proposition \\ref{fix5}, basically the missing step in \\cite{Gonza1} in proving Theorem \\ref{thmA}. We have opted for reproving here every property that we need from \\cite{Gonza1} for the sake of completeness as well as for correcting mistakes: while working on \\ref{fix5} below, the author realized that the statement of \\cite[Lemma 2.11]{Gonza1} is false. Nevertheless, this does not affect neither the main results of \\cite{Gonza1} nor the results that we present in this paper, and this comment is just intended as a warning to the interested reader.\n\n\\begin{defi}\\label{defiapprox}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be a telescopic transporter system associated to $\\mathcal{F}$ and containing $\\mathcal{L}$ as a full subcategory. An \\emph{approximation of $\\mathcal{G}$ by $p$-local finite groups} is a family $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ satisfying the following conditions.\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$.\n\n\\item For each $i \\geq 0$, $S_i$ is a finite $p$-group, $\\mathcal{F}_i$ is a saturated fusion system over $S_i$, and $\\mathcal{L}_i$ is a linking system associated to $\\mathcal{F}_i$. Furthermore, $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_i)$, and there are inclusions $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$ and $\\mathcal{L}_i \\subseteq \\widetilde{\\LL}$.\n\n\\item For each $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ and each $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, there are objects $P_i, Q_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ and morphisms $\\varphi_i \\in \\mathrm{Mor}_{\\mathcal{L}_i}(P_i, Q_i)$, such that $P = \\bigcup_{i \\geq M} P_i$ and $Q = \\bigcup_{i \\geq M} Q_i$, and $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nAlthough condition (i), or at least a weaker version of it, can be deduced from condition (iii) applied to $P = Q = S$ and to any $\\varphi \\in \\mathrm{Aut}_{\\mathcal{L}}(S)$, we prefer to include (i)  in the definition for the sake of clarity. We now show some basic properties of approximations of $p$-local compact groups by $p$-local finite groups.\n\n\\begin{lmm}\\label{finmorph}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be a finite approximation of $\\mathcal{G}$ by $p$-local finite groups with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. Then, for every finite subgroup $P\\leq S$ and every $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, $P\\leq S_i$, and $f$ is the composition of a morphism $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P, S_i)$ with the inclusion $S_i\\leq S$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $Q = f(P)$, which is also a finite subgroup of $S$, and let $\\gamma \\in \\mathrm{Iso}_{\\mathcal{F}}(P,Q)$ be the restriction of $f$ to its image. By property (i) in \\ref{defiapprox}, it is clear that there is some $M_0 \\in \\mathbb{N}$ such that $P, Q\\leq S_i$ for all $i \\geq M_0$. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist $W_0 = P, W_1, \\ldots, W_n = Q\\leq S$, $U_1, \\ldots, U_n \\in \\mathrm{Ob}(\\mathcal{L}) \\subseteq \\mathrm{Ob}(\\widetilde{\\LL})$, and morphisms $\\varphi_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(U_j)$, for $j = 1, \\ldots, n$, such that, for each $j$\n$$\nW_{j-1}, W_j\\leq U_j \\qquad \\mbox{and} \\qquad \\rho(\\varphi_j)(W_{j-1}) = W_j,\n$$\nand $\\gamma = \\rho(\\varphi_n) \\circ \\ldots \\circ \\rho(\\varphi_1)$. Combining properties (i) and (iii) in \\ref{defiapprox}, we see that for each $j = 1, \\ldots, n$ there exists some $M_j \\in \\mathbb{N}$ such that, for all $i \\geq M_j$, there exist $U_{j,i}, V_{j, i} \\in \\mathrm{Ob}(\\mathcal{L}_i)$, together with an isomorphism $\\varphi_{j.i} \\in \\mathrm{Iso}_{\\mathcal{L}_i}(U_{j,i}, V_{j,i})$, such that\n$$\nU_j = \\bigcup_{i \\geq M_j} U_{j,i} = \\bigcup_{i \\geq M_j} V_{j,i} \\qquad \\mbox{and} \\qquad \\4{\\varepsilon}_{V_{j,i}, U_j}(1) \\circ \\varphi_{j,i} = \\varphi_j \\circ \\4{\\varepsilon}_{U_{j,i}, U_j}(1).\n$$\nMoreover, since $W_{j-1}, W_j$ are finite subgroups, we may assume without loss of generality that $W_{j-1}\\leq U_{j,i}$ and $W_j\\leq V_{j, i}$ for all $i \\geq M_j$. Let $M = \\max\\{M_0, \\ldots, M_n\\}$. Then, for all $i \\geq 0$, it follows that\n$$\n\\gamma = \\rho_i(\\varphi_{n,i}) \\circ \\ldots \\circ \\rho_i(\\varphi_{1,i}) \\in \\mathrm{Mor}(\\mathcal{F}_i),\n$$\nand this finishes the proof.\n\\end{proof}\n\n\\begin{lmm}\\label{Quill}\n\nLet $\\mathcal{C}$ be a nonempty category that satisfies the following conditions:\n\\begin{enumerate}[(i)]\n\n\\item given objects $a_1, a_2$, there is an object $b$ and morphisms $a_1 \\Right1{g_1} b \\Left1{g_2} a_2$; and\n\n\\item given morphisms $g_1, g_2 \\colon a \\to b$, there is $h \\colon b \\to c$ such that $h \\circ g_1 = h \\circ g_2$.\n\n\\end{enumerate}\nThen, the nerve of $\\mathcal{C}$ is contractible.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is \\cite[Corollary 2]{Quillen}.\n\\end{proof}\n\n\\begin{lmm}\\label{approx-2}\n\nLet $\\mathcal{C}$ be a small category all of whose morphisms are epimorphisms and monomorphisms in the categorical sense, and let $\\mathcal{C}_0 \\subseteq \\mathcal{C}_1 \\subseteq \\ldots $ be a sequence of subcategories such that $\\mathcal{C} = \\bigcup_{i \\geq 0} \\mathcal{C}_i$. Then, $|\\mathcal{C}| \\simeq \\mathrm{hocolim \\,} |\\mathcal{C}_i|$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $I$ be the category of natural ordinals, with objects $\\mathrm{Ob}(I) = \\{i \\in \\mathbb{N}\\}$, and where the morphism set $\\mathrm{Mor}_I(i,j)$ is $\\{\\sigma_{i,j}\\}$ if $i\\leq j$, or empty otherwise. Define a functor\n$$\n\\Theta \\colon I \\Right3{} \\curs{\\mathbf{Cat}}\n$$\nby $\\Theta(i) = \\mathcal{C}_i$ and $\\Theta(\\sigma_{i,j}) = \\mathrm{incl} \\colon \\mathcal{C}_i \\to \\mathcal{C}_j$. The Grothendieck construction on $\\Theta$, namely $G(\\Theta)$, is the category with object set\n$$\n\\{(i, X) \\, | \\, i \\in \\mathrm{Ob}(I) \\mbox{ and } X \\in \\mathrm{Ob}(\\mathcal{C}_i)\\}.\n$$\nThe morphism sets $\\mathrm{Mor}_{G(\\Theta)}((i,X), (j, Y))$ are empty whenever $j < i$. Otherwise they consist of the pairs $(\\sigma_{i,j}, \\varphi)$, with $\\varphi \\in \\mathrm{Mor}_{\\mathcal{C}_j}(X,Y)$. By \\cite[Theorem 1.2]{Thomason}, we have an equivalence $\\mathrm{hocolim \\,} |\\mathcal{C}_i| \\simeq |G(\\Theta)|$.\n\nConsider now the projection functor $\\tau \\colon G(\\Theta) \\to \\mathcal{C}$ that sends an object $(i, X)$ to $\\tau(i, X) = X$, and a morphism $(\\sigma_{i,j}, \\varphi)$ to $\\tau(\\sigma_{i,j}, \\varphi) = \\varphi$. We claim that this functor induces an equivalence between the corresponding nerves. For each $X \\in \\mathcal{C}$, let $\\tau\/X$ be the category with object set\n$$\n\\mathrm{Ob}(\\tau\/X) = \\{((j, Y), \\varphi) \\,\\, | \\,\\, (j, Y) \\in \\mathrm{Ob}(G(\\Theta)) \\mbox{ and } \\varphi \\in \\mathrm{Mor}_{\\mathcal{C}}(Y, X)\\}.\n$$\nA morphism in $\\tau\/X$ from $((j,Y), \\varphi)$ to $((k, Z), \\psi)$ is $(\\sigma_{i,j}, \\gamma) \\in \\mathrm{Mor}_{\\mathcal{C}}((j,Y), (k, Z))$ such that $\\varphi = \\psi \\circ \\gamma$. By \\cite[Theorem A and Corollary 2]{Quillen}, it is enough to check that $\\tau\/X$ satisfies the conditions of Lemma \\ref{Quill}. Clearly, $\\tau\/X$ is nonempty, since $((i, X), 1_X) \\in \\tau\/X$ for some $i \\in \\mathbb{N}$ big enough. Let $((j,Y), \\varphi), ((k,Z), \\psi) \\in \\mathrm{Ob}(\\tau\/X)$, and let $m = \\max\\{i,j,k\\}$. Then condition (i) in Lemma \\ref{Quill} holds with\n$$\n((j,Y), \\varphi) \\Right3{(\\sigma_{j,m},\\varphi)} ((m, X), 1_X) \\Left3{(\\sigma_{k,m}, \\psi)} ((j,Y), \\psi).\n$$\nRegarding condition (ii), notice that $\\mathrm{Mor}_{\\tau\/X}(((j,Y), \\varphi), ((k, Z), \\psi))$ is either empty, or contains a single morphisms, since morphisms in $\\mathcal{C}$ are all epimorphisms and monomorphisms in the categorical sense. Thus, $|\\tau\/X|$ is contractible for all $X \\in \\mathcal{C}$, and the claim follows.\n\\end{proof}\n\n\\begin{lmm}\\label{approx0}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and suppose $\\mathcal{G}$ admits an approximation by $p$-local finite groups $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. Then, there is an equivalence $B\\mathcal{G} \\simeq (\\mathrm{hocolim \\,} |\\mathcal{L}_i|)^{\\wedge}_p$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nBy \\ref{defiapprox}, $\\widetilde{\\LL}$ contains $\\mathcal{L}$ as a full subcategory, and thus by \\ref{equinerv} the inclusion $\\mathcal{L} \\subseteq \\widetilde{\\LL}$ induces an equivalence $|\\mathcal{L}| \\simeq |\\widetilde{\\LL}|$. It is enough to show that $\\mathrm{hocolim \\,} |\\mathcal{L}_i| \\simeq |\\widetilde{\\LL}|$.\n\nSet $\\mathcal{L}^{\\circ} \\stackrel{def} = \\bigcup_{i \\geq 0} \\mathcal{L}_i \\subseteq \\widetilde{\\LL}$. By Lemma \\ref{epimono}, all morphisms in $\\widetilde{\\LL}$ are epimorphisms and monomorphisms in the categorical sense, and thus the same applies to $\\mathcal{L}^{\\circ}$. By Lemma \\ref{approx-2} it follows that\n$$\n\\mathrm{hocolim \\,} |\\mathcal{L}_i| \\simeq |\\mathcal{L}^{\\circ}|.\n$$\nThus, to finish the proof it is enough to show that the inclusion functor $\\iota \\colon \\mathcal{L}^{\\circ} \\to \\widetilde{\\LL}$ induces an equivalence of nerves. For each $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, the undercategory $\\iota\/P$ has object set\n$$\n\\mathrm{Ob}(\\iota\/P) = \\{(Q, \\varphi) \\, | \\, Q \\in \\mathrm{Ob}(\\mathcal{L}^{\\circ}) \\mbox{ and } \\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(Q,P)\\}.\n$$\nA morphism in $\\iota\/P$ from $(Q, \\varphi)$ to $(R, \\psi)$ is a morphism $\\gamma \\in \\mathrm{Mor}_{\\mathcal{L}^{\\circ}}(Q,R)$ such that $\\varphi = \\psi \\circ \\gamma$.\n\nWe show that $\\iota\/P$ satisfies the conditions of Lemma \\ref{Quill}, which implies that $|\\iota\/P|$ is contractible. Clearly, $\\iota\/P$ is nonempty. Let $(Q, \\varphi), (R, \\psi) \\in \\mathrm{Ob}(\\iota\/P)$. By property (iii) in Definition \\ref{defiapprox}, there is some $X \\in \\mathrm{Ob}(\\mathcal{L}^{\\circ})$, with $X\\leq P$, such that $\\varphi$ and $\\psi$ restrict to morphisms $\\varphi \\colon Q \\to X$ and $\\psi \\colon R \\to X$ in $\\mathcal{L}^{\\circ}$. Thus, condition (i) of \\ref{Quill} is satisfied with\n$$\n(Q, \\varphi) \\Right3{\\varphi} (X, \\varepsilon(1)) \\Left3{\\psi} (R, \\psi).\n$$\nRegarding condition (ii) in \\ref{Quill}, the set $\\mathrm{Mor}_{\\iota\/P}((Q, \\varphi), (R, \\psi))$ is either empty or contains a single morphism. Since the argument works for all $P \\in \\mathrm{Ob}(\\mathcal{L})$, it follows that $|\\mathcal{L}^{\\circ}| \\simeq |\\widetilde{\\LL}|$.\n\\end{proof}\n\n\\begin{rmk}\\label{approx-1}\n\nSuppose the $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ admits an approximation by $p$-local finite groups $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$. For each $i$, the fusion system $\\mathcal{F}_i$ is saturated, and we may consider its associated centric linking system $\\mathcal{T}_i$. Let also $\\mathcal{H}_i = \\mathrm{Ob}(\\mathcal{L}_i) \\cap \\mathrm{Ob}(\\mathcal{T}_i)$, and let $\\mathcal{L}_{\\mathcal{H}_i} \\subseteq \\mathcal{L}_i$ be the full subcategory with object set $\\mathcal{H}_i$. Since both $\\mathrm{Ob}(\\mathcal{L}_i)$ and $\\mathrm{Ob}(\\mathcal{T}_i)$ contain $\\mathrm{Ob}(\\mathcal{F}_i^{cr})$, it follows that $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathcal{H}_i$. Moreover, there is a commutative diagram\n$$\n\\xymatrix@R=1.2cm{\n\\ldots & \\mathcal{T}_i & \\mathcal{T}_{i+1} & \\ldots \\\\\n\\ldots \\ar[r] & \\mathcal{L}_{\\mathcal{H}_i} \\ar[u]^{\\iota_i} \\ar[d]_{\\mathrm{incl}} \\ar[r]^{\\mathrm{incl}} & \\mathcal{L}_{\\mathcal{H}_{i+1}} \\ar[u]_{\\iota_{i+1}} \\ar[d]^{\\mathrm{incl}} \\ar[r] & \\ldots \\\\\n\\ldots \\ar[r] & \\mathcal{L}_i \\ar[r]_{\\mathrm{incl}} & \\mathcal{L}_{i+1} \\ar[r] & \\ldots\n}\n$$\nwhere all the vertical arrows induce homotopy equivalences between the realizations of the corresponding nerves, by \\cite[Theorem B]{BCGLO1}. Thus, if we denote $B\\mathcal{G}_i = |\\mathcal{T}_i|^{\\wedge}_p$, then Lemma \\ref{approx0} implies that $B\\mathcal{G} \\simeq (\\mathrm{hocolim \\,} B\\mathcal{G}_i)^{\\wedge}_p$.\n\n\\end{rmk}\n\n\n\\subsection{Preliminary constructions}\\label{Sprelim}\n\nIn this subsection we establish the notation and basic facts necessary for the proof that every $p$-local compact group has an approximation by $p$-local finite groups, in the next subsection. \n\n\\begin{hyp}\\label{hyp1}\n\nFor the rest of this subsection, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{L}})$ be a finite retraction pair. Let also $\\widetilde{\\LL}$ be the associated telescopic transporter system, and let $\\Psi$ be a fine unstable Adams operation on $\\mathcal{L}$ (in the sense of \\ref{uAo}). By a slight abuse of notation, we denote by $\\Psi$ the corresponding extension of $\\Psi$ to $\\widetilde{\\LL}$ (see \\ref{extendL2}), which is again a fine unstable Adams operation. Set $\\Psi_0 = \\Psi$, and for all $i \\geq 0$, define\n\\begin{enumerate}[(a)]\n\n\\item $S_i = C_S(\\Psi_i) = \\{x \\in S \\, | \\, \\Psi_i(x) = x\\}$; and\n\n\\item $\\Psi_{i+1} = (\\Psi_i)^p$.\n\n\\end{enumerate}\n\n\\end{hyp}\n\n\\begin{lmm}\\label{SiS}\n\nThe following properties hold.\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$.\n\n\\item There is some $M_a \\in \\mathbb{N}$ such that $(S_i)^{\\star} = S$ for all $i \\geq M_a$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $T\\leq S$ be the maximal torus of $S$, and set $T_i = T \\cap S_i$. Notice that by definition $T_i$ is the subgroup of $T$ of elements fixed by $\\Psi_i$. By hypothesis, $\\Psi = \\Psi_0$ has degree $\\zeta \\in 1 + p^m \\Z^{\\wedge}_p$ for some $m > 0$, and $\\Psi_{i+1} = (\\Psi_i)^p$. Thus, we have $T_i \\lneqq T_{i+1}$ for all $i \\geq 0$, and $T = \\bigcup_{i \\geq 0} T_i$. By \\cite[Lemma 2.6]{JLL}, there is a subgroup $H\\leq S_0$ such that $S = H \\cdot T$. Thus, $H\\leq S_i$ for all $i$, and we get\n$$\nS = H \\cdot T = \\bigcup_{i \\geq 0} H \\cdot T_i \\subseteq \\bigcup_{i \\geq 0} S_i.\n$$\nThis proves part (i). To prove part (ii), suppose otherwise that no such $M_a \\in \\mathbb{N}$ exists, that is, $(S_i)^{\\star} \\lneqq S$ for all $i \\geq 0$. Since $\\mathrm{Ob}(\\mathcal{F}^{\\star})$ contains only finitely many conjugacy classes of elements, this means that there exist some $R \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ and some $M \\in \\mathbb{N}$ such that $R \\lneqq S$ and such that $(S_i)^{\\star} = R$ for all $i \\geq M$. Notice that this contradicts part (i): if $R \\lneqq S$, then there is some $x \\in S \\setminus R$. On the other hand, by part (i) we have $x \\in S_i$ for $i$ big enough, and thus $x \\in (S_i)^{\\star} = R$, hence a contradiction.\n\\end{proof}\n\nFor simplicity we may assume that $M_a = 0$. In particular, $S_i \\in \\mathrm{Ob}(\\widetilde{\\LL})$ for all $i \\geq 0$.\n\n\\begin{defi}\\label{Li}\n\nWith the conventions above, for each $i \\geq 0$ define $\\mathcal{L}_i$ as the category with object set $\\mathrm{Ob}(\\mathcal{L}_i) = \\{P\\leq S_i \\, | \\, P \\in \\mathrm{Ob}(\\widetilde{\\LL})\\}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) \\, | \\, \\Psi_i(\\varphi) = \\varphi\\}.\n$$\nDefine also $\\mathcal{F}_i$ as the fusion system over $S_i$ generated by the restriction of $\\4{\\rho} \\colon \\widetilde{\\LL} \\to \\mathcal{F}$ to $\\mathcal{L}_i$ (i.e. $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated). Finally, define functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{L}_i)}(S_i) \\Right3{\\varepsilon_i} \\mathcal{L}_i \\qquad \\mbox{and} \\qquad \\mathcal{L}_i \\Right3{\\rho_i} \\mathcal{F}_i\n$$\nas the obvious restrictions of the structural functors $\\4{\\varepsilon} \\colon \\mathcal{T}_{\\mathrm{Ob}(\\widetilde{\\LL})}(S) \\to \\widetilde{\\LL}$ and $\\4{\\rho} \\colon \\widetilde{\\LL} \\to \\mathcal{F}$.\n\n\\end{defi}\n\nDespite its simplicity, the following example illustrates why it is necessary to work with a telescopic linking system rather than a centric linking system.\n\n\\begin{expl}\\label{expl0}\n\nLet $T$ be a discrete $p$-torus, i.e. $T \\cong (\\Z\/p^{\\infty})^r$ for some $r \\geq 1$. Let also $\\mathcal{G} = (S, \\FF, \\LL)$ be the \\emph{trivial} $p$-local compact group associated to $T$. That is, $S = T$ and $\\mathcal{F} = \\mathcal{F}_T(T)$ is the fusion system over $T$ whose only morphisms are inclusions (since $T$ is abelian). This fusion system is obviously saturated, and has only one centric object, namely $T$ itself. Thus, $\\mathcal{L}$ has a single object, $T$, with $\\mathrm{Aut}_{\\mathcal{L}}(T) = T$. On the other hand, since $T = S$, we have $P^{\\bullet} = T$ for all $P\\leq T$, and the telescopic linking system $\\widetilde{\\LL}$ associated to $\\mathcal{L}$ in \\ref{expl1} is the actual transporter category of the group $T$. That is, $\\mathrm{Ob}(\\widetilde{\\LL}) = \\{P\\leq T\\}$, and $\\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) = N_T(P,Q) = T$ for all $P, Q\\leq T$. Let now $\\Psi$ be an unstable Adams operation as fixed in \\ref{hyp1}. An easy computation reveals that $C_T(\\Psi)$ must be a finite subgroup of $T$, and thus is not an object in $\\mathcal{L}$. In particular, without replacing $\\mathcal{L}$ by $\\widetilde{\\LL}$, the subcategories $\\mathcal{L}_i$ defined above would be empty for all $i \\geq 0$.\n\n\\end{expl}\n\n\\begin{prop}\\label{fix1}\n\nThe following holds.\n\\begin{enumerate}[(i)]\n\n\\item For each $P \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ there exists some $M_P \\in \\mathbb{N}$ such that $(P \\cap S_i)^{\\star} = P$ for all $i \\geq M_P$.\n\n\\item For each $\\varphi \\in \\mathrm{Mor}(\\widetilde{\\LL})$ there exists some $M_{\\varphi} \\in \\mathbb{N}$ such that $\\Psi_i(\\varphi) = \\varphi$ for all $i \\geq M_{\\varphi}$.\n\n\\item For each $i \\geq 0$ and each $x \\in S$, $x^{-1} \\cdot \\Psi_i(x) \\in T$.\n\n\\end{enumerate}\n\n\\end{prop}\n\n\\begin{proof}\n\nTo prove part (i), notice that $P^{\\star} = P = \\bigcup_{i \\geq 0} P \\cap S_i$ by Lemma \\ref{SiS} (i). Suppose that $(P \\cap S_i)^{\\star} \\lneqq P$ for all $i \\geq 0$. Since $\\mathcal{F}^{\\star}$ contains finitely many conjugacy classes of objects, this means that there is some $R \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ such that $(P \\cap S_i)^{\\star} = R \\lneqq P$ for all $i$ big enough, contradicting the identity $P = \\bigcup_{i \\geq 0} P \\cap S_i$. Part (iii) follows immediately by definition of unstable Adams operation, since the  $\\Psi_i \\in \\mathrm{Aut}(S)$ induces the identity on $S\/T$.\n\nFinally, part (ii) follows by construction of the unstable Adams operations $\\Psi_i$. More specifically, as stablished in \\ref{hyp1}, the unstable Adams operation $\\Psi$ satisfies the following property (see Remark \\ref{uAo1}, or the proof of \\cite[Theorem 4.1]{JLL} for a more detailed explanation): there is a (finite) set $\\mathcal{M}$ of morphisms in $\\mathcal{L}^{\\star}$ such that the following holds\n\\begin{enumerate}[(1)]\n\n\\item $\\Psi(\\varphi) = \\varphi$ for all $\\varphi \\in \\mathcal{M}$; and\n\n\\item every morphism $\\psi$ in $\\mathcal{L}$ (and hence in $\\widetilde{\\LL}$ by definition) decomposes as $\\psi = \\varepsilon(g) \\circ \\varphi$, where $\\varphi$ is (the restriction of) a morphism in $\\mathcal{M}$, and $g$ is an element of $S$.\n\n\\end{enumerate}\nMoreover, these properties depend only on $\\mathcal{L}^{\\star}$ containing finitely many $S$-conjugacy classes, and on $\\mathcal{L}$ being a centric linking system, but not on the functor $(-)^{\\star}_{\\mathcal{L}}$. Fix $\\psi \\in \\mathrm{Mor}(\\widetilde{\\LL})$, and let $\\psi = \\varepsilon(g) \\circ \\varphi$ be the corresponding decomposition, as described above. By assumption, $\\Psi(\\varphi) = \\varphi$, and thus $\\Psi_i(\\varphi) = \\varphi$ for all $i \\geq 0$. Also, $S = \\bigcup_{i \\geq 0} S_i$ by Lemma \\ref{SiS} (i), and thus there exists some $M_{\\varphi} \\in \\mathbb{N}$ such that $g \\in S_i$ for all $i \\geq M_{\\varphi}$. It follows that $\\Psi_i(\\psi) = \\Psi_i(\\varepsilon(g)) \\circ \\varphi = \\varepsilon(g) \\circ \\varphi = \\psi$, and part (ii) follows.\n\\end{proof}\n\n\\begin{prop}\\label{fix2}\n\nThere exists some $M_b \\in \\mathbb{N}$ such that, for all $i \\geq M_b$, the triple $(\\mathcal{L}_i, \\varepsilon_i, \\rho_i)$ is a transporter system associated to the fusion system $\\mathcal{F}_i$.\n\n\\end{prop}\n\n\\begin{proof}\n\nWe have to check the axioms in Definition \\ref{defitransporter}. Notice that $\\mathcal{L}_i$ is a finite category, and thus we do not have to deal with axiom (III). Axioms (A1), (B) and (C) follow immediately by definition of $\\mathcal{L}_i$ as a subcategory of $\\widetilde{\\LL}$. We deal with the remaining axioms of transporter systems in separate steps for the reader's convenience.\n\nFor each $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$, set\n$$\nE_i(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(P)).\n$$\nNote that, via the inclusion $\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\star})$, and by property (a) in Definition \\ref{compsyst} of the functor $(-)^{\\star}_{\\mathcal{L}}$, the subgroup $E_i(P)$ is mapped to a subgroup of $E(P^{\\star}) = \\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\star}) \\to \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star}))$. Indeed, if $\\varphi \\in E_i(P)$, then $f = \\rho_i(\\varphi) = \\mathrm{Id}$. Thus, $\\4{\\rho}((\\varphi)^{\\star}_{\\mathcal{L}}) = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id}$.\n\n\\textbf{Step 1.} Axiom (A2) is satisfied: for each $P, Q \\in \\mathrm{Ob}(\\mathcal{L}_i)$, the group $E_i(P)$ acts freely on $\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q)$ by right composition, and $\\rho_i \\colon \\mathrm{Mor}_{\\mathcal{L}_i}(P,Q) \\to \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q)$ is the orbit map of this action. Also, $E_i(Q)$ acts freely on $\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q)$ by left composition.\n\nThe freeness of both the left action by $E_i(P)$ and the right action of $E_i(Q)$ follows by Lemma \\ref{epimono}, which states that morphisms in $\\widetilde{\\LL}$ (and in particular in $\\mathcal{L}_i$ by definition) are monomorphisms and epimorphisms in the categorical sense. That $\\rho_i$ is the orbit map of the left conjugation action of $E_i(P)$ is now immediate.\n\n\\textbf{Step 2.} Axiom (I) is satisfied. In fact, since $\\mathcal{L}_i$ is a finite category, it is enough to show that there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, $\\mathcal{L}_i$ satisfies axiom (I') in Remark \\ref{rmktransp}: $\\varepsilon_i(S_i) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{L}_i}(S_i))$ or, equivalently, the group $\\mathrm{Out}_{\\mathcal{F}_i}(S_i)$ has trivial Sylow $p$-subgroup.\n\nFix a set $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\widetilde{\\LL}}(S)$ of representatives of the elements of $\\mathrm{Out}_{\\mathcal{F}}(S) = \\mathrm{Aut}_{\\widetilde{\\LL}}(S)\/S$. Then there exists some $M_b \\in \\mathbb{N}$ such that, for all $i \\geq M_b$, $(S_i)^{\\star} = S$ and $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\widetilde{\\LL}}(S_i)$ (by abuse of notation we consider $\\mathcal{N}$ as the restriction of its elements to $S_i$). Furthermore, by Proposition \\ref{fix1} (ii) we can assume that $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$ for all $i \\geq M_b$. Thus, there is a commutative diagram of group extensions\n$$\n\\xymatrix{\n\\4{\\varepsilon}_S(S)^{\\Psi_i} \\ar[r] \\ar[d]_{\\mathrm{res}} & \\mathrm{Aut}_{\\widetilde{\\LL}}(S)^{\\Psi_i} \\ar[r] \\ar[d]^{\\mathrm{res}} & \\mathrm{Out}_{\\mathcal{F}}(S) \\ar[d] \\\\\n\\varepsilon_i(S_i) \\ar[r] & \\mathrm{Aut}_{\\mathcal{L}_i}(S_i) \\ar[r] & \\mathrm{Out}_{\\mathcal{F}_i}(S_i)\n}\n$$\nwhere $G^{\\Psi_i} = \\{g \\in G \\, | \\, \\Psi_i(g) = g\\}$, for $G = \\varepsilon_S(S)$ or $G = \\mathrm{Aut}_{\\mathcal{L}}(S)$. Furthermore, note that the restrictions $\\mathrm{res} \\colon \\4{\\varepsilon}_S(S)^{\\Psi_i} \\to \\varepsilon_i(S_i)$ and $\\mathrm{res} \\colon \\mathrm{Aut}_{\\widetilde{\\LL}}(S)^{\\Psi_i} \\to \\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$ are isomorphisms by definition. Thus, for all $i \\geq M_b$ we have $\\mathrm{Out}_{\\mathcal{F}_i}(S_i) \\cong \\mathrm{Out}_{\\mathcal{F}}(S)$, and axiom (I') follows since $\\{1\\} \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Out}_{\\mathcal{F}}(S))$.\n\n\\textbf{Step 3.} Axiom (II) is satisfied: let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{L}_i}(P,Q)$, $P \\lhd \\4{P}\\leq S_i$, and $Q \\lhd \\4{Q}\\leq S_i$ be such that $\\varphi \\circ \\varepsilon_i(\\4{P}) \\circ \\varphi^{-1}\\leq \\varepsilon_i(\\4{Q})$; then there is some $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{L}_i}(\\4{P}, \\4{Q})$ such that $\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi$.\n\nFix some $i \\geq 0$, and let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{L}_i}(P,Q)$, $P \\lhd \\4{P}\\leq S_i$, and $Q \\lhd \\4{Q}\\leq S_i$ be as above, and notice that in this case we have $\\varepsilon_i(1) = \\4{\\varepsilon} \\colon X \\to \\4{X}$, where $X = P,Q$. Since $\\widetilde{\\LL}$ is a transporter system, there is some $\\4{\\varphi} \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(\\4{P}, \\4{Q})$ such that $\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi$. Applying $\\Psi_i$ to this equation we get\n$$\n\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi = \\Psi_i(\\varepsilon_i(1) \\circ \\varphi) = \\Psi_i(\\4{\\varphi}) \\circ \\varepsilon_i(1),\n$$\nand thus Lemma \\ref{epimono} implies that $\\Psi_i(\\4{\\varphi}) = \\4{\\varphi}$.\n\\end{proof}\n\nAgain, we may assume that $M_b = 0$ for simplicity.\n\n\\begin{cor}\\label{fix3}\n\nFor all $i \\geq 0$, the fusion system $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated and $\\mathrm{Ob}(\\mathcal{L}_i)$-saturated.\n\n\\end{cor}\n\n\\begin{proof}\n\nThe fusion system $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated by definition, and the $\\mathrm{Ob}(\\mathcal{L}_i)$-saturation follows by \\cite[Proposition 3.6]{OV}.\n\\end{proof}\n\n\n\\subsection{Existence of finite approximations}\\label{Ssapprox}\n\nWe are ready to prove that every $p$-local compact group has an approximation by $p$-local finite groups. In fact, we prove something stronger: every fine unstable Adams operation (in the sense of \\ref{uAo}) defines an approximation by $p$-local finite groups.\n\n\\begin{hyp}\\label{hyp2}\n\nFor the rest of this section, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair of \\ref{expl1}. Let also $\\widetilde{\\LL}$ be the associated telescopic linking system, and let $\\Psi$ be a fine unstable Adams operation on $\\widetilde{\\LL}$ (that is, $\\Psi$ is an unstable Adams operation whose degree $\\zeta \\neq 1$ is congruent to $1$ modulo $p$). Let also $\\{\\Psi_i\\}_{i \\geq 0}$ and $\\{S_i\\}_{i \\geq 0}$ be as defined in \\ref{hyp1}, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the associated family of finite transporter systems defined in \\ref{Li}. Finally, for all $i \\geq 0$ let\n\\begin{enumerate}[(a)]\n\n\\item $\\Gamma_i =\\{P\\leq S_i \\, | \\, P^{\\bullet} \\notin \\mathrm{Ob}(\\mathcal{L}^{\\bullet}) \\mbox{ and } P^{\\bullet} \\cap S_i = P\\}$; and\n\n\\item $\\Omega_i = \\{R \\in \\mathrm{Ob}(\\mathcal{F}^{\\bullet})\\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet}) \\, | \\, (R \\cap S_i)^{\\bullet} = R\\}$.\n\n\\end{enumerate}\nNote that $\\Psi_i$ is a fine unstable Adams operation for all $i$. Also, for each $i \\geq 0$ the sets $\\Gamma_i$ and $\\Omega_i$ are in one-to-one correspondence with each other for all $i \\geq 0$. The bijection is given in one direction by $P \\mapsto P^{\\bullet}$, and in the reverse direction by $R \\mapsto R \\cap S_i$. Also note that $\\Gamma_i \\cap \\mathrm{Ob}(\\mathcal{L}_i) = \\emptyset$ for all $i \\geq 0$.\n\n\\end{hyp}\n\n\\begin{prop}\\label{fix2-1}\n\nFor all $i \\geq 0$, $\\mathcal{L}_i$ is a linking system, and $P$ is $\\mathcal{F}_i$-centric for each $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$.\n\n\\end{prop}\n\n\\begin{proof}\n\nBy definition of $\\mathcal{L}_i$, for all $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$ we have $\\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) = \\varepsilon_P(Z(P^{\\bullet}))$ by property (1) in \\ref{expl1}, and thus $\\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\mathcal{F}_i}(P)) = \\varepsilon_P(Z(P^{\\bullet}) \\cap S_i) = \\varepsilon_i(Z(P))$. Furthermore, we have $\\varepsilon_i(C_{S_i}(P))\\leq \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\mathcal{F}_i}(P))$ by axiom (C) of transporter systems on $\\mathcal{L}_i$, and the statement follows immediately.\n\\end{proof}\n\n\\begin{prop}\\label{fix4}\n\nLet $i \\geq 0$ and let $P\\leq S_i$ be such that $P \\lneqq P^{\\bullet} \\cap S_i$. Then\n$$\n\\mathrm{Out}_{S_i}(P) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(P)) \\neq 1.\n$$\nIn particular, $P$ is not $\\mathcal{F}_i$-centric $\\mathcal{F}_i$-radical.\n\n\\end{prop}\n\n\\begin{proof}\n\nSuppose $P$ is $\\mathcal{F}_i$-centric. Since $P \\notin \\Gamma_i$, we have $P \\lneqq P^{\\bullet} \\cap S_i \\stackrel{def} = Q$. Notice that the functor $(-)^{\\bullet}$ induces an inclusion $\\mathrm{Aut}_{\\mathcal{F}_i}(P)\\leq \\mathrm{Aut}_{\\mathcal{F}_i}(Q)$. Consider the subgroup $A = \\{c_x \\in \\mathrm{Aut}_{\\mathcal{F}_i}(P) \\, | \\, x \\in N_Q(P)\\}$. Via the above inclusion, we have $A = \\mathrm{Aut}_{\\mathcal{F}_i}(P) \\cap \\mathrm{Inn}(Q)$. Since $P \\lneqq Q$, it follows that $P \\lneqq N_Q(P)$, and hence $\\mathrm{Inn}(P) \\lneqq A$, since $P$ is $\\mathcal{F}_i$-centric by hypothesis. The group $\\mathrm{Aut}_{\\mathcal{F}_i}(P)$, seen as a subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(Q)$, normalizes $\\mathrm{Inn}(Q)$, and thus $A \\lhd \\mathrm{Aut}_{\\mathcal{F}_i}(P)$, and\n$$\n\\{1\\} \\neq A\/\\mathrm{Inn}(P)\\leq \\mathrm{Out}_{S_i}(P) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(P)).\n$$\nThis finishes the proof.\n\\end{proof}\n\n\\begin{prop}\\label{fix5}\n\nAssume Hypothesis \\ref{hyp2}. Then, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, the following holds: if $P \\in \\Gamma_i$, then either $P$ is not $\\mathcal{F}_i$-centric or $P$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\n\n\\end{prop}\n\n\\begin{proof}\n\nWe start with some general observations, after which we deduce a certain condition ($\\ddagger$) which will imply the statement. The rest of the proof consists of a series of steps to show that ($\\ddagger$) holds.\n\nBy Proposition \\ref{3.2BLO3} (i), the set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet})$ contains finitely many $\\mathcal{F}$-conjugacy classes, and the same applies to $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) \\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. Let $\\mathcal{P} = \\{X_1, \\ldots, X_n\\}$ be a set of representatives of the $\\mathcal{F}$-conjugacy classes in $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) \\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. By Proposition \\ref{fix1} (i), there exists some $M' \\in \\mathbb{N}$ such that $\\mathcal{P} \\subseteq \\Omega_i$ for all $i \\geq M'$, and we can assume that $M' = 0$ without loss of generality.\n\nFor each $X \\in \\mathcal{P}$, fix also a set $\\mathcal{H}_X = \\{Y_1, \\ldots, Y_m\\}$ of representatives of the $T$-conjugacy classes in $X^{\\mathcal{F}}$. Again, there exists some $M'' \\in \\mathbb{N}$ such that $\\mathcal{H}_X \\subseteq \\Omega_i$ for all $i \\geq M''$, and once more we can assume $M'' = 0$ for simplicity.\n\nLet $i \\geq 0$, and let $Q\\leq S_i$. Then, $T_{Q^{\\bullet}} \\cap Q \\lhd Q$, and every automorphism in $\\mathcal{F}_i$ of $Q$ restricts to an automorphism of $T_{Q^{\\bullet}} \\cap Q$ (by the properties of $(-)^{\\bullet}$, see \\ref{3.2BLO3}). Set\n$$\nK_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_{Q^{\\bullet}} \\cap Q) \\times \\mathrm{Aut}(Q\/(T_{Q^{\\bullet}} \\cap Q))),\n$$\nwhich is a normal $p$-subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(Q)$ by Lemma \\ref{Kpgp}. Since $\\Gamma_i$ and $\\Omega_i$ are in one-to-one correspondence (see \\ref{hyp2}), in order to prove the statement it is enough to prove the following, slightly stronger statement for each $X \\in \\mathcal{P}$ and each $Y \\in \\mathcal{H}_X$.\n\\begin{itemize}\n\n\\item[($\\ddagger$)] There exists some $M_Y \\in \\mathbb{N}$ such that the following holds for all $i \\geq M_Y$: if $R \\in Y^T \\cap \\Omega_i$, then either $R \\cap S_i$ is not $\\mathcal{F}_i$-centric, or $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that $K_Q \\cap \\mathrm{Aut}_S(Q)$ contains some element which is not in $\\mathrm{Inn}(Q)$.\n\n\\end{itemize}\n\nIndeed, suppose that ($\\ddagger$) holds for all $X \\in \\mathcal{P}$ and all $Y \\in \\mathcal{H}_X$. We claim that the statement follows with $M = \\max\\{M_Y \\, | \\, X \\in \\mathcal{P} \\mbox{ and } Y \\in \\mathcal{H}_X\\}$. To prove this, let $P \\in \\Gamma_i$, and let $R = P^{\\bullet} \\in \\Omega_i$. Then, there exist some $X \\in \\mathcal{P}$ and some $Y \\in \\mathcal{H}_X$ such that $R \\in Y^T \\cap \\Omega_i$. Thus, ($\\ddagger$) applies to $Y$, and it follows that either $P = R \\cap S_i$ is not $\\mathcal{F}_i$-centric, or $P = R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that $K_Q \\cap \\mathrm{Aut}_S(Q)$ contains some element which is not in $\\mathrm{Inn}(Q)$, in which case we have\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\n\nFor the rest of the proof, fix $X \\in \\mathcal{P}$ and $\\mathcal{H}_X$ as above. Since this proof is rather long, we have divided it into several steps, for the reader's convenience. We also include a brief summary of the steps in the proof. In Step 1, we give a general tool to deduce that ($\\ddagger$) holds in some cases. In Step 2 we show that we may reduce to prove that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ with $T_Y$ fully $\\mathcal{F}$-normalized. In Step 3 we justify the reduction to assume that $A = T_X$ is normal in $\\mathcal{F}$. In Step 4 we show some properties regarding the quotient $\\mathcal{G}\/A = (S\/A, \\mathcal{F}\/A, \\mathcal{L}\/A)$. In Step 5 we introduce a certain subgroup $Z_Y$ for each $Y \\in \\mathcal{H}_X$, related to $C_{T\/A}(Y\/A)$, and prove some of its properties. In Step 6, we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $Z_Y \\not\\leq Y$. In Step 7 we prove that we may reduce to prove ($\\ddagger$) for all $Y \\in \\mathcal{H}_X$ such that $Z_Y$ is fully $\\mathcal{F}$-normalized. In Step 8 we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $C_S(Y) \\not\\leq Y$ (in particular, this applies to $Y \\in \\mathcal{H}_X$ such that $C_S(Y) \\not\\leq Y$ and such that $Z_Y$ is fully $\\mathcal{F}$-normalized). In step 9 we prove a technical property necessary for the last step of the proof. Finally, in Step 10 we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $Z_Y$ is fully $\\mathcal{F}$-normalized.\n\n\\textbf{Step 1.} Let $Y, Y' \\in \\mathcal{H}_X$, and suppose that ($\\ddagger$) holds for $Y$. Suppose in addition that there exists some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ such that $f(Y') = Y$. Then ($\\ddagger$) holds for $Y'$.\n\nLet $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ as above. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist subgroups $A_0 = Y'T, A_1, \\ldots, A_n = YT\\leq S$, objects $B_1, \\ldots, B_n \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$, and automorphisms $\\phi_k \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(B_k)$, for $k = 1, \\ldots, n$, such that\n$$\nA_{k-1}, A_k\\leq B_k \\qquad \\mbox{and} \\qquad \\rho(\\phi_k)(A_{k-1}) = A_k\n$$\nfor each $k = 1, \\ldots, n$, and such that $f = \\rho(\\phi_n) \\circ \\ldots \\rho(\\phi_1)$. By Proposition \\ref{fix1} (i) and (ii), there exists some $M_1 \\in \\mathbb{N}$ such that, for all $i \\geq M_1$ and all $k = 1, \\ldots, n$,\n$$\nB_k \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i) \\qquad \\mbox{and} \\qquad (\\phi_k)|_{B_k \\cap S_i} \\in \\mathrm{Aut}_{\\mathcal{L}_i}(B_k \\cap S_i).\n$$\nMoreover, since $T\\leq A_0$, it follows that $T\\leq A_k, B_k$ for all $k = 1, \\ldots, n$. Note that for each $k$, there exists some $t_k \\in T$ such that $t_k A_k t_k^{-1} \\in \\mathcal{H}_X$ (in particular, $t_0 = t_n = 1$). Since $T\\leq B_k$ for each $k$, we may replace $\\phi_k$ by $\\varepsilon(t_k) \\circ \\phi_k \\circ \\varepsilon(t_{k-1})$, and this way we may assume that $A_k \\in \\mathcal{H}_X$ for each $k$.\n\nSuppose now that ($\\ddagger$) holds for $Y$. That is, there exists some $M_Y \\in \\mathbb{N}$ (we may choose $M_Y \\geq M_1$) such that, for all $i \\geq M_Y$, the following holds: if $R \\in Y^T \\cap \\Omega_i$, then $R \\cap S_i$ satisfies the conclusion of ($\\ddagger$). Let $i \\geq M_Y$, and let $K \\in (Y')^T \\cap \\Omega_i$, and let $H = f(K)$. Since $f \\colon Y'T \\to YT$, it follows that $H \\in Y^T$. Moreover, $f(K \\cap S_i) = H \\cap S_i$ (since each $\\phi_k$ above is fixed by $\\Psi_i$). Thus, $K \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $H \\cap S_i$, and ($\\ddagger$) holds for $Y'$.\n\n\\textbf{Step 2.} We show that if ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $T_Y$ is fully $\\mathcal{F}$-normalized, then ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$.\n\nIndeed, let $Y' \\in \\mathcal{H}_X$, and suppose that $T_{Y'}$ is not fully $\\mathcal{F}$-normalized. Then, there exists some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(T_{Y'}), S)$ such that $\\gamma(T_{Y'})$ is fully $\\mathcal{F}$-normalized. Since $Y'\\leq N_S(T_{Y'})$, we may define $Y = \\gamma(Y')$. Since $\\mathcal{H}_X$ contains representatives of all the $T$-conjugacy classes in $X^{\\mathcal{F}}$, there is some $t \\in T$ such that $tYt^{-1} \\in \\mathcal{H}_X$. Thus, upon replacing $\\gamma$ by $c_t \\circ \\gamma$, we may assume that $Y \\in \\mathcal{H}_X$. Note that $T_{Y} = \\gamma(T_{Y'})$. Notice also that $T\\leq N_S(T_{Y'})$, and thus $\\gamma$ restricts to some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ such that $Y = \\gamma(Y')$. The claim follows by Step 1.\n\n\\textbf{Step 3.} Let $\\mathcal{M} = \\{A = T_Y\\leq T \\, | \\, Y \\in \\mathcal{H}_X \\mbox{ and $T_Y$ is fully $\\mathcal{F}$-normalized}\\}$. Since $\\mathcal{H}_X$ is finite, so is $\\mathcal{M}$. For each $A \\in \\mathcal{M}$, let $\\mathcal{H}_{X,A} = \\{U \\in \\mathcal{H}_X \\, | \\, T_U = A\\} \\subseteq \\mathcal{H}_X$. To prove the statement of \\ref{fix5}, it is enough to prove that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_{X,A}$, for a fixed $A \\in \\mathcal{M}$ at a time. Fix $A \\in \\mathcal{M}$, and let $\\mathcal{H}_{X,A}$ be as above. The main goal of this step is to justify the reduction to the case where $A$ is normal in $\\mathcal{F}$.\n\nSince $A$ is fully $\\mathcal{F}$-normalized, to prove that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_{X,A}$,  we can reduce to work on the normalizer $p$-local compact group $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ defined in \\ref{rmknorm}, instead of the whole $\\mathcal{G} = (S, \\FF, \\LL)$. Set for short $N = N_S(A)$, $\\mathcal{E} = N_{\\mathcal{F}}(A)$, and $\\mathcal{T} = N_{\\mathcal{L}}(A)$, and note that $T\\leq N$ since $A\\leq T$. Let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair for $\\mathcal{G}$ fixed in \\ref{hyp2}, and let $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$ be the finite retraction pair for $N_{\\mathcal{G}}(A)$ described in \\ref{expl3}. Recall that $P^{\\star} = P^{\\bullet}$ for all $P\\leq N$. Let also $\\widetilde{\\LL}$ and $\\4{\\mathcal{T}}$ be the corresponding associated telescopic transporter systems.\n\nWe start by stating and proving several general properties.\n\\begin{itemize}\n\n\\item[(3-a)] Let $U \\in \\mathcal{H}_{X,A}$ and let $R \\in U^T$. Then, $T_R = A$, $N_S(T_R) = N_S(A)$, and $N_S(R)\\leq N_S(A)$. Moreover, every automorphism of $R$ preserves $T_R$, and thus $\\mathrm{Aut}_{\\mathcal{F}}(R) = \\mathrm{Aut}_{\\mathcal{E}}(R)$.\n\n\\item[(3-b)] Since $A\\leq T$ is a subtorus, it follows that $\\Psi_i(A) = A$ for all $i \\geq 0$. Thus, each $\\Psi_i$ restricts to a fine unstable Adams operation (see \\ref{uAo}) on $N_{\\mathcal{G}}(A)$, which extends to $\\4{\\mathcal{T}}$ by \\ref{extendL2}. Let us denote by $\\Psi_i$ the resulting unstable Adams operation.\n\n\\end{itemize}\nConsider the family of transporter systems $\\{(N_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ associated to $(N, \\mathcal{E}, \\4{\\mathcal{T}})$ in \\ref{Li}. As pointed out in \\ref{expl3}, $\\4{\\mathcal{T}}$ is not a subcategory of $\\widetilde{\\LL}$, and thus it is hard to compare the fusion systems $\\mathcal{E}_i$ and $\\mathcal{F}_i$. However, if we restrict to the full subcategory $\\4{\\mathcal{T}}_{\\geq A} \\subseteq \\4{\\mathcal{T}}$ of subgroups $P\\leq N$ such that $A\\leq P^{\\star} = P^{\\bullet}$, then $\\4{\\mathcal{T}}_{\\geq A}$ is a subcategory of $\\widetilde{\\LL}$, by \\ref{expl3} (a). In particular we have the following.\n\\begin{itemize}\n\n\\item[(3-c)] For all $i \\geq 0$, there is an inclusion $\\4{\\mathcal{T}}_{\\geq A} \\cap \\mathcal{T}_i \\subseteq \\mathcal{L}_i$.\n\n\\end{itemize}\nFinally, note that $A^{\\bullet} = A$, since $A$ is the maximal torus of $U$, for some $U \\in \\mathcal{H}_{X,A}$, and $U = U^{\\bullet}$ by hypothesis. The following holds.\n\\begin{itemize}\n\n\\item[(3-d)] For each $i \\geq 0$, let $A_i = A \\cap S_i$. By \\ref{fix1} (i), there exists some $M_2 \\in \\mathbb{N}$ such that $(A_i)^{\\bullet} = A$ for all $i \\geq M_2$. For simplicity we may assume that $M_2 = 0$.\n\n\\item[(3-e)] For each $U \\in \\mathcal{H}_{X,A}$ and each $R \\in U^T \\cap \\Omega_i$, we have $R \\cap N_i = R \\cap S_i$ and $\\mathrm{Aut}_{N_i}(R \\cap S_i) = \\mathrm{Aut}_{S_i}(R \\cap S_i)$. This follows since $N_S(R \\cap S_i)\\leq N_S(R)\\leq N_S(A) = N$ by (3-a).\n\n\\item[(3-f)] Let $H,K\\leq N_i$ be such that $A\\leq H^{\\bullet}, K^{\\bullet}$. Then, for all $f \\in \\mathrm{Hom}_{\\mathcal{E}}(H,K)$, we have $(f)^{\\star}_{\\mathcal{E}} = (f)^{\\bullet}_{\\mathcal{F}}$. Similarly, if $H, K \\in \\mathrm{Ob}(\\4{\\mathcal{T}}_{\\geq A})$ and $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(H,K)$, then $H, K \\in \\mathrm{Ob}(\\widetilde{\\LL})$, and $(\\varphi)^{\\star}_{\\4{\\mathcal{T}}} = (\\varphi)^{\\bullet}_{\\widetilde{\\LL}}$. The first part follows by definition of $(-)^{\\star}_{\\mathcal{E}}$ in \\ref{expl3}, and the second part follows by \\ref{expl3} (c).\n\n\\end{itemize}\nIn fact, we deduce more. Let $W\\leq N_i$ be such that $A\\leq W^{\\bullet}$. Then, the following holds.\n\\begin{itemize}\n\n\\item[(3-g)] $\\mathrm{Hom}_{\\mathcal{E}_i}(W, N_i) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_i}(W, N_i)$.\n\n\\end{itemize}\nLet $f \\in \\mathrm{Hom}_{\\mathcal{E}_i}(W, N_i)$. Since $\\mathcal{E}_i$ is $\\mathrm{Ob}(\\mathcal{T}_i)$-generated, there exist objects $H_k, H_k' \\in \\mathrm{Ob}(\\mathcal{T}_i)$ and morphisms $\\gamma_k \\in \\mathrm{Hom}_{\\mathcal{T}_i}(H_k, H'_k)$, for $k = 1, \\ldots, n$, and such that, upon taking the necessary restrictions,\n$$\nf = \\4{\\rho}_i(\\gamma_n) \\circ \\ldots \\circ \\4{\\rho}_i(\\gamma_1).\n$$\nNotice that $(A_i)^{\\bullet} = A\\leq W^{\\bullet}$. Thus, $A\\leq W^{\\bullet}\\leq (H_1)^{\\bullet}$, and $A = \\4{\\rho}(\\gamma_1)(A)\\leq (H_1')^{\\bullet}$ since $\\gamma_1 \\in \\mathrm{Mor}(\\4{\\mathcal{T}})$. Inductively, it follows that $A\\leq (H_k)^{\\bullet}, (H'_k)^{\\bullet}$ for all $k$, and thus $H_k, H_k' \\in \\mathrm{Ob}(\\4{\\mathcal{T}}_{\\geq A})$. By (3-c) above, for all $k = 1, \\ldots, n$ we have $H_k, H'_k \\in \\mathcal{L}_i$, and $\\gamma_k \\in \\mathrm{Mor}(\\mathcal{L}_i)$. Thus $f \\in \\mathrm{Mor}(\\mathcal{F}_i)$, and (3-g) follows. In particular, if $U \\in \\mathcal{H}_{X,A}$, and $P \\in U^{\\mathcal{E}} \\cap \\Omega_i$, then\n$$\n\\mathrm{Hom}_{\\mathcal{E}_i}(P \\cap S_i, N_i) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_i}(P \\cap S_i, N_i).\n$$\n\nLet $U \\in \\mathcal{H}_{X,A}$. For each $P \\in U^{\\mathcal{E}} \\cap \\Omega_i$, note that $A \\cap S_i \\lhd P \\cap S_i$, and every automorphism of $P \\cap S_i$ in $\\mathcal{F}_i$ restricts to an automorphism of $A \\cap S_i$. Let $\\3{P} = P\/A$. Note that $P = (P \\cap S_i)A$ (since $P \\in \\Omega_i$), and thus we have $(P \\cap S_i)\/(A \\cap S_i) \\cong \\3{P}$. Set for short $Q = P \\cap S_i$, and let\n$$\nK_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(A \\cap S_i) \\times \\mathrm{Aut}(\\3{P}));\n$$\n$$\nK'_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{E}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{E}_i}(A \\cap S_i) \\times \\mathrm{Aut}(\\3{P})).\n$$\nBy Lemma \\ref{Kpgp}, we have $K_Q\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}_i}(Q))$ and $K'_Q\\leq O_p(\\mathrm{Aut}_{\\mathcal{E}_i}(Q))$. Moreover, by (3-g) there is an inclusion $\\mathrm{Aut}_{\\mathcal{E}_i}(Q)\\leq \\mathrm{Aut}_{\\mathcal{F}_i}(Q)$, and it follows that $K'_Q\\leq K_Q$. We claim that, in order to prove that ($\\ddagger$) holds for $U \\in \\mathcal{H}_{X,A}$, it is enough to prove that the following version of ($\\ddagger$) in terms of $\\mathcal{E}_i$ holds.\n\\begin{itemize}\n\n\\item[(3-h)] There exists some $M_U \\in \\mathbb{N}$ such that, for all $i \\geq M_U$ and all $R \\in U^T \\cap \\Omega_i$, either $R \\cap S_i$ is not $\\mathcal{E}_i$-centric, or $R \\cap S_i$ is $\\mathcal{E}_i$-conjugate to some $Q$ such that $K'_{Q} \\cap \\mathrm{Aut}_{N_i}(Q)$ contains some element that is not in $\\mathrm{Inn}(Q)$.\n\n\\end{itemize}\nIndeed, suppose that (3-h) holds for $U$, and let $i \\geq M_U$ and $R \\in U^T \\cap \\Omega_i$. Clearly, if $R \\cap S_i$ is not $\\mathcal{E}_i$-centric, then it is not $\\mathcal{F}_i$-centric by (3-g). Suppose that $R \\cap S_i$ is $\\mathcal{E}_i$-centric. Then, $R \\cap S_i$ is $\\mathcal{E}_i$-conjugate to some $Q$ such that $K'_Q \\cap \\mathrm{Aut}_{N_i}(Q)$ contains some element that is not in $\\mathrm{Inn}(Q)$. Then, $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $Q$ by (3-g), and ($\\ddagger$) holds for $U$ since $K'_Q \\cap \\mathrm{Aut}_{N_i}(Q)\\leq K_Q \\cap \\mathrm{Aut}_{S_i}(Q)$.\n\nThe above just shows that, in order to prove ($\\ddagger$) for $U \\in \\mathcal{H}_{X, A}$ with respect to $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$, it is enough to prove that the conclusion ($\\ddagger$) holds for $U$ with respect to $\\{(N_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$. For the rest of the proof, we assume that $A = T_X$ is normal in $\\mathcal{F}$, so $\\mathcal{H}_{X, A} = \\mathcal{H}_X$ (and $T_R = A$ for all $R \\in X^{\\mathcal{F}}$). Moreover, notice that we are only concerned about subgroups $H\\leq S$ such that $A\\leq H^{\\bullet}$, and for these subgroups there is no difference between the finite retraction pairs $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ and $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$, by (3-f). Thus, we may assume also that we are still working with $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$.\n\nTo finish this step, we prove the following.\n\\begin{itemize}\n\n\\item[(3-i)] If $A = T$ then ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$.\n\n\\end{itemize}\nIndeed, in this case we have $Y^T = \\{Y\\}$ for each $Y \\in \\mathcal{H}_X$. Since $\\mathcal{H}_X$ is finite, it is not hard to check in this case that in fact there exists some $M_X \\in \\mathbb{N}$ such that, for all $i \\geq M_X$ and all $Y \\in \\mathcal{H}_X$, the subgroup $Y \\cap S_i$ is not $\\mathcal{F}_i$-centric.\n\n\\textbf{Step 4.} Since $A = T_X$ is normal in $\\mathcal{F}$, consider the quotient $(S\/A, \\mathcal{F}\/A, \\widetilde{\\LL}\/A)$. By Lemma \\ref{quotient22}, $\\mathcal{F}\/A$ is a saturated fusion system over $S\/A$, and $\\widetilde{\\LL}\/A$ is a transporter system associated to $\\mathcal{F}\/A$ which contains all the centric subgroups of $\\mathcal{F}\/A$. In this step we prove several properties relating $(S, \\mathcal{F}, \\widetilde{\\LL})$ to $(S\/A, \\mathcal{F}\/A, \\widetilde{\\LL}\/A)$, which we will need in later steps.\n\nSet for short $\\3{S} = S\/A$, $\\3{\\mathcal{F}} = \\mathcal{F}\/A$ and $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$. Let also $\\widetilde{\\LL}_{\\geq A} \\subseteq \\widetilde{\\LL}$ be the full subcategory of subgroups that contain $A$, and let $\\tau \\colon \\widetilde{\\LL}_{\\geq A} \\to \\3{\\mathcal{T}}$ be the projection functor. By a slight abuse of notation, we also write $\\tau \\colon S \\to \\3{S}$ for the projection homomorphism. We adopt the notation $\\3{P}, \\3{Q}, \\ldots$ to denote subgroups of $\\3{S}$, $\\3{f}, \\3{f}', \\ldots$ to denote morphisms in $\\3{\\mathcal{F}}$, and $\\3{\\varphi}, \\3{\\psi}, \\ldots$ to denote morphisms in $\\3{\\mathcal{T}}$. In particular, $T\/A = \\3{T}\\leq \\3{S}$ denotes the maximal torus of $\\3{S}$. Also, for each $P\\leq S$ that contains $A$, we will write $\\3{P}$ instead of $\\tau(P) = P\/A$, unless there is risk of confusion.\n\nThe following is easily check to hold since $A\\leq T$ is normal in $\\mathcal{F}$.\n\\begin{itemize}\n\n\\item[(4-a)] Let $P, Q\\leq S$ be such that $A\\leq P,Q$. Then, $Q \\in P^{\\mathcal{F}}$ if and only if $\\3{Q} \\in \\3{P}^{\\3{\\mathcal{F}}}$. Similarly, $Q \\in P^T$ if and only if $\\3{Q} \\in \\3{P}^{\\3{T}}$, since $A\\leq T$.\n\n\\end{itemize}\n\nBy property (3-b), $\\Psi_i(A) = A$ for all $i \\geq 0$. By (3-i) we may assume that $A \\lneqq T$, and thus, for each $i \\geq 0$, the unstable Adams operation $\\Psi_i$ induces a fine unstable Adams (see \\ref{uAo}) operation $\\3{\\Psi}_i$ on $\\3{\\mathcal{T}}$. By definition, if $\\varphi \\in \\mathrm{Mor}(\\widetilde{\\LL})$ is such that $\\Psi_i(\\varphi) = \\varphi$ for some $i$, then $\\3{\\Psi}_i(\\3{\\varphi}) = \\3{\\varphi}$. The following is some sort of converse of this statement.\n\\begin{itemize}\n\n\\item[(4-b)] Let $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be such that $A\\leq P, Q$, and let $\\3{\\varphi} \\in \\mathrm{Mor}_{\\3{\\mathcal{T}}}(\\3{P}, \\3{Q})$ be such that $\\3{\\Psi}_i(\\3{\\varphi}) = \\3{\\varphi}$. Then, there exists some $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ such that $\\tau(\\varphi) = \\3{\\varphi}$ and such that $\\Psi_i(\\varphi) = \\varphi$. Similarly, let $\\3{x} \\in \\3{S}$ be such that $\\3{\\Psi}_i(\\3{x}) = \\3{x}$. Then there exists some $x \\in S$ such that $\\Psi_i(x) = x$ (i.e. $x \\in S_i$) and $\\tau(x) = \\3{x}$.\n\n\\end{itemize}\nLet $\\3{\\varphi}$ be as above, and let $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ be such that $\\tau(\\varphi) = \\3{\\varphi}$. Since $\\tau(\\varphi) = \\3{\\varphi} = \\3{\\Psi}_i(\\3{\\varphi}) = \\tau(\\Psi_i(\\varphi))$, it follows that $\\Psi_i(\\varphi) = \\varphi \\circ \\varepsilon_P(a)$, for some $a \\in A$ (see Definition \\ref{quotient1}). Consider the map $A \\to A$ defined by $t \\mapsto t^{-1} \\Psi_i(t)$. Since $A$ is abelian, this turns out to be a group homomorphism, which is in fact surjective, since $\\mathrm{Ker}(A)$ is the subgroup of fixed points of $A$, and this is a finite subgroup of $A$ for all $i \\geq 0$. In particular, there exists some $t \\in A$ such that $t^{-1} \\Psi_i(t) = a^{-1}$, and we get\n$$\n\\Psi_i(\\varphi \\circ \\varepsilon_P(t)) = \\Psi_i(\\varphi) \\circ \\varepsilon_P(\\Psi_i(t)) = \\varphi \\circ \\varepsilon_P(a) \\circ \\varepsilon_P(\\Psi_i(t)) = \\varphi \\circ \\varepsilon_P(t).\n$$\nA similar argument shows that, for each $\\3{x} \\in \\3{S}$ such that $\\3{\\Psi}_i(\\3{x}) = \\3{x}$, there is some $x \\in S_i$ such that $\\tau(x) = \\3{x}$.\n\nFor each $i \\geq 0$, let $\\3{S}_i\\leq \\3{S}$ be the subgroup of fixed points by $\\3{\\Psi}_i$. By (4-b), we deduce that $S_iA\/A = \\3{S}_i$, and thus $S_i\/(S_i \\cap A) \\cong \\3{S}_i$. Let $U \\in \\mathcal{H}_X$. By (4-a), $R \\in U^T$ if and only if $\\3{R} \\in \\3{U}^{\\3{T}}$.\n\\begin{itemize}\n\n\\item[(4-c)] For all $i \\geq 0$ and all $R \\in U^{T}$, $R \\in \\Omega_i$ if and only if $\\3{R}\\leq \\3{S}_i$.\n\n\\end{itemize}\nSuppose first that $R \\in \\Omega_i$, and let $P = R \\cap S_i$. Then, $R = PA$ (since $A = T_R$), and $\\3{R} = PA\/A\\leq S_iA\/A = \\3{S}_i$. Conversely, let $R \\in U^{T}$ be such that $\\3{R}\\leq \\3{S}_i$. Then, it follows by (4-b) that $P = R \\cap S_i$ contains representatives of all the elements in $\\3{R}$. Since $A_i = A \\cap S_i\\leq R \\cap S_i = P$ and $(A_i)^{\\bullet} = A$, it follows that $P^{\\bullet} \\geq R$. The inclusion $P^{\\bullet}\\leq R$ follows from $P = R \\cap S_i\\leq R$ and the fact that $R^{\\bullet} = R$. Thus $R \\in \\Omega_i$.\n\n\\textbf{Step 5.} For each $R \\in X^{\\mathcal{F}}$, recall the notation $\\3{R} = R\/A$. Set also\n$$\nZ_{\\3{R}} \\stackrel{def} = C_{\\3{T}}(\\3{R}) \\qquad \\mbox{and} \\qquad Z_R \\stackrel{def} = \\{t \\in N_T(R) \\, | \\, \\tau(t) \\in Z_{\\3{R}}\\}.\n$$\nNote that $A\\leq Z_R$ and $Z_{\\3{R}} = Z_R\/A$. The main goal of this step is to show the following: if $Z_U$ is fully $\\mathcal{F}$-normalized for some $U \\in \\mathcal{H}_X$, then $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized.\n\nIndeed, let $U \\in \\mathcal{H}_X$ be such that $Z_U$ is fully $\\mathcal{F}$-normalized. Note that $T\\leq N_S(Z_U)$, and\n$$\n\\3{T}\\leq N_{\\3{S}}(Z_{\\3{U}}) = N_S(Z_U)\/A.\n$$\nSuppose that $Z_{\\3{U}}$ is not fully $\\3{\\mathcal{F}}$-normalized, and let $\\3{\\gamma} \\in \\mathrm{Hom}_{\\3{\\mathcal{F}}}(N_{\\3{S}}(Z_{\\3{U}}), \\3{S})$ be such that $\\3{H} = \\3{\\gamma}(Z_{\\3{U}})$ is fully $\\3{\\mathcal{F}}$-normalized, and $|N_{\\3{S}}(Z_{\\3{U}})| < |N_{\\3{S}}(\\3{\\gamma}(Z_{\\3{U}}))|$. Then $\\3{\\gamma}$ lifts to a map $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(Z_U), S)$ such that $|N_S(Z_U)| = |\\gamma(N_S(Z_U))| < |N_S(\\gamma(Z_U))|$, since\n$$\n\\3{T}\\leq N_{\\3{S}}(\\3{\\gamma}(Z_{\\3{U}})) = N_S(\\gamma(Z_U))\/A,\n$$\nand this contradicts the maximality of $|N_S(Z_U)|$.\n\n\\textbf{Step 6.}  We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U \\not\\leq U$.\n\nLet $U \\in \\mathcal{H}_X$ be such that $Z_U \\not\\leq U$, and let $a \\in Z_U \\setminus U$. Then, by Lemma \\ref{SiS} there exists some $M_U \\in \\mathbb{N}$ such that $a \\in S_i$ for all $i \\geq M_U$. Fix some $i \\geq M_U$, and let $R \\in U^T \\cap \\Omega_i$. Since $R$ is $T$-conjugate to $U$, we have $Z_R = Z_U$. Let\n$$\nK_{R \\cap S_i} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_R \\cap S_i) \\times \\mathrm{Aut}(\\3{R})).\n$$\nBy Lemma \\ref{Kpgp} we know that $K_{R \\cap S_i}\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i))$.\n\nThe element $a \\in Z_U \\cap S_i$ fixed above satisfies $a \\in Z_R \\cap S_i$, since $Z_R = Z_U$, and in particular $a \\in N_{S_i}(R \\cap S_i)$. Moreover, $c_a \\in K_{R \\cap S_i} \\cap \\mathrm{Aut}_{S_i}(R \\cap S_i)$, since $a \\in T \\cap S_i$. In particular, if $R \\cap S_i$ is $\\mathcal{F}_i$-centric then $c_a \\notin \\mathrm{Inn}(R \\cap S_i)$, and this shows that ($\\ddagger$) holds for $U$.\n\n\\textbf{Step 7.} Suppose ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. Then we claim that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$.\n\nLet $U \\in \\mathcal{H}_X$ be such that $Z_U$ is not fully $\\mathcal{F}$-normalized. Then there exists some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(Z_U), S)$ such that $\\gamma(Z_U)$ is fully $\\mathcal{F}$-normalized. Since $U\\leq N_S(Z_U)$, we may set $V = \\gamma(U)$. Moreover, since $\\mathcal{H}_X$ contains representatives of all the $T$-conjugacy classes in $X^{\\mathcal{F}}$, we may assume that $V \\in \\mathcal{H}_X$. Finally, note that $T\\leq N_S(Z_U)$, since $Z_U\\leq T$. Thus, $Z_V = \\gamma(Z_U)$.\n\nSuppose now that ($\\ddagger$) holds for all $V \\in \\mathcal{H}_X$ such that $Z_V$ is fully $\\mathcal{F}$-normalized, and let $U \\in \\mathcal{H}_X$. By the above discussion, there exists some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(UT, S)$ such that $V = f(U) \\in \\mathcal{H}_X$ is such that $Z_V$ is fully $\\mathcal{F}$-normalized, and the claim follows by Step 1.\n\n\\textbf{Step 8.} We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $C_S(U) \\not\\leq U$. \n\nBy Step 6, if $Z_U \\not\\leq U$ then ($\\ddagger$) holds for $U$. Thus we may assume that $Z_U\\leq U$. Also, since $C_S(U) \\not\\leq U$, it follows that $C_{\\3{S}}(\\3{U}) \\not\\leq \\3{U}$. Notice that $C_{\\3{T}}(\\3{U}) = Z_{\\3{U}}\\leq \\3{U}$ and $\\3{U}$ is finite, which implies that $C_{\\3{S}}(\\3{U})$ is a finite group. Thus, there exists some $M_U \\in \\mathbb{N}$ such that $C_{\\3{S}}(\\3{U})\\leq \\3{S}_i$. Moreover, since $U \\in \\Omega_i$ by assumption, we have $\\3{U}\\leq \\3{S}_i$ by (4-c).\n\nLet $i \\geq M_U$ and let $R \\in U^T \\cap \\Omega_i$. Then $C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}$. By (4-a) we have $\\3{R} \\in \\3{U}^{\\3{T}}$, and by (4-c) we know that $\\3{R}\\leq \\3{S}_i$. Thus, by Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iii), we have\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R}) = Z_{\\3{R}} = Z_{\\3{U}}\n$$\nfor some $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$. Fix such $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$, and note that $\\3{x}$ conjugates $C_{\\3{S}}(\\3{R})$ to $C_{\\3{S}}(\\3{U})$, and in particular $C_{\\3{S}}(\\3{R}) \\not\\leq \\3{R}$. Moreover, since $Z_{\\3{U}} = C_{\\3{T}}(\\3{R})\\leq \\3{R}$, it follows that\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R})\\leq \\3{R} \\cap \\3{T}\\leq C_{\\3{T}}(C_{\\3{S}}(\\3{R})).\n$$\nThus, by \\ref{invar1} (i) we deduce that $C_{\\3{S}}(\\3{R})\\leq \\3{S}_i$.\n\nSet $X_R = C_S(R)A$. Then, by (4-b) $X_R \\cap S_i$ contains representatives of all the elements in $C_S(R)A\/A\\leq C_{\\3{S}}(\\3{R})\\leq \\3{S}_i$. Note that $C_S(R)A \\not\\leq R$, and thus $X_R \\cap S_i$ contains elements which are not in $R \\cap S_i$. If $(X_R \\cap S_i)\\setminus (R \\cap S_i)$ contains some element of $C_S(R)$, then clearly $R \\cap S_i$ is not $\\mathcal{F}_i$-centric. Suppose then that $(X_R \\cap S_i) \\cap C_S(R)\\leq R \\cap S_i$, and let\n$$\nK_{R \\cap S_i} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_R \\cap S_i) \\times \\mathrm{Aut}(\\3{R})),\n$$\nwhich is a normal $p$-subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i)$ by Lemma \\ref{Kpgp}. If $R \\cap S_i$ is $\\mathcal{F}_i$-centric, then, for every $x \\in (X_R \\cap S_i)\\setminus(R \\cap S_i)$, we have $c_x \\in K_{R \\cap S_i} \\cap \\mathrm{Aut}_{S_i}(R \\cap S_i)$ and $c_x \\notin \\mathrm{Inn}(R \\cap S_i)$, and ($\\ddagger$) holds for $U$.\n\n\\textbf{Step 9.} Recall the notation $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$, introduced in Step 4, which is a transporter system associated to $\\3{\\mathcal{F}}$. Fix $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. Then $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized by Step 5, and in particular it is fully $\\3{\\mathcal{F}}$-centralized. Thus we may consider the centralizer fusion system $\\3{\\mathcal{E}} = C_{\\3{\\mathcal{F}}}(Z_{\\3{U}})$ over $\\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$, and note that $\\3{T}\\leq \\3{Z}$. Since $\\3{\\mathcal{F}}$ is saturated, it follows that $\\3{\\mathcal{E}}$ is also saturated. The main goal of this step is to prove the following property.\n\\begin{itemize}\n\n\\item[(9-a)] Suppose that $Z_U\\leq U$. Then, there exists $V \\in \\mathcal{H}_X$ such that $C_S(V) \\not\\leq V$ and such that $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$.\n\n\\end{itemize}\n\nFirst, note that the following holds.\n\\begin{itemize}\n\n\\item[(9-b)] For each $\\3{R} \\in \\3{U}^{\\3{T}}$, we have $Z_{\\3{R}} = C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}$.\n\n\\item[(9-c)] Since $Z_{\\3{U}}\\leq \\3{T}$ is abelian, every $\\3{\\mathcal{E}}$-centric subgroup of $\\3{Z}$ must contain $Z_{\\3{U}}$. Thus, by Lemma \\ref{centricNFKA}, every $\\3{\\mathcal{E}}$-centric subgroup of $\\3{Z}$ is also an $\\3{\\mathcal{F}}$-centric subgroup of $\\3{S}$, and hence also an object in $\\3{\\mathcal{T}}$, since $\\3{\\mathcal{T}}$ contains all the $\\3{\\mathcal{F}}$-centric subgroups of $\\3{S}$.\n\n\\end{itemize}\nWe are ready now to prove (9-a). Let $K = \\mathrm{Aut}_U(Z_U)$, and let $N_S^K(Z_U) = \\{x \\in S \\, | \\, c_x \\in K\\}$. Then,\n$$\n\\mathrm{Aut}_S^K(Z_U) = K \\cap \\mathrm{Aut}_S(Z_U) = \\mathrm{Aut}_U(Z_U) = K \\cap \\mathrm{Aut}_{\\mathcal{F}}(Z_U) = \\mathrm{Aut}_{\\mathcal{F}}^K(Z_U).\n$$\nIn particular, $Z_U$ is fully $\\mathcal{F}$-centralized (since it is fully $\\mathcal{F}$-normalized), and clearly $\\mathrm{Aut}_S^K(Z_U) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{F}}^K(Z_U))$. By \\cite[Lemma 2.2]{BLO6} it follows that $Z_U$ is fully $K$-normalized in $\\mathcal{F}$ (see Section \\ref{Squotient}). Thus, the fusion system $N_{\\mathcal{F}}^K(Z_U)$ over $N_S^K(Z_U)$ (as defined in \\ref{definorm}) is saturated.\n\nNote that $U, T\\leq N_S^K(Z_U) = U C_S(Z_U)$. Since $U$ is not $\\mathcal{F}$-centric and $Z_U\\leq U$, it follows that $U$ is not $N_{\\mathcal{F}}^K(Z_U)$-centric by Lemma \\ref{centricNFKA}. Let $V$ be $N_{\\mathcal{F}}^K(Z_U)$-conjugate to $U$ and such that $C_{N_S^K(Z_U)}(V) \\not\\leq V$. Since $T\\leq N_S^K(Z_U)$, we may choose $V \\in \\mathcal{H}_X$. Then,\n$$\nC_{N_S^K(Z_U)}(V)\\leq C_S(V) \\not\\leq V.\n$$\nAlso, if $f \\in \\mathrm{Hom}_{N_{\\mathcal{F}}^K(Z_U)}(U,V)$, then by definition $f|_{Z_U} \\in \\mathrm{Aut}_U(Z_U)$. Hence, the induced morphism $\\3{f} \\in \\mathrm{Hom}_{\\3{\\mathcal{F}}}(\\3{U}, \\3{V})$ satisfies $\\3{f}|_{Z_{\\3{U}}} = \\mathrm{Id}$, and thus $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$. Note that in particular $Z_U\\leq Z_V$, although this may not be an equality.\n\n\\textbf{Step 10.} We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized.\n\nFix $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. By Step 6, we may assume that $Z_U\\leq U$, and thus $Z_{\\3{U}}\\leq \\3{U}$. By Step 5, we know that $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized (and in particular fully $\\3{\\mathcal{F}}$-centralized). Let $\\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$ for short, and let $\\3{\\mathcal{E}} = C_{\\3{\\mathcal{F}}}(Z_{\\3{U}})$ be the centralizer fusion system of $Z_{\\3{U}}$ in $\\3{\\mathcal{F}}$, which is saturated by Step 9. By (9-a), there exists $V \\in \\mathcal{H}_X$ such that $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$ and $C_S(V) \\not\\leq V$.\n\nNotice that the set $\\{T_K \\, | \\, K \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})\\}$ is finite since $\\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ contains finitely many $T$-conjugacy classes. Moreover, since $K^{\\bullet} = K$, it follows that $(T_K)^{\\bullet} = T_K$ for all $K \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. By Proposition \\ref{fix1} (i) there exists some $M_3 \\in \\mathbb{N}$ such that $(T_K \\cap S_i)^{\\bullet} = T_K = (T_K)^{\\bullet}$ for all $i \\geq M_3$ and all $T_K$ in the set above. Without loss of generality we may assume that $M_3 = 0$.\n\nFix $\\3{f} \\in \\mathrm{Hom}_{\\3{\\mathcal{E}}}(\\3{U}, \\3{V})$. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist sequences of subgroups $\\3{W}_0 = \\3{U}, \\3{W}_1, \\ldots, \\3{W}_n = \\3{V}\\leq\\3{Z}$ and $\\3{K}_1, \\ldots, \\3{K}_n \\in \\mathrm{Ob}((\\3{\\mathcal{E}})^c) \\subseteq \\mathrm{Ob}(\\3{\\mathcal{F}})$, and morphisms $\\3{\\gamma}_j \\in \\mathrm{Aut}_{\\3{\\mathcal{E}}}(\\3{K}_j)$ for each $j = 1, \\ldots, n$, such that\n$$\n\\3{W}_{j-1}, \\3{W}_j\\leq \\3{K}_j \\qquad \\mbox{and} \\qquad \\3{\\gamma}_j(\\3{W}_{j-1}) = \\3{W}_j\n$$\nfor each $j = 1, \\ldots, n$. For each $j = 1, \\ldots, n$, let $\\3{\\varphi}_j \\in \\mathrm{Aut}_{\\3{\\mathcal{T}}}(\\3{K}_j)$ be such that $\\3{\\rho}(\\3{\\varphi}_j) = \\3{\\gamma}_j$. Let also $K_j \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be the preimage of $\\3{K}_j$, and let $\\varphi_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(K_j)$ be such that $\\tau(\\varphi_j) = \\3{\\varphi}_j$. Note that $U\\leq K_1$ and $V\\leq K_n$.\n\nWe claim that we may assume that $K_j \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ for all $j$ without loss of generality. Indeed, we have $\\3{K}_j \\in \\mathrm{Ob}(\\3{\\mathcal{T}})$ by (9-c) (where $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$), and thus $K_j \\in \\mathrm{Ob}(\\widetilde{\\LL})$. This implies that $(K_j)^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ by definition of $\\widetilde{\\LL}$. Furthermore, we have $A\\leq (K_j)^{\\bullet}$, and it follows that $(K_j)^{\\bullet}\/A\\leq \\3{Z}$, since $(K_j)^{\\bullet}\\leq K_j T$ and $\\3{T}\\leq \\3{Z}$. By Proposition \\ref{fix1} (i) and (ii), there exists some $M_U \\in \\mathbb{N}$ such that\n$$\nK_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i) \\qquad \\mbox{and} \\qquad \\Psi_i(\\varphi_j) = \\varphi_j\n$$\nfor all $i \\geq M_U$ and all $j = 1, \\ldots, n$. In particular, this implies that $U \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $V \\cap S_i$. Note that this also implies that $\\3{\\Psi}_i(\\3{\\varphi}_j) = \\3{\\varphi}_j$ for all $i \\geq M_U$.\n\nLet $i \\geq M_U$, and let $R \\in U^T \\cap \\Omega_i$. By assumption, $\\3{U}\\leq \\3{S}_i$. By (4-a) we have $\\3{R} \\in \\3{U}^{\\3{T}}$, which implies that $C_{\\3{T}}(\\3{R}) = Z_{\\3{R}} = Z_{\\3{U}}$, and by (4-c) we know that $\\3{R}\\leq \\3{S}_i$. Thus, by Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iii), we have\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}\\leq \\3{R}\n$$\nfor some $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$. Fix such $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$, set $\\3{W}'_0 = \\3{R}$, and for all $j = 1, \\ldots, n$ set\n$$\n\\3{W}'_j = \\3{x}^{-1} \\cdot \\3{W}_j \\cdot \\3{x} \\qquad \\mbox{and} \\qquad \\3{K}'_j = \\3{x}^{-1} \\cdot \\3{K}_j \\cdot \\3{x}.\n$$\nSet also $\\3{\\varphi}_j' = \\3{\\varepsilon}(\\3{x})^{-1} \\circ \\3{\\varphi}_j \\circ \\3{\\varepsilon}(\\3{x}) \\in \\mathrm{Aut}_{\\3{\\mathcal{T}}}(\\3{K}'_j)$. Let also $W_j'$ and $K_j'$ be the corresponding preimages in $S$. The following holds.\n\\begin{itemize}\n\n\\item[(10-a)] Since $Z_{\\3{U}}$ is abelian and $\\3{K}_j, \\3{K}_j'$ are $\\3{\\mathcal{E}}$-centric, it follows that $Z_{\\3{U}}\\leq \\3{K}_j, \\3{K}_j'$ for each $j$.\n\n\\item[(10-b)] Since $\\3{x} \\in \\3{T}\\leq \\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$ and $\\3{K}_j\\leq \\3{Z}$, it follows that $\\3{K}'_j\\leq \\3{Z}$. Moreover, since $\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in Z_{\\3{U}}$ and $Z_{\\3{U}}\\leq C_{\\3{T}}(\\3{K}_j), C_{\\3{T}}(\\3{K}'_j)$, it follows from Lemma \\ref{invar1} (i) that $\\3{x} \\in N_{\\3{T}}(\\3{K}'_j \\cap \\3{S}_i, \\3{K}_j \\cap \\3{S}_i)$.\n\n\\item[(10-c)] Since $Z_{\\3{U}}\\leq \\3{K}_j, \\3{K}'_j$, $\\3{\\varphi}_j \\in \\mathrm{Mor}(\\3{\\mathcal{E}})$, and $\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in Z_{\\3{U}}$, it follows from axiom (C) of transporter systems for $\\3{\\mathcal{T}}$ that $\\3{\\varepsilon}(\\3{x}^{-1} \\Psi_i(\\3{x})) \\circ \\3{\\varphi_j}' = \\3{\\varphi}'_j \\circ \\3{\\varepsilon}(\\3{x}^{-1}\\Psi_i(\\3{x}))$. Thus, $\\3{\\Psi}_i(\\3{\\varphi}'_j) = \\3{\\varphi}'_j$ by Lemma \\ref{invar1} (ii). By (4-b), this implies that there is some $\\varphi'_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(K'_j)$ such that $\\Psi_i(\\varphi'_j) = \\varphi'_j$ and $\\tau(\\varphi'_j) = \\3{\\varphi}'_j$.\n\n\\end{itemize}\nLet $\\3{R'} = \\3{W}'_n \\in \\3{V}^{\\3{T}}$, and let $R'$ be its preimage in $S$, and note that $R' \\in V^T$ by (4-a). Note that $R\\leq K'_1$ and $R'\\leq K'_n$. Thus, if $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for each $j$, then $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $R' \\cap S_i$.\n\nIt remains to show that $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for each $j$. Recall that, by definition, $K_j' \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ if $(K'_j \\cap S_i)^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L})$. For each $j = 1, \\ldots, n$, let $T_j$ be the maximal torus of $K_j$. Note that $K_j'$ is $T$-conjugate to $K_j$, so in particular $K_j' \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. As noted above, we have $(T_j \\cap S_i)^{\\bullet} = T_j$ for all $i \\geq M_U$. Also, $T_j \\cap S_i\\leq K'_j \\cap S_i$ and\n$$\n(T_j \\cap S_i)^{\\bullet} = T_j\\leq (K'_j \\cap S_i)^{\\bullet}.\n$$\nIn particular, this implies that $K'_j \\cap S_i, T_j\\leq (K'_j \\cap S_i)^{\\bullet}$. Since $(K_j \\cap S_i)^{\\bullet} = K_j$, it follows that $K_j \\cap S_i$ contains representatives of all the elements in $\\3{K}_j \\cap \\3{S}_i$. Thus, by (10-b) we deduce that $K_j' \\cap S_i$ contains representatives of all the elements in $\\3{K}'_j \\cap \\3{S}_i$, and thus $K'_j = (K'_j \\cap S_i)T_j\\leq (K'_j \\cap S_i)^{\\bullet}$. It follows that $(K'_j \\cap S_i)^{\\bullet} = K_j'$, and thus $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$.\n\nBy the above, $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $R' \\cap S_i$. Moreover, $R' \\in V^T$, and $C_S(V) \\not\\leq V$ by assumption. By Step 8, ($\\ddagger$) holds for $V$, and thus the conclusion of ($\\ddagger$) holds for $R$ and $R'$. This shows that ($\\ddagger$) holds for $U$ as well. Repeating this step, we deduce that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized, and thus ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ by Step 7. This finishes the proof.\n\\end{proof}\n\n\\begin{thm}\\label{fix6}\n\nEvery $p$-local compact group admits an approximation by $p$-local finite groups.\n\n\\end{thm}\n\n\\begin{proof}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ and $\\Psi$ be as fixed in \\ref{hyp2}. Let also $\\Psi_0 = \\Psi$, and let $\\{\\Psi_i\\}_{i \\geq 0}$ be such that $\\Psi_{i+1} = \\Psi_i^p$. Let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the family of transporter systems defined in \\ref{Li}. For simplicity, we can assume that the degree of $\\Psi$ is high enough so that Lemma \\ref{SiS} and Propositions \\ref{fix2} and \\ref{fix5} hold (otherwise replace $\\Psi$ by an appropriate power of it), and we claim that $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ is an approximation of $\\mathcal{G}$ by $p$-local finite groups.\n\nCondition (i) in \\ref{defiapprox} is satisfied, by Lemma \\ref{SiS} (i). Also, $S_i$ is a finite $p$-group for all $i \\geq 0$, and $\\mathcal{L}_i$ is a linking system associated to $\\mathcal{F}_i$ by Proposition \\ref{fix2-1}, and there are inclusions $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$ and $\\mathcal{L}_i \\subseteq \\widetilde{\\LL}$ for all $i \\geq 0$. Thus, to show that condition (ii) in \\ref{defiapprox} is satisfied, it remains to check that $\\mathcal{F}_i$ is saturated and $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_i)$. By Propositions \\ref{fix4} and \\ref{fix5}, if $P\\leq S_i$ is $\\mathcal{F}_i$-centric but $P \\notin \\mathrm{Ob}(\\mathcal{L}_i)$, then $P$ is $\\mathcal{F}_i$-conjugate to some $Q$ such that\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\nMoreover, $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated and $\\mathrm{Ob}(\\mathcal{L}_i)$-saturated by Corollary \\ref{fix3}. Thus, the conditions of Theorem \\ref{5A} are satisfied: $\\mathcal{F}_i$ is saturated, and $\\mathrm{Ob}(\\mathcal{L}_i)$ contains all the centric radical subgroups of $\\mathcal{F}_i$. Thus condition (ii) in \\ref{defiapprox} is satisfied.\n\nFinally, we have to check condition (iii): for each $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ and each $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, there are objects $P_i, Q_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ and morphisms $\\varphi_i \\in \\mathrm{Mor}_{\\mathcal{L}_i}(P_i, Q_i)$, such that $P = \\bigcup_{i \\geq M} P_i$ and $Q = \\bigcup_{i \\geq M} Q_i$, and $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$. By Proposition \\ref{fix1} (i) and (ii), there is some $M \\in \\mathbb{N}$ such that $P \\cap S_i, Q \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for all $i \\geq M$, and the restriction $\\varphi_i = \\varphi|_{P \\cap S_i}$ is a morphism in $\\mathrm{Mor}_{\\mathcal{L}_i}(P \\cap S_i, Q \\cap S_i)$. Since $S = \\bigcup_{i \\geq 0} S_i$ by Lemma \\ref{SiS} (i), it follows that\n$$\nP = \\bigcup_{i \\geq 0} P_i \\qquad \\mbox{and} \\qquad Q = \\bigcup_{i \\geq 0} Q_i.\n$$\nThe condition $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$ is easily checked.\n\\end{proof}\n\n\\begin{rmk}\\label{fix7}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\mathcal{O}(\\mathcal{F})$ be its \\emph{orbit category}: the category $\\mathcal{O}(\\mathcal{F})$ with object set $\\mathrm{Ob}(\\mathcal{F})$, and with morphism sets\n\\begin{equation}\\label{orbitcat}\n\\mathrm{Mor}_{\\mathcal{O}(\\mathcal{F}_{\\mathcal{H}})}(P,Q) = \\mathrm{Inn}(Q)\\backslash \\mathrm{Hom}_{\\mathcal{F}}(P,Q).\n\\end{equation}\nNotice that the subcategory $\\mathcal{O}(\\mathcal{F}^{\\bullet c}) \\subseteq \\mathcal{O}(\\mathcal{F})$ has a finite (full) subcategory as a skeletal subcategory, which in addition contains a representative of each $\\mathcal{F}$-conjugacy class of centric radical subgroups. Let $\\widetilde{\\LL}$ be the associated telescopic linking system, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Denote by $\\mathcal{L}_i^{\\bullet} \\subseteq \\mathcal{L}^{\\bullet}$ the image of $\\mathcal{L}_i$ through the functor $(-)^{\\bullet} \\colon \\widetilde{\\LL} \\to \\mathcal{L}^{\\bullet}$ for each $i \\geq 0$. Then, by Definition \\ref{defiapprox} there is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, the category $\\mathcal{L}_i^{\\bullet}$ contains representatives of all the morphisms in $\\mathcal{O}(\\mathcal{F}^{\\bullet c})$ up to $S$-conjugation. In particular, this implies that\n\\begin{enumerate}[(a)]\n\n\\item the fusion system $\\mathcal{F}$ is generated by $\\mathcal{F}_i$ and $\\mathrm{Inn}(S)$; and\n\n\\item the linking system $\\widetilde{\\LL}$ is generated by $\\mathcal{L}_i^{\\bullet}$ and $S$ (and thus so is $\\mathcal{L}$).\n\n\\end{enumerate}\nMore precisely, property (b) means that every object in $\\mathcal{L}^{\\bullet}$ is $S$-conjugate to an object in $\\mathcal{L}_i^{\\bullet}$, and every morphism in $\\mathcal{L}^{\\bullet}$ is the composition of a morphism in $\\mathcal{L}_i^{\\bullet}$ with (the restriction of) a morphism in $\\varepsilon(S)\\leq \\mathrm{Aut}_{\\mathcal{L}}(S)$. Since $\\mathcal{L}^{\\bullet}$ is a deformation retract of $\\widetilde{\\LL}$, we can say that $\\widetilde{\\LL}$ is generated by $\\mathcal{L}_i^{\\bullet}$ and $S$.\n\n\\end{rmk}\n\n\n\\subsection{An example}\\label{Ssexample}\n\nIn this subsection we analyze our constructions in detail on a specific example: the $2$-local compact group associated to $SO(3)$. In particular, this example reveals that there are approximations by $p$-local finite groups that do not appear as fixed points of any (family of) fine unstable Adams operation.\n\nLet us first fix some notation and facts. The reader is referred to \\cite[Example 3.7]{Gonza2} for further details. Let $\\mathcal{G} = (S, \\FF, \\LL)$ be the $2$-local compact group associated to $SO(3)$. Then,\n$$\nS = \\gen{\\{t_n\\}_{n \\geq 1}, \\, x \\, | \\, \\forall n, \\, t_n^{2^n} = x^2 = 1, \\, t_{n+1}^2 = t_n, \\, x \\cdot t_n \\cdot x^{-1} = t_n^{-1}} \\cong D_{2^{\\infty}}.\n$$\nFor each $n \\geq 1$, set $T_n = \\gen{t_n}$, so that the maximal torus of $S$ is $T = \\gen{\\{t_n\\}_{n \\geq 1}}\\leq S$. Set also $V = \\gen{t_1, x} \\cong \\Z\/2 \\times \\Z\/2$. Then $\\mathcal{F}$ is generated by $\\mathrm{Aut}_{\\mathcal{F}}(S) = \\mathrm{Inn}(S)$ and $\\mathrm{Aut}_{\\mathcal{F}}(V) = \\mathrm{Aut}(V) \\cong \\Sigma_3$. Regarding the linking system $\\mathcal{L}$, the only centric radical subgroups (up to $S$-conjugation) are $S$ and $V$, with\n$$\n\\mathrm{Aut}_{\\mathcal{L}}(S) = S \\qquad \\mbox{and} \\qquad \\mathrm{Aut}_{\\mathcal{L}}(V) \\cong \\Sigma_4.\n$$\nLet $\\widetilde{\\LL}$ be the associated telescopic linking system. An easy computation shows that the only new objects in $\\widetilde{\\LL}$ are the subgroups $T_n$, for $n \\geq 2$, with $\\mathrm{Aut}_{\\widetilde{\\LL}}(T_n) = S$ for all $n \\geq 2$.\n\nLet now $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L})$ be an unstable Adams operation. As usual, set $\\Psi_0 = \\Psi$ and $\\Psi_{i+1} = (\\Psi_i)^2$. Then, for each $i \\geq 0$ we have $S_i = C_S(\\Psi_i) \\cong D_{2^{n_i}}$ for some $n_i \\in \\mathbb{N}$, and we may assume without loss of generality that $V\\leq S_i$ for all $i \\geq 0$. Notice that $S_i$ contains two different $S_i$-conjugacy classes of maximal elementary abelian subgroups, and $V$ is a representative of one of them. Fix a representative $W_i\\leq S_i$ of the other $S_i$-conjugacy class of maximal elementary abelian subgroups. Notice that, after embedding $S_i$ into $S_{i+1}$, the subgroup $W_i$ becomes $S_{i+1}$-conjugate to $V$.\n\nLet also $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the approximation by $2$-local finite groups associated to $\\{\\Psi_i\\}_{i \\geq 0}$. A careful inspection reveals that, in order to describe $(S_i, \\mathcal{F}_i, \\mathcal{L}_i)$ it is enough to specify the groups $\\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$, $\\mathrm{Aut}_{\\mathcal{L}_i}(V)$ and $\\mathrm{Aut}_{\\mathcal{L}_i}(W_i)$. By construction, we have\n$$\n\\mathrm{Aut}_{\\mathcal{L}_i}(S_i) = S_i \\qquad \\mbox{and} \\qquad \\mathrm{Aut}_{\\mathcal{L}_i}(V) \\cong \\Sigma_4,\n$$\nand we have to determine the group $\\mathrm{Aut}_{\\mathcal{L}_i}(W_i)$. Let $\\varphi \\in \\mathrm{Aut}_{\\mathcal{L}_i}(V)$ be an automorphism of order $3$ that conjugates $t_1$ to $x$. An easy computation in $S$ shows that there is some $t \\in N_T(W_i, V)$ such that $t^{-1} \\cdot \\Psi_i(t) = t_1$. By Lemma \\ref{invar1} (ii), it follows that $\\varepsilon(x)^{-1} \\circ \\varphi \\circ \\varepsilon(x)$ is not fixed by $\\Psi_i$, and thus\n$$\n\\mathrm{Aut}_{\\mathcal{L}_i}(W_i) = N_{S_i}(W_i) \\cong D_8.\n$$\nThese computations imply that, for all $i$, $(S_i, \\mathcal{F}_i, \\mathcal{L}_i)$ is the $2$-local finite group associated to $PGL_2(\\mathbb{F}_q)$, where $q$ is some power of some odd prime $p$.\n\nLet now $(S_i, \\mathcal{E}_i, \\mathcal{T}_i)$ be the $2$-local finite group associated to $PSL_2(\\mathbb{F}_q)$. This $2$-local finite group is determined by $\\mathrm{Aut}_{\\mathcal{T}_i}(S_i) = S_i$ and $\\mathrm{Aut}_{\\mathcal{T}_i}(V) \\cong \\Sigma_4 \\cong \\mathrm{Aut}_{\\mathcal{T}_i}(W_i)$. Clearly, $\\{(S_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ is an approximation of $(S, \\FF, \\LL)$ by $2$-local finite groups, but our computations above show that it cannot be the product of our constructions in (\\ref{Li}).\n\n\n\\section{Stable Elements Theorem for \\texorpdfstring{$p$}{p}-local compact groups}\\label{Sstable}\n\nIn this section we use the approximations constructed in the previous section to prove a version of the Stable Elements Theorem for $p$-local compact groups, which computes the cohomology of the classifying space of a $p$-local compact group with coefficients in a trivial $\\Z_{(p)}$-module as the \\emph{stable} elements of the cohomology of its Sylow $p$-subgroup with the same module of coefficients. The finite version of this result was proved in \\cite[Theorem 5.8]{BLO2} for coefficients in the trivial module $\\mathbb{F}_p$, and then generalized to any trivial $\\Z_{(p)}$-module in \\cite[Lemma 6.12]{BCGLO2}.\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a saturated fusion system. In this section we consider certain contravariant functors $A \\colon \\mathcal{F} \\to \\mathcal{C}$, where $\\mathcal{C}$ is either $\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of $\\Z_{(p)}$-modules, or $\\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of graded $\\Z_{(p)}$-modules.  We start with a brief discussion of some general properties of such functors. For each $X\\leq S$, let $\\iota_X \\colon X \\to S$ denote the inclusion homomorphism. This way, given a contravariant functor $A \\colon \\mathcal{F} \\to \\mathcal{C}$ and a subcategory $\\mathcal{E} \\subseteq \\mathcal{F}$, we define\n\\begin{equation}\\label{Aee}\nA^{\\mathcal{E}} \\stackrel{def} = \\{z \\in A(S) \\, | \\, A(\\iota_P)(z) = A(\\iota_Q \\circ f)(z), \\, \\forall P, Q \\in \\mathrm{Ob}(\\mathcal{E}) \\mbox{ and } \\forall f \\in \\mathrm{Hom}_{\\mathcal{E}}(P,Q)\\}.\n\\end{equation}\nGiven $\\mathcal{E}_1 \\subseteq \\mathcal{E}_2$ two subcategories of $\\mathcal{F}$, there is an obvious inclusion $A^{\\mathcal{E}_2} \\subseteq A^{\\mathcal{E}_1}$.\n\n\\begin{lmm}\\label{aux34}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$, and let $\\{(S_i, \\mathcal{F}_i)\\}_{i \\geq 0}$ be a family of finite fusion subsystems of $\\mathcal{F}$ with $\\mathcal{F}_i \\subseteq \\mathcal{F}_{i+1}$ for all $i$ (in particular, $S_i\\leq S_{i+1}$ are finite subgroups of $S$ for all $i$), and satisfying the following properties:\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$; and\n\n\\item for all $P\\leq S$ and for all $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P, S)$ there exists some $M_f \\in \\mathbb{N}$ such that, for all $i \\geq M_f$, $f|_{P \\cap S_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P \\cap S_i, S_i)$.\n\n\\end{enumerate}\nLet also $\\mathcal{C}$ be either $\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of $\\Z_{(p)}$-modules, or $\\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of graded-$\\Z_{(p)}$-modules, and let $A \\colon \\mathcal{F} \\to \\mathcal{C}$ be a contravariant functor satisfying the following property:\n\\begin{itemize}\n\n\\item[(\\textasteriskcentered)] For each $P\\leq S$, the natural map $A(P) \\to \\varprojlim_i A(P \\cap S_i)$ is an isomorphism.\n\n\\end{itemize}\nThen, upon setting $\\mathcal{F}^{\\circ} = \\bigcup_{i \\geq 0} \\mathcal{F}_i \\subseteq \\mathcal{F}$, there are equalities\n$$\nA^{\\mathcal{F}} = A^{\\mathcal{F}^{\\circ}} = \\varprojlim_i A^{\\mathcal{F}_i}\n$$\nas subsets of $A(S)$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nWe claim first that $\\mathcal{F}^{\\circ} \\subseteq \\mathcal{F}$ is the full subcategory of $\\mathcal{F}$ whose objects are the finite subgroups of $S$. Indeed, if $P\\leq S$ is a finite subgroup, then there exists some $M_P \\in \\mathbb{N}$ such that $P\\leq S_i$ for all $i \\geq M_P$, since $S = \\bigcup_{i \\geq 0} S_i$. Similarly, if $P, Q\\leq S$ are finite subgroups and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$, then by condition (ii) there exists some $M_f \\in \\mathbb{N}$ such that $P, Q\\leq S_i$, and $f = f|_{P \\cap S_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q)$ for all $i \\geq M_f$.\n\nBy (\\textasteriskcentered), we have $A(S) = \\varprojlim_i A(S_i)$, which implies that $A^{\\mathcal{F}^{\\circ}} = \\varprojlim_i A^{\\mathcal{F}_i}$. The inclusion $\\mathcal{F}^{\\circ} \\subseteq \\mathcal{F}$ implies that $A^{\\mathcal{F}} \\subseteq A^{\\mathcal{F}^{\\circ}}$ by (\\ref{Aee}). To show the reverse inclusion, let $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$ and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$. For $X = P, Q, S$, set $X_i = X \\cap S_i$, and notice that\n$$\nX = \\bigcup_{i \\geq 0} X_i \\qquad \\mbox{and} \\qquad A(X) = \\varprojlim_i A(X_i),\n$$\nby (i) and (\\textasteriskcentered) respectively. By condition (ii), there exists some $M_f \\in \\mathbb{N}$ such that,  for all $i \\geq M_f$, $f|_{P_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P_i, Q_i)$. Thus, if $z \\in A^{\\mathcal{F}^{\\circ}}$, then\n$$\nA(\\iota_{P_i})(z) = A(\\iota_{Q_i} \\circ f|_{P_i})(z)\n$$\nfor all $i \\geq M_f$, and thus $A(\\iota_P)(z) = A(\\iota_Q \\circ f)(z)$. Hence $A^{\\mathcal{F}^{\\circ}} \\subseteq A^{\\mathcal{F}}$.\n\\end{proof}\n\n\\begin{rmk}\n\nIf $\\mathcal{G} = (S, \\FF, \\LL)$ is a $p$-local compact group and $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ is an approximation of $\\mathcal{G}$ by $p$-local finite groups, then conditions (i) and (ii) in Lemma \\ref{aux34} follow from condition (i) in \\ref{defiapprox} and \\ref{finmorph}, respectively. Hence, in this case \\ref{aux34} applies to any functor $A \\colon \\mathcal{F} \\to \\mathcal{C}$ as above that satisfies condition (\\textasteriskcentered).\n\n\\end{rmk}\n\n\\begin{prop}\\label{stable1}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $M$ be a (finite) $\\Z_{(p)}$-module with trivial $S$-action. Let also $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups, and let $P\\leq S$. Then, there are isomorphisms\n$$\nH^{\\ast}(BP; M) \\cong \\varprojlim H^{\\ast}(B(P \\cap S_i); M) \\qquad \\mbox{and} \\qquad H^{\\ast}(B\\mathcal{G}; M) \\cong \\varprojlim H^{\\ast}(B\\mathcal{G}_i; M).\n$$\nIn particular, the functor $H^{\\ast}(-; M) \\colon \\mathcal{F} \\to \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$ satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}.\n\n\\end{prop}\n\n\\begin{proof}\n\nFix some $P\\leq S$, and set $P_i = P \\cap S_i$ for all $i \\geq 0$. Let $X$ be either $B\\mathcal{G}$ or $BP$, and similarly let $X_i$ be either $B\\mathcal{G}_i$ or $BP_i$, depending on which case we want to prove. Note that the following holds.\n\\begin{enumerate}[(i)]\n\n\\item If $X = BP$, then $X = \\mathrm{hocolim \\,} X_i$, since $P = \\bigcup_{i \\geq 0} P_i$ by hypothesis.\n\n\\item If $X = B\\mathcal{G}$, then $X \\simeq (\\mathrm{hocolim \\,} X_i)^{\\wedge}_p$ by Lemma \\ref{approx0} and Remark \\ref{approx-1}. In particular, $H^{\\ast}(X; M) \\cong H^{\\ast}(\\mathrm{hocolim \\,} X_i; M)$.\n\n\\end{enumerate}\nConsider the homotopy colimit spectral sequence for cohomology \\cite[XII.5.7]{BK}:\n$$\nE^{r,s}_2 = \\varprojlim \\!\\! \\phantom{i}^rH^s(X_i;M) \\Longrightarrow H^{r+s}(X;M).\n$$\nWe will see that, for $r \\geq 1$, $E_2^{r,s} = \\{0\\}$, which, in particular, will imply the statement.\n\nFor each $s$, let $H^s_i = H^s(X_i;M)$, and let $F_i$ be the induced morphism in cohomology (in degree $s$) by the map $|\\Theta_i| \\colon |\\mathcal{L}_i| \\to |\\mathcal{L}_{i+1}|$ induced by the inclusion $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$. The cohomology ring $H^{\\ast}(X_i;M)$ is noetherian by \\cite[Theorem 5.8]{BLO2}, and in particular $H^s_i$ is finite for all $s$ and all $i$. Thus, the inverse system $\\{H^s_i;F_i\\}$ satisfies the Mittag-Leffler condition \\cite[3.5.6]{Weibel}, and hence the higher limits $\\varprojlim^rH^s_i$ vanish for all $r \\geq 1$. This in turn implies that the differentials in the above spectral sequence are all trivial, and thus it collapses.\n\\end{proof}\n\nWe are ready to prove Theorem \\ref{thmB}, the Stable Elements Theorem for $p$-local compact groups, which we restate below.\n\n\\begin{thm}\\label{stable2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $M$ be a finite $\\Z_{(p)}$-module $M$ with trivial $S$-action. Then, the natural map\n$$\nH^{\\ast}(B\\mathcal{G}; M) \\Right3{\\cong} H^{\\ast}(\\mathcal{F};M) \\stackrel{def} = \\varprojlim_{\\mathcal{F}} H^{\\ast}(-; M) \\subseteq H^{\\ast}(BS; M)\n$$\nis an isomorphism.\n\n\\end{thm}\n\n\\begin{proof}\n\nLet $\\{\\mathcal{G}_i = (S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups, with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. By Remark \\ref{approx-1}, the space $B\\mathcal{G}_i \\stackrel{def} = |\\mathcal{L}_i|^{\\wedge}_p$ is the classifying space of a $p$-local finite group, and we can apply the Stable Elements Theorem for $p$-local finite groups: there is a natural isomorphism\n$$\nH^{\\ast}(B\\mathcal{G}_i; M) \\stackrel{\\cong} \\longrightarrow H^{\\ast}(\\mathcal{F}_i;M).\n$$\nBy Proposition \\ref{stable1} there are natural isomorphisms\n$$\nH^{\\ast}(B\\mathcal{G};M) \\cong \\varprojlim_i H^{\\ast}(B\\mathcal{G}_i; M) \\cong \\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\subseteq \\varprojlim_i H^{\\ast}(BS_i; M) \\cong H^{\\ast}(BS;M),\n$$\nand we have to show that $\\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\cong H^{\\ast}(\\mathcal{F};M)$. If we set $\\mathcal{F}^{\\circ} = \\bigcup_{i \\geq 0} \\mathcal{F}_i$, then Lemma \\ref{aux34} and Proposition \\ref{stable1} combined imply that there are isomorphisms\n$$\n\\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\cong H^{\\ast}(\\mathcal{F}^{\\circ};M) \\cong H^{\\ast}(\\mathcal{F};M),\n$$\nand this finishes the proof.\n\\end{proof}\n\n\\begin{rmk}\\label{stable21}\n\nThe reader may have noticed the difference between the original statement of the Stable Elements Theorem for $p$-local finite groups, \\cite[Theorem 5.8]{BLO3}, in terms of the orbit category of $\\mathcal{F}$ defined in (\\ref{orbitcat}), and our statement, in terms of $\\mathcal{F}$. In fact, a formulation of the Stable Elements Theorem in terms of $\\mathcal{F}$, rather than $\\mathcal{O}(\\mathcal{F})$, is already found in \\cite[6.12]{BCGLO2}, as well as in other papers, and it is rather straightforward to justify the equivalence of statements. Consider the projection functor $\\tau \\colon \\mathcal{F} \\to \\mathcal{O}(\\mathcal{F})$, which is the identity on objects. Since conjugation by elements of $S$ induces the identity on cohomology, we have a commutative triangle\n$$\n\\xymatrix{\n\\mathcal{F} \\ar[rrr]^{H^{\\ast}(-)} \\ar[d]_{\\tau} & & & \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}\\\\\n\\mathcal{O}(\\mathcal{F}) \\ar[rrru]_{H^{\\ast}(-)} & & &\n}\n$$\nand an induced morphism between the corresponding inverse limits\n$$\n\\varprojlim_{\\mathcal{O}(\\mathcal{F})} H^{\\ast}(-;\\mathbb{F}_p) \\Right2{} \\varprojlim_{\\mathcal{F}} H^{\\ast}(-;\\mathbb{F}_p).\n$$\nThis morphism is easily checked to be an isomorphism upon considering both groups as subgroups of stable elements in $H^{\\ast}(S; \\mathbb{F}_p)$, since every element of $H^{\\ast}(S; \\mathbb{F}_p)$ is stable by any morphism in $\\mathcal{F}_S(S)$.\n\n\\end{rmk}\n\nWe finish this section proving Theorem \\ref{thmD}, restated as Theorem \\ref{thmd} below, which states the existence of a certain spectral sequence associated to a strongly closed subgroup of a given saturated fusion system. We first fix some notation.\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$, let $R\\leq S$ be a strongly $\\mathcal{F}$-closed subgroup, and let $M$ be an $\\Z_{(p)}$-module with trivial $S$-action. For each $X\\leq S$, set $\\overline{X} = X\/(X \\cap R) \\cong XR\/R\\leq S\/R$. Then, each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ induces a morphism of extensions\n$$\n\\xymatrix{\nP \\cap R \\ar[r] \\ar[d]_{f_0} & P \\ar[d]^f \\ar[r] & \\overline{P} \\ar[d]^{\\overline{f}} \\\\\nQ \\cap R \\ar[r] & Q \\ar[r] & \\overline{Q},\n}\n$$\nand thus also homomorphisms\n$$\n\\gamma(f) \\colon H^n(\\overline{Q}; H^m(Q \\cap R;M)) \\Right2{f_0^{\\ast}} H^n(\\overline{Q}; H^m(P \\cap R; M))  \\Right2{\\overline{f}^{\\,\\ast}} H^n(\\overline{P}; H^m(P \\cap R;M))\n$$\nfor all $n, m \\geq 0$. This defines a contravariant functor\n\\begin{equation}\\label{frakx}\n\\mathfrak{X}^{n,m} \\colon \\mathcal{F} \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n\\end{equation}\nwhich satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}, since, for each $P\\leq S$ and each $i \\geq 0$, the group $H^n(\\overline{P \\cap S_i}; H^m(P \\cap S_i \\cap R; M))$ is finite and the Mittag-Leffler Condition \\cite[3.5.6]{Weibel} applies. Set\n$$\nH^n(S\/R; H^m(R; M))^{\\mathcal{F}} = (\\mathfrak{X}^{n,m})^{\\mathcal{F}}\n$$\nin order to match the notation of \\cite{Diaz}\n\n\\begin{thm}\\label{thmd}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $R\\leq S$ be a strongly $\\mathcal{F}$-closed subgroup, and let $M$ be a finite $\\Z_{(p)}$-module with trivial $S$-action. Then there is a first quadrant cohomological spectral sequence with second page\n$$\nE^{n,m}_2 = H^n(S\/R; H^m(R;M))^{\\mathcal{F}}\n$$\nand converging to $H^{n+m}(B\\mathcal{G}; M)$.\n\n\\end{thm}\n\n\\begin{proof}\n\nThe spectral sequence of the statement is constructed as an inverse limit of spectral sequences. For the reader's convenience, the proof is divided into smaller steps.\n\n\\textbf{Step 1.} Construction of inverse systems of spectral sequences. Let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Notice that $R_i \\stackrel{def} = R \\cap S_i$ is strongly $\\mathcal{F}_i$-closed for all $i \\geq 0$, since $R$ is strongly $\\mathcal{F}$-closed.\n\nFor each $i \\geq 0$, consider the group extension\n$$\nR_i \\Right2{} S_i \\Right2{} S_i\/R_i,\n$$\nand its associated Lyndon-Hochschild-Serre spectral sequence $\\{E(i)_k^{\\ast, \\ast}, d_k^i\\}_{k \\geq 2}$, with second page $E(i)_2^{n,m} = H^n(S_i\/R_i; H^m(R_i;M))$, and converging to $H^{n+m}(S;M)$. For all $k \\geq 2$ and all $i, n, m \\geq 0$, the inclusion homomorphism $\\iota_{i,i+1} \\colon S_i \\to S_{i+1}$ induces homomorphisms of spectral sequences for each $i \\geq 0$,\n$$\n\\gamma(\\iota_{i, i+1}) \\colon \\{E(i+1)_k^{\\ast,\\ast}, d_k^{i+1}\\}_{k \\geq 2} \\Right3{} \\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}\n$$\nand thus an inverse system of spectral sequences $\\{\\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}, \\gamma(\\iota_{i, i+1})\\}_{i \\geq 0}$.\n\nNote that $R_i$ is strongly $\\mathcal{F}_i$-closed, and thus for each $n, m \\geq 0$ there is a functor\n$$\n\\mathfrak{X}_i^{n,m} \\colon \\mathcal{F}_i \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\ndefined by similar arguments as those used in the definition the functor $\\mathfrak{X}^{n,m}$ in (\\ref{frakx}). Furthermore, we may consider the spectral sequence $\\{\\4{E}(i)_k^{\\ast, \\ast}, \\4{d}_k^i\\}_{k \\geq 2}$ of \\cite[Theorem 1.1]{Diaz}, whose second page is $\\4{E}(i)_2^{n,m} = H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i} = (\\mathfrak{X}_i^{n,m})^{\\mathcal{F}_i}$, and which converges to $H^{n+m}(B\\mathcal{G}_i;M)$. We may see this spectral sequence as the restriction of the spectral sequence $\\{E(i)_k^{\\ast, \\ast}, d_k^i\\}_{k \\geq 2}$. We claim that $\\gamma(\\iota_{i,i+1})$ restricts to a morphism of spectral sequences\n$$\n\\gamma(\\alpha_{i, i+1}) \\colon \\{\\4{E}(i+1)_k^{\\ast,\\ast}, \\4{d}_k^{i+1}\\}_{k \\geq 2} \\Right3{} \\{\\4{E}(i)_k^{\\ast,\\ast}, \\4{d}_k^i\\}_{k \\geq 2}.\n$$\n\nFor each $i,n, m \\geq 0$ and for $X\\leq S_i$, let $A^{n,m}_i(X) = \\mathrm{Hom}_X(\\mathcal{B}^n_{\\overline{X}} \\otimes \\mathcal{B}^m_X, M)$ be the double complex defined in \\cite[Section 3]{Diaz} (although we do not give an explicit description here, notice that the complex itself does not actually depend on $i$). With this notation, for each $k \\geq 2$, the group $\\4{E}(i)_k^{n,m}$ can be seen as $(\\xi(i)^{n,m}_k)^{\\mathcal{F}_i}$, for a certain functor\n$$\n\\xi(i)_k^{n,m} \\colon \\mathcal{F}_i \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\ndefined in terms of the double complexes $A^{n,m}_i(X)$ above. In particular, the functor $\\mathfrak{X}_i^{n,m}$ defined above corresponding to the case $k = 2$ (for the sake of brevity, the reader is referred to \\cite[Section 4]{Diaz} for details). Furthermore, we have a commutative triangle of functors\n$$\n\\xymatrix{\n\\mathcal{F}_i \\ar[rrr]^{\\xi(i)_k^{n,m}} \\ar[d]_{\\mathrm{incl}} & & & \\Z_{(p)}\\mathrm{\\mathbf{-Mod}} \\\\\n\\mathcal{F}_{i+1} \\ar[rrru]_{\\xi(i+1)_k^{n,m}} & & &\n}\n$$\n\nFix some $k \\geq 2$, and some $i, n, m \\geq 0$. For a subgroup $X\\leq S_i\\leq S_{i+1}$, let $\n\\iota_X \\colon X \\to S_i$ and $\\4{\\iota}_X \\colon X \\to S_{i+1}$ be the corresponding inclusion monomorphisms, and note that $\\4{\\iota}_X = \\iota_{i, i+1} \\circ \\iota_X$. Also, recall that $\\4{E}(i+1)_k^{n,m}$ is the subgroup of elements $z \\in E(i+1)_k^{n,m}$ such that\n$$\n\\xi(i+1)_k^{n,m}(\\4{\\iota}_P)(z) = \\xi(i+1)_k^{n,m}(\\4{\\iota}_Q \\circ f)(z)\n$$\nfor all $P,Q\\leq S_{i+1}$ and all $f \\in \\mathrm{Hom}_{\\mathcal{F}_{i+1}}(P,Q)$, and a similar, equation with $\\xi(i)_k^{n,m}$ replacing $\\xi(i+1)_k^{n,m}$, describes the elements of $\\4{E}(i)_k^{n,m}$.\n\nFix some $z \\in E(i+1)_k^{n,m}$, and set $w = \\gamma(\\iota_{i, i+1})(z) \\in E(i)_k^{n,m}$. Then, for all $P, Q\\leq S_i$ and all $f \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_{i+1}}(P,Q)$, the commutativity of the above triangle implies that\n$$\n\\begin{aligned}\n\\xi(i)_k^{n,m}(\\iota_P)(w) & = \\xi(i+1)_k^{n,m}(\\iota_{i, i+1} \\circ \\iota_P)(z) = \\xi(i+1)_k^{n,m}(\\4{\\iota}_P)(z) = \\\\\n & = \\xi(i+1)_k^{n,m}(\\4{\\iota}_Q \\circ f)(z) = \\xi(i+1)_k^{n,m}(\\iota_{i,i+1} \\circ \\iota_Q \\circ f)(z) = \\\\\n & = \\xi(i)_k^{n,m}(\\iota_Q \\circ f)(w).\n\\end{aligned}\n$$\nThus $w \\in \\4{E}(i)_k^{n,m}$ and the claim follows.\n\n\\textbf{Step 2.} The inverse limit spectral sequences and their convergence. Consider the inverse systems of spectral sequences defined in Step 1, $\\{\\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}, \\gamma(\\iota_{i, i+1})\\}_{i \\geq 0}$ and $\\{\\{\\4{E}(i)_k^{\\ast,\\ast}, \\4{d}_k^i\\}_{k \\geq 2}, \\gamma(\\alpha_{i, i+1})\\}_{i \\geq 0}$. For each $k \\geq 2$, and for each $n, m \\geq 0$, define\n$$\nE_k^{n,m} \\stackrel{def} = \\varprojlim_i E_k^{n,m}(i) \\qquad \\qquad \\4{E}_k^{n,m} \\stackrel{def} = \\varprojlim_i \\4{E}_k^{n,m}(i).\n$$\n\nFor each $k \\geq 2$ and each $i, n,m  \\geq 0$, the group $E(i)_k^{n,m}$ is finite by definition, and thus the higher limits of $\\{E(i)_k^{n,m}\\}_{i \\geq 0}$ all vanish, by the Mittag-Leffler Condition \\cite[3.5.6]{Weibel}. A similar conclusion applies to $\\{\\4{E}(i)_k^{n,m}\\}_{i \\geq 0}$, since it is a restriction of $\\{E(i)_k^{\\ast, \\ast}\\}_{i \\geq 0}$. Furthermore, the differentials $\\{d_k^i\\}_{i \\geq 0}$ and $\\{\\4{d}_k^i\\}_{i \\geq 0}$ induce differentials $d_k$ (on $E_k^{\\ast, \\ast}$) and $\\4{d}_k$ (on $\\4{E}_k^{\\ast, \\ast}$) respectively, and an easy computation shows that\n$$\n\\begin{array}{c}\n\\mathrm{Ker}(d_k) \\cong \\varprojlim_i \\mathrm{Ker}(d_k^i) \\qquad \\qquad \\Im(d_k) \\cong \\varprojlim_i \\Im(d_k^i)\\\\[4pt]\n\\mathrm{Ker}(\\4{d}_k) \\cong \\varprojlim_i \\mathrm{Ker}(\\4{d}_k^i) \\qquad \\qquad \\Im(\\4{d}_k) \\cong \\varprojlim_i \\Im(\\4{d}_k^i).\n\\end{array}\n$$\nHence, $\\{E_k^{\\ast, \\ast}, d_k\\}_{k \\geq 2}$ and $\\{\\4{E}_k^{\\ast, \\ast}, \\4{d}_k\\}_{k \\geq 2}$ are well defined spectral sequences. Moreover, their corresponding second and infinity pages are, respectively,\n$$\n\\begin{array}{c}\nE_2^{n,m} \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)) \\qquad \\4{E}_2^{n,m} \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i}\\\\[6pt]\nE_{\\infty}^{n,m} \\cong \\varprojlim_i H^{k}(S_i;M) \\cong H^{k}(S; M) \\qquad \\4{E}_{\\infty}^{n,m} \\cong \\varprojlim_i H^{k}(B\\mathcal{G}_i; M) \\cong H^{k}(B\\mathcal{G};M)\n\\end{array}\n$$\nwhere $k = n+m$, and where the last isomorphism on each position of the bottom row holds by Proposition \\ref{stable1}.\n\nIt remains to show that the corresponding second pages of the limit spectral sequences satisfy, respectively,\n$$\n\\begin{array}{l}\nE_2^{n,m} = \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)) \\cong H^n(S\/R; H^m(R;M)) \\\\[2pt]\n\\4{E}_2^{n,m} = \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i} \\cong H^n(S\/R; H^m(R;M))^{\\mathcal{F}}\n\\end{array}\n$$\nBy Proposition \\ref{stable1}, there are isomorphisms \n$$\nH^{\\ast}(R; M) \\cong \\varprojlim H^{\\ast}(R_i;M) \\quad \\mbox{and} \\quad H^{\\ast}(S\/R; H^{\\ast}(R;M))\\cong \\varprojlim H^{\\ast}(S_i\/R_i; H^{\\ast}(R;M)).\n$$\nFurthermore, \\cite[Proposition B.2.3]{Rubin} implies that there are isomorphisms\n$$\n\\begin{aligned}\nH^n(S\/R; H^m(R;M)) & \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R;M)) \\cong \\\\\n & \\cong \\varprojlim_i \\varprojlim_j H^n(S_i\/R_i; H^m(R_j;M)) \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)),\n\\end{aligned}\n$$\nsince $H^n(S_i\/R_i; H^m(R_j;M))$ is finite for all $i, n, m \\geq 0$. To finish the proof, we have to show that the $H^n(S\/R; H^m(R;M))^{\\mathcal{F}}$ is isomorphic to $\\varprojlim_i H^n(S_iR_i; H^m(R_i;M))^{\\mathcal{F}_i}$, and this follows from Lemma \\ref{aux34}, since the functor $\\mathfrak{X}^{n,m}$ in (\\ref{frakx}) satisfies the required condition (\\textasteriskcentered).\n\\end{proof}\n\n\n\\section{Mapping spaces}\\label{Smap}\n\nIn this section we describe the mapping space $\\operatorname{Map}\\nolimits(BP, B\\mathcal{G})$, where $P$ is a discrete $p$-toral group and $\\mathcal{G}$ is a $p$-local compact group, in terms of centralizers in $\\mathcal{G}$ of subgroups of $S$. Such mapping spaces where described in \\cite{BLO3} when $P\\leq S$ is centric, and in full generality in \\cite{BLO2} when $P$ is a finite $p$-group and $\\mathcal{G}$ is a $p$-local finite group. Our proof follows the same lines as the proof in \\cite[Theorem 6.3]{BLO2}.\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $H^{\\ast}(\\mathcal{F}) \\subseteq H^{\\ast}(BS)$ be the subring of stable elements for $\\mathcal{F}$. Let also $E\\leq S$ be an elementary abelian subgroup which is fully $\\mathcal{F}$-centralized, and let $j_E \\colon H^{\\ast}(\\mathcal{F}) \\to H^{\\ast}(BE)$ be the map induced by inclusion. Let $T_E$ be Lannes' $T$-functor (see \\cite{Lannes}), and let $T_E(H^{\\ast}(\\mathcal{F}); j_E)$ be the component in $T_E(H^{\\ast}(\\mathcal{F}))$ of $j_E \\in T_E^0(H^{\\ast}(\\mathcal{F})) \\cong \\mathrm{Hom}_{\\mathcal{K}}(H^{\\ast}(\\mathcal{F}), H^{\\ast}(BE))$, where $\\mathcal{K}$ is the category of unstable algebras over the mod $p$ Steenrod algebra.\n\n\\begin{lmm}\\label{mapping1}\n\nThere is an isomorphism\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\Right3{\\cong} H^{\\ast}(C_{\\mathcal{F}}(E)) \\stackrel{def} = \\varprojlim_{C_{\\mathcal{F}}(E)} H^{\\ast}(-),\n$$\nwhich is the restriction of the homomorphism $T_E(H^{\\ast}(BS);j_E) \\to H^{\\ast}(C_S(E))$ induced by the natural homomorphism $C_S(E) \\times E \\to S$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. For the reader's convenience, we have divided the proof into several shorter steps.\n\n\\textbf{Step 1.} Given a discrete $p$-toral group $X$ and a morphism $\\sigma \\colon E \\to X$, we claim that\n$$\nT_E(H^{\\ast}(BX);\\sigma^{\\ast}) \\cong H^{\\ast}(\\mathrm{Map}(BE, BX)_{B\\sigma}) \\cong H^{\\ast}(BC_X(\\sigma E)).\n$$\nIndeed, since $X$ is discrete $p$-toral, it follows that $BX$ is $p$-good and $H^{\\ast}(BX)$ is of finite type (i.e. finite in every dimension). Moreover, since both $E$ and $X$ are discrete groups, it follows that $\\mathrm{Map}(BE, BX)_{B\\sigma} \\simeq BC_X(\\sigma E)$. The claim follows immediately by \\cite[Proposition 3.4.4]{Lannes}.\n\n\\textbf{Step 2.} Next we claim that there is some $M \\in \\mathbb{N}$ such that $E$ is fully centralized in $\\mathcal{F}_i$, for all $i \\geq M$.\n\nTo prove this, fix representatives $V_0 = E, V_1, \\ldots, V_n\\leq S$ of the different $S$-conjugacy classes in $E^{\\mathcal{F}}$. Note that, since $E$ is abelian, we have $VC_S(V) = C_S(V)$ for each $V \\in E^{\\mathcal{F}}$. We will just write $C_S(V)$ instead of $VC_S(V)$ for simplicity. There is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, we have $V_0, \\ldots, V_n\\leq S_i$, and the following holds for each $j = 0, \\ldots, n$:\n\\begin{enumerate}[(a)]\n\n\\item $|C_{S_i}(V_j)|\\leq |C_{S_i}(E)|$; and\n\n\\item $|C_{S_i}(V_j)\/C_{T_i}(V_j)| = |C_S(V_j)\/C_T(V_j)|$.\n\n\\end{enumerate}\nIndeed, since $V_j$ is $\\mathcal{F}$-conjugate to $E$, there exists some $f_j \\in \\mathrm{Hom}_{\\mathcal{F}}(C_S(V_j), C_S(E))$ such that $f_j(V_j) = E$. Moreover, by Lemma \\ref{finmorph}, there is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$ and all $j = 0, \\ldots, n$, the restriction of $f_j$ to $C_S(V_j) \\cap S_i$ is a morphism in $\\mathcal{F}_i$. Property (a) follows immediately. Property (b) is easily checked, since $S\/T \\cong S_i\/T_i$ for all $i \\geq 0$ by assumption.\n\nFor each $i \\geq M$, set $T_i = T \\cap S_i$ for short. If $V\\leq S_i$ is $\\mathcal{F}_i$-conjugate to $E$, then $V$ is $S$-conjugate to some $V_j$, for some $j \\in \\{1, \\ldots, n\\}$. Fix $x \\in N_S(V, V_j)$. By Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iv), it follows that $x^{-1} \\Psi_i(x) \\in C_T(V)$. Since $T$ is abelian and $T_i\\leq T$, it follows that\n$$\nx^{-1} \\Psi_i(x) \\in C_T(C_{T_i}(V)) = T.\n$$\nThus, $C_{T_i}(V)$ is $S$-conjugate (by $x$) to a subgroup of $C_{T_i}(V_j)$, again by Lemma \\ref{invar1} (i), and a similar argument with $x^{-1}$ instead of $x$ shows that in fact the element $x$ conjugates $C_{T_i}(V)$ onto $C_{T_i}(V_j)$. Moreover, via the inclusion $C_{S_i}(V)\\leq C_S(V)$, the quotient $C_{S_i}(V)\/C_{T_i}(V)$ can be identified with a subgroup of $C_S(V)\/C_T(V)$, and\n$$\n|C_{S_i}(V)\/C_{T_i}(V)|\\leq |C_S(V)\/C_T(V)| = |C_S(V_j)\/C_T(V_j)| = |C_{S_i}(V_j)\/C_{T_i}(V_j)|.\n$$\nSince $C_{S_i}(V)$ and $C_{S_i}(V_j)$ are finite groups, it follows that\n$$\n\\begin{aligned}\n|C_{S_i}(V)| & = |C_{T_i}(V)| \\cdot |C_{S_i}(V)\/C_{T_i}(V)|\\leq \\\\\n &\\leq |C_{T_i}(V_j)| \\cdot  |C_{S_i}(V_j)\/C_{T_i}(V_j)| = |C_{S_i}(V_j)|\\leq |C_{S_i}(E)|,\n\\end{aligned}\n$$\nand $E$ is fully centralized in $\\mathcal{F}_i$.\n\n\\textbf{Step 3.} For each $P\\leq S$, set $\\mathcal{T}_P = \\mathrm{Rep}_{\\mathcal{F}}(E,P) = \\mathrm{Inn}(P)\\backslash \\mathrm{Hom}_{\\mathcal{F}}(E,P)$. Notice that $\\mathcal{T}_P$ is finite by \\cite[Lemma 2.5]{BLO3}. Consider the functor $\\4{T}_E \\colon \\mathcal{O}(\\mathcal{F}) \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, defined on objects by\n$$\n\\4{T}_E(P) = \\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}).\n$$\nWe claim that there is an isomorphism\n$$\nT_E(H^{\\ast}(\\mathcal{F});j_E) \\cong \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big).\n$$\n\nTo prove this, consider the orbit category $\\mathcal{O}(\\mathcal{F})$, defined in (\\ref{orbitcat}). By \\cite[Lemma 2.5]{BLO3}, all morphism sets in $\\mathcal{O}(\\mathcal{F})$ are finite. Recall also that the full subcategory $\\mathcal{F}^{\\bullet} \\subseteq \\mathcal{F}$ contains only finitely many $\\mathcal{F}$-conjugacy classes by \\cite[Lemma 3.2 (a)]{BLO3}, and thus the full subcategory $\\mathcal{O}(\\mathcal{F}^{\\bullet}) \\subseteq \\mathcal{O}(\\mathcal{F})$ contains a finite skeletal subcategory. Furthermore, $\\mathcal{F}^{\\bullet}$ contains all the $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ by \\cite[Corollary 3.5]{BLO3}.\n\nFix a finite skeletal subcategory $\\mathcal{O}_{sk}$ of $\\mathcal{O}(\\mathcal{F}^{\\bullet})$. The functor $T_E$ is exact and commutes with direct limits. As a consequence, it also commutes with inverse limits over finite categories, and we have\n$$\n\\begin{aligned}\nT_E(H^{\\ast}(\\mathcal{F})) & = T_E(\\varprojlim_{\\mathcal{O}(\\mathcal{F})} H^{\\ast}(-)) = \\\\\n & =  T_E(\\varprojlim_{\\mathcal{O}_{sk}} H^{\\ast}(-)) \\cong \\varprojlim_{\\mathcal{O}_{sk}} T_E(H^{\\ast}(-)) = \\varprojlim_{\\mathcal{O}(\\mathcal{F})} T_E(H^{\\ast}(-)),\n\\end{aligned}\n$$\nRestricting the above to $T_E(H^{\\ast}(\\mathcal{F});j_E)$, we obtain\n$$\n\\begin{aligned}\nT_E(H^{\\ast}(\\mathcal{F});j_E) & \\cong \\varprojlim_{P \\in \\mathcal{O}_{sk}} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) \\cong \\\\\n & \\cong \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) = \\varprojlim_{\\mathcal{O}(\\mathcal{F})} \\4{T}_E(-).\n\\end{aligned}\n$$\n\n\\textbf{Step 4.} Consider the functor $\\4{T}_E \\colon \\mathcal{O}(\\mathcal{F}) \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$ defined in Step 3. By precomposing with the projection functor $\\tau \\colon \\mathcal{F} \\to \\mathcal{O}(\\mathcal{F})$, we obtain a functor\n$$\n\\mathcal{F} \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\nwhich by abuse of notation we also denote by $\\4{T}_E$. This will not lead to confusion since both functors take the same values on objects (as $\\tau$ is the identity on objects), as well as the same value on any two morphisms in $\\mathcal{F}$ representing a given morphism in $\\mathcal{O}(\\mathcal{F})$. Note also that, for each $P \\in \\mathrm{Ob}(\\mathcal{F}) = \\mathrm{Ob}(\\mathcal{O}(\\mathcal{F}))$, the set $\\mathcal{T}_P$ does not depend on the choice of category between $\\mathcal{F}$ and $\\mathcal{O}(\\mathcal{F})$, and an argument similar to that used in Remark \\ref{stable21} implies that\n$$\n\\begin{aligned}\n\\varprojlim_{\\mathcal{O}(\\mathcal{F})} \\4{T}_E(-) & = \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) \\cong\\\\\n & \\cong \\varprojlim_{P \\in \\mathcal{F}} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) = \\varprojlim_{\\mathcal{F}} \\4{T}_E(-).\n\\end{aligned}\n$$\nFrom now on we consider $\\4{T}_E$ as a functor on $\\mathcal{F}$. In this step we prove that the functor $\\4{T}_E$ satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}: for each $P\\leq S$, the natural map $\\4{T}_E(P) \\to \\varprojlim_i \\4{T}_E(P \\cap S_i)$ is an isomorphism.\n\nFix $P\\leq S$, and let $\\Omega \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(E,P)$ be a set of representatives of the classes in $\\mathcal{T}_P$. Note that, by definition, there is an equality\n\\begin{equation}\\label{TEP}\n\\4{T}_E(P) \\stackrel{def} = \\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) = \\bigoplus_{f \\in \\Omega} T_E(H^{\\ast}(BP); f^{\\ast}).\n\\end{equation}\nFor each $i \\geq 0$, set $P_i = P \\cap S_i$. Since $\\Omega$ is a finite set, there exists some $M_P \\in \\mathbb{N}$ such that $f(E)\\leq P_i$ for all $i \\geq M_P$ and all $f \\in \\Omega$. For simplicity we may assume that $M_P = 0$.\n\nFor each $0\\leq i\\leq j$, consider the maps $\\mathcal{T}_{P_i} \\Right1{\\alpha_i} \\mathcal{T}_P$ and $\\mathcal{T}_{P_i} \\Right1{\\beta_{i,j}} \\mathcal{T}_{P_j}$, defined by $\\alpha_i(\\rho) = \\mathrm{incl}_{P_i}^P \\circ \\rho$ and $\\beta_{i,j}(\\rho) = \\mathrm{incl}_{P_i}^{P_j} \\circ \\rho$, respectively. Notice that $\\alpha_i = \\alpha_j \\circ \\beta_{i,j}$ for all $0\\leq i\\leq j$. Moreover, we claim that $\\mathcal{T}_P$ is the colimit of the system $\\{\\mathcal{T}_{P_i}, \\beta_{i,j}\\}$. Indeed, surjectivity of $\\operatornamewithlimits{colim} \\mathcal{T}_{P_i} \\to \\mathcal{T}_P$ follows from the discussion above. To prove injectivity, fix some $i \\geq 0$, and let $\\rho, \\rho' \\in \\mathcal{T}_{P_i}$ be such that $\\alpha_i(\\rho) = \\alpha_i(\\rho')$. Let also $f, f' \\in \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ be representatives of $\\rho$ and $\\rho'$ respectively. Then, by definition of $\\mathcal{T}_P$ there exists some $x \\in P$ such that\n$$\n\\mathrm{incl}_{P_i}^P \\circ f' = c_x \\circ \\mathrm{incl}_{P_i}^P \\circ f.\n$$\nSince $P = \\bigcup_{i \\geq 0} P_i$, it follows that $x \\in P_j$ for some $j \\geq i$, and thus $\\beta_{i,j}(\\rho) = \\beta_{i,j}(\\rho') \\in \\mathcal{T}_{P_j}$.\n\nFor each $i \\geq 0$ and each $\\rho \\in \\mathcal{T}_P$, set $\\mathcal{T}_{P_i}^{\\rho} = \\alpha_i^{-1}(\\rho)$. We also fix the following.\n\\begin{enumerate}[(a)]\n\n\\item For each $f \\in \\Omega$, let $f_i \\in \\mathrm{Hom}_{\\mathcal{F}}(E,P_i)$ be the restriction of $f$.\n\n\\item For each $\\rho \\in \\mathcal{T}_P$, fix a set $\\Omega_i^{\\rho} \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ of representatives of the classes in $\\mathcal{T}_{P_i}^{\\rho}$. In particular, if $f \\in \\Omega$ represents the class $\\rho \\in \\mathcal{T}_P$, then we choose $f_i$ as representative of its own class in $\\mathcal{T}_{P_i}^{\\rho}$. Let also $\\Omega_i = \\coprod_{\\rho \\in \\mathcal{T}_P} \\Omega_i^{\\rho}$. By definition there is an equality\n\\begin{equation}\\label{TEPi}\n\\4{T}_E(P_i) \\stackrel{def} = \\bigoplus_{\\gamma \\in \\mathcal{T}_{P_i}} T_E(H^{\\ast}(BP_i);\\gamma^{\\ast}) = \\bigoplus_{\\omega \\in \\Omega_i} T_E(H^{\\ast}(BP_i); \\omega^{\\ast}).\n\\end{equation}\n\n\\item For each $\\rho \\in \\mathcal{T}_P$, each $i \\geq 0$, and each $\\omega \\in \\Omega_i^{\\rho}$, fix an element $x_{\\omega} \\in P_{i+1}$ such that $c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega \\in \\Omega_{i+1}^{\\rho}$ (such an element must exist since $\\beta_{i,i+1}[\\omega] \\in \\mathcal{T}_{P_{i+1}}^{\\rho}$). In the particular case where $\\omega = f_i$ (see (a) above), we choose $x_{\\omega} = 1$. Although the element $x_{\\omega}$ clearly depends on $i$, we omit this dependence from the notation since it will be clear at all times which $i$ is involved.\n\n\\end{enumerate}\nNote that, since $x_{\\omega} \\in P_{i+1}$, we have\n\\begin{equation}\\label{samemap}\n\\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} = \\mathrm{Map}(BE, BP_{i+1})_{\\sigma},\n\\end{equation}\nwhere $\\4{\\omega} = \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$ and $\\sigma = c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$.\n\nFix $\\rho \\in \\mathcal{T}_P$. For each $i \\geq 0$ and each $\\omega \\in \\Omega_i^{\\rho}$, consider the homomorphism\n$$\n\\Gamma_{\\omega} \\colon C_{P_i}(\\omega E) \\times E \\Right2{} P_i,\n$$\ndefined by $\\Gamma_{\\omega}(a,y) = a \\cdot \\omega(y) = \\omega(y) \\cdot a$. Fix $i \\geq 0$ and $\\omega \\in \\Omega_i^{\\rho}$, and let $x_{\\omega}$ be as fixed in (c) above. Let also $\\4{\\omega} = \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$ and $\\sigma = c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$. Then there is a commutative diagram\n$$\n\\xymatrix@C=2cm{\nC_{P_i}(\\omega E) \\times E \\ar[d]_{(\\mathrm{incl}, \\mathrm{Id})} \\ar[r]^{\\Gamma_{\\omega}} & P_i \\ar[d]^{\\mathrm{incl}} \\\\\nC_{P_{i+1}}(\\4{\\omega} E) \\times E \\ar[d]^{\\cong}_{(c_{x_{\\omega}}, \\mathrm{Id})} \\ar[r]^{\\Gamma_{\\4{\\omega}}} & P_{i+1} \\ar[d]_{\\cong}^{c_{x_{\\omega}}}\\\\\nC_{P_{i+1}}(\\sigma E) \\times E \\ar[r]_{\\Gamma_{\\sigma}} & P_{i+1}\n}\n$$\nwhich in turn induces the commutative diagram below by first passing to classifying spaces and then applying adjunction.\n\\begin{equation}\\label{diagrams}\n\\vcenter{\n\\xymatrix@C=2cm{\nBC_{P_i}(\\omega E) \\ar[r]^{\\simeq} \\ar[d]_{B\\mathrm{incl}} & \\mathrm{Map}(BE, BP_i)_{\\omega} \\ar[d]^{B\\mathrm{incl}_{\\ast}}\\\\\nBC_{P_{i+1}}(\\4{\\omega} E) \\ar[r]^{\\simeq} \\ar[d]_{Bc_{x_{\\omega}}}^{\\cong} & \\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} \\ar[d]_{\\cong}^{(Bc_{x_{\\omega}})_{\\ast}}\\\\\nBC_{P_{i+1}}(\\sigma E) \\ar[r]_{\\simeq} & \\mathrm{Map}(BE, BP_{i+1})_{\\sigma}\n}\n}\n\\end{equation}\nNote that the horizontal maps in the diagram above are homotopy equivalences by Step 1. Moreover, recall from (\\ref{samemap}) that $\\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} = \\mathrm{Map}(BE, BP_{i+1})_{\\sigma}$.\n\nLet $\\rho \\in \\mathcal{T}_P$, and let $f \\in \\Omega$ be its representative. We claim that there is an isomorphism\n\\begin{equation}\\label{isoTP}\nH^{\\ast}(BC_P(f E)) \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E)) \\big),\n\\end{equation}\nwhere the limit is defined by the homomorphisms\n$$\nH_{\\omega} \\colon BC_{P_i}(\\omega E) \\Right2{B\\mathrm{incl}} BC_{P_{i+1}}(\\4{\\omega} E) \\Right2{Bc_{x_{\\omega}}} BC_{P_{i+1}}(\\sigma E).\n$$\nNote that the above limit does not depend on the choice of the elements $x_{\\omega}$ in (c), since a different choice would differ from $x_{\\omega}$ by an element in $C_{P_{i+1}}(\\sigma E)$.\n\nFor each $i \\geq 0$, let $f_i \\in \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ be the restriction of $f$ as fixed in (a) above, which is an element of $\\Omega_i^{\\rho}$ by (b), and let $\\3{\\Omega}_i^{\\rho} = \\Omega_i^{\\rho} \\setminus \\{f_i\\}$. For each $\\omega \\in \\Omega_i^{\\rho}$, set for short $A_{\\omega} = H^{\\ast}(BC_{P_i}(\\omega E))$. Then, there are short exact sequences\n$$\n0 \\to \\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\Right2{\\iota_i} \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\Right2{\\pi_i} A_{f_i} \\to 0\n$$\nfor all $i \\geq 0$. Let also $\\kappa_{i+1} \\colon \\bigoplus_{\\sigma \\in \\Omega_{i+1}^{\\rho}} A_{\\sigma} \\to \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega}$ be the morphism induced by the maps $H_{\\omega}$ described above, and let $\\3{\\kappa}_{i+1}$ be the restriction of $\\kappa_{i+1}$ to $\\bigoplus_{\\sigma \\in \\3{\\Omega}_{i+1}^{\\rho}} A_{\\sigma}$. In order to prove (\\ref{isoTP}) we use the above exact sequences to construct an exact sequence of inverse limits.\n\nRecall that $\\mathcal{T}_P$ is the colimit of the sets $\\mathcal{T}_{P_i}$. Thus, given $i \\geq 0$, there exists some $M \\in \\mathbb{N}$ such that $[\\mathrm{incl}_{P_i}^{P_{i+M}} \\circ \\omega] = [f_{i + M}] \\in \\mathcal{T}_{P_{i+M}}^{\\rho}$ for all $\\omega \\in \\Omega_i^{\\rho}$, where $f_{i+M}$ is the restriction of $f$ to $P_{i+M}$ as fixed in (a) above. For simplicity let us assume that $M = 1$. In terms of mapping spaces, this means that, composing with the inclusion map $BP_i \\to BP_{i+1}$, we have\n$$\n\\coprod_{\\omega \\in \\Omega_i^{\\rho}} \\mathrm{Map}(BE, BP_i)_{\\omega} \\Right2{} \\mathrm{Map}(BE, P_{i+1})_{f_{i+1}},\n$$\nand it follows that the morphism $\\3{\\kappa}_{i+1}$, defined in the previous paragraph, is simply the trivial homomorphism. The morphism $\\kappa_{i+1}$ also induces a morphism $A_{f_{i+1}} \\to A_{f_i}$, which is easily seen to coincide with the restriction homomorphism induced by the inclusion $C_{P_i}(f_iE)\\leq C_{P_{i+1}}(f_{i+1}E)$. Summarizing, we obtain commutative diagrams\n$$\n\\xymatrix@C=1.5cm{\n0 \\ar[r] & \\bigoplus_{\\sigma \\in \\3{\\Omega}_{i+1}^{\\rho}} A_{\\sigma} \\ar[d]_{\\3{\\kappa}_{i+1}= 0} \\ar[r]^{\\iota_{i+1}} & \\bigoplus_{\\sigma \\in \\Omega_{i+1}^{\\rho}} A_{\\sigma} \\ar[r]^{\\pi_{i+1}} \\ar[d]^{\\kappa_{i+1}} & A_{f_{i+1}} \\ar[r] \\ar[d]^{\\mathrm{res}_{i+1}} & 0 \\\\\n0 \\ar[r] & \\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\ar[r]_{\\iota_i} & \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\ar[r]_{\\pi_i} & A_{f_i} \\ar[r] & 0\n}\n$$\nThis produces a morphism of inverse systems, and hence, since the inverse limit functor is left exact, there is an exact sequence\n\\begin{equation}\\label{exactlim}\n0 \\to \\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\big) \\Right1{} \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\big) \\Right2{\\pi} \\varprojlim_i A_{f_i},\n\\end{equation}\nwhere $\\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E))\\big) = 0$ since the morphisms in the corresponding inverse system, namely $\\3{\\kappa}_{i+1}$, are trivial. Thus, $\\pi$ is a monomorphism, and it remains to prove the it is also surjective. Let $(x_i)_{i \\in \\mathbb{N}}$ be an element in $\\varprojlim_i H^{\\ast}(BC_{P_i}(f_i E))$. That is, $\\mathrm{res}_{i+1}(x_{i+1}) = x_i$ for all $i \\geq 0$. Let also\n$$\nX_i = \\pi_i^{-1}(\\{x_i\\}) \\subseteq \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E)).\n$$\nRestricting the maps $\\kappa_{i+1}$ to the sets $X_{i+1}$ produces an inverse system of sets, and the resulting inverse limit $X = \\varprojlim_i X_i$ is nonempty by \\cite[Proposition 1.1.4]{RZ}. Moreover, by definition of the maps $\\kappa_i$, any element in $X$ is a preimage of $(x_i)_{i \\in \\mathbb{N}}$, and $\\pi$ is surjective. Thus, we have\n$$\nH^{\\ast}(BC_P(f E)) \\cong \\varprojlim_i H^{\\ast}(BC_{P_i}(f_i E)) \\cong \\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E))\\big),\n$$\nwhere the leftmost isomorphism follows from Proposition \\ref{stable1}, and the rightmost isomorphism corresponds to $\\pi$ in (\\ref{exactlim}). This proves (\\ref{isoTP}).\n\n\nCombining Step 1 and the above discussion, we deduce the following.\n$$\n\\begin{aligned}\n\\varprojlim_i \\4{T}_E(P_i) & = \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} T_E(H^{\\ast}(BP_i); \\omega^{\\ast}) \\big) \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} H^{\\ast}(\\mathrm{Map}(BE, BP_i)_{\\omega}) \\big) \\cong \\\\\n & \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} H^{\\ast}(BC_{P_i}(\\omega E)) \\big) \\cong \\bigoplus_{f \\in \\Omega} H^{\\ast}(BC_P(f E)) \\cong \\\\\n & \\cong \\bigoplus_{f \\in \\Omega} H^{\\ast}(\\mathrm{Map}(BE, BP)_{f}) \\cong \\bigoplus_{f \\in \\Omega} T_E(H^{\\ast}(BP); f^{\\ast}) = \\4{T}_E(P).\n\\end{aligned}\n$$\nMore precisely, the first equality holds by (\\ref{TEPi}), while the last equality corresponds to (\\ref{TEP}), the first isomorphism in the first row follows from Step 1, the isomorphism between the last term in the first row and the first term in the second row follows from Step 1 together with the leftmost diagram in (\\ref{diagrams}), the middle isomorphism in the second row holds by (\\ref{isoTP}), and the rest follows again from Step 1.\n\n\n\\textbf{Step 5.} The isomorphism $T_E(H^{\\ast}(\\mathcal{F}); j_E) \\Right3{\\cong} H^{\\ast}(C_{\\mathcal{F}}(E))$. Consider the fusion system $C_{\\mathcal{F}}(E)$ over $C_S(E)$, and consider also the set of fusion subsystems $\\{C_{\\mathcal{F}_i}(E)\\}_{i \\geq 0}$. This setup satisfies the conditions of Lemma \\ref{aux34}: clearly, $C_S(E) = \\bigcup_{i \\geq 0} C_{S_i}(E)$, since $S = \\bigcup_{i \\geq 0} S_i$, and every morphism in $C_{\\mathcal{F}}(E)$ eventually restricts to a morphism in $C_{\\mathcal{F}_i}(E)$ for $i$ big enough, since the fusion systems $\\mathcal{F}_i$ are part of an approximation of $(S, \\FF, \\LL)$ by $p$-local finite groups. It follows that there is a sequence of isomorphisms\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\cong \\varprojlim_{\\mathcal{F}} \\4{T}_E(-) \\cong \\varprojlim_i \\big(\\varprojlim_{\\mathcal{F}_i} \\4{T}_E(-) \\big) \\cong \\varprojlim_i H^{\\ast}(C_{\\mathcal{F}_i}(E)) \\cong H^{\\ast}(C_{\\mathcal{F}}(E))\n$$\nwhere the first isomorphism follows from Steps 3 and 4 combined, the second isomorphism is a consequence of Lemma \\ref{aux34}, since we have checked in Step 4 that $\\4{T}_E$ satisfies condition (\\textasteriskcentered), the third isomorphism follows from \\cite[Lemma 5.7]{BLO2}, since we have shown in Step 2 that $E$ can be assumed to be fully $\\mathcal{F}_i$-centralized for all $i$, and the last isomorphism is a consequence of Proposition \\ref{stable1}.\n\\end{proof}\n\n\\begin{rmk}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Let also $E\\leq S$ be an elementary abelian subgroup which is fully centralized, and let $j_E$ be as above. Without loss of generality we may assume that $E\\leq S_i$ for all $i \\geq 0$, and that $E$ is fully $\\mathcal{F}_i$-centralized for all $i \\geq 0$. Let $j_{E,i} \\colon H^{\\ast}(\\mathcal{F}_i) \\to H^{\\ast}(BE)$ be the map induced by the inclusion $E\\leq S_i$, for all $i \\geq 0$. In this situation, we have just proved that\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\cong T_E(\\varprojlim_i H^{\\ast}(\\mathcal{F}_i); j_E) \\cong \\varprojlim_i T_E(H^{\\ast}(\\mathcal{F}_i);j_{E,i}).\n$$\nIn other words, the functor $T_E$ commutes with the inverse limit $\\varprojlim_i H^{\\ast}(\\mathcal{F}_i)$. We do not know of any general result about the functor $T_E$ commuting with infinite inverse limits.\n\n\\end{rmk}\n\nThe proof of Theorem \\ref{mapping} below requires the following result involving the quotient of a linking system by a normal discrete $p$-toral subgroup. The following is a generalization of \\cite[Lemma 5.6]{BLO2} to the compact case.\n\n\\begin{lmm}\\label{quotient2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be the telescopic linking system associated to $\\mathcal{L}$ in \\ref{expl1}. Let also $V\\leq S$ be a central subgroup in $\\mathcal{F}$ of order $p$, and let $(\\widetilde{\\LL}\/V, \\3{\\varepsilon}, \\3{\\rho})$ be the quotient transporter system, associated to the saturated fusion system $\\mathcal{F}\/V$. Finally, let $(\\widetilde{\\LL}\/V)^c \\subseteq \\widetilde{\\LL}\/V$ and $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ be the full subcategories with object sets\n$$\n\\begin{array}{l}\n\\mathrm{Ob}((\\widetilde{\\LL}\/V)^c) = \\{P\/V \\in \\mathrm{Ob}(\\widetilde{\\LL}\/V) \\, | \\, P\/V \\mbox{ is $\\mathcal{F}\/V$-centric}\\} \\\\[2pt]\n\\mathrm{Ob}(\\widetilde{\\LL}_0) = \\{P \\in \\mathrm{Ob}(\\widetilde{\\LL}) \\, | \\, P\/V \\in \\mathrm{Ob}((\\widetilde{\\LL}\/V)^c)\\}\n\\end{array}\n$$\nrespectively. Then the following holds.\n\\begin{enumerate}[(i)]\n\n\\item $(\\widetilde{\\LL}\/V)^c$ is a linking system associated to $\\mathcal{F}\/V$.\n\n\\item $BV \\to |\\widetilde{\\LL}_0|^{\\wedge}_p \\to |(\\widetilde{\\LL}\/V)^c|^{\\wedge}_p$ is a fibration sequence.\n\n\\item The inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nClearly, all the $\\mathcal{F}$-centric subgroups of $S$ must contain $V$, since $V$ is abelian and $\\mathcal{F}$-central. Thus, for all $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$, it follows by axiom (C) of transporter systems that the left and right actions of $V$ on $\\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ (via composition with $\\varepsilon_Q(V)$ and $\\varepsilon_P(V)$ respectively) are the same. Since the proof is rather long, it is divided into steps for the reader's convenience.\n\n\\textbf{Step 1.} For each $P\\leq S$ which contains $V$, we claim that\n$$\nP \\mbox{ fully $\\mathcal{F}$-normalized } \\Longrightarrow P\/V \\mbox{ fully $\\mathcal{F}\/V$-centralized } \\Longrightarrow \\Gamma_P\\leq \\mathrm{Aut}_S(P).\n$$\n\nLet $P\\leq S$ be such that $V\\leq P$, and note that $\\mathrm{Aut}_{\\mathcal{F}}(V) = \\{\\mathrm{Id}\\}$, since $V$ is central in $\\mathcal{F}$. Thus, the subgroup\n$$\n\\Gamma_P \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V))\n$$\nis a discrete $p$-toral normal subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ by Lemma \\ref{Kpgp}. Since $\\mathcal{F}$ is saturated, every subgroup of $S$ is $\\mathcal{F}$-conjugate to a fully $\\mathcal{F}$-normalized subgroup. Similarly, since $\\mathcal{F}\/V$ is saturated, every subgroup of $S\/V$ is $\\mathcal{F}\/V$-conjugate to a fully $\\mathcal{F}\/V$-centralized subgroup. Thus it is enough to show the following: if $P, Q\\leq S$ are $\\mathcal{F}$-conjugate subgroups such that $P\/V$ is fully $\\mathcal{F}\/V$-centralized and $Q$ is fully $\\mathcal{F}$-normalized, then $Q\/V$ is fully $\\mathcal{F}\/V$-centralized and $\\Gamma_P\\leq \\mathrm{Aut}_S(P)$.\n\nAs show above, the group $\\Gamma_Q$ is a normal discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$. Furthermore, since $Q$ is fully $\\mathcal{F}$-normalized we have $\\mathrm{Aut}_S(Q) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{F}}(Q))$, and thus $\\Gamma_Q\\leq \\mathrm{Aut}_S(Q)$. By axiom (II) of saturated fusion systems, every isomorphism $f \\in \\mathrm{Iso}_{\\mathcal{F}}(P,Q)$ extends to some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_f, N_S(Q))$, where\n$$\nN_f = \\{g \\in N_S(P) \\, | \\, f \\circ c_g \\circ f^{-1} \\in \\mathrm{Aut}_S(Q)\\}.\n$$\nSet $N_S^0(P) = \\{g \\in N_S(P) \\, | \\, c_g \\in \\Gamma_P\\}$, and notice that $N_S^0(P)\/V = C_{S\/V}(P\/V)$. We claim that $N_S^0(P)\\leq N_f$. To prove that, fix $g \\in N_S^0(P)$ and $a \\in Q$, and set $b = f^{-1}(a) \\in P$. By definition, $c_g(b) = bv$ for some $v \\in V$, and we have\n$$\n(f \\circ g \\circ f^{-1})(a) = (f \\circ c_g)(b) = f(bv) = f(b) v = a v,\n$$\nwhere $f(bv) = f(b) v$ since $V$ is central in $\\mathcal{F}$ and $V,\\leq P,Q$ (and thus the morphism $f$ restricts to the identity on $V$).\n\nThe above implies that $\\gamma$ restricts to $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S^0(P), N_S^0(Q))$, which in turn factors through a homomorphism\n$$\n\\3{\\gamma} \\in \\mathrm{Hom}_{\\mathcal{F}\/V}(C_{S\/V}(P\/V), C_{S\/V}(Q\/V)).\n$$\nSince $P\/V$ is fully $\\mathcal{F}\/V$-centralized, it follows that $\\3{\\gamma}$ is an isomorphism and $Q\/V$ is also fully $\\mathcal{F}\/V$-centralized. Furthermore, $\\gamma$ must be an isomorphism too, and thus $\\Gamma_P\\leq \\mathrm{Aut}_S(P)$.\n\n\\textbf{Step 2.} We show now that the category $(\\widetilde{\\LL}\/V)^c$ is a centric linking system associated to $\\mathcal{F}\/V$. First notice that $(\\widetilde{\\LL}\/V)^c$ is a transporter system associated to $\\mathcal{F}\/V$ by the above remarks, and thus we only have to check that, for each object $P\/V$ of $(\\widetilde{\\LL}\/V)^c$,\n$$\nE(P\/V) \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{(\\widetilde{\\LL}\/V)^c}(P\/V) \\to \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V)) = \\3{\\varepsilon}_{P\/V}(Z(P\/V)).\n$$\n\nFix $P\/V \\in \\mathrm{Ob}((\\widetilde{\\LL}\/V)^c)$, and consider the following commutative diagram\n$$\n\\xymatrix{\n\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\ar[rr]^{\\tau_P} \\ar[d]_{\\rho_P} & & \\mathrm{Aut}_{\\widetilde{\\LL}\/V}(P\/V) \\ar[d]^{\\3{\\rho}_{P\/V}} \\\\\n\\mathrm{Aut}_{\\mathcal{F}}(P) \\ar[rr]_{\\omega_P} & & \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V),\n}\n$$\nwhere $\\rho_{P}$ and $\\3{\\rho}_{P\/V}$ denote the corresponding structural functors in the transporter systems $\\widetilde{\\LL}$ and $\\widetilde{\\LL}\/V$, respectively. By definition, we have\n$$\n\\mathrm{Ker}(\\tau_P) = \\varepsilon_P(V) \\qquad \\qquad \\mathrm{Ker}(\\rho_P) = \\varepsilon_P(C_S(P)) \\qquad \\qquad \\mathrm{Ker}(\\omega_P) = \\Gamma_P,\n$$\nand an easy computation shows then that $E(P\/V) = \\tau_P(\\gen{\\varepsilon_P(C_S(P)), \\varepsilon_P(N_S^0(P))})$. In particular it follows that $E(P\/V)$ is a discrete $p$-toral group. To finish the proof, recall that $P\/V$ is $\\mathcal{F}\/V$-centric, and in particular it is fully $\\mathcal{F}\/V$-centralized. Thus $Z(P\/V) \\in \\operatorname{Syl}\\nolimits_p(E(P\/V))$, which implies that $E(P\/V) = Z(P\/V)$.\n\n\\textbf{Step 3.} $BV \\to |\\widetilde{\\LL}_0|^{\\wedge}_p \\to |(\\widetilde{\\LL}\/V)^c|^{\\wedge}_p$ is a fibration sequence. Using \\cite[Lemma 4.3 (a)]{BLO3} it is easy to check that each undercategory for the projection of $\\widetilde{\\LL}_0$ onto $(\\widetilde{\\LL}\/V)^c$ contains a category equivalent to $\\mathcal{B}(V)$ as a deformation retract. Thus, by Quillen's Theorem B, the map $|\\widetilde{\\LL}_0| \\to |(\\widetilde{\\LL}\/V)^c|$ has homotopy fiber $BV$. By \\cite[II.5.1]{BK}, the fibration sequence $BV \\to |\\widetilde{\\LL}_0| \\to |(\\widetilde{\\LL}\/V)^c|$ is still a fibration sequence after $p$-completion.\n\n\\textbf{Step 4.} The inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$. Notice that the functor $(-)^{\\bullet}$ restricts to a functor $(-)^{\\bullet}$ on $\\widetilde{\\LL}_0$. Indeed, let $P \\in \\mathrm{Ob}(\\widetilde{\\LL}^{\\bullet})\\setminus \\mathrm{Ob}(\\widetilde{\\LL}_0)$, and let $Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be such that $V\\leq Q$ and $Q^{\\bullet} = P$. Then $Q\/V\\leq P\/V$, and since $P\/V$ is not $\\mathcal{F}\/V$-centric by assumption, neither is $Q\/V$. Thus $Q \\notin \\mathrm{Ob}(\\widetilde{\\LL}_0)$. Conversely, if $Q \\in \\mathrm{Ob}(\\widetilde{\\LL}_0)$, then clearly $Q^{\\bullet}\\in \\mathrm{Ob}(\\widetilde{\\LL}_0)$.\n\nAlso, note that if $\\widetilde{\\LL}_0$ contains $\\mathcal{L}^{\\bullet}$ then the claim follows easily. Thus, fix some $P\\leq S$ in $\\mathcal{L}^{\\bullet}$ but not in $\\widetilde{\\LL}_0$. In particular, $P$ is $\\mathcal{F}$-centric, and $V\\leq P$, but $P\/V$ is not $\\mathcal{F}\/V$-centric. By replacing $P$ by a conjugate if necessary, we may assume that $C_{S\/V}(P\/V) \\neq Z(P\/V)$, and thus there is some $gV \\in S\/V$ such that $gV \\notin P\/V$ and $[gV, P\/V] = 1$. Equivalently, there is some $g \\in S$ such that $g \\notin P$ and $[g, P] = V$. Furthermore, $c_g \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$ cannot be an inner automorphism, because if this was the case then $c_g = c_x$ for some $x \\in P$, and $gx^{-1} \\in C_S(P) \\setminus P = \\emptyset$, since $P$ is $\\mathcal{F}$-centric. Thus $c_g$ is a nontrivial element of $\\mathrm{Ker}(\\mathrm{Out}_{\\mathcal{F}}(P) \\to \\mathrm{Out}_{\\mathcal{F}\/V}(P\/V))$, and thus $P$ is not $\\mathcal{F}$-radical. In other words, we have shown that $\\widetilde{\\LL}_0$ contains all $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups. By \\cite[Corollary A.10]{BLO6} the inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$.\n\\end{proof}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $P\\leq S$ be a fully $\\mathcal{F}$-centralized subgroup. Let also $C_{\\mathcal{G}}(P) = (C_S(P), C_{\\mathcal{F}}(P), C_{\\mathcal{L}}(P))$ be the centralizer $p$-local compact group of $P$ defined in \\ref{rmknorm}, with classifying space $BC_{\\mathcal{G}}(P)$, and let $\\mathcal{B}(P)$ be the category with a single object $\\circ_P$ and $P$ as automorphism group. By Lemma \\ref{centricNFKA}, if $Q\\leq C_S(P)$ is $C_{\\mathcal{F}}(P)$-centric then $QP$ is $\\mathcal{F}$-centric. Thus we can define a functor\n$$\n\\Gamma_{\\mathcal{L}, P} \\colon C_{\\mathcal{L}}(P) \\times \\mathcal{B}(P) \\Right3{} \\mathcal{L}\n$$\nby setting $\\Gamma_{\\mathcal{L},P}(Q, \\circ_P) = QP$ for each $C_{\\mathcal{F}}(P)$-centric subgroup of $C_S(P)$. Given a morphism $(\\varphi, g) \\in \\mathrm{Mor}_{C_{\\mathcal{L}}(P) \\times \\mathcal{B}(P)}((Q,\\circ), (R,\\circ))$, the functor $\\Gamma$ is defined as follows\n$$\n\\Gamma_{\\mathcal{L},P}(\\varphi, g) = \\varphi \\circ \\varepsilon_{QP}(g) = \\varepsilon_{RP}(g) \\circ \\varphi,\n$$\nwhere the last equality follows from condition (C) of transporter systems, since the underlying homomorphism of $\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(QP, RP)$ restricts to the identity on $P$ by definition of $C_{\\mathcal{L}}(P)$. By first realizing nerves and them $p$-completing, we obtain a map\n$$\nBC_{\\mathcal{G}}(P) \\times (BP)^{\\wedge}_p \\Right3{} B\\mathcal{G}\n$$\n(notice that $(BP)^{\\wedge}_p$ is not necessarily equivalent to $BP$ since $P$ is a discrete $p$-toral group). By taking adjoint first and then precomposing with the natural map $BP \\to (BP)^{\\wedge}_p$, we obtain a map\n$$\n\\Gamma'_{\\mathcal{L},P} \\colon |C_{\\mathcal{L}}(P)|^{\\wedge}_p \\Right3{} \\mathrm{Map}((BP)^{\\wedge}_p, B\\mathcal{G})_{\\mathrm{incl}} \\Right3{} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}.\n$$\n\n\\begin{thm}\\label{mapping}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $P$ be a discrete $p$-toral group, and $\\gamma: P \\to S$ be a group homomorphism such that $\\gamma(P)$ is fully $\\mathcal{F}$-centralized in $\\mathcal{F}$. Then,\n$$\n\\Gamma'_{\\mathcal{L}, \\gamma(P)} \\colon BC_{\\mathcal{G}}(\\gamma(P)) \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{B\\gamma}\n$$\nis a homotopy equivalence.\n\n\\end{thm}\n\n\\begin{proof}\n\nOur proof follows the same strategy as the proof of \\cite[Theorem 6.3]{BLO2}. The referee suggested an alternative to the cases 1-3 below, which we briefly discuss after the proof. By \\cite[Proposition 6.2]{BLO3}, for each $\\gamma \\in \\mathrm{Hom}(P, S)$,\n\\begin{equation}\\label{zero}\n\\mathrm{Map}(BP, B\\mathcal{G})_{B\\gamma} \\simeq \\mathrm{Map}(B\\gamma(P), B\\mathcal{G})_{\\mathrm{incl}}.\n\\end{equation}\nThus, it suffices to prove the statement when $P\\leq S$ is fully $\\mathcal{F}$-centralized and $\\gamma$ is the inclusion. The proof is divided into several cases for the reader's convenience.\n\n\\textbf{Case 1.} Suppose that $P$ is elementary abelian. In this case, by Theorem 0.5 \\cite{Lannes}, it is enough to prove that $\\Gamma_{\\mathcal{L},P}$ induces an isomorphism\n$$\nT_P(H^{\\ast}(B\\mathcal{G}); \\mathrm{incl}^{\\ast}) \\Right2{\\cong} H^{\\ast}(BC_{\\mathcal{G}}(P)).\n$$\nBy Theorem \\ref{stable2}, $H^{\\ast}(B\\mathcal{G}) \\cong H^{\\ast}(\\mathcal{F})$, and $H^{\\ast}(BC_{\\mathcal{G}}(P)) \\cong H^{\\ast}(C_{\\mathcal{F}}(P))$, and the above isomorphism follows from Lemma \\ref{mapping1}.\n\n\\textbf{Case 2.} Suppose that $P$ is a normal subgroup of $\\mathcal{F}$, that is, $N_{\\mathcal{F}}(P) = \\mathcal{F}$. Let $\\mathcal{L}_0 \\subseteq \\mathcal{L}$ be the full subcategory whose objects are the subgroups $Q\\leq S$ such that $C_Q(P)$ is $C_{\\mathcal{F}}(P)$-centric.\n\nIn order to prove the statement in this case, we first need to show that the inclusion of $\\mathcal{L}_0$ into $\\mathcal{L}$ induces an equivalence $|\\mathcal{L}_0|^{\\wedge}_p \\simeq |\\mathcal{L}|^{\\wedge}_p$. By \\cite[Corollary A.10]{BLO6}, it is enough to check that $\\mathrm{Ob}(\\mathcal{F}^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_0)$.\n\nLet $Q \\in \\mathrm{Ob}(\\mathcal{L})$ be an $\\mathcal{F}$-centric subgroup of $S$ which is not an object in $\\mathcal{L}_0$. We claim that $Q$ is not $\\mathcal{F}$-radical. Set $Q_0 = C_Q(P)$, and note that every element of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$ restricts to an automorphism of $Q_0$ since $P$ is normal in $\\mathcal{F}$. Furthermore $Q_0 \\lhd Q$, and we can define\n$$\nK = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(Q_0) \\times \\mathrm{Aut}(Q\/Q_0)) \\lhd \\mathrm{Aut}_{\\mathcal{F}}(Q).\n$$\nNote that $K$ is a discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$ by Lemma \\ref{Kpgp}. In order to prove that $Q$ is not $\\mathcal{F}$-radical, it is enough to check that $1 \\neq K \\not\\leq \\mathrm{Inn}(Q)$.\n\nBy assumption $Q_0 \\notin \\mathrm{Ob}(\\mathcal{L}_0)$, and thus $Q_0$ is not $C_{\\mathcal{F}}(P)$-centric. Thus, we may assume that $C_{C_S(P)}(Q_0) \\not\\leq Q_0$, since otherwise $Q$ can be replaced by an $\\mathcal{F}$-conjugate $R$ such that the corresponding subgroup $R_0 = C_R(P)$ is $C_{\\mathcal{F}}(P)$-conjugate to $Q_0$ and satisfies the desired condition. Set\n$$\nQ_1 \\stackrel{def} = C_S(Q_0P) = C_{C_S(P)}(Q_0).\n$$\nAs discussed above we have $Q_1 \\not\\leq Q_0 = C_Q(P)$, and thus, $Q_1 \\cap Q = C_Q(Q_0P)\\leq Q_0$, and $Q_1 \\not\\leq Q$. Also, $Q\\leq N_S(Q_1)$ by definition, and thus $Q_1Q\\leq S$ is a subgroup. Note that $Q \\lneqq Q_1Q$, and thus $Q \\lneqq N_{Q_1Q}(Q)$. Choose some $x \\in N_{Q_1Q}(Q)$ such that $x \\notin Q$. Then,\n$$\n[x, Q] = \\{xax^{-1}a^{-1} \\,\\, | \\,\\, a \\in Q\\}\\leq Q_1 \\cap Q\\leq Q_0,\n$$\nand hence $c_x \\in K$. Since $x \\notin Q$ and $Q$ is $\\mathcal{F}$-centric, it follows that $c_x \\notin \\mathrm{Inn}(Q)$, and thus $Q$ is not $\\mathcal{F}$-radical.\n\nLet $(\\mathcal{L}_0)_{P, Id}$ be the category with object set the pairs $(Q, \\alpha)$, for $Q$ in $\\mathcal{L}_0$ and $\\alpha \\in \\mathrm{Hom}_{\\mathcal{F}}(P, Q)$ and such that\n$$\n\\mathrm{Mor}_{(\\mathcal{L}_0)_{P, Id}}((Q,\\alpha), (R, \\alpha')) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(Q,R) \\mbox{ } | \\mbox{ } \\alpha' = \\rho(\\varphi) \\circ \\alpha\\}.\n$$\nThis is equivalent to the component of the object $(P,Id)$ in the category $\\mathcal{L}_0^P$ of \\cite[Proposition 6.2]{BLO3}, and hence there is a homotopy equivalence\n$$\n\\mathrm{Map} (BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\simeq |(\\mathcal{L})_{P, Id}|^{\\wedge}_p.\n$$\n\nAt this point, one can define functors\n$$\n\\xymatrix{\n(\\mathcal{L}_0)_{P,Id} \\ar @<  2pt> [r]^{\\sigma} & C_{\\mathcal{L}}(P) \\ar @< 2pt> [l]^{\\tau}\\\\\n}\n$$\nin the same way as they are constructed in Step 2 of the proof \\cite[Theorem 6.3]{BLO2}, and which are inverse to each other up to natural transformation. This implies that the composite\n$$\nBC_{\\mathcal{G}}(P) \\Right2{\\tau} |(\\mathcal{L}_0)_{P, Id}|^{\\wedge}_p \\Right2{\\simeq} \\mathrm{Map}(BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}\n$$\nis a homotopy equivalence, and by construction it is equal to $\\Gamma'_{\\mathcal{L},P}$.\n\n\\textbf{Case 3.} Suppose that $P$ is a finite subgroup of $S$. As an induction hypothesis, we can assume that the statement holds for all maps with source $BP'$, with $|P'| < |P|$, and all $p$-local compact groups.\n\nFix a subgroup $V\\leq  P \\cap Z(N_S(P))$ of order $p$, and note that in particular $N_S(P)\\leq C_S(V)$. By \\cite[Lemma 2.2 (b)]{BLO6}, there exists some $\\omega \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(V), S)$ such that $\\omega(V)$ is fully $\\mathcal{F}$-centralized. Hence, there is an inequality $|N_S(\\omega(P))| \\geq |N_S(P)|$ which is in fact an equality since we are assuming $P$ to be fully $\\mathcal{F}$-normalized.\n\nWe may replace $P$ and $V$ by $\\omega(P)$ and $\\omega(V)$, and assume that $V$ is fully $\\mathcal{F}$-centralized and $P$ is fully $\\mathcal{F}$-normalized. Furthermore, $P$ is fully normalized in the saturated fusion system $C_{\\mathcal{F}}(V)$, since $N_S(P) = N_{C_S(V)}(P)$.\n\nBy Case 1, the map $BC_{\\mathcal{G}}(V) \\to B\\mathcal{G}$, induced by the inclusion $C_{\\mathcal{L}}(V) \\subseteq \\mathcal{L}$, induces a homotopy equivalence $\\mathrm{Map}(BV, BC_{\\mathcal{L}}(V))_{\\mathrm{incl}} \\simeq \\mathrm{Map}(BV, B\\mathcal{G})_{\\mathrm{incl}}$, and hence also a homotopy equivalence\n$$\n\\mathrm{Map}((EP)\/V, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}((EP)\/V, B\\mathcal{G})_{\\mathrm{incl}}\n$$\nwhich is $P\/V$-equivariant (where the action of $P\/V$ is the action induced by the original action of $P$ on $EP$). This is still a homotopy equivalence after considering homotopy fixed point sets by \\cite[Remark 10.2]{DW0}, and thus we obtain another homotopy equivalence\n$$\n\\big[\\mathrm{Map}((EP)\/V, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}}\\big]^{h(P\/V)} \\Right2{\\simeq} \\big[\\mathrm{Map}((EP)\/V, B\\mathcal{G})_{\\mathrm{incl}}\\big]^{h(P\/V)}.\n$$\nNotice that $E(P\/V) \\times_{P\/V} (EP)\/V \\simeq BP$ with the given actions (here $P\/V$ is acting diagonally on $E(P\/V) \\times (EP)\/V$). Let $X = BC_{\\mathcal{G}}(V)$ or $B\\mathcal{G}$. By definition of homotopy fixed point sets, we have\n$$\n\\begin{aligned}\n\\big[\\mathrm{Map}((EP)\/V, X)\\big]^{h(P\/V)} & = \\mathrm{Map}_{P\/V}(E(P\/V), \\mathrm{Map}((EP)\/V,X)) \\simeq \\\\\n & \\simeq \\mathrm{Map}_{P\/V}(E(P\/V) \\times (EP)\/V, X),\n\\end{aligned}\n$$\nwhere the rightmost equivalence follows by adjunction. Furthermore, $P\/V$ acts trivially on $X$, and thus it follows that $\\mathrm{Map}_{P\/V}(E(P\/V) \\times (EP)\/V, X) \\simeq \\mathrm{Map}(E(P\/V) \\times_{P\/V} (EP)\/V, X) \\simeq \\mathrm{Map}(BP,X)$. Thus, the equivalence of homotopy fixed point sets above induces the following equivalence\n$$\n\\mathrm{Map}(BP, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}.\n$$\nWe can suppose that $\\mathcal{L} = C_{\\mathcal{L}}(V)$, and hence that $V$ is central in $\\mathcal{L}$. Let $\\mathcal{G}\/V$ be the quotient of $\\mathcal{G}$ by $V$, as described in Definition \\ref{quotient1}. In particular, $\\mathcal{F}\/V$ is a saturated fusion system on $S\/V$, and $\\mathcal{L}\/V$ is a transporter system.\n\nConsider the full subcategories $\\mathcal{L}_0 \\subseteq \\mathcal{L}$ and $(\\mathcal{L}\/V)^c \\subseteq \\mathcal{L}\/V$ whose objects are the subgroups $Q\\leq S$, respectively $Q\/V\\leq S\/V$, such that $Q\/V$ is $\\mathcal{F}\/V$-centric. In particular, $(\\mathcal{L}\/V)^c$ determines a centric linking system associated to $\\mathcal{F}\/V$, and there are homotopy equivalences $|\\mathcal{L}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$ and $|(\\mathcal{L}\/V)^c|^{\\wedge}_p \\simeq |\\mathcal{L}\/V|^{\\wedge}_p$.\n\nFinally, let also $\\mathcal{F}' = N_{\\mathcal{F}}(P)$, $\\mathcal{L}' = N_{\\mathcal{L}}(P)$, and define $\\mathcal{L}'\/V$, $\\mathcal{L}_0' \\subseteq \\mathcal{L}'$ in a similar way as done above. It follows by Lemma \\ref{quotient2} that there are fibration sequences\n$$\n\\xymatrix@R=2mm{\nBV \\ar[rr] & & |\\mathcal{L}_0|^{\\wedge}_p \\ar[rr]^{\\Phi} & & |(\\mathcal{L}\/V)^c|^{\\wedge}_p \\\\\nBV \\ar[rr] & & |\\mathcal{L}_0'|^{\\wedge}_p \\ar[rr]^{\\Phi'} & & |(\\mathcal{L}'\/V)^c|^{\\wedge}_p,\\\\\n}\n$$\nand hence also a homotopy pull-back square\n$$\n\\xymatrix{\n\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota} \\ar[rr]^{I_1} \\ar[d]_{\\Phi' \\circ -} & & \\mathrm{Map}(BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\ar[d]^{\\Phi \\circ -} \\\\\n\\mathrm{Map}(BP, |(\\mathcal{L}'\/V)^c|^{\\wedge}_p)_{\\Phi \\circ \\mathrm{incl}} \\ar[rr]_{I_2} & & \\mathrm{Map}(BP, |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{\\Phi \\circ \\mathrm{incl}},\n}\n$$\nwhere $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota}$ is the union of the connected components which map to the inclusion in $|\\mathcal{L}_0|^{\\wedge}_p$ and to $\\Phi \\circ \\mathrm{incl}$ in $|(\\mathcal{L}'\/V)^c|^{\\wedge}_p$, and $I_1$, $I_2$ are inclusions.\n\nBy (\\ref{zero}), together with the induction hypothesis, and since $P\/V$ has strictly smaller order than $P$ (recall that $P$ is finite by hypothesis), there are homotopy equivalences\n$$\n\\begin{aligned}\n|C_{(\\mathcal{L}\/V)^c}(P\/V)|^{\\wedge}_p & \\RIGHT6{\\Gamma'_{\\mathcal{L}\/V, P\/V}}{\\simeq} \\mathrm{Map}(B(P\/V), |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{\\mathrm{incl}} \\Right1{} \\\\\n & \\RIGHT6{- \\circ \\operatorname{proj}\\nolimits}{\\simeq} \\mathrm{Map}(BP, |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{f \\circ \\mathrm{incl}}\n\\end{aligned}\n$$\nand similarly for maps to $|(\\mathcal{L}'\/V)^c|^{\\wedge}_p$. Since $\\Gamma'_{\\mathcal{L}\/V, P\/V}$ is the composite of $\\Gamma'_{\\mathcal{L}'\/V, P\/V}$ with the inclusion by definition , this shows that the map $I_2$ in the diagram above is a homotopy equivalence, and hence so is $I_1$. In particular, $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota}$ is connected and contains the component of the inclusion. Thus, Case 3 follows from Case 2 applied to the mapping space $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\mathrm{incl}}$.\n\n\\textbf{Case 4.} Suppose that $P$ is an infinite discrete $p$-toral group. By Lemma \\ref{central2}, there is a sequence of subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$, and such that $P_n$ is fully $\\mathcal{F}$-centralized with $C_{\\mathcal{G}}(P_n) = C_{\\mathcal{G}}(P)$ for all $n \\geq 0$. We have a sequence of homotopy equivalences\n$$\n\\begin{aligned}\n\\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}} & = \\mathrm{Map}(\\mathrm{hocolim \\,} \\mbox{ } BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\\\\n& \\simeq \\operatornamewithlimits{holim} \\mbox{ } \\mathrm{Map}(BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\operatornamewithlimits{holim} \\mbox{ } BC_{\\mathcal{G}}(P_n) = BC_{\\mathcal{G}}(P),\n\\end{aligned}\n$$\nwhere the equivalence $\\mathrm{Map}(\\mathrm{hocolim \\,} \\mbox{ } BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\operatornamewithlimits{holim} \\mbox{ } \\mathrm{Map}(BP_n, B\\mathcal{G})_{\\mathrm{incl}}$ follows from \\cite[Proposition 2, page 187]{D-F}. This finishes the proof.\n\\end{proof}\n\nThe reader may think of replacing Cases 1 to 3 in the proof above by the following argument. Given a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ and a fully $\\mathcal{F}$-centralized finite subgroup $P\\leq S$, consider the centralizer $p$-local compact group $C_{\\mathcal{G}}(P) = (C_S(P), C_{\\mathcal{F}}(P), C_{\\mathcal{L}}(P))$ of $P$ in $\\mathcal{G}$. Set for short $Z = C_S(P)$, $\\mathcal{E} = C_{\\mathcal{F}}(P)$, and $\\mathcal{T} = C_{\\mathcal{L}}(P)$. Given a fine unstable Adams operation (see \\ref{uAo}), we may assume that $\\Psi(P) = P$, since $P$ is a finite subgroup of $S$, and thus $\\Psi$ restricts to a fine unstable Adams operation on $C_{\\mathcal{G}}(P)$.\n\nThis way, $\\Psi$ defines approximations of $\\mathcal{G}$ and $C_{\\mathcal{G}}(P)$ by $p$-local finite groups, namely $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ and $\\{(Z_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ respectively. Moreover, we may assume that $P\\leq S_i$ for all $i \\geq 0$. It is not hard to see that $Z_i = C_{S_i}(P)$ for all $i \\geq 0$, since $Z_i = Z \\cap S_i = C_S(P) \\cap S_i$. The main difficulty of this argument is that it is not clear whether $\\mathcal{E}_i$ corresponds to the centralizer fusion system of $P$ in $\\mathcal{F}_i$ for any $i$. Essentially, the main problem is that the finite retraction pairs for $\\mathcal{G}$ and $C_{\\mathcal{G}}(P)$ given in \\ref{expl1} do not agree with each other in general, and \\ref{expl3} does not apply to this situation, since in general $C_S(P)$ does not have finite index in $S$. One could drop this last condition, but at the price of dealing with a more complicated situation.\n\n\\begin{rmk}\n\nWith the above description of the homotopy type of the mapping spaces $\\operatorname{Map}\\nolimits(BP, B\\mathcal{G})$, one could now generalize the results of C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver \\cite{BCGLO1, BCGLO2}. More precisely, \\cite[Theorem A]{BCGLO1} has already been proved for $p$-local compact groups as \\cite[Theorem 4.2]{BLO6}, and \\cite[Theorem B]{BCGLO1} would follow easily now from our result above. Regarding \\cite{BCGLO2}, some results have already been extended to $p$-local compact groups in \\cite[Appendix B]{Gonza2}, and the rest would follow by the same arguments (with some minor modifications). We omit this for the sake of brevity of this paper.\n\n\\end{rmk}\n\n\n\\bibliographystyle{gtart}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAstronomical observations have been traditionally done with visible light. From~the times of Galileo times until now we have been able to expand the observation range to all the electromagnetic spectrum, from~radio to gamma rays.\nApart from photons, the~discovery of other particles has opened the possibility of using them as cosmic messengers to explore the Universe. For~instance, cosmic rays (CRs) were first discovered in the early 1910s. We know that CRs are ionized nuclei of extraterrestrial origin, and~that they are produced and accelerated in a broad energy range, reaching energies above 10$^{20}$ eV. The~spectrum of the CRs detected at Earth decreases with the energy, following approximately a power law E$^{-n}$ with n$\\sim$3. The~Sun is the main source of CRs below GeV energies. Up~to PeV energies, CR are believed to be dominated by Galactic sources, and~then, at~the highest energies, CRs are likely of extragalactic origin~\\cite{ParticleDataGroup:2020ssz}. As CRs are charged, their directionality is lost due to the Galactic magnetic field. This is probably one of the main reasons why the origin of the most energetic CRs is still unknown.\nWe have also detected neutrinos from extraterrestrial origin. First from the Sun~\\cite{Davis:1968cp}, and~then from a nearby supernova explosion in 1987~\\cite{Hirata:1988ad,IMB:1987klg,Baksan:1987} which can be considered as the birth of neutrino astronomy. By~that time, the first project to construct a neutrino telescope was already ongoing~\\cite{DUMAND:1989dxw}. However, only after decades of research and development, neutrino astronomy had its turning point in 2014 with the discovery of a high-energy cosmic neutrino flux by the IceCube collaboration~\\cite{IceCube:2014stg}.\nThe most recent cosmic messengers discovered are gravitational waves (GWs). The~path to the first GW detection was also not easy, and~it took a century from their theoretical prediction to the first confirmation in 2015 by the LIGO and VIRGO collaborations~\\cite{LIGOScientific:2016aoc}.\n\n\nMultimessenger astronomy originates as a consequence of astrophysical neutrino and GW detection techniques reaching maturity. Multimessenger astronomy is based on the observation of four cosmic messengers, namely, photons, CRs, neutrinos, and~GWs. The~detection in coincidence of all, or~some, of~these messengers allows the study of a source in a similar fashion as it has been done with multiwavelength electromagnetic observations. Moreover, the~fact of involving different types of particles adds extra information coming from interactions involving all fundamental forces of~nature.\n\nWe have divided this review into three sections. In~Section~\\ref{Milestones}, we will discuss the two events that marked the start of the multimessenger era, four years ago. In~Section~\\ref{Results}, we will review the most recent results and what we have learned from them. To~conclude, in~Section~\\ref{Future}, we will discuss the future prospects and challenges in the~field.\n \n\\section{Multimessenger Astronomy~Milestones}\n\\label{Milestones}\nExcluding the solar neutrino detection, the~observation by chance of neutrinos coming from the supernova explosion SN1987A is the first astronomical event producing a multimessenger (photon--neutrino) coincidence. However, there are two main events, both happening in 2017, that really marked the birth of a new field in astrophysics, multimessenger~astronomy.\n\nThe first event was the result of two neutron stars merging into a black hole. This produced a GW (GW170817) that was detected by the LIGO and Virgo~\\cite{LIGOScientific:2017vwq} Scientific Collaborations. Less than 2 seconds after the event, a~short gamma-ray burst (GRB) (GRB 170817A) was detected by the Fermi and INTEGRAL satellites. This coincidence triggered a campaign where several observatories followed-up the event in an unprecedented way~\\cite{LIGOScientific:2017ync}.\nThanks to this coincident detection it was possible to determine the location and the type of sources involved, bringing the first experimental evidence of a kilonova~\\cite{Metzger:2017}, a~type of transient event, theoretically predicted more than two decades ago~\\cite{Lixin:1998}, where nucleosynthesis of the heavy elements is produced.\n\nThe second event (IC-170922A) was triggered by a high-energy neutrino, of~about 300 TeV, detected by the IceCube observatory on 22 September 2017. Again, this event was extensively followed up by other observatories. \nIn this case, observations from the Fermi-LAT satellite were able to point out a blazar (TXS0506+056) in active state which was in spatial and temporal coincidence with the neutrino event. The~event was rejected to be produced by background fluctuations at 3$\\sigma$ level~\\cite{IceCube:2018dnn}.\nAfter this detection, archival analysis of IceCube data prior to the IceCube-170922A event unveiled a potential flare in neutrinos~\\cite{IceCube:2018cha}, between~September 2014 and March 2015, with~3.5$\\sigma$ statistical significance and independent of the 2017 neutrino alert. In~this case, no gamma-ray counterpart was observed.\nAn extensive multiwavelength monitoring of TXS0506+056 started after the coincident event in September 2017 showed a low state emission except for December 1st and 3rd, 2018 with a flare comparable to the one in 2017~\\cite{Satalecka:2021}. However, no neutrino excess was observed.\nIt is also important to mention that TXS0506+056 came out as the second most significant source (2.8$\\sigma$ pre-trial) in the point source search analysis done with the ANTARES neutrino telescope using a pre-selected list of sources~\\cite{Illuminati:2021b}, which makes the case of TXS0506+056~stronger.\n\n\\section{Recent~Results}\n\\label{Results}\nOnce it has been well established that a flux of high-energy neutrinos of cosmic origin exist~\\cite{IceCube:2014stg}, the~next step is to disentangle it and identify which are the sources. The~most recent all-sky searches performed by ANTARES~\\cite{Illuminati:2021b} and IceCube~\\cite{IceCube:2019cia} did not reveal any significant detection above the discovery threshold of 5$\\sigma$, with the excess near the galaxy NGC 1068 observed by IceCube being the most interesting spot with a post-trial significance of 2.9$\\sigma$.\nThis type of high-energy searches benefit from multimessenger astronomy thanks to including the sky coordinates and timing information from potential cosmic messenger counterparts. That was how the first evidence of a cosmic neutrino source, TXS0506+056, was~found. \n\nUnderstanding the multimessenger emission from TXS0506+056 has been challenging from the theoretical point of view, as it is difficult to get a good agreement between the observed neutrino signal and other wavelength observations. \nIf one tries to explain it with a single-zone model, i.e.,~both gammas and neutrinos coming from the same region, one finds out that a leptonic scenario with a radiatively subdominant hadronic component provides the only physically consistent single-zone picture~\\cite{Keivani:2018rnh}.\nA higher neutrino flux would be expected if the source hosts two physically distinct emitting regions (see for instance~\\cite{Xue:2019txw}). However, current observations cannot discriminate between single- or multi-zone emission models.\nRelated to this, a~compelling neutrino--radio correlation~\\cite{Plavin:2020mkf} has been recently discovered. The~authors proposed that neutrinos and gamma rays may be indeed produced in different regions~\\cite{Plavin:2020emb}. If~this is actually the case, X-ray and radio may be better wavelengths when looking for photon--neutrino correlations. \nThe correlation with radio blazars is also supported by ANTARES observations~\\cite{ANTARES:2020zng}.\nIn this regard, ANTARES has recently reported the results from an untriggered search from radio blazars with an interesting association coming from J0242+1101~\\cite{Illuminati:2021}.\n\nThanks to the IceCube alert system~\\cite{IceCube:2016cqr}, which is presently providing on the order of 10 (20) gold (bronze) alerts per year\\endnote{Gold (bronze) alerts are neutrino events with >50\\% (>30\\%) probability of being from astrophysical origin.}, more coincidences between neutrino and blazars have been found lately. For~instance, PKS 1502+106 blazar was coincident with a 300~TeV neutrino~\\cite{Rodrigues:2020fbu}. In~this case, the blazar was in a quiescent state at the time of the neutrino alert. However, no more neutrinos were detected. \nAnother example is 3HSP J095507.9, which was also coincident with a high-energy neutrino detected by IceCube~\\cite{Paliya:2020mqm,Giommi:2020viy}. However, for~this event a lot of sources lay around the best position provided by IceCube, preventing the identification of a potential source candidate. This also underscores that sub-degree angular resolution, achievable by future neutrino observatories, will be key when looking for spatial coincidences.\nFrom Fermi-LAT observations we know that blazars are the most abundant extragalactic gamma-ray sources, constituting roughly 80\\% of the entire extragalactic source population~\\cite{Fermi-LAT:2019pir}. However, current predictions based on stacking catalog searches performed with IceCube data estimate that neutrinos emitted by blazars, in~the range between around 10 TeV and 2 PeV, can only contribute up to 27\\% to the total neutrino diffuse flux~\\cite{IceCube:2016qvd}. \nMore multimessenger observations with next generation experiments are required to test current theoretical models, and~therefore provide valuable information to understand the particle production and acceleration in~blazars.   \n\nApart from blazars other neutrino candidates have been already identified thanks to multimessenger observations. This is the case of Tidal Disruption Events (TDEs), which are the result of a star being ripped apart when passing next to a supermassive black hole. TDEs were already hypothesized as possible neutrino sources, e.g., in~\\cite{Wang:2011}. However, it was not until 2019 when an IceCube neutrino, with~59\\% probability of being of astrophysical origin (IC-191001A), triggered an alert that was followed-up by the Zwicky Transient Facility~\\cite{Bellm:2019}. In~spatial coincidence with the neutrino alert a TDE (AT2019dsg) was observed. Given that TDEs are rare events, the~chance probability of finding this coincident event was estimated to be less than 0.5\\%~\\cite{Stein:2020xhk}.\nAfter the first neutrino-TDE coincidence, more recently another possible association has been detected (IC200530A with AT2019fdr) which has been considered by some scientists as evidence of an emerging trend. The~chance probability of finding this second event in coincidence was also small.\nBoth events have been followed up by ANTARES, however did not produce any significant neutrino {excess}~\\cite{ANTARES:2021jmp}.\nThe~non detection does not contradict the observation by IceCube, as~the sensitivity of ANTARES was above the neutrino flux prediction.\nOn the other hand, there are preliminary indications of an excess in GVD-Baikal data~\\cite{Allakhverdyan:2021}.\nCurrent estimations predict that TDEs contribute at least 2\\% but not more than 40\\% of the total neutrino flux~\\cite{Stein:2021}. Again, more observations are needed to confirm TDEs as sources of high-energy neutrinos and~determine their precise contribution to the diffuse neutrino~flux.\n\nGRBs are another type of sources that have long been predicted as good candidates to emit high-energy neutrinos, see, for instance, in~\\cite{Waxman:1997ti}. However, so far, all the searches have been unsuccessful~\\cite{Albert:2016eyr,IceCube:2016ipa}.\nRecently, some studies have tried to infer what would be the relative contribution of the different neutrino candidate sources, see, e.g., in~\\cite{Bartos:2021tok}. However, it seems that there is no clear indication of a dominant type of source producing cosmic neutrinos. Interestingly enough, the~same study suggest that there is room for unknown~sources.\n\n\\section{Future Prospects and~Challenges}\n\\label{Future}\n\\unskip\n\\subsection{Future Instruments and Instrument~Upgrades}\n\nRegarding neutrino telescopes there are three major projects that will be operating in the near future.\nKM3NeT~\\cite{KM3Net:2016zxf} is a research infrastructure being built in the Mediterranean sea. KM3NeT is composed of two detectors, first ARCA (Astroparticle Research with Cosmics in the Abyss) which is designed to be sensitive to high-energy neutrinos in the TeV-PeV range, and~therefore with astrophysics studies as the main goal. The~second detector is called ORCA (Oscillation Research with Cosmics in the Abyss) and it is sensitive to GeV neutrinos. The~ORCA detector will primarily be used for the study of neutrino properties.\nBoth instruments will use the same technology and detection principle, i.e.,~array of photomultipliers tubes (PMTs) in sea water, being the main difference the volume covered, and~therefore the PMT density.\nARCA is expected to be fully operational in 2027 and ORCA in~2025.\n\nMoreover, in water there is the GVD-Baikal project that is in construction in the Baikal Lake in Russia. GVD-Baikal is currently operational with 2304 optical modules arranged in eight~clusters of eight strings each. In~the present configuration, it has an effective volume of 0.4~km$^{3}$ for cascades with energy above 100 TeV~\\cite{Baikal:2021}. Current plans are to deploy six~additional clusters for the period from 2022 to 2024 which should provide an additional 0.3~km$^{3}$ effective volume.\n\nThe leading project in neutrino telescopes in the last decade has been IceCube~\\cite{IceCube:2021} which is a neutrino telescope installed in the South Pole.\nAfter 10 years of successful operation there are plans for two major upgrades. One, called IceCube-Gen2~\\cite{IceCube-Gen2:2020qha}, expected to be completed by the early 2030s, will significantly increase the IceCube effective volume and energy range sensitivity. The~other, called IceCube-Upgrade~\\cite{IceCube:2019xdf}, represents a fraction of a larger project called Precision IceCube Next Generation Upgrade (PINGU)~\\cite{IceCube:2016xxt}, which aims to increase the sensitivity to lower energies even more than with IceCube-DeepCore, mainly thanks to a higher string density, and~that will mostly study neutrino~properties.\n\nWe can also add to the list of future intended neutrino telescopes the Pacific Ocean Neutrino Experiment (P-ONE)~\\cite{PONE:2021}. The~P-ONE project is presently in research and development phase. The~goal of the collaboration is to install a multi-cubic-kilometer neutrino telescope in the Pacific Ocean, which is expected to be operational in the next~decade.\n\n\\textcolor{black}{Other neutrino experiments, foreseen to be operational by the end of this decade, like Hyper-Kamiokande~\\cite{Hyper-Kamiokande:2018ofw} and DUNE~\\cite{DUNE:2020lwj}, will be sensitive to lower neutrino energies (MeV-GeV). However, they can still contribute to get the whole picture of some astrophysical events. For~instance, the~explosion of a nearby supernova, where low-energy neutrinos are expected to be produced.}\n\nThere are also very exciting plans for next generation experiments aiming to detect other cosmic messengers. Just to mention a few of them, we have the Cherenkov Telescope Array (CTA)~\\cite{CTA:2021} with two planned sites (Northern and Southern Hemisphere), and~LHAASO~\\cite{LHAASO:2021}, fully operational since July 2021, detecting gamma rays. KAGRA~\\cite{KAGRA:2019}, detecting GWs, will join LIGO and Virgo for the next GW data taking run (O4), which is expected to start in late 2022. Finally, detecting ultra-high-energy CRs, there will be AugerPrime~\\cite{PierreAuger:2016qzd}, the~upgrade of the Pierre Auger Observatory.\nSuch a network of observatories, distributed in different locations around the globe (see Figure~\\ref{fig1}), will provide a full multimessenger coverage of the~sky.\n\n\\subsection{Alert Systems and~Strategies}\n\nConsidering the number of experiments currently being under construction and planned, it seems clear that an efficient communication between collaborations is crucial. Moreover, for~the case of pointing instruments, like CTA, this communication also needs to be fast. \n\\textcolor{black}{Neutrinos are actually very good messengers to trigger alerts because they are able to easily escape from sources. These neutrino alerts can give an early warning to other observatories of an incoming event, allowing for a prompt follow up of transient~phenomena.}\n\nThere are already ways to announce, in~real-time, interesting events to the astrophysics community, e.g., the~GammaRay Coordinates Network (GCN)~\\cite{GCN:2021}. In~addition to these announcements, there are also sites, like the Astronomers Telegram (ATEL)~\\cite{ATEL:2021}, where a brief report about recent observations made by the experiments is posted online.\nStill in beta testing phase there is a project called {Astro-COLIBRI}~\\cite{Reichherzer:2021pfe}\nwhose goal is to act as a central platform where a large set of information coming from different experiments is gathered. This can be accessed via web or smartphone interface. The~data will be immediately available and will contain relevant information such as the visibility of the event for a given observatory, the~false alarm rate, or~the probability of the event to be of astrophysical~origin.\n\n\\begin{figure}[H]\n\\includegraphics[width=0.95\\linewidth]{Figures\/MM_WorldMap.pdf}\n\\caption{Earth map indicating the location of a selection of multimessenger observatories that are either currently operating (circles) or planned (triangles). Satellites are depicted outside to the~map.}\n\\label{fig1}\n\\end{figure}   \n\nIn addition to these prompt alert systems across collaborations, there are alert follow-up programs like TaToO~\\cite{ANTARES:2015fce}, where the most promising ANTARES events trigger a prompt optical, radio, and X-ray follow-up on the sky region where the neutrino candidate comes from, looking for any potential transient counterpart within hours, days, and months since the event. This is done using a network of optical telescopes at different locations (at a rate of 25 alerts per year) and, for~the most energetic ones (around 6 alerts per year), the~XRT instrument aboard the Swift satellite and the Murchison Wide field Array radio telescope~\\cite{tatoo:2021}.\n\nAs another example of how data from different experiments is shared, we have the Astrophysical Multimessenger Observatory Network (AMON)~\\cite{AyalaSolares:2019iiy}. One of the main ideas of AMON is to use sub-threshold data from different experiments to exploit the fact that a combined detection is expected to increase the significance of the event. Therefore, an~event that by itself is not enough to claim a detection with a single experiment, and~could have been rejected, can actually become significant when detected in coincidence with other observatories. An~example of a recent analysis done through this network is~\\cite{Hugo:2021}, focused on gamma-ray and neutrino~coincidences.\n\n\\subsection{Future~Challenges}\nConcerning the multimessenger astronomy goals in the next few years, one thing that should be attainable very soon is a firm confirmation of a source of cosmic neutrinos above the discovery threshold of 5$\\sigma$. To~this end, the~selection of the most promising sources, to~reduce the amount of trials in the search, thanks to multimessenger observations will be crucial.\nThis is quite likely to be accomplished by more than one project which will provide an unbiased way of measuring the spectrum of the sources, bringing key information to understand the high-energy neutrino production and acceleration mechanisms in the source. \nAlso combined analyses are possible, as~has been already done in the past~\\cite{ANTARES:2020srt}.\n\nOne multimessenger observation that is most awaited, and~can probably be achieved thanks to the improved sensitivity of the future experiments, is the coincident detection of a GW event with high-energy neutrinos. \nThanks to recent GW observations, we have a better understanding of the link between neutron star mergers and short GRBs, and~the physics involved. We already discussed the particular case of the binary neutron star merger GW170817, which led to the GRB170817A coincidence. This type of event should produce a GW signature together with gamma rays and neutrinos. However, at~present, the~only confirmed coincident detection is the GW-gamma correlation, while no evidence of neutrino emission was found~\\cite{ANTARES:2017bia,Baikal:2019}.\nThe theoretical estimations of neutrino production~\\cite{Kimura:2018vvz} from GW170817 show that the expected flux should be already detectable, with~current neutrino telescopes, under~favorable circumstances, see Figure~\\ref{fig2}.\n\n\\begin{figure}[H]\n\\vspace{-9pt}\n\\includegraphics[width=\\linewidth]{Figures\/GW.pdf}\n\\caption{Fluence upper limits, per flavor, on~the high-energy neutrino emission from experimental data assuming a $\\pm$500s time window around the GW170817 {event}\n~\\cite{ANTARES:2017bia}. Baikal limits are from~\\cite{Baikal:2019}. KM3NeT preliminary sensitivity, computed for an optimal zenith angle, is from~\\cite{Palacios:2021}. Theoretical models for comparison are from~\\cite{Kimura:2018vvz}. Figure adapted from the work in ~\\cite{ANTARES:2017bia}.}\n\\label{fig2}\n\\end{figure}\n\nApart from the detection of high-energy neutrinos, the~detection of the prompt emission in gamma-rays by observatories like HAWC or LHAASO will be essential to understand how these energetic explosions~work.\n\nAnother open question to be addressed by multimessenger observations is the connection between ultra-high-energy CRs, gamma-rays, and~high-energy neutrinos. \nIntensity of gamma-rays, neutrinos, and UHECRs has been shown to be comparable (see \\mbox{Figure~\\ref{fig3}}), suggesting that they may be powered by the same sources. \nAs blazars are the most abundant extragalactic sources, they are by default the most promising candidates. However, blazars do not seem to fit the bill, as they are subdominant in the high-energy neutrino flux~\\cite{Oikonomou:2021}.\n\n\\begin{figure}[H]\n\\includegraphics[width=0.97\\linewidth]{Figures\/DF.pdf}\n\\caption{High-energy fluxes of gamma {rays}~\\cite{Fermi-LAT:2014ryh}, neutrinos~\\cite{IceCube:2020wum}, and~cosmic rays~\\cite{Auger:2017}. Figure adapted from~the work in \\cite{IceCube:2020wum}.}\n\\label{fig3}\n\\end{figure}\n\nApart from the usual suspects (e.g., GRBs and TDEs), the~case for star-forming galaxies as common sources has recently grown in popularity thanks to the work in~\\cite{Roth:2021lvk} where the authors claim that the diffuse gamma-ray flux detected by Fermi-LAT~\\cite{Fermi-LAT:2014ryh} is actually dominated by star-forming galaxies.\nAt the same time, the~data collected in the Pierre Auger Observatory showed an indication of anisotropy at 4.0$\\sigma$ level in the arrival direction of CRs with E > 39 EeV showing an excess from the direction of nearby starburst galaxies~\\cite{PierreAuger:2018qvk}. Claiming that this type of high-rate star-forming galaxies can be the cause of $\\sim$~10\\% of the ultra-high-energy CR flux.\nIt has been also shown that the contribution of the starburst galaxies to the neutrino diffuse flux is sub-dominant and constrained to be at the level of $\\sim$10\\%~\\cite{Lunardini:2019zcf}. Therefore, it remains unanswered for now whether there is a dominant type of source accelerating all cosmic~messengers.\n\n\n\nFinally, if~we restrict the search to our Galaxy, one of the questions that will be solved in the near future thanks to multimessenger observations is what are the Galactic CR sources and how those CR propagate through the Galaxy.\nRegarding this, it has been shown that a sub-PeV diffuse Galactic gamma-ray emission exists~\\cite{TibetASgamma:2021tpz}.\nBased on this result in~\\cite{Fang:2021ylv}, the authors showed that the Galactic neutrino contribution should constitute roughly 5--10$\\%$ of the IceCube diffuse flux and that, in~the 10--100 TeV range, the~expected Galactic neutrino flux should be comparable to the total neutrino diffuse flux. If~so, the~next-generation neutrino telescopes should be sensitive enough to detect it.\nIt is possible that part of the measured gamma-ray diffuse flux comes from individual sources, being the obvious candidates the very-high-energy sources detected by HAWC~\\cite{HAWC:2019tcx} and LHAASO~\\cite{Cao:2021}. \nCombined multimessenger observations should be able to confirm whether this is actually the case.\nIn fact, there have been claims for evidence for such a joint production, of~high-energy neutrinos and gamma rays, in~the Cygnus Coocon region, based on the correlation of a high-energy IceCube neutrino and a high-energy (>300 TeV) photon flare observed by the Carpet\u20132 experiment~\\cite{Carpet-3Group:2021ygp}. \n\n\\section{Outlook}\nMultimessenger astronomy is a new branch of astroparticle physics, for~which neutrinos are expected to play a key role. It took just a few events, detected in coincidence, to~demonstrate that combining different messengers has a great potential for~discoveries.\n\nQuestions like what is the origin of the ultra-high energy cosmic rays are likely to be answered thanks to multimessenger~observations.\n\nConsidering the numerous facilities planned for the near future, or~already taking data (KM3NeT, IceCube-Gen2, GVD-Baikal, P-ONE, CTA, LHAASO, KAGRA, AugerPrime, etc.), multimessenger astronomy has a bright future and is expected to revolutionize our understanding of the Universe in the next~decade. \n\n\n\\vspace{6pt} \n\\authorcontributions{{All authors contributed equally to this article. All authors have read and agreed to the published version of the manuscript.}\n\n\\funding{This work was funded by Generalitat Valenciana, Spain (CIDEGENT\/2018\/034 and CIDEGENT\/2020\/049 grants).}\n\n\\institutionalreview{{Not applicable.}\n\n\\informedconsent{{Not applicable.}\n\n\\dataavailability{{Not applicable.}}\n\n\n\\acknowledgments{The authors thank the Valencia Experimental Group on Astroparticle Physics (VEGA) and the Ministerio de Ciencia, Innovaci\u00f3n, Investigaci\u00f3n y Universidades (MCIU): Programa Estatal de Generaci\u00f3n de Conocimiento (ref. PGC2018-096663-B-C41) (MCIU\/FEDER). }\n\n\\conflictsofinterest{The authors declare no conflict of interest.}\n\n\n\n\\abbreviations{Abbreviations}{\nThe following abbreviations are used in this manuscript:\\\\\n\n\\noindent \n\\begin{tabular}{@{}ll}\nGW & Gravitational Wave\\\\\nCR & Cosmic Ray\\\\\nGRB & Gamma-Ray Burst\\\\\nTDE & Tidal Disruption Event\\\\\nPMT & Photomultipliers tubes\n\n\\end{tabular}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}\nIntroduction\n}\n\nThe antiferromagnetic (AF) Ising model on a triangular lattice is one of the most fundamental models for geometrically frustrated systems. \nWhen the interaction is restricted to the nearest-neighbor (NN) pairs, frustration in each triangle prevents the system from forming a long-range order (LRO) down to zero temperature, and the ground state has extensive degeneracy and associated residual entropy~\\cite{Wannier1950,Houtappel1950,Husimi1950}.\nThe degenerate ground state is extremely sensitive to perturbations.\nFor instance, an infinitesimal second-neighbor interaction lifts the degeneracy and induces a LRO in the ground state; a two-sublattice stripe order [Fig.~\\ref{fig:model}(a)] is selected as the ground state when the additional interaction is AF, while a three-sublattice ferrimagnetic (FR) order [Fig.~\\ref{fig:model}(b)] is selected for the ferromagnetic (FM) interaction.\n\nIn such a degenerate situation, thermal fluctuations also play an interesting role. \nIn general, there is a possibility that a high-entropic state is selected out of the ground state manifold by raising temperature ---this is called the order by disorder~\\cite{Villain1977}. \nFor the AF Ising model, a candidate for such an emergent state is a partially disordered (PD) state. \nThe PD state is peculiar coexistence of magnetically ordered moments and thermally-fluctuating paramagnetic moments.\nSuch possibility was first discussed by the mean-field study in the presence of second-neighbor FM interaction~\\cite{Mekata1977}; the mean-field study predicted that a three-sublattice PD phase with an AF ordering on the honeycomb subnetwork and paramagnetic moments at the remaining sites [Fig.~\\ref{fig:model}(c)] was induced at finite temperature from the degenerate manifold in the limit of vanishing second-neighbor interaction. \nAlthough such PD state was experimentally observed in several Co compounds~\\cite{Kohmoto1998,Niitaka2001} and theoretically shown to present in a stacked triangular lattice model~\\cite{Todoroki2004}, Monte Carlo (MC) simulations in two-dimensional triangular lattice models have indicated that PD is fragile and remains at most as a quasi-LRO; namely, in most cases, the PD state is taken over by another peculiar intermediate state, the Kosterlitz-Thouless (KT) state~\\cite{Wada1982,Fujiki1983,Landau1983,Takayama1983,Fujiki1984,Takagi1995}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig0v1.eps}\n   \\caption{(Color online).\n   Schematic pictures of (a) stripe order, (b) ferrimagnetic (FR) order, and (c) partial disorder (PD) on a triangular lattice.\n   The arrows show magnetically ordered sites and the open circles are thermally fluctuating paramagnetic sites.\n   }\n   \\label{fig:model}\n\\end{figure}\n\nOn the other hand, recently, the authors have studied Ising-spin Kondo lattice models on a triangular lattice~\\cite{Ishizuka2012} and kagome lattice~\\cite{Ishizuka2012-3} by MC simulation, and showed the presence of PD state in the purely two-dimensional models.\nIn these models, the interplay between localized moments and itinerant electrons plays a crucial role in the following points. \nFirst, the kinetic motion of electrons induces effective interactions known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism~\\cite{Ruderman1954,Kasuya1956,Yosida1957}. \nThe long-ranged and oscillating nature of the interactions drives keen competition between different magnetic states.\nFurthermore, the change of magnetic states affects the electronic state in a self-consistent manner through the spin-charge coupling; the system can gain the energy by forming some particular electronic state associated with magnetic ordering.  \nIn the previous study,  the authors suggested that the PD state is stabilized by the non-perturbative role of itinerant electrons~\\cite{Ishizuka2012}.\n\nIn this contribution, we present our comprehensive numerical results on the magnetic and electronic properties of the Ising-spin Kondo lattice model on a triangular lattice. \nTo further clarify the stabilization mechanism of PD, we analyze the evolution of band structure under the PD type magnetic texture on the basis of a simple mean-field argument. \nThe analysis suggests that the spin-charge coupling can stabilize the PD state by the Slater mechanism.\nBearing this mean-field picture in mind, we present and discuss the results of MC simulation in details. \nWe distinguish the two intermediate-temperature states, PD and KT-like states, from the two-sublattice stripe and three-sublattice FR LRO states, and identify the range of the phases by varying the electron filling and the strength of spin-charge coupling. \nAnalyzing the phase diagram and electronic states in comparison with the mean-field picture, we conclude that the two-dimensional PD state is stabilized through the Slater mechanism.\n\nThe organization of this paper is as follows.\nIn Sec.~\\ref{sec:model_and_method}, we introduce the model and method.\nThe definitions of physical quantities we calculated are also given.\nIn Sec.~\\ref{sec:mft}, we present the mean-field analyses on the band structure in the PD state.\nMC results are presented for magnetic properties in Sec.~\\ref{sec:mc} and for electronic properties in Sec.~\\ref{sec:estruct}.\nSection~\\ref{sec:summary} is devoted to summary.\n\n\\section{\\label{sec:model_and_method}\nModel and Method\n}\n\nIn this section, we introduce the model and method.\nThe model is given in Sec.~\\ref{sec:model} and the MC method is described in Sec.~\\ref{sec:method}.\nIn Sec.~\\ref{sec:pmoment}, we give the definitions of physical quantities that we used to elaborate the phase diagram and thermodynamic properties.\n\n\\subsection{\nModel\n\\label{sec:model}\n}\n\nWe consider a single-band Kondo lattice model on a triangular lattice with localized Ising spin moments.\nThe Hamiltonian is given by\n\\begin{eqnarray}\nH = -t \\! \\sum_{\\langle i,j \\rangle, \\sigma} \\! ( c^\\dagger_{i\\sigma} c_{j\\sigma} + \\text{H.c.} ) + J \\sum_{i}\\sigma_i^z S_i.\n\\label{eq:H}\n\\end{eqnarray}\nThe first term represents hopping of itinerant electrons, where $c_{i\\sigma}$ ($c^\\dagger_{i\\sigma}$) is\nthe annihilation (creation) operator of an itinerant electron with spin $\\sigma= \\uparrow, \\downarrow$ at\n$i$th site, and $t$ is the transfer integral.\nThe sum $\\langle i,j \\rangle$ is taken over nearest-neighbor (NN) sites on the triangular lattice.\nThe second term is the onsite interaction between localized spins and itinerant electrons, where $\\sigma_i^z = c_{i\\uparrow}^\\dagger c_{i\\uparrow} -  c_{i\\downarrow}^\\dagger c_{i\\downarrow}$ represents the $z$-component of itinerant electron spin, and $S_i = \\pm 1$ denotes the localized Ising spin at $i$th site; $J$ is the coupling constant (the sign of $J$ does not matter in the present model). \nHereafter, we take $t=1$ as the unit of energy, the lattice constant $a = 1$, and the Boltzmann constant $k_{\\rm B} = 1$.\n\n\\subsection{\nMonte Carlo simulation\n\\label{sec:method}\n}\n\nTo investigate thermodynamic properties of the model (\\ref{eq:H}), we adopted a MC simulation which is widely used for similar models~\\cite{Yunoki1998}.\nThe model belongs to the class of models in which fermions are coupled to classical fields.\nFor this class of models, the partition function is given by\n\\begin{eqnarray}\nZ={\\rm Tr}_f{\\rm Tr}_c \\exp[\\beta(H-\\mu\\hat{N_e})],\n\\end{eqnarray}\nwhere $\\beta=1\/T$ is the inverse temperature, $\\mu$ is the chemical potential, and $\\hat{N_e}$ is the total number operator for fermions.\nHere, ${\\rm Tr}_f$ is the trace over classical degree of freedom (in the current case, Ising spin configurations), and ${\\rm Tr}_c$ is the trace over itinerant fermions. \nIn the MC simulation, ${\\rm Tr}_f$ is calculated by using the Markov-chain MC sampling. \nMC updates are done by the usual single-spin flip on the basis of the standard METROPOLIS algorithm. \nThe MC weight is calculated by taking the fermion trace ${\\rm Tr}_c$ for each configuration of classical variables in the following form, \n\\begin{eqnarray}\nP(\\{S_i \\}) = \\exp[ -S_{\\rm eff}(\\{S_i \\}) ],\n\\label{eq:P}\n\\end{eqnarray}\nwhere $S_{\\rm eff}$ is the effective action calculated as\n\\begin{eqnarray}\nS_{\\rm eff}(\\{S_i \\}) = - \\sum_\\nu \\log[1 + \\exp\\{-\\beta(E_\\nu(\\{S_i \\})-\\mu)\\}].\n\\label{eq:S_eff}\n\\end{eqnarray}\nHere, $E_\\nu(\\{S_i \\})$ are the energy eigenvalues for the configuration $\\{S_i \\}$, which are readily calculated by the exact diagonalization as it is a one-particle problem in a static potential.\n\nThe calculations were conducted for the system sizes $N=12 \\times 12$, $15 \\times 15$, $12 \\times 18$, and $18 \\times 18$ under the periodic boundary conditions.\nThermal averages of physical quantities were calculated for typically 4300-9800 MC steps after 1700-5000 steps for thermalization. \nThe results are shown in the temperature range where the acceptance ratio is roughly larger than 1\\%.\nWe divide the MC measurements into five bins and estimate the statistical errors by the standard deviations among the bins.\n\n\\subsection{\nPhysical quantities\n\\label{sec:pmoment}\n}\n\nAs we will see later, the model (\\ref{eq:H}) exhibits phase transitions to various magnetic states including different types of three-sublattice orders: ferrimagnetic (FR) state [Fig.~\\ref{fig:model}(b)] and partially disordered (PD) state [Fig.~\\ref{fig:model}(c)].\nThese magnetic states, in principle, are distinguishable by the spin structure factor for the Ising spins,\n\\begin{eqnarray}\nS({\\bf q}) = \\frac{1}{N} \\sum_{i,j} \\langle S_i S_j \\rangle \\exp({\\rm i} {\\bf q}\\cdot{\\bf r}_{ij}),\n\\label{eq:Sq}\n\\end{eqnarray}\nwhere the braket denotes the thermal average in the grand canonical ensemble, and ${\\bf r}_{ij}$ is the position vector from $i$ to $j$th site. \nThe PD order is signaled by peaks of $S({\\bf q})$ at ${\\bf q}=\\pm(2\\pi\/3,-2\\pi\/3)$, while the FR order develops a peak at ${\\bf q}=0$ in addition to ${\\bf q}=\\pm(2\\pi\/3,-2\\pi\/3)$.\nNo Bragg peaks develop in the KT state as it is a quasi-LRO.\nHowever, in finite-size calculations, it is difficult to distinguish these phases solely by the structure factor, as the correlation length in the KT state is divergent and easily exceeds the system size at low temperature.\n\nFor distinguishing the FR, PD, and KT instabilities, it is helpful to use the pseudospin defined for each three-site unit cell:\n\\begin{eqnarray}\n\\tilde{\\bf S}_m = \n\\left(\n\\begin{array}{ccc}\n\\frac2{\\sqrt6} & -\\frac1{\\sqrt6} & -\\frac1{\\sqrt6} \\\\\n0              &  \\frac1{\\sqrt2} & -\\frac1{\\sqrt2} \\\\\n\\frac1{\\sqrt3} &  \\frac1{\\sqrt3} &  \\frac1{\\sqrt3} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nS_i  \\\\\nS_j  \\\\\nS_k  \\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nand its summation \n\\begin{eqnarray}\n\\tilde{\\bf M} = \\frac{3}{N} \\sum_m  \\tilde{\\bf S}_m \n\\end{eqnarray}\nwhere $m$ is the index for the three-site unit cells, and $(i,j,k)$ denote the three sites in the $m$th unit cell belonging to the sublattices (A,B,C), respectively~\\cite{Takayama1983,Fujiki1984}.\nThen, the three-sublattice PD state [Fig.~\\ref{fig:model}(c)] is characterized by a finite $\\tilde{\\bf M} = (\\tilde{M}_x,\\tilde{M}_y,\\tilde{M}_z)$ parallel to $(\\sqrt{3\/2},1\/\\sqrt2,0)$, $(0,\\sqrt2,0)$, or their threefold symmetric directions around the $z$-axis.\nOn the other hand, the three-sublattice FR state [Fig.~\\ref{fig:model}(b)] is characterized by a finite $\\tilde{\\bf M}$ along $(\\sqrt{2\/3},\\sqrt2,1\/\\sqrt3)$, $(2\\sqrt{2\/3},0,-1\/\\sqrt3)$, or their threefold symmetric directions around the $z$-axis.\nHence, the two states are distinguished by the azimuth of $\\tilde{\\bf M}$ in the $xy$-plane as well as $M_z$.\nIn the MC calculations, we measure \n\\begin{eqnarray}\nM_{xy} &=& \\langle (\\tilde{M}_x^2 + \\tilde{M}_y^2)^{1\/2} \\rangle, \n\\label{eq:Mxy} \\\\\nM_z &=& \\langle |\\tilde{M}_z| \\rangle,\n\\label{eq:Mz}\n\\end{eqnarray}\nand the corresponding susceptibilities,\n\\begin{eqnarray}\n\\chi_{xy} &=& \\frac{N}{T} (\\langle \\tilde{M}_x^2 + \\tilde{M}_y^2 \\rangle - M_{xy}^2 ), \\\\\n\\chi_z &=& \\frac{N}{T} (\\langle \\tilde{M}_z^2 \\rangle - M_z^2 ).\n\\end{eqnarray}\nWe also introduce the azimuth parameter of $\\tilde{{\\bf M}}$ defined by\n\\begin{eqnarray}\n\\psi = {\\cal M}^3 \n\\cos{6 \\phi_M},\n\\label{eq:psi}\n\\end{eqnarray}\nwhere $\\phi_M$ is the azimuth of $\\tilde{\\bf M}$ in the $xy$ plane and ${\\cal M} = \\frac38 M_{xy}^2$.\nThe parameter $\\psi$ has a negative value and $\\psi \\to -\\frac{27}{64}$ for the perfect PD ordering, while it becomes positive and $\\psi \\to 1$ for the perfect FR ordering; $\\psi=0$ for both paramagnetic and KT phases in the thermodynamic limit $N \\to \\infty$.\n\nIn addition, we calculate the spin entropy to distinguish the three-sublattice orderings.\nThe spin entropy per site is defined by\n\\begin{eqnarray}\n{\\cal S}(T) = -\\frac{1}{N} \\sum_{\\{S_i\\}} P(\\{S_i\\})\\log P(\\{S_i\\}),\n\\label{eq:Sdef}\n\\end{eqnarray}\nwhere $P(\\{S_i\\})$ is the probability for spin configuration $\\{S_i\\}$ to be realized, given in Eq.~(\\ref{eq:P}).\nIn the actual MC calculation, instead of directly calculating Eq.~(\\ref{eq:Sdef}), ${\\cal S}$ is evaluated by calculating its temperature derivative\n\\begin{eqnarray}\n\\frac{\\partial{\\cal S}(T)}{\\partial T} = \\frac{1}{NT^2} \\left\\{ \\langle S_{\\rm eff} H \\rangle - \\langle S_{\\rm eff}\\rangle \\langle H \\rangle \\right\\},\n\\label{eq:delSdelT}\n\\end{eqnarray}\nand integrating it as \n\\begin{equation}\n{\\cal S}(T) = \\int_0^T \\frac{\\partial{\\cal S}(T)}{\\partial T} dT = \\log 2 - \\int_T^\\infty \\frac{\\partial{\\cal S}(T)}{\\partial T} dT. \n\\label{eq:Sint}\n\\end{equation}\nIn Eq.~(\\ref{eq:delSdelT}), $S_{\\rm eff}$ is the effective action in Eq.~(\\ref{eq:S_eff}). \nIn the following calculations, we set the cutoff $T=1$ for the upper limit of the last integral in Eq.~(\\ref{eq:Sint}).\n\nOn the other hand, in order to identify the two-sublattice stripe order [Fig.~\\ref{fig:model}(a)], we calculate the order parameter\n\\begin{eqnarray}\nM_{{\\rm str}} = \\left[ \\sum_{{\\bf q}^{*}_{\\rm str}} \\left\\{ \\frac{S({\\bf q}^{*}_{\\rm str})}{N} \\right\\}^2 \\right]^{1\/2},\n\\label{eq:Mstr}\n\\end{eqnarray}\nand its susceptibility $\\chi_{\\rm str}$. \nHere, the sum is taken for the characteristic wave vectors of the stripe orders running in three different directions, ${\\bf q}^{*}_{\\rm str}= (\\pi,0)$ and $(\\pm\\frac12\\pi,\\frac{\\sqrt3}2\\pi)$.\n\nWe also examine the thermodynamic behavior of electronic states for itinerant electrons.\nThere, we computed the charge modulation defined by\n\\begin{eqnarray}\nn_{\\rm CO} = \\left\\{\\frac{N({\\bf q}^*_{\\rm CO})}{N}\\right\\}^{1\/2}\n\\label{eq:n_CO}\n\\end{eqnarray}\nat ${\\bf q}^*_{\\rm CO}=(-2\\pi\/3,2\\pi\/\\sqrt3)$, which corresponds to the wave numbers for the three-sublattice orders.\nHere, $N({\\bf q})$ is the charge structure factor for itinerant electrons,\n\\begin{eqnarray}\nN({\\bf q}) = \\frac{1}{N} \\sum_{i,j} \\langle n_i n_j \\rangle \\exp({\\rm i} {\\bf q}\\cdot{\\bf r}_{ij}),\n\\end{eqnarray}\nwhere $n_i = \\frac12\\sum_\\sigma c_{i\\sigma}^\\dagger c_{i\\sigma}$.\n\n\\section{\nMean-field band structure\n\\label{sec:mft}\n}\n\nBefore going to the MC results, we here discuss how one particle band structure is modulated by PD ordering in a mean-field picture. \nWe consider a three-sublattice LRO state, in which the localized spins give a mean-field local magnetic field to itinerant electrons.\nNamely, we consider a mean-field Hamiltonian given by\n\\begin{eqnarray}\n{\\cal H}^{\\rm MF} = \\sum_{\\bf k}\n\\begin{pmatrix}\n\\Delta_{\\mathrm{A}}\\sigma^z_\\alpha & \\tau_{\\bf k} & \\tau_{\\bf k}^\\ast \\\\\n \\tau_{\\bf k}^\\ast & \\Delta_{\\mathrm{B}}\\sigma^z_\\alpha & \\tau_{\\bf k} \\\\\n \\tau_{\\bf k} & \\tau_{\\bf k}^\\ast & \\Delta_{\\mathrm{C}}\\sigma^z_\\alpha\n\\end{pmatrix}\\label{eq:mfh}\n.\n\\end{eqnarray}\nHere, three rows correspond to the different sublattices A, B, and C in the three-site unit cell; $\\Delta_\\alpha$ is a mean field given by $J\\langle S_\\alpha\\rangle$ ($\\alpha=\\mathrm{A}, \\mathrm{B}, \\mathrm{C}$).\nThe sum is taken in the first Brillouin zone for the magnetic unit cell for three-sublattice order.\n$\\tau_{\\bf k}$ is the hopping term for itinerant electrons given by\n\\begin{eqnarray}\n\\tau_{\\bf k} = -t[e^{{\\rm i}k_x} + e^{{\\rm i}\\left(-\\frac{k_x}2 + \\frac{\\sqrt3}2 k_y\\right)} + e^{{\\rm i}\\left(-\\frac{k_x}2 - \\frac{\\sqrt3}2 k_y\\right)}]\n\\end{eqnarray}\nand $\\sigma^z_\\alpha$ corresponds to the $z$ component of itinerant electron spin in each sublattice $\\alpha$.\n\nThe band structure for a FR order, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=( \\Delta, \\Delta, -\\Delta)$, was recently studied by the authors~\\cite{Ishizuka2012-2}.\nThere, it was reported that the electronic structure in the FR order is semimetallic with forming Dirac nodes at the electron filling $n= \\frac{1}{2N}\\sum_{i\\sigma} \\langle c_{i\\sigma}^\\dagger c_{i\\sigma} \\rangle=1\/3$ for $J>t$.\n\n\\begin{figure}\n   \\includegraphics[width=0.92\\linewidth]{fig11v1.eps}\n   \\caption{(Color online).\n   Mean-field band structure calculated by Eq.~(\\ref{eq:mfh}) for the local magnetic field of PD type, $(\\Delta_{{\\rm A}},\\Delta_{{\\rm B}},\\Delta_{{\\rm C}})=(2,0,-2)$.\n   Each of the three bands shown is doubly degenerate, and there are totally six bands.\n   The gray hexagon on the basal plane shows the first Brillouin zone for the magnetic supercell.\n   }\n   \\label{fig:band1}\n\\end{figure}\n\nHere, we discuss the band structure for the PD case, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=(\\Delta,0,-\\Delta)$.\nThe band structure for $\\Delta=2$ is shown in Fig.~\\ref{fig:band1}.\nIn this case, all three bands shown in the figure are doubly degenerate and there are six bands in total.\nThe first Brillouin zone is shown by the gray shade in the bottom surface. \nThe result shows the presence of an energy gap at the Fermi level corresponding to $n=1\/3$, that opens between the lowest energy band and the middle band [see also Fig.~\\ref{fig:band2}(c)].\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig12v3.eps}\n   \\caption{(Color online).\n   Mean-field band structure along the symmetric lines in the local magnetic field of PD type, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=(\\Delta,0,-\\Delta)$: (a) $\\Delta=1\/3$, (b) $\\Delta=2\/3$, and (c) $\\Delta=2$.\n   The dashed horizontal lines indicate the Fermi level for $n=1\/3$.\n   }\n   \\label{fig:band2}\n\\end{figure}\n\nWe next look into the conditions for the energy gap formation in the mean-field PD band.\nFigure~\\ref{fig:band2} shows the results of band structure while varying $\\Delta$.\nThe results are plotted along the symmetric line in the Brillouin zone shown in the bottom surface in Fig.~\\ref{fig:band1}.\nFor small $\\Delta$, the system is metallic at $n=1\/3$, as shown in the case of $\\Delta=1\/3$ in Fig.~\\ref{fig:band2}(a); both electron and hole pockets are present at the Fermi level.\nThe pockets shrink as increasing $\\Delta$, and disappear at the same time at $\\Delta=2\/3$, as shown in Fig.~\\ref{fig:band2}(b).\nFor larger $\\Delta$, an energy gap opens between the lowest and middle bands, corresponding to $n=1\/3$, as stated above [Fig.~\\ref{fig:band2}(c)]. \nHence, $\\Delta_c=2\/3$ is the critical point for the metal-insulator transition in this mean-field PD state.\n\n\\begin{figure}\n   \\includegraphics[width=0.9\\linewidth]{fig13v2.eps}\n   \\caption{(Color online).\n   $\\Delta$ dependences of the mean-field energy gap and associated charge modulation $n_{\\rm CO}$ at $n=1\/3$.\n   }\n   \\label{fig:gap}\n\\end{figure}\n\nFigure~\\ref{fig:gap} shows $\\Delta$ depedences of the energy gap and associated charge modulation $n_{\\rm CO}$ [Eq.~(\\ref{eq:n_CO})] at $n=1\/3$.\nThe charge gap develops for $\\Delta > 2\/3$ and monotonically increases, approaching asymptotically a $\\Delta$-linear form as $\\Delta \\gg t$.\nThe charge modulation is induced by the inhomogeneity of local potential; the local charge density at B sites (the site corresponds to paramagnetic sites) becomes dilute compared to those at A and C sites (the magnetically ordered sites).\nIn the limit of $\\Delta \\gg t$, $n_{\\rm CO}$ approaches $n_{\\rm CO}=1\/\\sqrt{12}\\sim 0.289$.\n\nThe results above suggest a stabilization mechanism of PD which is absent in the localized spin only model.\nIn the previous studies on the Ising spin models~\\cite{Takayama1983,Fujiki1983,Wada1982} and an equivalent classical particle model~\\cite{Landau1983} on a triangular lattice, PD was shown to be unstable against thermal fluctuations and taken over by a KT state.\nIn the case of our model, however, as the KT state lacks a long-range periodic magnetic structure, it is expected that the KT state does not open an energy gap in the electronic state of itinerant electrons.\nTherefore, in contrast to the case of localized spin only models, there is a chance for the current model to stabilize the PD state by the Slater mechanism, that is, by forming an energy gap at the Fermi level with folding the Brillouin zone under a periodic magnetic order.\n\nIn addition, the formation of an energy gap for $\\Delta > 2\/3$ implies that, if the PD state is stabilized by the Slater mechanism, it should appear from a finite $J$, and not remain stable down to $J\\to0$.\nThis is in sharp contrast to magnetic ordering by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction~\\cite{Ruderman1954,Kasuya1956,Yosida1957}; as the RKKY interaction is given by the second-order perturbation in terms of $J\/t$, if the PD state is stabilized by the RKKY interaction, it should appear for an infinitesimal $J$.\nHence, the phase diagram in the small $J$ region gives an idea on how the PD state is stabilized.\nWe will discuss this point by showing the MC results while changing $J$ in the next section.\n\n\n\\section{\nMonte Carlo simulation\n\\label{sec:mc}\n}\nIn this section, we present the results of MC simulation introduced in Sec.~\\ref{sec:method}.\nWe first show the finite-temperature phase diagrams in Sec.~\\ref{sec:pdiag}, which include four magnetic phases: stripe, PD, FR, and KT-like states.\nThe details of numerical data for the PD state are elaborated in Sec.~\\ref{sec:pd}.\nThe results for stripe, KT-like, and FR states are discussed  in Sec.~\\ref{sec:stripe_ferri}.\n\n\\subsection{\nPhase diagrams\n\\label{sec:pdiag}\n}\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig2v3.eps}\n   \\caption{(Color online).\n   Phase diagrams of the model~(\\ref{eq:H}) while varying $n$ at (a) $J=1$ and (b) $J=2$.\n   The symbols show phase boundaries for the four phases: stripe, partially disordered (PD), KT-like (``KT\"), and ferrimagnetic (FR) phases.\n   PS represents a phase separation. The lines are guides for the eyes.\n   The strips at $T=0$ show the ground states obtained by comparing the energy of stripe and FR states.\n   }\n   \\label{fig:ndiag}\n\\end{figure}\n\nFigure~\\ref{fig:ndiag}(a) shows the phase diagram around the electron filling $n= 1\/3$ at $J=1$ obtained by MC calculations. \nThere are four dominant ordered phases ---stripe, FR, PD, and KT-like phases, in addition to an electronic phase separation (PS).\nThe strip at the bottom of the figure shows the ground state obtained by variational calculation comparing the ground state energy of the stripe and FR states (the details of variational calculation is given in Appendix~\\ref{sec:pseparation}).\nFor the relatively low filling of $n \\lesssim 0.29$, the stripe order with period two [Fig.~\\ref{fig:model}(a)] develops in the low temperature region.\nOn the other hand, for the higher filling of $n \\gtrsim 0.32$, the system exhibits the three-sublattice FR order at low temperature [Fig.~\\ref{fig:model}(b)]. \nMC data for the stripe and FR orders will be discussed in Sec.~\\ref{sec:stripe_ferri}.\nIn addition to these two states, the numerical results show two intermediate-temperature states depending on the electron filling $n$.\nFor $0.29 \\lesssim n \\lesssim 0.34$, we identify the intermediate phase as the three-sublattice PD state [Fig.~\\ref{fig:model}(c)]. \nThe details will be discussed in Sec.~\\ref{sec:pd}.\nMeanwhile, for $n \\gtrsim 0.34$, we find KT-like behavior similar to the one discussed in the Ising models~\\cite{Takayama1983,Fujiki1984,Wada1982,Fujiki1983,Landau1983}, as presented in Sec.~\\ref{sec:stripe_ferri}.\nIn these intermediate-temperature phases, the numerical data indicate a LRO for PD but a quasi-LRO in the KT-like region.\n\nA similar phase diagram is obtained at $J=2$, as shown in Fig.~\\ref{fig:ndiag}(b).\nIn this case also, the PD phase emerges in the intermediate-temperature region.\nHowever, in contrast to the case with $J=1$ where PD is found widely above the FR state as well as PS, the PD phase dominantly appears above the PS region between the stripe and FR states.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig3v4.eps}\n   \\caption{(Color online).\n   Phase diagram of the model~(\\ref{eq:H}) at $n=1\/3$ while varying $J$.\n   The notations are common to those in Fig.~\\ref{fig:ndiag}.\n   The boundary between PD and PS is difficult to determine by MC calculations, and supposed to be located at lower temperature than indicated by the gray arrows.\n   }\n   \\label{fig:jdiag}\n\\end{figure}\n\nWe also investigated the phase diagram of the model in Eq.~(\\ref{eq:H}) while varying $J$.\nFigure~\\ref{fig:jdiag} shows the numerically obtained phase diagram at $n=1\/3$.\nThe result shows that the PD state is stable in a wide range of $0.8 \\lesssim J \\lesssim 5.6$.\nThe transition temperature first rapidly increases as increasing $J$, while it turns to a gradual decrease after showing a peak at $J\\sim 2$.\n\nAn important observation in this constant-$n$ phase diagram is that the PD state does not survive down to $J \\to 0$, and it is taken over by the KT-like and FR phases in the small $J$ region.\nThe absence of PD state in the $J\\rightarrow 0$ limit implies that the RKKY interaction in the second-order perturbation theory is insufficient in stabilizing the PD state.\nMoreover, the emergence of PD for $J > J_c \\ne 0$ is consistently understood within the Slater mechanism discussed in Sec.~\\ref{sec:mft}; the MC result of $J_c\\sim 0.8$ is in good accordance with the mean-field argument of the critical value $\\Delta_c = 2\/3$.\nThe result clearly indicates that a non-perturbative effect of itinerant electrons plays a crucial role in stabilizing the PD state.\n\nIn the PD region in Fig.~\\ref{fig:jdiag}, our MC data do not show clear sign of further transition while decreasing temperature before the MC calculations become unstable.\nIn the low temperature region, however, it becomes difficult to determine the chemical potential $\\mu$ for $n=1\/3$.\nThe lowest temperature of MC calculations are shown in the phase diagram by the gray downward arrows. \nOn the other hand, the analysis of the ground state indicates that the ground state for $J \\lesssim 1.68$ is the FR state, while the region for $J \\gtrsim 1.68$ is PS between the stripe and FR states.\nIn addition, we observe the PS instability by carefully investigating the change of $n$ as a function of $\\mu$ at $J=5.4$ (see also Appendix~\\ref{sec:pseparation}). \nFrom these facts, we conclude that the PD for $J \\gtrsim 1.68$ is taken over by PS between the stripe and FR states.\nSince it is tedious to determine the PS boundary from $\\mu$-$n$ plot for all the values of $J$, we merely plot the lowest temperature we reached in our constant-$n$ calculations as the upper limit of temperature for the PS instability.\n\n\\subsection{\nPartial disorder\n\\label{sec:pd}\n}\n\n\\begin{figure*}\n   \\includegraphics[width=\\linewidth]{fig42v6.eps}\n   \\caption{(Color online).\n   MC results for (a1)-(c1) $M_{xy}$, $M_z$, and $\\psi$, (a2)-(c2) $\\chi_{xy}$ and $\\chi_z$, and (a3)-(c3) $\\cal{S}$ and its temperature derivative $\\partial {\\cal S}\/\\partial T$ at $n=1\/3$;\n   (a1)-(a3) $J=1$, (b1)-(b3) $J=2$, and (c1)-(c3) $J=4$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   $\\cal S$ is calculated from numerical integration of $\\partial {\\cal S}\/\\partial T$ by assuming ${\\cal S}(T=1)=\\log2$.\n   }\n   \\label{fig:mcpd}\n\\end{figure*}\n\nHere, we present the details of MC data for identifying the PD state.\nFigure~\\ref{fig:mcpd} shows $T$ dependences of MC results for different $J$ at $n=1\/3$.\nTo fix $n$, we tuned $\\mu$ for each temperature; the errors for $n$ at each temperature are controlled within 0.001.\nFigure~\\ref{fig:mcpd}(a1) is the result for the pseudomoments $M_{xy}$ and $M_z$ at $J=1$ [see the definitions in Eqs.~(\\ref{eq:Mxy}) and (\\ref{eq:Mz}), respectively].\n$M_{xy}$ shows two anomalies while decreasing temperature at $T_c^{\\rm (PD)} = 0.086(4)$ and $T_c^{\\rm (FR)} = 0.019(2)$.\nThe critical temperatures are determined by the peaks of the susceptibilities, $\\chi_{xy}$, and $\\chi_z$, as mentioned below.\nAt $T_c^{\\rm (PD)}$, $M_{xy}$ rapidly increases and approaches $\\sqrt{2}$ at lower temperature. \nIn addition, it shows a kink at $T_c^{\\rm (FR)}$ and further increase to $8\/3$ at lower temperature.\nMeanwhile, $M_z$ shows no anomaly at $T_c^{\\rm (PD)}$, while it shows a rapid increase to $1\/\\sqrt3$ at $T_c^{\\rm (FR)}$.\nCorrespondingly, $\\chi_{xy}$ and $\\chi_z$ in Fig.~\\ref{fig:mcpd}(a2) also show divergent peaks increasing with the system size;\npeaks of $\\chi_{xy}$ appear at both $T_c^{\\rm (PD)}$ and $T_c^{\\rm (FR)}$, while $\\chi_z$ shows a peak only at $T_c^{\\rm (FR)}$.\nThese results signal the presence of two successive phase transitions at $T_c^{\\rm (PD)} = 0.086(4)$ and $T_c^{\\rm (FR)} = 0.019(2)$.\nThe error bars are estimated by the range of temperature where the standard deviation of the MC data exceeds the difference of expectation value from the peak value.\nThe transition temperatures and error bars shown in Figs.~\\ref{fig:ndiag} and \\ref{fig:jdiag} are given by this criterion.\nMeanwhile, most of the calculations in Fig.~\\ref{fig:ndiag} were done by fixing $\\mu$ instead of $n$.\nHence, we also give the error bars in terms of $n$, as $n$ changes with $T$ in a fixed $\\mu$ calculation.\n\nTo determine the nature of low temperature phases at $n=1\/3$, we also computed the azimuth parameter $\\psi$ [Eq.~(\\ref{eq:psi})] shown in Fig.~\\ref{fig:mcpd}(a1).\nWhile increasing the system sizes, $\\psi$ apparently deviates from zero to a negative value below $T_c^{\\rm (PD)}$, indicating that the intermediate phase for $T_c^{\\rm (FR)} < T < T_c^{\\rm (PD)}$ has a PD type order.\nOn the other hand, $\\psi$ shows a sign change at $T_c^{\\rm (FR)}$, and rapidly increases to $\\psi=1$ at lower temperature.\nThis is a signature of the FR transition, which will be discussed in detail in Sec.~\\ref{sec:stripe_ferri}.\n\nThe emergence of PD is also seen in the results for the spin entropy $\\cal{S}$ and its temperature derivative [Eqs.~(\\ref{eq:Sint}) and (\\ref{eq:delSdelT}), respectively], as shown in Fig.~\\ref{fig:mcpd}(a3).\nIn the intermediate-temperature region for $T_c^{\\rm (FR)} < T < T_c^{\\rm (PD)}$, $\\cal{S}$ appears to approach $\\frac13\\log 2$ as decreasing temperature, which is the value expected for the ideal PD state where one out of three spins in the magnetic unit cell remains paramagnetic.\nThe remaining entropy is released rapidly at $T_c^{\\rm (FR)}$ and ${\\cal S} \\to 0$ at lower temperature due to the ordering of paramagnetic spins in the FR state.\n\nSimilar phase transitions to the PD state are observed in the wide range of $J$, as shown in Figs.~\\ref{fig:mcpd}(b) and \\ref{fig:mcpd}(c) at $J=2$ and $J=4$, respectively.\nIn these results, however, we could not confirm the presence of another phase transition at a lower temperature in the range of temperature we calculated, in contrast to the FR transition found in the case of $J=1$.\nAs the PD state retains a finite $\\cal{S}$, it is unlikely that this phase survives to $T\\rightarrow0$.\nHence, it is presumably taken over by other ordered phases or PS at a lower temperature. \nAs shown in Fig.~\\ref{fig:jdiag}, the ground state is deduced to be PS for the values of $J$ in Figs.~\\ref{fig:mcpd}(b)  and \\ref{fig:mcpd}(c).\nWe, therefore, expect that the PD state is taken over by PS below $T=0.02$ for $J \\gtrsim 2$. \nThe situation is indicated by the gray arrows in the phase diagram in Fig.~\\ref{fig:jdiag}, as discussed in Sec.~\\ref{sec:pdiag}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig6v1.eps}\n   \\caption{(Color online).\n   MC results for $S({\\bf q})$ along the ${\\bf q}=(q_x,0)$ line at $T=0.02$.\n   The calculations were done for the system size $N=18\\times18$.\n   }\n   \\label{fig:sq}\n\\end{figure}\n\nAnother point to be noted is the systematic change in $\\cal{S}$ in the PD state by changing $J$.\nWhile the result at $J=1$ appears to show plateau like behavior at ${\\cal S}\\sim \\frac13 \\log2$, the plateau value of ${\\cal S}$ in the PD state decreases while increasing $J$, as shown in Figs.~\\ref{fig:mcpd}(a3), \\ref{fig:mcpd}(b3), and \\ref{fig:mcpd}(c3).\nThe decrease in ${\\cal S}$ is presumably attributed to the development of spatial correlations between paramagnetic sites in the PD state; \nthe ideal value ${\\cal S} = \\frac13 \\log2$ is for completely uncorrelated paramagnetic spins, and correlations between them reduces the entropy.\nSuch development of correlatins are observed in the spin structure factor $S({\\bf q})$ defined in Eq.~(\\ref{eq:Sq}).\nFigure~\\ref{fig:sq} shows a profile of $S({\\bf q})$ calculated by MC simulation at $T=0.02$.\nThe peaks at ${\\bf q}=(4\\pi\/3,0)$ and $(8\\pi\/3,0)$ indicates that the system is in a three-sublattice ordered phase, while the absence of a sharp peak at ${\\bf q}=(0,0)$ indicates that there is no net magnetic moment; the result is consistent with PD order.\nWhen comparing the results at $J=2$ and $J=4$, the peak corresponding to the three-sublattice order gets sharper for $J=4$, while the height of the peak of $S({\\bf q})$ is almost the same. \nThis indicates that the PD order at $J=2$ shows more spin fluctuations than that at $J=4$, consistent with the trend of the plateau value of ${\\cal S}$.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig_p1v4.eps}\n   \\caption{(Color online).\n   MC results for $\\psi$ while varying $n$ at $T=0.08$ and $J=2$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:conT}\n\\end{figure}\n\nThus far, we showed the results at $n=1\/3$.\nNext, we show how the PD evolves while changing $n$.\nFigure~\\ref{fig:conT} shows the MC result of $\\psi$ as a function of $n$ at $T=0.08$ and $J=2$.\n$\\psi$ becomes negative around $n=1\/3$ and takes the lowest value at $n\\simeq1\/3$. \nThe data indicate that $\\psi$ is almost system size independent or rather slightly decreases as the system size increases in the finite range of $n$ around $n=1\/3$. \nHence, the PD state is stabilized not only at $n=1\/3$ but for a finite range of $0.31 \\lesssim n \\lesssim 0.34$ in the thermodynamic limit. \nThe range well agrees with that for the PD phase estimated from the peak of susceptibilities shown in Fig.~\\ref{fig:ndiag}(b).\n\nWith regard to the order of the PD transition, the PD transition in our MC results appears to be continuous, as shown in Fig.~\\ref{fig:mcpd}. \nHowever, it needs careful consideration, as we will discuss here.\nIt is known that the Ising model on a triangular lattice with AF NN interactions is effectively described by a six-state model, in which the low-energy states with three up-up-down and three up-down-down configurations in the three-site unit cell are described by six-state variables. \nThe PD state in our model also retains six low-energy states with different up-down-paramagnetic configurations, and hence, the transition to PD is expected to be classified in the framework of six-state models.\nHowever, from the argument of duality properties, it is prohibited that the six-state models exhibit a single second-order transition for changing temperature~\\cite{Cardy1980}.\nFor instance, a two-dimensional six-state clock model shows two KT transitions at finite temperature, without exhibiting true LRO for $T\\ne0$~\\cite{Jose1977,Challa1986}.\nOn the other hand, a six-state Potts model shows a weak first order transition to LRO, in which the correlation length reaches the order of 1000 sites at the critical point~\\cite{Buffenoir1993}.\nIn our PD case, the apparently second-order transition at $T_c^{\\rm (PD)}$ is not expected to be a single one, but is always followed by another transition to FR or PS at a lower temperature. \nThis appears not to violate the general argument for the six-state models, although it is not clear to what extent the argument applies, as the electronic PS never takes place in the localized spin models.\nHence, the PD transition can be of second order, as indicated in our numerical results. \nOf course, we cannot exclude the possibility of a weak first order transition, similar to that of the Potts model.\nIn this case, due to a long correlation length at the critical temperature, the system sizes used in our calculations are likely to be insufficient to distinguish the first order transition from second order one.\n\n\\subsection{\nOther magnetic orders\n\\label{sec:stripe_ferri}\n}\n\n\\begin{figure}\n   \\includegraphics[width=0.9\\linewidth]{fig41v5.eps}\n   \\caption{(Color online).\n   MC results for (a) $M_{\\rm str}$ and (b) its susceptibility $\\chi_{\\rm str}$ at $J=2$ and $n=0.27$.\n   The inset in (b) shows $T_c^{\\rm (str)}$ for different sizes and the solid line is the extrapolation which gives $T_c^{\\rm (str)}=0.051(13)$.\n   The calculations were done for the system sizes $N=12\\times 12$, $14\\times 14$, $12\\times 18$, $16\\times 16$, and $18\\times18$.\n   }\n   \\label{fig:mcstripe}\n\\end{figure}\n\nFigure~\\ref{fig:mcstripe} presents the results for the relatively low filling where the stripe order is stabilized at low temperature.\nFigure~\\ref{fig:mcstripe}(a) shows the order parameter for the stripe order, $M_{\\rm str}$ [Eq.~(\\ref{eq:Mstr})], and Fig.~\\ref{fig:mcstripe}(b) shows the corresponding susceptibility $\\chi_{\\rm str}$ at $J=2$ and $n=0.27$.\nA phase transition to the stripe phase is signaled by a rapid increase of $M_{{\\rm str}}$ and corresponding peak of $\\chi_{\\rm str}$;\nwe determine the transition temperature $T_c^{\\rm (str)}$ by the peak temperature of $\\chi_{\\rm str}$ for each system size, and plot them in the phase diagram in Fig.~\\ref{fig:ndiag}(a). \nThe error bars are estimated in a similar manner to the case of $T^{\\rm (PD)}_c$ and $T^{\\rm (FR)}_c$.\nWe also show the system-size extrapolation of $T_c^{\\rm (str)}$ in the inset of Fig.~\\ref{fig:mcstripe}(b).\nAlthough the data are rather scattered, we fit them by $f(N) = a + b\/N^c$ with fitting parameters $a$, $b$, and $c$. \nThe extrapolation clearly shows that the phase transition takes place at a finite temperature, as expected for the two-dimensional Ising order.\n\nThe stripe ordered phase is a peculiar magnetic state, in which the sixfold rotational symmetry of the lattice is spontaneously broken and reduced to twofold.\nDue to the symmetry breaking, the transport property is expected to show strong spatial anisotropy; e.g., the longitudinal conductivity will be large in the direction along the stripes, while suppressed in the perpendicular direction.\nThis is an interesting topic on the control of transport by magnetism and vice versa.\n\n\\begin{figure}\n   \\includegraphics[width=0.76\\linewidth]{fig43v2.eps}\n   \\caption{(Color online).\n   MC results for (a) $M_{xy}$, $M_z$, and $\\psi$, (b) $\\chi_{xy}$ and $\\chi_z$, and (c) $\\cal{S}$ and its temperature derivative $\\partial {\\cal S}\/\\partial T$ at $n=0.38$ and $J=2$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:mcferri}\n\\end{figure}\n\nFigure~\\ref{fig:mcferri} shows the results for the relatively high filling where the low temperature phase is FR, at $n=0.38$ and $J=2$.\nThe data indicate two successive transitions signaled by the peaks in $\\chi_{xy}$ and $\\chi_{z}$ at different temperature.\nThe peak of $\\chi_z$ corresponding to the increase of $M_z$ signals the phase transition to the FR phase at $T_c^{\\rm (FR)} = 0.098(4)$.\nAt the same time, $\\psi$ becomes finite below $T_c^{\\rm (FR)}$, and approaches 1, as expected for the FR ordering.\nSimilar behavior was observed at $T_c^{\\rm (FR)} = 0.019(2)$ in Figs.~\\ref{fig:mcpd}(a1) and \\ref{fig:mcpd}(a2).\nOn the other hand, at a higher $T_{\\rm KT} = 0.146(4)$, only $M_{xy}$ changes rapidly, and correspondingly, $\\chi_{xy}$ shows a peak.\n$M_{xy}$, however, shows a noticeable system-size dependence even below $T_{\\rm KT}$, in contrast with the results below $T_c^{\\rm (PD)}$. \nSimilar behavior was observed in the KT transition in Ising spin systems~\\cite{Takayama1983,Fujiki1984}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig45v2.eps}\n   \\caption{(Color online).\n   Extrapolation of $\\psi$ to $N\\to\\infty$ at different temperatures.\n   The solid lines for $T\\le 0.104$ is the linear fitting of data.\n   }\n   \\label{fig:psifit}\n\\end{figure}\n\nOn the other hand, $\\psi$ does not show an anomaly at $T_{\\rm KT}$, while it shows a sharp rise around $T_{c}^{{\\rm (FR)}}$, as shown in Fig.~\\ref{fig:mcferri}(a).\nThe value of $\\psi$ extrapolated to large $N$ converges to zero in the intermediate-temperature range.\nFigure~\\ref{fig:psifit} shows the extrapolation of $\\psi$ for $N\\to \\infty$.\nThe results indicate that, $\\psi$ remains to be zero at $N\\to\\infty$ for $T\\gtrsim0.104$, which is far below $T_{\\rm KT}=0.146(4)$.\nOn the other hand, the extrapolated value becomes finite for $T\\lesssim 0.104$, reflecting the FR order; the transition temperature is estimated as $\\tilde{T}_c^{\\rm (FR)} = 0.102(2)$, which is in accordance with $T_c^{\\rm (FR)} = 0.098(4)$.\n\nThe results above indicate that there is no sixfold symmetry breaking in $M_{xy}$ at $T_{\\rm KT}$, as seen in the KT phase in the Ising spin models~\\cite{Takayama1983}.\nHence, we consider that the higher-temperature transition at $T_{\\rm KT}$ is of KT type.\nNamely, the system exhibits two successive transitions from the paramagnetic phase to the KT-like phase at $T_{\\rm KT}$, and the KT-like phase to the low-temperature FR phase at $T_{c}^{{\\rm (FR)}}$. \nHere, we call the intermediate-temperature phase the KT-like phase, as it is difficult to confirm either the KT universality class by critical behavior or the quasi-LRO behavior within the system sizes we reached, as seen below.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig44v2.eps}\n   \\caption{(Color online).\n   MC results for the real-space spin correlation function $C(r)$ at $J=2$ and $n=0.38$.\n   The results are shown only for the sites with $C(r) > 0$.\n   The calculations were done for the system size $N=18\\times18$.\n   }\n   \\label{fig:mcsk}\n\\end{figure}\n\nThe signature of two successive transitions is also observed in the real-space spin correlation function $C(r)$.\nHere $C(r)$ is the averaged correlations between the Ising spins in distance $r$, defined by \n\\begin{eqnarray}\nC(r) = \\sum_{i,j} \\frac{1}{N_{\\rm p}(r)} \\langle S_i S_j \\rangle  \\delta(|{\\bf r}_{ij}| - r),\n\\end{eqnarray}\nwhere $N_{\\rm p}(r) =\\sum_{i,j} \\delta(|{\\bf r}_{ij}| - r)$ is the number of spin pairs with distance $r$, and $\\delta(x)$ is the delta function.\nThe MC data while varying temperature are shown in Fig.~\\ref{fig:mcsk}. \nAlthough the results are not conclusive due to the limitation on accessible system sizes, they appear to be consistent with the two transitions discussed above.\nFor $T \\lesssim T_{c}^{{\\rm (FR)}} = 0.098(4)$, the spin correlation appears to approach constant for large distance, well corresponding to the FR LRO developed in this low temperature region.\nOn the other hand, for $T \\gtrsim T_{\\rm KT} = 0.146(4)$, it becomes concave downward with a steep decrease with respect to the distance, which reflects an exponential decay in the high temperature paramagnetic state.\nIn the intermediate region for $T_c^{\\rm (FR)} \\lesssim T \\lesssim T_{\\rm KT}$, the spin correlation also decays with increasing distance.\nThe decay, however, is much slower and appears to obey an asymptotic power law, which is characteristic to the quasi-LRO in the KT state.\nIn principle, the critical exponents can be estimated from the asymptotic power-law behavior, but it is difficult to be conclusive in the current system sizes.\n\n\\section{\nElectronic structure of partially disordered state\n\\label{sec:estruct}\n}\n\nIn the previous section, we discussed the thermodynamic behavior of the localized spin degree of freedom, with emphasis on the emergence of peculiar PD state.\nIn this section, we focus on the behavior in the charge degree of freedom of itinerant electrons in the PD phase. \n\n\\begin{figure}\n   \\includegraphics[width=0.80\\linewidth]{fig5chargev4.eps}\n   \\caption{(Color online).\n   MC results for $n_{\\rm CO}$ at ${\\bf q}=(2\\pi\/3,-2\\pi\/3)$ at $n=1\/3$ and (a) $J=1$, (b) $J=2$, and (c) $J=4$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:mcnq}\n\\end{figure}\n\nFigure~\\ref{fig:mcnq} shows temperature dependence of the charge modulation $n_{\\rm CO}$ [Eq.~(\\ref{eq:n_CO})] at $n=1\/3$ for different $J$. \nFigure~\\ref{fig:mcnq}(a) is the result at $J=1$ for different system sizes.\nThe result shows an increase of $n_{\\rm CO}$ below $T \\simeq T_c^{\\rm (PD)} =0.086(4)$, indicating that the PD state is accompanied by charge modulation with period three.\nSimilar onsets of charge modulation at $T_c^{\\rm (PD)}$ are observed for larger $J$, as shown in Figs.~\\ref{fig:mcnq}(b) and \\ref{fig:mcnq}(c);\nthe amplitude of the modulation in the PD phase increases monotonically as $J$ increases.\nThe magnitude of the charge modulation is in the same order compared to the mean-field result in Fig.~\\ref{fig:gap}), while the growth is considerably suppressed by a factor of two to four.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig5dosv2.eps}\n   \\caption{(Color online).\n   MC results for DOS of itinerant electrons at $n=1\/3$ and $J=2$ for $N=18\\times 18$.\n   The Fermi level is set at $\\varepsilon = 0$.\n   The statistical errors are comparable to the width of the lines.\n   }\n   \\label{fig:mcdos}\n\\end{figure}\n\nWe next look into the electronic density of states (DOS) at different temperature. \nFigure~\\ref{fig:mcdos} shows the results for DOS while varying temperature at $J=2$ and $n=1\/3$.\nThe Fermi level is set at $\\varepsilon=0$.\nHere, DOS was calculated by counting the number of energy eigenvalues as the histogram with the energy interval of 0.0375.\nIn the paramagnetic region for $T \\gtrsim T_c^{\\rm (PD)} = 0.130(4)$, DOS is featureless near the Fermi level.\nOn the other hand, below $T_c^{\\rm (PD)}$, an energy gap develops at the Fermi level for $n=1\/3$.\nThe result shows that the PD state is an insulator, which supports the scenario that PD is stabilized by the Slater mechanism described in Sec.~\\ref{sec:mft}.\nSimilarly to the charge modulation, the energy gap in the MC results is largely suppressed compared to that obtained by the mean-field analysis in Fig.~\\ref{fig:gap}.\nThis appears to show the importance of appropriately taking into account of thermal fluctuations.\n\n\\section{\\label{sec:summary}\nSummary\n}\n\nTo summarize, by a combined analysis of the mean-field type calculation and Monte Carlo simulation, we have investigated the origin of the partial disorder in the Ising-spin Kondo lattice model in a two-dimensional triangular lattice. \nIn the mean-field type calculation, we have clarified that a local magnetic field of the partial disorder type induces a metal-insulator transition at 1\/3 filling at a critical value of the field. \nThe result suggests that the three-sublattice partial disorder can give rise to an energy gap, and therefore, it has a chance to be stabilized through the Slater mechanism. \nOn the other hand, in the Monte Carlo simulation, we have provided convincing numerical results on the emergence of partial disorder at finite temperatures where the stripe phase and the ferrimagnetic order compete with each other.\nThe Monte Carlo result shows that the partially disordered state appears above a nonzero value of the spin-charge coupling, and that it is insulating and accompanied by charge disproportionation.\nThe nonzero critical value of the spin-charge coupling and the opening of the charge gap are both qualitatively consistent with the mean-field analysis.\nThe results indicate that the partial disorder is stabilized by the Slater mechanism which is characteristic to itinerant magnets.\nOur results not only clarify the new mechanism of partial disorder in two dimensions but also pave the way for understanding of the interesting physics related to the peculiar coexistence of magnetic order and paramagnetic moments in itinerant electron systems.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig8v1.eps}\n   \\caption{\n   (Color online).\n   (a) The grand potetial $\\Omega$ and (b) electron filling $n$ with respect to the chemical potential $\\mu$, numerically calculated by exactly diagonalizing the one-body Hamiltonian for itinerant electrons.\n   The results are obtained at $J=2$ with $N_s=24\\times24$ site superlattice of $N=12\\times 12$ site unit cells.\n   The strip at the left side of (b) shows the ground state at the corresponding filling.\n   }\n   \\label{fig:nvc}\n\\end{figure}\n\nAn interesting extension of the current work would be to consider the effect of quantum fluctuation of localized spins.\nIn our result, the partial disorder remains stable down to very low temperature, implying that the paramagnetic spins are largely fluctuating and sensitive to perturbations at low temperatures.\nHence, an interesting possibility is that, by including quantum fluctuations, the partial disorder is further stabilized and remains stable even in the ground state.\nIndeed, a similar partial disorder was found in the ground state of the Kondo lattice model with quantum spins at half filling~\\cite{Motome2010}. \nTherefore, it is intriguing to examine the effect of quantum fluctuations on the present model with Ising spins. \nHowever, it is not straightforwardly calculated by the present Monte Carlo method.\nThe interesting problem is left for future study.\n\n\\begin{acknowledgements}\nThe authors are grateful to G.-W. Chern, H. Kawamura, M. Matsuda, S. Miyashita, and H. Yoshida for fruitful discussions.\nThe authors also thank S. Hayami and T. Misawa for helpful comments.\nPart of the calculations were performed on the Supercomputer Center, Insitute for Solid State Physics,\nUniversity of Tokyo. H.I. is supported by Grant-in-Aid for JSPS Fellows.\nThis research was supported by KAKENHI (No.19052008, 21340090, 22540372, and 24340076), Global COE Program ``the Physical Sciences Frontier\", the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe Bernoulli convolution has been around for over seventy years and has surfaced in several different areas of mathematics. This probability measure depends on a parameter $\\beta >1$ and is defined on the interval $\\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$, where $\\lfloor \\beta \\rfloor$ is the largest integer not exceeding $\\beta$. The {\\bf symmetric Bernoulli convolution} is the distribution of $\\sum_{k=1}^{\\infty} \\frac{b_k}{\\beta_k}$ where the coefficients $b_k$ take values in the set $\\{ 0 ,1, \\ldots, \\lfloor \\beta \\rfloor \\}$, each with probability $\\frac{1}{\\lfloor \\beta \\rfloor +1}$. If instead the values $0, 1, \\ldots, \\lfloor \\beta \\rfloor$ are not taken with equal probabilities, then the Bernoulli convolution is called {\\bf asymmetric} or {\\bf biased}. See \\cite{PSS00} for an overview of results regarding the Bernoulli convolution up to the year 2000. Recently attention has shifted to the multifractal structure of the Bernoulli convolution. Jordan, Shmerkin and Solomyak study the multifractal spectrum for typical $\\beta$ in \\cite{JSS11}, Feng considers Salem numbers $\\beta$ in \\cite{Fen12} and Feng and Sidorov look at the Lebesgue generic local dimension of the Bernoulli convolution in \\cite{FS11}. In this paper we are interested in the local dimension function of the Bernoulli convolution and to study the local dimension we use a new approach.\n\nIf a point $x \\in \\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ can be written as $x = \\sum_{k=1}^{\\infty} \\frac{b_k}{\\beta^k}$ with $b_k \\in \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}$ for all $k \\ge 1$, then this expression is called a $\\beta$-expansion of the point $x$. In \\cite{S03} (see also \\cite{DV05}) it is shown that Lebesgue almost every $x$ has uncountably many $\\beta$-expansions. In \\cite{DV05} a random transformation $K$ was introduced that generates for each $x$ all these possible expansions by iteration. The map $K$ can be identified with a full shift which allows one to define an invariant measure $\\nu_{\\beta}$ for $K$ of maximal entropy by pulling back the uniform Bernoulli measure. One obtains the Bernoulli convolution from $\\nu_{\\beta}$ by projection. In this paper we study the local dimension of the measure $\\nu_{\\beta}$. By projection, some of these results can be translated to the Bernoulli convolution. For now, our methods work only for a special set of $\\beta$'s called the generalised multinacci numbers. We have good hopes that in the future we can extend these methods to a more general class of $\\beta$'s.\n\nThe paper is organized as follows. In the first section we will give the necessary definitions. Next we study the local dimension of $\\nu_{\\beta}$. We give a formula for the lower and upper bound of the local dimension that holds everywhere using a suitable Markov shift. Moreover, we show that the local dimension exists and is constant a.e.~and we give this constant. We also show that on the set corresponding to points with a unique $\\beta$-expansion, the local dimension of $\\nu_{\\beta}$ takes a different value. Next we translate these results to a lower and upper bound for the local dimension of the symmetric Bernoulli convolution that holds everywhere. We then use a result from \\cite{FS11} to obtain an a.e.~value for the Bernoulli convolution in case $\\beta$ is a Pisot number. Finally we give the local dimension for points with a unique expansion. In the last section we consider one specific example of an asymmetric Bernoulli convolution, namely when $\\beta$ is the golden ratio. We give an a.e.~lower and upper bound for the local dimension of both the invariant measure for $K$ and the asymmetric Bernoulli convolution. This last section is just a starting point for more research in this direction.\n\n\n\\section{Preliminaries}\n\nThe set of $\\beta$'s we consider in this paper, the generalised multinacci numbers, are defined as follows. On the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ the {\\bf greedy} $\\beta$-transformation $T_{\\beta}$ is given by\n\\[ T_{\\beta} x = \\left\\{\n\\begin{array}{ll}\n\\beta x \\, (\\text{mod }1), & \\text{if } x \\in \\big[0,1),\\\\\n\\beta x - \\lfloor \\beta \\rfloor, & \\text{if } x \\in \\big[1, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big].\n\\end{array}\n\\right.\\]\nThe {\\bf greedy digit sequence} of a number $x \\in \\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ is defined by setting \n\\[ a_1=a_1(x) = \\left\\{\n\\begin{array}{ll}\nk, & \\text{if } x \\in \\big[\\frac{k}{\\beta}, \\frac{k+1}{\\beta} \\big), \\, k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor -1 \\},\\\\\n\\lfloor \\beta \\rfloor, & \\text{if } x \\in \\big[ \\frac{\\lfloor \\beta \\rfloor}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big],\n\\end{array}\n\\right.\\]\nand for $n \\ge 1$, $a_n=a_n(x) = a_1(T_{\\beta}^{n-1} x)$. Then $T_{\\beta}x = \\beta x - a_1(x)$ and one easily checks that $x = \\sum_{n=1}^{\\infty} \\frac{a_n}{\\beta^n}$. This $\\beta$-expansion of $x$ is called its {\\bf greedy $\\beta$-expansion}. A number $\\beta >1$ is called a {\\bf generalised multinacci number} if the greedy $\\beta$-expansion of the number 1 satisfies\n\\begin{equation}\\label{q:genmn}\n1 = \\frac{a_1}{\\beta} + \\frac{a_2}{\\beta^2} + \\cdots + \\frac{a_n}{\\beta^n},\n\\end{equation}\nwith $a_j \\ge 1$ for all $1 \\le j \\le n$ and $n \\ge 2$. (Note that $a_1 = \\lfloor \\beta \\rfloor$.) We call $n$ the {\\bf degree} of $\\beta$.\n\n\\begin{rem}{\\rm\nBetween 1 and 2  the numbers that satisfy this definition are called the multinacci numbers. The {\\bf $n$-th multinacci number} $\\beta_n$ satisfies\n\\[ \\beta^n_n = \\beta^{n-1}_n + \\beta^{n-2}_n + \\cdots + \\beta_n + 1,\\]\nwhich implies that $a_j=1$ for all $1 \\le j \\le n$ in (\\ref{q:genmn}). The second multinacci number is better known as the golden ratio.}\n\\end{rem}\n\n\\vskip .2cm\nFor the Markov shift we will construct later on, we need a suitable partition of the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. Consider the maps $T_k x = \\beta x -k$, $k =0, \\ldots, \\lfloor \\beta \\rfloor$. For each $x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, either there is exactly one $k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\}$ such that $T_k x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, or there is a $k$ such that both $T_k x$ and $T_{k+1} x$ are in $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. In this way the maps $T_k$ partition the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ into the following regions:\n\\[ \\begin{array}{ll}\n\\displaystyle E_0 = \\Big[0, \\frac{1}{\\beta} \\Big), & \\displaystyle E_{\\lfloor \\beta \\rfloor} = \\Big(  \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} +\\frac{\\lfloor \\beta \\rfloor-1}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\Big],\\\\\n\\\\\n\\displaystyle E_k = \\Big( \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} +\\frac{k-1}{\\beta}, \\frac{k+1}{\\beta}\\Big), & k \\in \\{ 0, 1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n\\\\\n\\displaystyle S_k = \\Big[ \\frac{k}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta-1)} + \\frac{k-1}{\\beta} \\Big], & k \\in \\{1, \\ldots, \\lfloor \\beta \\rfloor \\}.\n\\end{array}\\]\nSee Figure~\\ref{f:2.5} for a picture of the maps $T_k$ and the regions $E_k$ and $S_k$ in case $2 < \\beta < 3$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure1kleur}\n\\caption{The maps $T_0 x = \\beta x$, $T_1 x = \\beta x-1$ and $T_2 x = \\beta x -2$ and the intervals $E_0$, $S_1$, $E_1$, $S_2$ and $E_2$ for some $2< \\beta <3$.}\n\\label{f:2.5}\n\\end{figure}\n\n\\vskip .2cm\nWrite $\\Omega = \\{0,1 \\}^{\\mathbb N}$. The {\\bf random $\\beta$-transformation} is the map $K$ from the space $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ to itself defined as follows.\n\\[ K(\\omega, x) = \\left\\{\n\\begin{array}{ll}\n(\\omega, T_k x), & \\text{if } x \\in E_k, \\, k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n(\\sigma \\omega, T_{k-1+\\omega_1} x), & \\text{if }x \\in S_k, \\, k \\in \\{ 1, \\ldots, \\lfloor \\beta \\rfloor\\},\n\\end{array}\n\\right.\\]\nwhere $\\sigma$ denotes the left shift on sequences, i.e., $\\sigma (\\omega_n)_{n \\ge 1} = (\\omega_{n+1})_{n \\ge 1}$. The projection onto the second coordinate is denoted by $\\pi$. Let $\\lceil \\beta \\rceil$ denote the smallest integer not less than $\\beta$. The map $K$ is isomorphic to the full shift on $\\lceil \\beta \\rceil$ symbols. The isomorphism $\\phi: \\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big] \\to \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ uses the digit sequences produced by $K$. Let\n\\[b_1(\\omega,x) = \\left\\{\n\\begin{array}{ll}\nk, & \\text{if } x \\in E_k, \\, k \\in \\{0, 1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n& \\  \\text{or if } x \\in S_k \\text{ and } \\omega_1=1, \\, k \\in \\{1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\nk-1, & \\text{if } x \\in S_k \\text{ and } \\omega_1=0, \\, k \\in \\{ 1, \\ldots, \\lfloor \\beta \\rfloor \\}\n\\end{array}\n\\right.\\]\nand for $n \\ge 1$, set $b_n(\\omega,x) = b_1\\big(K^{n-1}(\\omega, x)\\big)$. Then\n\\[ \\phi(\\omega,x) = \\big(b_n (\\omega,x) \\big)_{n \\ge 1}.\\]\nThis map is a bimeasurable bijection from the set $Z=\\big\\{ (\\omega, x) \\, : \\, \\pi \\big(K^n (\\omega, x)\\big) \\in S \\, \\, \\text{i.o.} \\big\\}$ to its image. We have $\\phi\\circ K = \\sigma \\circ \\phi$. Let $\\mathcal F$ denote the $\\sigma$-algebra generated by the cylinders and let $m$ denote the uniform Bernoulli measure on $(\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}, \\mathcal F)$. Then $m$ is an invariant measure for $\\sigma$ and $\\nu_{\\beta} = m \\circ \\phi$ is invariant for $K$ with $\\nu_{\\beta}(Z)=1$. The projection $\\mu_{\\beta} = \\nu_{\\beta} \\circ \\pi^{-1}$ is the Bernoulli convolution on $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. For proofs of these facts and more information on the map $K$ and its properties, see \\cite{DK03} and \\cite{DV05}.\n\n\\vskip .2cm\nWe are interested in the local dimension of the measures $\\nu_{\\beta}$ and $\\mu_{\\beta}$. For any probability measure $\\mu$ on a metric space $(X,\\rho)$, define the {\\bf local lower} and {\\bf local upper dimension} functions by\n\\[ \\underline{d}(\\mu,x) = \\liminf_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}( x,r)\\big)}{\\log r} \\quad \\text{and} \\quad \\overline{d}(\\mu,x) = \\limsup_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}(x,r)\\big)}{\\log r},\\]\nwhere $B_{\\rho}(x,r)$ is the open ball around $x$ with radius $r$. If $\\underline{d}(\\mu,x) = \\overline{d}(\\mu,x)$, then the {\\bf local dimension} of $\\mu$ at the point $x \\in X$ exists and is given by\n\\[ d(\\mu,x)= \\lim_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}(x,r)\\big)}{\\log r}.\\]\nOn the sets $\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ and $\\Omega$ we define the metric $D$ by\n\\[ D \\big( \\omega, \\omega' \\big) = \\beta^{-\\min\\{k\\ge 0 \\, :\\, \\omega_{k+1} \\neq \\omega_{k+1}'\\}}.\\]\nWe will define an appropriate metric on the set $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ later.\n\n\n\n\\section{Local dimension for $\\nu_{\\beta}$}\n\nWe will study the local dimension of the invariant measure $\\nu_{\\beta}$ of the map $K$ for $\\beta$'s that are generalised multinacci numbers. It is proven in \\cite{DV05} that for these $\\beta$'s the dynamics of $K$ can be modeled by a subshift of finite type. So, on the one hand there is the isomorphism of $K$ with the full shift on $\\lceil \\beta \\rceil$ symbols and on the other hand there is an isomorphism to a subshift of finite type. It is this second isomorphism that allows us to code orbits of points $(\\omega, x)$ under $K$ in an appropriate way for finding local dimensions. We give the essential information here.\n\n\\medskip\nWe begin with some notation. We denote the {\\bf greedy map} by $T_{\\beta}$ as before, and the {\\bf lazy map} by $S_{\\beta}$. More precisely, \n\\[ T_{\\beta} x = \\left\\{\n\\begin{array}{ll}\nT_0 x, & \\text{if } x \\in E_0,\\\\\n\\\\\nT_k x, & \\text{if }x \\in  S_k\\cup E_k,\\\\\n& \\quad 1 \\le k \\le \\lfloor \\beta \\rfloor,\n\\end{array}\n\\right.\n\\quad \\text{and} \\quad S_{\\beta}x = \\left\\{\n\\begin{array}{ll}\nT_k x, & \\text{if } x \\in E_k\\cup S_{k+1},\\\\\n& \\quad 0 \\le k \\le \\lfloor \\beta \\rfloor-1,\\\\\n\\\\\nT_{\\lfloor \\beta \\rfloor} x, & \\text{if }x \\in E_{\\lfloor \\beta \\rfloor}.\n\\end{array}\n\\right.\\]\n\n\\noindent\nWe will be interested in the $K$-orbit of the points $(\\omega,1)$ and of their `symmetric counterparts' $(\\omega, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1}-1)$.\nProposition 2 (ii) in \\cite {DV05} tells us that the following set $F$ is finite:\n\\begin{multline}\\label{q:finite}\nF=\\Big\\{\\pi \\big(K^n(\\omega, 1)\\big), \\pi \\Big(K^n\\big(\\omega, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1}-1\\big)\\Big) \\, : \\, n \\ge 0,\\, \\omega \\in \\Omega\\Big\\}\\\\\n\\cup \\Big\\{\\frac{k}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} + \\frac{k}{\\beta} : k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\} \\Big\\}.\n\\end{multline}\nThese are the endpoints of the intervals $E_k$ and $S_k$ and their forward orbits under all the maps $T_k$. The finiteness of $F$ implies that the dynamics of $K$ can be identified with a topological Markov chain. To find the Markov partition, one starts by refining the partition given by the sets $E_k$ and $S_k$, using the points from the set $F$. Let $\\mathcal C$ be the interval partition consisting of the open intervals between the points from this set. Note that when we say {\\bf interval partition}, we mean a collection of pairwise disjoint open intervals such that their union covers the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ up to a set of $\\lambda$-measure 0, where $\\lambda$ is the one-dimensional Lebesgue measure. Write\n\\[ \\mathcal C=\\{C_1,C_2,\\ldots , C_L\\}.\\]\nLet $ S=\\bigcup_{1 \\le k \\le \\lfloor \\beta \\rfloor} S_k$. The property {\\bf p3} from \\cite{DV05} says that no points of $F$ lie in the interior of $S$, i.e., each $S_k$ corresponds to a set $C_j$ in the sense that for each $1 \\le k \\le \\lfloor \\beta \\rfloor$ there is a $1 \\le j \\le L$ such that $\\lambda(S_k \\setminus C_j)=0$. Let $s \\subset \\{1, \\ldots, L\\}$ be the set containing those indices $j$. Consider the $L\\times L$ adjacency matrix $A=(a_{i, j})$ with entries in $\\{0,1\\}$ defined by\n\\begin{equation}\\label{matrix}\na_{i,j}\\, =\\, \\left\\{ \\begin{array}{ll}\n1, & {\\mbox{ if }}\\; i\\not \\in s \\text{ and } \\lambda(C_j \\cap T_{\\beta}(C_i))=\\lambda(C_j),\\\\\n0, & \\text{ if }\\; i\\not \\in s \\text{ and } \\lambda(C_i\\cap T_{\\beta}C_j)= 0,\\\\\n1, & \\text{ if }\\; i \\in s \\text{ and } \\lambda(C_j \\cap T_{\\beta} C_i)=\\lambda(C_j) \\text{ or } \\lambda(C_j \\cap S_{\\beta}C_i)=\\lambda(C_j),\\\\\n0, & \\text{ if }\\; i \\in s \\text{ and } \\lambda(C_i\\cap T_{\\beta}C_j)= 0 \\text{ and } \\lambda(C_i\\cap S_{\\beta}C_j)= 0.\n\\end{array}\\right.\n\\end{equation}\n\\noindent Define the partition $\\mathcal P$ of $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ by\n\\[ \\mathcal P=\\big\\{\\Omega \\times C_j : j \\not \\in s \\big\\} \\cup \\big\\{ \\{\\omega_1=i\\} \\times C_j: i \\in \\{0,1\\}, \\, j \\in s \\big\\}.\\]\nThen $\\mathcal P$ is a Markov partition underlying the map $K$. Let $Y$ denote the topological Markov chain determined by the matrix $A$. That is, $Y=\\{(y_n)_{n \\ge 1}\\in \\{1,\\dots ,L\\}^{\\mathbb N}:a_{y_n, y_{n+1}}=1 \\}$. Let $\\mathcal Y$ denote the $\\sigma$-algebra on $Y$ determined by the cylinder sets, i.e., the sets specifying finitely many digits, and let $\\sigma_Y$ be the left shift on $Y$. We use Parry's recipe (\\cite{Par64}) to determine the Markov measure $Q$ of maximal entropy for $(Y, \\mathcal Y, \\sigma_Y)$. By results in \\cite{DV05} we know that $\\nu_{\\beta}$ is the unique measure of maximal entropy for $K$ with entropy $h_{\\nu_{\\beta}}(K)=\\log \\lceil \\beta \\rceil$. By the identification with the Markov chain we know that $h_Q(\\sigma_Y) = \\log \\lceil \\beta \\rceil$. One then gets that the corresponding transition matrix $(p_{i,j})$ for $Y$ satisfies $p_{i,j}=a_{i,j}\\frac{v_j}{\\lceil \\beta \\rceil v_i}$, where $(v_1,v_2, \\ldots,v_L)$ is the right probability eigenvector of $A$ with eigenvalue $\\lfloor \\beta \\rfloor +1$. From this we see that if $[j_1 \\cdots j_m]$ is an allowed cylinder in $Y$, then\n\\begin{equation}\\label{q:qmeasure}\nQ([j_1 \\cdots j_m])=\\frac{v_{j_m}}{\\lceil \\beta \\rceil^{m-1}}.\n\\end{equation}\nProperty {\\bf p5} from \\cite{DV05} says that for all $i \\in s$, $a_{i,1}=a_{i,L}=1$ and $a_{i,j}=0$ for all other $j$. By symmetry of the matrix $(p_{i,j})$, it follows that\n\\begin{equation}\\label{q:ps1L}\np_{i, 1} = p_{i,L} = \\frac12 \\quad \\text{for all } i \\in s.\n\\end{equation}\nLet\n\\[ X = \\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big] \\backslash \\big( \\bigcup_{n \\ge 0} K^{-n} F  \\big).\\]\nThen $\\nu_{\\beta} (X) =1$. The isomorphism $\\alpha: X \\to Y$ between $(K, \\nu_{\\beta})$ and $(\\sigma_Y, Q)$ is then given by \n$$\\alpha_j(\\omega, x) = k \\mbox{ if } K^{j-1}(\\omega, x) \\in C_k.$$ \nSee Theorem 7 in \\cite{DV05} for a proof of this fact. In Figure~\\ref{f:diagram} we see the relation between the different systems we have introduced so far.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure2}\n\\caption{The relation between the different spaces.}\n\\label{f:diagram}\n\\end{figure}\n\nTo study the local dimension of $\\nu_{\\beta}$, we need to consider balls in $X$ under a suitable metric. Define the metric $\\rho$ on $X$ by\n\\[ \\rho\\big((\\omega,x),(\\omega^{\\prime},x^{\\prime})\\big)=\\beta^{-\\min\\{k\\ge 0\\, : \\, \\omega_{k+1}\\not=\\omega^{\\prime}_{k+1} \\text{ or } \\alpha_{k+1}(\\omega,x)\\not=\\alpha_{k+1}(\\omega^{\\prime},x^{\\prime}) \\}}.\\]\nConsider the ball \n\\[ B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)=\\{(\\omega^{\\prime},x^{\\prime})\\, : \\, \\omega_i^{\\prime}=\\omega_i, \\mbox{ and } \\, \\alpha_i(\\omega^{\\prime},x^{\\prime})= \\alpha_i(\\omega,x), \\, i=1,\\cdots ,k\\}.\\]\nLet\n\\[ M_k(\\omega,x)=\\sum_{i=0}^{k-1}{\\bf 1}_{X \\cap \\Omega\\times S}\\big(K^i(\\omega,x)\\big) = \\# \\{ 1 \\le i \\le k \\, : \\, \\alpha_i(\\omega,x) \\in s \\}.\\]\nTo determine $\\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)$ for $r \\downarrow 0$ we will calculate $Q \\Big( \\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)\\Big)$. For all points $(\\omega',x')$ in the ball $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ the $\\alpha$-coding starts with $\\alpha_1(\\omega, x) \\cdots \\alpha_k(\\omega, x)$ and $\\omega'$ starts with $\\omega_1 \\cdots \\omega_k$. From the second part we know what happens the first $k$ times that the $K$-orbit of a point $(\\omega', x')$ lands in $\\Omega \\times S$. Since $M_k(\\omega, x)$ of these values have been used for $\\alpha_1(\\omega, x)\\cdots \\alpha_k(\\omega,x)$, there are $k-M_k(\\omega,x)$ unused values left. Note that $M_k(\\omega', x') = M_k(\\omega, x)=M_k$ for all $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$. Define the set\n\\[ Z = X \\cap \\bigcap_{n \\ge 1}\\bigcup_{i \\ge 1}  K^{-i} \\big(\\Omega \\times S\\big).\\]\nAll points in $Z$ land in the set $\\Omega \\times S$ infinitely often under $K$. Since $Z$ is $K$-invariant, by ergodicity of $K$ we have $\\nu_{\\beta} (Z) =1$. So, all points $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\cap Z$ make a transition to $S$ some time after $k$. Moreover, after this transition these points move to $C_1$ if $\\omega_{M_k+1} =1$ and to $C_L$ otherwise. The image of a point $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ under $\\alpha$ will thus have the form\n\\begin{multline*}\n\\alpha_1 \\cdots \\alpha_k \\, \\underbrace{a_{k+1} \\cdots a_{m_1-1}}_{\\not \\in s} \\, \\underbrace{a_{m_1}}_{\\in s} \\, \\underbrace{a_{m_1+1} \\cdots a_{m_2-1}}_{\\not \\in s} \\, \\underbrace{a_{m_2}}_{\\in s} \\\\\n\\cdots \\, \\underbrace{a_{m_{N-1}-1}\\cdots a_{m_N-1}}_{\\not \\in s} \\, \\underbrace{a_{m_N}}_{\\in s} \\, \\underbrace{a_{m_N+1}a_{m_N+2} \\cdots}_{\\text{tail}},\n\\end{multline*}\nwhere $a_{m_j+1} \\in \\{1,L\\}$, $m_{j+1}-m_j-2 \\ge 1$ and $N = k-M_k(\\omega, x)$. Note that by the ergodicity of $\\nu_{\\beta}$ we have\n\\[ Q \\Big( \\bigcup_{m \\ge 1} [a_1 \\cdots a_m] \\, : \\, a_m \\in s \\text{ and } a_i \\not \\in s , \\, i < m \\Big) = \\nu_{\\beta} \\Big( X \\cap \\bigcup_{m \\ge 1} K^{-m} (\\Omega \\times S) \\Big)=1.\\]\nSo, the transition from any state to $s$ occurs with probability 1. Then one of the digits $\\omega_j$, $M_k+1 \\le j \\le k$, specifies what happens in this event and by (\\ref{q:ps1L}) both possibilities happen with probability $\\frac12$. To determine the measure of all possible tails of sequences in $\\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)$, note that again by {\\bf p5} of \\cite{DV05} this tail always belongs to a point in $\\Omega \\times C_1$ or $\\Omega \\times C_L$. Since the $\\nu_{\\beta}$-measure of these sets is the same, the $Q$-measure of the set of all possible tails is given by $\\nu_{\\beta}(\\Omega \\times C_1)= \\mu_{\\beta}(C_1)$. Putting all this together gives\n\\begin{equation}\\label{q:Qball}\nQ \\Big( \\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)\\Big) = \\lceil \\beta \\rceil^{-(k-1)} v_{\\alpha_k(\\omega,x)} \\cdot \\underbrace{1 \\cdot \\frac{1}{2} \\cdot 1 \\cdot \\frac{1}{2} \\cdots 1\\cdot \\frac{1}{2}}_{k-M_k(\\omega, x) \\text{ times}} \\cdot \\, \\mu_{\\beta}(C_1),\n\\end{equation}\nand hence,\n\\begin{equation}\\label{q:nuball}\n \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big) = \\lceil \\beta \\rceil^{-(k-1)}v_{\\alpha_k(\\omega,x)}\\,2^{-(k-M_k(\\omega,x))}\\,\\mu_{\\beta}(C_1).\n\\end{equation}\nThis gives the following theorem.\n\\begin{thm}\\label{t:locdim}\nLet $\\beta >1$ be a generalised multinacci number. For all $(\\omega,x) \\in X$ we have\n\\begin{multline}\\label{q:locdim}\n\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\limsup_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k} \\Big] \\le \\underline d\\big(\\nu_{\\beta}, (\\omega,x)\\big)\\\\\n\\le \\overline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) \\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\liminf_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k} \\Big].\n\\end{multline}\n\\end{thm}\n\n\\begin{proof}\nLet $\\frac{1}{\\beta^{k+1}} < r \\le \\frac{1}{\\beta^k}$. Set $v_{min} = \\min\\{v_1, \\ldots, v_L\\}$ and $v_{max} = \\max\\{v_1, \\ldots, v_L\\}$. Then, by (\\ref{q:nuball}),\n\\[ \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\le  \\frac{(k-1)\\log \\lceil \\beta \\rceil}{k \\log \\beta} + \\frac{(k-M_k(\\omega, x))\\log 2}{k\\log\\beta} - \\frac{\\log \\big(v_{min}\\, \\mu_{\\beta}(C_1)\\big)}{k \\log \\beta}.\\]\nHence,\n\\[ \\overline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) = \\limsup_{k \\to \\infty} \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} + \\frac{\\log 2}{\\log \\beta}\\Big[ 1 - \\liminf_{k \\to \\infty} \\frac{M_k(\\omega,x)}{k} \\Big].\\]\nOn the other hand,\n\\[ \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\ge \\frac{(k-1)\\log \\lceil \\beta \\rceil}{(k+1) \\log \\beta} + \\frac{(k-M_k(\\omega, x))\\log 2}{(k+1)\\log\\beta} - \\frac{\\log \\big(v_{max}\\, \\mu_{\\beta}(C_1)\\big)}{(k+1) \\log \\beta}.\\]\nSince $M_{k+1}(\\omega, x)-1 \\le M_k(\\omega,x) \\le M_{k+1}(\\omega, x)$, we have that\n\\[ \\underline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) = \\liminf_{k \\to \\infty} \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\ge \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} + \\frac{\\log 2}{\\log \\beta}\\Big[ 1 - \\limsup_{k \\to \\infty} \\frac{M_k(\\omega,x)}{k} \\Big].\\]\nThis proves the theorem.\n\\end{proof}\n\n\\begin{rem}{\\rm From the proof of the previous theorem it follows that if $\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists, then $d\\big(\\nu_{\\beta}, (\\omega,x) \\big)$ exists and is equal to $\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big]$.\n}\\end{rem}\n\n\\begin{cor}\\label{c:locdimnubeta}\nLet $\\beta$ be a generalised multinacci number. The local dimension function $d\\big(\\nu_{\\beta}, (\\omega,x)\\big)$ is constant $\\nu_{\\beta}$-a.e.~and equal to\n\\[d\\big(\\nu_{\\beta}, (\\omega,x)\\big)=\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta} \\big( 1-\\mu_{\\beta}(S) \\big).\\]\n\\end{cor}\n\n\\begin{proof}\nSince $\\nu_{\\beta}$ is ergodic, the Ergodic Theorem gives that for $\\nu_{\\beta}$-a.e.~$(\\omega, x)$,\n\\[ \\lim_{k \\to \\infty} \\frac{M_k(\\omega, x)}{k} = \\nu_{\\beta}\\big(\\Omega \\times S\\big) = \\mu_{\\beta}(S).\\]\nThis gives the result.\n\\end{proof}\n\n\\medskip\nRecall that $\\phi$ maps points $(\\omega, x)$ to digit sequences $\\big( b_n(\\omega,x) \\big)_{n \\ge 1}$. It is easy to see that $x = \\sum_{n \\ge 1} \\frac{b_n(\\omega,x)}{\\beta^n}$ for each choice of $\\omega \\in \\Omega$. Note that a point $x$ has exactly one $\\beta$-expansion if and only if for all $n \\ge 0$, $\\pi\\big( K^n (\\omega,x) \\big) \\not \\in S$. Let $\\mathcal A_{\\beta} \\subset \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ be the set of points with a unique $\\beta$-expansion. Then $\\mathcal A_{\\beta} \\neq \\emptyset$, since $0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\in \\mathcal A_{\\beta}$ for any $\\beta>1$. By Proposition 2 from \\cite{DV05} all elements from $\\cup_{n \\ge 0}K^{-n}F$ will be in $S$ at some point and hence they will have more than one expansion. So, $\\mathcal A_{\\beta} \\subset X$. The next result also follows easily from Theorem~\\ref{t:locdim}. The measure $\\nu_{\\beta}$ is called {\\bf multifractal} if the local dimension takes more than one value on positive Hausdorff dimension sets. \n\n\\begin{cor}\nLet $\\beta$ be a generalised multinacci number. If $x \\in \\mathcal A_{\\beta}$, then $d\\big(\\nu_{\\beta}, (\\omega,x)\\big) =\\frac{\\log \\lceil \\beta \\rceil + \\log 2}{\\log \\beta}$ for all $\\omega \\in \\Omega$. The measure $\\nu_{\\beta}$ is multifractal.\n\\end{cor}\n\n\\begin{proof}\nIf $x \\in \\mathcal A_{\\beta}$, then $M_k(\\omega, x)=0$ for all $\\omega \\in \\Omega$ and $k \\ge 1$. Hence, by (\\ref{q:locdim}), $d\\big(\\nu_{\\beta}, (\\omega,x)\\big) =\\frac{\\log \\lceil \\beta \\rceil + \\log 2}{\\log \\beta}$. From standard results in dimension theory and our choice of metric it follows that $dim_H\\big(\\Omega \\times \\{x\\}\\big)=\\frac{\\log 2}{\\log \\beta}$ for all $x\\in \\mathcal A_{\\beta}$. Hence, $\\nu_{\\beta}$ is a multifractal measure.\n\\end{proof}\n\n\\begin{exa}\\label{r:goldenmean}\nWe give a example to show what can happen on points in $F$. Let $\\beta =\\frac{1+\\sqrt 5}{2}$ be the golden ratio. Then, $1 = \\frac{1}{\\beta} + \\frac{1}{\\beta^2}$. Figure~\\ref{f:gm} shows the maps $T_0$ and $T_1$ for this $\\beta$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure3kleur}\n\\caption{The maps $T_0 x = \\beta x$ and $T_1 x = \\beta x-1$ for $\\beta = \\frac{1+\\sqrt 5}{2}$. The region $S$ is colored yellow.}\n\\label{f:gm}\n\\end{figure}\nNote that $F = \\{ 0, \\frac{1}{\\beta}, 1, \\beta \\}$. The partition $\\mathcal C$ consists of only three elements and the transition matrix and stationary distribution of the corresponding Markov chain are\n\\[ P = \\left(\n\\begin{array}{ccc}\n1\/2 & 1\/2 & 0\\\\\n1\/2 & 0 & 1\/2\\\\\n0 & 1\/2 & 1\/2\n\\end{array} \\right) \\quad \\text{ and } \\quad v=(1\/3,1\/3,1\/3).\\]\nHence, $\\mu_{\\beta}(S)=1\/3$ and Corollary~\\ref{c:locdimnubeta} gives that for $\\nu_{\\beta}$-a.e.~$(\\omega,x) \\in \\Omega \\times [0, \\beta]$,\n\\[d\\big(\\nu_{\\beta},(\\omega,x)\\big)=\\frac{\\log 2}{\\log \\beta} \\Big[ 2-\\mu_{\\beta}(S) \\Big]=(2-1\/3) \\frac{\\log 2}{\\log \\beta}=\\frac{5\\log 2}{6 \\log \\beta}.\\]\n\n\\vskip .2cm\nNow consider the $\\alpha$-code of the points $(\\overline{10},1)$ and $(\\overline{01},1\/\\beta)$, where the bar indicates a repeating block:\n\\[\\alpha \\big((\\overline{10},1)\\big)=\\alpha \\big((\\overline{01},1\/\\beta)\\big)=(s,s,s,\\cdots),\\]\nwhich is not allowed in the Markov chain $Y$. Then for any point \n\\[(\\omega,x)\\in \\bigcup_{m=0}^{\\infty} K^{-m} \\big(\\{(\\overline{10},1),(\\overline{01},1\/\\beta)\\}\\big),\\]\none has $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ is a countable set for all $k$ sufficiently large. For the local dimension this implies that\n\\[ d\\big(\\nu_{\\beta},(\\omega,x)\\big) = \\lim_{r \\downarrow 0} \\frac{\\log 0}{\\log r} = \\infty.\\]\n\\end{exa}\n\n\n\\section{Local dimensions for the symmetric Bernoulli convolution}\n\nConsider now the Bernoulli convolution measure $\\mu_{\\beta}=\\nu_{\\beta}\\circ \\pi^{-1}$ on the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ for generalised multinacci numbers. In \\cite{FS11}, the local dimension of $\\mu_{\\beta}$ with respect to the Euclidean metric was obtained for all Pisot numbers $\\beta$. A {\\bf Pisot number} is an algebraic integer that has all its Galois conjugates inside the unit circle. It is well known that all multinacci numbers are Pisot numbers, but unfortunately not all generalised multinacci numbers are Pisot. In Remark~\\ref{r:pisot}(i) we list some classes of generalised multinacci numbers that are in fact Pisot numbers. Before stating the results from \\cite{FS11}, we introduce more notation. Let\n\\begin{equation}\\label{q:fs}\n\\mathcal N_k (x,\\beta)=\\#\\left\\{(a_1, \\ldots, a_k)\\in \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^k:\\exists (a_{k+n})_{n \\ge 1} \\text{ with } x=\\sum_{m=1}^{\\infty} \\frac{a_m}{\\beta^m} \\right\\}.\n\\end{equation}\nA straightforward calculation (see also Lemma 4.1 of \\cite{Kem12}) shows  that\n\\[ \\mathcal N_k (x,\\beta)=\\int_{\\Omega} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega),\\]\nwhere $m$ is the uniform Bernoulli measure on $\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ as before. In \\cite{FS11}, it was shown that if $\\beta$ is a Pisot number, then there is a constant $\\gamma=\\gamma(\\beta,m)$ such that \n\\begin{equation}\\label{q:gamma}\n\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma\n\\end{equation}\nfor $\\lambda$-a.e.~$x$ in $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, where $\\lambda$ is the one-dimensional Lebesgue measure. Using this, Feng and Sidorov obtained that for $\\lambda$-a.e.~$x$,\n\\[ d(\\mu_{\\beta},x)=\\frac{\\log \\lceil \\beta \\rceil -\\gamma}{\\log \\beta}\\]\n In fact, the result they obtained was stronger, but we will use their result in this form. We will show that one has the same value for the local dimension when the Euclidean metric on $\\mathbb R$ is replaced by the Hausdorff metric. To this end, consider the metric $\\bar \\rho$ on $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ defined by\n\\[ \\bar \\rho(x,y)=d_H\\big(\\pi^{-1}(x),\\pi^{-1}(y)\\big),\\]\nwhere $d_H$ is the Hausdorff distance given by\n\\begin{multline*}\nd_H \\big(\\pi^{-1}(x),\\pi^{-1}(y)\\big)\\\\\n=\\inf\\Big\\{\\epsilon >0:\\pi^{-1}(y)\\subset \\bigcup_{\\omega \\in \\Omega}B_{\\rho}\\big((\\omega,x),\\epsilon \\big) \\mbox{ and } \\pi^{-1}(x)\\subset \\bigcup_{\\omega \\in \\Omega}B_{\\rho}\\big((\\omega,y),\\epsilon\\big)\\Big\\}.\n\\end{multline*}\n\n\\begin{thm}\nLet $\\beta$ be a generalised multinacci number. Then for all $x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$,\n\\begin{multline*}\n\\frac{1}{\\log \\beta} \\Big[ \\log \\lceil \\beta \\rceil - \\limsup_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k} \\Big] \\le \\underline d(\\mu_{\\beta},x)\\\\\n\\le \\overline d(\\mu_{\\beta},x) \\le \\frac{1}{\\log \\beta} \\Big[ \\log \\lceil \\beta \\rceil - \\liminf_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k} \\Big].\n\\end{multline*}\n\\end{thm}\n\n\\begin{proof}\nLet $B_ {\\bar \\rho}(x,\\epsilon)=\\{y:\\bar \\rho(x,y)<\\epsilon\\}$. We want to determine explicitly the set $\\pi^{-1}\\big(B_{\\bar \\rho}(x,\\beta^{-k}) \\big)$. First note that for any $(\\omega,x)$, and any $k\\ge 0$, one has $(\\omega^{\\prime},y)\\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$, and $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)=B_{\\rho}\\big((\\omega^{\\prime},y),\\beta^{-k}\\big)$ for any $(\\omega^{\\prime},y)\\in [\\omega_1\\cdots\\omega_k]\\times [\\alpha_1(\\omega,x)\\cdots \\alpha_k(\\omega,x)]$. We denote the common set by $B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k}\\big)$. This implies that\n\\[\\pi^{-1}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big)=\\bigcup_{[\\omega_1\\cdots \\omega_k]}B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k}\\big),\\]\nwhere the summation on the right is taken over all possible cylinders of length $k$ in $\\Omega$. Again set $v_{min} = \\min\\{v_1, \\ldots, v_L\\}$ and $v_{max} = \\max\\{v_1, \\ldots, v_L\\}$. Then,\n\\begin{eqnarray*}\n\\mu_{\\beta}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) &=& \\sum_{[\\omega_1 \\cdots \\omega_k]}\\nu_{\\beta}\\big(B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k} \\big)\\big)\\\\\n& \\le & \\sum_{[\\omega_1 \\cdots \\omega_k]}\\lceil \\beta \\rceil^{-(k-1)}v_{\\alpha_k(\\omega,x)}\\,2^{-(k-M_k(\\omega,x))}\\, \\mu_{\\beta}(C_1)\\\\\n& \\le & \\lceil \\beta \\rceil^{-(k-1)}v_{max} \\, \\mu_{\\beta}(C_1) \\sum_{[\\omega_1 \\cdots \\omega_k]} 2^{M_k(\\omega,x)} 2^{-k}\\\\\n& = & \\lceil \\beta \\rceil^{-(k-1)}v_{max}\\, \\mu_{\\beta}(C_1) \\int_{\\Omega} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega)\\\\\n& = & \\lceil \\beta \\rceil^{-(k-1)} \\, v_{max}\\,   \\mu_{\\beta}(C_1) \\, {\\mathcal N}_k(x,\\beta).\n\\end{eqnarray*}\nNow taking logarithms, dividing by $\\log \\beta^{-k}$, and taking limits we get\n\\[ \\underline d(\\mu_{\\beta},x)\\ge \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} - \\frac{1}{\\log \\beta} \\limsup_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k}.\\]\nSimilarly we get that\n\\[ \\mu_{\\beta}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) \\ge \\lceil \\beta \\rceil^{-(k-1)} \\, v_{min} \\, \\mu_{\\beta}(C_1)  \\, {\\mathcal N}_{k}(x,\\beta),\\]\nwhich gives\n\\[ \\overline d(\\mu_{\\beta},x)\\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} - \\frac{1}{\\log \\beta} \\liminf_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k}. \\qedhere\\]\n\\end{proof}\n\n\\noindent By the results from \\cite{FS11} we have the following corollary.\n\\begin{cor}\nIf $\\beta$ is Pisot, then $d(\\mu_{\\beta},x)$ exists for $\\lambda$-a.e.~$x$ and is equal to $\\frac{\\log \\lceil \\beta \\rceil- \\gamma}{\\log \\beta}$, where $\\gamma$ is the constant from (\\ref{q:gamma}).\n\\end{cor}\n\n\\begin{rem}\\label{r:pisot}{\\rm\n(i) We give some examples of generalised multinacci numbers that are Pisot numbers. The generalised multinacci numbers in the interval $[1,2]$, i.e., the multinacci numbers, are all Pisot. In the interval $[2, \\infty)$ the numbers $\\beta$ that satisfy $\\beta^2 - k\\beta-1=0$, $k \\ge 2$, are all Pisot as well. Recall that if $1=\\frac{a_1}{\\beta} + \\cdots + \\frac{a_n}{\\beta^n}$ with $a_i \\ge 1$ for all $1 \\le i \\le n$, then $n$ is called the degree of $\\beta$. From Theorem 4.2 in \\cite{AG05} by Akiyama and Gjini we can deduce that all generalised multinacci numbers of degree 3 are Pisot numbers. Similarly, from Proposition 4.1 in \\cite{AG05} it follows that all generalised multinacci numbers of degree 4 with $a_4=1$ are Pisot. An example of a generalised multinacci number that is not Pisot is the number $\\beta$ satisfying\n\\[ 1=\\frac{3}{\\beta} + \\frac{1}{\\beta^2} + \\frac{2}{\\beta^3} + \\frac{3}{\\beta^4}.\\]\n(ii) In \\cite{Kem12} it is shown that for all $\\beta > 1$ and $\\lambda$-a.e.~$x$, $\\liminf_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k} \\ge \\mu_{\\beta}(S)\\log 2$, so we get\n\\[ \\overline d(\\mu_{\\beta},x)\\le \\frac{1}{\\log \\beta}\\Big[ \\log \\lceil \\beta \\rceil - \\mu_{\\beta}(S)\\log 2 \\Big].\\]\nKempton also remarks that this lower bound is not sharp.\n}\\end{rem}\n\n\\section{Asymmetric random $\\beta$-transformation: the golden ratio}\n\nIn the previous section, we considered the measure $\\nu_{\\beta}=\\nu_{\\beta,1\/2}$ which is the lift of the uniform Bernoulli measure $m=m_{1\/2}$ under the isomorphism $\\phi(\\omega, x)=\\big(b_n(\\omega,x)\\big)_{n \\ge 1}$. The projection of $\\nu_{\\beta}$ in the second coordinate is the symmetric Bernoulli convolution. In this section, we will investigate the asymmetric Bernoulli convolution in case $\\beta$ is the golden ratio.\n\n\\medskip\nLet $\\beta = \\frac{1+\\sqrt 5}{2}$ and consider the $(p,1-p)$-Bernoulli measure $m_p$ on $\\{0,1\\}^{\\mathbb N}$, i.e., with $m_p([0])=p$ and $m_p([1])=1-p$. Let $\\nu_{\\beta,p}=m_p\\circ \\phi$ on $\\Omega \\times [0, \\beta]$. Since $m_p$ is shift invariant and ergodic, we have that $\\nu_{\\beta,p}$ is $K$-invariant and ergodic. We first show that $\\nu_{\\beta,p}$ is a Markov measure with the same Markov partition as in the symmetric case (see Example~\\ref{r:goldenmean}), but the transition probabilities as well as the stationary distribution are different. This is achieved by looking at the $\\alpha$-code as well. The Markov partition is given by the partition $\\{E_0,S,E_1\\}$, and the corresponding Markov chain has three states $\\{e_0,s,e_1\\}$ with transition matrix\n\\[ P_p=\\left( \\begin{array}{ccc}\np & 1-p & 0 \\\\\np & 0 & 1-p \\\\\n0 & p & 1-p \\end{array} \\right)\\] \nand stationary distribution \n\\[ u=(u_{e_0},u_s,u_{e_1})=\\Big(\\frac{p^2}{p^2-p+1},\\frac{p(1-p)}{p^2-p+1},\\frac{(1-p)^2}{p^2-p+1} \\Big).\\]\nWe denote the corresponding Markov measure by $Q_p$, that is\n\\[ Q_p\\big([j_1 \\cdots j_k]\\big)=u_{j_1}p_{j_1,j_2}\\cdots p_{j_k,j_{k+1}},\\]\nand the space of realizations by \n\\[ Y=\\big\\{(y_1,y_2,\\ldots):y_i\\in \\{e_0,s,e_1\\}, \\mbox{ and } p_{y_i,y_{i+1}}>0\\big\\}.\\]\nConsider the map $\\alpha:\\Omega \\times [0,\\beta]$ of the previous section, namely\n\\[ \\alpha_j(\\omega,x) = \\left\\{ \\begin{array}{ll}\n         e_0, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times E_0$};\\\\\n        s, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times S$};\\\\\ne_1, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times E_1$}.\\end{array} \\right. \\] \nDefine $\\psi:Y\\to \\{0,1\\}^{\\mathbb N}$ by\n\\[ \\psi(y)_j = \\left\\{ \\begin{array}{ll}\n         0, & \\mbox{if $y_j=e_0$ or $y_jy_{j+1}=se_1$};\\\\\n         1, & \\mbox{if $y_j=e_1$ or $y_jy_{j+1}=se_0$}.\\end{array} \\right. \\] \nIt is easy to see that $\\psi\\circ \\alpha=\\phi$. We want to show that $Q_p\\circ \\alpha =\\nu_{\\beta,p}$. Since $\\nu_{\\beta,p}=m_p\\circ \\phi$, we show instead the following.\n\n\\begin{prop}\nWe have $m_p=Q_p\\circ \\psi^{-1}$.\n\\end{prop}\n\n\\begin{proof}\nIt is enough to check equality on cylinders. To avoid confusion, we denote cylinders in $\\{0,1\\}^{\\mathbb N}$ by $[i_1 \\cdots i_k]$ and cylinders in $Y$ by $[j_1 \\cdots j_k]$. We show by induction that\n\\[ \\psi^{-1}([i_1 \\cdots i_k])=[j_1 \\cdots j_k]\\cup [j^{\\prime}_1,\\ldots, j^{\\prime}_{k+1}],\\]\nwhere $j_k=e_{i_k}$, and $j^{\\prime}_kj^{\\prime}_{k+1}=se_{1-i_k}$, and\n\\[ Q_p([j_1 \\cdots j_k])+Q_p([j^{\\prime}_1 \\cdots  j^{\\prime}_{k+1}])=m_p\\big([i_1 \\cdots i_k]\\big) =p^{k-\\sum_{\\ell=1}^k i_{\\ell} }(1-p)^{\\sum_{\\ell=1}^k i_{\\ell}}.\\]\nConsider the case $k=1$. We have $\\psi^{-1}[0]=[e_0]\\cup [se_1]$ and $\\psi^{-1}[1]=[e_1]\\cup [se_0]$. Furthermore,\n\\[ Q_p([e_0]\\big)+Q_p([se_1])=\\frac{p^2}{p^2-p+1}+\\frac{p(1-p)^2}{p^2-p+1}=p=m_p([0]),\\]\nand\n\\[ Q_p([e_1])+Q_p([se_0])=\\frac{(1-p)^2}{p^2-p+1}+\\frac{p^2(1-p)}{p^2-p+1}=1-p=m_p([1])\\]\nas required.  Assume now the result is true for all cylinders $[i_1 \\cdots i_k]$ of length $k$, and consider a cylinder $[i_1 \\cdots  i_{k+1}]$ of length $k+1$. Then,\n\\[ \\psi^{-1}([i_1 \\cdots i_{k+1}])=[j_1 \\cdots j_{k+1}]\\cup [j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}],\\]\nwhere\n\\[ \\psi^{-1}([i_2 \\cdots i_{k+1}])=[j_2 \\cdots j_{k+1}]\\cup [j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}],\\]\nand \n\\[ j_1, j^{\\prime}_1= \\left\\{ \\begin{array}{ll}\n         e_{i_1}, & \\mbox{if $i_1=i_2$ or $i_1\\not= i_2$ and $j_2=s$},\\\\\n         s, & \\mbox{if $i_1\\not= i_2$ and $j_2\\not=s$},\\end{array} \\right. \\] \nBy the definition of $Q_p$, we have \n\\[ Q_p([j_1 \\cdots j_{k+1}])=\\frac{u_{j_1} p_{j_1,j_2}}{u_{j_2}} Q_p([j_2 \\cdots j_{k+1}]),\\]\nand\n\\[ Q_p([j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}])=\\frac{u_{j^{\\prime}_1} p_{j^{\\prime}_1,j^{\\prime}_2}}{u_{j^{\\prime}_2}} Q_p([j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}]).\\]\nOne easily checks that\n\\[ \\frac{u_{j_1} p_{j_1,j_2}}{u_{j_2}}=\\frac{u_{j^{\\prime}_1} p_{j^{\\prime}_1,j^{\\prime}_2}}{u_{j^{\\prime}_2}}=p^{1-i_1}(1-p)^{i_1} =\\left\\{ \\begin{array}{ll}\n         p, & \\mbox{if }i_1=0;\\\\\n        1-p, & \\mbox{if }i_1=1.\\end{array} \\right. \\] \nBy the induction hypothesis applied to the cylinder $[i_2 \\cdots i_{k+1}]$, we have $j_{k+1}=e_{i_{k+1}}$, and $j^{\\prime}_{k+1}j^{\\prime}_{k+2}=se_{1-i_{k+1}}$, and\n\\[ Q_p([j_2 \\cdots j_{k+1}])+Q_p([j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}])=m_p([i_2\\cdots i_{k+1}])\n=p^{k-\\sum_{\\ell=2}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=2}^{k+1} i_{\\ell}}.\\]\nThus,\n\\begin{eqnarray*}\nQ_p([j_1 \\cdots j_{k+1}])+Q_p([j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}]) & = &\np^{1-i_1}(1-p)^{i_1} p^{k-\\sum_{\\ell=2}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=2}^{k+1} i_{\\ell}}\\\\\n& = & p^{k+1-\\sum_{\\ell=1}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=1}^{k+1} i_{\\ell}}\\\\\n& = & m_p([i_1\\cdots i_{k+1}]).\n\\end{eqnarray*}\nThis gives the result.\n\\end{proof}\n\n\\noindent As before, let $M_k(\\omega,x)=\\sum_{i=0}^{k-1}{\\bf 1}_{\\Omega \\times S}(K_{\\beta}^i(\\omega,x))$.\n\\begin{thm} For $\\nu_{\\beta,p}$-a.e.~$(\\omega,x)$ for which $\\displaystyle\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists, one has\n\\[ d\\big( \\nu_{\\beta,p},(\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big),\\]\nwhere $H(p)=-p\\log p -(1-p)\\log(1-p)$.\n\\end{thm}\n\n\\begin{proof} We consider the same metric $\\rho$ as in the previous section, namely\n\\[ \\rho\\big((\\omega,x),(\\omega^{\\prime},x^{\\prime})\\big)=\\beta^{-\\min\\{k\\ge 0 \\, : \\, \n\\omega_{k+1}\\neq \\omega^{\\prime}_{k+1} \\text{ or } \\alpha_{k+1}(\\omega,x)\\neq \\alpha_{k+1}(\\omega^{\\prime},x^{\\prime})\\}}.\\]\nConsider a point $(\\omega,x)$ such that $\\displaystyle\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists. By the same reasoning that led to (\\ref{q:Qball}) we have\n\\begin{multline*}\n\\nu_{\\beta,p}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big)\\\\\n= Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} u_{e_{1-\\omega_k}}.\n\\end{multline*}\nLet $u_{\\max}=\\max(u_{e_0},u_{e_1})$ and $u_{\\min}=\\min(u_{e_0},u_{e_1})$, then $\\log \\nu_{\\beta,p}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big)$ is bounded from above by\n\\begin{gather*}\n\\log Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) + \\Big((k-M_k(\\omega,x)-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\Big)\\log p\\\\\n + \\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\log (1-p) +\\log u_{\\max},\n\\end{gather*}\nand is bounded from below by\n\\begin{gather*}\n\\log Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) + \\Big((k-M_k(\\omega,x)-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\Big)\\log p\\\\\n+ \\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\log (1-p) +\\log u_{\\min}.\n\\end{gather*}\nDividing by $-k\\log \\beta$ and taking limits, we have by the Shannon-McMillan-Breiman Theorem that\n\\[ \\lim_{k\\to\\infty}\\frac{\\log Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)])}{-k\\log \\beta}= \\frac{H(p)}{\\log \\beta},\\]\nand by the Ergodic Theorem we have\n\\[ \\lim_{k\\to\\infty}\\frac{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i}{-k\\log \\beta}= \\frac{-(1-p)\\big(1-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\big)}{\\log \\beta},\\]\nboth for $\\nu_{\\beta}$-a.e.~$(\\omega,x)$. Thus, both the upper and the lower bounds converge to the same value, implying that\n\\[ d\\big(\\nu_{\\beta,p}, (\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big). \\qedhere\\]\n\\end{proof}\n\n\\begin{cor}\nFor $\\nu_{\\beta,p}$-a.e.~$(\\omega,x)$ one has \n\\[ d\\big(\\nu_{\\beta,p}, (\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\nu_{\\beta,p}\\big(\\Omega \\times S\\big)\\Big) =\\frac{H(p)}{\\log \\beta} \\Big(2-\\frac{p(1-p)}{p^2-p+1}\\Big).\\]\n\\end{cor}\n\n\\vskip .5cm\nWe now turn to the study of the local dimension of the asymmetric Bernoulli convolution $\\mu_{\\beta,p}$, which is the projection in the second coordinate of $\\nu_{\\beta,p}$. Let ${\\mathcal N}_k(\\omega,x)$ be as given in equation (\\ref{q:fs}). In the symmetric case, it was shown that\n\\[\\mathcal N_{k}(x,\\beta)=\\int_{\\{0,1\\}^{\\mathbb N}} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega)=\\sum_{[\\omega_1 \\cdots \\omega_k]} 2^{M_k(\\omega,x)} 2^{-k}.\\]\nWe now give a similar formula for the asymmetric case.\n\n\\begin{lem} \\label{counting}\n$\\mathcal N_{k}(x,\\beta)=\\sum_{[\\omega_1 \\cdots \\omega_k]} p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i}$.\n\\end{lem}\n\n\\begin{proof} We use a similar argument as the one used for the symmetric case (see \\cite{Kem12}). Define\n\\[ \\Omega(k,x)=\\big\\{\\omega_1 \\cdots \\omega_{M_k(\\omega,x)}: \\omega\\in \\Omega \\big\\}.\\]\nIf $x$ has a unique expansion, then $\\Omega(k,x)$ consists of one element, the empty word. We now have $|\\Omega(k,x)|= \\mathcal N_k(x,\\beta)$, and \n\n\\begin{eqnarray*}\n& &\n \\sum_{[\\omega_1 \\cdots \\omega_k]} p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} \\\\\n& = & \\sum_{[\\omega_1 \\cdots \\omega_k]} \\frac{p^{k-\\sum_{i=1}^k \\omega_i}(1-p)^{\\sum_{i=1}^k \\omega_i}}{p^{M_k(\\omega,x)-\\sum_{i=1}^{M_k(\\omega,x)} \\omega_i}(1-p)^{\\sum_{i=1}^{M_k(\\omega,x)} \\omega_i}}\\\\\n& = & \\int_{\\Omega}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_{M_k(\\omega,x)}])} dm_p(\\omega) =  \\sum_{j=0}^k \\int_{\\{\\omega: M_k(\\omega,x)=j\\}}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_j])} dm_p(\\omega)\\\\\n& = & \\sum_{j=0}^k\\,\\,\\sum_{\\omega_1 \\cdots \\omega_j \\in \\Omega(k,x)}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_j])} m_p([\\omega_1 \\cdots \\omega_j]) =  |\\Omega(k,x)|=\\mathcal N_k(x,\\beta). \\quad \\quad \\qedhere\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{thm}\\label{bounds}\nFor $\\lambda$-a.e.~$x \\in [0, \\beta]$ one has\n\\[ \\frac{-\\big(\\log (\\max(p,1-p))+\\gamma \\big)}{\\log \\beta}\\le \\underline{d}(\\mu_{\\beta,p},x)\\le \\overline{d}(\\mu_{\\beta,p},x)\\le \\frac{-\\big(\\log (\\min(p,1-p))+\\gamma \\big)}{\\log \\beta},\\]\nwhere $\\displaystyle\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma$ is the constant from (\\ref{q:gamma}).\n\\end{thm}\n\n\\begin{proof} We use the same metric $\\bar \\rho$ on $[0,\\beta]$ as in the previous section. Then\n\\begin{multline*}\n\\mu_{\\beta,p}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) = \\sum_{[\\omega_1 \\cdots \\omega_k]}\\nu_{\\beta}(B_{\\rho}\\big([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k})\\big)\\\\ \n= \\sum_{[\\omega_1 \\cdots \\omega_k]}Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)]) p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} \\\\\n\\cdot(1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} u_{e_{1-\\omega_k}}.\n\\end{multline*}\nNow,\n\\[ Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)])=u_{\\alpha_1(\\omega,x)} p^{L_k(\\omega,x)}(1-p)^{k-L_k(\\omega,x)},\\]\nwhere\n\\[ L_k(\\omega,x)=\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)=e_0\\}+\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)\\alpha_{j+1}(\\omega,x)=e_1s\\},\\]\nand hence\n\\begin{multline*}\nk-L_k(\\omega,x)\\\\\n=\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)=e_1\\}+\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)\\alpha_{j+1}(\\omega,x)=e_0s\\}.\n\\end{multline*}\nLet $C_1=\\max(u_{e_0},u_s,u_{e_1})$ and $C_2=\\min(u_{e_0},u_s,u_{e_1})$. Then, from Lemma (\\ref{counting}) we have\n\\[ C_2\\big(\\min(p,1-p))\\big)^k \\mathcal N_k(x,\\beta)\\le \\mu_{\\beta,p}(B_{\\bar \\rho}(x,\\beta^{-k}))\\le C_1\\big(\\max(p,1-p))\\big)^k \\mathcal N_k(x,\\beta).\\]\nSince $\\beta$ is a Pisot number, $\\displaystyle\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma$ exists $\\lambda$-a.e.~(see \\cite{FS11}) and we have\n\\[ \\frac{-[\\log (\\max(p,1-p))+\\gamma]}{\\log \\beta}\\le \\underline{d}(\\mu_{\\beta,p},x)\\le  \\overline{d}(\\mu_{\\beta,p},x)\\le \\frac{-[\\log (\\min(p,1-p))+\\gamma]}{\\log \\beta}. \\qedhere\\]\n\\end{proof}\n\n\\begin{rem}{\\rm (i) If $p=1\/2$, then both sides of the inequality in Theorem (\\ref{bounds}) are equal to $\\displaystyle\\frac{\\log 2-\\gamma}{\\log \\beta}$ leading to \n\\[ d(\\mu_{\\beta,1\/2},x)= \\frac{\\log 2-\\gamma}{\\log \\beta}\\]\na.e.~as we have seen earlier.\\\\\n(ii) We now consider the extreme cases $x\\in \\{0,\\beta\\}$, the only two points with a unique expansion. We begin with $x=\\beta$. In this case\n\\[ Q_p([\\alpha_1(\\omega,\\beta) \\cdots \\alpha_k(\\omega,\\beta)])=Q_p([e_1 \\cdots e_1])=u_{e_1}(1-p)^k,\\]\nand $\\mathcal N_k(\\beta,\\beta)=1$, so that\n\\[ C_2(1-p)^k\\le \\mu_{\\beta,p}(B_{\\bar \\rho}(\\beta,\\beta^{-k}))\\le C_1(1-p)^k.\\]\nHence,\n\\[ d(\\mu_{\\beta,p}, \\beta)=\\frac{-\\log (1-p)}{\\log\\beta}\\]\nfor all $\\omega\\in \\Omega$. A similar argument shows that\n\\[ d(\\mu_{\\beta,p},0)=\\frac{-\\log (p)}{\\log\\beta}\\]\nfor all $\\omega\\in \\Omega$.\n}\\end{rem}\n\n\\bigskip\n\\footnotesize\n\\noindent\\textit{Acknowledgments.}\nThe second author was supported by NWO (Veni grant no. 639.031.140).\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{S:0}\nThe weak separation condition (WSC) was introduced by Lau and Ngai \\cite{Lau-Ngai_1999} to study the multifractal formalism of self-similar measures defined by iterated function systems of contractive similitudes with overlaps. Although strictly weaker than the well-known open set condition (OSC), the weak separation condition leads to many interesting results, such as the validity of the multifractal formalism (see, e.g. \\cite{Lau-Ngai_1999,Feng_2003,Feng_2005,Feng-Lau_2009, Shmerkin_2005,Ye_2005}),  equality of Hausdorff, box, and packing dimensions of self-conformal sets and the computation of these dimensions \\cite{Deng-Ngai_2011,Ferrari-Panzone_2011,Lau-Ngai-Wang_2009}, and conditions on absolute continuity of self-similar and self-conformal measures \\cite{Lau-Ngai-Rao_2001,Lau-Wang_2004,Lau-Ngai-Wang_2009}. Equivalent forms of the weak separation conditions have also been studied extensive (see, e.g. \\cite{Zerner_1996, Lau-Ngai-Wang_2009}).\n\nThe finite type condition (FTC) was introduced by Ngai and Wang \\cite{Ngai-Wang_2001} to calculate the Hausdorff dimension of self-similar sets with overlaps. It was generalized independently by Jin-Yau \\cite{Jin-Yau_2005} and Lau-Ngai \\cite{Lau-Ngai_2007} to include (OSC). Lau and Ngai proved that (FTC) implies (WSC) for IFSs of contractive similitudes. This result was generalized by Lau \\textit{et al.} \\cite{Lau-Ngai-Wang_2009} to conformal iterated function systems (CIFSs).\n\nIn 2009, Lau \\textit{et al.} \\cite{Lau-Ngai-Wang_2009} formulated (WSC) for conformal iterated function systems on $\\mathbb{R}^{n}$, and proved the equality of the Hausdorff, box and packing dimensions of the associated self-conformal sets. They also studied the absolute continuity of the associated self-conformal measures. The first goal of this paper is to extend results in \\cite{Lau-Ngai-Wang_2009} to Riemannian manifolds with nonnegative Ricci curvature. Our second goal is to generalize the method of computing the Haudorff dimension of self-similar sets in \\cite{Ngai-Wang_2001,Jin-Yau_2005,Lau-Ngai_2007} to Riemannian manifolds that are locally Euclidean.\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold.\nAssume that $U\\subset M$ is open and connected, and $W\\subset U$ is a compact set with $\\overline{W^{\\circ}}=W$, where $\\overline{W^{\\circ}}$ is the closure of the interior of $W$. Assume that $\\{S_{i}\\}_{i=1}^{N}$ is a conformal iterated function system (CIFS) on $U$ defined as in Section \\ref{S:1}. Then there exists a unique nonempty compact set $K\\subset W$, called the \\textit{attractor} or \\textit{self-conformal set}, satisfying $K=\\bigcup_{i=1}^{N}S_{i}(K)$ (see \\cite{Hutchinson_1981}).\n\nGiven a probability vector $(p_{1},\\dots,p_{N})$, i.e., $p_{i}>0$ for any $i\\in\\{1,\\dots,N\\}$ and $\\sum_{i=1}^{N}p_{i}=1$, there exists a unique Borel probability measure $\\mu$, called the \\textit{self-conformal measure}, such that $\\mu=\\sum_{i=1}^{N}p_{i}\\mu\\circ S_{i}^{-1}$ and $K={\\rm supp}(\\mu)$ (see \\cite{Hutchinson_1981}).\n\nLet $\\Sigma:=\\{1,\\dots,N\\}$, where $N\\in\\mathbb{N}$ and $N\\geq2$. Let $\\Sigma^{\\ast}:=\\bigcup_{k\\geq1}\\Sigma^{k}$, where $k\\in\\mathbb{N}$. For $\\mathbf{u}=(u_1,\\dots,u_k)\\in\\Sigma^{k}$, let $\\mathbf{u}^{-}:=(u_{1},\\dots,u_{k-1})$ and write $S_{\\mathbf{u}}:=S_{u_1}\\circ\\cdots\\circ S_{u_k}$, $p_{\\mathbf{u}}:=p_{u_1}\\cdots p_{u_k}$. Define\n$$r_{\\mathbf{u}}:=\\inf_{x\\in W}|\\det S'_{\\mathbf{u}}(x)|^{\\frac{1}{n}},~r:=\\min_{1\\leq i\\leq N}r_{i},~R_{\\mathbf{u}}:=\\sup_{x\\in W}|\\det S'_{\\mathbf{u}}(x)|^{\\frac{1}{n}},~R:=\\max_{1\\leq i\\leq N}R_{i}.$$\nIf $S=S_{\\mathbf{u}}$ for some $\\mathbf{u}\\in\\Sigma^{\\ast}$, we let $R_{S}=R_{\\mathbf{u}}$. For $0<b\\leq1$, let\n$$\\mathcal{W}_{b}:=\\{\\mathbf{u}=(u_1,\\dots,u_k):R_{\\mathbf{u}}\\leq b<R_{\\mathbf{u}^{-}}\\}\\quad\\text{and}\\quad\\mathcal{A}_{b}:=\\{S_{\\mathbf{u}}:\\mathbf{u}\\in\\mathcal{W}_{b}\\}.$$\nDenote the cardinality of a set $E$ by $\\#E$. We say that $\\{S_{i}\\}_{i=1}^{N}$ satisfies \\textit{the weak separation condition} (WSC) if there exists a constant $\\gamma\\in\\mathbb{N}$ and a subset $D\\subset W$, with $D^{\\circ}\\neq\\emptyset$, such that for any $0<b\\leq1$ and $x\\in W$,\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(D)\\}\\leq\\gamma.$$\nRemark that if the open set condition (OSC) holds, we can take $D$ to be an OSC set and let $\\gamma=1$ to show that (WSC) also holds.\nFor any $a>0$ and any bounded subsets $D\\subset W$ and $A\\subset M$, denote the diameter of $A$ by $|A|$, and let\n$$\\mathcal{A}_{a,A,D}:=\\{S\\in\\mathcal{A}_{a|A|}:S(D)\\cap A\\neq\\emptyset\\},\\quad\\gamma_{a,A}:=\\sup_{A\\subset M}\\#\\mathcal{A}_{a,A,D}.$$\nFor $S\\in\\mathcal{A}_{b}$, let $p_{S}:=\\sum\\{p_{\\mathbf{u}}:S_{\\mathbf{u}}=S,\\mathbf{u}\\in\\mathcal{A}_{b}\\}$.\n\nTheorems \\ref{thm(0.1)}--\\ref{thm(0.4)} generalize analogous results in \\cite{Lau-Ngai-Wang_2009}. In our proofs, the Lebesgue measure in \\cite{Lau-Ngai-Wang_2009} is changed to the more complicated Riemannian volume measure; properties such as the volume doubling property, need not hold. We assume that $M$ is a complete Riemannian  manifold with nonnegative Ricci curvature. Under this assumption, the Bishop-Gromov comparison theorem implies that $M$ is a \\textit{doubling space} (see, e.g. \\cite{Baudoin_2011,Berger_2003}), i.e., any $2r$-ball in $M$ can be covered by a finite union of a bounded number of $r$-balls,  a property that obviously holds on $\\mathbb{R}^{n}$. Another complication arises on manifolds; unlike Euclidean spaces, it is not easy to calculate the volume of a ball in Riemannian manifolds.  For Riemannian manifolds with nonnegative Ricci curvature, we use the Bishop-Gromov inequality  (see Lemma \\ref{lem(1.00)}), which says that the Riemannian volume of a ball can be controlled by a ball in $\\mathbb{R}^{n}$ with the same radius. This  is crucial in the proof of Theorem \\ref{thm(0.1)}.\n\n\nFor a set $K\\subset M$, let $\\dim_{{\\rm H}}(K),\\dim_{{\\rm B}}(K)$ and $\\dim_{{\\rm P}}(K)$ be the Hausdorff, box and packing dimensions, respectively. Let $\\mathcal{H}^{\\alpha}(K)$ and $\\mathcal{P}^{\\alpha}(K)$ be the Hausdorff and packing measures of $K$, respectively.\n\n\n\n\n\\begin{thm}\\label{thm(0.1)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold with non-negative Ricci curvature, and let $U\\subset M$ be open and connect. Assume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (WSC), and $K$ is the associated attractor. Then $\\alpha:=\\dim_{{\\rm H}}(K)=\\dim_{{\\rm B}}(K)=\\dim_{{\\rm P}}(K)$ and\n$0<\\mathcal{H}^{\\alpha}(K)\\leq\\mathcal{P}^{\\alpha}(K)<\\infty$.\n\\end{thm}\n\nLau \\textit{et al.} \\cite{Lau-Ngai-Rao_2001} formulated a sufficient condition for a self-similar measure defined by an IFS satisfying (WSC) to be singular. Later,  Lau and Wang \\cite{Lau-Wang_2004} established the necessary perfected the result on absolute continuity in \\cite{Lau-Ngai-Rao_2001}. We extend these results to manifolds.\n\n\n\n\n\\begin{thm}\\label{thm(0.2)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}. Let $K$ be the attractor\nwith $\\dim_{{\\rm H}}(K)=\\alpha$. Then a self-conformal measure $\\mu$ defined by $\\{S_{i}\\}_{i=1}^{N}$ is singular with respect to $\\mathcal{H}^{\\alpha}|_{K}$ if and only if there exist $0<b\\leq1$ and $S\\in\\mathcal{A}_{b}$ such that $p_{S}>R_{S}^{\\alpha}$.\n\\end{thm}\n\n\n\n\\begin{thm}\\label{thm(0.3)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}. If the self-conformal measure $\\mu$ is absolutely continuous with respect to $\\mathcal{H}^{\\alpha}|_{K}$, then the\nRadon-Nikodym derivative of $\\mu$ is bounded.\n\\end{thm}\n\nIn the proof of Theorem \\ref{thm(0.3)}, we use an analogue of the Lebesgue density theorem in metric spaces (see, e.g. \\cite[Theorem 2.9.8]{Federer_1969} and \\cite[Lemma 2.1(i)]{Bedferd-Fisher_1992}), applying to the collection of Borel sets that forms a Vitali relation (see \\cite{Federer_1969} and a brief summary in Section \\ref{S:2}). By \\cite[Definition 2.8.9 and Theorem 2.8.18]{Federer_1969}, we have a collection of open balls in Riemannian manifolds that forms a Vitali relation.\n\n\n We refer the reader to Section \\ref{S:3} for the definition of (FTC).\n\n\\begin{thm}\\label{thm(0.4)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}, and let $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$. If $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (FTC) on some open sets $\\Omega\\subset W$,\nthen $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC).\n\\end{thm}\n\nDenote the Riemannian distance in $M$ by $d(\\cdot,\\cdot)$. Let $W\\subset M$ be a compact set.\nWe say that $\\{S_{i}\\}_{i=1}^{N}$ is an \\textit{IFS of contractions} on $W$ if for any $i\\in\\{1,\\dots,N\\}$, there exists $\\rho_{i}\\in(0,1)$ such that for any $x,y\\in W$,\n\\begin{equation}\\label{eq:IFS_contraction}\nd(S_{i}(x),S_{i}(y))\\le \\rho_{i}d(x,y).\n\\end{equation}\nIf equality in \\eqref{eq:IFS_contraction} holds for all $i\\in\\{1,\\dots,N\\}$ and all $x,y\\in W$, then\nsay that $\\{S_{i}\\}_{i=1}^{N}$ is an \\textit{IFS of contractive similitudes} on $W$ and call $\\rho_i$ the \\textit{contraction ratio} of $S_{i}$.\n\n\nWe say that a Riemannian manifold $M$ is \\textit{locally Euclidean} if every point of $M$ has a neighborhood which is isometric to an open subset of a Euclidean space (see e.g. \\cite{Kobayashi-Nomizu}). By \\cite[Lemma 2 of Theorem 3.6]{Kobayashi-Nomizu}, contractive similitudes only exist in Riemannian manifolds that are locally Euclidean. In Section \\ref{S:4}, we obtain the following formula for computing the Hausdorff dimension formula of self-similar sets defined by a finite type IFS of contractive similitudes on a locally Euclidean Riemannian manifold, extending a result in \\cite{Jin-Yau_2005} and \\cite{Lau-Ngai_2007} to locally Euclidean Riemannian manifolds (see Theorem \\ref{thm(0.5)}).\n\n\\begin{thm}\\label{thm(0.5)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean, $W\\subseteq M$ be a compact subset, and $\\{S_{i}\\}_{i=1}^{N}$ be an IFS of contractive similitudes on $W$ with attractor $K$. Let $\\lambda_{\\alpha}$ be the spectral radius of the associated weighted incidence matrix $A_{\\alpha}$. If $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC), then $\\dim_{{\\rm H}}(K)=\\alpha$, where $\\alpha$ is the unique number such that $\\lambda_{\\alpha}=1$.\n\\end{thm}\n\n\nIn order to compute the Hausdorff dimension of certain attractors on Riemannian manifolds, it is necessary to study graph iterated function systems (see Example~\\ref{exam(5.4)}). We define graph-directed iterated function systems (GIFSs) and graph finite type condition (GFTC) on Riemannian manifolds in Section \\ref{S:41}. We obtain the following result for computing the Hausdorff dimension of graph self-similar sets (see Theorem \\ref{thm(41.1)}), extending a result in \\cite{Ngai-Wang-Dong_2010}.\n\n\n\\begin{thm}\\label{thm(41.1)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean. Assume that $G=(V,E)$ is a GIFS defined on $M$ satisfying (GFTC), and $K$ is the associated graph self-similar set. Let $\\lambda_{\\alpha}$ be the spectral radius of the associated weighted incidence matrix $A_{\\alpha}$. Then $\\dim_{{\\rm H}}(K)=\\alpha$, where $\\alpha$ is the unique number such that $\\lambda_{\\alpha}=1$.\n\\end{thm}\n\n\n\n\nThis paper is organized as follows. Section \\ref{S:1} introduces the definition of CIFSs, some properties of (WSC), and gives the proof of Theorem \\ref{thm(0.1)}. In Section \\ref{S:2}, we study the absolute continuity of self-conformal measures on Riemannian manifolds and prove Theorems \\ref{thm(0.2)} and \\ref{thm(0.3)}. Section \\ref{S:3} is devoted to the proof of Theorem \\ref{thm(0.4)}. In Section \\ref{S:4}, we study finite type IFSs of contractive similitudes and prove Theorem \\ref{thm(0.5)}. Section \\ref{S:41} is devoted to the proof of Theorem \\ref{thm(41.1)}. Finally, we present some examples of CIFSs and GIFSs satisfying (FTC) and (GFTC) on Riemannian manifolds, respectively.\n\n\\section{The weak separation condition}\\label{S:1}\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold, $U\\subset M$ be open and connected, and let $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$.\nRecall that a map $S:U\\longrightarrow U$ is called \\textit{conformal} if $S'(x)$ is a similarity matrix for any $x\\in U$. We say that $\\{S_{i}\\}_{i=1}^{N}$ is a \\textit{conformal iterated function system} (CIFS)\non $U$, if\n\\begin{itemize}\n\\item[$(a)$] for any $i\\in\\Sigma$, $S_{i}:U\\longrightarrow S_{i}(U)\\subset U$ is a conformal $C^{1+\\varepsilon}$ diffeomorphism for some $\\varepsilon\\in(0,1)$;\n\\item[$(b)$] $S_{i}(W)\\subset W$ for any $i\\in\\Sigma$;\n\\item[$(c)$] $0<|\\det S'_{i}(x)|<1$ for any $i\\in\\Sigma$ and $x\\in U$.\n\\end{itemize}\n\nSince $M$ is a manifold, we can find an open and connected set $U_{1}$ such that $\\overline{U_{1}}$ is compact and $W\\subset U_{1}\\subset \\overline{U_{1}}\\subset U$. According to \\cite{Patzschke_1997}, conditions $(a)$--$(c)$ together imply the \\textit{bounded distortion property} (BDP), without assuming any separation condition, i.e., there exists a constant $C_{1}\\geq1$ such that for any $\\mathbf{u}\\in\\Sigma^{*}$ and $x,y\\in U_{1}$,\n\\begin{equation}\\label{eq(1.1)}\nC_{1}^{-n}\\leq\\frac{|\\det S'_{\\mathbf{u}}(x)|}{|\\det S'_{\\mathbf{u}}(y)|}\\leq C_{1}^{n}.\n\\end{equation}\nIt follows that for any $\\mathbf{u}\\in\\Sigma^*$,\n\\begin{equation}\\label{eq:r_R_bound}\nC_{1}^{-1}\\leq\\frac{r_{\\mathbf u}}{R_{\\mathbf u}}\\leq\\frac{R_{\\mathbf u}}{r_{\\mathbf u}}\\leq C_{1}.\n\\end{equation}\nMoreover, there exists a constant $C_{2}\\geq1$ such that for any $\\mathbf{u}\\in\\Sigma^{\\ast}$ and $x,y\\in W$,\n\\begin{equation}\\label{eq(1.2)}\nC_{2}^{-1}R_{\\mathbf{u}}d(x,y)\\leq d(S_{\\mathbf{u}}(x),S_{\\mathbf{u}}(y))\\leq C_{2}R_{\\mathbf{u}}d(x,y).\n\\end{equation}\nLet $\\nu$ be the Riemannian volume measure, and let $A\\subset W$ be a measurable set. Denote the Jacobian determinant of a function $f$ by $\\mathbf{J}f$. Then\n\\begin{equation}\\label{eq(1.02)}\n\\nu\\big(S_{\\mathbf{u}}(A)\\big)=\\int_{A}\\big|\\mathbf{J}\\big(S_{\\mathbf{u}}(x)\\big)\\big|d\\nu=\\int_{A}\\big|\\det S'_{\\mathbf{u}}(x)\\big|d\\nu.\n\\end{equation}\n(see, e.g. \\cite[Proposition 8.1.10]{Abraham-Marsden-Ratiu_2007}).\n\nNote that for any $x\\in W$ and $\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}$,\n$$|\\det S'_{\\mathbf{u}\\mathbf{v}}(x)|=|\\det S'_{\\mathbf{u}}(S_{\\mathbf{v}}(x))\\cdot S'_{\\mathbf{v}}(x)|=|\\det S'_{\\mathbf{u}}(S_{\\mathbf{v}}(x))|\\cdot|\\det S'_{\\mathbf{v}}(x)|.$$\nHence $R_{\\mathbf{u}\\mathbf{v}}\\leq R_{\\mathbf{u}}R_{\\mathbf{v}}$ and $r_{\\mathbf{u}\\mathbf{v}}\\geq r_{\\mathbf{u}}r_{\\mathbf{v}}$. In particular, for $S=S_{\\mathbf{u}}\\in\\mathcal{A}_{b},\\mathbf{u}=(u_{1},\\dots,u_{k})\\in\\mathcal{W}_{b}$,\n$$\\begin{aligned}\nbr&<R_{u_{1},\\dots,u_{k-1}}r_{u_{k}}\\\\\n&\\leq C_{1}r_{u_{1},\\dots,u_{k-1}}r_{u_{k}}\\quad\\text{(by (\\ref{eq:r_R_bound}))}\\\\\n&\\leq C_{1}r_{\\mathbf{u}}\\\\\n&\\leq C_{1}R_{\\mathbf{u}},\n\\end{aligned}$$\ni.e.,\n\\begin{equation}\\label{eq(1.9)}\nb<\\frac{C_{1}}{r}R_{S}\\quad\\text{for all }S\\in\\mathcal{A}_{b}.\n\\end{equation}\n\n\n\\begin{lem}\\label{lem(1.1)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold, $U\\subset M$ be open and connected, and let $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$. Let $\\{S_{i}\\}_{i=1}^{N}$ be a CIFS on $U$ defined as above. Then the following hold.\n\\begin{itemize}\n\\item[$(a)$] For any $\\mathbf{u}\\in \\mathcal{W}_{b},~0<b\\leq1$, and any measurable set $A\\subset W$,\n$$\\bigg(\\frac{br}{C_{1}}\\bigg)^{n}\\nu(A)\\leq\\nu(S_{\\mathbf{u}}(A))\\leq b^{n}\\nu(A).$$\n\\item[$(b)$] For any $\\mathbf{u},\\mathbf{v}\\in \\mathcal{W}_{b},~0<b\\leq1$, and any measurable set $A\\subset W$,\n$$\\bigg(\\frac{r}{C_{1}}\\bigg)^{n}\\nu(S_{\\mathbf{v}}(A))\\leq\\nu(S_{\\mathbf{u}}(A))\\leq \\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\nu(S_{\\mathbf{v}}(A)).$$\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n$(a)$ Let $\\mathbf{u}=(u_{1},\\dots,u_{k})\\in \\mathcal{W}_{b}$. Then by the definition of $\\mathcal{W}_{b}$,\n\\begin{equation}\\label{eq(1.3)}\n\\sup_{x\\in W}|\\det S'_{\\mathbf{u}}(x)|\\leq b^{n}\\leq\\sup_{x\\in W}|\\det S'_{\\mathbf{u}^{-}}(x)|.\n\\end{equation}\nHence for any $x\\in W$,\n$$\\begin{aligned}\nb^{n}&\\geq |\\det S'_{\\mathbf{u}}(x)|=|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\cdot|\\det S'_{u_{k}}(x)|\\quad\\text{(by (\\ref{eq(1.3)}))}\\\\\n&\\geq r^{n}\\inf_{x\\in W}|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\\\\n&\\geq \\bigg(\\frac{r}{C_{1}}\\bigg)^{n}\\sup_{x\\in W}|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\quad\\text{(by (\\ref{eq(1.1)}))}\\\\\n&>\\bigg(\\frac{rb}{C_{1}}\\bigg)^{n}\\quad\\text{(by (\\ref{eq(1.3)}))}.\n\\end{aligned}$$\nIt follows that\n$$\\int_{A}b^{n}d\\nu\\geq\\int_{A}|\\det S'_{\\mathbf{u}}(x)|d\\nu\\geq\\int_{A}\\bigg(\\frac{rb}{C_{1}}\\bigg)^{n}d\\nu,$$\nwhich proves $(a)$ by (\\ref{eq(1.02)}).\n\n$(b)$ Making use of part $(a)$, we have\n$$\\nu(S_{\\mathbf{u}}(A))\\leq b^{n}\\nu(A)\\leq\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\nu(S_{\\mathbf{v}}(A)).$$\nSimilarly,\n$$\\nu(S_{\\mathbf{v}}(A))\\leq b^{n}\\nu(A)\\leq\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\nu(S_{\\mathbf{u}}(A)),$$\nwhich proves $(b)$.\n\\end{proof}\n\nLet $\\mathcal{F}:=\\{S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}:\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}\\}$. It is possible that $\\tau=S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}$ can be simplified to $S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1}$. Thus the domain of\n$\\tau$ is $S_{\\mathbf{u}'}(W)$ (containing $S_{\\mathbf{u}}(W)$). Denote the domain of $\\tau$ by ${\\rm Dom}(\\tau)$. The proof of the following lemma is similar to that of \\cite[Lemma 2.2]{Lau-Ngai-Wang_2009} and is omitted.\n\n\\begin{lem}\\label{lem(1.2)}\nAssume the same hypotheses of Lemma \\ref{lem(1.1)}. Then for any $\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}$ and any $x,y\\in{\\rm Dom}(S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1})$, we have\n$$\\frac{|\\det (S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1})'(x)|}{|\\det (S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1})'(y)|}\\leq C_{1}^{2n}.$$\n\\end{lem}\n\n\n\\begin{lem}\\label{lem(1.3)}\nAssume the same hypotheses of Lemma \\ref{lem(1.1)}. Let $\\tau=S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}=S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1}\\in\\mathcal{F}$ with ${\\rm Dom}(\\tau)=S_{\\mathbf{u}'}(W)$. Then the following hold.\n\\begin{itemize}\n\\item[$(a)$] For any measurable subset $A\\subset {\\rm Dom}(\\tau)$,\n$$\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\\leq \\nu(\\tau(A))\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A).$$\n\\item[$(b)$] For any $A,B$ belonging to some collection $\\mathcal{C}$ of measurable subsets of $W$, suppose $C\\geq1$ is a constant such that\n$$C^{-1}\\nu(B)\\leq\\nu(A)\\leq C\\nu(B).$$\nThen for any $A,B\\in\\mathcal{C}$ such that $A,B\\subset{\\rm Dom}(\\tau)$,\n$$C^{-1}C_{1}^{-2n}\\nu(\\tau(B))\\leq\\nu(\\tau(A))\\leq CC_{1}^{2n}\\nu(\\tau(B)).$$\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n$(a)$ For $x\\in {\\rm Dom}(\\tau)$, let $y=S_{\\mathbf{u}'}^{-1}(x)\\in W$. Then\n$$|\\det\\tau'(x)|=|\\det (S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1})'(x)|=|\\det S'_{\\mathbf{v}'}(S_{\\mathbf{u}'}^{-1}(x))|\\cdot|\\det (S_{\\mathbf{u}'}^{-1})'(x)|=\\frac{|\\det S'_{\\mathbf{v}'}(y)|}{|\\det S'_{\\mathbf{u}'}(y)|}.$$\nHence\n$$\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\leq\n|\\det\\tau'(x)|\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}.$$\nThus,\n$$\\int_{A}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}d\\nu\\leq\n\\int_{A}|\\det\\tau'(x)|d\\nu\\leq\\int_{A}\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}d\\nu,$$\nwhich proves $(a)$ by (\\ref{eq(1.02)}).\n\n$(b)$ Making use of part $(a)$ and (\\ref{eq(1.1)}), we have\n$$\\begin{aligned}\n\\nu(\\tau(A))&\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\n\\leq C\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(B)\\\\\n&\\leq C\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{2n}\n\\nu(\\tau(B))\\quad\\text{(by part $(a)$)}\\\\\n&\\leq CC_{1}^{2n}\\nu(\\tau(B))\\quad\\text{(by (\\ref{eq:r_R_bound}))}.\n\\end{aligned}$$\nOn the other hand,\n$$\\begin{aligned}\n\\nu(\\tau(A))&\\geq\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\n\\geq C^{-1}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(B)\\\\\n&\\geq C^{-1}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{2n}\n\\nu(\\tau(B))\\quad\\text{(by part $(a)$)}\\\\\n&\\geq C^{-1}C_{1}^{-2n}\\nu(\\tau(B))\\quad\\text{(by (\\ref{eq:r_R_bound}))}.\n\\end{aligned}$$\nThis proves $(b)$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem(1.30)} (Bishop-Gromov inequality (see, e.g. \\cite{Anderson_1990,Berger_2003,Bishop-Crittenden}))\\label{lem(1.00)}\nLet $M$ be a complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, and $B_{r}(x)$ be an $r$-ball in $M$. Then\n$$\\nu(B_{r}(x))\\leq c_{n}r^{n},$$\nwhere $c_{n}=\\pi^{\\frac{n}{2}}\/\\Gamma(\\frac{n}{2}+1)$ is the volume of the unit ball in $\\mathbb{R}^{n}$.\n\\end{lem}\n\nThe following proposition generalizes \\cite[Proposition 3.1]{Lau-Ngai-Wang_2009} to manifolds.\n\n\\begin{prop}\\label{prop(1.1)}\nLet $M$ be a complete $n$-dimensional orientable Riemannian manifold with non-negative Ricci curvature. Assume the same hypotheses of Lemma \\ref{lem(1.1)}. Then the following are equivalent:\n\\begin{itemize}\n\\item[$(a)$] $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC);\n\\item[$(b)$] there exists $a>0$ and a nonempty subset $D\\subset W$ such that $\\gamma_{a,D}<\\infty$;\n\\item[$(c)$] for any $a>0$ and any nonempty subset $D\\subset W$, $\\gamma_{a,D}<\\infty$;\n\\item[$(d)$] for any $D\\subset W$ there exists $\\gamma=\\gamma(D)$ (depending only on $D$) such that for any $0<b\\leq1$ and $x\\in W$,\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(D)\\}\\leq\\gamma.$$\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\n$(a)\\Rightarrow(b)$ It suffices to prove that there exists $\\gamma'\\in\\mathbb{N}$ and $D\\subset W$ with $D^{\\circ}\\neq\\emptyset$, such that for any $x\\in W$ and $0<b\\leq1$,\n$$\\#\\{S\\in\\mathcal{A}_{b}:S(D)\\cap B_{b}(x)\\neq\\emptyset\\}\\leq\\gamma'.$$\nTo obtain this equality, let $D$ be as in the definition of WSC, $S\\in\\mathcal{A}_{b}$ such that $S(D)\\cap B_{b}(x)\\neq\\emptyset$, and $x'\\in D$ such that\n$S(x')\\in B_{b}(x)$. Then for any $y\\in D$,\n$$\\begin{aligned}\nd(S(y),x)&\\leq d(S(y),S(x'))+d(S(x'),x)\\quad\\text{(triangle inequality)}\\\\\n&\\leq C_{2}R_{S}d(y,x')+b\\quad\\text{(by (\\ref{eq(1.2)}))}\\\\\n&\\leq (1+C_{2}|D|)b.\n\\end{aligned}$$\nLet $\\eta:=(1+C_{2}|D|)b$. We can rewrite the above as $S(D)\\subset B_{\\eta}(x)$. By $(a)$, each point in $W$ is covered by at most $\\gamma$ of the $S(D)$, where\n$S\\in\\mathcal{A}_{b}$. It follows that\n\\begin{equation}\\label{eq(1.4)}\n\\sum\\{\\nu(S(D)):S\\in\\mathcal{A}_{b},S(D)\\cap B_{b}(x)\\neq\\emptyset\\}\\leq\\gamma\\nu(B_{\\eta}(x)).\n\\end{equation}\nMaking use of Lemma \\ref{lem(1.1)}$(a)$, we have\n$$\\begin{aligned}\n\\bigg(\\frac{br}{C_{1}}\\bigg)^{n}\\nu(D)\\#\\{S\\in\\mathcal{A}_{b}:S(D)\\cap B_{b}(x)\\neq\\emptyset\\}\n&\\leq\\sum\\{\\nu(S(D)):S\\in\\mathcal{A}_{b},S(D)\\cap B_{b}(x)\\neq\\emptyset\\}\\\\\n&\\leq\\gamma\\nu(B_{\\eta}(x))\\quad\\text{(by (\\ref{eq(1.4)}))}\\\\\n&\\leq\\gamma c_{n}\\eta^{n}\\quad\\text{(by Lemma \\ref{lem(1.00)})}\\\\\n&:=\\gamma cb^{n},\n\\end{aligned}$$\nwhere $c:=c_{n}(1+C_{2}|D|)^{n}$.\nHence there exists $\\gamma'\\in\\mathbb{N}$ such that\n$$\\#\\{S\\in\\mathcal{A}_{b}:S(D)\\cap B_{b}(x)\\neq\\emptyset\\}\\leq\\frac{\\gamma c C_{1}^{n}}{r^{n}\\nu(D)}\\leq\\gamma'.$$\n\n\n$(b)\\Rightarrow(c)$ We prove the contrapositive. Assume $(c)$ is false. Then there exist $a_{0}>0$ and a nonempty subset $D_{0}\\subset W$ such that $\\gamma_{a_{0},D_{0}}=\\infty$. Hence there exists a sequence $\\{A_{k}\\}_{k=1}^{\\infty}$ of nonempty sets bounded in $M$ such that\n\\begin{equation}\\label{eq(1.5)}\n\\{S\\in\\mathcal{A}_{a_{0}|A_{k}|}:S(D_{0})\\cap A_{k}\\neq\\emptyset\\}\\geq k.\n\\end{equation}\nTo prove $(b)$ fails, we fix an arbitrary $a>0$ and nonempty subset $D\\subset W$. We will show $\\gamma_{a,D}=\\infty$. Let\n$$s:=\\sup_{x\\in D_{0},y\\in D}d(x,y)<\\infty.$$\nWe first claim that for any $S\\in\\mathcal{A}_{a_{0}|A_{k}|}$ and $\\delta_{k}:=s a_{0} C_{2}|A_{k}|$, $S(D_{0})\\cap A_{k}\\neq\\emptyset$ implies $S(D)\\cap (A_{k})_{\\delta_{k}}\\neq\\emptyset$, where $(A_{k})_{\\delta_{k}}=\\{x\\in M:{\\rm dist}(x,A_{k})\\leq\\delta_{k}\\}$ is the closed $\\delta_{k}$-neighborhood of $A_{k}$.\nTo prove the claim, we let $y\\in S(D_{0})\\cap A_{k}$. Then there exists $x\\in D_{0}$ such that $y=S(x)\\in S(D_{0})$. Now let $\\tilde{x}\\in D$ and $\\tilde{y}:=S(\\tilde{x})\\in S(D)$. It follows from (\\ref{eq(1.2)}) that\n$$d(\\tilde{y},y)=d(S(\\tilde{x}),S(x))\\leq C_{2}R_{S}d(\\tilde{x},x)\\leq s C_{2}R_{S}\\leq s a_{0}|C_{2}A_{k}|=\\delta_{k}.$$\nThis proves the claim. Note that $(A_{k})_{\\delta_{k}}$ is a set of diameter $2\\delta_{k}+|A_{k}|=(2sa_{0} C_{2}+1)|A_{k}|$. Since $M$ is a Riemannian manifold with non-negative Ricci curvature, $M$ has the doubling property. Hence we can cover $(A_{k})_{\\delta_{k}}$ by no more than $\\lambda$ sets of diameter $(a_{0}|A_{k}|)\/a$. Note that\n$\\mathcal{A}_{a_{0}|A_{k}|}=\\mathcal{A}_{(a_{0}|A_{k}|)\/a}$. It follows from (\\ref{eq(1.5)}) and the claim in the above that there exists $A_{k}^{\\ast}\\subset M$ with\n$|A_{k}^{\\ast}|=(a_{0}|A_{k}|)\/a$ such that\n$$\\{S\\in\\mathcal{A}_{a_{0}|A_{k}^{\\ast}|}:S(D_{0})\\cap A_{k}^{\\ast}\\neq\\emptyset\\}\\geq \\frac{k}{\\lambda}.$$\nSince $\\lambda$ is independent of $k$, we conclude that $\\gamma_{a,D}=\\infty$.\n\n\n$(c)\\Rightarrow(d)$ Let $D\\subset W$. Then for any $x\\in W$ and $0<b\\leq1$,\n$$\\begin{aligned}\n\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(D)\\}&\\leq\\#\\{S\\in\\mathcal{A}_{b},S(D)\\cap B_{b\/2}(x)\\neq\\emptyset\\}\\\\\n&=\\#\\mathcal{A}_{1,B_{b\/2}(x),D}\\\\\n&\\leq\\gamma_{1,D}<\\infty.\n\\end{aligned}$$\n\n$(d)\\Rightarrow(a)$ is trivial.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.1)}]  Use of the above lemmas and propositions, as in \\cite[Theorem 3.2]{Lau-Ngai-Wang_2009}; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\nAs an important consequence of Theorem \\ref{thm(0.1)}, we obtain the following estimate for $\\#\\mathcal{A}_{b}$ and a formula for $\\dim_{{\\rm H}}(F)$. Making use of (\\ref{eq(1.2)}), (\\ref{eq(1.9)}) and Proposition \\ref{prop(1.1)}, we obtain the following analogue of \\cite[Corollary 3.3]{Lau-Ngai-Wang_2009}. The proof is similar and is omitted.\n\n\\begin{coro}\\label{coro(1.1)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ and $\\alpha$ be as in Theorem \\ref{thm(0.1)}. Then there exists a constant $C_{3}>0$ such that for any $0<b\\leq1$,\n\\begin{equation}\\label{eq(1.10)}\nC_{3}^{-1}b^{-\\alpha}\\leq\\#\\mathcal{A}_{b}\\leq C_{3}b^{\\alpha}.\n\\end{equation}\nConsequently,\n$$\\alpha=\\dim_{{\\rm H}}(K)=\\dim_{{\\rm B}}(K)=\\dim_{{\\rm P}}(K)=-\\lim_{b\\rightarrow0^{+}}\\frac{\\log\\#\\mathcal{A}_{b}}{\\log b}.$$\n\\end{coro}\n\n\n\n\n\n\\section{Absolute continuity of self-conformal measures}\\label{S:2}\nThroughout this section, we let $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold with non-negative Ricci curvature, $U\\subset M$ be open and connected, and $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$. The proof of the following proposition is similar to that of \\cite[Proposition 4.1]{Lau-Ngai-Wang_2009} and is omitted.\n\n\\begin{prop}\\label{prop(2.1)}\nAssume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (WSC). Then for any finite subset $\\Lambda\\subset\\Sigma^{\\ast}$, the family $\\{S_{\\mathbf{v}}:\\mathbf{v}\\in\\Lambda\\}$ also satisfies (WSC).\n\\end{prop}\n\n\n\nFor $\\mathbf{u}\\in\\Sigma^{\\ast}$, let $[\\mathbf{u}]:=\\{\\mathbf{u}'\\in\\Sigma^{\\ast}:S_{\\mathbf{u}}=S_{\\mathbf{u}'}\\}$. The following lemma can be proved by using Corollary \\ref{coro(1.1)} and the argument in \\cite[Lemma 4.2]{Lau-Ngai-Wang_2009}.\n\n\\begin{lem}\\label{lem(2.1)}\nAssume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (WSC). Let $\\{p_{i}\\}_{i=1}^{N}$ be the associated probability weights, and let $K$ be the attractor\nwith $\\dim_{{\\rm H}}(K)=\\alpha$. For $0<b\\leq1$ and $\\Lambda\\subset\\mathcal{W}_{b}$, let\n$$\\widetilde{\\Lambda}:=\\bigg\\{\\mathbf{u}\\in\\Lambda:\\sum_{\\mathbf{u}'\\in[\\mathbf{u}]\n\\cap\\Lambda}p_{\\mathbf{u}'}>\\frac{b^{\\alpha}}{4C_{3}}\\bigg\\},$$\nwhere $C_{3}$ is as in Corollary \\ref{coro(1.1)}. Then $P(\\Lambda)>1\/2$ implies $P(\\widetilde{\\Lambda})>1\/4$.\n\\end{lem}\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm(0.2)}] The proof of Theorem \\ref{thm(0.2)} follows by using Proposition \\ref{prop(1.1)}, Lemma \\ref{lem(2.1)}, and a technique in the proofs of \\cite[Theorem 1.1]{Lau-Ngai-Rao_2001} and \\cite[Theorem 1.1]{Lau-Wang_2004}; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe first give the definition of Vitali relation (see \\cite[Definition 2.8.16]{Federer_1969}). Let $X$ be a metric space.\nAny subset of\n$$\n\\{(x,A): x\\in A\\subset X\\}\n$$\nis called a \\textit{covering relation}. Let $\\mathbf{C}$ be a covering relation and $Z\\subset  X$, we let\n$$\n\\mathbf{C}(Z):=\\{A\\subset X:(x,A)\\in \\mathbf{C}\\text{ for some }x\\in Z\\}.\n$$\nWe say $\\mathbf{C}$ is \\textit{fine at $x$} if\n$$\n\\inf\\{|A|:(x,A)\\in \\mathbf{C}\\}=0.\n$$\nWe say $\\mathbf{C}(Z)$ is \\textit{fine on $Z$} if for any $x\\in Z$, $\\mathbf{C}(x)$ is fine at $x$. Let $\\phi$ be a regular Borel measure on $X$. A covering relation $\\mathbf{V}$ is called a\n$\\phi$ \\textit{Vitali relation} if $\\mathbf{V}(X)$ is a family of Borel sets, $\\mathbf{V}$ is fine on $X$, and moreover, whenever $\\mathbf{C}\\subset \\mathbf{V}, Z\\subset X$, and $\\mathbf{C}$ is fine on $Z$, then $\\mathbf{C}(Z)$ has a countable disjoint subfamily covering $\\phi$ almost all of $Z$.\n\nFor an $n$-dimension Riemannian manifold  $M$, by making use of \\cite[Definition 2.8.9 and Theorem 2.8.18]{Federer_1969}, we see that $\\{B_{r}(x):x\\in M,0<r<\\infty\\}$ is a $\\phi$ Vitali relation in $M$. Let $f$ be the Radon-Nikodym derivative of $\\mu$ with respect to $\\phi$. By \\cite[Theorem 2.9.8]{Federer_1969} or \\cite[ Lemma 2.1(i)]{Bedferd-Fisher_1992}, we have\n$$\\lim_{r\\rightarrow0}\\frac{1}{\\phi(B_{r}(x))}\\int_{B_{r}(x)}f d\\phi=\\lim_{r\\rightarrow0}\\frac{\\mu(B_{r}(x))}{\\phi(B_{r}(x))}=f(x)$$\nfor $\\phi$ almost all $x\\in K$. Assume that $A\\subset K$, and $f=\\chi_{A}$. It follows that\n\\begin{equation}\\label{eq(2.4)}\n\\lim_{r\\rightarrow0}\\frac{\\phi(A\\cap B_{r}(x))}{\\phi(B_{r}(x))}\n=\\lim_{r\\rightarrow0}\\frac{1}{\\phi(B_{r}(x))}\\int_{B_{r}(x)}\\chi_{A} d\\phi=1\n\\end{equation}\nfor $\\phi$ almost all $x\\in A$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.3)}]\nLet $\\phi:=\\mathcal{H}^{\\alpha}|_{K}$ and $f$ be the Radon-Nikodym derivative of $\\mu$ with respect to $\\phi$. Suppose that $f$ is unbounded. For any $Q>0$, let $A=A(Q):=\\{t\\in K:f(t)>Q\\}$. Then for any $\\delta>0$, by (\\ref{eq(2.4)}), there exist $x\\in K$ and $b>0$ such that\n\\begin{equation}\\label{eq(2.5)}\n\\phi\\big(A\\cap B_{b\\delta}(x)\\big)>\\frac{1}{2}\\phi(B_{b\\delta}(x)).\n\\end{equation}\nThen\n\\begin{equation}\\label{eq(2.6)}\\begin{aligned}\n\\mu(B_{b\\delta}(x))&=\\int_{B_{b\\delta}(x)}f(t)d\\phi(t)\\\\\n&\\geq \\int_{A\\cap B_{b\\delta}(x)}f(t)d\\phi(t)\\\\\n&\\geq Q\\phi(A\\cap B_{b\\delta}(x))\\\\\n&>\\frac{1}{2}Q\\phi(B_{b\\delta}(x))\\quad\\text{(by (\\ref{eq(2.5)}))}.\n\\end{aligned}\\end{equation}\nLet $\\delta:=C_{2}|K|$ in the above inequality. Note that $x\\in K=\\bigcup_{S\\in\\mathcal{A}_{b}}S(K)$, and hence there exists $S\\in\\mathcal{A}_{b}$ such that $x\\in S(K)$.\nMaking use of (\\ref{eq(1.2)}), we have\n$$|S(K)|\\leq C_{2}R_{S}|K|\\leq b\\delta,$$\nwhich implies that $S(K)\\subset B_{b\\delta}(x)$. For $S=S_{u_{1}\\cdots u_{k}}\\in\\mathcal{A}_{b}$, we have $R_{S}=R_{\\mathbf{u}}\\leq b<R_{\\mathbf{u}^{-}}$. Hence by (\\ref{eq(1.1)}){\\color{blue},}\n\\begin{equation}\\label{eq(2.7)}\nbr<R_{u_{1}\\cdots u_{k-1}}r_{u_{k}}\\leq C_{1}r_{u_{1}\\cdots u_{k-1}}r_{u_{k}}\\leq C_{1}r_{\\mathbf{u}}\\leq C_{1}R_{\\mathbf{u}}.\n\\end{equation}\nCombining these derivations with (\\ref{eq(1.2)}) and (\\ref{eq(2.7)}), we have\n\\begin{equation}\\label{eq(2.8)}\n\\phi(B_{b\\delta}(x))\\ge\\phi(S(K))=\\mathcal{H}^{\\alpha}(S(K))\\geq C_{2}^{-\\alpha}R_{S}^{\\alpha}\\mathcal{H}^{\\alpha}(K)>\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}b^{\\alpha}\\phi(K).\n\\end{equation}\nCombining \\eqref{eq(2.6)} and \\eqref{eq(2.8)}, we obtain\n\\begin{equation}\\label{eq(2.9)}\n\\mu(B_{b\\delta}(x))>\\frac{1}{2}Q\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}b^{\\alpha}\\phi(K).\n\\end{equation}\nOn the other hand,\n$$\\begin{aligned}\n\\mu(B_{b\\delta}(x))&=\\sum_{S\\in\\mathcal{A}_{b},S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset}p_{S}\\mu\\circ S^{-1}(B_{b\\delta}(x))\\\\\n&\\leq\\sum_{S\\in\\mathcal{A}_{b},S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset}p_{S}.\n\\end{aligned}$$\nLet $S\\in\\mathcal{A}_{b}$ such that $p_S$ is the maximum among all the summands in the above summation. Then\n\\begin{equation}\\label{eq(2.10)}\n\\mu(B_{b\\delta}(x))\\leq p_{S}\\#\\{S\\in\\mathcal{A}_{b}:S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset\\}\\leq p_{S}\\gamma_{\\frac{1}{2\\delta},K}.\n\\end{equation}\nWe choose $Q$ such that\n\\begin{equation}\\label{eq(2.11)}\n\\frac{1}{2}Q\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}\\phi(K)>\\gamma_{\\frac{1}{2\\delta},K}.\n\\end{equation}\nIt follows from \\eqref{eq(2.9)}--\\eqref{eq(2.11)} that there exists $S\\in\\mathcal{A}_{b}$ such that\n$$b^{\\alpha}\\gamma_{\\frac{1}{2\\delta},K}<\\mu(B_{b\\delta}(x))\\leq p_{S}\\gamma_{\\frac{1}{2\\delta},K}.$$\nHence\n$$R_{S}^{\\alpha}\\leq b^{\\alpha}<p_{S}.$$\nIt now follows from Theorem \\ref{thm(0.2)} that $\\mu$ is singular with respect to $\\phi$, a contradiction. This proves the theorem.\n\\end{proof}\n\n\n\\section{The finite type condition}\\label{S:3}\n\n\nIn this section we extend (FTC) to Riemannian manifolds and prove Theorem~\\ref{thm(0.4)}. We follow the set-up in \\cite{Lau-Ngai_2007, Lau-Ngai-Wang_2009}. We begin by defining a \\textit{partial order} $\\preceq$ on $\\Sigma^{\\ast}$. For $\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}$, we denote by $\\mathbf{u}\\preceq\\mathbf{v}$ if $\\mathbf{u}$ is an initial segment of $\\mathbf{v}$ or $\\mathbf{u}=\\mathbf{v}$. We denote by $\\mathbf{u}\\npreceq\\mathbf{v}$ if $\\mathbf{u}\\preceq\\mathbf{v}$ does not hold. Let\n$\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ be a sequence of index sets such that for any $k\\geq0$, $\\mathcal{M}_{k}$ is a finite subset of $\\Sigma^{\\ast}$.\nWe say that $\\mathcal{M}_{k}$ is an \\textit{antichain} if for any $\\mathbf{u},\\mathbf{v}\\in\\mathcal{M}_{k}$, $\\mathbf{u}\\npreceq\\mathbf{v}$ and $\\mathbf{v}\\npreceq\\mathbf{u}$. Let\n$$\\underline{m}_{k}=\\underline{m}_{k}(\\mathcal{M}_{k}):=\\min\\{|\\mathbf{u}|:\\mathbf{u}\\in\\mathcal{M}_{k}\\},\n\\quad \\overline{m}_{k}=\\overline{m}_{k}(\\mathcal{M}_{k}):=\\max\\{|\\mathbf{u}|:\\mathbf{u}\\in\\mathcal{M}_{k}\\},$$\nwhere $|\\mathbf{u}|$ is the length of $\\mathbf{u}$.\n\n\\begin{defi}\\label{defi(3.1)}\nWe say that $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ is a \\textit{sequence of nested index sets} if it satisfies the following conditions:\n\\begin{itemize}\n\\item[$(a)$] both $\\{\\underline{m}_{k}\\}$ and $\\{\\overline{m}_{k}\\}$ are nondecreasing, and\n$\\displaystyle{\\lim_{k\\rightarrow\\infty}\\underline{m}_{k}=\\lim_{k\\rightarrow\\infty}\\overline{m}_{k}}=\\infty$;\n\\item[$(b)$] for any $k\\geq0$, $\\mathcal{M}_{k}$ is an antichain in $\\Sigma^{\\ast}$;\n\\item[$(c)$] for any $\\mathbf{v}\\in\\Sigma^{\\ast}$ with $|\\mathbf{v}|>\\overline{m}_{k}$, there exists $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\mathbf{u}\\preceq\\mathbf{v}$;\n\\item[$(d)$] for any $\\mathbf{v}\\in\\Sigma^{\\ast}$ with $|\\mathbf{v}|<\\underline{m}_{k}$, there exists $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\mathbf{v}\\preceq\\mathbf{u}$;\n\\item[$(e)$] there exists a positive integer $L$, independent of $k$, such that for any $\\mathbf{u}\\in\\mathcal{M}_{k}$ and $\\mathbf{v}\\in\\mathcal{M}_{k+1}$ with\n$\\mathbf{u}\\preceq\\mathbf{v}$, we have $|\\mathbf{v}|-|\\mathbf{u}|\\leq L$.\n\\end{itemize}\n\\end{defi}\n\nNote that $\\mathcal{M}_{k}$ can intersect $\\mathcal{M}_{k+1}$, and $\\{\\Sigma^{k}\\}_{k=0}^{\\infty}$ is an example of a sequence of nested index sets.\nFor each integer $k\\geq0$, let $\\mathcal{V}_{k}$ be the set of \\textit{$k$-th level vertices} defined as\n$$\\mathcal{V}_{0}:=\\{(\\mathbf{u},0)\\}\\quad\\text{and}\\quad\\mathcal{V}_{k}\n:=\\{(S_{\\mathbf{u}},k):\\mathbf{u}\\in\\mathcal{M}_{k}\\}\\text{ for any }k\\geq1.$$\nWe write $\\omega_{{\\rm root}}:=(\\mathbf{u},0)$ and call it the \\textit{root vertex}. Let $\\mathcal{V}:=\\bigcup_{k\\geq0}\\mathcal{V}_{k}$ be the set of all vertices. For\n$\\omega=(S_{\\mathbf{u}},k)$, we define $S_{\\omega}:=S_{\\mathbf{u}}$. Let $W\\subset M$ be a compact set. For an IFS $\\{S_{i}\\}_{i=1}^{N}$ on $W$, let $\\Omega\\subset W$ be a nonempty open set that is invariant under $\\{S_{i}\\}_{i=1}^{N}$. Such a set exists if $\\{S_{i}\\}_{i=1}^{N}$ are contractions on $W$. We say that two $k$-th level vertices $\\omega,\\omega'\\in\\mathcal{V}_{k}$ are \\textit{neighbors} if $S_{\\omega}(\\Omega)\\cap S_{\\omega'}(\\Omega)\\neq\\emptyset$. Let\n$$\\Omega(\\omega):=\\{\\omega':\\omega'\\in\\mathcal{V}_{k}\\text{ is a neighbor of }\\omega\\},$$\nwhich is called the \\textit{neighborhood} of $\\omega$.\n\n\\begin{defi}\\label{defi(3.2)}\nFor any two vertices $\\omega\\in\\mathcal{V}_{k}$ and $\\omega'\\in\\mathcal{V}_{k'}$, let\n$$\\tau:=S_{\\omega'}S_{\\omega}^{-1}:\\bigcup_{\\sigma\\in\\Omega(\\omega)}S_{\\sigma}(W)\\longrightarrow W.$$\nWe say $\\omega$ and $\\omega'$ are \\textit{equivalent}, i.e., $\\omega\\sim\\omega'$, if the following conditions hold\n\\begin{itemize}\n\\item[$(a)$] $\\{S_{\\sigma'}:\\sigma'\\in\\Omega(\\omega')\\}=\\{\\tau S_{\\sigma}:\\sigma\\in\\Omega(\\omega)\\}$;\n\\item[$(b)$] for $\\sigma\\in\\Omega(\\omega)$ and $\\sigma'\\in\\Omega(\\omega')$ such that $S_{\\sigma'}=\\tau S_{\\sigma}$, and for any positive integer $k_{0}\\geq1$,\n$\\mathbf{u}\\in\\Sigma^{\\ast}$, $\\mathbf{u}$ satisfies $(S_{\\sigma}\\circ S_{\\mathbf{u}},k+k_{0})\\in\\mathcal{V}_{k+k_{0}}$ if and only if it satisfies\n$(S_{\\sigma'}\\circ S_{\\mathbf{u}},k'+k_{0})\\in\\mathcal{V}_{k'+k_{0}}$.\n\\end{itemize}\n\\end{defi}\nIt is easy to see that $\\sim$ is an equivalence relation. Denote the equivalent class of $\\omega$ by $[\\omega]$, and call it the \\textit{neighborhood types} of $\\omega$.\n\\par Let $\\omega=(S_{\\mathbf{u}},k)\\in\\mathcal{V}_{k}$ and $\\sigma=(S_{\\mathbf{v}},k+1)\\in\\mathcal{V}_{k+1}$. Suppose that there exists $\\mathbf{w}\\in\\Sigma^{\\ast}$ such that\n$$\\mathbf{v}=(\\mathbf{u},\\mathbf{w}).$$\nThen we connect a \\textit{directed edge} from $\\omega$ to $\\sigma$, and denote this as $\\omega\\stackrel{\\mathbf{w}}{\\longrightarrow}\\sigma$. We call $\\omega$ a \\textit{parent} of $\\sigma$ and $\\sigma$ an \\textit{offspring} of $\\omega$. Define a graph $\\mathcal{G}:=(\\mathcal{V},\\mathcal{E})$, where $\\mathcal{E}$ is the set of all directed edges. We first remove from $\\mathcal{G}$ all but the smallest (in the lexicographic order) directed edges going to a vertex. After that, we remove all vertices that do not have any offspring, together with all vertices and edges leading only to them. The resulting graph is called the \\textit{reduced graph}. Denote it by $\\mathcal{G}_{R}:=(\\mathcal{V}_{R},\\mathcal{E}_{R})$, where $\\mathcal{V}_{R}$ and $\\mathcal{E}_{R}$ are the sets of all vertices and all edges, respectively.\n\nThe proof of the following proposition is similar to that of \\cite[Proposition 2.4]{Lau-Ngai_2007}; we omit the details.\n\n\\begin{prop}\\label{prop(3.1)}\nLet $\\omega$ and $\\omega'$ be two vertices in $\\mathcal{V}$ with offspring $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{m}$ and $\\mathbf{u}'_{1},\\dots,\\mathbf{u}'_{s}$ in $\\mathcal{G}_{R}$, respectively. Suppose $[\\omega]=[\\omega']$. Then\n$$\\big\\{[\\mathbf{u}_{i}]:1\\leq i\\leq m\\big\\}=\\big\\{[\\mathbf{u}'_{i}]:1\\leq i\\leq s\\big\\}$$\ncounting multiplicity. In particular, $m=s$.\n\\end{prop}\n\n\\begin{defi}\\label{defi(3.3)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be an IFS on $W$ consisting of injective contractions, and let $\\mathcal{V}\/_{\\sim}:=\\{[\\omega]:\\omega\\in\\mathcal{V}\\}$. We say that $\\{S_{i}\\}_{i=1}^{N}$ satisfies the \\textit{finite type condition} (FTC) if there exists a nonempty invariant open set $\\Omega\\subset W$ with respect to some sequence of nested index sets $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ and such that\n$$\\#\\mathcal{V}\/_{\\sim}<\\infty.$$\nSuch a set $\\Omega$ is called \\textit{a finite type condition set (FTC set)}.\n\\end{defi}\n\nObviously, if $\\{S_{i}\\}_{i=1}^{N}$ satisfies (OSC), then $\\#\\mathcal{V}\/_{\\sim}=1$, and thus $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC). We assume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS in the rest of this section.\n\n\\begin{lem}\\label{lem(3.1)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be a CIFS on a compact subset $W\\subset M$. Assume that $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC) with $\\Omega\\subset W$ being an FTC set. Then there\nexists a constant $C_{4}\\geq1$ such that for any two neighboring vertices $\\omega_{1}$ and $\\omega_{2}$, we have\n$$C_{4}^{-1}\\leq\\frac{\\nu(S_{\\omega_{1}}(\\Omega))}{\\nu(S_{\\omega_{2}}(\\Omega))}\\leq C_{4}.$$\n\\end{lem}\n\\begin{proof}\nLet $\\mathcal{T}$ be a neighborhood type, and $\\omega$ be a vertex such that $[\\omega]=\\mathcal{T}$. Let\n$$\\Omega(\\omega)=\\{\\omega_{0},\\omega_{1},\\dots,\\omega_{m}\\},$$\nwhere $\\omega_{0}=\\omega$. Substituting $S_{\\omega_{0}}(\\Omega)=A$ and $S_{\\omega_{i}}S_{\\omega_{0}}^{-1}=\\tau$ into Lemma \\ref{lem(1.3)}$(a)$ and using \\eqref{eq:r_R_bound}, we see that there exists a constant $c_{1}\\geq1$ such that for any $i\\in\\{0,1,\\dots,m\\}$,\n\\begin{equation}\\label{eq(3.1)}\nc_{1}^{-1}\\nu(S_{\\omega_{0}}(\\Omega))\\leq\\nu(S_{\\omega_{i}}(\\Omega))\\leq c_{1}\\nu(S_{\\omega_{0}}(\\Omega)).\n\\end{equation}\nLet $\\omega\\sim\\omega'$, $\\tau=S_{\\omega'}S_{\\omega}^{-1}\\in \\mathcal{F}$, and\n$$\\Omega(\\omega')=\\{\\omega'_{0},\\omega'_{1},\\dots,\\omega'_{m}\\},$$\nwhere $\\omega'_{0}=\\omega'$. Without loss of generality, for any $i\\in\\{0,1,\\dots,m\\}$ we can assume $S_{\\omega'_{i}}=\\tau S_{\\omega_{i}}$. It follows from the definition of $\\tau$ that\n$S_{\\omega_{i}}(\\Omega)\\subset{\\rm Dom}(\\tau)$. Making use of (\\ref{eq(3.1)}) and substituting $S_{\\omega_{i}}(\\Omega)=A$, $S_{\\omega_{0}}(\\Omega)=B$ and $S_{\\omega'_{i}}S_{\\omega_{i}}^{-1}=\\tau$ into Lemma \\ref{lem(1.3)}$(b)$, we see that for any $i\\in\\{0,1,\\dots,m\\}$,\n$$c_{1}^{-1}C_{1}^{-2n}\\nu(S_{\\omega'_{0}}(\\Omega))\\leq\\nu(S_{\\omega'_{i}}(\\Omega))=\\nu(\\tau S_{\\omega_{i}}(\\Omega))\\leq c_{1}C_{1}^{2n}\\nu(S_{\\omega'_{0}}(\\Omega)).$$\nHence the lemma holds for any two neighboring vertices $\\omega_{1},\\omega_{2}$ with one of them being of type $\\mathcal{T}$. Since there are only finitely\nmany distinct neighborhood types, the result follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem(3.2)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be a CIFS on a compact subset $W\\subset M$. Then for any $\\mathbf{u}\\in\\Sigma^{k}$ and $\\Omega\\subset W$, we have\n\\begin{equation}\\label{eq(3.01)}\nr^{kn}\\leq\\frac{\\nu(S_{\\mathbf{u}}(\\Omega))}{\\nu(\\Omega)}\\leq R^{kn}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nLet $x\\in W$. Then by the definition of $R$, we have\n$$\\begin{aligned}\n|\\det S'_{\\mathbf{u}}(x)|&=|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\cdot|S'_{u_{k}}(x)|\\\\\n&\\leq R_{\\mathbf{u}^{-}}^{n}R_{u_{k}}^{n}\\\\\n&\\leq R_{u_{1}}^{n}\\cdots R_{u_{k}}^{n}\\\\\n&\\leq R^{kn}.\n\\end{aligned}$$\nFor any set $\\Omega\\subset W$, making use of (\\ref{eq(1.02)}), we have\n$$\\nu(S_{\\mathbf{u}}(\\Omega))=\\int_{\\Omega}|\\det S'_{\\mathbf{u}}(x)|d\\nu(x)\\leq R^{kn}\\nu(\\Omega).$$\nThis proves the right side of (\\ref{eq(3.01)}). On the other hand, if $x\\in W$, then by the definition of $r$, we have\n$$\\begin{aligned}\n|\\det S'_{\\mathbf{u}}(x)|&=|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\cdot|S'_{u_{k}}(x)|\\\\\n&\\geq r_{\\mathbf{u}^{-}}^{n}r_{u_{k}}^{n}\\\\\n&\\geq r_{u_{1}}^{n}\\cdots r_{u_{k}}^{n}\\\\\n&\\geq r^{kn}.\n\\end{aligned}$$\nConsequently,\n$$\\nu(S_{\\mathbf{u}}(\\Omega))=\\int_{\\Omega}|\\det S'_{\\mathbf{u}}(x)|d\\nu(x)\\geq r^{kn}\\nu(\\Omega).$$\nThis proves the left side of (\\ref{eq(3.01)}).\n\\end{proof}\n\n\nWe now prove Theorem \\ref{thm(0.4)}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.4)}]\nFor $0<b\\leq1$, $x\\in W$, let\n$\\mathcal{S}(\\Omega):=\\{S\\in\\mathcal{A}_{b}:x\\in S(\\Omega)\\}.$\nList all elements of $\\mathcal{S}(\\Omega)$ as $S_{\\mathbf{u}_{1}},\\dots,S_{\\mathbf{u}_{m}}$. For any $j\\in\\{1,\\dots,m\\}$, note that $\\mathbf{u}_{j}\\in\\mathcal{M}_{k}$ for some $k$. Let $\\mathbf{u}_{j}=(\\widetilde{\\mathbf{u}}_{j},\\widetilde{\\mathbf{v}}_{j})$, where\n$\\widetilde{\\mathbf{u}}_{j}\\in\\mathcal{M}_{k_{j}}$ is the longest initial segment of $\\mathbf{u}_{j}$. Without\nloss of generality, we assume\n$$k_{1}=\\min\\{k_{j}:1\\leq j\\leq m\\}=k.$$\nThen $\\widetilde{\\mathbf{u}}_{1}\\in\\mathcal{M}_{k}$. For any $j\\in\\{1,\\dots,m\\}$, let $\\mathbf{u}'_{j}$ be the initial segment of $\\mathbf{u}_{j}$ such that\n$\\mathbf{u}'_{j}\\in\\mathcal{M}_{k}$. Note that $\\mathbf{u}'_{1}=\\widetilde{\\mathbf{u}}_{1}$. Since $x\\in S(\\Omega)$ for any $S\\in\\mathcal{S}_{\\Omega}$, we have\n$$\\omega_{2}=(S_{\\mathbf{u}'_{2}},k),\\dots,\\omega_{m}=(S_{\\mathbf{u}'_{m}},k)$$\nare neighbors of $\\omega_{1}=(S_{\\mathbf{u}'_{1}},k)$. It follows from the definition of (FTC) that there exists a positive integer $c_{2}$ independent of $x,b$ such that\n\\begin{equation}\\label{eq(3.2)}\n\\#\\{\\omega_{1},\\dots,\\omega_{m}\\}\\leq c_{2}.\n\\end{equation}\nBy Lemma \\ref{lem(3.1)}, for any $j\\in\\{2,\\dots,m\\}$, we have\n\\begin{equation}\\label{eq(3.3)}\nC_{4}^{-1}\\leq\\frac{\\nu(S_{\\mathbf{u}'_{1}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{j}}(\\Omega))}\\leq C_{4}.\n\\end{equation}\nFor any $j\\in\\{2,\\dots,m\\}$, making use of Lemma \\ref{lem(1.1)}$(b)$,  we have\n\\begin{equation}\\label{eq(3.4)}\n\\bigg(\\frac{r}{C_{1}}\\bigg)^{n}\\leq\\frac{\\nu(S_{\\mathbf{u}_{j}}(\\Omega))}{\\nu(S_{\\mathbf{u}_{1}}(\\Omega))}\\leq \\bigg(\\frac{C_{1}}{r}\\bigg)^{n}.\n\\end{equation}\nIt follows from (\\ref{eq(3.3)}) and (\\ref{eq(3.4)}) that\n\\begin{equation}\\label{eq(3.5)}\nC_{4}^{-1}\\bigg(\\frac{r}{C_{1}}\\bigg)^{n}\\leq\n\\frac{\\nu(S_{\\mathbf{u}_{j}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{j}}(\\Omega))}\n\\cdot\\frac{\\nu(S_{\\mathbf{u}'_{1}}(\\Omega))}{\\nu(S_{\\mathbf{u}_{1}}(\\Omega))}\n\\leq C_{4}\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}.\n\\end{equation}\nFor each $j\\in\\{1,\\dots,m\\}$, write $\\mathbf{u}_{j}=\\mathbf{u}'_{j}\\mathbf{v}_{j}$. Let $S_{\\mathbf{I}}$ be the identity map, and $\\tau=S_{\\mathbf{u}'_{j}}S_{\\mathbf{I}}^{-1}$. Then\n\\begin{equation}\\label{eq(3.6)}\n\\begin{aligned}\n\\frac{\\nu(S_{\\mathbf{u}_{j}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{j}}(\\Omega))}\n&=\\frac{\\nu(\\tau S_{\\mathbf{v}_{j}}(\\Omega))}{\\nu(\\tau(\\Omega))}\\\\\n&\\leq\\frac{R_{\\mathbf{u}'_{j}}^{n}\\nu(S_{\\mathbf{v}_{j}}(\\Omega))}{r_{\\mathbf{u}'_{j}}^{n}\\nu(\\Omega)}\n\\quad\\text{(by Lemma \\ref{lem(1.3)}$(a)$)}\\\\\n&\\leq C_{1}^{n}\\frac{\\nu(S_{\\mathbf{v}_{j}}(\\Omega))}{\\nu(\\Omega)}\\quad\\text{(by \\eqref{eq:r_R_bound})}.\n\\end{aligned}\n\\end{equation}\nIt follows that for any $j\\in\\{1,\\dots,m\\}$,\n\\begin{equation}\\label{eq(3.7)}\n\\begin{aligned}\n\\frac{\\nu(S_{\\mathbf{u}_{1}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{1}}(\\Omega))}\n&\\leq C_{4}\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\frac{\\nu(S_{\\mathbf{u}_{j}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{j}}(\\Omega))}\n\\quad\\text{(by (\\ref{eq(3.5)}))}\\\\\n&\\leq C_{4}\\bigg(\\frac{C_{1}^{2}}{r}\\bigg)^{n}\\frac{\\nu(S_{\\mathbf{v}_{j}}(\\Omega))}{\\nu(\\Omega)}\\quad\\text{(by (\\ref{eq(3.6)}))}\\\\\n&\\leq C_{4}\\bigg(\\frac{C_{1}^{2}}{r}\\bigg)^{n}R^{|\\mathbf{v}_{j}|n}\\quad\\text{(by (\\ref{eq(3.01)}))}.\n\\end{aligned}\n\\end{equation}\nOn the other hand, similar to (\\ref{eq(3.7)}) and by (\\ref{eq(3.01)}), we have\n\\begin{equation}\\label{eq(3.8)}\n\\frac{\\nu(S_{\\mathbf{u}_{1}}(\\Omega))}{\\nu(S_{\\mathbf{u}'_{1}}(\\Omega))}\\geq C_{1}^{-n}\\frac{\\nu(S_{\\mathbf{v}_{1}}(\\Omega))}{\\nu(\\Omega)}\n\\geq C_{1}^{-n}r^{|\\mathbf{v}_{j}|n}.\n\\end{equation}\nRecall that $\\mathbf{u}'_{1}\\in\\mathcal{M}_{k}$ is the longest initial segment of $\\mathbf{u}_{1}=\\mathbf{u}'_{1}\\mathbf{v}_{1}$. In view of Definition \\ref{defi(3.1)}$(c)$, there exists $\\mathbf{v}'_{1}\\in\\Sigma^{\\ast}$ such that $\\mathbf{u}'_{1}\\mathbf{v}_{1}=\\mathbf{u}'_{1}\\mathbf{v}_{1}\\mathbf{v}'_{1}\\in\\mathcal{M}_{k+1}$. By Definition \\ref{defi(3.1)}$(e)$, we have\n\\begin{equation}\\label{eq(3.9)}\n|\\mathbf{v}_{1}|\\leq|\\mathbf{v}_{1}|+|\\mathbf{v}'_{1}|\\leq L.\n\\end{equation}\nCombining (\\ref{eq(3.7)}), (\\ref{eq(3.8)}) and (\\ref{eq(3.9)}) show that there exists a constant $c_{3}>0$ such that for any $j\\in\\{1,\\dots,m\\}$,\n\\begin{equation}\\label{eq(3.10)}\nc_{3}\\leq R^{|\\mathbf{v}_{j}|}.\n\\end{equation}\nIn particular, we can take $c_{3}=r^{L+1}\/(C_{1}^{3}C_{4}^{1\/n})$. Let $c_{4}:=\\lfloor\\log c_{3}\/\\log R\\rfloor+1$. Then $|\\mathbf{v}_{j}|\\leq c_{4}$. Combining these and (\\ref{eq(3.2)}) yields\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(\\Omega)\\}\\leq c_{2}N^{c_{4}},$$\nwhich implies that $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC).\n\\end{proof}\n\n\n\\section{Hausdorff dimension of self-similar sets}\\label{S:4}\n\nIn this section, we assume that $M$ is a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean, i.e., every point of $M$ has a neighborhood which is isometric to an open subset of a Euclidean space.\nLet $W\\subset M$ be compact, and let $\\{S_{i}\\}_{i=1}^{N}$ be an IFS satisfying (FTC) of contractive similitudes on some open sets $\\Omega\\subset W$ with attractor $K\\subset W$. Recall that $\\rho_i$ denotes the contraction ratio of $S_{i}$. We define\n$$\\rho:=\\min\\{\\rho_{i}:1\\leq i\\leq N\\}, \\quad \\rho_{\\max}:=\\max\\{\\rho_{i}:1\\leq i\\leq N\\}.$$\nDenote the neighborhood types of $\\{S_{i}\\}_{i=1}^{N}$ by $\\{\\mathcal{T}_{1},\\dots,\\mathcal{T}_{q}\\}$. Fix a vertex $\\omega\\in\\mathcal{V}_{R}$ such that $[\\omega]\\in\\mathcal{T}_{i}$, where $i\\in\\{1,\\dots,q\\}$. Let $\\sigma_{1},\\dots,\\sigma_{m}$ be the offspring of $\\omega$ in $\\mathcal{G}_{R}$, and let $\\mathbf{w}_{k}$ be the unique edge in $\\mathcal{G}_{R}$ connecting $\\omega$ to $\\sigma_{k}$ for $1\\leq k\\leq m$. Define a \\textit{weighted incidence matrix} $A_{\\alpha}=(A_{\\alpha}(i,j))_{i,j=1}^{q}$ as\n$$A_{\\alpha}(i,j):=\\sum_{k=1}^{m}\\{\\rho_{\\mathbf{w}_{k}}^{\\alpha}:\n\\omega\\stackrel{\\mathbf{w}_{k}}{\\longrightarrow}\\sigma_{k},[\\sigma_{k}]=\\mathcal{T}_{j}\\}.$$\nWe remark that the definition of $A_{\\alpha}$ is independent of the choice of $\\omega$ above. We denote by $\\omega\\rightarrow_{R}\\sigma$ if $\\omega,\\sigma\\in\\mathcal{V}_{R}$ and $\\sigma$ is an offspring of $\\omega$ in $\\mathcal{G}_{R}$. We define an (infinite) \\textit{path} in $\\mathcal{G}_{R}$ to be an infinite sequence $(\\omega_{0},\\omega_{1},\\dots)$ such that for any $k\\geq0$,\n$$\\omega_{k}\\in\\mathcal{V}_{k}\\quad\\text{and}\\quad\\omega_{k}\\rightarrow_{R}\\omega_{k+1},$$\nwhere $\\omega_{0}=\\omega_{{\\rm root}}$. Let $\\mathbb{P}$ be the set of all paths in $\\mathcal{G}_{R}$. If the vertices $\\omega_{0}=\\omega_{{\\rm root}},\\omega_{1},\\dots,\\omega_{k}$ are such that\n$$\\omega_{j}\\rightarrow_{R}\\omega_{j+1}\\text{ for }1\\leq j\\leq k-1,$$\nthen we call the set\n$$I_{\\omega_{0},\\omega_{1},\\dots,\\omega_{k}}=\\{(\\sigma_{0},\\sigma_{1},\\dots)\\in\\mathbb{P}:\\sigma_{j}=\\omega_{j}\n\\text{ for any }0\\leq j\\leq k\\}$$\na \\textit{cylinder}. Since the path from $\\omega_{0}$ to $\\omega_{k}$ is unique in $\\mathcal{G}_{R}$, we {\\color{blue}}let\n$$I_{\\omega_{k}}:=I_{\\omega_{0},\\omega_{1},\\dots,\\omega_{k}}.$$\nFor any cylinder $I_{\\omega_{k}}$, where $\\omega_{k}\\in\\mathcal{V}_{k}$ and $[\\omega_{k}]=\\mathcal{T}_{i}$, let\n$$\\hat{\\mu}(\\omega_{{\\rm root}})=a_{1}=1\\quad\\text{and}\\quad\\hat{\\mu}(\\omega_{k})=\\rho_{\\omega_{k}}^{\\alpha}a_{i},$$\nwhere $[a_{1},\\dots,a_{q}]^{T}$ is a $1$-eigenvector of $A_{\\alpha}$, normalized so that $a_{1}=1$. We will show that $\\hat{\\mu}$ is a measure on $\\mathbb{P}$ in the following. Note that for two cylinders $I_{\\omega}$ and $I_{\\omega}'$ with $\\omega\\in\\mathcal{V}_{k},\\omega'\\in\\mathcal{V}_{\\ell}$ and $k\\leq\\ell$, $I_{\\omega}\\cap I_{\\omega}'\\neq\\emptyset$ iff either $\\omega'=\\omega$ in the case $k=\\ell$ or $\\omega'$ is a descendant of $\\omega$ in the case $k<\\ell$. Whatever, $I_{\\omega}'\\subset I_{\\omega}$. Let $\\omega\\in\\mathcal{V}_{R}$ and $\\mathcal{D}:=\\{\\sigma_{k}\\}_{k=1}^{m}$ denote the set of all offspring of $\\omega$ in $\\mathcal{G}_{R}$. For $k\\in\\{1,\\dots,m\\}$, let $\\omega\\stackrel{\\mathbf{w}_{k}}{\\longrightarrow}_{R}\\sigma_{k}$. Then\n$$\\begin{aligned}\n\\sum_{\\sigma\\in\\mathcal{D}}\\hat{\\mu}(I_{\\sigma})\n&=\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\\hat{\\mu}(I_{\\sigma})\\\\\n&=\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\\rho_{\\sigma}^{\\alpha}a_{j}\n\\\\\n&=\\rho_{\\omega}^{\\alpha}\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\n\\rho_{\\mathbf{w}_{k}}^{\\alpha}a_{j}\\\\\n&=\\rho_{\\omega}^{\\alpha}\\sum_{j=1}^{q}A_{\\alpha}(i,j)a_{j}\\\\\n&=\\rho_{\\omega}^{\\alpha}a_{i}=\\hat{\\mu}(I_{\\omega}).\n\\end{aligned}$$\nCombining these with $\\hat{\\mu}(\\mathbb{P})=\\hat{\\mu}(\\omega_{{\\rm root}})=1$ shows that $\\hat{\\mu}$ is indeed a measure on $\\mathbb{P}$. Define $f:\\mathbb{P}\\longrightarrow W$ by letting $f(\\omega_{0},\\omega_{1},\\dots)$ be the unique point in $\\bigcap_{k=0}^{\\infty}S_{\\omega_{k}}(K)$. It is clear that $f(\\mathbb{P})=K$. Let $\\widetilde{\\mu}:=\\hat{\\mu}\\circ f^{-1}$. Then $\\widetilde{\\mu}$ is a measure on $K$.\n\nFor any bounded Borel set $F\\subset M$, let\n\\begin{equation}\\label{eq(4.1)}\n\\mathcal{B}(F):=\\{I_{\\omega_{k}}=I_{\\omega_{k},\\dots,\\omega_{k}}:\n|S_{\\omega_{k}}(\\Omega)|\\leq|F|<|S_{\\omega_{k-1}}(\\Omega)|\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}.\n\\end{equation}\n\n\\begin{lem}\\label{lem(4.1)}\nThere exists a constant $C_{5}>0$, independent of $k$, such that for any bounded Borel set $F\\subset M$, we have $\\#\\mathcal{B}(F)\\leq C_{5}$.\n\\end{lem}\n\\begin{proof}\nDefine\n$$\\begin{aligned}\n\\widetilde{\\mathcal{B}}(F):&=\\{\\omega_{k}\\in\\mathcal{V}_{k}:\n|S_{\\omega_{k}}(\\Omega)|\\leq|F|<|S_{\\omega_{k-1}}(\\Omega)|\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}\\\\\n&=\\{\\omega_{k}\\in\\mathcal{V}_{k}:\n\\rho_{\\omega_{k}}\\leq|F|\/|\\Omega|<\\rho_{\\omega_{k-1}}\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}.\n\\end{aligned}$$\nSince $I_{\\omega_{k}}$ is one-to-one with $\\omega_{k}$, we have $\\#\\mathcal{B}(F)=\\#\\widetilde{\\mathcal{B}}(F)$. Let $b:=|F|\/|\\Omega|$ and $\\omega_{k}\\in\\widetilde{\\mathcal{B}}(F)$. Then there exists a unique $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\omega_{k}=(S_{\\mathbf{u}},k)$. Let $\\mathbf{u}'\\preccurlyeq\\mathbf{u}$ such that $S_{\\mathbf{u}'}\\in\\mathcal{A}_{b}$. Then\n$$\\rho_{\\mathbf{u}'}\\leq b=|F|\/|\\Omega|<\\rho_{\\omega_{k-1}}.$$\nThus $\\mathbf{u}'\\in\\mathcal{M}_{k-1}$ or $\\mathcal{M}_{k}$. Combining these and Definition \\ref{defi(3.1)}$(e)$, we have $|\\mathbf{u}|-|\\mathbf{u}'|\\leq L$.\nNote that $F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset$ implies that $F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset$. Since $S_{\\mathbf{u}'}\\in\\mathcal{A}_{b}$, we have\n$$|S_{\\mathbf{u}'}(\\Omega)|=\\rho_{\\mathbf{u}'}|\\Omega|\\leq b\\Omega.$$\nLet $\\delta:=2b|\\Omega|$ and fix any $x_{0}\\in F$. Then\n$$S_{\\mathbf{u}'}(\\Omega)\\subset B_{\\delta}(x_{0}).$$\nSince (FTC) implies (WSC), there exists a constant $\\gamma>0$ (independent of $b$) such that for all $x\\in U$,\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(\\Omega)\\}\\leq\\gamma.$$\nNote that the contraction ratio of $S_{\\mathbf{u}}$ is $\\rho_{\\mathbf{u}}=|\\det S_{\\mathbf{u}}'(x)|^{\\frac{1}{n}}$ for any $x\\in W$. Let $A\\subset W$ be a measurable set. Then by Lemma \\ref{lem(1.1)}, we have\n$$\\nu(S_{\\mathbf{u}}(A))\\geq(b\\rho)^{n}\\nu(A).$$\nCombining these we have\n$$\\begin{aligned}\n(b\\rho)^{n}\\nu(\\Omega)\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}&\\leq\n\\sum\\{\\nu(S_{\\mathbf{u}'}(\\Omega)):F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\\\\n&\\leq \\gamma\\nu(B_{\\delta}(x_{0}))\\\\\n&\\leq\\gamma c_{n}\\delta^{n}\\quad(\\text{by Lemma \\ref{lem(1.30)}})\\\\\n&:=\\gamma c_{1}b^{n},\n\\end{aligned}$$\nwhere $c_{n}$ is the volume of the unit ball in $\\mathbb{R}^{n}$ and $c_{1}:=c_{n}2^{n}|\\Omega|^{n}$. Thus,\n$$\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\leq\\frac{\\gamma c_{1}}{\\rho^{n}\\nu(\\Omega)}:=c_2.$$\nHence\n$$\\#\\mathcal{B}(F)=\\#\\widetilde{\\mathcal{B}}(F)\\leq\nN^{L}\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\leq c_{2}N^{L}.$$\nThe lemma follows by letting $C_{5}:=c_{2}N^{L}$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.5)}]  Use of Lemma \\ref{lem(4.1)} and the properties of\nmeasures $\\widetilde{\\mu}$ on $K$, as in \\cite[Theorem 1.2]{Lau-Ngai_2007}; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Hausdorff dimension of graph self-similar sets}\\label{S:41}\n\nIn this section, we define graph self-similar sets on Riemannian manifolds, and derive a formula for computing the Hausdorff dimension of such sets. We assume that $M$ is a complete $n$-dimensional Riemannian manifold that is locally Euclidean.\n\nLet $G=(V,E)$ be a graph, where $V=\\{1,\\dots,t\\}$ is the set of vertices and $E$ is the set of all directed edges. We assume that there is at least one edge between two vertices. It is possible that the initial and terminal vertices are same. A \\textit{directed path} in $G$ is a finite string $\\mathbf{e}=e_{1}\\cdots e_{p}$ of edges in $E$ such that the terminal vertex of each $e_{i}$ is the initial vertex of the next edge $e_{i+1}$. For such a path, denote the \\textit{length} of $\\mathbf{e}$ by $|\\mathbf{e}|=p$. For any two vertices $i,j\\in V$, and any positive integer $p$, $E^{i,j}$ be the set of all directed edges from $i$ to $j$, let $E_{p}^{i,j}$ be the set of all directed paths of length $p$ from $i$ to $j$, $E_{p}$ be the set of all directed paths of length $p$, $E^{*}$ be the set of all directed paths, i.e.,\n$$E_{p}:=\\bigcup_{i,j=1}^{p}E_{p}^{i,j}\\quad\\text{and}\\quad E^{*}:=\\bigcup_{p=1}^{\\infty}E_{p}.$$\nFor any edge $e\\in E$, we assume that there corresponds a contractive similitude $S_{e}$ with ratio $\\rho_{e}$ on $M$. For $\\mathbf{e}=e_{1}\\cdots e_{p}\\in E^{*}$, let\n$$S_{\\mathbf{e}}=S_{e_{1}}\\circ\\cdots\\circ S_{e_{p}}\\quad\\text{and}\\quad \\rho_{\\mathbf{e}}=\\rho_{e_{1}}\\cdots \\rho_{e_{p}}.$$\nThen there exists a unique family of nonempty compact sets $K_{1},\\dots,K_{t}$ satisfying\n$$K_{i}=\\bigcup_{j=1}^{t}\\bigcup_{e\\in E^{i,j}}S_{e}(K_{j}),\\quad i\\in\\{1,\\dots,t\\},$$\n(see e.g. \\cite{Mauldin-Williams_1988,Edgar-Mauldin_1992,Ngai-Wang-Dong_2010,Das-Ngai_2004}).\nDefine\n$$K:=\\bigcup_{i=1}^{t}K_{i}.$$\nWe call $K$ the \\textit{graph self-similar set} defined by $G=(V,E)$, and call $G=(V,E)$ the \\textit{graph-directed iterated function system} (GIFS) associated with $\\{S_{e}\\}_{e\\in E}$.\n\nSubstituting $E^{*}$ for $\\Sigma^{*}$ in Definition \\ref{defi(3.1)}, we define a sequence of nested index sets $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ of directed paths. Note that $\\{E_{k}\\}_{k=1}^{\\infty}$ is an example of a sequence of nested index sets of directed paths. Fix a sequence $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ of nested index sets. For $i,j\\in\\{1,\\cdots,t\\}$, we partition $\\mathcal{F}_{k}$ to $\\mathcal{F}_{k}^{i,j}$ as\n$$\\mathcal{F}_{k}^{i,j}:=\\mathcal{F}_{k}\\cap\\bigg(\\bigcup_{p\\geq1}E_{p}^{i,j}\\bigg)=\\{\\mathbf{e}=e_{1}\\cdots e_{p}\\in \\mathcal{F}_{k}:\\mathbf{e}\\in E_{p}^{i,j}~\\text{for some }p\\geq1\\}.$$\nNote that $\\mathcal{F}_{k}=\\bigcup_{i,j=1}^{t}\\mathcal{F}_{k}^{i,j}$. For $i,j\\in\\{1,\\cdots,t\\}$, $k\\geq1$, let $\\mathbb{V}_{k}$ be the set of \\textit{$k$-th level vertices} defined as\n$$\\mathbb{V}_{k}:=\\{(S_{\\mathbf{e}},i,j,k):\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j},1\\leq i,j\\leq t\\}.$$\nFor $\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j}$, we call $(S_{\\mathbf{e}},i,j,k)$ (or simply $(S_{\\mathbf{e}},k)$) a \\text{vertex}. For a vertex $\\omega=(S_{\\mathbf{e}},i,j,k)\\in\\mathbb{V}_{k}$ with $k\\geq1$, let\n$$S_{\\omega}=S_{\\mathbf{e}}\\quad\\text{and}\\quad \\rho_{\\omega}=\\rho_{\\mathbf{e}}.$$\nLet $\\mathcal{F}_{0}=\\{1,\\dots,t\\}$ and $\\mathbb{V}_{0}=\\{\\omega^{1}_{{\\rm root}},\\dots,\\omega^{t}_{{\\rm root}}\\}$, where $\\omega^{i}_{{\\rm root}}=(I,i,i,0)$ and $I$ is the identity map on $M$. Then we say $\\mathbb{V}_{0}$ is the set of \\textit{root vertices}, and $\\{\\mathcal{F}_{k}\\}_{k=0}^{\\infty}$ is a sequence of nested index sets if $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ is. Let $\\mathbb{V}:=\\bigcup_{k\\geq0}\\mathbb{V}_{k}$ be the set of all vertices, and $\\pi:\\bigcup_{k\\geq0}\\mathcal{F}_{k}\\longrightarrow \\mathbb{V}$ be defined as\n$$\\pi(\\mathbf{e}):=\\begin{cases}(S_{\\mathbf{e}},i,j,k),\\quad\\text{if }\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j}, k\\geq1,\\\\\n\\omega^{i}_{{\\rm root}},\\quad\\text{if }\\mathbf{e}=i\\in\\mathcal{F}_{0}.\\end{cases}$$\n\nLet $\\omega\\in\\mathbb{V}_{k}$ and $\\omega'\\in\\mathbb{V}_{k+1}$. Suppose that there exist directed paths $\\mathbf{e}\\in\\mathcal{F}_{k},\\mathbf{e}'\\in\\mathcal{F}_{k+1}$ and $\\mathbf{k}\\in E^{*}$ such that $\\pi(\\mathbf{e})=\\omega$, $\\pi(\\mathbf{e}')=\\omega'$ and $\\mathbf{e}'=\\mathbf{e}\\mathbf{k}$. Then we connect a \\textit{directed edge} $\\mathbf{k}$ from $\\omega$ to $\\omega'$, and denote this as $\\omega\\stackrel{\\mathbf{k}}{\\longrightarrow}\\omega'$. We call $\\omega$ a \\textit{parent} of $\\omega'$ and $\\omega'$ an \\textit{offspring} of $\\omega$.\nDefine a graph $\\mathbb{G}:=(\\mathbb{V},\\mathbb{E})$, where $\\mathbb{E}$ is the set of all directed edges of $\\mathbb{G}$. Let $\\mathbb{G}_{R}:=(\\mathbb{V}_{R},\\mathbb{E}_{R})$ be the \\textit{reduced graph} of $\\mathbb{G}$, defined as in Section \\ref{S:3} similarly, where $\\mathbb{V}_{R}$ and $\\mathbb{E}_{R}$ are the sets of all vertices and all directed edges, respectively.\n\nLet $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$, where $\\Omega_{i}\\subset M$ is a nonempty bounded open set for any $i\\in\\{1,\\dots,t\\}$. We say that $\\mathbf{\\Omega}$ is \\textit{invariant} under the GIFS $G=(V,E)$ if\n$$\\bigcup_{e\\in E^{i,j}}S_{e}(\\Omega_{j})\\subset\\Omega_{i},\\quad i,j\\in\\{1,\\cdots,t\\}.$$\nSince $S_{e}$ is a contractive similitude for any $e\\in E^{i,j}$, such a family always exists. Fix an invariant family $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$ of $G=(V,E)$. Let $\\omega=(S_{\\mathbf{e}},i,j,k)\\in\\mathbb{V}_{k}$ with $\\mathbf{e}\\in E_{q}^{i,j}$ and $\\omega=(S_{\\mathbf{e}'},i',j',k)\\in\\mathbb{V}_{k}$ with $\\mathbf{e}'\\in E_{s}^{i',j'}$, where $q,s>0$ are integers. We say that two vertices $\\omega$, $\\omega'$ are \\textit{neighbors} (with respect to $\\mathbf{\\Omega}$) if\n$$i=i'\\quad\\text{and}\\quad S_{\\mathbf{e}}(\\Omega_{j})\\cap S_{\\mathbf{e}'}(\\Omega_{j'})\\neq\\emptyset.$$\nLet\n$$\\mathcal{N}(\\omega):=\\{\\omega':\\omega'\\in\\mathbb{V}_{k}\\text{ is a neighbor of }\\omega\\},$$\nwhich is called the \\textit{neighborhood} of $\\omega$ (with respect to $\\mathbf{\\Omega}$).\n\\begin{defi}\\label{defi(41.1)}\nFor any two vertices $\\omega=(S_{\\mathbf{e}_{\\omega}},i_{\\omega},j_{\\omega},k)\\in\\mathbb{V}_{k}$ and $\\omega'=(S_{\\mathbf{e}_{\\omega'}},i_{\\omega'},j_{\\omega'},k')\\in\\mathbb{V}_{k'}$, let $\\sigma=(S_{\\mathbf{e}_{\\sigma}},i_{\\omega},j_{\\sigma},k)\\in\\mathcal{N}(\\omega)$ and $\\sigma'=(S_{\\mathbf{e}_{\\sigma'}},i_{\\omega'},j_{\\sigma'},k')\\in\\mathcal{N}(\\omega')$. Assume that\n$$\\tau=S_{\\omega'}S_{\\omega}^{-1}:\\bigcup_{\\sigma\\in\\mathcal{N}(\\omega)}S_{\\sigma}(\\Omega_{j_{\\sigma}})\n\\longrightarrow \\bigcup_{i=1}^{t}\\Omega_{i}$$\ninduces a bijection $f_{\\tau}:\\mathcal{N}(\\omega)\\longrightarrow\\mathcal{N}(\\omega')$ defined by\n\\begin{equation}\\label{eq(41.1)}\nf_{\\tau}(\\sigma)=f_{\\tau}(S_{\\mathbf{e}_{\\sigma}},i_{\\omega},j_{\\sigma},k)=(\\tau\\circ S_{\\mathbf{e}_{\\sigma'}},i_{\\omega'},j_{\\sigma'},k').\n\\end{equation}\nWe say $\\omega$ and $\\omega'$ are \\textit{equivalent}, i.e., $\\omega\\sim\\omega'$, if the following conditions hold:\n\\begin{itemize}\n\\item[$(a)$] $\\#\\mathcal{N}(\\omega)=\\#\\mathcal{N}(\\omega')$ and $j_{\\sigma}=j_{\\sigma'}$ in \\eqref{eq(41.1)};\n\\item[$(b)$] for $\\sigma\\in\\mathcal{N}(\\omega)$ and $\\sigma'\\in\\mathcal{N}(\\omega')$ such that $f_{\\tau}(\\sigma)=\\sigma'$, and for any positive integer $k_{0}\\geq1$, a directed path $\\mathbf{e}\\in E^{*}$ satisfies $(S_{\\sigma}\\circ S_{\\mathbf{e}},k+k_{0})\\in\\mathbb{V}_{k+k_{0}}$ if and only if it satisfies\n$(S_{\\sigma'}\\circ S_{\\mathbf{e}},k'+k_{0})\\in\\mathbb{V}_{k'+k_{0}}$.\n\\end{itemize}\n\\end{defi}\nIt is easy to check that $\\sim$ is an equivalence relation. Denote the equivalent class of $\\omega$ by $[\\omega]$, and call it the \\textit{neighborhood types} of $\\omega$ (with respect to $\\mathbf{\\Omega}$).\n\nFor a graph $\\mathbb{G}=(\\mathbb{V},\\mathbb{E})$, we can prove that $\\mathbb{G}$ satisfies Proposition \\ref{prop(3.1)} as in \\cite[Propositin 2.5]{Ngai-Wang-Dong_2010}; we omit the details. We now define the graph finite type condition.\n\n\n\\begin{defi}\\label{defi(41.2)}\nLet $M$ be a complete $n$-dimensional Riemannian manifold that is locally Euclidean, let $G=(V,E)$ be a GIFS, and let $\\{S_{e}\\}_{e\\in E}$ be a family of contractive similitudes defined on $M$.\nIf there exists an invariant family of nonempty bounded open sets $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$ with respect to some sequence of nested index sets $\\{\\mathcal{F}_{k}\\}_{k=0}^{\\infty}$ such that\n$$\\#\\mathbb{V}\/_{\\sim}<\\infty,$$\nthen we say that $G=(V,E)$ satisfies the \\textit{graph finite type condition} (GFTC).\nWe call such an invariant family $\\mathbf{\\Omega}$ a \\textit{graph finite type condition family} of $G$.\n\\end{defi}\n\n\nBy assuming a GIFS satisfies (GFTC), we get a formula for $\\dim_{{\\rm H}}(K)$.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm(41.1)}] The proof is similar to that of \\cite[Theorem 1.1]{Ngai-Wang-Dong_2010} and the definition of a weighted incidence matrix is the same as that in Section \\ref{S:4}; we omit the details.\n\\end{proof}\n\n\n\\section{Examples}\\label{S:5}\nIn this section, we construct some examples of CIFSs and GIFSs on Riemannian manifolds satisfying (FTC) and (GFTC), respectively.\n\nLet $\\{f_{i}\\}_{i=1}^{N}$ be a contractive CIFS on a compact set $W_0\\subset\\mathbb{R}^{n}$, i.e., $f_{i}$ is $C^{1+\\varepsilon}$ where $0<\\varepsilon<1$, and there exists an open and connected set $U_{0}\\supset W_{0}$ such that for any $i\\in\\{1,\\dots,N\\}$, $f_{i}$ can be extended to an injective conformal map $f_{i}:U_{0}\\longrightarrow U_{0}$. Let $M$ be a complete $n$-dimensional smooth Riemannian manifold. Assume that there exists a diffeomorphism\n$$\\varphi:U\\longrightarrow U_{0},$$\nwhere $U\\subset M$ is open and connected.\nDefine\n\\begin{equation}\\label{eq(5.1)}\nS_{i}:=\\varphi^{-1}\\circ f_{i}\\circ\\varphi:U\\longrightarrow S_{i}(U)\\quad\\text{for any }i\\in\\{1,\\dots,N\\}.\n\\end{equation}\nBy \\cite[Proposition 7.1]{Ngai-Xu_2022}, $\\{S_{i}\\}_{i=1}^{N}$ is a contractive CIFS on $U$.\n\n\nFix a sequence of nested index sets $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$. Let $\\widetilde{\\mathcal{V}}$ and $\\mathcal{V}$ be the sets of all vertices with respect to $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ of $\\{f_{i}\\}_{i=1}^{N}$ and $\\{S_{i}\\}_{i=1}^{N}$, respectively.\nFor $\\tilde{\\omega}=(f_{\\mathbf{u}},k)\\in\\widetilde{\\mathcal{V}}_{k}$ and $\\omega=(S_{\\mathbf{u}},k)\\in\\mathcal{V}_{k}$, where $\\mathbf{u}\\in\\mathcal{M}_{k}$, $k\\geq0$, we define $f_{\\tilde{\\omega}}:=f_{\\mathbf{u}}$ and $S_{\\omega}:=S_{\\mathbf{u}}$. It follows from \\eqref{eq(5.1)} that\n\\begin{equation}\\label{eq(5.01)}\nS_{\\omega}=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}\\circ\\varphi.\n\\end{equation}\n\n\\begin{prop}\\label{prop(5.1)}\nLet $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,\\dots,N\\}$. If $\\{f_{i}\\}_{i=1}^{N}$ satisfies (FTC), then $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC).\n\\end{prop}\n\\begin{proof}\nFor any $\\tilde{\\omega},\\tilde{\\omega}'\\in\\widetilde{\\mathcal{V}}$ and $\\tilde{\\omega}\\sim\\tilde{\\omega}'$, by Definition \\ref{defi(3.2)}, there exist $\\tilde{\\sigma}\\in\\Omega(\\tilde{\\omega})$ and $\\tilde{\\sigma}'\\in\\Omega(\\tilde{\\omega}')$ such that\n\\begin{equation}\\label{eq(5.2)}\nf_{\\tilde{\\omega}'}^{-1}\\circ f_{\\tilde{\\sigma}'}=f_{\\tilde{\\omega}}^{-1}\\circ f_{\\tilde{\\sigma}}.\n\\end{equation}\nFor any $\\omega,\\omega'\\in\\mathcal{V}$, $\\sigma\\in\\Omega(\\omega)$ and $\\sigma'\\in\\Omega(\\omega')$, we have\n$$\\begin{aligned}\nS_{\\omega'}^{-1}\\circ S_{\\sigma'}&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}'}^{-1}\\circ\\varphi\\circ\\varphi^{-1}\\circ f_{\\tilde{\\sigma}'}\\circ\\varphi\\quad\\text{(by~(\\ref{eq(5.01)}))}\\\\\n&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}^{-1}\\circ f_{\\tilde{\\sigma}}\\circ\\varphi\\quad\\text{(by~(\\ref{eq(5.2)}))}\\\\\n&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}^{-1}\\circ\\varphi\\circ\\varphi^{-1}\\circ f_{\\tilde{\\sigma}}\\circ\\varphi\\\\\n&=S_{\\omega}^{-1}\\circ S_{\\sigma}.\n\\end{aligned}$$\nIt follows that $\\omega\\sim\\omega'$. Since $\\{f_{i}\\}_{i=1}^{N}$ satisfies (FTC),\nwe have $\\#\\widetilde{\\mathcal{V}}\/_{\\sim}<\\infty$. Thus, $\\#\\mathcal{V}\/_{\\sim}<\\infty$. This proves the proposition.\n\\end{proof}\n\nLet\n$$\\mathbb{S}^{n}:=\\left\\{(x_{1},\\dots,x_{n+1})\\in \\mathbb{R}^{n+1}:\\sum_{i=1}^{n+1}x_{i}^{2}=1\\right\\},\\quad\n\\mathbb{D}^{n}:=\\left\\{(x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}:\\sum_{i=1}^{n}x_{i}^{2}<1\\right\\}.$$\nLet $\\mathbb{S}^{n}_{+}$ be the upper hemisphere of $\\mathbb{S}^{n}$, and  define the stereographic projection $\\varphi:\\mathbb{S}^{n}_{+}\\rightarrow \\mathbb{D}^{n}$ as\n$$\n\\varphi(x_{1},\\dots,x_{n+1})=\\frac{1}{1+x_{n+1}}(x_{1},\\dots,x_{n}):=(y_{1},\\dots,y_{n}).\n$$\nThen\n$$\\varphi^{-1}(y_{1},\\dots,y_{n})=\\frac{1}{|\\boldsymbol{y}|^{2}+1}\\big(2y_{1},\\dots,2y_{n},1-|\\boldsymbol{y}|^{2}\\big),$$\nwhere $|\\boldsymbol{y}|^{2}=y_{1}^{2}+\\cdots+y_{n}^{2}$.\n\nFor convenience, we consider the case of $n=2$. We will give some actual examples of Proposition \\ref{prop(5.1)}. The first example below satisfies (OSC), while the other two satisfy (FTC) but not (OSC).\n\n\\begin{exam}\\label{exam(5.1)}\nLet $\\{f_{i}\\}_{i=1}^{3}$ be a Sierpinski gasket on $\\mathbb{R}^{2}$, i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$f_{1}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{2}\\bigg),\\quad\nf_{2}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(-\\frac{1}{4},0\\bigg),\\quad\nf_{3}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},0\\bigg).$$\nLet $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3\\}$. Then $\\{S_{i}\\}_{i=1}^{3}$ is a CIFS satisfying (OSC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(a)).\n\\end{exam}\n\n\\begin{exam}\\label{exam(5.2)}\nLet $\\{f_{i}\\}_{i=1}^{4}$ be a CIFS defined as in \\cite{Lau-Ngai_2007} satisfying (FTC), i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$\\begin{aligned}\nf_{1}(\\boldsymbol{x})&=\\rho\\boldsymbol{x}+\\bigg(\\frac{1}{2}\\rho,0\\bigg),\\qquad\nf_{2}(\\boldsymbol{x})=r\\boldsymbol{x}+\\bigg(\\rho-\\rho r+\\frac{1}{2}r,0\\bigg),\\\\\nf_{3}(\\boldsymbol{x})&=r\\boldsymbol{x}+\\bigg(1-\\frac{1}{2}r,0\\bigg),\\qquad\nf_{4}(\\boldsymbol{x})\n=r\\boldsymbol{x}+\\bigg(\\frac{1}{2}r,1-r\\bigg),\n\\end{aligned}$$\nwhere $0<\\rho,r<1$ and $\\rho+2r-\\rho r\\leq1$. Let $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3,4\\}$. By Proposition \\ref{prop(5.1)}, $\\{S_{i}\\}_{i=1}^{4}$ is a CIFS satisfying (FTC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(b)).\n\\end{exam}\n\n\\begin{exam}\\label{exam(5.3)}\nLet $\\{f_{i}\\}_{i=1}^{3}$ be a golden Sierpinski gasket defined as in \\cite{Ngai-Wang_2001} satisfying (FTC), i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$\nf_{1}(\\boldsymbol{x})=\\rho \\boldsymbol{x},\\quad\nf_{2}(\\boldsymbol{x})=\\rho \\boldsymbol{x}+\\big(\\rho^{2},0\\big),\\quad\nf_{3}(\\boldsymbol{x})=\\rho^{2} \\boldsymbol{x}+\\big(\\rho,\\rho\\big),\n$$\nwhere $\\rho=\\big(\\sqrt{5}-1\\big)\/2$. Let $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3\\}$. By Proposition \\ref{prop(5.1)}, $\\{S_{i}\\}_{i=1}^{3}$ is a CIFS satisfying (FTC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(c)).\n\\end{exam}\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n   \\begin{tikzpicture}[scale=0.2,domain=0:180,>=stealth]\n    \\coordinate (org) at (0,0);\n    \\draw (0,0) circle[radius=8];\n    \\draw (org) ellipse (8cm and 3cm);\n    \\draw[help lines,dashed](9.5,0)--(0,0);\n    \\draw[help lines,dashed](0,-9.5)--(0,0);\n    \\draw[help lines,dashed](5.8,7.25)--(0,0);\n    \\draw[white] plot ({8*cos(\\x)},{3*sin(\\x)});\n    \\foreach\\i\/\\text in{{-6.8,-8.5}\/y,{-10.5,0}\/x,{0,10.5}\/{}}\n    \\draw[help lines,->] (org)node[above right]{}--(\\i)node[above]{$\\text$};\n    \\draw[help lines,->]node[left] at (0,10){$z$};\n    \\draw[thin,black] (-4,0)--(4,0);\n    \\draw[thin,black] (-4,0)--(-2.3,-2.875);\n    \\draw[thin,black] (4,0)--(-2.3,-2.875);\n    \\draw[thin,black] (-3.15,-1.4375)--(0.85,-1.4375);\n    \\draw[thin,black] 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\\draw[thin,blue](0,8) to [out=-59, in=86](1.35,2.17);\n    \\draw[fill=blue!30,opacity=.5] (1.35,2.17) to [out=158, in=49](-4.3,1.05) to [out=-60, in=118](-2.31,-2.82) to [out=58, in=-134](1.35,2.17);\n    \\draw[fill=blue!30,opacity=.5] (0,8) to [out=182, in=51](-6.39,4.81) to [out=-56, in=121](-4.3,1.1) to [out=84, in=-148](0,8);\n    \\draw[fill=blue!30,opacity=.5] (6.36,4.85) to [out=129.2, in=-2.7](0,8) to [out=-59, in=86](1.35,2.17) to [out=43, in=-167.8](6.36,4.85);\n     \\foreach \\i\/\\tex in {0\/(a)}\n     \\draw(0,-10)node[below]{\\tex};\n     \\foreach \\i\/\\tex in {0\/}\n     \\draw(0,-19)node[below]{\\tex};\n   \\end{tikzpicture}\n   \\quad\n  \\begin{tikzpicture}[scale=.2,domain=0:180,>=stealth]\n    \\coordinate (org) at (0,0);\n    \\draw (0,0) circle[radius=8];\n    \\draw (org) ellipse (8cm and 3cm);\n    \\draw[help lines,dashed](9.5,0)--(0,0);\n    \\draw[help lines,dashed](0,-9.5)--(0,0);\n    \\draw[help lines,dashed](5.8,7.25)--(0,0);\n    \\draw[white] plot 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color=olive!60!white!30,bottom color=red!30,opacity=.5]\n    (3.2,0)--(2.58,-0.8)--(1.08,-0.8)--(1.7,0)--(3.2,0);\n     \\draw[top color=olive!60!white!30,bottom color=red!30,opacity=.5]\n    (-4.7,-0.8)--(-3.1,-0.8)--(-2.5,0)--(-4,0)--(-4.68,-0.8);\n     \\draw[top color=olive!60!white!30,bottom color=red!30,opacity=.5]\n    (1.6,-1)--(3.14,-1)--(2.52,-1.8)--(0.9,-1.8)--(1.6,-1);\n   \\draw[thin,blue] (-6.45,4.75) to [out=-100, in=110](-5.8,-1.5) to [out=45, in=150] (3.5,0.6) to [out=63, in=-138] (6.45,4.75) to [out=129.1, in=-2](0,8) to [out=181, in=50](-6.45,4.75);\n    \\draw[thin,cyan,dashed](0,-8)--(-5.8,-1.5);\n    \\draw[thin,cyan,densely dashed](0,-8)--(-6.45,4.75);\n    \\draw[thin,cyan,densely dashed](0,-8)--(-5,3.6);\n   \n    \\draw[thin,cyan,densely dashed](0,-8)--(3.5,0.6);\n    \\draw[thin,cyan,densely dashed](0,-8)--(6.45,4.75);\n   \n   \n    \\draw[thin,blue] (-4.6,6.55) to [out=-140, in=52] (-6.45,4.75) to [out=-102, in=89.5] (-6.65,2.25) to [out=46, in=-146] (-5,3.6) to [out=90, in=-105] (-4.6,6.55);\n    \\draw[fill=blue!30,opacity=.5](-4.6,6.55) to [out=-140, in=52] (-6.45,4.75) to [out=-102, in=89.5] (-6.65,2.25) to [out=46, in=-146] (-5,3.6) to [out=90, in=-105] (-4.6,6.55);\n\n   \n   \n    \\draw[thin,cyan,densely dashed](0,-8)--(2.74,4.2);\n    \\draw[thin,blue] (1.4,1.37) to [out=-11,in=152] (3.5,0.6) to [out=63,in=-126](4.83,2.95) to [out=142,in=-22](2.74,4.2)to[out=-125,in=72](1.4,1.37);\n    \\draw[fill=blue!30,opacity=.5](1.4,1.37) to [out=-11,in=152] (3.5,0.6) to [out=63,in=-126](4.83,2.95) to [out=142,in=-22](2.74,4.2)to[out=-125,in=72](1.4,1.37);\n\n   \n   \n    \\draw[thin,cyan,densely dashed](0,-8)--(4.15,4.95);\n    \\draw[thin,blue](4.15,4.95) to [out=-32,in=137] (5.565,3.87) to [out=51,in=-135] (6.45,4.75) to [out=127,in=-42] (5.05,6.2) to [out=-132,in=60] (4.15,4.95);\n    \\draw[fill=blue!30,opacity=.5](4.15,4.95) to [out=-32,in=137] (5.565,3.87) to [out=51,in=-135] (6.45,4.75) to [out=127,in=-42] (5.05,6.2) to [out=-132,in=60] (4.15,4.95)\n\n   \n   \n    \\draw[thin,cyan,dashed](0,-8)--(1.9,5);\n    \\draw[thin,cyan,densely dashed](0,-8)--(4.3,3.9);\n    \\draw[thin,blue](3.1,7.375) to [out=-125,in=70] (1.9,5) to [out=-20,in=150] (4.3,3.9) to [out=55,in=-130] (5.65,5.65) to [out=138,in=-21] (3.1,7.375);\n    \\draw[fill=blue!30,opacity=.5](3.1,7.375) to [out=-125,in=70] (1.9,5) to [out=-20,in=150] (4.3,3.9) to [out=55,in=-130] (5.65,5.65) to [out=138,in=-21] (3.1,7.375)\n   \\foreach \\i\/\\tex in {0\/(b)}\n     \\draw(0,-10)node[below]{\\tex};\n     \\foreach \\i\/\\tex in {0\/\\text{}}\n     \\draw(0,-19)node[below]{\\tex};\n   \\end{tikzpicture}\n   \\quad\n  \\begin{tikzpicture}[scale=.2,domain=0:180,>=stealth]\n    \\coordinate (org) at (0,0);\n    \\draw (0,0) circle[radius=8];\n    \\draw (org) ellipse (8cm and 3cm);\n    \\draw[help lines,dashed](9.5,0)--(0,0);\n    \\draw[help lines,dashed](0,-9.5)--(0,0);\n    \\draw[help lines,dashed](5.8,7.25)--(0,0);\n    \\draw[white] plot 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\\draw[thin,cyan,dashed](0,-8)--(-2.3,-2.875);\n    \\draw[thin,cyan,densely dashed](0,-8)--(8,0);\n    \\draw[thin,blue] (-8,0) to [out=90, in=-180](0,8) to [out=0, in=90] (8,0) to [out=140, in=55] (-2.3,-2.875)\n    to [out=125, in=-5] (-8,0);\n    \\draw[thin,cyan,densely dashed](0,-8)--(-5.4,-0.51);\n    \\draw[thin,cyan,densely dashed](0,-8)--(-3.8,7.05);\n    \\draw[thin,cyan,densely dashed](0,-8)--(3.8,7.05);\n    \\draw[thin,blue](-5.4,-0.51) to [out=45, in=150] (2.2,0.7);\n    \\draw[thin,blue](3.8,7.05) to [out=-175, in=75] (-5.4,-0.51);\n    \\draw[thin,blue](-3.8,7.05) to [out=-20, in=110] (2.2,0.7);\n \\draw[fill=blue!30,opacity=.5](-5.4,-0.51) to [out=45, in=150] (2.2,0.7) to [out=-158, in=53] (-2.3,-2.875) to [out=126, in=-19](-5.4,-0.51);\n \\draw[fill=blue!30,opacity=.5](8,0) to [out=90, in=0] (0,8) to [out=-180, in=26] (-3.8,7.05) to [out=-20, in=110] (2.2,0.7) to [out=20, in=140] (8,0);\n \\draw[fill=blue!30,opacity=.5](-8,0) to [out=90, in=180] (0,8) to [out=0, in=154] (3.8,7.05) to [out=-175, in=75] (-5.4,-0.51) to [out=162, in=-5] (-8,0);\n  \\foreach \\i\/\\tex in {0\/(c)}\n     \\draw(0,-10)node[below]{\\tex};\n     \\foreach \\i\/\\tex in {0\/}\n     \\draw(0,-19)node[below]{\\tex};\n  \\end{tikzpicture}\n  \\vspace{-3.2em}\n\\caption{Figures (a), (b) and (c) are respectively the IFSs in Examples \\ref{exam(5.1)}, \\ref{exam(5.2)} and \\ref{exam(5.3)}.} \\label{fig.1}\n\\end{center}\n\\end{figure}\n\n\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold that is locally Euclidean.\nNow, we construct an example of GIFS on $M$ satisfying (GFTC) to illustrate Theorem \\ref{thm(41.1)}. Let $G=(V,E)$ be a GIFS of contractive similitudes defined on $\\mathbb{R}^{n}$, and $\\{O_{i}\\}_{i=1}^{t}$, where $O_{i}\\subset\\mathbb{R}^{n}$, be an invariant family of nonempty bounded open sets under $G=(V,E)$. Let $O:=\\bigcup_{i=1}^{t}O_{i}$. For any edge $e\\in E$, there corresponds a contractive similitude $f_{e}:O\\longrightarrow O$. Assume that there exists a diffeomorphism\n$$\\varphi:O_{i}\\longrightarrow \\Omega_{i}\\quad\\text{for any }i\\in\\{1,\\dots,t\\},$$\nwhere $\\Omega_{i}\\subset M$ is open and connected. Let $\\Omega:=\\bigcup_{i=1}^{t}\\Omega_{i}$.\nFor any edge $e\\in E$, define\n\\begin{equation}\\label{eq(5.3)}\nS_{e}:=\\varphi^{-1}\\circ f_{e}\\circ\\varphi:\\Omega\\longrightarrow \\Omega.\n\\end{equation}\nAs in \\cite[Proposition 7.1]{Ngai-Xu_2022}, $\\{S_{e}\\}_{e\\in E}$ is a family of contractive maps on $M$. If for any $e\\in E$, $S_{e}$ is a similitude, then $G=(V,E)$, along with $\\{\\Omega_{i}\\}_{i=1}^{t}$ and $\\{S_{e}\\}_{e\\in E}$, forms a GIFS on $M$. The proof of the following proposition is similar to that of Proposition \\ref{prop(5.1)}; we omit it.\n\n\n\\begin{prop}\\label{prop(5.2)}\nUse the above notation and setup. Let $M$ be a complete $n$-dimensional smooth Riemannian manifold that is locally Euclidean. Let $G=(V,E)$ be a GIFS defined on $\\mathbb{R}^{n}$ satisfying (GFTC) with $\\{O_{i}\\}_{i=1}^{t}$ being a GFTC-family and $\\{f_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nFor any $e\\in E$, let $S_{e}$ be a similitude defined as in (\\ref{eq(5.3)}). Then the GIFS $G=(V,E)$ defined on $M$ satisfies (GFTC) with $\\{\\Omega_{i}\\}_{i=1}^{t}$ being a GFTC-family and $\\{S_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes. Moreover, for such two GIFSs connected by a diffeomorphism, the neighborhood types, weighted incidence matrices, and the Hausdorff dimension of the corresponding graph self-similar sets are the same.\n\\end{prop}\n\n\n\nAssume that $G=(V,E)$ is a GIFS of contractive similitudes defined on $M$ satisfying (GFTC). Let $\\mathcal{T}_{1},\\dots,\\mathcal{T}_{q}$ be all the distinct neighborhood types with $\\mathcal{T}_{i}=[\\omega_{{\\rm root}}^{i}]$ for any $i\\in\\{1,\\dots,t\\}$. Fix a vertex $\\omega\\in\\mathbb{V}_{R}$ such that $[\\omega]\\in\\mathcal{T}_{i}$, where $i\\in\\{1,\\dots,q\\}$.\nLet $\\sigma_{1},\\dots,\\sigma_{m}$ be the offspring of $\\omega$ in $\\mathbb{G}_{R}$, let $\\mathbf{k}_{\\ell}$ be the unique edge in $\\mathbb{G}_{R}$ connecting $\\omega$ to $\\sigma_{\\ell}$ for $1\\leq \\ell\\leq m$, and let\n$$C_{ij}:=\\{\\sigma_{\\ell}:1\\leq \\ell\\leq m,~[\\sigma_{\\ell}]=\\mathcal{T}_{j}\\}.$$\nNote that for two edges $\\mathbf{k}_{\\ell}$ and $\\mathbf{k}_{\\ell'}$ connecting $\\omega$ to two distinct $\\sigma_{\\ell}$ and $\\sigma_{\\ell'}$ satisfying $[\\sigma_{\\ell}]=[\\sigma_{\\ell}']=\\mathcal{T}_{j}$, the contraction ratios $\\rho_{\\sigma_{\\ell}}$ and $\\rho_{\\sigma_{\\ell'}}$ may be different. We can partition $C_{ij}:=C_{ij}(1)\\cup\\cdots\\cup C_{ij}(n_{ij})$ by using $\\rho_{\\sigma_{\\ell}}$, where for $s=1,\\dots,n_{ij}$,\n$$C_{ij}(s):=\\{\\sigma_{\\ell}\\in C_{ij}:\\rho_{\\sigma_{\\ell}}=\\rho_{ijs}\\},$$\nand the $\\rho_{ijs}$ are distinct. Thus, for any entry $A_{\\alpha}(i,j)$ of the weighted incidence matrix,\n$$A_{\\alpha}(i,j)=\\sum_{s=1}^{n_{ij}}\\#C_{ij}(s)\\rho_{ijs}^{\\alpha}.$$\nMoreover, we can write symbolically\n$$\\mathcal{T}_{i}\\longrightarrow\\sum_{j=1}^{q}\\sum_{s=1}^{n_{ij}}\\#C_{ij}(s)\\mathcal{T}_{j}(\\rho_{ijs}),$$\nwhere the $\\mathcal{T}_{j}(\\rho_{ijs})$ are defined in an obvious way.\nWe say that $\\mathcal{T}_{i}$ generates $\\#C_{ij}(s)$ neighborhoods of type $\\mathcal{T}_{j}$ with contraction ratio $\\rho_{ijs}$.\n\nFor $\\boldsymbol{x}\\in[0,1]\\times[0,1]$, we consider the following iterated function system with overlaps:\n$$\nh_{1}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{4}\\bigg),\\ \\\nh_{2}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{1}{4}\\bigg),\\ \\\nh_{3}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},\\frac{1}{4}\\bigg),\\ \\\nh_{4}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{3}{4}\\bigg).\n$$\nIterations of $\\{h_i\\}_{i=1}^4$ induce iterations on the $2$-torus $\\mathbb{T}^2=\\mathbb R^2\/\\mathbb Z^2$, generating an attractor on $\\mathbb{T}^2$. We are interested in computing the Hausdorff dimension of the attractor. However, the relations induced by $\\{h_i\\}_{i=1}^4$ on $\\mathbb{T}^2$ are not well-defined functions, making it awkward to apply the theory of IFSs developed in Section~\\ref{S:4}. To overcome this difficulty, we will use the GIFS framework and Theorem~\\ref{thm(41.1)}, as shown in the following example.\n\n\n\\begin{exam}\\label{exam(5.4)}\nLet $\\{h_i\\}_{i=1}^4$ be defined as above, and let $\\mathbb{T}^2=\\mathbb{S}^{1}\\times\\mathbb{S}^{1}$ be a $2$-torus, viewed as $[0,1]\\times[0,1]$ with opposite sides identified. Let $\\mathbb{T}^2$ be endowed with the Riemannian metric induced from $\\mathbb{R}^2$. Consider the IFS $\\{g_{i}\\}_{i=1}^{4}$ on $[0,1]\\times[0,1]$ under the Euclidean metric, where for $\\boldsymbol{x}\\in[0,1]\\times[0,1]$,\n$$\ng_{1}(\\boldsymbol{x})=h_1(\\boldsymbol{x}),\\quad\ng_{2}(\\boldsymbol{x})=h_2(\\boldsymbol{x}),\\quad\ng_{3}(\\boldsymbol{x})=h_3(\\boldsymbol{x}),\n$$\nand\n$$g_{4}(\\boldsymbol{x})=\\begin{cases}\\frac{1}{2}\\boldsymbol{x}+\\big(\\frac{1}{4},\\frac{3}{4}\\big),\\quad \\boldsymbol{x}\\in[0,1]\\times[0,\\frac{1}{2}],\\\\ \\frac{1}{2}\\boldsymbol{x}+\\big(\\frac{1}{4},-\\frac{1}{4}\\big),\\quad \\boldsymbol{x}\\in[0,1]\\times[\\frac{1}{2},1]\\end{cases}$$\n(see Figure \\ref{fig.3}(a)). $\\{g_{i}\\}_{i=1}^{4}$ induces four relations $\\{S_{i}\\}_{i=1}^{4}$ on $\\mathbb{T}^2$. We are interested in the Hausdorff dimension of the attractor generated by $\\{S_{i}\\}_{i=1}^{4}$.\nNote that the image of $S_4$ is a connected rectangle in $\\mathbb{T}^2$, but the image of $g_4$ is divided into two rectangles in $\\mathbb{R}^2$. It is easy to see that $\\{S_{i}\\}_{i=1}^{4}$ are not well-defined functions and are not contractive under the metric of $\\mathbb{T}^{2}$. As a result, we need to use the framework of a GIFS.\nConsider the GIFS $G=(V,E)$ defined on $\\mathbb{R}^{2}$ associated to $\\{g_{i}\\}_{i=1}^{4}$ with $V=\\{1,2\\}$ and $E=\\{e_{1},\\dots,e_{8}\\}$, where $\\mathbf{O}=\\{O_{1},O_{2}\\}$ with $O_{1}=(0,1)\\times(0,1\/2)$ and $O_{2}=(0,1)\\times(1\/2,1)$ is the invariant family, and\n$$e_{1},e_{2},e_{3}\\in E^{1,1},\\quad e_{4}\\in E^{1,2},\\quad e_{5},e_{6},e_{7}\\in E^{2,2},\\quad e_{8}\\in E^{2,1}$$\n(see Figure \\ref{fig.2}). The associated similitudes are defined as\n$$f_{e_{1}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{4}\\bigg),~~\nf_{e_{2}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{1}{4}\\bigg),~~\nf_{e_{3}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},\\frac{1}{4}\\bigg),~~\nf_{e_{4}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},-\\frac{1}{2}\\bigg),~~\n$$\n$$f_{e_{5}}=\\frac{1}{2}\\boldsymbol{x},~~\nf_{e_{6}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},0\\bigg),~~\nf_{e_{7}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},0\\bigg),~~\nf_{e_{8}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{3}{4}\\bigg).~~\n$$\nLet $\\Omega_{1}$ and $\\Omega_{2}$, viewed as $(0,1)\\times(0,1\/2)$ and $(0,1)\\times(1\/2,1)$, be the lower and upper pieces of the interior of $\\mathbb{T}^{2}$, respectively. Obviously, for $i=1,2$, and any $e\\in E$, there exists a diffeomorphism\n$\\varphi:O_{i}\\longrightarrow \\Omega_{i}$ such that $S_{e}$, defined as in \\eqref{eq(5.3)}, is a contractive similitude.\nLet $K$ and $K_{0}$ be the graph self-similar set of $G=(V,E)$ generated by $\\{S_{e}\\}_{e\\in E}$ and $\\{f_{e}\\}_{e\\in E}$, respectively. Then $\\dim_{{\\rm H}}(K)=\\dim_{{\\rm H}}(K_{0})=\\log(2+\\sqrt{2})\/\\log2=1.77155\\dots$.\n\\end{exam}\n\\begin{figure}[htbp]\n\\begin{center}\n\\tikzset{every picture\/.style={line width=0.75pt}}\n\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw    (170.44,123.61) .. controls (191.13,144.03) and (315.33,147.16) .. (351.66,123.43) ;\n\\draw [shift={(353.78,121.94)}, rotate = 142.77] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (164.83,118.04) .. controls (164.13,119.22) and (162.61,119.61) .. (161.44,118.91) .. controls (160.26,118.21) and (159.88,116.7) .. (160.57,115.52) .. controls (161.27,114.35) and (162.79,113.96) .. (163.96,114.65) .. controls (165.14,115.35) and (165.52,116.87) .. (164.83,118.04) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (364.66,118.71) .. controls (363.96,119.88) and (362.45,120.27) .. (361.27,119.58) .. controls (360.1,118.88) and (359.71,117.36) .. (360.41,116.19) .. controls (361.1,115.01) and (362.62,114.62) .. (363.79,115.32) .. controls (364.97,116.02) and (365.36,117.54) .. (364.66,118.71) -- cycle ;\n\\draw    (170.11,112.28) .. controls (196.73,89.81) and (320.37,97.44) .. (350.28,111.92) ;\n\\draw [shift={(352.78,113.28)}, rotate = 211.87] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (158.78,109.28) .. controls (92.94,41.46) and (229.06,41.9) .. (167.05,107.49) ;\n\\draw [shift={(165.11,109.5)}, rotate = 314.63] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (358.44,109.28) .. controls (292.61,41.46) and (428.08,41.9) .. (366.05,107.49) ;\n\\draw [shift={(364.11,109.5)}, rotate = 314.63] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (370.27,114.04) .. controls (438.68,48.82) and (437.64,183.9) .. (372.62,121.28) ;\n\\draw [shift={(370.63,119.32)}, rotate = 45.15] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (366.67,124.69) .. controls (432.19,192.81) and (297.1,192.37) .. (359.44,127.08) ;\n\\draw [shift={(361.39,125.08)}, rotate = 134.9] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (154.9,120.5) .. controls (86.64,185.87) and (87.37,50.79) .. (152.53,113.26) ;\n\\draw [shift={(154.52,115.22)}, rotate = 225.02] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\\draw    (166.01,124.02) .. controls (231.52,192.15) and (96.44,191.7) .. (158.78,126.41) ;\n\\draw [shift={(160.73,124.42)}, rotate = 134.9] [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.08]  [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle    ;\n\n\\draw (122,65.9) node [anchor=north west][inner sep=0.75pt]    {$e_{1}$};\n\\draw (82.5,112) node [anchor=north west][inner sep=0.75pt]    {$e_{2}$};\n\\draw (122,157) node [anchor=north west][inner sep=0.75pt]    {$e_{3}$};\n\\draw (248,82) node [anchor=north west][inner sep=0.75pt]    {$e_{4}$};\n\\draw (248,145) node [anchor=north west][inner sep=0.75pt]    {$e_{8}$};\n\\draw (390,65.9) node [anchor=north west][inner sep=0.75pt]    {$e_{5}$};\n\\draw (428.5,112) node [anchor=north west][inner sep=0.75pt]    {$e_{6}$};\n\\draw (390,157) node [anchor=north west][inner sep=0.75pt]    {$e_{7}$};\n\\draw (158,132) node [anchor=north west][inner sep=0.75pt]    {$\\mathbf{1}$};\n\\draw (358,132) node [anchor=north west][inner sep=0.75pt]    {$\\mathbf{2}$};\n\n\\end{tikzpicture}\n\\vspace{-1.1em}\n\\caption{The GIFS in Examples \\ref{exam(5.4)}.} \\label{fig.2}\n\\end{center}\n\\end{figure}\n\\begin{proof}\nFor convenience, we write $f_{e_{i}}=f_{i}$ for any $i\\in\\{1,\\dots,8\\}$. Let $\\mathcal{F}_{k}=E_{k}$ for $k\\geq1$, and let $\\mathcal{T}_{1}$ and $\\mathcal{T}_{2}$ be the neighborhood types of the root neighborhoods $[O_{1}]$ and $[O_{2}]$, respectively. Iterations of the root vertices are shown in Figure \\ref{fig.3}(b, c). All neighborhood types are generated after three iterations.\nTo construct the weighted incidence matrix in the reduced graph $\\mathbb{G}_{R}$, we note that\n$$\\mathbb{V}_{1}=\\{(f_{1},1),\\dots,(f_{8},1)\\}.$$\nDenote by $\\omega_{1},\\dots,\\omega_{8}$ the vertices in $\\mathbb{V}_{1}$ according to the above order. Then $[\\omega_{4}]=\\mathcal{T}_{2}$ and $[\\omega_{8}]=\\mathcal{T}_{1}$. Let $\\mathcal{T}_{3}:=[\\omega_{1}]$, $\\mathcal{T}_{4}:=[\\omega_{2}]$, $\\mathcal{T}_{5}:=[\\omega_{3}]$, $\\mathcal{T}_{6}:=[\\omega_{5}]$, $\\mathcal{T}_{7}:=[\\omega_{6}]$, $\\mathcal{T}_{8}:=[\\omega_{7}]$. Then\n$$\\mathcal{T}_{1}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{3}+\n\\mathcal{T}_{4}+\\mathcal{T}_{5}$$\nand\n$$\\mathcal{T}_{2}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{6}\n+\\mathcal{T}_{7}+\\mathcal{T}_{8},$$\nwhere we write $\\mathcal{T}_{i}=\\mathcal{T}_{i}(\\frac{1}{2})$ for convenience. Since $f_{13}=f_{21}$, the edge $e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{1}$ has three offspring, i.e.,\n$$(f_{11},2),(f_{12},2),(f_{14},2)\\in\\mathbb{V}_{2},$$\nwhich are of neighborhood types $\\mathcal{T}_{3},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{1},1)$ gives\n$$\\mathcal{T}_{3}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{3}+\\mathcal{T}_{4}.$$\nSince $f_{23}=f_{31}$, the edge $e_{2}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{2}$ has three offspring, i.e.,\n$$(f_{21},2),(f_{22},2),(f_{24},2)\\in\\mathbb{V}_{2}.$$\nNote that $[(f_{22},2)]=\\mathcal{T}_{4}$ and $[(f_{24},2)]=\\mathcal{T}_{2}$. Let $\\omega_{9}:=(f_{22},2)$ and $\\mathcal{T}_{9}:=[\\omega_{9}]$. Then\n$$\\mathcal{T}_{4}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{9}.$$\nNote that $\\omega_{3}$ has four offspring, i.e.,\n$$(f_{31},2),(f_{32},2),(f_{33},2),(f_{34},2)\\in\\mathbb{V}_{2},$$\nwith $[(f_{32},2)]=\\mathcal{T}_{4}$, $[(f_{33},2)]=\\mathcal{T}_{5}$ and $[(f_{34},2)]=\\mathcal{T}_{2}$. Let $\\omega_{10}:=(f_{31},2)$ and $\\mathcal{T}_{10}:=[\\omega_{10}]$. Then\n$$\\mathcal{T}_{5}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{5}+\\mathcal{T}_{10}.$$\nSince $f_{213}=f_{221}$, the edge $e_{2}e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{9}$ has three offspring, i.e.,\n$$(f_{211},2),(f_{212},2),(f_{214},2)\\in\\mathbb{V}_{3},$$\nwhich are of neighborhood types $\\mathcal{T}_{10},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{22},2)$ gives\n$$\\mathcal{T}_{9}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{10}.$$\nSince $f_{313}=f_{321}$, the edge $e_{3}e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{10}$ has three offspring, i.e.,\n$$(f_{311},2),(f_{312},2),(f_{314},2)\\in\\mathbb{V}_{3},$$\nwhich are of neighborhood types $\\mathcal{T}_{10},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{31},2)$ gives\n$$\\mathcal{T}_{10}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{10}.$$\nLet $\\mathcal{T}_{11}:=[(f_{65},2)]$ and $\\mathcal{T}_{12}:=[(f_{75},2)]$. Using the same argument, it can be checked directly that\n$$\\mathcal{T}_{6}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{6}+\\mathcal{T}_{7},~~\n\\mathcal{T}_{7}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{11},$$\n$$\\mathcal{T}_{8}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{8}+\\mathcal{T}_{12},~~\n\\mathcal{T}_{11}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{12},~~\n\\mathcal{T}_{12}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{12}.$$\nHence the weighted incidence matrix is\n$$\nA_{\\alpha}=\\Big(\\frac{1}{2}\\Big)^{\\alpha}\\left({\\begin{array}{cccccccccccc}\n0&1&1&1&1&0&0&0&0&0&0&0\\\\\n1&0&0&0&0&1&1&1&0&0&0&0\\\\\n0&1&1&1&0&0&0&0&0&0&0&0\\\\\n0&1&0&1&0&0&0&0&1&0&0&0\\\\\n0&1&0&1&1&0&0&0&0&1&0&0\\\\\n1&0&0&0&0&1&1&0&0&0&0&0\\\\\n1&0&0&0&0&0&1&0&0&0&1&0\\\\\n1&0&0&0&0&0&1&1&0&0&0&1\\\\\n0&1&0&1&0&0&0&0&0&1&0&0\\\\\n0&1&0&1&0&0&0&0&0&1&0&0\\\\\n1&0&0&0&0&0&1&0&0&0&0&1\\\\\n1&0&0&0&0&0&1&0&0&0&0&1\n\\end{array}}\\right)=:\\Big(\\frac{1}{2}\\Big)^{\\alpha}\\widetilde{A}_{\\alpha},\n$$\nand the maximal eigenvalue of $\\widetilde{A}_{\\alpha}$ is $2+\\sqrt{2}$.\nThe GIFS $G=(V,E)$ defined on $\\mathbb{R}^{2}$ satisfies (GFTC) with $\\{O_{i}\\}_{i=1}^{2}$ being a GFTC-family and $\\{f_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nBy Proposition \\ref{prop(5.2)}, the GIFS $G=(V,E)$ defined on $\\mathbb{T}^{2}$ satisfies (GFTC) with $\\{\\Omega_{i}\\}_{i=1}^{2}$ being a GFTC-family and $\\{S_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nBy Theorem \\ref{thm(41.1)}, $\\dim_{{\\rm H}}(K)=\\dim_{{\\rm H}}(K_{0})=\\log(2+\\sqrt{2})\/\\log2=1.77155\\dots$.\n\\end{proof}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{0.png}\\label{fig.(a)}}\n\\qquad\\qquad\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{1.png}\\label{fig.(b)}}\n\\\\\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{2.png}\\label{fig.(c)}}\n\\qquad\\qquad\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{G9.png}\\label{fig.(d)}}\n\n\\caption{(a) $\\{g_{i}\\}_{i=1}^{4}$. (b) The first iteration of $G=(V,E)$ on $\\mathbb{R}^{2}$ associated to $\\{g_{i}\\}_{i=1}^{4}$. (c) The second iteration. (d) The corresponding graph self-similar set.}\\label{fig.3}\n\\end{figure}\n\n\n\\begin{rema}\\label{rema(5.1)}\nUnlike $\\mathbb{T}^{2}$, the family of contractive similitudes associated to a GIFS defined on $\\mathbb{R}^{2}$ can be characterized explicitly. In fact, Proposition \\ref{prop(5.2)} is not needed in the proof of Example \\ref{exam(5.4)}. We may use Theorem \\ref{thm(41.1)} and a similar arguement as in the proof of Example \\ref{exam(5.4)} to obtain the same result.\n\\end{rema}\n\n\n\\noindent\\textbf{Acknowledgements} The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.\\\\\n\n\n\\noindent\\textbf{Declaration}\\\\\n\n\n\\noindent\\textbf{Competing interests} The authors hereby declare that there is no conflict of interest regarding this work.\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNonequilibrium systems are common in nature because they are, in general, subject to thermal gradients, chemical potential gradients or may be triggered by time-dependent forces. Heat transport is one such example of nonequilibrium systems where the heat carriers could be electrons, phonons, magnons, etc. To study heat transport in phononic systems, one considers a finite junction part, which can be an insulator, connected with two heat baths that are maintained at different temperatures. In the past decade, the main focus was on the calculation of the steady state heat current or heat flux flowing through the junction part from the leads \n\\cite{Caroli,Meir,Rego-Kirczenow-1998,Segal,Mingo-Yang-2003,Yamamoto-2006,Dhar1,Wang-prb06,Wang-pre07,WangJS-europhysJb-2008}. For diffusive systems, the answer is given by Fourier's law \\cite{Dhar, Lepri, Lebowitz} which is true only in the linear response regime, i.e., when the temperature difference between the baths is small. However for harmonic or ballistic systems, the heat current is given by a Landauer-like formula \\cite{Rego-Kirczenow-1998,Dhar1,WangJS-europhysJb-2008} which was first derived for electronic transport. Landauer formula on the contrary to the Fourier's law is true for arbitrary temperature differences between the leads. No such explicit expression for current is known for transient states.  In recent times, several works \\cite{eduardos-paper,eduardos-paper1} followed to answer what happens to current in the transient regime. This is an important question both from the theoretical and experimental points of view.  \n\nMuch attention has been given to phonon transport, in particular on thermal devices and on controlling heat flow \\cite{baowen-review}. With the advent of technology it is now possible to study transport problems and observe a single mode of vibration in small systems with few degrees of freedom \\cite{oconell}. These systems shows strong thermal fluctuations which play an important role because thermal fluctuations can lead to instantaneous heat transfer from colder to hotter lead. It is therefore necessary to talk about the statistical distribution of heat flux for these systems. In the electronic literature the distribution $P(Q_L)$ of the charge $Q_L$, flowing from the left lead to the junction part, was answered by calculating the corresponding CGF, ${\\cal Z}(\\xi)= \\langle e^{i \\xi Q_L} \\rangle$, and is given by the celebrated Levitov-Lesovik formula \\cite{Levitov,Levitov1,Levitov-PRB}. This methodology is also known as the {\\it full counting statistics} \\cite{otherworks,otherworks1,otherworks2,Klich,Pilgram,Bagrets,Gogolin,Urban,Gutman,fluct-theorems}\nin the field of electronic transport. Experimentally the electron counting statistics has been measured in quantum-dot systems \\cite{Flindt-etal-2009,Gustavsson-etal-2006}. However few experiments have been done for phonons \\cite{measurement}. In the phononic case Saito and Dhar \\cite{Saito-Dhar-2007} gave an explicit expression of the CGF.  Ren et al.\\ gave a result for two-level systems \\cite{ren-jie}. Full counting statistics of energy fluctuations in a driven quantum resonator is studied by Clerk \\cite{AAClerk-2011}. The main focus in these papers was on the long-time limit and SSFT \\cite{noneq-fluct,fluct-theorem-1,fcs-bijay,kundu,Esposito-review-2009, Hanggi2}. Using the NEGF method \\cite{schwinger-keldysh,rammer86} and two-time measurement \\cite{Esposito-review-2009, Hanggi2,Hanggi3,Hanggi4} concept, Wang~et al.\\ \\cite{fcs-bijay} gave an explicit expression for the CGF which is valid for both transient and steady state regimes.   \n\nIn this paper, we extend our previous work in Ref.~\\onlinecite{fcs-bijay} and derive the CGF in a more general scenario, i.e., in the presence of both the temperature difference and the time-dependent driving force. We analyze the cumulants of heat $Q_L$ for three different initial conditions of the density operator and study the effects on both transient and steady state regimes. We also derive the CGF for the joint probability distribution of left and right lead heat $P(Q_L,Q_R)$ which help us to obtain the correlations between $Q_L$ and $Q_R$. By calculating CGF for $P(Q_L,Q_R)$ we can immediately obtain the CGF for the total entropy that flows to the leads. We present analytical expressions of the CGF's in the steady state and discuss the SSFT. Our method can be easily generalized for multiple heat baths.\n\nThe plan of the paper is as follows. We start in Sec.~II by introducing our model. Then in Sec.~III we define current and corresponding quantum heat operator followed by the definition of CGF for $Q_L$ using the two-time measurement concept, in Sec.~IV. In Sec.~V we derive the CGF using Feynman's path integral method and in Sec.~VI we use Feynman's diagrammatic technique to derive the CGF. We discuss the steady state result and fluctuation theorems in Sec.~VII. In Sec. VIII and IX we discuss how to calculate the CGF's numerically in transient regime and give numerical results for one-dimensional (1D) linear chain model, connected with Rubin heat baths, for three different initial conditions of the density operator. Then in Sec.~X we obtain the CGF for joint probability distribution of heat transferred $P(Q_L,Q_R)$ and discuss correlations and total entropy flow. In. Sec.~XI we give the long-time limit expression for the driven part of the full CGF. In Sec.~XII we discuss another definition of generating function due to Nazarov and discuss the corresponding long-time limit.  We found that using this definition, the generating function \ndoes not obey the Gallavotti-Cohen (GC) fluctuation symmetry. \nWe conclude with a short discussion in Sec.~XIII. Few appendices give some details of technique nature. In particular, an electron system of a tight-binding model is treated using our method.\n  \n\n\\section{The model}\nOur model consists of a finite harmonic junction part, which we denote by $C$, coupled to two heat baths, the left ($L$) and the right ($R$), kept at two different temperatures $T_L$ and $T_R$, respectively. To model the heat baths, we consider an infinite collection of coupled harmonic oscillators. We take the three systems to be decoupled initially and to be described by the Hamiltonians,\n\\begin{equation}\n{\\cal H}_\\alpha = \\frac{1}{2} p_\\alpha^T p_\\alpha + \\frac{1}{2} u_\\alpha^T K^\\alpha u_\\alpha, \\quad \\alpha=L, C, R,\n\\end{equation}\nfor the left, right, and the finite central region. The leads are assumed to be semi-infinite. Masses are absorbed by defining $u = \\sqrt{m}\\,\nx$. $u_\\alpha$ and $p_\\alpha$ are column vectors of coordinates and momenta. $K^\\alpha$ is the spring constant matrix of region $\\alpha$.\nCouplings of the center region with the leads are turned on either adiabatically from time $t=-\\infty$, or switched on abruptly at $t=0$.\nThe interaction Hamiltonian takes the form \n\\begin{equation}\n{\\cal H}_{{\\rm int}} = u_L^T \\, V^{LC} \\,u_{C} + u_R^T\\, V^{RC}\\, u_{C}.\n\\end{equation}\nFor $t>0$, an external time-dependent force is applied only to the center atoms, which is of the form\n\\begin{equation}\n{\\cal V}_C(t) =-f^T(t)\\,u_C,\n\\end{equation}\nwhere $f(t)$ is the time-dependent force vector. The driving force couples only with the position operators of the center. The force can be in the form of electromagnetic field. Coupling of this form helps us to obtain analytical solution for the CGF of heat flux.\nSo the full Hamiltonian for $t>0$ (in the Schr\\\"odinger picture) is\n\\begin{equation}\n{\\cal H}(t) = {\\cal H}(0^{-}) + {\\cal V}_C(t) = {\\cal H}_C + {\\cal H}_L+ {\\cal H}_R+ {\\cal H}_{\\rm int}+ {\\cal V}_C(t).\n\\end{equation}\nIn the next section we will define current operator and the corresponding heat operator based on this Hamiltonian.\n \n\\section{Definition of current and heat operators}\nIt is possible to define the current operator ${\\cal I}$ depending on where we want to measure the current. Here we consider the current flowing from the left lead to the center system and ${\\cal I}_{L}$ is defined (in Heisenberg picture) as\n\\begin{equation}\n{\\cal I}_{L}(t) = - \\frac{d{\\cal H}^{H}_L(t)}{dt}=\\frac{i}{\\hbar}[{\\cal H}^{H}_L(t), {\\cal H}_H(t)]= p_L^{T}(t)\\,V^{LC}\\,u_C(t),\n\\label{current}\n\\end{equation}\nwhere ${\\cal H}_H(t)$ is the (time-dependent) Hamiltonian in the Heisenberg picture at time $t$. The corresponding heat operator can be written down as\n\\begin{equation}\n\\label{eq-hatQ}\n{\\cal Q}_{L}(t)=\\int_0^t {\\cal I}_{L}(t')\\,dt' = {\\cal H}_L(0)-{\\cal H}^{H}_L(t),\n\\end{equation}\nwhere ${\\cal H}_L \\big[={\\cal H}_L(0)\\big]$ is the Schr\\\"odinger operator of the free left lead and \n\\begin{equation}\n{\\cal H}^{H}_L(t)= {\\cal U}(0,t)\\, {\\cal H}_L\\, {\\cal U}(t,0),\n\\end{equation}\nand ${\\cal U}(t,t')$ is the evolution operator corresponding to the full Hamiltonian ${\\cal H}(t)$ and satisfies the Schr\\\"odinger equation\n\\begin{equation}\ni\\hbar \\,{ \\partial {\\cal U}(t,t') \\over \\partial t} = {\\cal H}(t)\\, {\\cal U}(t,t').\n\\end{equation}\nThe formal solution of this equation is (assuming $t \\geq t'$)\n\\begin{equation}\n{\\cal U}(t,t')=T \\exp\\left\\{ - \\frac{i}{\\hbar} \\int_{t'}^t {\\cal H}(\\bar{t})\\, d\\bar{t} \\right\\},\n\\label{eq-unitary}\n\\end{equation}\nwhere $T$ is the time-order operator where time increases from right to left. Also ${\\cal U}^{\\dagger}(t,t')={\\cal U}(t',t)$. $Q$ of non-calligraphic font will be a classical variable. \n\nIn the following section we derive the CGF based on this definition of heat operator and using two-time measurement scheme.\n\n\\section{Definition of the generating function for heat operator}\nOur primary interest here is to calculate the moments or cumulants of the heat energy transferred in a given time interval $t_M$. Hence, it is advantageous to calculate the generating function instead of calculating moments directly. Since ${\\cal Q}_{L}$ is a quantum operator, there are subtleties as to how exactly the generating function should be defined. Naively we may use $\\langle e^{i \\xi {\\cal Q}_{L}} \\rangle $. But this definition fails the fundamental requirement of positive definiteness of the probability distribution.\n\nHere we will give two different definitions that are used to calculate the generating function for such problem. The first definition comes from the idea of two-time measurements and based on this concept the CGF can be written down as\n\\begin{equation}\n{\\cal Z}(\\xi)=\\langle e^{i \\xi {\\cal H}_L} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\rangle'\n\\label{eq-Z-two-time}\n\\end{equation}\nwhich we will discuss in great detail in this section.\n\nThe second definition of the CGF is\n\\be\n{\\cal Z}_1(\\xi) = \\langle \\bar{T} e^{i\\xi {\\cal Q}_{L}\/2} T e^{i\\xi {\\cal Q}_{L}\/2} \\rangle,\n\\label{eq-Z1-Nazarov}\n\\ee\nwhere $\\bar{T}$ is the anti-time order operator. The time (or anti-time) order is meant to apply to the integrand when the\nexponential is expanded and ${\\cal Q}_{L}$ is expressed as integral over ${\\cal I}_{L}$ as in Eq.~(\\ref{eq-hatQ}). This definition is used by Nazarov et al.\\ \\cite{otherworks,otherworks1} mostly for the electronic transport case. In the last section we will show how this generating function can be derived starting from ${\\cal Z}(\\xi)$ under a particular approximation and will also give explicit expression for ${\\cal Z}_{1}(\\xi)$ in the long-time limit. \n\nIn the following we will discuss the idea of two-time measurement and derive the corresponding CGF ${\\cal Z}(\\xi)$.\n \n\\subsection{Two-time measurement}\nThe heat operator in Eq.~(\\ref{eq-hatQ}) depends on the left-lead Hamiltonian ${\\cal H}_{L}$ at time 0 and $t$. The concept of two-time measurement implies the measurement of a certain operator (in this case ${\\cal H}_{L})$ at two different times. Here the measurement is in the sense of quantum measurement\nof von Neumann \\cite{neumann}.   \n\nLet us first assume that the full system is in a pure state $|\\Psi_0\\rangle$ at $t=0$.\nWe want to do measurement of the energy associated with the operator\n${\\cal H}_L$.  According to quantum mechanics, the result of a measurement can\nonly be an eigenvalue of the (Schr\\\"odinger) operator ${\\cal H}_L$ and the wave function\ncollapses into an eigenstate of ${\\cal H}_L$.  Let\n\\begin{equation}\n{\\cal H}_L | \\phi_a \\rangle = a | \\phi_a \\rangle,\\quad \\Pi_a = |\\phi_a\\rangle \\langle \\phi_a  |,\n\\end{equation}\nwhere $\\Pi_a$ is the projector into the state $|\\phi_a\\rangle$ satisfying $\\Pi_a^2 = \\Pi_a$,\nand $\\sum_a \\Pi_a = 1$.  We assume the eigenvalues are discrete (this is always so\nif the lattice system is finite). After the measurement at time $t=0$, the wave function\nis proportional to $\\Pi_a | \\Psi_0 \\rangle$ if the result of the measurement is the energy $a$ and the probability of such event happen is $\\langle \\Psi_0 | \\Pi_a^2 | \\Psi_0 \\rangle$. Let's propagate this state to time $t$ and do a second measurement of the lead\nenergy, finding that the result is $b$.  The wave function now becomes proportional to $\\Pi_b \\, {\\cal U}(t,0) \\, \\Pi_a \\,  | \\Psi_0 \\rangle$.\nThe joint probability of getting $a$ at time $0$ and $b$ at time $t$ is\nthe norm (inner product) of the above (unnormalized) state. \n\nIf the initial state is in a mixed state, we add up the initial probability classically, i.e., if \n\\begin{equation}\n\\rho(0) = \\sum_k w_k | \\Psi_0^k \\rangle\\langle \\Psi_0^k |,\\quad w_k > 0, \\quad \\sum_k w_k = 1,\n\\end{equation}\nthe joint probability distribution of two-time measurement output is\n\\begin{eqnarray}\nP(b,a) &=& \\sum_k w_k \\langle \\Psi_0^k | \\,\\Pi_a\\, {\\cal U}(0,t) \\,\n\\Pi_b \\,{\\cal U}(t,0) \\,\\Pi_a \\,| \\Psi_0^k \\rangle  \\nonumber \\\\\n&=& {\\rm Tr} \\bigl[ \\Pi_a \\,\\rho(0) \\, \\Pi_a \\, {\\cal U}(0,t) \\, \\Pi_b \\, {\\cal U}(t,0)\\bigr].\n\\end{eqnarray}\nBy definition, we see that $P(b,a)$ is a proper probability in the sense that $P(b,a) \\ge 0$ and $\\sum_{a,b} P(b,a) = 1$.\nThen the generating function for $Q_L=a-b$ is defined as\n\\begin{eqnarray}\n{\\cal Z}(\\xi) &=& \\langle e^{i\\xi (a-b)} \\rangle =  \\sum_{a,b} e^{i \\xi(a-b)} P(b,a) \\nonumber \\\\\n&=&  \\sum_{a,b} e^{i \\xi (a-b)} {\\rm Tr} \\bigl[\\Pi_a \\, \\rho(0) \\, \\Pi_a \\, {\\cal U}(0,t) \\, \\Pi_b \\, {\\cal U}(t,0) \\bigr] \\nonumber \\\\\n& = & \\langle e^{i \\xi {\\cal H}_L} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\rangle' \\nonumber \\\\\n\\label{eqZ2-def}\n& = &  \\langle e^{i \\xi {\\cal H}_L\/2} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\, e^{i \\xi {\\cal H}_L\/2 } \\rangle'.\n\\end{eqnarray}\nwhere we define \\cite{neumann}\n\\begin{equation}\n\\rho'(0) = \\sum_{a} \\Pi_a\\, \\rho(0) \\Pi_a.\n\\label{projected}\n\\end{equation}\nwhich we call as the {\\it projected} density matrix.\n\nIf the initial state at $t=0$ is a product state  i.e., $\\rho(0)=\\rho(-\\infty)=\\rho_L \\otimes \\rho_C \\otimes \\rho_R $, where the left, center and right density\nmatrices are in equilibrium distributions corresponding to the \nrespective temperatures:\n$\\rho_\\alpha={e^{-\\beta_\\alpha {\\cal{H}}_\\alpha}}\/{{\\rm Tr} [e^{-\\beta_\\alpha\n    {\\cal H}_\\alpha}]}$ for $\\alpha=L,C,R$ and $\\beta_{\\alpha}=1\/(k_{\\rm B}\nT_{\\alpha})$, then the projection operators $\\Pi_a$ do not play any role and $\\langle....\\rangle'={\\rm Tr}\\Bigl[\\rho(-\\infty)\\cdots \\Bigr]=\\langle....\\rangle$. \n\nHere we will derive the CGF for three different initial conditions:\n\\begin{itemize}\n\\item{Product initial state, i.e., $\\rho(-\\infty)$, which corresponds to sudden switch-on of the coupling between the leads and the center.}\n\\item{steady state as the initial state, i.e., $\\rho(0)$, which we can obtain, starting with the decoupled Hamiltonians at $t=-\\infty$, switch on the couplings between the center region and the leads, adiabatically upto time $t=0$.}\n\\item{{\\it projected} density matrix $\\rho'(0)$ considering $\\rho(0)$ as the steady state, i.e., taking the effects of measurements into account.}\n\\end{itemize}   \n\nIn the following sections we will analytically show that the CGF's corresponding to different initial conditions reach the same steady state in the long-time limit and hence is independent of initial distributions. However for short time transient behavior depends significantly on initial conditions and also the measurements do play an important role. \n\n\\section{calculation for ${\\cal Z}(\\xi)$ for initial states $\\rho(0)$ and $\\rho'(0)$}\nIn this section we will give detail derivation for ${\\cal Z}(\\xi)$, using Feynman path-integral formalism, for two different initial density operators $\\rho(0)$ and $\\rho'(0)$. \n\n\\subsection{Removing the projection $\\Pi_a$ at $t=0$}\nThe projection by $\\Pi_a$ at $t=0$ Eq.~(\\ref{projected}) to the density matrix creates a problem\nfor formulation in path integrals.  We can remove it following Ref.~\\onlinecite{Esposito-review-2009} by putting it into part of an evolution of ${\\cal H}_L$, just like the factor associated with the generating function variable $\\xi$, with a price we have to pay, introducing another integration variable $\\lambda$. The key observation is that we can represent the projector by the Dirac\n$\\delta$ function\n\\begin{eqnarray}\n\\Pi_a & \\propto & \\delta(a-{\\cal H}_L)  \\nonumber \\\\\n&=& \\int_{-\\infty}^\\infty \\frac{d\\lambda}{2\\pi} \\,e^{-i\\lambda(a-{\\cal H}_L)}.\n\\label{eq-delta}\n\\end{eqnarray}\nFor this to make sense, we assume the spectrum of the energy of\n${\\cal H}_L$ is continuous, which is valid if we take the large size limit first.\nIdentifying $\\Pi_a$ as $\\delta(a-{\\cal H}_L)$ with a continuous variable\n$a$ introduces an constant proportional to the Dirac $\\delta(0)$ to $\\rho'(0)$, since\n$\\Pi_a$ is now normalized as $\\Pi_a \\Pi_b = \\delta(a-b) \\Pi_a$.\nHowever, this constant can be easily fixed by the condition\n${\\cal Z}(0)=1$.  So using $\\Pi_a = \\delta(a -{\\cal H}_L)$ will not cause\ndifficulty.  \n\nSubstituting the Fourier integral representation into $\\rho'$ we obtain\n\\begin{eqnarray}\n\\rho'(0) &\\propto&\\int da\\, \\Pi_a\\, \\rho(0) \\Pi_a  \\\\\n&=& \\int \\frac{d\\lambda}{2\\pi} e^{i\\lambda {\\cal H}_L} \\rho(0) e^{-i\\lambda {\\cal H}_L}.\n\\end{eqnarray}\nUsing the symmetric form of ${\\cal Z}$, Eq.~(\\ref{eqZ2-def}), we have\n\\begin{eqnarray}\n{\\cal Z}(\\xi)&\\propto& \\int \\frac{d\\lambda}{2\\pi} {\\rm Tr} \\bigl\\{ \n\\rho(0) \\,{\\cal U}_{\\xi\/2-\\lambda}(0,t)\\, {\\cal U}_{-\\xi\/2-\\lambda}(t,0) \\bigr\\}  \\nonumber \\\\\n&=&  \\int \\frac{d\\lambda}{2\\pi}\\; {\\cal Z}(\\xi, \\lambda),\n\\label{eq-Uxilambda}\n\\end{eqnarray}\nwhere ${\\cal U}_{x}(t,t')$ is the modified evolution operator of an effective Hamiltonian given by\n\\begin{equation}\n{\\cal H}_{x}(t) = e^{i x {\\cal H}_L} {\\cal H}(t) e^{-i x {\\cal H}_L},\n\\end{equation}\nwhere $x$ is a real parameter which in this case is $\\xi\/2 -\\lambda$ and $-\\xi\/2 -\\lambda$. Finally ${\\cal U}_{x}(t,t')$ is given by ($t \\geq t'$)\n\\begin{eqnarray}\n{\\cal U}_{x}(t,t') &=& e^{i x {\\cal H}_L} {\\cal U}(t,t') e^{-i x {\\cal H}_L} \\nonumber \\\\\n&=& \\sum_{n=0}^\\infty \\left(-\\frac{i}{\\hbar} \\right)^n \\int_{t'}^t dt_1 \\int_{t'}^{t_1} dt_2 \\cdots \\int_{t'}^{t_{n-1}}dt_n\n\\nonumber \\\\ \n&& \\times e^{i x {\\cal H}_L} {\\cal H}(t_1) {\\cal H}(t_2) \\cdots {\\cal H}(t_n) e^{-i x {\\cal H}_L} \\nonumber \\\\\n\\label{eq-Ulambda}\n&=& T \\exp\\left\\{ - \\frac{i}{\\hbar} \\int_{t'}^t {\\cal H}_x(t') dt' \\right\\}.\n\\end{eqnarray}\nIt is important to note that substituting $\\lambda=0$ in ${\\cal Z}(\\xi,\\lambda)$ gives us the initial density matrix $\\rho(0)$. \n\nNow we will give an explicit expression of the modified Hamiltonian ${\\cal H}_x$ which helps us to calculate the CGF using path integral.\n\n\n\\subsection{The expression for ${\\cal H}_{x}$}\nThe modified Hamiltonian is the central quantity for calculating CGF. It is the Heisenberg evolution of the full Hamiltonian ${\\cal H}(t)$ (in Schr\\\"odinger picture) with respect to ${\\cal H}_L$. Since ${\\cal H}_L$ commutes with\nevery term $\\tilde{\\cal H}$ where ${\\cal H}(t)= {\\tilde {\\cal H}} + u_L^T V^{LC} u_C$, except the coupling term $u_L^T V^{LC} u_C$, we can write\n\\begin{eqnarray}\n{\\cal H}_{x}(t) &=& e^{i x {\\cal H}_L} {\\cal H}(t) e^{-i x {\\cal H}_L} \\nonumber \\\\\n& = &  e^{i x {\\cal H}_L} \\bigl({\\tilde {\\cal H}} + u_L^T V^{LC} u_C \\bigr) e^{-i x {\\cal H}_L} \\nonumber \\\\\n& = & {\\cal H}(t) + \\bigl( u_L(\\hbar x) - u_L\\bigr)^T V^{LC} u_C,\n\\end{eqnarray}\nwhere $u_L(\\hbar x) = e^{i x {\\cal H}_L} u_L e^{-i x {\\cal H}_L}$ is the free left lead\nHeisenberg evolution to time $t = \\hbar x$.  $u_L(\\hbar x)$ can be obtained\nexplicitly as\n\\begin{equation}\nu_L(\\hbar x) = \\cos(\\sqrt{K_L} \\hbar x) u_L + \\frac{1}{\\sqrt{K_L}} \\sin( \\sqrt{K_L} \\hbar x) p_L.\n\\end{equation} \nThe matrix $\\sqrt{K_L}$ is well-defined as the matrix $K_L$ is positive definite. $u_L$ and $p_L$ are the initial conditions at $t=0$.\nThe final expression for ${\\cal H}_x(t)$ is \n\\begin{equation}\n{\\cal H}_x(t) = {\\cal H}(t) + \\bigl[ u_L^T {\\cal C}(x) + p_L^T {\\cal S}(x) \\bigr]u_C,\n\\label{modified}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\label{eq-C}\n{\\cal C}(x) &=& \\bigl(\\cos(\\hbar x \\sqrt{K_L}) -I\\bigr) V^{LC}, \\\\\n\\label{eq-S}\n{\\cal S}(x) &=& (1\/\\sqrt{K_L}) \\sin( \\hbar x \\sqrt{K_L}) V^{LC}.\n\\end{eqnarray}\nThe effective Hamiltonian now has two additional term with respect to the full ${\\cal H}(t)$. The term $u_L^T {\\cal C}(x)u_C $ is like the harmonic coupling term which modifies the coupling matrix $V^{LC}$. \n\n\nIn the following we calculate the two parameter generating function ${\\cal Z}(\\xi,\\lambda)$ using Eq.~(\\ref{eq-Uxilambda}).\n\n\\subsection{Expression for ${\\cal Z}(\\xi,\\lambda)$}\nThe expression for ${\\cal Z}(\\xi,\\lambda)$ can be written down on the contour as (see Fig.~\\ref{fig1})\n\\begin{equation}\n{\\cal Z}(\\xi,\\lambda) = {\\rm Tr} \\Bigl[ \\rho(0) T_c e^{-\\frac{i}{\\hbar} \\int_C {\\cal H}^{x}(\\tau) d\\tau} \\Bigr].\n\\label{eq-Z-contour} \n\\end{equation} \nwhere $T_c$ is the contour-ordered operator which orders operators according to their contour time argument, earlier contour time places an operator to the right. The contour function $x(\\tau)$ is defined as 0 whenever $t < 0$ or $t>t_M$, and when $0 < t < t_M$, i.e., within the measurement time interval, for upper branch of the contour $x^{+}(t) = -\\xi\/2 - \\lambda$, and for lower branch $x^{-}(t) = \\xi\/2 - \\lambda$. \n \nFor the moment, let us forget about the other lead and concentrate only on the left lead and center.\nThe effect of other lead simply modifies the self-energy of the\nleads additively, according to Feynman and Vernon \\cite{Feynman-Vernon-1963}. Using Feynman path integral technique we can write\n\\begin{equation}\n{\\cal Z}(\\xi,\\lambda)= \\int {\\cal D}[u_C] {\\cal D}[u_L] \\rho(-\\infty) e^{(i\/\\hbar) \n\\int_K d\\tau ( {\\cal L}_C + {\\cal L}_L + {\\cal L}_{LC} ) }.\n\\label{eq-Z-keldysh}\n\\end{equation}\nNote that in Eq.~(\\ref{eq-Z-contour}), the \ncontour $C$ is from 0 to $t_M$ and back, while that in Eq.~(\\ref{eq-Z-keldysh}) is on the\nKeldysh contour $K$, that is, from $-\\infty$ to $t_M$ and back to take into account\nof adiabatic switch on, replacing $\\rho(0)$ by $\\rho(-\\infty)$.   Their relation\nis\n\\begin{equation}\n\\rho(0) = {\\cal U}(0, -\\infty) \\rho(-\\infty) {\\cal U}(-\\infty, 0).\n\\end{equation}\nWe can identify the Lagrangian's as \n\\begin{eqnarray}\n{\\cal L} &=& {\\cal L}_L + {\\cal L}_C + {\\cal L}_{LC}, \\nonumber \\\\\n{\\cal L}_L &=& \\frac{1}{2}\\dot{u}_L^2 - \\frac{1}{2} u_L^T K^L u_L, \\nonumber \\\\\n{\\cal L}_C &=& \\frac{1}{2} \\dot{u}_C^{2}  +f^T u_C -\\, \\frac{1}{2} u_C^T \\bigl(K^C-{\\cal S}^T{\\cal S}\\bigr) u_C, \\nonumber \\\\\n{\\cal L}_{LC} &=& - \\dot{u}^T_L {\\cal S} u_C - u_L^T \\bigl(V^{LC} + {\\cal C}\\bigr) u_C.\n\\label{Lagrangian}\n\\end{eqnarray}\nFor notational simplicity, we have dropped the argument $\\tau$.\nThe vector or matrices $f$, ${\\cal C}$, and ${\\cal S}$ are parametrically dependent on\nthe contour time $\\tau$.  They are zero except on the interval\n$0 < t < t_M$.  $f$ is the same on the upper and lower branches,\nwhile ${\\cal C}$ and ${\\cal S}$ take different values depending on $x(\\tau)$.\n\nNow the lead part can be integrated out by performing Gaussian integral \\cite{Feynman-Vernon-1963}. Since the\ncoupling between the lead and the center is linear, it is plausible that the result will be\na quadratic form in the exponential, i.e., another Gaussian.  To find exactly\nwhat it is, we convert the path integral back to the\ninteraction picture (with respect to ${\\cal H}_L$) operator form and evaluate the expression by standard perturbative expansion. The only difference is that now the coupling with the center involving both $u_L$ and $\\dot{u}_L$. The result for the influence\nfunctional is given by \\cite{Stockburger}\n\\begin{eqnarray}\nI_L[u^C(\\tau)] &\\equiv & \\int {\\cal D}[u_L] \\rho_L(-\\infty) \ne^{\\frac{i}{\\hbar} \\int d\\tau ({\\cal L}_L + {\\cal L}_{LC}) } \\nonumber  \\\\\n&=& {\\rm Tr}\\Bigl[\\frac{e^{-\\beta_L H_L}}{Z_L} T_c e^{ -\\frac{i}{\\hbar} \\int d\\tau {\\cal V}_I(\\tau) } \\Bigr] \\nonumber \\\\\n&=& e^{-\\frac{i}{2\\hbar} \\int d\\tau \\int d\\tau' u_C^T(\\tau) \\Pi(\\tau, \\tau') u_C(\\tau') }.\\\n\\end{eqnarray}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.5\\columnwidth]{fig1.eps\n\\caption{\\label{fig1}The complex-time contour in the Keldysh formalism. The path of the contour begins at time $t_{0}$, goes to time $t_M$, and then goes back to time $t=t_{0}$. $\\tau$ and $\\tau'$ are complex-time variables along the contour. $t_{0}=-\\infty$ and $0$ corresponds to Keldysh contour K and C, respectively.} \n\\end{figure}\nIn the influence functional, the contour function $u_C(\\tau)$ is not a \ndynamical variable but a parametric function. ${\\cal V}_I(\\tau)$ is the interaction picture operator with respect to the Hamiltonian ${\\cal H}_L$ and is given by\n\\begin{eqnarray}\n{\\cal V}_I(\\tau) &=& p_L^T {\\cal S} u_C + u_L^T(V^{LC} + {\\cal C}) u_C+\\frac{1}{2} u_C^T {\\cal S}^T {\\cal S} u_C \\nonumber \\\\\n\\label{eq-SSterm} \n&=&u_L^T\\bigl(\\tau + \\hbar x(\\tau)\\bigr)V^{LC}u_C + \\frac{1}{2} u_C^T {\\cal S}^T {\\cal S} u_C.\n\\end{eqnarray}  \n\nThe important influence functional self-energy on contour is given by\n\\begin{eqnarray}\n\\Pi(\\tau,\\tau')= \\Sigma_L^A(\\tau,\\tau') &+& \\Sigma_L(\\tau,\\tau') + {\\cal S}^T{\\cal S}\\,\\delta(\\tau, \\tau'), \\\\\n\\Sigma_L^A(\\tau,\\tau') + \\Sigma_L(\\tau,\\tau') &= & V^{CL} g_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr) V^{LC} \\nonumber \\\\\n\\label{eq-SAL}\n&=&  \\Sigma_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr),\n\\label{eq-shifted-self-energy}\n\\end{eqnarray}\nwhere we obtain a shifted self-energy $\\Sigma_L\\big(\\tau+\\hbar x(\\tau),\\tau'+\\hbar x(\\tau')\\big)$ which is the usual self-energy of the lead in contour time with arguments\nshifted by $\\hbar x(\\tau)$ and $\\hbar x(\\tau')$. We define the self-energy $\\Sigma^A_L$ as the difference between the shifted self-energy and the usual one $\\Sigma_L(\\tau,\\tau')$. $\\Sigma^{A}_L$ turns out to be a central quantity for this problem as we will show that, the CGF ${\\cal Z}$ can be concisely expressed in terms of the center Green's function $G_{0}$ and $\\Sigma^{A}_L$.\n\nSubstituting the explicit expression for the influence functionals of both the left and right leads to the path integral expression given in Eq.~(\\ref{eq-Z-keldysh}), we have\n\\begin{eqnarray}\n{\\cal Z}(\\xi,\\lambda)&=& \\int {\\cal D}[u_C] \\rho_C(-\\infty) e^{(i\/\\hbar) \n\\int d\\tau  {\\cal L}_C } I_L[u_C] I_R[u_C]\\nonumber \\\\\n&=& \\int {\\cal D}[u_C] \\rho_C(-\\infty) e^{\\frac{i}{\\hbar}S_{\\rm eff}},\n\\end{eqnarray}\nwhere the effective action is given by\n\\begin{eqnarray}\nS_{\\rm eff} &=& \\int d\\tau \\Bigl[ \n\\frac{1}{2} \\dot{u}_C^2 - \\frac{1}{2} u_C^T K^C u_C \n+ f^T u_C \\Bigr]  \\\\\n& \\!\\!-\\!& \\frac{1}{2}\\! \\int d\\tau \\!\\int d\\tau'\\! u_C^T(\\tau) \\bigl(\n\\Sigma(\\tau, \\tau') + \\Sigma_L^A(\\tau, \\tau') \\bigr) u_C(\\tau'), \\nonumber\n\\end{eqnarray}\nwhere $\\Sigma = \\Sigma_L +\\Sigma_R$, taking into account the effect of both the leads.  The ${\\cal S}^T{\\cal S}$ term in $I_{L}[u_C]$ cancels exactly with\nthe one in ${\\cal L}_C$. We can perform an integration by part on the $\\dot{u}^2$ term, assuming that the surface term\ndoes not matter (since it is at $t=-\\infty$), we can write the expression in a standard quadratic form\n\\begin{eqnarray}\nS_{\\rm eff} &=& \\frac{1}{2} \\int d\\tau \\int d\\tau' u_C^T(\\tau) D(\\tau, \\tau') u_C(\\tau') \\nonumber \\\\\n&& \\qquad  + \\int f^T(\\tau) u_C(\\tau) d\\tau. \n\\end{eqnarray}\n$D(\\tau,\\tau')$ is the differential operator and is given by\n\\begin{eqnarray}\nD(\\tau, \\tau') &=& -I \\frac{\\partial^2}{\\partial \\tau^2} \\delta(\\tau, \\tau')\n- K^C  \\delta(\\tau, \\tau') \\nonumber \\\\\n&& -\\Sigma(\\tau, \\tau') - \\Sigma_L^A(\\tau, \\tau') \\nonumber \\\\\n&=& D_0(\\tau,\\tau') - \\Sigma^A_L(\\tau,\\tau').\n\\end{eqnarray}\nThe above equation defines the Dyson equation on Keldysh contour. The generating function is obtained by doing another Gaussian integration and is of the following form\n\\begin{equation}\n{\\cal Z} \\propto {\\rm det}(D)^{-1\/2} e^{-\\frac{i}{2\\hbar} f^T D^{-1} f}.\n\\label{eq-Z-det}\n\\end{equation}\n(The meaning of the determinant will be explained in Appendix~C). \nWe define the Green's function $G$ and $G_{0}$ by $DG = 1$, and $D_0 G_0 = 1$, or more precisely\n\\begin{equation}\n\\int D(\\tau, \\tau'') G(\\tau'', \\tau') d\\tau'' = I \\delta(\\tau, \\tau'),\n\\end{equation}\nand similarly for $G_0$. $G$ can be written in terms of $G_{0}$ in the following Dyson equation form\n\\begin{eqnarray}\n\\label{eq-Dyson-full}\nG(\\tau, \\tau') &=& G_0(\\tau,\\tau') \\\\\n&&\\> + \\int \\int d\\tau_1d \\tau_2 \nG_0(\\tau, \\tau_1) \\Sigma_L^A(\\tau_1,  \\tau_2) G(\\tau_2, \\tau'). \\nonumber\n\\end{eqnarray}\n\nWe view the differential operator\n(integral operator) $D$ and $D^{-1}$ as matrices that are\nindexed by space $j$ and contour time $\\tau$. $f$ is a column vector.  The\nexponential factor term can also be written as a trace, \n$f^T D^{-1} f = {\\rm Tr}_{(j,\\tau)} ( G f f^T)$.\nWe can fix the proportionality constant by noting that\n${\\cal Z}(\\xi=0, \\lambda=0) =1$.  Since when $\\xi=0$, $\\lambda=0$, we have\n$x=0$ and thus $\\Sigma_L^A(\\tau,\\tau') = \\Sigma_L(\\tau+x, \\tau'+x') - \\Sigma_L(\\tau, \\tau') = 0$, so $D=D_0$.  The properly normalized\nCGF is \n\\begin{equation}\n\\label{eq-Zxilam}\n{\\cal Z}(\\xi, \\lambda) = {\\rm det}\\bigl( D_0^{-1}D)^{-1\/2}e^{-\\frac{i}{2\\hbar} f^T D^{-1} f}.\n\\end{equation}\nWe don't need to do anything for the exponential factor because of the following reason.\nWe note\n\\begin{eqnarray}\nf^T G_0 f &=& \\int \\int d\\tau d\\tau' f(\\tau)^T G_0(\\tau, \\tau') f(\\tau') \\\\\n&=& \\sum_{\\sigma,\\sigma'} \\int \\int \\sigma dt\\, \\sigma' dt' f(t)^T G_0^{\\sigma\\sigma'}(t,t') f(t').\\nonumber\n\\end{eqnarray}\nSince the driven force $f$ does not depend on the branch indices, i.e., \n$f^{+}(t) = f^{-}(t)$, we can take the summation inside and obtain\n\\begin{equation}\n\\sum_{\\sigma\\sigma'} \\sigma\\sigma' G^{\\sigma\\sigma'} = G_0^t + G_0^{\\bar t} - G_0^> - G_0^< = 0.\n\\end{equation}\n\nFinally making use of the formulas for operators or matrices \n${\\rm det}(M) = e^{{\\rm Tr} \\ln M}$, and \n$\\ln (1 - y) = -\\sum_{k=1}^\\infty \\frac{y^k}{k}$ we can write the CGF in terms of $\\Sigma^{A}_L$ for the {\\it projected} initial condition case as,\n\\begin{eqnarray}\n\\ln {\\cal Z}(\\xi)&=& \\lim_{\\lambda \\to \\infty}\\ln {\\cal Z}(\\xi,\\lambda) \\nonumber \\\\\n &=& \\lim_{\\lambda \\to \\infty} \\left\\{-\\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln ( 1 - G_0 \\Sigma_L^A) - \\frac{i}{2\\hbar} {\\rm Tr}_{j,\\tau}( G f f^T) \\right\\} \\nonumber \\\\\n&=& \\lim_{\\lambda \\to \\infty} \\sum_{n=1}^\\infty \\frac{1}{2n}\n{\\rm Tr}_{(j,\\tau)} \\Bigl[(G_0 \\Sigma_L^A)^n \\Bigr] \\nonumber  - \\frac{i}{2\\hbar} f^T G f \\nonumber \\\\\n&=& \\frac{1}{2} {\\rm Tr}_{(j,\\tau)}(G_0 \\Sigma_L^A) + \\frac{1}{4} {\\rm Tr}_{(j,\\tau)}(G_0 \\Sigma_L^A G_0 \\Sigma_L^A) + \\cdots \\nonumber \\\\\n&& -\\frac{i}{2\\hbar} f^T G_0 \\Sigma^A G_0 f + \\cdots.\n\\label{projected_state}\n\\end{eqnarray}\nThis expression for CGF is valid for any transient time $t_M$ present in the self-energy\n$\\Sigma^A_{L}$ and is the starting point for the calculation in transient regime.  The notation ${\\rm Tr}_{(j,\\tau)}$ means trace both in\nspace index $j$ and contour time $\\tau$ (see Appendix~C). In order to obtain ${\\cal Z}(\\xi)$ from ${\\cal Z}(\\xi,\\lambda)$ we have to take the limit $\\lambda \\rightarrow \\infty$ because ${\\cal Z}(\\xi, \\lambda)$ approaches a constant as $| \\lambda| \\to \\infty$ and hence\nthe value of the integral is dominated by the value at infinity. Since $\\Sigma^{A}_{L}(\\tau,\\tau')=0$ for $\\xi=0$ we have the correct normalization ${\\cal Z}(0)=1$.\n\nSimilarly, for the steady state initial condition $\\rho(0)$ the CGF is given by \n\\begin{equation}\n\\ln {\\cal Z}(\\xi)=\\lim_{\\lambda \\rightarrow 0} \\ln {\\cal Z}(\\xi,\\lambda)\n\\end{equation}\nThe difference in this two cases is in the matrix $\\Sigma^{A}_L$. \n\nSimilar relations also exist if we want to calculate the CGF for right lead heat operator ${\\cal Q}_R$. In this case one has to do two-time measurement on the right lead corresponding to the Hamiltonian ${\\cal H}_R$. The final formula for the CGF remains the same except $\\Sigma^{A}_L$ should be replaced by $\\Sigma^{A}_R$.\n \nNow in order to calculate the cumulants $\\langle \\langle Q_{\\alpha}^{n} \\rangle \\rangle$ with $\\alpha=L,R$  we need to go to the real time using Langreth's rule \\cite{rammer86}. In this case, it is more convenient to work with a Keldysh rotation (see Appendix~C) for the contour ordered functions while keeping\n${\\rm Tr}(ABC \\cdots D)$ invariant.  The effect of the Keldysh rotation is to change any given matrix ${\\cal D}^{\\sigma\\sigma'}(t,t')$,\nwith $\\sigma, \\sigma'=\\pm$ for branch indices, to,\n\\begin{eqnarray}\n\\label{keldysh-rotation}\n\\breve{{\\cal D}} &=& \n\\left( \\begin{array}{cc}\n                             {\\cal D}^r & {\\cal D}^K \\\\\n                             {\\cal D}^{\\bar K} & {\\cal D}^a \n\\end{array} \\right)  \\\\\n &=&\n\\frac{1}{2} \\left( \\begin{array}{cc}\n              {\\cal D}^t - {\\cal D}^< + {\\cal D}^> - {\\cal D}^{\\bar t}, & {\\cal D}^t + {\\cal D}^{\\bar t} + {\\cal D}^< + {\\cal D}^> \\\\\n              {\\cal D}^t + {\\cal D}^{\\bar t} - {\\cal D}^< - {\\cal D}^>, & {\\cal D}^< - {\\cal D}^{\\bar t} + {\\cal D}^{t} - {\\cal D}^{>}  \n                         \\end{array} \\right). \\nonumber\n\\end{eqnarray}\nIn this case we define the quantities \n${\\cal D}^r$, ${\\cal D}^a$, ${\\cal D}^K$, and ${\\cal D}^{\\bar{K}}$ as above.  In particular,\n${\\cal D}^{K} \\neq {\\cal D}^< + {\\cal D}^>$, as one usually might thought it is. \n\nUsing the above definition for the center Green's function $G_0$ we get \n\\begin{equation}\n\\breve{G_0} = \n\\left( \\begin{array}{cc}\n                             G_0^r & G_0^K \\\\\n                             0 & G_0^a \n\\end{array} \\right). \n\\end{equation}\nThe $G_0^{\\bar K}$ component is 0 due to the standard relation\namong Green's functions.  But the $\\bar K$ components are not zero for\n$\\Sigma_L^A$ and $G_0$, as we will compute later.  \n\nIt is useful computationally to work in Fourier space even if \nthere is no time translational invariance.  We define the two-frequency\nFourier transform by\n\\begin{equation}\n\\breve{A}[\\omega, \\omega'] =\n\\int_{-\\infty}^{+\\infty}\\!\\!\\! dt \\int_{-\\infty}^{+\\infty}\\!\\!\\! dt' \n\\breve{A}(t,t') e^{i(\\omega t + \\omega' t')}.\n\\label{two-time-FT}\n\\end{equation}\nSince $\\breve{G}_0$ is time-translationally invariant then,\n\\begin{equation}\n\\breve{G}_0[\\omega, \\omega'] = 2\\pi \\delta(\\omega+\\omega') \\breve{G}_0[\\omega],\n\\label{G0}\n\\end{equation}\nis ``diagonal'', where the single argument Fourier transform is similarly defined,\n\\begin{equation}\n\\label{eq-Fourier}\nA[\\omega] = \\int_{-\\infty}^{+\\infty} A(t,0) e^{i\\omega t} dt.\n\\end{equation}\n(The expressions for different components of $\\breve{G}_{0}[\\omega]$ and $\\breve{\\Sigma}[\\omega]$ are given in Appendix A). Using $\\breve{G}_0[\\omega]$, we can save one integration due to the $\\delta$ function, and have\n\\begin{eqnarray}\n\\label{eq-Zsteady}\n\\ln {\\cal Z}(\\xi) &=& - \\frac{1}{2} {\\rm Tr}_{j,\\sigma,\\omega}\n\\ln\\Bigl[ 1 - \\breve{G}_0[\\omega] \\breve{\\Sigma_L^A}[\\omega, \\omega'] \\Bigr]\n\\nonumber \\\\\n&& - \\frac{i}{2 \\hbar} {\\rm Tr}_ {j,\\sigma,\\omega} \\Bigl[\\breve{G}[\\omega,\\omega'] \\, \\breve{{\\cal F}}[\\omega', \\omega]\\Bigr],\n\\end{eqnarray}\nwhere $ \\breve{G}_0[\\omega] \\breve{\\Sigma_L^A}[\\omega, \\omega']$ is viewed\n\nas a matrix indexed by $\\omega$ and $\\omega'$.  The trace is performed on\nthe frequency as well as the usual space and branch components. (The meaning of trace in frequency domain is discussed in Appendix~C). $\\breve{{\\cal F}}$ is given by\n\\begin{equation}\n\\label{force-matrix}\n\\breve{{\\cal F}}[\\omega,\\omega'] = \n\\left( \\begin{array}{cc}\n                             0 & 2 f[\\omega]f[\\omega']^T \\\\\n                             0 & 0 \n\\end{array} \\right).  \n\\end{equation}\nIn the next section we derive the CGF for the product initial condition using Feynman diagrammatic technique. Because of the special form of the initial density matrix the calculation for the CGF simplifies greatly in this case. \n  \n\\section{Product state $\\rho(-\\infty)$ as initial state}\nIn this section, we derive the CGF starting with a product initial state, i.e., the density matrix at time $t=0$ is given\nby  $\\rho(-\\infty)=\\rho_C   \\otimes \\rho_L \\otimes \\rho_R$.  Since this density matrix commutes with the projection operator $\\Pi_a$, the initial projection does not play any role in this case. Working in the interaction picture with respect to the decoupled Hamiltonian ${\\cal H}(-\\infty)= \\sum_i {\\cal H}_i$, the \ninteraction part of the Hamiltonian on the contour $C=[0,t_M]$ is \n\\begin{eqnarray}\n{\\cal V}^{x}_I(\\tau) &=& -f^T(\\tau) u_C(\\tau) + u_R(\\tau) V^{RC} u_C(\\tau) \\nonumber \\\\\n&&+\\, u_L\\bigl(\\tau+\\hbar x(\\tau)\\bigr)V^{LC}u_C(\\tau).\n\\end{eqnarray}\nIn the last term for $u_L$, the argument is shifted by $\\hbar x$ where \n$x^+(t) = - \\xi\/2$, $x^{-}(t) = \\xi\/2$ for $0< t < t_M$.\n\nThe density matrix remains unaffected by the transformation to the interaction picture, because it commutes with ${\\cal H}(-\\infty)$. The CGF can now be written as \n\\begin{equation}\n{\\cal Z}(\\xi) = {\\rm Tr}\\Bigl[ \\rho(-\\infty) T_c \\,e^{-\\frac{i}{\\hbar} \\int_C {\\cal V}^{x}_I(\\tau)\\, d\\tau}\\Bigr].\n\\end{equation}\nExpanding the exponential, we generate various terms of product of\n$u_\\alpha$.  These terms can be decomposed in pairs according to\nWick's theorem \\cite{rammer86}. Since the system is decoupled, each type of $u$ comes\nin an even number of times for a non-vanishing contributions because \n$\\langle u_C\\rangle = 0$, $\\langle u_C u_L\\rangle = 0$ and we know\n\\begin{equation}\n-\\frac{i}{\\hbar} \\langle T_C u_{\\alpha}(\\tau) u_{\\alpha'}(\\tau')^T \\rangle_{\\rho(-\\infty)} = \\delta_{\\alpha,\\alpha'} g_\\alpha(\\tau, \\tau').\n\\end{equation}\nWe use Feynman diagrammatic technique to sum the series.  since ${\\cal V}_I$ contains only two-point couplings,\nthe graphs are all ring type.  The combinatorial factors can be worked\nout as $1\/(2n)$ for a ring containing $n$ vertices.\nWe use a very general theorem which says $\\ln {\\cal Z}$ contains only \nconnected graphs, and the disconnected graphs cancel exactly when we\ntake the logarithm. The final result can be expressed as\n\\begin{equation}\n\\ln {\\cal Z}(\\xi) = - \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - g_C \\Sigma^{\\rm tot} \\Big] \n-\\frac{i}{2\\hbar}  f^T G f,  \n\\end{equation}\nwhere\n\\begin{equation}\n\\Sigma^{\\rm tot} = \\Sigma_L(\\tau+x,\\tau'+x') + \\Sigma_R(\\tau,\\tau') = \\Sigma + \\Sigma_L^A,\n\\end{equation} \nand $\\Sigma$ is the total self-energy due to both the leads. $G(\\tau,\\tau')$ obeys the following Dyson's equation\n\\begin{eqnarray}\nG(\\tau,\\tau') &=& g_C(\\tau,\\tau')  \\\\ \n&&\\> + \\int \\int  d\\tau_1 d\\tau_2  g_C(\\tau,\\tau_1) \\Sigma^{\\rm tot}(\\tau_1,\\tau_2) G(\\tau_2,\\tau'). \\nonumber \n\\end{eqnarray}   \n\n\nThe above expression for CGF can be written down more explicitly, by\ngetting rid of the vacuum diagrams. Let us define a new type of Dyson's equation\n\\begin{eqnarray}\n\\label{eq-Dyson-product}\nG_0(\\tau, \\tau') &=& g_C(\\tau,\\tau') \\\\\n&&\\> + \\int \\!\\int d\\tau_1d \\tau_2\\, \ng_C(\\tau, \\tau_1) \\Sigma(\\tau_1,  \\tau_2) G_0(\\tau_2, \\tau'), \\nonumber\n\\end{eqnarray}\nwhere $g_C$ is the contour ordered Green's function of the isolated center. (The Green's functions for an isolated single \nharmonic oscillator is given is appendix A).\nThis expression looks formally the same as before except that $G_0$\nsatisfies a Dyson equation defined on the contour from 0 to $t_M$ and\nback, while $G$ is defined on the Keldysh contour from $-\\infty$ to\n$t_M$. Using this definition we can write\n\\begin{eqnarray}\n1 - g_C \\Sigma^{\\rm tot} &=& 1 - g_C (\\Sigma + \\Sigma_L^A) \\nonumber \\\\\n&=& (1 - g_C \\Sigma)\\, (1 - G_0 \\Sigma_L^A).\n\\end{eqnarray}\nThe two factors above are in matrix (and contour time) multiplication.\nUsing the relation between trace and determinant, \n$\\ln \\det(M) = {\\rm Tr} \\ln M$, and the fact, $\\det(AB) = \\det(A) \\det(B)$,\nwe find that the two terms give two factors for ${\\cal Z}$, and the factor\ndue to $1 - g_C \\Sigma$ is exactly 1.  We have then\n\\begin{equation}\n\\label{eq-Zprod}\n\\ln {\\cal Z}(\\xi) = - \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - G_0 \\Sigma_L^{A} \\Big]\n-\\frac{i}{2\\hbar}  f^T G f,\n\\end{equation}\nwhere the $G(\\tau,\\tau')$ can now be expressed in terms of $G_{0}(\\tau,\\tau')$ as\n\\begin{equation}\nG^{-1} = G_0^{-1} - \\Sigma_L^A.\n\\end{equation}\nwhich is similar in form to Eq.~(\\ref{eq-Dyson-full}).\n\nThe expression for $\\ln {\\cal Z}(\\xi)$ is consistent with the earlier result,\nEq.~(\\ref{projected_state}), in the long-time limit. So we can conclude that the long-time limit is the same independent of the initial distributions. \n\nTo compute the cumulants $\\langle \\langle Q^n \\rangle \\rangle$, we\nneed to take derivative with respect to $\\xi$ $n$-times to $\\ln {\\cal Z}$.\nNote that the shifted self-energy for $0 < t < t_M$ is (for all three initial conditions)\n\\begin{eqnarray}\n\\Sigma_A^t (t,t') &=& 0, \\nonumber \\\\\n\\Sigma_A^{\\bar t}(t,t') &=& 0, \\nonumber \\\\\n\\Sigma_A^{<}(t,t') &=& \\Sigma_L^{<}(t-t'-\\hbar \\xi) -  \\Sigma_L^{<}(t-t'), \\nonumber \\\\\n\\Sigma_A^{>}(t,t') &=& \\Sigma_L^{>}(t-t'+\\hbar \\xi) -  \\Sigma_L^{>}(t-t').\n\\label{shifted-product}\n\\end{eqnarray}\nWe note $\\Sigma_L^A(\\xi=0)=0$. The derivatives at $\\xi=0$ can be obtained as\n\\begin{eqnarray}\n{\\partial^n \\Sigma_A^{<} \\over\n\\partial \\xi^n}\\Big|_{\\xi=0} &=& (-\\hbar)^n \\Sigma_L^{<,(n)}(t-t'), \\nonumber \\\\\n{\\partial^n \\Sigma_A^{>} \\over\n\\partial \\xi^n}\\Big|_{\\xi=0} &=&  \\hbar^n \\Sigma_L^{>,(n)}(t-t'),\n\\end{eqnarray}\nwhere the superscript $(n)$ means derivatives with respect to the argument of\nthe function $n$ times. In the following sections we first show the explicit expression of the CGF in the long-time limit and then discuss the steady state fluctuation theorem.  \n\n\n\\section{Long-time limit and Steady state fluctuation theorem}\nFor the long-time limit calculation we can use either Eq.~(\\ref{eq-Zsteady}) or Eq.~(\\ref{eq-Zprod}).        \nFor convenience of taking the large time limit, i.e., $t_M$ large, we prefer\nto set interval to $(-t_M\/2, t_M\/2)$. In this way, when $t_M \\to \\infty$,\nthe interval becomes the full domain and Fourier transforms to all the\nGreen's functions and self-energy can be performed (where the translational\ninvariance is restored). Applying the convolution theorem to the\ntrace formula in Eq.~(\\ref{eq-Zprod}), we find that there is one more time integral left\nwith integrand independent of $t$.  This last one can be set from $-t_M\/2$ to $t_M\/2$, obtaining an overall factor\nof $t_M$ and we have\n\\begin{equation}\n{\\rm Tr}_{(j,\\tau)}(AB \\cdots D) = t_M \n\\int \\frac{d\\omega}{2\\pi}{\\rm Tr} \\Bigl[\\breve {A}(\\omega)\\breve{B}(\\omega) \\cdots \\breve{D}(\\omega)\\Bigr].\n\\end{equation}\n \nIn the long-time limit, the shift given to the argument in $\\Sigma_L^A$ depends on the branches, and the \ntwo arguments $(t,t')$ becomes $t-t'$ and we have \n\\begin{eqnarray}\n\\Sigma_A^{\\sigma\\sigma'}(t,t') &=& \\Sigma^{\\sigma\\sigma'}_L(t\\! +\\!x^\\sigma\\!-\\! t'\\!-\\!x^{\\sigma'}) - \\Sigma^{\\sigma\\sigma'}_L(t\\!-\\!t'), \\\\\n\\Sigma_A^t &=& \\Sigma_A^{\\bar t} = 0,\\nonumber \\\\\n\\Sigma_A^<(t) &=& \\Sigma_L^{<}(t-\\hbar \\xi) -\\Sigma_L^{<}(t),\\\\\n \\Sigma_A^>(t) &=& \\Sigma_L^{>}(t+\\hbar \\xi)-\\Sigma_L^{>}(t). \\nonumber \n\\label{steady}\n\\end{eqnarray}\nFourier transforming the greater and lesser self-energy, we get\n\\begin{eqnarray}\n\\label{eq-a}\n\\Sigma_A^{>}[\\omega] = \\Sigma^{>}_L[\\omega] \\bigl(e^{-i\\hbar \\omega \\xi} - 1 \\bigr) = a, \\\\\n\\label{eq-b}\n\\Sigma_A^{<}[\\omega] = \\Sigma^{<}_L[\\omega] \\bigl(e^{i\\hbar \\omega \\xi} - 1 \\bigr) = b.\n\\end{eqnarray}\nWe note that $\\Sigma_L^A$ is supposed to depend on both $\\xi$ and $\\lambda$.\nHowever in the long-time limit, the $\\lambda$ dependence drops out which makes the steady state result independent of the initial distribution.  \n\nFinally, we can express the generating function as\n\\begin{eqnarray}\n\\ln {\\cal Z}(\\xi) &= & - t_M \\int \\frac{d\\omega}{4\\pi} {\\rm Tr} \\ln \\Bigl[1 - \\breve{G}_0 [\\omega]\\breve{\\Sigma}_L^A [\\omega]\\Bigr] \\nonumber \\\\\n&& \\quad - \\frac{i}{\\hbar}  \\int \\frac{d\\omega}{4\\pi}\n{\\rm Tr} \\Bigl[ \\breve{G}[\\omega] \\breve{\\cal F}[\\omega,-\\omega]\\Bigr],\n\\label{eq-lnZxi-1}\n\\end{eqnarray}\nwhere $\\breve{G}[\\omega]$ is obtained by solving the Dyson equation\nin frequency domain and in the long-time obeys time-translational invariance. So the full CGF can be written as the sum of contributions due to driving force and due to temperature difference between the leads, i.e.,\n\\begin{equation}\n\\ln {\\cal Z}(\\xi) = \\ln {\\cal Z}^{s}(\\xi) + \\ln {\\cal Z}^{d}(\\xi). \n\\end{equation}\nIn the following and subsequent sections we discuss about ${{\\cal Z}^{s}(\\xi)}$ and we will return to ${{\\cal Z}^{d}(\\xi)}$ in Sec. XI. \n\nIn order to obtain the explicit expression for $\\ln {\\cal Z}^{s}(\\xi)$ we need to compute the matrix product\n\\begin{eqnarray}\n\\breve{G}_0[\\omega] \\breve{\\Sigma}_L^A[\\omega]  &=& \n\\frac{1}{2} \\left(  \\begin{array}{cc}\nG_0^r & G_0^{K} \\\\\n0 & G_0^a  \n\\end{array}\n\\right)\n\\left(  \\begin{array}{cc}\na-b & a+b \\\\\n-(a+b) & b-a  \n\\end{array}\n\\right).\n\\end{eqnarray}\nTo simplify the expression, we rewrite the term ${\\rm Tr} \\ln (1-M)$ as\na determinant and use the formula (assuming A to be an invertible matrix)  \n\\begin{equation}\n{\\rm det} \\left( \\!\\!\\begin{array}{cc}\nA & B \\\\\nC & D \\end{array}\\!\\!\n\\right) = {\\rm det}(A) \\det(D - C A^{-1}B)\n\\end{equation}\nto reduce the dimensions of the determinant matrix by half. \nThe steady state solution for ${{\\cal Z}^{s}(\\xi)}$ is given by\n\\begin{eqnarray}\n\\ln {\\cal Z}^{s}(\\xi)&=&-t_M \\int \\frac{d\\omega}{4\\pi} \\,\\ln \\det \\Bigl\\{ I - G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\Big[  (e^{i\\xi \\hbar \\omega}\\! -\\! 1) f_L \\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+ ( e^{-i\\xi\\hbar \\omega} \\!-\\! 1) f_R + (\ne^{i\\xi \\hbar \\omega} \\!+\\! e^{-i\\xi\\hbar\\omega} \\!-\\!2 ) f_L f_R \\Big]\\Bigr\\}.\\qquad \n\\label{eq-steady1}\n\\end{eqnarray}\nwith $f_{\\alpha}=1\/(e^{\\beta_{\\alpha} \\hbar \\omega} - 1)$, $\\alpha=L,R,$ the Bose-Einstein distribution function and $\\Gamma_{\\alpha}[\\omega]=i\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big)$. If we consider the full system as a one-dimensional linear chain, then because of the special form of $\\Gamma_{\\alpha}$ matrices (only one entry of the $\\Gamma$ matrices are non-zero) it can be easily shown that\n\\begin{equation}\n\\det [I-\\bigl(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R \\bigr) \\Xi(\\xi)]=1-{\\cal T}[\\omega]\\Xi(\\xi)\n\\end{equation}\nwhere $\\Xi(\\xi)$ is any arbitrary function of $\\xi$ and ${\\cal T}[\\omega]={\\rm Tr}(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R)$ is known as the transmission function and is given by the Caroli formula \\cite{Caroli,WangJS-europhysJb-2008}. The generating function ${{\\cal Z}^{s}(\\xi)}$ in the steady state obeys the following symmetry  \n\\begin{equation}\n{{\\cal Z}^{s}}(\\xi)={{\\cal Z}^{s}}\\big(-\\xi + i\\,{\\cal A}\\big),\n\\end{equation}\nwhere ${\\cal A}=\\beta_R-\\beta_L$ is known as thermodynamic affinity. This relation is also known as Gallavotti-Cohen (GC) symmetry \\cite{noneq-fluct}. The immediate consequence of this symmetry is that the probability distribution for heat transferred $Q_{L}$ which is given by the Fourier transform of the CGF, i.e., $P(Q_L)=\\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} d\\xi \\, {\\cal Z}(\\xi) \\, e^{-i \\xi Q_L}$ obeys the following relation in the large $t_M$ limit,\n\\begin{equation}\nP_{t_M}(Q_L)=e^{{\\cal A}Q_L}\\,P_{t_M}(-Q_L).\n\\end{equation} \nThis relation is known as the steady state fluctuation theorem and was first derived by Saito and Dhar \\cite{Saito-Dhar-2007} in the phononic case. This theorem quantifies the ratio of positive and negative heat flux and second law violation. \n\nThe cumulants $\\langle \\langle {Q}^{n} \\rangle \\rangle $ can be obtained by taking derivative of $\\ln {\\cal Z}^{s}(\\xi)$ with respect to $i\\xi$ and setting $\\xi=0$. The first cumulant is given by \n\\begin{equation}\n\\frac{\\langle \\langle Q \\rangle \\rangle}{t_M} = \\int_{-\\infty}^{\\infty} \\frac{d\\omega}{4 \\pi} \\, \\hbar\\, \\omega \\,{\\cal T}(\\omega)(f_L-f_R),\n\\end{equation}\nwhich is known as the Landauer-like formula in thermal transport. Similarly the second cumulant $\\langle \\langle {Q}^{2} \\rangle \\rangle= \\langle {Q}^{2} \\rangle - \\langle {Q}\\rangle ^{2}$, which describes the fluctuation of the heat transferred, can be written as \\cite{Saito-Dhar-2007, Huag, Buttiker},\n\\begin{eqnarray}\n\\frac{\\langle \\langle Q^{2} \\rangle \\rangle}{t_M} &=&\\int_{-\\infty}^{\\infty} \\frac{d\\omega}{4 \\pi} \\, (\\hbar \\omega)^{2} \\Bigl \\{{\\cal T}^{2}(\\omega)\\, (f_L-f_R)^{2} \\nonumber \\\\\n&&+ {\\cal T}(\\omega) \\, (f_L+f_R+2\\,f_L f_R ) \\Bigr\\}.\n\\end{eqnarray}\nOur formalism can be easily generalized for multiple heat baths and for $N$ leads connected with the center $C$, we can generalize the above formula as\n\\begin{eqnarray}\n&&\\ln {\\cal Z}^{s}(\\xi)=\\!-\\!t_M \\!\\int \\frac{d\\omega}{4\\pi} \\,\\ln \\det \\Bigl\\{ I \\!-\\!\\sum_{m} G_{0}^r \\Gamma_L G_{0}^a \\Gamma_m \\Big[(e^{i\\xi \\hbar \\omega}\\! -\\! 1) \\times \\nonumber \\\\\n&& f_L\\!\\!+ ( e^{-i\\xi\\hbar \\omega} \\!-\\! 1) f_m \\!+\\! (\\!\ne^{i\\xi \\hbar \\omega} \\!\\!+\\! e^{-i\\xi\\hbar\\omega}\\! \\!-\\!2 )\\! f_L \\!f_m \\Big]\\Bigr\\}.\\qquad .\n\\end{eqnarray}\nIn the following section we will discuss the how to numerically calculate the CGF in the transient case for projected density matrix $\\rho'(0)$. We also discuss about solving the Dyson equation given in Eq.~(\\ref{eq-Dyson-product}).\n\n\n\\section{Transient Region}\nThe central quantity to calculate the CGF numerically is the shifted self-energy $\\Sigma_L^{A}$ which is given by \n\\begin{equation}\n\\Sigma_L^A(\\tau,\\tau') =\\Sigma_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr)- \\Sigma_L\\big(\\tau,\\tau'\\big).\n\\end{equation}\nHere $\\tau$ is a contour variable which runs over Keldysh contour $K=(-\\infty,\\infty)$ and back, for the initial conditions $\\rho(0)$ and $\\rho'(0)$, whereas for $\\rho(-\\infty)$, $\\tau$ runs over the contour $C=[0,t_M]$ (see Fig.~1). The contour function $x(\\tau)$ is 0 whenever $t < 0$ or $t>t_M$, and for $0 < t < t_M$, $x^{+}(t) = -\\xi\/2 - \\lambda$, and $x^{-}(t) = \\xi\/2 - \\lambda$. \nDepending on the values of $t$, $t'$, and $\\lambda$ ($\\lambda \\rightarrow 0$  and $\\lambda \\rightarrow \\infty$ corresponds to steady state initial state and {\\it projected} initial state, respectively) $\\Sigma_L^A$ will have different functional form. If $ 0 < t,t' < t_M$ then $\\Sigma_L^A$'s are given by Eq.~(\\ref{shifted-product}). This is the region which dominates in the long-time limit and gives steady state result. If both $t$ and $t'$ lies outside the measurement time, i.e., $t,t'<0$ or $t,t'>t_M$ then $\\Sigma_L^A$ is zero. \n\nThe main computational task for a numerical evaluation of the cumulants\nis to compute the matrix series $-\\ln(1-M) = M + \\frac{1}{2} M^2 + \\cdots$.\nIt can be seen due to the nature of $\\Sigma_L^A$ that for the product initial\nstate, exact $n$ terms upto $M^n$ is required for the $n$-th culumants, as the infinite series terminates due to $\\Sigma_L^A(\\xi=0) = 0$.\nNumerically, we also observed for the projected state $\\rho'(0)$, exactly\n$3n$ terms is required (although we don't have a proof) if calculation is\nperformed in time domain. \n\nThe computation can be performed in time as well as in the frequency domain. However for projected and steady state initial condition since $G_{0}[\\omega]$ is time translational invariant it is advantageous to work in the frequency domain. But for the product state there is no such preference as $G_{0}$ in Eq.~(\\ref{eq-Dyson-product}) is not time translational invariant and one has to solve it numerically. \n\nIn the following we first discuss how to calculate $\\Sigma_{L}^{A}[\\omega,\\omega']$ for projected initial state, defined in Eq.~(\\ref{eq-Zsteady}) and then we will discuss how to solve the Dyson equation for the product initial condition case, given in Eq.~(\\ref{eq-Dyson-product}).\n\n\\subsection{calculation of $\\Sigma_L^{A}(\\omega,\\omega')$ }\nTo calculate $\\Sigma_L^{A}(\\omega,\\omega')$ for projected initial state $\\rho'(0)$ we define two types of theta functions $\\theta_{1}(t,t')$ and $\\theta_{2}(t,t')$.\n$\\theta_{1}(t,t')$ is non-zero when \n\\begin{equation}\n0 \\leq t \\leq t_M, \\,\\,{\\rm and} \\quad t' \\leq 0 \\quad {\\rm or} \\quad t'\\geq t_M,\n\\end{equation}\nor \n\\begin{equation}\n0 \\leq t' \\leq t_M, \\,\\,{\\rm and} \\quad t \\leq 0 \\quad {\\rm or} \\quad t \\geq t_M, \n\\end{equation}\nand $\\theta_{2}(t,t')$ is non-zero only in the regime where $ 0 \\le t,t' \\le t_M$ . For the regions where $\\theta_1(t,t')$ is non-zero the expression for $\\Sigma_L^{A}$ after taking the limit $\\lambda \\rightarrow \\infty$ is, (assuming all correlation functions decays to zero as $t \\rightarrow \\pm \\infty$)\n\\begin{equation}\n\\Sigma_{A}^{t,\\bar{t},<,>}(t,t')=-\\Sigma_{L}^{t,\\bar{t},<,>}(t-t').\n\\end{equation}\nSo using theta functions we may write $\\Sigma_L^{A}(t,t')$ in the full t,t' domain as \n\\begin{eqnarray}\n\\Sigma_{A}^{t,\\bar{t}}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{t,\\bar{t}}(t-t') \\nonumber \\\\\n\\Sigma_{A}^{<}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{<}(t-t')+ \\theta_2 (t,t') \\times \\nonumber \\\\\n&&\\>\\big[\\Sigma_{L}^{<}(t-t'-\\hbar \\xi) -  \\Sigma_{L}^{<}(t-t')\\big] \\nonumber \\\\\n\\Sigma_{A}^{>}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{>}(t-t')+ \\theta_2 (t,t') \\times \\nonumber \\\\\n&&\\>\\big[\\Sigma_{L}^{>}(t-t'+\\hbar \\xi) -  \\Sigma_{L}^{>}(t-t')\\big]\n\\end{eqnarray}\nBy doing Fourier transform it can be easily shown that\n\\begin{equation}\n\\Sigma_{A}^{t,\\bar{t}}[\\omega,\\omega']=-\\int_{-\\infty}^{\\infty}\\!\\!\\! \\frac{d\\omega_{c}}{2 \\pi} \\, \\theta_{1}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{t,\\bar{t}}(\\omega_{c})\n\\end{equation}\nand \n\\begin{eqnarray}\n\\Sigma_{A}^{>,<}[\\omega,\\omega']&\\!=\\!& \\!-\\!\\int_{-\\infty}^{\\infty} \\!\\!\\!\\frac{d\\omega_{c}}{2 \\pi}\\theta_{1}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{>,<}(\\omega_{c}) \\\\\n\\!+\\!\\int_{-\\infty}^{\\infty}\\! \\frac{d\\omega_{c}}{2 \\pi}\\!\\!\\!&&\\!\\theta_{2}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{>,<}(\\omega_{c}) \n(e^{i \\omega_{c}\\eta \\xi}\\!-\\!1), \\nonumber\n\\end{eqnarray}\nwhere $\\eta=\\pm 1$. The positive sign is for $\\Sigma_{A}^{<}$ and negative sign for  $\\Sigma_{A}^{>}$.\n\nThe theta functions are now given by\n\\begin{eqnarray}\n\\theta_{1}(\\omega_{a},\\omega_{b})&=&f(\\omega_{a}).g(\\omega_{b})+f(\\omega_{b}).g(\\omega_{a}), \\nonumber \\\\\n\\theta_{2}(\\omega_{a},\\omega_{b})&=&f(\\omega_{a}).f(\\omega_{b}),\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nf(\\omega)&=&\\frac{e^{i \\omega t_{M}}-1}{i \\omega}, \\nonumber \\\\\ng(\\omega)&=& \\frac{1}{i \\omega + \\epsilon} -\\frac{e^{i \\omega t_{M}-\\eta t_{M}}}{i \\omega -\\epsilon},\n\\end{eqnarray}\nwith $\\epsilon \\rightarrow 0^{+}$. The theta functions are of immense importance which carries all information about the measurement time $t_{M}$.\n\nIn the limit $t_{M} \\rightarrow \\infty$, the region  $ 0 \\le t,t' \\le t_M$ dominates and corresponding theta function, i.e., $\\theta_2(\\omega,\\omega')$ reduces to \n\\begin{equation}\n\\theta_{2}(\\omega-\\omega_{c},\\omega'+\\omega_{c}) \\approx \\delta(\\omega-\\omega_{c}) \\delta(\\omega'+\\omega_{c}),\n\\end{equation}\nand is responsible for obtaining the steady state result.\n\nTo calculate all the cumulants we only need to take derivative of $\\Sigma_{A}(\\omega,\\omega')$ with respect to $i \\xi$ since $G_{0}$ does not have any $\\xi$ dependence. Also $\\Sigma^{A}$ has $\\xi$ dependence only for $ 0 \\le t,t' \\le t_M$ and hence the derivatives are given by \n\\begin{eqnarray}\n\\frac{\\partial^{n}\\Sigma_{A}^{>,<}}{{\\partial}(i \\xi)^{n}}[\\omega,\\omega']&=&\\int_{-\\infty}^{\\infty} \\frac{d\\omega_{c}}{2 \\pi} (\\eta \\hbar \\omega_{c})^{n} \\theta_{2}\\bigl[\\omega-\\omega_{c},\\omega'+\\omega_{c}\\bigr]\\nonumber \\\\\n&&\\Sigma_{L}^{>,<}(\\omega_{c}) e^{i \\omega_{c}\\eta \\xi}.\n\\end{eqnarray}  \nHere $n$ refers to the order of the derivative. \n\n\\subsection{Dyson equation on contour C}\n\nLet us now discuss about solving the Dyson's equation for $G_{0}$ given in Eq.~(\\ref{eq-Dyson-product}) for product initial state $\\rho(-\\infty)$. In order to compute the matrix $\\breve{G}_0(t,t')$ we have to calculate two components $G_{0}^{r}$ and $G_{0}^{K}$ which are written in the integral form by applying Langreth's rule \\cite{Huag,rammer86}\n\\begin{eqnarray}\nG_{0}^{r}(t,t')&=&g_{C}^{r}(t\\!-\\!t')  \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1)\\,\\Sigma^{r}(t_1\\!-\\!t_2)G_{0}^{r}(t_2,t'), \\nonumber \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&G_{0}^{K}(t,t')=g_{C}^{K}(t\\!-\\!t') \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1)\\,\\Sigma^{r}(t_1\\!-\\!t_2) G_{0}^{K}(t_2,t') \\nonumber \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1) \\Sigma^{K}(t_1\\!-\\!t_2) G_{0}^{a}(t_2,t')\\nonumber \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{K}(t\\!-\\!t_1) \\Sigma^{a}(t_1\\!-\\!t_2)G_{0}^{a}(t_2,t'). \\nonumber \n\\end{eqnarray}\n\nNote that the argument for center Green's function $g_{C}$ and lead self-energy $\\Sigma$ are written as time difference $t-t'$ because they are Green's functions for isolated center part and leads respectively and hence are calculated at equilibrium. The analytical expressions for $\\Sigma$ and $g_{C}$ are known in frequency domain and are given in Appendix~A. To determine their time-dependence we numerically calculate their inverse Fourier transforms using trapezoidal rule \\cite{NR}. Then in order to solve above equations for any $t_M$  we discretize the time variable into $N$ total intervals of incremental length $\\Delta t=t_M\/N$ and thus converting the integral into a sum. After discretization, the above equations can be written in the matrix form which are indexed by space $j$ and discrete time $t$, as \n\\begin{eqnarray}\n\\tilde{G}_{0}^{r}&=& \\tilde{g}_C^{r} + \\tilde{g}_C^{r} \\tilde{\\Sigma}^{r} \\tilde{G}_{0}^{r}, \\nonumber \\\\\n\\tilde{G}_{0}^{K}&=& \\tilde{G}_{0}^{r}\\tilde{\\Sigma}^{K}\\tilde{G}_{0}^{a}+ ( I + \\tilde{G}_{0}^{r} \\tilde{\\Sigma}^{r}) \\tilde{g}_C^{K} (I+\\tilde{\\Sigma}^{a} \\tilde{G}_{0}^{a}).\n\\end{eqnarray}\nSo $\\tilde{G}_{0}^{r}$ can be obtained by doing an inverse of the matrix $(I-\\tilde{g}_{C}^{r} \\tilde{\\Sigma}^{r})$ and then multiplying by $\\tilde{g}_{C}^{r}$. $\\tilde{G}_{0}^{r}$ in this case also obeys time-translational invariance, so it can also be obtained by direct inverse Fourier transform. $\\tilde{G}_{0}^{a}$ can be obtained by taking transpose of $\\tilde{G}_{0}^{r}$. Once $\\tilde{G}_{0}^{r}$ and $\\tilde{G}_{0}^{a}$ are obtained we use the second equation to calculate $\\tilde{G}_{0}^{K}$ which is simply multiplying matrices.\n\nSimilarly $\\Sigma_L^{A}$ in Eq.~(\\ref{shifted-product}) are obtained by doing inverse Fourier transforms of the lead self-energy. We follow the same steps independently to calculate the cumulants for $Q_R$. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2.eps}\n\\caption{(Color online) The cumulants $\\langle \\langle {Q}_{L}^{n} \\rangle \\rangle$ and $\\langle \\langle {Q}_{R}^{n} \\rangle \\rangle$ for $n$=1, 2, 3, and 4 for one-dimensional linear chain connected with Rubin baths, for the projected initial state $\\rho'(0)$. The black and red curves corresponds to $\\langle \\langle {Q}_{L}^{n} \\rangle \\rangle$ and  $\\langle \\langle {Q}_{R}^{n} \\rangle \\rangle$ respectively. The temperatures of the left and the right lead are 310 K and 290 K, respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig3.eps}\n\\caption{(Color online) Same as in Fig.~2 except for product initial state $\\rho(-\\infty)$. The temperatures of the left, the center and the right lead are 310 K, 300 K and 290 K, respectively.}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4.eps}\n\\caption{(Color online) Same as in Fig.~2 except for steady state initial state $\\rho(0)$.}\n\\end{figure}\n\n\\section{Numerical results}\nWe now present some numerical results. In Fig.~2 and 3, we show the results for first four cumulants for both ${\\cal Q}_L$ and ${\\cal Q}_R$ (measurement is on the right lead) for 1D linear chain connected with Rubin baths, starting with the projected initial state $\\rho'(0)$ and product state $\\rho(-\\infty)$ respectively. \nRubin baths \\cite{Rubin, Weiss} mean in our case a uniform linear\nchain with spring constant $K$ and a small onsite $K_0$ for all the atoms.\nOnly one atom is considered as the center.  The atoms of the left and right side\nof the center are considered baths.  We use $K=1$ eV\/(u\\AA$^2)$  and the onsite potential $K_0=0.1$ eV\/(u\\AA$^2)$ in all our calculations.\nFirst of all, cumulants greater than two are nonzero, which confirms that the distribution for $P(Q_L)$ or $P(Q_R)$ is not Gaussian.  The generic features are almost the same in both the cases. However the fluctuations are larger for the product initial state $\\rho(-\\infty)$ as this state corresponds to the sudden switch on of the couplings between the leads and the center and hence the state is far away from the correct steady state distribution. On the contrary, for the initial state $\\rho'(0)$ the fluctuations are relatively small. For $\\rho'(0)$ due to the effect of the measurement, at starting time energy goes into the leads, which is quite surprising. But for $\\rho(-\\infty)$ although initial measurement do not play any role, energy still goes into the leads. This can also be shown analytically (see Appendix B). At the starting time the behavior of both ${\\cal Q}_L$ and ${\\cal Q}_R$ are very similar and can be understood since both the left and right leads are identical and the effect of temperature difference is not present. However at longer times the odd cumulants starts differing and finally grows linearly with time $t_M$ and agrees with the corresponding long-time predictions.\n\nIn Fig.~4 we show the results for the steady state initial condition, i.e., $\\rho(0)$ which can be obtained by mapping the projection operators as identity operator, i.e., taking the limit $\\lambda \\rightarrow 0$. So in this case measurement effect is ignored and the dynamics starts with the actual steady state for the full system. The first cumulant increases linearly from the starting, $\\langle Q \\rangle = t I$ and the slope gives the correct prediction with the Landauer-like formula. However, high order cumulants still have transient behavior. In this case the whole system achieve steady state much faster compared with the other two cases.\n\n\n\\section{correlation between left and right lead heat}\n\\subsection{Product initial state}\nIn this section, we derive the CGF for the joint probability distribution $P(Q_L,Q_R)$ for the product initial state $\\rho(-\\infty)$. In order to calculate the CGF we need to measure both ${\\cal H}_L$ and ${\\cal H}_R$ at time 0 and at time $t_M$. Since the Hamiltonians for the left and the right lead commute at the same instance of time i.e., $\\big[{\\cal H}_L, {\\cal H}_R \\big]=0$, such type of measurements are allowed in quantum mechanics and also Nelson's theorem \\cite{Nelson} gurentee's that $P(Q_L,Q_R)$ is a well-defined probability distribution. The immediate consequence of deriving such CGF is that, the correlations between the left and the right lead heat can be obtained and it is also possible to calculate the CGF for total entropy flow (defined below) to the reservoirs. To calculate the CGF we need two counting fields $\\xi_L$ and $\\xi_R$ and the CGF in this case can be written down as \\cite{Esposito-review-2009}\n\\begin{equation}\n\\mathcal {Z}(\\xi_L,\\xi_R)=\\langle e^{i\\,\\xi_L\\,{\\cal H}_L+i\\,\\xi_R\\,{\\cal H}_R}\\,\\, e^{-i\\,\\xi_L\\,{\\cal H}^{H}_L(t)-i\\,\\xi_R\\,{\\cal H}^{H}_R(t)} \\rangle',\n\\end{equation}\nwhere the average is defined as \n\\begin{equation}\n\\langle \\cdots \\rangle'=\\sum_{a,c}\\Pi^{L}_a \\, \\Pi^{R}_c \\, \\rho(0)\\, \\Pi^{L}_a \\,\\Pi^{R}_c.\n\\end{equation}\n$\\Pi^{L}_a$ and $\\Pi^{R}_c$ are the projectors onto the eigenstates of ${\\cal H}_L$ and ${\\cal H}_R$ with eigenvalues $a$ and $c$ respectively, corresponding to the measurements at $t=0$. Here we will consider only the product state $\\rho(-\\infty)$, then initial projections $\\Pi^{L}_a$ and $\\pi^{R}_c$ do not play any role. We can proceed as before and finally the CGF can be written down as\n\\begin{equation}\n \\ln {\\cal Z}(\\xi_L,\\xi_R) = \\sum_{k=1}^\\infty \\frac{1}{2k}\n{\\rm Tr}_{(j,\\tau)} \\Bigl[\\big( G_{0} (\\Sigma_L^{A}+\\Sigma_R^A) \\big)^k \\Bigr],\n\\label{eq-lead-lead}\n\\end{equation}\ni.e., in this case we need to shift the contour-time arguments for both left and right lead self-energies. In the long-time limit ${\\cal Z}(\\xi_L,\\xi_R)$ becomes a function of difference of counting field $\\xi_L$ and $\\xi_R$, i.e., $\\xi_L-\\xi_R$. The explicit expression for the CGF in the long-time limit is \n\\begin{eqnarray}\n&&\\ln {\\cal Z}(\\xi_L-\\xi_R)=-t_M \\int\\frac{d\\omega}{4\\pi} \\ln \\det \\Bigl\\{ I - G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\nonumber \\\\ &&\\big[(e^{i(\\xi_L-\\xi_R) \\hbar \\omega}\\! -\\! 1) f_L  \n+( e^{-i(\\xi_L-\\xi_R)\\hbar \\omega} \\!-\\! 1) f_R  \\nonumber \\\\\n&&+(e^{i(\\xi_L-\\xi_R) \\hbar \\omega} \\!+\\! e^{-i(\\xi_L-\\xi_R)\\hbar\\omega} \\!-\\!2 ) f_L f_R \\big]\\Bigr\\}.\\qquad\n\\label{eq-lnZxi}\n\\end{eqnarray}\nwhere $G_{0}$ obeys the same type of Dyson equation as in Eq.~(\\ref{eq-Dyson-product}). This CGF in the steady state obeys the same type of GC fluctuation symmetry, which in this case is given by \n\\begin{equation}\n{\\cal Z}(\\xi_L-\\xi_R)={\\cal Z}(-\\xi_L+\\xi_R +i {\\cal A}).\n\\end{equation}\nNow performing Fourier transform of the CGF, the joint probability distribution is given by $P(Q_L,Q_R)=P(Q_L)\\,\\delta(Q_L+Q_R)$. The appearance of the delta function is a consequence of the energy conservation in the steady state, i.e., $I_L=-I_R$. In the steady state knowing probability distribution either for ${\\cal Q}_L$ or ${\\cal Q}_R$ is sufficient to know the joint probability distribution.\n\nThe cumulants can be obtained from the CGF by taking derivatives with respect to both $\\xi_L$ and $\\xi_R$, i.e.,$\\langle \\langle {Q}_L^{n} {Q}_R^{m} \\rangle \\rangle =\\partial^{n+m} \\ln {\\cal Z}\/\\partial (i\\xi_L)^n \\partial (i\\xi_R)^m,$ substituting $\\xi_L=\\xi_R=0.$ In the steady state the cumulants obey $\\langle \\langle {Q}_L^{n} {Q}_R^{m} \\rangle \\rangle= (-1)^{m} \\langle \\langle Q_L^{m+n} \\rangle \\rangle = (-1)^{n} \\langle \\langle Q_R^{m+n} \\rangle \\rangle$. The first cumulant give us the left and right lead correlation $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle=\\langle {Q}_{L} {Q}_{R} \\rangle - \\langle {Q}_{L} \\rangle \\langle {Q}_{R} \\rangle$ and in the steady state is equal to $-\\langle \\langle {Q}_L^{2} \\rangle \\rangle$.\n\nIn Fig.~5 we plot the first three cumulants for one dimensional linear chain connected with Rubin bath where the center consists of only one atom. Initially the cumulant $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle $ is positively correlated as both $Q_L$ and $Q_R$ are negative, however in the longer time since $Q_L=-Q_R$ the correlation becomes negative. We also give plots for  $\\langle \\langle {Q}_L^{2} {Q}_R \\rangle \\rangle$ (black online) and  $\\langle \\langle {Q}_R^{2} {Q}_L \\rangle \\rangle$ (red online) which are in the long-time limit \nnegative and positively correlated respectively and match with the long-time predictions.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5.eps}\n\\caption{(Color online) First three cumulants of the correlations between left and right lead heat flux for one dimensional linear chain connected with Rubin baths, starting with product initial state $\\rho(-\\infty)$. The left graph corresponds to $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle$ and the right graph corresponds to cumulants $\\langle \\langle {Q}_L^{2} {Q}_R \\rangle \\rangle$ (Black curve) and  $\\langle \\langle {Q}_R^{2} {Q}_L \\rangle \\rangle$ (Red curve). The left, center and right lead temperatures are 310 K, 290 K and 300 K respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6.eps}\n\\caption{The cumulants of entropy production $\\langle \\langle \\sigma^{n} \\rangle \\rangle$ for $n$=1, 2, 3, 4 for one dimension linear chain connected with Rubin baths, for product initial state $\\rho(-\\infty)$. The left, center and right lead temperatures are 510 K, 400 K, and 290 K respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\n\n\\subsection{Entropy flow to the reservoir}\nFrom the two parameter ($\\xi_L,\\xi_R$) CGF one can also obtain the total entropy that flows into the leads. The total entropy flow to the reservoirs can be defined as \\cite{ep1,ep2}\n\\begin{equation}\n{\\cal \\sigma}=-\\beta_{L}{\\cal Q}_{L} -\\beta_{R}{\\cal Q}_{R}.\n\\end{equation}\nIn order to calculate this CGF we just make the substitutions $\\xi_L \\rightarrow -\\beta_L  \\mu$ and $ \\xi_R \\rightarrow - \\beta_R  \\mu$ in Eq.~(\\ref{eq-lead-lead}).\nIn the long-time limit the expression for entropy-production is similar to $\\ln {\\cal Z}(\\xi_L,\\xi_R)$ with $\\xi_L-\\xi_R$ replaced by ${\\cal A}$ and becomes an explicit function of thermodynamic affinity $\\beta_R-\\beta_L$ \\cite{fluct-theorems}. The CGF in this case satisfies the following symmetry\n\\begin{equation}\n{\\cal Z}(\\mu)={\\cal Z}(-\\mu + i)\n\\end{equation}\nIn Fig.~6 we give results for the first four cumulants of the entropy flow. All cumulants are positive and in the long-time limit give correct predictions. \n\n\n\n\\section{Long-time result for $\\ln {\\cal Z}^d(\\xi)$}\n\nIn this section we derive the explicit expression for the long-time limit of the CGF $\\ln {\\cal Z}^d(\\xi)$ which is given by (Eq.~\\ref{eq-lnZxi-1})\n\\begin{equation}\n\\ln {\\cal Z}^d(\\xi)=- \\frac{i}{\\hbar}  \\int \\frac{d\\omega}{4\\pi}\n{\\rm Tr} \\bigl[ \\breve{G}[\\omega] \\breve{{\\cal F}}[\\omega,-\\omega]\\bigr].\n\\end{equation}\nwhere $G[\\omega]$ obeys the Dyson equation given in Eq.~(\\ref{eq-Dyson-full}). It is possible to write down  $\\breve{G}[\\omega]$ in terms of $\\breve{G}_{0}$ and $\\breve{\\Sigma}_L^{A}$ as $\\breve{G}[\\omega]=\\big(I-\\breve{G}_{0}\\breve{\\Sigma}^{A}_L\\big)^{-1} \\breve{G}_{0}[\\omega]$. This equation can be solved analytically. Next we assume that the product of $f(t)$ and $f(t')$ is a time-translationally invariant function, i.e., $f(t)f^{T}(t')=F(t-t')$ in order to get rid of $t+t'$ dependence term. In the Fourier domain this means $f[\\omega]f^{T}[\\omega']=2\\pi F[\\omega] \\delta(\\omega+\\omega')$. So from Eq.~(\\ref{force-matrix}) the matrix element ${\\cal F}_{12}$ is given by $\\breve{{\\cal F}}[\\omega,-\\omega]_{12} \\propto \\delta(0) F[\\omega]$. We write $\\delta(0)=t_M\/2\\pi$. Using these results the CGF can be expressed as\n\n\\begin{equation}\n \\ln {\\cal Z}^{d}(\\xi)= {i t_M}  \\int \\frac{d\\omega}{4\\pi \\hbar} \\, \\frac{1} {{\\cal N}(\\xi)}{\\rm Tr} \\Big[G_{0}^{r}[\\omega] (a+b) G_{0}^{a}[\\omega] F[\\omega] \\Big],\n\\end{equation}\nwhere $a$ and $b$ are defined in Eq.~(\\ref{eq-a}) and Eq.~(\\ref{eq-b}). Using the expressions for the self-energy the CGF reduces to \n\\begin{equation}\n \\ln {\\cal Z}^{d}(\\xi)=\\!\\!\\int \\!\\frac{d\\omega}{4 \\pi \\hbar}\\,\\frac{\\cal{K}(\\xi)}{{\\cal N}(\\xi)}\\, {\\rm Tr} \\Big[G_{0}^{r}[\\omega] \\Gamma_{L}[\\omega] G_{0}^{a}[\\omega]F[\\omega] \\Big],\n\\label{eq-driven-CGF}\n\\end{equation}\nwith \n\\begin{equation}\n{\\cal{K}}(\\xi)=(e^{-i\\xi\\hbar \\omega}\\!-\\!1 )+f_L(e^{i\\xi\\hbar\\omega}\\!+\\! e^{-i\\xi\\hbar \\omega}\\!-\\!2),\n\\end{equation} \nand \n\\begin{eqnarray}\n{\\cal {N}}(\\xi)&=& {\\rm det} \\Big[I -\\big(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R\\big) \\Bigl \\{ (f_L e^{i\\xi \\hbar \\omega}\\!-\\!1) + f_R \\nonumber \\\\ \n&&( e^{-i\\xi\\hbar \\omega}\\!-\\!1) + (e^{i\\xi \\hbar \\omega}\\!+\\! e^{-i\\xi\\hbar\\omega}\\!-\\!2 ) f_L f_R  \\Bigr\\}.\n\\end{eqnarray}\nIt is important to note that ${\\cal K}(\\xi)$ depends only on left lead temperature and satisfies the symmetry ${\\cal K}(\\xi)={\\cal K}(-\\xi-i\\beta_L)$. So we can immediately write ${\\cal Z}^{d}(-i\\beta_L)=1$ and this relation is completely independent of the information about the right lead. If we consider the two leads at the same temperature ($\\beta_L=\\beta_R=\\beta$), this form of symmetry is then closely related to the Jarzynski equality (JE) \\cite{JE,talkner2008} and ${\\cal Z}^{d}(-i\\beta)=1$ is one special form of JE. However since ${\\cal N}(\\xi)$ does not satisfy this particular symmetry of $\\xi$ at thermal equilibrium (it obeys the GC summery when the leads are at different temperatures) and the CGF $\\ln{\\cal Z}^{d}(\\xi)$ doesn't satisfy any such symmetry relation and hence JE is not satisfied. This does not violate JE as our definition of ${\\cal Z}^{d}(\\xi)$ is different from the one used to derive JE.\n  \nLet us now come back to the general scenario with leads at different temperatures and give the explicit expression of first and second cumulant by taking derivative of $\\ln {\\cal Z}^{d}(\\xi)$ with respect to $i\\xi$. \n\nThe first cumulant or moment is given by \\cite{driven-bijay}\n\\begin{equation}\n\\frac{\\langle \\langle Q_{d} \\rangle \\rangle}{t_M} = - \\int \\frac{d\\omega}{4 \\pi} \\, \\omega \\, {\\cal S}[\\omega],\n\\end{equation}\nwhere we define ${\\cal S}[\\omega]$ as the transmission function for the driven case and is given by\n\\begin{equation}\n{\\cal S}[\\omega]={\\rm{Tr}} \\bigl[G_{0}^{r} \\Gamma_{L} G_{0}^{a} F \\bigr].\n\\end{equation}\nFrom the expression of ${\\cal S}[\\omega]$ it is clear that the average energy current due to driven force is independent of $\\hbar$ and since it contains $G_{0}^{r,a}$ and $\\Gamma_L$, which are independent of temperature we can conclude that the energy current is independent of the temperature of the heat baths in the ballistic transport case. However the second cumulant and similarly the higher ones do depend on temperature of the baths. The second cumulant can be written as\n\\begin{eqnarray}\n\\frac{\\langle \\langle  Q_{d}^{2} \\rangle \\rangle}{t_M} &=& \\int \\frac{d\\omega}{4 \\pi \\hbar} \\, (\\hbar \\omega)^{2} \\, {\\cal S}[\\omega] \\Bigl[(1+2\\,f_{L})- \\nonumber \\\\\n&& 2 \\,{\\cal T}(\\omega) (f_L-f_R) \\Bigr].\n\\label{second-driven-cumulant}\n\\end{eqnarray}\nSimilarly all the higher cumulants can be obtained from the CGF and we can conclude that the distribution $P(Q_d)$ is not Gaussian.\n\n\n\\subsection{Classical limit}\nIn this section we will give the classical limit of the steady state expression for the CGF $\\ln {\\cal Z}^{s}(\\xi)$ and $\\ln {\\cal Z}^{d}(\\xi)$ given in Eq.~(\\ref{eq-steady1}) and Eq.~(\\ref{eq-driven-CGF}).\n\nFirst of all we note that retarded and advanced Green's functions, i.e., $G_{0}^{r}$ and $G_{0}^{a}$ are similar both for quantum and classical case, so they stay the same when $\\hbar \\to 0$. We know that in the classical limit $f_{\\alpha} \\rightarrow  \\frac{k_{B}T_{\\alpha}}{\\hbar \\omega}$ and also $e^{ix}=1+ix + \\frac{(ix)^{2}}{2} + \\cdots$, where $x=\\xi \\hbar \\omega$. Using this we obtain from Eq.~(\\ref{eq-steady1}) the classical limit of ${\\cal Z}^{s}(\\xi)$. \n\\begin{eqnarray}\n\\ln{\\cal Z}^{s}_{\\rm cls}(\\xi)&=& \\frac{t_M}{4\\pi} \\int d\\omega\\,\\ln \\det \\Big[I-\\big(G_{0}^{r} \\Gamma_{L} G_{0}^{a} \\Gamma_{R}\\big) \\times \\nonumber \\\\\n&& \\, k_B T_L \\, k_B T_R \\,i\\xi (i\\xi +{\\cal A})\\Big].\n\\end{eqnarray}\nThis result reproduces that of Ref.~\\cite{kundu}.\nIn the classical case also the CGF obeys the GC symmetry, i.e., it remains invariant under the transformation $i\\xi \\rightarrow -i\\xi -{\\cal A}$.\n\nLet us now get the classical limit for $\\ln {\\cal Z}^{d}(\\xi)$ using Eq.~(\\ref{eq-driven-CGF}). Following above relations the function ${\\cal K}(\\xi)$ in the limit $\\hbar \\rightarrow 0$ reduces to\n\\begin{equation}\n{\\cal K}_{\\rm cls}(\\xi)= -\\hbar \\omega \\,\\Big(i\\xi + \\frac{\\xi^2}{\\beta_L}\\Big).\n\\end{equation} \nThe transmission function ${\\cal S}[\\omega]$ stays the same as it is independent of temperature and $\\hbar$. So in the classical limit $\\ln {\\cal Z}^{d}(\\xi)$ reduces to \n\\begin{equation}\n\\ln {\\cal Z}^{d}_{\\rm cls}(\\xi)= t_M \\int \\frac{d\\omega}{4 \\pi} \\, \\omega \\, {\\cal S}[\\omega] \\, \\frac{\\Big(i\\xi + \\frac{\\xi^2}{\\beta_L}\\Big)}{{\\cal N}_{\\rm cls}(\\xi)},\n\\end{equation}\nwhere \n\\begin{eqnarray}\n{\\cal N}(\\xi)_{\\rm cls}&=&\\det\\Big[I-\\big(G_{0}^{r} \\Gamma_{L} G_{0}^{a} \\Gamma_{R}\\big) \\,k_B T_L \\, k_B T_R \\nonumber \\\\\n&&i\\xi (i\\xi +{\\cal A})\\Big].\n\\end{eqnarray}\nHere we can easily see that ${\\cal Z}^{d}(-i\\beta_L)=1$.\n\nWe can also derive the fluctuation dissipation theorem from Eq.(\\ref{second-driven-cumulant}) if we assume the leads are at the same temperature, i.e., $\\beta_{L}=\\beta_{R}=\\beta$ then we can write the second cumulant $\\langle \\langle Q_{d}^{2} \\rangle \\rangle$ as \n\\begin{equation}\n\\frac{\\langle \\langle Q_{d}^{2} \\rangle \\rangle}{t_M} = \\int \\frac{d\\omega}{4 \\pi \\hbar} \\, (\\hbar \\omega)^{2} \\,{\\cal S}[\\omega] (1+2\\,f_{L}).\n\\end{equation}\nIn the high-temperature limit using $f_{L} \\rightarrow \\frac{k_{B}T_{L}}{\\hbar \\omega}$ and we obtain\n\\begin{equation}\n\\langle \\langle Q_{d}^{2} \\rangle \\rangle =\\frac{2}{\\beta_{L}} \\langle Q_{d} \\rangle.\n\\end{equation}\n\nIn the next section we discuss Nazarov's generating function and give long-time limit expression. \n\n\\section{Nazarov's definition of generating function}\nIn this section we will derive another definition of CGF given by Eq.~(\\ref{eq-Z1-Nazarov}), starting from the CGF, derived using two-time measurement concept, i.e., Eq.~(\\ref{eq-Z-two-time}). Eq.~(\\ref{eq-Z1-Nazarov}) can be obtained from Eq.~(\\ref{eq-Z-two-time}) in the small $\\xi$ approximation as follows. In the small $\\xi$ approximation the modified Hamiltonian given in Eq.~(\\ref{modified}) takes the following form \n\\be\n{\\cal H}_{x}(t)={\\cal H}(t)+\\hbar x {\\cal I}_L(0),\n\\ee\nbecause $\\lim_{x\\rightarrow 0}{\\cal C}(x)=0$ and $\\lim_{x\\rightarrow 0}{\\cal S}(x)=\\hbar x V^{LC}$. ${\\cal I}_L$ is defined in Eq.~(\\ref{current}).\nSo the modified unitary operator becomes\n\\be\n{\\cal U}_{x}(t,0)=T e^{-\\frac{i}{\\hbar} \\int_{0}^t [{\\cal H}(\\bar{t})+\\hbar x {\\cal I}_L(0)] d\\bar{t}}.\n\\ee\nWe can consider $\\hbar x {\\cal I}_L(0)$ as the interaction Hamiltonian and write the full unitary operator ${\\cal U}_x$ as a product of two unitary operators as following \n\\begin{equation}\n{\\cal U}_{x}(t,0)={\\cal U}(t,0) \\, {\\cal U}_{x}^{I}(t,0),\n\\label{u}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n{\\cal U}(t,0)&=& T e^{-\\frac{i}{\\hbar} \\int_{0}^t {\\cal H}(t') \\, dt'}, \\nonumber \\\\\n{\\cal U}_{x}^{I}(t,0) &=& T e^{-\\frac{i}{\\hbar} \\int_{0}^t  \\hbar x {\\cal I}_L(t') dt'},\n\\label{u1}\n\\end{eqnarray}\nwith ${\\cal I}_L(t')={\\cal U}^{\\dagger}(t',0)\\,{\\cal I}_L(0)\\,{\\cal U}(t',0)$ is the current operator in the Heisenberg picture. It is important to note that ${\\cal U}$ is the usual unitary operator which evolves with the full Hamiltonian ${\\cal H}(t)$ in Eq.~(\\ref{eq-unitary}) and has no $\\xi$ dependence.\n\nIf we use product state $\\rho(-\\infty)$ as the initial state the CGF is given by \n\\begin{equation}\n{\\cal Z}(\\xi)={\\rm Tr}\\big[\\rho(-\\infty) \\, {\\cal U}_{\\xi\/2}(0,t)\\, {\\cal U}_{-\\xi\/2}(t,0)\\big].\n\\end{equation} \nIn the small $\\xi$ approximation and using the expressions for ${\\cal U}_{x}$ we can write the CGF as\n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\lim_{\\xi \\to 0} {\\cal Z}(\\xi)= {\\rm Tr}\\big[\\rho(-\\infty) \\, {\\cal U}_{\\xi\/2}^{I}(0,t)\\, {\\cal U}_{-\\xi\/2}^{I}(t,0)\\big],\n\\end{equation}\nwhere we use the property of unitary operator, i.e., ${\\cal U}^{\\dagger}(t,0) {\\cal U}(t,0)=1$. Finally using the definition of heat operator ${\\cal Q}_L$ given in Eq.~(\\ref{eq-hatQ}) and the CGF can be written down as \n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\Big \\langle {\\bar T} e^{i\\xi {\\cal Q}_{L}(t)\/2} \\, T e^{i\\xi {\\cal Q}_{L}(t)\/2}\\Big \\rangle,\n\\end{equation}\nwhich is the same as in Eq.~(\\ref{eq-Z1-Nazarov}).\n\nIn the following we will give the long-time limit expression for this CGF.\n \nIn order to calculate the CGF, it is important to go to the interaction picture with respect to the Hamiltonian ${\\cal H}_{0}={\\cal H}_L+{\\cal H}_C+ {\\cal H}_R$, as we know how to calculate Green's functions for operators which evolves with ${\\cal H}_{0}$ and treat the rest part as the interaction ${\\cal V}_{x}={\\cal H}_{\\rm int}+ \\hbar x {\\cal I}_{L}(0)$.  So the CGF on contour $C=\\big[0,t_M \\big]$ can be written as\n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\Big \\langle T_c e^{-\\frac{i}{\\hbar} \\int {\\cal V}_{x}^{I}(\\tau) d\\tau } \\Big \\rangle,\n\\end{equation}\nwhere ${\\cal V}_{x}^{I}(\\tau)$ is now given by \n\\begin{eqnarray}\n{\\cal V}_{x}^{I}(\\tau)&=&u_{L}^T(\\tau) V^{LC} u_{C}(\\tau)+ u_{R}^T(\\tau) V^{RC} u_{C}(\\tau) \\nonumber \\\\\n&&+\\hbar x(\\tau) p_{L}(\\tau) V^{LC} u_{C}(\\tau),\n\\end{eqnarray} \nwhere $p_L=\\dot{u}_L$. The time-dependence $\\tau$ is coming from the free evolution with respect to ${\\cal H}_{0}$. $x(\\tau)$ has the similar meaning as before, i.e., on the upper branch of the contour $x^{+}(t)=-\\xi\/2$ and on the lower branch $x^{-}(t)=\\xi\/2$. Now using the same idea as before, we expand the series, use Wick's theorem and finally the CGF can be expressed as\n\\begin{equation}\n\\ln {\\cal Z}(\\xi)=- \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - G_0 \\Sigma_L^{A} \\Big]. \n\\end{equation}\nHere $G_{0}$ is the same as before and is given by Eq.~(\\ref{eq-Dyson-product}). However the shifted self-energy $\\Sigma_{L}^{A}$ in this case is different and is given by (in contour-time argument) \n\\begin{eqnarray}\n\\Sigma_{L}^{A}(\\tau,\\tau')&=&\\hbar\\, x(\\tau)\\, \\Sigma_{p_L u_L}(\\tau,\\tau')+\\hbar\\, x(\\tau')\\, \\Sigma_{u_L p_L}(\\tau,\\tau') \\nonumber \\\\\n&&+\\hbar^{2}\\, x(\\tau)\\,x(\\tau')\\,\\Sigma_{p_L p_L}(\\tau,\\tau').\n\\end{eqnarray}\nThe notation $\\Sigma_{A B}(\\tau,\\tau')$ means\n\\begin{equation}\n\\Sigma_{A B}(\\tau,\\tau')= \\bigl(-\\frac{i}{\\hbar}\\bigr) V^{CL} \\,  \\langle\\, T_{c} A(\\tau) B^{T}(\\tau')\\,\\rangle \\, V^{LC}.\n\\end{equation}\nThe average here is with respect to equilibrium distribution of the left lead. It is possible to express the correlation functions such as $\\Sigma_{p_L u_L}(\\tau,\\tau')$ in terms of the $\\Sigma_{u_L,u_L}(\\tau,\\tau')=\\Sigma_L(\\tau,\\tau')$ correlations. $\\Sigma_{p_L u_L}(\\tau,\\tau')$ and $\\Sigma_{u_L p_L}(\\tau,\\tau')$ is simply related with $\\Sigma_L(\\tau,\\tau')$ by the contour-time derivative whereas for  $\\Sigma_{p_L p_L}(\\tau,\\tau')$ the expression is \n\\begin{equation}\n\\Sigma_{p_L p_L}(\\tau,\\tau')= \\frac{\\partial^{2} \\Sigma_{u_L u_L}(\\tau,\\tau')}{\\partial \\tau \\partial \\tau'} + \\delta(\\tau,\\tau') \\Sigma_{L}^{I}.\n\\end{equation}\nWhere $\\Sigma_L^{I}=V^{CL} V^{LC}$. Now in the frequency domain different components of $\\Sigma_L^{A}$ takes the following form\n\\begin{eqnarray}\n\\Sigma_A^{t}[\\omega]&=& \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\Sigma_{L}^{t}[\\omega]+ \\frac{\\hbar^{2}\\xi^{2}}{4}\\Sigma_{L}^{I}, \\nonumber \\\\\n\\Sigma_A^{\\bar{t}}[\\omega]&=& \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\Sigma_{L}^{\\bar{t}}[\\omega]- \\frac{\\hbar^{2}\\xi^{2}}{4}\\Sigma_{L}^{I}, \\nonumber \\\\\n\\Sigma_A^{<}[\\omega]&=& \\big( i \\hbar \\xi \\omega - \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\big) \\Sigma_{L}^{<}[\\omega], \\nonumber \\\\\n\\Sigma_A^{>}[\\omega]&=& \\big(-i \\hbar \\xi \\omega - \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\big) \\Sigma_{L}^{>}[\\omega]. \n\\end{eqnarray}\n\nFinally using the relations between the self-energy (see Appendix A), in the long-time limit the CGF can be written down as,\n\\begin{eqnarray}\n\\ln {\\cal Z}_{1}(\\xi) &=& - t_M \\int \\frac{d\\omega}{4\\pi} \\ln \\Big[1-(i\\xi \\hbar \\omega){\\cal T}[\\omega]\\,(f_L-f_R) \n\\nonumber \\\\ &&- \\frac{(i\\xi \\hbar \\omega)^{2}}{4}\\Big({\\cal T}[\\omega](1+2f_L)(1+2f_R)-G_{0}^{a}\\Sigma_{L}^{r} \\nonumber \\\\\n&&+G_{0}^{r}\\Sigma_{L}^{a} -G_{0}^{r}\\Gamma_{L}G_{0}^{a}\\Gamma_{L}\\Big)+ {\\cal J}(\\xi^{2},\\xi^{4})\\Big],\n\\end{eqnarray}\nwhere ${\\cal J}(\\xi^2,\\xi^4)$ is given by\n\\begin{eqnarray}\n{\\cal J}(\\xi^2,\\xi^4)&=&-\\frac{\\hbar^2 \\xi^2}{4}\\big(G_{0}^{a}+G_{0}^{r}\\big)\\Sigma_{L}^{I} - \\frac{1}{4} \\frac{(i\\xi \\hbar \\omega)^2}{2} \\frac{\\hbar^2 \\xi^2}{2} \\nonumber \\\\ \n&&+\\big(G_{0}^{r}\\Sigma_{L}^{a}G_{0}^{a}\\Sigma_L^{I}+G_{0}^{r}\\Sigma_{L}^{I}G_{0}^{a}\\Sigma_L^{r}\\big) + \\frac{1}{4} \\frac{(i\\xi \\hbar \\omega)^4}{4} \\nonumber \\\\\n&&G_{0}^{r}\\Sigma_{L}^{a}G_{0}^{a}\\Sigma_L^{r} + \\frac{1}{4} \\frac{(\\hbar^{4} \\xi^{4})}{4} G_{0}^{r}\\Sigma_{L}^{I}G_{0}^{a}\\Sigma_L^{I}.\n\\end{eqnarray}\n\nThis CGF does not obey the GC fluctuation symmetry. However it gives the correct first and second cumulant as it should because the definition of first and second cumulant turn out to be the same for both the generating functions ${\\cal Z}(\\xi)$ and ${\\cal Z}_{1}(\\xi)$ and is given by\n\\begin{eqnarray}\n&&\\langle \\langle Q \\rangle \\rangle=\\langle Q \\rangle = \\frac{\\partial \\ln {\\cal Z}(\\xi)}{\\partial {(i\\xi)}}=\\frac{\\partial \\ln {\\cal Z}_1(\\xi)}{\\partial {(i\\xi)}}= \\int_{0}^{t}  dt_1 \\langle {\\cal I}_L(t_1) \\rangle, \\nonumber \\\\\n&&\\langle \\langle Q^{2} \\rangle \\rangle =\\langle Q^{2} \\rangle- \\langle Q \\rangle^{2} = \\frac{\\partial^{2} \\ln {\\cal Z}(\\xi)}{\\partial {(i\\xi)^{2}}}=\\frac{\\partial^{2} \\ln {\\cal Z}_1(\\xi)}{\\partial {(i\\xi)^{2}}}\\nonumber \\\\\n&& \\> = \\int_{0}^{t} dt_1  \\int_{0}^{t} dt_2  \\langle {\\cal I}_L(t_1) {\\cal I}_L(t_2) \\rangle-\\Big[\\int_{0}^{t}\\! dt_1\\! \\langle {\\cal I}_L(t_1) \\rangle \\Big]^{2}. \n\\end{eqnarray}\nExpressions for higher cumulants are different for the two generating functions and hence the final expressions for the CGF's are completely different from each other.  \n\n\\section{Conclusion}\nIn summary, we present an elegant way of deriving the CGF for heat ${\\cal Q}_{L,R}$  transferred from the leads to the center for driven linear systems using the two-time measurement concept and with the help of the NEGF technique. The CGF is written in terms of the Green's function of the center and the  self-energy $\\Sigma_{L}^A$ of the leads. \nThe counting of the energy is related to the shifting in time for the self-energy.\nThis expression is valid in both transient and steady state regimes, where the information about the measurement time $t_M$ is contained in $\\Sigma_{L}^A$.  The form of the expression,\n$-(1\/2) {\\rm Tr} \\ln (1 - G_{0} \\Sigma_L^A)$, is the same whether we use a product initial state or a projected initial state, except that the meaning of the Green's function has to be adjusted accordingly. We consider three different initial conditions and show numerically for 1D linear chains connected with Rubin baths, that transient behaviors significantly differs from each other but eventually leads to the same steady state distribution in the long-time limit. We give explicit expressions of the CGF in the steady state invoking the symmetry of translational invariance in time. The CGF obeys the GC symmetry. We also give the steady state expression for the CGF in the presence of time-dependent driving forces. We obtain a two parameter CGF which is useful for calculating the correlations between heat flux and also the total entropy which flows to the leads. Our calculations can be easily generalized to arbitrary dimensions with any number of heat baths. We will show in the appendix that our method can be extended for the electronic calculations where we derive the CGF for a tight-binding model. It will be interesting to derive the CGF by taking magnetic field contribution into the Hamiltonian and also to study the cumulants in the presence of nonlinear interactions such as phonon-phonon interactions or electron phonon interactions.\n\n\n\\section*{Acknowledgments}\nWe are grateful to Juzar Thingna, Meng Lee Leek, Zhang Lifa, and Li Huanan for insightful discussions. This work is supported in part by a URC research grant R-144-000-257-112 of National University of Singapore.\n\n\n\n\\section*{Appendix}\n\\subsection{Expressions for different type of Green's functions}\nHere we give the explicit expressions for the center Green's function $G_{0}[\\omega]$ in the steady state, for a harmonic system which is connected with the leads. These formulas are required to derive the analytical form of the CGF given in Eq.~(\\ref{eq-steady1}). For the basic definitions of different types of Green's functions we refer to Ref.~\\onlinecite{WangJS-europhysJb-2008}. \n\nThe retarded Green's function $G_{0}^{r}[\\omega]$ is given by\n\\begin{equation}\nG_{0}^{r}[\\omega]=\\Big[(\\omega+i\\eta)^{2}-K^{C}-\\Sigma_{L}^{r}[\\omega]-\\Sigma_{R}^{r}[\\omega]\\Big]^{-1}.\n\\end{equation}\nHere $\\eta$ is an infinitesimal positive number which is required to satisfy the condition of causality i.e.,$G_{0}^r(t)=0$ for $t <0 $.\nThe advanced Green's function is $G_{0}^{a}[\\omega]=\\big[G_{0}^{r}[\\omega]\\big]^{\\dagger}$. The Keldysh Green's function $G_{0}^{K}[\\omega]$ can be obtained by solving the corresponding Dyson equation, Eq.~(\\ref{eq-Dyson-product}), and is given by\n\\begin{equation}\nG_{0}^{K}[\\omega]=G_{0}^{r}[\\omega]\\Sigma^{K}[\\omega]G_{0}^{a}[\\omega],\n\\end{equation}\nwhere $\\Sigma^{K}=\\Sigma_L^{K}+\\Sigma_R^{K}$ and $\\Sigma_{\\alpha}^{K}=\\Sigma_{\\alpha}^{<}+ \\Sigma_{\\alpha}^{>}$ with $\\alpha=L,R$. Alternatively, $G_{0}^{K}=G_{0}^< +G_{0}^>$. Another important identity is\n\\begin{equation}\nG_{0}^{r}[\\omega]-G_{0}^{a}[\\omega]=-i\\, G_{0}^{r}[\\omega] \\big(\\Gamma_L[\\omega]+\\Gamma_R[\\omega]\\big)G_{0}^{a}[\\omega],\n\\end{equation}\nwhere $\\Gamma_{\\alpha}[\\omega]=i\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big)$, and $\\alpha=L,R$. The self-energy for the leads are given by\n\\begin{eqnarray}\n\\Sigma_{\\alpha}^{<}[\\omega]&=&f_{\\alpha}[\\omega]\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big), \\nonumber \\\\\n\\Sigma_{\\alpha}^{>}[\\omega]&=&(1+f_{\\alpha}[\\omega])\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big).\n\\end{eqnarray}\nwhere $f_{\\alpha}[\\omega]=1\/\\bigl(e^{\\beta_{\\alpha} \\hbar \\omega_{\\alpha}}-1\\bigr)$ is the Bose distribution function. \n\nExplicit expressions for $G_{0}^{r}[\\omega]$ and $\\Sigma_{L}^{r}[\\omega]$ can be obtained for 1D homogeneous linear chain, with inter particle force constant $K$ and onsite spring constant $K_{0}$ and which is divided into three parts: the center, the left and the right. The classical equation of motion for the atoms in all three regions is\n\\begin{equation}\n\\ddot{u}_j=K u_{j-1} + \\bigl (-2K -K_{0}\\bigr )u_{j} + Ku_{j-1},\n\\end{equation}\nwhere the index $j$ runs over all the atoms in the full system.\n\nThe retarded Green's function $G_{0}^{r}[\\omega]$ can be obtained by solving \\cite{Wang-pre07} $[(\\omega+i\\eta)^{2}-\\tilde{K}]G_{0}^{r}=I$, where matrix $\\tilde{K}$ which is infinite in both directions and is $2K+K_{0}$ on the diagonals and $-K$ on the first off-diagonals.  The solution is translationally invariant in space index and is given by \n\\be\nG_{0,jk}^{r}[\\omega]=\\frac{\\lambda^{|j-k|}}{K(\\lambda-\\frac{1}{\\lambda})},\n\\ee \nwith $\\lambda=-\\frac{\\Omega}{2K}\\pm \\frac{1}{2K}\\sqrt{\\Omega^{2}-4K^{2}}$ and $\\Omega=(\\omega+i\\eta)^{2}-2K-K_{0}$, choosing between plus and minus sign by $|\\lambda|\\le 1$. \n\nThe surface Green's function $g_L^{r}[\\omega]$ can be similarly obtained in frequency domain and is given in terms of the self-energy $\\Sigma_{L}^{r}[\\omega]=-K \\lambda$. Since in equilibrium only one Green's function is independent, knowing $\\Sigma_{L}^{r}[\\omega]$ is sufficient to obtain all other Green's functions.\n\nHere we also give the expressions for Green's functions $g_C$ in time and frequency domain for an isolated single harmonic oscillator with frequency $\\omega_{0}$ (we have omitted the subscript $C$ in $g_C$) \\cite{Zeng,Brouwer}\n\\begin{eqnarray}\ng^{r}(t) & = & -\\theta(t) \\, \\frac{\\sin{\\omega_{0}t}}{\\omega_{0}},\\nonumber \\\\\ng^{r}[\\omega] & = & \\frac{1}{(\\omega+i\\eta)^{2}-\\omega_{0}^{2}},\\nonumber \\\\\ng^{<}(t) & = & \\frac{-i}{2\\omega_{0}}\\left[(1+f)e^{i\\omega_{0}t}+fe^{-i\\omega_{0}t}\\right],\\nonumber \\\\\ng^{<}[\\omega] & = & \\frac{-i\\pi}{\\omega_{0}}\\left[\\delta(\\omega+\\omega_{0})(1+f)+\\delta(\\omega-\\omega_{0})f\\right],\n\\end{eqnarray} \nwhere $f=f(\\omega_{0})=\\frac{1}{e^{\\beta\\hbar\\omega_{0}}-1}$. Other components can be obtained by exploiting the symmetry between the Green's functions such as $g^a(-t)=g^r(t)$ for $t > 0$ hence $g^r[\\omega]=g^a[-\\omega]$. The greater component is related with the lesser component via $g^>(t)=g^<(-t)$ which in the frequency domain satisfy $g^>[\\omega]=g^<[-\\omega]$.\n\n\n\n\n\\subsection{Current at short time for product initial state $\\rho(-\\infty)$}\nUsing the definition of current operator given in Eq.~(\\ref{current}) the energy current flowing from the left lead to the center is (here we assume that there is no driving force $f(t)$) \n\\begin{equation}\n\\langle {\\cal I}_{L}(t)\\rangle =-\\langle \\frac{d{\\cal H}_L(t)}{dt} \\rangle =\\frac{i}{\\hbar} \\langle \\big[{\\cal H}_{L}(t),{\\cal H} \\big] \\rangle ,\n\\end{equation}\nwhere the average is with respect to $\\rho(-\\infty)$. If $t$ is small we can expand ${\\cal H}_{L}(t)$ in a Taylor series and is given by ${\\cal H}_{L}(t)={\\cal H}_{L}(0)+t \\dot{{\\cal H}}_{L}(0) +\\cdots $ \n\nNow since $\\big[\\rho(-\\infty), {\\cal H}_{L}(0) \\big]=0$, then it immediately follows that $\\langle \\big[{\\cal H}_{L}(0),{\\cal H}\\big]\\rangle =0$ by using the cyclic property of trace. So in linear order of $t$ the current is given by\n\\begin{equation}\n\\langle {\\cal I}_{L}(t)\\rangle=t \\frac{i}{\\hbar} \\langle \\big[{\\dot{\\cal H}}_{L}(0),{\\cal H}\\big]\\rangle= -t \\frac{i}{\\hbar}\\langle \\big[p_{L}^{T}V^{LC}u_C, {\\cal H}\\big]\\rangle.\n\\end{equation}\nThe only term of full ${\\cal H}$ that will contribute to the  is ${\\cal H}_{LC}=u_{L}^T V^{LC} u_{C}$. \n\nNow using the relation that $\\big[p_L,u_L\\big]=-i\\hbar$, for one-dimensional linear chain we can write\n\\begin{equation}\n\\langle {\\cal I}_{L}(t) \\rangle =-t \\, K^{2} \\langle (u^{C}_{1})^{2} \\rangle = -t \\, K^{2} \\frac{\\hbar}{\\omega_{0}} \\Big(f_{C}(\\omega_{0})+\\frac{1}{2}\\Big).\n\\end{equation}\nwhere $u^{C}_{1}$ is the first particle in the center which is connected with the first particle of the left lead with force constant $K$. Now since the average is with respect to $\\rho(-\\infty)$,$\\langle (u^{C}_{1})^{2} \\rangle $ can be easily computed. Here $f_{C}(\\omega_{0})$ is the Bose distribution function of the particle with characteristic frequency $\\omega_{0}$. So we can see that for short time the current is negative, i.e, it goes into the lead. It is now easy to see that similar expression should also hold for $\\langle {\\cal I}_{R}(t)\\rangle$.  The negative sign in currents means that the energy flows into the leads initially irrespect to the temperature of the leads. This is consistent with the numerical results obtained by Cuansing et al.\\ \\cite{eduardos-paper,eduardos-paper1}.\n\n\\subsection{\\label{apdC} Convolution, trace, and determinant on Keldysh contour}\nHere we discuss the meaning of convolution, trace and determinant on the Keldysh contour which we used to derive the CGF's for heat flux. We define the convolution on contour in the following way.\n\\begin{eqnarray}\nA B  \\cdots D &\\rightarrow& \\sum_{j_2,j_3, \\cdots, j_n}\\int d\\tau_2 \\cdots \\int d\\tau_n A_{j_1,j_2}(\\tau_1, \\tau_2)\\nonumber \\\\\n&& B_{j_2,j_3}(\\tau_2, \\tau_3) \\cdots D_{j_n,j_{n+1}}(\\tau_n, \\tau_{n+1}), \n\\end{eqnarray}\nFrom the convolution we define trace by substituting $\\tau_{n+1}=\\tau_1$, $j_{n+1}=j_1$ and integrate also over $\\tau_1$, sum over $j_1$ i.e., \n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D)&=&\\int d\\tau_1 \\int d\\tau_2 \\cdots \\int d\\tau_n \\\\\n&& {\\rm Tr}_j\\bigl[ A(\\tau_1, \\tau_2) B(\\tau_2, \\tau_3) \\cdots D(\\tau_n, \\tau_1) \\bigr], \\nonumber  \n\\end{eqnarray}\nChanging from contour to real-time integration from $-\\infty$ to $+\\infty$, i.e., using $\\int d\\tau = \\sum_{\\sigma} \\int \\sigma dt $ we have\n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D) = \\!\\!\\!\\sum_{\\sigma_1,\\sigma_2, \\cdots, \\sigma_n}\\!\\!\\! \\int dt_1 \\int dt_2 \\cdots \\int dt_n \\qquad  \\\\\n{\\rm Tr}_j \\bigl[ A^{\\sigma_1\\sigma_2}(t_1, t_2) \\sigma_2 B^{\\sigma_2\\sigma_3}(t_2, t_3) \\cdots \\sigma_n D^{\\sigma_n\\sigma_{n+1}}(t_n, t_{1}) \\bigr]. \\nonumber   \n\\end{eqnarray}\nLet us absorb the extra $\\sigma$ into the definition of branch components,\ni.e., define\n\\begin{equation}\n\\bar{A}_{\\sigma\\sigma'} = \\sigma A^{\\sigma\\sigma'}, \\quad{\\rm or}\\quad\n\\bar{A} = \\sigma_z A,\n\\end{equation}\nwhere $A$ is viewed as $2\\times 2$ block matrix with the usual\n$+$, $-$ component, \n\\begin{equation}\nA = \\left( \\begin{array}{cc}\n                             A^{++} & A^{+-} \\\\\n                             A^{-+} & A^{--} \n                         \\end{array} \\right) = \n\\left( \\begin{array}{cc}\n                             A^t & A^< \\\\\n                             A^> & A^{\\bar t} \n                         \\end{array} \\right),\n\\end{equation}\nand $\\sigma_z$ is defined as\n\\begin{eqnarray}\n\\sigma_z &=& \\left( \\begin{array}{cc}\n                             1 & 0 \\\\\n                             0 & -1 \n                         \\end{array} \\right),\n\\end{eqnarray}\nthen it can be easily seen that \n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D)&=& \\int dt_1 \\int dt_2 \\cdots \\int dt_n  {\\rm Tr}_{j} \\bigl[{\\bar A}(t_1,t_2) \\nonumber \\\\\n&&{\\bar B}(t_2,t_3) \\cdots {\\bar D}(t_n,t_1) \\bigr] \\nonumber \\\\\n&&={\\rm Tr}_{t,j,\\sigma}(\\bar{A}\\bar{B} \\cdots \\bar {D}).\n\\end{eqnarray}\nThen we can do a rotation, where the rotation matrix is given by \n\\begin{eqnarray}\nO &=& \\frac{1}{\\sqrt{2}} \\left(\n\\begin{array}{cc}\n                             1 & 1 \\\\\n                            -1 & 1 \n\\end{array} \\right), \\quad OO^T = I.\n\\end{eqnarray}\nand we define for any matrix $A$, the rotated matrix as \n\\begin{equation}\n\\breve{A} = O^T \\sigma_z A O = O^{T} \\bar{A} O.\n\\end{equation}\nThis is known as Keldysh rotation. The effect of Keldysh rotation is given in Eq.~(\\ref{keldysh-rotation}). Since this is an orthogonal transformation the trace remains invariant and hence we can write\n\\begin{eqnarray}\n{\\rm Tr}_{t,j,\\sigma}(\\bar{A}\\bar{B} \\cdots \\bar {D})&=& {\\rm Tr}_{t,j,\\sigma}(\\breve{A}\\breve{B} \\cdots \\breve{D}).\n\\end{eqnarray}\nIf we now go to the frequency domain using the definition of two-time Fourier transform given in Eq.~(\\ref{two-time-FT}) \nthen we can compute the trace in frequency domain as \n\\begin{eqnarray}\n{\\rm Tr}_{(j,\\tau)}(AB \\cdots D) &=& \\int\\! \\frac{d\\omega_1}{2\\pi} \\!\n\\int\\! \\frac{d\\omega_2}{2\\pi}\\! \\cdots\n\\int\\! \\frac{d\\omega_n}{2\\pi} \\!\n{\\rm Tr} \\bigl\\{ \\nonumber \\\\\n && \\breve{A}[\\omega_1, -\\omega_2] \n\\breve{B}[\\omega_2, -\\omega_3] \\cdots \\breve{D}[\\omega_n, -\\omega_1] \\bigr\\}\n\\nonumber \\\\ \n&=& {\\rm Tr}_{j,\\sigma,\\omega} ( \\breve{A} \\breve{B} \\cdots \\breve{D} ).\n\\end{eqnarray}\nThe last line above define what we mean by trace over frequency domain given in Eq.~(\\ref{eq-Zsteady}).\nUnlike trace in time domain, the second argument of the each of the variables\nneed a minus sign. \n\nLet us now define what do we mean by 1 on contour. In the sense of convolution we define 1 as\n\\begin{equation}\nA \\, 1\\, D =A \\, D\n\\end{equation}\nwhich means\n\\begin{equation}\n\\int d\\tau_1 \\!\\! \\int d\\tau_2\\, A(\\tau,\\tau_1) I \\delta(\\tau_1,\\tau_2) D(\\tau_2,\\tau')= \\int d\\tau_1 A(\\tau,\\tau_1) D(\\tau_1,\\tau'). \n\\end{equation}\nNote that $\\delta(\\tau,\\tau')$ in the real time has the following form\n\\begin{equation}\n\\delta^{\\sigma,\\sigma'}(t,t')=\\sigma \\delta_{\\sigma,\\sigma'} \\delta(t-t').\n\\end{equation}\nThe inverse on the contour is defined as\n\\begin{equation}\n\\int d\\tau_1 A(\\tau,\\tau_1) B(\\tau_1,\\tau') = I \\delta(\\tau,\\tau'),\n\\end{equation}\nwhere the identity matrix $I$ takes care about the space index. Similar to the above we go to the real time and multiply the above equation with the branch index $\\sigma$ and we can write,\n\\begin{equation} \n\\int dt_1 {\\bar A}(t,t_1) {\\bar B}(t_1,t') = I \\bar{\\delta}(t-t').\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\bar{\\delta}(t-t')&=&\\sigma \\delta^{\\sigma,\\sigma'}(t,t')=\\sigma^{2} \\delta_{\\sigma,\\sigma'} \\delta(t-t') \\nonumber \\\\\n&&=\\delta_{\\sigma,\\sigma'} \\delta(t-t')\n\\end{eqnarray}\nIf we now discretize the time and write $\\delta(t_i,t_{i'})=\\delta_{i,i'}\/\\Delta t$ with $\\Delta t= |t_i-t_{i'}|$ then we have\n\\begin{equation}\n\\tilde{A} \\tilde{B} =\\tilde{I}.\n\\end{equation}\nwith $\\tilde{A}= A \\Delta t$ and similarly for other matrices. \n\nWith similar notions we can now write different types of Dyson's equation given in Eq.~(\\ref{eq-Dyson-full},\\ref{eq-Dyson-product}) as following. In contour time we have \n\n\\begin{eqnarray}\nG_0(\\tau, \\tau') &=& g_C(\\tau,\\tau') \\\\\n&&\\> + \\int \\!\\int d\\tau_1d \\tau_2\\, \ng_C(\\tau, \\tau_1) \\Sigma(\\tau_1,  \\tau_2) G_0(\\tau_2, \\tau'), \\nonumber\n\\end{eqnarray}\nIn real time following the above arguments we write\n\\begin{eqnarray}\n\\bar{G}_{0}(t,t') &=& \\bar{g}_C(t,t') \\\\\n&&\\> + \\int \\!\\int dt_1 dt_2\\, \n{\\bar g}_C(t, t_1) {\\bar \\Sigma}(t_1, t_2) {\\bar G}_0(t_2, t'), \\nonumber\n\\end{eqnarray}\nAfter Keldysh rotation we can write \n\\begin{eqnarray}\n\\breve{G}_{0}(t,t') &=& \\breve{g}_C(t,t') \\\\\n&&\\> + \\int \\!\\int dt_1 dt_2\\, \n{\\breve g}_C(t, t_1) {\\breve \\Sigma}(t_1, t_2) {\\breve G}_0(t_2, t'). \\nonumber\n\\end{eqnarray}\nFinally in the discretize time $t$ we write \n\\begin{equation}\n\\tilde{G}_{0}=\\tilde{g}_C + \\tilde{g}_C \\tilde{\\Sigma} \\tilde{G}_{0},\n\\end{equation}\nwhich is a matrix equation. Similar equations can also be written down for Eq.~(\\ref{eq-Dyson-full}).\n \nNow we define determinant via the relation $\\det(A)=\\exp({\\rm Tr} \\ln A)$, i.e, the determinant is defined in terms of trace. In order for $\\ln A$ to be defined we have to assume a Taylor expansion. For example we can define $\\ln(1+M)=M-M^2\/2 + M^3\/3 + \\cdots $ where 1 means $\\delta_{jj'}\\delta(\\tau,\\tau')$ in contour space.\n\n\\subsection{A quick derivation of the Levitov-Lesovik formula for electrons using NEGF}\nThe generating function for the non-interacting electrons was first derived by Levitov and Lesovik \\cite{Levitov,Levitov1} using Landauer type of wave scattering approach.  Klich \\cite{Klich} and Sch\\\"onhammer \\cite{otherworks2} re-derived the formula using a trace and determinant relation to reduce the problem from many-body problem to a single particle Hilbert space problem. Esposito et al.\\ gave an approach using the superoperator nonequilibrium Green's function \\cite{Esposito-review-2009}. A more rigorous treatment is given in Ref.~\\onlinecite{Bernard}.\n\nOur method for calculating CGF can be easily extended for the electron case. Here we will derive the CGF for the joint probability distribution for particle and energy without time-dependent driving force. \nThe Hamiltonian of the whole system can be written as (using tight-binding model)\n\\begin{equation}\n{\\cal H}^{e}=\\sum_{\\alpha=L,C,R} c_{\\alpha}^{\\dagger} h^{\\alpha} c_{\\alpha} + \\sum_{\\alpha=L,R} \\big(c_{\\alpha}^{\\dagger} V_{e}^{\\alpha C} c_{C} + {\\rm h.c.}\\big)\n\\end{equation}\nwhere $c_{\\alpha}$ is a column vector consisting of all the annihilation operator of region $\\alpha$. $c_{\\alpha}^{\\dagger}$ is a row vector of  the corresponding creating operators. $h^{\\alpha}$ is the single particle Hamiltonian matrix. $V_{e}^{\\alpha C}$ has similar meaning as $V^{\\alpha C}$ in the phonon Hamiltonian and $V_{e}^{\\alpha C}=(V_{e}^{C\\alpha})^{\\dagger}$. \n\nWe are interested in calculating the generating function corresponding to the particle operator ${\\cal N}_L$ and energy operator ${\\cal H}_L$ of the left-lead where ${\\cal H}_L= c_{L}^{\\dagger} h^{L} c_L$ and ${\\cal N}_L= c_{L}^{\\dagger} c_L$ \\cite{Hanggi1}. One can easily generalize the formula for right lead also as we did in the phonon case. For electrons ${\\cal N}_L$ and ${\\cal H}_L$ can be measured simultaneously because they commute, i.e., $\\big[{\\cal H}_L, {\\cal N}_L \\big]=0$. In order to calculate the CGF we introduce two counting fields $\\xi_p$ and $\\xi_e$ for particle and energy respectively. Here we will consider the product initial state (with fixed temperatures and chemical potentials for the leads) and derive the long-time result.\n\nSimilar to the phonon case we can write the CGF as \n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= \\Big \\langle e^{i\\big(\\xi_e {\\cal H}_L + \\xi_p {\\cal N}_L \\big)}\\,e^{-i\\big(\\xi_e {\\cal H}^{H}_L + \\xi_p {\\cal N}^{H}_L \\big)} \\Big \\rangle,\n\\end{equation}\nwhere superscript $H$ means the operators are in the Heisenberg picture at time $t$. In terms of modified Hamiltonian the CGF can be expressed as\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= \\Big \\langle {\\cal U}_{(\\frac{\\xi_e}{2},\\frac{\\xi_p}{2})} (0,t) \\, {\\cal U}_{(-\\frac{\\xi_e}{2},-\\frac{\\xi_p}{2})} (t,0) \\Big \\rangle,\n\\end{equation} \nwhere \n\\begin{eqnarray}\n{\\cal U}_{x,y}(t,0)&=& e^{i x {\\cal H}_L + i y {\\cal N}_L}\\, {\\cal U}(t,0)\\,e^{-i x {\\cal H}_L - i y {\\cal N}_L} \\nonumber \\\\\n&&= e^{-\\frac{i}{\\hbar} {\\cal H}_{x,y} t}\n\\end{eqnarray}\nwith $x=\\xi_e\/2$ and $y=\\xi_p\/2$ and ${\\cal U}(t,0)=e^{-i {\\cal H} t}$. ${\\cal H}_{x,y}$ is the modified Hamiltonian which evolves with both ${\\cal H}_L$ and ${\\cal N}_L$ and is given by\n\\begin{eqnarray}\n{\\cal H}_{x,y}&=& e^{i x {\\cal H}_L + i y {\\cal N}_L}\\, {\\cal H} \\,e^{-i x {\\cal H}_L - i y {\\cal N}_L} \\nonumber \\\\\n&&= {\\cal H}_L + {\\cal H}_C + {\\cal H}_R + \\big(e^{iy} c_{L}^{\\dagger}(\\hbar x) V_{e}^{LC} c_C +{\\rm h.c.}\\big)\\nonumber \\\\\n&& + \\big(c_R^{\\dagger} V_{e}^{RC} c_{C} + {\\rm h.c.}\\big),\n\\end{eqnarray}\nwhere we have used the fact that\n\\begin{eqnarray}\ne^{i x {\\cal H}_L} c_{L}(0) e^{-i x {\\cal H}_L} &=& c_{L}(\\hbar x), \\nonumber \\\\\ne^{i y {\\cal N}_L} c_{L}(0) e^{-i y {\\cal N}_L} &=& e^{-i y} c_{L}.\n\\end{eqnarray}\nSo the evolution with ${\\cal H}_L$ and ${\\cal N}_L$ is to shift the time-argument and produce a phase for $c_{L},c_{L}^{\\dagger}$ respectively. Next we go to the interaction picture of the modified Hamiltonian ${\\cal H}_{x,y}$ with respect to ${\\cal H}_{0}= \\sum_{\\alpha=L,C,R} {\\cal H}_{\\alpha}$ and the CGF then can be written on the contour running from 0 to $t_M$ and back as,\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= {\\rm Tr}\\Big[\\rho(-\\infty) T_{c} e^{-\\frac{i}{\\hbar} \\int d\\tau {\\cal V}_{x,y}^{I}(\\tau)} \\Big],\n\\end{equation} \nwhere ${\\cal V}_{x,y}^{I}(\\tau)$ is written in contour time. \n\\begin{eqnarray}\n{\\cal V}_{x,y}^{I}(\\tau)&=&\\big ( e^{i y} c_{L}^{\\dagger}(\\tau+\\hbar x) V_{e}^{LC} c_C (\\tau)+ {\\rm h.c.} \\big)+ \\nonumber \\\\\n&& \\big(c_{R}(\\tau)^{\\dagger} V_{e}^{RC} c_C(\\tau) + {\\rm h.c.}\\big).\n\\end{eqnarray}\nNow we can expand the exponential in the generating function and use Feynman diagrams to sum the series and finally the CGF can be shown to be\n\\begin{equation}\n\\ln {\\cal Z}(\\xi_e,\\xi_p)={\\rm Tr}_{j,\\tau} \\ln \\Big[1-G^{e}_{0} \\Sigma_{L,e}^{A} \\Big],\n\\end{equation}\nwhere we define the shifted self-energy for the electron case as\n\\begin{equation}\n\\Sigma_{L,e}^{A}(\\tau,\\tau')=e^{i (y(\\tau')-y(\\tau))} \\Sigma_{L,e}(\\tau+\\hbar x, \\tau'+\\hbar x') -\\Sigma_{L,e}(\\tau,\\tau').\n\\end{equation}\nThe counting of the electron number is associated with factor of a phase, while\nthe counting of the energy is related to translation in time.\nNote that the CGF does not have the characteristic $1\/2$ pre-factor as compared to the phonon case because $c$ and $c^{\\dagger}$ are independent\nvariables. In the long-time limit following the same steps as we did for phonons, the CGF can be written down as (after doing Keldysh rotation) \n\\begin{equation}\n\\ln {\\cal Z}(\\xi_e,\\xi_p)=t_M \\int \\frac{dE}{2\\pi \\hbar} {\\rm Tr} \\ln \\big(I- \\breve{G}^{e}_{0}(E)\\breve{\\Sigma}_{L,e}^{A}(E)\\big).\n\\end{equation}\nIn the energy $E$ domain different components of the shifted self-energy are\n\\begin{eqnarray}\n\\Sigma_{A}^{t}(E)&=&\\Sigma_{A}^{\\bar{t}}(E)=0, \\nonumber \\\\\n\\Sigma_{A}^{<}(E)&=& \\big( e^{i (\\xi_p + \\xi_e E)} -1 \\big) \\Sigma_{L}^{<}(E), \\nonumber \\\\\n\\Sigma_{A}^{>}(E)&=& \\big( e^{-i (\\xi_p + \\xi_e E)} -1 \\big) \\Sigma_{L}^{>}(E). \n\\end{eqnarray}\nFinally the CGF can be simplified as\n\\begin{eqnarray}\n\\ln {\\cal Z}&=&t_M \\int \\frac{dE}{2\\pi\\hbar} \\,\\ln \\det \\Bigl\\{ I + G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\Big[(e^{i\\alpha}\\! -\\! 1)f_L \\nonumber \\\\ \n&&+ ( e^{-i\\alpha} \\!-\\! 1) f_R - (\ne^{i\\alpha} \\!+\\! e^{-i\\alpha} \\!-\\!2 ) f_L f_R \\Big]\\Bigr\\}.\\qquad .\n\\end{eqnarray}\nwhere $\\alpha=\\xi_p+ \\xi_e E$ and $f_\\alpha$ is the Fermi distribution. Note the difference of the signs in the CGF as compared to the phonons.  If we replace $\\alpha$ by $(E-\\mu_L) \\xi$, the resulting formula is for the counting of the heat \n${\\cal Q}_L = {\\cal H}_L - \\mu_L {\\cal N}_L$ transferred, where $\\mu_L$ is the chemical potential of the left lead.\nThe CGF obeys the following fluctuation symmetry \\cite{fluct-theorem-1}\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= {\\cal Z}\\big(-\\xi_e + i(\\beta_R-\\beta_L), -\\xi_p -i (-\\beta_R \\mu_R -\\beta_L \\mu_L)\\big).\n\\end{equation}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n  The century-old general relativity (GR) theory still successfully passes \n  all local experimental tests. However, there are many reasons to consider \n  this theory not as an ultimate theory of gravity but only as a reasonable \n  approximation well working in a large but finite range of length and energy \n  scales. Among such reasons are the old problem of unifying gravity with other \n  physical interactions and the difficulties in attempts to quantize GR. \n  Other reasons for dealing with modifications of GR are the well-known problems \n  experienced by the theory itself: its prediction of space-time singularities\n  in the most physically relevant solutions, actually showing situations where \n  the theory does not work any more, and its inability to explain the\n  main observable features of the Universe without introducing so far invisible  \n  forms of matter, dark matter (DM) and dark energy (DE), which \n  add up to as much as 95\\,\\% of the energy content of the Universe.\n   \n  The existing modifications and extensions of classical GR can be divided into \n  two large classes. The first one changes the geometric content of the theory\n  and includes, in particular, $f(R)$ theories, multidimensional theories and \n  non-Riemannian geometries. The second class introduces new fundamental, \n  non-geometric fields and includes, in particular, scalar-tensor theories, \n  Horndeski theory \\cite{horn} and vector-tensor theories. Of much interest are the \n  cases where  it is possible to establish connections between different representatives \n  of the same class or even different classes of theories (possibly the most well-known \n  example of such a connection is the equivalence of $f(R)$ theories with a \n  certain subclass of scalar-tensor theories, see, e.g., \\cite{f(R),odin}). In the present paper \n  we discuss such an equivalence between large families of models of k-essence theories \n  and Rastall's non-conservative theories with a scalar field as a source of gravity. \n\n  The k-essence theories, introducing a non-stan\\-dard form of the kinetic term \n  of a scalar field \\cite{k1, k2}, evidently belong to the class of theories with\n  non-standard fundamental fields coupled to gravity. They proved to be a way of \n  obtaining both early inflation and the modern accelerated expansion of the \n  Universe \\cite{k2, k3, k4} driven by a scalar kinetic term instead of a \n  potential. Notably, a kind of k-essence structure also appears in string theories,\n  for example, in the Dirac-Born-Infeld action, where the kinetic term of the \n  scalar field has a structure similar to that of the Maxwell-like term in Born-Infeld \n  electrodynamics \\cite{dbi}.\n\n  Rastall's theory \\cite{rastall} is one more generalization of GR, which relaxes the\n  conservation laws expressed by the zero divergence of the stress-energy tensor (SET) \n  $T\\mN$ of matter. In this theory, the quantity $\\nabla_\\nu T\\mN$ is linked to the \n  gradient of the Ricci scalar, and in this way Rastall's theory may be viewed as \n  a phenomenological implementation of some quantum effects in a curved background.\n  Rastall's theory leads to results of interest in cosmology, e.g., the evolution of \n  small DM fluctuations is the same as in the $\\Lambda$CDM model, but DE is able to \n  cluster. This might potentially provide an evolution of DM inhomogeneities in the non-linear \n  regime different from the standard CDM model \\cite{cosmo}. The whole success of the \n  $\\Lambda$CDM model is reproduced at the background and linear perturbation levels, \n  but new effects are expected in the non-linear regime, where the $\\Lambda$CDM model\n  faces some difficulties \\cite{cusp, satt}. It has also been shown  \n  \\cite{oliver} that Rastall's theory with the canonical SET of a scalar field, in the context of  \n  cosmological perturbations,  is only consistent if matter is present. An interesting \n  observation in this analysis is that scalar field coupling with gravity leads to equations \n  very similar to those in some classes of Galileon theories.\n\n  A consideration of static, spherically symmetric solutions in k-essence theories with a power law\n  kinetic function \\cite{denis}\n  and similar solutions in Rastall's theory in the presence of a free or self-interacting scalar field\n  \\cite{we-16} has shown that some exact solutions of these two theories describe quite the same  \n  geometries, although the properties of the scalar fields are different. \n  Also, a no-go theorem concerning the possible emergence of Killing horizons, proved in the \n  k-essence framework \\cite{denis}, has its counterpart in the Rastall-scalar field system\n  \\cite{we-16}. These similarities indicate a deeper relationship between the two theories,\n  to be analyzed in this paper. We will show that in the absence of other matter than the \n  (possibly self-interacting) scalar fields, the two theories lead to completely coinciding \n  geometries (we will call this {\\sl k-R duality}) under the assumptions that the relevant \n  quantities depend on a single spatial or temporal coordinate and that the k-essence theory \n  is specified by a power-law function; then, there emerges a simple \n  relation between the numerical parameters of the theories. If there are other forms of matter, \n  the situation is more involved and depends on how the non-conservation of the SET is \n  distributed between different matter contributions in Rastall's theory. We will discuss two \n  variants of such non-conservation in the case of isotropic spatially flat cosmological models\n  and show that k-R duality generically takes place. \n   \n  The paper is organized as follows. In the next section we discuss vacuum solutions of \n  k-essence and Rastall theories. In Section 3, isotropic cosmological models are analyzed \n  with a matter source in the form of a perfect fluid. Some considerations on the speed of \n  sound are presented in Section 4, while Section 5 is devoted to some special values of \n  the numerical parameters of both theories. Some concrete cosmological configurations \n  with dustlike matter are discussed in Section 6. Our conclusions are presented in Section 7.\n\n\\section{Scalar-vacuum space-times}\n\n\\subsection{k-essence}\n\n  The k-essence theories can be defined as \\GR\\ with generalized forms \n  of scalar fields minimally coupled to gravity. In the absence of matter \n  nonminimally coupled to gravity, the most general Lagrangian is\n\\begin{eqnarray} \\lal \t\t\t\\label{L-k}\n\t{\\cal L} = \\sqrt{-g}[R + F(X,\\phi) + L_m],\n\\end{eqnarray}\n  with\n\\begin{eqnarray}                             \\label{X}\n\tX = \\eta\\phi_\\mu\\phi^\\mu,\n\\end{eqnarray}\n  where $\\phi_\\mu = \\partial_\\mu \\phi$, $F(X,\\phi)$ is an arbitrary function, and\n  $\\eta = \\pm 1$ is used to make $X$ positive since otherwise in the cases\n  like general power-law dependence $F$ will be ill-defined for $X < 0$;\n  $L_m$ is the Lagrangian density of other kinds of matter having no direct \n  coupling to the curvature or the $\\phi$ field.\n  We are using the system of units where $c = 8\\pi G =1$\n\n Variation of the Lagrangian (\\ref{L-k}) with respect to the metric and\n the scalar field leads to the field equations\n\\begin{eqnarray} \\lal                                   \t\t    \\label{EE}\n\tG\\mN \\equiv R\\mN - {\\fract{1}{2}}\\delta\\mN R = -T\\mN [\\phi] - T\\mN[m],\n\\\\[5pt] \\lal                                                       \\label{T-phi}\n\tT\\mN [\\phi] \\equiv \\eta F_X \\phi_\\mu \\phi^\\nu\n\t\t- \\frac{1}{2} \\delta\\mN F ,\n\\\\[5pt] \\lal    \t\t\t\t    \t\t \\label{eq-phi}\n       \\eta\\nabla_\\alpha (F_X \\phi^\\alpha) - {\\fracd{1}{2}}\\, F_\\phi = 0,\n\\end{eqnarray}\n  where $G\\mN$ is the Einstein tensor, $F_X = \\partial F\/\\partial X$, $F_\\phi = \\partial F\/\\partial\\phi$,\n  and $T\\mN[m]$ is the SET of matter due to $L_m$.\n\n  Now, let us make the following assumptions: \\\\[5pt] {}\n{\\bf (i)} the k-essence Lagrangian is\n\\begin{equation}                                              \\label{L-k1}\n          F (X, \\phi) = F_0 X^n - 2V(\\phi),\n\\end{equation}\n  where $n = {\\rm const} \\ne 0$ and $V(\\phi)$ is an arbitrary function (the potential).\n\\\\[5pt] {}\n{\\bf (ii)} $\\phi = \\phi(u)$, where $u$ is one of the coordinates, which may be temporal or spatial.\n\\\\[5pt] {}\n{\\bf (iii)} The metric has the form \n\\begin{equation}                  \\label{ds}\n           ds^2 = \\eta {\\,\\rm e}^{2\\alpha(u)} du^2 + h_{ik} dx^i dx^k,  \n\\end{equation} \n  where $i, k$ are the numbers of coordinates other than $u$, and the determinant of $h_{ik}$\n  has the factorized structure\n\\begin{equation}                                     \\label{h_ik}\n            \\det (h_{ik}) = {\\,\\rm e}^{2\\sigma(u)} h_1(x^i).\n\\end{equation}\n\n  In this case, we have $X = {\\,\\rm e}^{-2\\alpha(u)} \\phi_u^2$ (the index $u$ means $d\/du$), \n  and the SET of the $\\phi$ field has the following nonzero components:\n\\begin{eqnarray} \\lal            \\label{uu-k}\n              T^u_u [\\phi] =  (n - {\\fract{1}{2}}) F_0 X^n + V,\n\\\\[5pt] \\lal                \\label{ii-k}\n              T^\\ui_i [\\phi] = - {\\fract{1}{2}} F_0 X^n + V          \n\\end{eqnarray}\n  (there is no summing over an underlined index). The scalar field equation has the form\n\\begin{eqnarray} \\lal                     \\label{eq-k1}\n             {\\,\\rm e}^{-2n\\alpha} \\phi_u^{2n-2} \\big[(2n-1)\\phi_{uu} + \\sigma_u \\phi_u  \n\\nonumber\\\\ \\lal  \\hspace*{1cm}\n                               - (2n-1) \\alpha_u\\phi_u \\big] = - \\frac {1}{nF_0} \\frac {d V}{d\\phi}.\n\\end{eqnarray}\n\n\\subsection{Rastall's theory with a scalar field}\n\n  Rastall's theory of gravity is characterized by the following equations \\cite{rastall}:\n\\begin{eqnarray} \\lal                                                                        \\label{EE-Ra}\n\tR\\mN - \\frac{\\lambda}{2}\\delta\\mN R = - T\\mN,\n\\\\[5pt] \\lal \n\t\\nabla_\\nu T\\mN = \\frac{\\lambda - 1}{2}\\partial_\\mu R,\n\\end{eqnarray}\n  where $\\lambda$ is a free parameter and $T\\mN$ is the SET of matter. \n  At $\\lambda = 1$, GR is recovered.\n\n  These equations can be rewritten as\n\\begin{eqnarray} \\lal                     \t\\label{EE-R1}\n\tG\\mN = - \\biggr\\{T\\mN - \\frac{b - 1}{2}\\delta\\mN T\\biggl\\} \\equiv - \\tT\\mN,\n\\\\[5pt] \\lal \\hspace*{-0.5em}                              \\label{cons-R} \n          \\nabla_\\nu T\\mN =  \\frac{b {-} 1}{2}\\partial_\\mu T,\\quad \n                  b: = \\frac{3\\lambda - 2}{2\\lambda - 1}, \\quad T = T^\\alpha_\\alpha.\n\\end{eqnarray}\n  In this parametrization, GR is recovered if $b = 1$.\n\n  Let us consider matter in the form of a minimally coupled scalar field $\\psi$, so that  \n\\begin{equation}                           \\label{SET-psi}\n            T\\mN[\\psi] = \\epsilon (\\psi_\\mu \\psi^\\nu - {\\fract{1}{2}} \\delta\\mN \\psi_\\alpha \\psi^\\alpha)\n                                 + \\delta\\mN W(\\psi),  \n\\end{equation}\n  where $\\epsilon = \\pm 1$, indicating an ordinary ($+1$) or phantom ($-1$) nature \n  of the $\\psi$ field, $\\psi_\\mu \\equiv \\partial_\\mu\\psi$, and $W(\\psi)$ is a potential. \n  The scalar field equation follows from \\rf{cons-R} and has the form\n\\begin{equation}                                          \\label{e-psi-R}\n            \\Box\\psi + (b - 1)\\frac{\\psi^\\mu \\psi^\\nu \\nabla_\\mu \\psi_\\nu}\n                     {\\psi^\\alpha \\psi_\\alpha} = - \\epsilon (3 - 2b) \\frac {dW}{d\\psi}.\n\\end{equation}\n  \n  Let us now, in full similarity with what was done for k-essence theory, assume that \n  $\\psi = \\psi(u)$ and the metric has the form \\rf{ds}. Then the nonzero components \n  of the modified scalar field SET in the right-hand side of \\eqn{EE-R1} are\n\\begin{eqnarray} \\lal                         \\label{uu-R}\n             \\tT^u_u [\\psi] = {\\fract{1}{2}} \\epsilon b \\eta {\\,\\rm e}^{-2\\alpha} \\psi_u^2 + (3-2b) W(\\psi),\n \\\\[5pt] \\lal                           \\label{ii-R}\n\t    \\tT^{\\ui}_i [\\psi] = {\\fract{1}{2}} \\epsilon (b-2) \\eta {\\,\\rm e}^{-2\\alpha} \\psi_u^2 + (3-2b) W(\\psi),\n\\end{eqnarray}  \n  while the scalar field equation \\rf{e-psi-R} takes the form\n\\begin{equation}                                     \\label{epsi-R2}\n            {\\,\\rm e}^{-2\\alpha}\\big[b \\psi_{uu} + \\psi_u (\\sigma_u - b\\alpha_u)\\big]\n                                  = - \\epsilon \\eta (3-2b) W_\\psi,\n\\end{equation}\n  where $W_\\psi \\equiv dW\/d\\psi$ and, as before, $\\eta = \\mathop{\\rm sign}\\nolimits g_{uu}$.\n\n\\subsection{Comparison}\n\n  We assume that in the k-essence system there is no other matter than the scalar \n  field $\\phi$ and in the Rastall system there is no other matter than the scalar $\\psi$. \n  Let us find out under which conditions the right-hand sides of the \\EE s \\rf{uu-k}\n  and \\rf{ii-k} coincide with those of the effective \\EE s of Rastall's theory,\n  \\rf{uu-R} and \\rf{ii-R}. This will guarantee that the solutions for the metric are \n  also the same. \n\n  To begin with, we identify the potentials:\n\\begin{equation}         \\label {VW}\n                 V(\\phi) = (3-2b) W(\\psi).\n\\end{equation} \n  Next, we equate the ratios of the kinetic parts of $T\\mN[\\phi]$ and $\\tT\\mN[\\psi]$,\n  to obtain \n\\begin{equation}             \\label{nb}\n      \\frac{2n-1}{-1} = \\frac b {b-2} \\ \\ \\Rightarrow\\  \\  (2-b) n = 1.  \n\\end{equation}\n  Then, equating the kinetic parts themselves, we find that\n\\begin{eqnarray} \\lal              \\label{phips}\n           \\psi^2_u = \\epsilon\\eta n F_0 \\phi_u^{2n}{\\,\\rm e}^{2(1-n)\\alpha}.\n\\end{eqnarray} \n\n  Under the three conditions \\rf{VW}--\\rf{phips}, the metric field equations of the \n  two theories completely coincide, therefore their sets of solutions are also identical.\n  Substituting \\rf{phips} to \\rf{epsi-R2}, one can easily verify that under these conditions \n  the scalar field equations  \\rf{eq-k1} and \\rf{epsi-R2} are also equivalent.\n\n  This general result covers many static symmetries (spherical, plane, cylindrical, etc.),\n  homogeneous cosmologies (FRW, all Bianchi types, \\KS) and even inhomogeneous \n  ones if their metrics are of the form \\rf{ds}, \\rf{h_ik}.\n\n  Here and in most of the paper we consider the generic values of the parameters $n$ and \n  $b$ and exclude from consideration their special values that require a separate analysis, \n  such as, for example, $b=0$, $b=3\/2$ and $n = 1\/2$. Some remarks on these special cases \n  will be made in Section 5.   \n\n\\section{Cosmology with matter}\n  \n  When, besides the scalar field, matter is present, it is better, for evident technical reasons, \n  to restrict ourselves from the beginning to a certain type of metrics. We will consider \n  cosmological FLRW spatially flat metrics\n\\begin{equation}                     \\label{ds1}\n\t  ds^2 = dt^2 - a(t)^2[dx^2 + dy^2 + dz^2],\n\\end{equation}\n  so that in \\rf{ds} and \\rf{h_ik} we have $\\eta=1$, ${\\,\\rm e}^\\alpha =1$, and ${\\,\\rm e}^\\sigma = a(t)^3$.\n  Matter will be taken in the form of a perfect fluid, so that \n\\begin{equation}               \\label{Tm}\n\t  T\\mN [m] = \\mathop{\\rm diag}\\nolimits (\\rho, -p, -p, -p), \n\\end{equation}  \n  where $\\rho$ is the density and $p$ is the pressure.\n\n\\subsection{k-essence cosmology}\n  \n In the FLRW metric \\rf{ds1} and with $\\phi=\\phi(t)$, the field equations \\rf{EE}--\\rf{eq-phi} \n with matter (where we denote $\\rho = \\rho_k,\\ p=p_k$) take the form\n\\begin{eqnarray} \\lal          \\label{tt-k1}\n  \t3H^2 = {\\fract{1}{2}}(2n - 1) F_0\\dot\\phi^{2n} + V(\\phi) + \\rho_k,\n\\\\[5pt] \\lal             \\label{ii-k1}\n          2\\dot H + 3H^2 = - {\\fract{1}{2}} F_0 \\dot\\phi^{2n} + V(\\phi) - p_k,\n\\\\[5pt] \\lal             \\label{phi-k1} \n         \\dot\\phi^{2n-2} [(2n-1)\\ddot\\phi + 3H\\dot\\phi] = - \\frac{1}{nF_0}V_\\phi,\n\\end{eqnarray}\n  where $H =\\dot a\/a$ is the Hubble parameter and $V_\\phi \\equiv dV\/d\\phi$. The SET \n  of matter satisfies the conservation law $\\nabla_\\nu T\\mN[m] =0$, whence\n\\begin{equation}               \\label{cons}\n\t\\dot\\rho_k + 3H (\\rho_k + p_k) =0.\n\\end{equation} \n  \n\\subsection{Rastall cosmology with a scalar field and matter}\n\\def\\trho{{\\widetilde \\rho}}\n\\def\\tp{{\\widetilde p}}\n\n  The Rastall equations have the form \\rf{EE-R1} and \\rf{cons-R}, where now  \n  $T\\mN$ is the total energy-momentum tensor,\n\\begin{eqnarray}\n              T\\mN = T\\mN [\\psi] + T\\mN [m],\n\\end{eqnarray}\n  with $T\\mN[\\psi]$ given by \\rf{SET-psi} and $T\\mN[m]$ by \\rf{Tm}. \n  In Eqs.\\, \\rf{EE-R1}, the modified energy-momentum tensor $\\tT\\mN$ is then a sum\n  of $\\tT\\mN[\\psi]$ given by \\rf{uu-R} and \\rf{ii-R} and $\\tT\\mN[m]$ with the components\n\\begin{eqnarray} \\lal        \\label {redef}\n\t  \\tT^t_t [m] = {\\fracd{1}{2}} [(3-b) \\rho + 3 (b-1) p] \\equiv  \\trho,\n\\nonumber\\\\ \\lal \n\t  \\tT^\\ui_i [m] = {\\fracd{1}{2}} [(1-b)\\rho + (3b-5)p] \\equiv - \\tp \n\\end{eqnarray} \n  (we preserve the notation $\\rho$ and $p$ without indices for matter in Rastall gravity). \n  Hence the Rastall equations read\n\\begin{eqnarray} \\lal               \\label{tt-Ra}\n           3 H^2 = {\\fracd{1}{2}} \\epsilon b \\dot\\psi{}^2 + (3-2b) W + \\trho,\n\\\\[5pt] \\lal                   \\label{ii-Ra}\n\t 2\\dot H + 3H^2 = {\\fracd{1}{2}} \\epsilon (b-2)  \\dot\\psi{}^2 + (3-2b) W - \\tp,\n\\end{eqnarray}\n  while the equation for $\\psi$ depends of further assumptions on how the nonconservation \n  of the full SET according to \\rf{cons-R} is distributed between $\\psi$ \n  and matter. One can notice that \n\\begin{equation}                 \\label{NEC-R}\n\t\\trho + \\tp = \\rho + p,\n\\end{equation}\n  and \n\\begin{equation}                              \\label{trace}\n        \\trho - \\rho = p - \\tp = \\frac{1-b}{2} (\\rho - 3p).   \n\\end{equation}\n\n  Let us consider two (of an infinite number of) alternatives in incorporating matter \n  to Rastall's theory with a scalar field:  \n\\begin{description}\n\\item[R1:] \n  The SETs of $\\psi$ and matter obey \\rf{cons-R} each separately, so there is\n  no mixing between the two sources of gravity;\n\\item[R2:]\n   The SET of matter is conservative, so that \n\\begin{equation} \n            \\dot\\rho + 3H (\\rho+p) =0.               \n\\end{equation}\n\\end{description}\n\n\\subsection{Case R1: No mixing of scalar field and matter} \n\n  In this case we have\n\\begin{eqnarray} \\lal              \\label{cons-i1}\n          \\nabla_\\nu T\\mN[\\psi] =  \\frac{b - 1}{2}\\partial_\\mu T[\\psi],\n\\\\[5pt] \\lal                    \\label{cons-i2}\n          \\nabla_\\nu T\\mN[m] =  \\frac{b - 1}{2}\\partial_\\mu T[m],\n\\end{eqnarray}\n  The first of these conditions leads to the scalar field equation \\rf{epsi-R2}, which in  \n  the present case reads\n\\begin{equation}               \\label{epsi-i} \n\tb \\ddot\\psi + 3H \\dot\\psi = -\\epsilon (3-2b) W_\\psi.\n\\end{equation}\n  With \\rf{NEC-R}, the condition \\rf{cons-i2} has the form \n\\begin{equation}               \\label{cons-i}\n\t\\dot\\trho + 3H (\\rho + p) =0.\n\\end{equation} \n  The full set of equations consists of \\rf{tt-Ra}, \\rf{ii-Ra}, \\rf{epsi-i}, and \\rf{cons-i}, \n  with the definitions \\rf{redef}.\n  \n  From \\rf{NEC-R} it follows that if matter satisfies the null energy condition (NEC),\n  then the same is true for the ``effective'' density and pressure ($\\trho$ and $\\tp$) in \n  Rastall's theory. However, from \\rf{redef} it can be verified that the positivity of \n  the energy density (or pressure) is not guaranteed in the k-essence case if it is \n  imposed in the Rastall theory, and vice versa.\n\n  It is easy to see that the right-hand sides of Eqs.\\, \\rf{tt-Ra} and \\rf{ii-Ra} coincide with\n  those of \\rf{tt-k1} and \\rf{ii-k1} if, in addition to the relationships \\rf{VW}--\\rf{phips}\n  for scalar variables, we identify \n\\begin{equation}                                      \\label{id-m}\n                \\rho_k = \\trho, \\qquad p_k = \\tp.\n\\end{equation}\n  The correctness of this identification is confirmed by the identity of the conservation \n  laws \\rf{cons} and \\rf{cons-i}. Thus, as in the vacuum case, the parameters \n  $n$ and $b$ are related by \\rf{nb}, that is, $n(2-b) =1$, and the scalar fields $\\phi$ and \n  $\\psi$ are related by \\eqn{phips} which now reads\n\\begin{equation}                                              \\label{psif}\n               \\dot\\psi{}^2 = \\epsilon n F_0 \\dot\\phi{}^{2n}.\n\\end{equation}\n  \n\\subsection{Case R2: Conservative matter}\n \n  We now have $\\nabla_\\nu T\\mN [m] = 0$. This condition is particularly important \n  for the structure formation in the universe for the case of a pressureless fluid \n  since ordinary matter must agglomerate.\n\n  In this case, for the scalar field SET we have \n\\begin{eqnarray}\n         \\nabla_\\nu T\\mN [\\psi] ={\\fracd{1}{2}} (b - 1) (\\partial_\\mu T[\\psi] + \\partial_\\mu T[m]),\n\\end{eqnarray}\n  which leads to the scalar field equation\n\\begin{eqnarray} \\lal              \\label{epsi-ii} \n\t\\dot\\psi(b \\ddot\\psi + 3H \\dot\\psi) = -\\epsilon (3-2b) W_\\psi \\dot\\psi\n\\nonumber\\\\ \\lal  \\hspace*{1in}\n\t+ {\\fracd{1}{2}}\\epsilon (b-1)(\\dot\\rho - 3\\dot p). \n\\end{eqnarray}\n  The full set of equations consists of \\rf{tt-Ra}, \\rf{ii-Ra}, \\rf{epsi-ii}, and \\rf{cons}, \n  with the definitions \\rf{redef}. Note that \\eqn{epsi-ii} mixes the scalar field and the \n  matter fluid even though the fluid is conserved as in GR.\n\n  This conservation makes us identify the matter SET components in the k-essence\n  and Rastall theories: $\\rho = \\rho_k$, $p = p_k$. As a result, identification of the other \n  parts of the total SET is only partly the same as in the previous case. \n\n  Identifying, as before, the potentials according to \\rf{VW} (that is, $V = (3-2b)W$)\n  and comparing the Friedmann-like equations \\rf{tt-Ra} and \\rf{ii-Ra} with \n  their k-essence counterparts \\rf{tt-k1} and \\rf{ii-k1}, we obtain, as before,  \n\\begin{equation}                                  \\label{psif-ii}\n                   \\epsilon\\dot\\psi^2 = nF_0\\dot\\phi^{2n}.\n\\end{equation}\n  The correctness of this identification is verified by substituting \\rf{psif-ii} into \n  the scalar field equation \\rf{epsi-ii}: indeed, since we have now, due to \\rf{trace}, \n\\begin{equation}                                   \\label{nb-ii}\n                   \\epsilon\\dot\\psi^2 \\Big[b - \\frac{2n-1}{n}\\Big] = (b-1) (\\rho-3p),  \n\\end{equation}\n  this substitution leads precisely to the scalar field equation \\rf{phi-k1} of the \n  k-essence theory.\n\n  It is important that in the case of conservative matter, a comparison \n  between the two theories does not lead to a direct relationship like \n  \\rf{nb} between their numerical parameters $n$ and $b$. Instead, we have \n  the equality \\rf{nb-ii}, from which \\rf{nb} is restored only in the special case \n  $\\rho=3p$ (zero trace of the matter SET, radiation).\n\n\\subsection{Further consequences of matter conservation}\n\n  The relation \\rf{nb-ii} creates a connection between the temporal behavior of $\\psi$ and the \n  matter content. Indeed, inserting \\rf{nb-ii} to \\rf{epsi-ii} with zero or constant potential, \n  we find\n\\begin{equation}\n                  \\dot F + \\frac{6n}{2n - 1}HF = 0,\n\\end{equation}\n  where $F = \\dot\\psi^2$. This leads to\n\\begin{equation}             \\label{sol1}\n                 \\dot\\psi^2 = \\psi_0a^{-6n\/(2n - 1)},\n\\end{equation}\nwhere $\\psi_0$ is an integration constant.\n\n  From Eq.\\, \\rf{nb-ii} it is clear that the matter density and pressure  must also evolve by a \n  power law as functions of the scale factor. Hence, only an equation of state (EoS) of the type \n  $p = w\\rho$, with $w = {\\rm const}$, is possible. In this case, \n\\begin{equation}\n          \\rho = \\rho_0 a^{-3(1 + w)},      \\qquad \\rho_0 = {\\rm const},     \n\\end{equation}\n  implying the relation between $w$ and $n$\n\\begin{equation}                                \\label{rela}\n               w = \\frac{1}{2n - 1}.\n\\end{equation}\n  We see that a substitution of  $p = w\\rho$ into the scalar field equation relates the EoS factor\n  $w$ with the k-essence power $n$, while the Rastall constant $b$ remains arbitrary. \n  Moreover, Eqs.\\, \\rf{tt-k1} and \\rf{ii-k1} show that the pure k-essence scalar field $\\phi$ behaves \n  as a perfect fluid with the same EoS factor \\rf{rela} (see also \\cite{carla}).\n\n  In other words, assuming a zero or constant potential $V = (3-2b)W$ and conservative matter \n  in the Rastall framework, we obtain that {\\sl the k-R duality is only possible if matter \n  is a perfect fluid with the linear EoS $p = w\\rho$, coinciding with the effective EoS\n  of the scalar field $\\phi$. }\n\n  With $p = w\\rho$ \\eqn{nb-ii} gives \n\\begin{eqnarray} \\lal\n      \\epsilon \\dot\\psi^2 = k\\rho,\n\\qquad\n\tk = \\frac{n(b- 1)(1 - 3w)}{bn-2n+1}.\n\\end{eqnarray}\n  Inserting this to the Friedmann-like equation \\rf{tt-Ra}, we obtain in terms of $n$ or $w$\n\\begin{eqnarray}\n            3H^2 \\al =\\al V + \\frac{\\rho}{2n - 1}\n\t\\biggr\\{\\frac{2nb(b - 1)(n - 2)}{bn -2n + 1}\n\\nonumber\\\\ \\lal  \\hspace*{1cm}\\cm\n\t           + 2(2b - 3) + 2n(3 - b)\\biggr\\}\n\\\\[5pt] {} \n           \\al =\\al   V + \\frac{\\rho}2 \\biggl\\{\\frac{b(b-1)(1+w)(1-3w)}{(b-2)(1+w)+2w}\n\\nonumber\\\\ \\lal  \\hspace*{1cm}\\cm\n                 + 3-b + 3w(b-1)\\biggr\\}.\n\\end{eqnarray}\n  The right-hand side must be positive. Therefore, given $n$ and $V$ (or alternatively $w$ \n  and $V$), we obtain a restriction on $b$. For example, if $V=0$ and $w = 0$ \n  (dust, $n \\to \\infty$),\\footnote\n\t{This relation makes sense even if the Lagrangian formulation becomes ill-defined,\n\t see \\cite{carla}.} \n  we have either $b < 3\/2$ or $b > 2$ (provided $\\rho > 0$). For $w = 1$ (stiff matter, $n = 1$), \n  there is no restriction on $b$, and we obtain $H^2 = V = {\\rm const} >0$, hence a de Sitter \n  expansion, $a(t) \\propto{\\,\\rm e}^{Ht}$. In this case, stiff matter precisely cancels the contribution \n  from the scalar field $\\psi$ or $\\phi$. \n\n  If we introduce a variable potential or a more complex equation of state, the situation becomes \n  much more involved. It must be stressed that the EoS $p = w\\rho$ with $w={\\rm const}$\n  covers most of the interesting cases in cosmology. Moreover, we expect that the \n  perturbative behavior may be very different in the two theories even in this case.\n\n  There emerge two more natural questions. First, we have obtained that in k-essence theory\n  there are simultaneously a scalar field and a perfect fluid with the same EoS and hence the same \n  time evolution of their densities and pressures. Can we unify them by, for example, redefining\n  the scalar field? A probable answer is ``no'' because these two kinds of matter are expected to\n  behave quite differently at the perturbative level.\n\n  Another question is: how is it possible to have a completely definite situation in k-essence \n  theory but an arbitrariness in the parameter $b$ in the dual solution of Rastall's theory? An \n  answer is that this arbitrariness is compensated by the corresponding non-conservative \n  behavior of the scalar field $\\psi$.     \n\n\\section{Perturbations and the speed of sound}\n\n  A power law k-essence model with $V = 0$ is equivalent in the cosmological \n  framework (such that $(\\partial_\\mu \\phi)^2 > 0$) to a perfect fluid with the equation \n  of state $p = w \\rho$, where the constant $w$ is related to the power $n$:\n\\begin{equation}      \\label{w}\n\tw = \\frac{1}{2n - 1}.\n\\end{equation}\n  In a fluid, adiabatic perturbations propagate as a sound with the speed $v_s$ such that\n\\begin{equation}             \\label{v-fl}\n\tv_s^2 = dp\/d\\rho = w.\n\\end{equation}\n  \n  Scalar field perturbations for general Lagrangians of the form $F(X,\\phi)$\n  have been treated in detail in \\cite{neven, k3}. It has been shown there that\n  a k-essence theory implies \n\\begin{equation}\n             v_s^2 = \\frac{F_X}{F_X + 2X F_{XX}},\n\\end{equation}\n  and this expression is valid even if there is an arbitrary potential term $V(\\phi)$.\n  In particular, for the theory \\rf{L-k1}, where $F(X) = F_0 X^n - 2V(\\phi)$, we find again\n\\begin{equation}                  \\label{ss1}\n              v_s^2 = \\frac{1}{2n - 1}\n\\end{equation}\n  in full agreement with \\rf{v-fl}. Thus there is a complete equivalence between \n  a perfect fluid and k-essence without a potential not only for a cosmological \n  background but even on the perturbative level as far as adiabatic perturbations\n  are concerned. In particular, if $w < 0$, corresponding to $n < 1\/2$, the model \n  is perturbatively unstable since it implies $v_s^2 < 0$. This is true both for a perfect \n  fluid and for k-essence. Moreover, although the presence of a potential \n  changes the scalar field dynamics, the propagation speed of its perturbations, coinciding \n  with the derived speed of sound \\cite{neven}, is still the same as with $V=0$.  \n  \n  In Rastall's theory things may be different. The speed of sound for a scalar field is \n  given by \\cite{oliver, liddle}\n\\begin{equation}                    \\label{ss2}\n                 v_s^2 = \\frac{2 - b}{b}.\n\\end{equation}\n  In scalar vacuum and in the R1 case (matter obeys the non-conservation equation \n  \\rf{cons-i1}), we have the relation \\rf {nb}, $(2-b)n = 1$, which makes (\\ref{ss1}) \n  and (\\ref{ss2}) identical. Furthermore, the fluids in the corresponding models \n  obey different equations of state, see \\rf{id-m}. However, in the Rastall model we \n  can still characterize the fluid by the ``effective'' density and pressure, $\\trho$ and \n  $\\tp$, the SET written in their terms is conservative, hence the squared speed of \n  sound of the Rastall fluid is equal to $d\\tp\/d\\trho = d p_k\/d\\rho_k$.\n  Thus we can conclude, even without performing a complete perturbation analysis, \n  that the models belonging to the two theories coincide not\n  only at the background level but also at the level of adiabatic perturbations.\n\n  In the case R2 (conserved matter), we have another relation \\rf{nb-ii} between the \n  parameters $b$ and $n$, without such a simple connection. As a consequence, in principle \n  it is possible that an unstable model in a k-essence model may correspond to a stable model \n  in Rastall's theory, or vice versa, since, as shown above, Eq.\\, \\rf{nb} does not hold,\n  $b$ being now essentially independent of $n$ up to some possible restrictions on their range. \n  In fact, in this case, even non-adiabatic perturbations may appear, due to the coupling\n  between the scalar field and matter. \n\n\\section{Some special cases}    \n\n\\subsection{$n=1\/2$}    \n  \n  In this case, the k-essence scalar field equation takes the form\n\\begin{equation}                         \\label{n-half}\n\t\t3H = -2 F_0^{-1} V_\\phi. \n\\end{equation}\n  Thus if $V={\\rm const}$, we have $H=0$, hence $a = {\\rm const}$, and flat space-time is obtained. \n  One can certainly obtain $H=0$ in Rastall gravity under special assumptions, but the question\n  of k-R duality looks meaningless in this trivial case. \n  \n  If $V=V(\\phi)$, the $\\phi$ field has no dynamics of its own, but \\eqn{n-half} expresses it\n  in terms of $H$, and the Friedmann equations \\rf{tt-k1} and \\rf{ii-k1} are meaningful.\n  \n  The Rastall counterpart in the scalar-vacuum and R1 cases is then obtained with $b=0$, \n  $V = 3W$ (according to \\rf{nb} and \\rf{VW}) and \n\\begin{equation}                                                                          \\label{psif-half}\n\t\t2\\epsilon \\dot\\psi{}^2 = F_0 \\dot\\phi. \n\\end{equation}\n\n  In the case R2 (conserved matter), $b$ remains arbitrary, but k-R duality still holds.\n  Indeed, if we substitute \\rf{psif-half} into \\eqn{epsi-ii} and use the Friedmann-like \n  equations \\rf{tt-Ra}, \\rf{ii-Ra} to calculate $\\dot\\rho-3\\dot p$, we obtain \\eqn{n-half}.\n\n  The main feature in this case is that nontrivial solutions with $n = 1\/2$ and k-R duality\n  are only achieved in the presence of a variable potential.\n\n\\subsection{$b=3\/2$}\n\n  With this value of $b$, the potential disappears from Rastall's gravity, hence the k-R duality \n  implies $V=0$, and we deal with zero potentials. In other respects, the situation is  \n  described as in the general case. \n\n  A feature of interest is that with $b = 3\/2$ \\eqn{redef} leads to $\\trho = 3\\tp$. \n  Therefore, in the scalar-vacuum and R1 cases, the dual k-essence counterpart of \n  this Rastall model contains matter with $\\rho_k= 3p_k$ (see \\rf{id-m}).  \n  Due to \\rf{nb}, in addition, $n=2$, so that the $\\phi$ field also behaves as radiation.\n\n  In the case R2 (conserved matter), the relations \\rf{id-m} and \\rf{nb} are no more \n  valid, and the general description is applicable.  \n      \n\\subsection{$b=2$}\n\n  Equations \\rf{w} and \\rf{ss2} give zero values of pressure and the speed of sound of\n  a scalar field in Rastall's theory. The corresponding expression \\rf{ss1} in k-essence \n  theory leads to $n \\to \\infty$ according to the general relation \\rf{nb}.\n \n  If there is conserved matter (case R2), then (unless this conserved matter is pure\n  radiation, $\\rho = 3p$) \\eqn{nb} is no more valid, so that the speeds of sound \n  of scalar fields are different in the two theories. It means that k-R duality does not\n  exist for perturbations even though it does exist for the isotropic background.\n\n\\subsection{$b=0$}\n\n  In the scalar-vacuum and R1 cases we return to the above description for $n = 1\/2$.\n  \n  With conserved matter (case R2), the Rastall scalar field takes the form\n\\begin{equation}                 \\label{psi-b0}\n         3H \\epsilon \\dot\\psi{}^2 = -3 \\dot W -{\\fracd{1}{2}} (\\dot\\rho-3\\dot p),\n\\end{equation}\n  looking like a constraint equation since it contains only the first-order derivative.\n  However, the k-R duality still works, as before: thus, a substitution of \\rf{psif} \n  (what is important, with arbitrary $n \\ne 0$) and $\\rho-3p$ from Eqs.\\, \\rf{tt-Ra}, \n  \\rf{ii-Ra} into \\rf{psi-b0} leads to \\rf{phi-k1}, which is a second-order equation \n  unless $n = 1\/2$.  \n\n\\section{Examples}\n\n  Let us now consider some specific examples of the equivalence discussed above,\n  assuming a zero or constant potential and dust as a possible matter contribution.\n\n\\subsection{Scalar vacuum}\n\n  Consider scalar vacuum with zero potential. The k-essence equations give\n\\begin{eqnarray}                                    \\label{sfe}\n\t\\dot\\phi \\al =\\al  \\phi_0  a^{- 3\/(2n - 1)},    \\qquad  \\phi_0 = {\\rm const},\n\\\\[5pt] {}\n          H^2 \\al =\\al  \\frac{2n - 1}{6}F_0 \\phi_0^{2n}a^{-6n\/(2n - 1)}.\n\\end{eqnarray}\n  In terms of cosmic time we obtain\n\\begin{eqnarray}\n               a \\al =\\al a_0  t^{2\/[3(1 + w)]},  \\qquad  a_0 = {\\rm const},\n\\\\[5pt] {}\n\t  \\dot\\phi \\al =\\al  \\phi_1  t^{- 2w\/(1 + w)},\n\\end{eqnarray}\n  where $\\phi_1$ is a combination of the previous constants, and we \n  have written $w = 1\/(2n - 1)$, thus identifying the k-essence with a perfect \n  fluid with the EoS $p = w\\rho$. \n\n  In Rastall's theory, the dual solution contains the same $a(t)$, while the scalar field is given by\n\\begin{equation}                              \\label {sfr}\n                 \\dot\\psi \\propto a^{- 3(1 + w)\/2} = a^{-3\/b} \\propto t^{-1},\n\\end{equation}\n  where now we should put $w = (2-b)\/b$. We notice that while the k-essence scalar field \n  behavior depends on the EoS parameter $w$, the Rastall scalar is simply $\\psi = \\log t + {\\rm const}$.\n\n  Addition of a constant potential, equivalent to a cosmological constant, does not change the \n  scalar field evolution laws \\rf{sfe} and \\rf{sfr} in terms of $a$ but makes the time \n  dependences more complex, not to be considered here.\n\n  In the presence of matter, as we saw above, the form of k-R duality depends on how matter \n  couples to the scalar field. \n\n\\subsection{Dust and Rastall-R1 models}\n\n  Suppose that in k-essence theory, in addition to the scalar field $\\phi$, there is \n  pressureless fluid (dust), so that\n\\begin{equation}                            \\label{dust-k} \n              p_k =0, \\qquad  \\rho_k = \\rho_1 a^{-3}, \\qquad   \\rho_1={\\rm const}.\n\\end{equation}\n  then, in the R1 version of Rastall's theory, according to \\rf{id-m}, we have the conditions\n\\begin{eqnarray}                           \\label{ex1}\n\t {\\fracd{1}{2}}\\Big[(3 - b)\\rho + 3(b-1)p\\Big] \\al =\\al \\trho = \\rho_k,\n\\nonumber\\\\ {}\n          {\\fracd{1}{2}} \\Big[(b-1)\\rho + (5-3b)p \\Big] \\al =\\al \\tp =  0,\n\\end{eqnarray}\n  leading to the following relations for the density and pressure \n\\begin{eqnarray}                               \\label{ex2}\n          \\rho \\al =\\al \\frac{5-3b}{2(3-2b)} \\rho_k,\n\\nonumber\\\\ {}\n\tp \\al =\\al \\frac{1-b}{5-3b} \\rho = \\frac{1-b}{2(3-2b)} \\rho_k,\n\\end{eqnarray}\n  both evolving as $\\rho \\propto p \\propto a^{-3}$. Thus in Rastall cosmology the \n  fluid acquires pressure (except for the GR value $b = 1$). The scalar fields in both \n  models satisfy the same relations as in the vacuum case, valid for any values of \n  $n$ and $b$ such that $n(b-2)=1$. \n\n  Adding a constant potential $V = (3-2b)W$ does not change the relations \\rf{ex1}\n  and \\rf{ex2} and introduces an effective cosmological constant. We then obtain  \n  a three-component model with matter, a cosmological constant and a scalar field \n  whose behavior is determined by $n$ or, equivalently, by $w = 1\/(2n-1)$. In the  \n  dual Rastall model, we have an effective pressure even though in the k-essence \n  model $p_k=0$. \n\n  If, on the contrary, we introduce matter with $p=0$ in Rastall's (R1) theory, \n  then in the dual k-essence model we have \n\\begin{equation}\n              \\rho_k = \\trho = {\\fracd{1}{2}} (3-b)\\rho, \\qquad   p_k = \\tp = {\\fracd{1}{2}} (b-1)\\rho, \n\\end{equation}\n  and their evolution law agreeing with \\eqn{cons-i} reads\n\\begin{equation}\n              \\rho_k \\propto \\rho \\propto a^{-6\/(3-b)} = a^{-3(1+w_k)},                 \n\\end{equation}\n  where the EoS parameter $w_k$ of the fluid in the k-essence model is \n\\begin{equation}\n              w_k = \\frac{b-1}{3-b} = \\frac{n-1}{n+1}       \n\\end{equation}\n  (not to be confused with $w = 1\/(2n-1)$ characterizing the $\\phi$\n  field behavior); as before, the relation $n(2-b) =1$ holds. The model thus obtained\n  is quite different from the one with dust introduced in k-essence theory.\n\n\\subsection{Dust and Rastall-R2 models}\n\n  Let us again assume \\eqn{dust-k} but now consider version R2 of Rastall's theory,\n  so that now $\\rho \\propto a^{-3}$ and \n\\begin{equation}                    \\label{nb-ex}\n\t\\epsilon \\dot\\psi^2\\Big[b - \\frac{2n - 1}{n}\\Big] = (b - 1)\\rho \\propto a^{-3}.\n\\end{equation}\n  Then for $b \\neq 1$ we find according to \\rf{psif-ii}\n\\begin{eqnarray}\n\t\\dot\\psi &\\propto& a^{- 3\/2},\n\\nonumber\\\\ {}\n           \\dot\\phi^n &\\propto& a^{-3\/2} \\ \\ \\ \\Rightarrow\\  \\ \\ \\dot\\phi \\propto a^{-3\/(2n)}.\n\\end{eqnarray}\n  Combining this with the relation $\\epsilon \\dot\\psi{}^2 = nF_0 \\dot\\phi{}^{2n}$ and the \n  field equation \\rf{phi-k} with $V_\\phi=0$, we find that this situation corresponds to\n  the limit $n \\to \\infty$, which is, however, well defined. In this way we obtain \n  $a \\propto t^{2\/3}$ as in the pure dust model of GR. \n  For the scalar fields it follows in this limit\n\\begin{eqnarray} \\lal \n        \\dot\\phi ={\\rm const} \\ \\ \\Rightarrow\\  \\ \\phi \\propto t,\n\\nonumber\\\\ \\lal \n        \\dot\\psi \\propto a^{-3\/2} \\propto 1\/t.\n\\end{eqnarray}\n  The condition for $b$ is obtained from \\rf{nb-ex}: writing $\\dot\\psi = \\psi_0\/t $ and\n  $\\rho = \\rho_0\/t^2$, we arrive at\n\\begin{equation}\n\t\\epsilon\\psi_0^2 (b - 2) = (b - 1)\\rho_0.\n\\end{equation}\n  Thus the value of $b$ is determined by the relative contributions of matter and the scalar field. \n  Moreover, the speed of sound of the scalar field now does not follow the adiabatic relation \n  verified in the R1 case.\n\n  A cosmological constant can be easily introduced in the form of $V = (3-2b) W = {\\rm const}$. \n  The scalar field again follows the law \\rf{nb-ex}, and the whole configuration reduces to the\n  $\\Lambda$CDM model where $\\Lambda$ is given by the constant potential and the matter \n  component consists of the scalar field and ordinary matter. All background relations of the \n  $\\Lambda$CDM model are preserved in this case, but the degeneracy between the scalar \n  field and usual matter is broken at the perturbative level. Due to the fact that the \n  $\\Lambda$CDM model is subject to problem at the perturbative level in the non-linear regime \n  (see, e.g., \\cite{cusp,satt}), such a more complex configuration in k-essence and Rastall \n  models may lead to interesting results, to be studied in the future.\n\n\\section{Conclusion}\n\n  We have studied the conditions of equivalence between the k-essence and Rastall theories of \n  gravity in the presence a scalar field (k-R duality). These two theories have actually emerged in \n  very different contexts, the k-essence theory being based on a generalization of the kinetic term \n  of a scalar field, suggested by some fundamental theories, while Rastall's theory is a \n  non-conservative theory of gravity which can be seen as a possible phenomenological \n  implementation of quantum effects in gravitational theories. Such equivalence has been \n  revealed in the case of static spherically symmetric models \\cite{denis, we-16}, and it \n  has been more explicitly stated here for all cases where the metric and scalar fields essentially\n  depend on a single coordinate, and the k-essence theory is specified by a power-law function \n  of the usual kinetic term, to which a potential term can be added. This generalization covers \n  diverse static and cosmological models, including all homogeneous cosmologies.\n\n  We have discussed cosmological configurations with scalar fields and matter in the form of \n  a perfect fluid whose evolution in Rastall's theory can follow one of two possible laws: one (R1) \n  assumes no mixing between matter and the scalar field, each of them separately obeying \n  the non-conservation law \\rf{cons-R}, and the other (R2) ascribes the whole non-conservation \n  to the scalar field while matter is conservative ($\\nabla_\\nu T\\mN[m] =0$). Let us summarize\n  the main results obtained in this context:\n\\begin{enumerate}\n\\item \n\tk-R duality has been established for version R1 of Rastall's theory with an arbitrary \n \tEoS of matter. It has been found that the EoS of matter is different in the mutually dual\n  \tk-essence and Rastall models; however, it is argued that the respective speeds of \n\tsound are the same. Since the speeds of sound characterizing the scalar fields ($\\phi$ in \n\tk-essence theory and $\\psi$ in Rastall's) also coincide, we conclude that k-R duality\n         is maintained not only for the cosmological backgrounds but also for adiabatic \n         perturbations.\n\\item  \n\tFor version R2 of Rastall's theory, it has been found that k-R duality exists only with \n\tfluids having the EoS $p = w\\rho,\\ w={\\rm const}$, which is the same for k-essence and \n\tRastall models. Moreover, in the k-essence model the scalar field obeys the same \n\teffective EoS. However, on the perturbative level the mutually dual models \n\tbehave, in general, differently. \n\\item \n\tSome special cases have been discussed, showing how there emerge some restrictions \n\ton the free parameters of each theory. \n\\item \n\tAn example has been considered in which a cosmological model completely equivalent \n         to the $\\Lambda$CDM model of GR is obtained at the background level, but different \n\tfeatures must appear at the perturbative level. \n\\end{enumerate}\n\n  The equivalence between the two theories discussed here is somewhat surprising because of\n  their basically different origin. A curious aspect is that the k-essence theory has a Lagrangian\n  formulation unlike Rastall's theory. It is possible that the equivalence studied here may lead to\n  a restricted Lagrangian formulation of Rastall's theory in the minisuperspace in terms of \n  metric functions depending on a single variable. If this is true, it might suggest how to recover \n  a complete Lagrangian formulation for Rastall's theory in a more general framework.\n\n\\subsection*{Acknowledgments} \n\n  We thank CNPq (Brazil) and FAPES (Brazil) for partial financial support.\n  KB thanks his colleagues from UFES for kind hospitality and collaboration.\n  The work of KB was performed within the framework of the Center \n  FRPP supported by MEPhI Academic Excellence Project \n  (contract No. 02.a03. 21.0005, 27.08.2013),\n  within the RUDN University program 5-100, and RFBR project No. 16-02-00602. \n\n\\small\n\n\\subsection*{Acknowledgment} #1}\n\n\t\n\n\\def.95{.95}\n\\def.95{.95}\n\\def.95{.95}\n\\def.05{.05}\n\\def.95{.95}\n\\def.95{.95}\n\n\\newcommand{\\EFigure}[2]{\\begin{figure} \\centering  \n        \\framebox[85mm]{\\epsfxsize=80mm\\epsfbox{#1}}\n        \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure}}\n\\newcommand{\\REFigure}[2]{\\begin{figure} \\centering\n        \\framebox[85mm]{\\epsfysize=80mm\\rotate[r]{\\epsfbox{#1}}}\n        \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure}}\n\\newcommand{\\WEFigure}[2]{\\begin{figure*} \\centering\n        \\framebox[178mm]{\\epsfxsize=170mm\\epsfbox{#1}}\n        \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure*}}\n\n\t\n\n\\def\\Jl#1#2{#1 {\\bf #2},\\ }\n\n\\def\\ApJ#1 {\\Jl{Astroph. J.}{#1}}\n\\def\\CQG#1 {\\Jl{Class. Quantum Grav.}{#1}}\n\\def\\DAN#1 {\\Jl{Dokl. AN SSSR}{#1}}\n\\def\\GC#1 {\\Jl{Grav. Cosmol.}{#1}}\n\\def\\GRG#1 {\\Jl{Gen. Rel. Grav.}{#1}}\n\\def\\JETF#1 {\\Jl{Zh. Eksp. Teor. Fiz.}{#1}}\n\\def\\JETP#1 {\\Jl{Sov. Phys. JETP}{#1}}\n\\def\\JHEP#1 {\\Jl{JHEP}{#1}}\n\\def\\JMP#1 {\\Jl{J. Math. Phys.}{#1}}\n\\def\\NPB#1 {\\Jl{Nucl. Phys. B}{#1}}\n\\def\\NP#1 {\\Jl{Nucl. Phys.}{#1}}\n\\def\\PLA#1 {\\Jl{Phys. Lett. A}{#1}}\n\\def\\PLB#1 {\\Jl{Phys. Lett. B}{#1}}\n\\def\\PRD#1 {\\Jl{Phys. Rev. D}{#1}}\n\\def\\PRL#1 {\\Jl{Phys. Rev. Lett.}{#1}}\n\n\t\n\n\\def&\\nhq{&\\hspace*{-0.5em}}\n\\def&&\\nqq {}{&&\\hspace*{-2em} {}}\n\\defEq.\\,{Eq.\\,}\n\\defEqs.\\,{Eqs.\\,}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\begin{eqnarray} \\lal{\\begin{eqnarray} &&\\nqq {}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\nonumber \\end{eqnarray}{\\nonumber \\end{eqnarray}}\n\\def\\nonumber\\\\ {}{\\nonumber\\\\ {}}\n\\def\\nonumber\\\\[5pt] {}{\\nonumber\\\\[5pt] {}}\n\\def\\nonumber\\\\ \\lal {\\nonumber\\\\ &&\\nqq {} }\n\\def\\nonumber\\\\[5pt] \\lal {\\nonumber\\\\[5pt] &&\\nqq {} }\n\\def\\\\[5pt] {}{\\\\[5pt] {}}\n\\def\\\\[5pt] \\lal {\\\\[5pt] &&\\nqq {} }\n\\def\\al =\\al{&\\nhq =&\\nhq}\n\\def\\al \\equiv \\al{&\\nhq \\equiv &\\nhq}\n\\def\\sequ#1{\\setcounter{equation}{#1}}\n\n\n\\def\\displaystyle{\\displaystyle}\n\\def\\textstyle{\\textstyle}\n\\def\\fracd#1#2{{\\displaystyle\\frac{#1}{#2}}}\n\\def\\fract#1#2{{\\textstyle\\frac{#1}{#2}}}\n\\def{\\fracd{1}{2}}{{\\fracd{1}{2}}}\n\\def{\\fract{1}{2}}{{\\fract{1}{2}}}\n\n\n\\def{\\,\\rm e}{{\\,\\rm e}}\n\\def\\partial{\\partial}\n\\def\\mathop{\\rm Re}\\nolimits{\\mathop{\\rm Re}\\nolimits}\n\\def\\mathop{\\rm Im}\\nolimits{\\mathop{\\rm Im}\\nolimits}\n\\def\\mathop{\\rm arg}\\nolimits{\\mathop{\\rm arg}\\nolimits}\n\\def\\mathop{\\rm tr}\\nolimits{\\mathop{\\rm tr}\\nolimits}\n\\def\\mathop{\\rm sign}\\nolimits{\\mathop{\\rm sign}\\nolimits}\n\\def\\mathop{\\rm diag}\\nolimits{\\mathop{\\rm diag}\\nolimits}\n\\def\\mathop{\\rm dim}\\nolimits{\\mathop{\\rm dim}\\nolimits}\n\\def{\\rm const}{{\\rm const}}\n\\def\\varepsilon{\\varepsilon}\n\\def\\epsilon{\\epsilon}\n\n\\def\\ \\Rightarrow\\ {\\ \\Rightarrow\\ }\n\\newcommand{\\mathop {\\ \\longrightarrow\\ }\\limits }{\\mathop {\\ \\longrightarrow\\ }\\limits }\n\\newcommand{\\aver}[1]{\\langle \\, #1 \\, \\rangle \\mathstrut}\n\\newcommand{\\vars}[1]{\\left\\{\\begin{array}{ll}#1\\end{array}\\right.}\n\\def\\sum\\limits{\\sum\\limits}\n\\def\\int\\limits{\\int\\limits}\n\n \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGraphs are ubiquitous in the real world, which can easily express various and complex relationships between objectives. In recent years, extensive studies have been conducted on deep learning methods for graph-structured data. There are several approaches on analyzing the graph, including network embedding \\cite{perozzi2014deepwalk, tang2015line, grover2016node2vec}, which only uses the graph structure, and graph neural networks (GNNs), which consider graph structure and node features simultaneously. GNNs have shown powerful ability on modeling the graph-structured data in a variety of graph learning tasks such as node classification \\cite{2018Large, hamilton2017inductive, yang2016revisiting, kipf2016semi, velivckovic2017graph, wu2019simplifying}, link prediction \\cite{zhang2017weisfeiler, zhang2018link, cai2020multi} and graph classification \\cite{gilmer2017neural, lee2019self, ma2019graph, xu2018powerful, ying2018hierarchical, zhang2018end}. GNNs have also been applied to a range of applications, including social analysis \\cite{qiu2018deepinf, li2019encoding}, recommender systems \\cite{ying2018graph, monti2017geometric}, traffic prediction \\cite{guo2019attention, li2019predicting}, drug discovery \\cite{zitnik2017predicting} and fraud detection \\cite{liu2019geniepath}. \n\nGNNs usually have different design paradigms, which include the spectral graph convolutional networks \\cite{bruna2013spectral, defferrard2016convolutional, kipf2016semi},  message passing framework \\cite{gilmer2017neural, hamilton2017inductive}, and neighbor aggregation via recurrent neural networks \\cite{li2015gated, dai2018learning}. By using the idea of message passing framework, GNNs are to design various graph convolutional layers to update each node representation by aggregating the node representations from its neighbors. \n\nHowever, most GNNs only consider the immediate neighbors for each node, which impedes their ability to extract the information of high-order neighbors. More layers usually lead to the performance degradation, which is caused by over-fitting and over-smoothing, of which the former is due to the increasing number of parameters when fitting a limited dataset whereas the latter is the inherent issue of the graph learning. How to make use of the high-order information of neighbors as well as achieving better performance remains a challenge. We need more insights to understand what GCN does and why over-smoothing occurs.\n\\begin{figure*}[t]\n\t\\centering  \n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{original.pdf}}\\hspace{-4mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{mlp.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-2.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-6.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-8.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+8.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+16.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+32.pdf}}\n\t\\caption{t-SNE Visualization of learned node representations, which include original features, MLP, different layers of GCN and different hops of GCN+ on \\textit{Cora}. Colors represent node classes.}\n\t\\label{model_vs}\n\\end{figure*}\n\nSeveral studies \\cite{li2018deeper, xu2018representation, klicpera2018predict, chen2020simple, liu2020towards} have noticed over-smoothing, that is after multiple propagations, the final output of vanilla multi-layer GCN converges to a vector which only carries the information of the degree of graph and the node features are indistinguishable. Fig. \\ref{model_vs} shows the node representations of vanilla multi-layer GCN on a small citation network \\textit{Cora}. We can observe that 2-layer GCN learns a meaningful embeddings which distinguish the different classes whereas more layers degrade the performance and lead to indistinguishable features.\n\nDifferent from previous studies, we interpret the current graph convolutional operations from an optimization perspective, and argue that over-smoothing is mainly caused by the naive first-order approximation of the solution to the optimization problem. By solving it and applying the first-order approximation, we get the standard GCN kernel. This suggests that the original GCN kernel can be viewed as a simplified version of the solution. We argue that this simplification loses necessary information which is crucial to tackle the over-smoothing to some extent. Based on this observation, two metrics are proposed to measure the smoothness of connected and disconnected pairwise node features respectively. Furthermore, we set three constraints: (a) the embedding learned by GCNs should not be too far off of the original features; (b) the connected nodes should have similar embeddings; (c) the disconnected nodes are assumed to have different embeddings. \n\nAs a result, we build a universal theoretical framework of GCN from an optimization perspective which smooths the node features and regularizes the (disconnected) node feature simultaneously. We consider two different cases of our framework, where the first case contains the current popular GCN \\cite{kipf2016semi}, SGC \\cite{wu2019simplifying} and PPNP \\cite{klicpera2018predict}, and the second case regularizes the pairwise distance of disconnected nodes.\n\nThe contributions of this work are summarized as follow:\n\\begin{itemize}\n\t\\item We provide a universal theoretical framework of GCN from an optimization perspective where the popular GCNs can be viewed as a special case of it. Furthermore, we derive a novel convolutional kernel named GCN+, which relieves the over-smoothing inherently and has lower parameter amount.\n\t\\item We propose two quantitative metric to measure the smoothness and over-smoothness of the final nodes representations, which provides new insight to analyze the over-smoothing.\n\t\\item We conduct extensive experiments on several public real-world datasets. Our results demonstrate the superior performance of GCN+ over state-of-the-art baseline methods.\n\\end{itemize} \n\n\\section{Notations}\nGiven an undirected graph $G=(V,E,X)$, $V$ is node set with $|V|=n$, $E$ is edge set. Let $A\\in \\mathbb{R}^{n \\times n}$ denote the adjacency matrix, where $A_{ij}=1$ if there is an edge between node $i$ and node $j$ otherwise 0. Let $D\\in \\mathbb{R}^{n \\times n}$ denote the diagonal degree matrix where $D_{ii}=\\sum_{j}A_{ij}$. Each node is associated with $d$ features, and $X \\in \\mathbb{R}^{n\\times d}$ is the feature matrix of nodes. each row of $X$ is a signal defined over nodes. The graph Laplacian matrix is defined as $L=D-A$. Let $\\tilde{A}=A+I$ and $\\tilde{D}=D+I$ denote the adjacency and degree matrices of the self-loop graph respectively. We denote $\\tilde{A}_{\\textit{sym}}=\\tilde{D}^{-1\/2} \\tilde{A} \\tilde{D}^{-1\/2}$ and $\\tilde{A}_{\\textit{rw}}=\\tilde{D}^{-1} \\tilde{A}$. Assume that each node $v_i$ is associated with a class label $y_i \\in Y$ where $Y$ is a set of $c$ classes. Let $N(v)$ denote the neighbors of $v$ in graph, that is $N(v)=\\{u\\in V|\\{u,v\\}\\in E\\}$ and $\\tilde{N}(v)=N(v) \\cup \\{v\\}$.\n$L'$ is the Laplacian matrix of the graph $G'(V',E',X)$, which is the complement of $G$, that means $G'$ has the same nodes as $G$ whereas if $\\{u,v\\}\\in E$, then $\\{u,v\\}\\notin E'$. Let $A'$ and $D'$ denote the corresponding adjacency and degree matrix respectively. We have $A'+A=J_n-I_n$ and $D'+D=(n-1)I$ where $J_n$ is a matrix whose element are all 1. Let $\\text{num}(E)$ and $\\text{num}(E')$ denote the numbers of edges in $G$ and $G'$ respectively, we have $\\text{num}(E)+\\text{num}(E')=\\frac{n(n-1)}{2}$.\n\\section{Perspectives of GCN}\nHere we provide three views to derive or understand the vanilla GCNs.\n\\subsection{Spectral Graph  Convolution}\n\\citet{bruna2013spectral} define the spectral convolutions on graph by applying a filter $g_\\theta$ in the Fourier domain to a graph signal. ChebNet \\cite{defferrard2016convolutional} suggests that the graph convolutional operation can be further approximated by the $k$-th order Chebyshev polynomial of Laplacian. \\citet{kipf2016semi} simplify the ChebNet and obtains a reduced version of ChebNet by the renormalization trick:\n\\begin{equation} \\label{gcn}\nH^{(l+1)}\\!\\!=\\!\\!\\sigma(\\!\\tilde{A}_{\\textit{sym}}H^{(l)}W^{(l)})\\!\\!=\\!\\!\\sigma(\\tilde{D}^{-\\frac{1}{2}}\\tilde{A}\\tilde{D}^{-\\frac{1}{2}}H^{(l)}W^{(l)}),\n\\end{equation}\nwhere $\\sigma$ denote the activation function such as ReLU. $W^{(l)}$ is a layer-specific trainable weight matrix. $H^{(l)}$ is the feature matrix of $l$-th layer and $H^{(0)}=X$.\n\n\\subsection{Message Passing}\nMessage passing \\cite{gilmer2017neural} means that a node on the graph aggregates the message from neighbors and update its embedding:\n\\begin{equation} \\label{mpnn}\nh_v^{(l)}=U_l\\bigg(h_v^{(l-1)}, \\sum_{u \\in N(v)}M_l\\big(h_u^{(l-1)}, h_v^{(l-1)}, e_{uv}\\big)\\bigg),\n\\end{equation}\nwhere $M_l(\\cdot)$ and $U_l(\\cdot)$ are message aggregation function and vertex update function, respectively. $h_v^{(l)}$ denotes the hidden state of node $v$ at $l$-th layer, and $e_{uv}$ is the edge features.\n\nIn this way, GCN layer can be decomposed into two steps, including the neighbors' message aggregation and update:\n\\begin{equation} \\label{gcn-mpnn}\nh_v^{(l)}=\\sigma\\bigg(W^{(l)}\\sum_{u\\in \\tilde N (v)} \\frac{h^{(l-1)}_u}{\\sqrt{|N(v)||N(u)|}}\\bigg).\n\\end{equation}\n\nHere a GCN layer can be viewed as a weighted average of all neighbors' message where the weighting is proportional to the inverse of the number of neighbors.\n\n\\subsection{Graph Regularized  Optimization} \\label{GRO}\nLet $\\bar{X} \\in \\mathbb{R}^{n\\times d}$ denote the final node embeddings matrix, and $\\bar{x}_i$ is the $i$-th row of $\\bar{X}$. We consider the following optimization problem:\n\\begin{equation} \\label{gcn-opt}\nf = \\min_{\\bar X} \\bigg(\\sum_{i \\in V}\\|\\bar{x}_i -x_i\\|_{\\tilde{D}}^2 + \\alpha \\sum_{\\{i,j\\} \\in E}\\|\\bar{x}_i -\\bar{x}_j\\|_2^2\\bigg),\n\\end{equation}\nwhere $(x,y)_{\\tilde D}=\\sum_{i\\in V}d(i)x(i)y(i)$, if $x=y$, we have$\\quad \\|x\\|_{\\tilde D}=\\sqrt{(x,x)_{\\tilde D}}$.\n\nThe first term in the above optimization problem is the fitting constraint, which means the output features (also called embeddings) should not be too far off of the input features, while the second term is the smoothness constraint, which means the connected nodes should have similar embeddings. $\\alpha > 0$ is a hyperparameter to balance the importance of two objections. It is worth noting that there is no limit to the specific transformation from $X$ to $\\bar{X}$.\n\nBefore solving the optimization problem, we have the following lemma.\n\\begin{lemma}\n\t$\\tilde{A}_{\\textit{rw}}$ and $\\tilde{A}_{\\textit{sym}}$ always have the same eigenvalues $|\\lambda|\\leq1$.\n\\end{lemma}\n\n\\begin{corollary} \\label{inverse}\n\t$(I_n-\\alpha\\tilde{A}_{\\textit{rw}})$ and $(I_n-\\alpha\\tilde{A}_{\\textit{sym}})$ are invertible if $\\alpha \\in[0,1)$. \n\\end{corollary}\n\n\\begin{lemma}\n\tGiven a graph with adjacency matrix $A$, the powers of $A$ give the number of walks between any two vertices.\n\\end{lemma}\n\n\\begin{corollary}  \\label{high-order}\n\t$A^k$ includes the information of high-order neighbors.\n\\end{corollary}\nNext, we derive the closed-form solution of Eq. \\ref{gcn-opt}. Specifically, we rewrite  Eq. \\ref{gcn-opt} as \n$$\nf=\\min_{\\bar X} \\bigg(\\text{Tr}\\big((\\bar X-X)(\\bar X-X)^T\\tilde{D}\\big) + \\alpha\\text{Tr}({\\bar X}^TL \\bar X)\\bigg).\n$$\nDifferentiating $f$ with respect to $\\bar{X}$, we have \n$$\n\\frac{df}{d\\bar{X}}=\\tilde{D}(\\bar{X}-X)+\\alpha L\\bar{X}=0.\n$$\nNotice Corollary \\ref{inverse}, we have\n$$\n\\bar{X}=(1-\\mu) (I-\\mu\\tilde {A}_{\\textit{rw}})^{-1}X,\n$$\nwhere $\\mu=\\frac{\\alpha}{1+\\alpha}$.\n\nActually, the solution is also the personalized PageRank \\cite{page1999pagerank}' s limiting distribution. If we set $\\mu=0.5$, we get $\\bar X=(2I-\\tilde {A}_{\\textit{rw}})^{-1}X$, \nand $\\tilde {A}_{rw}X$ is the first-order Taylor approximation. \nBy replacing $\\tilde {A}_{rw}$ with $\\tilde {A}_{\\textit{sym}}$, we get standard graph convolution kernel. In other words, we lose the information from high-order neighbors, which is contained in the error series of the Taylor expansion. (See Corollary \\ref{high-order}).\n\nIn a nutshell, we obtains the well-known kernel or resemble form of the graph convolution from different ways. \n\\section{Over-smoothing in Vanilla Deep GCN}\n\nNeural network usually performs better when stack more layers while graph neural network does not benefit from the depth. On the contrary, more layers often result in significant degradation in performance. \n\nPrevious work illustrates the over-smoothing by computing the limiting distribution of $A_{k}$ when $k \\rightarrow \\infty$, Actually, this is not identical with vanilla deep GCN, which contains non-linear transformation among different layers. Litter work considers the non-linearity in multi-layer GCN. \\citet{oono2019graph} extend the linear analysis to the non-linearity firstly, which considers the ReLU activation function. They suggest that the node features of a $k$-layer GCNs will converge to a subspace and incur information loss, which makes the node feature indistinguishable.\n\nAt first, one main reason we introduce the deep architecture in GCN is that we want to use the long-range neighbor's information. We argue that vanilla deep GCN is not the correct way to capture this information. However, It does not mean that deep architecture is useless. \\citet{chen2020simple} and \\citet{liu2020towards} have shown that more layers can boost the performance of GCN on several datasets and tasks.\n\nTo quantify the over-smoothing in vanilla deep GCN, we compute the overall pairwise distance of node embeddings as follows:\n\\begin{equation} \n\\nonumber\n\\begin{aligned}\n\\!M_{\\textit{\\!overall}}&\\!=\\!\\!\\!\\sum_{i,j\\in\\!V}\\!\\!\\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2\\!=\\!\\!\\!\\!\\sum_{\\{i,j\\}\\in\\!E}\\!\\!\\! \\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2\\!\\!+\\!\\!\\!\\!\\sum_{\\{i,j\\}\\notin\\!E}\\!\\!\\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2 \\\\\n&= \\text{Tr}({\\bar X}^TL \\bar X)+ \\text{Tr}({\\bar X}^TL' \\bar X).\\\\\n\\end{aligned}\n\\end{equation}\n\\begin{figure}[t]\n\t\\subfigure[Overall distance]{\n\t\t\\label{overall}\n\t\t\\includegraphics[width=0.22\\textwidth]{m_overall.pdf}}\\hspace{-1mm}\n\t\\subfigure[Fraction of diatsnce]{\n\t\t\\label{fraction}\n\t\t\\includegraphics[width=0.24\\textwidth]{smooth.pdf}}\n\t\\caption{ $M_{\\textit{\\!overall}}, M_{\\textit{\\!smooth}}$ and $M_{\\textit{\\!non-smooth}}$ of the output node embeddings of Vanilla GCN with increasing layers on \\textit{Cora}.}\n\t\\label{boxplot}\n\\end{figure}\n\nFig. \\ref{overall} depicts the pairwise distance distribution of vanilla GCN with increasing layers on \\textit{Cora}. We can see that $M_{overall}$ decreases as the model goes deeper. Revisit the two parts of $M_{overall}$, we propose two fine quantitative metrics to measure the over-smoothing of graph representation.\n\\begin{equation} \n\\begin{aligned}\nM_{\\textit{smooth}} &= D_{\\textit{smooth}}\/D_{\\textit{overall}},\\\\\nM_{\\textit{non-smooth}} &= D_{\\textit{non-smooth}} \/D_{\\textit{overall}},\\\\\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation} \n\\begin{aligned}\nD_{\\textit{smooth}} &= \\text{Tr}({\\bar X}^TL \\bar X)\/\\text{num}(E) ,\\\\\nD_{\\textit{non-smooth}} &= \\text{Tr}({\\bar X}^TL' \\bar X)\/\\text{num}(E'),\\\\\nD_{\\textit{overall}}&=D_{\\textit{smooth}}+D_{\\textit{non-smooth}}.\n\\end{aligned}\n\\end{equation}\n\nHere, $\\text{num}(E)$ and $\\text{num}(E')$ in the denominator are used to eliminate the impact of unbalanced edge numbers in $G$ and $G'$. $M_{\\textit{smooth}}$ measures the smoothness of the graph representation of connected pair nodes while $M_{\\textit{non-smooth}}$ measures the smoothness of the graph representation of disconnected pair nodes. \n\nFig. \\ref{fraction} compares $M_{\\textit{smooth}}$ and $M_{\\textit{non-smooth}}$. We see that $M_{\\textit{smooth}}$ contributes to quite a few parts of the overall distance, which seems counter-intuitive. We will discuss this two metrics of GCN+ in Section \\ref{oversmooth-of-gcn+} again.\n\n\\section{A General Framework of GCN}\nRecall the graph regularized optimization problem, we add a negative term to constrain the sum of distances between disconnected pairs as follow:\n\\begin{equation} \\label{full-opt}\n\\nonumber\n\tf \\!\\!=\\!\\min_{\\bar X}\\!\\!\\bigg(\\!\\sum_{i \\in V}\\|\\bar{x}_i \\!-\\!x_i\\|_{\\tilde{D}}^2 \\!+\\! \\alpha\\!\\!\\!\\!\\! \\sum_{\\{i,j\\} \\in E} \\! \\|\\bar{x}_i \\!-\\!\\bar{x}_j\\|_2^2 \\!-\\!\\beta\\!\\!\\!\\!\\! \\sum_{\\{i,j\\} \\notin E} \\!\\|\\bar{x}_i \\!-\\!\\bar{x}_j\\|_2^2\\!\\bigg)\\!,\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are hyperparameters to balance the importance of the corresponding terms.\n\n\nWe consider two cases: $\\beta=0$ and $\\beta \\neq0$.\n\\subsection{Case 1:$\\beta=0$}\nIn this situation, $\\bar{X}=(1-\\mu)(I_n-\\mu\\tilde{A}_{\\textit{rw}})^{-1}X$ where $\\mu=\\frac{\\alpha}{1+\\alpha}\\in(0,1)$. Directly calculating such an intractable expression is not only computationally inefficient but also results in a dense $\\mathbb{R}^{n \\times n}$matrix. It would lead to a high computational complexity and memory requirement when we apply such operator on large graphs. We can achieve linear computational complexity via power iteration.\n\nWe use $\\tilde{A}$ to denote $\\tilde{A}_{\\textit{sym}}$ and $\\tilde{A}_{\\textit{rw}}$. Here we consider a more general expression $(1-\\mu)(I_n-\\mu \\tilde{A})^{-1}H$ where $H=H=f_{\\theta}(X)$.\n\\begin{theorem} \\label{theorem1}\n\t$(I_n-\\mu\\tilde{A})$ is invertible. Consider the following iterative scheme\n\t\\begin{equation} \\label{iterative1}\n\t\\begin{aligned}\n\tZ^{(0)}&=H, \\\\\n\tZ^{(k)}&=\\mu\\tilde{A}Z^{(k-1)}+(1-\\mu) H,\n\t\\end{aligned}\n\t\\end{equation} where $\\mu \\in (0,1)$.\n\tWhen $k \\rightarrow \\infty$, \n\t\\begin{equation}\nZ^{(\\infty)}=(1-\\mu)(I_n-\\mu\\tilde{A})^{-1}H.\n\t\\end{equation} \n\\end{theorem}\n\n\\begin{proof}\n\tUsing corollary \\ref{inverse}, we can see that $(I_n-\\mu\\tilde{A})$ is invertible. Combining the two equation of \\ref{iterative1}, we have \n\t$$\n\tZ^{(k)}=\\bigg(\\mu^k \\tilde{A}^k+(1-\\mu) \\sum_{i=0}^{k-1} \\mu^i \\tilde{A}^i\\bigg)H.\n\t$$\n\tNotice that\n\t\\begin{equation} \n\t\\begin{aligned}\n\t\\lim_{k \\rightarrow \\infty}\\mu^k \\tilde{A}^k&=0, \\\\\n\t\\lim_{k \\rightarrow \\infty}\\sum_{i=0}^{k-1} \\mu^i \\tilde{A}^i&=\\big(I-\\mu\\tilde{A}\\big)^{-1}.\n\t\\end{aligned}\n\t\\end{equation}\n\tHence, the proof is finished. \n\n\\end{proof}\n\nActually, the prevalent GCN, SGC and APPNP can be viewed as the special variant of Case 1.\n\\subsection{Case 2:$\\beta \\neq 0$} \n\nIn this situation, $\\bar{X}=Q^{-1}$ if $Q=\\big((1+\\alpha+\\beta)I-(\\alpha+\\beta)\\tilde D^{-1}\\tilde A-\\beta n\\tilde D^{-1}+\\beta \\tilde D^{-1} \\mathbf J_n\\big)$ is invertible when  we choose a suitable $\\beta$. We will introduce the conditions later.\n\n\nFirst we use the  first-order Taylor approximation of above convolutional kernel ($\\text{GCN}^*$) directly without any tricks such as Batch Normalization \\cite{ioffe2015batch} or residual connection \\cite{he2016deep} on two small citation datasets \\textit{Cora} and \\textit{Citeseer}. We compare the performance of the vanilla deep GCN and $\\text{GCN}^*$ as the model layer increases. Fig. \\ref{figure1} shows the result of GCN and $\\text{GCN}^*$.  Dashed lines illustrate the performance of GCN, which shows that deep GCN suffers from performance drop. We can see that the performance decay with $\\text{GCN}^*$ kernel is much slower.\n\n\\citet{oono2019graph} have proved that the node feature of vanilla $k$-layer GCN will converges to an invariant subspace which only carry the information of the connected component and node degree. The convergence speed is proportional to the $\\lambda^k$, where $\\lambda$ is the supremum of eigenvalue of $\\tilde{A}$. In GCN*, $\\lambda>1$(see the proof of Theorem \\ref{theorem2}), which implies that $\\lambda^k$ is large, thus the information loss and over-smoothing are relieved.\n\nAlthough the modified graph kernel relieves over-smoothing to some extent, more layers do not boost the performance, which is not our focus. However the above result demonstrates that it is an efficient way to tackle the over-smoothing issue. We can achieves linear computational complexity via power iteration similar to Case 1.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{cora}\n\t\\caption{Performance comparison of vanilla deep GCN vs. $\\text{GCN}^*$ with increasing layers on two small datasets.}\n\t\\label{figure1}\n\\end{figure}\n\n\\begin{theorem} \\label{theorem2}\n\t$(I_n-\\mu \\hat{A})$ is invertible when $\\beta<\\frac{1}{n}$ where $\\mu=\\frac{\\alpha+\\beta}{1+\\alpha+\\beta}$ and $\\hat{A}=\\tilde{A}+\\frac{\\beta n\\tilde D^{-1}-\\beta \\tilde D^{-1} \\mathbf J_n}{\\alpha+\\beta}$.  Consider the following iterative scheme\n\t\\begin{equation} \\label{iterative2}\n\t\\begin{aligned}\n\tZ^{(0)}&=H, \\\\\n\tZ^{(k)}&=\\mu\\hat{A}Z^{(k-1)}+ (1-\\mu)H,\n\t\\end{aligned}\n\t\\end{equation} where $\\mu \\in (0,1)$.\n\tWhen $k \\rightarrow \\infty$, \n\t\\begin{equation}\n\tZ^{(\\infty)}=(1-\\mu)(I_n-\\mu\\hat{A})^{-1}H.\n\t\\end{equation} \n\\end{theorem}\n\\begin{proof}\n\tLet $M=\\frac{\\beta n\\tilde D^{-1}-\\beta \\tilde D^{-1} \\mathbf J_n}{\\alpha+\\beta}=\\frac{\\beta \\tilde D^{-1} }{\\alpha+\\beta}(nI-J_n)$. Note that $(nI-J_n)$ has the largest eigenvalue $n$. Suppose that $\\lambda$ is the eigenvalue of $M$, we have $\\lambda \\leq \\frac{\\beta n}{\\alpha+\\beta} $. Then eigenvalue of $ \\hat{A}$ is less than $1+\\frac{\\beta n}{\\alpha+\\beta}$. \n\t $(I_n-\\mu \\hat{A})$ is invertible iff $\\frac{1}{\\mu}$ is not an eigenvalue of $ \\hat{A}$. Note that $\\frac{1}{\\mu}=\\frac{1+\\alpha+\\beta}{\\alpha+\\beta}=1+\\frac{1}{\\alpha+\\beta}$, when $\\beta<\\frac{1}{n}$ we have $\\frac{1}{\\mu}>1+\\frac{\\beta n}{\\alpha+\\beta}$, hence $\\frac{1}{\\mu}$ cannot be an eigenvalue of $\\hat{A}$ and $(I_n-\\mu \\hat{A})$ is invertible. The proof of the iterative scheme follows the similar procedure of case 1 with a slight difference, as it is trivial, we omit the proof.\n\\end{proof}\n\n\\subsection{Why GCN+ relieve the over-smoothing?}\nWe have no assumptions on the specific transformation from $X$ to $\\bar{X}$. In our implementation, the mathematical expression of GCN+ is defined as \n\\begin{equation} \\label{gcn+}\n\\begin{aligned}\nZ^{(0)}&=H=\\sigma(XW_1), \\\\\nZ^{(k)}&=\\mu\\hat{A}Z^{(k-1)}+ (1-\\mu)H,\\\\\nX_{out}&=\\text{softmax}(Z^{(k)}W_2),\\\\\n\\end{aligned}\n\\end{equation} \nwhere $W_1\\in \\mathbb{R}^{d \\times m}$ and $W_2\\in \\mathbb{R}^{m \\times c}$ are learnable weight matrices, $k$ is the dimension of the hidden layers. \n\n\nWe interpret the anti-oversmoothing of GCN+ from two ways. First, note that in the power iterative scheme, a fraction of initial node features $H$ is always preserved in each iteration, which can be viewed as a flexible version of residual connection. In addition, we can also understand GCN+ from the frequency of graph signal. In Section \\ref{GRO} , we have shown that the original GCN is corresponding to the first-order Taylor approximation of the optimization solution, that means we lost the high frequency part of the signal which contains the high-order information. Actually we omit the error series when we approximate the GCN.\n\n\nRecall the current representative methods: DAGNN and JKNet, which shows promising improvement than the original GCN. The core formulas of them are as follows:\n\\begin{equation} \n\\nonumber\n\\begin{aligned}\n\\text{DAGNN:}&\\\\\n&Z=\\text{stack}(H, Z^{(1)}, ..., Z^{(k)}), \\quad Z^{(k)}=\\hat{A}^{(k)}H,\\\\\n\\text{JKNet:\\quad}&\\\\\n&Z=\\text{Aggr}(Z^{(1)}, ...,Z^{(k)}), \\\\\n\\end{aligned}\n\\end{equation} \nwhere \\text{Aggr} includes \\textit{Concatenation}, \\textit{Max-pooling} and \\textit{LSTM-attention}.\n\nActually, DAGNN and JKNet both make use of the information which from the immediate and high-order neighbors while GCN+ also benefit from this. Moreover, we provide the theoretical and empirical evidence of GCN+.\n\n\\subsection{Parameters Amount}\nIt is worth noting that the power iterative schemes are parameter-free in two versions of GCN+,  which is similar to APPNP \\cite{klicpera2018predict}. In particular,  GCN+ ($\\beta=0$) adopts the same scheme as APPNP, where we re-implement it and achieve more impressive results. \n\\section{Experiments}\nIn this section, we evaluate the performance of GCN+ on several benchmark datasets against various graph neural networks on semi-supersized node classification tasks.\n\t\n\\subsection{Experimental Setup}\n\\subsubsection{Datasets}\nWe conduct extensive experiments on the node-level tasks on two kinds commonly used networks: Planetoid: \\textit{Cora}, \\textit{CiteSeer}, \\textit{Pubmed} \\cite{sen2008collective} and recent \\textit{Open Graph Benchmark} (OGB) \\cite{hu2020open}:\\textit{ogb-arxiv}, \\textit{ogb-proteins}. The statistics of datasets are summarized in Table \\ref{dataset}.  It is worth nothing that OGB includes enormous challenging and large-scale datasets than Planetoid. We refer readers to \\cite{hu2020open} for more details about OGB datasets.\n\\begin{table} \n\t\\small\n\t\\setlength{\\tabcolsep}{1mm}{\n\t\\begin{tabular}{{cccccc}}\n\t\t\\toprule\n\t\tDataset & Nodes  & Edges    & Classes & Features  &Metric\\\\ \n\t\t\\midrule\n\t\tCora  & 2708     & 5429      &  7   &  1433 & Accuracy \\\\ \n\t\tCiteseer  & 3327     & 4732      &  7   &  2703  & Accuracy \\\\ \n\t\tPubmed  & 19717     & 44338      &  3   &  500  & Accuracy \\\\ \n\t\togb-arxiv  & 169343     & 1166243      &  40   &  128  & Accuracy \\\\ \n\t\togb-proteins & 132534     & 39561252      &  112   &  8 & ROC-AUC  \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}}\n\\caption{Dataset statistics.} \n\\label{dataset}\n\\end{table}\n\\subsubsection{Implementations}\nWe choose the optimizer and hyperparameters of GNN models as follows. We use the Adam optimizer \\cite{kingma2014adam} to train all the GNN models with a maximum of 1500 epochs. We set the number of hidden units to 64 on \\textit{Cora}, \\textit{Citeseer} and \\textit{Pubmed} , 256 on \\textit{ogb-arxiv} and \\textit{ogb-proteins}. For SGC, we vary number of layer in \\{1, 2, ..., 10, 15, ..., 60\\} and for GCN and GAT in \\{2, 4, ..., 10, 15, ..., 30\\}. For $\\alpha$ in APPNP, we search it from \\{0.1, 0.2, 0.3, 0.4, 0.5\\}. For DAGNN and JKNet, we search layers from \\{2, 3, ..., 10\\}. For learning rate, we choose from \\{0.001, 0.005, 0.01\\}. For dropout rate, we choose from \\{0.1, 0.2, 0.3, 0.4, 0.5\\}. We perform a grid search to tune the hyperparameters for other models based on the accuracy on the validation set. We run each experiment 10 times and report the average.  \n\nIn practice, we use Pytorch \\cite{paszke2019pytorch} and Pytorch Geometric \\cite{Fey\/Lenssen\/2019} for an efficient GPU-based implementation of GCN+.\nAll experiments in this study are conducted on NVIDIA TITAN RTX 24GB GPU.\n\n\\begin{table*}[t]\n\t\\small\n\t\\centering\n\t\\setlength{\\tabcolsep}{1.5mm}{\n\t\\begin{tabular}{ccccccc}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{model} & \\multicolumn{2}{c}{\\textit{Cora}} & \\multicolumn{2}{c}{\\textit{Citeseer}} & \\multicolumn{2}{c}{\\textit{Pubmed}}                             \\\\\n\t\t& Fixed      & Random      & Fixed        & Random        & Fixed        & Random \\\\\n\t\t\\midrule\n\t\tMLP & $61.6\\pm0.6$   & $59.8\\pm2.4$& $61.0\\pm1.0$  & $58.8\\pm2.2$  & $74.2\\pm0.7$   &  $70.1\\pm2.4$    \\\\\n\t\tGCN\\cite{kipf2016semi} & $81.3\\pm0.8$   & $79.1\\pm1.8$& $71.1\\pm0.7$  & $68.2\\pm1.6$  & $78.8\\pm0.6$   &  $77.1\\pm2.7$    \\\\\n\t\t GAT\\cite{velivckovic2017graph}  &$83.1\\pm0.4$ &$80.8\\pm1.6$ &$70.8\\pm0.5$ &$68.9\\pm1.7$ & $79.1\\pm0.4$  &$77.8\\pm2.1$   \\\\\n\t\t SGC\\cite{wu2019simplifying} & $81.1\\pm0.5$   & $80.4\\pm0.3$& $71.9\\pm0.3$  & $71.8\\pm0.3$  & $78.9\\pm0.0$   &  $77.8\\pm0.6$    \\\\\n\t\t JKNet\\cite{xu2018representation}& $80.7\\pm0.9$ &$79.2\\pm0.9$ &$70.1\\pm0.6$&$68.3\\pm1.8$ & $78.1\\pm0.6$ & $77.9\\pm0.9$    \\\\\n\t\t APPNP\\cite{klicpera2018predict} &$83.3\\pm0.5$ &$81.9\\pm1.4$ &$71.8\\pm0.4$ &$69.8\\pm1.7$ & $80.1\\pm0.2$  &$79.5\\pm2.2$     \\\\\n\t\t DAGNN\\cite{liu2020towards}  &$84.4\\pm0.5$ &$\\textbf{83.7}\\pm\\textbf{1.4}$ &$73.3\\pm0.6$ &$71.2\\pm1.4$ & $\\textbf{80.5}\\pm\\textbf{0.5}$  &$80.1\\pm1.7$     \\\\\n\t\t\\midrule\n\t\t GCN+($\\beta=0$)  &$85.2\\pm0.5$ &$83.3\\pm1.1$ &$73.3\\pm0.5$ &$72.3\\pm0.7$ & $80.4\\pm0.6$  &$80.1\\pm0.6$     \\\\\n\t\t GCN+($\\beta \\ne 0$)   &$\\textbf{85.6}\\pm\\textbf{0.4}$ &$83.6\\pm1.3$ &$\\textbf{73.5}\\pm\\textbf{0.4}$ &$\\textbf{72.5}\\pm\\textbf{0.9}$ & $80.5\\pm0.6$  &$\\textbf{80.3}\\pm\\textbf{0.7}$     \\\\\n\t\t\\bottomrule  \n\t\\end{tabular}}\n\\caption{Summary of classification accuracy(\\%) on Planetoid datasets of semi-supervised node classification.}\n\\label{Plantoid-semi}\n\\end{table*}\n\n\\begin{table}[t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{3.5mm}{\n\t\\begin{tabular}{{ccccc}}\n\t\t\\toprule\n\t\tDataset & \\textit{ogb-arxiv}  & \\textit{ogb-proteins}  \\\\ \n\t\t\\midrule\n\t\tGCN  &  $71.74\\pm0.29$           &  $72.51\\pm0.35$     \\\\ \n\t\tGraphSAGE  &  $71.49\\pm0.25$          &  $77.68\\pm0.20$      \\\\ \n\t\t\\midrule\n\t\tGCN+($\\beta=0$)  & $71.85\\pm0.23$&$78.63\\pm0.28$  \\\\ \n\t\tGCN+($\\beta \\ne0$)  &$\\textbf{71.95}\\pm\\textbf{0.28}$ & $\\textbf{79.07}\\pm\\textbf{0.34}$  \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\caption{Summary of classification performance(\\%) on OGB datasets. For  \\textit{ogb-arxiv}, it indicates accuracy and for \\textit{ogb-proteins}, it indicates ROC-AUC.} \n\t\\label{ogb-semi}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{constant}\n\t\\caption{$M_{\\textit{\\!non-smooth}}$ of GCN+ with increasing hops on \\textit{Cora}.}\n\t\\label{oversmooth}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{sys_rw}\n\t\\caption{Performance comparison of different propagation matrices $\\tilde{A}_{sym}$ vs. $\\tilde{A}_{rw}$ in GCN+ with increasing hops on \\textit{Cora}.}\n\t\\label{choice}\n\\end{figure}\n\\begin{figure*}[t]\n\t\\centering\n\t\\subfigure[$\\alpha=9$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_9.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=4$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_4.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=2$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_2.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=1$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_1.pdf}}\\hspace{-1mm}\n\t\\caption{Test accuracy of different propagation steps and $\\alpha$ on \\textit{Cora}.}\n\t\\label{alpha and beta}\n\\end{figure*}\n\n\n\\subsection{Comparison with SOTA}\nThe evaluate metric of various datasets are listed in Table \\ref{dataset}. Actually it is commonly used to evaluate the model by the community.\n\\subsubsection{Planetoid}\nWe use standard fixed and random training\/validation\/testing splits. Specifically, we use 20 labeled nodes per class as the training set, 500 nodes as the validation set, and 1000 nodes as the test set for all models. For fixed split, we follow the experimental setup in \\cite{yang2016revisiting}.  We compare Multiplayer Perception (MLP) ,GCN \\cite{kipf2016semi}, GAT \\cite{velivckovic2017graph}, SGC \\cite{wu2019simplifying}, DAGNN \\cite{liu2020towards} and APPNP \\cite{klicpera2018predict} with GCN+. Although DropEdge \\cite{rong2019dropedge}, PairNorm \\cite{zhao2019pairnorm} are proposed to tackle over-smoothing issue recently, our baseline methods don't include them as they do not help to boost the performance on node classification task. \nTable \\ref{Plantoid-semi} compares the average test accuracy of 10 runs for each model on Planetoid dataset. As shown, GCN+ outperforms better than the representative baselines. Note that the shallow model APPNP achieves better performance than GCN and GAT. Recent deeper model named DAGNN shows competitive result and robustness on these datasets and GCN+ performs slightly better than it.\n\\subsubsection{OGB}\nWe adopt the setting of \\cite{hu2020open}, which is more challenging and realistic. We consider the following representative models GCN \\cite{kipf2016semi}, GraphSAGE \\cite{hamilton2017inductive} and GCNII \\cite{chen2020simple} as our baselines. In particular, we use the reported metric of the leaderboards of OGB team, which provide an open benchmark on several tasks and datasets. \n\nTable \\ref{ogb-semi} compares the average test accuracy\/ROC-AUC on OGB datasets. As shown, GCN+ outperforms the GCN and GraphSAGE. It is clear that our proposed GCN+ outperform SOTA in two middle scale datasets.\n\nIn summary, GCN+ achieves superior performance on several benchmarks, which shows that considering the information of high-order neighbors makes sense and we need more reasonable way to deepen GCNs or make use of the high-order neighbors. Note that GCN+ ($\\beta \\ne0$) is slightly better than GCN+ ($\\beta =0$) which is benefit from the third term of Eq. (\\ref{full-opt}). \n\\subsection{Over-smoothing Analysis} \\label{oversmooth-of-gcn+}\nWe employ the two proposed metrics to measure the node embeddings learned by GCN+. The results on \\textit{Cora} are shown in Fig. \\ref{oversmooth}. We can observe that as the number of hops increases, the $M_{\\textit{smooth}}$ values nearly remains a small constant which is lower than vanilla deep GCN. This implies that GCN+ use the information of long-range neighbors and do not suffer from over-smoothing. \n\nFig. \\ref{model_vs} also compares the final output embeddings of GCN+ with multiple hops, which shows different behaviors with GCN. GCN+ relieves the over-smoothing and learns the meaningful embeddings with the increasing hops.\n\n\\subsection{Hyperparameter Analysis}\nIn the previous sections, we use $\\tilde{A}$ to refer the $\\tilde{A}_{\\textit{sym}}$ and $\\tilde{A}_{\\textit{rw}}$. Here we compare the different choices of propagation matrix $\\tilde{A}$. Fig. \\ref{choice} depicts the test accuracy achieved by varying the hops of different propagation matrices. The result illustrates that \n$\\tilde{A}_{\\textit{sym}}$ is slightly better than $\\tilde{A}_{\\textit{rw}}$. \n\nWe consider three hyperparameter of GCN+, that is $\\alpha$, $\\beta$ and number of power iteration steps $k$. Fig. \\ref{alpha and beta} compares the effect of these hyperparameters on \\textit{Cora}. We can see that $k=16,32$ is suitable and more steps does not boost the performance significantly. For \\textit{Cora}, when $\\alpha=9$ (that means the fraction of retained initial node features is 0.1.), GCN+ achieve the best performance. The value of $\\alpha$ varies by different datasets. More results and details listed in the supplementary material.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\subsection{Graph Neural Networks}\nGraph neural networks (GNNs) have been extensively studied for the past years. There are different views on designing new architecture, including the spectral-based, spatial-based and other types, such as understand the GNN using dynamic system \\cite{xhonneux2019continuous}. Numerous methods are proposed to model the graph-structure data and apply on a wide range of applications. Besides the GCNs, there are also other types of GNNs, such as attention-based GNN \\cite{velivckovic2017graph} which use multiple attention to aggregate information from neighbors, autoencoder-based GNN \\cite{kipf2016variational}, which use a GCN encoder and decoder to learn meaningful embeddings, and dynamic GNNs \\cite{seo2018structured, hajiramezanali2019variational, yan2020sgrnn} which learn the node embedding over time.\n\\subsection{Deep GCN and Over-smoothing}\nMost GNNs are shallow models as deep architecture suffers from over-smoothing. Several studies explore deep GCNs. \\citet{xu2018representation} introduce Jumping Knowledge Networks, which uses residual connection to combine the output of each layer. \\citet{klicpera2018predict} use Personalized PageRank, which consider the information of root node to replace the graph convolution operator to solve the over-smoothing. DropEdge \\cite{rong2019dropedge} suggests that randomly removing the edge of original graph impede over-smoothing. PairNorm \\cite{zhao2019pairnorm} is another scheme which uses a normalization layer to scale the node features after the convolution layer. \\citet{li2019deepgcns} build on ideas from ResNet to train very deep GCNs. \\citet{li2020deepergcn} further propose MsgNorm, which boosts the performance on several datasets. \\citet{yang2020revisiting} present NodeNorm to scale the node features. \\cite{chen2020simple} propose a deep GCN models which use initial residual connection and identity mapping. \n\nA few work analyzes the cause and behaviors of over-smoothing theoretically. \\citet{oono2019graph} investigate the asymptotic behaviors of GCNs as the layer size tends to infinity and reveals the information loss in deep GCNs. \\citet{cai2020note} further extend analysis of \\cite{oono2019graph} from linear GNNs to the nonlinear architecture.\n\n\n\\section{Conclusion}\nWe summarize the existing different views on the mechanism of GCNs, which help us understand and design the graph convolutional kernel. We further provide a general optimization framework named GCN+. Based on this framework, we derive two forms of GCN+ and propose two metrics to measure the smoothness of output node representations. Extensive empirical studies on several real-world datasets demonstrate that GCN+ compares favorably to state of the art with a small amount of parameters. For future work, we will consider different optimization objectives which encode the graph structure and node features adaptively. As we do not limit the transformation from $X$ to $\\bar{X}$, another reasonable formulas can be further explored.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\nLocalization is an essential part of autonomous driving. For past years, there has been a wide variety of works in different approaches like \\textit{Simultaneous Localization and Mapping} (SLAM) \\cite{cadena2016past}, localization in previously built HD maps \\cite{pauls2020monocular}, and geo-referencing using aerial imagery \\cite{hu2019accurate}. A data association process becomes necessary for all these approaches to find correspondences between landmarks in maps and detections from the onboard sensors. The typical probabilistic models used in localization, such as factor-graphs or Bayesian filters, need these correspondences for inference. Hence, these associations are crucial for pose estimation.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=220pt]{figures\/01_overview.pdf}\n\\caption{An overview of a complete localization system, where the data representation and data association play a key role. Our research focuses on evaluating these modules (inside red dashes) by comparing the presented DA-LMR and the DC-SAC with state-of-the-art approaches.}\n\\label{fig:overview}\n\\end{figure}\n\nThe landmarks and detections used in the process usually depend on the application and the localization approach. In SLAM or localization in HD maps, lane markings, poles, traffic signs, and traffic lights have been used. While these maps usually provide high local accuracy, they also suffer from drift leading to global inconsistencies due to accumulated errors. To avoid this effect, a so-called geo-referencing is performed, where local detections from sensors are aligned with landmarks from aerial imagery. In that context, traffic signs and lights are unobservable, and poles are hard to detect. Here, lane markings are the potentially best observable landmarks for geo-referencing approaches. These lane markings are usually represented in maps as polylines and dashes.\n\nThe question that arises at this point is how to represent polylines and dashes in the data association process. In most works in the literature, there is no evaluation on how lane representations affect the data association, and therefore, the final result of localization. For this reason, in this paper, we focus on evaluating different data association methods using different data representations (yellow and green modules in Fig. \\ref{fig:overview}). For this, we propose the \\textit{Delta-Angle Lane Marking Representation} (DA-LMR), which represents polylines and dashes as points where an additional dimension represents a proportional value of the differential angle concerning the adjacent line segments. In addition, for data association, we propose a heuristic version of RANSAC, called \\textit{Distance-Compatible Sample Consensus} (DC-SAC), that limits possible samples to distance compatible ones.\n\nTo summarize, the main contributions are the following:\n\\begin{itemize}\n    \\item DA-LMR, a new way of representing lane markings for data association.\n    \\item A formalization of the data association method DC-SAC, used in our previous work \\cite{munoz2020targetless}.\n    \\item An evaluation of the possible combinations of data representation and data association approaches, including both state-of-the-art, and those proposed in this paper.\n\\end{itemize}\n\nOne of the possible applications of the data representation and data association proposed in this work is the geo-referencing localization using aerial imagery. For this reason, we present this work assuming a 2D world representation.\n\n\\section{RELATED WORK}\n\\label{sec:related_work}\nIn this section, we briefly review existing data association methods and existing data representations for lane markings.\n\n\\subsection{Data Association}\nWe review the data association methods distinguishing two categories: \\textit{pose estimation} and \\textit{graph-theoretic}. \n\n\\textit{Pose estimation} methods search for the transformation that minimizes the error between both a set of detections and a set of landmarks or maximizes its number of associations inliers. One of the most widely used \\textit{pose estimation} method for data association in localization is Iterative Closest Point (ICP). SLAM approaches such as \\cite{weichen2020hector,hu2019mapping,soatti2018implicit}, rely on the ICP for LiDAR scan matching. Currently, state-of-the-art Lidar Odometry And Mapping (LOAM) approaches also implement ICP for the data association process \\cite{shan2018lego,shan2020lio}. Nevertheless, this is a local method; hence, it requires a accurate initialization to converge in a global minimum. Further, it is more sensitive to suffer from drift in high outlier scenarios. We do not consider ICP a potential solution for data association as in the context of geo-referencing using aerial imagery no sufficiently accurate pose is available. The random Sample Consensus (RANSAC) method \\cite{fischler1981random} is a global data association approach common in the literature. RANSAC follows a strategy that samples hypothesis transformations depending on the data structure. Different authors use RANSAC for localization \\cite{yang2010ransac,jiao2020globally} (include aerial imagery-based \\cite{hu2019accurate,ramalingam2011pose}), and mapping \\cite{cunningham2012fully}.\n\n\\textit{Graph-theoretic} methods are based on searching a maximum number of associations being compatible. One common measure of compatibility is Distance Compatibility (DC). This problem is also known in the literature as the maximum clique (MC) problem \\cite{wu2015review}. In the past decades, this approach has been used in localization \\cite{bailey2000data,mangelson2018pairwise}, including loop closures \\cite{frey2019efficient,tian2021kimera}. This is also a global data association approach, and we consider it potentially suitable for localization in aerial imagery. In past years, there has been research focused on a weighed version of the MC problem \\cite{wu2012multi,benlic2013breakout}. An interesting variant is CLIPPER \\cite{lusk2020clipper}, where the authors implement a weighed MC problem using a continuous relaxation for the optimization process. We can also consider the proposed DC-SAC as a variant of this kind of association.\n\nIt is worth noting that there are other data association methods that we don't include in this work because we consider them strongly coupled with the SLAM problem. That is the case of the Hungarian \\cite{welte2020improved}, Nearest Neighbor (NN) \\cite{castellanos1999spmap}, Joint Compatibility Branch and Bound (JCBB) \\cite{neira2001data,yang2020improved}, probabilistic data association based on Mixture Models \\cite{zhang2019hierarchical,doherty2020probabilistic}, and Multi-Hypothesis based data association \\cite{hsiao2019mh,jiang2021imhs}.\n\n\\subsection{Data Representations for polylines}\n\\label{sec:related_work_dr}\nThe most common approach presented in the literature to represent data in the data association process is 2D points for 3-DOF localization \\cite{heidenreich2015laneslam,vivacqua2017self}. To represent lane markings as 2D points, a previous sampling of polylines becomes necessary. When this process is computed for both landmarks and detections, it is called Point-to-Point (P2P) association. Another strategy is the Point-to-Line association (P2L) \\cite{poggenhans2018precise,welte2019estimating}. In this case, the authors project point detections onto the map polylines in each \\textit{pose estimation} iteration. However, after projection, the distance is measured between points. Then, we can consider the polylines (landmarks) sampled and converted into 2D points and, roughly speaking, the representation becomes the P2P strategy. Hence, we consider both strategies, P2P and P2L, as unique  2D points-based representations. In other works \\cite{hu2019accurate,kummerle2019accurate}, the authors represent the data as lines, also with a previous sampling. In these cases, the authors use a distance metric for line segments, such as Hausdorff distance \\cite{yang2021matching,quehl2017good}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=130pt]{figures\/02_invariance.pdf}\n\\caption{Depiction of direction invariance of the delta-angle calculation: in \\textit{red}, the $i$-th point, and in \\textit{blue}, the $j$-th point are in inverse direction. In both cases, the delta-angle represents the same value.}\n\\label{fig:delta_angle}\n\\end{figure}\n\n\\section{DELTA ANGLE LANE MARKINGS REPRESENTATION (DA-LMR)}\nAs discussed in the previous section, the most common representations of lane markings in the process of data association are point-based. Nevertheless, transforming lines and polylines into isolated points entails a loss of information. In a polyline, how the angle of each point changes in a curve provides additional information about the geometric structure of that curve. Following this assumption, we present the \\textit{Delta-Angle Lane Marking Representation} (DA-LMR). In the following subsections, we explain how to derive this representation for both polylines and dashes.\n\n\\subsection{DA-LMR for polylines}\nWe can describe a polyline as a set of 2D points $\\mathbf{P}^{2D} = (\\mathbf{p}^{2D}_0, \\mathbf{p}^{2D}_1, ..., \\mathbf{p}^{2D}_N)$, where $\\mathbf{p}^{2D}_i = (x_i, y_i)$, and where each point has a connection with its adjacent ones. Alternatively, given this connection, we can describe the polyline as a set of vectors $\\mathbf{V} = (\\vec{\\mathbf{v}}_{0}, \\vec{\\mathbf{v}}_{1}, ..., \\vec{\\mathbf{v}}_{N-1})$, where each vector is $\\vec{\\mathbf{v}}_{i} = (x_{i+1} - x_{i}, y_{i+1} - y_{i})$. This vectorial representation provides information about the orientation of each $i$-th point in a polyline. However, representing the orientation in a global reference frame can lead to problems due to the Euler angle wrap-around limitations. To obtain a compact and invariant angle representation, we use the differential angle between adjacent vectors $\\vec{\\mathbf{v}}_{i-1}$ and $\\vec{\\mathbf{v}}_{i}$. Then, for each $i$-th point in a polyline, we can obtain the delta-angle as follows:\n\n$$\n\\Delta\\alpha_{i} = \\arccos{\\left( \\frac{\\vec{\\mathbf{v}}_{i-1} \\cdot \\vec{\\mathbf{v}}_{i}}{\\left\\lVert \\vec{\\mathbf{v}}_{i-1}\\right\\rVert \\left\\lVert\\vec{\\mathbf{v}}_{i}\\right\\rVert} \\right)}. \n\\eqno{(1)}\n$$\n\nIt is worth noting, that the points $i = 0$ and $i = N$ does not have adjacent information, and therefore we assign them a default value of $\\Delta\\alpha_{0} = \\Delta\\alpha_{N} = 0$. The delta-angle calculation defined in (1) is a non-oriented angle calculation. This entails that the representation is also invariant to the direction in the polyline, i.e., we can sort the polyline from $0$ to $N$ or from $N$ to $0$, obtaining the same result. In Fig. \\ref{fig:delta_angle}, we show an example of this direction invariance. In practice, this is important as the representation is independent of the direction from which it is observed.\n\nAs the final step of DA-LMR, we use delta-angle information as an extra dimension in the 2D point representation to obtain a compact description suitable for data association methods. Now, the polyline is converted in a set of 3D points $\\mathbf{P}^{3D} = (\\mathbf{p}^{3D}_0, \\mathbf{p}^{3D}_1, ..., \\mathbf{p}^{3D}_N)$, where\n\n$$\n\\mathbf{p}^{3D}_i = (x_i, y_i, \\Delta\\alpha_i w).\n\\eqno{(2)}\n$$\n\nHere $w$ is a weight that tunes how strongly the delta-angle describes the data structuring. Note that when $w \\xrightarrow{} 0$, the representation tends to a traditional 2D representation. In Fig. \\ref{fig:dalmr_example}, we show an example of the DA-LMR in contrast to line segments and 2D points. In the case of DA-LMR, we represent the delta angle as the size of the circles. Given this additional dimension, DA-LMR is more informative than previously mentioned representations.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=220pt]{figures\/03_polylines.pdf}\n\\caption{Comparison of different representations of the same line: in \\textit{black}, line segments representation as polyline, in \\textit{blue}, 2D point representation, and in \\textit{red}, the proposed DA-LMR. We depict the delta angle for DA-LMR by the size of the circles.}\n\\label{fig:dalmr_example}\n\\end{figure}\n\n\\subsection{DA-LMR for dashes}\nFor dashed lines, it is not trivial to determine adjacent lines. Hence, for that case, we move the representation formulation to the distance measurement step in the data association process. This distance could be used for the error measurement in \\textit{pose estimation} data association, or otherwise, for distance compatibility measurement in \\textit{graph-theoretic} data association. \n\nWe can describe dashes as vectors where $\\vec{\\mathbf{v}}_i = (x_{i_e}-x_{i_s}, y_{i_e}-y_{i_s})$, and where subscripts $i_s$ and $i_e$ denotes start point and end point, respectively for $i$-th vector. Then, given a pair of dashes $\\vec{\\mathbf{v}}_i$ and $\\vec{\\mathbf{v}}_j$, we can formulate the distance measurement for DA-LMR as follows:\n\n$$\nd_{ij} = \\sqrt{(x_j-x_i)^2 + (y_j-y_i)^2 + (\\Delta\\alpha_{ij} w)^2}.\n\\eqno{(3)}\n$$\n\nWhere $(x_i, y_i)$ and $(x_j, y_j)$ are the centroid of each dash respectively. And where\n\n$$\n\\Delta\\alpha_{ij} = \\arccos{\\left( \\frac{\\vec{\\mathbf{v}}_{i} \\cdot \\vec{\\mathbf{v}}_{j}}{\\left\\lVert \\vec{\\mathbf{v}}_{i}\\right\\rVert \\left\\lVert\\vec{\\mathbf{v}}_{j}\\right\\rVert} \\right)}. \n\\eqno{(4)}\n$$\n\nIn Fig. \\ref{fig:dalmr_dashes}, we show an example of two dashes intersecting in its centroid but misaligned. In this case, if we use a 2D point representation, we obtain a distance measurement $d^{pt}_{ij} = 0$, which doesn't describe the misalignment. Using DA-LMR, we obtain a distance measurement $d^{dalmr}_{ij} = \\Delta\\alpha_{ij} w$ that uncovers the inconsistency.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=130pt]{figures\/04_dashes.pdf}\n\\caption{Depiction of two dashes intersecting in its centroid. If we compute the distance measurement (3) for a 2D point representation, the result is $d^{pt}_{ij} = 0$. In contrast, if we compute (3) for DA-LMR, the result is $d^{dalmr}_{ij} = \\Delta\\alpha_{ij} w$, showing the misalignment.}\n\\label{fig:dalmr_dashes}\n\\end{figure}\n\n\\section{DC-SAC DATA ASSOCIATION}\n\\textit{Distance-Compatible Sample Consensus} (DC-SAC) could be seen as a heuristic version of the RANSAC method. This is a formalization of a previous work \\cite{munoz2020targetless} for camera-LiDAR calibration. In \\cite{quan2020compatibility}, the authors propose a similar approach but focus on dense point clouds and geometric descriptors.\n\nDA-LMR is a \\textit{pose estimation} method that samples the possible transformations. How to sample the data for a $\\Delta_T \\in SE(2)$ transformation candidate is the key difference between RANSAC and DC-SAC. \n\nGiven a set of landmarks $\\mathcal{L}$ and a set of detections $\\mathcal{D}$, RANSAC chooses a pair of points randomly from each set. Thereafter, it estimates $\\Delta_T \\in SE(2)$ by minimizing \n\n$$\ne(\\Delta_T) = \\left\\lVert\\mathbf{p}^{d}_1 - \\mathbf{p}^{l}_1\\right\\rVert + \\left\\lVert\\mathbf{p}^{d}_2 - \\mathbf{p}^{l}_2\\right\\rVert.\n\\eqno{(5)}\n$$\n\nSuperscripts $l$ and $d$ mean landmarks and detections, respectively. Then, all the points in $\\mathcal{D}$ are transformed using the estimated $\\Delta_T^*$, obtaining a new set $\\mathcal{D}^*$. Thereafter, an error $\\epsilon$ is computed by the sum of the distance measurement between each detection in $\\mathcal{D}^*$ and the nearest neighbor in $\\mathcal{L}$. RANSAC iterates this process obtaining a new $\\epsilon$ in each iteration until the error satisfies the condition $\\epsilon < \\xi$, where $\\xi$ is a configurable threshold. The threshold limits the accuracy of the method. For example, a high value for $\\xi$ can miss potentially optimal hypotheses producing inaccurate results. In contrast, a low-value configuration of $\\xi$ can dramatically increase computation time, possibly making the method unusable.\n\nIn the case of DC-SAC, we sample a pair of points in $\\mathcal{D}$. However, in contrast to RANSAC, we randomly choose a pair of points in $\\mathcal{L}$ that are distance compatible, i.e., that satisfies\n\n$$\n\\gamma > \\left\\lvert \\left\\lVert\\mathbf{p}^{d}_1 - \\mathbf{p}^{d}_2\\right\\rVert - \\left\\lVert\\mathbf{p}^{l}_1 - \\mathbf{p}^{l}_2\\right\\rVert \\right\\rvert.\n\\eqno{(6)}\n$$\n\nWhere $\\gamma$ is a configurable value. The constraint proposed in (6) could be applied in a general way for data association. Additionally,  as we present this work in localization, we can propose other constraints in the hypothesis space $\\mathcal{H}$ depending on the uncertainty of the prior. Given the distribution of $SE(2)$ localization, we can get the standard deviation values for each dimension $(\\sigma_x, \\sigma_y, \\sigma_\\theta)$. Then, by considering $\\sigma_x$ and $\\sigma_y$, we can limit the area of sampling points in $\\mathcal{L}$. By using $\\sigma_\\theta$ and given $\\vec{\\mathbf{v}}_l = (\\mathbf{p}^{l}_1 - \\mathbf{p}^{l}_2)$ and $\\vec{\\mathbf{v}}_d = (\\mathbf{p}^{d}_1 - \\mathbf{p}^{d}_2)$, we can define an orientation constraint between pairs of samples as follows:\n\n$$\n\\theta_t > \\arccos{\\left( \\frac{\\vec{\\mathbf{v}}_{l} \\cdot \\vec{\\mathbf{v}}_{d}}{\\left\\lVert \\vec{\\mathbf{v}}_{l}\\right\\rVert \\left\\lVert\\vec{\\mathbf{v}}_{d}\\right\\rVert} \\right)}. \n\\eqno{(7)}\n$$\n\nWhere $\\theta_t$ is a configurable value that depend on $\\sigma_\\theta$. The above-presented heuristic constraints reduce the size of $\\mathcal{H}$ dramatically. In this way, we can search the full space $\\mathcal{H}$ for the minimum value of $\\epsilon$ instead of iterating until $\\epsilon < \\xi$. This is the best scenario for converging to a potentially optimal solution. In the next section, we demonstrate experimentally how DC-SAC produces more accurate results than RANSAC by configuring a value of $\\xi$ where both methods run with similar computation time.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n    \\begin{axis}[\n    name=plot,\n    xlabel={$\\sigma$ (m)}, ylabel={\\%},\n    xmin=0.1, xmax = 0.5, ymin=70, ymax=100,\n    width=230pt,height=200pt,\n    grid=both,\n    grid style={line width=.1pt, draw=gray!10},\n    major grid style={line width=.2pt,draw=gray!50},\n    legend pos=south west,\n    legend style={nodes={scale=0.5, transform shape}}\n    ]\n    \n    \\addplot[color={rgb:red,4;green,0;yellow,5},mark=pentagon*,very thick] table{.\/data\/05_clipper_lsr_f1.txt};\n    \\addlegendentry{LSR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dotted,very thick] table{.\/data\/05_clipper_lsr_p.txt};\n    \\addlegendentry{LSR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dashed,very thick] table{.\/data\/05_clipper_lsr_r.txt};\n    \\addlegendentry{LSR (Recall)}\n\n    \\addplot[color={rgb:red,1;green,2;blue,5},mark=triangle*,very thick] table{.\/data\/05_clipper_2dpr_f1.txt};\n    \\addlegendentry{PR (F1 score)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dotted,very thick] table{.\/data\/05_clipper_2dpr_p.txt};\n    \\addlegendentry{PR (Precision)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dashed,very thick] table{.\/data\/05_clipper_2dpr_r.txt};\n    \\addlegendentry{PR (Recall)}\n\n    \\addplot[color={rgb:red,4;green,0;yellow,1},mark=square*,very thick] table{.\/data\/05_clipper_dalmr_f1.txt};\n    \\addlegendentry{DA-LMR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dotted,very thick] table{.\/data\/05_clipper_dalmr_p.txt};\n    \\addlegendentry{DA-LMR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dashed,very thick] table{.\/data\/05_clipper_dalmr_r.txt};\n    \\addlegendentry{DA-LMR (Recall)}\n\n    \\end{axis}\n    \\end{tikzpicture}\n    \\caption{Comparison in F1 score (\\textit{solid}), precision (\\textit{dotted}) and recall (\\textit{dashed}) metrics between LSR, PR, and DA-LMR for CLIPPER \\cite{lusk2020clipper} as data association method.}\n    \\label{fig:clipper_p}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n    \\begin{axis}[\n    name=plot,\n    xlabel={$\\sigma$ (m)}, ylabel={\\%},\n    xmin=0.1, xmax = 0.5, ymin=70, ymax=100,\n    width=230pt,height=200pt,\n    grid=both,\n    grid style={line width=.1pt, draw=gray!10},\n    major grid style={line width=.2pt,draw=gray!50},\n    legend pos=south west,\n    legend style={nodes={scale=0.5, transform shape}}\n    ]\n    \n    \\addplot[color={rgb:red,4;green,0;yellow,5},mark=pentagon*,very thick] table{.\/data\/06_mc_lsr_f1.txt};\n    \\addlegendentry{LSR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dotted,very thick] table{.\/data\/06_mc_lsr_p.txt};\n    \\addlegendentry{LSR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dashed,very thick] table{.\/data\/06_mc_lsr_r.txt};\n    \\addlegendentry{LSR (Recall)}\n\n    \\addplot[color={rgb:red,1;green,2;blue,5},mark=triangle*,very thick] table{.\/data\/06_mc_2dpr_f1.txt};\n    \\addlegendentry{PR (F1 score)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dotted,very thick] table{.\/data\/06_mc_2dpr_p.txt};\n    \\addlegendentry{PR (Precision)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dashed,very thick] table{.\/data\/06_mc_2dpr_r.txt};\n    \\addlegendentry{PR (Recall)}\n\n    \\addplot[color={rgb:red,4;green,0;yellow,1},mark=square*,very thick] table{.\/data\/06_mc_dalmr_f1.txt};\n    \\addlegendentry{DA-LMR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dotted,very thick] table{.\/data\/06_mc_dalmr_p.txt};\n    \\addlegendentry{DA-LMR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dashed,very thick] table{.\/data\/06_mc_dalmr_r.txt};\n    \\addlegendentry{DA-LMR (Recall)}\n\n    \\end{axis}\n    \\end{tikzpicture}\n    \\caption{Comparison in F1 score (\\textit{solid}), precision (\\textit{dotted}) and recall (\\textit{dashed}) metrics between LSR, PR, and DA-LMR for Max Clique \\cite{konc2007improved} as data association method.}\n    \\label{fig:mc_p}\n\\end{figure}\n\n\\section{EVALUATION}\n\nThe evaluation step aims to compare the presented DA-LMR with other data representations as well as the presented DC-SAC with other data association methods. It is difficult to perform a quantitative comparison using maps and real onboard sensor data since it is hard to obtain ground truth. Hence, in Section \\ref{sec:quantitative}, we generate synthetic detections $\\mathcal{D}^s$ from the original landmarks $\\mathcal{L}$ of a real map by cropping, transforming, and adding noise. In Section \\ref{sec:qualitative}, we show qualitative results using the previously used $\\mathcal{L}$ and real detections $\\mathcal{D}^r$ from stereo cameras in a real autonomous vehicle.\n\n\\subsection{Quantitative results}\n\\label{sec:quantitative}\n\nFor quantitative evaluation, we use a map that contains continuous and dashed lane markings. This map was labeled by hand from aerial imagery from the city of Karlsruhe, Germany. We define the set of landmarks $\\mathcal{L}$ by sampling the map polylines with steps of $\\SI{1}{\\meter}$, and we use this labeled data as ground truth. Then, using a pre-recorded trajectory in the mapped area as a reference, we apply a sliding window to $\\mathcal{L}$. In each iteration, we crop a set of data that forms a new  $\\mathcal{D}^s_i$, where the subscript $i$ indicates the iteration of the sliding window, and the superscript $s$ denotes synthetic. Thereafter, we apply a random 3-DOF rigid body transformation $\\mathbf{t} = (t_x, t_y)$ and $\\mathbf{R} = f(r_\\theta)$ to each $\\mathcal{D}^s_i$. Where $t_x$ and $t_y$ are uniformly distributed random variables with range $(\\SI{-5}{\\meter}, \\SI{5}{\\meter})$, and where $r_\\theta$ is also uniformly distributed random variable with range $(\\SI{-5}{\\degree}, \\SI{5}{\\degree})$. Also, for each experiment, we produce additive Gaussian noise with a range of $\\sigma$ between $\\SI{0.1}{\\meter}$ and $\\SI{0.5}{\\meter}$ with steps of $\\SI{0.1}{\\meter}$. Finally, we include outliers, where the number of outliers is $\\SI{10}{\\percent}$ of the samples in $\\mathcal{D}^s_i$. It is worth noting that it is not recommendable to execute the data association process in the complete set $\\mathcal{L}$. Hence, assuming we have a prior localization, we generate a set $\\mathcal{L}_i$ bigger than $\\mathcal{D}^s_i$ in each iteration of the sliding window that covers an area around it.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n    \\begin{axis}[\n    name=plot,\n    xlabel={$\\sigma$ (m)}, ylabel={\\%},\n    xmin=0.1, xmax = 0.5, ymin=90, ymax=100,\n    width=230pt,height=195pt,\n    grid=both,\n    grid style={line width=.1pt, draw=gray!10},\n    major grid style={line width=.2pt,draw=gray!50},\n    legend pos=south west,\n    legend style={nodes={scale=0.5, transform shape}}\n    ]\n    \n    \\addplot[color={rgb:red,4;green,0;yellow,5},mark=pentagon*,very thick] table{.\/data\/07_ransac_lsr_f1.txt};\n    \\addlegendentry{LSR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dotted,very thick] table{.\/data\/07_ransac_lsr_p.txt};\n    \\addlegendentry{LSR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dashed,very thick] table{.\/data\/07_ransac_lsr_r.txt};\n    \\addlegendentry{LSR (Recall)}\n\n    \\addplot[color={rgb:red,1;green,2;blue,5},mark=triangle*,very thick] table{.\/data\/07_ransac_2dpr_f1.txt};\n    \\addlegendentry{PR (F1 score)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dotted,very thick] table{.\/data\/07_ransac_2dpr_p.txt};\n    \\addlegendentry{PR (Precision)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dashed,very thick] table{.\/data\/07_ransac_2dpr_r.txt};\n    \\addlegendentry{PR (Recall)}\n\n    \\addplot[color={rgb:red,4;green,0;yellow,1},mark=square*,very thick] table{.\/data\/07_ransac_dalmr_f1.txt};\n    \\addlegendentry{DA-LMR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dotted,very thick] table{.\/data\/07_ransac_dalmr_p.txt};\n    \\addlegendentry{DA-LMR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dashed,very thick] table{.\/data\/07_ransac_dalmr_r.txt};\n    \\addlegendentry{DA-LMR (Recall)}\n\n    \\end{axis}\n    \\end{tikzpicture}\n    \\caption{Comparison in F1 score (\\textit{solid}), precision (\\textit{dotted}) and recall (\\textit{dashed}) metrics between LSR, PR, and DA-LMR for RANSAC \\cite{hu2019accurate} as data association method.}\n    \\label{fig:ransac_p}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n    \\begin{axis}[\n    name=plot,\n    xlabel={$\\sigma$ (m)}, ylabel={\\%},\n    xmin=0.1, xmax = 0.5, ymin=90, ymax=100,\n    width=230pt,height=195pt,\n    grid=both,\n    grid style={line width=.1pt, draw=gray!10},\n    major grid style={line width=.2pt,draw=gray!50},\n    legend pos=south west,\n    legend style={nodes={scale=0.5, transform shape}}\n    ]\n    \n    \\addplot[color={rgb:red,4;green,0;yellow,5},mark=pentagon*,very thick] table{.\/data\/08_dcsac_lsr_f1.txt};\n    \\addlegendentry{LSR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dotted,very thick] table{.\/data\/08_dcsac_lsr_p.txt};\n    \\addlegendentry{LSR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,5},dashed,very thick] table{.\/data\/08_dcsac_lsr_r.txt};\n    \\addlegendentry{LSR (Recall)}\n\n    \\addplot[color={rgb:red,1;green,2;blue,5},mark=triangle*,very thick] table{.\/data\/08_dcsac_2dpr_f1.txt};\n    \\addlegendentry{PR (F1 score)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dotted,very thick] table{.\/data\/08_dcsac_2dpr_p.txt};\n    \\addlegendentry{PR (Precision)}\n    \\addplot[color={rgb:red,1;green,2;blue,5},dashed,very thick] table{.\/data\/08_dcsac_2dpr_r.txt};\n    \\addlegendentry{PR (Recall)}\n\n    \\addplot[color={rgb:red,4;green,0;yellow,1},mark=square*,very thick] table{.\/data\/08_dcsac_dalmr_f1.txt};\n    \\addlegendentry{DA-LMR (F1 score)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dotted,very thick] table{.\/data\/08_dcsac_dalmr_p.txt};\n    \\addlegendentry{DA-LMR (Precision)}\n    \\addplot[color={rgb:red,4;green,0;yellow,1},dashed,very thick] table{.\/data\/08_dcsac_dalmr_r.txt};\n    \\addlegendentry{DA-LMR (Recall)}\n\n    \\end{axis}\n    \\end{tikzpicture}\n    \\caption{Comparison in F1 score (\\textit{solid}), precision (\\textit{dotted}) and recall (\\textit{dashed}) metrics between LSR, PR, and DA-LMR for the proposed DC-SAC as data association method.}\n    \\label{fig:dcsac_p}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=500pt]{figures\/08_mosaic.pdf}\n\\caption{Examples of data association results. In \\textit{red}, we show the $\\mathcal{L}_i$, while in \\textit{blue}, we show $\\mathcal{D}^r_i$. \\textit{Green} lines indicate the association between landmarks and detections. The size of circles is proportional to the Z-axis value.}\n\\label{fig:mosaic}\n\\end{figure*}\n\nAt this point, given $\\mathcal{L}_i$ and $\\mathcal{D}^s_i$, we run a data association process for each data representation and each $\\sigma$ value of the range mentioned above. We use the representations shown Fig. \\ref{fig:dalmr_example}, i.e., line segments representation (LSR), 2D points representation (PR), and the proposed DA-LMR with $w = \\SI{5}{\\frac{\\meter}{\\radian}}$. For metrics, we use Euclidean distance for PR and DA-LMR and Hausdorff distance for LSR. Finally, we repeat the process for the different data association methods: CLIPPER \\cite{lusk2020clipper}, Max Clique \\cite{konc2007improved}, RANSAC \\cite{hu2019accurate}, and the proposed DC-SAC.\n\nIn Fig. \\ref{fig:clipper_p}, we show a comparison in F1 score, precision and recall metrics between the representation approaches using CLIPPER \\cite{lusk2020clipper} as data association method. The statistics are accumulated for all $i$ values, i.e., for the complete pre-recorded trajectory. For this experiment, we configure the distance compatibility (6) parameter as $\\gamma = 3 \\sigma$\\footnote{It is worth noting that the parameter $\\gamma$ only defined for DC-SAC is also used in graph-theoretic approaches because such methods are based on distance compatibility constraints too.}. As we can see in Fig. \\ref{fig:clipper_p}, DA-LMR improves the results of PR for both precision and recall. LSR shows the best results in precision. However, it generates the worst results in recall. In other words, LSR can associate less data than the other, but with better precision. \n\nIn the case of Max Clique \\cite{konc2007improved}, for this experiment, we also configure the distance compatibility (6) parameter as $\\gamma = 3 \\sigma$. We can see in Fig. \\ref{fig:mc_p} that the behavior of the different representations is similar to the previous. But in this case, LSR seems more sensitive against the noise in recall statistics than PR and DA-LMR.\n\nFor RANSAC \\cite{hu2019accurate}, we configure $\\xi$ with a value that produces a computational time similar to the DC-SAC method. The results for RANSAC are strongly correlated with the threshold $\\xi$. For this reason, the results are stable against the noise. We can see this behavior in Fig. \\ref{fig:ransac_p} for PR and DA-LMR, where DA-LMR improves the results of PR. However, LSR shows different behavior and presents less stability against noise again.\n\nIn Fig. \\ref{fig:dcsac_p}, we compare the data representation approaches using the presented DC-SAC data association method and configure $\\gamma = 3 \\sigma$. In this case, we can see the best results in precision with DA-LMR. While in recall, LSR shows the best results, but the difference is relatively small.\n\nIn general, LSR shows good results in low-noise scenarios, but it is more sensitive to noise disturbances. The PR and DA-LMR show more robustness against noise, and DA-LMR improves the results of PR in all cases.\n\nComparing the data association methods, we found the \\textit{graph-theoretic} ones (CLIPPER and Max Clique) more sensitive against noise. Especially the CLIPPER method. Max Clique shows good results for low-noise, but for high noise, the precision is similar to RANSAC and the recall much worst. In contrast, we found the \\textit{pose estimation} methods more robust against noise. RANSAC shows reasonably good results. In precision it is similar to the worst case of Max Clique, but in recall shows more stability. DC-SAC shows, in general, the best results in precision and recall and, like RANSAC, high robustness against noise.\n\nFinally, in Table \\ref{tab:time_consuming}, we show the evaluation of the computational time for each data association and each data representation depending on the samples number $\\left\\lvert \\mathcal{L}_i \\right\\rvert + \\left\\lvert \\mathcal{D}^s_i \\right\\rvert$. The experiments were performed on an i7-7700HQ CPU with 16 GB of RAM in a C++ compiled code. We can see that RANSAC and DC-SAC show similar results for the configuration described in this section and are reasonable. In the case of Max Clique, it improves the time consumption for low sample scenarios, but for $>200$ samples, it is slower than SAC-based methods. The CLIPPER method is the slowest of all. For 400 samples, the CPU cannot even run a complete process. Comparing data representation, LSR shows slower behavior than PR and DA-LMR for SAC-based methods. However, for \\textit{graph-theoretic} approaches, the behavior is different. In that case, LSR is faster than the others representations. In all cases, DA-LMR is faster than PR.\n\n\\begin{table}[ht]\n\\caption{Computation time evaluation in seconds.}\n\\label{tab:time_consuming}\n\\begin{center}\n\\begin{tabular}{c c c c c c c}\n\\hline\nSamples & DR & \\textbf{DC-SAC} & \\textbf{RANSAC} & \\textbf{MC} & \\textbf{CLIPPER} \\\\\n\\hline\n\\hline\n100 & LSR & 0.09 & 0.06 & 0.02 & 0.17 \\\\\n    & PR & 0.06 & 0.06 & 0.03 & 0.69 \\\\\n    & DA-LMR & 0.06 & 0.06 & 0.03 & 0.31 \\\\\n\\hline\n200 & LSR & 0.91 & 1.12 & 0.27 & 4.79 \\\\\n    & PR & 0.67 & 0.69 & 0.53 & 21.16 \\\\\n    & DA-LMR & 0.61 & 0.59 & 0.55 & 10.50 \\\\\n\\hline\n300 & LSR & 4.58 & 5.15 & 2.86 & 41.32 \\\\\n    & PR & 3.27 & 2.97 & 4.86 & 151.3 \\\\\n    & DA-LMR & 2.91 & 2.83 & 4.35 & 55.62 \\\\\n\\hline\n400 & LSR & 18.55 & 19.72 & 12.77 & - \\\\\n    & PR & 12.88 & 13.05 & 18.61 & - \\\\\n    & DA-LMR & 11.91 & 12.64 & 18.53 & - \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nAs we obtain the highest values on precision and recall, we consider that the combination of the proposed data representation approach DA-LMR, and the presented data association DC-SAC, is the best combination among the evaluated for a localization application and specially for geo-referencing using aerial imagery.\n\n\\subsection{Qualitative results}\n\\label{sec:qualitative}\nIn Section \\ref{sec:quantitative}, we compared different data association methods with different data representations, and we concluded that DA-LMR and DC-SAC are the preferable combination. In this section, we show some qualitative results with the chosen combination. For the evaluation, we use the same map as in the previous section and the same pre-recorded trajectory. We use the same $\\mathcal{L}_i$ as a set of landmarks for each sliding window iteration. However, in this case, we use real data for detections ($\\mathcal{D}^r_i$) obtained while driving the pre-recorded trajectory. Detections were obtained using stereo cameras from our\nexperimental vehicle \\textit{BerthaOne} \\cite{tacs2017making}. Then, given $\\mathcal{L}_i$ and $\\mathcal{D}^r_i$, we iteratively run the process of DA-LMR and DC-SAC for the complete trajectory.\n\nIn Fig. \\ref{fig:mosaic}, we show some examples of data association results. In \\textit{red}, we show the $\\mathcal{L}_i$, while in \\textit{blue}, we show $\\mathcal{D}^r_i$. \\textit{Green} lines indicate the association between landmarks and detections. The size of circles is proportional to the delta angles. The results look very reasonable for its implementation in a complete localization approach.\n\n\\section{CONCLUSIONS}\nWe proposed DA-LMR, a robust data representation for lane markings in data association processes for localization. As we demonstrated experimentally, this representation provides richer information of geometrical data structure than others, such as line segments and 2D points. Furthermore, we proposed DC-SAC, a data association method that can improve the results by eliminating potentially non-optimal hypotheses in the space using distance compatibility constraints. We also demonstrate experimentally how the presented method improves the results of other state-of-the-art ones, such as RANSAC, Max Clique, and CLIPPER.\n\nIn future work, we will apply the combination of DA-LMR and DC-SAC for a geo-referencing localization using aerial imagery. Although we have emphasized this kind of localization in this work, we also plan to apply the presented method to localization in HD maps.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe approximate Nonnegative Matrix Factorization (NMF) of\nnonnegative matrices is a data analysis technique only recently\nintroduced \\cite{leeseung1999, sullivan2000}. Roughly speaking the\nproblem is to find, for a given nonnegative matrix $V \\in\n\\mathbb{R}_+^{m\\times n}$, and an assigned $k$, a pair of nonnegative\nmatrices $W\\in\\mathbb{R}_+^{m\\times k}$ and $H\\in\\mathbb{R}_+^{k\\times n}$ such\nthat, in an appropriate sense, $V \\approx WH$. In\n\\cite{leeseung1999} EM like algorithms for the construction of a\nfactorization have been proposed. The algorithms have been later\nderived in \\cite{leeseung2001} by using an {\\em ad-hoc} auxiliary\nfunction, a common approach in deriving EM algorithms. In\n\\cite{sullivan2000} the connection with the classic alternating\nminimization of the I-divergence \\cite{ct1984} has been pointed\nout but not fully investigated. In this paper we pose the NMF\nproblem as a minimum I-divergence problem that can be solved by\nalternating minimization and derive, from this point of view, the\nalgorithm proposed in \\cite{leeseung1999}. There are alternative\napproaches to approximate nonnegative matrix factorization. For\ninstance, recently, see~\\cite{laa2004}, results have been obtained\nfor the approximate factorization (w.r.t. the Frobenius norm) of\n{\\em symmetric} nonnegative matrices.\n\nAlthough only recently introduced the NMF has found many\napplications as a data reduction procedure and has been advocated\nas an alternative to Principal Components Analysis (PCA) in cases\nwhere the positivity constraint is relevant (typically image\nanalysis). The title of \\cite{sullivan2000} is a clear indication\nof this point of view, but a complete analysis of the relations\nbetween NMF and PCA is still lacking.\nOur interest in NMF stems from the system theoretic problem of\napproximate realization (or order reduction) of Hidden Markov\nModels. Partial results have already been obtained \\cite{fs2002}.\n\nThis paper is organized as follows. In section~\\ref{sec:problem}\nwe pose the approximate nonnegative matrix factorization problem,\ndefine the I-divergence between matrices and discuss the solution\nproposed in \\cite{leeseung1999, leeseung2001}. In\nsection~\\ref{section:lift} we pave the way for the alternating\nminimization algorithm presenting the properly lifted version of\nthe minimization problem and solving the two partial minimizations\nin the style of Csisz\\'ar and Tusn\\'ady \\cite{ct1984}. In\nsection~\\ref{section:altmin} we construct the alternating\nminimization algorithm and compute the iteration gain. One of the\nadvantages of working with the lifted problem is that it sheds a\nnew light also on the derivation of the algorithm via auxiliary\nfunctions given in \\cite{leeseung2001}. In\nsection~\\ref{section:auxfunc} we will use the results of\nsection~\\ref{section:lift} to construct a very natural auxiliary\nfunction to solve the original problem. A discussion of the\nconvergence properties of the algorithm is given in\nsection~\\ref{section:convprop}. In the concluding\nsection~\\ref{section:otherprob} we establish a connection between\nthe approximate NMF problem and the Archetypal Analysis algorithm\nof Cutler and Breiman \\cite{cb1994}. The present paper is an\nextended version of~\\cite{MTNS2004}.\n\n\\section{Preliminaries and problem statement}\\label{sec:problem}\n\nThe NMF is a long standing problem in linear algebra \\cite{haz,\nphs}. It can be stated as follows. Given $V \\in \\mathbb{R}_+^{m\\times n}$,\nand $1 \\le k \\le \\min\\{m,n\\}$, find a pair of matrices\n$W\\in\\mathbb{R}_+^{m\\times k}$ and $H\\in\\mathbb{R}_+^{k\\times n}$ such that $V =\nWH$.  The smallest $k$ for which a factorization exists is called\nthe positive rank of $V$, denoted ${\\rm prank}(V)$. This\ndefinition implies that ${\\rm rank}(V) \\le {\\rm prank}(V) \\le\n\\min\\{m,n\\}$. It is well known that ${\\rm prank}(V)$ can assume\nall intermediate values, depending on $V$. Examples for which\nnonnegative factorizations do not exist, and examples for which\nfactorization is possible only for $k > {\\rm rank}(V)$ have been\nconstructed in the literature \\cite{haz}. The ${\\rm prank}$ has\nbeen characterized only for special classes of matrices \\cite{phs}\nand algorithms for the construction of a NMF of a general positive\nmatrix are not known.\n\nThe \\emph{approximate} NMF has been recently introduced in\n\\cite{leeseung1999} independently from the exact NMF problem. The\nset-up is the same, but instead of exact factorization it is\nrequired that $V \\approx WH$ in an appropriate sense. In\n\\cite{leeseung1999}, and in this paper, the approximation is to be\nunderstood in the sense of minimum I-divergence. For two\nnonnegative numbers $p$ and $q$ the I-divergence is defined as\n\\begin{equation*}\nD(p||q)= p\\log\\frac{p}{q}-p+q,\n\\end{equation*}\nwith the conventions $0\/0=0$, $0\\log 0=0$ and $p\/0=\\infty$ for\n$p>0$. From the inequality $x\\log x\\geq x-1$ it follows that\n$D(p||q)\\geq 0$ with equality iff $p=q$. For two nonnegative\nmatrices $M=(M_{ij})$ and $N=(N_{ij})$, of the same size, the\nI-divergence is defined as\n\\begin{equation*}\nD(M||N)= \\sum_{ij}D(M_{ij}||N_{ij}).\n\\end{equation*}\nAgain it follows that $D(M||N)\\geq 0$ with equality iff $M=N$. For\nnonnegative vectors or tensors of the same size a similar\ndefinition applies.\n\n\\noindent The problem of approximate NMF is to find for given $V$\nand a fixed number $k$ (often referred to as the {\\em inner size}\nof the factorization)\n\\begin{equation}\\label{eq:dmin}\n\\arg \\min_{W,H}D(V||WH).\n\\end{equation}\nThe function $D: (W,H) \\to D(V||WH)$ will sometimes be referred to\nas the {\\em objective function}. The {\\em domain} of $D$ is the\nset of pairs $(W,H)$ with nonnegative entries. The {\\em interior}\nof the domain is the subset of pairs $(W,H)$ with positive ($>0$)\nentries, whereas pairs on the {\\em boundary} have at least one\nentry equal to zero.\n\nAlthough the objective function $(W,H)\\mapsto D(V||WH)$ is easily\nseen to be convex in $W$ and $H$ separately, it is not jointly\nconvex in the two variables. Hence $(W,H)\\mapsto D(V||WH)$ may\nhave several (local) minima and saddle points, that may prevent\nnumerical minimization algorithms to converge to the global\nminimizer. However $D(V||WH)$ cannot have a local maximum in an\ninterior point $(W_0,H_0)$, because then also $W\\mapsto\nD(V||WH_0)$ would have a local maximum in $W_0$, which contradicts\nconvexity. Local maxima at the boundary are not a priori excluded.\n\n\\medskip\nIt is not immediately obvious that the approximate NMF problem\nadmits a solution. The following result is therefore relevant.\n\\begin{prop}\\label{prop:exist}\nThe minimization problem~(\\ref{eq:dmin}) has a solution.\n\\end{prop}\nThe proof of this proposition is deferred to\nsection~\\ref{section:altmin}.\n\n\\medskip \\noindent Notice that, increasing the inner size from $k$ to\n$k+1$, the optimal value of the objective function decreases. This\nfollows from the fact that one can trivially embed the\nfactorization problem with inner size $k$ into the problem with\ninner size $k+1$ simply adding a zero last column to the optimal\n$W$ and an arbitrary last row to the optimal $H$ of the problem\nwith inner size $k$. Unfortunately, unlike the SVD of a matrix,\nthe best approximations with increasing $k$ are not embedded one\ninto another. For increasing $k$ the computations are to be\ncarried out anew.\n\nAlthough, according to proposition~\\ref{prop:exist}, a solution to\nthe minimization problem exists, it will certainly not be unique.\nIn order to rule out too many trivial multiple solutions, we\nimpose the condition that $H$ is row stochastic, so\n$\\sum_jH_{lj}=1$ for all $l$. This is not a restriction. Indeed,\nfirst we exclude without loss of generality the case where $H$ has\none or more zero rows, since we would then in fact try to minimize\nthe I-divergence with inner size smaller than $k$. Let $h$ be the\ndiagonal matrix with elements $h_i=\\sum_j H_{ij}$, then\n$WH=\\tilde{W}\\tilde{H}$ with $\\tilde{W}=Wh$, $\\tilde{H}=h^{-1}H$\nand $\\tilde{H}$ is by construction row stochastic. The convention\nthat $H$ is row stochastic still does not rule out non-uniqueness.\nThink e.g.\\ of post-multiplying $W$ with a permutation matrix\n$\\Pi$ and pre-multiplying $H$ with $\\Pi^{-1}$.\n\n\n\nLet $e_n$ ($e_n^\\top$) be the column (row) vector of size $n$\nwhose elements are all equal to one. Given $k$, the (constrained)\nproblem we will look at from now on is\n\\begin{equation}\\label{maxc}\n\\min_{W,H: He_m=e_k} D(V||WH).\n\\end{equation}\nFor the sake of brevity we will often write $e$ for a vector of\n$1$'s of generic size. The constraint in the previous problem will\nthen read as $He=e$.\n\n\\medskip\nTo carry out the minimization numerically, Lee and\nSeung~\\cite{leeseung1999,leeseung2001} proposed the following\niterative algorithm. Denoting by $W^t$ and $H^t$ the matrices at\nstep $t$, the update equations are\n\\begin{align}\nW^{t+1}_{il} & =  W^t_{il} \\sum_j\n\\frac{H^t_{lj}V_{ij}}{(W^tH^t)_{ij}}\\label{eq:wbar2}\\\\\nH^{t+1}_{lj} & =  H^t_{lj} \\sum_i\n\\frac{W^t_{il}V_{ij}}{(W^tH^t)_{ij}} \\Big\/ \\sum_{ij}\n\\frac{W^t_{il}H^t_{lj}V_{ij}}{(W^tH^t)_{ij}}.\\label{eq:hbar2}\n\\end{align}\nThe initial condition $(W^0, H^0)$ will always be assumed to be in\nthe interior of the domain. Only a partial justification for this\nalgorithm is given in \\cite{leeseung2001}, although the update\nsteps~(\\ref{eq:wbar2}) and~(\\ref{eq:hbar2}) are like those in the\nEM algorithm, known from statistics, see~\\cite{em}. Likewise the\nconvergence properties of the algorithm are unclear. In the next\nsection the minimization problem will be cast in a different way\nto provide more insight in the specific form of the update\nequations and on the convergence properties of the algorithm.\n\n\\medskip\nWe will now show that the $V$ matrix in the approximate NMF\nproblem can always be taken as a probability matrix $P$ i.e. such\nthat $P_{ij}\\ge 0, \\sum_{ij}P_{ij}=1$. This will pave the way for\nthe probabilistic interpretation of the exact and approximate NMF\nproblems to be given later.\n\nLet $P =\\frac{1}{e^\\top Ve}V$, $Q_- =\\frac{1}{e^\\top We}W$,\n$w=e^\\top We$ and $Q_+ = H$. Notice that $e^\\top Pe=e^\\top Q_-e=1$\nand $Q_+e=e$. Using the definition of divergence and elementary\ncomputations, we obtain the decomposition\n\\[\nD(V||WH)=e^\\top Ve\\, D(P||Q_-Q_+)+ D(e^\\top Ve||w).\n\\]\nHence, since the number $e^\\top V e$ is known, minimizing\n$D(V||WH)$ w.r.t. $(W,H)$ is equivalent to minimizing\n$D(P||Q_-Q_+)$ w.r.t.\\ $(Q_-, Q_+)$ and $D(e^\\top Ve||w)$ w.r.t.\\\n$w$. The minimizers of the three problems satisfy the relations\n$W^*=e^\\top Ve\\, Q_-^*$, $H^* = Q_+^*$, and $w^*=e^\\top Ve$.\nMinimizing $D(V||WH)$ is therefore equivalent to minimizing\n$D(P||Q_-Q_+)$. This enables us to give the problem a\nprobabilistic interpretation. Indeed,\n\\begin{equation}\\label{eq:divnew}\nD(P||Q_-Q_+)=\\sum_{ij} D(P_{ij}||(Q_-Q_+)_{ij}) =\n\\sum_{ij}P_{ij}\\log\\frac{P_{ij}}{(Q_-Q_+)_{ij}},\n\\end{equation}\nwhich is the usual I-divergence (Kullback-Leibler distance)\nbetween (finite) probability measures. This will be exploited in\nlater sections.  From now on we will always consider the following\nproblem. Given the probability matrix $P$ and the integer $k$ find\n\\[\n\\min_{Q_-, Q_+ : Q_+e=e} D(P||Q_-Q_+).\n\\]\nFor typographical reasons we often, but not always, denote the\nentries of $P$ by $P(ij)$ instead of $P_{ij}$ and likewise for\nother matrices. \\medskip\\\\\nThe minimization algorithm is easily seen to be {\\em invariant\nunder the previous normalizations}. Let $Q_-^t=\\frac{W^t}{e^\\top\nW^te}$ and $Q_-^t=H^t$. Substitute the definitions of $(P, Q_-^t,\nQ_+^t)$ into~(\\ref{eq:wbar2}) and~(\\ref{eq:hbar2}) and use the\neasily verified fact that $e^\\top W^te=e^\\top Ve$ for $t\\geq 1$ to\nobtain the update equations in the new notations\n\\begin{align}\nQ_-^{t+1}(il) & = Q_-^t(il) \\sum_j\n\\frac{Q_+^t(lj)P(ij)}{(Q_-^tQ_+^t)(ij)}\\label{eq:q-bar2}\\\\\nQ_+^{t+1}(lj) & = Q_+^t(lj) \\sum_i\n\\frac{Q_-^t(il)P(ij)}{(Q_-^tQ_+^t)(ij)} \\Big\/ \\sum_{ij}\n\\frac{Q_-^t(il)Q_+^t(lj)P(ij)}{(Q_-^tQ_+^t)(ij)}.\\label{eq:q+bar2}\n\\end{align}\n\\rm\n\n\\section{Lifted version of the problem}\\label{section:lift}\n\nIn this section we lift the I-divergence minimization problem to\nan equivalent minimization problem where the `matrices' (we should\nspeak of {\\em tensors}) have three indices.\n\n\\subsection{Setup}\nLet be given a probability matrix $P$ (i.e. $P(ij) \\ge 0, \\,\\,\n\\sum_{ij}P(ij)=1$) and an integer $k\\le \\min\\{m,n\\}$. We introduce\nthe following sets\n\\begin{align*}\n\\mbox{{\\boldmath $\\mathcal{P}$}}  = & \\left\\{\\mathbf{P}\\in\\mathbb{R}^{m\\times k\\times n}_+ \\,:\n\\,\\, \\sum_l\\mathbf{P}(ilj)=P(ij)\\right\\}, \\\\ \\\\\n\\mbox{{\\boldmath $\\mathcal{Q}$}}  = & \\big\\{\\mathbf{Q}\\in\\mathbb{R}^{m\\times k\\times n}_+ \\, :\n\\,\\, \\mathbf{Q}(ilj)=Q_-(il)Q_+(lj),  \\big.\\\\  & \\,\\, \\big.\n\\qquad\\qquad\\qquad\\qquad  Q_- ,\\,\\, Q_+ \\ge 0,\\,\\,\\, Q_+e=e, \\,\\,\n e^\\top Q_-e=1 \\big\\}, \\\\ \\\\\n\\mathcal{Q}  = & \\left\\{ Q\\in \\mathbb{R}^{m\\times n}_+ \\, : \\,\\,\nQ(ij)=\\sum_l\\mathbf{Q}(ilj) \\quad {\\rm for \\,\\,  some} \\,\\,\n\\mathbf{Q}\\in \\mbox{{\\boldmath $\\mathcal{Q}$}} \\right\\}.\n\\end{align*}\n\n\\medskip\n\\noindent The interpretation of the sets $\\mbox{{\\boldmath $\\mathcal{P}$}}, \\mbox{{\\boldmath $\\mathcal{Q}$}}, \\mathcal{Q}$\nis given next.\n\n\\medskip\n\\noindent Suppose one is given random variables $(Y_-,X,Y_+)$,\ntaking values in $\\{1,\\dots ,m\\}\\times \\{1,\\dots\n,k\\}\\times\\{1,\\dots ,n\\}$. For convenience we can think of the\nr.v.'s as defined on the canonical measurable space\n$(\\Omega,\\mathcal{F})$, where $\\Omega$ is the set of all triples\n$(i,l,j)$ and $\\mathcal{F}$ is $2^\\Omega$. For $\\omega=(i,l,j)$ we\nhave the identity mapping $(Y_-,X,Y_+)(\\omega)=(i,l,j)$. If\n$\\mathbb{R}$ a given probability measure on this space, then the\ndistribution of the triple $(Y_-,X,Y_+)$ under $\\mathbb{R}$ is\ngiven by the {\\em tensor} $\\mathbf{R}$ defined by\n\\begin{equation}\\label{eq:probtens}\n\\mathbf{R}(ilj)=\\mathbb{R}(Y_-=i,X=l,Y_+=j).\n\\end{equation}\nConversely, a given tensor $\\mathbf{R}$ defines a probability\nmeasure $\\mathbb{R}$ on $(\\Omega,\\mathcal{F})$. We will use the\nnotation $D$ both for I-divergence between tensors and matrices\nand for the Kullback-Leibler divergence between probabilities. If\n$\\mathbf{P}$, $\\mathbf{Q}$ are tensors related to probability\nmeasures $\\mathbb{P}$ and $\\mathbb{Q}$ like in~(\\ref{eq:probtens})\nwe obviously have\n$D(\\mathbf{P}||\\mathbf{Q})=D(\\mathbb{P}||\\mathbb{Q})$.\n\n\\medskip \\noindent The sets $\\mbox{{\\boldmath $\\mathcal{P}$}}, \\mbox{{\\boldmath $\\mathcal{Q}$}}$ correspond to subsets of the\nset of all measures on $(\\Omega,\\mathcal{F})$. In particular $\\mbox{{\\boldmath $\\mathcal{P}$}}$\ncorresponds to the subset of all measures whose $Y=(Y_-, Y_+)$\nmarginal coincides with the given $P$, while $\\mbox{{\\boldmath $\\mathcal{Q}$}}$ corresponds to\nthe subset of measures under which $Y_-$ and $Y_+$ are\nconditionally independent given $X$. The first assertion is\nevident by the definition of $\\mbox{{\\boldmath $\\mathcal{P}$}}$. To prove the second assertion\nnotice that if\n$\\mathbb{Q}(Y_-=i,X=l,Y_+=j)=\\mathbf{Q}(ilj)=Q_-(il)Q_+(lj)$, then\nsumming over $j$ one gets $\\mathbb{Q}(Y_-=i,X=l)=Q_-(il)$ (since\n$Q_+e=e$) and similarly $\\mathbb{Q}(Y_+=j|X=l)= Q_+(lj)$. It\nfollows that $\n\\mathbb{Q}(Y_-=i,X=l,Y_+=j)=\\mathbb{Q}(Y_-=i,X=l)\\mathbb{Q}(Y_+=j|X=l)$\n which is equivalent to\n\\[\n\\mathbb{Q}(Y_-=i,Y_+=j|X=l)=\\mathbb{Q}(Y_-=i|X=l)\\mathbb{Q}(Y_+=j|X=l)\n\\]\ni.e. $Y_-, Y_+$ are conditionally independent given $X$.\n\n\\medskip\n\\noindent Finally the set $\\mathcal{Q}$ is best interpreted\nalgebraically as the set of $m\\times n$ probability matrices that\nadmit exact NMF of size $k$.\n\n\\medskip\nThe following observation (taken from~\\cite{ps}) motivates our\napproach.\n\n\\begin{lem}\\label{lemma:61}\n$P$ admits exact factorization of inner size $k$ iff\n$\\mbox{{\\boldmath $\\mathcal{P}$}}\\cap\\mbox{{\\boldmath $\\mathcal{Q}$}}\\neq\\emptyset$.\n\\end{lem}\n{\\bf Proof.} If $\\mbox{{\\boldmath $\\mathcal{P}$}}\\cap\\mbox{{\\boldmath $\\mathcal{Q}$}}\\neq\\emptyset$ then there exists a\nmatrix  $\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$ which also belongs to $\\mbox{{\\boldmath $\\mathcal{P}$}}$, therefore\n$P = Q_-Q_+$. Conversely, if we have $P=Q_-Q_+$ with inner size\n$k$, then the tensor $\\mathbf{P}$ given by\n$\\mathbf{P}(ilj)=Q_-(il)Q_+(lj)$ clearly belongs to $\\mbox{{\\boldmath $\\mathcal{P}$}}$. As in\nsection~\\ref{sec:problem} we can w.l.o.g.\\ assume that $Q_+e=e$,\nso that $\\mathbf{P}$ belongs to $\\mbox{{\\boldmath $\\mathcal{Q}$}}$ as well.~\\hfill $\\square$\n\n\\medskip\n\\noindent We are now ready to give a natural probabilistic\ninterpretation to the exact NMF problem. The probability matrix\n$P$ admits exact NMF $P=Q_-Q_+$ iff there exists at least one\nmeasure on $(\\Omega,\\mathcal{F})$ whose $Y=(Y_-,Y_+)$ marginal is\n$P$ and at the same time making $Y_-$ and $Y_+$ conditionally\nindependent given $X$.\n\n\\bigskip\n\\noindent Having shown that the exact NMF factorization $P=Q_-Q_+$\nis equivalent to $\\mbox{{\\boldmath $\\mathcal{P}$}}\\cap\\mbox{{\\boldmath $\\mathcal{Q}$}}\\neq\\emptyset$ it is not surprising\nthat the approximate NMF, corresponding to $\\mbox{{\\boldmath $\\mathcal{P}$}}\\cap\\mbox{{\\boldmath $\\mathcal{Q}$}}\n=\\emptyset$, can be viewed as a double minimization over the sets\n$\\mbox{{\\boldmath $\\mathcal{P}$}}$ and $\\mbox{{\\boldmath $\\mathcal{Q}$}}$.\n\\begin{prop}\\label{prop:pqq}\nLet $P$ be given. The function $(\\mathbf{P},\\mathbf{Q})\\mapsto\nD(\\mathbf{P}||\\mathbf{Q})$ attains a minimum on $\\mbox{{\\boldmath $\\mathcal{P}$}}\\times \\mbox{{\\boldmath $\\mathcal{Q}$}}$\nand it holds that\n\\begin{equation*}\n\\min_{Q\\in\\mathcal{Q}}D(P||Q)=\\min_{\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}},\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}}D(\\mathbf{P}||\\mathbf{Q}).\n\\end{equation*}\n\\end{prop}\n\n\\noindent The proof will be given in\nsubsection~\\ref{subsection:pm}.\n\n\\begin{remark}\\label{remark:lessk}\n{\\em Let $\\mathbf{P}^*$ and $\\mathbf{Q}^*$ be the minimizing\nelements in proposition~\\ref{prop:pqq}. If there is $l_0$ such\nthat $\\sum_{ij}\\mathbf{P}^*(il_0 j)=0$, then all\n$\\mathbf{Q}^*(il_0j)$ are zero as well. Similarly, if there is\n$l_0$ such that $\\sum_{ij}\\mathbf{Q}^*(i l_0 j)=0$, then all\n$\\mathbf{P}^*(il_0j)$ are zero as well. In each (and hence both)\nof these cases the optimal approximate factorization $Q^*_-Q^*_+$\nof $P$ is of inner size less than $k$ (delete the column\ncorresponding to $l_0$ from $Q^*_-$ and the corresponding row of\n$Q^*_+$).}\n\\end{remark}\n\n\\subsection{Two partial minimization\nproblems}\\label{subsection:pm}\n\nIn the next section we will construct the algorithm for the\nsolution of the double minimization problem\n$$\n\\min_{\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}},\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}}D(\\mathbf{P}||\\mathbf{Q}),\n$$\nof proposition~\\ref{prop:pqq}, as an alternating minimization\nalgorithm over the two sets $\\mbox{{\\boldmath $\\mathcal{P}$}}$ and $\\mbox{{\\boldmath $\\mathcal{Q}$}}$. This motivates us to\nconsider here two partial minimization problems. In the first one,\ngiven $\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$ we minimize the I-divergence\n$D(\\mathbf{P}||\\mathbf{Q})$ over $\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}}$. In the second\nproblem, given $\\mathbf{P} \\in\\mbox{{\\boldmath $\\mathcal{P}$}}$ we minimize the I-divergence\n$D(\\mathbf{P}||\\mathbf{Q})$ over $\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$.\n\n\\medskip\nLet us start with the first problem. The unique solution\n$\\mathbf{P}^*=\\mathbf{P}^*(\\mathbf{Q})$ can easily be computed\nanalytically and is given by\n\\begin{equation}\\label{eq:p*}\n\\mathbf{P}^*(ilj)=\\frac{\\mathbf{Q}(ilj)}{Q(ij)}\\, P(ij),\n\\end{equation}\nwhere $Q(ij)=\\sum_l\\mathbf{Q}(ilj)$. We also adopt the convention\nto put $\\mathbf{P}^*(ilj)=0$ if $Q(ij)=0$, which ensures that,\nviewed as measures, $\\mathbf{P}^*\\ll\\mathbf{Q}$.\n\n\\medskip\nNow we turn to the second partial minimization problem. The unique\nsolution $\\mathbf{Q}^*=\\mathbf{Q}^*(\\mathbf{P})$ to this problem\ncan also be easily computed analytically and is given by\n\\begin{align}\nQ^*_-(il) & = \\sum_j\\mathbf{P}(ilj)\\label{eq:q-} \\\\\nQ^*_+(lj) & = \\frac{\\sum_i\n\\mathbf{P}(ilj)}{\\sum_{ij}\\mathbf{P}(ilj)},\\label{eq:q+}\n\\end{align}\nwhere we assign arbitrary values to the $Q^*_+(lj)$ (complying\nwith the constraint $Q_+e=e$) for those $l$ with\n$\\sum_{ij}\\mathbf{P}(ilj)=0$.\n\n\\bigskip \\noindent The two partial minimization problems and their\nsolutions have a nice probabilistic interpretation.\n\n\\medskip \\noindent In the first minimization problem, one is given a distribution\n$\\mathbf{Q}$, which makes the pair $Y=(Y_-, Y_+)$ conditionally\nindependent given $X$, and finds the best approximation to it in\nthe set $\\mbox{{\\boldmath $\\mathcal{P}$}}$ of distributions with the marginal of $Y$ given by\n$P$. Let $\\mathbf{P}^*$ denote the optimal distribution of\n$(Y_-,X,Y_+)$. Equation~(\\ref{eq:p*}) can then be interpreted, in\nterms of the corresponding measures, as\n$$\n\\mathbb{P}^*(Y_-=i,X=l,Y_+=j) = \\mathbb{Q}(X=l|Y_-=i,Y_+=j)P(ij).\n$$\nNotice that the conditional distributions of $X$ given $Y$ under\n$\\mathbb{P}^*$ and $\\mathbb{Q}$ are the same. We will see below\nthat this is not a coincidence.\n\n\\medskip\n\\noindent In the second minimization problem, one is given a\ndistribution $\\mathbf{P}$, with the marginal of $Y$ given by $P$\nand finds the best approximation to it in the set $\\mbox{{\\boldmath $\\mathcal{Q}$}}$ of\ndistributions which make $Y=(Y_-, Y_+)$ conditionally independent\ngiven $X$. Let $\\mathbf{Q}^*$ denote the optimal distribution of\n$(Y_-,X,Y_+)$. Equations~(\\ref{eq:q-}) and~(\\ref{eq:q+}) can then\nbe interpreted, in terms of the corresponding measures, as\n\\[\n\\mathbb{Q}^*(Y_-=i,X=l)=\\mathbb{P}(Y_-=i,X=l)\n\\]\nand\n\\[\n\\mathbb{Q}^*(Y_+=j|X=l)=\\mathbb{P}(Y_+=j|X=l).\n\\]\nWe see that the optimal solution $\\mathbb{Q}^*$ is such that the\nmarginal distributions of $(X,Y_-)$ under $\\mathbb{P}$ and\n$\\mathbb{Q}^*$ coincide as well as the conditional distributions\nof $Y_+$ given $X$ under $\\mathbb{P}$ and $\\mathbb{Q}^*$. Again,\nthis is not a coincidence, as we will explain below.\n\n\\medskip\n\\begin{remark}\n{\\em As a side remark we notice that the minimization of\n$D(\\mathbf{Q}||\\mathbf{P})$ over $\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}}$ for a given\n$\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$ yields the same solution $\\mathbf{P}^*$. A\nsimilar result does not hold for the second minimization problem.\nThis remark is not relevant for what follows.}\n\\end{remark}\n\n\n\\noindent We can now state the so called {\\em Pythagorean rules}\nfor the two partial minimization problems. This terminology was\nintroduced by Csisz\\'ar \\cite{c1975}.\n\\begin{lem}\\label{lemma:pyth}\nFor fixed $\\mathbf{Q}$ and $\\mathbf{P}^*=\\mathbf{P}^*(\\mathbf{Q})$\nit holds that, for any $\\mathbf{P} \\in\\mbox{{\\boldmath $\\mathcal{P}$}}$,\n\\begin{equation}\\label{eq:pythp}\nD(\\mathbf{P}||\\mathbf{Q})=D(\\mathbf{P}||\\mathbf{P}^*)+D(\\mathbf{P}^*||\\mathbf{Q}),\n\\end{equation}\nmoreover\n\\begin{equation}\\label{eq:p0q0}\nD(\\mathbf{P}^*||\\mathbf{Q})=D(P||Q),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:qq}\nQ(ij)=\\sum_l\\mathbf{Q}(ilj).\n\\end{equation}\nFor fixed $\\mathbf{P}$ and $\\mathbf{Q}^*=\\mathbf{Q}^*(\\mathbf{P})$\nit holds that, for any $\\mathbf{Q} \\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$,\n\\begin{equation}\\label{eq:pythq}\nD(\\mathbf{P}||\\mathbf{Q})=D(\\mathbf{P}||\\mathbf{Q}^*)+D(\\mathbf{Q}^*||\\mathbf{Q}).\n\\end{equation}\n\\end{lem}\n{\\bf Proof.} To prove the first rule we compute\n\\begin{eqnarray*}\n\\lefteqn{D(\\mathbf{P}||\\mathbf{P}^*)+ D(\\mathbf{P}^*||\\mathbf{Q})}\\\\\n& = &\n\\sum_{ilj}\\mathbf{P}(ilj)\\log\\frac{\\mathbf{P}(ilj)Q(ij)}{\\mathbf{Q}(ilj)P(ij)}\n+\n\\sum_{ilj}\\mathbf{Q}(ilj)\\frac{P(ij)}{Q(ij)}\\log \\frac{P(ij)}{Q(ij)} \\\\\n& = &\n\\sum_{ilj}\\mathbf{P}(ilj)\\log\\frac{\\mathbf{P}(ilj)}{\\mathbf{Q}(ilj)}\n+\n\\sum_{ilj}\\mathbf{P}(ilj)\\log\\frac{Q(ij)}{P(ij)} \\\\\n& & \\mbox{} + \\sum_{ij}Q(ij)\\frac{P(ij)}{Q(ij)}\\log\n\\frac{P(ij)}{Q(ij)} = D(\\mathbf{P}||\\mathbf{Q}).\n\\end{eqnarray*}\nThe first rule follows. To prove the relation~(\\ref{eq:p0q0})\ninsert equation~(\\ref{eq:p*}) into $D(\\mathbf{P}^*||\\mathbf{Q})$\nand sum over $l$ to get\n\\[\nD(\\mathbf{P}^*||\\mathbf{Q})=\\sum_{ilj}P(ij)\\frac{\\mathbf{Q}(ilj)}{Q(ij)}\\log\n\\frac{P(ij)}{Q(ij)} = D(P||Q).\n\\]\n\n\\noindent To prove the second rule we first introduce some\nnotation. Let $\\mathbf{P}(il\\cdot)=\\sum_j \\mathbf{P}(ilj)$,\n$\\mathbf{P}(\\cdot lj)=\\sum_i\\mathbf{P}(ilj)$ and\n$\\mathbf{P}(j|l)=\\mathbf{P}(\\cdot lj)\/\\sum_j\\mathbf{P}(\\cdot lj)$.\nFor $\\mathbf{Q}$ we use similar notation and observe that\n$\\mathbf{Q}(il\\cdot)=Q_-(il)$, and\n$\\mathbf{Q}(j|l)=Q_+(lj)\/\\sum_jQ_+(lj)$, and\n$Q^*_-(il)=\\mathbf{P}(il\\cdot)$ and $Q^*_+(lj)=\\mathbf{P}(j|l)$.\nWe now compute\n\\begin{align*}\nD(\\mathbf{P}||\\mathbf{Q})-D(\\mathbf{P}||\\mathbf{Q}^*)\n&=\\sum_{ilj}\\mathbf{P}(ilj) \\left( \\log\n\\frac{\\mathbf{P}(il\\cdot)}{Q_-(il)}+\\log\\frac{\\mathbf{P}(j|l)}{Q_+(lj)} \\right)\\\\\n& = \\sum_{il}\\mathbf{P}(il\\cdot)\\log\n\\frac{\\mathbf{P}(il\\cdot)}{Q_-(il)} + \\sum_{lj}\\mathbf{P}(\\cdot\nlj)\\log\\frac{\\mathbf{P}(j|l)}{Q_+(lj)} \\\\\n& = D(\\mathbf{Q}^*||\\mathbf{Q}).\n\\end{align*}\nThe second rule follows. \\hfill$\\square$\n\n\\bigskip \\noindent With the aid of the relation~(\\ref{eq:p0q0}) we can now\nprove\nproposition~\\ref{prop:pqq}. \\medskip\\\\\n{\\bf Proof of proposition~\\ref{prop:pqq}.} With\n$\\mathbf{P}^*=\\mathbf{P}^*(\\mathbf{Q})$, the optimal solution of\nthe partial minimization over $\\mbox{{\\boldmath $\\mathcal{P}$}}$, we have\n\\begin{align*}\nD(\\mathbf{P}||\\mathbf{Q})& \\geq D(\\mathbf{P}^*||\\mathbf{Q}) \\\\\n& = D(P||Q) \\\\\n& \\geq  \\min_{Q\\in\\mathcal{Q}}D(P||Q).\n\\end{align*}\nIt follows that\n$\\inf_{\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}},\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}}D(\\mathbf{P}||\\mathbf{Q})\\geq\\min_{Q\\in\\mathcal{Q}}D(P||Q)$.\n\\\\\nConversely, let $\\mathbf{Q}$  in $\\mbox{{\\boldmath $\\mathcal{Q}$}}$ be given and let $Q$ be\ndefined by $Q(ij)=\\sum_l\\mathbf{Q}(ilj)$ . From\n\\begin{align*}\nD(P||Q) & = D(\\mathbf{P}^*(\\mathbf{Q})|| \\mathbf{Q})\\\\\n& \\geq\n\\inf_{\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}},\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}}D(\\mathbf{P}||\\mathbf{Q}),\n\\end{align*}\nwe obtain\n\\[\n\\min_{Q\\in\\mathcal{Q}}D(P||Q)\\geq\\inf_{\\mathbf{P}\\in\\mbox{{\\boldmath $\\mathcal{P}$}},\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}}D(\\mathbf{P}||\\mathbf{Q}).\n\\]\nFinally we show that we can replace the infima by minima. Let\n$Q^*_-$ and $Q^*_+$ be such that $(Q_-,Q^+)\\mapsto D(P||Q_-Q^+)$\nis minimized (their existence is guaranteed by\nproposition~\\ref{prop:exist}). Let $\\mathbf{Q}^*$ be a\ncorresponding element in $\\mbox{{\\boldmath $\\mathcal{Q}$}}$ and\n$\\mathbf{P}^*=\\mathbf{P}^*(\\mathbf{Q}^*)$. Then\n$D(\\mathbf{P}^*||\\mathbf{Q}^*)=D(P||Q^*_-Q^*_+)$ and the result\nfollows.\n\\hfill$\\square$\n\n\\bigskip\nFor a probabilistic derivation of the solutions of the two partial\nminimization problems and of their corresponding Py\\-tha\\-go\\-rean\nrules, we use a general result (lemma~\\ref{lemma:uv} below) on the\nI-divergence between two joint laws of any random vector $(U,V)$.\nWe denote the law of $(U,V)$ under arbitrary probability measures\n$\\mathbb{P}$ and $\\mathbb{Q}$ by $\\mathbb{P}^{U,V}$ and\n$\\mathbb{Q}^{U,V}$. The conditional distributions of $U$ given $V$\nare summarized by the matrices $\\mathbb{P}^{U|V}$ and\n$\\mathbb{Q}^{U|V}$, with the obvious convention\n$\\mathbb{P}^{U|V}(ij)=\\mathbb{P}(U=j|V=i)$ and likewise for\n$\\mathbb{Q}^{U|V}$.\n\\begin{lem}\\label{lemma:uv}\nIt\nholds that\n\\begin{equation}\\label{eq:duv}\nD(\\mathbb{P}^{U,V}||\\mathbb{Q}^{U,V})=\\mathbb{E}_\\mathbb{P}\nD(\\mathbb{P}^{U|V}||\\mathbb{Q}^{U|V}) +\nD(\\mathbb{P}^V||\\mathbb{Q}^V),\n\\end{equation}\nwhere\n\\[\nD(\\mathbb{P}^{U|V}||\\mathbb{Q}^{U|V}) = \\sum_j\nP(U=j|V)\\log\\frac{P(U=j|V)}{Q(U=j|V)}.\n\\]\nIf moreover $V=(V_1,V_2)$, and $U, V_2$ are conditionally\nindependent given $V_1$ under $\\mathbb{Q}$, then the first term on\nthe RHS of~(\\ref{eq:duv}) can be written as\n\\begin{equation}\\label{eq:duv1} \\mathbb{E}_\\mathbb{P}\nD(\\mathbb{P}^{U|V}||\\mathbb{Q}^{U|V}) =\\mathbb{E}_\\mathbb{P}\nD(\\mathbb{P}^{U|V}||\\mathbb{P}^{U|V_1}) + \\mathbb{E}_\\mathbb{P}\nD(\\mathbb{P}^{U|V_1}||\\mathbb{Q}^{U|V_1}).\n\\end{equation}\n\\end{lem}\n{\\bf Proof.} It follows from elementary manipulations.\n\\hfill $\\square$\\medskip\\\\\nThe first minimization problem can be solved probabilistically as\nfollows. Given $\\mathbf{Q}$ we are to find its best\napproximation within $\\mbox{{\\boldmath $\\mathcal{P}$}}$. Let $\\mathbb{Q}$ correspond to the\ngiven $\\mathbf{Q}$ and $\\mathbb{P}$ correspond to the generic\n$\\mathbf{P} \\in \\mbox{{\\boldmath $\\mathcal{P}$}}$. Choosing $U=X$, $V=Y=(Y_-,Y_+)$ in lemma\n\\ref{lemma:uv}, and remembering that $\\mathbb{P}^Y$ is determined\nby $P$ for all $\\mathbf{P} \\in \\mbox{{\\boldmath $\\mathcal{P}$}}$, equation (\\ref{eq:duv}) now\nreads\n\\begin{equation}\\label{eq:pq1}\nD(\\mathbf{P}||\\mathbf{Q})=\\mathbb{E}_{\\mathbb{P}}\nD(\\mathbb{P}^{X|Y}||\\mathbb{Q}^{X|Y}) + D(P||Q),\n\\end{equation}\nwhere the matrix $Q$ is as in~(\\ref{eq:qq}).\nThe problem is equivalent to the minimization\nof $\\mathbb{E}_{\\mathbb{P}}D(\\mathbb{P}^{X|Y}||\\mathbb{Q}^{X|Y})$\nw.r.t. $\\mathbf{P} \\in \\mbox{{\\boldmath $\\mathcal{P}$}}$, which is attained (with value $0$) at\n$\\mathbb{P}^*$ with $\\mathbb{P}^{*\\, X|Y}=\\mathbb{Q}^{X|Y}$ and\n$\\mathbb{P}^{*Y}= P$. To derive probabilistically the\ncorresponding Pythagorean rule, we apply~(\\ref{eq:duv}) with\n$\\mathbb{P}^*$ instead of $\\mathbb{Q}$. We obtain, using\n$\\mathbb{P}^Y=\\mathbb{P}^{*Y}$,\n\\begin{equation}\\label{eq:pq8}\nD(\\mathbb{P}^{X,Y}||\\mathbb{P}^{*^{X,Y}})=\n\\mathbb{E}_\\mathbb{P}D(\\mathbb{P}^{X|Y}||\\mathbb{P}^{*^{X|Y}}).\n\\end{equation}\nSince also\n\\begin{equation}\\label{eq:pq2}\n\\mathbb{E}_\\mathbb{P}D(\\mathbb{P}^{X|Y}||\\mathbb{Q}^{X|Y})=\\mathbb{E}_\\mathbb{P}\nD(\\mathbb{P}^{X|Y}||\\mathbb{P}^{*^{X|Y}}),\n\\end{equation}\nwe combine equations~(\\ref{eq:pq8}) and~(\\ref{eq:pq2}) and insert\nthe result into~(\\ref{eq:pq1}). Recognizing the fact that\n$D(\\mathbf{P}||\\mathbf{P}^*)=D(\\mathbb{P}^{X,Y}||\\mathbb{P}^{*^{X,Y}})$,\nand using $D(\\mathbf{P}^*||\\mathbf{Q})=D(P||Q)$\naccording to (\\ref{eq:p0q0}), we then identify~(\\ref{eq:pq1}) as\nthe first Pythagorean rule~(\\ref{eq:pythp}).\n\\medskip\\\\\nThe treatment of the second minimization problem follows a similar\npattern. Given $\\mathbf{P}$ we are to find its best\napproximation within $\\mbox{{\\boldmath $\\mathcal{Q}$}}$.  Let $\\mathbb{P}$ correspond to the\ngiven $\\mathbf{P}$ and $\\mathbb{Q}$ correspond to the generic\n$\\mathbf{Q} \\in \\mbox{{\\boldmath $\\mathcal{Q}$}}$. Choosing $U=Y_+$, $V_1=X$ and $V_2=Y_-$ in\nlemma \\ref{lemma:uv}, and remembering that under any\n$\\mathbf{Q}\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$ the r.v. $Y_-, Y_+$ are conditionally\nindependent given $X$, equation (\\ref{eq:duv}) refined with\n(\\ref{eq:duv1}) now reads\n\\begin{align*}\nD(\\mathbf{P}||\\mathbf{Q})= &\n\\mathbb{E}_\\mathbb{P}D(\\mathbb{P}^{Y_+|X,Y_-}||\\mathbb{P}^{Y_+|X})\\\\\n& \\mbox{}\n+\\mathbb{E}_\\mathbb{P}D(\\mathbb{P}^{Y_+|X}||\\mathbb{Q}^{Y_+|X})+D(\\mathbb{P}^{Y_-,X}||\\mathbb{Q}^{Y_-,X}).\n\\end{align*}\nThe problem is equivalent to the minimizations of the second and\nthird I-divergences on the RHS w.r.t. $\\mathbf{Q} \\in \\mbox{{\\boldmath $\\mathcal{Q}$}}$, which\nare attained (both with value $0$) at $\\mathbb{Q}^*$ with\n$\\mathbb{Q}^{*\\, Y_+|X}=\\mathbb{P}^{Y_+|X}$ and\n$\\mathbb{Q}^{*Y_-,X}= \\mathbb{P}^{Y_-,X}$. Note that $X$ has the\nsame distribution under $\\mathbb{P}$ and $\\mathbb{Q}^*$. To derive\nprobabilistically the corresponding Pythagorean rule we notice\nthat\n\\begin{equation}\\label{eq:pq3}\nD(\\mathbf{P}||\\mathbf{Q})-D(\\mathbf{P}||\\mathbf{Q}^*) =\n\\mathbb{E}_{\\mathbb{Q}^*}D(\\mathbb{Q}^*{^{Y_+|X}}||\\mathbb{Q}^{Y_+|X})+D(\\mathbb{Q}^{*^{Y_-,X}}||\\mathbb{Q}^{Y_-,X}).\n\\end{equation}\nIn the right hand side of~(\\ref{eq:pq3}) we can, by conditional\nindependence, replace\n$\\mathbb{E}_{\\mathbb{Q}^*}D(\\mathbb{Q}^*{^{Y_+|X}}||\\mathbb{Q}^{Y_+|X})$\nwith\n$\\mathbb{E}_{\\mathbb{Q}^*}D(\\mathbb{Q}^*{^{Y_+|X,Y_-}}||\\mathbb{Q}^{Y_+|X,Y-})$.\nBy yet another application of~(\\ref{eq:duv}), we thus see that\n$D(\\mathbf{P}||\\mathbf{Q})-D(\\mathbf{P}||\\mathbf{Q}^*)=D(\\mathbf{Q}^*||\\mathbf{Q})$,\nwhich is the second Pythagorean rule~(\\ref{eq:pythq}).\n\n\\section{Alternating minimization algorithm}\\label{section:altmin}\n\nThe results of the previous section are aimed at setting up an\nalternating minimization algorithm for obtaining $\\min_Q D(P||Q)$,\nwhere $P$ is a given nonnegative matrix. In view of\nproposition~\\ref{prop:pqq} we can lift this problem to the\n$\\mbox{{\\boldmath $\\mathcal{P}$}}\\times \\mbox{{\\boldmath $\\mathcal{Q}$}}$ space. Starting with an arbitrary $\\mathbf{Q}^0\n\\in\\mbox{{\\boldmath $\\mathcal{Q}$}}$ with positive elements, we adopt the following alternating\nminimization scheme\n\\begin{equation}\\label{eq:altalgo}\n \\to \\mathbf{Q}^t \\to \\mathbf{P}^t\\to \\mathbf{Q}^{t+1}\\to \\mathbf{P}^{t+1}\n\\to\n\\end{equation}\nwhere $\\mathbf{P}^t=\\mathbf{P}^*(\\mathbf{Q}^t)$,\n$\\mathbf{Q}^{t+1}=\\mathbf{Q}^*(\\mathbf{P}^t)$.\n\\medskip\\\\\nTo relate this algorithm to the one of section~\\ref{sec:problem}\n(formulas (\\ref{eq:q-bar2}) and (\\ref{eq:q+bar2})) we combine two\nsteps of the alternating minimization at a time.\nFrom~(\\ref{eq:altalgo}) we get\n$$\\mathbf{Q}^{t+1}=\\mathbf{Q}^*(\\mathbf{P}^*(\\mathbf{Q}^t)).$$\n\\noindent Computing the optimal solutions according\nto~(\\ref{eq:p*}),~(\\ref{eq:q-}) and (\\ref{eq:q+}) one gets from\nhere the formulas (\\ref{eq:q-bar2}) and (\\ref{eq:q+bar2}) of\nsection~\\ref{sec:problem}.\n\n\\medskip\n\\noindent The Pythagorean rules allow us to easily compute the\nupdate gain $D(P||Q^t)-D(P||Q^{t+1})$ of the algorithm.\n\\begin{prop}\\label{prop:gain}\nThe update gain at each iteration of the\nalgorithm~(\\ref{eq:altalgo}) in terms of the matrices $Q^t$ is\ngiven by\n\\begin{equation}\\label{eq:gain}\nD(P||Q^{t}) - D(P||Q^{t+1}) =\nD(\\mathbf{P}^{t}||\\mathbf{P}^{t+1})+D(\\mathbf{Q}^{t+1}||\\mathbf{Q}^t).\n\\end{equation}\n\\end{prop}\n{\\bf Proof.} The two Pythagorean rules from lemma~\\ref{lemma:pyth}\nnow take the forms\n\\begin{align*}\nD(\\mathbf{P}^t||\\mathbf{Q}^t) & =\nD(\\mathbf{P}^t||\\mathbf{Q}^{t+1})+D(\\mathbf{Q}^{t+1}||\\mathbf{Q}^t),\n\\\\\nD(\\mathbf{P}^{t}||\\mathbf{Q}^{t+1}) & =\nD(\\mathbf{P}^{t}||\\mathbf{P}^{t+1})+D(\\mathbf{P}^{t+1}||\\mathbf{Q}^{t+1}).\n\\end{align*}\nAddition  of these two equations\nresults in\n\\begin{align*}\nD(\\mathbf{P}^{t}||\\mathbf{Q}^t) & =\nD(\\mathbf{P}^{t}||\\mathbf{P}^{t+1})+D(\\mathbf{P}^{t+1}||\\mathbf{Q}^{t+1})+D(\\mathbf{Q}^{t+1}||\\mathbf{Q}^t),\n\\end{align*}\nand since $D(\\mathbf{P}^{t}||\\mathbf{Q}^t) = D(P||Q^{t})$ from\n(\\ref{eq:p0q0}), the result follows. \\hfill$\\square$\n\n\n\\begin{remark}\n{\\em If one starts the algorithm with matrices $(Q^0_-, Q^0_+)$ in\nthe interior of the domain, the iterations will remain in the\ninterior. Suppose that, at step $n$, the update gain is zero.\nThen, from~(\\ref{eq:gain}), we get that\n$D(\\mathbf{Q}^{t+1}||\\mathbf{Q}^t)=0$. Hence the tensors\n$\\mathbf{Q}^{t+1}$ and $\\mathbf{Q}^t$ are identical. From this it\nfollows by summation that $Q^{t+1}_-=Q^t_-$. But then we also have\nthe equality $Q^t_-(il)Q^{t+1}_+(lj)=Q^t_-(il)Q^{t}_+(lj)$ for all\n$i,l,j$. Since all $Q^t_-(il)$ are positive, we also have\n$Q^{t+1}_+=Q^t_+$. Hence, the updating formulas strictly decrease\nthe objective function until the algorithm reaches a fixed point.}\n\\end{remark}\n\\rm\nWe close this section with the proof of\nproposition~\\ref{prop:exist} in which we use the result of\nproposition~\\ref{prop:gain}.\n\\medskip\\\\\n{\\bf Proof of proposition~\\ref{prop:exist}.} We first prove that\nthere exists a pair of matrices $(W,H)$ with $He_m=e_k$ and\n$We_k=Ve_n$ for which $D(V||WH)$ is finite. Put\n$W=\\frac{1}{k}Ve_ne_k^\\top$ and $H=\\frac{1}{e_m^\\top Ve_n}e_k\ne_m^\\top V$. Note that indeed $He_m=e_k$ and $We_k=Ve_n$ and that\nall elements of $W$ and $H$, and hence those of $WH$, are\npositive, $D(V||WH)$ is therefore finite.\n\nNext we show that we can restrict ourselves to minimization over a\ncompact set $\\mathcal{K}$ of matrices. Specifically, we will show\nthat for all positive matrices $W$ and $H$, there exist positive\nmatrices $W'$ and $H'$ with $(W',H')\\in \\mathcal{K}$ such that\n$D(V||W'H')\\leq D(V||WH)$. We choose for arbitrary $W^0$ and $H^0$\nthe matrices $W^1$ and $H^1$ according to~(\\ref{eq:wbar2})\nand~(\\ref{eq:hbar2}). It follows from proposition~\\ref{prop:gain}\nthat indeed $D(V||W^1H^1)\\leq D(V||W^0H^0)$. Moreover, it is\nimmediately clear from~(\\ref{eq:wbar2}) and~(\\ref{eq:hbar2}) that\nwe have $W^1e=Ve$ and $H^1e=e$. Hence, it is sufficient to confine\nsearch to the compact set $\\mathcal{L}$ where $He=e$ and $We=Ve$.\n\nFix a pair of indices $i,j$. Since we can compute the divergence\nelementwise we have the trivial estimate\n\\[\nD(V||WH)\\geq V_{ij}\\log \\frac{V_{ij}}{(WH)_{ij}}-V_{ij}+(WH)_{ij}.\n\\]\nSince for $V_{ij}>0$ the function $d_{ij}:x\\to V_{ij}\\log\n\\frac{V_{ij}}{x}-V_{ij}+x$ is  decreasing on $(0,V_{ij})$, we have\nfor any sufficiently small $\\varepsilon>0$ (of course $\\varepsilon < V_{ij}$)\nthat $d_{ij}(x)>d_{ij}(\\varepsilon)$ for $x\\leq \\varepsilon$ and of course\n$\\lim_{\\varepsilon\\to 0}d_{ij}(\\varepsilon)=\\infty$. Hence to find the minimum\nof $d_{ij}$, it is sufficient to look at $x\\geq \\varepsilon$. Let $\\varepsilon_0\n>0$ and such that $\\varepsilon_0 <\\min\\{V_{ij}:V_{ij}>0\\}$. Let $\\mathcal{G}$ be\nthe set of $(W,H)$ such that $(WH)_{ij}\\geq \\varepsilon_0$ for all $i,j$\nwith $V_{ij}>0$. Then $\\mathcal{G}$ is closed. Take now\n$\\mathcal{K}=\\mathcal{L}\\cap \\mathcal{G}$, then $\\mathcal{K}$ is\nthe compact set we are after. Let us observe that $\\mathcal{K}$ is\nnon-void for sufficiently small $\\varepsilon_0$. Clearly the map\n$(W,H)\\mapsto D(V||WH)$ is continuous on $\\mathcal{K}$ and thus\nattains its minimum. \\hfill\\ $\\square$\n\n\\section{Auxiliary functions}\\label{section:auxfunc}\n\nAlgorithms for recursive minimization can often be constructed by\nusing {\\em auxiliary functions}. For the problem of minimizing the\ndivergence $D(V||WH)$, some such functions can be found\nin~\\cite{leeseung2001} and they are analogous to functions that\nare used when studying the EM algorithm, see~\\cite{wu}. The choice\nof an auxiliary function is usually based on {\\em ad hoc}\nreasoning, like for instance finding a Lyapunov function for\nstudying the stability of the solutions of a differential\nequation. We show in this section that the lifted version of the\ndivergence minimization problem leads in a natural way to useful\nauxiliary functions. Let us first explain what is meant by an\nauxiliary function.\n\nSuppose one wants to minimize a function $x\\mapsto F(x)$, defined\non some domain. The function $(x,x')\\mapsto G(x,x')$ is an\nauxiliary function for $F$ if\n\\begin{align*}\nG(x,x')  & \\geq  F(x'), \\,\\,\\, \\forall x,x',\\\\\nG(x,x) & =  F(x), \\quad \\forall x.\n\\end{align*}\nIf we define (assuming that the $\\arg\\min$ below exists and is\nunique)\n\\begin{equation}\\label{eq:update}\nx'=x'(x)=\\arg\\min G(x,\\cdot),\n\\end{equation}\nthen we have\n\\[\nF(x')\\leq G(x,x') \\leq G(x,x)=F(x),\n\\]\nand hence the value of $F$ decreases by replacing $x$ with $x'$. A\nrecursive procedure to find the minimum of $F$ can be based on the\nrecipe~(\\ref{eq:update}) by taking $x=x^t$ and $x'=x^{t+1}$. To be\nuseful an auxiliary function $G$ must allow for a simple\ncomputation or characterization of $\\arg\\min G(x,\\cdot)$.\n\n\\medskip\nWe consider now the minimization of $D(P||Q)$ and its lifted\nversion, the minimization of $D(\\mathbf{P}||\\mathbf{Q})$ as in\nsection~\\ref{section:lift}. In particular, with reference to the\nalternating minimization scheme~(\\ref{eq:altalgo}), with the\nnotations of section~\\ref{section:altmin}, we know that\n$\\mathbf{Q}^{t+1}$ is found by minimizing $\\mathbf{Q}'\\mapsto\nD(\\mathbf{P}^*(\\mathbf{Q}^t)||\\mathbf{Q}')$. This strongly\nmotivates the choice of the function\n\\[ (\\mathbf{Q},\\mathbf{Q}') \\mapsto G(\\mathbf{Q},\\mathbf{Q}')=D(\\mathbf{P}^*(\\mathbf{Q})||\\mathbf{Q}')\n\\]\nas an auxiliary function for minimizing $D(P||Q)$ w.r.t. $Q$.\n\n\\medskip\nUsing the decomposition of the divergence in\nequation~(\\ref{eq:duv}) we can rewrite $G$ as\n\\begin{equation}\\label{eq:div29}\nG(\\mathbf{Q},\\mathbf{Q}')=D(\\mathbb{P}^{*^Y}||\\mathbb{Q}'^Y)+\\mathbb{E}_{\\mathbb{P}^*}D(\\mathbb{P}^{*^{X|Y}}||\\mathbb{Q}'^{X|Y}).\n\\end{equation}\nSince $\\mathbb{P}^{*X|Y} = \\mathbb{Q}^{X|Y}$, and $\\mathbb{P}^{*Y}\n= P$ we can rewrite~(\\ref{eq:div29}) as\n\\begin{equation}\\label{eq:div30}\nG(\\mathbf{Q},\\mathbf{Q}')=D(P||\\mathbb{Q}'^Y)+\\mathbb{E}_PD(\\mathbb{Q}^{X|Y}||\\mathbb{Q}'^{X|Y}).\n\\end{equation}\nFrom~(\\ref{eq:div30}) it follows that\n$G(\\mathbf{Q},\\mathbf{Q}')\\geq D(P||Q')$, and that\n$G(\\mathbf{Q},\\mathbf{Q})= D(P||Q)$, precisely the two properties\nthat define an auxiliary function for $D(P||Q)$.\n\n\\medskip\\noindent\nIn~\\cite{leeseung2001} one can find two auxiliary functions for\nthe original minimization problem $D(V||WH)$. One function is for\nminimization over $H$ with fixed $W$, the other for minimization\nover $W$ with fixed $H$. To show the connection with the function\n$G$ defined above, we first make the dependence of $G$ on\n$Q_-,Q_+,Q'_-,Q'_+$ explicit by writing\n$G(\\mathbf{Q},\\mathbf{Q}')$ as $G(Q_-,Q_+,Q'_-,Q'_+)$.\n\n\\medskip \\noindent The auxiliary function for minimization with fixed\n$Q_-$ can then be taken as \\[Q'_+\\mapsto\nG^+_{\\mathbf{Q}}(Q'_+)=G(Q_-,Q_+,Q_-,Q'_+),\\]\n\n\\noindent whereas the auxiliary function for minimization with\nfixed $Q_+$ can be taken as \\[Q'_-\\mapsto\nG^-_{\\mathbf{Q}}(Q'_-)=G(Q_-,Q_+,Q'_-,Q_+)\\]\n\n\\medskip \\noindent The functions $G^+_{\\mathbf{Q}}$ and $G^-_{\\mathbf{Q}}$\ncorrespond to the auxiliary functions in~\\cite{leeseung2001},\nwhere they are given in an explicit form, but where no rationale\nfor them is given.\n\nFor the different auxiliary functions introduced above, we will\nnow compute the update gains and compare these expressions with\n(\\ref{eq:gain}).\n\\begin{lem}\\label{lemma:auxdiff}\nConsider the auxiliary functions $G$ ,$G^-_{\\mathbf{Q}}$ ,\n$G^+_{\\mathbf{Q}}$ above. Denote by $Q'_-$ and $Q'_+$ the\nminimizers of the auxiliary functions in all three cases. The\nfollowing equalities hold\n\\begin{align}\nD(P||Q_-Q_+)-G^-_{\\mathbf{Q}}(Q'_-)  & =\nD(\\mathbb{Q}'^{Y_-,X}||\\mathbb{Q}^{Y_-,X}) \\label{eq:auxdiff1}\\\\\nD(P||Q_-Q_+)-G^+_{\\mathbf{Q}}(Q'_+)  & =\n\\mathbb{E}_{\\mathbb{P}^*}D(\\mathbb{Q}'^{Y_+|X}||\\mathbb{Q}^{Y_+|X})\\label{eq:auxdiff2}\\\\\nD(P||Q_-Q_+)-G(Q_-,Q_+,Q'_-,Q'_+) & =\nD(\\mathbb{Q}'^{Y_-,X}||\\mathbb{Q}^{Y_-,X}) \\nonumber \\\\\n&  \\,\\, \\mbox{~~~}\n+\\mathbb{E}_{\\mathbb{Q}'}D(\\mathbb{Q}'^{Y_+|X}||\\mathbb{Q}^{Y_+|X}).\n\\label{eq:auxdiff3}\n\\end{align}\n\\end{lem}\n{\\bf Proof.} We prove~(\\ref{eq:auxdiff3}) first. The other two\nfollow from this. A simple computation, valid for any $Q_-$a nd $Q_+$, yields\n\\begin{align}\n\\lefteqn{D(P||Q_-Q_+)-G(Q_-,Q_+,Q'_-,Q'_+)} \\\\\n&\n=\\sum_{ij}P(ij)\\sum_l\\frac{\\mathbf{Q}(ilj)}{Q(ij)}\\left(\\log\\frac{Q'_-(il)}{Q_-(il)}\n+\\log\\frac{Q'_+(lj)}{Q_+(lj)}\\right) \\nonumber\\\\\n& =\n\\sum_{il}\\big(\\sum_j\\frac{P(ij)\\mathbf{Q}(ilj)}{Q(ij)}\\big)\\log\\frac{Q'_-(il)}{Q_-(il)}\n +\n\\sum_{lj}\\big(\\sum_i\\frac{P(ij)\\mathbf{Q}(ilj)}{Q(ij)}\\big)\\log\\frac{Q'_+(lj)}{Q_+(lj)}\\label{eq:dg}\n\\end{align}\nNow we exploit the known formulas~(\\ref{eq:q-bar2})\nand~(\\ref{eq:q+bar2}) for the optimizing $Q'_-$ and $Q'_+$. The\nfirst term in~(\\ref{eq:dg}) becomes in view of~(\\ref{eq:q-bar2})\n(or, equivalently, in view of~(\\ref{eq:p*}) and~(\\ref{eq:q-}))\n\\[\n\\sum_{il}Q'_-(il)\\log\\frac{Q'_-(il)}{Q_-(il)},\n\\]\nwhich gives the first term on the RHS of~(\\ref{eq:auxdiff3}). Similarly, the\nsecond term in~(\\ref{eq:dg}) can be written in view\nof~(\\ref{eq:q+bar2}) as\n\\[\n\\sum_{l}\\big(\\sum_{ij}\\mathbf{Q}'(ilj)\\big)\\sum_{j}Q'_+(lj)\\log\\frac{Q'_+(lj)}{Q_+(lj)},\n\\]\nwhich yields the second term on the RHS of\nformula~(\\ref{eq:auxdiff3}). Formulas~(\\ref{eq:auxdiff1})\nand~(\\ref{eq:auxdiff2}) are obtained similarly, noticing that\noptimization of $G^+_{\\mathbf{Q}}$ and $G^-_{\\mathbf{Q}}$\nseparately yield the same $Q'_+$, respectively $Q'_-$, as those\nobtained by minimization of $G$.\n \\hfill$\\square$\n\\begin{remark}\n{\\em Notice that although for instance $G^-_{\\mathbf{Q}}(Q'_-)\\geq\nD(P||Q'_-Q'_+)$ for all $Q'_-$ and $Q'_+$, we have for the optimal\n$Q'_-$ that $G^-_{\\mathbf{Q}}(Q'_-)\\leq D(P||Q_-Q_+)$.\n}\n\\end{remark}\n\\begin{cor}\\label{cor:corgain}\nThe update gain of the\nalgorithm~(\\ref{eq:q-bar2}),~(\\ref{eq:q+bar2}) can be represented\nby\n\\begin{align}\\label{eq:corgain}\nD(P||Q^t)-D(P||Q^{t+1})  =\n& D(\\mathbb{Q}^{{t+1}^{Y_-,X}}||\\mathbb{Q}^{t^{Y_-,X}}) \\nonumber\\\\\n& +\n\\mathbb{E}_{\\mathbb{Q}^{t+1}}D(\\mathbb{Q}^{{t+1}^{Y_+|X}}||\\mathbb{Q}^{t^{Y_+|X}})\n\\nonumber\\\\\n& + \\mathbb{E}_PD(\\mathbb{Q}^{t^{X|Y}}||\\mathbb{Q}^{{t+1}^{X|Y}}).\n\\end{align}\n\\end{cor}\n{\\bf Proof.} Write\n\\begin{align*}\n\\lefteqn{D(P||Q^t)-D(P||Q^{t+1})=} \\\\ & \\mbox{~~~~}\nD(P||Q^t)-G(\\mathbf{Q}^t,\\mathbf{Q}^{t+1})+G(\\mathbf{Q}^t,\\mathbf{Q}^{t+1})-D(P||Q^{t+1})\n\\end{align*}\nand use equations~(\\ref{eq:div29}) and~(\\ref{eq:auxdiff3}).\n\\hfill$\\square$\n\\medskip\\\\\nWe return to the update formula~(\\ref{eq:gain}). A computation\nshows the following equalities.\n\\begin{align}\nD(\\mathbf{P}^t||\\mathbf{P}^{t+1})= &\n\\mathbb{E}_{P}D(\\mathbb{Q}^{t^{X|Y}}||\\mathbb{Q}^{{t+1}^{X|Y}})\\label{eq:gain1}\n\\\\\nD(\\mathbf{Q}^{t+1}||\\mathbf{Q}^t) = &\nD(\\mathbb{Q}^{{t+1}^{Y_-,X}}||\\mathbb{Q}^{{t}^{Y_-,X}}) \\nonumber\\\\\n& +\n\\mathbb{E}_{\\mathbb{Q}^{t+1}}D(\\mathbb{Q}^{{t+1}^{Y_+|X}}||\\mathbb{Q}^{{t}^{Y_+|X}}).\\label{eq:gain2}\n\\end{align}\nIn equation~(\\ref{eq:gain1}) we recognize the second term in the\nauxiliary function, see~(\\ref{eq:div30}).\nEquation~(\\ref{eq:gain2}) corresponds to\nequation~(\\ref{eq:auxdiff3}) of lemma~\\ref{lemma:auxdiff} and\nwe see that formula~(\\ref{eq:gain}) is indeed the same\nas~(\\ref{eq:corgain}) .\n\\medskip\\\\\nThe algorithm~(\\ref{eq:q-bar2}),~(\\ref{eq:q+bar2}) is to be\nunderstood by using these two equations simultaneously. As an\nalternative one could first use~(\\ref{eq:q-bar2}) to obtain\n$Q^{t+1}_-$ and, instead of using $Q^t_-$, feed this result\ninto~(\\ref{eq:q+bar2}) to obtain $Q^{t+1}_+$. If we do this, we\ncan express the update gain of the first partial step, like in the\nproof of corollary~\\ref{cor:corgain}, by adding the result of\nequation~(\\ref{eq:auxdiff1}) to the second summand\nof~(\\ref{eq:div30}), with the understanding that $\\mathbb{Q}'$ is\nnow given by the $Q^{t+1}(ij)Q^t(lj)$. The update gain of the\nsecond partial step is likewise obtained by combining the result\nof~(\\ref{eq:auxdiff2}) and the second summand of~(\\ref{eq:div30}),\nwith the understanding that now $\\mathbb{Q}$ is to be interpreted\nas given by the $Q^{t+1}(ij)Q^t(lj)$. Of course, as another\nalternative, the order of the partial steps can be reversed.\nClearly, the expressions for the update gains for these cases also\nresult from working with the auxiliary functions\n$G^-_{\\mathbf{Q}}$ and $G^+_{\\mathbf{Q}}$, the\nequations~(\\ref{eq:auxdiff1}) and~(\\ref{eq:auxdiff2}) and\nproceeding as in the proof of corollary~\\ref{cor:corgain}.\n\n\\section{Convergence properties}\\label{section:convprop}\n\nIn this section we study the convergence properties of the\ndivergence minimization algorithm (\\ref{eq:q-bar2}),\n(\\ref{eq:q+bar2}).\n\nThe  next theorem states that the sequences generated by the\nalgorithm converge for every (admissible) initial value. Of course\nthe limits will in general depend on the initial value.\n\n\n\\begin{thm}\\label{thm:conv}\nLet $Q^t_-(il)$, $Q_+^t(lj)$ be generated by the algorithm\n(\\ref{eq:q-bar2}), (\\ref{eq:q+bar2}) and $\\mathbf{Q}^t$ the\ncorresponding tensors. Then the $Q^t_-(il)$ converge to  limits\n$Q_-^\\infty(il)$ and the $\\mathbf{Q}^t$ converges to a limit\n$\\mathbf{Q}^\\infty$ in $\\mbox{{\\boldmath $\\mathcal{Q}$}}$. The $Q_+^t(lj)$ converge to limits\n$Q_+^\\infty(lj)$ for all $l$ with $\\sum_iQ_+^\\infty(il)>0$.\n\\end{thm}\n{\\bf Proof.} We first show that the $Q^t_-$ and $Q^t_+$ form\nconvergent sequences. We start with equation~(\\ref{eq:gain}). By\nsumming over $n$ we obtain\n\\[\nD(P||Q^0)-D(P||Q^t)=\\sum_{k=1}^{t-1}\\Big(D(\\mathbf{P}^{s}||\\mathbf{P}^{s+1})\n+ D(\\mathbf{Q}^{s+1}||\\mathbf{Q}^s)\\Big).\n\\]\nIt follows that\n$\\sum_{k=1}^{\\infty}D(\\mathbf{P}^{s}||\\mathbf{P}^{s+1})$ and\n$\\sum_{k=1}^{\\infty}D(\\mathbf{Q}^{s+1}||\\mathbf{Q}^s)$ are finite.\nNow we use that fact that for any two probability measures, the\nKullback-Leibler divergence $D(\\mathbb{P}||\\mathbb{Q})$ is greater\nthan or equal to their Hellinger distance\n$H(\\mathbb{P},\\mathbb{Q})$, which is the $L^2$ distance between\nthe square roots of corresponding densities w.r.t.~some dominating\nmeasure, see~\\cite[p.~368]{shiryaev}. In our case we have\n$H(\\mathbb{Q}^{s},\\mathbb{Q}^{s+1})=\\sum_{ilj}\n(\\sqrt{\\mathbf{Q}^{s+1}(ilj)}-\\sqrt{\\mathbf{Q}^{s}(ilj)})^2$. So\nwe obtain that\n\\[\n\\sum_{k=1}^{\\infty}H(\\mathbf{Q}^{s+1},\\mathbf{Q}^s)<\\infty.\n\\]\nWe therefore have that, pointwise, the tensors $\\mathbf{Q}^t$ form\na Cauchy sequence and hence have a limit $\\mathbf{Q}^\\infty$. We\nwill show that $\\mathbf{Q}^\\infty$ belongs to $\\mbox{{\\boldmath $\\mathcal{Q}$}}$. Since the\n$\\mathbf{Q}^t(ilj)$ converge to limits $\\mathbf{Q}^\\infty(ilj)$,\nby summation we have that the  marginals\n$Q^t_-(il)=\\mathbf{Q}^t(il\\cdot)$ converge to limits\n$\\mathbf{Q}^\\infty(il\\cdot)$ (we use the notation of the proof of\nlemma~\\ref{lemma:pyth}), and likewise we have convergence of the\nmarginals $\\mathbf{Q}^t(\\cdot lj)$ to $\\mathbf{Q}^\\infty(\\cdot\nlj)$ and $\\mathbf{Q}^t(\\cdot l\\cdot)$ to $\\mathbf{Q}^\\infty(\\cdot\nl\\cdot)$. Hence, if $\\mathbf{Q}^\\infty(\\cdot l\\cdot)>0$, then  the\n$Q^t_+(lj)$ converge to $Q^\\infty_+(ij):=\\mathbf{Q}^\\infty(\\cdot\nlj)\/\\mathbf{Q}^\\infty(\\cdot l\\cdot)$ and we have\n$\\mathbf{Q}^\\infty(ilj)=\\mathbf{Q}^\\infty(il\\cdot)Q^\\infty_+(ij)$.\nNow we analyze the case where $\\mathbf{Q}^\\infty(\\cdot\nl_0\\cdot)=0$ for some $l_0$. Since in this case both\n$\\mathbf{Q}^\\infty(il_0j)$ and $\\mathbf{Q}^\\infty(il_0\\,\\cdot)$\nare zero, we have still have a factorization\n$\\mathbf{Q}^\\infty(il_0j)=Q^\\infty_-(il_0)Q^\\infty_+(l_0j)$, where\nwe can assign to the $Q^\\infty_+(l_0j)$ arbitrary values. Let $L$\nbe the set of $l$ for which $\\sum_iQ^\\infty_-(il)>0$. Then\n$Q^\\infty(ij)=\\sum_{l\\in L}Q^\\infty_-(il)Q^\\infty_+(lj)$ and the\n$Q^t$ converge to $Q^\\infty$. This proves the theorem.\n\\hfill$\\square$\n\\begin{remark}\\label{remark:strange}\n{\\em Theorem~\\ref{thm:conv} says nothing of the convergence of the\n$Q_+^t(lj)$ for those $l$ where $\\sum_iQ^\\infty_-(il)=0$. But\ntheir behavior is uninteresting from a factorization point of\nview. Indeed, since the $l$-th column of $Q^\\infty_-$ is zero, the\nvalues of the $l$-th row of $Q^\\infty_+$ are not relevant, since\nthey don't appear in the product $Q^\\infty_-Q^\\infty_+$. As a\nmatter of fact, we now deal with an approximate nonnegative\nfactorization with a lower inner size. See also\nremark~\\ref{remark:lessk}. }\n\\end{remark}\n\\noindent In the next theorem we characterize the properties of\nthe fixed points of the algorithm. Recall from\nsection~\\ref{sec:problem} that the objective function has no local\nmaxima in the interior of the domain.\n\n\\begin{thm}\\label{thm:fixed}\nIf $(Q_-,Q_+)$ is a limit point of the\nalgorithm~(\\ref{eq:q-bar2}),~(\\ref{eq:q+bar2}) in the interior of\nthe domain, then it is a stationary point of the objective\nfunction $D$. If $(Q_-,Q_+)$ is a limit point  on the boundary of\nthe domain corresponding to an approximate factorization where\nnone of the columns of $Q_-$ is zero ($\\sum_iQ_-(il)>0$ for all\n$l$), then all partial derivatives $\\frac{\\partial D}{\\partial\nQ_-(il)}$ and $\\frac{\\partial D}{\\partial Q_+(lj)}$ are\nnonnegative.\n\\end{thm}\n{\\bf Proof.} By computing the first order partial derivatives of\nthe objective function, using the middle term of\nequation~(\\ref{eq:divnew}), we can rewrite the update\nequations~(\\ref{eq:q-bar2}),~(\\ref{eq:q+bar2}) as\n\\begin{equation}\\label{eq:qn-}\nQ_-^{t+1}(il) = Q_-^t (il) \\left(-\\frac{\\partial D^t}{\\partial\nQ_-(il)} + 1 \\right)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:qn+}\nQ_+^{t+1}(lj) \\left(\\sum_{i}Q_-^{t+1}(il) \\right) = Q_+^t(lj)\n\\left(-\\frac{\\partial D^t}{\\partial Q_+(lj)}+\\sum_i\nQ_-^t(il)\\right).\n\\end{equation}\nwhere $\\frac{\\partial D^t}{\\partial Q_-(il)}$ stands for the\npartial derivative $\\frac{\\partial D}{\\partial Q_-(il)}$ evaluated\nat $(Q_-^t,Q_+^t)$ and likewise for $\\frac{\\partial D^t}{\\partial\nQ_+(lj)}$.\n\n\n\nLet $(Q_-,Q_+)$ be a limit point of the algorithm.\nEquations~(\\ref{eq:qn-}) and~(\\ref{eq:qn+})  become\n\\[\nQ_-(il)=Q_-{il} \\left(-\\frac{\\partial D}{\\partial Q_-(il)} +\n1\\right)\n\\]\n\\[\nQ_+(lj)\\left(\\sum_{i}Q_-(il)\\right)=Q_+(lj)\\left(-\\frac{\\partial\nD}{\\partial Q_+(lj)}+\\sum_i Q_-(il)\\right).\n\\]\nIt follows that we then have the relations\n\\[\nQ_-(il)\\,\\,\\frac{\\partial D}{\\partial Q_-(il)}=0\n\\] and\n\\[ Q_+(lj)\\,\\,\\frac{\\partial D}{\\partial Q_+(lj)}=0. \\]\nWe first consider $Q_-$. Suppose that for some $i$ and $l$ we have\n$Q_-(il)>0$, then necessarily $\\frac{\\partial D}{\\partial\nQ_-(il)}=0$. Suppose now that for some $i,l$ we have $Q_-(il)=0$\nand that $\\frac{\\partial D}{\\partial Q_-(il)}<0$. Of course, by\ncontinuity, this partial derivative will be negative in a\nsufficiently small neighborhood of this limit point. Since we deal\nwith a limit point of the algorithm, we must have infinitely often\nfor the iterates that $Q_-^{t+1}(il)<Q_-^t(il)$.\nFrom~(\\ref{eq:qn-}) we then conclude that in these points we have\n$\\frac{\\partial D}{\\partial Q_-(il)}>0$. Clearly, this contradicts\nour assumption of a negative partial derivative, since eventually\nthe iterates will be in the small neighborhood of the limit point,\nwhere the partial derivative is positive. Hence, we conclude that\n$\\frac{\\partial D}{\\partial Q_-(il)}\\geq 0$,  if $Q_-(il)=0$. The\nproof of the companion statement for the $Q_+(lj)$ is similar. If\n$Q_+(lj)>0$, the corresponding partial derivative is zero. Let $l$\nbe such that\n $Q_+(lj)=0$ and suppose that\nwe have that $\\frac{\\partial D}{\\partial Q_+(lj)}<0$. If we run\nthe algorithm, then $\\frac{\\partial D^t}{\\partial\nQ_+(lj)}\/\\sum_iQ^{t+1}_-(il)$ converges to a negative limit,\n whereas $\\sum_iQ^{t}_-(il)\/\\sum_iQ^{t+1}_-(il)$ converges to one.\n Hence there is $\\eta>0$ such that eventually\n$\\frac{\\partial D^t}{\\partial\nQ_+(lj)}\/\\sum_iQ^{t+1}_-(il)<-2\\eta\/3$ and\n$\\sum_iQ^{t}_-(il)\/\\sum_iQ^{t+1}_-(il)>1- \\eta\/3$. Hence\neventually we would have, see~(\\ref{eq:qn+}),\n\\[\nQ_+^{t+1}(lj) -Q_+^{t}(lj) = Q_+^t(lj) \\left(-\\frac{\\frac{\\partial\nD^t}{\\partial Q_+(lj)}}{\\sum_i Q_-^{t+1}(il)}+\\frac{\\sum_i\nQ_-^t(il)}{\\sum_i Q_-^{t+1}(il)} -1\\right)>\\eta\/3,\n\\]\nwhich contradicts convergence of $Q_+^{t}(lj)$ to zero.\n\\hfill$\\square$\n\n\\begin{remark}\n{\\em If it happens that  a limit point $Q_-$ has a zero $l$-th\ncolumn, then it can easily be shown that the partial derivatives\n$\\frac{\\partial D}{\\partial Q_+(lj)}$ of $D$ are zero. Nothing can\nbe said of the values of the partial derivatives $\\frac{\\partial\nD}{\\partial Q_-(il)}$ for such $l$. But, see also\nremark~\\ref{remark:strange}, this case can be reduced to one with\na lower inner size factorization, for which the assertion of\ntheorem~\\ref{thm:fixed} is valid. }\n\\end{remark}\n\\rm\n\\begin{cor}\nThe limit points of the algorithm with $\\sum_iQ_-(il)>0$ for all\n$l$ are all Kuhn-Tucker points for minimization of $D$ under the\ninequality constraints $Q_-\\geq 0$ and $Q_+\\geq 0$.\n\\end{cor}\n{\\bf Proof.} Consider the Lagrange function $L$ defined by\n\\[\nL(Q_-,Q_+)=D(P||Q_-Q_+)-\\lambda\\cdot Q_--\\mu\\cdot Q_+, \\] where\nfor instance the inner product $\\lambda\\cdot Q_-$ is to be read as\n$\\sum_{il}\\lambda_{il}Q_-(il)$ for $\\lambda_{il}\\in\\mathbb{R}$.\nLet us focus on a partial derivative $\\frac{\\partial L}{\\partial\nQ_-(il)}$ in a fixed point of the algorithm. The treatment of the\nother partial derivatives is similar. From the proof of\ntheorem~\\ref{thm:fixed} we know that in a fixed point we have\n$Q_-(il)\\frac{\\partial D}{\\partial Q_-(il)}=0$. Suppose that\n$Q_-(il)>0$, then $\\frac{\\partial D}{\\partial Q_-(il)}=0$ and the\nKuhn-Tucker conditions for this variable are satisfied with\n$\\lambda_{il}=0$. If $Q_-(il)=0$, then we know from\ntheorem~\\ref{thm:fixed} that $\\frac{\\partial D}{\\partial\nQ_-(il)}\\geq 0$. By taking $\\lambda_{il}=\\frac{\\partial\nD}{\\partial Q_-(il)}\\geq 0$, we see that also here the Kuhn-Tucker\nconditions are satisfied. \\hfill$\\square$\n\\begin{remark}\n{\\em Wu~\\cite{wu} has a number of theorems that characterize the\nlimit points of the closely related EM algorithm, or generalized\nEM algorithm. These are all consequence of a general convergence\nresult in Zangwill~\\cite{zangwill}. The difference of our results\nwith his is, that we also {\\em have to} consider possible limit\npoints on the boundary, whereas Wu's results are based on the\nassumption that all limit points lie in the interior of the\ndomain.}\n\\end{remark}\\rm\n\n\\section{Relation with other minimization problems}\\label{section:otherprob}\n\nOther data analysis methods proposed in the literature enforce\nsome form of positivity constraint and it is useful to investigate\nthe connection between NMF and these methods. An interesting\nexample is the so called Archetypal Analysis (AA) technique\n\\cite{cb1994}. Assigned a matrix $X \\in \\mathbb{R}^{m\\times n}$ and an\ninteger $k$, the AA problem is to find, in the convex hull of the\ncolumns of $X$, a set of $k$ vectors whose convex combinations can\noptimally represent $X$. To understand the relation between NMF\nand AA we choose the $L_2$ criterion for both problems. For any\nmatrix $A$ and positive definite matrix $\\Sigma$ define\n$||A||_\\Sigma = (\\rm{tr} (A^T \\Sigma A))^{1\/2} $. Denote $||A||_I\n= ||A||$. The solution of the NMF problem is then\n$$\n(W, H) = \\arg \\min_{W, H} || V - WH ||\n$$\nwhere the minimization is constrained to the proper set of\nmatrices. The solution to the AA problem is given by the pair of\ncolumn stochastic matrices $(A, B)$ of respective sizes $k \\times\nn$ and $m \\times k$ such that $||X - XBA||$ is minimized (the\nconstraint to column stochastic matrices is imposed by the\nconvexity). Since $||X - XBA|| = ||I - BA||_{X^TX}$ the solution\nof the AA problem is\n$$\n(A, B) = \\arg \\min_{A, B} || I - BA ||_{X^TX}.\n$$\nAA and NMF can therefore be viewed as special cases of a more\ngeneral problem which can be stated as follows. Given any matrix\n$P \\in \\mathbb{R}_+^{m\\times n}$, any positive definite matrix $\\Sigma$,\nand any integer $k$, find the best nonnegative factorization $P\n\\approx Q_1Q_2$ (with $Q_1 \\in\\mathbb{R}_+^{m\\times k}, \\,\\, Q_2\n\\in\\mathbb{R}_+^{k\\times n}$) in the $L_2$ sense, {\\it i.e.}\n$$\n(Q_1, Q_2) =  \\arg \\min_{Q_1, Q_2} ||P - Q_1Q_2||_\\Sigma.\n$$\n{\\bf Acknowledgement.} An anonymous referee is gratefully\nacknowledged for helping us to improve the quality of the\npresentation and for suggesting to us to investigate the boundary\nbehavior of the algorithm, similar to what has been reported\nin~\\cite{laa2004}. \\rm\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nEvery year, millions of children enter a new public school. \nFrequently, they have to choose between a number of different schools, whose capacity is limited, such that not all wishes can be accommodated perfectly.\nA school choice mechanism is a procedure that collects the students' (or the parents') preferences over schools and determines a matching of students to schools.\nOne such mechanism is the \\emph{Boston mechanism}, which is frequently used in practice, but has also been heavily criticized for its manipulability.\nIn this paper, we consider a setting with no priority structure, in which we study two variants of the Boston mechanism in terms of incentives and efficiency.\n\\subsection{Strategyproofness versus Efficiency}\n\\label{sec:introduction:SPvsEFF}\nBy assuming no priority structure, the school choice problem becomes equivalent to the \\emph{one-sided matching problem}, where indivisible objects must be allocated to self interested agents and monetary transfers are not permitted.\nAs mechanism designers, we are interested in mechanisms that perform well with respect to incentives, efficiency, and fairness.\nUnfortunately, previous research on this problem has revealed that it is impossible to achieve the optimum on all dimensions simultaneously \\citep{Zhou1990}.\n\nStrategyproof mechanisms, such as Random Serial Dictatorship ($\\mathrm{RSD}$), are appealing because they make truthful reporting a dominant strategy for all agents.\nParticipation in the mechanism becomes an easy task as there is no need for deliberation about the best response, thus reducing cognitive costs for the agents and (likely) endowing the mechanism with correct information about agents' preferences.\nWhile strategyproofness is certainly a desirable property, it also imposes severe\nrestrictions.\nIn particular, strategyproofness is incompatible with ordinal efficiency and symmetry \\citep{Bogomolnaia2001}, and it is also incompatible with rank efficiency \\citep{Featherstone2011b}.\n\nThe Boston mechanism is an example of a manipulable mechanism that is frequently used in school choice settings \\citep{Abdulkadiroglu2003SchoolChoice,KojimaUnver2014BostonSchoolChoice}.\nUnder the Boston mechanism, agents submit their preferences in the form of rank ordered lists. \nThe mechanism then lets agents ``apply'' to their reported first choice and the objects are distributed amongst applicants according to some (possibly random) priority ordering. If an agent does not obtain its first choice, it ``applies'' to its second choice in the second round, etc., until all agents have received an object or all objects are exhausted.\nDespite the manipulability of the Boston mechanism, advocates of this mechanism argue that it may lead to ex-ante welfare gains because agents may coordinate on equilibria that they find more desirable under their particular preference intensities \\citep{Abdulkadiroglu2010ExpandingChoiceCADA}.\nOn the other hand, the existence of manipulation opportunities may result in disadvantages for unsophisticated participants \\citep{Miralles2008CaseForTheBostonMechanism} and inferior overall welfare \\citep{ErginSonmez2006}.\n\\citet{CasaGuell2013BCNEvidence} presented evidence from Barcelona, indicating that under a particular variant of the Boston mechanism, parents are so afraid of being matched to an undesirable school that the matching essentially degenerates to the matching determined purely by the neighborhood priority structure.\nIn recent years, some school districts have adopted alternative procedures based on the Deferred Acceptance mechanism (which is strategyproof for students), e.g., Boston and Chicago \\citep{Pathak2013AERComparingMechanismsByVulnerability}.\n\nThis paper remains agnostic with respect to this debate, i.e., we do not argue in favor or against the use of non-strategyproof mechanisms in school choice or other matching domains.\nInstead, we closely examine two variants of the Boston mechanism and compare them to each other and to the strategyproof $\\mathrm{RSD}$\\ mechanism in terms of incentives and efficiency.\nOur analysis provides new insights into the subtle trade-offs between strategyproofness and efficiency that a choice between the two variants of the Boston mechanism and $\\mathrm{RSD}$\\ entails.\n\\subsection{An Adaptive Variant of the Boston Mechanism}\n\\label{sec:introduction:abm}\nUnder the Boston mechanism as outlined above, an agent may apply to an object that has already been exhausted in a previous round.\nIn that case the agent effectively ``wastes'' one round in which it cannot compete with other agents for other objects that are still available.\nThis ``na\\\"{i}ve'' variant of the Boston mechanism ($\\mathrm{NBM}$) is susceptible to a particular type of manipulation: suppose an agent knows that its 2nd choice will be exhausted by other agents in the first round;\nunless it ranks its second choice first the agent has no chance of obtaining that 2nd choice.\nThus, by ranking its $2$nd choice last instead of second, the agent can improve its chances of obtaining a better object either in round $2$ or in any later round without foregoing any chances of the getting its $2$nd choice.\nObviously, $\\mathrm{NBM}$\\ is highly manipulable. \n\nA simple remedy for this particular problem is to let agents skip exhausted objects and instead let them apply to their best \\emph{available choice} in each round.\nThis smalle change creates a slightly different Boston mechanism, which has been largely overlooked in the literature so far. We call this mechanism the \\emph{adaptive Boston mechanisms} ($\\mathrm{ABM}$).\nLike $\\mathrm{NBM}$, $\\mathrm{ABM}$\\ is also used in practice, e.g., for the allocation of students to middle schools in Freiburg, Germany: \nparents apply to their preferred school;\nif the child is rejected, the parents receive a list of schools that still have capacity available;\nwhen the parents then apply to a school from this list, this procedure effectively implements $\\mathrm{ABM}$.\nIn this paper, we formalize and study this mechanism in detail.\n\\subsection{Understanding Trade-offs}\n\\label{sec:introduction:understand_tradeoff}\nThe traditional approach to understanding the trade-off between strategyproofness and efficiency offered by various mechanisms is to verify (or falsify) that one mechanism satisfies a certain property while another mechanism does not.\nSuch ``binary'' properties include strategyproofness, different forms of efficiency, and the existence of equilibria with desirable outcomes.\nHowever, as this paper will illustrate, this approach does not always work, i.e., a comparison of mechanisms via the traditional methods may yield inconclusive results. \n\nFor example, traditional methods do not reveal the efficiency advantages of $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ over $\\mathrm{RSD}$. \nAll three mechanisms are ex-post efficient, but neither ordinally nor rank efficient.\nRegarding incentives, a \\emph{comparison by vulnerability to manipulation} \\citep{Pathak2013AERComparingMechanismsByVulnerability} does not differentiate between $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ in our setting.\nWe therefore apply a set of new methods:\nfirst, regarding incentives, we successfully employ \\emph{partial strategyproofness}, a new, scalable concept to quantify the degree of strategyproofness of non-strategyproof mechanisms \\citep{Mennle2014}.\nSecond, regarding efficiency, we consider existing notions of dominance for allocations and extend these notions towards the comparison of mechanisms.\nIn combination with computational experiments, our methods shed new light on the trade-off between strategyproofness and efficiency when choosing between the two Boston mechanisms and $\\mathrm{RSD}$.\n\\subsection{Overview of Contributions}\n\\label{sec:introduction:contributions}\nIn this paper, we study $\\mathrm{RSD}$, $\\mathrm{NBM}$, and $\\mathrm{ABM}$\\ in a setting with no priority structure.\nWe analyze their incentive and efficiency properties, and we obtain the following results:\n\\begin{enumerate}[(1)]\n\t\\item \\textbf{Price of Strategyproofness:} we show that $\\mathrm{NBM}$\\ imperfectly rank dominates $\\mathrm{RSD}$, i.e., if the outcomes are comparable by rank dominance, then the outcome of $\\mathrm{NBM}$\\ is at least weakly preferable to that of $\\mathrm{RSD}$. This efficiency advantage can be interpreted as the \\emph{cost of strategyproofness} when choosing $\\mathrm{RSD}$\\ over $\\mathrm{NBM}$. \t\n\t\\item \\textbf{Partial Strategyproofness of $\\mathbf{ABM}$:} we formalize the adaptive Boston mechanism and show that it is partially strategyproof, i.e., strategyproof on a domain of uniformly relatively bounded indifference. In contrast, $\\mathrm{NBM}$\\ is not even weakly strategyproof. This clear distinction between $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ in terms of incentives is novel, since the comparison by vulnerability to manipulation \\citep{Pathak2013AERComparingMechanismsByVulnerability} does not differentiate between the two mechanisms in our setting.\n\t\\item{\\textbf{Imperfect Rank Dominance of ABM over RSD in Large Markets:}} we show that $\\mathrm{ABM}$\\ essentially imperfectly rank dominates $\\mathrm{RSD}$, i.e., while $\\mathrm{RSD}$\\ may rank dominate $\\mathrm{ABM}$\\ at certain type profiles, we find that this is almost never the case. We show that the share of type profiles at which $\\mathrm{RSD}$\\ rank dominates $\\mathrm{ABM}$\\ becomes arbitrarily small in large markets. Numerical evidence suggests that the efficiency advantages of $\\mathrm{ABM}$\\ over $\\mathrm{RSD}$\\ are similar in magnitude to those of $\\mathrm{NBM}$\\ over $\\mathrm{RSD}$. \t\n\t\\item{\\textbf{Price of Partial Strategyproofness:}} we show that a formal imperfect rank dominance comparison of $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ is inconclusive, i.e., $\\mathrm{NBM}$\\ is not clearly preferable to $\\mathrm{ABM}$. However, numerical evidence suggests that, conditional on comparability, $\\mathrm{NBM}$\\ is ``usually'' more efficient than $\\mathrm{ABM}$, which can be interpreted as the \\emph{cost of partial strategyproofness} when choosing $\\mathrm{ABM}$\\ over $\\mathrm{NBM}$.\n\\end{enumerate}\nIn summary, we find that choosing between $\\mathrm{NBM}$, $\\mathrm{ABM}$, and $\\mathrm{RSD}$\\ remains a question of trade-offs between efficiency and incentives.\n$\\mathrm{ABM}$\\ presents a viable alternative to $\\mathrm{NBM}$\\ when partial strategyproofness is desirable and full strategyproofness is not a hard requirement.\n\\section{Related Work}\n\\label{sec:related_work}\nThe Boston mechanism has received significant attention, because it is frequently used for the allocation of students to public schools in many school districts around the world.\nThe mechanism has been heavily criticized for its manipulability and the resulting loss in welfare and fairness:\nfor the case of strict priorities,\n\\citet{Abdulkadiroglu2003SchoolChoice} found that the Boston mechanism is neither strategyproof nor stable, and they suggested the Student Proposing Deferred Acceptance mechanism \\citep{GaleShapley1962CollegeAdmissionStability} as an alternative that is strategyproof for students.\n\\citet{ErginSonmez2006} showed that with full information, the Boston mechanism has undesirable equilibrium outcomes.\nExperimental studies by \\citet{Chen2006SchoolChoiceExperimentalStudy} and \\citet{PaisPinter2008SCandInforamtionExp} revealed that the Boston mechanism is indeed manipulated more frequently than alternative, strategyproof mechanisms.\n\n\\citet{KojimaUnver2014BostonSchoolChoice} provided an axiomatic characterization of this mechanism for the case of strict priorities.\nHowever, they also pointed out that the assumption of \\emph{strict} priorities is usually violated in school choice problems.\nSome recent work has considered coarse priorities, uncovering a number of surprising properties of the Boston mechanism:\n\\citet{Abdulkadiroglu2010ExpandingChoiceCADA} demonstrated that in a setting with \\emph{coarse} priorities and perfectly correlated preferences, the Boston mechanism can lead to higher ex-ante welfare than $\\mathrm{RSD}$.\nSimilarly, simulations conducted by \\citet{Miralles2008CaseForTheBostonMechanism} illustrate that for a setting without priorities, equilibria of the Boston mechanism can have desirable ex-ante welfare properties.\nIn this paper, we consider a setting with no priority structure and give a systematic description of the efficiency advantages of the Boston mechanism over the strategyproof alternative $\\mathrm{RSD}$\\ (which is equivalent to Student Proposing Deferred Acceptance if there is only one priority class and ties are broken using single uniform tie breaking).\n\n\\citet{Miralles2008CaseForTheBostonMechanism} also found that the manipulability of the Boston mechanism can lead to lower welfare for unsophisticated participants, and he noted that an adaptive adjustment could mitigate this effect.\nFor the case of strict priorities, \\citet{Dur2013} characterized the adaptive Boston mechanism and showed that it is less vulnerable to manipulation than the traditional Boston mechanism.\nIn this paper, we formalize the adaptive Boston mechanism in settings without priorities and study its properties.\nFor coarse priorities, a characterization of either the traditional or the adaptive Boston mechanism remains an open research question.\n\nIn the absence of priorities, $\\mathrm{RSD}$\\ is known to be strategyproof, ex-post efficient, and anonymous.\n\\citet{AbdulSonmez1998RSDandTheCore} showed that it is equivalent to the Core from Random Endowments mechanism for house allocation, and \\citet{Bade2013RSDOneAndOnly} recently extended their result: she showed that any mechanism that randomly assigns agents to roles in some strategyproof, ex-post efficient, deterministic mechanism is equivalent to $\\mathrm{RSD}$.\nHowever, it remains an open conjecture whether $\\mathrm{RSD}$\\ is the \\emph{unique} mechanism that is ex-post efficient, strategyproof, and anonymous.\n\n\\citet{Bogomolnaia2001} introduced the Probabilistic Serial mechanism and showed that it satisfies ordinal efficiency, a strict refinement of ex-post efficiency.\nHowever, they also showed that no mechanism can be ordinally efficient, strategyproof, and symmetric.\nRank efficiency, introduced by \\citet{Featherstone2011b}, is an even stronger efficiency requirement, but it is incompatible with strategyproofness, independent of any fairness requirements.\nIn this paper, we find that both variants of the Boston mechanism are ex-post efficient, but neither ordinally nor rank efficient.\nHowever, we show that both variants of the Boston mechanism provide significant efficiency gains over $\\mathrm{RSD}$.\n\nRecently, \\citet{Pathak2013AERComparingMechanismsByVulnerability} introduced a general framework to compare different mechanisms by their vulnerability to manipulation.\nThis framework is applicable for strict priority structures, but must be slightly extended for the study of mechanisms with coarse priorities in this paper.\nUnfortunately, even the weakest conceivable extension of the comparison concept does not differentiate between $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ in terms of manipulability.\nIn \\citep{Mennle2014} we have recently introduced \\emph{partial strategyproofness}, which yields a parametric measure for the degree of strategyproofness of non-strategyproof mechanisms.\nIn this paper, we use this new concept to compare the incentives of $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$.\nWe find that $\\mathrm{ABM}$\\ is partially strategyproof while $\\mathrm{NBM}$\\ is not.\n\\section{Model}\n\\label{sec:model}\n\\subsection{Preferences}\n\\label{sec:model:settings_preferences}\nA \\emph{setting} $(N,M,\\mathbf{q})$ consists of a set $N$ of $n$ \\emph{agents}, a set $M$ of $m$ \\emph{objects}, and a vector $\\mathbf{q} = (q_1, \\ldots, q_m)$ of \\emph{capacities}, i.e., there are $q_j$ units of object $j$ available. We assume that supply satisfies demand, i.e., $n \\leq \\sum_{j \\in M} q_j$, since we can always add dummy objects.\nAgents are endowed with \\emph{von Neumann-Morgenstern (vNM) utilities} $u_i, i \\in N,$ over the objects. If $u_i(a) > u_i(b)$, we say that agent $i$ \\emph{prefers object $a$ over object $b$}, which we denote by $a \\succ_i b$. We exclude indifferences, i.e., $u_i(a) = u_i(b)$ implies $a=b$. A utility function $u_i$ is \\emph{consistent} with \\emph{preference ordering} $\\succ_i$ if $a \\succ_i b$ whenever $u_i(a) > u_i(b)$. Given a preference ordering $\\succ_i$, the corresponding \\emph{type} $t_{i}$ is the set of all utilities that are consistent with $\\succ_i$, and $T$ is the set of all types, called the \\emph{type space}. We use types and preference orderings synonymously.\n\\subsection{Allocations}\n\\label{sec:model:allocations}\nAn \\emph{allocation} is a (possibly probabilistic) assignment of objects to agents. It is represented by an $n\\times m$-matrix $x = (x_{i,j})_{i \\in N, j \\in M}$ satisfying the \\emph{fulfillment constraint} $\\sum_{j\\in M} x_{i,j} = 1$, the \\emph{capacity constraint} $\\sum_{i\\in N} x_{i,j} \\leq q_j$, and $x_{i,j} \\in [0,1]$ for all $i\\in N,j \\in M$. The entries of the matrix are interpreted as probabilities, where agent $i$ gets object $j$ with probability $x_{i,j}$. An allocation is \\emph{deterministic} if $x_{i,j} \\in \\{0,1\\}$ for all $i \\in N, j \\in M$. The Birkhoff-von Neumann Theorem and its extensions \\citep{Budishetal2013DesignRandomAllocMechsTheoryAndApp} ensure that, given any allocation, we can always find a lottery over deterministic assignments that implements these marginal probabilities. Finally, let $\\mathcal{X}$ denote the space of all allocations.\n\\subsection{Mechanisms}\n\\label{sec:model:mechanisms}\nA \\emph{mechanism} is a mapping $ f: T^n \\rightarrow \\mathcal{X}$ that chooses an allocation based on a profile of reported types. We let $f_i(t_i,t_{-i})$ denote the allocation that agent $i$ receives if it reports type $t_i$ and the other agents report $t_{-i} = (t_1,\\ldots, t_{i-1},t_{i+1},\\ldots,t_n) \\in T^{n-1}$.\nNote that mechanisms only receive type profiles as input.\nThus, we consider \\emph{ordinal} mechanisms, where the allocation is independent of the actual vNM utilities.\nIf $i$ and $t_{-i}$ are clear from the context, we may abbreviate $f_i(t_i,t_{-i})$ by $f(t_i)$.\nThe expected utility for $i$ is given by the dot product $\\left\\langle u_i,f(t_i)\\right\\rangle$, i.e.,\n\\begin{equation}\n\t\\mathbb{E}_{f_i(t_i,t_{-i})}(u_i) = \\sum_{j \\in M} u_i(j) \\cdot f_i(t_i,t_{-i})(j)  =\\left\\langle u_i,f(t_i)\\right\\rangle.\n\\end{equation}\n\\subsection{A Note on Priorities}\n\\label{sec:model:priorities}\nThe classical school choice problem can be considered a hybrid between the \\emph{two-sided matching} (or \\emph{marriage}) problem and the \\emph{one-sided matching} (or \\emph{allocation}) problem. In two-sided matching, agents on both sides of the market have preferences over the agents on the other side, while in one-sided matching, objects are completely indifferent to whom they are allocated.\nIn school choice, \\emph{priorities} take the role of the schools' preferences, and they are usually not strict, i.e., many students may have the same priority, so that ties must be broken randomly.\n\nIn this paper, we do not model priorities explicitly.\nInstead, we can incorporate priorities implicitly in the formulation of the mechanism.\nThe variants of the Boston mechanisms defined in Sections \\ref{sec:nbm} and \\ref{sec:abm} choose a single priority ordering uniformly at random, i.e., all students are part of a single, large priority class, and ties are broken in the same way at all schools. This is consistent with \\citep{Miralles2008CaseForTheBostonMechanism} and \\citep{Abdulkadiroglu2010ExpandingChoiceCADA}, and ``orthogonal'' to the case of strict priorities.\nNonetheless, one of our main results (Theorem \\ref{thm:rsd_not_rd_nbm}) generalizes to settings with some priority requirements.\nThe definition of the Boston mechanisms for diverse priorities would not differ significantly from ours, but we leave the study of their properties in the more general environment to future research.\n\\section{Preliminaries}\n\\label{sec:preliminaries}\nIn this section, we review common notions of strategyproofness, efficiency, and fairness, and we define a number of new concepts that we need for the comparison of mechanisms.\n\\subsection{Strategyproofness Concepts}\n\\label{sec:preliminaries:sp}\nWe begin with a revision of the standard strategyproofness requirement.\n\\begin{definition}[Strategyproofness] \\label{def:SP} A mechanism is \\emph{strategyproof} if for any agent $i\\in N$, any type profile $\\mathbf{t}=(t_i,t_{-i})\\in T^n$, any misreport $t_{i}' \\in T$, and any utility $u_i \\in t_i$ we have\n\\begin{equation}\n\t\\left\\langle u_i , f(t_i) - f(t'_i)\\right\\rangle \\geq 0.\n\\label{eq:incentive_constraint_SP}\n\\end{equation}\n\\end{definition}\nStrategyproofness is particularly attractive because it makes reporting truthfully a dominant strategy for all agents. This removes the need for agents to deliberate about potentially beneficial manipulations, \nand agents do not need to exert effort to coordinate on certain equilibria.\n\nIf agents are only interested in manipulations that improve the outcome (for them) in a first order-stochastic dominance sense (e.g., if they are not aware of their own preference intensities), then weak strategyproofness becomes useful.\nWeak strategyproofness was first used by \\citet{Bogomolnaia2001} to describe the incentive properties of the Probabilistic Serial mechanism.\n\\begin{definition}[Weak Strategyproofness] \\label{def:wsp} A mechanism is \\emph{weakly strategyproof} if for any type profile $\\mathbf{t}=(t_i,t_{-i}) \\in T^n$, the outcome from truthful reporting is never strictly first order-stochastically dominated by the outcome from any misreport for agent $i$.\n\\end{definition}\nWeak strategyproofness is a very weak requirement for multiple reasons. \nIn particular, it does not even guarantee that there exists a single utility profile for which truthful reporting is a best response (in terms of expected utility) \\citep{Mennle2014}.\n\nIn \\citep{Mennle2014} we have introduced \\emph{partial strategyproofness}, a scalable requirement that bridges the gap between strategyproofness and weak strategyproofness.\nIntuitively, partially strategyproof mechanisms are strategyproof, but only on a particular domain restriction, i.e., the agents can still have any type, but their underlying utility functions are constrained. First we define this domain restriction.\n\\begin{definition}[Uniformly Relatively Bounded Indifference] \\label{def:URBI} A utility function $u$ satisfies \\emph{uniformly relatively bounded indifference with respect to bound $r \\in [0,1]$ ($\\mathrm{URBI}(r)$)} if for any objects $a,b$ with $u(a) > u(b)$ we have\n\\begin{equation}\nr\\cdot (u(a)-\\min(u)) \\geq u(b)-\\min(u).\n\\label{eq:urbi_constraint}\n\\end{equation}\nDenote by $\\mathrm{URBI}(r)$\\ the set of all utility functions that satisfy uniformly relatively bounded indifference with respect to bound $r$.\n\\end{definition}\nA mechanism is partially strategyproof if it is strategyproofness in the domain restricted by the $\\mathrm{URBI}(r)$\\ constraint. Formally:\n\\begin{definition}[$r$-partial Strategyproofness] \\label{def:PSP} Given a setting $(N,M,\\mathbf{q})$ and a bound $r \\in [0,1]$, mechanism $f$ is \\emph{$r$-partially strategyproof in the setting $(N,M,\\mathbf{q})$} if for any agent $i\\in N$, any type profile $\\mathbf{t} = (t_i,t_{-i}) \\in T^n$, any misreport $t_i' \\in T$, and any utility $u_i \\in \\mathrm{URBI}(r) \\cap t_i$, we have\n\\begin{equation}\n\t\\left\\langle u_i , f(t_i) - f(t'_i)\\right\\rangle \\geq 0.\n\t\\label{eq:incentive_constraint_urbi_psp}\n\\end{equation}\n\\end{definition}\nSometimes, we simply want to state that a mechanism is $r$-partially strategyproof for some non-trivial $r > 0$ without explicitly naming the parameter $r$. In this case, we simply say that the mechanism is \\emph{partially strategyproof}.\n\nThe partial strategyproofness concept has an axiomatic characterization.\nFor details, please see Appendix \\ref{app:review_psp} or \\citep{Mennle2014}. In the following we only briefly review the two axioms we need throughout this paper. \n\nEach axiom restricts the way in which a misreport from the neighborhood of the agent's true type, i.e., a misreport involving only a single swap, can affect the allocation of the reporting agent.\n\\begin{axiom}[Swap Monotonic] \\label{ax:SM} A mechanism $f$ is \\emph{swap monotonic} if for any agent $i \\in N$, any type profile $\\mathbf{t} = (t_i,t_{-i}) \\in T^n$, and any type $t_i' \\in N_{t_i}$ (i.e., in the neighborhood of $t_i$) with $a_k \\succ a_{k+1}$ under $t_i$ and $a_{k+1} \\succ a_k$ under $t_i'$, one of the following holds:\n\\begin{enumerate}[1)]\n\t\\item $i$'s allocation is unaffected by the swap, i.e.,\n\t\t$f(t_i) = f(t_i')$, or\n\t\\item $i$'s allocation for $a_k$ strictly decreases and its allocation for $a_{k+1}$ strictly increases, i.e.,\n\t\\begin{equation} f(t_i)(a_k) > f(t_i')(a_k) \\;\\;and\\;\\; f(t_i)(a_{k+1}) < f(t_i')(a_{k+1}). \\end{equation}\n\\end{enumerate}\n\\end{axiom}\n\\begin{axiom}[Upper Invariance] \\label{ax:UI} A mechanism $f$ is \\emph{upper invariant} if for any agent $i \\in N$, any type profile $\\mathbf{t} = (t_i,t_{-i}) \\in T^n$, and any type $t_i' \\in N_{t_i}$ with $a_k \\succ a_{k+1}$ under $t_i$ and $a_{k+1} \\succ a_{k}$ under $t_i'$, $i$'s allocation for the upper contour set $U(a_k,t_i)$ is unaffected by the swap, i.e., $ f(t_i)(j) = f(t_i')(j) $ for all $j \\in U(a_k,t_i)$.\n\\end{axiom}\n\\emph{Lower invariance} is defined analogously by replacing the upper contour set by the lower contour set.\n\nIn \\citep{Mennle2014}, we have shown that a mechanism is strategyproof if and only if it is swap monotonic, upper invariant, and lower invariant. Furthermore, by dropping lower invariance, the class of partially strategyproof mechanisms arises, i.e., a mechanism is partially strategyproof if and only if it is swap monotonic and upper invariant. \n\nFor any partially strategyproof mechanism, we can consider the largest admissible indifference bound $r$ as a single-parameter measure for ``how strategyproof'' the mechanism is.\n\\begin{definition} \\label{def:dosp} For a setting $(N,M,\\mathbf{q})$, the \\emph{degree of strategyproofness (DOSP)} of a mechanism $f$ is given by\n\\begin{equation}\n\t\\rho_{(N,M,\\mathbf{q})}(f) = \\max\\{r \\in [0,1] : f\\text{ is }r\\text{-partially strategyproof in }(N,M,\\mathbf{q})\\}.\n\\end{equation}\n\\end{definition}\nIn \\citep{Mennle2014} we have also shown that the DOSP is computable and consistent with the vulnerability to manipulation concept \\citep{Pathak2013AERComparingMechanismsByVulnerability}.\n\\subsection{Dominance and Efficiency}\n\\label{sec:preliminaries:dominance_efficiency}\nIn matching markets, welfare usually cannot be expressed as the sum of the agents' cardinal utilities, since the welfare of individual agents may not be comparable. \nInstead, we consider whether or not an allocation can be improved upon, i.e., whether it is \\emph{efficient}.\n\nFirst, we review ex-post (Pareto) dominance and efficiency.\n\\begin{definition}[Ex-post Dominance] \\label{def:expost_dom} Given a type profile $\\mathbf{t} =(t_i,t_{-i}) \\in T^n$, a deterministic allocation $x$ \\emph{ex-post dominates} another deterministic allocation $y$ \\emph{at} $\\mathbf{t}$ if all agents weakly prefer their allocation under $x$ to their allocation under $y$. The dominance is \\emph{strict} if at least one agent strictly prefers its allocation under $x$.\n\\end{definition}\n\\begin{definition}[Ex-post Efficiency] \\label{def:expost_eff} Given a type profile $\\mathbf{t} =(t_i,t_{-i}) \\in T^n$, a deterministic allocation $x$ is \\emph{ex-post efficient at} $\\mathbf{t}$ if it is not strictly ex-post dominated by any other deterministic allocation at $\\mathbf{t}$. Finally, a probabilistic allocation is \\emph{ex-post efficient} if it has a lottery decomposition consisting only of ex-post efficient deterministic allocations.\n\\end{definition}\nEx-post efficiency is ubiquitous in the matching literature and can be considered a baseline requirement for mechanism design.\nHowever, even when considering ex-post efficient probabilistic allocations, some unambiguous welfare gains may be possible: agents may wish to trade probability shares at some objects in such a way that all agents are weakly better off and some strictly (see \\citep{Bogomolnaia2001} for an example).\nThis kind of \\emph{ex-interim improvement}, i.e., before the lottery is implemented, is formalized by the following ordinal dominance and efficiency concepts.\n\\begin{definition}[Ordinal Dominance] \\label{def:ordinal_dominance} For a type $t : a_1 \\succ \\ldots \\succ a_m$ and two allocation vectors $v = v_{j \\in M}$ and $ w = w_{j \\in M}$, the allocation vector $v$ \\emph{first order-stochastically dominates} $w$ if for all ranks $k \\in \\{1,\\ldots,m\\}$\n\\begin{equation}\n\\sum_{j \\in M: j \\succ a_k} v_{j} \\geq \\sum_{j \\in M: j \\succ a_k} w_{j}.\n\\label{eq:fosd}\n\\end{equation}\nFor a type profile $\\mathbf{t}$, an allocation $x$ \\emph{ordinally dominates} another allocation $y$ \\emph{at} $\\mathbf{t}$ if for all agents $i \\in N$ their respective allocation vector $x_i$ first order-stochastically dominates $y_i$ (for type $t_i$). $x$ \\emph{strictly} ordinally dominates $y$ if in addition, inequality (\\ref{eq:fosd}) is strict for some agent $i \\in N$ and some rank $k \\in \\{1,\\ldots,m\\}$.\nWe let $\\succeq_O$ and $\\succ_O$ denote weak and strict ordinal dominance, respectively.\n\\end{definition}\n\\begin{definition}[Ordinal Efficiency] \\label{def:ordinal_eff}\nFor a type profile $\\mathbf{t}$, an allocation $x$ is \\emph{ordinally efficient at} $\\mathbf{t}$ if it is not strictly ordinally dominated by any other allocation at $\\mathbf{t}$.\n\\end{definition}\nA refinement of ordinal efficiency is rank efficiency. Rank efficiency \\citep{Featherstone2011b} captures the intuition that allocating two first choices and one second choice is preferable to allocating one first and two second choices.\n\\begin{definition}[Rank Dominance] \\label{def:rank_dom} For a type\n$t \\in T$ and rank $k \\in \\{1,\\ldots,m\\}$ let $\\mathrm{ch}(k,t)$ denote the $k$th choice object of an agent of type $t$.\nThe \\emph{rank distribution} of an allocation $x$ at type profile $\\mathbf{t}$ is the vector $\\mathbf{d}^x = (d_1^x,\\ldots,d_m^x)$ with\n\\begin{equation}\nd_k^x = \\sum_{i \\in N} x_{i,\\mathrm{ch}(k,t_i)} \\mathrm{\\;for\\;}k\\in \\{1,\\ldots,m\\},\n\\label{eq:def_rank_dist}\n\\end{equation}\ni.e., $d_k^x$ is the expected number of $k$th choices allocated under $x$ with respect to type profile $\\mathbf{t}$.\n\nAn allocation $x$ \\emph{rank dominates} another allocation $y$ at $\\mathbf{t}$ if the rank distribution $\\mathbf{d}^x$ first order-stochastically dominates $\\mathbf{d}^y$. $x$ \\emph{strictly rank dominates} $y$ if in addition, inequality (\\ref{eq:def_rank_dist}) is strict for some rank $r\\in\\{1,\\ldots,m\\}$.\nLet $\\succeq_R$ and $\\succ_R$ denote weak and strict rank dominance, respectively.\n\\end{definition}\n\\begin{definition}[Rank Efficiency] \\label{def:rank_eff}\nAn allocation $x$ is \\emph{rank efficient} at $\\mathbf{t}$ if it is not strictly rank dominated by any other allocation at $\\mathbf{t}$.\n\\end{definition}\nRank efficient allocations can be interpreted as maximizing some form of social value: suppose there is a value $v_k$ associated with giving any agent its $k$th choice object.\nA vector $\\mathbf{v}=(v_1,\\ldots, v_m)$ with $v_1 > v_2 > \\ldots > v_m$ is called \\emph{rank valuation}.\nAn allocation $x$ is rank efficient if and only if there exists a rank valuation $v$ such that\n\\begin{equation}\nx \\in \\arg\\max_{y \\in \\mathcal{X}} \\sum_{k = 1}^m \\sum_{i \\in N} v_k y_{i,\\mathrm{ch}(k,t_i)}.\n\\end{equation}\n\nSo far, we have only discussed dominance concepts for allocations. Dominance can easily be extended to mechanisms by requiring the allocations resulting from one mechanism to always dominate those from the other mechanism.\n\\begin{definition}[Dominance for Mechanisms] \\label{def:perf_dom} Consider two mechanisms $f$ and $g$. \nMechanism $g$ \\emph{weakly ordinally dominates} $f$ if $g(\\mathbf{t}) \\succeq_O f(\\mathbf{t})$ at $\\mathbf{t}$ for all type profiles $\\mathbf{t} \\in T^n$.\n$g$ \\emph{strictly ordinally dominates} $f$ if in addition $g(\\mathbf{t}) \\succ_O f(\\mathbf{t})$ at $\\mathbf{t}$  for some type profiles $\\mathbf{t} \\in T^n$. \n\\emph{Weak} and \\emph{strict rank dominance} are defined analogously.\nWe denote weak and strict ordinal dominance for mechanisms by $\\succeq_{O}$ and $\\succ_{O}$, and weak and strict rank dominance for mechanisms by $\\succeq_{R}$ and $\\succ_{R}$, respectively, simply extending the notation from allocations.\n\\end{definition}\nTo determine whether a mechanism can be improved upon with respect to efficiency, one must check whether there exists another mechanism that is unambiguously more efficient, but otherwise has the same desirable properties. This is captured by the \\emph{efficient frontier}.\n\\begin{definition}[Efficient Frontier] \\label{def:frontier} For any set of properties $\\mathcal{E}$ and a dominance concept $\\succ \\in \\{\\succ_O,\\succ_R\\}$ (in the sense of Definition \\ref{def:perf_dom}), a mechanism is on the \\emph{efficient frontier (with respect to $\\succ$, subject to $\\mathcal{E}$)}, if it is not $\\succ$-dominated by any mechanism that also satisfies $\\mathcal{E}$.\n\\end{definition}\n\nUnfortunately, the dominance concept from Definition \\ref{def:perf_dom} may be overly restrictive for the comparison of many popular mechanisms, i.e., essentially all common mechanisms are not comparable in this way (e.g., Random Serial Dictatorship, Probabilistic Serial, variants of the Boston mechanism). Therefore, we introduce \\emph{imperfect dominance} for mechanisms, which only requires that one mechanism should produce dominant allocations \\emph{whenever the resulting allocations are comparable}.\n\\begin{definition}[Imperfect Dominance for Mechanisms] \\label{def:imp_dom} Consider two mechanisms $f$ and $g$. $g$ \\emph{weakly imperfectly ordinally dominates} $f$ if\n$g(\\mathbf{t}) \\succeq_O f(\\mathbf{t})$ at $\\mathbf{t}$ for all type profiles $\\mathbf{t} \\in T^n$ where $g(\\mathbf{t})$ and $f(\\mathbf{t})$ are comparable by the dominance relation.\n$g$ \\emph{strictly imperfectly ordinally dominates} $f$ if in addition $g(\\mathbf{t}) \\succ_O f(\\mathbf{t})$ at $\\mathbf{t}$ for some type profiles $\\mathbf{t} \\in T^n$. \\emph{Weak} and \\emph{strict imperfect rank dominance} are defined analogously.\nWe denote weak and strict imperfect ordinal dominance by $\\succeq_{IO}$ and $\\succ_{IO}$, and weak and strict imperfect rank dominance by $\\succeq_{IR}$ and $\\succ_{IR}$, respectively.\n\\end{definition}\n\\subsection{Fairness}\n\\label{sec:preliminaries:fairness}\nIn the absence of strict priorities, \\emph{anonymity} is a common fairness requirement, which all mechanisms we study in this paper satisfy.\n\\begin{definition}[Anonymity] A mechanism is \\emph{anonymous} if the allocation only depends on the reports of the agents, but not their names, i.e., for any bijection $\\phi:N\\rightarrow N$ and all $i \\in N$\n\\begin{equation}\n\tf_{\\phi(i)}(t_{\\phi(1)},\\ldots,t_{\\phi(n)}) = f_{i}(t_1,\\ldots,t_n).\n\\end{equation}\n\\end{definition}\nAnonymity implies \\emph{symmetry} (also called \\emph{equal treatment of equals}), which requires that agents with the same report also receive the same allocation. However, the inverse is not true.\n\nAnother important fairness requirement in school choice is the \\emph{absence of justified envy}: after the implementation of the lottery, agent $i$ \\emph{envies} another agent $i'$ if $i$ prefers the object that $i'$ received to its own object.\nThe envy is \\emph{justified} if $i$ has a higher priority at that object than $i'$.\nSince in this paper, we consider mechanisms in the absence of strict priorities, an agent cannot have higher priority at any object ex-ante, and thus, justified envy is no concern.\n\nNonetheless, we can consider the absence of justified envy \\emph{ex-post}, i.e., after the implementation of the lottery and with respect to the particular tie-breaker.\nIn this ex-post sense, Random Serial Dictatorship eliminates justified envy, while the Boston mechanisms may not. However, the Boston mechanisms do eliminate \\emph{observable justified envy}: \nif agents are not informed about the (randomly chosen) priority ordering, they only learn their relative priority with respect to other agents against whom they have competed (and won or lost) at some objects. \nFrom the perspective of an agent $i$, if any agent $i'$ received an object that $i$ prefers to its own allocation, then $i'$ either applied to that object in an earlier round, in which case the relative priority of $i$ and $i'$ is unobservable, \nor $i'$ applied to the object in the same round as $i$ and $i'$ had the higher priority. Thus, for any agent that $i$ envies, that agent either has a higher priority or the priority is unobservable to $i$.\n\\subsection{The Random Serial Dictatorship Mechanism}\n\\label{sec:preliminaries:rsd}\nWe now define the Random Serial Dictatorship mechanism.\n\\begin{definition} \\label{def:rsd} The \\emph{Random Serial Dictatorship ($\\mathrm{RSD}$) mechanism} works as follows:\n\\begin{enumerate}\n\t\\item Collect all type reports $\\hat{\\mathbf{t}}=(\\hat{t}_1,\\ldots,\\hat{t}_n) \\in T^n $ from the agents.\n\t\\item Draw an ordering $\\pi$ of all agents uniformly at random from the space of all possible bijections $\\pi: N \\rightarrow \\{1,\\ldots,n\\}$. $\\pi(i) < \\pi(i')$ means that $i$ has priority over $i'$.\n\t\\item Step 1: the agent with the highest priority gets a copy of the object it likes best and is removed together with that copy of the object.\n\t\\item Step $k$: in each subsequent step, the agent with the highest priority (amongst the remaining agents) gets the object it likes best out of all objects that still have capacity left.\n\t\\item The process ends when all agents have been allocated an object.\n\\end{enumerate}\n\\end{definition}\nLet $\\mathrm{SD}_{\\pi}(\\hat{\\mathbf{t}})$ denote the allocation if a Serial Dictatorship ($\\mathrm{SD}$) is applied to reports $\\hat{\\mathbf{t}}$ with fixed priority ordering $\\pi$.\n$\\mathrm{RSD}$\\ can be viewed as producing a probabilistic allocation, because the priority ordering over the agents is unknown when the agents report their types. The probabilistic allocation is given by the convex combination over all possible deterministic allocations\n\\begin{equation}\n\\mathrm{RSD}(\\hat{\\mathbf{t}}) = \\sum_{\\pi:N\\rightarrow\\{1,\\ldots,n\\}\\mathrm{\\ bijection}} \\frac{1}{n!} \\mathrm{SD}_{\\pi}(\\hat{\\mathbf{t}}).\n\\label{eq:prob_alloc_rsd}\n\\end{equation}\n\\begin{remark} \\label{rem:SD_supports_EE} Any ex-post efficient deterministic allocation is supported by $\\mathrm{SD}$\\ with respect to a particular priority ordering $\\pi$ of the agents \\citep{Featherstone2011b}.\nThis implies that $\\mathrm{RSD}$\\ performs a lottery over \\emph{all} ex-post efficient deterministic allocations, where each allocation is weighted by the number of different priority orderings $\\pi$ for which $\\mathrm{SD}$\\ produces that allocation.\n\\end{remark}\n\\begin{remark} \\label{rem:RSD_equiv_DA}\nWith a single, uniform tie-breaker, the Student Proposing Deferred Acceptance mechanism \\citep{GaleShapley1962CollegeAdmissionStability} is equivalent to $\\mathrm{RSD}$.\n\\end{remark}\n\\section{The Na\\\"{i}ve Boston Mechanism}\n\\label{sec:nbm}\nIn this section, we define and study the traditional (na\\\"{i}ve) Boston mechanism ($\\mathrm{NBM}$) in the absence of priorities and with single uniform tie-breaking.\nIt is well known that this mechanism is ex-post efficient, but not even weakly strategyproof.\nWe show that it is neither ordinally nor rank efficient, nor is it on the efficient frontier.\nDespite these negative findings, our first main result (Theorem \\ref{thm:rsd_not_rd_nbm}) establishes that $\\mathrm{NBM}$\\ does have efficiency advantages over $\\mathrm{RSD}$.\n\\subsection{Formalization of NBM}\n\\label{sec:nbm:formalization}\n\\begin{definition} \\label{def:nbm} The \\emph{na\\\"{i}ve Boston mechanism ($\\mathrm{NBM}$)}\\footnote{The mechanism is called the \\emph{Boston mechanism}, despite the fact that it is no longer used in Boston. Some researchers argue that calling it the \\emph{Immediate Acceptance mechanism} would be more consistent with the naming of the \\emph{Deferred Acceptance mechanism}.} works as follows:\n\\begin{enumerate}\n\t\\item Collect all type reports $\\hat{\\mathbf{t}}=(\\hat{t}_1,\\ldots,\\hat{t}_n) \\in T^n $ from the agents.\n\t\\item Draw an ordering $\\pi$ of all agents uniformly at random from the space of all possible bijections $\\pi: N \\rightarrow \\{1,\\ldots,n\\}$. $\\pi(i) < \\pi(i')$ means that $i$ has priority over $i'$.\n\t\\item Round 1: all agents apply to their reported first choice object. At every object, if the object's capacity is sufficient to satisfy all applications to this object, all the applicants get a copy of the object. Otherwise, the agents with the highest priority get copies of the object until it is exhausted. All remaining agents do not receive a copy and enter the next round.\n\t\\item Round $k$: an agent who did not receive an object in any of the previous $k-1$ rounds applies to its reported $k$th choice object. Analogous to round 1, the agents with the highest priorities are allocated copies of the objects for which they applied. If an object has more applicants than remaining capacity, the remaining agents enter the next round.\n\t\\item The process ends when all agents have been allocated an object.\n\\end{enumerate}\n\\end{definition}\nLet $\\mathrm{NBM}_{\\pi}(\\hat{\\mathbf{t}})$ denote the allocation if $\\mathrm{NBM}$\\ is applied to reports $\\hat{\\mathbf{t}}$ with priority ordering $\\pi$.\nLike $\\mathrm{RSD}$, $\\mathrm{NBM}$\\ produces probabilistic allocations, because it randomizes over priority orderings. The probabilistic allocation is given by the convex combination\n\\begin{equation}\n\\mathrm{NBM}(\\hat{\\mathbf{t}}) = \\sum_{\\pi:N\\rightarrow\\{1,\\ldots,n\\}\\mathrm{\\ bijection}} \\frac{1}{n!} \\mathrm{NBM}_{\\pi}(\\hat{\\mathbf{t}}).\n\\label{eq:prob_alloc_nbm}\n\\end{equation}\n\n\\subsection{Manipulability and Upper Invariance of NBM}\n\\label{sec:nbm:incentives}\n$\\mathrm{NBM}$\\ is known to be manipulable in a first order-stochastic dominance sense.\n\\begin{proposition}\\label{prop:nbm_not_wsp} $\\mathrm{NBM}$\\ is not weakly strategyproof.\n\\end{proposition}\n\\begin{proof} Consider a setting $(N,M,\\mathbf{q})$ with $N = \\{1,\\ldots,4\\}$, $M = \\{a,b,c,d\\}$, objects have unit capacities, and the type profile is\n\\begin{eqnarray*}\n\tt_1 & : & a \\succ b \\succ c \\succ d, \\\\\n\tt_2,t_3 & : & a \\succ c \\succ d \\succ b, \\\\\n\tt_4 & : & b \\succ \\ldots.\n\\end{eqnarray*}\nAgent 1's allocation is $(1\/3,0,0,2\/3)$ for the objects $a$ through $d$, respectively. If agent 1 swaps $b$ and $c$ in its report, its allocation will be $(1\/3,0,1\/3,1\/3)$, which first order-stochastically dominates the allocation under truthful reporting.\n\\end{proof}\n\nDespite this obvious manipulability, $\\mathrm{NBM}$\\ does have one attractive incentive property: it satisfies upper invariance, i.e., changing the order of two adjacent objects in an agent's type report ($t: a \\succ b$ to $t':b \\succ a$, say) will not influence the allocation of any object that this agent strictly prefers to $a$.\n\\begin{proposition} \\label{prop:nbm_ui} $\\mathrm{NBM}$\\ is upper invariant.\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:nbm:nbm_ui_and_abm_ui})] Consider each priority ordering $\\pi$ separately. \nWhen swapping two objects $a$ and $b$, observe that if an object in the upper contour set is allocated, the swap has no effect, and if an object from the lower contour set is allocated, the swap will not lead to an allocation from the upper contour set. Therefore, the probabilities for the objects in the upper contour set remain unchanged.\n\\end{proof}\n\\begin{proposition} \\label{prop:nbm_not_sm_li} $\\mathrm{NBM}$\\ is neither swap monotonic nor lower invariant.\n\\end{proposition}\n\\begin{proof} The example in the proof of Proposition \\ref{prop:nbm_not_wsp} shows that $\\mathrm{NBM}$\\ has neither property.\n\\end{proof}\n\n\\subsection{Efficiency of NBM}\n\\label{sec:nbm:efficiency_failure}\nIn this section, we check whether $\\mathrm{NBM}$\\ satisfies any existing efficiency concepts or lies on the efficient frontier, subject to upper invariance.\nFirst, we show that like $\\mathrm{RSD}$, $\\mathrm{NBM}$\\ is ex-post efficient. \nHowever, also like $\\mathrm{RSD}$, $\\mathrm{NBM}$\\ is neither ordinally nor rank efficient, nor does it lie on the efficient frontier. This motivates a direct comparison of $\\mathrm{NBM}$\\ to $\\mathrm{RSD}$.\n\\subsubsection{Ex-post Efficiency of NBM}\n\\label{sec:nbm:efficiency_failure:ex_post}\nFirst, we state the well-known fact that $\\mathrm{NBM}$\\ is ex-post efficient.\n\\begin{proposition} \\label{prop:nbm_ex_post} $\\mathrm{NBM}$\\ is ex-post efficient at all type profiles.\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:nbm:nbm_ex_post_and_abm_ex_post})] For each type profile $\\mathbf{t}$ and each priority ordering $\\pi$, we construct a priority order $\\pi'$ such that $\\mathrm{NBM}_{\\pi}(\\mathbf{t}) = \\mathrm{SD}_{\\pi'}(\\mathbf{t})$. \nThe result follows, because $\\mathrm{SD}$ is ex-post efficient for all priority orders $\\pi$. \n\\end{proof}\n\\subsubsection{Failure of Ordinal and Rank Efficiency}\n\\label{sec:nbm:efficiency_failure:ordinal_and_rank}\nSince rank efficiency implies ordinal efficiency, ordinal inefficiency implies rank inefficiency. Thus, the following Proposition \\ref{prop:nbm_not_OE} shows ordinal inefficiency and rank inefficiency of $\\mathrm{NBM}$\\ at the same time.\n\\begin{proposition} \\label{prop:nbm_not_OE} $\\mathrm{NBM}$\\ and $\\mathrm{RSD}$\\ are neither ordinally efficient nor rank efficient.\n\\end{proposition}\n\\begin{proof}\nConsider the setting with $N = \\{1,\\ldots,4\\}$, $M = \\{a,b,c,d\\}$, unit capacities, and the type profile\n\\begin{eqnarray*}\n\tt_1 & : & a \\succ b \\succ c \\succ d, \\\\\n\tt_2 & : & a \\succ b \\succ d \\succ c, \\\\\n\tt_3\t& : & b \\succ a \\succ c \\succ d, \\\\\n\tt_4 & : & b \\succ a \\succ d \\succ c.\n\\end{eqnarray*}\nThe allocations are\n\\begin{equation}\n\\mathrm{RSD}(\\mathbf{t}) = \\left(\\begin{array}{cccc}\n\\frac{5}{12} & \\frac{1}{12} & \\frac{5}{12} & \\frac{1}{12} \\\\\n\\frac{5}{12} & \\frac{1}{12} & \\frac{1}{12} & \\frac{5}{12} \\\\\n\\frac{1}{12} & \\frac{5}{12} & \\frac{5}{12} & \\frac{1}{12} \\\\\n\\frac{1}{12} & \\frac{5}{12} & \\frac{1}{12} & \\frac{5}{12} \t\n\\end{array}\\right) \\text{ and }\n\\mathrm{NBM}(\\mathbf{t}) = \\left(\\begin{array}{cccc}\n\\frac{1}{2} & 0 & \\frac{3}{8} & \\frac{1}{8} \\\\\n\\frac{1}{2} & 0 & \\frac{1}{8} & \\frac{3}{8} \\\\\n0 & \\frac{1}{2} & \\frac{3}{8} & \\frac{1}{8} \\\\\n0 & \\frac{1}{2} & \\frac{1}{8} & \\frac{3}{8} \t\n\\end{array}\\right),\n\\end{equation}\nwhich are both ordinally dominated by\n\\begin{equation}\n\\mathrm{PS}(\\mathbf{t}) = \\left(\\begin{array}{cccc}\n\\frac{1}{2} & 0 & \\frac{1}{2} & 0 \\\\\n\\frac{1}{2} & 0 & 0 & \\frac{1}{2} \\\\\n0 & \\frac{1}{2} & \\frac{1}{2} & 0 \\\\\n0 & \\frac{1}{2} & 0 & \\frac{1}{2} \t\n\\end{array}\\right).\n\\end{equation}\nSince ordinal dominance implies rank dominance, failure of ordinal efficiency implies failure of rank efficiency.\n\\end{proof}\n\\subsubsection{Failure of Efficient Frontier}\n\\label{sec:nbm:efficiency_failure:frontier}\nIn the proof of Proposition \\ref{prop:nbm_not_OE}, the Probabilistic Serial mechanism ordinally dominates $\\mathrm{NBM}$\\ at a particular type profiles, i.e., it is more efficient. \nSince $\\mathrm{PS}$\\ is ordinally efficient, it is a particularly interesting candidate mechanism which might ordinally dominate $\\mathrm{NBM}$\\ \\emph{at all type profiles}.\nHowever, this is not the case, i.e., at some type profile, $\\mathrm{PS}$\\ does not ordinally dominate $\\mathrm{NBM}$.\nInstead, $\\mathrm{NBM}$\\ may even \\emph{rank dominate} $\\mathrm{PS}$\\ at some type profile, as the following Proposition \\ref{prop:nbm_rank_dom_ps} shows.\n\\begin{proposition} \\label{prop:nbm_rank_dom_ps} $\\mathrm{PS}$\\ does not ordinally dominate $\\mathrm{NBM}$\\ at all type profiles, and $\\mathrm{NBM}$\\ rank dominates $\\mathrm{PS}$\\ at some type profile.\n\\end{proposition}\n\\begin{proof}\nConsider the setting with $N = \\{1,2,3\\}$, $M = \\{a,b,c\\}$, unit capacities, and the type profile\n\\begin{eqnarray*}\n\tt_1 & : & a \\succ b \\succ c, \\\\\n\tt_2 & : & a \\succ b \\succ c, \\\\\n\tt_3\t& : & b \\succ a \\succ c.\n\\end{eqnarray*}\nThe allocations for $\\mathrm{NBM}$\\ is\n\\begin{equation}\n\\mathrm{NBM}(\\mathbf{t}) = \\left(\\begin{array}{ccc}\n\\frac{1}{2} & 0 & \\frac{1}{2} \\\\\n\\frac{1}{2} & 0 & \\frac{1}{2} \\\\\n0 & 1 & 0\n\\end{array}\\right),\n\\end{equation}\nwhich has rank distribution $\\mathbf{d} = (2,0,1)$. Probabilistic Serial on the other hand yields\n\\begin{equation}\n\\mathrm{PS}(\\mathbf{t}) = \\mathrm{RSD}(\\mathbf{t}) = \\left(\\begin{array}{ccc}\n\\frac{1}{2} & \\frac{1}{6} & \\frac{1}{3} \\\\\n\\frac{1}{2} & \\frac{1}{6} & \\frac{1}{3} \\\\\n0 & \\frac{2}{3} & \\frac{1}{3}\n\\end{array}\\right),\n\\end{equation}\nwhich has a strictly worse rank distribution $\\mathbf{d} = (5\/3,1\/3,1)$. Thus, $\\mathrm{NBM}$ strictly rank dominates $\\mathrm{PS}$ at $\\mathbf{t}$, and therefore $\\mathrm{PS}$ does not ordinally dominate $\\mathrm{NBM}$ even weakly at $\\mathbf{t}$.\n\\end{proof}\nSince $\\mathrm{NBM}$\\ is not ordinally dominated by the particularly interesting candidate $\\mathrm{PS}$, it could lie on the efficient frontier.\nHowever, we find that this is not the case, i.e., it is possible to construct a mechanism that is upper invariant and ordinally dominates $\\mathrm{NBM}$.\n\\begin{proposition} \\label{prop:nbm_not_frontier} $\\mathrm{NBM}$\\ is not on the efficient frontier with respect to ordinal dominance, subject to upper invariance.\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:nbm:nbm_not_frontier_and_abm_not_frontier})]\nWe construct a mechanism $\\mathrm{NBM}^+$\\ that is essentially equal to $\\mathrm{NBM}$, except for a certain set of type profiles. For these type profiles, $\\mathrm{NBM}^+$\\ chooses the allocation selected by $\\mathrm{PS}$. \nIt is easy to show that $\\mathrm{NBM}^+$\\ ordinally dominates $\\mathrm{NBM}$, and the dominance may be strict.\nTo show that $\\mathrm{NBM}^+$\\ satisfies upper invariance, we argue that under any swap, the mechanism's changes to the allocation are consistent with the axiom. In particular, this is true for transitions between the special type profiles where $\\mathrm{NBM}^+$\\ and $\\mathrm{NBM}$\\ choose different allocations and those type profiles where the allocations of both mechanisms are equal.\n\\end{proof}\n\\subsection{NBM versus RSD - The Price of Strategyproofness}\n\\label{sec:nbm:nbm_ird_RSD}\nSo far we have not identified any efficiency advantage of $\\mathrm{NBM}$\\ over $\\mathrm{RSD}$. \nIn this section, we present our first main result, which shows in what sense $\\mathrm{NBM}$\\ is in fact more efficient than $\\mathrm{RSD}$.\n\\subsubsection{Imperfect Rank Dominance of NBM over RSD}\n\\label{sec:nbm:nbm_ird_RSD:nbm_ird_rsd}\n\\begin{theorem} \\label{thm:rsd_not_rd_nbm} $\\mathrm{NBM}$\\ strictly imperfectly rank dominates $\\mathrm{RSD}$, i.e., $\\mathrm{NBM}$\\ strictly rank dominates $\\mathrm{RSD}$\\ at some type profiles, but is never strictly rank dominated by $\\mathrm{RSD}$\\ at any type profile.\n\\end{theorem}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:nbm:rsd_not_rd_nbm})]\nProposition \\ref{prop:nbm_rank_dom_ps} has already established that $\\mathrm{NBM}$\\ may strictly rank dominate $\\mathrm{RSD}$.\nFor any fixed priority order $\\pi$ we show that if $\\mathrm{SD}_{\\pi}$ and $\\mathrm{NBM}_{\\pi}$ allocate the same number of first choices, they will in fact allocate these first choices to the same agents. We can remove these agents and the corresponding objects and proceed by induction, carefully handling the case when some object has capacity zero. Averaging across all priority orderings, we find that if $\\mathrm{RSD}$\\ rank dominates $\\mathrm{NBM}$, the allocation from both mechanism must be the same, and therefore $\\mathrm{RSD}$\\ never strictly rank dominates $\\mathrm{NBM}$.\n\\end{proof}\nImperfect rank dominance of $\\mathrm{NBM}$\\ over $\\mathrm{RSD}$\\ by Theorem \\ref{thm:rsd_not_rd_nbm} can be viewed as a lower bound on the price of strategyproofness, i.e., by choosing $\\mathrm{RSD}$\\, we forego some unambiguous improvements of the rank distribution that could be attained by using $\\mathrm{NBM}$\\ instead.\n\\begin{remark} \\label{rem:extension_priorities} Theorem \\ref{thm:rsd_not_rd_nbm} remains valid even if some priorities are imposed:\nlet $\\Pi$ be the set of all possible priority orderings, i.e., bijections $\\pi: N \\rightarrow \\{1,\\ldots,n\\}$, and consider any probability distribution $\\mathds{P}$ on $\\Pi$.\nMechanisms $\\mathrm{RSD}_{\\mathds{Pi}}$ and $\\mathrm{NBM}_{\\mathds{Pi}}$ can be defined analogously to Definitions \\ref{def:rsd} and \\ref{def:nbm}, but each mechanism draws the priority ordering $\\pi$ from $\\mathds{P}$ instead of the uniform distribution.\nThe proof of Theorem \\ref{thm:rsd_not_rd_nbm} considers every priority ordering separately, and thus, $\\mathrm{RSD}_{\\mathds{Pi}}$ never strictly ordinally dominates $\\mathrm{NBM}_{\\mathds{Pi}}$.\n\nIn this way it is possible to give priority to some agent $i$ over another agent $i'$, e.g., by setting $\\mathds{P}(\\pi) = 0$ for all $\\pi$ with $\\pi(i) > \\pi(i')$. Thus, Theorem \\ref{thm:rsd_not_rd_nbm} remains valid if some groups of agents (e.g., minorities) are given preference \\emph{at all objects}. \nMore complex priority structures where priorities can vary across different objects (e.g., walk zone priorities) lie outside the scope of this result.\n\\end{remark}\nWe conclude that $\\mathrm{NBM}$\\ is more efficient than the strategyproof baseline mechanism $\\mathrm{RSD}$.\nThis result differs from prior attempts to compare Boston mechanisms to other mechanism, which relied on stylized assumptions about agents' preference.\nOur results highlight that choosing between $\\mathrm{NBM}$\\ and $\\mathrm{RSD}$\\ involves a non-trivial decision about the trade-off between strategyproofness and efficiency.\n\\subsubsection{Simulation Results}\n\\label{sec:nbm:nbm_ird_RSD:simulation}\nTo determine how strongly $\\mathrm{NBM}$\\ imperfectly rank dominates $\\mathrm{RSD}$, we conducted simulations: for settings with $n=m \\in \\{3, \\ldots, 10\\}$ and $q_j = 1 $ for all $j \\in M$, we sampled 100'000 type profiles uniformly at random for each value of $n$. We then determined the rank dominance relation between the resulting allocations under $\\mathrm{NBM}$\\ and $\\mathrm{RSD}$\\ at the sampled type profiles.\nFigure \\ref{fig:nbm_rsd} shows how often $\\mathrm{NBM}$\\ strictly rank dominates (red bars) or has the same rank distribution as $\\mathrm{RSD}$\\ (green bars), and how often the two mechanisms are incomparable (blue bars). We see that the share where $\\mathrm{NBM}$\\ strictly rank dominates $\\mathrm{RSD}$\\ shrinks for an increasing number of objects, but only slowly.\n\\begin{figure}%\n\\includegraphics[width=\\columnwidth]{NBM_RSD_1.pdf}%\n\\caption{Share of type profiles by rank dominance relation between $\\mathrm{NBM}$ and $\\mathrm{RSD}$, for settings with unit capacity, $n=m$ agents, and 100'000 sample type profiles for each $n$.}%\n\\label{fig:nbm_rsd}%\n\\end{figure}\n\\section{The Adaptive Boston Mechanism}\n\\label{sec:abm}\nThis section contains our second main contribution: we define the adaptive Boston mechanism ($\\mathrm{ABM}$) and study its properties.\nThe main take-away is that $\\mathrm{ABM}$\\ has significantly better incentive properties than $\\mathrm{NBM}$, but has almost the same efficiency advantages over $\\mathrm{RSD}$.\n\nLike $\\mathrm{NBM}$, $\\mathrm{ABM}$\\ is manipulable. However, we show that, in contrast to $\\mathrm{NBM}$, it is partially strategyproof, i.e., strategyproof on the domain of uniformly bounded indifference.\nLike $\\mathrm{NBM}$, $\\mathrm{ABM}$\\ is ex-post efficient, but neither ordinally nor rank efficient.\nWe also show that it is not on the efficient frontier, subject to partial strategyproofness.\nThus, we are in the same situation as with $\\mathrm{NBM}$, where traditional efficiency measures cannot be used to distinguish between $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$.\nTherefore, we compare $\\mathrm{ABM}$\\ to $\\mathrm{RSD}$\\ in terms of rank dominance.\nRather surprisingly, we find that $\\mathrm{RSD}$\\ may rank dominate $\\mathrm{ABM}$\\ at certain type profiles. However, we also show that these cases are extremely rare, i.e., $\\mathrm{ABM}$\\ imperfectly rank dominates $\\mathrm{RSD}$\\ in the limit as markets become large. Numerically, we find evidence that the efficiency advantages of $\\mathrm{ABM}$\\ over $\\mathrm{RSD}$\\ are almost the same as those of $\\mathrm{NBM}$\\ over $\\mathrm{RSD}$\\ and that the theoretical possibility that $\\mathrm{RSD}$\\ may rank dominate $\\mathrm{ABM}$\\ is negligible.\n\\subsection{Formalization of ABM}\n\\label{sec:abm:formalization}\nThe adaptive Boston mechanism is designed to eliminate the obvious manipulations described in the introduction. Instead of applying to their $k$th choice object in the $k$th round, in each round agents apply to the object that they prefer most out of the objects that are still available. Formally:\n\\begin{definition} \\label{def:abm} The \\emph{adaptive Boston mechanism ($\\mathrm{ABM}$)} works as follows:\n\\begin{enumerate}\n\t\\item Collect all type reports $\\hat{\\mathbf{t}}=(\\hat{t}_1,\\ldots,\\hat{t}_n) \\in T^n $ from the agents.\n\t\\item Draw an ordering $\\pi$ of all agents uniformly at random from the space of all possible bijections $\\pi: N \\rightarrow \\{1,\\ldots,n\\}$. $\\pi(i) < \\pi(i')$ means that $i$ has priority over $i'$.\n\t\\item Round 1: all agents apply to their reported first choice object. At every object, if the object's capacity is sufficient to satisfy all applications to this object, all the applicants get a copy of the object. Otherwise, the agents with the highest priority get copies of the object until it is exhausted. All remaining agents do not receive a copy and enter the next round.\n\t\\item Round $k$: an agent who did not receive an object in any of the previous $k-1$ rounds applies to the object which it reported to prefer most out of the objects with non-zero remaining capacity. Analogous to round 1, the agents with the highest priorities are allocated copies of the objects for which they applied. If an object has more applicants than capacity, the remaining agents enter the next round.\n\t\\item The process ends when all agents have been allocated an object.\n\\end{enumerate}\n\\end{definition}\nLet $\\mathrm{ABM}_{\\pi}$ denote the adaptive Boston mechanism when the mechanism is used for a single, fixed priority order $\\pi$, in which case the resulting allocation is deterministic. We have\n\\begin{equation}\n\\mathrm{ABM}(\\hat{\\mathbf{t}}) = \\sum_{\\pi:N\\rightarrow\\{1,\\ldots,n\\}\\mathrm{\\ bijection}} \\frac{1}{n!} \\mathrm{ABM}_{\\pi}(\\hat{\\mathbf{t}}).\n\\label{eq:prob_alloc_abm}\n\\end{equation}\n\\subsection{Partial Strategyproofness of ABM}\n\\label{sec:abm:psp}\nIn this section, we show that $\\mathrm{ABM}$\\ is partially strategyproof and we asses its degree of strategyproofness.\n\\subsubsection{Partial Strategyproofness Result for ABM}\n\\label{sec:abm:psp:result}\nLike $\\mathrm{NBM}$, $\\mathrm{ABM}$\\ it is upper invariant, but not strategyproof. However, we also show that $\\mathrm{ABM}$\\ is swap monotonic. Thus, $\\mathrm{ABM}$\\ is partially strategyproof, while $\\mathrm{NBM}$\\ is not even weakly strategyproof.\n\\begin{proposition} \\label{prop:abm_ui} $\\mathrm{ABM}$\\ is upper invariant.\n\\end{proposition}\nThe proof of Proposition \\ref{prop:abm_ui} is essentially the same as for Proposition \\ref{prop:nbm_ui}.\n$\\mathrm{ABM}$\\ is also swap monotonic, as Theorem \\ref{thm:abm_sm} shows.\n\\begin{theorem} \\label{thm:abm_sm} $\\mathrm{ABM}$\\ is swap monotonic.\n\\end{theorem}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:abm:abm_sm})] We consider each priority ordering $\\pi$ separately. \nFor each $\\pi$, we consider the reaction of the mechanism to a swap from $t_i:a \\succ b$ to $t_i':b \\succ a$ by some agent $i$.\nWe show that\n\\begin{itemize}\n\t\\item if $\\pi$ leads to an allocation of $b$ to $i$ under $t$, it will still lead to an allocation of $b$ under $t'$, i.e., the probability that $b$ is allocated to $i$ weakly increases,\n\t\\item if $\\pi$ does not lead to an allocation of $a$ to $i$ under $t$, it will not lead to an allocation of $a$ to $i$ under $t'$ either, i.e., the probability that $a$ is allocated to $i$ weakly decreases,\n\t\\item if the allocation changes at all under some $\\pi$, we can construct another ordering $\\pi'$, such that under $\\pi'$ a change of report from $t$ to $t'$ leads to a change in allocation from $a$ to $b$.\n\\end{itemize}\nThus, the probabilities for $a$ and $b$ change strictly if the allocation changes at all, which corresponds to the requirement of swap monotonicity.\n\\end{proof}\n\\begin{corollary} \\label{cor:abm_psp} $\\mathrm{ABM}$\\ is partially strategyproof.\n\\end{corollary}\nCorollary \\ref{cor:abm_psp} follows immediately if we consider that $\\mathrm{ABM}$\\ is swap monotonic (Theorem \\ref{thm:abm_sm}) and upper invariant (Proposition \\ref{prop:abm_ui}) and hence partially strategyproof (Fact \\ref{app_fact:char_psp} in Appendix \\ref{app:review_psp}).\n\\subsubsection{Degree of Strategyproofness of ABM}\n\\label{sec:abm:psp:dosp}\nSince $\\mathrm{ABM}$\\ is partially strategyproof, the degree of strategyproofness measure\n\\begin{equation}\n\t\\rho_{(N,M,\\mathbf{q})}(\\mathrm{ABM}) = \\max\\{r \\in [0,1] : \\mathrm{ABM}\\text{ is }r\\text{-partially strategyproof in }(N,M,\\mathbf{q})\\}\n\\end{equation}\nis positive for any setting.\nWe are interested in learning ``how strategyproof'' $\\mathrm{ABM}$\\ is in different settings.\nUnfortunately, $\\rho_{(N,M,\\mathbf{q})}(\\mathrm{ABM})$ becomes small when the number of objects increases (assuming unit capacity and an equal number of objects and agents), which means intuitively that $\\mathrm{ABM}$\\ becomes ``less strategyproof.''\n\\begin{proposition} \\label{prop:dosp_abm_to_zero} For settings where $n = m, q_j=1$ for all $j \\in M$\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty}\\rho_{(N,M,\\mathbf{q})}(\\mathrm{ABM}) = 0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:abm:dosp_abm_to_zero})] We construct a particular sequence of type profiles.\nIn each step, we consider a certain agent and show that this agent can exchange very little probability at its first choice for substantial probability at its second choice.\nThis trade-off is beneficial for the agent, unless the relative difference in valuation between the first and second choice is very high.\n\\end{proof}\nAlthough the degree of strategyproofness becomes small for large numbers of objects and agents, numerical evidence suggests that for a fixed number of objects $m$ and evenly distributed capacity ($q_j = q$ for all $j \\in M$), the degree of strategyproofness does not decrease as more agents (and more capacity) are added to the setting, as long as the capacity is evenly distributed. This can be seen in Table \\ref{tbl:rho_abm} and is formalized by the following conjecture.\n\\begin{table}%\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\t\t\t\t\t\t\t\t\t\t\t\t& \\multicolumn{3}{c|}{Capacity $q$} \\\\\n\\hline\n\tNumber of objects $m$ & $1$ & $2$ & $3$ \\\\\n\\hline\n\\hline\n\t$3$\t\t\t\t\t\t\t\t\t\t& $0.5$ & $0.5$ & $0.5$ \\\\\n\\hline\n\t$4$ \t\t\t\t\t\t\t\t\t& $0.33$ & $0.33$ & ? \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Degree of strategyproofness of ABM with balanced capacities and $n =m q$ agents.}\n\\label{tbl:rho_abm}\n\\end{table}\n\\begin{conjecture} \\label{conj:rho_abm_const} For $q \\neq q'$ and two settings $(N,M,\\mathbf{q})$ and $(N',M',\\mathbf{q}')$ with\n\\begin{itemize}\n\t\\item $M = M'$,\n\t\\item $\\mathbf{q} = (q,\\ldots,q)$ and $\\mathbf{q}' = (q',\\ldots,q')$,\n\t\\item $n = \\# N = q m$ and $n' = \\#N' = q' m$,\n\\end{itemize}\nwe have\n\\begin{equation}\n\t\\rho_{(N,M,\\mathbf{q})}(\\mathrm{ABM}) = \t\\rho_{(N',M',\\mathbf{q}')}(\\mathrm{ABM}).\n\\end{equation}\n\\end{conjecture}\n\\subsection{Efficiency of ABM}\n\\label{sec:abm:efficiency_failure}\nAnalogous to our findings for $\\mathrm{NBM}$, the traditional methods fail to differentiate between $\\mathrm{ABM}$\\ and $\\mathrm{RSD}$\\ in terms of efficiency. \n$\\mathrm{ABM}$\\ is ex-post efficient, but neither ordinally nor rank efficient, nor on the efficient frontier, even if we consider only partially strategyproof mechanisms.\n\\begin{proposition} \\label{prop:abm_ex_post} $\\mathrm{ABM}$\\ is ex-post efficient at all type profiles.\n\\end{proposition}\nThe proof of Proposition \\ref{prop:abm_ex_post} is analogous to the proof of Proposition \\ref{prop:nbm_ex_post}.\n\\begin{proposition} \\label{prop:abm_not_OE} $\\mathrm{ABM}$\\ is neither ordinally efficient nor rank efficient.\n\\end{proposition}\nProposition \\ref{prop:abm_not_OE} follows from the same example as Proposition \\ref{prop:nbm_not_OE}, because in the specific setting and type profile the results of $\\mathrm{ABM}$\\ and $\\mathrm{NBM}$\\ coincide.\n\\begin{proposition} \\label{prop:abm_rank_dom_ps}\n$\\mathrm{PS}$\\ does not ordinally dominate $\\mathrm{ABM}$\\ at all type profiles, and $\\mathrm{ABM}$\\ rank dominates $\\mathrm{PS}$\\ at some type profile.\n\\end{proposition}\nProposition \\ref{prop:abm_rank_dom_ps} follows from the same example as Proposition \\ref{prop:nbm_rank_dom_ps}.\n\\begin{proposition} \\label{prop:abm_not_frontier}\n$\\mathrm{ABM}$\\ is not on the efficient frontier with respect to ordinal dominance, subject\nto partial strategyproofness\n\\end{proposition}\nThe proof is analogous to the proof of Proposition \\ref{prop:nbm_not_frontier}.\n\\begin{remark} The mechanism $\\mathrm{ABM}^+$\\ constructed in the proof of Propositions \\ref{prop:nbm_not_frontier} \/ \\ref{prop:abm_not_frontier} even satisfies partial strategyproofness.\nIt remains an open question for future research whether compared to $\\mathrm{ABM}$, $\\mathrm{ABM}^+$\\ has the same, higher, or lower degree of strategyproofness.\n\\end{remark}\n\\subsection{ABM versus RSD - Imperfect Rank Dominance in Large Markets}\n\\label{sec:abm:abm_ird_rsd}\nIn the previous section, we have shown that the efficiency of $\\mathrm{ABM}$\\ is not differentiable from the efficiency of $\\mathrm{RSD}$\\ by traditional methods.\nThus, we again consider a direct comparison of $\\mathrm{ABM}$\\ and $\\mathrm{RSD}$\\ in terms of imperfect rank dominance.\nSurprisingly, $\\mathrm{ABM}$\\ does not imperfectly rank dominate $\\mathrm{RSD}$, i.e., there exists at least one type profile at which $\\mathrm{RSD}$\\ actually produces a rank dominant allocation.\nHowever, we recover imperfect rank dominance of $\\mathrm{ABM}$\\ over $\\mathrm{RSD}$\\ for large markets, and numerical evidence suggests that the violation of imperfect dominance is insignificant, even in finite markets.\n\\subsubsection{Failure of Imperfect Rank Dominance}\n\\label{sec:abm:abm_ird_rsd:failure_ird}\nThe following Proposition yields the surprising result that $\\mathrm{RSD}$\\ may rank dominate $\\mathrm{ABM}$.\n\\begin{proposition} \\label{prop:abm_not_ird_rsd} $\\mathrm{ABM}$\\ does not imperfectly rank dominate $\\mathrm{RSD}$.\n\\end{proposition}\n\\begin{proof} We present a counter example, i.e., a setting and type profile for which $\\mathrm{RSD}$\\ strictly rank dominates $\\mathrm{ABM}$.\nConsider a setting with 6 agents, 6 objects in unit capacity, and the type profile\n\\begin{eqnarray*}\nt_1, \\ldots ,t_4 & : & a \\succ b \\succ c \\succ d \\succ e \\succ f, \\\\\nt_5, t_6 & : & e \\succ b \\succ a \\succ d \\succ f \\succ c.\n\\end{eqnarray*}\nConsider the ordering $\\pi = \\mathrm{id}$ of the agents. With this ordering, $\\mathrm{RSD}$\\ will allocate objects $a$ through $d$ to agents $1$ through $4$. Agents $5$ and $6$ get objects $e$ and $f$. Thus, $\\mathrm{RSD}$\\ allocates 2 first, 1 second, 1 third, 1 fourth, and 1 fifth choice. For the same ordering, $\\mathrm{ABM}$\\ will allocate $a,b,c$ to agents $1, 2, 3$, respectively. Agents $5$ and $6$ will get $e$ and $d$, which leaves agent $4$ with $f$. Observe that with this ordering, $\\mathrm{ABM}$\\ allocates 2 first, 1 second, 1 third, 1 fourth, no fifth, and 1 sixth choice. This is strictly rank dominated by the rank efficient allocation chosen by $\\mathrm{RSD}$.\n\nThe allocation under $\\mathrm{ABM}$\\ is\n\\begin{equation}\n\\mathrm{ABM}(\\mathbf{t}) = \\frac{1}{60}\\left(\\begin{array}{cccccc}\n15 & 12 & 15 & 3 & 0 & 15 \\\\\n15 & 12 & 15 & 3 & 0 & 15 \\\\\n15 & 12 & 15 & 3 & 0 & 15 \\\\\n15 & 12 & 15 & 3 & 0 & 15 \\\\\n0 & 6 & 0 & 24 & 30 & 0 \\\\\n0 & 6 & 0 & 24 & 30 & 0\n\\end{array}\\right),\n\\end{equation}\nand the allocation under $\\mathrm{RSD}$\\ is\n\\begin{equation}\n\\mathrm{RSD}(\\mathbf{t}) = \\frac{1}{60}\\left(\\begin{array}{cccccc}\n15 & 12 & 15 & 8 & 0 & 10 \\\\\n15 & 12 & 15 & 8 & 0 & 10 \\\\\n15 & 12 & 15 & 8 & 0 & 10 \\\\\n15 & 12 & 15 & 8 & 0 & 10 \\\\\n0 & 6 & 0 & 14 & 30 & 10 \\\\\n0 & 6 & 0 & 14 & 30 & 10\n\\end{array}\\right).\n\\end{equation}\nThe fact that all allocations chosen by $\\mathrm{ABM}$\\ are rank inefficient, but the allocations chosen by $\\mathrm{RSD}$\\ are sometimes rank efficient implies that $\\mathrm{RSD}$\\ strictly rank dominates $\\mathrm{ABM}$\\ for this type profile.\n\\end{proof}\nThe example in the proof of Proposition \\ref{prop:abm_not_ird_rsd} raises the question how frequently $\\mathrm{RSD}$\\ rank dominates $\\mathrm{ABM}$.\nInterestingly, complete enumeration (using a computer) has revealed that $\\mathrm{RSD}$\\ does not rank dominate $\\mathrm{ABM}$\\ for any setting with less than 6 objects and unit capacities.\nIn Section \\ref{sec:abm:abm_ird_rsd:house_allocation}, we will show via simulation that even for more objects, such examples are pathological, i.e., out of 500'000 sampled type profiles (100'000 for each $n \\in \\{6,\\ldots,10\\}$), there was not a single occurrence.\n\nWith this intuition in mind, we ask what the rank dominance relation looks like in large markets.\nWe consider two independent notions of how settings ``get large.'' In Section \\ref{sec:abm:abm_ird_rsd:school_choice} we employ the first notion, which is adopted from \\citep{Kojima2010IncentivesInTheProbabilisticSerial}, where the capacities of a fixed number of objects increase. This approximates school choice settings, where the capacity of public schools frequently exceeds 100 seats.\nFor the second notion, which we use in Section \\ref{sec:abm:abm_ird_rsd:house_allocation}, we let the number of objects increase and they all have unit capacity. This is more adequate for house allocation problems, where every ``house'' is different and can be given to only one agent.\n\\subsubsection{Limit Result for Large School Choice Markets}\n\\label{sec:abm:abm_ird_rsd:school_choice}\nIn this section, we present our first limit result, showing that as the capacity of the objects increases the share of type profiles where $\\mathrm{RSD}$\\ rank dominates $\\mathrm{ABM}$\\ becomes arbitrarily small.\n\nBefore we formulate our result, we introduce first-choice-maximizing allocations and provide an intuition of why the convergence holds.\nIn words, an allocation $x$ is \\emph{first-choice-maximizing} at type profile $\\mathbf{t}$ if it can be represented as a lottery over deterministic allocations that give the maximum number of first choices, i.e., where\n\\begin{equation}\n\td^x_1 = \\sum_{i \\in N} x_{i,\\mathrm{ch}(1,t_i)} = \\max_{y \\in \\mathcal{X}} d^y_1.\n\\end{equation}\n\n\\begin{remark} \\label{rem:rsd_fcm} We already observed that $\\mathrm{RSD}$\\ puts positive probability on \\emph{all} ex-post efficient, deterministic allocations, while $\\mathrm{ABM}$\\ assigns different probabilities, which may be zero.\nOne key observation is that $\\mathrm{ABM}$\\ maximizes the expected number of allocated first choices.\nThus, $\\mathrm{ABM}$\\ gives no weight to any allocation that does not yield the maximum possible number of first choices.\nIn contrast, if at type profile $\\mathbf{t}$ there exists any ex-post efficient deterministic allocation that is not first-choice-maximizing, then $\\mathrm{RSD}$\\ does not allocate the maximum expected number of first choices, i.e., it is not first-choice-maximizing.\n\\end{remark}\nUsing the observation from Remark \\ref{rem:rsd_fcm}, we can now prove the following Proposition \\ref{prop:RSD_not_first_choice_max_school_choice}, which in turn yields the limit result, Theorem \\ref{thm:rsd_not_rd_abm_in_the_limit_school_choice}.\n\\begin{proposition}\\label{prop:RSD_not_first_choice_max_school_choice} For any fixed number of objects $m \\geq 3$ and any $\\epsilon > 0$, there exists a $q_{\\min} \\in \\mathds{N}$, such that for any capacity vector $\\mathbf{q} = (q_1, \\ldots, q_m)$ with $q_j \\geq q_{\\min}$ for all $j \\in M$ and $n = \\sum_{j \\in M} q_j$ agents, and for $\\mathbf{t}$ chosen uniformly at random, the probability that $\\mathrm{RSD}(\\mathbf{t})$ is first-choice-maximizing is smaller than $\\epsilon$.\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:abm:prop:RSD_not_first_choice_max_school_choice})]\nConsider a fixed type profile $\\mathbf{t}$.\nAs discussed previously, $\\mathrm{RSD}$\\ is first-choice-maximizing only if all ex-post efficient deterministic allocations with respect to $\\mathbf{t}$ are first-choice-maximizing.\nTo prove the proposition, we show that an ex-post efficient allocation that is not first-choice-maximizing can almost always be found (for sufficiently high capacities).\nIntuitively, we exploit the following: if objects $j,j' \\in M$ are the first choice of more than $q_j$ and $q_{j'}$ agents, respectively, then we can construct a priority ordering $\\pi$ that ranks the agents with first choice $j$ first. Then under $\\mathrm{SD}_{\\pi}$ some $q_j$ agents get their first choice $j$, but it turns out to be ``unlikely'' that none of the remaining agents with first choice $j$ have $j'$ as their second choice.\n\\end{proof}\nThus, $\\mathrm{RSD}$\\ is almost never first-choice-maximizing as the capacities of the objects grow.\nSince $\\mathrm{ABM}$\\ is first-choice-maximizing, it will allocate strictly more first choices than $\\mathrm{RSD}$\\ at almost all type profiles.\nThus, even weak rank dominance of $\\mathrm{RSD}$\\ over $\\mathrm{ABM}$\\ almost always breaks already at the first choice.\n\\begin{theorem} \\label{thm:rsd_not_rd_abm_in_the_limit_school_choice} For any $m$ and any sequence of settings $(N^k,M^k,\\mathbf{q^k})_{k \\geq 0}$ with\n\\begin{itemize}\n\t\\item $\\# M^k = m$ (i.e., constant number of objects),\n\t\\item $ \\min_{j \\in M^k} q_j^k \\rightarrow \\infty$ as $k \\rightarrow \\infty$ (i.e., growing capacities),\n\t\\item $n_k = \\# N^k = \\sum_{j \\in M^k} q_j$ (i.e., total supply matches total demand),\n\\end{itemize}\nwe have\n\\begin{equation}\n\t\\frac{\\#\\{\\mathbf{t} \\in T^{n_k} | \\mathrm{RSD}(\\mathbf{t}) \\succeq_{R} \\mathrm{ABM}(\\mathbf{t}) \\text{ in the setting }(N^k,M^k,\\mathbf{q}^k) \\}}{\\#\\{\\mathbf{t}\\in T^{n_k}\\}} \\rightarrow 0 \\;\\;\\;(k \\rightarrow \\infty).\n\\end{equation}\n\\end{theorem}\n\n\n\\subsubsection{Limit Result for Large House Allocation Markets}\n\\label{sec:abm:abm_ird_rsd:house_allocation}\nIn this section, we present our second limit result for large house allocation markets: we show that as the number of objects with unit capacity and agents increases, the share of type profiles where RSD rank dominates ABM becomes arbitrarily small.\nAs in the previous section, we use the fact that $\\mathrm{RSD}$\\ is almost never first-choice-maximizing to prove the convergence result.\n\\begin{proposition} \\label{prop:rsd_not_first_choice_max_in_limit_house_allocation} For any $\\epsilon > 0$, there exists $n \\in \\mathds{N}$, such that for any setting $(N,M,\\mathbf{q})$ with $\\#M = \\#N \\geq n $ and $q_j =1 $ for all $j \\in M$, and for $\\mathbf{t}$ chosen uniformly at random, the probability that $\\mathrm{RSD}(\\mathbf{t})$ is first-choice-maximizing is smaller than $\\epsilon$.\n\\end{proposition}\n\\begin{proof}[Proof outline (formal proof in Appendix \\ref{app:proofs:abm:prop:rsd_not_first_choice_max_in_limit_house_allocation})] The proof idea is similar to the proof of Proposition \\ref{prop:RSD_not_first_choice_max_school_choice}, but the formal analysis requires advanced combinatorics. \nWe derive an upper bound for the probability that all agents with over-demanded first choices have second choices such that they do not exhaust some other agent's first choice. We then show that this upper bound converges to zero.\n\\end{proof}\nTheorem \\ref{thm:rsd_not_rd_abm_in_the_limit_house_allocation} follows from Proposition \\ref{prop:rsd_not_first_choice_max_in_limit_house_allocation} in the same way in which Theorem \\ref{thm:rsd_not_rd_abm_in_the_limit_school_choice} follows from Proposition \\ref{prop:RSD_not_first_choice_max_school_choice}.\n\\begin{theorem} \\label{thm:rsd_not_rd_abm_in_the_limit_house_allocation}\nFor settings with $n = m$ and $q_j = 1$, $\\mathrm{ABM}$\\ is not rank-dominated by $\\mathrm{RSD}$\\ in the limit, i.e.,\n\\begin{equation}\n\t\\frac{\\#\\{\\mathbf{t} \\in T^n | \\mathrm{RSD}(\\mathbf{t}) \\succeq_{R} \\mathrm{ABM}(\\mathbf{t}) \\}}{\\#\\{\\mathbf{t}\\in T^n\\}} \\rightarrow 0 \\;\\;\\;(n \\rightarrow \\infty).\n\\end{equation}\n\\end{theorem}\n\\subsubsection{Simulation Results}\n\\label{sec:abm:abm_ird_rsd:simulation}\nWe conducted simluatons similar to those in Section \\ref{sec:nbm:nbm_ird_RSD:simulation} to compare $\\mathrm{ABM}$\\ and $\\mathrm{RSD}$: \nfor settings with $n=m \\in \\{3, \\ldots, 10\\}$ and $q_j = 1 $ for all $j \\in M$, we sampled 100'000 type profiles uniformly at random for each value of $n$. We then determined the rank dominance relation between the resulting allocations under $\\mathrm{ABM}$\\ and $\\mathrm{RSD}$\\ at the sampled type profiles.\nFigure \\ref{fig:abm_rsd} shows how often $\\mathrm{ABM}$\\ strictly rank dominates (red bars) or has the same rank distribution (green bars) as $\\mathrm{RSD}$, and how often the two mechanisms are incomparable (blue bars).\nNote that the results are very similar to the findings for $\\mathrm{NBM}$\\ and $\\mathrm{RSD}$\\ (Figure \\ref{fig:nbm_rsd}), i.e., the share of profiles with comparability shrinks, but the share of those profiles where $\\mathrm{ABM}$\\ strictly rank dominates $\\mathrm{RSD}$\\ shrinks very slowly.\nFurthermore, the case where $\\mathrm{RSD}$\\ rank dominates $\\mathrm{ABM}$\\ never occurred in the entire sample. This suggests that such cases are extremely rare and should not matter for an evaluation of $\\mathrm{ABM}$\\ against $\\mathrm{RSD}$.\nWe conclude that $\\mathrm{ABM}$\\ is ``more efficient'' than $\\mathrm{RSD}$. \n\\section{NBM versus ABM - The Price of Partial Strategyproofness}\n\\label{sec:price_of_psp}\nIn Section \\ref{sec:nbm}, we have established the \\emph{price of strategyproofness} in terms of efficiency:\n$\\mathrm{RSD}$\\ is less efficient than $\\mathrm{NBM}$, a price we have to pay if we want to achieve full strategyproofness using $\\mathrm{RSD}$. \nWe will now assess the \\emph{price of partial strategyproofness} by comparing $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ in terms of rank dominance.\nContrary to intuition, this comparison is not unambiguous in favor of $\\mathrm{NBM}$: while $\\mathrm{NBM}$\\ strictly rank dominates $\\mathrm{ABM}$\\ at some type profiles, there also exist type profiles at which the opposite holds.\nFurthermore, we also present numerical evidence that the two mechanisms are usually not comparable by rank dominance, but if they are comparable, then dominance of $\\mathrm{NBM}$\\ over $\\mathrm{ABM}$\\ occurs more frequently than the opposite case.\nTherefore, when choosing $\\mathrm{ABM}$\\ in favor of $\\mathrm{NBM}$, the fact that $\\mathrm{NBM}$\\ rank dominates $\\mathrm{ABM}$\\ more often than vice versa can be considered a price that we pay for partial strategyproofness. Of course, since neither mechanism is on the efficient frontier, there may be other, more desirable trade-offs that are not accessible via $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$.\n\\begin{figure}%\n\\includegraphics[width=\\columnwidth]{ABM_RSD_1.pdf}%\n\\caption{Share of type profiles by rank dominance relation between $\\mathrm{ABM}$ and $\\mathrm{RSD}$, for settings with unit capacity, $m=n$ agents, and 100'000 sampled type profiles for each $n$.}%\n\\label{fig:abm_rsd}%\n\\end{figure}\n\\subsection{Failure of Imperfect Rank Dominance}\n\\label{sec:price_of_psp:failure_ird}\nWe first attempt a direct comparison between $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ via imperfect rank dominance. Interestingly, $\\mathrm{NBM}$\\ is not always the more efficient mechanism, i.e., while there exists type profiles at which $\\mathrm{NBM}$\\ strictly rank dominates $\\mathrm{ABM}$\\ (Proposition \\ref{prop:nbm_rank_dom_abm}), there also exist type profiles where the opposite holds (Proposition \\ref{prop:abm_rank_dom_nbm}).\n\\begin{proposition} \\label{prop:nbm_rank_dom_abm} $\\mathrm{NBM}$\\ strictly rank dominates $\\mathrm{ABM}$\\ at some type profiles.\n\\end{proposition}\n\\begin{proof} Consider the setting with $N = \\{1,\\ldots,4\\}$, $M = \\{a,b,c,d\\}$, unit capacities, and the type profile\n\\begin{eqnarray*}\n\tt_1 & : & a \\succ b \\succ c \\succ d, \\\\\n\tt_2,t_3 & : & a \\succ c \\succ d \\succ b, \\\\\n\tt_4 & : & b \\succ \\ldots,\n\\end{eqnarray*}\nwhich is the same as Example \\ref{ex:nbm_more_manip}.\nThe allocations from $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ are\n\\begin{equation}\n\\mathrm{NBM}(\\mathbf{t}) = \\left(\\begin{array}{cccc}\n\\frac{1}{3} & 0 & 0 & \\frac{2}{3} \\\\\n\\frac{1}{3} & 0 & \\frac{1}{2} & \\frac{1}{6} \\\\\n\\frac{1}{3} & 0 & \\frac{1}{2} & \\frac{1}{6} \\\\\n0 & 1 & 0 & 0 \t\n\\end{array}\\right)\n\\text{ and }\n\\mathrm{ABM}(\\mathbf{t}) = \\left(\\begin{array}{cccc}\n\\frac{1}{3} & 0 & \\frac{1}{3} & \\frac{1}{3} \\\\\n\\frac{1}{3} & 0 & \\frac{1}{3} & \\frac{1}{3} \\\\\n\\frac{1}{3} & 0 & \\frac{1}{3} & \\frac{1}{3} \\\\\n0 & 1 & 0 & 0 \t\n\\end{array}\\right).\n\\end{equation}\nThe rank distribution under $\\mathrm{NBM}$\\ is $\\mathbf{d} =(2,1,0,1)$, which dominates $\\mathbf{d}' = (2,2\/3,1\/3,1)$, the rank distribution under $\\mathrm{ABM}$.\n\\end{proof}\nThe next result is more surprising: it shows that $\\mathrm{ABM}$\\ may strictly rank dominate $\\mathrm{NBM}$.\nNote that the result holds for strict priorities as well.\n\\begin{proposition} \\label{prop:abm_rank_dom_nbm} $\\mathrm{ABM}$\\ strictly rank dominates $\\mathrm{NBM}$\\ at some type profiles.\n\\end{proposition}\n\\begin{proof} Consider the setting with $N = \\{1,\\ldots,5\\}$, $M = \\{a,b,c,d,e\\}$, unit capacities, and the type profile\n\\begin{eqnarray*}\n\tt_1,t_2 & : & a \\succ b \\succ c \\succ d \\succ e, \\\\\n\tt_3,t_4 & : & a \\succ d \\succ c \\succ e \\succ b, \\\\\n\tt_5 & : & b \\succ \\ldots.\n\\end{eqnarray*}\nThe allocations from $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ are\n\\begin{equation}\n\\mathrm{NBM}(\\mathbf{t}) = \\frac{1}{60}\\left(\\begin{array}{ccccc}\n15 & 0 & 25 & 0 & 20 \\\\\n15 & 0 & 25 & 0 & 20 \\\\\n15 & 0 & 5 & 30 & 10 \\\\\n15 & 0 & 5 & 30 & 10 \\\\\n0 & 60 & 0 & 0 & 0\n\\end{array}\\right)\n\\text{ and }\n\\mathrm{ABM}(\\mathbf{t}) = \\frac{1}{60}\\left(\\begin{array}{ccccc}\n15 & 0 & 30 & 0 & 15 \\\\\n15 & 0 & 30 & 0 & 15 \\\\\n15 & 0 & 0 & 30 & 15 \\\\\n15 & 0 & 0 & 30 & 15 \\\\\n0 & 60 & 0 & 0 & 0\n\\end{array}\\right).\n\\end{equation}\nThe rank distribution under $\\mathrm{NBM}$\\ is $\\mathbf{d} =(2,1,1,1\/3,2\/3)$, which is dominated by $\\mathbf{d}' = (2,1,1,1\/2,1\/2)$, the rank distribution under $\\mathrm{ABM}$.\n\\end{proof}\n\\begin{remark} \\label{rem:strengthen_dur}\nRecall that a \\emph{rank valuation} is a vector that specifies the value of allocating first, second, third, etc. choices to agents.\nIf an allocation $x$ rank dominates another allocation $y$, then the aggregate rank value is higher under $x$ than under $y$ for any rank valuation $v$, i.e.,\n\\begin{equation}\n\t\\sum_{i \\in M}\\sum_{k \\in \\{1,\\ldots,m\\}} v_k \\cdot x_{i,\\mathrm{ch}(k,t_i)} \\geq \\sum_{i \\in M}\\sum_{k \\in \\{1,\\ldots,m\\}} v_k \\cdot y_{i,\\mathrm{ch}(k,t_i)}.\n\\end{equation}\n\\citet{Dur2013} showed that for the case of strict priorities and for \\emph{a particular rank valuation}, $\\mathrm{ABM}$\\ can lead to a higher aggregate rank value than $\\mathrm{NBM}$.\nProposition \\ref{prop:abm_rank_dom_nbm} strengthens this result, since rank dominance of $\\mathrm{ABM}$\\ over $\\mathrm{NBM}$\\ implies that $\\mathrm{ABM}$\\ has higher aggregate rank value for any rank valuation.\n\\end{remark}\n\\subsection{Simulation Results}\n\\label{sec:price_of_psp:simulation}\nSo far, we have found that $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ are incomparable by imperfect rank dominance, because $\\mathrm{NBM} \\succ_{R} \\mathrm{ABM}$ for some type profiles, but $\\mathrm{ABM} \\succ_{R} \\mathrm{NBM}$ for others.\nThis raises the question whether one of the cases is more common.\nNumerically we find that $\\mathrm{NBM}$\\ is ``usually'' more efficient. \n\nAt each type profile $\\mathbf{t}$, one of the following cases can occur:\n\\begin{enumerate}\n\t\\item $\\mathrm{NBM}(\\mathbf{t})$ strictly rank dominates $\\mathrm{ABM}(\\mathbf{t})$,\n\t\\item $\\mathrm{NBM}(\\mathbf{t})$ and $\\mathrm{ABM}(\\mathbf{t})$ have the same rank distribution,\n\t\\item $\\mathrm{ABM}(\\mathbf{t})$ strictly rank dominates $\\mathrm{NBM}(\\mathbf{t})$,\n\t\\item $\\mathrm{NBM}(\\mathbf{t})$ and $\\mathrm{ABM}(\\mathbf{t})$ are incomarable by rank dominance.\n\\end{enumerate}\nFigure \\ref{fig:nbm_abm} shows how often each of these cases occurs when sampling type profiles uniformly at random in settings with unit capacities ($q_j = 1$ for all $j \\in M$)  and various $n(=m)$.\nThe results suggest that the two mechanisms are usually not comparable (blue bars).\nHowever, conditioned on comparability, $\\mathrm{NBM}$\\ rank dominates $\\mathrm{ABM}$\\ more often than vice versa (red vs. yellow bars), and the share of profiles where the rank distributions coincide shrinks.\n\n\\begin{remark}\nEven though $\\mathrm{NBM}$\\ outperforms $\\mathrm{ABM}$\\ in this comparison, the share of type profiles for which $\\mathrm{ABM}$\\ dominates $\\mathrm{NBM}$\\ is substantially larger than the share of profiles where $\\mathrm{RSD}$\\ dominates $\\mathrm{ABM}$: conditional on comparability, $\\mathrm{ABM}$\\ dominates $\\mathrm{NBM}$\\ (yellow bars) in about $0.8\\%$ of the cases for $n \\in \\{7,8,9,10\\}$. Thus, while dominance of $\\mathrm{RSD}$\\ over $\\mathrm{ABM}$\\ is negligible, the out-performance is less pronounced for $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$.\n\\end{remark}\n\\begin{figure}%\n\\includegraphics[width=\\columnwidth]{NBM_ABM_1.pdf}%\n\\caption{Share of type profiles by rank dominance relation between $\\mathrm{NBM}$ and $\\mathrm{ABM}$, for settings with unit capacity, $m=n$ agents, and 100'000 sampled type profiles for each $n$.}%\n\\label{fig:nbm_abm}%\n\\end{figure}\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper, we have studied the traditional (na\\\"{i}ve) and a new (adaptive) variant of the Boston mechanism in the absence of priorities.\nOur findings demonstrate the trade-off between efficiency and strategyproofness that is achievable by these mechanisms.\n\nFirst, we have shown that the na\\\"{i}ve Boston mechanism ($\\mathrm{NBM}$) strictly imperfectly rank dominates $\\mathrm{RSD}$, and we have presented numerical evidence suggesting that strict dominance occurs for a considerable share of type profiles.\nOn the other hand, $\\mathrm{RSD}$\\ is strategyproof, while $\\mathrm{NBM}$\\ is not even weakly strategyproof.\nThus, the efficiency advantages of $\\mathrm{NBM}$\\ over $\\mathrm{RSD}$\\ can be interpreted as a lower bound for the \\emph{price of strategyproofness} in terms of efficiency.\n\nSecond, we have introduced the adaptive Boston mechanism ($\\mathrm{ABM}$).\nWe have shown that it satisfies partial strategyproofness, a relaxed notion of strategyproofness, which $\\mathrm{NBM}$\\ fails to satisfy. Furthermore, $\\mathrm{ABM}$\\ has almost the same efficiency advantages over $\\mathrm{RSD}$\\ as $\\mathrm{NBM}$: \nthe imperfect rank dominance of $\\mathrm{ABM}$\\ over $\\mathrm{RSD}$\\ has similar magnitude, and the share of type profiles where $\\mathrm{RSD}$\\ rank dominants $\\mathrm{ABM}$\\ converges to 0 as the markets get large.\n\nThird, $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ are incomparable by imperfect rank dominance. \nVia simulations we have shown that - conditioned on comparability - $\\mathrm{NBM}$\\ frequently rank dominates $\\mathrm{ABM}$\\ (but not always). \nThese efficiency advantages of $\\mathrm{NBM}$\\ over $\\mathrm{ABM}$\\ can be viewed as the \\emph{price of partial strategyproofness} one pays when choosing $\\mathrm{ABM}$\\ over $\\mathrm{NBM}$.\n\nThroughout the paper, it has become apparent that traditional analysis methods frequently fail to differentiate between two matching mechanisms.\nWhile a comparison by \\emph{vulnerability to manipulation} is inconclusive for $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$, except in the most basic case (see Appendix \\ref{app:manipulability_comparison}), \\emph{partial strategyproofness} clearly captures the better incentive properties of $\\mathrm{ABM}$. Similarly, all mechanisms we considered are ex-post efficient, but neither ordinally nor rank efficient. Furthermore, $\\mathrm{NBM}$\\ and $\\mathrm{ABM}$\\ are not even on the efficient frontier. Nonetheless, a comparison by \\emph{imperfect rank dominance}, \\emph{limit arguments}, and \\emph{numerical analysis} revealed a hierarchy between $\\mathrm{NBM}$, $\\mathrm{ABM}$, and $\\mathrm{RSD}$: $\\mathrm{NBM}$\\ has the most appeal in terms of rank dominance, but $\\mathrm{ABM}$\\ is still more appealing than $\\mathrm{RSD}$;\non the other hand, $\\mathrm{NBM}$\\ exhibits the worst incentive properties, while $\\mathrm{ABM}$\\ is at least partially strategyproof, and $\\mathrm{RSD}$\\ is fully strategyproof.\n\nIn summary, we have found that a decision between $\\mathrm{NBM}$, $\\mathrm{ABM}$, and $\\mathrm{RSD}$\\ requires a non-trivial trade-off between strategyproofness and efficiency.\nOur new methods facilitate the comparison in situations where traditional methods fail to differentiate, but where a decision is nonetheless essential.\nFurthermore, $\\mathrm{ABM}$\\ constitutes an attractive design alternative to $\\mathrm{NBM}$\\ when incentive properties are a concern: $\\mathrm{ABM}$\\ has almost the same efficiency advantages over the strategyproof alternative $\\mathrm{RSD}$, but it has significantly better incentive properties than $\\mathrm{NBM}$.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince their discovery, single-walled carbon nanotubes (CNTs) have attracted major research interest due to their extraordinary mechanical, chemical and electronic properties \\cite{saito98}.\nThey are metallic or semiconducting depending on their chirality and as-synthesized material is normally a mixture of both types. \nFor many applications, however, purified samples of only a certain type are in high demand. Purified semiconducting tubes are required, for instance, to achieve a large on\/off ratio and high carrier mobility in thin-film field effect transistors \\cite{chen05,lee11,bisri12,sangwan12,ding14,brady14} and\nhigh power conversion efficiency for photovoltaics \\cite{bindl10,holt10}. Moreover, for optoelectronic applications working in a specific wavelength range, the sorting of semiconducting CNTs according to diameter is of great importance.\n\nIn view of such demands, methods for the selective synthesis of CNTs of a certain electronic type or chiralites have been developed \\cite{ding09,sanchez14}. A low-cost mass production of selected CNTs is yet to be achieved, however, and post-synthesis methods are often relied on \\cite{hersam08}.\nA promising post-synthesis selection method discovered recently is based on the\nphysisorption of polymers on the surface of CNTs, which has the advantage of leaving the electronic properties of CNT nearly unperturbed \\cite{hersam08}. There is a relatively long history of using polymers to disperse CNTs in aqueous or organic solutions \\cite{star01,oconnell01}. A recent finding is that, by using suitable \npolymers, CNTs can be selectively dispersed either for a specific diameter\nrange or for certain chiral angles \\cite{hersam08}. Among those tested, the $\\pi$-conjugated polymer\ngroup of polyfluorene derivatives shows the ability to selectively disperse semi-conducting\nCNTs \\cite{nish07,chen07,gao11,ozawa11,tange12,gomulya13,mistry13,berton14,fukumaru14}. \nIn particular, the di-octyl substituted polyfluorene (PFO) used with\ntoluene as solvent prefers to disperse small-diameter semi-conducting nanotubes with\nchiral angles bigger than about 20 degrees \\cite{nish07}. With longer side-chains, larger-diameter tubes can be dispersed but the chiral angle preference is gradually lost \\cite{gomulya13,ding14}. More recently, copolymers of polyfluorene with anthracene or pyridine groups were found to selectively disperse large-diameter semiconducting nanotubes with high purity and high yields \\cite{mistry13,berton14}. This fits well with the need to fabricate photoelectronic devices working in the infrared wavelength range \\cite{avouris08}. \nLarge-diameter nanotubes also benefit from a diminishing contact resistance and higher carrier mobilities.\nPurified semiconducting CNTs have been used to fabricate high-performance field-effect transistors with high carrier mobilities and large on-off ratios \\cite{lee11,bisri12,sangwan12,ding14,brady14}. \n\nIntensive experimental and numerical works have been undertaken to study the conformation of polymers adsorbed on the CNT surface. For DNAs and some bio-macromolecules \\cite{zheng03}, which prefer to take a helical conformation even in the free state, a helically wrapped configuration on CNT was naturally expected. Studies on the adsorption conformation of linear conjugated polymers are less conclusive. For instance, poly(aryleneethynylene)s (PAE) were found to align linearly along the CNT when dispersed with toluene \\cite{chen02}. The similar poly(p-phenyleneethynylene) polymer, poly[p-{2,5-bis(3-propoxysulfonicacidsodiumsalt)}phenylene]ethynylene, was found to form helically wrapped structures in aqueous dispersion \\cite{kang09}. Imaged {\\it via} SEM, it was shown that when dispersed in chloroform, poly(3-hexyl-thiophene) (P3HT) forms a helically wrapped structure on the surface of multi-walled CNTs \\cite{giulianini09}. Recently, regioregular poly(3-alkylthiophene)s (rr-P3ATs) were used with toluene as solvent to \nenrich semi-conducting CNTs, and MD simulations showed that P3ATs take quasi-linear conformations, as adsorbed on CoMocat CNTs \\cite{wang14}. In 2014 Shea \\textit{et al.} reported the first experimental study on the adsorption configuration of PFOs on CNTs by using photoluminescence energy transfer and anisotropy measurements \\cite{shea14}. Their data, however, are open to interpretation \\cite{supp}. \n\n\nIn the past, efforts were also made to theoretically explain the selection mechanism. For DNAs, the intrinsic helical nature was believed to play a crucial role in their selective adsorption on CNTs \\cite{zheng03}.\nFor aromatic polymers, Nish {\\it et al.} \\cite{nish07} found that PFOs on CNT surfaces form n-fold symmetric structures with their backbones aligned along the tube axis.\nThe magnitude of the binding energy between CNTs and polymers was shown to increase with the tube diameter, a trend that was later confirmed by several authors \n\\cite{ozawa11,gomulya13,fukumaru14}. \nIf the stability of adsorption complexes, as indicated by the binding energy, would determine the dispersibility of CNTs, the above results \\cite{nish07,ozawa11,gomulya13,fukumaru14} would imply that large-diameter CNTs are more easily dispersed than small-diameter ones.\nThis, however, is in contradiction to experimental observations that PFO prefers to disperse small-diameter CNTs \\cite{nish07,ozawa11,fukumaru14}. \nFurthermore, helically wrapped PFO structures on CNTs were used to explain the chirality preference of PFO \\cite{gao11}. We will show below, however, that such helical structures are not dynamically stable.\nRecently, a coarse-grained model was developed and used together with statistical mechanic arguments to explain the diameter preference of several pyridine-containing copolymers \\cite{berton14}. But it is unclear how well the method can be transferred to other systems. \nDespite these advances, it is fair to say that a thorough understanding of the diameter and chirality selectivity of the polymer adsorption method is still lacking.\n\n\\section{Results and Discussion}\nThis article focuses\non understanding the diameter selectivity of the polymer adsorption method since the band gap and related electronic\/optical properties of semiconducting CNTs are mainly determined by the diameter \\cite{white93}. \nIn particular, we propose that diameter selectivity results from a competition between the adsorption of polymers on the CNT surface and the bundling of individual CNTs (see Fig. \\ref{fgr:fig3}). \nOur results on four relevant polymers are in excellent agreement with \nexperimentally observed diameter preferences \\cite{nish07,mistry13,berton14} and, thus, resolve a controversy on the nature of the mechanism that underlies the diameter selection process.\nDespite the complexity of the competitive dispersion of CNTs, the success of our simple energetic model regarding diameter selectivity relies on its correct representation of some key factors including steric effects\/coverage.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1.0]{scheme-competition2-b-resize.pdf}\n\\caption{The proposed mechanism of diameter-selective dispersion: a competition between bundling of carbon nanotubes (CNTs) and adsorption of polymers on the surface of CNTs. The initial state of the process, given by individual CNTs and individual polymers, is a transition state (on the top of a potential energy hill) created by sonication. \n}\n\\label{fgr:fig3}\n\\end{figure}\n\n{\\bf Simulations:}\nTo study the CNT-dispersion process, we performed classical molecular dynamics (MD) simulations using force fields.\nTip sonication treatment is known to generate high, local energy densities that break bundles into individual CNTs \\cite{mason,huang12}. \nFor the dilute polymer concentrations used in typical dispersion processes, the polymers exist as individual molecules \\cite{justino11}. \nTherefore, isolated, individual CNTs and polymers were assumed as the initial configuration. Solvent molecules of toluene were usually not explicitly included here. \nWe tested that their inclusion \ndid not significantly change the results but mostly slowed down the adsorption dynamics.\nFour representative types of polymers were considered in this study: the homopolymer of polyfluorene with side-chain length C8 (PFO) or C6 (PFH), and copolymers with anthracene group poly[(9,9-dihexylfluorenyl-2,7-diyl)-co-(9,10-anthracene)] (PFH-A),\nor pyridine groups poly[9,9-didodecylfluorene-2,7-diyl-alt-pyridine-2,6-diyl] (PFD-Py). The chemical structures of these four polymers are presented in the Supporting Information.\nFurthermore, 13 CNTs with diameters in the range from 0.8 to 1.4 nm\nwere considered. Such diameters are typically obtained in high-pressure CO conversion (HiPco) or pulsed laser vaporization (PLV) synthesis.\nFurther information on our simulation methods can be found in the Method section.\n\n\n{\\bf Adsorption complexes:}\nThe geometries of adsorption complexes were obtained by MD simulations using many different initial configurations, temperatures, and CNT diameters. \nThe simulations always lead to an almost linear alignment of PFO chains on the CNT surface, even after using initial conditions that promote the formation of helically wrapped structures.\nThrough geometry optimizations, we found that a multitude of such helically wrapped structures \\cite{gao11}, with different pitches and surface coverages, are local minima on the potential energy landscape (see Fig.~\\ref{fgr:fig1} (a) and S2) \\cite{supp}. \nHowever, if they were subject to MD simulations\nunwrapping proceeds gradually and after a sufficiently long run, a linearly aligned structure, as shown in Fig. \\ref{fgr:fig1} (b), was always obtained. We conclude that helically wrapped adsorption complexes are metastable \\cite{supp}. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1.0]{fig1-nv2-e.pdf}\n\\caption{The geometry of adsorption complexes: (a) A helically wrapped configuration is metastable, {\\it i.e.},  a local minimum on the potential energy landscape. \n(b) A snapshot of the much more stable, linearly aligned configuration of PFO on a (8,6) CNT.\n}\n\\label{fgr:fig1}\n\\end{figure}\n\n\n{\\bf Binding energy and stability of adsorption complexes:}\nA standard measure used to characterize the stability of the adsorption complex is the binding energy. It is defined as the difference between the potential energy of an adsorption complex and the sum of its constituent molecules. For the adsorption of polymers on CNT, it reads\n\\begin{equation}\n E^{binding}_{CNT\\mhyphen Polymer}=E_{CNT\\mhyphen Polymer}-E_{CNT}-E_{Polymer}.\\label{eqn:ebin}\n\\end{equation}\nThe binding energy for the adsorption of a single polymer chain on a CNT is shown in Fig.~\\ref{fgr:fig2} (a). Note that the magnitude of the binding energy increases with the tube diameter in agreement with previous results \\cite{nish07,ozawa11,gomulya13,fukumaru14}. \nThis is caused by a better contact between polymers and CNT owing to the increasingly flatter surface of large-diameter CNTs \\cite{supp}.\nThe side chain contributes a large part, about two-thirds, to the total binding energy of PFO. Consistently, the binding energy for PFH is smaller due to a shorter side-chain length \\cite{lee11,gomulya13,wang14}.\nThe magnitude of the binding energy of \nPFH-A is smallest, which means that, for all the tested polymers with similar length, it is the easiest to remove from a CNT surface. This is in qualitative agreement with our recent experimental observation that PFH-A can be washed away from thin films deposited using dispersed CNTs (unpublished results). In contrast, PFO cannot be washed away in the same manner. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1.0]{fig3-H-150227.pdf}\n\\caption{The binding energy $E^{binding}_{CNT\\mhyphen Polymer}$ (Eq.~\\ref{eqn:ebin}) of adsorption complexes for \n(a) a single polymer chain and (b) the maximal coverage of the CNT surface. The magnitude of the binding energy increases with the nanotube diameter.\nTherefore, the binding energy $E^{binding}_{CNT\\mhyphen Polymer}$ alone cannot explain why polymers selectively disperse CNTs with specific diameters. \nNote the discontinuities of $E^{binding}_{CNT\\mhyphen Polymer}$ in (b) that are due to abrupt changes in the surface coverage of the CNTs.\n}\n\\label{fgr:fig2}\n\\end{figure}\n\n{\\bf Surface coverage of CNTs by polymers and binding energy of CNT-polymer complexes:}\nTo avoid the rebundling of CNTs after sonication, it is necessary to sufficiently cover the CNT surface with polymers.\n\nWe concentrate here on the situations where there is an excess of polymers and maximal coverage of the CNT surface is expected.\nBinding energies for the maximal coverage of CNTs by polymers are shown in Fig.~\\ref{fgr:fig2} (b). \nNote the discontinuities in the binding energy \nthat are due to a sudden change in the number of polymers needed for maximal surface coverage \\cite{nish07}. The positions of the discontinuities are different from those reported previously in the literature \\cite{nish07,ozawa11,gomulya13,fukumaru14} because our MD simulations lead to different surface coverages \nthan the geometry optimizations performed in those works \\cite{supp}. \nThese discontinuities have a direct relation to the diameter preference of polymers, as will be discussed below. \n\n{\\bf Polymer-assisted dispersion as a competition between adsorption and bundling:}\nThe binding energy $E^{binding}_{CNT\\mhyphen Polymer}$ alone cannot\nexplain the selectivity of the polymer adsorption method, because \nits magnitude simply increases with the diameter (see Fig. \\ref{fgr:fig2}). \nThis would imply that large-diameter CNTs are more easily dispersed than small-diameter ones. \nHowever, this is in clear contrast to the experimental observations discussed above \\cite{nish07,mistry13}. The binding energy between polymer-wrapped CNTs could explain well the polymer-assisted dispersion of CNTs in certain solvents but not the selectivity on CNTs.\nThe key factor for understanding the selection mechanism\nis competition between the bundling of CNTs on the one hand and the adsorption of polymers on the CNT surface on the other (see Fig. \\ref{fgr:fig3}). \nThis reasoning is based on the observation that CNT dispersions in toluene without polymers are not stable and the CNTs eventually rebundle. \nFor this competition to take place, the initial state to be considered is a transition state consisting of individual polymers and individual CNTs. This transition state is experimentally realized by sonication, an integral work step of all selection methods.\nTherefore, the selectivity of CNTs is determined by the \\textit{difference} between the binding energy for CNT bundling and the binding energy for polymer adsorption, which reads \n\\begin{equation}\n\\Delta E^{binding}= E^{binding}_{CNT\\mhyphen Polymer} -  E^{binding}_{CNT\\mhyphen CNT}. \\label{eqn:ediff}\n\\end{equation}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1.0]{fig4-H-150227.pdf}\n\\caption{The energetics of CNT bundling: (a) Weighted average $\\bar E_i$ (Eq.~\\ref{eqn:ecntpair}) of the pair binding energy of the interaction between a CNT of a given diameter and another CNT, arbitrarily selected from the sample. (b) Average number of neighbors $N_{avg}$ of a CNT with a given diameter in a bundle with mixed diameters (fractional numbers are a result of the non-uniform diameter distribution). \n(c) Binding energy $E^{binding}_{CNT\\mhyphen CNT} = \\bar E_i N_{avg}$ of CNT bundling. The variation of $E^{binding}_{CNT\\mhyphen CNT}$ with the diameter follows the same trend as $E^{binding}_{CNT\\mhyphen Polymer}$ in Fig.~\\ref{fgr:fig2}, and only the competition between adsorption and bundling leads to selectivity. \n}\n\\label{fgr:fig5}\n\\end{figure}\n\n{\\bf Binding energy of CNT bundles:}\nAs-produced CNTs are normally a mixture of different diameters. This polydisperse nature makes a direct simulation of bundling computationally very expensive.\nTo overcome the difficulties, we first calculated the average pair binding energy $\\bar E_i$ of the interaction between a CNT of a given diameter and another CNT, arbitrarily selected from a sample of mixed CNTs. \nIt reads \n\\begin{equation}\n\\bar E_i= \\sum_j w_j  E_{ij}, \\label{eqn:ecntpair}\n\\end{equation}\nwhere $E_{ij}$ is the pair binding energy between CNT species $i$ and $j$, and $w_j$ is the population weight (abundance) of CNT species $j$ in the sample.\nAs shown in Fig. \\ref{fgr:fig5} (a), the magnitude of $\\bar E_i$ increases with the CNT diameter, due to the increase in the contact area between CNTs. \nNext, we estimate the average number of neighbors $N_{avg}$ of a CNT in bundles. A simple approach would be to ignore the polydispersity and assume that all CNTs have just six neighbors. But our method is to\nconsider the surface of a CNT of a given diameter to be covered by CNTs having the average diameter of the considered sample, {\\it i.e.} HiPco or PLV.\nTherefore, the average number of neighbors can be a non-integer. The estimates of $N_{avg}$ for two CNT samples are presented in Fig.~\\ref{fgr:fig5} (b).\nFinally, the binding energy for CNT bundling $E^{binding}_{CNT\\mhyphen CNT}$ can be calculated as \n\\begin{equation}\nE^{binding}_{CNT\\mhyphen CNT} = \\bar E_i N_{avg}.\n\\label{eqn:ecntbundlg}\n\\end{equation}\nAs shown in Fig. \\ref{fgr:fig5} (c) for both samples, the magnitude of the binding energy of CNT bundles increases with the CNT diameter.\n\n{\\bf Binding energy difference and diameter selectivity:}\nAs discussed already, for both CNT bundles and CNT-polymer complexes, the magnitude of the binding energy increases with the tube diameter. \nIn the binding energy difference $\\Delta E^{binding}$, the two trends \nnearly compensate for each other and only their competition leads to the preference for certain diameters, \nwhich are reflected in the location of the minima of $\\Delta E^{binding}$ in Fig.~\\ref{fgr:fig6}. \n\nConsider first the adsorption of PFO on HiPco CNTs in Fig \\ref{fgr:fig6}. Except for the CNT with the smallest diameter, $\\Delta E^{binding}$ increases with tube diameter. \nSince the number of polymers needed for the maximal surface coverage of the CNTs changes from three to four, $E^{binding}_{CNT\\mhyphen Polymer}$ abruptly changes at 0.83 nm (see inset of Fig.~\\ref{fgr:fig2} (b)), causing $\\Delta E^{binding}$ to have a minimum at about the same diameter. \nThe behavior of $\\Delta E^{binding}$ explains \n(i) the preference of PFO to disperse HiPco CNTs in the diameter range 0.8--0.95 nm, a fact that has been repeatedly reported by different groups \\cite{nish07,ozawa11,fukumaru14}, and\n(ii) why CNTs with diameter smaller than 0.8 nm are not well-dispersed by PFO. These two insights explain the dominance of (8,6) CNTs (d = 0.95 nm) in HiPco CNT dispersions and the elimination of (6,5) CNTs (d = 0.75 nm) in CoMoCAT CNT dispersions \\cite{nish07}. \n\nFor PFH, with side-chains two carbon atoms shorter than PFO, more polymer chains are needed to cover the surface of a CNT. Therefore, the discontinuity in $\\Delta E^{binding}$\nis upshifted to 1.03 nm. This explains why,\nfor HiPco CNTs using PFH instead of PFO, the dominant CNTs in the dispersion become (8,7) (d = 1.02 nm) and (9,7) (d = 1.09 nm) (see Fig.1b of Nish {\\it et al.} \\cite{nish07}).\n\nFor the copolymer PFD-Py, the \nminimum of \n$\\Delta E^{binding}$ is at about 1.25 nm. This agrees with recent experimental findings that, for HiPco CNTs, PFD-Py prefers to disperse CNTs with diameters of about 1.23 nm (see Fig.1n of Berton {\\it et al.} \\cite{berton14}).\n \nThe $\\Delta E^{binding}$ of PFH-A increases continuously \nin the considered diameter range (0.8--1.4 nm)\nand no minimum is discernible.\nMistry {\\it et al.} performed a systematic study on the selectivity of PFH-A on CNTs synthesized {\\it via} laser vaporization of graphite at different temperatures and found that\nit always prefers to disperse CNTs with the smallest diameters in the sample \\cite{mistry13}. The absence of a minimum in that range is again consistent with the experiments even though the simulations are based on HiPco CNTs. Further discussions on the selectivity of PFH-A can be found in the Supporting Information.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1.0]{fig5-150305d.pdf}\n\\caption{Diameter selectivity as competition between CNT bundling and polymer adsorption. \nThe diameter preferences of specific polymers for HiPco CNTs in our simulations are defined by the minima of the corresponding binding energy difference $\\Delta E^{binding}$ (Eq.~\\ref{eqn:ediff}). \nThey are in excellent agreement with experimental results indicated by the shadowed regions\n\\cite{nish07,berton14,mistry13}. \n}\n\\label{fgr:fig6}\n\\end{figure}\n\nTo summarize, for \nthe polymers PFO, PFH and PFD-Py, the minima of the \nbinding energy difference $\\Delta E^{binding}$ \nmatch perfectly the experimentally reported diameters that are dominantly dispersed by those polymers.\nThis excellent agreement strongly suggests \nthat the mechanism of diameter selectivity is a competition between CNT bundling and polymer adsorption.\n\nIt is interesting to note that the sign of $\\Delta E^{binding}$ is negative for PFO, PFH and PFD-Py. This indicates a preference for the formation of CNT-polymer adsorption complexes over CNT-CNT bundling.\nTherefore, for long sonication times, SWNTs of all diameters can be dispersed in principle. \nWith increasing sonication time, the amount of dispersed CNTs will increase and the selectivity will gradually diminish.\nTherefore, an optimal sonication time should be experimentally determined, providing a compromise between yield and purity.\nPositive values of $\\Delta E^{binding}$ for PFH-A mean that \nthe rebundling should happen more frequently than adsorption,\nwhich implies a potential lower yield of the dispersion process using PFH-A.\n\nNote also that, due to the possibility of partial adsorption and other \"imperfect\" packing configurations the transition in Fig. 3(b) may turn out to be not so abrupt, see also the radial distribution functions shown in Fig.S6 and the corresponding discussions in the Supporting Information. Therefore, in experiments a range of diameters is often selected by a certain polymer.\n\nOne can also view the sonication-assisted dispersion process as a reversible reaction,\n\\begin{equation}\nCNT@CNT + PFO \\rightleftharpoons PFO@CNT . \\label{eqn:ereac}\n\\end{equation} \nIn this language, the initial configuration of isolated CNTs and polymers corresponds to a transition state, which is achieved with the aid of ultrasonication treatment \\cite{mason,huang12}. \nThe adsorption of polymers on the CNT surface and the bundling of CNTs are the \nrate-determining steps for forward and backward reactions, respectively. The two binding energies are the (negative) activation energies for reactions in the two directions. \nFor this reversible reaction, the binding energy difference only estimates energetic contributions to\nthe reaction rates, neglecting \nentropic contributions, reaction orders and the concentrations of the reactants.\n\nThe focus of the current study is on the diameter selectivity of CNTs by aromatic polymers. For the chirality selection, the match\/mismatch between the atomic structures of polymers and CNTs will be quite crucial. The popular implementations of van der Waals' interaction, as used here, were found unsuitable for the purpose and the anisotropic intermolecular potentials turn out to be a better alternative \\cite{kc05}. For the sorting of CNTs with respect to electronic properties, {\\it ab initio} quantum simulations with the electronic interactions being included would be more appropriate. All these issues deserve their own separate publications. The mentioned success of our simple energetic model implies that the key factors determining the diameter selectivity of (semi-conducting) CNTs were properly represented. The model can certainly be further improved by including: (i) entropy factors for a proper estimate of Gibbs free energy, which is of direct relevance to \nthe reaction kinetics, and (ii) the effect of explicit solvent.\nCalculations with explicit solvents and estimates of entropic contributions are provided in the Supporting Information.\n\n\nIn conclusion, we explain the diameter selectivity of polymer adsorption methods to be the result of a competition between the bundling of CNTs and the adsorption of polymers on the CNT surface. \nThe preference of certain diameters corresponds to local minima of the binding energy difference between these two processes. Such minima occur due to abrupt changes in the CNT's coverage with polymers at certain diameters.\nFor all tested polymers including two homopolymers of polyfluorene with different side-chain lengths and two copolymers with anthracene or pyridine groups, our simulation results are in excellent agreement with the experimental findings regrading the diameter selectivity. Interestingly, even the influence of a fine-tuning of side-chain length on the selectivity was correctly captured in our method. Our insights resolve a long-standing controversy regarding the understanding of CNT selection schemes and \nare important for the further development of dispersion\/extraction methods, {\\it i.e.}, they enable MD simulations to be used for the screening of polymer candidates, tailoring of polymer structures, and obtaining further scientific insights. The proposed mechanism is general enough to be valid for other (sonication-aided) dispersion processes, for instance, the exfoliation of layered materials \\cite{nicolosi13}, and the dispersion of CNTs by DNAs and mononucleotides \\cite{zheng03,zheng09,ju08,johnson08}.\n\n\n\\section{Methods}\n\n\nThe adsorption of polymers on single-walled CNTs and the bundling of CNTs were studied with classical molecular dynamics (MD) simulations by using the CP2K \\cite{cp2k} and Gromacs packages \\cite{gromacs}. MD simulations were performed in NVT ensemble at $T=300K$ using the Nose-Hoover or Langevin thermostats.\nThe standard CHARMM force field parameters for the intra-molecular interaction \\cite{charmm} were benchmarked against the density functional method (DFT) MD simulations with Grimme dispersion corrections DFT-D3 \\cite{grimme10} and the BLYP exchange-correlation functional \\cite{blyp} in CP2K and classical MD simulations with the MM3 force field \\cite{mm3} using the Tinker package \\cite{tinker}. The torsion angle parameter, describing amongst others the twist of the backbones of polymers, was modified to match the results of DFT first principles MD simulation. \nThe inter-molecular interactions include an electrostatic part due to partial charges on atoms and a dispersion force part modeled by a Lennard-Jones potential as usual in standard CHARMM force field implementations \\cite{charmm}.  \n\nFor the adsorption of polymers on the CNT surface, the polymer backbones were initially aligned parallel to the CNT axis. Multiple chains of polymers were arranged in an n-fold symmetric structure surrounding the tube. The initial distance between the backbone and the CNT surface was set to 1 to 1.5 nm depending on the CNT diameter and the number of polymers. For the binding energy calculation of CNT pairs, the two CNTs were placed in parallel with the initial distance between the surfaces of 0.6 nm. The time step for the integration of Newton's equation of motion was 1 fs. The duration of MD simulations ranged from 1 ns to 20 ns. For the calculation of thermodynamic averages, the equilibration time, ranging between 0.2 and 2 ns, was not considered.\nTo check the stability of self-constructed helical adsorption structures, geometry optimizations were performed using the CP2K package. The criteria of convergence are $3\\times 10^{-3}$ $Bohr$ for the geometry change and $4.5\\times 10^{-4}$ $Hartree\/Bohr$ for the change in the force. \n\nIt is known that the solvent plays an important role in the selective dispersion of CNTs by polymers \\cite{huang08}. However, here we are interested in studying the effect of different polymers in combination with the same weakly polarized solvent, toluene.\nMoreover, our tests showed that the explicit inclusion of solvent molecules of toluene in MD simulations did not change the structures of adsorption but significantly slowed down the dynamics of the adsorption process. Therefore, to enable MD simulations within reasonable times, solvent molecules were not explicitly included in our production runs. Our additional MD simulations with explicit solvents showed that, the adsorption configurations of the polymer backbones and sidechains, which are in contact with CNT surface, hardly change with the inclusion of solvents. Only the sidechains, which are not in contact with CNT surface, tend to point outwards to the solvent instead of folding back and aligning along CNT surface as in vacuum. The second-layer of polymers moves a bit away from the CNT surface. Therefore, with the inclusion of solvents the values of binding energies may change by some degree but the surface coverage of CNTs and the related positions of the abrupt changes in Fig.3(b) will be unaffected. Further details on the effect of explicit solvent can be found in the Supporting Information. \nStudies showed that a PFO octamer already has the same selectivity as a PFO polymer. Furthermore, the stability of the oligomer-CNT complex increases strongly with the chain length of the oligomer \\cite{berton12}. \nTo meet the capacity of the available computing resources in our simulation, we used 30-nm-long polymer \nchains, which consist of 32, 22 and 30 monomers of PFO\/PFH, PFD-Py and PFH-A, respectively. \nCNT segments of length between 30 and 36 nm were used, their length varying with the chirality.  \n\nTo obtain the binding energy of a polymer-CNT complex, three MD simulations were\nperformed for the adsorption complex, the isolated CNT and the polymer, respectively.\nThe mean value of the potential energy was determined from the corresponding trajectories\nand the binding energy was then calculated. This procedure is different from most cases in the literature where the binding energy was calculated from ($T=0$ K) single-point calculations of the optimized structures. \n\nIt is worth pointing out that, for the adsorption of polymers on a CNT surface, the binding energy can be measured per unit length of a polymer chain, or per unit length of CNT covered completely by polymers. The former describes how hard it is to remove a polymer chain from the CNT surface while the latter is suitable for characterizing the competition for the adsorption on a CNT surface. \n\n\n\n\\begin{acknowledgement}\nWe acknowledge fruitful discussions with Gotthard Seifert. HLY thanks Jia Gao and Elton Carvalho for communications on the preparation of helically wrapped complexes. This work was partially funded by the European Union (ERDF) {\\it via} the FP7 project CARbon nanoTube phOtONic devices on silicon (CARTOON) and is supported by Dresden Center for Computational Materials Science (DCCMS). We also acknowledge the support by the\nGerman Research Foundation (DFG) within the Cluster of\nExcellence ``Center for Advancing Electronics Dresden''\n(cfAED).\nWe acknowledge the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for computational resources.\n\\end{acknowledgement}\n\n\n\\begin{suppinfo}\nCalibration of CHARMM force field parameters, stability of the helically wrapped configurations for PFO on CNTs, the determination of the surface coverage of CNT by PFO, variation of the binding energy of PFO-CNT complexes with CNT diameter, the population distributions of HiPco \\cite{chen07} and PLV \\cite{tange12} CNTs with respect to the diameter, the effect of solvent on the adsorption configurations, and the influence of entropic effect on diameter selectivity.\n\\end{suppinfo}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S1}\nLet $\\mathbb{F}_q^n$ be the $n$-dimensional vector space over $\\mathbb{F}_q$, the unique finite field with $q$ elements; $q$ is necessarily a prime power. The set of all subspaces of $\\mathbb{F}_q^n$ is the \\emph{projective space}\\footnote{This terminology is not standard. In other branches of mathematics the term projective space defines the collection of all lines passing through the origin of a vector space.} $\\mathbb{P}_q(n)$ which can be formally defined as\n\\begin{equation*}\n\t\\mathbb{P}_q(n) := \\{V : V \\le \\mathbb{F}_q^n\\},\n\\end{equation*}\nwhere $\\le$ signifies the usual vector space inclusion. The collection of all subspaces in $\\mathbb{P}_q(n)$ with a fixed dimension $k$ is called the \\emph{Grassmannian} of dimension $k$ for all $0 \\le k \\le n$, and is denoted as $\\mathbb{G}_q(n, k)$. In terms of notation, $\\mathbb{G}_q(n, k) := \\{V: V \\le \\mathbb{F}_q^n, \\dim V = k\\}$. Clearly, $\\mathbb{P}_q(n) = \\bigcup\\limits_{k=0}^{n} \\mathbb{G}_q(n, k)$. The \\emph{subspace distance} between two subspaces $X$ and $Y$ in $\\mathbb{P}_q(n)$ is defined as\n\\begin{equation*}\n\td_S(X, Y) := \\dim (X+Y) - \\dim (X \\cap Y),\n\\end{equation*}\nwhere $X+Y$ denotes the smallest subspace containing both $X$ and $Y$. It was proved in \\cite{KK, AAK} that the projective space $\\mathbb{P}_q(n)$ is a metric space under the action of the subspace distance metric. A \\emph{code} in the projective space $\\mathbb{P}_q(n)$ is a subset of $\\mathbb{P}_q(n)$.\n\nCodes in projective spaces have recently gained attention since they were proved to be useful for error and erasure-correction in \\emph{random network coding} \\cite{KK}. An $(n, M, d)$ code in $\\mathbb{P}_q(n)$ is a collection of $M$ number of subspaces of $\\mathbb{F}_q^n$ such that the minimum subspace distance between any two of them is $d$. K\\\"{o}tter and Kschischang showed that an $(n, M, d)$ code can correct any combination of $t$ errors and $\\rho$ erasures during the communication of packets through a volatile network as long as $2(t+\\rho) < d$ \\cite{KK}. Subsequently codes in $\\mathbb{P}_q(n)$ were studied extensively \\cite{EV, HKK, BP, GR}. Such codes are also referred to as \\emph{subspace codes}.\n\nHowever, designing and studying code structures in $\\mathbb{P}_q(n)$ is considered relatively trickier than study of classical error-correction in the \\emph{Hamming space} $\\mathbb{F}_q^n$. This is because unlike in $\\mathbb{F}_q^n$, the volume of a \\emph{sphere} is not independent of the choice of its \\emph{center} in the projective space $\\mathbb{P}_q(n)$. Thus, standard geometric intuitions often do not hold in $\\mathbb{P}_q(n)$. In other words, $\\mathbb{F}_q^n$ is \\emph{distance-regular} while $\\mathbb{P}_q(n)$ is not. This implies that a different framework is required to study codes in projective spaces than the approach taken for block codes in classical error-correction, e.g. in \\cite{MS}. The problem of lack of distance-regularity in $\\mathbb{P}_q(n)$ is, however, tackled to some extent by considering codewords of a fixed dimension. Such class of subspace codes are commonly known as \\emph{constant dimension codes}. A constant dimension code in $\\mathbb{P}_q(n)$ is a subset of $\\mathbb{G}_q(n, k)$ for some $0 \\le k \\le n$. The fact that $\\mathbb{G}_q(n, k)$ is distance-regular is exploited to construct various classes of constant dimension codes \\cite{SE, SE2, GY, XF, TMBR, KoK, ER}.\n\nA lattice framework for studying both binary block codes and subspace codes was discussed in \\cite{MB}. The authors of \\cite{BEV} established that codes in projective spaces are $q$-analogs of binary block codes defined within Hamming spaces using a framework of lattices. A \\emph{lattice} is a partially ordered set wherein both least upper bound and greatest lower bound of any pair of elements exist within the set and are unique. We denote the set of all subsets of the canonical $n$-set $[n] := \\{1, \\ldots, n\\}$ as $\\mathcal{P}(n)$, also known as the \\emph{power set} of $[n]$. The lattices corresponding to the block codes in $\\mathbb{F}_2^n$ and the subspace codes in $\\mathbb{P}_q(n)$ are the \\emph{power set lattice} $(\\mathcal{P}(n), \\cup, \\cap, \\subseteq)$ and the \\emph{linear lattice} $(\\mathbb{P}_q(n), +, \\cap, \\le)$, respectively. Here the notation $\\subseteq$ represents set inclusion.\n\nA lattice is called \\emph{modular} if the modularity condition holds for all elements in it (see Def.~\\ref{LD5}). Many of the well-known coding spaces including both $\\mathbb{F}_q^n$ and $\\mathbb{P}_q(n)$ are examples of a modular lattice. This motivated the work of Kendziorra and Schmidt where they generalized the model of subspace codes introduced in \\cite{KK} to codes in modular lattices \\cite{KS}. There are two significant types of semimodular lattices, viz. \\emph{geometric} lattices and \\emph{distributive} lattices that have inspired a rich variety of literature, e.g. \\cite{B, RS}. While $\\mathbb{F}_2^n$ is a geometric distributive lattice, $\\mathbb{P}_q(n)$ is an example of a geometric lattice which is modular but non-distributive. There have been quite a few attempts to characterize distributive lattices, such as in \\cite{LS, S}. However, no known characterization of geometric distributive lattices exists to the best of our knowledge.\n\nThe notion of ``linearity\" and ``complements\" in $\\mathbb{P}_q(n)$ are not as straightforward as they are in the Hamming space $\\mathbb{F}_2^n$. This is owed to the fact that $\\mathbb{F}_2^n$ is a vector space with respect to the bitwise XOR-operation whereas $\\mathbb{P}_q(n)$ or $\\mathbb{G}_q(n, k)$ are not vector spaces with respect to the usual vector space addition. Therefore, the subspace distance metric is not \\emph{translation invariant} over $\\mathbb{P}_q(n)$ or $\\mathbb{G}_q(n, k)$. Braun et al. addressed this problem in \\cite{BEV} and defined linearity and complements in subsets of $\\mathbb{P}_q(n)$ by elucidating key features from the equivalent notion in $\\mathbb{F}_2^n$.\n\nThe maximum size of a linear code in $\\mathbb{P}_2(n)$ was conjectured to be $2^n$ by Braun et al. \\cite{BEV}. A particular case of this problem was proved by Pai and Rajan where the ambient space $\\mathbb{F}_2^n$ is included as a codeword \\cite{PS}. The maximal code achieving the upper bound was identified as a \\emph{code derived from a fixed basis}. The authors of \\cite{PS} observed that such a code is basically embedding of a distributive lattice into the linear lattice, which is geometric. This motivated them to conjecture a generalized statement which can already be found in literature, e.g. in \\cite[Ch.~IX, Sec.~4, Ex.~1]{B}.\n\\begin{problem}\n\t\\label{Pr1}\n\tThe size of the largest distributive sublattice of a gemetric lattice of height $n$ must be $2^n$.\n\\end{problem}\nLattice-theoretic connection of other classes of linear codes in $\\mathbb{P}_q(n)$ was investigated thoroughly in \\cite{BK}. The findings of \\cite{BK} include the discovery that the only class of linear subspace codes that have a sublattice structure of the corresponding linear lattice must be geometric distributive. Thus it is an interesting problem to find out a unique characterization of geometric distributive lattices should it exist.\n\nIn this paper, we determine the unique criterion for a geometric lattice to be distributive. We in fact prove a more generalized version of this statement. This helps us to bring out the unique characterization of class of geometric distributive lattices. We then use this characterization to solve a few problems involving linear codes and complements in $\\mathbb{P}_q(n)$. Problem~\\ref{Pr1} is also solved by applying the said characterization.\n\nThe rest of the paper is organized as follows. In Section~\\ref{S2} we give a few requisite definitions concerning lattices and formally define linear codes and complements in the projective space $\\mathbb{P}_q(n)$. Section~\\ref{UAL} concerns with the study of uniquely atomistic lattices; in particular we show that any such finite lattice is modular. The \\emph{unique-decomposition theorem} that gives the unique characterization of finite geometric distributive lattices is derived in Section~\\ref{S3} after proving a sequence of results regarding modular lattices and distributive lattices. Section~\\ref{S4} is attributed to various applications of the unique-decomposition theorem in lattice theory that include determining the maximum size of a distributive sublattice of a finite geometric lattice and counting the \\emph{Whitney numbers} of a geometric distributive lattice. In particular, we consider a few problems about linearity and complements in the linear lattice. An important finding is that any distributive sublattice of $\\mathbb{P}_q(n)$ can be used to construct a linear code closed under intersection. Concluding remarks and interesting open problems are listed in Section~\\ref{S5}.\n\\paragraph{Notation.}\n$\\mathbb{F}_q^n$ represents the unique vector space of dimension $n$ over $\\mathbb{F}_q$. The set of all subspaces of $\\mathbb{F}_q^n$ is denoted as $\\mathbb{P}_q(n)$. The usual vector space sum of two disjoint subspaces $X$ and $Y$, called the \\emph{direct sum} of $X$ and $Y$, is written as $X \\oplus Y$. For any subset $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$, the collection of all $i$-dimensional members of $\\mathcal{U}$ will be denoted as $\\mathcal{U}_i$; $\\mathcal{U}_i := \\{X: X \\in \\mathcal{U}, \\dim X = i\\}$. The notation $\\langle \\mathcal{S}\\rangle$ for any subset $\\mathcal{S}$ of vectors in $\\mathbb{F}_q^n$ will denote the linear span of all the vectors in $\\mathcal{S}$. $\\triangle$ denotes the \\emph{symmetric difference} operator, which can be defined for two sets $S$ and $T$ as\n\\begin{equation*}\n\tS \\triangle T := (S \\cup T) \\backslash (S \\cap T).\n\\end{equation*}\n\\section{Preliminaries}\n\\label{S2}\n\\subsection{An Overview of Lattices}\nWe will go through some standard definitions and results concerning lattices that can be found in the existing literature, e.g. in \\cite{B}.\n\\begin{definition}\n\t\\label{LD1}\n\tFor a set $P$, the pair $(P, \\preceq)$ is called a \\emph{poset} if there exists a binary relation $\\preceq$ on $P$, called the \\emph{order relation}, that satisfies the following for all $x, y, z \\in P$:\n\t\\begin{itemize}\n\t\t\\item[(i)] (Reflexivity) $x \\preceq x$;\n\t\t\\item[(ii)] (Antisymmetry) If $x \\preceq y$ and $y \\preceq x$, then $x = y$; and\n\t\t\\item[(iii)] (Transitivity) If $x \\preceq y$ and $y \\preceq z$, then $x \\preceq z$.\n\t\\end{itemize}\n\tThe \\emph{dual} of a poset $P$ is the poset $P^{*}$ defined on the same set as $P$ such that $y \\preceq x$ in $P^{*}$ if and only if $x \\preceq y$ in $P$.\n\\end{definition}\nThe notation $x \\preceq y$ is read as ``$x$ is less than $y$'' or ``$x$ is contained in $y$''. If $x \\preceq y$ such that $x \\ne y$, then we write $x \\prec y$. In the sequel a poset $(P, \\preceq)$ will be denoted as $P$ when the order relation $\\prec$ is obvious from the context.\n\\begin{definition}\n\t\\label{LD2}\n\tAn upper bound (lower bound) of a subset $S$ of a poset $P$ is an element $p \\in P$ containing (contained in) every $s \\in S$. A least upper bound (greatest lower bound) of $S \\subseteq P$ is an element of $P$ contained in (containing) every upper bound (lower bound) of $S$.\n\\end{definition}\nA least upper bound or a greatest lower bound of a poset, should it exist, is unique according to the antisymmetry property of the order relation $\\preceq$. The least upper bound and the greatest lower bound of a poset $P$ are denoted as $\\sup P$ and $\\inf P$, respectively.\n\\begin{definition}\n\t\\label{LD3}\n\tA \\emph{lattice} $(L, \\vee, \\wedge)$ is a poset $L$ such that $\\sup \\{x, y\\}$ and $\\inf \\{x, y\\}$ exist for all $x, y \\in L$. The notation for the $\\sup \\{x, y\\}$ and the $\\inf \\{x, y\\}$ are $x \\vee y$ (``$x$ \\emph{join} $y$'') and $x \\wedge y$ (``$x$ \\emph{meet} $y$''), respectively.\n\\end{definition}\nOnce again, a lattice $(L, \\vee, \\wedge)$ will be denoted as $L$ whenever the join $\\vee$ and meet $\\wedge$ operations are obvious from the context. In this work we will consider only finite lattices, i.e. when the underlying poset is finite. The unique greatest element and the unique least element of a lattice will be denoted as $I$ and $O$, respectively, unless specified otherwise.\n\\begin{definition}\n\t\\label{LD4}\n\tA \\emph{sublattice} of a lattice $L$ is a subset $S \\subseteq L$ such that $x \\vee y, x \\wedge y \\in S$ for all $x, y \\in S$.\n\\end{definition}\nThe \\emph{Hasse diagram} of a finite poset completely describes the order relations of that poset. If $x \\prec y$ in the poset $P$ such that there exists no $z \\in P$ satisfying $x \\prec z \\prec y$, then $y$ is said to \\emph{cover} $x$; we denote this as $x \\lessdot y$. In the Hasse diagram of a lattice, two elements are joined if and only if one of them covers the other; $y$ is written above $x$ if $y$ covers $x$. Hence, $x \\prec y$ if and only if there exists a path from $x$ moving up to $y$.\n\nThe power set $\\mathcal{P}(m)$ of a finite set $[m]$ and the projective space $\\mathbb{P}_q(n)$ are examples of lattices. The Hasse diagram associated with the lattice of $(\\mathbb{P}_2(2), +, \\cap)$ is shown here (Fig.~\\ref{F1}). This particular lattice is known as $M_3$.\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.6]\n\t\t\\node (A1) at (0,3) {$\\mathbb{F}_2^2$};\n\t\t\\node (A2) at (-3,0) {$\\langle \\{(0, 1)\\}\\rangle$};\n\t\t\\node (A3) at (0,0) {$\\langle \\{(1, 0)\\}\\rangle$};\n\t\t\\node (A4) at (3,0) {$\\langle \\{(1, 1)\\}\\rangle$};\n\t\t\\node (A5) at (0,-3) {$\\{0\\}$};\n\t\n\t\n\t\t\\draw (A5) -- (A2) -- (A1) -- (A3) -- (A5) -- (A4) -- (A1);\n\t\\end{tikzpicture}\n\t\\caption{$M_3$ lattice representing $(\\mathbb{P}_2(2), +, \\cap)$}\n\t\\label{F1}\n\\end{figure}\n\\begin{definition}\n\t\\label{LD5}\n\tA finite lattice $(L, \\vee, \\wedge)$ is \\emph{semimodular} if the following holds for all $x, y \\in L$:\n\t\\begin{equation*}\n\t\tx \\wedge y \\lessdot x, y \\quad \\Rightarrow \\quad x, y \\lessdot x \\vee y.\n\t\\end{equation*}\n\tA lattice is \\emph{modular} if both the lattice and its dual are semimodular. It can be proved that a finite lattice $L$ is modular if for any $x, y, z \\in L$ the following holds:\n\t\\begin{equation*}\n\t\tx \\preceq z \\Rightarrow x \\vee (y \\wedge z) = (x \\vee y) \\wedge z.\n\t\\end{equation*}\n\\end{definition}\nThe smallest finite lattice that is non-modular is called $N_5$ (Fig.~\\ref{F0}). $N_5$ plays a crucial role in characterizing modular lattices as we will see later.\n\nThe elements of a lattice which cover the least element of the lattice are known as \\emph{atoms}. A lattice with a least element is \\emph{atomic} if for every non-zero element $a$ there exists an atom $p$ such that $p \\preceq a$. An atomic lattice is called \\emph{atomistic} if any element is a join of atoms. A lattice that is \\emph{uniquely} atomistic is defined in the following way.\n\\begin{definition}\n\t\\label{LDA}\n\tA lattice is \\emph{uniquely atomistic} if each element therein is uniquely expressible as join of its atoms. If $L$ is a uniquely atomistic lattice with $\\{x_1, \\ldots, x_m\\}$ as the set of all atoms in $L$ then for any $x \\in L$ there exists a unique subset $S_x \\subseteq [m]$ such that $x = \\bigvee\\limits_{i \\in S_x} x_i$. We denote this relation as $x = \\sup S_x$ when the choice of $m$ is clear from the context.\n\\end{definition}\nAtoms play an important role in defining geometric lattices.\n\\begin{definition}\n\t\\label{LD6}\n\tA finite lattice that is both semimodular and atomistic is called \\emph{geometric}.\n\\end{definition}\nBoth the linear lattice $(\\mathbb{P}_q(n), +, \\cap)$ and the power set lattice $(\\mathcal{P}(m), \\cup, \\cap)$ are examples of a geometric lattice. From Section~\\ref{S4} onwards all lattices considered will be geometric. The variety of modular lattices that will play a key role in this work are the distributive lattices which are defined next.\n\\begin{definition}\n\t\\label{LD7}\n\tA lattice $L$ is \\emph{distributive} if the following two equivalent conditions hold for any $x, y, z \\in L$:\n\t\\begin{eqnarray}\n\t\tx \\vee (y \\wedge z) &=& (x \\vee y) \\wedge (x \\vee z); \\nonumber \\\\\n\t\tx \\wedge (y \\vee z) &=& (x \\wedge y) \\vee (x \\wedge z). \\nonumber\n\t\\end{eqnarray}\n\\end{definition}\nThe power set lattice $\\mathcal{P}(m)$ is an example of a distributive lattice. However, the linear lattice $\\mathbb{P}_q(n)$ is modular but not distributive. In general, any distributive lattice is modular, and the $M_3$ lattice is pivotal in characterizing modular non-distributive lattices. Similarly a modular lattice can be defined by non-inclusion of the lattice $N_5$. The following theorem is due to Dedekind and Birkhoff.\n\\begin{theorem}(\\cite{G}, Page~59)\n\t\\label{TL1}\n\tA lattice is modular if and only if it does not contain a sublattice isomorphic to $N_5$. A modular lattice is non-distributive if and only if it contains a sublattice isomorphic to $M_3$.\n\\end{theorem}\n\\begin{definition}\n\t\\label{LD8}\n\tA real valued function $v: L \\rightarrow \\mathbb{R}$ on a lattice $L$ is called a \\emph{positive isotone valuation} if the following conditions hold for all $x, y \\in L$:\n\t\\begin{itemize}\n\t\t\\item[(i)] (Valuation) $v(x \\vee y) + v(x \\wedge y) = v(x) + v(y)$; \n\t\t\\item[(ii)] (Isotone) $x \\preceq y \\Rightarrow v(x) \\le v(y)$;\n\t\t\\item[(iii)] (Positive) $x \\prec y \\Rightarrow v(x) < v(y)$.\n\t\\end{itemize}\n\\end{definition}\nThe distance function induced by an isotone valuation $v$ is defined as $d_v(x, y) := v(x \\vee y) - v(x \\wedge y)$.\n\\begin{theorem}(\\cite{B})\n\t\\label{TL2}\n\tFor an isotone valuation $v$ defined on a lattice $L$, the function $d_v(x, y) := v(x \\vee y) - v(x \\wedge y)$ is a metric if and only if $v$ is positive.\n\\end{theorem}\nA totally ordered subset of a lattice is called a \\emph{chain}. Given two elements $x$ and $y$ in a lattice $L$, a chain of $L$ between $x$ and $y$ is a chain $\\{x_1, \\ldots, x_l\\}$ such that $x = x_0 \\prec x_1 \\prec \\cdots \\prec x_l = y$. The \\emph{length} of this chain is $l$. The \\emph{height} of an element $x \\in L$ is the maximum length of all chains between $O$ and $x$, and is denoted by $h_L(x)$. We often use the notation $h(x)$ when $L$ is obvious from the context. The \\emph{height of the lattice} $L$ is the number $h_L(I)$ where $I$ is the greatest element in $L$.\n\nFor modular lattices, the following is a consequence of Theorem~\\ref{TL2}.\n\\begin{theorem}[Page~41, Theorem~16, \\cite{B}]\n\t\\label{TL3}\n\tIf $h$ is the height function defined on a finite modular lattice $L$ then $h$ is a positive isotone valuation and $d_h$ is a metric on $L$.\n\\end{theorem}\nBy definition, $h_L(x) = 0$ if and only if $x$ is the least element in $L$; similarly, $h_L(x) = 1$ if and only if $x$ is an atom in $L$.\n\\begin{definition}\n\t\\label{LD9}\n\tThe total number of elements with a given height $k$ of a lattice $L$ with a height function $h$ defined on $L$ is called the \\emph{Whitney number}, denoted as $W_k(L)$. In terms of notation, $W_k(L) := |\\{x \\in L: h_L(x) = k\\}|$.\n\\end{definition}\n\\subsection{Complements and Linearity in Projective Spaces}\nThe notions of complements and linearity in the projective space $\\mathbb{P}_q(n)$ are not straightforward as they are in the Hamming space $\\mathbb{F}_2^n$. Braun et al. introduced the definition of both in \\cite{BEV} by extracting key properties of the same in $\\mathbb{F}_2^n$. We begin with a formal definition of the complement mapping in $\\mathbb{P}_q(n)$.\n\\begin{definition}\n\t\\label{D1}\n\tFor any subset $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$, a function $f: \\mathcal{U} \\rightarrow \\mathcal{U}$ is a \\emph{complement} on $\\mathcal{U}$ if $f$ satisfies the following conditions:\n\t\\begin{itemize}\n\t\t\\item[(i)] $X \\cap f(X) = \\{0\\}$ and $X \\oplus f(X) = \\mathbb{F}_q^n$ for all $X \\in \\mathcal{U}$;\n\t\t\\item[(ii)] There exists a unique $f(X) \\in \\mathcal{U}_{n-k}$ for each $X \\in \\mathcal{U}_k$ for all $0 \\le k \\le n$;\n\t\t\\item[(iii)] $f(f(X)) = X$ for all $X \\in \\mathcal{U}$; and\n\t\t\\item[(iv)] $d_S(X, Y) = d_S (f(X), f(Y))$ for all $X, Y \\in \\mathcal{U}$.\n\t\\end{itemize}\n\\end{definition}\nIt was proved before that a complement function does not exist in the entirety of $\\mathbb{P}_q(n)$ \\cite[Theorem~10]{BEV}. The largest size of a subset of $\\mathbb{P}_q(n)$ wherein a complement can be defined still remains an open problem. However, we will tackle that question in Section~\\ref{S4} with the additional constraint that a subset has distributive sublattice structure. The following is an upper bound on the number of one-dimensional subspaces in a subset with a complement defined on it.\n\\begin{proposition}({\\cite{BEV}, Proposition~1})\n\t\\label{CP1}\n\tSuppose there exists a complement on the subset $\\mathcal{U} \\subseteq \\mathbb{P}_2(n)$. Then $|\\mathcal{U}_1| \\le 2^{n-1}$.\n\\end{proposition}\nWe will later investigate the same for distributive sublattices of $\\mathbb{P}_q(n)$ for all prime powers $q$.\n\nBraun et al. defined linearity in $\\mathbb{P}_2(n)$ by identifying a subset that is a vector space over $\\mathbb{F}_2$ with respect to some randomly chosen linear operation such that the corresponding subspace distance metric is translation invariant within the chosen subset. Later this definition was generalized for all prime powers \\cite{PS, BK2}.\n\\begin{definition}\n\t\\label{D2}\n\tA subset $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$ is called a \\emph{linear code} if $\\{0\\} \\in \\mathcal{U}$ and there exists a function $\\boxplus: \\mathcal{U} \\times \\mathcal{U} \\rightarrow \\mathcal{U}$ such that\n\t\\begin{itemize}\n\t\t\\item[(i)] $(\\mathcal{U}, \\boxplus)$ is an abelian group;\n\t\t\\item[(ii)] $X \\boxplus \\{0\\} = X$ for all $X \\in \\mathcal{U}$;\n\t\t\\item[(iii)] $X \\boxplus X = \\{0\\}$ for all $X \\in \\mathcal{U}$; and\n\t\t\\item[(iv)] $d_S(X, Y) = d_S(X \\boxplus W, Y \\boxplus W)$ for all $X, Y, W \\in \\mathcal{U}$.\n\t\\end{itemize}\n\\end{definition}\nThe first three conditions stated in the above definition makes any linear code in $\\mathbb{P}_q(n)$ a vector space over $\\mathbb{F}_2$. It was conjectured in \\cite{BEV} that a linear code in $\\mathbb{P}_2(n)$ can be as large as $2^n$ at most.\n\nThe linear addition of two disjoint codewords in a linear code yields their usual vector space sum.\n\\begin{lemma}(\\cite{BEV}, Lemma~8)\n\t\\label{LL1}\n\tFor two codewords $X$ and $Y$ of a linear code $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$, $X \\boxplus Y = X + Y$ if $X \\cap Y = \\{0\\}$.\n\\end{lemma}\nA linear code is said to be \\emph{closed under intersection} if it is closed with respect to vector space intersection: The subspace $X \\cap Y$ is a codeword of $\\mathcal{U}$ for any two codewords $X$ and $Y$ if $\\mathcal{U}$ is a linear code closed under intersection. The following is a method to construct such class of linear codes.\n\\begin{theorem}(\\cite{BK2}, Theorem~7)\n\t\\label{LT1}\n\tSuppose there exists a linearly independent subset $\\mathcal{E} = \\{e_1, \\ldots, e_r\\}$ of $\\mathbb{F}_q^n$ over $\\mathbb{F}_q$. Let $\\{\\mathcal{E}_1, \\ldots, \\mathcal{E}_m\\}$ be a partition of $\\mathcal{E}$. Define $\\mathcal{E}_{\\mathcal{I}} := \\bigcup\\limits_{i \\in \\mathcal{I}} \\mathcal{E}_i$ for any nonempty subset $\\mathcal{I} \\subseteq [m]$ and $\\mathcal{E}_{\\phi} := \\phi$. The code $\\mathcal{U} = \\{\\langle \\mathcal{E}_{\\mathcal{I}}\\rangle: \\mathcal{I} \\subseteq [m]\\}$ is linear and closed under intersection.\n\\end{theorem}\nA linear code thus constructed is also referred to as a \\emph{code derived from a partition of a linearly independent set}. The particular case when $r = m = n$ in Theorem~\\ref{LT1} is referred to as a \\emph{code derived from a fixed basis}, and was first introduced in \\cite{PS}. It is known that any linear code closed under intersection can only be constructed in the way described in Theorem~\\ref{LT1} \\cite[Theorem~8]{BK2}. The lattice structure of such class of linear codes was studied in \\cite{BK}.\n\\begin{theorem}(\\cite{BK}, Theorem~18)\n\t\\label{LT2}\n\tA linear code in $\\mathbb{P}_q(n)$ that is closed under intersection forms a distributive sublattice of the linear lattice $\\mathbb{P}_q(n)$.\n\\end{theorem}\nThe maximum size of a linear code closed under intersection was investigated in \\cite{BK2} and it revealed that the maximal case is unique.\n\\begin{theorem}(\\cite{BK2})\n\t\\label{LT3}\n\tThe maximum size of a linear code closed under intersection in $\\mathbb{P}_q(n)$ is $2^n$. The bound is reached if and only if the code is derived from a fixed basis.\n\\end{theorem}\nWe will exploit the lattice theoretic connection of linear codes further in Section~\\ref{S4} using the unique decomposition of geometric distributive lattices.\n\\section{Uniquely Atomistic Lattices}\n\\label{UAL}\nWe will prove modularity of uniquely atomistic lattices in this section via\na series of results. First a few elementary lemmas follow from definition.\n\\begin{lemma}\n\t\\label{UAL1}\n\tIf $x = \\sup S$ and $y = \\sup T$ are two elements of a uniquely atomistic lattice $L$, then $x \\vee y = \\sup (S \\cup T)$.\n\\end{lemma}\n\\begin{proof}\n\tBy definition, if the set of all atoms in $L$ is $\\{x_1, \\ldots, x_m\\}$ then $x = \\bigvee\\limits_{i \\in S} x_i$ and $y = \\bigvee\\limits_{j \\in T} x_j$. Since the join-operation $\\vee$ is associative, it follows that $x \\vee y = \\bigvee\\limits_{k \\in S \\cup T} x_k = \\sup (S \\cup T)$.\n\\end{proof}\n\\begin{lemma}\n\t\\label{UAL2}\n\tSuppose $L$ is an uniquely atomistic lattice. For any distinct $x, y \\in L$ we must have\n\t\\begin{equation*}\n\t\tx \\wedge y = \\sup (S_1 \\cap S_2),\n\t\\end{equation*}\n\twhere $x = \\sup S_1$ and $y = \\sup S_2$.\n\\end{lemma}\n\\begin{proof}\n\tBy definition of a lattice, $x \\vee (x \\wedge y) = x$. We can write $x \\wedge y = \\sup S_3$ for some finite set $S_3$ as $L$ is uniquely atomistic. That $x = \\sup S_1$ and $y = \\sup S_2$ implies that $\\bigvee\\limits_{i \\in S_1 \\cup S_3} x_i = \\bigvee\\limits_{j \\in S_1} x_j$ by Lemma~\\ref{UAL1}. By unique atomisticity, we get $S_3 \\subset S_1$ since $x \\ne x \\wedge y$. Similarly, $S_3 \\subset S_2$. Thus $S_3 \\subseteq S_1 \\cap S_2$.\n\t\n\tAssume that $S_3 \\ne S_1 \\cap S_2$, i.e. $S_3 \\subset S_1 \\cap S_2$. But that means $\\sup (S_1 \\cap S_2)$ is a lower bound of $x$ and $y$ and $x \\wedge y \\prec \\sup (S_1 \\cap S_2)$, a contradiction. Thus the statement follows.\n\\end{proof}\n\\begin{lemma}\n\t\\label{UAL3}\n\tSuppose $x = \\sup S$ and $y = \\sup T$ are elements of a uniquely atomistic lattice $L$. If $x \\prec y$ then $S \\subset T$.\n\\end{lemma}\n\\begin{proof}\n\tAs $x \\prec y$, we have $x \\wedge y = x$. From unique atomisticity of $L$ and Lemma~\\ref{UAL2} it can be observed that $S \\cap T = S$. The rest follows because $S \\ne T$.\n\\end{proof}\nWe will now establish that modularity is inherent in any finite uniquely atomistic lattice.\n\\begin{theorem}\n\t\\label{UAT}\n\tA finite lattice $L$ is modular if $L$ is uniquely atomistic.\n\\end{theorem}\n\\begin{proof}\n\tAccording to Theorem~\\ref{TL1} it is enough to show that there exists no sublattice of $L$ which is isomorphic to $N_5$. Let the set of all atoms in $L$ be $\\{x_1, \\ldots, x_m\\}$. We proceed by contradiction.\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.6]\n\t\t\t\\node (A1) at (0,2) {$\\mathcal{M}$};\n\t\t\t\\node (A2) at (-2,1) {$a_1$};\n\t\t\t\\node (A3) at (-2,-1) {$a_2$};\n\t\t\t\\node (A4) at (2,0) {$b$};\n\t\t\t\\node (A5) at (0,-2) {$\\mu$};\n\t\t\n\t\t\n\t\t\t\\draw (A5) -- (A3) -- (A2) -- (A1) -- (A4) -- (A5);\n\t\t\\end{tikzpicture}\n\t\t\\caption{$N_5$ lattice}\n\t\t\\label{F0}\n\t\\end{figure}\t\t\n\t\n\tAssume that there exists a sublattice of $L$ isomorphic to $N_5$ as shown in Fig.~\\ref{F0}. By unique atomisticity of $L$, we can write the following for some fixed subsets $\\mathcal{I}_1, \\mathcal{I}_2, \\mathcal{K} \\subseteq [m]$:\n\t\\begin{equation*}\n\t\ta_1 = \\bigvee\\limits_{i \\in \\mathcal{I}_1} x_i = \\sup \\mathcal{I}_1; \\quad a_2 = \\bigvee\\limits_{j \\in \\mathcal{I}_2} x_j = \\sup \\mathcal{I}_2; \\quad b = \\bigvee\\limits_{k \\in \\mathcal{K}} x_k = \\sup \\mathcal{K}.\n\t\\end{equation*}\n\tSince $\\mathcal{M} = a_1 \\vee b$ and $\\mu = a_2 \\wedge b$, we obtain from Lemmas~\\ref{UAL1} and \\ref{UAL2} that $\\mathcal{M} = \\sup (\\mathcal{I}_1 \\cup \\mathcal{K})$ and $\\mu = \\sup (\\mathcal{I}_2 \\cap \\mathcal{K})$. Similarly, from $\\mathcal{M} = a_2 \\vee b$ and $\\mu = a_1 \\wedge b$ we yield $\\mathcal{M} = \\sup (\\mathcal{I}_2 \\cup \\mathcal{K}), \\mu = \\sup (\\mathcal{I}_1 \\cap \\mathcal{K})$. Combining both, we have the following:\n\t\\begin{eqnarray}\n\t\t\\mathcal{I}_1 \\cup \\mathcal{K} &=& \\mathcal{I}_2 \\cup \\mathcal{K}; \\label{UAE1} \\\\\n\t\t\\mathcal{I}_1 \\cap \\mathcal{K} &=& \\mathcal{I}_2 \\cap \\mathcal{K}. \\label{UAE2}\n\t\\end{eqnarray}\n\tFrom \\eqref{UAE2} it follows that $(\\mathcal{I}_1 \\backslash \\mathcal{I}_2) \\cap \\mathcal{K} = \\phi$. Since $a_2 \\prec a_1$, thus $\\mathcal{I}_2 \\subset \\mathcal{I}_1$ by Lemma~\\ref{UAL3}; i.e. $\\mathcal{I}_1 \\backslash \\mathcal{I}_2$ is nonempty. For any $l \\in \\mathcal{I}_1 \\backslash \\mathcal{I}_2$ we must have $l \\in \\mathcal{I}_2 \\cup \\mathcal{K}$ from \\eqref{UAE1}, i.e. $l \\in \\mathcal{K}$. This implies that $\\mathcal{I}_1 \\backslash \\mathcal{I}_2 \\subseteq \\mathcal{K}$, or in other words $\\mathcal{I}_1 \\backslash \\mathcal{I}_2 = (\\mathcal{I}_1 \\backslash \\mathcal{I}_2) \\cap \\mathcal{K} = \\phi$. This is a contradiction to $\\mathcal{I}_2 \\subset \\mathcal{I}_1$, hence proved.\n\\end{proof}\n\\begin{corollary}\n\t\\label{UAC}\n\tAny finite uniquely atomistic lattice is geometric.\n\\end{corollary}\n\\begin{proof}\n\tFollows directly from Definition~\\ref{LD6} and Theorem~\\ref{UAT}.\n\\end{proof}\n\\begin{remark}\n\tThe converse of the statement of Theorem~\\ref{UAT} is not true, i.e. a modular lattice need not always be uniquely atomistic. E.g. the $M_3$ lattice is not uniquely atomistic. From Fig.~\\ref{F1} one can see that $\\mathbb{F}_2^2 = \\langle \\{0, 1\\}\\rangle + \\langle \\{1, 0\\}\\rangle = \\langle \\{0, 1\\}\\rangle + \\langle \\{1, 1\\}\\rangle$, where $+$ denotes the usual vector space addition.\n\\end{remark}\nTheorem~\\ref{UAT} brings us to a position where we can prove the unique-decomposition theorem in the next section.\n\n\\section{The Unique-decomposition Theorem}\n\\label{S3}\nIn this section we will establish the unique criterion needed for an atomistic lattice to be distributive and use that to characterize finite geometric distributive lattices. Atoms of geometric distributive lattices play an important part in the unique characterization. The first step towards that is observing that the greatest lower bound of two atoms in any lattice is the least element of that lattice.\n\\begin{lemma}\n\t\\label{L1}\n\tFor any two distinct atoms $x_1, x_2$ in a lattice $(L, \\vee, \\wedge)$, we have $x_1 \\wedge x_2 = O$, where $O$ is the least element of $L$.\n\\end{lemma}\n\\begin{proof}\n\tSuppose $x_1 \\wedge x_2 \\neq O$. By definition, $x_1 \\wedge x_2 \\preceq x_1$. As $x_1$ is an atom in $L$, hence $O \\lessdot x_1$, which means $x_1 \\wedge x_2 = x_1$ by our supposition. Similarly we obtain $x_1 \\wedge x_2 = x_2$. Since $x_1, x_2$ are distinct, this is a contradiction and the result follows.\n\\end{proof}\n\nNext we will prove the generalization of the above lemma for any finite number of atoms in a distributive lattice.\n\\begin{lemma}\n\t\\label{L2}\n\tLet $\\{x_1, \\ldots, x_m\\}$ be a set of atoms in a distributive lattice $M$. Then for all $i \\in [m]$,\n\t\\begin{equation*}\n\t\tx_i \\wedge (\\bigvee\\limits_{j \\in [m] \\backslash \\{i\\}} x_j) = O.\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tBy distributivity in $M$ we can write,\n\t\\begin{equation*}\n\t\tx_i \\wedge (\\bigvee\\limits_{j \\in [m] \\backslash \\{i\\}} x_j) = \\bigvee\\limits_{j \\in [m] \\backslash \\{i\\}} (x_i \\wedge x_j).\n\t\\end{equation*}\n\tAccording to Lemma~\\ref{L1}, $x_i \\wedge x_j = O$ for $i \\ne j$, and the statement is proved.\n\\end{proof}\n\nThe height of join of two atoms in a modular lattice (if the height function is defined) is the sum of heights of the individual atoms, as illustrated in the following lemma.\n\\begin{lemma}\n\t\\label{L3}\n\tSuppose $L$ is a modular lattice and $h$ is the height function defined on $L$. For any two atoms $x$ and $y$ in $L$ the following holds true:\n\t\\begin{equation*}\n\t\th(x \\vee y) = h(x) + h(y).\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tThe height function $h$ is a valuation which according to Definition~\\ref{LD8} implies that $h(x \\vee y) = h(x) + h(y) - h(x \\wedge y)$. As $x$ and $y$ are atoms in $L$, Lemma~\\ref{L1} dictates that $x \\wedge y = O$. That $h(O) = h(x \\wedge y) = 0$ which proves the rest.\n\\end{proof}\n\nWe are now going to generalize Lemma~\\ref{L3} for any $m \\ge 2$ number of atoms if the lattice is also distributive.\n\\begin{lemma}\n\t\\label{L3a}\n\tConsider a set $\\{x_1, \\ldots, x_m\\}$ of $m \\ge 2$ atoms in a distributive lattice $L$. If $h$ is the height function defined on $L$ then\n\t\\begin{equation*}\n\t\th(\\bigvee\\limits_{i \\in [m]} x_i) = \\sum\\limits_{i \\in [m]}h(x_i).\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tThe proof is by induction. The base case for $m = 2$ is covered by Lemma~\\ref{L3}. Suppose the statement holds true for any $(m-1)$ atoms in $L$, i.e., $h(\\bigvee\\limits_{i \\in [m-1]} x_i) = \\sum\\limits_{i \\in [m-1]}h(x_i)$. Since the join operation $\\vee$ is associative over the elements of $L$, we can write $\\bigvee\\limits_{i \\in [m]} x_i = x_m \\vee (\\bigvee\\limits_{j \\in [m-1]} x_j)$. The height of $\\bigvee\\limits_{i \\in [m]} x_i$ can therefore be expressed as:\n\t\\begin{equation*}\n\t\th(\\bigvee\\limits_{i \\in [m]} x_i) = h(x_m) + h(\\bigvee\\limits_{i \\in [m-1]} x_i) - h(x_m \\wedge (\\bigvee\\limits_{i \\in [m-1]} x_i)).\n\t\\end{equation*}\n\tAs $h(x_m \\wedge (\\bigvee\\limits_{i \\in [m-1]} x_i)) = 0$ according to Lemma~\\ref{L2}, the rest follows from the induction hypothesis.\n\\end{proof}\n\nConsequence of Lemma~\\ref{L3a} is that the number of atoms in a distributive sublattice of a finite modular lattice of height $n$ cannot exceed $n$. We formally state the result.\n\\begin{proposition}\n\t\\label{P2}\n\tIf $L$ is a finite modular lattice of height $n$ then the number of atoms in a distributive sublattice $M$ of $L$ can be at most $n$. The maximum number of atoms is reached if and only if all atoms of $M$ are also atoms in $L$ and the greatest element of $L$ is join of the atoms in $M$.\n\\end{proposition}\n\\begin{proof}\n\tSuppose $\\{x_1, \\ldots, x_m\\}$ be the set of atoms in $M$. If $I$ is the greatest element in $L$ then certainly $\\bigvee\\limits_{i \\in [m]} x_i \\preceq I$. As $h$ is an isotone valuation, Definition~\\ref{LD8}(ii) implies that $h_L(\\bigvee\\limits_{i \\in [m]} x_i) \\le h_L(I)$. Observe that $h_L(y) \\ge h_M(y)$ for all $y \\in M$. Since $h_L(I) = n$, by Lemma~\\ref{L3a} we have the following inequality that proves the claim:\n\t\\begin{equation}\n\t\t\\label{E0}\n\t\tm = \\sum\\limits_{i \\in [m]} h_M(x_i) = h_M(\\bigvee\\limits_{i \\in [m]} x_i) \\le h_L(\\bigvee\\limits_{i \\in [m]} x_i) \\le h_L(I) = n.\n\t\\end{equation}\n\tSince $h$ is positive isotone, it is evident from \\eqref{E0} that $m = n$ if and only if $I = \\bigvee\\limits_{i \\in [m]} x_i$ and $h_M(\\bigvee\\limits_{i \\in [m]} x_i) = h_L(\\bigvee\\limits_{i \\in [m]} x_i)$. That $x_i$'s are atoms in $L$ for all $i \\in [m]$ if and only if $x_1 \\prec x_1 \\vee x_2 \\prec \\cdots \\prec \\bigvee\\limits_{j \\in [m-1]} x_j \\prec \\bigvee\\limits_{i \\in [m]} x_i$ is a maximal chain in $L$ proves the rest.\n\\end{proof}\n\nAny element in a geometric lattice can be expressed as a join of atoms (Definition~\\ref{LD6}). However, such representation is not unique. To elaborate, if $\\{x_1, \\ldots, x_m\\}$ is the set of atoms in a geometric lattice $L$ then there may exist $y \\in L$ such that $y = \\bigvee\\limits_{i \\in \\mathcal{I} \\subseteq [m]} x_i = \\bigvee\\limits_{j \\in \\mathcal{J} \\subseteq [m]} x_j$ for two different subsets $\\mathcal{I}, \\mathcal{J} \\subseteq [m]$. We will now show that representation of elements as join of atoms is unique if and only if the lattice is also distributive. In fact we will prove the following which is a more generalized statement.\n\\begin{theorem}[Unique-decomposition Theorem]\n\t\\label{T1}\n\tA finite atomistic lattice is distributive if and only if it is uniquely atomistic.\n\n\\end{theorem}\n\\begin{proof}\n\tLet $M$ be an atomistic lattice with $\\{x_1, \\ldots, x_m\\}$ as its set of all atoms. First we prove that $M$ is uniquely atomistic, i.e. $\\bigvee\\limits_{i \\in \\mathcal{I} \\subseteq [m]} x_i \\in M$ is uniquely determined by $\\mathcal{I} \\subseteq [m]$ when $M$ is distributive.\n\t\n\tThe proof is by contradiction. Suppose the claim is false, i.e. there exist subsets $\\mathcal{I}, \\mathcal{J} \\subseteq [m]$ such that $\\mathcal{I} \\ne \\mathcal{J}$ and $\\bigvee\\limits_{i \\in \\mathcal{I}} x_i = \\bigvee\\limits_{j \\in \\mathcal{J}} x_j$. Since $\\mathcal{I}$ and $\\mathcal{J}$ are distinct, at least one of them is not contained within the other. Without loss of generality, suppose $\\mathcal{I} \\nsubseteq \\mathcal{J}$. Then the difference set $\\mathcal{I} \\backslash \\mathcal{J}$ is nonempty, i.e. there exists an integer $l \\in \\mathcal{I} \\backslash \\mathcal{J}$. Thus we can write as per our supposition:\n\t\\begin{equation}\n\t\t\\label{E1}\n\t\tx_l \\wedge (\\bigvee\\limits_{i \\in \\mathcal{I}} x_i) = x_l \\wedge (\\bigvee\\limits_{j \\in \\mathcal{J}} x_j).\n\t\\end{equation}\n\t\n\tThe individual terms can be decomposed further. Since $l \\in \\mathcal{I}$, by distributivity in $M$ we obtain $x_l \\wedge (\\bigvee\\limits_{i \\in \\mathcal{I}} x_i) = (x_l \\wedge (\\bigvee\\limits_{i \\in \\mathcal{I} \\backslash \\{l\\}} x_i)) \\bigvee (x_l \\wedge x_l) = O \\vee x_l = x_l$. The penultimate step follows from Lemma~\\ref{L2}. Similar technique yields $x_l \\wedge (\\bigvee\\limits_{j \\in \\mathcal{J}} x_j) = O$ as $l \\notin \\mathcal{J}$. Hence \\eqref{E1} suggests $x_l = O$. However, this is a contradiction since $l \\in \\mathcal{I} \\subseteq [m]$, i.e. $x_l$ is an atom. We conclude that our initial assumption was wrong and the representation $\\bigvee\\limits_{i \\in \\mathcal{I}} x_i$ is uniquely determined by $\\mathcal{I}$.\n\t\n\tNow it remains to prove that $M$ is distributive if it is uniquely atomistic. Once again we proceed by contradiction. That $M$ is modular follows at once from Theorem~\\ref{UAT}. Suppose $M$ is modular non-distributive. By Theorem~\\ref{TL1} $M$ must contain a sublattice isomorphic to $M_3$. In other words there exist $y_1, y_2, y_3 \\in M$ such that $y_1 \\vee y_2 = y_2 \\vee y_3 = y_3 \\vee y_1$ and $y_1 \\wedge y_2 = y_2 \\wedge y_3 = y_3 \\wedge y_1$, where $y_i \\npreceq y_j$ for $i \\neq j$ (See Fig.~\\ref{F2}). Suppose $\\mathcal{M} = y_1 \\vee y_2$ and $\\mathfrak{m} = y_1 \\wedge y_2$. By imposition of unique atmosticity, $y_j = \\bigvee\\limits_{i \\in \\mathcal{J}_j} x_i$ for $j = 1, 2, 3$, where $\\mathcal{J}_j \\subseteq [m]$ uniquely determines $y_j$. This implies the following:\n\t\\begin{equation}\n\t\t\\label{E1a}\n\t\t\\mathcal{M} = \\bigvee\\limits_{i \\in \\mathcal{J}_1 \\cup \\mathcal{J}_2} x_i = \\bigvee\\limits_{j \\in \\mathcal{J}_2 \\cup \\mathcal{J}_3} x_j = \\bigvee\\limits_{k \\in \\mathcal{J}_3 \\cup \\mathcal{J}_1} x_k.\n\t\\end{equation}\n\n\n\n\n\tSince $M$ is uniquely atomistic, applying Lemma~\\ref{UAL1} to \\eqref{E1a} implies that $\\mathcal{J}_1 \\cup \\mathcal{J}_2 = \\mathcal{J}_2 \\cup \\mathcal{J}_3 = \\mathcal{J}_3 \\cup \\mathcal{J}_1$.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.7]\n\t\t\t\\node (A1) at (0,2) {$\\mathcal{M}$};\n\t\t\t\\node (A2) at (-2,0) {$y_1$};\n\t\t\t\\node (A3) at (0,0) {$y_2$};\n\t\t\t\\node (A4) at (2,0) {$y_3$};\n\t\t\t\\node (A5) at (0,-2) {$\\mathfrak{m}$};\n\t\t\t\\node[right=0pt of A1,inner xsep=0pt] {$= y_1 \\vee y_2 = y_2 \\vee y_3 = y_3 \\vee y_1$};\n\t\t\t\\node[right=0pt of A5,inner xsep=0pt] {$= y_1 \\wedge y_2 = y_2 \\wedge y_3 = y_3 \\wedge y_1$};\n\t\t\t\\draw (A5) -- (A2) -- (A1) -- (A3) -- (A5) -- (A4) -- (A1);\n\t\t\\end{tikzpicture}\n\t\t\\caption{$M_3$-sublattice in $M$}\n\t\t\\label{F2}\n\t\\end{figure}\n\t\n\tOn the other hand $\\mathfrak{m} = y_1 \\wedge y_2$, where $y_1 = \\sup \\mathcal{J}_1$ and $y_2 = \\sup \\mathcal{J}_2$. If $\\mathcal{L} \\subseteq [m]$ uniquely determines $\\mathfrak{m}$, i.e. $\\mathfrak{m} = \\sup \\mathcal{L}$, then Lemma~\\ref{UAL2} implies that $\\mathcal{L} = \\mathcal{J}_1 \\cap \\mathcal{J}_2$. Similarly we can deduce for $y_2 \\wedge y_3$ and $y_3 \\wedge y_1$ which indicates that $\\mathcal{J}_1 \\cap \\mathcal{J}_2 = \\mathcal{J}_2 \\cap \\mathcal{J}_3 = \\mathcal{J}_3 \\cap \\mathcal{J}_1$.\n\t\n\tWe can now express $\\mathcal{J}_1$ as $\\mathcal{J}_1 = \\mathcal{J}_1 \\cap (\\mathcal{J}_1 \\cup \\mathcal{J}_2) = \\mathcal{J}_1 \\cap (\\mathcal{J}_2 \\cup \\mathcal{J}_3) = (\\mathcal{J}_1 \\cap \\mathcal{J}_2) \\cup (\\mathcal{J}_1 \\cap \\mathcal{J}_3) = \\mathcal{J}_1 \\cap \\mathcal{J}_2$, i.e. $\\mathcal{J}_1 \\subseteq \\mathcal{J}_2$. This means $y_1 \\preceq y_2$, which contradicts our initial assumption. Hence, $M$ is distributive.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{UDT}\n\tA geometric lattice is distributive if and only if it is uniquely atomistic.\n\\end{corollary}\n\\begin{proof}\n\tA geometric lattice is finite atomistic by definition, which concludes the proof.\n\\end{proof}\n\\begin{remark}\n\tThe semimodularity of a geometric lattice is not required for it to be distributive.\n\\end{remark}\nThe consequence of Corollary~\\ref{UDT} is that any element of a geometric distributive lattice can be uniquely decomposed as join of its atoms. The next statement also follows from Theorem~\\ref{T1}:\n\\begin{corollary}\n\t\\label{P3}\n\tThe greatest element in a geometric distributive lattice is the join of all of its atoms.\n\\end{corollary}\n\\begin{proof}\n\tLet $\\{x_1, \\ldots, x_m\\}$ be the set of all atoms in a geometric distributive lattice $M$. We aim to show that the greatest element in $M$ is $g := \\bigvee\\limits_{i \\in [m]} x_i$. To that end, say $y \\in M$ is the greatest element in $M$. By Theorem~\\ref{T1} we can write $y = \\bigvee\\limits_{i \\in \\mathcal{I}} x_i$ for some $\\mathcal{I} \\subseteq [m]$. However, by associativity of the join operation $\\vee$ in $M$, $g$ can also be decomposed as $g = (\\bigvee\\limits_{j \\in [m] \\backslash \\mathcal{I}} x_j) \\vee (\\bigvee\\limits_{i \\in \\mathcal{I}} x_i) = (\\bigvee\\limits_{j \\in [m] \\backslash \\mathcal{I}} x_j) \\vee y$, which implies $y \\preceq g$; thus the only possibility is $y = g$, which concludes the proof.\n\\end{proof}\nIn the following section we will see a few applications of the Unique-decomposition theorem, mainly in the context of linearity and complements in $\\mathbb{P}_q(n)$.\n\n\\section{Applications of the Unique-decomposition Theorem}\n\\label{S4}\nThe unique-decomposition theorem, akin to unique decomposition in context of linear subspace codes \\cite[Proposition~12]{BK2}, lays the path for determining the maximum size of a distributive sublattice of a finite geometric lattice. We also characterize the extremal case.\n\\begin{theorem}\n\t\\label{T2}\n\tThe size of any distributive sublattice of a finite geometric lattice of height $n$ can be at most $2^n$. The bound is reached if and only if each of the atoms in the sublattice is also an atom in the geometric lattice and the greatest element of the geometric lattice belongs to the distributive sublattice.\n\\end{theorem}\n\\begin{proof}\n\tSuppose $M$ is a distributive sublattice of a finite geometric lattice $L$ with the height function $h$ defined on $L$ such that $h(I) = n$, where $I$ is the greatest element of $L$. We require to show that $|M| \\le 2^n$.\n\t\n\tBy supposition $M$ is geometric. Let $\\{x_1, \\ldots, x_m\\}$ be the set of all atoms in $M$. We define a mapping $\\Phi$ from $\\mathcal{P}(m)$ to $M$ as below:\n\t\\begin{eqnarray*}\n\t\t\\Phi : \\mathcal{P}(m) &\\longrightarrow& M \\nonumber \\\\\n\t\t\\mathcal{I} &\\mapsto& \\bigvee\\limits_{i \\in \\mathcal{I}} x_i.\n\t\\end{eqnarray*}\n\tBy definition the map $\\Phi$ is well-defined. Suppose there exist $\\mathcal{I}, \\mathcal{J} \\in \\mathcal{P}(m)$ such that $\\Phi(\\mathcal{I}) = \\Phi(\\mathcal{J})$, i.e., $\\bigvee\\limits_{i \\in \\mathcal{I}} x_i = \\bigvee\\limits_{j \\in \\mathcal{J}} x_j$. As $M$ is geometric distributive, Corollary~\\ref{UDT} dictates that $\\mathcal{I} = \\mathcal{J}$; thus $\\Phi$ is injective. To check that $\\Phi$ is also surjective, any $y \\in M$ can be expressed as $y = \\bigvee\\limits_{i \\in \\mathcal{S} \\subseteq [m]} x_i$ for some $\\mathcal{S} \\subseteq [m]$ as $M$ is geometric; the choice of $\\mathcal{S}$ is unique according to Corollary~\\ref{UDT}, hence $y = \\Phi(\\mathcal{S})$. Therefore $\\Phi$ is a bijective map, which implies that $|M| = |\\mathcal{P}(m)| = 2^m$. Combining this with Proposition~\\ref{P2} yields $|M| \\le 2^n$.\n\n\n\n\n\t\n\tFor the extremal case we must have $m = n$. The rest then follows from Proposition~\\ref{P2}.\n\\end{proof}\n\\begin{remark}\n\tAn atom in a sublattice $M$ of a geometric lattice $L$ may not be an atom in $L$. E.g. consider the lattice $\\mathcal{P}(4)$, the set of all subsets of $\\{1, 2, 3, 4\\}$. It is a geometric lattice with atoms $\\{1\\}, \\{2\\}, \\{3\\}$ and $\\{4\\}$. The sublattice $M = \\{\\phi, \\{1, 2\\}, \\{3, 4\\}, \\{1, 2, 3, 4\\}\\}$ of $\\mathcal{P}(4)$ has atoms $\\{1, 2\\}$ and $\\{3, 4\\}$, none of which is an atom in $\\mathcal{P}(4)$. On the other hand, a proper sublattice of a geometric lattice $L$ does not contain all atoms of $L$.\n\\end{remark}\n\\begin{remark}\n\tIrrespective of the choice of the lattice $M$, the mapping $\\Phi$ in Theorem~\\ref{T2} always maps the ground set $[m]$ to $I$ and the empty subset $\\phi$ to $O$.\n\\end{remark}\nClass of distributive sublattices of maximum size in a finite geometric lattice can be characterized in an alternative way.\n\\begin{corollary}\n\t\\label{C1}\n\tThe size of a distributive sublattice $M$ of a finite geometric lattice $L$ of height $n$ is $2^n$ if and only if $M$ contains $n$ atoms of $L$.\n\\end{corollary}\n\\begin{proof}\n\tBy Proposition~\\ref{P2} $M$ can have at most $n$ atoms, and for the extremal case all of them belong to $L$. Since $M$ is geometric distributive, proof technique of Theorem~\\ref{T2} suggests that $|M| = |\\mathcal{P}(n)| = 2^n$.\n\t\n\tConversely, if $|M| = 2^n$ then according to Theorem~\\ref{T2} each atom of $M$ is also an atom in $L$ and $I \\in M$. Suppose $\\{x_1, \\ldots, x_m\\}$ is the set of all atoms in $M$; thus $h_L(x_i) = 1$ for all $i \\in [m]$. By Corollary~\\ref{P3}, $\\bigvee\\limits_{i \\in [m]} x_i$ is the greatest element in $M$, which implies that $I = \\bigvee\\limits_{i \\in [m]} x_i$. By Lemma~\\ref{L3a}, $m = \\sum\\limits_{i \\in [m]}h_L(x_i) = h_L(\\bigvee\\limits_{i \\in [m]} x_i) = n$, which settles the proof.\n\n\n\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{LT3}]\n\tThe upper bound follows at once from Theorem~\\ref{LT2} and Theorem~\\ref{T2}. Corollary~\\ref{C1} implies that the maximal case occurs if and only if the number of one-dimensional codewords is $n$, i.e. the code is derived from a fixed basis.\n\\end{proof}\nWe now consider the Whitney numbers of a geometric distributive lattice.\n\\begin{corollary}\n\t\\label{C2}\n\tThe Whitney numbers of a distributive sublattice $L$ of a finite geometric lattice of height $n$ are bounded by $W_k(L) \\le \\binom{n}{k}$ for all $k \\in \\{0, 1, \\ldots, n\\}$. Equality occurs if and only if $L$ contains $n$ atoms.\n\\end{corollary}\n\\begin{proof}\n\tLet the set of atoms in $L$ be $\\{x_1, \\ldots, x_m\\}$. The case of $k = 0$ is obvious since $h(O) = 0$. As  shown in proof of Theorem~\\ref{T2}, there exists a bijection from $L$ to $\\mathcal{P}(m)$ that sends $\\bigvee_{i \\in \\mathcal{I}} x_i$ to $\\mathcal{I} \\subseteq [m]$. From Lemma~\\ref{L3a} it follows that $W_k(L) = \\binom{m}{k}$ for all $k \\in [m]$. As $m \\le n$ by Proposition~\\ref{P2}, the result follows. The bound is achieved if and only if $m = n$.\n\\end{proof}\n\\begin{remark}\n\tThe statement in Corollary~\\ref{C2} is similar in nature to the main result in Pai and Rajan's paper \\cite[Theorem~2]{PS}.\n\\end{remark}\nIn the sequel we will consider the linear lattice $\\mathbb{P}_q(n)$ instead of a geometric lattice in general. $\\mathbb{P}_q(n)$ is non-distributive geometric. There remain a few unanswered questions regarding linearity and complements in $\\mathbb{P}_q(n)$ that can be resolved by applying the unique-decomposition theorem. It was shown before that a linaer code in $\\mathbb{P}_q(n)$ that is closed under intersection necessarily is a distributive sublattice of the geometric lattice $\\mathbb{P}_q(n)$ \\cite{BK}. Using the unique-decomposition theorem we now prove the converse.\n\\begin{theorem}\n\t\\label{T3}\n\tA subset $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$ is a distributive sublattice of the corresponding linear lattice $\\mathbb{P}_q(n)$ if and only if $\\mathcal{U}$ is a linear code closed under intersection.\n\\end{theorem}\n\\begin{proof}\n\tSuppose $\\mathcal{U}$ is a distributive sublattice of $\\mathbb{P}_q(n)$ and $\\{X_1, \\ldots, X_m\\}$ is the set of all atoms in $\\mathcal{U}$. Obviously $\\mathcal{U}$ is geometric distributive, which according to Lemma~\\ref{L2} implies that\n\t\\begin{equation}\n\t\t\\label{E3}\n\t\tX_i \\cap (\\sum_{j \\in [m] \\backslash \\{i\\}} X_j) = \\{0\\}, \\qquad \\forall i \\in [m].\n\t\\end{equation}\n\tApplying the unique-decomposition theorem it is easy to see that any $Y \\in \\mathcal{U}$ can be uniquely expressed as $Y = \\sum\\limits_{i \\in \\mathcal{I}} X_i$ for some fixed $\\mathcal{I} \\subseteq [m]$. If we choose arbitrary bases $B_i$ that span $X_i$ over $\\mathbb{F}_q$ for all $i \\in [m]$ then \\eqref{E3} implies that the set $\\{B_1, \\ldots, B_m\\}$ is a partition of $B := \\bigcup_{i=1}^{m} B_i$, a linearly independent subset of $\\mathbb{F}_q^n$ over $\\mathbb{F}_q$. Expressing any $Y = \\sum\\limits_{i \\in \\mathcal{I}} X_i$ as $Y = \\langle B_{\\mathcal{I}}\\rangle$ where $B_{\\mathcal{I}} := \\cup_{i \\in \\mathcal{I}} B_i$, we can say by Theorem~\\ref{LT1} that $\\mathcal{U} = \\{\\langle B_{\\mathcal{I}}\\rangle: \\mathcal{I} \\subseteq [m]\\}$ is a linear code closed under intersection with linear addition $\\boxplus$ of two codewords $Y_1 = \\sum\\limits_{j \\in \\mathcal{I}_1} X_j$ and $Y_2 = \\sum\\limits_{l \\in \\mathcal{I}_2} X_l$ defined as:\n\t\\begin{equation*}\n\t\tY_1 \\boxplus Y_2 := \\langle B_{\\mathcal{I}_1 \\triangle \\mathcal{I}_2}\\rangle = \\sum\\limits_{j \\in \\mathcal{I}_1 \\triangle \\mathcal{I}_2} X_j.\n\t\\end{equation*}\n\t\n\tThe converse is basically the statement of Theorem~\\ref{LT2}.\n\\end{proof}\nThe maximum size of a subset of $\\mathbb{P}_q(n)$ wherein a complement function can be defined is hitherto unknown. We next investigate any such subset of $\\mathbb{P}_q(n)$ that has a distributive sublattice structure.\n\\begin{theorem}\n\t\\label{T4}\n\tA subset $\\mathcal{U} \\subseteq \\mathbb{P}_q(n)$ is a distributive sublattice of the linear lattice $\\mathbb{P}_q(n)$ with a complement function defined on $\\mathcal{U}$ if and only if $\\mathcal{U}$ is a linear code closed under intersection with $\\mathbb{F}_q^n \\in \\mathcal{U}$.\n\\end{theorem}\n\\begin{proof}\n\tSuppose $\\mathcal{U}$ is a distributive sublattice of $\\mathbb{P}_q(n)$ and $f : \\mathcal{U} \\rightarrow \\mathcal{U}$ is a complement on $\\mathcal{U}$. It follows directly from Theorem~\\ref{T3} that $\\mathcal{U}$ is a linear code closed under intersection. For any $X \\in \\mathcal{U}$, the direct sum of $X$ and $f(X)$ is $X \\oplus f(X) = \\mathbb{F}_q^n$. Since $X \\cap f(X) = \\{0\\}$ by definition of $f$, it follows from Lemma~\\ref{LL1} that $X \\boxplus f(X) = X \\oplus f(X) = \\mathbb{F}_q^n \\in \\mathcal{U}$.\n\t\n\tConversely, if $\\mathcal{U}$ is a linear code closed under intersection with $\\mathbb{F}_q^n \\in \\mathcal{U}$ then the function $f$ defined as $f(X) := X \\boxplus \\mathbb{F}_q^n$ for all $X \\in \\mathcal{U}$ serves as a complement function on $\\mathcal{U}$. $\\mathcal{U}$ is a distributive sublattice of $\\mathbb{P}_q(n)$ according to Theorem~\\ref{LT2}.\n\\end{proof}\n\\begin{corollary}\n\t\\label{C3}\n\tThe largest distributive sublattice of $\\mathbb{P}_q(n)$ on which a complement function can be defined is a code derived from a fixed basis.\n\\end{corollary}\n\\begin{proof}\n\tBy Theorem~\\ref{T4} a distributive sublattice of $\\mathbb{P}_q(n)$ on which a complement function can be defined is a linear code closed under intersection containing $\\mathbb{F}_q^n$. Theorem~\\ref{LT3} indicates that the code has a maximum size if and only if it is derived from a fixed basis.\n\\end{proof}\nThe maximum size of a sublattice of $\\mathbb{P}_q(n)$ wherein a complement function can be defined is, however, unknown. It needs to be investigated first whether a complement function can exist in a non-distributive sublattice of $\\mathbb{P}_q(n)$.\n\\section{Conclusion}\n\\label{S5}\nIn this paper we have derived the unique criterion required for an atomistic lattice to be distributive and used that to characterize all finite geometric distributive lattices. Using the unique characterization we were able to prove that the size of a distributive sublattice of a finite geometric lattice of height $n$ is always upper bounded by $2^n$, which was conjectured in \\cite{PS}. We also applied the characterization to the linear lattice $\\mathbb{P}_q(n)$, which is geometric, to obtain certain results regarding linearity and complements in $\\mathbb{P}_q(n)$.\n\nTheorem~\\ref{T3} states that any distributive sublattice of $\\mathbb{P}_q(n)$ can be used to construct a linear code in $\\mathbb{P}_q(n)$ that is closed under intersection. This result is similar to the fact that certain $d$-intersecting families in $\\mathbb{G}_q(n, 2d)$ can always be used to construct \\emph{equidistant} linear codes with constant distance $2d$ \\cite{PB}. It might be interesting to find a more generalized statement that encapsulates the essence of both these results.\n\n\\section*{Acknowledgement}\n\nThe author would like to thank Prof. Navin Kashyap and Dr. Arijit Ghosh for their insightful comments.\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Catalogue construction}\n\nWhilst semi-analytic models have proven useful in modelling and predicting numerous properties of the galaxy population (Kauffmann, White \\& Guiderdoni \\cite{kwg}; Cole et al \\cite{coleetal94}; Somerville \\& Primack \\cite{somervilleprimack} they provide only limited information on the spatial distribution of galaxies, and so it has been difficult to study in detail the clustering properties of galaxies using these models. Recently, semi-analytic models have been combined with N-body simulations to provide the required spatial information (Kauffmann, Nusser \\& Steinmetz \\cite{kns}; Governato et al \\cite{fabio}; Kauffmann et al \\cite{gketal}). We have used a similar technique to study the clustering of galaxies in CDM universes within well constrained semi-analytic models. The main difference between our technique and that of Kauffmann et al \\cite{gketal} is that whilst they extract the merging history of each dark matter halo directly from the simulation, we construct this history using the extended Press-Schechter theory. We find that this gives the same statistical results as the Kauffmann et al method and allows us to resolve merger trees to much smaller masses.\n\nTo construct a catalogue of galaxies containing spatial information we use the following procedure: (i) take the output from a dissipationless N-body simulation of dark matter and use a group finding algorithm (here we use the friends-of-friends algorithm with the standard linking length of $b=0.2$) to locate bound, virialised haloes of dark matter of 10 or more particles (such groups have been shown to be stable by Kauffmann et al \\cite{gketal}); (ii) determine the mass of each group, the position and velocity of its centre of mass and the positions and velocities of randomly selected particles within the group; (iii) for each group use a semi-analytic model of galaxy formation constrained to match the local B and K-band luminosity functions to determine the population of galaxies living within the dark matter halo; (iv) attach the central galaxy of the halo to the centre of mass of the group and attach any satellite galaxies to randomly selected particles within the halo so that galaxies trace mass within a given dark matter halo (which may not be exactly true in reality because of processes such as dynamical friction).\n\nThis results in a galaxy catalogue which can be analysed to determine the clustering properties of galaxies of any given luminosity, morphology, colour and so on. In fact by using this technique it is possible to produce catalogues of galaxies complete with spatial information (or alternatively redshifts, and angular coordinates) with any observationally motivated selection criteria. Furthermore, by identifying dark matter halos on the past lightcone of an observer a full kock galaxy redshift survey can be constructed. The luminosity functions determined from the galaxy catalogue are shown in Figure \\ref{fig1}. Although they become incomplete at the faint end because of the limited resolution of the N-body simulation there is good agreement between the bright ends and the observed luminosity functions. Since we only consider the clustering of galaxies for which our catalogue is complete the resolution limit is not important. We find however that an accurate match to the bright end of the luminosity function is important as it strongly constrains the resulting two-point correlation function. An example of the information produced by our model is given in Figure \\ref{fig2}, which shows a slice through a $\\Lambda$CDM N-body simulation upon which the positions of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ (we define the Hubble constant to be ${\\mathrm H}_0 = 100 h$ km s$^{-1}$ Mpc$^{-1}$) have been overlaid as circles. It can be seen quite clearly that the galaxies trace the mass to some extent. To determine exactly how well galaxies trace the underlying dark matter we estimate the two-point correlation function of these galaxies.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\psfig{file=benson-fig1a.ps,width=80mm} & \\psfig{file=benson-fig1b.ps,width=80mm}\n\\end{tabular}\n\\caption{The local B and K-band luminosity functions from our model compared to various observational determinations. Note the good agreement between model (solid line) and observations (symbols) at the bright end. Our galaxy catalogues become incomplete at the faint end of the luminosity functions (faintwards of $M_{\\mathrm B} - 5 \\log h \\approx -17.5$ in the B-band and $M_{\\mathrm K} - 5 \\log h \\approx -20.5$ in the K-band) due to the limited resolution of the N-body simulation. Our catalogues are constructed using only galaxies for which a complete sample is available within the model.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\hspace{0.in}\\psfig{file=benson-fig2.ps,width=100mm}\n\\caption{A slice through a $\\Lambda$CDM N-body simulation. The volume shown is 141 $\\times$ 141 $\\times$ 8 $h^{-1}$ Mpc. Dark matter is shown by the greyscale, with the darker areas being the most dense. Overlaid are the positions of all galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ indicated by circles.}\n\\label{fig2}\n\\end{figure}\n\n\\section{The two-point correlation function}\n\n\\begin{figure}[t]\n\\centering\n\\hspace{0.in}\\psfig{file=benson-fig3.ps,width=100mm}\n\\caption{The correlation function of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ in a $\\Lambda$CDM cosmology. Points with errorbars show the APM galaxy correlation function of Baugh \\cite{APM}. The dotted line shows the dark matter correlation function whilst the solid line shows the correlation function of galaxies in our model (the dashed lines to either side indicate the Poisson errors on this correlation function).}\n\\label{fig3}\n\\end{figure}\n\nShown in Figure \\ref{fig3} is the two-point correlation function of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ in our $\\Lambda$CDM model (which has $\\Omega _0 = 0.3$, $\\Lambda = 0.7$, $h = 0.7$ and $\\sigma _8 = 0.9$). This is compared to the correlation function of the underlying dark matter and to the observationally determined correlation function of galaxies in the APM survey from Baugh \\cite{APM}. Note that all of the curves shown here are real space correlation functions. The galaxy correlation function shows many interesting properties. Firstly it is biased with respect to the mass correlation function, and furthermore this bias is scale dependent. On small scales there is in fact an antibias, which is exactly what is needed to reconcile the theory with the observed galaxy correlation function. The observed and model galaxy correlation functions agree over a wide range of scales, both showing approximate power law behaviour. The scale-dependant bias seen in these models arises from a complex interplay of effects. On large scales the bias is due to the intrinsic bias of dark matter halos in a CDM universe as described by Mo \\& White \\cite{mowhite}. On smaller scales the bias is controlled by the way halos are populated with galaxies, specifically the variations in number of galaxies per halo for halos of a given mass. The Lagrangian radius exclusion of halos also affects the small scale bias. These issues are explored in greater detail by Benson et al \\cite{meetal}, who also study $\\Omega _0 = 1$ models and find that such models fail to reproduce the observed clustering of galaxies.\n\n\\acknowledgements{}\n\nAJB's attendance at the X$^{\\mathrm th}$ Rencontres de Blois was funded in part by a grant from the European Commision.\n\n\n\\begin{bloisbib}\n\\bibitem{APM} Baugh C. M., 1996, \\mnras {280} {267}\n\\bibitem{meetal} Benson A. J., Cole S., Frenk C. S., Baugh C. M., Lacey C. G., 1998, {\\it in preparation}\n\\bibitem{coleetal94} Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 1994, \\mnras {271} {781}\n\\bibitem{coleetal98} Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 1998, {\\it in preparation}\n\\bibitem{colless} Colless M., Boyle B., 1997, {\\it astro-ph\/9710268}\n\\bibitem{gardner} Gardner J. P., Sharples R. M., Frenk C. S., Carrasco B. E., 1997, \\apj {480} {L99}\n\\bibitem{glazebrook} Glazebrook K. et al, 1995, {\\it astro-ph\/9503116}\n\\bibitem{fabio} Governato F. et al., 1998, \\nat {392} {359}\n\\bibitem{kwg} Kauffmann G., White S. D. M., Guiderdoni B., 1993, \\mnras {264} {201}\n\\bibitem{kns} Kauffmann G., Nusser A., Steinmetz M., 1997, \\mnras {286} {795}\n\\bibitem{gketal} Kauffmann G., Colberg J. M., Diaferio A., White S. D. M., 1998, {\\it astro-ph\/9805283}\n\\bibitem{loveday} Loveday J., Peterson B. A., Efstathiou G., Maddox S. J., 1992, \\apj {390} {338}\n\\bibitem{mowhite} Mo H. J., White S. D. M., 1996, \\mnras {282} {347}\n\\bibitem{arat} Ratcliffe A., Shanks T., Parker Q. A., Fong R., 1998, \\mnras {296} {173}\n\\bibitem{somervilleprimack} Somerville R., Primack J., 1998, {\\it astro-ph\/9802268}\n\\bibitem{szokoly} Szokoly G.P., Subbarao M.U., Conolly A.J., Mobasher B., 1998, {\\it astro-ph\/9801132}\n\\bibitem{zucca} Zucca E. et al, 1997, \\aa {326} {477}\n\\end{bloisbib}\n\\vfill\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe (original) Caldero-Chapoton map $X$ is an important object in\ncluster theory.  The arguments of $X$ are certain objects of a cluster category, and the values are the corresponding elements of a cluster algebra.  The map $X$ expresses\nthat the cluster category is a categorification of the\ncluster algebra, see \\cite{CC}, \\cite{CK}, \\cite{CK2}, \\cite{FK},\n\\cite{Palu}.  For example, Figure \\ref{fig:AR_quiver} shows the\nAuslander-Reiten (AR) quiver of $\\mathsf{C}( A_5 )$, the cluster category of\nDynkin type $A_5$, with a useful ``coordinate system''.  Figure\n\\ref{fig:frieze} shows the AR quiver again, with the values of $X$ on\nthe indecomposable objects of $\\mathsf{C}( A_5 )$.  The values are Laurent\npolynomials over $\\mathbb{Z}$; indeed, the cluster algebra consists of such\nLaurent polynomials.\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    *+[blue]{\\{\\, 5,7 \\,\\}} \\ar[dr] && \\{\\, 6,8 \\,\\} \\ar[dr] && *+[blue]{\\{\\, 1,7 \\,\\}} \\ar[dr] && \\{\\, 2,8 \\,\\} \\ar[dr] && \\{\\, 1,3 \\,\\} \\ar@{.}[dd] \\\\\n    & \\{\\, 5,8 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 1,6 \\,\\} \\ar[dr] \\ar[ur] && *+[red]{\\{\\, 2,7 \\,\\}} \\ar[dr] \\ar[ur] && \\{\\, 3,8 \\,\\} \\ar[dr] \\ar[ur] \\\\\n    \\{\\, 4,8 \\,\\} \\ar[dr] \\ar[ur] \\ar@{.}[uu] \\ar@{.}[dd] && \\{\\, 1,5 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 2,6 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 3,7 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 4,8 \\,\\} \\ar@{.}[uu] \\ar@{.}[dd] \\\\\n    & \\{\\, 1,4 \\,\\} \\ar[dr] \\ar[ur] && *+[red]{\\{\\, 2,5 \\,\\}} \\ar[dr] \\ar[ur] && \\{\\, 3,6 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 4,7 \\,\\} \\ar[dr] \\ar[ur] \\\\\n    \\{\\, 1,3 \\,\\} \\ar[ur] && *+[blue]{\\{\\, 2,4 \\,\\}} \\ar[ur] && \\{\\, 3,5 \\,\\} \\ar[ur] && \\{\\, 4,6 \\,\\} \\ar[ur] && *+[blue]{\\{\\, 5,7 \\,\\}} \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of the cluster category $\\mathsf{C}( A_5\n  )$.  The dotted lines should be identified with opposite\n  orientations.  The red vertices show the direct summands of a rigid\n  object $R$, and the red and blue vertices show the direct summands\n  of a cluster tilting object $T$.} \n\\label{fig:AR_quiver}\n\\end{figure}\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    z \\ar[dr]&& \\frac{ ux + uy + yz + z }{ uyz } \\ar[dr]&& u \\ar[dr]&& \\frac{ y + 1 }{ u } \\ar[dr]&& \\frac{ uvx + vz + xy + y }{ vxy } \\\\\n    & \\frac{ ux + yz + z }{ uy } \\ar[dr] \\ar[ur]&& \\frac{ ux + uy + z }{ yz } \\ar[dr] \\ar[ur]&& y \\ar[dr] \\ar[ur]&& \\frac{ uvx + vyz + vz + xy + xy^2 + y+y^2 }{ uvxy } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ uvx + vyz + vz + y + y^2 }{ uxy } \\ar[dr] \\ar[ur]\\ar@{.}[uu] \\ar@{.}[dd] && \\frac{ ux + z }{ y } \\ar[dr] \\ar[ur]&& \\frac{ x + y }{ z } \\ar[dr] \\ar[ur]&& \\frac{ vz + xy +  y }{ vx } \\ar[dr] \\ar[ur]&& \\frac{ uvx + vyz + vz + y + y^2 }{ uxy } \\ar@{.}[uu] \\ar@{.}[dd] \\\\\n    & \\frac{ uvx + vz + y }{ xy } \\ar[dr] \\ar[ur]&& x \\ar[dr] \\ar[ur]&& \\frac{ vz + x + x^2 + xy + y }{ vxz } \\ar[dr] \\ar[ur]&& \\frac{ vz + y }{ x } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ uvx + vz + xy + y }{ vxy }\\ar[ur] && v \\ar[ur]&& \\frac{ x + 1 }{ v } \\ar[ur]&& \\frac{ vz + x + y }{ xz } \\ar[ur]&& z \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of $\\mathsf{C}( A_5 )$ with values of\n  the original Caldero-Chapoton map $X$.  The map depends on the\n  cluster tilting object $T$ shown by red and blue vertices in Figure\n  \\ref{fig:AR_quiver}.}\n\\label{fig:frieze}\n\\end{figure}\n\nIt is a salient property of $X$ that it is a {\\em frieze} in\nthe sense of \\cite{AD}, that is, if $\\tau c \\rightarrow b \\rightarrow\nc$ is an AR triangle then\n\\[\n  X( \\tau c )X( c ) - X( b ) = 1,\n\\]\nsee \\cite[theorem]{DG} and \\cite[prop.\\ 3.10]{CC}.  In the case of\n$\\mathsf{C}( A_5 )$, this means that for each ``diamond'' in the AR quiver,\nof the form\n\\begin{equation}\n\\label{equ:diamond}\n\\vcenter{\n  \\xymatrix @-1.2pc {\n    & b_1 \\ar[dr] & \\\\ \\tau c \\ar[ur] \\ar[dr] & & c \\lefteqn{,} \\\\ & b_2 \\ar[ur] &\n                    }\n        }\n\\end{equation}\nwe have\n\\[\n  X( \\tau c )X( c ) - X( b_1 )X( b_2 ) = 1.\n\\]\n\nThe definition of $X$ depends on a cluster tilting object $T$.  For\ninstance, the $X$ shown in Figure \\ref{fig:frieze} depends on the $T$\nwhich has the indecomposable summands shown by red and blue vertices in\nFigure \\ref{fig:AR_quiver}.\n\nThis paper is about a modified Caldero-Chapoton map $\\rho$ which is\nmore general than $X$ in two respects: it depends on a rigid object\n$R$ and has values in a general commutative ring $A$.  An object $R$\nis rigid if $\\operatorname{Hom}( R,\\Sigma R ) = 0$.  This is much weaker than being\ncluster tilting: recall that $T$ is cluster tilting if $\\operatorname{Hom}( T ,\n\\Sigma t ) = 0 \\Leftrightarrow t \\in \\operatorname{add} T \\Leftrightarrow \\operatorname{Hom}( t ,\n\\Sigma T ) = 0$.  Our first main result gives conditions under which\n$\\rho$ is a {\\em generalised frieze}, in the sense that if $\\tau c\n\\rightarrow b \\rightarrow c$ is an AR triangle then\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b ) \\in \\{\\, 0,1 \\,\\}.\n\\]\nOur second main result is that the conditions can be satisfied if\n$A$ is chosen to be a Laurent polynomial ring over the integers.\n\nGeneralised friezes with values in the integers were introduced by\ncombinatorial means in \\cite{BHJ}, and it was shown in \\cite{HJ} that\nthey can be recovered from a modified Caldero-Chapoton map.  The\ntheory of \\cite{HJ} and the original Caldero-Chapoton map are both special cases of the theory developed here.\n\nFor example, consider $\\mathsf{C}( A_5 )$ again and let $R$ be the rigid\nobject which has the indecomposable summands shown by red vertices in\nFigure \\ref{fig:AR_quiver}.  Our results imply that we can choose $A =\n\\mathbb{Z}[ u^{ \\pm 1 } , v^{ \\pm 1 } , z^{ \\pm 1 } ]$, and Figure\n\\ref{fig:generalised_frieze} shows the AR quiver of $\\mathsf{C}( A_5 )$ with\nthe values of $\\rho$ on the indecomposable objects.\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    z \\ar[dr]\\ar@{.}[dd] && \\frac{ u+z }{ uz } \\ar[dr] && u \\ar[dr]&& \\frac{ 1 }{ u } \\ar[dr]&& \\frac{ 1+uv+vz }{ v } \\ar@{.}[dd] \\\\\n    \\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ u+z }{ u } \\ar[dr] \\ar[ur]&& \\frac{ u+z }{ z } \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& 1 \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ 1+uv+vz }{ uv } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ 1+uv+vz }{ u } \\ar[dr] \\ar[ur]\\ar@{.}[dd] && u+z \\ar[dr] \\ar[ur]&& \\frac{ 1 }{ z } \\ar[dr] \\ar[ur] && \\frac{ 1+vz }{ v } \\ar[dr] \\ar[ur]&& \\frac{ 1+uv+vz }{ u } \\ar@{.}[dd] \\\\\n    & 1+uv+vz \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& 1 \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ 1+vz }{ vz } \\ar[dr] \\ar[ur]&& 1+vz \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture} \\\\\n    \\frac{ 1+uv+vz }{ v } \\ar[ur]&& v \\ar[ur]&& \\frac{ 1 }{ v } \\ar[ur]&& \\frac{ 1+vz }{ z } \\ar[ur] && z \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of $\\mathsf{C}( A_5 )$ with values of\n  the modified Caldero-Chapoton map $\\rho$.  The map depends on the\n  rigid object $R$ shown by red vertices in Figure\n  \\ref{fig:AR_quiver}.  The values form a generalised frieze, the grey\n  diamonds indicating where $\\rho( \\tau c )\\rho( c ) - \\rho( b_1\n  )\\rho( b_2 ) = 1$.}\n\\label{fig:generalised_frieze}\n\\end{figure}\nIn this case, the generalised frieze property means that for\neach ``diamond'' in the AR quiver, of the form \\eqref{equ:diamond}, we\nhave\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b_1 )\\rho( b_2 )\n  \\in \\{\\, 0,1 \\,\\}.\n\\]\nThe solid grey diamonds in Figure \\ref{fig:generalised_frieze} indicate where the displayed expression is equal to $1$.\n\nLet us explain how $\\rho$ is defined.  Let $k$ be an algebraically\nclosed field, $\\mathsf{C}$ an essentially small $\\operatorname{Hom}$-finite $k$-linear\ntriangulated category.  Assume that $\\mathsf{C}$ has split idempotents and\nhas a Serre functor.  Note that these are the only assumptions on\n$\\mathsf{C}$ which is hence permitted to be a good deal more general than a\ncluster category.  Let $\\Sigma$ denote the suspension functor of $\\mathsf{C}$\nand write $\\mathsf{C}( -,- )$ instead of $\\operatorname{Hom}_{ \\mathsf{C} }( -,- )$.  Let $R$ be a\nrigid object of $\\mathsf{C}$, assumed to be basic for reasons of simplicity,\nwith endomorphism algebra $E = \\operatorname{End}_{ \\mathsf{C} }( R )$.  There is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{G}{\\longrightarrow} & \\mathsf{mod}\\,E, \\\\[2mm]\n    c   & \\longmapsto                   & \\mathsf{C}( R,\\Sigma c ).\n  \\end{array}\n\\]\nLet $A$ be a commutative ring and let\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{mod}\\,E ) \\rightarrow A\n\\]\nbe two maps, where $\\operatorname{obj}\\,\\mathsf{C}$ is the set of objects of $\\mathsf{C}$ and\n$\\operatorname{K}_0$ denotes the Grothendieck group of an abelian category.\n\nThe modified Caldero-Chapoton map is the map $\\rho : \\operatorname{obj}\\,\\mathsf{C}\n\\rightarrow A$ defined as follows.\n\\[\n  \\rho( c )\n  = \\alpha( c )\n    \\sum_e \\chi \\big( \\operatorname{Gr}_e ( Gc ) \\big) \\beta( e )\n\\]\nHere $c \\in \\mathsf{C}$ is an object, the sum is over $e \\in \\operatorname{K}_0( \\mathsf{mod}\\,E\n)$, and $\\chi$ denotes the Euler characteristic defined by \\'{e}tale\ncohomology with proper support.  By $\\operatorname{Gr}_e$ is denoted the\nGrassmannian of submodules $M \\subseteq Gc$ with $\\operatorname{K}_0$-class $[ M ] =\ne$. \n\nThe original Caldero-Chapoton map is the special case where $R$ is a\ncluster tilting object and $\\alpha$ and $\\beta$ are two particular\nmaps; see Remark \\ref{rmk:original_CC}.  The modified Caldero-Chapoton\nmap of \\cite{HJ} is the special case where $A = \\mathbb{Z}$ and $\\alpha$ and\n$\\beta$ are identically equal to $1$.  In general, $\\rho$ is only\nlikely to be interesting if the maps $\\alpha$ and $\\beta$ are chosen\ncarefully, and we formalise this by saying that $\\alpha$ and $\\beta$\nare {\\em frieze-like for the AR triangle} $\\Delta = \\tau c \\rightarrow\nb \\rightarrow c$ if they satisfy the technical conditions in\nDefinition \\ref{def:good}.  Observe that the conditions are trivially\nsatisfied if $\\alpha$ and $\\beta$ are identically equal to $1$.  Our\nfirst main result is the following.\n\n\n\\bigskip\n{\\bf Theorem A. }\n{\\em \nIf $\\alpha$ and $\\beta$ are frieze-like for each AR triangle in $\\mathsf{C}$,\nthen the modified Caldero-Chapoton map $\\rho : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow\nA$ is a generalised frieze in the sense of \\cite[def.\\ 3.4]{HJ}.  That\nis,\n\\begin{enumerate}\n\n  \\item  $\\rho( c_1 \\oplus c_2 ) = \\rho( c_1 )\\rho( c_2 )$,\n\n\\medskip\n\n  \\item  if $\\Delta  = \\tau c \\rightarrow b \\rightarrow c$ is an AR\n    triangle then $\\rho( \\tau c )\\rho( c ) - \\rho( b ) \\in \\{\\, 0,1\n    \\,\\}$.\n\n\\end{enumerate}\n}\n\\bigskip\n\nThis follows from Proposition \\ref{pro:exponential} and Theorem\n\\ref{thm:A} which even show\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b )\n  = \\left\\{\n      \\begin{array}{cl}\n        0 & \\mbox{ if $G( \\Delta )$ is a split short exact sequence, } \\\\[2mm]\n        1 & \\mbox{ if $G( \\Delta )$ is not a split short exact sequence. }\n      \\end{array}\n    \\right.\n\\]\nNote that $G( \\Delta )$ is never split exact when $R$ is a\ncluster tilting object, that is, in the case of the original\nCaldero-Chapoton map.\n\nOur second main result is that one can find frieze-like maps $\\alpha$\nand $\\beta$, and hence generalised friezes, with values in Laurent\npolynomials.\n\n\n\\bigskip\n{\\bf Theorem B. }\n{\\em\nAssume that $\\mathsf{C}$ is $2$-Calabi-Yau and that the basic rigid object\n$R$ has $r$ indecomposable summands and is a direct summand of a\ncluster tilting object.\n\nThen there are maps $\\alpha$ and $\\beta$ with values in $\\mathbb{Z}[ x_1^{\n  \\pm 1 }, \\ldots, x_r^{ \\pm 1 } ]$, using all the variables $x_1$,\n$\\ldots$, $x_r$, which are frieze-like for each AR triangle in $\\mathsf{C}$.\n\nHence there is a modified Caldero-Chapoton map $\\rho : \\operatorname{obj} \\mathsf{C}\n\\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1 } ]$, using\nall the variables $x_1$, $\\ldots$, $x_r$, which is a generalised\nfrieze. \n}\n\\bigskip\n\nThis is established in Definition \\ref{def:alpha_and_beta}, Theorem\n\\ref{thm:B}, and Remark \\ref{rmk:app}.  We leave it vague for now what\nit means to ``use all the variables $x_1, \\ldots, x_r$'', but see\nRemark \\ref{rmk:app}.  In fact, it is sometimes possible to get values\nin Laurent polynomials in more than $r$ variables.  For example, the\nbasic rigid object $R$ defined by the red vertices in Figure\n\\ref{fig:AR_quiver} has $r = 2$ indecomposable summands, but the\ncorresponding $\\rho$ has values in $\\mathbb{Z}[ u^{ \\pm 1 }, v^{ \\pm 1 },\nz^{ \\pm 1 } ]$ as shown in Figure \\ref{fig:generalised_frieze}.\n\nIt is natural to ask if the $\\mathbb{Z}$-algebra generated by the values\nof $\\rho$ is an interesting object.  In particular, one can ask how it\nis related to cluster algebras and how it is affected by mutation of\nrigid objects as defined in \\cite[sec.\\ 2]{MP}; see the questions in\nSection \\ref{sec:questions}.\n\nThe paper is organised as follows: Section \\ref{sec:conditions}\ndefines what it means for $\\alpha$ and $\\beta$ to be frieze-like and\nproves Theorem A.  Section \\ref{sec:construction} proves Theorem B.\nSection \\ref{sec:example} shows how to obtain the example in Figure\n\\ref{fig:generalised_frieze}.  Section \\ref{sec:questions} poses some\nquestions. \n\n\n\n\n\\section{The frieze-like condition on the maps $\\alpha$ and $\\beta$\nimplies that $\\rho$ is a generalised frieze}\n\\label{sec:conditions}\n\n\nThis section proves Theorem A in the introduction.  It is a consequence of Proposition\n\\ref{pro:exponential} and Theorem \\ref{thm:A}.\n\nWe start by setting up items to be used in the rest of the paper.\nInstead of a basic rigid object $R$ we will work with a rigid\nsubcategory $\\mathsf{R}$.  This is more general since we can set $\\mathsf{R} =\n\\operatorname{add}\\,R$ when $R$ is given, but not every $\\mathsf{R}$ has this form, see\n\\cite[sec.\\ 6]{JP}.  The higher generality means that the definitions\nof $G$ and $\\beta$ are different from those in the introduction.\n\n\n\\begin{Setup}\n\\label{set:blanket}\nLet $k$ be an algebraically closed field and let $\\mathsf{C}$ be an\nessentially small $k$-linear $\\operatorname{Hom}$-finite triangulated category with\nsplit idempotents.  Hence $\\mathsf{C}$ is a Krull-Schmidt category.\n\nAssume that $\\mathsf{C}$ has a Serre functor.  Hence it has AR triangles by\n\\cite[thm.\\ A]{RVdB}, and the Serre functor is $\\Sigma \\circ \\tau$\nwhere $\\Sigma$ is the suspension functor and $\\tau$ is the AR\ntranslation.\n\nLet $\\mathsf{R}$ be a full subcategory of $\\mathsf{C}$ which is closed under direct\nsums and summands, is functorially finite, and rigid, that is, $\\mathsf{C}(\n\\mathsf{R},\\Sigma \\mathsf{R} ) = 0$.\n\nThe category of $k$-vector spaces is denoted $\\mathsf{Mod}\\,k$ and the\ncategory of $k$-linear contravariant functors $\\mathsf{R} \\rightarrow\n\\mathsf{Mod}\\,k$ is denoted $\\mathsf{Mod}\\,\\mathsf{R}$.  It is a $k$-linear abelian category\nand its full subcategory of objects of finite length is denoted\n$\\mathsf{fl}\\,\\mathsf{R}$.\n\nLet $A$ be a commutative ring and let\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\rightarrow A\n\\]\nbe maps which are ``exponential'' in the sense that\n\\begin{align*}\n  \\alpha( 0 ) = 1 \\;\\; , \\;\\;\n  & \\alpha( c \\oplus d ) = \\alpha( c )\\alpha( d ), \\\\[2mm]\n  \\beta( 0 ) = 1 \\;\\; , \\;\\;\n  & \\beta( e + f ) = \\beta( e )\\beta( f ).\n\\end{align*}\n\\end{Setup}\n\n\n\\begin{bfhpg}\n[The modified Caldero-Chapoton map]\n\\label{bfhpg:CC}\nThere is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{G}{\\longrightarrow} & \\mathsf{Mod}\\,\\mathsf{R}, \\\\[2mm]\n    c   & \\longmapsto                   & \\mathsf{C}( -,\\Sigma c ) |_{ \\mathsf{R} }.\n  \\end{array}\n\\]\nThe modified Caldero-Chapoton map is defined by the following\nformula.\n\\begin{equation}\n\\label{equ:CC}\n  \\rho( c )\n  = \\alpha( c )\n    \\sum_e \\chi \\big( \\operatorname{Gr}_e ( Gc ) \\big) \\beta( e )\n\\end{equation}\nThe sum is over $e \\in \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )$, and $\\operatorname{Gr}_e( Gc )$ is the\nGrassmannian of submodules $M \\subseteq Gc$ where $M$ has finite\nlength in $\\mathsf{Mod}\\,\\mathsf{R}$ and class $[ M ] = e$ in $\\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )$.\nThe notation is otherwise as explained in the introduction.\n\n\nThe formula may not make sense for each $c \\in \\mathsf{C}$, but it does make\nsense if $Gc$ has finite length in $\\mathsf{Mod}\\, \\mathsf{R}$ since then $\\operatorname{Gr}_e( Gc\n)$ is finite-dimensional and non-empty for only finitely many values\nof $e$; see \\cite[1.6 and 1.8]{JP}.  When the formula makes\nsense, it defines an element $\\rho( c ) \\in A$.  Note that\n\\[\n  \\rho( 0 ) = 1.\n\\]\n\n\\end{bfhpg}\n\n\n\\begin{Proposition}\n\\label{pro:exponential}\nLet $a,c \\in \\mathsf{C}$ be objects such that $Ga,Gc$ have finite length in\n$\\mathsf{Mod}\\, \\mathsf{R}$.  Then $G( a \\oplus c )$ has finite length in $\\mathsf{Mod}\\,\n\\mathsf{R}$ and\n\\[\n  \\rho( a \\oplus c ) = \\rho( a )\\rho( c ).\n\\]\n\\end{Proposition}\n\n\\begin{proof}\nThe statement about the length of $G( a \\oplus c )$ is clear.\n\nIt is immediate that\n\\begin{equation}\n\\label{equ:product}\n  \\rho( a )\\rho( c )\n  = \\alpha( a \\oplus c )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi \\big( \\operatorname{Gr}_e ( Ga ) \\times \\operatorname{Gr}_f ( Gc ) \\big)\n    \\Big) \\beta( g )\n  = ( {\\textstyle *} ).\n\\end{equation}\nIn \\cite[sec.\\ 2]{HJ} we considered a pair of morphisms $a \\rightarrow\nb \\rightarrow c$ in $\\mathsf{C}$ and introduced auxiliary spaces $X_{ e,f }$.\nIf we set $a \\rightarrow b \\rightarrow c$ equal to the canonical\nmorphisms $a \\rightarrow a \\oplus c \\rightarrow c$, then \\cite[lem.\\\n2.4(i+v)]{HJ} and \\cite[rmk.\\ 2.5]{HJ} mean that we can compute as\nfollows.\n\\[\n  ( {\\textstyle *} )\n  = \\alpha( a \\oplus c )\n    \\sum_g \\Big( \\sum_{ e+f = g } \\chi( X_{ e,f } ) \\Big) \\beta( g )\n  = \\alpha( a \\oplus c )\n    \\sum_g \\chi \\Big( \\operatorname{Gr}_g \\big( G( a \\oplus c ) \\big) \\Big)\\beta( g )\n  = \\rho( a \\oplus c )\n\\]\n\\end{proof}\n\n\n\\begin{Definition}\n[Frieze-like $\\alpha$ and $\\beta$]\n\\label{def:good}\nLet \n\\[\n  \\Delta = \\tau c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$ and assume that $Gc$ and $G( \\tau c )$ have \nfinite length in $\\mathsf{Mod}\\, \\mathsf{R}$.  We say that {\\em $\\alpha$ and $\\beta$\nare frieze-like for $\\Delta$} if the following hold.\n\\begin{enumerate}\n\n  \\item  If $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$\n    is a split short exact sequence, then\n\\[\n  \\alpha( b ) = \\alpha( c \\oplus \\tau c ).\n\\]\n\n\\medskip\n\n  \\item  If $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$\n    is a non-split short exact sequence, or if $c = \\Sigma^{-1 }r\n    \\in \\Sigma^{ -1 }\\mathsf{R}$, then\n\\[\n  \\alpha( b ) = \\alpha( c \\oplus \\tau c ) \n\\;\\;,\\;\\;\n  \\alpha( c \\oplus \\tau c )\\beta \\big( [ Gc ] \\big) = 1.\n\\]\n\n\\medskip\n\n  \\item  If $c = r \\in \\mathsf{R}$, then\n\\[\n  \\alpha( c \\oplus \\tau c ) = 1\n\\;\\;,\\;\\;\n  \\alpha( b ) = \\beta \\big( [S_r] \\big) \n\\]\nwhere $S_r \\in \\mathsf{Mod}\\,\\mathsf{R}$ is the simple object supported at $r$, see\n\\cite[prop.\\ 2.2]{AusRepII}.\n\n\\end{enumerate}\n\\end{Definition}\n\n\n\\begin{Remark}\nNote that if $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ then $G( \\Delta )$\nis a (split or non-split) short exact sequence, see \\cite[lem.\\\n1.12(iii)]{HJ}. \n\\end{Remark}\n\n\n\\begin{Theorem}\n\\label{thm:A}\nLet \n\\[\n  \\Delta = \\tau c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$.  Assume that $Gc$ and $G( \\tau c )$ have \nfinite length in $\\mathsf{Mod}\\, \\mathsf{R}$ and that $\\alpha$ and $\\beta$ are\nfrieze-like for $\\Delta$.  Then $Gb$ has finite length in $\\mathsf{Mod}\\, \\mathsf{R}$\nand\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b )\n  =\n  \\left\\{\n    \\begin{array}{cl}\n      0 & \\mbox{ if $G( \\Delta )$ is a split short exact sequence, } \\\\[2mm]\n      1 & \\mbox{ if $G( \\Delta )$ is not a split short exact sequence. }\n    \\end{array}\n  \\right.\n\\]\n\\end{Theorem}\n\n\\begin{proof}\nIt is clear from the definition of $G$ that it is a homological\nfunctor, so $G( \\Delta )$ is an exact sequence (albeit not necessarily\nshort exact).  The statement about the length of $Gb$ follows.\n\nWe split into cases.  First some preparation:\nsetting $a = \\tau c$ in Equation \\eqref{equ:product} gives\n\\begin{equation}\n\\label{equ:product2}\n  \\rho( \\tau c )\\rho( c )\n  = \\alpha( c \\oplus \\tau c )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi \\big( \\operatorname{Gr}_e ( G( \\tau c ) ) \\times \\operatorname{Gr}_f ( Gc ) \\big)\n    \\Big) \\beta( g )\n    = ( {\\textstyle *} ).\n\\end{equation}\nMoreover, we use the morphisms $a \\rightarrow b \\rightarrow c$ and\nauxiliary spaces $X_{ e,f }$ from \\cite[sec.\\ 2]{HJ} again, this time\nsetting $a \\rightarrow b \\rightarrow c$ equal to the AR triangle\n$\\tau c \\rightarrow b \\rightarrow c$.  We can then use the results of\n\\cite[sec.\\ 2]{HJ}.\n\nCase (i):  $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$ is\na split short exact sequence.  \n\nWe start from Equation \\eqref{equ:product2} and\ncompute as follows:\n\\[\n  ( {\\textstyle *} )\n  \\stackrel{\\rm (a)}{=}\n    \\alpha( b )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi ( X_{ e,f } )\n    \\Big) \\beta( g ) \\\\\n  \\stackrel{\\rm (b)}{=}\n    \\alpha( b )\n    \\sum_g \\chi \\big( \\operatorname{Gr}_g( Gb ) \\big) \\beta( g ) \\\\\n  = \\rho(b),\n\\]\nwhere (a) is by Definition \\ref{def:good}(i) and \\cite[lem.\\\n2.4(i+v)]{HJ}, and (b) is by \\cite[rmk.\\ 2.5]{HJ}.\n\nCase (ii): $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$ is a\nnon-split short exact sequence.\n\nWe start from Equation \\eqref{equ:product2} and compute as follows:\n\\begin{align*}\n  ( {\\textstyle *} )\n  & =\n    \\alpha( c \\oplus \\tau c )\n    \\Big\\{\n      \\chi \\big( \\operatorname{Gr}_0 ( G(\\tau c) ) \\times \\operatorname{Gr}_{[Gc]} ( Gc ) \\big)\n      \\beta \\big( [Gc] \\big)  \\\\\n      & \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n        + \\sum_g \\Big(\n                   \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                                \\chi \\big(\n                                        \\operatorname{Gr}_e ( G(\\tau c) )\n                                        \\times \\operatorname{Gr}_f ( Gc )\n                                      \\big)\n                 \\Big) \\beta( g ) \n    \\Big\\} \\\\\n  & \\stackrel{\\rm (c)}{=}\n    \\alpha( c \\oplus \\tau c ) \n    \\Big\\{ \n      \\beta \\big( [Gc] \\big) \n      + \\sum_g \\Big(\n                 \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                   \\chi \\big( X_{ e,f } \\big)\n               \\Big) \\beta( g )  \n    \\Big\\} \\\\\n  & \\stackrel{\\rm (d)}{=}\n    1 + \\alpha(b) \\sum_g \\Big(\n                           \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                           \\chi \\big( X_{ e,f } \\big)\n                         \\Big) \\beta( g ) \\\\\n  & \\stackrel{\\rm (e)}{=}\n    1 + \\alpha(b) \\sum_g \\Big(\n                           \\sum_{ e+f=g }\n                           \\chi \\big( X_{ e,f } \\big)\n                         \\Big) \\beta( g ) \\\\\n  & \\stackrel{\\rm (f)}{=}\n    1 + \\alpha(b) \\sum_g \\chi \\big( \\operatorname{Gr}_g( Gb ) \\big) \\beta( g ) \\\\\n  & = 1 + \\rho( b ),\n\\end{align*}\nwhere (c) follows from \\cite[lem.\\ 2.4(ii)+(iv)+(v)]{HJ}, (d) is by\nDefinition \\ref{def:good}(ii), (e) is by \\cite[lem.\\ 2.4(iii)]{HJ},\nand (f) is by \\cite[rmk.\\ 2.5]{HJ}.\n\nCase (iii): $c = \\Sigma^{ -1 }r \\in \\Sigma^{ -1 }\\mathsf{R}$.  Then\n$G( \\Delta )$ is not a split short exact sequence, but\n\\begin{equation}\n\\label{equ:Pr}\n  G( \\Delta )\n  = G( \\tau c \\rightarrow b \\rightarrow c )\n  = 0 \\rightarrow \\operatorname{rad}\\,P_r \\rightarrow P_r\n\\end{equation}\nby \\cite[lem.\\ 1.12(i)]{HJ}, where $P_r = \\mathsf{C}( -,r ) \\,|_{ \\mathsf{R} }$ is\nthe indecomposable projective object of $\\mathsf{Mod}\\,\\mathsf{R}$ associated with\n$r$, see \\cite[1.5]{HJ}.  In particular, we have $G( \\tau c ) = 0$ whence\n$\\rho( \\tau c ) = \\alpha( \\tau c )$, and this gives the first equality\nin the following computation.\n\\begin{align*}\n  \\rho( \\tau c )\\rho( c )\n  & = \\alpha( \\tau c )\\alpha( c )\n      \\sum_f \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f ) \\\\\n  & = \\alpha( c \\oplus \\tau c )\n      \\sum_f \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f ) \\\\\n  & = \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\chi \\big( \\operatorname{Gr}_{ [Gc] }( Gc ) \\big) \\beta \\big( [Gc] \\big)\n        + \\sum_{ f \\neq [Gc] } \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (g)}{=}\n    \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\beta \\big( [Gc] \\big)\n        + \\sum_{ f \\neq [Gc] } \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (h)}{=}\n    \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\beta \\big( [Gc] \\big)\n        + \\sum_{ f } \\chi \\big( \\operatorname{Gr}_f( Gb ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (j)}{=}\n    1 + \\alpha(b) \\sum_{ f } \\chi \\big( \\operatorname{Gr}_f( Gb ) \\big) \\beta( f ) \\\\\n  & = 1 + \\rho( b )\n\\end{align*}\nTo see (g), note that for $M' \\subseteq Gc$ we have\n\\begin{equation}\n\\label{equ:1.2}\n  [ M' ] = [ Gc ] \\Leftrightarrow M' = Gc\n\\end{equation}\nby \\cite[eq.\\ (1.2)]{HJ}.  Hence $\\operatorname{Gr}_{ [Gc] }( Gc )$ has only a\nsingle point whence $\\chi \\big( \\operatorname{Gr}_{ [Gc] }( Gc ) \\big) = 1$.\nTo see (h), note that Equation \\eqref{equ:Pr} says that $Gc$ is an\nindecomposable projective object with radical $Gb$.  So the proper\nsubmodules of $Gc$ are precisely all the submodules of $Gb$, whence\nEquation \\eqref{equ:1.2} implies\n\\[\n  \\operatorname{Gr}_f( Gb )\n  = \\left\\{\n      \\begin{array}{cl}\n        \\operatorname{Gr}_f( Gc ) & \\mbox{ for $f \\neq [ Gc ]$, } \\\\[2mm]\n        \\emptyset   & \\mbox{ for $f = [ Gc ]$ }\n      \\end{array}\n    \\right.\n\\]\nand (h) follows.  Finally, (j) holds by Definition\n\\ref{def:good}(ii). \n\nCase (iv): $c = r \\in \\mathsf{R}$.  Then $G( \\Delta )$ is not a split short\nexact sequence, but we have\n\\[\n  G( \\Delta )\n  = G( \\tau c \\rightarrow b \\rightarrow c )\n  = I_r \\rightarrow \\operatorname{corad}\\,I_r \\rightarrow 0\n\\]\nby \\cite[lem.\\ 1.12(ii)]{HJ}, where $I_r = \\mathsf{C}( -,\\Sigma\\tau r ) \\,|_{\n  \\mathsf{R} }$ is the indecomposable injective object of $\\mathsf{Mod}\\,\\mathsf{R}$\nassociated with $r$, see \\cite[1.10]{HJ}, and $\\operatorname{corad}$ denotes the\nquotient by the socle.  Now proceed dually to Case (iii), replacing\nDefinition \\ref{def:good}(ii) by Definition \\ref{def:good}(iii).\n\\end{proof}\n\n\n\n\n\\section{A construction of frieze-like maps $\\alpha$ and $\\beta$ with\nvalues in Laurent polynomials}\n\\label{sec:construction}\n\n\nThis section proves Theorem B in the introduction.  It is a consequence of Definition\n\\ref{def:alpha_and_beta}, Theorem \\ref{thm:B}, and Remark\n\\ref{rmk:app}. \n\n\n\\begin{Setup}\n\\label{set:2}\nWe continue to work under Setup \\ref{set:blanket} and \nhenceforth add the assumption that $\\mathsf{C}$ is a $2$-Calabi-Yau category with a\ncluster tilting subcategory $\\mathsf{T}$ which belongs to a cluster\nstructure in the sense of \\cite[sec.\\ II.1]{BIRS}, and which satisfies\n$\\mathsf{R} \\subseteq \\mathsf{T}$.\n\n\nNote that the AR translation of $\\mathsf{C}$ is\n\\[\n  \\tau = \\Sigma\n\\]  \nand that the Serre functor is $\\Sigma^2$.\n\\end{Setup}\n\n\n\\begin{Remark}\nWhen $\\mathsf{R}$ is a rigid subcategory of $\\mathsf{C}$, it is often possible to\nfind a cluster tilting subcategory $\\mathsf{T}$ with $\\mathsf{R} \\subseteq \\mathsf{T}$.\nNot always, however: if $\\mathsf{C}$ is a cluster tube, then such a $\\mathsf{T}$\ncannot be found since cluster tubes have no cluster tilting\nsubcategories, see \\cite[cor.\\ 2.7]{BMV}.\n\\end{Remark}\n\n\n\\begin{bfhpg}\n[Mutation and exchange triangles]\nLet ${\\mathsf{ind}}\\, \\mathsf{T}$ denote the set of (isomorphism classes of)\nindecomposable objects of $\\mathsf{T}$.  Each $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ has a\nmutation $t^{ {\\textstyle *} }$ which is the unique indecomposable object in\n$\\mathsf{C}$ such that $\\mathsf{T}$ remains a cluster tilting subcategory if $t$ is\nreplaced by $t^{ {\\textstyle *} }$.  There are distinguished triangles\n\\begin{equation}\n\\label{equ:exchange_triangles1}\n  t^{ {\\textstyle *} } \\rightarrow a \\rightarrow t\n\\;\\;,\\;\\;\n  t \\rightarrow a' \\rightarrow t^{ {\\textstyle *} }\n\\end{equation}\nwith $a, a' \\in \\operatorname{add}\\big( ( {\\mathsf{ind}}\\, \\mathsf{T} ) \\setminus t \\big)$, known\nas exchange triangles, see \\cite[sec.\\ II.1]{BIRS}.  \n\\end{bfhpg}\n\n\n\\begin{Definition} \n[The subgroup $N$]\n\\label{def:N}\nThe split Grothendieck group of an additive category is denoted by\n$\\operatorname{K}_0^{ \\operatorname{split} }$.  It has a relation $[ a \\oplus b ] = [ a ] + [ b ]$\nfor each pair of objects $a,b$, where $[ a ]$ is the $\\operatorname{K}_0^{ \\operatorname{split}\n}$-class of $a$.\n\nDefine a subgroup of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ as follows.\n\\begin{equation}\n\\label{equ:N}\n  N = \\bigg\\langle\\, [a] - [a'] \n      \\,\\bigg|\n        \\begin{array}{l}\n          \\mbox{ $s^{ {\\textstyle *} } \\rightarrow a \\rightarrow s$ \\;,\\;\n                 $s \\rightarrow a' \\rightarrow s^{ {\\textstyle *} }$\n                 are exchange } \\\\[2mm]\n          \\mbox{ triangles with \n                 $s \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$ }\n        \\end{array}\n      \\,\\bigg\\rangle\n\\end{equation}\nLet\n\\[\n  Q : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \\rightarrow \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N\n\\]\ndenote the canonical surjection.\n\\end{Definition}\n\n\n\\begin{bfhpg}\n[Simple objects, $\\operatorname{K}$-theory, and the homomorphism $\\overline{ \\theta }$]\n\\label{bfhpg:simples_and_K}\nThe inclusion functor $i : \\mathsf{R} \\rightarrow \\mathsf{T}$ induces an exact functor\n\\[\n  i^{ {\\textstyle *} } : \\mathsf{Mod}\\,\\mathsf{T} \\longrightarrow \\mathsf{Mod}\\,\\mathsf{R}\n\\;\\;,\\;\\;\n  i^{ {\\textstyle *} }( M ) = M \\circ i.\n\\]\nEach indecomposable object $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ gives rise to a simple\nobject $\\overline{S}_t \\in \\mathsf{Mod}\\,\\mathsf{T}$, and each $r \\in {\\mathsf{ind}}\\, \\mathsf{R}$\ngives rise to a simple object $S_r \\in \\mathsf{Mod}\\,\\mathsf{R}$, see \\cite[prop.\\\n2.3(b)]{AusRepII}.  It is not hard to show\n\\[\n  i^{ {\\textstyle *} }\\overline{S}_t \n  = \\left\\{\n      \\begin{array}{cl}\n        S_t & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{R}$, } \\\\[2mm]\n        0   & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$. }\n      \\end{array}\n    \\right.\n\\]\nSince $i^{ {\\textstyle *} }$ is exact and sends simple objects to simple objects\nor $0$, is preserves finite length so restricts to an exact functor\n\\[\n  i^{ {\\textstyle *} } : \\mathsf{fl}\\,\\mathsf{T} \\rightarrow \\mathsf{fl}\\,\\mathsf{R}.\n\\]\nLet\n\\[\n  \\kappa : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \\rightarrow \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )\n\\]\nbe the induced homomorphism.  The source is a free group on the\nclasses $[ \\overline{S}_t ]$ for $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ and the target\nis a free group on the classes $[ S_r ]$ for $r \\in {\\mathsf{ind}}\\, \\mathsf{R}$.\nThe homomorphism $\\kappa$ is surjective and given by\n\\begin{equation}\n\\label{equ:kappa}\n  \\kappa\\big( [ \\overline{S}_t ] \\big)\n  = \\left\\{\n      \\begin{array}{cl}\n        [ S_t ] & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{R}$, } \\\\[2mm]\n        0       & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$. }\n      \\end{array}\n    \\right.\n\\end{equation}\n\nThere is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{\\overline{G}}{\\longrightarrow} & \\mathsf{Mod}\\,\\mathsf{T}, \\\\[2mm]\n    c   & \\longmapsto                              & \\mathsf{C}( -,\\Sigma c ) |_{ \\mathsf{T} },\n  \\end{array}\n\\]\nand $i^{ {\\textstyle *} } \\overline{G} = G$ where $G$ is the functor from\nSubsection \\ref{bfhpg:CC}.\n\nWe define a homomorphism as follows,\n\\begin{equation}\n\\label{equ:overlinetheta}\n  \\overline{\\theta} : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \n    \\rightarrow \\operatorname{K}_0^{\\operatorname{split}}( \\mathsf{T} )\n  \\;\\;,\\;\\; \\overline{ \\theta }\n            \\big( [ \\overline{S}_t ] \\big) = [ a ] - [ a' ],\n\\end{equation}\nwhere $a, a'$ come from the exchange triangles\n\\eqref{equ:exchange_triangles1}, see \\cite[1.5(ii)]{JP}. \n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[The homomorphism $\\theta$]\nIt is clear from Equations \\eqref{equ:N}, \\eqref{equ:kappa}, and \\eqref{equ:overlinetheta}\nthat there is a unique homomorphism $\\theta$ which makes the following\nsquare commutative.\n\\begin{equation}\n\\label{equ:square}\n\\vcenter{\n  \\xymatrix {\n    \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \\ar@{->>}_{\\kappa}[d] \\ar^-{ \\overline{ \\theta }}[r] & \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \\ar@{->>}^{Q}[d]\\\\\n    \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\ar_-{ \\theta }[r] & \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\\\\n                     }\n        }\n\\end{equation}\n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[Index and coindex]\n\\label{bfhpg:ind}\nFor $c \\in \\mathsf{C}$ there is a distinguished triangle $t_1 \\rightarrow t_0\n\\rightarrow c$ with $t_0, t_1 \\in \\mathsf{T}$ by \\cite[sec.\\ 1]{DK}, and the\nindex $\\operatorname{ind} c = [t_0] - [t_1]$ is a well-defined element of $\\operatorname{K}_0^{\n\\operatorname{split} }( \\mathsf{T} )$.  Similarly there is a distinguished triangle $c\n\\rightarrow \\Sigma^2 t^0 \\rightarrow \\Sigma^2 t^1$ with $t^0, t^1 \\in\n\\mathsf{T}$, and the coindex $\\operatorname{coind} c = [t^0] - [t^1]$ is a well-defined\nelement of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$. \n\\end{bfhpg}\n\n\n\\begin{Definition}\n[The maps $\\alpha$ and $\\beta$]\n\\label{def:alpha_and_beta}\nRecall that $A$ is a commutative ring.  Let\n\\begin{equation}\n\\label{equ:epsilon_exponential}\n  \\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow A\n\\end{equation}\nbe a map which is ``exponential'' in the sense that\n\\[\n  \\varepsilon( 0 ) = 1\n\\;\\;,\\;\\;\n  \\varepsilon( e+f ) = \\varepsilon( e )\\varepsilon( f ).\n\\]\nDefine\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\rightarrow A\n\\]\nby\n\\begin{equation}\n\\label{equ:alpha_beta}\n  \\alpha( c ) = \\varepsilon Q( \\operatorname{ind} c )\n\\;\\;,\\;\\;\n  \\beta( e ) = \\varepsilon \\theta( e ).\n\\end{equation}\nIt is easy to see that $\\alpha$ and $\\beta$ satisfy the conditions in\nSetup \\ref{set:blanket}, and Equation \\eqref{equ:CC} now defines a\nmodified Caldero-Chapoton map $\\rho$ with values in $A$. \n\nRemark \\ref{rmk:app} has further comments to the choice of $\\varepsilon$ which is crucial to the properties of $\\rho$. \n\\end{Definition}\n\n\n\\begin{Remark}\n\\label{rmk:original_CC}\nThe definition of $\\alpha$ and $\\beta$ is motivated by the original\nCaldero-Chapoton map which is recovered as follows when $\\mathsf{R} = \\mathsf{T}$:\nin this case, $N = 0$ and $Q$ is the identity while $\\theta =\n\\overline{ \\theta }$, so Equations \\eqref{equ:alpha_beta} read\n\\[\n  \\alpha( c ) = \\varepsilon ( \\operatorname{ind} c )\n\\;\\;,\\;\\;\n  \\beta( e ) = \\varepsilon \\overline{ \\theta }( e ).\n\\]\nThe group $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is free on the classes $[ t ]$ for\n$t \\in \\operatorname{ind}\\,\\mathsf{T}$.  Let $A$ be the Laurent polynomial ring on\ngenerators $x_t$ for $t \\in \\operatorname{ind}\\,\\mathsf{T}$ and set $\\varepsilon \\big( [ t\n] \\big) = x_t$ for $t \\in \\operatorname{ind}\\,\\mathsf{T}$.  Then the map $\\rho$ from\nEquation \\eqref{equ:CC} is the original Caldero-Chapoton map, see\n\\cite[1.8]{JP}. \n\\end{Remark}\n\n\n\\begin{Lemma}\n\\label{lem:coind}\nThe map $\\overline{\\theta}$ from Equation \\eqref{equ:overlinetheta}\nsatisfies the following. \n\\begin{enumerate}\n\n  \\item  If $\\overline{G}c$ has finite length in $\\mathsf{Mod}\\, \\mathsf{T}$ then\n\\[\n  \\overline{\\theta} \\big( [ \\overline{G}c ] \\big)\n  = - ( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c).\n\\]\n\n\\smallskip\n\n  \\item Let $\\Sigma c \\stackrel{ \\varphi }{ \\longrightarrow } b\n  \\longrightarrow c$ be an AR triangle in $\\mathsf{C}$.  If $\\overline{ G }(\n  \\Sigma c )$ and $\\overline{ G }c$ have finite length in $\\mathsf{Mod}\\, \\mathsf{T}$\n  then\n\\[\n  \\operatorname{ind} b\n  = \\left\\{\n      \\begin{array}{cl}\n        - \\overline{ \\theta } \\big( [ \\overline{ G }c ] \\big)\n          & \\mbox{ if $c \\not\\in \\mathsf{T}$, } \\\\[2mm]\n        \\overline{ \\theta } \\big( [ \\overline{S}_t ] \\big)\n          & \\mbox{ if $c = t \\in \\mathsf{T}$. }\n      \\end{array}\n    \\right.\n\\]\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nFirst note that \\cite[lem.\\ 2.1(2) and prop.\\ 2.2]{Palu} apply to the present setup by \\cite[1.3]{JP}.\n\n(i)  Combine \\cite[1.5]{JP} with \\cite[lem.\\ 2.1(2)]{Palu}.\n\n(ii)  Observe that\n\\begin{equation}\n\\label{equ:Sigmab1}\n  \\operatorname{ind} b\n  = \\overline{ \\theta } \\big( [ \\operatorname{Ker} \\overline{ G }\\varphi ] \n                              - [ \\overline{ G }c ] \\big).\n\\end{equation}\nThis can be seen by combining part (i) of the lemma with \n\\cite[prop.\\ 2.2]{Palu}.\n\n\nThe case $c \\not\\in \\mathsf{T}$: then $\\mathsf{C}( t,b ) \\rightarrow \\mathsf{C}( t,c\n)$ is surjective because $\\Sigma c \\stackrel{ \\varphi }{\n\\longrightarrow } b \\longrightarrow c$ is an AR triangle.  The long\nexact sequence \n$\\mathsf{C}( t,b )\n  \\longrightarrow \\mathsf{C}( t,c )\n  \\longrightarrow \\mathsf{C}( t,\\Sigma^2 c )\n  \\stackrel{ ( \\Sigma \\varphi )_* }{ \\longrightarrow }\n    \\mathsf{C}( t,\\Sigma b )$\nshows that $( \\Sigma \\varphi )_{ {\\textstyle *} } : \\mathsf{C}( t,\\Sigma^2 c )\n\\rightarrow \\mathsf{C}( t,\\Sigma b )$ is injective.  This implies that\n$\\overline{G}\\varphi$ is injective whence Equation \\eqref{equ:Sigmab1}\ngives $\\operatorname{ind} b = - \\overline{ \\theta } \\big( [ \\overline{ G }c ]\n\\big)$ as desired.\n\nThe case $c = t \\in \\mathsf{T}$: then \n\\[\n  \\overline{ G }( \\Sigma c\n  \\stackrel{ \\varphi }{ \\longrightarrow } b\n  \\longrightarrow c )\n  = \\overline{ I }_t\n  \\stackrel{ \\overline{ G }\\varphi }{ \\longrightarrow }\n  \\operatorname{corad} \\overline{ I }_t\n  \\rightarrow\n  0\n\\]\nby \\cite[lem.\\ 1.12(ii)]{HJ}.  Here $\\overline{ I }_t = \\mathsf{C}(\n-,\\Sigma^2 t )|_{ \\mathsf{T} }$ is the indecomposable injective object of\n$\\mathsf{Mod}\\,\\mathsf{T}$ associated with $t$, and $\\operatorname{corad}$ denotes the quotient by\nthe socle.  Hence $\\operatorname{Ker} \\overline{ G }\\varphi = \\overline{ S }_t$ and\n$\\overline{ G }c = 0$, whence Equation \\eqref{equ:Sigmab1} reads $\\operatorname{ind}\nb = \\overline{ \\theta } \\big( [ \\overline{ S }_t ] \\big)$ as desired.\n\\end{proof}\n\n\n\\begin{Theorem}\n\\label{thm:B}\nLet \n\\[\n  \\Delta \n   = \\Sigma c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$ such that $\\overline{ G }c$ and\n$\\overline{ G }( \\Sigma c )$ have finite length in $\\mathsf{Mod}\\, \\mathsf{T}$.\n\nThen $Gc$ and $G( \\Sigma c )$ have finite length in $\\mathsf{Mod}\\, \\mathsf{R}$, and\nthe maps $\\alpha$ and $\\beta$ from Definition \\ref{def:alpha_and_beta}\nare frieze-like for $\\Delta$.\n\\end{Theorem}\n\n\\begin{proof}\nThe statement on lengths holds because $G = i^{ {\\textstyle *} }\\overline{ G }$\nand $i^{ {\\textstyle *} }$ preserves finite length, see Subsection\n\\ref{bfhpg:simples_and_K}.  We must now check the conditions of\nDefinition \\ref{def:good}.\n\nFirst, we show that\n\\begin{equation}\n\\label{equ:alpha2}\n  \\alpha( c \\oplus \\Sigma c )\\beta \\big( [ Gc ] \\big) = 1,\n\\end{equation}\nin particular establishing the second equation in Definition\n\\ref{def:good}(ii).  Equation \\eqref{equ:alpha_beta} gives\n\\begin{equation}\n\\label{equ:alpha10}\n  \\alpha( c \\oplus \\Sigma c )\n  = \\varepsilon Q \\big( \\operatorname{ind} ( c \\oplus \\Sigma c ) \\big)\n  = \\varepsilon Q( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c ).\n\\end{equation}\nOn the other hand, combining Equation \\eqref{equ:alpha_beta}, the fact\nthat $[ Gc ] = [ i^{ {\\textstyle *} }\\overline{ G }c ] = \\kappa[ \\overline{ G }c\n]$, the commutative square \\eqref{equ:square}, and Lemma\n\\ref{lem:coind}(i) gives\n\\[\n  \\beta \\big( [ Gc ] \\big)\n  = \\varepsilon \\theta \\big( [ Gc ] \\big)\n  = \\varepsilon \\theta \\kappa \\big( [ \\overline{ G }c ] \\big)\n  = \\varepsilon Q \\overline{ \\theta } \\big( [ \\overline{ G }c ] \\big)\n  = \\varepsilon Q \\big( - ( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c ) \\big).\n\\]\nMultiplying the last two equations proves Equation \\eqref{equ:alpha2}. \n\nSecondly, if $c = t \\in \\mathsf{T}$ then it is direct from the\ndefinition of index and coindex that\n\\[\n  \\operatorname{ind} c = [ t ]\n\\;\\;,\\;\\;\n  \\operatorname{ind} \\Sigma c = - [ t ].\n\\]\nInserting into Equation \\eqref{equ:alpha10} gives\n\\begin{equation}\n\\label{equ:alpha3}\n  c \\in \\mathsf{T}\n  \\; \\Rightarrow \\;\n  \\alpha( c \\oplus \\Sigma c ) = 1,\n\\end{equation}\nin particular establishing the first equation in Definition\n\\ref{def:good}(iii).\n\nThirdly, suppose $c \\not\\in \\mathsf{R}$.  We will show\n\\begin{equation}\n\\label{equ:alpha}\n  \\alpha( b ) = \\alpha( c \\oplus \\Sigma c ),\n\\end{equation}\nestablishing Definition \\ref{def:good}(i) as well as the first\nequation in Definition \\ref{def:good}(ii).\n\nThe case $c = t \\in \\mathsf{T}$: Note that $c \\not\\in \\mathsf{R}$ implies\n$\\overline{ \\theta } \\big( [ \\overline{ S }_t ] \\big) \\in N$ by\nEquations \\eqref{equ:overlinetheta} and \\eqref{equ:N}.  Using\nEquation \\eqref{equ:alpha_beta} and Lemma \\ref{lem:coind}(ii)\ntherefore gives \n\\[\n  \\alpha( b )\n  = \\varepsilon Q ( \\operatorname{ind} b )\n  = \\varepsilon Q \\overline{ \\theta }\\big( [ \\overline{ S }_t ] \\big)\n  = \\varepsilon( 0 )\n  = 1.\n\\]\nCombining with Equation \\eqref{equ:alpha3} shows Equation\n\\eqref{equ:alpha}. \n\nThe case $c \\not\\in \\mathsf{T}$: combining the two parts of Lemma\n\\ref{lem:coind} shows $\\operatorname{ind} b = \\operatorname{ind} c + \\operatorname{ind} \\Sigma c$.  Applying\n$\\varepsilon Q$ shows $\\alpha( b ) = \\alpha( c )\\alpha( \\Sigma c )$\nwhich is equivalent to Equation \\eqref{equ:alpha}.\n\nFinally, suppose $c = r \\in \\mathsf{R}$.  We show that\n\\[\n  \\alpha( b ) = \\beta \\big( [ S_r ] \\big),\n\\]\nestablishing the second equation in Definition \\ref{def:good}(iii).\nLemma \\ref{lem:coind}(ii) says \n\\[\n  \\operatorname{ind} b = \\overline{ \\theta } \\big( [ \\overline{ S }_r ] \\big).\n\\]\nApplying $\\varepsilon Q$ gives the first of the following equalities. \n\\[\n  \\alpha( b )\n  = \\varepsilon Q \\overline{ \\theta } \\big( [ \\overline{ S }_r ] \\big)\n  = \\varepsilon \\theta \\kappa \\big( [ \\overline{ S }_r ] \\big)\n  = \\varepsilon \\theta \\big( [ S_r ] \\big)\n  = \\beta \\big( [ S_r ] \\big)\n\\]\nThe other equalities are by the commutative diagram \\eqref{equ:square}\nand Equations \\eqref{equ:kappa} and \\eqref{equ:alpha_beta}.\n\\end{proof}\n\n\n\\begin{Remark}\n\\label{rmk:app}\nThe maps $\\alpha$ and $\\beta$ from Definition\n\\ref{def:alpha_and_beta}, and hence the modified Caldero-Chapoton map\n$\\rho$ from Equation \\eqref{equ:CC}, depend on the map $\\varepsilon :\n\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow A$.  The possible choices of\n$\\varepsilon$ are determined by the structure of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T}\n) \/ N$ which we do not know in general.\n\nHowever, let us suppose that ${\\mathsf{ind}}\\, \\mathsf{R}$ and ${\\mathsf{ind}}\\, \\mathsf{T}$ are\nfinite, with $r$, respectively $r+s$, objects.  This is the situation from\nTheorem B in the introduction if we set $R$, respectively $T$, equal\nto the direct sum of the indecomposable objects in $\\operatorname{ind}\\,\\mathsf{R}$,\nrespectively $\\operatorname{ind}\\,\\mathsf{T}$.\nThen we can set $A = \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1\n} ]$ and use all the variables $x_1, \\ldots, x_r$, thereby proving\nTheorem B.\n\nNamely, $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is a free abelian group\non $r+s$ generators, one per object in ${\\mathsf{ind}}\\, \\mathsf{T}$, and the\nsubgroup $N$ has $s$ generators, one per object in ${\\mathsf{ind}}\\, \\mathsf{T}\n\\setminus {\\mathsf{ind}}\\, \\mathsf{R}$.  So $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$ has a\nquotient group $F$ which is free abelian of rank $( r+s ) - s = r$.\nThe desired map\n$\\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1\n}, \\ldots, x_r^{ \\pm 1 } ]$ can be obtained by sending each generator\nof $F$ to a generator of $\\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1 }\n]$.\n\nNote that $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ may have a quotient group which is\nfree abelian of rank $n > r$, see the example in Section\n\\ref{sec:example}.  In this case, the above method means that\nwe can even set $A = \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{ \\pm 1 } ]$ and\nuse all the variables $x_1, \\ldots, x_n$.\n\\end{Remark}\n\n\n\n\n\\section{Example: a modified Caldero-Chapoton map on the cluster\ncategory of Dynkin type $A_5$}\n\\label{sec:example}\n\n\nThis section shows how to obtain the example in Figure\n\\ref{fig:generalised_frieze} in the introduction.\n\n\\begin{Setup}\nLet the category $\\mathsf{C}$ of Setups \\ref{set:blanket} and \\ref{set:2} be\n$\\mathsf{C}( A_5 )$, the cluster category of Dynkin type $A_5$.  There is a\nbijection between ${\\mathsf{ind}}\\, \\mathsf{C}$ and the diagonals of a $8$-gon, see\n\\cite[secs.\\ 2 and 5]{CCS}.  We let ${\\mathsf{ind}}\\, \\mathsf{R}$, respectively\n${\\mathsf{ind}}\\, \\mathsf{T}$, be given by the red diagonals, respectively all the\nred and blue diagonals, in Figure \\ref{fig:8gon}.  These data satisfy\nour assumptions, see \\cite[sec.\\ 1]{BMRRT}.\n\\begin{figure}\n\\[\n  \\begin{tikzpicture}[auto]\n    \\node[name=s, shape=regular polygon, regular polygon sides=8,\n    minimum size=5cm, draw] at (0,0) {}; \n    \\node[name=t, shape=regular polygon, regular polygon sides=8, minimum size=5.8cm] at (0,0) {}; \n    \\draw[shift=(t.corner 1)] node {$1$};\n    \\draw[shift=(t.corner 2)] node {$2$};\n    \\draw[shift=(t.corner 3)] node {$3$};\n    \\draw[shift=(t.corner 4)] node {$4$};\n    \\draw[shift=(t.corner 5)] node {$5$};\n    \\draw[shift=(t.corner 6)] node {$6$};\n    \\draw[shift=(t.corner 7)] node {$7$};\n    \\draw[shift=(t.corner 8)] node {$8$};\n    \\draw[very thick, blue] (s.corner 1) to (s.corner 7);\n    \\draw[very thick, blue] (s.corner 2) to (s.corner 4);\n    \\draw[very thick, red] (s.corner 2) to (s.corner 5);\n    \\draw[very thick, red] (s.corner 2) to (s.corner 7);\n    \\draw[very thick, blue] (s.corner 5) to (s.corner 7);\n  \\end{tikzpicture} \n\\]\n\\caption{The diagonals of the $8$-gon correspond to the indecomposable\nobjects of $\\mathsf{C}( A_5 )$.  The red diagonals define ${\\mathsf{ind}}\\, \\mathsf{R}$ and\nall the red and blue diagonals define ${\\mathsf{ind}}\\, \\mathsf{T}$.}   \n\\label{fig:8gon}\n\\end{figure}\n\\end{Setup}\n\n\n\\begin{bfhpg}\n[Some properties of $\\mathsf{C}$]\nWe denote diagonals and their corresponding indecomposable objects by\npairs of vertices, so $\\{\\, 2,7 \\,\\}$ is both a red diagonal in Figure\n\\ref{fig:8gon} and an object of ${\\mathsf{ind}}\\, \\mathsf{R}$.  The AR quiver of\n$\\mathsf{C}$ and the objects of $\\operatorname{ind}\\,\\mathsf{R}$, respectively $\\operatorname{ind}\\,\\mathsf{T}$, are\nshown in Figure \\ref{fig:AR_quiver} in the introduction. \n\nAt the level of objects, the suspension functor $\\Sigma$ is given by\n$\\Sigma \\{\\, i,j \\,\\} = \\{\\, i-1,j-1 \\,\\}$.  Note that vertex numbers\nare taken modulo $8$.\n\nIf $x, y \\in {\\mathsf{ind}}\\, \\mathsf{C}$ then\n\\begin{equation}\n\\label{equ:C7_Homs}\n  \\mathsf{C}( x,\\Sigma y )\n  =\n  \\left\\{\n    \\begin{array}{cl}\n      k & \\mbox{ if the diagonals corresponding to $x$ and $y$ cross, } \\\\[2mm]\n      0 & \\mbox{ if not. }\n    \\end{array}\n  \\right.\n\\end{equation}\nIf $i,k,j,\\ell$ are four vertices in anticlockwise\norder on the polygon, then $\\{\\, i,j \\,\\}$ and $\\{\\, k,\\ell \\,\\}$ are\ncrossing diagonals, and there are the following non-split distinguished\ntriangles,\n\\begin{equation}\n\\label{equ:exchange_triangles}\n  \\{\\, i,j \\,\\}\n  \\rightarrow \\{\\, i,\\ell \\,\\} \\oplus \\{\\, j,k \\,\\}\n  \\rightarrow \\{\\, k,\\ell \\,\\}\n  \\;\\; , \\;\\;\n  \\{\\, k,\\ell \\,\\}\n  \\rightarrow \\{\\, i,k \\,\\} \\oplus \\{\\, j,\\ell \\,\\}\n  \\rightarrow \\{\\, i,j \\,\\},\n\\end{equation}\nwhere a pair of neighbouring vertices must be interpreted as $0$.\n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[$\\operatorname{K}$-theory]\nThe category $\\mathsf{T}$ has the following indecomposable objects.\n\\[\n  \\{\\, 1,7 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,4 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,5 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,7 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 5,7 \\,\\}\n\\]\nTheir $\\operatorname{K}_0^{ \\operatorname{split} }$-classes are free generators of $\\operatorname{K}_0^{ \\operatorname{split}\n}( \\mathsf{T} )$.  To save parentheses, the classes are denoted $[ 1,7 ]$\netc.  The objects in ${\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$ are\n\\[  \n    \\{\\, 1,7 \\,\\} \\;\\;,\\;\\; \\{\\, 2,4 \\,\\} \\;\\;,\\;\\; \\{\\, 5,7 \\,\\},\n\\]\nand Equation \\eqref{equ:exchange_triangles} means that they sit in the\nfollowing exchange triangles.\n\\[\n  \\xymatrix @R=1ex {\n    \\{\\, 2,8 \\,\\} \\ar[r] & 0 \\ar[r] & \\{\\, 1,7 \\,\\}\n    & \\{\\, 1,7 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 2,8 \\,\\} \\\\\n    \\{\\, 3,5 \\,\\} \\ar[r] & 0 \\ar[r] & \\{\\, 2,4 \\,\\}\n    & \\{\\, 2,4 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 3,5 \\,\\} \\\\\n    \\{\\, 2,6 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 5,7 \\,\\}\n    & \\{\\, 5,7 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 2,6 \\,\\}\n            }\n\\]\nAccordingly, the subgroup $N$ of Definition \\ref{def:N} is\n\\[\n  N = \\big\\langle\\,\n        - [ 2,7 ] \\;,\\; - [ 2,5 ] \\;,\\; [ 2,7 ] - [ 2,5 ]\n      \\,\\big\\rangle \\\\[2mm]\n    = \\big\\langle\\,\n        [ 2,5 ] \\;,\\; [ 2,7 ]\n      \\,\\big\\rangle,\n\\]\nand $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$ is the free abelian group generated\nby\n\\[\n  [ 1,7 ] + N\n  \\;\\;,\\;\\; [ 2,4 ] + N\n  \\;\\;,\\;\\; [ 5,7 ] + N.\n\\]\n\nThe category $\\mathsf{fl}\\, \\mathsf{R}$ has the simple objects\n\\[\n  S_{\\{\\, 2,5 \\,\\}} \\;\\;,\\;\\; S_{\\{\\, 2,7 \\,\\}}\n\\]\nwhose $\\operatorname{K}_0$-classes are free generators of $\\operatorname{K}_0( \\mathsf{fl}\\, \\mathsf{R} )$, and\n$\\mathsf{fl}\\, \\mathsf{T}$ has the simple objects\n\\[\n  \\overline{ S }_{\\{\\, 1,7 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,4 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,5 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,7 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 5,7 \\,\\}}\n\\]\nwhose $\\operatorname{K}_0$-classes are free generators of $\\operatorname{K}_0( \\mathsf{fl}\\, \\mathsf{T} )$.  To\nsave parentheses, the simple objects will be denoted $S_{ 2,5 }$,\nrespectively $\\overline{ S }_{ 1,7 }$, etc.\n\\end{bfhpg}\n\n\n\\begin{Definition}\nLet the map\n\\[\n  \\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N\n                \\rightarrow\n                \\mathbb{Z}[ u^{ \\pm 1 } , v^{ \\pm 1 } , z^{ \\pm 1 } ]\n\\]\nbe given by\n\\begin{equation}\n\\label{equ:epsilon}\n  \\varepsilon \\big( [ 1,7 ] + N \\big) = u   \n\\;\\;,\\;\\;\n  \\varepsilon \\big( [ 2,4 ] + N \\big) = v   \n\\;\\;,\\;\\;\n  \\varepsilon \\big( [ 5,7 ] + N \\big) = z.  \n\\end{equation}\nEquation \\eqref{equ:alpha_beta} defines the maps $\\alpha$ and $\\beta$, \nand the modified Caldero-Chapton map $\\rho$ is defined by Equation\n\\eqref{equ:CC}.\n\\end{Definition}\n\n\n\\begin{Example}\nLet us compute $\\rho \\big( \\{\\, 4,6 \\,\\} \\big)$.\n\nEquation \\eqref{equ:alpha_beta} gives\n\\[\n  \\alpha \\big( \\{\\, 4,6 \\,\\} \\big)\n  = \\varepsilon Q( \\operatorname{ind} \\{\\, 4,6 \\,\\} )\n  = ( {\\textstyle *} ).\n\\]\nNow $\\{\\, 4,6 \\,\\} = \\Sigma \\{\\, 5,7 \\,\\}$ so $\\operatorname{ind} \\{\\, 4,6 \\,\\} =\n\\operatorname{ind} \\Sigma \\{\\, 5,7 \\,\\} = - [ 5,7 ]$, where the last equality is\ndirect from the definition of index because $\\{\\, 5,7 \\,\\} \\in \\mathsf{T}$,\nsee Subsection \\ref{bfhpg:ind}.  Hence Equation \\eqref{equ:epsilon}\ngives\n\\[\n  ( {\\textstyle *} )\n  = \\varepsilon Q \\big( -[ 5,7 ] \\big)\n  = \\varepsilon \\big( -[ 5,7 ] + N \\big)\n  = z^{ -1 }.\n\\]\n\n\nWe have $G \\big( \\{\\, 4,6 \\,\\} \\big) = \\mathsf{C}( -,\\Sigma \\{\\, 4,6 \\,\\}\n)|_{ \\mathsf{R} }$.  Moreover, $\\mathsf{R} = \\operatorname{add} \\big\\{\\, \\{\\, 2,5 \\,\\} , \\{\\, 2,7\n\\,\\} \\,\\big\\}$, and it is direct from Equation\n\\eqref{equ:C7_Homs} that $G \\big( \\{\\, 4,6 \\,\\} \\big)$ is supported\nonly at $\\{\\, 2,5 \\,\\}$ where it has the value $k$.  That is,\n\\[\n  G \\big( \\{\\, 4,6 \\,\\} \\big) = S_{ 2,5 }.\n\\]\nIt follows that the only non-empty Grassmannians appearing in Equation\n\\eqref{equ:CC} when computing $\\rho \\big( \\{\\, 4,6 \\,\\} \\big)$ are\n$\\operatorname{Gr}_0 \\big( G\\{\\, 4,6 \\,\\} \\big)$ and $\\operatorname{Gr}_{ [ S_{ 2,5 } ] } \\big(\nG\\{\\, 4,6 \\,\\} \\big)$, and it is clear that each is a point so has\nEuler characteristic $1$.\n\nFinally, Equations \\eqref{equ:kappa} and \\eqref{equ:alpha_beta} and\ndiagram \\eqref{equ:square} give \n\\[\n  \\beta \\big( [ S_{ 2,5 } ] \\big)\n  = \\varepsilon \\theta \\big( [ S_{ 2,5 } ] \\big)\n  = \\varepsilon \\theta \\kappa \n    \\big( [ \\overline{ S }_{ 2,5 } ] \\big)\n  = \\varepsilon Q \\overline{ \\theta }\n    \\big( [ \\overline{ S }_{ 2,5 } ] \\big)\n  = ( {\\textstyle *} {\\textstyle *} ).\n\\]\nEquation \\eqref{equ:exchange_triangles} gives exchange triangles\n\\[\n  \\xymatrix @R=1ex {\n    \\{\\, 4,7 \\,\\} \\ar[r] & \\{\\, 2,4 \\,\\} \\oplus \\{\\, 5,7 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\}\n    & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 4,7 \\,\\}\n                   }\n\\]\nand Equation \\eqref{equ:overlinetheta} gives $\\overline{ \\theta }\n\\big( [ \\overline{ S }_{ 2,5 } ] \\big) = [ 2,4 ] + [ 5,7 ] -\n[ 2,7 ]$ whence Equation \\eqref{equ:epsilon} gives\n\\[\n  ( {\\textstyle *} {\\textstyle *} )\n  = \\varepsilon Q\n    \\big( [ 2,4 ] + [ 5,7 ] - [ 2,7 ] \\big)\n  = vz.\n\\]\n\nHence Equation \\eqref{equ:CC} says\n\\begin{align*}\n  \\rho \\big( \\{\\, 4,6 \\,\\} \\big)\n  & = \\alpha \\big( \\{\\, 4,6 \\,\\} \\big)\n      \\sum_e \\chi \\big( \\operatorname{Gr}_e( G\\{\\, 4,6 \\,\\} ) \\big) \\beta( e ) \\\\\n  & = z^{ -1 } \\cdot \\Big(\n      \\chi \\big( \\operatorname{Gr}_0( S_{ 2,5 } ) \\big) \n         \\beta( 0 )\n      + \\chi \\big( \\operatorname{Gr}_{[ S_{ 2,5 } ]}( S_{ 2,5 } ) \\big) \n         \\beta \\big( [S_{ 2,5 }] \\big)\n                    \\Big) \\\\\n  & = z^{ -1 } \\cdot ( 1 + vz ) \\\\\n  & = \\frac{ 1+vz }{ z }.\n\\end{align*}\n\nSimilar computations for the other indecomposable objects finally\nproduce the generalised frieze in Figure \\ref{fig:generalised_frieze}\nin the introduction.\n\\end{Example}\n\n\n\n\n\\section{Questions}\n\\label{sec:questions}\n\n\nWe end the paper with some questions.\n\n\\begin{enumerate}\n\n\\item  The group $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is free abelian on ${\\mathsf{ind}}\\,\n  \\mathsf{T}$, and the subgroup $N$ of Definition \\ref{def:N} is generated by\n  all expressions $[a] - [a']$ where $s^{ {\\textstyle *} } \\rightarrow a\n  \\rightarrow s$ and $s \\rightarrow a' \\rightarrow s^{ {\\textstyle *} }$ are\n  exchange triangles with $s \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$.\n\n\\medskip\n\\noindent\nWhat is the rank $n$ of the quotient $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$?\nNote that when $n$ is finite, it is the largest integer such that the\nmethod of Remark \\ref{rmk:app} results in a modified Caldero-Chapoton\nmap $\\rho : \\operatorname{obj} \\mathsf{C} \\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{ \\pm\n  1 } ]$ using all the variables $x_1, \\ldots, x_n$.\n\n\\medskip\n\n\\item  Consider the $\\mathbb{Z}$-subalgebra of $\\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{\n    \\pm 1 } ]$ generated by the values of the modified Caldero-Chapoton map $\\rho$. \n\n\\medskip\n\\noindent\n  What is its relation to the cluster algebra?\n\n\\medskip\n\n\\item  Let $T$ be a cluster tilting object and use it to define a\n  Caldero-Chapoton map $X$.  If $T$ is subjected to cluster mutation,\n  then the values of $X$ change in a well-understood way, see\n  \\cite[proof of cor.\\ 5.4]{Palu}.\n\n\\medskip\n\\noindent\n  There is a notion of mutation of rigid objects due to\n  \\cite[sec.\\ 2]{MP}.  What happens to the values of the modified\n  Caldero-Chapoton map under such mutation?\n\n\\end{enumerate}\n\n\n\n\n\\medskip\n\\noindent\n{\\bf Acknowledgement.}\nThis paper is a direct continuation of \\cite{HJ}.   Both papers grew\nout of \\cite{BHJ} with Christine Bessenrodt, and we are grateful to\nher for the fruitful collaboration.\n\nWe thank the referee for several interesting comments and Robert Marsh\nfor answering a question about rigid subcategories.\n\nPart of this work was done while Peter J\\o rgensen was visiting the\nLeibniz Universit\\\"{a}t Hannover.  He thanks Christine Bessenrodt,\nThorsten Holm, and the Institut f\\\"{u}r Algebra, Zahlentheorie und\nDiskrete Mathematik for their hospitality.  He gratefully acknowledges\nsupport from Thorsten Holm's grant HO 1880\/5-1, which falls under the\nresearch priority programme SPP 1388 {\\em Darstellungstheorie} of the\nDeutsche Forschungsgemeinschaft (DFG).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOntologies are witnessing an increasing popularity outside specialized AI\ncommunities. While this is mostly due to Semantic Web\napplications~\\cite{ber01}, we must also credit their ability to\ncope with taxonomies and part-whole relationships, to\nhandle  heterogeneous attributes, and their provision for various\nautomated reasoning services --- see, e.g.,~\\cite{staab2013handbook}.  \nThese features have been recognized since long time in system engineering,\nthe community encompassing all areas of research devoted to design,\nimplementation, monitoring and diagnosis of technical processes.\nFor instance, in the operations and maintenance sub-community, the use\nof ontologies is explicitly advocated\\footnote{See, e.g., the MIMOSA\nopen standard architecture at \\url{www.mimosa.org}.}.  \nAlso standards like ISO 13374\n(\\textit{Condition monitoring and diagnostics of \nmachines -- Data processing, communication and presentation}) suggest\nthe use of ontologies for several tasks, mostly related to data\nconceptualization. However, the adoption of ontologies\nfaces some challenges, mostly due to speed and reliability constraints\nimposed by industrial settings.  \n\nHere we investigate this issue by considering four contributions of\nours to application domains wherein ontologies provide key\ncapabilities in system engineering. The first case study is\nabout an on-board rolling-stock condition analyzer, i.e., a\nsystem to perform fault detection and\nclassification~\\cite{AmbrosiGT09}. The second one is about monitoring \nan intermodal logistic system~\\cite{CasuCT13}. The third one is about an\nontology-based framework to generate diagnostic-decision support\nsystems~\\cite{CicalaLOT16}. Finally, a fourth case study is an \napplication to computer-automated design of elevator\nsystems.\nIn the following, we briefly introduce each case study, giving details\nabout its context, underlying motivation and intended objectives. The \nultimate goal of the paper is to discuss and compare the results\nobtained to assess the effectiveness of ontologies in such\napplication domains.  \n\n\\noindent\n\\textit{Ontologies for condition analysis.}\nWe introduced an ontology-based condition analyzer\n(CA)~\\cite{AmbrosiGT09} in the context of the EU project  \nIntegrail\\footnote{More details about Integrail\nat \\url{http:\/\/www.integrail.eu\/}.}.      \nOur CA collects signals from control logic installed on locomotives,\nand it leverages an ontology to correlate\nobserved data, symptoms and faults. \nThe CA must mate two competing\nneeds: $(i)$ railway regulations\nrequire hardware which is highly reliable, and whose performances are\nthus far even from desktop workstations; $(ii)$\nontology-related tools, e.g., description logic reasoners,\nhave relatively large memory, processor and storage footprints.\nIn this experience, the main \ngoal was thus to check whether reasoning with ontologies can provide\nuseful diagnostic feedback in a resource-restricted scenario. \n\n\\noindent\n\\textit{Ontologies for system monitoring.}\nIn~\\cite{CasuCT13} we provided strong evidence of practical uses\nfor ontologies in complex systems engineering by implementing\na monitor for \\emph{Intermodal Logistics Systems} (ILSs), i.e.,\nsystems supporting the movement of containerized goods. In\nparticular, we considered combination of rail and road transport,\nwhere rail transport is provided by short-distance shuttle trains,\nand network coverage is\nachieved through connections at specialized terminals. In this\nexperience, the main goal was to gather data about terminal operations\nand compute global performances indicators, where access to data is\nmediated by an ontology --- ontology-based data access\n(OBDA)~\\cite{cal05}. Here, unlike the CA case study, \nthe ability to handle large amount of data is crucial, but reasoning\nis limited to SPARQL query answering.\n\n\\noindent\n\\textit{Ontologies for diagnostic support system generation.}\nDiagnostic Decision Support Systems (DDSSs) help humans to \ninfer the health status of physical systems. In~\\cite{CicalaLOT16} we\nintroduced DiSeGnO --- for ``Diagnostic Server Generation\nthrough Ontology'' ---  to generate customized DDSSs.\nAs in the ILS monitoring case study, since it is\nexpected that large quantities of data should be handled, \nthe ontology language is\nrestricted to those designed for tractable reasoning --- see,\ne.g.,~\\cite{cal05}. In this case, ontology-based  \nreasoning is not leveraged, as DiSeGnO generates relational databases\nfrom the domain ontology and then computes diagnostic rules\nwith \\textsc{Ptolemy II}{}~\\cite{Ptolemy}, an open-source software simulating\nactor-based models.  \n\n\\noindent\n\\textit{Ontologies for computer-automated design.}\nAs mentioned in~\\cite{ByeOPHS16}, the first scientific report of\nintelligent computer-automated design (CautoD) is the paper by\nKamentsky and Liu~\\cite{KamentskyL63}, who created a computer program\nfor designing character-recognition logic circuits satisfying given\nhardware  constraints. In mechanical design --- see,\ne.g.,~\\cite{rao2012mechanical} --- the term usually refers to\ntechniques that mitigate the effort in exploring alternative\nsolutions for structural implements. In our \\textsc{LiftCreate}{}\nCautoD program for elevator systems\\footnote{Part of the \\textsc{AiLift}{}\nsoftware suite \\url{www.ailift.it}.}, ontologies  \nsupport intelligent design creation and optimization by managing\ndetailed part-whole taxonomies, wherein different relations among\ncomponents can be expressed. This case study provides thus yet another\napplication of ontologies, mostly oriented to intelligent computation\nand data persistency.\n\nOverall, the case studies considered witness the great flexibility\nthat ontologies provide in handling diverse application scenarios,\nfrom condition analysis of locomotives, to automated design of\nelevators, considering both cases wherein they provide the basis for\nlogic reasoning services, or just advanced data-modeling capabilities.  \nThe rest of the paper is structured as follows. In\nSections~\\ref{sec:condition}, \\ref{sec:ilog}, \\ref{sec:ondaBrief}\nand \\ref{sec:elevator} we sketch the design, the implementation and\nthe results obtained in the case studies described above.\nSection~\\ref{sec:concl} concludes the paper by summarizing the results\nand providing some discussion thereof. \n\n\n\n\\section{Rolling stock condition analysis}\n\\label{sec:condition}\n\\begin{figure}[t!]\n\\begin{center}\n  \\includegraphics[scale=0.27]{OntologyTraction.png}\n  \\caption{\\label{fig:e414ont}A portion of the E414 ontology regarding\n    traction faults. Concepts are nodes and\n    object properties are edges: white nodes are SP3A\n    concepts, grey ones are E414-specific concepts.}\n\\end{center}\n\\end{figure}\n\nThe CA prototype described in~\\cite{AmbrosiGT09} focuses on \nfault detection on Trenitalia E414 locomotive.\nThe main task \nof the CA is to perform fault classification according to \npriority for maintenance, and impact on mission-related and\nsafety-related aspects.\nHere, we focus on traction groups as an example of\nsubsystem that can generate a faulty condition.\nOur ontology for the E414 locomotive is written in OWL 2 language\nand it builds on the SP3A core ontology \n--- see~\\cite{AmbrosiGT09} for details. \nIn particular, the E414 ontology leverages the SP3A concepts of {\\sc\n  ObservationData}, i.e., process variables, and {\\sc Observation},\ni.e., sequences of observation data from which individuals of class\n{\\sc Symptom} and {\\sc Fault} arise. {\\sc Symptom} individuals\nare related to {\\sc Observation} individuals via the {\\sc\n  refersToObservation} property and to {\\sc Fault} individuals via the\n{\\sc refersToFault} property. {\\sc Fault} is a concept whose\nindividuals are defined in terms of the necessary {\\sc hasSymptom}\nrelationship with {\\sc Symptom} individuals.\nTwo subclasses of {\\sc Fault} are defined: {\\sc\n  PriorityFault} and {\\sc   NonPriorityFault}, with obvious\nmeaning.\nIn Figure~\\ref{fig:e414ont} we show a portion of the E414 ontology\nrelated to  traction faults, where concepts have been specialized \nin subclasses whose individuals correspond to actual signals and\nsubsystems. Fault and symptom classification is obtained by\na Description Logic (DL) reasoner considering the patterns \nobserved. For instance, in the case of {\\sc\n  TractionHighTemperatureObservation}, three ranges of temperatures\nare defined that correspond to ``interesting'' patterns: from 70 to 80\ndegrees, from 80 to 130 degrees, and over 130 degrees. It is \npostulated that observations falling in the second and in the\nthird ranges are to be considered mission critical, while the ones in\nthe first category are only maintenance critical.\n\n\n\n\nA detailed description of the CA architecture can be found\nin~\\cite{AmbrosiGT09}. Here we provide some intuition on how the\nanalyzer works considering high temperatures in the traction groups. \nWhen the temperature of a group is higher than 70 \ndegrees for at least 3 consecutive samples read from the field bus,\nthe CA starts tracking a potential anomalous pattern.\nOnce such a pattern is detected, the corresponding\nindividuals in the classes \\textsc{TractionObservationData} \nand \\textsc{TractionHighTemperatureObservation}\nare recorded. {\\sc Symptom} individuals\nare built along with all the properties required by the ontology\nspecification. For example, if an observation of the class\n\\textsc{TractionHighTemperatureObservation} has been created,   \na specific individual \n\\textsc{TractionHighTemperatureObservation} is related to  a new\n\\textsc{Symptom} individual by the\n\\textsc{refersToObservation} property.\n{\\sc Fault} individuals for each {\\sc Symptom} individual are created \ntogether with the {\\sc causedBySymptom} property.\n{\\sc Fault} as well as {\\sc Symptom} individuals are built of generic\ntype, leaving their classification to the DL reasoner. Once\nthe classification is done, the CA publishes the results, transmitting\nthem to external agents. As an example, let us assume that $i$ is an\nindividual of the class \\textsc{TractionHighTemperatureObservation}\nwhose property \\textsc{isAt} is set to the constant\n\\textsc{\\_130degrees}, $s$ is the \\textsc{Symptom} individual related\nto $i$, and $f$ is the \\textsc{Fault} individual related to $s$. \nThe E414 ontology postulates that all symptoms such that the\ncorresponding observation is an instance of\n\\textsc{TractionHighTemperatureObservation} related by \\textsc{isAt}\nto the constant \\textsc{\\_130degrees}\nare also an instance of \\textsc{TractionTotalMissionImpactSymptom},\nwhich is a subclass of \\textsc{Symptom}. Therefore, a reasoner \ncan infer that $s$ belongs to \n\\textsc{MissionRelatedSymptom}\n\n\n\\begin{table}[t!]\n\\caption{\\label{tab:results} Results with \n  (a) lazy and (b) eager implementations of the CA.}\n\\begin{center}\n\\tiny\n    \\begin{tabular}{ | c | c | c | c | }\n    \\hline\t\t\t\n      Scenario & Memory Consumption [MB] & CPU Time [ms]  & Amortized CPU Time [ms] \\\\\n    \\hline\n      1a & 38 & 90 & ND \\\\\n      2a & 74 & 25373 & 25373 \\\\\n      3a & 106 & 1053656 & 210731 \\\\\n      4a & OUT OF MEMORY & 3253637 & 191390 \\\\\n    \\hline\n      1b & 37 & 90 & ND \\\\\n      2b & 72 & 21506 & 21506 \\\\\n      3b & 104 & 86938 & 17387 \\\\\n      4b & 105 & 279523 & 16442 \\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\\end{table}\n\nOut of the three sets of experiments performed in~\\cite{AmbrosiGT09},\nwe report just those to ensure that the CA\nimplementation fits the constraints. To this end, we ran several tests\nusing different fault scenarios\\footnote{Sets  \n  of multidimensional time series (3600 samples at 1Hz) corresponding\n  to 52  process variables are generated.\n \n \n \n  Simulations run on EN50155-compliant embedded devices \n  with 1GHz Socket 370 FC-PGA Celeron Processor with 256MB \n  of main memory and a 1GB SSD running Linux Blue Cat (kernel 2.6) and\n  Sun Java Virtual Machine implementation (JRE 1.6). The DL reasoner is \\textsc{Pellet}{}~\\cite{sirin07}.}.\nTable~\\ref{tab:results} shows the results obtained by running the CA\non four different scenarios --- the first includes no\nfault, the second includes only one fault, the third includes five\ncontemporary faults, and the last 17 contemporary faults --- \nusing two different configurations.\nConfiguration (a) is ``lazy'', i.e., it keeps all the individuals,\nwhile configuration (b) is ``eager'', i.e., it  \ndeletes individuals as soon as possible.\nAs we can see in Table~\\ref{tab:results}, the eager version results in\na great improvement over the lazy one, both in terms of memory\nconsumption and in terms of computation time.\nIn particular, in the second column of Table~\\ref{tab:results} we can\nnotice that the eager version performs reasonably well, even in the\nfourth test case (worst-case scenario). In the same scenario,\nthe lazy version exceeds the amount of available memory.\nAs we can see in the rightmost column of Table~\\ref{tab:results}, the\namortized computation time over a single scenario\ndecreases with the number of concurrent\nobservations detected in the round.\nManaging a round of samples without detected observations\ntakes only 90 ms, which leaves enough time for other activities, and\nallows the CA to process all the incoming signals in due course.\n\n\n\n\n\n\n\n\n\n\n\\section{Monitoring of intermodal systems}\n\\label{sec:ilog}\n\\begin{figure}[t!]\n\\centerline{\\scalebox{.26}{\\includegraphics{iLogDot}}}\n\\caption{\\label{fig:iLog_dot} ILS ontology describing the design of the\n  OBDA solution. Ellipses denote concepts with datatype properties;\n  directed edges are object properties; dotted edges are concept\n  inclusions.} \n\\end{figure}\n\nIn~\\cite{CasuCT13} we provided evidence that ontology-based data\naccess (OBDA)~\\cite{cal05}  \nis of practical use in the context of \\emph{Intermodal Logistics\n  Systems} (ILSs). The investigation focuses on the opportunity to build a monitoring information\nsystem (MIS) using OBDA instead of relational databases\n(RDBs). The application scenario is an ILS relying on a\nlogic akin to computer networks, i.e., frequent short-distance trains\nwith a fixed composition and a predefined daily schedule to cover some\ngeographical area. \\emph{Intermodal Transport Units} (ITUs) enter\nthe network at some terminal and travel to their destination according\nto a predefined route, usually boarding more than one train along the\nway. Terminals collect ITUs from areas of approximately 150Km in\nradius in order to minimize road transport. The MIS is a key\nenabler to minimize delivery time, maximize rolling-stock and network\nutilization and, ultimately, reduce the economic overhead of\ntransportation for the final customer. The main goal of the MIS is to\ncompute \\emph{Key Performance Indicators} (KPIs) to perform tactical and \nstrategical decision making about the network.\n\nIn Figure~\\ref{fig:iLog_dot} we present a graphical outline of the\nontology at the heart of our OBDA solution to monitor the ILS.\nThe ontology --- ILS ontology in the following --- is compliant with\nthe OWL 2 QL profile described in the official W3C's recommendation\nas \\emph{``[the sub-language of OWL 2] aimed at applications that use \n  very large volumes of instance data, and where query answering is\n  the most important reasoning task.''}. Given the ILS application\ndomain, OWL 2 QL guarantees that conjunctive query answering and\nconsistency checking can be implemented efficiently\nwith respect to the size of data and ontology, respectively. The\nrestrictions that OWL 2 QL implies did not hamper the modeling\naccuracy of our ILS ontology. In Figure~\\ref{fig:iLog_dot} we can\npinpoint classes related to freight forwarding such as \n\\textbf{Customer}, i.e., companies forwarding their goods through the\nnetwork, \\textbf{RequestForWork}, i.e., the main document witnessing\nthat a given customer has issued a request for transporting a number\nof ITUs, \\textbf{TransportOrder}, i.e., the ``bill of transit''\nassociated to each ITU, as well as entities related to physical\nelements such as \\textbf{ITU}, \\textbf{Terminal} and \\textbf{Train}. Also\n``logical'' entities are modeled such as \\textbf{Route}, i.e., a\nsequence of terminals and railway connections serviced regularly by\none or more scheduled trains and \\textbf{ScheduledStop}, i.e.,\nterminals associated to a given route with a given\nschedule. \\textbf{Event} is the main monitoring entity, as the\ncalculation of most KPIs relies on the exact recording of events at\nspecific locations. \n\n\\begin{figure}[t!]\n\\begin{center}\n  \\scalebox{.27}{\\includegraphics{q3heavy.pdf}}\n\\end{center}\n\\caption{\\label{fig:results} Computation time of a KPI with different\n    query processors: SQL (square), \\textsc{ARQ}{} (circle), \\textsc{Pellet}{}\n    (hourglass), \\textsc{Quest}{} (triangle). In each plot, the $x$ axis\n    displays the number of simulation days from 1 to 15, the $y$ axis\n    displays the CPU time (in milliseconds on a logarithmic\n    scale).} \n\\end{figure}\n\nTo assess OBDA performances, in~\\cite{CasuCT13} we obtained  different\nartificial utilization scenarios by changing \nthe number of ITUs shipped daily from each terminal. Considering  \ntypical usage patterns, we postulated that a\nprovision of 10 to 50 ITUs is to be shipped daily from each terminal,\nwith 40 to 50 ITUS corresponding to a heavy utilization.\nScenarios are simulated for an increasing number of days to\nevaluate scalability, and all of them share common settings as far as\nnumber of train travels, number of cars per train, and timetabling are\nconcerned. Unexpected delays as well as the number of customers per\nterminal follow a probabilistic model --- see~\\cite{CasuCT13} for more\ndetails. In Figure~\\ref{fig:results} we display the results\\footnote{\n  All results are obtained on a family of identical\n  Intel-based PCs, featuring a Core2Duo 2.13 GHz CPU, 4GB of RAM and\n  running Ubuntu Linux 10.04 (64 bit edition).} obtained\nin the case of an heavy utilization scenario to compute a specific\nKPI, namely the average number of ITUs unloaded per hour.\nThe performance of four different query-answering systems\nare reported: a SQL query on a native RDB implementation, and a SPARQL\nquery on the ontology store. The SPARQL query can be answered by three\ndifferent systems, namely \\textsc{ARQ}{} (the default query processor in the\n\\textsc{Jena}{} library), \\textsc{Pellet}{} (the same DL reasoner that we consider in\nSection~\\ref{sec:condition}) and \\textsc{Quest}{}~\\cite{mur12}. The latter is the only\nreasoner exploiting the fact that SPARQL queries can be compiled\non-the-fly into SQL queries for an equivalent RDB representation of\nthe ontology stored in the main memory. As we can observe in\nFigure~\\ref{fig:results}, OBDA-based solutions show higher overall\ncomputation times than the RDB-based solution --- from 1 to 2 orders\nof magnitude --- together with an apparently growing trend associated\nto the time span of the simulation. However, as we have shown\nin~\\cite{CasuCT13}, a trend test performed on the results obtained\nwith the best OBDA solutions for various KPIs, displays no statistically\nsignificant increase in the CPU time required to answer various\nqueries with respect to the number of days. Considering that for most\nKPIs we can adopt an ``eager'' solution similar to that considered in\nSection~\\ref{fig:results}, we can conclude that OBDA is practically\nfeasible for monitoring medium-to-large scale systems.\n\n\n\n\\section{Diagnostic support systems generation}\n\\label{sec:ondaBrief}\n\\begin{figure}[!t]\n\\centering\n\\scalebox{0.34}{\\includegraphics{ondaFramework.png}}\n\\caption{\\label{fig:ondaModel}Functional architecture and work-flow of DiSeGnO framework.}\n\\end{figure}\n\nIn~\\cite{CicalaLOT16} we introduced an approach to compile ontology-based\ndescriptions of equipment into diagnostic decision support systems\n(DDSSs). The tool DiSeGnO, whose functional architecture and work-flow\nis sketched in Figure~\\ref{fig:ondaModel}, fulfills  this task in\nthree phases: in the \\textsf{USER} phase, a domain ontology and\ndiagnostic rules model are designed by the user; in the\n\\textsf{DiSeGnO} phase, the system reads and analyzes the  ontology\nand the rules to output the actual DDSS; in the \\textsf{DDSS} phase,\ninput web services receive data from the observed physical system and\nrecord them in the generated data  store. According to the ISO 13374-1\nstandard a DDSS consists of six modules of which DiSeGnO implements three:\n\\textit{Data Manipulation} to  perform signal analysis and \ncompute meaningful descriptors, \\textit{State Detection}\nto check conformity to reference patterns, and \\textit{Health\n  Assessment} to diagnose faults and rate the current  \nhealth of the equipment or process.\nAs shown in Figure~\\ref{fig:ondaModel}, the ontology description is\ncreated by a system architect in the \\textsf{USER} phase. \nThe ontology must be written using OWL 2 QL language\\footnote{While\n  this can be accomplished in several ways,  the tool\n  \\textsc{prot\\'eg\\'e}{}~\\cite{gennari2003evolution} is suggested  because it \nis robust, easy to use, and it provides, either directly or through\nplug-ins, several add-ons that facilitate ontology design and\ntesting.} as in the case study shown in Section~\\ref{sec:ilog}. \nThe diagnostic computation model must be a sound\nactor diagram generated by \\textsc{Ptolemy II}{}~\\cite{Ptolemy} which describes\nthe processing to be applied to incoming data in order to generated\ndiagnostic events --- here we focus on the ontology part, but more\ndetails on the rule handling part can be found in~\\cite{CicalaLOT16}.\nThe \\textsf{DiSeGnO} phase contains the actual DDSS generation\nsystem which consists of the \\textsf{Data Store Generator}, i.e.,\na piece of software that creates a relational  database \nby mapping the domain ontology to suitable tables, \nand the \\textsf{Web Services Generator}, i.e., a module  \nthat creates interface services for incoming and outgoing events.\nFinally, in the \\textsf{DDSS} phase, \ndata is acquired and stored in the internal database,\nthe rules engine processes data and generates diagnostic events which\nare then served to some application.  \n\n\\begin{figure}[t!]\n\\centering\n\\scalebox{0.25}{\\includegraphics{ontology.pdf}}\n\\caption{\\label{fig:hvac_onto} Domain ontology for HVAC monitoring.\n  Formalism is the same as in Figure~\\ref{fig:iLog_dot}.}\n\\end{figure}\n\nAn example of a DiSeGno-compliant equipment description \nis shown in Figure~\\ref{fig:hvac_onto}. The ontology is related to a \nHeating Ventilation and Air Conditioning (HVAC) appliance and it is\ndivided into  a \\emph{static} and a \\emph{dynamic} part. In the static\npart, which is not updated while monitoring, the ontology contains a\ndescription of the observed physical system. In the\nHVAC ontology we have \\textbf{System} and\n\\textbf{DataSource},  \nrelated by the \\textbf{isInSystem} property. \\textbf{hasSubsystem}\nrelationship indicates that one \\textbf{System} could be composed  by\none or more \\textbf{SystemComponent} which are themselves subclasses\nof  \\textbf{System}. Finally, \\textbf{DataSource} is the \nclass of elements that can generate diagnostic-relevant information.   \nThe dynamic part describes \\emph{events}, including both\nthe ones generated by the  observed system and its components, and\nthose output by the DDSS. An event is always associated to a\ntime-stamp and it can be either \\emph{incoming} to the DDSS from the\nobserved system, or \\emph{outgoing} from the DDSS\\footnote{This\n  distinction is fundamental, because DiSeGnO must know which \nevents have to be associated with input and output web services,\nrespectively.}.\nThe main concepts in the dynamic part of the HVAC\nontology are \\textbf{DDSS} which \\textbf{receives} instances of\n\\textbf{IncomingEvent} and \\textbf{sends} instances of\n\\textbf{OutgoingEvent}. Notice that \\textbf{IncomingEvent} instances\nare connected to \\textbf{DataSource} instances by the role\n\\textbf{generates}, denoting that all incoming events\nare generated by some data source.\nAlso every \\textbf{OutgoingEvent} instance, i.e., every\ndiagnostic event, \\textbf{relatesTo} some instance of\n\\textbf{DataSource}, because the end user must be able to\nreconstruct which data source(s) provided information that caused\ndiagnostic rules to fire a given diagnostic event.\n\\textbf{OutgoingEvent} specializes to \\textbf{AlarmEvent}, \\textbf{FaultEvent}\nand \\textbf{DescriptorEvent}. Every \\textbf{OutgoingEvent} instance is\nconnected to one of \\textbf{DiagnosticIndicator} instances, \ni.e. \\textbf{Alarm}, \\textbf{Fault} and \\textbf{Descriptor} sub-concepts,\n by \\textbf{reports} relation, in order to have a reference message\n about the diagnostic rules.\n\n\n\\section{Computer-automated design of elevators}\n\\label{sec:elevator}\n\\begin{figure}[t!]\n\\begin{center}\n\\scalebox{.22}{\\includegraphics{Elevator.png}}\\\\\n\\scalebox{.26}{\\includegraphics{HydraulicElevator.png}}\n\\end{center}\n\\caption{\\label{fig:elev_onto} Ontologies describing the\n  implements handled by \\textsc{LiftCreate}{} (top) and \n  the components of \\textsf{OnePistonHydraulicElevator}\n  (bottom). Concepts are rectangles, concept inclusion is denoted by\n  solid arrows, and HAS-A object properties are denoted by diamond-based arrows.}  \n\\end{figure}\n\nOur latest ontology-based application is in the field of\ncomputer-automated design (CautoD) which differs\nfrom ``classical'' computer-aided design (CAD) in that it\nis oriented to replace some of the designer's capabilities and not just\nto support a traditional work-flow with computer graphics\nand storage capabilities. Nevertheless, CautoD programs most\noften include CAD facilities to visualize technical drawings related\nto the implements of interest. \nIn particular, our \\textsc{LiftCreate}{} program is oriented to automating\ndesign of elevators, taking the designer from the very first\nmeasurements to a complete project which guarantees feasibility within\na specific normative framework. \\textsc{LiftCreate}{} works in three steps. In\nthe first step,  the user is asked to enter relevant parameters\ncharacterizing the project, and an overall ``design philosophy'' to be\nimplemented. For instance, if the size of the elevator's shaft is\nknown and fixed in advance, \\textsc{LiftCreate}{} can generate solutions which\nmaximize payload,  door size, or car size. A design philosophy is just\na set of heuristics which, e.g., prioritize door size over other\nelements, still keeping into account hard constraints, e.g., payload\nand car size should not fall below some threshold. In the second\nphase, \\textsc{LiftCreate}{} retrieves components from a database of parts and\nexplores the (combinatorial) space of potential solutions, either\nusing heuristic search techniques, or resorting to optimizations\ntechniques --- like those suggested, e.g., in~\\cite{ByeOPHS16}. In the\nthird phase, a set of feasible designs is proposed to the user, sorted\naccording to decreasing relevance considering the initial design\nphilosophy. For instance, if door size is to be maximized, the first\nalternatives shown to the user are those with the widest doors,\nperhaps at the expense of payload or car size.\n\nThe main issue with \\textsc{LiftCreate}{} work-flow is that even simple\nversions of elevators consists of a large number of components,\nincluding car frame, car, doors (car and landing doors), emergency\nbrakes, pistons or cables, motors and control logic. In order to\nexplore the space of potential designs, components cannot be\nsolely available as drawing elements, like in classical CAD solutions,\nbut they must be handled as first class data inside \\textsc{LiftCreate}{}\nlogic. This aspect required us to organize the taxonomy related to\ndifferent kinds of elevators and, for each elevator kind, to structure\nthe components in a part-whole hierarchy. In\nFigure~\\ref{fig:elev_onto} we show a fragment of the taxonomy for\nelevators and an example of part-whole structure for a specific\nelevator kind. In particular, in\nFigure~\\ref{fig:elev_onto} (top), we see that \\textsc{LiftCreate}{} classifies\n\\textbf{Elevator} individuals in two main subclasses corresponding to\nhydraulic-based (\\textbf{HydraulicElevator}) and rope-based\n(\\textbf{RopeElevator}) designs. Both subclasses feature additional\npartitions to handle specific design requirements, e.g., rope\nelevators can be provided with a reduction gearbox or not, and the\ndrive can be direct of reeved. For one leaf class of the taxonomy,\nnamely \\textbf{OnePistonDirectHydraulicElevator}, in\nFigure~\\ref{fig:elev_onto} (bottom) we show the detailed part-whole\ndiagram, from which we learn that, e.g., the only peculiar aspects of\nsuch subclass is to have only one \\textbf{Piston}, whereas the\nremaining components are common to \\textbf{HydraulicElevator} or\n\\textbf{Elevator}. Also we can see that the car frame is specific of hydraulic elevators (\\textbf{CarFrameHydra}) and it is comprised of several parts, including\n\\textbf{CarRails}, \\textbf{Buffer} and \\textbf{Ropes}. The\nrelationships encoded in such part-whole hierarchy are\ninstrumental to \\textsc{LiftCreate}{} when it comes to handle drawing, storage\nand retrieval of designs, but also to reason about the various\ntrade-offs of a design when searching in the space of potential solutions.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\nConsidering the experiences herein outlined, we summarize some\nlessons learned in applying ontologies for systems engineering. First\nand foremost, while ontologies provide an effective tool for\nconceptualizing scenarios as diverse as those considered, \nsome ontology-based tools, e.g., DL reasoners, are untenable\nunless small-to-medium scale systems are considered. In the case of\nE414 ontology reasoning with an expressive ontology required us to\nimplement strategies to ``forget'' data to avoid cluttering\nthe reasoner. In the ILS ontology, where SPARQL queries for KPIs are\nthe only reasoning requested and the usage of OWL 2 QL profile banned\nexpressive but hard-to-compute constructs, scaling \nstill requires discarding data using a recency approach. On the other hand, \nin DiSeGnO and \\textsc{LiftCreate}{}, ontologies merely provide means for\nconceptualizing data and, as such, flexibility is gained without\nsacrificing performances. The second take-home message is that\nsublanguages of OWL 2 are adequate for most modeling purposes. \nWith the only exception of E414 ontology, the ones herein\nconsidered fit OWL 2 QL constraints which allowed us\nto combine in a  natural way subclassing (``IS-A'' relationships) with \nother kind of object properties (including ``HAS-A''). However, the fact that OWL 2 QL  \nontologies can be compiled to relational databases --- as in the case\nof DiSeGnO --- or handled trough an object-persistency module --- as\nin the case of \\textsc{LiftCreate}{} --- makes their use transparent to other\nsystem components. Third, and final point, with the exception of ILS\nmonitoring, none of our applications required the integration of\ndifferent data sources which is indeed one of the main tasks which\nontologies are advocated for. Nevertheless, our experience witnesses\nthat even in single-source data modeling, ontologies provide an\nexcellent mean to bridge the gap between domain experts and\ncomputer software designers. \n\n\n\n\n\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nProprioception, or the ability to perceive the relative positioning of\nneighboring body parts as well as the muscle effort deployed to\nproduce it, is a fundamental human sense. Together with vision and\ntactile sensing, it plays a unique and important role in human hand\nperception \\cite{blanchard2013differential}. For robotic manipulation,\nproprioception is translated as the combination of joint position and\ntorque sensing (assuming a hand comprised exclusively of revolute joints).\n\nCompared to vision and tactile sensing, both of which have been\nstudied extensively in the context of manipulation, we believe that\ngrasping with proprioception is still an\nimportant area to advance. On one hand, vision and tactile sensing\nhave intrinsic limitations, such as occlusion for vision and hardware\ncomplexity for tactile sensing. On the other hand, when all these\nsenses are available, they can still complement each other.  Demonstrating\nmanipulation capabilities based exclusively on proprioception becomes\na useful exercise: we believe the more a hand can do with only one\nsensing modality, the more versatile it will be when multi-modal\nsensory information gets integrated.\n\nAs we show here, proprioception is promising in providing the hand\nwith the ability to adapt to the previously unknown shape of the\nobject, and to execute stable grasps. We note that there are multiple\nways for hands to adapt to an object: while fully-actuated\nhands use sensor information (such as proprioception here) to perform\nactive adaptation, underactuated hands are good examples of passive\nadaptation without the use of sensing. However, the former have the\nadvantage of versatility: proprioceptive grippers can provide\ncompliance similar to passive mechanisms, but can also change behavior\nat runtime and selectively execute different types of grasps. Of\ncourse, the price paid for the additional versatility is the increased\ncomplexity of the sensory setup.\n\nIn this study, we explore the problem of \\textit{grasping using only\nproprioceptive feedback}, without any contact information or\nknowledge of object pose and properties. To the best of our knowledge,\nwe are the first to show that a robot hand can perform all the\nfollowing tasks using proprioception exclusively:\n\\begin{itemize}\n\\item execution of fingertip grasps for unknown objects;\n\\item execution of enveloping grasps for unknown objects;\n\\item on-demand transitions between fingertip and enveloping grasps.\n\\end{itemize}\nOur main contributions are to provide methods for the tasks above,\nand, in the process, demonstrate their effectiveness by\nexperiment. Our results indicate that the proprioceptive gripper is\nmore versatile in the range of fingertip grasps it can perform,\ncompared to our two baselines: an emulated underactuated gripper\ncommanding fixed torques to the joints, as well as a physically\nconstructed underactuated gripper. In addition, our gripper also\ndisplays the ability to execute enveloping grasps and to transition to\nthem from fingertip grasps. Both examples of increased versatility\nwere achieved using proprioception as the only available sensing\nmodality.\n\n\\section{Related Work}\n\nResearchers have been exploring real-time sensing and control as\nan alternative to vision-based planning in manipulation. Assuming\nobject information is available, model-based controllers can perform\ngrasping or in-hand manipulation. For example, Yoshikawa et\nal. presented studies (e.g. \\cite{yoshikawa2000control}) on hybrid\nforce-position control for manipulation.  Arimoto et\nal. \\cite{arimoto2000dynamics} derived the dynamics of a dual-finger\ngripper and proposed a controller which can regulate the object\nposition and orientation. Caccavale et al. \\cite{caccavale2013grasp}\nproposed an impedance controller to keep track of desired object\ntrajectory and ensure the grasp quality simultaneously. Unlike our\napproach, these methods require complete information of the\nhand-object system.\n\nWhen the models of the objects are not available, researchers either\nrelied on assumption about the contacts, or used sensor-based\ntechniques for grasping. For example, Schneider and\nCannon \\cite{schneider1992object} studied object impedance control\nusing multiple manipulators. Arimoto et al. \\cite{arimoto2005two} and Yoshida et al. \\cite{yoshida2007blind}\nstudied ``blind grasping'' using two fingertips. However, these studies\nassume the contacts only happen at the end points or the end hemispheres\nof the fingertips. Wang et al. \\cite{wang2007switching} proposed a\ncontroller that can search appropriate finger contact locations using\nhaptic feedback and can switch between control modes for different\nsurfaces. Platt et al. \\cite{platt2010null} presented a study on\nchanging the contact configuration by following the gradient of\ngrasping objective functions using six-axis loadcell data. Hsiao et\nal. \\cite{hsiao2010contact} proposed a contact-reactive method using\ntactile sensing to deal with uncertainty. However, these methods\nrequire contact sensing methods, such as tactile sensors or in-finger\nload cells. In contrast, our approach does\n  not make any assumptions about contact location or state, and does not\nrequire tactile sensing data.\n\nTorque measurement is often used for grasp control. Researchers have\ndeveloped several robotic hands with force or torque sensing. For\nexample, the Robonaut Hand \\cite{lovchik1999robonaut} and the DLR Hand\nII \\cite{butterfass2001dlr} have strain gauges or force-torque sensors\nembedded in their fingers. The hand of the DOMO robot\n\\cite{edsinger2004domo} and the hand of the Obrero robotic platform\n\\cite{torres2005obrero}\nmake use of the Series Elastic Actuators (SEA), which are a type of\nactuators with elastic components in series with the motor to sense\nthe torque \\cite{pratt1995series}. Furthermore, the DLR Hand-Arm\nSystem \\cite{grebenstein2011dlr} incorporates the Variable Impedance\nActuators, which are SEAs whose spring stiffnesses are actively\ncontrolled. These hardware designs offer high performance, but at the\ncost of high complexity and large overall packages.\n\nAs an alternative, researchers have developed underactuated hands that do\nnot require sophisticated sensing and control, and this types of hands are good baselines to compare against. Underactuated hands can\nadapt to the object and make a grasp by the virtue of\ncarefully-designed torque ratios between joints. The Harvard Hand\n\\cite{dollar2010highly}, iHY Hand \\cite{odhner2014compliant}, Robotiq\nHand \\cite{birglen2004kinetostatic}, and Velo Gripper\n\\cite{ciocarlie2014velo} are good examples in this\ncategory. However, even though underactuation simplifies control, it\ngenerally does not provide as much dexterity as full actuation. Many\nof the hands above can only perform certain types of grasps, or lack the flexibility to choose the\nconfiguration after making the grasp.\n\n\n\n\\section{Hardware Platform}\n\nWhile joint position sensing is ubiquitous for fully-actuated robot\nhands, torque sensing and control is not common in commercially available\nmanipulators. This compelled us to design our own hardware testbed. We\nimplemented torque sensing and control with Series Elastic Actuators\n(SEA), a method known for high-fidelity torque control, shock\nprotection, and human-safety~\\cite{pratt1995series}. \n\n\\subsubsection{SEA Module}\n\nSimilar to the design from Ates et al. \\cite{ates2014servosea}, we developed a simple and compact SEA module (Fig.~\\ref{fig:sea}). A position-driven servo (gray) is used as the driving motor, which receives position commands and returns the current position (measured by the built-in potentiometer), i.e., $\\theta_{motor}$  can be measured. A torsion spring (orange) is used as the elastic component, connecting the motor shaft (purple) and the pulley shaft (blue). A Spectra cable is tied on the pulley to transmit the force to the finger joint. An absolute magnetic encoder (in green) is mounted on the end of the pulley shaft to measure $\\theta_{pulley}$. In steady-state, the force in the tendon can be calculated as the product of spring stiffness and deflection divided by pulley radius: \n\\begin{equation} \\label{eq:sea}\nF_{tendon} = K_{spring} \\cdot (\\theta_{pulley} - \\theta_{motor}) \/ R_{pulley}\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = 80 mm]{images\/hand.pdf}\\\\\n\\caption{Schematics of the gripper.}\n\\label{fig:hand}\n\\vspace{-4mm}\n\\end{figure}\n\n\n\n\\subsubsection{Gripper Design}\n\nThe gripper consists of two fingers and each finger has two links, shown in Fig.~\\ref{fig:hand}. \nIn each joint, flexion is\npowered by the tendon (shown as colored dash lines in\nFig.~\\ref{fig:hand}) connected to the SEA pulley, and the extension is\ndriven by a restoring spring. The tendon connected to the\ndistal joint (the red dash line) goes right through the axis of the\nproximal joint so that the torques of proximal and distal joints are\nfully decoupled.\n\n\n\\subsubsection{SEA-level Control}\n\nThere are three SEA-level control modes: motor position control, pulley position control and torque control, and they can be switched online. \nWe note that we built joint-level or hand-level controllers on top of these SEA-level controllers. For example, joint position and torque control can be achieved by SEA pulley position and torque control with a simple linear conversion.\n\n\n\n\\section{Fingertip Grasping}\n\nFingertip grasps commonly refer to grasps where only the most\ndistal links of each finger make contact with the object. We note that it does not necessarily mean the contacts are located in the very end of the fingers, so contact locations are still unknown. This type of\ngrasp is important not only for precision tasks, but also for cases\nwhere a more stable enveloping grasp is not immediately available\nbecause of the environment (an object laying on a table, against a\nbackdrop, etc.). In this section, we introduce a control algorithm\nwhich can perform stable fingertip grasps for unknown objects.\n\n\\subsection{Problem Statement}\n\nThe objective of our algorithm is to\nincrease the torques applied at the joints (and implicitly the contact forces) in a ``stable'' fashion after making initial contacts. In other words, we need to find an\nincrease in joint torques that produces no net wrench on the\nobject. For a hand with multi-link fingers, this is not straightforward given that the joint torques need to be coordinated in the absence of information on object\nshape and contact locations. It is necessary to note that we are not\nsolving contact planning problem, and we wish to develop reactive control strategies that do not require pre-planning.\n\nOur insight is that proprioception alone can characterize grasp stability. For an SEA-powered proprioceptive gripper, an\nunbalanced net wrench will produce object movement against the\ncompliant elements, which we can measure. Therefore, \\textit{we\n  formulate the goal of grasp stability as the one of minimizing\n  object movement while applying forces}. Since we do not have a direct measure of object\nmovement in Cartesian coordinates by using only proprioception, we use the change of joint position in joint space (measured by SEA)\nduring grasping as a proxy for object movement.\n\nFor intuition, consider the analogies shown in Fig.~\\ref{fig:analogy}:\nFigure (a) shows a (simplified) scenario in which the motors (red dots) are\ndriving the torsion springs, and the springs are pushing the fingers\n(blue dots) to make a grasp (gray dots). (b) shows a simpler\none-dimensional abstraction using linear springs, with similar\ncolor-coding as in (a). Here, we actively control the positions\n$x_{m1}$ and $x_{m2}$ (which translate to motor positions\n$\\theta_{motor}$ on the real gripper) to apply forces, and measure\n$x_{j1}$ and $x_{j2}$ (which translate to pulley positions\n$\\theta_{pulley}$ linearly mapped to joint positions). We aim to keep\n$x_{j1}$ and $x_{j2}$ constant as we squeeze.\n\n\\subsection{MIMO Grasping Controller}\n\nOur key insight is that the problem of grasping unknown objects can be solved even in the absence of\ncontact information, by using proprioception as inputs for a\nmulti-input-multi-output (MIMO) proportional-integral (PI)\ncontrol. Without knowledge of object geometry, contact locations and contact states, it\nis impossible to fully model the dynamic system analytically. However,\na proprioceptive platform still provides sensory access to the\nvariables that characterize the grasp stability, and the PI control framework provides ways to regulate these variables even though the analytical relationship is not constructed . We\nthus aim to use a feedback scheme operating exclusively in\nthe sensory space of the robot, without explicitly modeling the physics of the gripper and the object.\n\nFig.~\\ref{fig:mimo} shows the block diagram of the MIMO control loop. Here, the controller is constructed on top\nof the low-level sensing, so we consider the joint angles and torques\nare already obtained from SEA measurements. The reference vector\n$\\bm{u}$ consists of desired joint angle values ($\\theta^{p1}_{des}$,\n$\\theta^{d1}_{des}$, $\\theta^{p2}_{des}$, $\\theta^{d2}_{des}$, where\nthe superscripts $p$ represent proximal and $d$ represent distal\njoints) and reference joint torques ($\\tau^{p1}_{des}$,\n$\\tau^{d1}_{des}$, $\\tau^{p2}_{des}$, $\\tau^{d2}_{des}$ ). The feedback vector $\\bm{y}$ has the same structure, but contains actual measured\nmeasured values. The desired joint angles (first half of the reference\nvector $\\bm{u}$) are extracted by a feedforward matrix $\\bm{F}$ and\nused as a feedforward term. The error between the reference $\\bm{u}$\nand feedback $\\bm{y}$ is fed into a MIMO PI block (a combination of\nmany P and PI controllers) which is a $4\\times8$ matrix. The output of\nthe PI block and the feedforward term are summed up as motor position\ncommand $\\bm{c}$ (a $ 4 \\times 1$ vector) and sent to the motors:\n\\begin{equation} \\label{eq:mimo}\n\\bm{c}(t)=\\bm{F u}(t)+ {\\bm{K}_{p}}  \\bm{e}(t)+ {\\bm{K}_{i}} \\int_{0}^{t}{ \\bm{e}(\\tau) d\\tau} \n\\end{equation}\nHere,\n$\n\\bm{F}=\\left[ \\begin{matrix} {\\bm{I}_{4\\times 4}} & {\\bm{O}_{4\\times 4}}\\end{matrix} \\right]\n$\nis the feedforward matrix,\n$\\bm{K}_p$ ($4\\times 8$ matrix with all entries being non-zero) and \n$\\bm{K}_i=\\left[ \\begin{matrix} {\\bm{K}_{4\\times 4}} & {\\bm{O}_{4\\times 4}}\\end{matrix} \\right]$ (where $\\bm{K}$ is a matrix with all entries being non-zero)\nare proportional and integral gain matrices, and $\\bm{e}$ is the $ 8 \\times 1$ error vector between $\\bm{y}$ and $\\bm{u}$. After that, the actual motor position vector $\\bm{d}$ goes to the black-box system of the gripper and the unknown object.\n\nIn the reference vector $\\bm{u}$, the desired joint angle values  ($\\theta^{p1}_{des}$, $\\theta^{d1}_{des}$, $\\theta^{p2}_{des}$, $\\theta^{d2}_{des}$) \nare equal to those in the initial touch configuration, while the reference torques ($\\tau^{p1}_{des}$, $\\tau^{d1}_{des}$, $\\tau^{p2}_{des}$, $\\tau^{d2}_{des}$ ) \nare chosen using the maximum motor torques. A special design of this controller is that we require the joint angles to be regulated exactly to the set points, but do not require the torques to be so. We allow and make use of the steady-state error of pure proportional control (we note that entries in the right half of the integral gain $\\bm{K}_i$ are set to be zeros). In this way, the reference torques do not need to be a legal set of torques that result in equilibrium --- actually, we are not able to design such a legal set of torques due to the absence of contact or object information. We let the law of dynamics decide the steady-state values for torques, and let the system balance itself automatically. The effectiveness of increasing joint torques is shown in section VI.\n\nFrom a practical standpoint, the tuning process of the MIMO PI controller is not\nas complicated as it would seem based on the number of parameters. First, due to gripper symmetry, the number of parameters is cut by half. Second, we formulate\nevery gain as a product of a baseline value ($b_i$ in (\\ref{eq:kp})(\\ref{eq:ki}) )  and a weight\ncoefficient ($w_i$ in (\\ref{eq:kp})(\\ref{eq:ki}) ). The baseline values are set to be the same if the input\nentries corresponding to those gains have same physical\ndimensionality, and the weight coefficients are tuned based on its\nrelative importance. Third, conventional tuning heuristics for the gains of single-input-single-output systems also apply here. The structures of the gain matrices are as follows:\n\\begin{equation} \\label{eq:kp}\n\\bm{K}_p=\\left[ \\begin{smallmatrix}\n{w_1 b_1} & {w_2 b_2}  & {w_2 b_1}  &  {w_2 b_2}  & {w_3 b_3}  & {w_4 b_4} & {w_4 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_1 b_2}  & {w_2 b_1}  &  {w_2 b_2}  & {w_4 b_3}  & {w_3 b_4} & {w_4 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_2 b_2}  & {w_1 b_1}  &  {w_2 b_2}  & {w_4 b_3}  & {w_4 b_4} & {w_3 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_2 b_2}  & {w_2 b_1}  &  {w_1 b_2}  & {w_4 b_3}  & {w_4 b_4} & {w_4 b_3} & {w_3 b_4} \\\\\n\\end{smallmatrix} \\right]\n\\end{equation}\n\\begin{equation} \\label{eq:ki}\n\\bm{K}_i=\\left[ \\begin{smallmatrix}\n{w_1 b_5} & {w_2 b_6}  & {w_2 b_5}  &  {w_2 b_6}  & ~~0~~ & ~~0~~ & ~~0~~ & ~~0~~ \\\\\n{w_2 b_5} & {w_1 b_6}  & {w_2 b_5}  &  {w_2 b_6}  &   0 & 0  & 0  &     0\\\\\n{w_2 b_5} & {w_2 b_6}  & {w_1 b_5}  &  {w_2 b_6}  &   0  & 0  & 0  &    0 \\\\\n{w_2 b_5} & {w_2 b_6}  & {w_2 b_5}  &  {w_1 b_6}  & 0 & 0 & 0 & 0 \\\\\n\\end{smallmatrix} \\right]\n\\end{equation}\nWe pick $b_1 = 0.2$, $b_2 = 0.5$, $b_3 = 4.0$, $b_4 = 8.0$, $b_5 = 1.0$, $b_6 = 1.0$, $w_1 = 1.0$, $w_2 = 0.3$, $w_3 = 1.0$, $w_4 = 0.5$ for our hardware.\n\n\n\\section{Enveloping Grasping and Transitions}\n\nAn enveloping grasp is the one where both distal and proximal links\nmake contact with the object around its circumference. This type of\ngrasp is generally considered more stable than a fingertip grasp\nbecause it can resist disturbances in a wider range of directions. The\ninability to envelop is also one of the main shortcomings of simple\nparallel grippers. In contrast, some of the more recent underactuated\nhands are optimized explicitly for effective enveloping grasps of a\nwide range of objects (e.g. \\cite{ciocarlie2014velo}).\n\nWhen using our proprioceptive gripper, we found that stable\nenveloping grasps for unknown objects are easier to obtain than fingertip grasps. The\nmechanism is generally fully constrained and all the links are\ncounterbalancing each other. A simple joint torque control scheme, or the MIMO Grasping Controller, can fulfill this task.\n\nA very important ability of this gripper, further underlining its\nversatility, is to \\textit{transition} between grasp types when holding unknown objects. After\nexecuting a stable fingertip grasp (using the MIMO Grasping\nController), the gripper can switch to joint torque control with the\ntorque ratio (between distal and proximal joints) being 0.5 to 1.0,\nthus bringing the object into the hand and creating an enveloping\ngrasp. We illustrate this behavior with several experiments in the\nfollowing section.\n\n\n\n\n\\section{Experiments and Results}\nIn this section, we demonstrate the merits of the\nproprioception-enabled gripper by several experiments. We validated\nits capability of performing fingertip grasps, enveloping grasps, and\nthe transition from the former to the latter. We note that in this\nstudy we only consider a two-dimensional scenario, in which the\nobjects are confined to move only in the plane of the fingers.\n\n\\subsection{Fingertip Grasp}\nThe goal of this experiment is to test the hypothesis that the MIMO Grasping Controller is effective in fingertip grasping for unknown objects. We compare against two baselines: a (fully-actuated) gripper running a Fixed Torque Ratio Controller, and a physical underactuated gripper. \n\nThe Fixed Torque Ratio Controller, where the torques applied to proximal and distal joints always follow a certain ratio, can be thought of as an emulation of a common type of underactuated grippers (tendon-pulley-driven, without special designs such as stoppers or clutches). When this kind of grippers make grasps, the configuration-dependent torques from the extension springs can be ignored, as they are usually much smaller than the flexing torques from the tendons. Therefore, the net joint torques in proximal and distal joint have a configuration-independent and design-time-fixed ratio, which is the ratio between the joint pulley radii.\n\nFurthermore, we understand that this emulation is subject to limited control bandwidth and may have unrealistic behavior compared to its physical counterpart. Thus, we also built a physically underactuated gripper testbed for comparison. This testbed has same specs as the fully-actuated proprioceptive gripper, except that the proximal and distal joints are driven by a single tendon wrapping around the joint pulleys. In this design, we can alter the torque ratio by physically changing the pulleys between experiments. We perform torque control for proximal joints in our experiments, thus the torques on the distal joints are defined by the physically determined ratios. However, in this setup, we lose the ability to measure joint positions by SEA readings because, in underactuated mechanisms, joint positions are determined not only by actuator positions but also by contact forces which here are unknown. \n\n\n\n\\subsubsection{Experiment Protocol}\n\nOur experiment proceeds as follows. We execute the grasping in two phases. In the first phase (approaching and touching), the fingers are set in torque control mode with very low reference torques so that they stop when they touch the object. In the second phase(squeezing), the gripper executes the MIMO Grasping Controller or the Fixed Torque Ratio Controller in the fully-actuated testbed, or the  joint torque control on proximal joints in the underactuated testbed for comparison.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=55 mm]{images\/experiment_setup.pdf}\\\\\n\\caption{Object sizes, object locations and initial touch poses.}\n\\label{fig:expsetup}\n\\vspace{-3mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\hspace{-6mm}\\includegraphics[width=100 mm]{images\/pose_photo.pdf}\n\\caption{Pose changes in different controllers.}\n\\label{fig:pose}\n\\vspace{-3mm}\n\\end{figure}\n\nThere are a lot of factors that may influence the performance. To have a well-rounded comparison, we swept the following dimensions:\n\\begin{itemize}\n\\item \\textit{Controllers.} The torque ratio is a key parameter for both the Fixed Torque Ratio Controller, and the physically underactuated gripper. We tested the MIMO Grasping Controller against the other two baselines with three different ratios between the distal and proximal joint: 0.3, 0.4 and 0.5.\n\\item \\textit{Objects.} We selected four objects for the test: a big cylinder (diameter: 67mm), a big box (side length: 57mm), a small cylinder (diameter: 47mm) and a small box (side length: 39 mm). All objects have negligible friction with the table.\n\\item \\textit{Object locations.} We swept three locations along the center line of the gripper within the range of fingertip grasp: 100 mm, 120 mm and 140 mm from the palm.\n\\item \\textit{Initial touch poses.} We tested three different distal joint angles for the initial touch: 0, 30 and 60 degrees. We note that we only include this dimension for MIMO Grasping Controller and Fixed Torque Ratio Controller test, and not for the physically underactuated hand because the distal joint angles of initial touch cannot be explicitly controlled in runtime.\n\\item \\textit{Friction coefficient.} We tested the controllers with\n  two fingertip materials: rubber (high friction, $\\mu = 1.2$) and\n  vinyl plastic (low friction, $\\mu = 0.4$).\n\\end{itemize}\n\nTo sum up, we swept all five dimensions and conducted 360 grasping experiments. Fig.~\\ref{fig:expsetup} shows the three object locations (shown as the crosshairs), three initial touch poses (colored fingers), and the sizes of the objects relative to the gripper (orange shapes).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics{images\/torque_increase.pdf}\n\\caption{Increases of torque magnitude during squeezing}\n\\label{fig:trq}\n\\vspace{-3mm}\n\\end{figure}\n\n\\subsubsection{Performance Metric}\n\n\\begin{itemize}\n\\item \\textit{Success rate.} The success rate is our primary performance metric. We define a ``success\" if the gripper finally settles down in equilibrium with the object in hand after squeezing. This definition includes three scenarios: (1) the gripper keeps the object in fingertips near initial touch pose, without converting to enveloping grasp or reaching joint limits, (2) the gripper holds the object but reconfigures to an enveloping grasp, and (3) the gripper keeps the object in fingertips but reaches a mechanical joint limit (thus the joint torque ratio changes). These cases are all considered successful but still need to be distinguished. Case (1) is the most desirable, while (2) and (3) mean the grasp is not stable at initial pose and relies on reconfiguration to be balanced.\n\n\\item \\textit{Gripper pose change.} We believe it is also useful to keep the object in the same pose as when first contact is made. We thus use a secondary performance metric that evaluates how much the object moves in the hand during the squeezing process, with less movement considered better. Without access to object pose in Cartesian space, we measure this as the change in gripper pose between initial touch and final grasp (Euclidian distance in four-dimensional joint space). This metric is only calculated and averaged for the successful cases. Besides, it is not computed for physically underactuated gripper because the joint angles during grasping are not accessible for the reasons mentioned above.\n\\end{itemize}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{images\/result.pdf}\n\\caption{Experiment results of fingertip grasping compared to Fixed Torque Ratio Controller.}\n\\label{fig:result}\n\\vspace{-3mm}\n\\end{figure*}\n\n\n\\subsubsection{Results}\nFig.~\\ref{fig:pose} shows the photos of some typical scenarios in the experiments. (a) and (b) shows the resting pose and the initial touch. (c) shows the successful grasp near initial touch pose. The other images show cases in which the object was kept in fingertip grasp but a joint limit was reached (d), the grasp was transformed into an enveloping one (e), and the object was squeezed out of the hand (f). \n\nFig.~\\ref{fig:trq} shows that all controllers are effective in increasing joint torques. The horizontal axis shows different controllers (or grippers), the vertical axis is the magnitude of the four-dimensional torque vector ($\\tau^{p1}_{meas}$, $\\tau^{d1}_{meas}$, $\\tau^{p2}_{meas}$, $\\tau^{d2}_{meas}$) which indicates how ``strong'' the grasp is, and bar colors distinguish between initial touch and final grasp. We can see that there is a significant increase in the torque magnitude, and the torque levels in different cases are similar.\n\n\nThe results of the experiments comparing against Fixed Torque Ratio Controller are visualized as multiple bar charts in Fig.~\\ref{fig:result}. In each plot, the bar colors show four different controllers,\nthe vertical axis is one of the performance metrics \nand the horizontal axis represents another dimension which is different in each plot (from (a) (f) to (d) (i): different objects, object locations, initial poses, and friction coefficients). In each bar in the first row showing the success rate, the pure-color area, the dotted area, and the line-shaded area represent, respectively, successful fingertip grasp without reaching joint limit, successful grasps but converted to enveloping, as well as successful fingertip grasp but joint angles reached limits.\n\nSimilarly, the results comparing against physically underactuated grippers are shown in Fig.~\\ref{fig:result_ua}. Here, the initial touch pose dimension is not available, so there are three dimensions (from (a) to (c): different objects, object locations, and friction coefficients). Also, the pose change metric is not available due to the absence of joint angle information. All other plotting rules are the same as Fig.~\\ref{fig:result}.\n\n\n\n\nAs shown in Fig.~\\ref{fig:result} (e)(j) and Fig.~\\ref{fig:result_ua} (d), the overall success rates are 91.67\\% for MIMO Grasping Controller, 72.22\\%, 76.39\\%, 66.67\\% for Fixed Torque Ratio Controller with torque ratio of 0.3, 0.4, 0.5, respectively, and 87.50\\%, 75.00\\%, 87.50\\% for physically underactuated gripper with torque ratio of 0.3, 0.4, 0.5, respectively. Even when the overall success rates are close (for example, Fig.~\\ref{fig:result_ua} (d)), the types of the resulting grasps are significantly different. Besides, the gripper pose change metric for the MIMO controller is 9.96, compared to 41.55, 31.93, and 81.14 (degrees) respectively for the Fixed Torque Ratio controllers. \n\n\n\\subsection{Enveloping Grasps and Transitions}\n\nWe performed a second experiment to show that this gripper can perform enveloping grasp with either MIMO Grasping Controller or Fixed Torque Ratio Controller. We tested on two objects (big cylinder and big box), two object locations (60mm and 80mm from the palm), and two controllers mentioned above. The success rate is 100\\%.\n\nThe last experiment is to show the performance of the transition from fingertip grasp to enveloping grasp. We first created fingertip grasps using the MIMO Grasping Controller, and then switched to Fixed Torque Ratio Controller with a ratio of 0.5. We tested on two objects (big cylinder and big box), three object locations (100, 120 and 140mm), three initial poses (0, 30 and 60 degrees) with the low friction fingertips. We found the success rate was 83.33\\%.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.8 \\textwidth]{images\/result_ua.pdf}\\\\\n\\caption{Experiment results of fingertip grasping compared to physically underactuated gripper.}\n\\label{fig:result_ua}\n\\vspace{-2mm}\n\\end{figure*}\n\n\\section{Discussion and Conclusion}\n\nOverall, the results of the previous section support our hypotheses: the proprioceptive gripper running MIMO Grasping Controller is effective at executing stable fingertip grasps in a variety of situations and outperforms the baselines. Furthermore, the proprioceptive gripper exhibits versatility in being able to perform multiple types of grasps and also to transition between them on-demand.\n\nBased on Fig.~\\ref{fig:result} and \\ref{fig:result_ua}, the MIMO Grasping Controller outperforms the baselines in fingertip grasping, and usually succeeds without transforming to an enveloping grasp or reaching joint limits. In contrast, the emulated and physical underactuated gripper often transform to an enveloping grasp or reach joint limits, thus relying on gripper reconfiguration. The second row of Fig.~\\ref{fig:result} ((f) to (j)) gives similar intuition: for the Fixed Torque Ratio Controller, most cases have a large pose change. While the end-result is stable, it is different from the originally intended grasp. This might be unimportant or detrimental depending on the application.\n\nIt is also interesting to notice that, in different conditions, the optimal torque ratio for the emulated or physical underactuated gripper is different. We take this to mean that there is no one clearly preferable pre-set torque ratio, which could be physically implemented in a mechanical design, in order to obtain ideal performance in all these cases. In contrast, the proprioception-enabled gripper has the flexibility to alter torques at run-time.\n\nLooking at how specific variables affect performance we can gain additional insights. From Fig.~\\ref{fig:result} and~\\ref{fig:result_ua} (a) and (b) we can see that the success rates for Fixed Torque Ratio Controller are low if the objects are small and close to the palm. This is because the emulated or physical underactuated gripper tends to transform the initial unstable fingertip grasps to enveloping when contacts are close to distal joints, but cannot cage the object if it is small because the distal links are fighting against each other --- a common issue for underactuated grippers. In contrast, the MIMO Grasping Controller does not suffer from this because it does not perform the conversion.\n\n\nIn the transitioning experiment, the high success rate shows the\nproprioceptive gripper can indeed perform the conversion between grasp types\n\\textit{on-demand}. Though underactuated grippers also\noccasionally perform such transitions, they occur unintentionally\nand without giving the user an option to select the desired type of\ngrasp.\n\nIt is important to also highlight the limitations of this\nstudy. Due to high dimensionality of the brute-force sweep in\nour experiment, we cannot afford to cover a larger range with a finer\nresolution for each dimension. In particular, we are unable to explore\nmore possibilities for physically implemented torque ratios. The\nevaluation of the controller is primarily experimental and would\nbenefit from additional stability analysis, carried out for example\nfor representative cases and grasps.\n\nOverall, we claim that proprioceptive manipulators, using active\nsensing and control such as the MIMO Grasping Controller, represent a\npromising way towards more versatile grasping and manipulation for\nunknown objects. Future work will include the extension of the operation to three-dimensional cases, optimization \/ learning of the control gains, and the inclusion of hand\nposition to our set of actively controlled variables. We are aiming to\nfurther explore these possibilities.\n\n\\section*{APPENDIX}\n\n\n\n\n\n\n\\bibliographystyle{bib\/IEEEtran}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nOver the past decade, advances in data collection and increasing access to computational resources have led to a revolution in the use of data-driven techniques for the solution of complex inverse problems. One such problem is that of turbulence, the multiscale nature of which causes extreme computational demands for most practical systems. As a result, turbulence requires the use multiple modeling approximations for the higher wavenumbers which remain unsupported by computational degrees of freedom. One such modeling approach is that of large eddy simulation (LES) \\cite{sagaut2006large}, which attempts to simulate the evolution of the smaller wavenumbers while the unresolved frequencies are modeled by an algebraic or differential equation. As such, the basic premise of LES is extendable to many partial differential equation systems with quadratic non-linearities. The procedure of modeling these smaller scales is often denoted \\emph{closure} due to insufficient knowledge about higher-order wavenumber interactions with the coarse-grained system \\cite{berselli2006mathematics} and remains vital for the accurate computation of many applications \\cite{hickel2014subgrid,yu2016dynamic,zhou2018structural}. From an LES point of view, the closure problem may be considered to be dominated by commutative errors in the calculation of the non-linear term as well as the defects associated with commutative errors stemming from the dynamic term. In this study, we focus on the former.\n\nThere are two main schools of thought when it comes to the LES of the Navier-Stokes equations. The first of these promotes the use of explicit closures. Explicit LES argues for the utilization of closures in the form of sub-grid models specified as algebraic or differential equations for the unresolved scales. These are built on intuitive reasoning of how the losses of coarse graining the Navier-Stokes equations may be incorporated into an LES deployment. Some of the most notable sub-grid closure strategies are those given by the eddy-viscosity hypothesis. Within the context of the Navier-Stokes equations, it is generally accepted that the finer scales are dissipative at the Kolmogorov length scales \\cite{kolmogorov1941local} and therefore, most turbulence models seek to specify a sub-grid dissipation \\cite{frisch1995turbulence}. Most sub-grid models can be traced back to the seminal work of Smagorinsky \\cite{smagorinsky1963general}, where a model was proposed based on the concepts of an effective eddy-viscosity determined by an \\emph{a priori} specified mixing length and a $k^{-5\/3}$ scaling recovery for the kinetic energy content in the wavenumber domain. Similar hypotheses have also been used for two-dimensional turbulence \\cite{leith1968diffusion} (often utilized as a test-bed for geophysical scenarios, for instance see works by Pearson \\textit{et al.}\\cite{pearson2018log,pearson2017evaluation}), for approximating the $k^{-3}$ cascade in two-dimensional turbulence and generally have their roots in dimensional analysis related to the cascade of enstrophy. These models may also be classified as \\emph{functional} due to the phenomenological nature of their deployment and represent the bulk of explicit LES turbulence models used in practical deployments. Explicit LES closures may also be specified through the specification of a low-pass spatial filter to account for the unresolved scales \\cite{bardina1980improved,stolz1999approximate,layton2003simple,mathew2003explicit} where phenomenology is bypassed but ansatz are provided for the bulk dissipative nature of the smaller scales through the control of a characteristic filter-width. In either scenario, (i.e., whether structural or functional), the choice of the phenomenology (or dissipation control parameter) plays a key role in the successful calculation of accurate \\emph{a posteriori} statistics. In contrast, the implicit LES (or ILES) approach utilizes numerical dissipation to model the unresolved scales in a turbulent flow \\cite{grinstein2007implicit,el2017investigation,Margolin2018}. In essence, the predominantly dissipative effects of the smallest scales are replicated through an artificial numerical dissipation via a biased discretization used in the calculation of the non-linear advective term \\cite{thornber2007implicit,debonis2013solutions}. The ILES approach is popular due to reduced algorithmic complexity and represents a union of turbulence modeling and shock capturing mechanisms but is often criticized due to the difficulties involved in quantifying the correct amount of dissipation in a turbulent flow evolution. This results in ILES methods often proving robust and stable but overly dissipative. In this work, we propose a machine learning algorithm to enable selective dissipation within an ILES deployment through the use of explicit LES concepts during the training of the learning framework. \n\n\n\nThe past few years have seen a rapid increase in the use of data-driven techniques for the spatio-temporal modeling of dynamical systems \\cite{schmidt2009distilling,bright2013compressive,xiao2015non,ma2015using,gautier2015closed,brunton2016discovering,schaeffer2017learning,raissi2017machine,mohan2018deep,raissi2018hidden,rudy2018deep,san2018neural,wan2018data,kim2018deep,muravleva2018application,jin2018prediction}. When it comes to turbulence, some widely used strategies for inference include symbolic regression \\cite{weatheritt2016novel,weatheritt2017development,weatheritt2017hybrid}, where functional model-forms for Reynolds-averaged Navier-Stokes (RANS) deployments were generated through evolutionary optimization against high-fidelity data. Other techniques incorporating Bayesian ideologies have also been used, for instance by Xiao \\textit{et al.}\\cite{xiao2016quantifying} where an iterative ensemble Kalman method was used to assimilate prior data for quantifying model form uncertainty in RANS models. In Wang \\textit{et al.}\\cite{wang2017physics,wang2017comprehensive} and Wu \\textit{et al.}\\cite{wu2018data}, random-forest regressors were utilized for RANS turbulence-modeling given direct numerical simulation (DNS) data. In Singh and Duraisamy \\cite{singh2016using} and Singh \\textit{et al.}\\cite{singh2017machine}, an ANN was utilized to predict a non-dimensional correction factor in the Spalart-Allmaras turbulence model through a field-inversion process using experimental data.  Bypassing functional formulations of a turbulence model (a focus of this study) was also studied from the RANS point of view by Tracey \\textit{et al.} \\cite{tracey2015machine}. Ling and Templeton \\cite{ling2015evaluation} utilized support vector machines, decision trees and random forest regressors for identifying regions of high RANS uncertainty. A deep-learning framework where Reynolds-stresses would be predicted in an invariant subspace was developed by Ling \\textit{et al.} \\cite{ling2016reynolds}. Machine learning of invariance properties has also been discussed in the context of turbulence modeling \\cite{ling2016machine}. The reader is directed to a recent review by Duraisamy \\textit{et al.}\\cite{duraisamy2018turbulence}, for an excellent review of turbulence modeling using data-driven ideas.\n\nAs shown above, the use of data-driven ideologies and in particular artificial neural networks (ANNs) has generated significant interest in the turbulence modeling community for addressing long-standing challenges (also see \\cite{sotgiu2018turbulent,zhang2018machine,zhu2019machine,zhang2019application,raissi2019deep} for recent progress). One motivation for the popularity of ANNs is that a multilayered ANN may be optimally trained to universally approximate any non-linear function \\cite{hornik1989multilayer}. In addition, the deployment of ANNs is amenable to integration within existing computational frameworks. Greater accessibility to data and ever-improving computing capabilities has also motivated the development of advanced ANN architectures for large-scale learning of complicated physical phenomena such as turbulence. Within the context of LES (and associated with the scope of this paper) there are several investigations into sub-grid modeling using data-driven techniques. In one of the first studies of the feasibility of using learning from DNS based high-fidelity data, Sarghini \\textit{et al.}\\cite{sarghini2003neural} utilized ANNs for estimating Smagorinsky model-form coefficients within a mixed sub-grid model for a turbulent channel flow. ANNs were also used for wall-modeling by Milano and Koumotsakos \\cite{milano2002neural} where it was used to reconstruct the near wall field and compared to standard proper-orthogonal-decomposition techniques. An alternative to ANNs for sub-grid predictions was proposed by King \\textit{et al.}\\cite{king2016autonomic} where \\emph{a priori} optimization was utilized to minimize the $L^2$-error between true and modeled sub-grid quantities in a least-squares sense using a parameter-free Volterra series. Maulik and San \\cite{maulik2017neural} utilized an extreme-learning-machine (a variant of a single-layered ANN) to obtain maps between low-pass spatially filtered and deconvolved variables in an \\emph{a priori} sense. This had implications for the use of ANNs for turbulence modeling without model-form specification. A more in-depth investigation was recently undertaken by Fukami \\textit{et al.}\\cite{fukami2018super} where convolutional ANNs were utilized for reconstructing from downsampled snapshots of turbulence. Maulik \\textit{et al.} \\cite{maulik2018deconvolution} also deployed a data-driven convolutional and deconvolutional operation to obtain closure terms for two-dimensional turbulence. Gamahara and Hattori \\cite{gamahara2017searching}, utilized ANNs for identifying correlations with grid-resolved quantities for an indirect method of model-form identification in turbulent channel flow. The study by Vollant \\textit{et al.} \\cite{vollant2017subgrid} utilized ANNs in conjuction with optimal estimator theory to obtain functional forms for sub-grid stresses. In Beck \\textit{et al.}\\cite{beck2018neural}, a variety of neural network architectures such as convolutional and recurrent neural networks are studied for predicting closure terms for decaying homogeneous isotropic turbulence. A least-squares based truncation is specified for stable deployments of their model-free closures. Model-free turbulence closures are also specified by Maulik \\textit{et al.}\\cite{maulik2018deconvolution,maulik2019subgrid} and Wang \\textit{et al.}\\cite{wang2018investigations}, where sub-grid scale stresses are learned directly from DNS data and deployed in \\emph{a posteriori} assessments. King \\textit{et al.}\\cite{king2018deep} studied generative-adversarial networks and the LAT-NET \\cite{hennigh2017lat} for \\emph{a priori} recovery of statistics such as the intermittency of turbulent fluctuations and spectral scaling. A detailed discussion of the potential benefits and challenges of deep learning for turbulence (and fluid dynamics in general) may be found in the article by Kutz \\cite{kutz2017deep}.\n\nWhile a large majority of the LES-based frameworks presented above utilize a least-squares error minimization technique for constructing maps to sub-grid stresses \\emph{directly} for theoretically optimal LES \\cite{langford1999optimal,moser2009theoretically,labryer2015framework}, this work is novel in that it utilizes sub-grid statistics (pre-computed from DNS data) to train a classifier. This classifier determines whether a location requires dissipation or not through \\emph{a priori} experience in the learning phase. Once classified, the non-linear term at this particular point is evaluated using one of two schemes. If it is determined that the point requires no sub-grid closure, a symmetric and second-order accurate, energy and enstrophy conserving Arakawa-scheme \\cite{arakawa1981potential} is utilized for the non-linear term computation. If dissipation is necessary, an upwinding scheme is utilized instead. Therefore this study may be interpreted as a machine learning framework for devising hybrid schemes for non-linear term computation with a view to reconstructing turbulence statistics in a superior fashion. Therefore, this study is similar to that employed by Ling and Kurzawski \\cite{ling2017data} for adaptively determining RANS corrections. We note that the classification framework devised in this study is also deployed in an aligned work to switch between functional and structural explicit LES hypotheses spatio-temporally \\cite{maulik2018online} thus proving that high-fidelity DNS statistics may be qualitatively utilized to inform modeling strategies through conditional probability predictions. The article shall describe how the proposed framework is effective in moderating the larger dissipation of an upwinded-scheme through assessments on the Kraichnan turbulence test-case. \n\n\\section{Turbulence modeling equations}\n\nThe governing equations for two-dimensional turbulence are given by the Navier-Stokes equations in the vorticity-stream function formulation. In this formulation, our non-dimensional governing equation for incompressible flow may be represented as\n\\begin{align}\n\\label{eq1}\n\\frac{\\partial \\omega}{\\partial t} + J(\\omega,\\psi) = \\frac{1}{Re} \\nabla^2 \\omega,\n\\end{align}\nwhere $Re$ is the Reynolds number, $\\omega$ and $\\psi$ are the vorticity and stream function respectively connected to each other through the Poisson equation given by\n\\begin{align}\n\\label{eq2}\n\\nabla^2 \\psi = - \\omega.\n\\end{align}\nIt may be noted that the Poisson equation implicitly ensures a divergence-free flow evolution. The non-linear term (denoted the Jacobian) is given by\n\\begin{align}\n\\label{eq3}\nJ(\\omega,\\psi) = \\frac{\\partial \\psi}{\\partial y} \\frac{\\partial \\omega}{\\partial x} - \\frac{\\partial \\psi}{\\partial x} \\frac{\\partial \\omega}{\\partial y}.\n\\end{align}\nThe stream function and the two-dimensional velocity components are related as \n\\begin{align}\n\\label{eq3a}\nu &= \\frac{\\partial \\psi}{\\partial y}, \\quad v = -\\frac{\\partial \\psi}{\\partial x}.\n\\end{align}\n\nA reduced-order implementation of the aforementioned governing laws (i.e., an LES) is obtained through\n\\begin{align}\n\\label{eq4}\n\\frac{\\partial \\bar{\\omega}}{\\partial t} + J(\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{Re} \\nabla^2 \\bar{\\omega},\n\\end{align}\nwhere the overbarred variables are now evolved on a grid with far fewer degrees of freedom. Due to the reduction in supported frequencies, the non-linear Jacobian fails to capture inter-eddy interactions at different wavenumbers. If it is assumed that the finer scales of vorticity are generally dissipative in nature for two-dimensional turbulence (based on Kraichnan's cascade of enstrophy \\cite{kraichnan1967inertial}), dissipative models may be embedded into the coarse-grained evolution of the vorticity evolution equation to recover some portion of the effect of the finer scales. Explicit LES closures embed dissipation into the vorticity evolution in the form of eddy-viscosity phenomenology or through structural arguments of scale-similarity. However ILES manipulates the computation of the non-linear Jacobian term to add numerical dissipation to mimic that of the unresolved frequencies. The latter framework, while numerically robust, suffers from difficulties associated with \\emph{directed} dissipation where it is often very easy to be over-dissipative in regions where sub-grid dissipation may not be as pronounced. In this article, we introduce a hybrid ILES framework that focuses upwinding at areas where high probability of sub-grid dissipation necessity is detected.\n\n\n\\section{Non-linear Jacobian computation}\n\nThe study utilizes two types of non-linear term computation schemes. Our first choice is symmetric, second-order accurate and conserves energy and enstrophy to minimize numerical dissipation. This is given by the well-known second-order Arakawa scheme \\cite{arakawa1981potential} as detailed below. The non-linear term in Equation \\ref{eq4} may be numerically calculated on a coarse grid using \n\\begin{align}\nJ^A (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{3} \\left( J_1 (\\bar{\\omega}, \\bar{\\psi}) + J_2 (\\bar{\\omega}, \\bar{\\psi}) + J_3 (\\bar{\\omega}, \\bar{\\psi}) \\right)\n\\end{align}\nwhere $J^A(\\bar{\\omega},\\bar{\\psi})$ will henceforth refer to the Arakawa discretization. The individual terms on the right hand side of the above equation are given as \n\\begin{align}\n\\begin{split}\n J_1 (\\bar{\\omega},\\bar{\\psi}) & = \\frac{1}{4 \\Delta x \\Delta y} \\left[ (\\bar{\\omega}_{i+1,j}-\\bar{\\omega}_{i-1,j}) (\\bar{\\psi}_{i,j+1} - \\bar{\\psi}_{i,j-1}) \\right. \\\\ \n& \\left.  - (\\bar{\\omega}_{i,j+1}-\\bar{\\omega}_{i,j-1}) (\\bar{\\psi}_{i+1,j} - \\bar{\\psi}_{i-1,j}) \\right],\n\\end{split}\n\\end{align}\n\n\\begin{align}\n\\begin{split}\n & J_2 (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{4 \\Delta x \\Delta y} \\left[ \\bar{\\omega}_{i+1,j} (\\bar{\\psi}_{i+1,j+1}-\\bar{\\psi}_{i+1,j-1}) \\right. \\\\ \n & \\left. - \\bar{\\omega}_{i-1,j} (\\bar{\\psi}_{i-1,j+1}-\\bar{\\psi}_{i-1,j-1})  - \\bar{\\omega}_{i,j+1} (\\bar{\\psi}_{i+1,j+1}-\\bar{\\psi}_{i-1,j+1})\\right. \\\\\n & \\left. + \\bar{\\omega}_{i,j-1} (\\bar{\\psi}_{i+1,j-1}-\\bar{\\psi}_{i-1,j-1}) \\right],\n\\end{split}\n\\end{align}\n\n\\begin{align}\n\\begin{split}\n& J_3 (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{4 \\Delta x \\Delta y} \\left[ \\bar{\\omega}_{i+1,j+1} (\\bar{\\psi}_{i,j+1} - \\bar{\\psi}_{i+1,j}) \\right. \\\\\n& \\left. - \\bar{\\omega}_{i-1,j-1} (\\bar{\\psi}_{i-1,j}-\\bar{\\psi}_{i,j-1}) - \\bar{\\omega}_{i-1,j+1} (\\bar{\\psi}_{i,j+1}-\\bar{\\psi}_{i-1,j}) \\right. \\\\\n& \\left. + \\bar{\\omega}_{i+1,j-1} (\\bar{\\psi}_{i+1,j}-\\bar{\\psi}_{i,j-1}) \\right].\n\\end{split}\n\\end{align}\nThe aforementioned scheme is utilized when our proposed classifier recognizes that no dissipation is necessary. \n\nA numerically dissipative computation of the non-linear term allows for that stabilization of noise accumulation at the grid cut-off wavenumbers. Although there are many different methodologies for upwind based dissipation with varying degrees of complexity, in this article, we utilize a conventional upwind-biased scheme as detailed in the following \\cite{hoffmann2000computational}. Our ILES Jacobian is computed as \n\\begin{align}\n\\begin{split}\nJ^I(\\bar{\\omega},\\bar{\\psi}) =& \\bar{u}_{i,j} \\frac{\\bar{\\omega}_{i+1,j} - \\bar{\\omega}_{i-1,j}}{2 \\Delta x} + \\frac{1}{2} (\\bar{u}^{+} \\bar{\\omega}_x^{-} + u^{-} \\bar{\\omega}_x^{+}) \\\\\n& + \\bar{v}_{i,j} \\frac{\\bar{\\omega}_{i,j+1} - \\bar{\\omega}_{i,j-1}}{2 \\Delta y} + \\frac{1}{2} (\\bar{v}^{+} \\bar{\\omega}_y^{-} + \\bar{v}^{-} \\bar{\\omega}_y^{+}),\n\\end{split}\n\\end{align}\nwhere \n\\begin{alignat}{2}\n\\bar{u}^{-} &= \\min(\\bar{u}_{i,j},0), \\quad \\bar{u}^{+} &= \\max(\\bar{u}_{i,j},0), \\\\\n\\bar{v}^{-} &= \\min(\\bar{v}_{i,j},0), \\quad \\bar{v}^{+} &= \\max(\\bar{v}_{i,j},0).\n\\end{alignat}\nIn addition,\n\\begin{align}\n\\begin{split}\n\\bar{\\omega}_x^{-} &= \\frac{\\bar{\\omega}_{i-2,j} - 3 \\bar{\\omega}_{i-1,j} + 3 \\bar{\\omega}_{i,j} - \\bar{\\omega}_{i+1,j} }{3 \\Delta x}, \\\\\n\\bar{\\omega}_x^{+} &= \\frac{\\bar{\\omega}_{i-1,j} - 3 \\bar{\\omega}_{i,j} + 3 \\bar{\\omega}_{i+1,j} - \\bar{\\omega}_{i+2,j} }{3 \\Delta x}, \\\\\n\\bar{\\omega}_y^{-} &= \\frac{\\bar{\\omega}_{i,j-2} - 3 \\bar{\\omega}_{i,j-1} + 3 \\bar{\\omega}_{i,j} - \\bar{\\omega}_{i,j+1} }{3 \\Delta y}, \\\\\n\\bar{\\omega}_y^{+} &= \\frac{\\bar{\\omega}_{i,j-1} - 3 \\bar{\\omega}_{i,j} + 3 \\bar{\\omega}_{i,j+1} - \\bar{\\omega}_{i,j+2} }{3 \\Delta y}.\n\\end{split}\n\\end{align}\nNote that velocity components are recovered using\n\\begin{align}\n\\begin{split}\n\\bar{u}_{i,j} &= \\frac{\\bar{\\psi}_{i,j+1}-\\bar{\\psi}_{i,j-1}}{2 \\Delta y} \\\\\n\\bar{v}_{i,j} &= -\\frac{\\bar{\\psi}_{i+1,j}-\\bar{\\psi}_{i-1,j}}{2 \\Delta x},\n\\end{split}\n\\end{align}\nwhere the second-order accurate reconstruction of the velocity leads to overall second-order accuracy for non-linear Jacobian reconstruction using the upwinded procedure outlined above. We also note that our Poisson equation given by Equation \\ref{eq2} is solved using a spectrally-accurate scheme. \n\nWith the choice of one of the two aforementioned schemes, a point in space-time may or may not have an artificial dissipation imparted to it numerically. However, we mention the caveat that switching between these two schemes would mean that the kinetic energy and enstrophy preserving property of the Arakawa scheme is lost. \n\n\\section{Machine learning for scheme selection}\n\nWe now discuss the procedure of utilizing DNS data for learning to classify one of the two dissipation scenarios. Of these two options, one is given by the choice of the Arakawa scheme and the other by our upwinded computation of the Jacobian (i.e., when the classification framework has determined that the point does not require sub-grid dissipation or vice-versa respectively). This switching between scenarios is spatio-temporally dynamic. We proceed by outlining our training strategy through the utilization of DNS data. Five equidistant snapshots of DNS data at $Re=32000$ (i.e., at $t=0,1,2,3,4$) and at $N^2 = 2048^2$ degrees of freedom (from 40000 available snapshots) are utilized to compute the grid-filtered variables (denoted FDNS) (at $N^2 = 256^2$ degrees of freedom) through the application of a spectral cut-off filter. Perfect closure values \n\\begin{align}\n\\Pi = J(\\bar{\\omega},\\bar{\\psi})-\\overline{J(\\omega,\\psi)}\n\\end{align}\nare then obtained (the reader is directed to \\citep{maulik2019subgrid} for details related to the calculation of these quantities). Note here, that the Kraichnan turbulence problem is transient with the evolution of vorticity represented in Figure \\ref{Fig1} representing different closure needs over time evolution. \n\nWe proceed by introducing the \\emph{a priori} eddy-viscosity given by\n\\begin{align}\n\\nu_e^a = \\frac{\\Pi}{\\nabla^2 \\bar{\\omega}}\n\\end{align}\nwhere the right-hand side of the above equation may be calculated from DNS snapshots. The \\emph{a priori} eddy-viscosity is centered at zero (corresponding to where closure modeling is unnecessary) and spreads out in the negative and positive directions (a hallmark of isotropic turbulence). We segregate this \\emph{a priori} estimate of sub-grid effects into three categories as follows. The \\emph{a priori} eddy-viscosities calculated from the DNS data are compared with a Gaussian distribution where values lying less than a distance of 1\\% of the standard-deviation from the mean (which is zero) are labeled as those requiring no dissipation (due to the low strength of the \\emph{a priori} eddy-viscosity). For posterity, we label these points as $k=1$. Positive values lying beyond this range are labeled as those requiring sub-grid dissipation and are labeled $k=2$. Negative values less than 1\\% of the standard-deviation are also considered to require no dissipation and are labeled $k=3$. This three-category segregation stems from a learning hypothesis that seeks to identify regions in a flow evolution that require structural, functional or no-closure modeling hypothesis. We link labels of negative or nearly-zero eddy-viscosities to the Arakawa classification and positive eddy-viscosities to the upwinded classification. The positive eddy-viscosity prediction would indicate that the sub-grid term at a point is predominantly dissipative in nature at which point the numerical dissipation of the upwinded scheme would be utilized. We note here that the concept of an \\emph{a priori} eddy-viscosity lies firmly within the explicit LES hypothesis. The classifier is therefore instrumental in moderating ILES deployments through a decision making process that recognizes the dissipative (or forcing) nature of the sub-grid quantities. \n\nWe note that the choice of 1\\% as the decision parameter for switching between hypothesis is motivated by a sensitivity study that showed the highest classification accuracy for the ANN framework. Larger choices of this hyper-parameter would result in a classifier that would be prone to classify most points in the `no-model' zone. However, we clarify that the choice of this value is also correlated with the architecture of the ANN. A potential extension of the proposed hypothesis is to combine architecture search algorithms with varying value of the decision hyper-parameters for larger classification accuracies. In addition, the three-category framework is derived from an aligned study \\cite{maulik2018online} where sub-grid models are determined according to negative, positive and nearly-zero \\emph{a priori} eddy-viscosities and utilizes the same learning. This enables use to determine a unified framework for switching between turbulence model hypotheses as well as numerical dissipation scenarios. However, we would like to emphasize that, for the purpose of switching between the Arakawa and upwinded Jacobian computation, a simple two-class framework would also suffice. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{Figure_1-eps-converted-to.pdf}\n\\caption{Time evolution of the Kraichnan turbulence case with DNS ($N^2 = 2048^2$) contours for vorticity of $t=1$ (top-left), $t=2$ (top-right), $t=3$ (bottom-left), $t=4$ (bottom-right). One can discern the dissipation of vorticity as the system evolves.}\n\\label{Fig1}\n\\end{figure*}\n\nA one-hot labeling of our eddy-viscosity classes is utilized for a classification deployment and a schematic for this hypothesis segregation and labeling is shown in Figure \\ref{Segregation}. The labels indicate the conditional probability of a point belonging to each possible class. As such, the training labels are given by a value of 1 for the particular class that a point belongs to and zeros for other choices. This is because there is no ambiguity in the class a training sample belongs to. Each label for the \\emph{a priori} eddy-viscosity is also associated with a corresponding input kernel of grid-resolved quantities. This kernel is given by a local stencil of vorticity and stream function. There are 9 inputs each for vorticity and stream function given by a query of the field quantity at a point on the coarse grid, 4 adjacent points in each dimension ($x,y$) and the 4 diagonally adjacent points. Each sample of our training data thus consists of 18 inputs of vorticity and stream function and outputs given by one-hot labels for the choice of closure modeling strategy. We then utilize an ANN to establish a relationship between these inputs and outputs. Mathematically, if our input vector $\\mathcal{P}$ resides in a $P$-dimensional space and our desired output $\\mathcal{Q}$ resides in a $Q$-dimensional space, this framework establishes a map $\\mathbb{M}$ as follows:\n\\begin{align}\n\\label{eq6}\n\\mathbb{M} : \\{ \\mathcal{P}_1, \\mathcal{P}_2, \\hdots, \\mathcal{P}_P\\} \\in \\mathbb{R}^P \\rightarrow \\{ \\mathcal{Q}_1, \\mathcal{Q}_2, \\hdots, \\mathcal{Q}_Q\\} \\in \\mathbb{R}^Q.\n\\end{align}\nAccordingly, the framework utilized in this article leads to the following relation:\n\\begin{align}\n\\label{eq7}\n\\mathbb{M} : \\{ \\textbf{p} \\} \\in \\mathbb{R}^{18}  \\rightarrow \\{ P(\\textbf{q}|\\textbf{p})\\} \\in \\mathbb{R}^3,\n\\end{align}\nwhere \n\\begin{align}\n\\begin{gathered}\n\\textbf{p}_{i,j} = \\{ \\bar{\\omega}_{i,j}, \\bar{\\omega}_{i,j+1}, \\bar{\\omega}_{i,j-1}, \\hdots, \\bar{\\omega}_{i-1,j-1}, \\\\ \\bar{\\psi}_{i,j}, \\bar{\\psi}_{i,j+1}, \\bar{\\psi}_{i,j-1}, \\hdots, \\bar{\\psi}_{i-1,j-1} \\}\n\\end{gathered}\n\\end{align}\nis our input vector for each query of the machine learning framework and where \n\\begin{align}\nP(\\textbf{q}|\\textbf{p})_{i,j} = \\{ P(J^k(\\bar{\\omega},\\bar{\\psi})_{i,j}| \\textbf{p}_{i,j})\\},\n\\end{align}\nis the conditional probability of a Jacobian computation (given by a connection to the explicit closure hypothesis). Note that $i,j$ refer to the spatial indices on the coarse-grid (i.e., the point of deployment). The indices $k=1$ and $k=3$ refer to the Arakawa non-linear Jacobian computation and $k=2$ refers to the upwinded computation instead (see Figure \\ref{Segregation}). Our optimal map $\\mathbb{M}$ is then trained by minimizing the categorical cross-entropy loss-function\n\\begin{align}\nE(\\textbf{w}) = -\\sum_{n=1}^{N} \\sum_{k=1}^{K} \\{ t_{nk} \\log(y_{nk}) + (1-t_{nk})\\log(1-y_{nk})\\},\n\\end{align}\nwhere $\\textbf{w}$ are the variable weight and bias parameters of the network, $N$ refers to the total number of samples and $K=3$ is the total number of classification scenarios (i.e., negative, positive or nearly-zero \\emph{a priori} eddy-viscosities). Here, $t_{nk}$ refers to the true label of class $k$ and sample $n$ and $y_{nk}$ refers to a corresponding prediction of the learning framework. One-hot encoding ensures that $t_{nk}$ values are always binary \\cite{Bishop:2006:PRM:1162264} and the outputs of the ANN may be interpreted as conditional-probabilities. Our optimal architecture is given by five 40-neuron hidden layers (obtained via grid-search hyper-parameter tuning). All hidden layers utilize ReLU units to impart non-linearity to the layer-wise transformations. For reference, our architecture is trained using the open-source deep learning software Tensorflow and is optimized with the use of ADAM, a popular gradient-descent based optimizer \\cite{kingma2014adam}. Figure \\ref{Loss_history} shows the progress to convergence for our framework with our optimally trained network displaying approximately 79\\% accuracy in classifying points to their correct labels. To summarize this section, we train a deep ANN to estimate probabilities of negative, positive or nearly-zero eddy-viscosities which are utilized to decide the choice of the Jacobian computation. We clarify that the decision to deploy a particular hypothesis is obtained by utilization of the classification scenario which has the highest conditional probability.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={3cm 16cm 6cm 1cm},clip,width=\\columnwidth]{Schematic-eps-converted-to.pdf}\n\\caption{Hypothesis segregation and one-hot labeling for our proposed framework. The learning predicts conditional probabilities for the three segregated \\emph{a priori} eddy-viscosity classes which are utilized for Jacobian calculation decisions spatio-temporally.}\n\\label{Segregation}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Loss_history-eps-converted-to.pdf}\n\\caption{Learning rate and convergence of our classification framework training. 2000 epochs were sufficient for converged validation loss.}\n\\label{Loss_history}\n\\end{figure}\n\n\\section{Results}\n\n\\subsection{\\emph{A posteriori} deployment}\n\nIn this section, we detail the results from an \\emph{a posteriori} deployment of the classification framework (denoted ML henceforth) for the Kraichnan test-case. In the LES evolution of the problem, a considerably coarser grid is used (at $N^2=256^2$). We remark that the forward deployment of our framework needs to overcome the challenge of numerical errors and is a robust test of the generalizability and robustness of our learning. Our LES results are assessed using angle-averaged kinetic energy spectra and through structure functions of vorticity. In addition, qualitative comparisons are also provided through visual examinations of the vorticity contours. We remark that the LES deployment is performed from $t=0$ to $t=4$ which spans the training regime data obtained from DNS. In what follows we note that DNS refers to a high-fidelity evolution of the governing equations (i.e., at $N^2=2048^2$ degrees of freedom), UNS refers to results obtained using the Arakawa scheme alone and ILES refers to a simulation where the non-linear Jacobian at all points in space and time are upwinded. Figure \\ref{Fig2} shows the \\emph{a posteriori} performance of the proposed framework at $Re=32000$ in terms of energy spectra predictions. The reader may find an exact definition of the kinetic-energy spectra in Maulik and San \\cite{maulik2017stable}. We note that the training data was obtained for the same Reynolds number as well. The prediction of the proposed framework is seen to agree remarkably well with DNS. It is apparent that the switching of schemes using the classifier has obtained an optimal balance between both techniques.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_2-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=32000$ and at $t=4$ in terms of angle-averaged kinetic energy spectra. Comparisons with DNS, the Arakawa scheme (UNS) and the upwinded scheme (ILES) show that ML provides directed dissipation adequately.}\n\\label{Fig2}\n\\end{figure}\n\nVorticity contours for LES resolution assessments are shown in Figure \\ref{Fig3}, where it is apparent that the proposed framework optimally balances the energy-conserving and dissipative natures of the Arakawa and upwinded schemes respectively. This is verified by qualitative examination with FDNS contours obtained by spectrally filtering the DNS snapshot for $Re=32000$ at $t=4$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{Figure_3-eps-converted-to.pdf}\n\\caption{Contours for the vorticity at LES resolution and at $t=4$. In the top-left, we have predictions from the ML approach. The top-right field has been obtained using ILES, the bottom-left field is obtained from UNS and the bottom right shows FDNS contours obtained by spectral cut-off filtering of DNS. }\n\\label{Fig3}\n\\end{figure*}\n\nA second statistically significant quantity of interest studied in this investigation is the vorticity structure function \\cite{grossmann1992structure} given by\n\\begin{align}\nS_\\omega^x = \\langle |\\omega(x+r,y) - \\omega(x,y)|^2 \\rangle \\\\\nS_\\omega^y = \\langle |\\omega(x,y+r) - \\omega(x,y)|^2 \\rangle,\n\\end{align}\nwhere the angle-brackets indicate ensemble averaging and $x,y$ indicate a position on the grid with $r$ being a certain distance from this location. Figures \\ref{Fig4} and \\ref{Fig5} show the structure functions obtained from \\emph{a posteriori} deployments of the UNS, ILES and ML frameworks compared against those obtained from the final time FDNS snapshot. It is clear that the proposed framework balances between UNS and ILES deployments well to recover appropriate trends. We can thus claim that our learning is appropriate for hybrid deployments of dissipative and conservative frameworks for two-dimensional turbulence. Before moving on, we would like to point out to the reader here that the proposed methodology for closure does not require any post-processing prior to deployment in the forward simulation as utilized in several data-driven turbulence modeling studies\\cite{beck2018neural,maulik2019subgrid}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_4-eps-converted-to.pdf}\n\\caption{\\emph{A posteriori} vorticity structure functions in $x$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.}\n\\label{Fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_5-eps-converted-to.pdf}\n\\caption{\\emph{A posteriori} vorticity structure functions in $y$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.}\n\\label{Fig5}\n\\end{figure}\n\n\n\\subsection{Validation of learning}\n\nIn this section, we proceed with a rigorous validation of our learning for deployment in regimes that are not a part of the training data. This is to ensure that the framework has truly learnt a classification based on the underlying physical hypothesis used for data segregation and is not memorizing data. This ensures that our classifier can be used in a more generalizable fashion. Figure \\ref{Fig6} shows kinetic energy spectra obtained from the forward deployment of the ML framework for a $Re=64000$ which represents a classification task that the framework has not previously seen (although the physics of the test-case remains similar). As observed, the proposed method performs quite well in this out-of-training data range as well. We note that a similar resolution ($N^2=256^2$) is utilized for this deployment. In contrast, Figure \\ref{Fig7} shows the performance of the ML technique for a reduced resolution of $N^2=128^2$ but utilizing the same Reynolds number of 32000. The kinetic energy spectra show a successful stabilization of the flow evolution at this reduced resolution although some forcing to the large scales is observed. This suggests that the classification framework may be improved by sampling from different resolutions. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_6-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=64000$ and at $t=4$ in terms of energy spectra. This represents deployment of our learning at a different Reynolds number than that used for generating training data.}\n\\label{Fig6}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_7-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=32000$, $t=4$ and at $N^2 = 128^2$ in terms of energy spectra. This represents deployment of our learning at a different resolution than that used for generating training data.}\n\\label{Fig7}\n\\end{figure}\n\n\n\\section{Concluding remarks}\n\nIn this article, we have proposed a neural network based classifier that enables us to take decisions on the choice of non-linear term computation in the LES evolution of the Kraichnan turbulence test-case. The classifier outputs conditional probabilities for the presence (or absence) of eddy-viscosities within three different ranges during deployment and is used to switch between the Arakawa and upwind computation of the non-linear Jacobian for a hybrid upwinded deployment that optimally directs dissipation on the coarse-grained flow field. Our machine learning framework is trained by calculating \\emph{a priori} eddy-viscosities which are projected onto a Gaussian distribution and segregated into three categories. Each category is devised to capture a unique behavior of the underlying sub-grid terms with negative and nearly-zero eddy-viscosity classes signifying absence of sub-grid dissipation. An optimally trained classifier is then utilized to identify if a point requires sub-grid dissipation based on if it is placed in the positive eddy-viscosity category. If so, the upwind Jacobian is calculated for imparting numerical dissipation. \n\nWe perform \\emph{a posteriori} assessments on the Kraichnan turbulence test-case through statistical quantities such as the angle-averaged kinetic energy spectra and the vorticity structure functions. It is observed that the proposed framework is successful in balancing the dissipative nature of the upwind scheme and the energy-conserving Arakawa scheme to give excellent agreement with DNS statistics. Validation for out-of-training regimes also indicate that the framework is able to learn the link between grid-resolved quantities at a coarse resolution and the nature of the sub-grid forcing. \n\nOur conclusions therefore point toward the possibility of using classifiers for the unified deployment of numerical schemes with varying dissipation through the decision making process described above. A key strength of our hypothesis stems from the fact that an ILES deployment is moderated by concepts drawn from the explicit LES ideology (i.e., that of an \\emph{a priori} eddy-viscosity). The successful deployment of our method thus points towards the possibility of deploying directed numerical dissipation that preserves the statistics of turbulence without sacrificing the shock-capturing ability of many non-oscillatory schemes. Our future work lies in that particular direction. \n\n\n\\begin{acknowledgements}\nThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290. OS gratefully acknowledges their support. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nWe study the jet kinematics and jet ridge line evolution of the Caltech-Jodrell Bank flat-spectrum (CJF) sample. The sample consists of 293 radio-loud active galaxies \\citep{Taylor1996}, spanning a large redshift range ($z_{avg}=1.254$; $z_{max}=3.889$). Each source has at least 3 epochs of radio interferometric (VLBA) observations and has been imaged and studied kinematically \\citep{Britzen2007b,Britzen2008}.\n\n\\section{Radio Jets in the CJF}\n\\subsection{Apparent Speeds and $\\mathbf{\\gamma}$-ray Properties}\n We correlate the kinematic information \\citep{Britzen2008} with the $\\gamma$-ray properties of the sample. In total 25 sources have been detected in $\\gamma$-rays. 18 CJF sources are included in the 3 month bright AGN source list of the Fermi-LAT \\citep{Abdo2009}, with 7 additional sources detected by EGRET \\citep{Hartman1999}. The maximum of the apparent speed distribution for the $\\gamma$-ray non-detected sources is $3\\le\\beta_{tot,max}\\le5$. For the detected sources the distribution shows two maxima ($\\beta_{tot,max;1}=0$ and $\\beta_{tot,max;2}=15$). The detected sources dominate the high speed tail of the total distribution (Fig. \\ref{fig:gamma_speeds}).\n\\begin{figure*}[!ht]\n\\begin{center}\n  \\includegraphics[width=0.7\\textwidth]{Karouzos_M_fig1.eps}\n   \\caption{Distribution of the maximum total apparent speed $\\beta_{tot,max}$ for $\\gamma$-ray detected (shaded) and non-detected (non-shaded) sources (data from \\citealt{Britzen2008}, \\citealt{Hartman1999}, and \\citealt{Abdo2009}).}\n  \\label{fig:gamma_speeds}\n\\end{center}\n\\end{figure*}\n\\subsection{AGN Jet Ridge Lines}\nWe define the ridge line of a jet as the line that linearly connects all component positions at a given epoch. We build the jet ridge lines for all CJF sources (for S5 1803+784, see Britzen et al. 2009, A\\&A, in press) and study their properties and evolution across all available epochs.\n\n\\subsubsection{Ejection Angle Evolution}\nWe define the angle between the innermost jet component and the core, $theta$, as the approximate ejection angle of the jet (the core being at the beginning of the coordinates system). We study the change of this angle across all available epochs. Most sources exhibit a maximum ejection angle change $<5^{\\circ}$. 5.1\\% of the sources show ejection angle changes $>15^{\\circ}$, with two sources at $\\Delta\\theta_{max}\\approx45^{\\circ}$.\n\\subsubsection{Jet Ridge Lines and $\\mathbf{\\gamma}$-ray properties}\nWe quantify the jet ridge line evolution as the total component position change across all epochs. $\\gamma$-ray variable sources show stronger ridge line evolution than non-variable ones. Sources with stronger ridge line evolution tend to be more luminous in the $\\gamma$-rays.\n\n\\section{Conclusions}\nWe find that $\\gamma$-ray detected sources show higher apparent speeds than non-detected ones. $\\gamma$-variable sources show stronger jet ridge line evolution. $5.1\\%$ show considerable jet ejection angle changes ($>15^{\\circ}$). Such jet ejection angle changes might be connected to precessing or helical jets.\n\\acknowledgements\nM. Karouzos was supported for this research through a stipend from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics. M.K. wants to thank C.S. Chang for proofreading this poster and for insightful comments on various points of this work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\nNowadays, Computer Science has become predominant in many fields of science.\nData storage, data analysis and visualization are only a few examples where it plays a fundamental role.\nThe amount of available data is increasing every day which encourages the improvement of computers and the exploration of more complex problems.\nTherefore, there is an enormous flow of information in terms of quantity and dimensionality.\nSound, images and videos are common example of large multidimensional data.\nMany areas of science need methods to reliably extract specific information from these growing amount of data.\nThis is usually done through analysis, visualization or a combination of both.\n\nComputer Vision (CV) is a field addressing the problem of analyzing visual multimedias by automatic processing.\nImage classification and pattern recognition are two examples of problems studied in Computer Vision.\nMany state-of-the-art solutions for these problems are combining the advance of Machine Learning to find data-driven models, instead of hand-designing algorithms, which usually requires expert knowledge of the domain.\nAmong these methods, {\\em Convolutional Neural Networks} (CNN), are a variant of {\\em Neural Network}s (NN) used in vision tasks and are now widely employed for Computer Vision with great success\\cite{krizhevsky2012imagenet}\\cite{rowley1998neural}\\cite{prechelt1994proben1}.\nThey are trained in a supervised way to perform a systematic processing task, for instance to classify images into a number of fixed categories (e.g.: handwritten digits between 0 and 9).\nTheir {\\em features} are the internal representation of the input inside NN that is being optimized during training to give better results.\nIn common classification tasks, the features are meant to lie in a low dimensional space and should provide a more compact representation of the input.\nTherefore, these NN can be seen as two embedded parts: a dimensionality reduction part that extracts important informations and a classifier that takes a decision based on these features.\nIn practical applications, image distortions of various kind are present in the data: shearing, noise, camera viewpoint, spatial positioning and lighting condition.\nIn the case of classification, distortions can require putting extra efforts to train a robust model able to classify correctly the input in the presence of such deformations.\nMost implementations decide to nullify the distortion signal with data-augmentation which assigns the original label on distorted samples.\nThis can be done when the application does not require this information.\nHowever, there are practical cases, for instance in Biomedical Imaging and Face Detection, which may require to have a feature representation able to quantify and possibly predict the distortions.\n\nThe first step of this work is to better understand NN embeddings and explore if conventional tools can be used to extract distortion features reliably.\nMoreover, visual inspection of the feature space using a human-friendly representation can help to gain insights and to improve models.\nOne way to analyze the features is to apply dimensionality reduction techniques to project highly-dimensional points into lower dimensional space.\nMany differences characterize dimensionality reduction methods such as: properties, flexibility and final goals.\nA brief overview is describing: PCA, MDS, LLE, Isomap and the more recent SNE.\nHowever many of them lack flexibility by representing only linearity, while the others don't preserve hierarchical structures accurately\\cite{t-SNE}.\nResearches have shown that t-SNE, a variant of SNE, has greatly improved on this regard: it is capable of keeping global clusters and fine-grained coherence at small scales.\nMoreover it was used multiple times successfully to visualize the feature space of NN\\cite{donahue2013decaf}\\cite{yu2014visualizing}.\n\nNonetheless, several limitations of t-SNE are encountered in this work.\nIt cannot be controlled directly because t-SNE is only given unlabelled points like unsupervised methods.\nTherefore, as shown later, the resulting embedding can have an unexpected shape unfit for the target application.\nSecondly, this method works directly on the representations of the points without creating a mapping that computes the relation between inputs and outputs.\nThus it is impossible to compute the representation of points that were not part of the optimization process at the beginning.\nIt would be necessary to recompute the whole representation from scratch with these new points but t-SNE is expensive.\nIn this work, a first try is made by combining CNN and t-SNE to produce an embedding quantifying distortions.\nHowever, the results are not predictable and difficult to interpret.\nDistortions are the primary factor of clustering, while classes are not well separated.\nMoreover, problems were faced to scale this method on larger datasets.\nt-SNE could be probably improved to leverage external prior-informations such as labels, but because of the major shortcomings faced, we opted for another solution.\n\nAn alternative dimensionality reduction method consists of using the NN to directly compute an embedding similar to t-SNE but trained in a supervised manner.\nThe network is used to learn a mapping from the image space to a lower embedding which does not have the previous shortcomings.\nAs NN have the capacity to represent any non-linear continuous functions\\cite{csaji2001approximation}, it is suitable to compute a very powerful dimensionality reduction method.\n\nTo achieve this goal, {\\em Dimensionality Reduction by Learning an Invariant Mapping} ({\\em DrLIM}) combines concepts similar to the ones employed in t-SNE but expressed in an equivalent adapted formulation for NN\\cite{hadsell2006dimensionality}.\nThe key elements are: the contrastive loss function with the Siamese network architecture for training and a pairing strategy directly describing the embedding.\nThe current formulation allows to express a one-dimensional relation between two pair of images based on the similarity: similar pairs of images should be close together whereas dissimilar pairs should be far from each other.\nIn this work, an extension of DrLIM is proposed with a generalized loss function.\nIt allows to express richer similarity relations between image pairs for training.\nMoreover, each of these similarities is expressed in the embedding by dimensions selected in advance and is associated feature components.\n\nTherefore, thanks to this extension, it becomes possible to have a better control on the final embedding of a NN.\nThe experiments with DrLIM are reproduced on the data-augmentation of the MNIST dataset\\cite{lecun1998mnist} with translations or rotations and likewise on the NORB dataset\\cite{lecun2004learning}.\nThe resulting embeddings are studied in details with multiple comparisons to the original work on DrLIM.\nWe find it is possible to embed distortion information in one extra dimension while preserving the characteristics of DrLIM on MNIST.\nWe also find that this new solution can represent the DrLIM's coherent and cyclic space of the camera viewpoint on NORB.\nThe successful application on these two datasets demonstrates the effectiveness of this method.\n\n\\section{Thesis Outline}\nIn Chapter~\\ref{chap:related}, the previous papers related to this work are presented.\nAn overview of the prior art is describing dimensionality reduction methods and the advance of Computer Vision with NN.\nThen, the reference paper introducing DrLIM models is discussed.\n\nIn Chapter~\\ref{chap:neural_network}, a summary is given for the background knowledge necessary to understand the networks used in this work.\nThe general architecture of NN, their similarity with CNN and some important definitions are also discussed.\n\nThis is followed by Chapter~\\ref{chap:dim_red} where the standard dimensionality reduction methods are reviewed in more details.\nThe major advantages and drawbacks of t-SNE, compared to other methods are explained.\nLater in the chapter, the basic theoretical knowledge behind t-SNE optimisation problem is provided.\nThen the necessary tools are explained to achieve dimensionality reduction with NN and how to extend the contrastive loss to N-dimensional similarity.\n\nIn Chapter~\\ref{chap:results}, the experiment environment is established in terms of technical choices (software, network architecture and datasets).\nThis chapter also includes all the practical results, the hypothesises and also the early conclusions of our work.\n\nIn Chapter~\\ref{chap:conclusion}, some final remarks are made about the results and we encourage to continue from our contribution by providing some propositions of future works.\n\n\n\\chapter{Related Work}\n\\label{chap:related}\n\nMore than 20 years ago, researchers discovered that NN is a highly flexible model architecture to solve many classification problems.\nMost of today's NN are using convolutions in CNN which are a powerful and specialized variant for visual tasks like MNIST \\cite{mnist_web} and ImageNet \\cite{krizhevsky2012imagenet}, face detection as well \\cite{rowley1998neural} and many more \\cite{prechelt1994proben1}.\nThis architecture is more efficient for Computer Vision problems, partly because the convolutions share their weights in the first layers to extract spatial cues.\n\nCurrent researches are mainly focused on advancing into more complex classification problems using Deep Learning.\nThis field suggests improving the current results by deepening the architecture in terms of number of layers to express more abstract and higher-level concepts.\nUnfortunately, more layers require more computations than before due to the increase of parameters but researchers started to overcome this challenge.\nWays to speed up the training and the classification appeared and were greatly beneficial for the development of this field in recent years\\cite{ciresan2011flexible}\\cite{schmidhuber2015deep}\\cite{nasse2009face}.\nThe use of GPUs parallelism and cloud computing allowed scaling up to much deeper architecture (with many more parameters) than before\\cite{coates2013deep}.\nBoth researchers and professional programmers built various frameworks to train and to use NN with many different goals such as: performance, accessibility or composability.\nThe most populars are: Theano\\cite{bastien2012theano}, Caffe\\cite{jia2014caffe}, Torch7\\cite{collobert2011torch7}, Pylearn2\\cite{goodfellow2013pylearn2}.\nThe efficiency and modularity proposed by Caffe were leveraged in this work for several reasons described later.\n\nEven though CNN models are now widely used in Computer Vision with great success thanks to these frameworks, current research exposed the ignorance of these networks behaviors.\nSome papers work on reliable ways to fool networks using adverserial attacks which exploits their unintuitive properties\\cite{szegedy2013intriguing}.\nMost publications do not try to formally justify their good results because of the non-linear and complex relations between units.\nMoreover, this problem is greater with the addition of distortions in the inputs.\nIt is not possible to directly look at the high-dimensional embedding, because humans cannot easily plot nor interpret so many dimensions.\nThe manifestation of this relation can be looked visually using methods to produce an alternative human-friendly representation.\n\nDimensionality reduction methods is a popular tool in such a case\\cite{dai2014document}\\cite{taylor2011learning} and several methods exist that are specialized for visualization.\nStandard methods, like PCA and MDS\\cite{cox2000multidimensional}, are used for reducing dimensions usually as a preprocessing step and they are limited to linearity in the data.\nTheir primary characteristic is to maximize the variance of the original data which is important for reconstruction but not necessarily for visualization.\nOther non-linear dimensionality reduction methods, for instance: Isomap\\cite{tenenbaum2000global}, LLE\\cite{roweis2000nonlinear} and SNE\\cite{SNE}, solve an optimisation problem with a loss function which improve their flexibility for purposes like visualization and preserve local structure\/cluster.\nHowever, recent papers show that SNE has better potential to keep global clusters and local details at the same time unlike the other previously cited methods\\cite{SNE}.\n\nSNE has introduced a better maximization problem to preserve point neighborhood and general clusters with an important emphasis on distances\\cite{SNE}.\nHowever it suffers from ``center-crowdedness'' and faces difficulties in optimisation\\cite{t-SNE}.\nThis is the reason why recent works are now using t-SNE, a variant of SNE, to reduce CNN embeddings into a 2D human-friendly manifold.\nThis method exhibits an easier optimisation formulation and preserves more structures at diverse scales.\nMore convincing examples were created with t-SNE directly applied on popular datasets\\cite{van2009new} like MNIST\\cite{t-SNE}.\nA more recent work proposed an alternative method, called Barnes-Hut SNE, that approximates t-SNE with a much faster algorithm but it wasn't useful for this work\\cite{van2013barnes}.\n\nMany more details in CNN embeddings emerge by using t-SNE.\nFor instance, how samples are grouped depending on multiple factors: essentially based on similarity of the digits (strokes, thickness and shapes) and natural variance (rotation).\nMany papers use dimensionality reduction to create a human viewable representation of the output embedding \\cite{donahue2013decaf}\\cite{yu2014visualizing}\\cite{yaotiny}.\nOnly a few papers actually try to formalize the input-output relations\\cite{goodfellow2009measuring}.\nSome papers proposes to learn transformation-invariant embeddings, by means of modelling distortions directly into their models\\cite{gens2014deep} or using data-augmentation\\cite{hadsell2006dimensionality}.\n\nThe latter proposes to create a dimensionality reducing mapping, called {\\em Dimensionality Reduction by Learning an Invariant Mapping} ({\\em DrLIM}), by training a CNN using a new loss function inspired by Energy-Based modeling, called the {\\em contrastive loss}.\nThis paper will be mentioned in this work as the {\\em reference paper} or DrLIM's paper.\nTheir results on the MINST dataset\\cite{lecun1998mnist} shows that CNN can learn a mapping that distinguish labels, and group alike digits even when they are translated artificially.\nThey experimented with the NORB dataset\\cite{lecun2004learning} as well with intriguing results: images are mapped on a 3D cylinder whose axes quantify the orientation in 2D and the azimuth angle in 1D.\nThe network was successfully trained to ignore lighting conditions, which is a strongly non-linear distortion.\nThey use the Siamese training architecture to present image pairs which are optimized to be close if similar or far otherwise\\cite{bromley1993signature}\\cite{chopra2005learning}.\nThere are several advantages of such a method: the cost penalty is very low and present only during training.\nThe usage of standard CNN also allows more freedom for experimenting with different architectures and the proposed loss function is effective while simple.\n\nHowever the input-output relations of distorted images is usually damped instead of being quantified in a predictable way.\nThe distortion information should be kept because they can be valuable later on.\nBesides, the reference paper provides subjective comments of the embedding coherence (descriptions and figures) but they do not offer nor propose a formal way to measure and compare its quality objectively.\n\nIn this work, a few steps are presented towards predictable embedding with respect to distortions and a simple qualitative measure is presented to compare similar methods.\n\n\n\\chapter{Theory: Neural Networks}\n\\label{chap:neural_network}\nThe general layout of neural networks and their mathematical foundations is now introduced.\nThe important differences between {\\em Neural Networks} (NN) and {\\em Convolutional Neural Networks} (CNN) are discussed and a justification is given why CNN is so predominant in Computer Vision tasks.\nAn intuitive explanation of embeddings is given and will be used later for dimensionality reduction.\n\n\\section{NN and CNN Classifiers}\n\nLet us quickly recall the definition of a Neural Network and its workings for classification.\nBasically, NN are composed of $L$ layers which are made of inter-connected neurons (see figure~\\ref{fig:neural_network} for an example).\nA connection $l_{i \\rightarrow j}$ in the $k$th layer is weighted by a parameter $w^k_{i,j}$ which is learned.\nClassification is done by feed-forwarding the input onto the first layer whose output is propagated layers by layers through the network.\nThe output of one layer is forwarded onto the next through their neuronal connections.\nThis process is repeated until the last layer, whose output is considered as the network output.\nEach layer does a single particular computation over its input.\nUsual NN have a repetition of the following pattern: one layer computing inner-products with their neuronal weights then adding biases, followed by a non-linear layer using an activation function.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/NN.jpg}\n    \\end{center}\n    \\caption{An example of a Neural Network with three layers -- Source: neuralnetworksanddeeplearning.com}\n    \\label{fig:neural_network}\n\\end{figure}\n\nThe figure~\\ref{fig:artificial_neurons} presents the computation involving a particular neuron and its activation.\nLet us define the $j$th neuron in the $k$th layer for the inner-product case, then its output $z^k_j$ is defined:\n\\begin{eqnarray}\n    z^k_j = \\sum_{i=1}^{U^{k-1}} w^k_{i,j} z^{k-1}_i + b^k_j\n\\end{eqnarray}\nwhere $U^k$ is the number of neurons in layer $k$.\nThen we define the $j$th neuron in the $k$th layer for the activation case:\n\\begin{eqnarray}\n    z^k_j = g(z^{k-1}_j, w^k_{j})\n\\end{eqnarray}\nwhere $g$ is the activation function with its parameter.\nMost of them use the sigmoid activation function and connect each neuron to all the next layer neurons.\nTraining the weights generally involves gradient descent to minimize the loss function of such networks.\nThis loss represents the error between the prediction of the network (its output) versus the expected output (in classification: the class label).\nThe training involves an iterative process where: the input is first feed-forwarded into the network to compute the error, then this latter is back-propagated up to the first layer, while each unit weights is adjusted to minimize the unit error.\nThis process should be repeated until the error on the validation set has converged to a local optimum.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/800px-ArtificialNeuronModel_english.jpg}\n    \\end{center}\n    \\caption{A neuron computing an inner-product with a link to its activation neuron -- Source: Wikipedia}\n    \\label{fig:artificial_neurons}\n\\end{figure}\n\nOne particular architecture is the CNN, a special case of NN with certain restrictions.\nCNN models learn very effective solutions for many Computer Vision problems in image classification.\nThe three key ideas of CNN are described below: local receptive fields, weight sharing and subsampling.\nIn CNN, a convolutional layer can model receptive fields by computing multiple trainable 2D kernels which is convoluted with its entire input whose result is called its feature maps.\nA kernel convolution can be seen as the replication of a single unit along the dimensions of the input (a 2D grid for images), thus all weights are constrained to be equal by definition.\nA convolutional layer contains multiple units where each has its own kernel.\nTherefore different kernels are applied to the image where a single kernel convolution is called a feature map.\nSuch layer naturally computes filters which can be seen like data-driven feature extractors.\nThe benefit of weight sharing in convolutional layers is to reduce the number of global parameters to learn general purpose filters which can increase generalization.\nUsually, CNNs have subsampling layers right after convolutional layers to reduce the dimensionality of the feature maps so that more concise and high-level information are extracted.\nMost CNN architectures puts multiple convolutional layers connected to the input to extract visual features after which they have regular inner-product layers (see figure~\\ref{fig:convnet}).\nCNN can be seen as two parts: a trainable feature extractor made by the convolutional network followed by a neural network classifier.\nAlthough CNNs have more constraints, it has been shown they generally outperform NN in multiple Computer Vision problems because they learn more invariance\\cite{simard2003best}\\cite{mnist_web}\\cite{lawrence1997face}\\cite{krizhevsky2012imagenet}.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics[width=\\textwidth]{thesis_figures\/conv-net2.jpg}\n    \\end{center}\n    \\caption{Architecture of a CNN where the computation flow is explicitly shown -- Source: code.flickr.net}\n    \\label{fig:convnet}\n\\end{figure}\n\nMoreover, research found interesting structures in the last-layer embedding of classification NN using t-SNE.\nThe last layer tends to cluster samples of the same class together while separating the rest\\cite{donahue2013decaf}\\cite{yu2014visualizing}.\nThis is explained by the fact that deeper layers extract more high-level information, which is necessary to separate classes. Moreover the prediction is a direct product with the last layer, which encourages the network to have a simple structure directly clustered by classes.\n\n\n\\chapter{Theory: Dimensionality reduction}\n\\label{chap:dim_red}\nDimensionality reduction is an important method in machine learning that maps points in a high-dimensional space into a space with a lower number of dimensions.\nThis lower subspace is later called an {\\em embedding}.\nRepresentations suitable for human visualization should keep important relations between points of the original space.\nIn the experiments, we will apply dimensionality reduction on the neural networks' outputs.\nThis allows to quickly compare qualitatively the impact of controlled distortions applied on images over the networks' outputs.\nThe structure inferred by distances between points such as clusters, intra-cluster neighbors and outsiders, reveals important properties of neural network models.\nThus, accurate low-dimensional representations of embeddings that preserve local distances are desirable to compare different models.\nIn the following sections, dimensionality reduction methods are introduced where some are disregarded because they were not justified for this project, then the one used here, called t-SNE, is introduced in details.\n\n\\section{Optimization Problem}\nVarious algorithms differentiate themselves by several properties: their goal (e.g.: interpolation, compression, visualization), by what they preserve (e.g.: variance, distances), how they model point correlation (e.g.: linearly or not) or whether they can model new points (e.g.: a representation or a mapping).\n\nFirst, the primary factor for the selection of a method is guided by the requirements, which is here to visualize.\nThus it should produce a 2D or 3D embedding, preferably in 2D, for easily interpretable scatter plots.\nSecondly, we need an optimization problem that keeping certain relations between points such that clusters are equivalently represented.\nThis can be implemented by preserving relative distances to close neighborhood.\nConsidering the previous works discussed above, t-SNE is the best candidate in the current state-of-the-art.\nThe definition of the objective functions, the probabilities and the intuitive properties behind this method are now established.\n\nLet us define the dataset $D$ formed by points in the input-space $X$ of dimension $N$: $D = \\left\\{ x_1, x_2, \\dots, x_n \\right\\}$, each $x_i \\in \\mathbb{R}^N$, and one representation $D^\\ast$ in the output-space $Y$ of dimension $M$: $D^\\ast = \\left\\{ y_1, y_2, \\dots, y_n \\right\\}$, each $y_i \\in \\mathbb{R}^M$.\nThe process of dimensionality reduction is to find the best representation $D^\\ast$ that best preserves the most ``important'' information between $x_i$ and $y_i$ for each point.\nSNE uses two Gaussian distributions for each point expressing the neighbor distances in $X$ and its equivalent in $Y$.\nThe Kullback-Leibler divergence is used to compute the objective function, which represents the mismatch of these two distribution for each pair:\n\\begin{eqnarray}\n    L = \\sum_{i=1}^n KL(P_i || Q_i) = \\sum_{i=1}^n \\sum_{j=1}^n p_{j|i} \\log\\left(\\frac{p_{j|i}}{q_{j|i}}\\right)\n\\end{eqnarray}\nThe probability for a pair of point $x_i$ electing $x_j$ in $X$ follows a Gaussian is as follows:\n\\begin{eqnarray}\n    p_{j|i} = \\frac{\\exp(-|x_i - x_j|^2 \/ 2 \\sigma_i^2)}{\\sum_{k \\not = i} \\exp(-|x_i - x_k|^2 \/ 2 \\sigma_i^2 )}\n   \n   \n\\end{eqnarray}\nThe equivalent probability $q_{j|i}$ in $Y$ is the same except that $\\sigma = 0$ and $x$ is replaced by $y$.\nTherefore, a closer pair implies a higher neighbor-election probability because the distance is low.\nThis cost function gives an asymmetrical importance to the distances: nearby points in $X$ are greatly penalized if they are far in $Y$; whereas a small cost is incurred for pairs far in $X$ but close in $Y$.\n\nAs said SNE suffers from crowdedness problems in the middle of the embedding and the optimisation is harder due to the asymmetrical nature of the objective function.\nBoth problems were addressed in t-SNE which give very good results in practice.\nThe two major differences with t-SNE is the symmetrization of the cost function and replaces the distributions of the embedding by student variants.\nIn t-SNE, a single objective function is minimized:\n\\begin{eqnarray}\n    L = \\sum_{i=1}^n KL(P || Q) = \\sum_{i=1}^n \\sum_{j=1}^n p_{ij} \\log\\left(\\frac{p_{ij}}{q_{ij}}\\right)\n\\end{eqnarray}\nwhere:\n\\begin{eqnarray}\n    p_{ij} = \\frac{p_{j|i} + p_{i|j}}{2 n}\n\\end{eqnarray}\nwhich forces outliers points to contribute more to the loss.\nAnd:\n\\begin{eqnarray}\n    q_{ij} = \\frac{1 \/ (1 + |y_i - y_j|^2)}{\\sum_{k \\not = l} 1\/(1 + |y_k - y_l|^2)}\n\\end{eqnarray}\nwhich replaces the $q$ distribution in SNE by a Student with a heavier tail: distances in high dimensional spaces spread across more dimensions; in low dimensional space, the accurate equivalent distance needs to be much higher per dimension (thus more points end up further in $Y$, a heavier tail than in $X$).\nAs before, a point cannot elect itself: $p_{ii} = q_{ii} = 0$ and probabilities are symmetric for both distributions: $p_{ij} = p_{ji}$ and $q_{ij} = q_{ji}$.\nAs stated previously the input space $X$ has much more dimensions were distances can be expressed than the 2 dimensions of $Y$.\nIn summary, t-SNE uses Kullback-Leibler divergence to minimize the mismatch of these two spaces by means of probabilities, therefore the chance of important local structures (frequent patterns) being preserved is higher than with other methods (mosts do not express this goal through an objective function).\nGlobal structures is encouraged by the coherence of the local structures as the divergence decreases and the system stabilizes.\n\n\\section{CNN for Dimensionality Reduction}\nUsual dimensionality reductions like t-SNE are helpful for many visualizing tasks but it has important drawbacks as well.\nThe most important ones are: the computational cost, the incapacity of mapping new points and the indirect control over the resulting embedding.\nCurrent implementations of t-SNE are still rare, unpolished and require tricks to make them tractable in practice (e.g.: reducing first with PCA).\nAs there is currently no way with t-SNE to map new points (not in $D$), it is necessary to optimize the whole system from scratch.\nBesides, t-SNE parameters like perplexity and learning rate are not simple to chose.\nFortunately there are new promising alternatives directly harnessing the power of NN.\nSuch models are introduced in the following section.\n\nCNN are mostly used for classification but it can be optimized for other purposes as well.\nInstead of the SoftMax loss function used in classification, a special training architecture with a suitable loss can be used to optimize topological constraints.\nFor example to create an embedding with particular properties in the output layer.\nMoreover the ideas presented by reduction methods like t-SNE can be formulated in different terms to be applicable to NN.\nIn this work, the most important idea is to keep similar points together and dissimilar far away from each other.\n\nThe Siamese network combined with a contrastive loss is a good practical solution to train such networks\\cite{bromley1993signature}\\cite{chopra2005learning}.\nLet us define $G_W$, the function that computes the network output, with parameters $W$ (see figure~\\ref{fig:siamese_network}).\nThen the Siamese network put two weight-sharing instances of $G_W$ side by side, each having their own input.\nOn top of this Siamese network is placed the ``cost module'', the contrastive loss, which will compute a loss proportional to the difference between the two output.\nThe complete network takes a pair of images as input, each image fed into a single $G_W$ instance, and the output is computed over their outputs.\nThe idea is to optimize the metric between points represented by $G_W$ and to spread contributions between the weights because they are shared.\nAt the end, when the network is used, only a single instance of $G_W$ is required to feed an input and $G_W$ gives the dimensionality-reduced point.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics[width=0.5\\textwidth]{thesis_figures\/siamese_network.jpg}\n    \\end{center}\n    \\caption{Schematic representation of a Siamese network -- Source:~\\cite{bromley1993signature}}\n    \\label{fig:siamese_network}\n\\end{figure}\n\nThe contrastive loss function takes its root from Energy-Based Models (figure~\\ref{fig:contrastive_spring}).\nThe loss is constructed as follows: an attractive term is used to assemble similar points together.\nInsufficient alone because it does not prevent degenerate solutions where all points are merged.\nAn opposing term is added to push dissimilar pairs to allow the system to converge towards a structured embedding.\n\nMore formally, the definition of the contrastive loss for a pair is as follows:\n\\begin{eqnarray}\n    L = \\frac{1}{2} Y (D_W)^2 + \\frac{1}{2} (1-Y) \\max(0, m - D_W)^2\n\\end{eqnarray}\nwhere $Y \\in \\{0,1\\}$ is the label: $1$ for similar pairs, $0$ otherwise; $D_W \\in \\mathbb{R}^N$ is the difference between the two networks' outputs and the parameter $m \\in \\mathbb{R}$ defines the minimal distance between dissimilar points.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics[width=0.5\\textwidth]{thesis_figures\/contrastive_spring.jpg}\n    \\end{center}\n    \\caption{A schematic representation of the contrastive loss using physical springs. Consult \\cite{hadsell2006dimensionality} for more details. (a) shows the points connected to similar points with {\\em attract-only} springs. (b) the loss function and its gardient associated with similar pairs. (c) the point connected only with dissimilar points inside the circle of radius $m$ with {\\em m-repulse-only} springs. (d) shows the loss function and its gradient associated with dissimilar pairs. (e) shows the situation where a point is pulled by other points in different directions, creating equilibrium -- Source \\cite{hadsell2006dimensionality}}\n    \\label{fig:contrastive_spring}\n\\end{figure}\n\nIn the contrastive loss, a label means whether a pair should be close in the embedding but the definition of similarity is left open for the application.\n\n\\section{CNN for Predictable Reduction}\nThe method above can learn an embedding which groups points or separates them based on the pairing strategy but there is still room for improvement.\nThe shape of the embedding is determined by the dataset labels which helps to form local structures.\nHowever, there is no guarantee that this system will converge to an intended global structure with so many dimensions, such as: the ordering of the deformations (e.g.: from lowest to highest) or their distribution (e.g.: a line, plane or circle).\nSecondly, the training required to make the embedding converge to a desirable structure can happen or not depending on the training time or specific initialization seeds, and this is partially due to the lack of constraints in the task.\nWhen the number of embedding dimensions, $M$, is higher than the dimension of the pairing, $1$, the model can become unpredictable because more dimensions give more freedom in the way to represent these pairs.\n\nTo improve this situation, a solution is to add more constraints on such dimensions directly into the optimisation process.\nWe propose to give more than one information per pair and this allows to structure the embedding directly through the loss function.\nThis new method allows to control separately the usage of each dimension to express simultaneously different properties of the dataset.\nTo accomplish that, a generalization to the contrastive loss is proposed to work on training pairs with $p$-dimensional labels and an embedding with $M$ dimensions, where $p \\leq N$ by definition.\nIn this framework, an embedding dimension is allocated to one particular component of the label and one component can be expressed through one or more embedding dimensions.\nThe allocation of the embedding dimensions and the pairing strategy are left open to the use-case.\n\nThis new loss function is now introduced with its formal definition.\nLet us define $M$ as the number of embedding dimensions, $p$ the dimensions of the labels and $p \\leq M$.\nWe define $D \\in \\mathbb{N}_+^p$ as the number of embedding dimensions assigned for each label component.\nOnly problems where each dimension is assigned to a single label and no dimension is left unconstrained are considered in this work: $\\sum_i^p D_i = M$.\nThen it follows that the definition of the generalized $p$-dimensional contrastive loss for a pair is:\n\\begin{eqnarray}\n    L = \\frac{1}{2} \\sum_{i=1}^p \\left( Y_i (D_{Wi})^2 + (1-Y_i) \\max(0, m_i - D_{Wi})^2 \\right)\n\\end{eqnarray}\nwhere $Y \\in \\{0,1\\}^p$ with $Y_i$ being the $i$th component of $Y$, $D_{Wi} \\in \\mathbb{R}^{D_i}$ is the difference between the two points in the sub-embedding for dimensions of the $i$th component, and $m_i \\in \\mathbb{R}$ is the minimal distance for dimension $i$.\n\n\n\\chapter{Methodology and Results}\n\\label{chap:results}\n\nIn this chapter, the following sections describe the general environment used to make the experiments, the settings to reproduce them and their outcomes.\nThe different models are introduced in details with their architecture, then the implementation using Caffe and Python is explained.\nThe generation of the datasets by combining samples from MNIST or NORB is also discussed.\nFinally, the idea of quantitative measures is presented and used in the results for comparisons with the previous work.\nThe source code is available on GitHub at: \\url{https:\/\/github.com\/axel-angel\/master-project}.\n\n\\section{Models and parameters}\n\nAs previously said, the work in the reference paper \\cite{hadsell2006dimensionality} is closely followed here.\nTherefore, we employ the same two models to experiment on the respective datasets MNIST and NORB.\n\nThe {\\bf first model} is {\\em LeNet 5}, a multi-layer neural networks that is characterized by an architecture designed for handwritten characters as illustrated in figure~\\ref{fig:siamese_cnn}.\nIn the experiments with the MNIST dataset, a variant is used with some minor changes described below.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/siamese_cnn.jpg}\n    \\end{center}\n    \\caption{Architecture of the CNN used for the experiments on MNIST (based on LeNet 5). The network contains two convolutions layer which are connected by a subsampling layer and the final output is computed by a fully-connected layer. The computation flow is illustrated for a region of an image. -- Source: \\cite{hadsell2006dimensionality}}\n    \\label{fig:siamese_cnn}\n\\end{figure}\n\nThis network architecture is comprised of a total of 7 layers with trainable parameters.\nThe first part of this network contains two pairs of convolution-pooling layers, where each convolution shares its weights defined as a kernel.\nAs explained earlier, each convolution is applied to the entire image to create one feature map and the ensemble of convolutions of a layer creates the feature map of this layer.\nThe first convolutional layer has 20 different trainable $5 \\times 5$ kernels, which is followed by a trainable $2 \\times 2$ max-pooling layer (with pixel stride of $2$).\nThe second convolutional layer has 50 trainable $5 \\times 5$ kernels, followed again by a $2 \\times 2$ max-pool layer (same stride).\nThese two pairs of layers form the convolutional network ({\\em convnet}) part of this network which can be seen as {\\em feature extractors}.\nThey will extract features encoding high-level cues (e.g.: shape of strokes: straight or curved) in this application to discriminate between digits.\n\nThe second part of this network is made of two fully-connected layers which are connected through a non-linear activation layer.\nThe first layer has 500 trainable units computing an inner-product, followed by a non-trainable Rectified Linear Unit (ReLU) activations\\cite{nair2010rectified}, followed by 10 trainable output units computing an inner-product which gives the digit class (1 versus rest).\nThese three layers form the neural network part used for classification based on the prior features.\n\nThe {\\bf second model} is only made of two fully-connected inner-product layers.\nThis network is much simpler than the one above because it is trained on a subset of the NORB dataset, which exhibits very few variability compared to MNIST.\nIndeed, only a single 3D object will be used and its shape in 3D is easily recognizable from any angle.\nBack to the network, its two layers are made of 20 and 3 trainable units respectively, without non-linearity between them.\n\nIn the following experiments, the variant of LeNet is trained in two different ways: for digit classification to analyze its ``natural'' embedding then with a Siamese training architecture with the contrastive loss function to analyze a ``constrained'' embedding.\nIn the second version, a single fully-connected layer is left in the NN part and reshape its number of units to match the embedding dimensions (e.g.: $2$ or $3$).\nThe motivation is that the classification task is built on top of the network stack generating this embedding but in the second experiment this output needs to be projected, like in the reference paper.\nRegarding this second model, a ``constrained'' embedding is directly trained with the Siamese architecture for NORB.\n\nIn the case of digit classification, the model is trained using Stochastic Gradient Descend (SGD) with a learning rate of 0.01 optimizing a SoftMax loss function.\nSiamese models are trained with SGD with a learning rate of 0.01 or 0.001 optimizing the contrastive loss.\n\nFor this work, the {\\bf Caffe} Deep Learning framework was chosen to make models.\nThere are several reasons such as: simplicity to express, train and manipulate networks and the inclusion of several practical architectures: LeNet on MNIST, Siamese on MNIST, AlexNet on ImageNet.\nMoreover Caffe is well optimized in C++ for CPU and CUDA for GPU, flexible with official bindings for Python and MatLab and its community is very active and helpful.\n\nThe provided LeNet is easily adapted to match the architecture design discussed above for MNIST.\nAs the second model is very simple, it was trivial to write its definition ourself.\nThe training stage needs many important parameters related to the SGD solver: learning rate, batch size, momentum, weight decay (gamma and power) and number of epoch.\nThe default parameters were mostly used as defined in Caffe because the community already fine-tuned them.\nThe experiments are limited to $10'000$ iterations for MNIST and $100'000$ for NORB, with a batch size of $64$.\nAll networks were trained using the Caffe built-in solver started using the {\\tt caffe train} command.\nThe alternative loss function was implemented in a new loss layer inspired by the built-in contrastive loss.\nThe implementation was straightforward to make in Python with NumPy and the Caffe bindings to integrate this layer into the network architecture.\n\nThe set of tools to generate distorted training sets were created ourself for the t-SNE experiments and to generate the distorted pairs training sets for the two Siamese experiments.\nThe scikit-learn Python library ({\\em sklearn}) implements most of the standard algorithms for machine learning including: PCA and t-SNE\\cite{pedregosa2011scikit}.\nThe performance of t-SNE depends heavily on the number of input dimensions and the usage of PCA on the datasets was necessary and followed other papers suggestions\\cite{t-SNE}.\nLikewise sklearn recommendation is to reduce to $50$ dimensions with PCA before applying t-SNE\\footnote{t-SNE documentation page: \\url{http:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.manifold.TSNE.html}}.\n\nFor the visualization of the resulting embeddings, a solution based on the web library {\\em CanvasJS} was developed for this project.\nIts main advantage is to allow to interactively visualize with scatter plots directly from the output of t-SNE or the neural networks.\nSeveral features were already provided: coloring points, fast plotting for interactivity but some were added by ourselves: zooming and moving the viewport, display the image of any sample and filtering\/highlighting capabilities for the experiments.\nAll these functionalities provide the necessary tools to inspect the results and to come with the presented insights.\n\n\\subsection{Datasets}\nTo perform the experiments, two different datasets were used: {\\bf MNIST}\\cite{lecun1998mnist} (a popular handwritten digit dataset) and {\\bf NORB}\\cite{lecun2004learning} (a popular dataset of photographed 3D objects).\nEach dataset is briefly discussed then a few characteristics is established regarding their variability and their usual usage in Computer Vision.\n\nThe {\\bf MNIST} dataset ({\\em Modified National Institute of Standards and Technology}) is a gathering of multiple databases of handwritten digits.\nOne of the goal of MNIST is to provide a unified benchmark for digit recognition and it is widely used in Computer Vision for many years.\nIt is composed of $60'000$ training and $10'000$ testing images of digits between $0$ and $9$.\nA few examples of the training set are illustrated in figure~\\ref{fig:mnist}.\nThe samples are heavily post-processed: uniform black background, white-shaded digits with bold strokes.\nThe digits are centered such that the gravity center is in the middle and size-normalized so one digit lies inside a restrained sub-region of the $28 \\times 28$ bitmap.\nThe variability of this dataset lies in the different strokes, handwritting style, natural rotations, thickness, curve roundedness and such.\nThe MNIST dataset is considered nowadays extremely simple due to the lack of natural variability and because state-of-the-art achieved extremely good results.\nHowever it is still a good subject of experiments to try and validate new ideas.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/mnist.jpg}\n    \\end{center}\n    \\caption{A few samples taken from the MNIST training set to illustrate the dataset. Images were concatenated to save space.}\n    \\label{fig:mnist}\n\\end{figure}\n\nThe {\\bf NORB} dataset ({\\em NYU Object Recognition Benchmark}) is a collection of photos of toys taken from continuous poses.\nThis dataset has two variants and the normalized-uniform set was retained.\nIt is also intended for benchmark usages, in ``large-scale invariant object categorization''.\nThe set is composed of $48'600$ processed photos of toys, as illustrated by figure~\\ref{fig:norb}.\nThe dataset is made by combining: $9$ elevation views, $18$ azimuth views, $6$ illumination conditions and $5$ toys categories, each containing $10$ toys.\nThe categories are: humans, animals, airplanes, trucks and cars.\nThe usual training task is to recognize the toy category whichever viewpoint or illumination is captured.\nThe DrLIM paper only selected a particular toy of a plane for its experiments and they infer the coherent representation of the camera viewpoint in 3D.\nThis same toy is used in this work to make the results comparable with DrLIM.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/norb.jpg}\n    \\end{center}\n    \\caption{A few samples taken from the NORB training set to illustrate the dataset. Images were concatenated to save save.}\n    \\label{fig:norb}\n\\end{figure}\n\nThe experiments require to process the images in a systematic and coherent way.\nPython scripts were used to automatically derive the training and testing sets in a predictable way based on the original dataset (MNIST and NORB).\nThe training architecture apply sequentially SGD over the samples in batch and loops over the whole dataset until the number of iterations reaches zero.\nThe generation of these datasets depends on multiple parameters related to the experiment.\nIn the case of the t-SNE visualization, a distorted dataset was created containing each sample plus its distorted versions varying in strength, where each image has only a single transformation at a time.\nThe samples are all distorted using the same set of intensities, quantified as follows: translations and shearing using pixel displacement, rotations using positive and negative angles and blurring with averaging radius (examples in figure~\\ref{fig:mnist_transfo_tsne}.\nIn the case of Siamese training, paired datasets were created containing two distorted images with their similarity (1 or 0).\nImages in a pair can have different distortions and different classes (digit for MNIST, elevation\/azimuth for NORB) and still be paired (label 1), depending on the experiment.\nAn example of pairs for MNIST is illustrated in figure~\\ref{fig:mnist_pairs}.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/mnist_transfo_tsne.jpg}\n    \\end{center}\n    \\caption{Illustration of some distortions applied to a random sample included in the data-augmented dataset used in the t-SNE visualization experiment. The following distortions are shown: translations, rotations, blurring and linear deformation}\n    \\label{fig:mnist_transfo_tsne}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/mnist_pairs.jpg}\n    \\end{center}\n    \\caption{Illustration of some pairs generated for a random sample with its distortions, neighbors and non-neighbors. The pairs are included in the training dataset for the Siamese network.}\n    \\label{fig:mnist_pairs}\n\\end{figure}\n\n\\section{Evaluation Metrics}\nQualitative results can provide insights regarding the model behavior or the quality of embeddings but it has drawbacks.\nFirst it comes from human judgement which requires manual efforts, thus it cannot be systematic and automatic.\nSecondly the subjective nature of qualitative measures is subject to different interpretation or can be biased towards certain aspect that people value differently.\nWe think qualitative judgements are unreliable and should not be used alone to asses the work in NN.\nHowever, DrLIM results are lacking objective measures, for example the final loss or any sense of scale of dimensions to relate in the plots nor does it mention any kind of measure to quantify the quality.\nBy lacking any form of measures, we think that DrLIM paper does not promote the continuation of its work because there is no common way to measure ``good''.\nEvaluation metrics are very important to permit objective comparisons between several results to see how they perform.\n\nThus we propose a very simple quantitative measure that allows to compare DrLIM and the new models.\nModels are optimized with respect to their loss functions and indeed the loss is an objective measure that can be directly used to compare models on certain conditions.\nGiven the same loss, if its parameters are the same then it is logically fair to compare them.\nTherefore, the contrastive loss of DrLIM is used on its test set using the original pairing strategy.\nHowever, DrLIM paper does not provide the margin which is problematic to reproduce their work.\nIn the following, replicated versions of the DrLIM's models are trained ourself and thus a common margin is chosen per dataset for the two models.\n\n\\section{Results and Discussion}\n\nThis chapter is concluded by the presentation of the results and insights, including the plots and quantitative measures.\nThe first experiment is the visualization of the LeNet embedding on the MNIST classification task.\nThe structural differences are compared between models between trained with and without data-augmentation.\nHowever the resulting embedding has an unexpected structures that does not fit the needed requirements.\nThen, this method is left aside to learn directly optimized embedding with models using the contrastive loss.\nThe DrLIM's methods are applied by replicating the experiments on MNIST and NORB.\nFollowed by a throughout detailed discussion of the new methods in the same conditions to compare the two.\n\n\n\\subsection{t-SNE on LeNet}\n\nIn this experiment, two models for classification are trained on MNIST which is once unmodified, the second time it is data-augmented with translations.\nThe {\\em data augmentation} process grows a dataset artificially by applying various distortions to train invariant models.\nIn this case, it contains each original digit plus many translated copies which are all labeled identically.\nAfter the models are trained, their underlying embedding are inspected by using t-SNE.\nThe goal of this experiments is two folds.\nFirst we would like to understand classifier embeddings and its interaction with t-SNE in a practical manner: testing various data-augmentation methods to see how it impacts the model, its accuracy and how t-SNE adapts.\nSecondly, this experiment tests whether it is possible to get an embedding with the prerequisites properties of predictability with a state-of-the-art method specialized for projecting in 2D.\n\nTo create the first model, the training is done on the MNIST classification task to predict digits without any modification.\nAt this point, all images of the training set are used without data-augmentation, thus there is little invariance.\nThe error rate of this model is 1.0\\% which is computed over the MNIST test set.\nAccording to the published results on the MNIST website, this is a reasonable value (LeNet-5: 0.95)\\cite{mnist_web}.\nIn the following visualization with t-SNE, the classification layers of this this network are removed (SoftMax and 10-outputs inner-product).\nTherefore the last remaining layer is a ReLU which follows the last inner-product of 500 dimensions.\nThe hypothesis at this stage is as follows: this last layer expresses high-level concepts of digits.\nTo understand the behavior of the system, a few distorted samples are included in the test set: for each digit, five samples are selected and their distortions included.\nThis modified test set is feed-forwarded into the network, apply PCA, then apply t-SNE to create the final embedding.\nThe whole process takes around 45 minutes on a single core.\nThe resulting embedding is illustrated in the figures~\\ref{fig:mnist_nda_tsne} and ~\\ref{fig:mnist_nda_tsne2}.\nMost of the original digits are well separated on the first figure where each cluster represents a single class.\nThe major exception is the 1s spreading on a thin vertical curve over a small part of he 6s.\nThe hypothesis seems validated by the structure on this plot: class separation and coherence inside clusters.\nMost of the distorted samples on figure~\\ref{fig:mnist_nda_tsne2} are part of their respective clusters however they start to drift towards other classes as their distortion increases and the most extreme examples are at the opposite side.\nAt this point, it should be noted that the model is very sensitive to translations and many reside in the wrong clusters.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/mnist_nda_tsne.jpg}\n    \\caption{The t-SNE representation of the CNN embedding for the MNIST test set using the interactive tool.\n    This figure shows a general view of the non-distorted samples.\n    Some random samples are shown for each cluster.\n    Each color represent a digit class.\n    The original digits are well separated into clusters for each class.\n    The major exception is the 1s spreading on a thin vertical curve over a small part of he 6s.\n    Inside clusters, digits are spread coherently to their characteristics.\n    }\n    \\label{fig:mnist_nda_tsne}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/mnist_nda_tsne2.jpg}\n    \\caption{The t-SNE representation of the CNN embedding for the MNIST test set using the interactive tool.\n    This figure shows a close view of the cluster for 0s including this time the distorted samples.\n    A unique color is given to each digit class as before plus one unique color for each sample's distortions.\n    Translated samples quickly end up outside the digit cluster whereas other distortions like rotations are structured into curved lines.\n    Most illustrated images are rotations of a few selected samples.\n    }\n    \\label{fig:mnist_nda_tsne2}\n\\end{figure}\n\nIn the next experiment, the same process is applied with a translation-invariant model.\nOne simple way to achieve invariance is to train the model over a data-augmented training set to classify correctly translated digits as well.\nThe augmentation consists of a range of translations applied to all samples which are included into the training set.\nIn this case, all translations in $x$ between $-5$ to $+5$ plus translations in $y$ between $-10$ and $+10$ are generated.\nThe error rate on the original test set increased up to $3.5\\%$, but there is no comparison for this value.\nHowever this model has a strong translation invariance because its error rate is only $8.5\\%$ on the data-augmented test set (it contains extreme translations).\nThe second hypothesis is as follows: As the features contain many different characteristics, it should also detect translations which was heavily present in the training set.\nThe embeddings of this new model can be now compared (see figure~\\ref{fig:mnist_da_tsne}) versus the one above.\nFrom a general perspective, translations are dominating the global structure of the clusters.\nMany centered classes are still in their own clusters but most of the translated digits are grouped disregarding their class.\nHowever the ``translated'' clusters exhibits coherence: sub-clusters represent digit classes but they are heavily overlapping.\nFor example, the right-displaced digits are exclusively appearing in the top-left corner where all digits are present and mostly stacked together in the center.\nThis is actually a general problem with this plot: many clusters appear but their delimitations are fuzzy which suggests it would be hard to quantify translations and digits.\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics[width=\\textwidth]{thesis_figures\/mnist_da_tsne.jpg}\n    \\end{center}\n    \\caption{The t-SNE representation of the CNN embedding for the data-augmented MNIST test set using the interactive tool.\n    A unique color is attributed to each digit class and one for each sample's distortions (as before).\n    The illustrated digits were manually picked to represent local trends.\n    Global structures are dominated by translations but most of the translated digits are grouped disregarding their class.\n    Local structures exhibit partial coherence by packing digit classes together but the overlapping is too important to ignore.\n    }\n    \\label{fig:mnist_da_tsne}\n\\end{figure}\n\nFrom these two experiments, important knowledge emerges about this classification tasks, the data-augmentations and t-SNE as well.\nManual experiments confirmed that LeNet is sensitive to the employed distortions and can easily misclassify if no form of data-augmentation is used during training.\nAs previously said, nearly no natural distortion is present in the original dataset and this is partly due to the heavy prior processing applied to the MNIST dataset.\nHowever small rotations are still present in MNIST which explains why the new model is less prone to errors on artificially rotated samples.\nData-augmentation is an effective way to reduce the error rate on certain type of distortions but it has a cost.\nIn this case, t-SNE embedding looks ``overcrowded'' and most clusters are not well separated anymore.\nWe think it is due to the important difference of energy in the pixel space, e.g.: the center of mass is translated.\nOne explanation is that the network learned to separate this source of variance and acquired position-specific features to distinguish translated digits which impacts characteristics of the last layer.\nThen t-SNE is impacted by clustering translations first, then digits as its second major structure.\nOne important insight is that t-SNE seem to represent hierarchical structures with decreasing importance thus the most discriminative features are weighted more.\n\nThese results are unexpected: the clusters are not well separated and they are lacking in coherence even though the task require to separate the digits.\nThis experiment was meant to create an invariant model for classification but the representation does not naturally split digits and translations into different axes.\nIt should be noted again that t-SNE is oriented towards visualization and the goals of visualization are not formally and universally well defined.\nThe current formulation of this method is to preserve distances between the features which is not sufficient for the application.\nThere is currently no way to add prior information like supervised methods to guide the optimization to associate particular dimensions to particular variance in the data.\nA possible solution is to modify the objective function of t-SNE but it has the important limitations previously discussed.\nThe lack of end-to-end optimization in this method is also important and this can be solved by using layers handling the dimensionality reducing into a single NN.\n\nIn the following part, results of an alternative method is presented based on end-to-end and supervised training of a single CNN.\n\n\\subsection{Contrastive LeNet (MNIST)}\nIn the two following experiments, Siamese models were trained based on DrLIM settings on both MNIST and NORB so that the results are comparable between the reference paper and the networks with the extension as described below for two-label pairs.\nThe goal of DrLIM is to train a model where points are close together if they are ``visually'' similar without considering distortions depending on the dataset.\nThe implementation of their pairing strategy is as follows.\nFirst they group each sample into a common ``neighborhood'' relation: the 5 most similar ones using the Euclidean distance in pixel space.\nHowever in practice, neighbors with different digit class would appear and DrLIM's paper did not mention if they considered legit or not.\nIn this work, only neighborhood with the same class is used because a manual look unveils that they are in fact quite dissimilar.\nThe sample and its $n$ translations are all paired with both its 5 neighbors and all its $n$ translations (this sums up to $n \\times 5n$ pairs per sample).\nAll other pairs are considered dissimilar: when the two samples are not part of the same common translated neighborhood.\nThis strategy uses the regular contrastive loss function in 1D with a single similarity per pair.\n\nThe objective is comparable in the sense that staying invariant is desirable but only in certain dimensions priorly picked.\nThe new implementation can still quantify distortions by expression them in the other dimensions.\nOne important advantage is to allow more control, especially in how dimensions are used.\nAs said, a single model is trained which is less expensive and it learns reusable features for multiple pairings.\nThe extension of the contrastive loss function can be used with $p$ labels for each pair.\nThe definitions of $p$ and $D$ for the two Siamese experiments are described and justified later.\n\nIn the case of MNIST, a 3-dimensional embedding space is used: where the first two dimensions express neighborhood similarity and the last dimension expresses the distortion similarity.\nThis case only needs two dimensions but to make it comparable to DrLIM's 2D embedding, a 2D neighborhood space will be used as well and distortions are separated into its own space.\nMaking comparisons between 2D space (DrLIM) and 1D neighborhood (this work) would be unfair due to distances increasing very quickly as the number of dimensions increases.\nIn this settings: $M=3$ (embedding dimensions), $p = 2$ (two labels) and $D = \\left\\{ 2, 1 \\right\\}$ (2D + 1D).\nThe network is trained using a pair strategy inspired by DrLIM but with important differences.\nThe first label is called the ``neighborhood'' similarity and considers: a sample with its translations to be similar, the sample with its neighbors also similar if they have the same translations, and the sample with any non-neighbor sample to be dissimilar with or without translation.\nThe second label is called the ``transformation'' similarity and considers: a sample with any other sample with the same translation similar, and any pair without the same translation dissimilar.\nPairs not mentioned in this descriptions above are not included in the datasets.\nMoreover the size and the time for training is reduced with a low impact by randomly taking a subset of dissimilar pairs to balance the label ratio instead of including all of them.\n\nIn the case of NORB, a 3D embedding is used as well.\nThe first two dimensions are allocated for the cyclic azimuth (horizontal angle of the viewpoint) and the last dimension for the non-cyclic elevation (vertical angle).\nDrLIM's work defined similar pairs when the two images are from contiguous elevation or azimuth, this work will continue likewise.\nThe justification of DrLIM's dimension allocation is as follows: the azimuth viewpoint is cyclic and the shape, to represent such a structure without overlapping, is an ellipsis and this requires at least two dimensions.\nThe elevation viewpoint is not cyclic because NORB has only a smaller subset of values, thus only a single dimension is allocated.\nA direct comparison with DrLIM's embedding is possible in this case as the models work on the same problem.\nOnce again: $M=3$ (embedding dimensions), $p = 2$ (two labels) and $D = \\left\\{ 2, 1 \\right\\}$ (2D + 1D).\nThe network is trained based on DrLIM's pairing strategy where the azimuth and elevation are separated into two labels (instead of having them merged).\n\nThe first dataset is a subset of MNIST containing only 4s and 9s digits which are used to derive the training and testing set based on the 2D pairing strategy.\nTherefore the resulting dataset is composed of each sample plus its translations (\u00b13, \u00b16 pixels) which is then paired.\nThe models are trained until the loss on the respective test set has attained a local minima, which takes around 10 minutes.\nAfter around $100$ iterations, both models have reasonably converged to a stable solution (see figure~\\ref{fig:mnist_cl2d_loss}).\nHowever, the new model has reached a lower solution than DrLIM: comparing them directly is not possible because the two datasets are different but the difference in the loss in this model is more important and converge as quickly.\nIntuitively this means that the new problem formulation is easier to solve and the model finds a ``better'' solution, in the sense it could satisfy more the new constraints described by the loss.\nThe resulting embeddings are shown in figure ~\\ref{fig:mnist_cl_drlim} for DrLIM and figures ~\\ref{fig:mnist_cl2d_1} and ~\\ref{fig:mnist_cl2d_2} for this model.\nThe DrLIM embedding reproduced in this work is very similar to the expected result: a coherent grouping based on translation-invariant similarity but the separation between digits is more pronounced.\nThe DrLIM embedding can be considered as a top-down projection of the new 3D embedding where the additional vertical dimension quantify the displacement.\nThe different intensity of translations are well separated into their own cluster as demonstrated by a meticulous manual inspection and remarkably they are ordered vertically by their intensity.\nOnce again, the space inside clusters is well organized by digit features like DrLIM.\n\n\\begin{figure*}[h]\n    \\centering\n    \\begin{subfigure}{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/final_loss_testset_2bv7.jpg}\n       \n    \\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/final_loss_testset_3Dc3.jpg}\n       \n    \\end{subfigure}\n    \\caption{The model losses for 500 iterations of DrLIM and the new model on MNIST.\n    The two losses cannot be compared directly but the speed of convergence and the final difference are key.\n    Both models show sign of convergence after $100$ iterations but the new model has reached a simpler solution due to the different pairing strategy chosen in this work.\n    An important point to emphasis is that DrLIM is in 2D whereas the new model is in 3D.\n    }\n    \\label{fig:mnist_cl2d_loss}\n\\end{figure*}\n\nIn order to compare the two methods, quantitative measures are presented where the contrastive loss is computed on the DrLIM paired test set as described in the methodology.\nThe evolution of this measure during training is plotted in figure~\\ref{fig:loss_mnist_test_common} and the final losses after 10'000 iterations are: $0.160023$ (DrLIM) and 0.145418 (the new model).\nThe two loss series shown in this figure are statistically indistinguishable according to a t-test with a null hypothesis of $\\alpha = 0.05$.\nTherefore it can be concluded that the new model learns an equivalent representation in its 2D digit-projection to DrLIM with a slightly different formulation and it can expresses distortions without loosing information in its 3rd dimension at the same time.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics{thesis_figures\/mnist_cl_drlim.jpg}\n    \\caption{DrLIM embedding for the MNIST translation-augmented test set where some random samples are shown for illustration.\n    This figure is very similar to the own present in DrLIM paper and also exhibits coherent groupings based on translation-agnostic similarity.\n    }\n    \\label{fig:mnist_cl_drlim}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/mnist_cl2d.jpg}\n    \\caption{The new model embedding on the MNIST translation-augmented test set projected into its last two dimensions.\n    Two samples of different class were picked at random and their distortions are marked as triangles in the figure.\n    This 3D embedding has an additional vertical dimension compared to DrLIM that quantifies the displacement and the other top-down projection of this embedding would give an identical plot as above for DrLIM.\n    Clusters are formed for each translation intensity and each digit class in a well separated manner and remarkably they are logically ordered.\n    }\n    \\label{fig:mnist_cl2d_1}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/mnist_cl2d2.jpg}\n    \\caption{The new model embedding on the MNIST translation-augmented test set projected into its last two dimensions.\n    General view where many random sample's distortions are displayed to represent the local tendencies.\n    The space inside each cluster appears to be also coherent.\n    }\n    \\label{fig:mnist_cl2d_2}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/final_loss_test2bv7.jpg}\n    \\end{center}\n    \\caption{Comparison of quantitative measures between DrLIM and the new model.\n    The measures are taken during training iterations over the translation-augmented MNIST test set.\n    The two loss series are statistically indistinguishable and it can be concluded that both models learned an equivalent representation whereas the new model can also expresses distortions without noticeable impact.\n    }\n    \\label{fig:loss_mnist_test_common}\n\\end{figure}\n\nThe new model was tested on a data-augmentation of MNIST based on rotations as well to show it works on non-linear deformations as well.\nClusters are still arranged in a linear fashion although rotations are non-linear but the cluster separations are smaller, see figure~\\ref{fig:mnist_cl2d_rotate}.\nThe results are similar to the previous experiment and the figure is provided for demonstration as it will not be discussed further.\n\n\\begin{figure}[h]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/mnist_cl2d_rotate.jpg}\n    \\end{center}\n    \\caption{The new model embedding on the MNIST rotation-augmented test set.\n    Clusters are remarkably arranged in a linear fashion although rotations are non-linear.\n    }\n    \\label{fig:mnist_cl2d_rotate}\n\\end{figure}\n\n\\subsection{Contrastive LeNet (NORB)}\nOnce again, the network is trained following the new conventions, this time on a subset of the NORB dataset with a single plane.\nAs previously described, this subset is composed of all viewpoint of a single plane which is later called the 1-plane NORB dataset.\nThis subset is split into two parts: 660 training samples and 312 test samples.\nThese two sets are both separately paired according to the strategy described in the methodology to form the training and testing sets.\nThe models are trained until the loss on the respective test set has attained a reasonable local minima, which takes around 10 minutes.\nAfter $2'000$ iterations, both models have reasonably converged to a stable solution (see figure~\\ref{fig:norb_cl2d_loss}).\nTo get the most convincing results, a margin $m=1$ was picked for DrLIM and $m=10$ for the new model, therefore their loss on the test set should not be compared directly, instead the quantitative measures below should be used instead.\nHowever, it can seen that DrLIM in figure \\ref{fig:norb_drlim_embedding} has an easier formulation this time (smaller margin) and the embedding is very similar to the one presented in the reference paper.\nThe most important features of the embedding to represent NORB are: the cyclic structures for the azimuth angle, the continuity along the two axes disregarding lighting illumination and the sharp separation along the axes.\nIndeed, this embedding is a thin cylinder whose long axis represent the elevation and the radius is the azimuth angle.\nHowever, the cylinder is not perfect: multiples parts are more flat along the elevation axis (like a plane) and the radius is fuzzier than presented in the reference paper.\nThe major difference with the new solution, as shown in figures \\ref{fig:norb_cl2d_embedding_1} and \\ref{fig:norb_cl2d_embedding_2}, is the alignment along the axes, a more stable radius and a better separation inside the cylinder.\nThe predictability of the solution is improved by these properties compared to DrLIM.\n\n\\begin{figure*}[h]\n    \\centering\n    \\begin{subfigure}{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/final_loss_testset_3d.jpg}\n       \n    \\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/final_loss_testset_2D1g2.jpg}\n       \n    \\end{subfigure}\n    \\caption{The model losses for 10'000 iterations of DrLIM and the new model $m=10$ on NORB.\n    After $2'000$ iterations, both models have reasonably converged to a stable solution.\n    DrLIM model has a very low loss and the new model faces problems to correctly represent the azimuth dimensions.\n    }\n    \\label{fig:norb_cl2d_loss}\n\\end{figure*}\n\n\\begin{figure*}[h]\n    \\centering\n    \\begin{subfigure}{0.65\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/norb_drlim1.jpg}\n    \\end{subfigure}\n    \\begin{subfigure}{0.65\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{thesis_figures\/norb_drlim2.jpg}\n    \\end{subfigure}\n    \\caption{The DrLIM embedding in 3D on the NORB 1-plane complete dataset after $100'000$ iterations.\n    The colors indicate the elevation angle.\n    The result found that a thin cylinder whose long axis represent the elevation and the radius is the azimuth angle.\n    Such results are remarkable because the model found this shape on its own.\n    However it is not perfect, some parts are flat along the elevation axis and the radius is fuzzy.\n    }\n    \\label{fig:norb_drlim_embedding}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/norb_cl2d.jpg}\n    \\caption{The new model embedding on the NORB 1-plane complete after $100'000$ iterations.\n    2D projection of the first two axes.\n    Colors indicate the azimuth angle.\n    The major difference with DrLIM is the alignment along the plot axes and a better separation inside the cylinder.\n    The mapping is also invariant to the lighting conditions.\n    }\n    \\label{fig:norb_cl2d_embedding_1}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{thesis_figures\/norb_cl2d2.jpg}\n    \\caption{The new model embedding on the NORB 1-plane complete after $100'000$ iterations.\n    2D projection of the last two axes.\n    Colors indicate the elevation angle.\n    The relation between elevation and azimuth is well organized in the embedding.\n    }\n    \\label{fig:norb_cl2d_embedding_2}\n\\end{figure}\n\nTo compare the two models, the contrastive loss was computed on the test set of the 1-plane NORB with the same parameters ($m=1$) as with MNIST.\nThe loss evolution is shown in figure \\ref{fig:loss_norb_test_common} where a version of the new model was also added with $m=1$.\nThe embedding of this model has many aspects of the $m=10$ model and the main difference is instead of having a round shape for the azimuth, it is similar to a heart which is also cyclic but less desirable.\nA major difference can be found with the $m=10$  model compared to both DrLIM and the $m=1$ model.\nBy definition of the loss function, the model is not penalized to put dissimilar pairs further than the margin, therefore it means the $m=10$ model has decided it is better to put similar neighbors further than the other two models.\nMoreover, the loss suggests that the $m=1$ model is superior but $m=10$ looks better, although it has a higher loss.\nUnfortunately, the visual quality is not reflected by the loss function in this case.\n\n\\begin{figure}[h]\n    \\begin{center}\n        \\includegraphics{thesis_figures\/final_loss_testv3.jpg}\n    \\end{center}\n   \n    \\caption{Quantitative measures to compare between DrLIM and the new model during training iterations on the 1-plane NORB test set.\n    The $m=10$ model put less importance to the proximity of similar neighbors than the other two models.\n    This plot suggests that the $m=1$ model is better but $m=10$ is qualitatively better, although it has a higher loss.\n    The loss function does not seem to reflect the visual quality in this case.\n    }\n    \\label{fig:loss_norb_test_common}\n\\end{figure}\n\n\\subsection{Remarks}\nTo conclude this chapter, a few remarks about the general results and some difficulties during the experiments should be noted.\nThe primary goal of the experiments using contrastive losses is to show more predictability can be expected when given more control, over each dimension and over the clustering, with a simple extension of the original formulation.\nOur qualitative judgement is positive about the success of the experiments.\nThe first experiment demonstrated that models based on DrLIM can include more informations concerning the distortions with a minor impact over the original feature space.\nIn our second experiment it was shown that these models can also be guided to represent the NORB space by using dimensions in a certain way with minimal efforts.\nMoreover some difficulties were faced to reproduce DrLIM, especially on NORB due to the lack of directives for parameters and due to the initial randomness of model initialization which changes the final embedding consequently.\nDuring these experiments, one if not the most important factor for training a Siamese network comes from the pairing strategy.\nOn this regard, many small factors can have a big impact over the final results such as: the ratio of similar\/dissimilar pairs, their ordering (shuffled or not) and the addition\/removal of certain pairs.\nThe new models are easier to train because their convergence to the above results required nearly no effort: the default weights and margin were safely used and the models converged to the expected solutions unlike DrLIM in our experiments.\n\nHowever there is still room for improvements, especially for the quantitative measures which does not always reflect the structure quality quantified by a visual inspection.\nIt is important to keep in mind that the visual quality is not necessarily representative of its real usability as features for further layers in a NN.\nNonetheless, it shows that the new model can create quite interesting and human-friendly representation by simple methods.\n\n\n\\chapter{Conclusion}\n\\label{chap:conclusion}\n\nNeural Networks are widely used in many applications of Computer Vision and their usage is growing every day.\nThe recent usage of NN in dimensionality reduction is an important step to create better embeddings.\nIndeed they learn powerful mappings whose relation seems more deeply tied to the dataset.\nIn particular they can learn invariance to non-linear distortions while preserving relations between the other distortions.\nIn the experiments, NN were found to be much faster and reliable for our application than even the state-of-the-art methods such as t-SNE and the only prior-knowledge is the labels or the viewpoint angles.\nMoreover, many tricks developed during the past years of researches on NN have helped in this task: CNN architectures for images, Siamese networks and the contrastive loss function to train the new specialized networks.\nThanks to these findings and the current advance in Deep Learning, this work benefited from their improvement in the automatic derivation of better features to represent the data.\n\nIn this thesis, the performance of t-SNE on data-augmented datasets were presented with its advantages and shortcomings due to its unsupervised nature.\nAnother solution was then chosen based on NN because they have many advantages, in particular that they can represent very complex mappings.\nTo achieve more predictable embeddings, an extension to DrLIM was designed in this work.\nThen, several NN models were experimented on different datasets to compare the state-of-the-art with this new solution which demonstrated similar results, while being more general.\nIn particular, with the new method it is possible to quantify distortions in a consistent way, by just looking at the components of the learned feature representation.\nThe qualitative results were presented using plots which were analyzed to understand the models.\nBut quantitative measures were also introduced based on the loss function to compare more objectively with the previous work.\nAs said, there is still room for improvements in this regard as current measures does not seem completely reliable to measure the quality.\nMoreover, a better metric to give a more accurate score was a subject of our research to compare clustering and the predictability of the new models.\nHowever, such a solution is non-trivial to find and this would require comparing the new models with a simple baseline, which, to the best of our knowledge, does not exist at this day.\n\nThe current direction of dimensionality reduction started with DrLIM looks very similar to regression problems and this is also the case for the proposed generalization for N-dimensions.\nHowever, regression is more constrained than the current formulation because: (1) in this work, clusters are not constrained to be at a particular position, but only their relative positions matter whereas a regression based approach would set them in absolute positions; (2) the margin formulation allows to put extra gap to make space for clusters without cost whereas regression penalizes in every directions.\nThus, the current formulation allows more freedom in placement than regression and a possibility is that regression would perform worse due to the addition of these artificial constraints but the opposite may be true.\nFuture work could compare the new models with regression models to see if further restriction would actually help instead.\n\nAnother aspect that needs to be explored in future research is to precisely test the predictability of the embeddings.\nFor instance, the learned features could be used in a larger classification network to predict the distortion intensity and the digit class in MNIST.\nAnother useful contribution would find a measure that quantifies the ``predictability'' of the new model for certain application.\nOne simple example would be to use the accuracy of the classification model above but it would not represent the effective usability in practical applications which combine both distortions and application-specific features.\nA better experiment is to train a regression model, for instance, to predict the distorted inputs' features given a single input's features.\nThe accuracy of such a model would serve as a score for predictability.\n\n\nAn application is possible in Biomedical Imaging for classification tasks (e.g.: segmentation) which need to detect orientation-dependent structures in images.\nThe classification of such images begins by a normalization step with a rotation and a translation to normalize the structure of interest into a centered slice.\nThis slice is used to extract orientation-dependant features which are then fed into a classifier like a CNN.\nIt is important to understand that the normalization step to extract features is costly especially for 3D images and even specialized FPGAs were designed to compute 3D expensive rotations.\nThe presented contribution allows theoretically to skip this normalization step to directly create enhanced features.\nIn this case, the model learns features predictable with respect to rotations and the features for other distortions are easily derived and classified.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Galerkin-Finite Element (FE) method applied to partial differential equations (PDEs) with advection terms dominating over diffusion ones may suffer of numerical instability \\cite{hughes2012finite,Quarteroni_2017}. These numerical instabilities cause the numerical solution to exhibit oscillations that increase in amplitude with a local increment of the transport dominance over diffusion, i.e. as soon as the local P\\`eclet number becomes larger than one. In order to eliminate (or at least mitigate) numerical instabilities, a general employed strategy is to consider the generalized Galerkin method, i.e. the Galerkin method with additional stabilization terms \\cite{Quarteroni_2017}. Examples of stabilization methods to reduce numerical oscillations in the advection dominated regimes are for instance the upwind and streamline-diffusion method, both methods that are not strongly consistent. Instead, examples of strongly consistent methods are the Galerkin-Least-Squares, the Streamline Upwind Petrov-Galerkin (SUPG), and the Douglas-Wang methods \\cite{Roos_1996,Brooks_1982,Quarteroni_2017}. In particular, for these strongly consistent stabilization methods, the formulation depends on the residual of the PDE (in strong formulation), other than on a parameter -- called \\textit{stabilization parameter} -- whose definition is crucial for the success of the stabilization strategies. Determining the values of such stabilization parameter is not straightforward, especially for 2D and 3D problems. In addition, a universal formulation of such parameter is lacking, especially as it is strongly dependent on data of the model and the numerical setting used for the FE approximation of the PDEs, e.g. low and high order FE methods, spectral element methods, isogeometric methods, etc. \\cite{Schwab, spectralbook, cottrell2009isogeometric}. Some of the formulations for the stabilization parameter have been derived from analytical considerations on the differential problem in 1D, others are suitable only for a specific numerical approximation method, while the most comprehensive ones incorporate some (empirical) dependence on numerical setting, as e.g. the order of the FE \\cite{Bochev,Brooks_1982, CODINA200061,CODINA2008264,CODINA20072413,franca1992stabilized,galeao2004finite,REBOLLO2015406,ScovazziRossi,SCOVAZZI2007923,tezduyar2000finite}.\n\nIn this work, we propose using Machine Learning (ML) \\cite{mitchell1997machine,Yegnanarayana,goodfellow2016deep} to make computers learn autonomously the values of the stabilization parameter for the FE approximation of advection dominated PDEs. Artificial Neural Newtorks (ANNs) \\cite{university1988continuous,Kutyniok_2021}, are widely popular in ML and Deep Learning for a wide array of applications: they are indeed very versatile tools that are increasingly finding their way in scientific computing \\cite{Neittaanm\u00e4ki202227}, especially in the context of numerical approximation of PDEs \\cite{Mishra_2018}. For instance, as substitute to standard numerical methods, ANNs can be employed as meshless methods in Physics Informed Neural Networks (PINNs) to directly approximate the solution of the PDE as it is trained by minimizing the (strong) residual of the PDE \\cite{raissi2017machine,raissi2018hidden,raissi2019physics}. ANNs are also largely employed in a data-driven fashion in the context of model order reduction for parametric PDEs \\cite{fresca2021comprehensive, regazzoni2019machine, hesthaven2018non, guo2019data, zancanaro2021hybrid} and to enhance the stability properties of numerical methods for PDEs \\cite{discacciati2020controlling}. In fluid dynamics modelling, ANNs are massively adopted for flow features extractions, modelling, optimization and control. For flow features extractions through clustering and classification to classify wake topologies \\cite{annurev_wake}; for modeling fluid dynamics by reconstructing specific flows such as the near wall field in a turbulent flow \\cite{annurev_turbulent_wall} and for flow optimization \\cite{annurev_gliding}, and control  for aerodynamics applications \\cite{annurev_control}. ANNs are also used in a data-driven framework as a manner for providing alternative closure models for RANS' stress tensor \\cite{annurev_rans}, for sub-grid scale models in LES turbulence models \\cite{janssensadvancing, janssenthesis, xie2019artificial, zhou2019subgrid, xie2020modeling}, or for model learning input-output relationships in complex physical processes \\cite{REGAZZONI2020113268}. \n\nIn this work, we use ANNs to learn the optimal stabilization parameter in advection dominated PDEs that are discretized by means of FE method: the goal is to enhance the accuracy of the SUPG FE method by optimally selecting the stabilization parameter under different data of the PDE and numerical settings of the FE method. We found that the proposed ANN-enhanced stabilization method allows to improve accuracy and stabilization properties of the numerical solution compared to those results obtained by analytical expressions of the stabilization parameter. The numerical results obtained shed light also on the possibility to apply the presented strategy to learn closure laws for stabilization and turbulence models of fluid dynamics, as for instance to learn the stabilization parameters in the Variational Multiscale--LES model \\cite{forti2015semi, bazilevs2007variational}.\n\nThis work is organized as follows: in Section~\\ref{section:advection-diffusion}, we recall the SUPG stabilization method for advection-diffusion equations; in Section~\\ref{sec:strategy}, we present our numerical strategy to compute an optimal SUPG stabilization parameter through a feed-forward fully connected ANN. In Section~\\ref{sec:validation} we validate our method by comparing the ANN results with those obtained with the 1D advection-diffusion problem from which the expression of the theoretical stabilization parameter has been derived. In Section~\\ref{sec:results} we first show the ANN's training and we present our numerical results on the 2D advection-diffusion problem used for training and we finally generalize our findings using the ANN's prediction on a different advection-diffusion problem. Finally, in Section~\\ref{sec:conclusions} we draw our conclusions highlighting possible future developments.\n\n\\section{The SUPG method for advection-diffusion problems}\n\\label{section:advection-diffusion}\nWe briefly recall the advection-diffusion differential problem and the SUPG stabilization method for the advection dominated regime.\n\nLet $\\Omega \\in \\mathbb{R}^d, \\; d = 1, 2, 3$ be the physical domain with $\\partial \\Omega$ being its boundary. We consider the following problem in the unknown function $u$:\n\\begin{equation}\n    \\label{eq:transport_diffusion}\n    \\left\\{ \\; \\begin{aligned}\n        - \\nabla \\cdot (\\mu \\nabla u) + \\boldsymbol{\\beta} \\cdot \\nabla u &= f & \\qquad \\text{in} \\; \\Omega, \\\\\n        u &= g & \\qquad \\text{on} \\; \\partial \\Omega,\n    \\end{aligned} \\right.\n\\end{equation}\nwhere $\\mu$, $\\boldsymbol{\\beta}$, and $f$ are assigned functions or constants, with $\\mu \\in L^\\infty(\\Omega)$, $\\bm \\beta \\in [L^\\infty(\\Omega)]^d$, with $\\nabla \\cdot \\bm \\beta \\in L^2(\\Omega)$, and $f \\in L^2(\\Omega)$. \nThe Dirichlet datum on the boundary is $g\\in H^{1\/2}(\\partial \\Omega)$.\n\nLet $V_g = \\{ v \\in H^1(\\Omega): \\, v|_{\\partial \\Omega} = g \\}$ and $V_0 = H_0^1(\\Omega)$, the weak formulation of Eq.~\\eqref{eq:transport_diffusion} reads\n\\begin{equation}\n    \\label{eq:weak_bilinear}\n    \\text{find} \\; u \\in V_g \\ : \\ a(u,v) = F(v) \\quad \\text{for all } v \\in V_0.\n\\end{equation}\nwith the bilinear form $a(u,v)$ and the linear functional $F(v)$  respectively defined as:\n\\begin{equation*}\n    \\label{eq:bilinearForm}\n    \\begin{aligned}\n        a(u,v) &:= \\int_\\Omega \\mu \\, \\nabla u \\cdot \\nabla v \\, d\\Omega + \\int_\\Omega v \\, \\boldsymbol{\\beta} \\cdot \\nabla u \\, d\\Omega, \\\\\n        F(v) &:= \\int_\\Omega f \\, v \\, d\\Omega.\n    \\end{aligned}\n\\end{equation*}\nWe consider a family of function spaces $V_h \\subset V$ (either for $V_g$ and $V_0$) dependent on a parameter $h$ such that $\\dim(V_h) = N_h < \\infty$.  Let $X_r^h = \\{ v^h \\in C^0(\\overline{ \\Omega}):\\,  v^h|_K \\in \\mathbb{P}_r,\\,  \\, \\text{for all } \\, K \\in \\mathcal T_h\\}$ be the function space of the FE discretization  with piecewise Lagrange polynomials of degree $r \\geq 1$, $\\mathcal T_h$ a triangulation of $\\Omega$ and $h$ the characteristic size of the mesh, comprised of elements $K \\in \\mathcal T_h$. By setting $V_h = \\mathring{X}_h^r \\bigcap V_0$, the Galerkin FE method applied to Eq.~\\eqref{eq:bilinearForm} reads\n\\begin{equation}\n    \\label{eq:galerkin_1}\n    \\text{find} \\; u_h \\in V_{g,h} \\ : \\ a(u_h,v_h) = F(v_h) \\quad \\text{for all } v_h \\in V_{h},\n\\end{equation}\nwhere $V_{g,h}= \\mathring{X}_h^r \\bigcap V_g$.\nThe standard Galerkin-FE method in Eq.~\\eqref{eq:galerkin_1}, can generate numerical oscillations on $u_h$ if the problem is dominated by the advection term. In particular, these numerical instabilities can arise if the local P\\'eclet number is $\\Peclet_h > 1$, where\n\\begin{equation}\n    \\label{eq:peclet}\n    \\Peclet_h = \\frac{|\\boldsymbol{\\beta}| h}{2 \\mu}.\n\\end{equation}\nThe {generalized Galerkin} method (\\cite{QV94}) allows eliminating or mitigating numerical oscillations by adding  stabilization terms to the standard Galerkin formulation as\n\\begin{equation}\n    \\label{eq:generalized_galerkin}\n    \\text{find} \\; u_h \\in V_{g,h} \\ : \\ a(u_h,v_h) + b_h(u_h,v_h) = F(v_h) \\quad \\text{for all } v_h \\in V_{h}.\n\\end{equation}\nIn the SUPG method, the additional stabilization term reads:\n\\begin{equation}\n    \\label{eq:supg}\n    b_h(u_h,v_h) = \\int_\\Omega R(u_h) \\; \\tau \\; \\frac{1}{2} \\left( \\nabla \\cdot (\\boldsymbol{\\beta} v_h) + \\boldsymbol{\\beta} \\cdot \\nabla v_h \\right) \\, d\\Omega,  \n\\end{equation}\nwhere $R(u_h)$ stands as the residual in strong formulation of Eq.~\\eqref{eq:transport_diffusion}, which is defined as:\n\\begin{equation*}\n   \n    R(u_h) := - \\nabla \\cdot (\\mu \\nabla u_h) + \\boldsymbol{\\beta} \\cdot \\nabla u_h - f.\n\\end{equation*}\nThe term $\\tau$, appearing in Eq.~\\eqref{eq:supg}, is the \\textit{stabilization parameter}, which is the focus of this work. A universal and optimal definition of $\\tau$ in terms of the problem data, numerical settings like the mesh size and FE degree is lacking. An extensive review of stabilization parameters for the SUPG method is reported in \\cite{john2007spurious}. The most common choices for $\\tau$ come from an analytic derivation made for the advection-diffusion problem in 1D with $f=0$ and approximated by means of linear finite elements, which reads:\n\\begin{equation}\n    \\label{eq:tau_theory}\n    \\widetilde{\\tau}_1 = \\frac{h}{2 |\\boldsymbol{\\beta}|} \\, \\xi(\\Peclet_h),\n\\end{equation}\nwhere $\\xi(\\theta)$ is the upwind function:\n\\begin{equation*}\n    \\xi(\\theta) = \\coth({\\theta}) - \\frac{1}{\\theta},   \\quad \\theta>0.\n\\end{equation*}\nThe stabilization parameter $\\widetilde{\\tau}_1$ is generally defined locally, i.e. mesh element by element. If a uniform mesh is used, as we consider in this paper, then the value of $h$ is uniform over $\\mathcal T_h$; if in addition $\\boldsymbol{\\beta}$ is a constant, then this implies that $\\tau$ is uniform over $\\mathcal T_h$. The choice of $\\tau$ made in Eq.~\\eqref{eq:tau_theory} represents an optimal choice of the stabilization parameter as it yields a nodally exact numerical solution for the 1D advection-diffusion problem if $f = 0$ and the FE polynomial order is $r = 1$ \\cite{Quarteroni_2017, galeao2004finite}. \nThus, the stabilization parameter of Eq.~\\eqref{eq:tau_theory} may not be fully effective to provide the optimal stabilization for a general advection-diffusion problem, e.g. to guarantee a nodally exact solution in 2D\/3D or when using FE degrees larger than $r=1$. \nA commonly used generalization of the formula of Eq.~\\eqref{eq:tau_theory} to higher FE degrees ($r>1$) is presented in \\cite{galeao2004finite} and reads:\n\\begin{equation}\n    \\label{eq:tau_theory_r}\n    \\widetilde{\\tau}_r = \\frac{h}{2 |\\boldsymbol{\\beta}| r} \\, \\xi \\left( \\frac{\\Peclet_h}{r} \\right).\n\\end{equation}\nDifferently from Eq. \\eqref{eq:tau_theory}, the stabilization parameter $\\widetilde{\\tau}_r$ in Eq.~\\eqref{eq:tau_theory_r} takes into account the contribute of higher polynomial degrees $r$. Still, this formula is not optimal as it does not guarantee a nodally exact numerical solution. \nOur goal is to find a general and optimal expression of the stabilization parameter holding for advection-diffusion problems and FE approximations of degree $r\\geq 1$ to be dependent on: the dimension $d$, the FE degree $r$, the mesh size $h$, the forcing term $f$, the diffusion coefficient $\\mu$ and the transport coefficient $\\beta$.\n\n\\section{ANN-based approach for determining the optimal stabilization parameter}\n\\label{sec:strategy}\n\nWe present our approach to determine the optimal stabilization parameter $\\tau$ for the SUPG stabilization method by using an ANN.  We consider as {features} (inputs) of our ANN: the degree of finite element $r$, the mesh size $h$ and the global P\\'eclet number $\\mathbb Pe_g := | \\boldsymbol{\\beta}| L \/ (2\\mu)$ of the advection-diffusion problem, where $L$ is the characteristic length of the problem. As we consider $\\boldsymbol{\\beta}$ to be uniform and fixed, using $\\Peclet_g$ as input corresponds to varying the value of the diffusion coefficient $\\mu$. The features (input) of the ANN read:\n\\begin{equation}\n\\mathbf x^{(i)} = \\left [ r, \\, h,\\,  \\Peclet_g \\right ]; \n\\label{input}\n\\end{equation}\nthe \\textit{target} (output) of interest is the optimal SUPG stabilization parameter that we denote with $\\tau^*$:\n\\begin{equation}\n\\mathbf y^{(i)} = \\left [ \\tau^* \\right ].\n\\label{output}\n\\end{equation}\nThe first step consist in generating the dataset, i.e. pairs of inputs and outputs to be used for training the ANN (\\textit{data generation} step). This consists in choosing repeatedly and randomly values of the features $\\mathbf x^{(i)}$ in given ranges. These features are used to feed an \\textit{optimization problem} that, through an optimizer and a suitable error measure, provide an optimal stabilization parameter $\\mathbf y^{(i)}$, i.e. the target of the given feature. Such optimization problem considers as error measure $E(\\tau)$: the mismatch between the numerical solution $u_h$ and the exact one $u_\\mathrm{ex}$ over the nodes of the FE mesh. $E(\\tau)$ reads as:\n\\begin{equation}\n    \\label{eq:td_error_definition}\n    E(\\tau) = \\sum_{k=1}^{K_h} | \\, e(\\bm x_k, \\tau) \\, |, \\qquad e(\\bm x_k, \\tau) = u_h(\\bm x_k;\\tau) - u(\\bm x_k).\n\\end{equation}\nbeing $\\bm x_k$ the $k$-th node of the FE mesh $\\mathcal T_h$ and $K_h$ its number. To find the minimum of $E(\\tau)$ for different problem configurations we solve, for each instance of the input parameters $\\mathbf x^{(i)}$, the following optimization problem:\n\\begin{equation}\n    \\label{eq:td_findtau}\n    \\text{find} \\; \\tau^* \\ : \\quad \\min_\\tau \\, E(\\tau).\n\\end{equation}\nThus, the dataset generated consists of $m$ pairs of inputs-outputs $(\\mathbf x^{(i)}, \\mathbf y^{(i)})$, with $i=1, \\dots, m$. The latter is used for \\textit{training} the ANN. The latter will be then used to predict the optimal stabilization parameter $\\mathbf y^{(j)}= \\tau_\\mathrm{ANN}$ to be used for the FE approximation of the advection-diffusion problem in the settings provided by $\\mathbf x^{(j)}$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\singlefigureXL]{figures\/ad\/sketch_strategy_az.png}\n\\caption{Representation of the strategy used for learning the the relation among $\\tau$ and the advection-diffusion parameters and numerical settings}\n\\label{fig:td_scheme}\n\\end{figure}\n\n\nWe report in Figure~\\ref{fig:td_scheme} a sketch of the procedure used to build the ANN for the prediction of the optimal stabilization parameter for any new feature $\\mathbf x^{(j)}$.\n\n\n\n\\section{Numerical validation}\n\\label{sec:validation}\nIn this section, we validate the proposed numerical strategy by means of a 1D advection-diffusion problem from which the expression of $\\widetilde{\\tau}_1$ in Eq.~\\eqref{eq:tau_theory} has been derived. In particular, we consider the advection-diffusion problem in Eq.~\\eqref{eq:transport_diffusion} with $\\Omega = (0, 1)$, $f = 0$ and with Dirichlet BCs $ u = 0$ on $x = 0$ and $ u = 1$ on $x = 1$. It admits the following exact solution:\n\\begin{equation*}\n   \n    u(x) = \\frac{\\exp{\\left( \\frac{\\beta}{\\mu} x \\right)} - 1}{\\exp{\\left( \\frac{\\beta}{\\mu} \\right)} - 1}.\n\\end{equation*}\nWe recall that the stabilization parameter $\\widetilde{\\tau}_1$ of Eq.~\\eqref{eq:tau_theory} yields a nodally exact numerical solution if Eq.~\\eqref{eq:transport_diffusion} is solved by linear FE ($r = 1$).\n\n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/minimumErr_Pe12.5_r1-eps-converted-to.pdf}\n         \\caption{$r = 1$}\n         \\label{fig:td_minimumErr_r1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/minimumErr_Pe12.5_r3-eps-converted-to.pdf}\n         \\caption{$r = 3$}\n         \\label{fig:td_minimumErr_r3}\n    \\end{subfigure}\n    \\caption{Error (\\ref{eq:td_error_definition}) of numerical solutions with SUPG method vs. the stabilization parameter $\\tau$ for the 1D problem in Section~\\ref{sec:validation} with $h = 1\/20$ and $\\Peclet_h = 12.5$; comparison between the theoretical value $\\widetilde{\\tau}_r$ of Eq.~\\eqref{eq:tau_theory_r} and the optimal one $\\tau^*$}\n    \\label{fig:td_minimumErr}\n\\end{figure}\n\nWe plot in Figure~\\ref{fig:td_minimumErr} the error measure $E(\\tau)$ against the value of the stabilization parameter $\\tau$, with $h = 1\/20$, $\\Peclet_h = 12.5$ and by using $r = 1$ (Figure \\ref{fig:td_minimumErr_r1}) and $r = 3$ (Figure \\ref{fig:td_minimumErr_r3}). We observe that $E(\\tau)$ shows a minimum in $\\tau^*$, thus suggesting the possibility to use an optimization algorithm to solve the problem. Specifically, we employ the L-BFGS-B optimization algorithm from {\\tt SciPy}, an open source library for {\\tt Python} \\cite{scipy}.  Instead, the advection-diffusion problem has been solved using the FE open source library {\\tt FEniCS} \\cite{fenics}, by applying the SUPG method on a uniform mesh.\nParticularly, the optimal value $\\tau^*$ found for the case $r=1$ corresponds to the one provided by the theory in Eq.~\\eqref{eq:tau_theory}. For the case $r=3$, we observe a optimal value of $\\tau^*$ for which the error is minimized: in this case, it does not corresponds to the stabilization parameter $\\widetilde{\\tau}_r$ provided by the theory in Eq.~\\eqref{eq:tau_theory_r}: as a matter of fact, the latter arises from empirical considerations to extend the parameter $\\widetilde{\\tau}_1$ in Eq.~\\eqref{eq:tau_theory} to polynomials degrees $r>1$. However, this does not ensure a nodally exact numerical solution. \n\nWe solve the optimization problem for different values of the local P\\'eclet number $1 \\leq \\Peclet_h \\leq 250$ and we plot in Figure~\\ref{fig:td_1d_comparison} the optimal stabilization parameter $\\tau^*$ against the local P\\'eclet number for $r = 1, 2$, and $3$.\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/1D_comparison_r1-eps-converted-to.pdf}\n         \\caption{$r = 1$}\n         \\label{fig:td_1d_comparison_r1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/1D_comparison_r2-eps-converted-to.pdf}\n         \\caption{$r = 2$}\n         \\label{fig:td_1d_comparison_r2}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/1D_comparison_r3-eps-converted-to.pdf}\n         \\caption{$r = 3$}\n         \\label{fig:td_1d_comparison_r3}\n    \\end{subfigure}\n    \\caption{Comparison of the optimal stabilization parameter $\\tau^*$ (full line) and the theoretical one $\\widetilde{\\tau}_r$ of Eq.~(\\ref{eq:tau_theory_r}) (dashed line) against $\\Peclet_h$, with $h = 1\/20$. Results are referred to the 1D advection-diffusion problem described Section~\\ref{sec:validation}}\n    \\label{fig:td_1d_comparison}\n\\end{figure}\n\nFigure~\\ref{fig:td_1d_comparison_r1} shows that, by employing a linear FE space $r=1$, the optimal stabilization parameter $\\tau^*$ obtained by minimizing $E(\\tau)$ exactly matches the theoretical one $\\widetilde{\\tau}_1$ for every value of $\\Peclet_h$, thus confirming the findings of Figure~\\ref{fig:td_minimumErr_r1}. This also serves as validation of the optimization procedure that we proposed. By recalling that $\\widetilde{\\tau}_1$ of Eq.~\\eqref{eq:tau_theory} guarantees the numerical solution to be exact at nodes for this particular advection-diffusion problem and $r=1$, we can infer that the optimization procedure is meaningful and it can be therefore exploited further, for example for $r>1$. On the other hand, Figures~\\ref{fig:td_1d_comparison_r2} and~\\ref{fig:td_1d_comparison_r3} show instead a mismatch between the theoretical $\\widetilde{\\tau}_r$ of Eq.~\\eqref{eq:tau_theory_r} and the optimal one $\\tau^*$, thus confirming the findings reported in Figure~\\ref{fig:td_minimumErr_r3}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/1D_error_r2-eps-converted-to.pdf}\n         \\caption{$r = 2$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/1D_error_r3-eps-converted-to.pdf}\n         \\caption{$r = 3$}\n    \\end{subfigure}\n    \\caption{Comparison of the error of Eq.(\\ref{eq:td_error_definition}) obtained with the optimal $\\tau^*$ (full line) and theoretical $\\widetilde{\\tau}_r$ (dashed line) for varying $\\Peclet_h$ with $h = 1\/20$. Results referred to the 1D advection-diffusion problem of Section~\\ref{sec:validation}}\n    \\label{fig:td_1d_error}\n\\end{figure}\n\nIn order to better appreciate the differences in the numerical solutions due to the choice of the stabilization parameters, we report, in Figure~\\ref{fig:td_1d_error}, a comparison of the error $E(\\tau)$ obtained by means of the optimal $\\tau^*$ and theoretical $\\widetilde{\\tau}_r$ stabilization parameters for $r=2$ and $3$. Moreover, we report in Figure~\\ref{fig:td_boundary_layer_r3} a comparison among the exact solution $u$, the SUPG-stabilized numerical solution $u^*_h$ obtained with the optimal stabilization parameter $\\tau^*$, and the numerical solution $\\widetilde{u}_h$ obtained with theoretical one $\\widetilde{\\tau}_r$. Specifically, we consider FE of degree $r=3$ and two different values of $\\Peclet_h$. In both these cases, the optimal parameter $\\tau^*$ leads to a more accurate solution with respect to using the theoretical parameter $\\widetilde{\\tau}_r$. In particular, when $\\Peclet_h$ is ``small\", $\\widetilde{\\tau}_r$ leads to overshooting in the numerical solution, while if $\\Peclet_h$ is ``large\" the theoretical stabilization parameter leads to undershooting. Conversely, the optimal parameter $\\tau^*$ accurately intercepts the exact solution at the nodes.\n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/solution_Pe2.5_r3-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 2.5$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/solution_Pe12.5_r3-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 12.5$}\n    \\end{subfigure}\n    \\caption{Boundary layers of the numerical solutions $u^*_h$ and $\\widetilde{u}_h$ obtained with the SUPG method with optimal $\\tau^*$ and theoretical one $\\widetilde{\\tau}_r$ for $h = 1\/20$ and $r = 3$, respectively; comparison with the exact solution $u$ ($u^*_h$ is nodally exact at the node in $x=0.95$). Results are referred to the 1D advection-diffusion problem of Section~\\ref{sec:validation}}\n    \\label{fig:td_boundary_layer_r3}\n\\end{figure}\n\n\\newpage\n\n\\section{Numerical results}\n\\label{sec:results}\nWe first introduce the training set of our problem and we detail the setup of the ANN that we use in this work. Then, we show the prediction of the stabilization parameter by means of the ANN on different advection-diffusion problems.\n\n\\subsection{Training the ANN for a 2D advection-diffusion problem}\n\\label{sec:training_2D}\nWe apply the strategy presented so far to a 2D advection-diffusion problem to generate the dataset for the training of the ANN. Specifically, we consider in Eq.~\\eqref{eq:transport_diffusion}: $\\Omega = (0, 1)^2$, $f = 0$, and $\\boldsymbol{\\beta} = (1,1)$. We prescribe the following exact solution on the whole boundary $\\partial \\Omega$:\n\\begin{equation}\n    \\label{eq:td_2d_exact}\n    u(x, y) = \\frac{e^{(x \/ \\mu)} - 1}{e^{(1 \/ \\mu)} - 1} + \\frac{e^{(y \/ \\mu)} - 1}{e^{(1 \/ \\mu)} - 1}.\n\\end{equation}\n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{0.32\\textwidth}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/mesh10-eps-converted-to.pdf}\n         \\caption{$h = \\sqrt{2} \/ 10$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.32\\textwidth}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/mesh20-eps-converted-to.pdf}\n         \\caption{$h = \\sqrt{2} \/ 20$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.32\\textwidth}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/mesh40-eps-converted-to.pdf}\n         \\caption{$h = \\sqrt{2} \/ 40$}\n    \\end{subfigure}\n    \\caption{Structured meshes used for the FE approximation of the 2D advection-diffusion problem}\n    \\label{fig:td_2d_mesh}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.65\\textwidth]{figures\/ad\/dataset_ann2-eps-converted-to.pdf}\n    \\caption{Visualization of the dataset used for the ANN training: target $\\tau^*$ against feature $\\Peclet_g$ colored by feature $r=1$ (red), $2$ (blue), and $3$ (green) for different values of the feature $h$ (increasing values of $h$ from bottom to top) as listed in Table~\\ref{tab:dataset}}\n    \\label{fig:td_ann2_dataset}\n\\end{figure}\n\nWe generate in $\\Omega$ a structured mesh $\\mathcal T_h$ of triangles with {\\tt FEniCS} \\cite{fenics}, as shown in Figure~\\ref{fig:td_2d_mesh}.\nWe generate the dataset by repeatedly solving the optimization problem of Eq.~(\\ref{eq:td_findtau}) for varying set of features as described in Section~\\ref{sec:strategy}; specifically, we choose the features as reported in Eq.~\\eqref{input}. The complete dataset contains $m = \\, 900$ examples with $r = \\left \\{1, 2, 3 \\right \\}$, $h = \\left \\{ \\frac{\\sqrt{2}}{10}, \\frac{\\sqrt{2}}{20}, \\frac{\\sqrt{2}}{40} \\right \\}$ and values of $\\Peclet_g$ randomly chosend in the range $[7, 70'710]$ (uniform distribution), which yields $\\mu \\in [10^{-5}, 10^{-1}]$. We summarize the details for generating the dataset in Table~\\ref{tab:dataset}. Figure~\\ref{fig:td_ann2_dataset} provides an overview of the dataset used to the training of the ANN, specifically the features $\\mathbf x^{(i)}=[r, \\, h,\\, \\mathbb Pe_g]$ and the target $\\mathbf y^{(i)}=[\\tau^*]$. In this case, the characteristic length $L$ is fixed and set equal to $L=1$.\n\n\n\n\\begin{table}[t!]\n\\begin{tabular}{cccccc}\n\\hline\n\\# Data & \\# Training & \\# Validation & \\multirow{2}{*}{$r$} & \\multirow{2}{*}{$h$} & \\multirow{2}{*}{$\\Peclet_g$} \\\\\nset ($m$) & set & set & & & \\\\\n\\hline\n\\multirow{2}{*}{$900$} & \\multirow{2}{*}{$720\\, (80 \\%)$} & \\multirow{2}{*}{$180\\, (20 \\%)$} & \\multirow{2}{*}{$\\left \\{1, 2, 3 \\right \\}$} &\\multirow{2}{*}{  $\\left \\{ \\frac{\\sqrt{2}}{10}, \\frac{\\sqrt{2}}{20}, \\frac{\\sqrt{2}}{40} \\right \\} $}\n& randomly \\\\\n& & & & & in $[7, 70'710]$.\n\\end{tabular}\n\\caption{Details of the dataset used for the ANN training}\n\\label{tab:dataset}\n\\end{table}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/ad\/ann.png}\n    \\caption{Architecture of the feed-forward fully-connected ANN}\n    \\label{fig:ann_architecture}\n\\end{figure}\n\nWe train a fully-connected feed-forward ANN on the generated dataset by using the open source library {\\tt Keras} \\cite{keras} built on top of {\\tt TensorFlow} \\cite{tensorflow}. We divide the dataset into two parts: a training dataset that takes $80\\%$ of the examples to be used for the ANN training and a validation dataset that takes the remaining $20\\%$. We choose the loss function as the mean squared error that measures, for each training feature, the squared mismatch between the prediction of the ANN $\\widehat{y}^{(j)}$ and the actual target $y^{(j)}$. Specifically, the loss function is defined as\n\\begin{equation}\n    \\mathcal J = \\frac{1}{2m} \\sum_{j = 1}^m \\left ( \\widehat{y}^{(j)} - y^{(j)}\\right )^2.\n    \\label{cost}\n\\end{equation}\nWe normalize the features by subtracting their mean and dividing by their standard deviation in order to help the weights to better adapt to the different scales of the features. Moreover, the targets and the feature $\\Peclet_g$ need special care as built on a logarithmic scale; specifically, the features are normalized by applying base $10$ logarithm. The normalized features and targets read:\n\\begin{equation*}\n   \n    \\widetilde{\\mathbf x} = \\left[ \n    \\begin{array}{c}\n        \\frac{r - \\overline{r}}{\\sigma_r}, \\; \\frac{h - \\overline{h}}{\\sigma_h}, \\; \\frac{\\log_{10}(\\Peclet_g) - \\overline{\\log_{10}(\\Peclet_g)}}{\\sigma_{\\log_{10}(\\Peclet_g)}}\n    \\end{array} \\right], \\qquad\n    \\widetilde{\\mathbf y} = \\left[ \n    \\begin{array}{c}\n        - \\log_{10} (\\tau^*)\n    \\end{array} \\right],\n\\end{equation*}\nwhere $\\overline{r}$, $\\overline{h}$, $\\overline{\\log_{10}(\\Peclet_g)}$ are the mean of the training features $r$, $h$ and the logarithm of $\\Peclet_g$ respectively, while $\\sigma_r$, $\\sigma_h$ and $\\sigma_{\\log_{10}(\\Peclet_g)}$ are their standard deviations.\n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_ann2_predictions_r1-eps-converted-to.pdf}\n         \\caption{$\\tau_{\\mathrm{ANN}}$ for $r = 1$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_theory_predictions_r1-eps-converted-to.pdf}\n         \\caption{$\\widetilde{\\tau}_r$ for $r = 1$}\n    \\end{subfigure}\n   \n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_ann2_predictions_r2-eps-converted-to.pdf}\n         \\caption{$\\tau_{\\mathrm{ANN}}$ for $r = 2$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_theory_predictions_r2-eps-converted-to.pdf}\n         \\caption{$\\widetilde{\\tau}_r$ for $r = 2$}\n    \\end{subfigure}\n   \n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_ann2_predictions_r3-eps-converted-to.pdf}\n         \\caption{$\\tau_{\\mathrm{ANN}}$ for $r = 3$}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_theory_predictions_r3-eps-converted-to.pdf}\n         \\caption{$\\widetilde{\\tau}_r$ for $r = 3$}\n    \\end{subfigure}\n    \\caption{Stabilization parameter $\\tau_{\\mathrm{ANN}}$ predicted by the ANN and theoretical one $\\widetilde{\\tau}_r$ for varying $\\Peclet_g$ and $h$ at different FE degrees $r=1,2$, an $3$ for the 2D advection-diffusion problem of Section~\\ref{sec:training_2D}}\n    \\label{fig:td_2d_predictions_comparison}\n\\end{figure}\n\n\n\nIn order to find the best performing ANN architecture, we carried out a study by testing different numbers of hidden layers, nodes per layer, optimization algorithm, its learning rate, and batch size. Finally, we choose an ANN with $3$ hidden layers, $64$ nodes per layer and an output layer with a single node, all using a rectified linear unit (\\textit{ReLU}) activation function. We display the ANN architecture in Figure~\\ref{fig:ann_architecture}. We trained the ANN with the SGD optimization algorithm, a constant learning rate of $0.01$, and mini-batch size of $32$ examples. Moreover, we employed a momentum of $0.9$ in the optimization algorithm to update the weights. The trained ANN is available in the {\\tt GitLab} repository \\cite{ann_repo}.\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_ann2_predictions_r4-eps-converted-to.pdf}\n         \\caption{ANN's predicted $\\tau$ at $r = 4$}\n    \\end{subfigure}\n   \n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/td_theory_predictions_r4-eps-converted-to.pdf}\n         \\caption{Theoretical $\\tau$ at $r = 4$}\n    \\end{subfigure}\n    \\caption{Stabilization parameter $\\tau_{\\mathrm{ANN}}$ predicted by the ANN and theoretical one $\\widetilde{\\tau}_r$ for varying $\\Peclet_g$ and $h$ with FE degree $r=4$ for the 2D advection-diffusion problem of Section~\\ref{sec:training_2D}}\n    \\label{fig:td_2d_predictions_comparison_r4}\n\\end{figure}\n\n\\subsection{Predictions of the stabilization parameter by ANN}\nNow, we compare the predictions of the ANN with the theoretical stabilization parameter $\\widetilde{\\tau}_r$ of Eq.~\\eqref{eq:tau_theory_r} \\cite{Quarteroni_2017,galeao2004finite}. We show in Figure~\\ref{fig:td_2d_predictions_comparison} (left) the stabilization parameter $\\tau_{\\mathrm{ANN}}$ predicted by the ANN by varying mesh size $h$ and global P\\'eclet number $\\mathbb Pe_g$. For comparison, we report in Figure~\\ref{fig:td_2d_predictions_comparison}(right) the corresponding theoretical stabilization parameter $\\widetilde{\\tau}_r$. We observe that the overall behavior of the trained ANN's predictions are qualitatively similar of the theoretical stabilization parameter $\\widetilde{\\tau}_r$. Nevertheless, it can be inferred that the values of the stabilization parameters are almost completely matched with linear FE ($r=1$), while they quantitatively differ for $r=2$ and $r=3$. This was expected as $\\widetilde{\\tau}_r$ for $r>1$ is an empirical extension of the formula for the case $r=1$.\n\nThe ANN allows to make predictions with features outside the range of values for which it has been trained, that is for unseen values of such features. With this aim, we report in Figure~\\ref{fig:td_2d_predictions_comparison_r4} the comparison of the ANN's predictions $\\tau_\\mathrm{ANN}$ with $\\widetilde{\\tau}_r$ for the FE degree $r = 4$. This comparison shows a clear difference between the theoretical and ANN's $\\tau$ for $r = 4$ even though the general trend is maintained similar. \n\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/sol1_compare-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 2$, $h = \\sqrt{2}\/10$, $r = 1$}\n         \\label{fig:td_2d_solutions_extracted1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/sol2_compare-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 500$, $h = \\sqrt{2}\/20$, $r = 3$}\n         \\label{fig:td_2d_solutions_extracted2}\n    \\end{subfigure}\n    \\caption{Comparison of the solutions $u$ (blue), $\\widetilde{u}_h$ (green), and $u_h^\\mathrm{ANN}$ (red) of the 2D advection-diffusion problem along the line $(1-h,y)$ for any $y \\in [0,1]$}\n    \\label{fig:td_2d_solutions_extracted}\n\\end{figure}\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{ccc|cc|cc|}\n&  &  & \\multicolumn{2}{c|}{$\\widetilde{\\tau}_r$ ($e=\\widetilde u_h - u$)}\n& \\multicolumn{2}{c|}{${\\tau}_\\mathrm{ANN}$ ($e=u^\\mathrm{ANN}_h - u$)}\n\\\\\n$r$ & $h$ & $\\Peclet_h$ & $||e||_{L^2(\\Omega)}$ & $||e||_{H^1(\\Omega)}$ & $||e||_{L^2(\\Omega)}$ & $||e||_{H^1(\\Omega)}$ \n\\\\\n\\hline\n1 & $\\sqrt{2}\/10$ & 2 & $9.56 \\cdot 10^{-2}$ & $7.93 \\cdot 10^{-1}$ & $6.49 \\cdot 10^{-2}$ & $5.31 \\cdot 10^{-1}$\\\\\n2 & $\\sqrt{2}\/10$ & 2 & $6.48 \\cdot 10^{-2}$ & $5.31 \\cdot 10^{-1}$ & $6.50 \\cdot 10^{-2}$ & $5.33 \\cdot 10^{-1}$\\\\\n3 & $\\sqrt{2}\/10$ & 2 & $6.50 \\cdot 10^{-2}$ & $5.33 \\cdot 10^{-1}$ & $6.49 \\cdot 10^{-2}$ & $5.32 \\cdot 10^{-1}$\\\\\n1 & $\\sqrt{2}\/20$ & 500 & $2.99 \\cdot 10^{-4}$ & $5.30 \\cdot 10^{-3}$ & $3.11 \\cdot 10^{-5}$ & $5.50 \\cdot 10^{-4}$\\\\\n2 & $\\sqrt{2}\/20$ & 500 & $2.71 \\cdot 10^{-2}$ & $4.86 \\cdot 10^{-1}$ & $2.50 \\cdot 10^{-2}$ & $4.48 \\cdot 10^{-1}$\\\\\n3 & $\\sqrt{2}\/20$ & 500 & $1.05 \\cdot 10^{-2}$ & $1.90 \\cdot 10^{-1}$ & $1.68 \\cdot 10^{-3}$ & $3.46 \\cdot 10^{-2}$\\\\\n\\end{tabular}\n\\caption{Comparison of the errors in norms $L^2$ and $H^1$ between the numerical solutions ($u_h^\\mathrm{ANN}$ and $\\widetilde{u}_h$) and the exact one $u$ of the 2D advection-diffusion problem used for the training (Section~\\ref{sec:training_2D}) for different values of the features}\n\\label{tab:errors}\n\\end{table}\n\n\n\\subsubsection{Predictions for the problem used in the ANN's training}\nWe compare the numerical solutions $u^\\mathrm{ANN}_h$ and $\\widetilde u_h$ obtained by means of the SUPG stabilization method with the parameter $\\tau_\\mathrm{ANN}$ predicted by the ANN and the theoretical one $\\widetilde{\\tau}_r$, respectively; the comparison also involves the exact solution $u$~(\\ref{eq:td_2d_exact}) of the 2D advection-diffusion problem used for the training of the ANN. In particular, we display the comparison of the former 2D solutions in Figure~\\ref{fig:td_2d_solutions_extracted} along the line $(1-h,y)$ for any $y \\in [0,1]$ with: (a) $\\Peclet_h = 2$, $r=1$, and $h=\\sqrt{2}\/10$ (Figure~\\ref{fig:td_2d_solutions_extracted1}); (b) $\\Peclet_h = 500$, $r=3$, and and $h=\\sqrt{2}\/20$ (Figure~\\ref{fig:td_2d_solutions_extracted2}). We notice that \nin both the cases, the numerical solution $u_h^\\mathrm{ANN}$ involving the stabilization parameter $\\tau_\\mathrm{ANN}$ provides more accurate results than with the theoretical stabilization paramter $\\widetilde{\\tau}_r$. In particular, in the case (a), $\\widetilde{u}_h$ involves a much smoother boundary layer and overshoots the exact solutions $u$, conversely to the nearly nodally exact numerical solution $u_h^\\mathrm{ANN}$. In the case~(b),\n$u_h^\\mathrm{ANN}$ provides a much better representation of the bundary layer, without the undershooting of the solution $u$ exhibited by $\\widetilde u_h$. Furthermore, we report in Table ~\\ref{tab:errors} absolute errors between the numerical solutions ($u_h^\\mathrm{ANN}$ and $\\widetilde{u}_h$) and the exact one $u$. The errors, computed in $L^2(\\Omega)$ and $H^1(\\Omega)$ norms, show that the ANN-based SUPG stabilization method very often produces more accurate results than the ones obtained with theoretical stabilization parameter $\\widetilde{\\tau}_r$, especially for $r=3$.\n\n\n\\subsubsection{Predictions for an unseen problem}\n\\label{sec:newadvection-diffusion}\nWe check the model generalization of the ANN trained for the 2D advection-diffusion problem with exact solution~$u$ of Eq.~\\ref{eq:td_2d_exact} by predicting $\\tau_\\mathrm{ANN}$ for an unseen 2D advection-diffusion problem. In particular, we consider the advection-diffusion problem of Eq.~\\eqref{eq:transport_diffusion} in $\\Omega=(0,1)^2$, with $f=1$ and $\\boldsymbol{\\beta} = (1, 1)$.  We prescribe the following exact solution on the whole boundary $\\partial \\Omega$:\n\\begin{equation}\n    \\label{eq:td_2d_exact_new}\n    u(x, y) = \\frac{1}{2}(x+y) + \\frac{1 - \\frac{1}{2} (e^{(x \/ \\mu)} + e^{(y \/ \\mu)})}{e^{(1 \/ \\mu)} - 1}.\n\\end{equation}\n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/sol1_compare_new-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 2$, $h = \\sqrt{2}\/10$, $r = 1$}\n         \\label{fig:td_2d_solutions_extracted_new_1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/sol2_compare_new-eps-converted-to.pdf}\n         \\caption{$\\Peclet_h = 500$, $h = \\sqrt{2}\/20$, $r = 3$}\n         \\label{fig:td_2d_solutions_extracted_new_2}\n    \\end{subfigure}\n    \\caption{Comparison of the solutions $u$~(\\ref{eq:td_2d_exact_new}) (blue), $\\widetilde{u}_h$ (green), and $u_h^\\mathrm{ANN}$ (red) of the \\emph{unseen} 2D advection-diffusion problem along the line $(1-h,y)$ for any $y \\in [0,1]$}\n    \\label{fig:td_2d_solutions_extracted_new}\n\\end{figure}\n\nWe compare in Figure~\\ref{fig:td_2d_solutions_extracted_new} the numerical solutions $u^\\mathrm{ANN}_h$ and $\\widetilde u_h$ with the novel exact solution $u$. As for the previous numerical tests, the stabilization parameter $\\tau_\\mathrm{ANN}$ predicted by the network provides more accurate numerical solutions $u_h^\\mathrm{ANN}$ than with the theoretical stabilization parameter $\\widetilde{\\tau}_r$. In particular, the boundary layers and overall solution behaviours are better represented.\n\n\nWe better investigate assess the former qualitative consideration, by computing the error $E(\\tau)$ associated with the numerical solutions with SUPG stabilization for the parameters $\\tau_\\mathrm{ANN}$ and $\\widetilde{\\tau}_r$; different values of $\\Peclet_g$ and $r$ are considered. In particular, Figure~\\ref{fig:td_error_comparison_new} compares $E(\\tau)$ against $\\Peclet_g$ (in logarithmic scale) for different values of $r$: the error obtained by using $\\tau_\\mathrm{ANN}$ is always lower than the one achieved with $\\widetilde{\\tau}_r$. These results indicate that $\\tau_\\mathrm{ANN}$, although trained on a specific advection-diffusion problem, can be used to make predictions of the optimal $\\tau$ for an unseen advection-diffusion problem in place of the theoretical stabilization parameter. \n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/error2_r1-eps-converted-to.pdf}\n         \\caption{$r = 1$}\n    \\end{subfigure}\n   \n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/error2_r2-eps-converted-to.pdf}\n         \\caption{$r = 2$}\n    \\end{subfigure}\n    \\par\\bigskip\n    \\begin{subfigure}[b]{\\doublefigure}\n         \\includegraphics[width=\\textwidth]{figures\/ad\/error2_r3-eps-converted-to.pdf}\n         \\caption{$r = 3$}\n    \\end{subfigure}\n    \\caption{Comparison of errors $E(\\tau)$ in Eq.~(\\ref{eq:td_error_definition}) obtained with $\\tau_\\mathrm{ANN}$ (blue, full line) and $\\widetilde{\\tau}_r$ (orange, ashed line) applied to the useen 2D advection-diffusion problem with the exact solution $u$~(\\ref{eq:td_2d_exact_new}); $E(\\tau)$ against $\\Peclet_g$ with $h = \\sqrt{2}\/10$ and FE degrees $r=1,2$, and $3$}\n    \\label{fig:td_error_comparison_new}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this work, we presented an approach based on Machine Learning and ANN to compute the optimal stabilization parameter to be used in the SUPG FE approximation of advection-diffusion problems. Indeed, albeit the expression of the stabilization parameter is available for 1D problems and FE of degree $r=1$, its extension to more general advection-diffusion problems (2D and 3D) and FE degrees $r>1$ is still lacking.\n\nWe validated our approach against the 1D case, for which the ANN-stabilization parameter matches the already optimal, theoretical one for $r=1$, leading to nodally exact numerical solutions Instead, for higher polynomials degrees $r>1$, remarkable differences are observed among the theoretical and optimal stabilization parameter: we observed better accuracy and stabilization properties of the numerical solution with the optimized stabilization parameter with respect to the theoretical one. \n\nThen, we generated the dataset on a 2D advection-diffusion problem, and we used it to train a fully-connected feed-forward ANN. We applied the predictions of the network on the original advection-diffusion problem for unseen input values and we also apply it to an unseen advection-diffusion problem to check for model generalization.\nOur numerical results showed that the proposed ANN-based approach provides more accurate numerical solutions than using the theoretical stabilization parameter for the SUPG method.\n\n\nThis work represents a step towards the enhancements of stabilization methods for the FE approximation of advection-dominated differential problems. In particular, this work is limited to few features for the ANN-based stabilization parameter and provides as output a single optimal stabilization parameter meant for the whole mesh FE $\\mathcal T_h$. Natural extensions of this work will therefore involve local stabilization parameters, i.e. element by element over the mesh $\\mathcal T_h$, to better account for non-uniform meshes, differential problems with varying coefficients, and capturing the local behaviour of the solution. We envision extending the proposed Machine Learning approach to SUPG and variational multiscale stabilization methods for the FE approximation of the Navier-Stokes equations. \n\n\n\\section*{Acknowledgements}\nThe authors acknowledge Prof. C. Canuto, Politecnico di Torino, for the fruitful discussions and useful suggestions.  L.D. and A.Z. are members of the INdAM Research group GNCS. L.D. has been partially funded by the research project PRIN 2020, n.20204LN5N5, by MIUR.\n\n\n\\printbibliography\n\\clearpage\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHigh mass X-ray binaries (HMXBs) are among the brightest X-ray \nsources in the sky.  They consist of an accreting compact object (a neutron star \nor black hole) in orbit with an early type star (O or B star).\nThey represent an important stage in the evolution of binary stars with\nearly-type components, and they dominate the X-ray output of galaxies with\nyoung stellar populations.  HMXBs emit X-rays \nvia accretion onto the compact object by gas from the strong wind from the companion star, \nor from Roche lobe overflow. The X-rays interact with the accretion flow and stellar \nwind, producing observable signatures in timing and spectra.  This provides\na means to study the hot star wind and the accretion flow and the interaction\nof X-rays with these structures.  One consequence of this interaction is the\nproduction of polarized X-rays via scattering.  Polarization provides unique\ninformation into the geometry of the stars, wind and scattering region.\nIn this paper, we \nexplore the polarization signatures associated with this interaction.\n\nThere are $\\simeq$10 HMXB sources \nwith properties of their stellar winds and binary orbit which are well enough determined\nfor quantitative modeling of the interaction of the accretion flow with X-rays \\citep{Kape98}.\nThe number of known HMXBs is many times greater when sources with less well constrained properties are included\n\\citep{Liu06,Walt15}; these include sources discovered by INTEGRAL which are seen primarily at hard X-ray energies\nowing to obscuration, and transient sources which are detected primarily via long time-baseline\nmonitoring.    The most thoroughly studied systems  include the  objects Cyg X-1, Vela X-1, and Cen X-3. \nThe majority of HMXBs contain pulsating neutron stars.  If so, orbital eclipses, X-ray pulse timing and primary radial velocities\n\\citep{vand07} provides strong  constraints on the orbital \nseparations and masses.  Simple estimates suggest that the \naccretion rate onto the compact object can be provided by the wind from the \ncompanion star in fewer than half the sources,  while Roche lobe overflow is \nrequired in the remainder \\citep{Cont78, Whit85,Kape98}.   Though they are outnumbered \nby low mass X-ray binaries in our galaxy, these sources dominate the X-ray \noutput from galaxies with larger star formation rates \\citep{Fabb06}.\nTheir mass transfer can be very rapid; in a system in which the primary has a radiative envelope\nand fills its Roche lobe, the primary radius will shrink as a consequence of mass transfer,\nbut if the mass transfer is conservative the Roche lobe will shrink more rapidly, potentially\nleading to a common envelope phase.  If the primary is an evolved star with a convective envelope\nthis outcome is more probable \\citep{Taam00}.  Common envelope evolution when the primary is an evolved star\nis likely to lead to ejection of much of the envelope of the primary, leaving a binary consisting of a\nneutron star and the core of the primary.   \nWhether due to nuclear evolution or to unstable mass transfer, their evolutionary lifetimes \nmust be short ($\\leq 10^5$ yrs).    \n\nThe wind from the companion absorbs and scatters the X-rays from the accreting \ncompact object in HMXBs, and the column densities traversed by the X-rays \nrange from $\\sim 10^{21} - 10^{24}$cm$^{-2}$.  The X-ray source luminosities \nrange from $\\sim 10^{35} - 10^{38}$ erg s$^{-1}$.  In sources where the X-ray \nsource is luminous and the wind is weak, the wind is ionized throughout most \nof the region between the two stars.  In systems with more massive winds \nand weaker X-ray sources X-ray-ionization effects are a perturbation \non the wind properties.\nKey questions about HMXBs include the nature of the compact object: its mass, \nvariability and intrinsic radiation pattern.  Also of interest are the properties of \nthe companion star wind and its uniformity in density and temperature.\nThe X-ray source can provide a probe for the \nstudy of wind regions which are only weakly affected by ionization or the \ngravity of the compact object.   \n\n\nMany of the X-ray properties of HMXBs are affected by the stellar wind and accretion flow. \nThis includes the variability around the orbit, which shows gradual eclipse transitions due \nto photoelectric absorption in the wind \\citep{Clar88}, enhanced absorption at late \norbital phases due to a wake or stream trailing the compact object \\citep{Kall82, Wata06}.  \nThe wind and accretion flow also provide torque to the compact object and regulate its \nspin or angular momentum.   The structure of the wind and accretion flow is \nuncertain; winds from comparable single O or B stars have radiatively driven winds \\citep{Lame99}\nwhich are affected by instabilities \\citep{Owocki88}, shocks \\citep{LucyWhite80}  and clumping \\citep{Cohen06};\nobservations provide the most reliable means for understanding these processes, via fitting \nof parameterized  models.  In addition, the compact object affects the gas flow, via its gravity\nand also X-ray heating and ionization.  \n\nThe next new astrophysical window will be the advent of measurements \nof X-ray polarization \\citep{Jaho14,Weis13}.  Among other things, polarization allows for potentially \nsensitive tests of the geometry of astrophysical sources, on scales which are \nfar too small to be imaged directly.  \nThere is only one source in the sky whose X-ray polarization is known, the \nCrab nebula \\citep{Weis78}.  Owing to visibility constraints, it is likely \nthat the first astronomical X-ray polarimetry observations will be of objects \nwhich have never before been observed with this technique.   This motivates \nthorough and accurate modeling of the polarization properties of the brightest \nand (otherwise) best understood X-ray sources, for use as calibrators and test sources for X-ray polarimetry. \nHMXBs are among the sources best suited for this.  Their orbital elements are relatively well understood\nand their orbital variability provides a predictably changing view with respect to an important source\nof polarization: the strong stellar wind from the companion star.   \nX-ray polarimetry provides a means to probe the structure and the physical processes \noccuring in and near compact objects, on length scales too small to be imaged directly.\nIn spite of scanty astrophysical detections so far \\citep{Weis78} it is of interest to \nconsider the possible signals and diagnostic use of X-ray polarization in known classes \nof cosmic sources, as it is likely that instruments with improved sensitivity\nwill become available eventually.\n\nPolarization properties of HMXBs have been discussed previously in the context \nof scattering of optical and UV from the primary star \\citep{Brow77, Brow78, Rudy78}. \nThese authors predicted that the linear polarization has a characteristic variability around the \nbinary orbit, associated with scattering in the stellar envelope which is distorted by \nthe gravity of the compact object.  This variability can be used as a diagnostic of the \nshape of the star, and also of the inclination of the binary orbit.   These effects are observed \nat the predicted level in some systems; their absence in other cases suggests that the optical light \nmay be affected by contributions from structures whose shape is not symmetric around the line \nof centers.  Related work has been carried out by \\citet{Alma99,Igna09,Nofi14}.\nUseful formalism for the calculation of polarization has been provided by \\citet{Matt96}.\nThe polarization properties of the accretion columns in X-ray pulsars have been calculated by\n\\citep{Mesz88}. Predictions of X-ray polarization by the large scale structures in binaries has been carried out for\ndistorted stars \\citep{Ange69}; for Compton scattering within the accretion column of magnetic cataclysmic variables\n\\citep{Mcna08}, following up on pioneering work by \\cite{Matt04}.\n\nX-rays have an important advantage over optical light which is that the source is almost certainly \ncompact compared with the size of the binary system.  That is, we expect X-rays to be emitted from \na region smaller than the Alfven shell, defined as the distance from the compact object where \nthe pressure due to a dipole magnetic field balances the ram pressure of accretion.  \nFor fields $\\sim 10^{12}$ Gauss this region has a size $\\sim 10^8$ cm \\citep{Lamb73}.\nFar outside this region the influence of the field on the dynamics is negligible.\nThe Alfven shell size can be compared with the primary star radius and orbital separation\nwhich are  $\\sim 10^{12}$ cm.  This disparity in sizes greatly simplifies the geometrical \nconsiderations associated with the use of X-ray polarization since the X-ray source is\neffectively a point source when considering the effect of the scattering by the stellar wind.\n Calculating the polarization from these structures is straightforward, \ngiven certain simplifying assumptions, though it is important to take into account the \neffects of atomic absorption and resonance line scattering, which in turn depends on the \ngas dynamics,  along with electron scattering.   \n\n\nThe polarization properties of HMXBs will be affected by the intrinsic polarization \nof the compact X-ray source, and by the polarization imprinted by the stellar \nwind and accretion flow.  These can be distinguished by their differing  \nvariability behavior:  the compact source generally varies on a pulsation  timescale for\nsources containing a pulsar, which is $\\sim$ seconds -- minutes.\nThe very few sources which contain black holes also have their strongest intrinsic variability\non comparably short timescales.  Wind variability, on the other hand,\nis associated with the orbital timescale ($\\sim$ days) or possibly on the wind flow timescale which is\n$\\geq$hours.   In some HMXBs the X-ray source is luminous enough to ionize the wind almost completely, \nso the light  observed during and near eclipse, and its polarization, is \ndominated by electron scattering.\n\nIn this paper \nwe present a general discussion of the behavior of X-ray polarization from HMXBs \nas a function of the parameters of the system and the viewing position.  \nWe present calculations of the polarization \nsignature for various simple analytic models for the wind density and ionization structure,\nand discuss the dependence of these on parameters:  wind optical depth, inclination, and\nviewing direction or orbital phase.   \nWe also explore the effects of wind hydrodynamics, i.e. departures from spherical symmetry due\nto the effects of the X-ray source gravity and heating.  We utilize sample three-dimensional\ndynamical models.   We take the \nintrinsic polarization of the compact object as unpolarized and explore the combined effects of\ngeometry and scattering physics on the predicted linear polarization in the 0.1 - 10 keV X-ray band.\n\n In section \\ref{sec1} we present numerical calculations \nfor spherical winds using only electron scattering.  The effects of resonance line scattering\nare discussed in section \\ref{sec3}.\nIn section \\ref{sec3b} we present models which include photoelectric absorption \nand an ensemble of resonance scatterers.\nIn section \\ref{sec4} we present models utilizing three-dimensional hydrodynamic models for the \nwind density and velocity field.  We use these to derive approximate predictions for the polarization\nlevels expected for several well known systems in section \\ref{sec5}.  The Appendix presents\nsimple analytic estimates of the polarization for idealized HMXB conditions;\nmany of these mirror earlier results.   The computational techniques and level of detail we employ \nare similar to those of our previous work \\citep{DoraKall10} on the polarization from warm absorbers in \nSyefert galaxies.\n\n\n\\section{Spherical Winds}\n\\label{sec1}\n\nThe basic geometry of an HMXB is illustrated in figure \\ref{fig1}.\nThis shows  a schematic of an HMXB with a typical orbital separation and a sample \nX-ray ionized zone.  It also shows the geometry we use when calculating the\npolarization:  the X-ray source is at the origin and the primary orbits in a plane\ninclined by an angle $i$ to the line of sight.  The orbital phase is described by the angle $\\Theta_V$ relative\nto the line of centers.  The system is viewed along the y axis.\n\nIn this section we will consider the simple case in which the wind density is spherically\nsymmetric about the primary star, and the only interaction between the X-rays and\nthe wind is electron scattering.  If so, the scattering phase function follows the\nRayleigh form \\citep{ChandraRadTransfer}.  The scattered emissivity at each point\ndepends only on the local gas density and on the flux from the X-ray source.\nWe assume the X-ray source radiates isotropically with a constant total luminosity\n$L_0$ and is unpolarized.\n\nWe calculate the polarization signatures in the form of the three Stokes parameters for linearly polarized \nlight  using the formal solution to the equation of transfer \\citep{Miha78}.\n\n\\begin{equation}\n\\label{eq1}\n\\left\\{\n\\begin{matrix}\nL(\\varepsilon)  \\\\\nQ(\\varepsilon)  \\\\\nU(\\varepsilon) \n\\end{matrix}\n\\right\\}\n= \\int dV \\kappa(\\varepsilon,{\\bf r}) S(\\varepsilon,{\\bf r}) e^{-\\tau(\\varepsilon,{\\bf r})}\n\\left\\{\n\\begin{matrix}\n1 + \\cos^2\\chi  \\\\\n\\sin^2\\chi \\cos(2\\gamma) \\\\ \n\\sin^2\\chi \\sin(2\\gamma)  \n\\end{matrix}\n\\right\\}\n\\end{equation}\n\n\\noindent where $\\varepsilon$ is the photon energy, $S(\\varepsilon,{\\bf r})$,  is the source function, $\\kappa(\\varepsilon,{\\bf r})$\nis the opacity,  $\\tau(\\varepsilon,{\\bf r})=\\int{\\kappa(\\varepsilon,{\\bf r}) d\\zeta}$ is the optical\ndepth from a point ${\\bf r}$ to a distant viewer,\n$\\chi$ is the scattering angle and $\\gamma$ is the angle between the scattering \nplane and a reference direction on the sky.  Here and in what follows we describe the Stokes parameters in terms of the\nluminosity seen by a distant observer, i.e. the total energy (in ergs s$^{-1}$ sr$^{-1}$) radiated by the\nsystem in that direction.  Equation \\ref{eq1} defines the scattered luminosity, $L$\n(polarized plus unpolarized); observations are also affected by an\nunscattered component, $L_u(\\varepsilon)=L_0(\\varepsilon)e^{-\\tau(\\varepsilon,{\\bf r_x})}$\nwhere ${\\bf r_x}$ is the position of the X-ray source.  For the purpose of evaluating these\nquantities we adopt a cylindrical coordinate system with the X-ray source at the origin \nand the viewing direction along the $\\hat{\\bf y}$ axis.  The angle $\\gamma$ corresponds \nto the azimuthal angle on the plane of the sky and is measured relative\nto a line which is the intersection between the orbital plane and the plane of the sky.\nIn the case of pure electron scattering the source function is $S=L_0\/(4\\pi r_x^2)$ where \n$r_x$ is the distance from the X-ray source and  the opacity is $\\kappa(\\varepsilon,{\\bf r})=n({\\bf r}) \\sigma_{Th}$.\n\nSome useful results are presented in the Appendix.  The fractional polarization relevant to observation is\n$P=\\sqrt{(Q^2+U^2)}\/(L+L_u)$.  This quantity is zero when the system is viewed at inclination\n$i=\\pi\/2$ and at either conjunction, i.e. orbital phase 0.5 or 1, $\\Theta_V=0$ or $\\pi$.\nThe polarization at $i=\\pi\/2$ is a maximum at quadrature, phase 0.25 or 0.75.\nWe can also define the polarization of the scattered radiation only,\ni.e. $P_s=\\sqrt{(Q^2+U^2)}\/L$, and the value of this quantity at quadrature and $i=\\pi\/2$ depends only on\nthe orbital separation and on the distribution of gas density in the wind.\nThe value of $P$ is less than the value of $P_s$ by a factor proportional to the Thomson depth \nwhenever the X-ray source is not eclipsed. \nAt $i=0$, i.e. when the system is viewed face-on, the polarization rotates with orbital phase at a rate\ntwice the orbital rotation rate.\n\nAs an illustration of these results, we calculate the polarization produced by a  single-scattering\ncalculation by numerically evaluating equation \\ref{eq1}.  The wind velocity is assumed to be radial\nrelative to the star with a speed given by $v(r)=v_0+(v_{\\infty}-v_0)(1-r_{*}\/r)$\nwhere the terminal velocity is $v_{\\infty}$=1000 km s$^{-1}$ and $v_0$=100 km s$^{-1}$, and\n$r_{*}=10^{12}$ cm.  The wind mass loss rate is $\\dot M_{wind}=10^{-8} M_{\\odot} {\\rm yr}^{-1}$.\nThe orbital separation is $a=1.5 r_{\\*}$.\n\nIn the remainder of this paper we discuss quantitative calculations of polarization produced by wind scattering.  These\ninclude calculations based on both the very simple spherical wind with parameters given in the previous paragraph\nand also on hydrodynamic calculations of the wind density structure which make no assumptions about spherical symmetry.\nIn the former case, it is illuminating to estimate the accretion rate onto the compact object if the accretion is supplied\nsolely by the wind.  This can be done using simple estimates based on  \\citet{Bond44}  and \\citet{Davi73}, i.e.\n$\\dot M_{acc}=\\pi R_{acc}^2 n_x v_x m_H$ where the accretion radius is $R_x=2GM\/v_x$, $M$ is the mass of the compact\nobject and $v_x$ is the speed of the wind at the X-ray source.  This corresponds to\n$\\dot M_{acc}\/\\dot M_{wind}= 1.8 \\times 10^{-4} (M\/M_\\odot)^2 v_{x8}^{-4} a_{12}^{-2}$  where $v_{x8}=v_x\/(1000 {\\rm km s}^{-1})$,\n $a_{12}=a\/(10^{12} {\\rm cm})$ or an X-ray luminosity $L_x\\simeq 1.8 \\times 10^{35} {\\rm erg s}^{-1}$ using the\nparameters given in the previous paragraph and assuming an efficiency of converting accreted mass into energy of 0.1.\nThis luminosity is less than the time averaged luminosities of most HMXBs by factors $\\sim$5 -- 100, which likely reflects\nthe fact that wind mass loss rates may be greater, additional mass can be supplied by Roche lobe overflow, and the\nwind and accretion flow dynamics can be enhanced by the influence of the compact object gravity and ionization.\nThe results in what follows, those which are based on the simple spherical wind approximation, must\ninterpreted subject to this caveat.\n\nAn additional caveat which applies to essentially all of the results presented in this paper is the assumption of single\nscattering.  The validity of this assumption depends on the wind optical depth from the X-ray source;\nmultiple scattering is important when this quantity approaches or exceeds unity.  For parameters similar\nto those discussed so far, this quantity for a distant observer corresponds to\n$\\tau_{Th} \\sim n_x a \\simeq 2 \\times 10^{-4} \\dot M_8 a_{12}^{-1} v_{x8}^{-1}$,\nwhere $\\dot M_8= \\dot M\/(10^{-8}M_{\\odot}{\\rm yr}^{-1})$.  Comparing this expression\nwith the accretion luminosity estimate given above implies that multiple scattering can be important when\nthe accretion luminosity exceeds $\\sim10^{38} {\\rm erg s}^{-1}$ for a 1 $M_\\odot$ compact object.   The likely effect of\nmultiple scattering is  to produce smaller net polarization than for single scattering, since the second and subsequent\nscatterings have a wider range of scattering angles and planes than for single scattering.  If the\ndepth is moderate, i.e. $\\leq$ 10, there will still be a significant fraction of photons which reach the\nobserver after a single scattering.  If so the net polarization will likely be less than predicted here,\nby factors of order unity, and our results should be modified to include multiple scattering effects for such sources.  \n\nMaps of polarization projected against the sky\nare shown in figure \\ref{fig2}.  This shows contours of constant intensity\nas solid colors separated by solid black curves, along with lines corresponding to polarization vectors,\nwhich appear as dashed curves.  These are plotted vs position in units of $10^{12}$ cm.\nThese illustrate many of the results presented in the Appendix:  At high inclination, $i=\\pi\/2$,\nand at orbital phase angles 0 and $\\pi$ (orbital phases 0 and 0.5) polarization vectors are perfectly\ncircumferential, contours of constant intensity are also circular and the net polarization is zero.\nAt orbital phase angles $\\pi\/2$ and $3\\pi\/2$ (orbital phases 0.25 and 0.75) the net polarization is\nmaximum.  \nAt inclination $i=0$ the star always influences the polarization,\nso that the net polarization fraction is constant, but the position rotates with the orbit.\n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=0, scale=0.5]{fig1.ps\n\\caption{\\label{fig1} Schematic of high mass X-ray binary (HMXB) as viewed from above the orbital plane,\n showing typical orbital separation and companion \nstar size. Coordinates are labeled according to their use in the Appendix.}\n\\end{figure*} \n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=0, scale=0.5]{fig2.ps\n\\caption{\\label{fig2} Map of intensity and polarization vectors projected on the sky\n  for four orbital phases and two inclinations.  Colors correspond to scattered intensity\n  according to the color bar, in which the labeled quantities correspond to log$_{10}$ of the specific\n  intensity.  Short black lines correspond to polarization vectors.\n  Spherically symmetric wind is assumed.} \n\\end{figure*} \n\nMore insight comes from the Stokes quantities as a function of the orbital phase.  These\nare shown in figure \\ref{fig3} for inclinations $i=0$ and $i=\\pi\/2$.  U is green while Q is shown\nin red; solid is $i=\\pi\/2$ and dashed is $i=0$.  This further illustrates the results from the\nprevious figure:  at $i=0$ the polarization oscillates between Q and U along orbital phase, and the\nvector sum is constant.  At $i=\\pi\/2$ U is always small (in this convention) and Q is a maximum\nat phase angles $\\pi\/2$ and $3\\pi\/2$.\nAnother way of displaying the same thing is shown in\nfigure \\ref{fig4}, which shows U plotted vs Q, for $i=0$ in green, and $i=\\pi\/2$ in red.  This\nshows that the trajectory of U vs. Q is circular at $i=0$ and becomes linear at $i=\\pi\/2$.  As shown by\n\\citet{Brow78}, this trajectory is an ellipse for intermediate inclinations, and the cases\nin figure \\ref{fig3} are the extremes of eccentricity.   This has been suggested as a means\nfor measuring inclination \\citep{Rudy78}.   \n\n\\begin{figure*}[p] \n\\includegraphics*[angle=90, scale=0.3]{fig3.ps\n\\caption{\\label{fig3} Stokes parameters for spherical wind, Q (red ) and U (green)  vs. phase\n  for inclinations $i=0$ (dashed)  and $i=\\pi\/2$ (solid).  The total intensity I  is shown as dashed black for\n  $i=0$.  At $i=0$ the polarization oscillates between\nQ and U along orbital phase, and the vector sum is constant.\nAt $i=\\pi\/2$ U is always small (in this convention) and Q is a maximum \nat phase angles $\\pi\/2$ and $3\\pi\/2$.}\n\\end{figure*} \n\n\n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=90, scale=0.4]{fig4.ps\n\\caption{\\label{fig4} Stokes parameters for spherical wind, Q vs U $i=0$ (green)\n  and $i=\\pi\/2$ (red).  This again illustrates the circular path of the net polarization\ndirection with orbital phase at $i=0$ and the constancy of the direction at $i=\\pi\/2$.}\n\\end{figure*} \n\nFigure \\ref{fig5} shows the scattered polarization fraction vs. orbital phase,\ni.e. $P_s=\\sqrt{(Q^2+U^2)}\/L$.  This shows that the\nmaximum polarization fraction is $\\simeq$10$\\%$ for the parameters chosen here, when the\npolarized component is compared with the total scattered component.  This is comparable to the\nresult in the Appendix, calculated for a thin shell at the star rather than for an extended wind.\nWe emphasize that this value depends primarily on geometric quantities:  the extent of the wind, and\norbital separation relative to the stellar radius.  It does not depend on the wind\ndensity or column density since it is a comparison of scattered quantities.  The difference between\n$i=0$ and $i=\\pi\/2$ is clearly apparent:  the former  produces approximately constant polarization fraction\nand the latter oscillates between 0 and a value comparable to the $i=\\pi\/2$ value.\nFigure \\ref{fig6} shows the polarization fraction measured relative to the total radiation, scattered\nplus direct, i.e. i.e. $P=\\sqrt{(Q^2+U^2)}\/(L+ L_u)$.  This illustrates the diluting effect of the direct\nradiation for this model, which has $\\tau_{Th}\\simeq 0.25$.  Maximum linear polarization at $i=\\pi\/2$occurs\njust following the eclipse transition, where the departures from circular symmetry on the sky are not negligible,\nand where the direct radiation is blocked by the primary star.  At mid-eclipse the polarization at $i=\\pi\/2$ is\nzero due to symmetry.\nFigure \\ref{fig7} shows the polarization angles from the same set of models.  The\nangle sweeps through 180 degrees twice per orbital period for $i=0$, while the\nangle is constant (though undefined near conjunctions) for $i=\\pi\/2$.\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.4]{fig5.ps\n\\caption{\\label{fig5}  Polarization fraction of scattered radiation i.e.\n  $P_s=\\sqrt{(Q^2+U^2)}\/L$ for spherical wind, $i=0$ (red)\n  and $i=\\pi\/2$ (black).}\n\\end{figure*} \n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.4]{fig6.ps\n\\caption{\\label{fig6}  Polarization fraction relative to total radiation, scattered plus direct,\n  for spherical wind, $i=0$ (red)\n  and $i=\\pi\/2$ (black).}\n\\end{figure*} \n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.5]{fig7.ps\n\\caption{\\label{fig7} Polarization angle for spherical wind, $i=0$ (red)\n  and $i=\\pi\/2$ (black).}\n\\end{figure*} \n\n\n\\section{Resonance Line Scattering: Single Line}\n\\label{sec3}\n\nThe models discussed in section \\ref{sec1} include solely electron scattering;\nthey do not include resonant scattering in bound-bound transitions.  This\nprocess (when associated with UV and optical transitions) is the dominant driving\nmechanism for the winds from early type stars.  Here we consider\nscattering in the X-ray band.   In the absence of X-ray ionization, the ions\nwhich are most abundant in early-type star winds are in charge states from\n$\\sim$1 -- 5 times ionized, and do not have many strong X-ray resonance\nline transitions.   Ionization by a compact X-ray source can produce\nions of arbitrary charge state, depending on the X-ray flux and gas density,\nand so resonance scattering can affect the scattered  X-ray intensity and\npolarization.  The ionization structure in HMXB winds has been explored by\n\\citet{Hatc77} who showed that the surfaces of constant ionization in a spherical\nwind are either nested spheres surrounding the X-ray source, or else open surfaces\nenclosing the primary star.  The ionization distribution depends on a parameter\n$q=\\xi\/\\xi_x$ where $\\xi_x=L_x\/(n_xa^2)$ where $L_x$ is the ionizing X-ray luminosity,\n$n_x$ is the gas density at the X-ray source and $a$ is the orbital separation.\nThis is a particular example\nof the scaling of ionization in optically thin photoionized gas, which depends on\nthe ionization parameter $\\xi=L_x\/(n(r) r_x^2)$ \\citep{Tart69,KallmanBautista01}.\nThe properties of resonance scattering as applied to X-ray polarization\nin HMXBs are dominated by the fact that,\nowing to the restricted regions where ionization is suitable,\nmost lines can only have strong opacity\nover a fraction of the wind.  The regions where this occurs most often resemble\nspherical shells surrounding the X-ray source.\n\nResonance scattering differs from electron scattering in its sensitivity to the velocity\nstructure of the wind.  At a given observed photon energy, scattering can occur only\nover a resonant surface with shape determined by the wind velocity law; in a\nspherical wind with monotonic velocity law these surfaces are open surfaces of revolution\nsymmetric about the line of sight to the X-ray source.\n\nResonance scattering affects polarization by \nredistributing the polarization among the components\nof the Stokes vector according to the phase matrix which is a linear\n combination of the Rayleigh and isotropic  phase matrices; coefficients \nof the matrix depend on the angular momentum quantum \nnumbers of the initial and final states of the transition, $j$ \\citep{LeeBlandfordWestern94}.  \nThe matrices describing this process have been calculated by \\citet{Hamilton47}.  \nIn practice most of the strongest X-ray resonance lines have initial and final values\n$j_{lower}=1\/2$ and $j_{upper}=3\/2$, which correspond to the Rayleigh phase matrix,\nand we adopt this in our calculations.  Similar assumptions were employed in \n\\citet{DoraKall10}.\n\nPolarization also depends on the scattering angle, and the resonance\ncondition constrains the scattering geometry, thereby imposing a relation\nbetween the energy of the scattered photon  and its polarization.\nResonance scattering can have a cross section which is much  greater than\nelectron scattering, but only over the relatively narrow energy band spanned by the\nresonance line, including the effects of Doppler broadening associated with the\nwind motion.  Within this band, the polarization can be greater than that produced by\nelectron scattering, owing to the greater cross section, given suitable geometry.  \n\nIn the remainder of this section we illustrate these effects by generalizing\nthe single scattering spherical wind calculations to include resonance\nscattering.  We consider a single resonance line, chosen to crudely resemble\na line such as O VIII L$\\alpha$.  The parent ion is assumed to exist over\na range of ionization parameter $1\\leq {\\rm log}(\\xi) \\leq 3$.\nWe calculate the optical depth using the Sobolev expression \\citep{Castor}:\n\n\\begin{equation}\n\\label{equation2}\n  \\tau_{line}(\\varepsilon,{\\bf r})=\\frac{\\pi e^2}{mc} f n({\\bf r})x_{ij}y_j \\frac{\\lambda r}{v(r)(1+\\frac{z^2}{r^2}\\left(\\frac{d{\\rm ln}v(r)}{d{\\rm ln}r}-1\\right)}\n  \\end{equation}\n\n\\noindent where $f$ is the oscillator strength, $n({\\bf r})$ is the gas number density, $x_{ij}$ is the ion fraction, $y_j$ is the\nelement abundance, $z$ is the position along the line of sight, and $\\lambda$ is the line wavelength.  For the source\nfunction we adopt the same expression used for electron scattering $S=L_0\/(4\\pi r_x^2)$ since thermalization is unimportant and the\nsize of the X-ray source is small compared with the other length scales of interest for HMXBs.\n\n\n\nResults of our simple single line calculation are shown in figure \\ref{fig8}.\nThis shows the line profile as a function of orbital phase for a system viewed edge on ($i=\\pi\/2$).\nFor each phase we display the luminosity, with the transmitted luminosity shown in black and the\nscattered luminosity shown in green (bottom panel) and polarization fraction (top panel).\nThe polarization angle is not shown because there is no significant dependence on energy\nor orbital phase when the system is viewed at high inclination; the polarization is always\nperpendicular to the orbital plane.  The horizontal axis is energy in units of the wind terminal\nvelocity.\n\nAt phase 0 (superior conjunction of the X-ray source) the shape of the profile in\nluminosity is similar to P-Cygni profiles familiar from UV resonance\nlines in hot stars:  the outflow produces blue-shifted absorption (shown as negative energy in these\nfigures) of the continuum.  This absorption is offset from the zero of energy owing to the fact that\nthe continuum source in this case is the compact X-ray source, and the wind speed at the X-ray source\nis approximately half the terminal speed.  In addition, the X-ray source creates an ionized region\nwithin which line scattering cannot occur.  As a result,\nthe absorption occurs in a relatively narrow region of energy near the energy corresponding to the\nwind terminal velocity.  The scattered emission, on the other hand, is essentially symmetric in energy, since\nscattered emission comes from a region which is not necessarily along the line of sight to the X-ray source.  \nProjection effects make the scattered emission appear at all energies $|\\varepsilon| \\leq \\varepsilon_0 v_\\infty\/c$\nwhere $\\varepsilon_0$ is the line rest energy.  The scattered emission is unpolarized at phase 0 for the same\nreason as in the pure electron scattering case.\nAt phase 0.25 (quadrature) the wind is viewed perpendicular to its velocity vector at the X-ray source,\nand therefore the absorption covers a large fraction of the energy spanned by the wind.\nThe scattered emission is polarized up to $\\sim$50$\\%$; the degree of polarization is symmetric around\nthe center of the line, owing to the fact that the wind velocity structure is symmetric around the line\nof centers.  At phase 0.5 (inferior conjunction of the X-ray source)\nthere is no transmitted flux (due to occultation by the star) and the emission is predominantly red-shifted\nowing to the location of the X-ray ionized zone in the receding part of the wind.  In this case,\neven though the flux is all scattered, the polarization fraction is negligible owing to the circular\nsymmetry of the scattering region of the wind as viewed in the plane of the sky.\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.5]{fig8.ps\n\\caption{\\label{fig8} Spectrum, polarization fraction and angle in a resonance line vs. energy\n  in units of the terminal velocity at orbital phase 0.25 for inclination $i=\\pi\/2$}\n  \\end{figure*} \n\n\n\\section{Absorption Effects}\n\nThe ionization balance in an HMXB wind is determined at each point by the\nX-ray flux and by the gas density.   The X-ray flux in turn depends\non the effects of geometrical dilution and on attenuation. \nIn order to do so, we utilize results from {\\sc xstar} \\citep{KallmanBautista01} at each point\nin the wind in order to determine the opacity.   This is done along\nradial rays originating from the X-ray source, so that the opacity\nat each point is calculated using an ionization parameter which corresponds\nto the flux transmitted along the corresponding ray.\n\nThis single stream\ntransfer treatment is similar to the transfer treatment used by  {\\sc xstar}\nwith one difference:  {\\sc xstar} uses the local spectral energy distribution (SED)\nto calculation the ionization along a ray; here we use the same unattenuated\nSED everywhere for the ionization calculation (so we can use a stored table),\nbut we calculate the ionization parameter self-consistently, i.e. by calculating the \ntransmitted flux at each position and then using that value to calculate the ionization parameter.   \nThis simplification allows for more efficient computation without unduly sacrificing physical realism.\nThis approximation will be least accurate when the column density is highest, i.e.\n$\\geq 10^{23}$ cm$^{-2}$.  In such regions, the ionization is expected to be low,\nowing to attenuation, and X-ray scattering will be relatively unimportant.  Our\napproximation will tend to produce less attenuation at a given column density\nthan a self-consistent calculation would.\n\n\\section{Resonance Line Scattering:  Ensemble of Lines}\n\\label{sec3b}\n\nX-ray ionization creates an ensemble of resonance lines in an HMXB wind from the many\ntrace elements such as C, N, O, Ne, Mg, Si, S, and Fe.  At\nany point in the wind the ionization depends most sensitively on the ionization parameter,\ndefined in the previous section.  Figure \\ref{fig9} shows the distribution of ionization\nparameter produced in a wind with a density distribution which is spherically symmetric\naround the primary star, as discussed so far. \nWe adopt a spherical wind with mass loss rate $10^{-5} M_\\odot {\\rm yr}^{-1}$, corresponding to a\ntotal wind column density of $3 \\times 10^{23}$ cm$^{-2}$, and an X-ray source luminosity of $10^{38}$ erg s$^{-1}$.\nEach spatial region with a distinct ionization parameter will have a different distribution of ion abundances,\nand hence a unique distribution of resonance line opacity.  These can be calculated\nunder the assumption of ionization equilibrium.  Figure \\ref{fig10} \nshow examples of the spectrum and the polarization fraction produced by\nsuch a distribution from the spherical wind shown in figure \\ref{fig9}.\nIonization and line opacities were calculated as a function\nof ionization parameter using the {\\sc xstar} code \\citep{KallmanBautista01}.\nThese were binned into 1000 energy bins over energy from 0.1 eV to 10$^4$ eV.\nSince the Doppler shifts associated with the wind speed is less than the energy bin size,\nwe adopt an approximate form for the source function each resonance line:\n\n\\begin{equation}\n  S(\\varepsilon,{\\bf r})=\\frac{L_0}{4\\pi r_x^2}  \\frac{v_{\\infty}-v_0}{\\Delta \\varepsilon c\/\\varepsilon}\n\\end{equation}\n\n\\noindent where the quantity\n$(v_{\\infty}-v_0)\/(\\Delta \\varepsilon c\/\\varepsilon)$ takes into account the fact that the line\ndoes not conver the entire energy bin.  This treatment assumes that the line optical depths are not large,\nwhich is an adequate approximation for our situation.  \nDepolarization effects at large Sobolev optical depths associated with multiple scatterings are not taken into account.\nThese  calculations serve to illustrate the magnitude of the polarization effects expected from\nresonance scattering in HMXBs.  Simulations suitable for quantitatively diagnosing the wind or the X-ray source properties\nwill require reexamination of these assumptions.\n\nResults of spectra and polarization fractions are shown in figure \\ref{fig10}.\nThese demonstrate that the resonance lines cover a significant fraction of the X-ray energy\nband and that they can scatter a significant flux of X-rays, and create polarization fractions\nas high as 0.5, much greater than would be produced by electron scattering alone.\nA difference between resonance scattering and electron scattering is that resonance\nscattering in a given line occurs within a relatively narrow spatial region where the parent\nion is most abundant.  For X-ray lines, these are most likely to be approximately spherical\nsurface surrounding the X-ray source.  This region tends to produce a small polarization which\nis almost independent of the viewing angle.   Thus resonance scattering produces\nweaker modulation  of the polarization with orbital phase than does electron scattering.\n\nIt is also worth noting that the effects of scattering can be either polarizing or not polarizing,\ndepending on the scattering angle.  Also, lines which appear in absorption in the\nspectrum can have enhanced polarization in their troughs owing to the presence of scattered light\nin the residual intensity.  In the spectrum shown in figure \\ref{fig10} the polarization\nin the absorption lines is generally lower than in the emission features, or in the adjacent\ncontinuum.  This is due to the fact that the scattered light in the residual intensity is\nforward scattered, and so is not polarized because of the scattering angle.  \n\n\\begin{figure*}[p] \n\\includegraphics*[angle=90, scale=0.5]{fig9.ps\n\\caption{\\label{fig9}Distribution of ionization parameter in the orbital plane\n  for a spherical wind with $\\dot M_{wind}=10^{-5} M_\\odot {\\rm yr}^{-1}$\n   illuminated by a 10$^{38}$ erg s$^{-1}$ X-ray source.  Contours are\n  labeled with log$\\xi$.}\n\\end{figure*} \n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.5]{fig10.ps\n\\caption{\\label{fig10} Spectrum and polarization fraction vs energy\nfor spherical wind in units log(E\/keV) at orbital phase 0.25 for inclination $i=\\pi\/2$}\n  \\end{figure*} \n\n\n\n\n\n\n\n\\section{Hydrodynamic Models}\n\\label{sec4}\n\nThe stellar wind in HMXBs is not spherically symmetric. Physical processes \nwhich affect the wind and accretion flow in HMXBs include: the three-dimensional \ngeometry of the flow (i.e. the flow both in and out of the \norbital plane), rotational forces, the influence of gravity and radiation \npressure from both stars in the binary, transport of the radiation \nfrom the stars into the flow and the reprocessed radiation out of the \nflow, X-ray heating and ionization, and departures from thermal equilibrium \ndue to advection and adiabatic heating and cooling.  The dynamics\nof X-ray heated winds have been discussed by \\citet{Fran80}, and \\citet{Hatc77}, \nand multi-dimensional models have been \ncalculated by \\citet{Blon90, Blondin94, Mauc08}\n\nIn this section we illustrate the effects of the wind hydrodynamics on the \npolarization. We use two sample wind hydrodynamic models calculated using \na numerical model similar to that described in \\citet{Blon09}\nbut extended to three dimensions.  The hydrodynamic models \nare computed on one hemisphere of a spherical grid, assuming reflection \nsymmetry about the orbital plane.  A non-uniform grid of 448 (r) by \n128 ($\\theta$) by 512($\\phi$) zones is used with highest resolution near the \nsurface of the primary star and in the vicinity of the accreting neutron \nstar.  This model calculates the wind dynamics in three dimensions taking \ninto account the radiative driving by the UV\/optical radiation from \nthe primary, and also the gravity of the star and the compact X-ray source. \nA fixed X-ray luminosity of 10$^{36}$ erg\/s is used to calculate X-ray heating \nusing the approximate formulae given in Blondin (1994) and the effects \nof X-ray ionization on the dynamics via changes in the UV radiation \nforce multiplier.  The two models have identical dimensions:\nprimary radius 2.4 $\\times 10^{12}$ cm and orbital separation  3.6 $\\times 10^{12}$ cm.\nThey differ in their mass loss rates, which are 4 $\\times 10^{-7} M_\\odot {\\rm yr}^{-1}$ and\n1.7 $\\times 10^{-6} M_\\odot {\\rm yr}^{-1}$.  The maximum wind speed in\nboth models is approximately 1600 km s$^{-1}$.\nA plot showing the density contours and velocity vectors\nis shown in figure \\ref{fig11}.  This\nclearly shows the influence of the gravity of the compact object in creating a region of higher\ndensity and non-radial wind flow in the vicinity of the X-ray source.\n\nThe distorted stellar wind of the hydrodynamic models is illustrated with velocity \nvectors in the orbital plane and a series of transparent density isosurfaces.\nRight panel is the model with a higher mass loss rate (Mdot = 1.7e-6); left panel\nis a lower mass loss rate (Mdot = 4.0e-7).  The lowest density isosurface corresponds\nto a density of 4e9 cm$^{-3}$ in both panels.  Velocity vectors are not shown for values\nless than 800 km\/s.\n\nThe high mass loss rate model has a denser spherical component of the wind, but a\nless pronounced accretion wake.  The low mass loss rate model has a larger volume\nof wind moving at low velocity due to photoionization of the wind in the vicinity \nof the X-ray source.  The result is\na denser wind coming off the primary along the line of centers of the binary system\nand a larger, denser accretion wake that wraps more tightly around the primary star.\n\n\n\nWe apply the same calculation of X-ray scattering to the hydrodynamic wind\nas was done in section \\ref{sec3}. We assume an X-ray source\nluminosity of $10^{36}$ erg s$^{-1}$ and a $\\Gamma$=2 power law ionizing spectrum.\nThe distribution of ionization parameter in the orbital plane is shown in figure\n\\ref{fig12}.  This shows a spiral structure, owing to a corresponding structure\nin the gas density.\n\nFigures \\ref{fig13}  shows the spectra and polarization produced\nby the hydrodynamic wind model with   $\\dot M=4.0 \\times 10^{-7} M_\\odot {\\rm yr}^{-1}$\nat orbital phases 0, 0.25 and 0.5.  Comparison\nwith figure \\ref{fig10} shows that the hydrodynamic model produces a spectrum which is \n similar to that from \na smooth spherical wind. However, the degree of polarization is significantly \ngreater, due to the departure from spherical symmetry created by the X-ray \nsource. In particular, the spectrum during eclipse is greater, owing to the \nfact that the wake structure extends beyond the disk of the primary star \nduring eclipse. The spherical wind produces fractional polarization averaged \nover the 0.1 \u2013 10 keV energy band, of at most 10$\\%$ during eclipse. The spherical \nwind produces polarization values which are much less than this at mid-eclipse. \nThe hydrodynamic wind produces fractional polarization of approximately 21$\\%$ at mid-eclipse.\n\nPolarization calculations for hydrodynamic models have been carried out for\nthe two mass loss rates shown in figure \\ref{fig11}, and for several choices\nof ionizing X-ray luminosity.  Note that the hydrodynamic models assume a fixed X-ray \nluminosity (used to calculate X-ray heating and ionization within the simulation) \nof 10$^{36}$ erg\/s, independent of the X-ray luminosity used to calculate the \nspectrum and polarization.  Moreover, the prescribed X-ray luminosity is \nnot generally consistent with the mass accretion rate derived from the hydrodynamic \nsimulation, and thus these models are not fully self-consistent. Nonetheless, \nthey serve to illustrate the behavior of the polarization and its dependence \non wind density and X-ray luminosity.  \n\nValues for polarization fractions during\neclipse are shown in table \\ref{table1}.  These are averages of the linear polarization\nover energy in the range from 0.1 -- 10 keV.  Polarization values reflect competing effects.\nFor highly ionized gas, the wind is essentially transparent and the polarization is due to\nThomson scattering.  The polarization fraction is proportional to the Thomson depth (when small)\nand also to departures from circular symmetry of the scattering material in the plane of the sky.\nPartially ionized gas can produce larger polarization, owing to the greater cross section associated\nwith resonance scattering, although each line has very limited spectral range and the ensemble of\nlines for any given model seldom has a width which exceeds $\\Delta \\varepsilon\/\\varepsilon \\sim 0.1$.  On the other hand,\npartially ionized  and near-neutral gas also is affected by photoelectric absorption.  This limits\nthe size of the X-ray scattering region, and tends to reduce the net polarization.\nTable \\ref{table1} shows that, for the limited range of parameters spanned by our models,\nmore highly ionized models tend to produce greater polarization.  That is, the effects of photoelectric\nabsorption offset the effects of resonance scattering in partially ionized gas, leading to very weak\ndependence of the polarization on X-ray luminosity when the luminosity is not large.\nWe expect that at very low X-ray luminosities this effect will be even stronger, since the\npartially ionized zone containing the resonance scattering gas will shrink and become more round,\nand photoelectric absorption will remove a larger fraction of photons.\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=0, scale=0.4]{fig11a.ps\n\\includegraphics*[angle=0, scale=0.4]{fig11b.ps\n\\caption{\\label{fig11} Density contour and vector plot of wind hydrodynamic\n  models.   Right panel is model with\n  $\\dot M=1.7 \\times 10^{-6} M_\\odot {\\rm yr}^{-1}$\n  left panel is model with\n  $\\dot M=4.0 \\times 10^{-7} M_\\odot {\\rm yr}^{-1}$.  Spatial scale is\n  the same for both panels.  Yellow contour corresponds to a density of\n  $6 \\times 10^8 {\\rm cm}^{-3}$. }\n\\end{figure*} \n\n\\begin{figure*}[p] \n\\includegraphics*[angle=90, scale=0.5]{fig12.ps\n\\caption{\\label{fig12}Distribution of ionization parameter in the orbital plane\n  for hydrodynamic wind illuminated by a 10$^{36}$ erg s$^{-1}$ X-ray source.  Contours are  labeled with log$\\xi$.}\n\\end{figure*} \n\n\\begin{figure*}[p] \n\\includegraphics*[angle=270, scale=0.5]{fig13.ps\n\\caption{\\label{fig13}  Spectrum and polarization fraction vs energy\n  at phase 0, 0.25, 0.5 for hydrodynamic wind with $\\dot M=4.0 \\times 10^{-7} M_\\odot {\\rm yr}^{-1}$\n  illuminated by a 10$^{36}$ erg s$^{-1}$ X-ray source.}\n\\end{figure*} \n\n\n\\begin{figure*}[p] \n\\includegraphics*[angle=90, scale=0.6]{fig14.ps\n\\caption{\\label{fig14} Diagram showing estimated polarization fractions in lines and continuum at conjunction vs. quadrature for\nthe objects in table \\ref{table2}.}\n\\end{figure*} \n\n\n\\section{Discussion}\n\\label{sec5}\n\nOur results so far on the polarization produced by wind scattering in HMXBs can be summarized as follows:\nPolarization depends on inclination: high inclination produces variable polarization fraction plus\nconstant angle; low inclination produces constant polarization fraction plus variable polarization position angle.\nThe maximum attainable continuum polarization fraction scales approximately proportional to electron scattering optical depth, for\ndepths less than unity; this is partly a consequence of our single scattering assumption and must be suitably modified\nif multiple scattering is important.\nAt high inclination, polarization fraction of the scattered radiation ($P_S$) at phase 0 and 0.5 is less than \nat phases 0.25 or 0.75; for a spherical wind the phase 0 and 0.5 polarization is zero.\nPolarization fraction of the total radiation, including unscattered, is small for the parameters considered\nhere for all orbital phases out of eclipse.  A spherical wind thus produces narrow intervals of high polarization\nduring eclipse but away from phase 0.\nResonance line optical depths are greater than for electron scattering, and so can produce greater\nlinear polarization.  On the other hand, the optical depths are often large, and the\nX-ray scattering regions tend to be small, i.e. nearly circular around the compact object, producing\nsmaller orbital phase modulation.  The hydrodynamics of the interaction between the wind and the compact object breaks\nthe spherical symmetry and increases the net polarization.  \n\nPredictions of the polarization signatures for particular known HMXBs require\nhydrodynamic simulations for each system incoporating known physical parameters:  the\nsizes and masses of the components, the primary wind mass loss rate and the X-ray source\nluminosity and spectrum.  These would then need to be analyzed using a transfer calculation\nsuch as the single scattering calculations we have presented.\nIn this paper we have presented generic models for the wind densities and X-ray source properties,\nboth for spherical winds and incoporating the hydrodynamic effects of the\ntwo gravitating centers.  These generic models can be used to infer, in a very simple way, the\npolarization signatures expected from various sources.\n\nTable \\ref{table2} shows the known parameters of the 5 brightest and best-studied HMXBs.   The\nrelevant quantities are the source X-ray luminosity, primary type, mass loss rate, and average\nX-ray luminosity.  These quantities are taken from \\citet{Cont78} and from\n\\citet{Kape98}.  We do not include several systems which generally have smaller observed X-ray fluxes\nor less well constrained properties.\n\nWith these quantities we can use our spherical wind models to\ncrudely predict the polarization fractions and orbital phase modulation of the polarization for these\nsources.  We do this in the following way:  we use the results in table \\ref{table1}\nto construct an approximate scaling of mid-eclipse net polarization with the wind mass loss rate\nand X-ray luminosity.   We then apply this to the known HMXBs in table 1.   The results\nare shown graphically in figure \\ref{fig14}.  The most relevant quantities for each of the table 1 HMXBs, X-ray\nluminosity and wind mass-loss rate, are plotted on the axes and the values for each object are plotted\nas points.  Contours of constant predicted mid-eclpise polarization are shown as solid curves, and labeled.  This\nshows that high polarizations are expected for some objects, those with the strongest winds and weakest\nX-rays generally.  This demonstrates that polarization fractions in the range\nfrom 5 -- 30 $\\%$ are expected at mid-eclipse for these systems.  More detailed information is contained in the\nspectra and the time variation of the net polarization during eclipse; interpretation of these signals\nrequires modeling which is tailored to each particular system.\n\n\\acknowledgements  Support was provided through grant 10-ATP10-0171 through the NASA astrophysics theory program. \n\n\n\\begin{deluxetable}{crrr}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Hydrodynamic Model Results.  Values for the polarization fraction are given for several hydrodynamic\n  models described in the text. \\label{table1}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Model}&\\colhead{$\\dot M$}&\\colhead{L$_{x}$}&\\colhead{P$_{\\phi=0.5}$} }\n\\startdata\n  &$M_\\odot {\\rm yr}^{-1}$ & erg s$^{-1}$&\\\\\n1& 4 $\\times 10^{-7}$ & 10$^{38}$&0.25\\\\\n2& 4 $\\times 10^{-7}$ & 10$^{37}$&0.25\\\\\n3& 4 $\\times 10^{-7}$ & 10$^{39}$&0.29\\\\\n4& 1.7 $\\times 10^{-6}$ & 10$^{38}$&0.22\\\\\n\\enddata\n\\end{deluxetable}\n\n\\begin{deluxetable}{crrr}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Sample HMXBs and Properties Needed for Predicting Wind Polarization.  \\label{table2}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{object}&\\colhead{Sp. type}&\\colhead{$\\dot M_{wind}$}&\\colhead{L$_x$} }\n\\startdata\n  &  &$M_\\odot {\\rm yr}^{-1}$&erg s$^{-1}$\\\\\nVela X-1&B0.5 Iab&7 $\\times 10^{-6}$&0.05\\\\\nCen X-3&O6.5 II-III&2 $\\times 10^{-7}$&0.3\\\\\nCyg X-1&O9.7 Iab&2.5 $\\times 10^{-6}$&0.1\\\\\n4U1700-37&O6.5 Iaf+&1 $\\times 10^{-5}$&0.01\\\\\nsmc x-1&B0.6 Iab&5 $\\times 10^{-7}$&3 \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{secintro}\n\\IEEEPARstart{T}HE Internet of Things (IoT) is the interconnection of network-enabled devices communicating with each other over the Internet ~\\cite{Da14,Oteafy17,Deng18,Adhatarao18}. Hence, the purpose of IoT is to integrate the physical world and the virtual world to attain a self-sustaining system ~\\cite{Da14,Oteafy17,Deng18,Adhatarao18}. To achieve this integration in the near future, research into the application of IoT for the creation of smart-cities, smart-homes, smart-energy, smart-transportation and many more is growing ~\\cite{Da14,Oteafy17,Deng18,Adhatarao18}. In IoT, access and interaction between devices are seamless due to the use of unique identifiers, sensors and communication technologies ~\\cite{Ciuonzo12,Ciuonzo14,Da14,Oteafy17,Deng18,Adhatarao18}. The evolution of IoT started from wireless sensor networks (WSN) and developed into different heterogeneous networks interacting with each other over the internet ~\\cite{Ciuonzo12,Ciuonzo14,Rossi16,Deng18}. A wireless network of spatially distributed autonomous devices acting as sensors to monitor and record physical or environmental conditions (e.g.s humidity, pressure, temperature, power-line voltages, and vital body functions) is called a WSN ~\\cite{Raghavendra06,Yick08,Ciuonzo12,Ciuonzo14,Rossi16,Yetgin17}. These sensor nodes may range from a few hundreds to thousands in a sensor network ~\\cite{Raghavendra06,Yick08,Yetgin17}. Each sensor node is equipped with an antenna, microcontroller, interfacing electric circuit, and an energy source ~\\cite{Raghavendra06,Yick08,Yetgin17}. \n\nWhile a WSN and other sensor networks operate in their own private networks, IoT integrates the various sensor networks into an IoT network by providing them access to the internet ~\\cite{Adhatarao18}. For an IoT network, by the use of routing schemes, a gateway supports communication between the sensor nodes and a central system where actions are taken based on the information received from the sensor network ~\\cite{Raghavendra06,Yick08,Ciuonzo12,Ciuonzo14,Rossi16,Yetgin17}. The gateway of a sensor network within an IoT network communicates with the central system over the internet. The routing schemes facilitates either direct or relayed communication between a gateway and a particular sensor node ~\\cite{Raghavendra06,Yick08,Yetgin17}. Relays not only facilitate information forwarding, but may lead to an increase in a communication system's throughput ~\\cite{Bai15,Simmons16,Guo16,Song17}. Relays, however, depend on their resources (i.e., power and computational components) to aid the processing and transfer of signals ~\\cite{Bi15,Lu15,Guo16}. However, sensor nodes in IoT networks have limited resources, especially power ~\\cite{Raghavendra06,Yick08,Yetgin17,Guo16,Adhatarao18}. Energy harvesting (EH) can be used to reduce the strain on the power resource of the relaying sensor nodes. By harvesting energy from a portion of the received radio frequency (RF) signal at the relay node, the harvested energy can be a source of power for information signal relaying ~\\cite{Ulukus15,Ding15,Guo16}. \n\nThe technique used for EH from RF signals is known as wireless power transfer (WPT) ~\\cite{Zhou16,Song17,Mahama17}. WPT involves the wireless transfer of electrical energy from a power source to a load ~\\cite{Mahama17}. EH is the process of scavenging energy from an external energy source ~\\cite{Zhou16,Song17,Mahama17}. Typical traditional energy sources include hydro, wind, solar and wind, however, these sources of energy are less stable due unruly factors such as weather ~\\cite{Mahama17}. Therefore, RF energy signals can serve as a more stable source of electrical energy for self-sustaining cellular systems ~\\cite{Zhou16,Song17,Mahama17}. This implies that, EH from RF via WPT can be used in sensor network to aid sensor nodes with power constraints when operating as relays ~\\cite{Guo16}. WPT implementation in cellular communication systems is accomplished using two main approaches, namely, wireless powered communication networks (WPCN) and simultaneous wireless information and power transfer (SWIPT) \\cite{Tang18,Asiedu18,Mahama17,Lee16,Yin17,Lee18}. SWIPT involves the transmission of wireless information signal and wireless power signal concurrently ~\\cite{Mukhlif18,Jameel17,Perera17}. Time switching (TS) and power splitting (PS) are the two main techniques by which SWIPT is implemented ~\\cite{Mukhlif18,Jameel17,Perera17}. The successive transmission of wireless information signal and wireless power signal is the technique used to accomplish WPCN ~\\cite{Ramezani17,Niyato16}. \n\nA few investigations on dual-hop SWIPT relay node system configurations are presented in ~\\cite{Mahama17,Asiedu18,Do17} and ~\\cite{Liu17}. Using outage probability and throughput as performance matrices, the research in ~\\cite{Mahama17} optimizes the PS ratio for a single amplify-and-forward (AF) relay node facilitating communication between two nodes. In ~\\cite{Mahama17}, it is assumed that each relay node's battery is charged with the energy it harvests from a portion of the received RF signal. In addition, all communicating nodes were equipped with a single antenna. The work in ~\\cite{Mahama17} also included an extension into AF-SWIPT relay node selection. In ~\\cite{Asiedu18}, the authors  considered a single antenna AF dual-hop configuration with multiple relay nodes. Both the power control factors and the PS ratios for each relay node were optimized with the objective of maximizing the achievable rate. The work presented in ~\\cite{Do17} focused on a non-orthogonal multiple access (NOMA) system where a MIMO base station (BS) communicates with a single antenna cell-center user and a single antenna cell-edge user. The authors proposed three cooperative downlink (DL) transmission schemes based on hybrid SWIPT and BS antenna selection. The hybrid SWIPT powered relaying between the BS and cell-edge user is done by the cell-center user. The authors in ~\\cite{Do17} acquired closed-form solutions for outage probability for their proposed schemes. The schemes were then compared to both orthogonal multiple access and non-NOMA systems in their simulation results. In addition, the authors presented a discussion on the diversity gains and complexity requirements of the various proposed schemes. A two-hop multi-antenna decode-and-forward (DF) SWIPT relay scenario is investigated in ~\\cite{Liu17}. The PS ratio and power allocation at the DF-SWIPT relay node were optimized to maximize the end-to-end system throughput. The authors in ~\\cite{Liu17} proposed an optimal clustering algorithm and a greedy clustering algorithm which grouped the multiple antennas of the DF-SWIPT relay node into information detection and energy harvesting antenna sets. \n\nSWIPT schemes in multi-hop relay systems are researched in ~\\cite{Chen17}, ~\\cite{Xu17}, ~\\cite{Mao15}, and ~\\cite{Liu19}. The work in ~\\cite{Chen17} presents a novel multi-hop relay transmission strategy, where energy is harvested by the source and the relay nodes from the co-channel interference. The PS ratio for the source and AF-SWIPT multi-hop relay nodes were optimized based on a given outage probability threshold. In ~\\cite{Chen17}, the authors identified the maximum number of AF-SWIPT multi-hop nodes that can support information transfer between a source node and a destination node. The research in ~\\cite{Xu17} studied the coexistence of primary users (PUs) and secondary users (SUs). The PUs and SUs interacted in an energy harvesting cognitive radio network implemented using time division multiple access (TDMA) technique. The multi-hop relaying scenario in ~\\cite{Xu17} occurs during data transmission between the SU nodes. The SUs harvest energy from the PU's RF signals. By employing an iterative algorithm, the end-to-end throughput of the SUs is maximized based on the time and power resource optimization  in ~\\cite{Xu17}. A DF-SWIPT and an AF-SWIPT multi-hop relay configurations considering both the TS ratio and the PS ratio techniques are investigated in ~\\cite{Mao15}. The authors in ~\\cite{Mao15} aimed to find the maximum number of nodes which support communication between a source and a destination given a rate threshold constraint. A SWIPT AF multi-hop system with multiple-input-multiple-output (MIMO) relay nodes is investigated in ~\\cite{Liu19}. In ~\\cite{Liu19}, the achievable rate was maximized by optimizing the source and relay beamforming vectors, and the power resource of each node. \n \nIn this paper, we investigate a DF sensor network, where a single antenna source node communicates with a single antenna destination node through multi-hop single antenna relay nodes. By adopting the PS ratio scheme, the multi-hop relay nodes operate in the SWIPT mode. Each relay node uses a portion of the RF signal it receives for EH, and it is saved in a supercapacitor. Unlike ~\\cite{Chen17} and ~\\cite{Xu17} which harvest energy from their co-channel interference and primary users, our model considers energy harvesting from only a portion of the RF signal received from the previous node. The harvested energy in the supercapacitor is used to both decode the rest of the RF signal, and forward the decoded information signal to the next node. An example in which our system model occurs is when the BS is requesting data from the destination wireless sensor node ~\\cite{Raghavendra06,Sabor17,Djiroun17,Sandeep17}. The BS sends the request command to the destination wireless sensor via routing through the DF wireless sensor relay nodes. Another application of our system model involves device-to-device (D2D) communication between nodes facilitated by energy harvesting. Instead of relaying information, each node harvests energy from the RF signal it received from a previous node to power its own information signal transmission to the next node. Unlike the work in ~\\cite{Mao15}, where the maximum number of hops is determined based on an available source power and throughput constraint, we solve the DF system throughput maximization and the source power minimization problems. In addition, we found a closed-form solution for determining the maximum number of DF-SWIPT relay nodes which can support communication between the source and the destination nodes given an SNR threshold constraint and an available power source value. Our close-form solution differs from the approach presented in ~\\cite{Chen17} and ~\\cite{Mao15}, which uses an algorithm to determine the maximum number of relay nodes. The main contributions of this paper are as follows.\n\n\\begin{itemize}\n\\item First, we seek to identify the minimum amount of source transmit power which supports communication between the source node and destination node via the DF-SWIPT multi-hop relay nodes. By tackling this problem, we reduce the strain (i.e., the depletion of the source power resource) on the source power during routing of information in the IoT network. From our source power minimization problem, we derive closed-form solutions for determining the minimum required transmit power at the source and the PS ratios for the DF-SWIPT relay nodes. \n\\item We further consider the maximization of the minimum system achievable throughput also based on an already specified number of DF-SWIPT relay nodes and a source transmit power constraint. The maximization of the minimum system achievable rate is to aid in improving the system rate with different individual quality-of-service (QoS) constraint (i.e., the SNR thresholds) within the IoT network. We then derive closed-form solutions for the optimal PS ratios for each DF-SWIPT relay node, and the optimum system achievable throughput.\n\\item Using the optimal solutions for the source transmit power minimization problem and the optimal system achievable throughput problem, we show in this work that with a general SNR threshold constraint for all nodes within the IoT network, our two optimization solutions become equivalent. We treat equivalency of our solution due to the general SNR threshold consideration as a special scenario. \n\\item From the special case, we present a closed-form solution to determine the number of relay nodes which support communication between the source node and destination node for a homogeneous sensor network. This closed-form solution can be used in a routing algorithm for our proposed system model. The prediction of the DF-SWIPT number can aid in resolving coverage, connectivity and routing issues in the IoT sensor networks ~\\cite{Yetgin17,Mhatre04}.\n\\item Using the closed-form optimal PS ratio solution, we propose a centralized and a distributed method by which the PS ratio of each node can be determined in a real-world sensor network. In addition, we compare our two PS ratio determination methods in terms of complexity.\n\\item Since in practical systems channel estimation may not be perfect, within the simulation section of this paper we discuss the effects of imperfect channel estimation on the performance of our discussed optimization problems. \n\\end{itemize}\nFrom the remarks of authors in ~\\cite{Yick08} and ~\\cite{Yetgin17} concerning major issues on sensor networks, we can state the following advantages of implementing our proposed system model in a sensor network\\footnote{The current issues as stated in ~\\cite{Yick08} and ~\\cite{Yetgin17} concerning sensor networks are with the limitations on the node powers, the computational power of each sensor node, the storage capacity of each sensor node, and the QoS requirement for the system.}. The application of our proposed sensor network scenario and optimization solutions presented in this work can be implemented in large or small sensor node network types. This is due to the low computational complexity of the closed-form solutions presented in this paper which is important considerations for sensor networks. Another advantage of this work is the possibility of lengthening network lifetime for a sensor network using energy harvesting. We also covered QoS requirement constraint (i.e. throughput), and resource limitation constraint (i.e. transmit power, computational and data storage capabilities) in this work. To the best of our knowledge, this is the first work to analyze both the source transmit power minimization problem and the optimization of the system achievable throughput problem for a DF-SWIPT multi-hop relay system model. We compare the optimal closed-form solutions to the fixed PS ratio scheme in our simulation results. The results affirmed that the optimal scheme outperformed the suboptimal scheme in terms of the minimum source power and the system achievable throughput. \n\nThe rest of the paper is organized as follows: Section ~\\ref{secsystem_model} presents the system model and optimization problem statement. Closed-form solutions for the optimization problems are discussed in Section ~\\ref{secoptimization}. Section ~\\ref{secsimulation} discusses our numerical results, and conclusions are drawn in Section ~\\ref{secconclusion}.\n\n\\emph{Notations}: $n \\sim \\mathcal{CN}(0,\\delta^{2})$ denotes a circularly symmetric complex Gaussian random variable, $n$, with zero mean and a variance of $\\delta^{2}$. $\\mathop{\\mathbb{E}}_{X}[f(X)]$ is the expectation operation over random variable $X$. $f(X)$ and $R(X)$ represent a general function and a rate function respectively which are dependent on the variable $X$.  \n\\section{System Model and Problem Formulation}\n\\label{secsystem_model}\nA sensor network consisting of a source, $K$ multi-hop DF-SWIPT relay nodes, and a destination is investigated in this paper as shown in Fig. ~\\ref{figSystems}. Each node is equipped with a single antenna. The source is a BS which may consist of the data gateway and the external\/central systems of the sensor network ~\\cite{Raghavendra06,Sabor17,Ciuonzo14,Rossi16,Djiroun17,Sandeep17}. The multi-hop DF-SWIPT relay nodes are the wireless sensor nodes found in the mesh network. This is due to the wireless sensor nodes being able to act as repeaters and relays ~\\cite{Raghavendra06,Sabor17,Djiroun17,Sandeep17}. The destination node is also a wireless sensor node within the network ~\\cite{Raghavendra06,Sabor17,Djiroun17,Sandeep17}. To reduce the strain on the relay node's resource (i.e., the power resource) during the relaying process, the relay nodes operate in the SWIPT mode. During the SWIPT mode, each relay utilizes their EH interfacing electrical circuit and a supercapacitor to facilitate the EH process from a portion of the RF signal it receives. Each relay node then facilitates the information decoding and forwarding process of the rest of the RF signal with the harvested energy.\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.3in]{systemmodel}\n\\end{center}\n\\caption{Multi-hop DF relay systems with SWIPT architecture.}\n\\label{figSystems}\n\\end{figure}\n\nThe SWIPT architecture of our DF relay nodes in the sensor network system model is presented in Fig. ~\\ref{figSWIPTarch}. The source, $\\mathcal{S}$, and the destination, $\\mathcal{D}$, already possess their own energy source for communication. Since each DF-SWIPT relay node uses a supercapacitor, each relay node dissipates all its harvested energy for information decoding (ID) and retransmission. Each DF-SWIPT relay node undergoes EH via the PS technique and operates in the half-duplex mode. We assume that the source node knows the channel state information (CSI) for all nodes communicating, while each DF-SWIPT relay node and destination node have knowledge of only the CSI for their communicating channels\\footnote{The CSI for the sensor network can be acquired during the training phase for channel gain estimation when sensor nodes send pilots to each other and the BS (i.e., gateway and central system unit) ~\\cite{Zhao04,Taricco12,Wang18}.}. It is assumed that, there is no direct link between the source and the destination nodes. Also, we assume that there is no direct link between the relay nodes. For example, from Fig. ~\\ref{figSystems}, no direct link exists between $\\mathcal{R}_{1}$ and $\\mathcal{R}_{3}$. The assumption of no direct link holds for the worst case scenario, that is, when the distance between nodes is large. This assumption can be justified by virtue of routing from using a routing algorithm in the sensor network ~\\cite{Liu19,Qiao13,Wu18}. The detailed operation of the considered system model will now be presented.\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.4in]{SWIPTArchitecture}\n\\end{center}\n\\caption{Multi-hop DF relay node SWIPT architecture.}\n\\label{figSWIPTarch}\n\\end{figure}\n\nThe received RF signal at node $k$ from the previous node is given as\n\\begin{equation}\ny_{k}=\\sqrt{E_{k-1}}h_{k}\\hat{x}_{k}+n_{k}, \\text{ }k=1,\\ldots,K+1,\n\\end{equation}\nwhere $h_{k}$ is the channel coefficient between the current node and the previous node, and $n_{k}\\sim \\mathcal{CN}(0,\\delta^{2}_{k})$ represents the antenna noise at the current node. $\\hat{x}_{k}$ stands for information signal from the preceding node. $K+1$ denotes the total number of subsequent nodes after the source node (i.e. the total number of DF-SWIPT relay nodes and the destination node). The channel $h_{k}$ is modeled as $h_{k}=\\sqrt{\\xi_{k}}\\tilde{h}_{k}$, where $\\xi_{k}$ is the large-scale fading coefficient and $\\tilde{h}_{k}\\sim \\mathcal{CN}(0,1)$ represents the small-scale fading component with Rayleigh distribution ~\\cite{Zhang13,Lee17}. The large-scale fading coefficient is modeled as $\\xi_{k}=C_{k}\\Big(\\frac{d_{k}}{d_{0}}\\Big)^{-\\alpha_{k}}$, where $C_{k}$ is the constant attenuation for a reference distance $d_{0}$ ~\\cite{Zhang13,Lee17}. $\\alpha_{k}$ and $d_{k}$ indicate the pathloss exponent, and the distance between the transmit and receive nodes respectively ~\\cite{Zhang13,Lee17}.\n\nNext, the received RF signal at node $k$ is then split into two based on the PS ratio, $\\rho_{k}$, for EH and ID. The EH and ID signals at node $k$ are respectively written as\n\\begin{equation}\n\\label{eq1}\ny^{EH}_{k}=\\sqrt{\\rho_{k}}\\Big(\\sqrt{E_{k-1}}h_{k}\\hat{x}_{k}+n_{k}\\Big),\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq2}\ny^{ID}_{k}=\\sqrt{1-\\rho_{k}}\\Big(\\sqrt{E_{k-1}}h_{k}\\hat{x}_{k}+n_{k}\\Big)+z_{k},\n\\end{equation}\nwhere $z_{k}\\sim \\mathcal{CN}(0,\\sigma^{2}_{k})$ is the additional noise introduced by the ID circuitry.\n\nFrom (~\\ref{eq1}), the harvested energy, $E_{k}$, at node $k$ is expressed as\n\\begin{equation}\n\\begin{aligned}\n\\label{eq3}\nE_{k}&\\quad=\\beta_{k}\\mathop{\\mathbb{E}}_{\\hat{x}_{k},n_{k}}\\Big[\\vert y^{EH}_{k}\\vert^{2}\\Big]\\backsimeq\\beta_{k}\\rho_{k}E_{k-1}\\vert h_{k}\\vert^{2}\\\\&\\quad=E_{0}\\Gamma_{k}\\prod^{k}_{j=1}\\rho_{j}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\n\\end{equation}\nwhere $\\Gamma_{k}\\triangleq\\prod^{k}_{j=1}\\beta_{j}\\vert h_{j}\\vert^{2}$, $0<\\beta_{k}\\leq 1$ is the energy conversion efficiency of the $k$th node, and $E_{0}$ is the source transmit power. Also, from (~\\ref{eq2}), the achievable rate, $R_{k}$, at node $k$ becomes\n\\begin{equation}\n\\begin{aligned}\n\\label{eq4}\nR_{k}&\\quad=\\log_{2}\\Bigg(1+\\frac{E_{k-1}\\vert h_{k} \\vert^2(1-\\rho_{k})}{(1-\\rho_{k})\\delta^{2}_{k}+\\sigma^{2}_{k}}\\Bigg)\\\\&\\quad\\backsimeq\\log_{2}\\Bigg(1+\\frac{E_{k-1}\\vert h_{k} \\vert^2}{\\sigma^{2}_{k}}(1-\\rho_{k})\\Bigg)\\\\&\\quad=\\log_{2}\\Bigg(1+\\frac{E_{0}\\Gamma_{k}}{\\sigma^{2}_{k}\\beta_{k}}\\prod^{k-1}_{j=1}\\rho_{j}(1-\\rho_{k})\\Bigg),\\\\&\\quad \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\n\\end{equation}\nwhere (~\\ref{eq4}) results from $\\delta^{2}_{k}\\ll \\sigma^{2}_{k}$, that is, the antenna noise is negligible compared to the ID circuit noise power ~\\cite{Liu13,Lee018,Zhang13}. \n\nIn this paper, we consider two different optimization methods for the proposed multi-hop DF-SWIPT networks by jointly optimizing the PS ratio $\\{\\rho_{k}\\}^{K}_{k=1}$ at the relay nodes. First, we aim to minimize the source transmit power, $E_{0}$, under the individual SNR constraint and PS ratio for each of multi-hop links as\\footnote{To further emphasize the importance of solving the source power minimization problem, we will shortly present two possible scenarios in which our source power minimization problem is applicable as well as the DF multi-hop relay systems. The first deals with the system model where several node pairs perform SWIPT D2D communication with neighboring nodes (i.e., $\\mathcal{R}_{k-1}\\text{-to-}\\mathcal{R}_{k}$ and $\\mathcal{R}_{k}\\text{-to-}\\mathcal{R}_{k+1}$, $k=1,\\ldots,K$ using Fig. ~\\ref{figSystems} as a reference). Here, within the $\\mathcal{R}_{k-1}\\text{-to-}\\mathcal{R}_{k}$ D2D nodes, $\\mathcal{R}_{k}$ uses SWIPT PS ratio technique to decode and harvest energy from the RF signal it receives from $\\mathcal{R}_{k-1}$. $\\mathcal{R}_{k}$ then uses its harvested energy to forward its own information signal to $\\mathcal{R}_{k+1}$ in the preceding D2D communication (i.e., $\\mathcal{R}_{k}\\text{-to-}\\mathcal{R}_{k+1}$). The second application involves a SWIPT PS ratio D2D communication between the BS and $\\mathcal{R}_{1}$ node. $\\mathcal{R}_{1}$ then transfer energy to $\\mathcal{R}_{2}$ to recharge its battery. The D2D WET occurs between $\\mathcal{R}_{k}\\text{-to-}\\mathcal{R}_{k+1}$ for all subsequent nodes after $\\mathcal{R}_{1}$ to recharge their batteries.}\n\\begin{equation}\n\\label{eq7}\n\\begin{aligned}\n& \\underset{E_{0},\\{\\rho_{k}\\}^{K}_{k=1}}{\\text{min}}\\text{ }E_{0}\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & \\frac{E_{0}\\Gamma_{k}}{\\sigma^{2}_{k}\\beta_{k}}\\prod^{k-1}_{j=1}\\rho_{j}(1-\\rho_{k})\\geq \\bar{\\gamma}_{k}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n& \\text{ } \n\\begin{aligned}  \n& & & & & & & & & & 0 \\leq \\rho_{k}\\leq 1, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n&\\text{ } \n\\begin{aligned}  \n& & & & & & & & & & E_{0} \\geq 0,\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nwhere $\\bar{\\gamma}_{k}$ is the SNR threshold constraint of node $k$. \n\nNext, we consider the maximization of the overall DF system rate which can be formulated as\n\\begin{equation}\n\\label{eq5}\n\\begin{aligned}\n& \\underset{\\{\\rho_{k}\\}^{K}_{k=1}}{\\text{max}}\\text{ }\\text{ }\nR\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & &0 \\leq \\rho_{k}\\leq 1, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nwhere $R=\\underset{\\{1\\leq k\\leq K+1\\}}{\\text{min}}\\text{ }\\text{ }R_{k}$, and $R_{k}$ is defined in equation (~\\ref{eq4}). With problem (~\\ref{eq5}), the source transmit power is fixed. It is not considered as a constraint because we assume the source will transmit the RF signal with full power. \n\nProblem (~\\ref{eq7}) is always feasible for any given set of the SNR constraints $\\bar{\\gamma}_{k}$ since the initial energy $E_{0}$ is included as an optimization variable in (~\\ref{eq7}). For instance, it is obvious that the problem in (~\\ref{eq7}) is feasible when $\\gamma_{k}=0, \\text{ }\\forall k$. Also, even though some $\\bar{\\gamma}_{k}$ become large, we can always find a feasible PS ratio $\\rho_{k}\\in [0,1]$ by setting $E_{0}$ to a sufficiently large number. In addition, it is not difficult to show that the feasibility of the problem in (~\\ref{eq5}) is always guaranteed since no constraints on the system throughput $R$ is included. Hence, both (~\\ref{eq7}) and (~\\ref{eq5}) are always feasible in practice. In the next section, we provide the globally optimal $\\{\\rho^{\\star}_{k}\\}^{K}_{k=1}$, $E^{\\star}_{0}$ and $R^{\\star}$ solutions for both problems. \n\n\\section{Multi-Hop Relay Joint Optimal Design}\n\\label{secoptimization}\nIn this section, we solve the source transmit power minimization problem and the minimum system rate maximization problems in subsections ~\\ref{subsecpowermin} and ~\\ref{subsecrateminmax} respectively. We also discuss the physical (i.e., real-world) implementation of our optimal solutions and its influence on power constraint, computational constraint and QoS constraint in subsection ~\\ref{subsecsecapplic}.\n\\subsection{Source Transmit Power Minimization}\n\\label{subsecpowermin}\nIn this subsection, the optimum values for the source transmit power and the PS ratio are presented in the Theorem ~\\ref{thmoptimal_min}. We then provide a few remarks based on the optimal solutions for the source transmit power and the PS ratio of each DF-SWIPT relay node. \n\\begin{theorem} \\label{thmoptimal_min}\nFor the joint optimization problem given in (~\\ref{eq7}), the optimal source transmit power, $E^{\\star}_{0}$, is deduced as $$E^{\\star}_{0}=\\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}_{k}\\sigma^{2}_{k}\\beta_{k}}{\\Gamma_{k}},$$ with the optimal PS ratio, $\\rho^{\\star}_{k}$, at node $k$ defined as\n$$\\rho^{\\star}_{k}=\\begin{cases}\n1-\\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}\\frac{\\frac{\\bar{\\gamma}_{k}\\beta_{k}}{\\Gamma_{k}}}{\\sum^{K+1}_{j=1}\\frac{\\bar{\\gamma}_{j}\\beta_{j}}{\\Gamma_{j}}},\\text{ }\\text{ }k=1,\\ldots,K \\\\\n0,\\text{ }\\text{ }\\text{  }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }k=K+1.\n\\end{cases}$$\n\\end{theorem}\n\n\\textit{Proof:} See Appendix ~\\ref{app1}.\n\nFrom Theorem ~\\ref{thmoptimal_min}, we can infer that $E^{\\star}_{0}$ is dependent on $\\beta_{k}$, $\\sigma_{k}$, $\\Gamma_{k}$, and $\\bar{\\gamma}_{k}$, and independent of $\\rho_{k}$. This implies that, the optimal source power can be determined at the source node without knowledge of each relay node's PS ratio. \n\nSecondly, we can further state that, increasing the number of DF-SWIPT relay nodes, increases the minimum source transmit power needed to support communication. Using an example to illustrate this deduction, for simplicity, we assume $\\beta_{k}$, $\\vert h_{k}\\vert^{2}$, $\\bar{\\gamma}_{k}$ and $\\sigma^{2}_{k}$ are the same for all DF-SWIPT relay nodes, that is, all the relays have similar properties\\footnote{Similar properties here mean all the relays have the same channel structure, equal distances between each node, energy harvesting efficiency, and noise variance. This implies that, $\\beta_{K+1}=1$, $\\beta_{1}=\\beta_{2}=\\ldots=\\beta_{K}=\\beta$, $\\vert h_{1}\\vert^{2}=\\vert h_{2}\\vert^{2}=\\ldots=\\vert h_{K+1}\\vert^{2}=\\vert h \\vert^{2}$, $\\bar{\\gamma}_{1}=\\bar{\\gamma}_{2}=\\ldots=\\bar{\\gamma}_{K+1}=\\bar{\\gamma}$, and $\\sigma^{2}_{1}=\\sigma^{2}_{2}=\\ldots=\\sigma^{2}_{K+1}=\\sigma^{2}$. Therefore, at the $k$th RF signal hop, we have $\\Gamma_{k}$ becoming $\\Gamma^{k}$ because $\\Gamma_{1}=\\beta\\vert h \\vert^{2}=\\Gamma$, $\\Gamma_{2}=\\beta^2\\vert h \\vert^{4}=\\Gamma^{2}$, and so on.}. $E^{\\star}_{0}$ becomes $E^{\\star}_{0}\\thickapprox \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{k}}$. As the number of nodes after the source node increases, $\\Gamma^{k}$ reduces further below $1$, hence, leading $E^{\\star}_{0}$ also increases.\n\nFrom the inspection of $\\rho^{\\star}_{k}$ in Theorem ~\\ref{thmoptimal_min}, it can be established that $\\rho^{\\star}_{k}$ is dependent on the product of all previous nodes' PS ratio $\\rho_{j}$ (i.e., $j=1,\\ldots,k-1$) and independent of the optimal minimum source power. This implies that, the current $\\rho^{\\star}_{k}$ can not be determined without knowledge of the previous relay nodes' PS ratios. Since $\\rho_k \\propto \\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}$, this means the current DF-SWIPT relay node harvests less energy from the RF signal it receives from the previous node. Therefore, more portion of the received signal will be dedicated to ID at the current DF-SWIPT node. \n\\subsection{Achievable System Rate Optimization}\n\\label{subsecrateminmax}\nIn this subsection, we will first reformulate the system rate optimization problem. From the reformulated problem, propose a theorem for finding the optimum system rate and the PS ratio of each relay node. Now, the system rate optimization problem for the DF-SWIPT relaying protocol is formulated as\n\\begin{equation}\n\\label{eq20}\n\\begin{aligned}\n& \\underset{\\{\\rho_{k}\\}^{K}_{k=1}}{\\text{max}}\\text{ }\\underset{\\{1\\leq k\\leq K+1\\}}{\\text{min}}\\text{ }\nR_{k}\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & &0 \\leq \\rho_{k}\\leq 1, \\text{ }k=1,\\ldots,K+1, \n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nwhere $R_{k}=\\log_{2}(1+\\gamma_{k})$, and $\\gamma_{k}$ represents the received SNR at node $k$ represented in equation (~\\ref{eq4}). As $\\gamma_{k}$ increases or decreases, the rate $R_{k}$ also increases and decreases. Hence, the system rate maximization problem (~\\ref{eq20}) can be rewritten as   \n\\begin{equation}\n\\label{eq21}\n\\begin{aligned}\n& R^{\\star}=\n\\log_{2}\\Bigg(1+\\underset{\\{\\rho_{k}\\}^{K}_{k=1}}{\\text{max}}\\text{ }\\underset{\\{1\\leq k\\leq K+1\\}}{\\text{min}}\\text{ }\\gamma_{k}\\Bigg)\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & &0 \\leq \\rho_{k}\\leq 1, \\text{ }k=1,\\ldots,K+1.\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nBy introducing a new system SNR constraint variable, $\\hat{\\gamma}$ (i.e., $\\hat{\\gamma}\\leq \\text{min}\\text{ }\\{\\gamma_{k}\\}^{K+1}_{k=1}$), the optimization problem (~\\ref{eq21}) is redefined as \n\\begin{equation\n\\label{eq22}\n\\begin{aligned}\n& \\underset{\\{\\rho_{k}\\}^{K}_{k=1},\\hat{\\gamma}}{\\text{max}}\\text{ }\\hat{\\gamma}\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & & \\frac{E_{0}\\Gamma_{k}}{\\sigma^{2}_{k}\\beta_{k}}\\prod^{k-1}_{j=1}\\rho_{j}(1-\\rho_{k})\\geq \\hat{\\gamma}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n& \\text{ } \n\\begin{aligned}  \n& & & & & & & & & & 0 \\leq \\rho_{k}\\leq 1, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nFrom problem (~\\ref{eq22}), it can be deduced that $\\hat{\\gamma}$ is the minimum achievable SNR threshold of the system, hence, the solution to (~\\ref{eq22}) should give us the optimal $\\rho_{k}$ solution for which each hop SNR constraint is not less than $\\hat{\\gamma}$ ~\\cite{Xu17,Liu17}. By solving (~\\ref{eq22}), we obtain the following theorem.\n\n\\begin{theorem} \\label{thmoptimal_minmax\nThe optimal rate, $R^{\\star}$, of the DF-SWIPT system is expressed as \n$$R^{\\star}=\\log_{2}\\Bigg(1+\\frac{E_{0}}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{\\Gamma_{k}}}\\Bigg),$$ \nand the optimal PS ratio, $\\rho^{\\star}_{k}$, deduced as \n$$\\rho^{\\star}_{k}=\\begin{cases}\n1-\\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}\\frac{\\frac{\\beta_{k}}{\\Gamma_{k}}}{\\sum^{K+1}_{j=1}\\frac{\\beta_{j}}{\\Gamma_{j}}},\\text{ }\\text{ }\\text{  } k=1,\\ldots,K \\\\\n0,\\text{ }\\text{ }\\text{  }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{  }k=K+1.\n\\end{cases}$$\n\\end{theorem}\n\n\\textit{Proof:} See Appendix ~\\ref{app2}.\n\nFrom Theorem ~\\ref{thmoptimal_minmax}, we make a few assertions. First, the current optimal PS ratio in (~\\ref{eq34}) is dependent on the product of the PS ratios of all the preceding relay nodes. This means that the current node's PS ratio is smaller than the previous node's PS ratio. Hence, there may be more information decoded at the current node as compared to the previous node. Also, the optimal PS ratio is independent of the achievable SNR threshold. The achievable SNR threshold is also independent of the optimal PS ratio but depends on $\\beta_{k}$, $\\sigma_{k}$, $\\Gamma_{k}$, and $E_{0}$. The optimal SNR threshold for the system reduces with increasing number of DF-SWIPT relay nodes between the source and the destination. \n\n\\subsection{Implementation and Analysis}\n\\label{subsecsecapplic}\nIn this subsection, we discuss how the proposed protocols can be implemented. We discuss a special scenario where both the source power minimization and the minimum system rate maximization are equivalent. We also present a simple closed-form solution for the determination of the number of relay nodes need to support communication between the source and the destination. This solution is for the special scenario where the sensor network is homogeneous and the inter-node distance are equivalent (i.e., the distance between nodes are equal). The closed-form solution for determining the number of relay nodes can be used in a routing algorithm for our proposed system model. For the protocol implementation, we will delve into how the PS ratio can be calculated for each relay node based on the computation constraints of each sensor node. \n\nFor the source power minimization problem, if each relay node has the same SNR threshold constraint (i.e., $\\bar{\\gamma}_{1}=\\bar{\\gamma}_{2}=\\ldots=\\bar{\\gamma}_{K}=\\bar{\\gamma}_{K+1}$), then the PS ratio solution for both problems (~\\ref{eq8}) and (~\\ref{eq20}) are the same. That is, for both cases, the optimal PS ratio is expressed as\n\\begin{equation}\n\\label{eqgenrho}\n\\rho^{\\star}_{k}=\\begin{cases}\n1-\\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}\\frac{\\frac{\\beta_{k}}{\\Gamma_{k}}}{\\sum^{K+1}_{j=1}\\frac{\\beta_{j}}{\\Gamma_{j}}},\\text{ }\\text{ }k=1,\\ldots,K \\\\\n0,\\text{ }\\text{ }\\text{  }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }k=K+1.\n\\end{cases}\n\\end{equation}\nIn addition, (~\\ref{eq18}) and (~\\ref{eq32}) become equivalent. This implies that, given either a QoS constraint (i.e., the system required SNR threshold) or a source power constraint (i.e., the minimum available source power), we can calculate either the minimum source power or the maximum system achievable rate, respectively, from the generalized equation,\n\\begin{equation}\n\\label{eqgeneraleq}\n\\frac{E_{0}}{\\hat{\\gamma}}=\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{\\Gamma_{k}}.\n\\end{equation} \n\nKnowing $E_{0}$ and $\\hat{\\gamma}$, we can estimate the number of relay nodes needed to support communication between the source node and the destination node from (~\\ref{eqgeneraleq}). By manipulating the (~\\ref{eqgeneraleq}), we can obtain the estimated maximum number of relay nodes as\n\\begin{equation}\n\\label{eqnodenumb}\nK\\thickapprox \\begin{cases}\n\\frac{\\ln\\Big[1-\\Big(\\frac{E_{0}}{\\bar{\\gamma}\\sigma^{2}\\beta}+1\\Big)\\Big(1-\\frac{1}{\\Gamma}\\Big)\\Big]}{-\\ln \\Gamma}-1,\\text{ }\\text{ }\\Gamma > 1 \\\\\n\\frac{\\ln\\Big[1+\\Big(\\frac{E_{0}}{\\bar{\\gamma}\\sigma^{2}\\beta}+1\\Big)\\Big(\\frac{1}{\\Gamma}-1\\Big)\\Big]}{-\\ln \\Gamma}-1,\\text{ }\\text{ }\\Gamma <1,\n\\end{cases}\n\\end{equation}\nwhere $\\Gamma =0.5 \\beta C\\Big(\\frac{d}{d_{0}}\\Big)^{-\\alpha}$, and the variables $C$, $d$, $\\alpha$ are the inter-node attenuation constant, distance, and pathloss exponent. The $\\Gamma$ and $K$ are predetermined at the source node and used in a routing algorithm. The detailed derivation of (~\\ref{eqnodenumb}) is presented in Appendix ~\\ref{app3}. \n\nThe two possible methods for determining the PS ratio are by a centralized method and a distributed method. For the centralized method, since the source node knows all the CSI for all communicating channels in the network, it can calculate the PS ratio for each DF-SWIPT node. The source node calculates the PS ratio for each node using equations (~\\ref{eq19}) and (~\\ref{eq34}) for the source power minimization case and the system throughput maximization case, respectively. After calculating all the PS ratios, $\\{\\rho^{\\star}_{k}\\}^{K}_{k=1}$, it transmits the PS ratios with its information signal to the first DF-SWIPT node. With the centralized system, the relay node $k$ must transmit its decoded information along with $\\{\\rho^{\\star}_{j}\\}^{K}_{j=k+1}$ to the next relay node. However, with the distributed system, since each relay node knows its own CSI for the channels it communicates on, it can calculate the PS ratio for the next node. In the distributed system, the source node calculates the PS ratio of the first relay node as \n\\begin{equation}\n\\rho_{1}=1-\\tilde{\\psi}_{1},\n\\end{equation}\nwhere $\\tilde{\\psi}_{1}=\\frac{1}{\\vert h_{1} \\vert^2\\sum^{K+1}_{j=1}\\frac{\\beta_{j}}{\\Gamma_{j}}}$. The source node then transmits its information signal, $\\rho_{1}$ and $\\tilde{\\psi}_{1}$ to the first relay node. The $k$th relay node calculates the $k+1$ relay's PS ratio as \n\\begin{equation}\n\\rho_{k+1}=1-\\tilde{\\psi}_{k+1},\n\\end{equation} \nwhere $\\tilde{\\psi}_{k+1}=\\frac{1}{\\rho_{k}\\beta_{k}\\vert h_{k+1} \\vert^2}\\tilde{\\psi}_{k}$, and $k=1,\\ldots,K-1$. The current $k$ DF-SWIPT node transmits its decoded information, $\\rho_{k+1}$ and $\\tilde{\\psi}_{k+1}$ to the next $k+1$ DF-SWIPT node. The advantage of the centralized system is with the relay nodes not having any computational burden concerning the PS ratio calculation. However, the first few relay nodes would have a large amount of data bits to process and forward depending on the number of preceding relay nodes' PS ratios transmitted to it. The DF process may be affected if the relay nodes do not have enough memory (i.e., computational processing power). But, with the distributed system, each node receives fewer information bits to DF compared to the centralized system. The drawback of the distributed system is the need for the DF-SWIPT relay node to compute variables $\\rho_{k+1}$ and $\\tilde{\\psi}_{k+1}$ before retransmission to the next node.\n\nA comparison of the centralized and distributed methods is summarized in Table ~\\ref{tabsummary}.  For the computational complexity comparison, let $\\mathcal{O}(J)$ represent a single arithmetic operation. For the centralized system, the source node performs $\\mathcal{O}((K+1)J)$ arithmetic operations, that is, it calculates the $E_{0}$ and $\\{\\rho_{k}\\}^{K}_{k=1}$ values. The relay nodes do not perform any arithmetic operations in the centralized system. However, with the distributed method, the source node performs $\\mathcal{O}(J)$ arithmetic operations to determine $E_{0}$, $\\tilde{\\psi}_{1}$ and $\\rho_{1}$, while the $k-$th relay node performs $\\mathcal{O}(J)$ arithmetic operations to determine $\\tilde{\\psi}_{k+1}$ and $\\rho_{k+1}$. \n\n\\begin{table*}\n\\centering\n\\caption{Comparison of PS Ratio Determination Methods}\n\\label{tabsummary}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nPS ratio Scheme & \\multicolumn{2}{c|}{Centralized Method} & \\multicolumn{2}{c|}{Distributed Method} \\\\ \\hline\\hline\nCommunicating Node & Source & $k-$th Relay Node  & Source & $k-$th Relay Node\\\\  \\hline\nComputational Complexity & $\\mathcal{O}((K+1)J)$ & --  & $\\mathcal{O}(J)$ & $\\mathcal{O}(J)$\\\\  \\hline\nTransmit Bits & $K (i_{0}\\log_{2} K + B) + F $  & $(K-k) \\{i_{0} \\log_{2}(K-k) + 2B\\} + F $ & $2B + F$ & $ 2B + F $ \\\\  \\hline\nCSI Requirement & Global CSIs & -- & Global CSIs & Local CSIs  \\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\nNext, we discuss the number of bits needed to be transmitted at each node. First, we assume the actual transmitted information bit, real number bits (i.e. either $\\rho_{k+1}$ or $\\tilde{\\psi}_{k+1}$ value), and relay index bits broadcast from the current node to the next node to be processed are defined as $F$, $B$, and $i_{0}$, respectively. With the centralized system, the source node transmits the information signal, the $K$ relay indexes and $K$ PS ratios to the first relay node as $K (i_{0}\\log_{2} K + B) + F$ bits. Each $k-$th relay node then transmit its decoded information signal, the $K-k$ relay indexes and $K-k$ PS ratios to the next relay node. For the distributed method, each node including the source node transmits its information signal, and the next node's PS ratio and $\\tilde{\\psi}_{k+1}$ value to the next relay node as $2B + F$ bits.\n\n\\section{Simulation Results}\n\\label{secsimulation}\nThis section provides simulation results to demonstrate the system performance of multi-hop DF-SWIPT sensor network. Unless otherwise stated, the following parameters are utilized for the simulations: for the model channel model presented in paragraph three of Section ~\\ref{secsystem_model}, for the large-scale fading component, the attenuation constant $C_{0}=-10$dB, the pathloss exponent $\\alpha = 3$ and the inter-node distance is set as $d=d_{1}=d_{2}=\\ldots=d_{K}=2$m. Please, note that by considering $d_{1}=d_{2}=\\ldots=d_{K}=2$m, the distance between the source node and the destination node increases with increasing number of DF-SWIPT relay nodes. Hence, the total distance between the source node and destination node for $K$ DF-SWIPT relay nodes is $2\\times (K+1)\\times d$. The antenna noise variance $\\sigma^{2}_{1}=\\sigma^{2}_{2}=\\ldots=\\sigma^{2}_{K+1}=-80$dBm, and the energy conversion efficiency $\\beta_{1}=\\beta_{2}=\\ldots=\\beta_{K}=0.7$. A suboptimal naive scheme with a fixed PS ratio ($\\rho_{k}=0.5$, $\\forall k$) is adopted for comparison with the optimal DF-SWIPT scheme. The simulation results are obtained over $10^{4}$ random channel realizations. \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.4in]{results5}\n\\end{center}\n\\caption{Comparison of various schemes based on harvested power at each relay node, $K=3$ and $d_{k}=2$m.}\n\\label{figresults5}\n\\end{figure}\n \nFig. ~\\ref{figresults5} shows a comparison of the energy harvested by the optimal DF-SWIFT scheme and suboptimal DF-SWIFT scheme at each node. The plot of each relay node's harvested energy is presented for $K=3$, a set of SNR threshold constraints (i.e. $\\bar{\\gamma}=-10$dB, $0$dB, $10$dB and $20$dB). For each SNR threshold in the figure, the optimal source transmit power is calculated for $K=3$, and used in determining how much energy is harvested at each node. From the figure, it can be seen that the fixed $\\rho_{k}$ scheme harvests lesser energy at each node as compared to the optimal PS ratio. There is a reduction in the amount of harvested energy as the RF signal moves from one node to the other. This may lead to a subset of all nodes being able to harvest enough energy to support communication between the source and destination for the suboptimal scheme as shown in the first plot of Fig. ~\\ref{figresults5}(a). Therefore, to achieve the same QoS, the source needs to transmit more power for the suboptimal scheme. Hence, the suboptimal scheme puts a higher strain on the source power. \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{Results11}\n\\end{center}\n\\caption{The maximum number of DF-SWIPT nodes against increasing source transmit power, $\\bar{\\gamma}=20$dB and $d_{k}=2$m.}\n\\label{figresults11}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{Results10}\n\\end{center}\n\\caption{The maximum number of DF-SWIPT nodes against increasing system required SNR, $E_{0,max}=50$dBm and $d_{k}=2$m.}\n\\label{figresults10}\n\\end{figure}\nFigs. ~\\ref{figresults11} and ~\\ref{figresults10} show the implementation of (~\\ref{eqnodenumb}) in determining the number of relay nodes. Fig. ~\\ref{figresults11} shows the plot of the number of relay nodes against varying $E_{0}$, while, Fig. ~\\ref{figresults10} is a plot on the number of relay nodes against varying $\\bar{\\gamma}$. From both plots, we can observe that the closed-form solution for determining the number of nodes gives an excellent approximation of the number of relay nodes.\n\\subsection{Transmit Power Minimization}\n\\label{subsecsulenergymin}\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results1}\n\\end{center}\n\\caption{Average minimum transmit power with respect to varying $\\bar{\\gamma}$ plot for $K=2$ and $3$, and $d_{k}=2$m.}\n\\label{figresults1}\n\\end{figure}\nFig. ~\\ref{figresults1} shows a graph of the minimum source transmit power, $E_{0}$, against the SNR threshold constraint range, $\\bar{\\gamma}$, for different number of relay nodes, i.e., $K=2$ and $K=3$. The optimal PS scheme outperforms the fixed PS suboptimal scheme in terms of the $E_{0}$ needed to support the source to destination communication. The optimal scheme achieves a $15$dBm gain over the suboptimal scheme when $K=2$ and $20$dBm gain with $K=3$. An increase in the SNR threshold generally results in a corresponding increase in the minimum source power required for successful communication. Also, a rise in the number of relay nodes results in an increase in the minimum source power. This conclusion is consistent with the analysis of Theorem ~\\ref{thmoptimal_min}. It is observed that there is a loss of about $60$dBm and $50$dBm of the source transmit power in the optimal and the suboptimal PS schemes, respectively, with just a single increase in the number of relay nodes. This increase in $E_{0}$ is due to the increase in the distance and the number of relay nodes between the source and the destination nodes.\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results3}\n\\end{center}\n\\caption{Average minimum transmit power  for increasing distance between nodes with $K=2$ and $3$, and $\\bar{\\gamma}=-10$dB.}\n\\label{figresults3}\n\\end{figure}\n \nFig. ~\\ref{figresults3} shows a plot of the minimum source power , $E_{0}$, against the inter-node distance, $d_{k}$. The average minimum amount of source transmit power needed to support the end-to-end communication in the sensor network rises with increasing $d_{k}$ for both the optimal and suboptimal PS schemes. There is a constant difference in the $E_{0}$ for the optimal and suboptimal for all values of $d_{k}$. A performance degradation of about $15$dBm and $25$dBm occurs between the optimal and suboptimal schemes for both $K=2$ and $K=3$, respectively as $d_{k}\\geq 2$m. By increasing the number of relays, there is a significant rise in the $E_{0}$ needed to facilitate communication. This phenomenon is due to the increase in distance between the source and destination nodes which in-turn influences the signal attenuation over the increasing inter-node distance. \n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results4}\n\\end{center}\n\\caption{Average minimum transmit power against rate threshold for $K=2$ and $3$.}\n\\label{figresults4}\n\\end{figure}\nA plot of the average minimum $E_{0}$ against the system rate threshold constraint is presented in Fig. ~\\ref{figresults4}. Similarly, by increasing the system rate threshold, the system's minimum source transmit power, $E_{0}$, requirement increases. From the system rate threshold of $1$bps\/Hz upward, there is an improvement of about $15$dBm in terms of $E_{0}$ for the optimal PS scheme, and a $20$dBm improvement for the suboptimal PS scheme.\n\nThis behavior is well appreciated in Fig. ~\\ref{figresults2}, where we plot the minimum source power against the number of relay nodes. Here, we set $d_{k}=2$m. The distance between the source and destination node increases from $4$m to $14$m for $K=1$ and $K=6$, respectively. Increasing the DF-SWIPT nodes produces a constant increase in the minimum source transmit power needed to support the system QoS. \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results2}\n\\end{center}\n\\caption{Average minimum transmit power against the number of DF-SWIPT relay nodes at  $\\bar{\\gamma}=-10$ and $0$dB, and $d_{k}=2$m.}\n\\label{figresults2}\n\\end{figure}\n\\subsection{System Rate Maximization}\n\\label{subsecsulratemaxmin}\nFig. ~\\ref{figresults7} shows the impact of the source transmit power on the achievable rate for different numbers of relay nodes. The optimal PS scheme has better achievable rate as compared to the suboptimal PS scheme. The lesser the number of relay nodes, the higher the achievable rate. This is because the distance between the source and destination node also reduces. For the $E_{0}$ range of $30$dBm to $50$dBm, the optimal and suboptimal schemes have a performance gap of about $4$bps\/Hz and $6$bps\/Hz for $K=2$ and $K=3$, respectively. There is a performance improvement for both schemes of about $5$bps\/Hz when relay node is reduced (i.e. from $K=3$ to $K=2$).\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results7}\n\\end{center}\n\\caption{Average achievable system rate against available source power, $K=2$ and $3$, and $d_{k}=2$m.}\n\\label{figresults7}\n\\end{figure}\n\nTo further appreciate the effect of how increasing the number of DF-SWIPT relay nodes has on the achievable rate, we consider Fig.~\\ref{figresults8}. The plot shows the average achievable rate against the number of relays for $E_{0}=30$ and $60$dBm. From Fig. ~\\ref{figresults8}, the achievable rate deteriorates with increasing number of relay nodes. For $K>5$, the achievable rate approaches zero for both schemes. A similar behavior can be seen in Fig. ~\\ref{figresults9} where the achievable rate reduces with an increase in the inter-node distance. For a fixed number of relays, as the distance between the relays increases, the performance gap between the suboptimal and optimal schemes narrows. This is due to the widening of the distance between the source and destination node by a total distance of $d_{k}\\times(K+1)$. \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{results8}\n\\end{center}\n\\caption{Average achievable system rate against available source power, $E_{0}=30$ and $60$dBm, and $d_{k}=2$m.}\n\\label{figresults8}\n\\end{figure}\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.2in]{Results9}\n\\end{center}\n\\caption{Average achievable system rate against available source power, $E_{0}=30$dBm, and $K=2$ and $3$.}\n\\label{figresults9}\n\\end{figure}\n\n\n\\subsection{Imperfect Channel State Information}\n\\label{subsecimpcsicon}\nWe now consider imperfect channel state information (ICSI) for calculating the optimal PS ratio since channel estimation in practical systems may not be accurate. This inaccuracy may be due: (i) inherent delay occurring between the channel estimation and actual data transmission, and (ii) limited feedback in frequency division multiplexing or imperfect reciprocity in time division multiplexing ~\\cite{Yoo06,Lee09,Xiang12}. The source node is assumed to have ICSI of each relay node ~\\cite{Yoo06,Lee09,Xiang12}. With the ICSI, the channel estimation error is modeled as $\\tilde{h}_{k}=\\hat{h}_{k}+e_{k}$, where $\\hat{h}_{k}\\sim \\mathcal{CN}(0,1-\\sigma^{2}_{E})$ and $e_{k}\\sim \\mathcal{CN}(0,\\sigma^{2}_{E})$ represent the estimated and the error channel coefficients, respectively. Here, $\\sigma^{2}_{E}$ is the estimation error variance, which is assumed to be $0$, $0.1$, $0.2$, and $0.3$ for our simulation. \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.35in]{ResultsImp1}\n\\end{center}\n\\caption{The effect of estimated channel errors for $K=2$ and $3$, and $d_{k}=2$m.}\n\\label{figresultsimp1}\n\\end{figure}\n\nThe effect of ICSI on our achievable rate problem is presented in Fig. ~\\ref{figresultsimp1}. As expected, when $\\sigma^{2}_{E}$ increases, the maximum  achievable rate reduces. It can be observed from Fig. ~\\ref{figresultsimp1} that the fixed PS ratio scheme remains unchanged for all $\\sigma^{2}_{E}$ considered in the simulation, since it does not exploit the channel state information to compute the PS ratio. In addition, by reducing the estimation error variance, the curves in both plots approach their perfect CSI curves (i.e., $\\sigma^{2}_{E}=0$) for optimal schemes. This result shows that the proposed rate maximization solution can be used in practical environment. \n\n\\section{Conclusion}\n\\label{secconclusion}\nThis paper investigated a DF-SWIPT multi-hop relay configuration in an IoT sensor network scenario. The DF-SWIPT relay nodes facilitated communication between a source and a destination node. All the nodes possess a single antenna. Two optimization problems have been studied in this work. We first optimized the PS ratio with the aim of minimizing the source transmit power. We then optimized the PS ratio with the objective of maximizing the achievable rate. Closed-form solutions were found for the minimum source power, maximum system achievable rate and the PS ratio at each DF-SWIPT relay node. From our research, we found a single equation linking the source energy to the system achievable rate. We have also shown that the source power minimization problem is equivalent to the system throughput maximization. The optimal PS ratio scheme was compared to a fixed PS ratio suboptimal scheme. The optimal scheme outperformed the suboptimal scheme for both the source power minimization problem and the system throughput maximization problem in the simulation section.  \n\nIn this work, we consider the simple case of single antenna system for our initial research analysis and presentation of our DF-SWIPT proposed communication model. A multi-antenna systems as in ~\\cite{Liu19} and imperfect channel estimation at each node can be considered as extension of this work. Unlike the works presented in ~\\cite{Mao15} and ~\\cite{Liu16} which considered both AF and DF relay configuration, we focus on only DF relay configuration in this paper. The AF relay configuration can be considered as an extension of this work. Another approach or extension which can be researched is the use of rechargeable battery instead of a supercapacitor as considered in ~\\cite{Mahama17} and ~\\cite{Lee016}. We also considered DF-SWIPT multi-hop relay system using the PS SWIPT technique in this work. Hence, the TS SWIPT technique can be considered as an extension of the  work presented in this paper. \n\n\n\\appendices \n\\section{Proof of Theorem 1}\n\\label{app1}\nFirst of all, problem (~\\ref{eq7}) is non-convex with respect to both $E_{0}$ and $\\{\\rho_{k}\\}^{K}_{k=1}$. By introducing four new variable definitions, we reformulate problem (~\\ref{eq7}) into an equivalent convex problem as follows; \n\\begin{equation}\n\\label{eq8}\n\\begin{aligned}\n& \\underset{Q,\\{A_{k}\\}^{K+1}_{k=1}}{\\text{minimize}}\\text{ }\\frac{1}{Q}\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & & A_{k-1}-A_{k} \\geq Q\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n& \\text{ } \n\\begin{aligned}  \n& & & & & & & & & & A_{k} \\leq A_{k-1}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n&\\text{ } \n\\begin{aligned}  \n& & & & & & & & & & Q \\geq 0,\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\nwhere $A_{k}\\triangleq\\prod^{k}_{j=1}\\rho_{j}$, $A_{0}\\triangleq 1$, $A_{K+1}\\triangleq 0$, and $Q\\triangleq \\frac{1}{E_{0}}$. It is obvious that, if the first constraint is satisfied $k=1,\\ldots,K+1$, then the second constraint is also satisfied. Thus, we can remove the second constraint $k=1,\\ldots,K+1$.\n\nThe Lagrangian of problem (~\\ref{eq8}) is defined as\n\\begin{equation}\n\\label{eq9}\n\\begin{aligned}\n\\mathcal{L}\\Big\\{Q,\\{A_{k}\\}^{K+1}_{k=1},\\{\\lambda_{k}\\}^{K+1}_{k=0}\\Big\\}&\\quad=\\frac{1}{Q}+Q\\sum^{K+1}_{k=1}\\lambda_{k-1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}\\\\&\\quad +\\sum^{K+1}_{k=1}(\\lambda_{k-1}-\\lambda_{k})A_{k}-\\lambda_{0},\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda_{k}\\geq 0$ and $\\lambda_{K+1}\\geq 0$ are the dual variables corresponding to the constraints $A_{k-1}-A_{k} \\geq Q\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}, k=1,\\ldots,K+1$ and $A_{K+1}$, respectively. \nFollowing (~\\ref{eq9}), we can express the KKT conditions as \n\\begin{equation}\n\\label{eq11a}\n-\\frac{1}{Q^{\\star 2}}+\\sum^{K+1}_{k=1}\\lambda^{\\star}_{k-1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}=0,\n\\end{equation}\n\\begin{equation}\n\\label{eq12a}\n\\sum^{K+1}_{k=1}\\lambda^{\\star}_{k-1}\\Bigg(Q^{\\star}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}-(A^{\\star}_{k-1}-A^{\\star}_{k})\\Bigg)=0. \n\\end{equation}\nIf we have $\\lambda^{\\star}_{k-1}-\\lambda^{\\star}_{k}>0$, then  the optimal $A^{\\star}_{k}$ minimizing the Lagrangian is given by $A^{\\star}_{k}=-\\infty$, and the dual function is unbounded. Also, if $\\lambda^{\\star}_{k-1}-\\lambda^{\\star}_{k}<0$, then we have $A^{\\star}_{k}=\\infty$, which also yields an unbounded dual function. Therefore, $\\lambda^{\\star}_{k-1}-\\lambda^{\\star}_{k}=0, k=1,\\ldots,K+1$, this implies that, $\\lambda^{\\star}_{0}=\\lambda^{\\star}_{1}=\\ldots=\\lambda^{\\star}_{K+1}$. Hence, the KKT conditions are rewritten as\n\\begin{equation}\n\\label{eq11}\n-\\frac{1}{Q^{\\star 2}}+\\lambda^{\\star}_{0}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}=0,\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\label{eq12}\n\\lambda^{\\star}_{0}\\Bigg(Q^{\\star}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}-(A^{\\star}_{k-1}-A^{\\star}_{k})\\Bigg)=0,\\text{ } k=1,\\ldots,K+1,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\label{eq12b}\n-\\lambda^{\\star}_{0}A^{\\star}_{K+1}=0.\n\\end{aligned}\n\\end{equation}\nThe optimal minimum source power, $E^{\\star}_{0}$, can be acquired from (~\\ref{eq11}) as\n\\begin{equation}\n\\label{eq14}\nE^{\\star}_{0}=\\frac{1}{Q^{\\star}}=\\sqrt{\\lambda_{0}^{\\star}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}}.\n\\end{equation}\n\nFrom the complementary slackness condition for $k=1$ in (~\\ref{eq12}), we have the following two cases: \n\\begin{equation}\n\\label{eq15}\n\\begin{aligned}\n&\\quad\\textit{Case 1:}\\text{ }\\lambda^{\\star}_{0}=0,\\text{ and }1-A^{\\star}_{1}>Q^{\\star}\\frac{\\sigma^{2}_{1}\\beta_{1}\\bar{\\gamma}_{1}}{\\Gamma_{1}},\\\\&\\quad\\textit{Case 2:}\\text{ }\\lambda^{\\star}_{0}>0,\\text{ and }1-A^{\\star}_{1}=Q^{\\star}\\frac{\\sigma^{2}_{1}\\beta_{1}\\bar{\\gamma}_{1}}{\\Gamma_{1}}.\n\\end{aligned}\n\\end{equation}\nFor case 1, $\\frac{1}{Q}=\\sqrt{\\lambda^{\\star}_{0}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}}=\\infty$, this implies $E_{0}=0$. Obviously, this is not the feasible solution for the problem in equation (~\\ref{eq8}) with arbitrary given $\\bar{\\gamma}_{k}>0$. Thus, we focus on case 2, which yields $\\frac{1}{Q}=\\sqrt{\\lambda^{\\star}_{0}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}}<\\infty$, hence $\\lambda^{\\star}_{0}$ is always positive. Since $\\lambda^{\\star}_{0}=\\lambda^{\\star}_{1}=\\ldots=\\lambda^{\\star}_{K+1}$, all the optimal dual variables are positive and equal. From the complementary slackness condition in (~\\ref{eq12}), it follows that\n\\begin{equation}\n\\label{eq16}\n\\begin{aligned}\n&\\quad Q^{\\star}\\frac{\\sigma^{2}_{k+1}\\beta_{k+1}\\bar{\\gamma}_{k+1}}{\\Gamma_{k+1}}=(A^{\\star}_{k}-A^{\\star}_{k+1}), A^{\\star}_{K+1}=0,\\\\&\\quad \\Bigg(\\lambda^{\\star}_{0}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}\\Bigg)^{-\\frac{1}{2}}\\frac{\\sigma^{2}_{k+1}\\beta_{k+1}\\bar{\\gamma}_{k+1}}{\\Gamma_{k+1}}=(A^{\\star}_{k}-A^{\\star}_{k+1}),\\\\&\\quad \\text{ }k=1,\\ldots,K+1.\n\\end{aligned}\n\\end{equation}\nSince $\\sum^{K}_{k=0}(A^{\\star}_{k}-A^{\\star}_{k+1})=1$, the optimal dual variable $\\lambda^{\\star}_{0}$ is given by\n\\begin{equation}\n\\label{eq17}\n\\lambda^{\\star}_{0}=\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}, \n\\end{equation}\nwhich is obtained from (~\\ref{eq12a}). Now, substituting (~\\ref{eq17}) into (~\\ref{eq14}) yields \n\\begin{equation}\n\\label{eq18}\nE^{\\star}_{0}=\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\bar{\\gamma}_{k}}{\\Gamma_{k}}.\n\\end{equation}\nBy inserting $E^{\\star}_{0}$ into SNR constraint of the optimization problem (~\\ref{eq7}), $\\rho^{\\star}_{k}$ is derived as \n\\begin{equation}\n\\label{eq19}\n\\rho^{\\star}_{k}=\\begin{cases}\n1-\\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}\\frac{\\frac{\\bar{\\gamma}_{k}\\beta_{k}}{\\Gamma_{k}}}{\\sum^{K+1}_{j=1}\\frac{\\bar{\\gamma}_{j}\\beta_{j}}{\\Gamma_{j}}},\\text{ }\\text{ }\\text{  }k=1,\\ldots,K \\\\\n0,\\text{ }\\text{ }\\text{  }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{  }k=K+1.\n\\end{cases}\n\\end{equation}\nThe $\\rho^{\\star}_{k}$ is found by simply making $\\rho_{k}$ the subject at equality. $\\blacksquare$\n\\section{Proof of Theorem 2}\n\\label{app2}\nSimilar to the proof of Theorem ~\\ref{thmoptimal_min}, we apply the change of variables where $A_{k}\\triangleq\\prod^{k}_{j=1}\\rho_{j}$, $A_{0}\\triangleq 1$ and $A_{K+1}\\triangleq 0$, since problem (~\\ref{eq22}) is also non-convex with respect to $\\hat{\\gamma}$ and $\\{\\rho_{k}\\}^{K}_{k=1}$. Then, the non-convex problem in (~\\ref{eq22}) can be equivalently transformed to the following convex formulation\n\\begin{equation}\n\\label{eq23}\n\\begin{aligned}\n& \\underset{\\{A_{k}\\}^{K+1}_{k=1}}{\\text{maximize}}\\text{ }\\hat{\\gamma}\\\\\n& \\text{subject to} \n\\begin{aligned}  \n& & & A_{k-1}-A_{k} \\geq \\hat{\\gamma}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}, \\text{ }k=1,\\ldots,K+1,\n\\end{aligned}\\\\\n& \\text{ } \n\\begin{aligned}  \n& & & & & & & & & & A_{k} \\leq A_{k-1}, \\text{ }k=1,\\ldots,K+1.\n\\end{aligned}\n\\end{aligned}\n\\end{equation}\n\nBy solving the first, we solve the second constraint of problem (~\\ref{eq23}), therefore, the second constraint is removed (i.e., it is not considered in the Lagrangian and solution). The Lagrangian for problem (~\\ref{eq23}) is thus expressed as\n\\begin{equation}\n\\label{eq24}\n\\begin{aligned}\n\\mathcal{L}\\Big\\{\\hat{\\gamma},\\{A_{k}\\}^{K+1}_{k=1},\\{\\lambda_{k}\\}^{K+1}_{k=0}\\Big\\}&\\quad= \\hat{\\gamma}+\\hat{\\gamma}\\sum^{K+1}_{k=1}\\lambda_{k-1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}\\\\&\\quad+\\sum^{K+1}_{k=1}(\\lambda_{k-1}-\\lambda_{k})A_{k}-\\lambda_{0},\n\\end{aligned}\n\\end{equation}\nwith its KKT conditions given as \n\\begin{equation}\n\\label{eq25}\n1+\\sum^{K+1}_{k=1}\\lambda^{\\star}_{k-1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}=0,\n\\end{equation}\n\\begin{equation}\n\\label{eq26}\n\\lambda^{\\star}_{k-1}-\\lambda^{\\star}_{k}=0,\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq27}\n\\sum^{K+1}_{k=1}\\lambda^{\\star}_{k-1}\\Bigg(\\hat{\\gamma}^{\\star}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}-(A^{\\star}_{k-1}-A^{\\star}_{k})\\Bigg)=0.\n\\end{equation}\n\nFrom (~\\ref{eq26}), we deduce that $\\lambda^{\\star}_{k-1}=\\lambda^{\\star}_{k}$, hence, $\\lambda^{\\star}_{0}=\\lambda^{\\star}_{1}=\\ldots=\\lambda^{\\star}_{K+1}$. The KKT conditions are modified to give,\n\\begin{equation}\n\\label{eq28}\n1+\\lambda^{\\star}_{0}\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}=0,\n\\end{equation}\n\\begin{equation}\n\\label{eq29}\n\\lambda^{\\star}_{0}\\Bigg(\\hat{\\gamma}^{\\star}\\frac{\\sigma^{2}_{k}\\beta_{k}}{E_{0}\\Gamma_{k}}-(A^{\\star}_{k-1}-A^{\\star}_{k})\\Bigg)=0,\\text{ }\n-\\lambda^{\\star}_{0}A^{\\star}_{K+1}=0.\n\\end{equation}\nThe optimal $\\lambda^{\\star}_{0}$ can be derived from (~\\ref{eq28}) as\n\\begin{equation}\n\\label{eq31}\n\\lambda^{\\star}_{0}=-\\frac{1}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\hat{\\gamma}^{\\star}}{E_{0}\\Gamma_{k}}}=-\\frac{E_{0}}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\hat{\\gamma}^{\\star}}{\\Gamma_{k}}}.\n\\end{equation}\nNoting that $\\lambda^{\\star}_{0}=\\lambda^{\\star}_{1}=\\ldots=\\lambda^{\\star}_{K+1}$, the optimal dual variables are negative and equal based on (~\\ref{eq31}). To find the optimal $\\hat{\\gamma}^{\\star}$, (~\\ref{eq31}) is substituted into (~\\ref{eq29}) to produce ,\n\\begin{equation}\n\\begin{aligned}\n\\label{eq33}\n&\\quad\\frac{E_{0}}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\hat{\\gamma}^{\\star}}{\\Gamma_{k}}}\\sum^{K+1}_{k=1}(A^{\\star}_{k-1}-A^{\\star}_{k})=\\Bigg(\\frac{\\sum^{K+1}_{k=1}\\frac{\\hat{\\gamma}^{\\star}\\sigma^{2}_{k}\\beta_{k}E_{0}}{E_{0}\\Gamma_{k}}}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}\\hat{\\gamma}^{\\star}}{\\Gamma_{k}}}\\Bigg)\\\\&\\quad=1.\n\\end{aligned}\n\\end{equation} \nSince $\\sum^{K+1}_{k=1}(A^{\\star}_{k-1}-A^{\\star}_{k})=1$, the optimal $\\hat{\\gamma}^{\\star}$ is obtained as \n\\begin{equation}\n\\label{eq32}\n\\hat{\\gamma}^{\\star}=\\frac{E_{0}}{\\sum^{K+1}_{k=1}\\frac{\\sigma^{2}_{k}\\beta_{k}}{\\Gamma_{k}}}.\n\\end{equation}\nBy placing $\\hat{\\gamma}^{\\star}$ into the SNR constraint of problem (~\\ref{eq23}), making $\\rho_{k}$ the subject at equality, the optimal $\\rho^{\\star}_{k}$ can be found as \n\\begin{equation}\n\\label{eq34}\n\\rho^{\\star}_{k}=\\begin{cases}\n1-\\frac{1}{\\prod^{k-1}_{j=1}\\rho_{j}}\\frac{\\frac{\\beta_{k}}{\\Gamma_{k}}}{\\sum^{K+1}_{j=1}\\frac{\\beta_{j}}{\\Gamma_{j}}},\\text{ }\\text{ }k=1,\\ldots,K \\\\\n0,\\text{ }\\text{ }\\text{  }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }k=K+1.\n\\end{cases}\n\\end{equation}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=3.4in]{proofmin}\n\\end{center}\n\\caption{A graphical explanation of how the optimal system rate is achieved: (a) shows the achievable rate by each of the DF-SWIPT relay nodes and the destination node, (b) the node rates are rearranged in ascending order and the minimum rate's PS ratio is optimized, (c) the first and second node adjust their individual rates based on their PS ratio until they achieve the same rate,(d) each successive node's rate is adjusted based on their PS ratio until the whole system achieves the optimal system rate.}\n\\label{figproofmin}\n\\end{figure}\n\nWe will now show that $\\hat{\\gamma}^{\\star}$ is the optimal SNR threshold with the aid of both simple arithmetic and the diagrams presented in Fig. ~\\ref{figproofmin}. We know that each of the $K+1$ nodes (i.e., the DF-SWIPT relay nodes and the destination node) achieve different individual rates, \\{$R(\\rho_{k})\\}^{K+1}_{k=1}$ depending on their current $\\rho_{k}$ value as shown in Fig. ~\\ref{figproofmin}(a). These rates can be sorted in an ascending order as $\\hat{R}(\\rho_{\\bar{1}})\\leq\\hat{R}(\\rho_{\\bar{2}})\\leq\\ldots\\leq\\hat{R}(\\rho_{\\overline{K+1}})$, where $\\bar{k}=\\bar{1},\\bar{2},\\ldots,\\overline{K+1}$ is the new ordering number for each of the $K+1$ nodes based on the achieved rates. Also, its obvious that a node's rate is a function of its $\\rho_{\\bar{k}}$. A node's rate increases or decreases by reducing or increasing its $\\rho_{\\bar{k}}$, respectively. The system throughput at this first stage is constrained by $\\hat{R}(\\rho_{\\bar{1}})$ since it is the minimum achievable rate as seen in Fig. ~\\ref{figproofmin}(b). In Fig. ~\\ref{figproofmin}(c), by reducing $\\rho_{\\bar{1}}$ and increasing $\\rho_{\\bar{2}}$, the system can achieve a state where $\\hat{R}(\\rho_{\\bar{1}})=\\hat{R}(\\rho_{\\bar{2}})\\leq\\ldots\\leq\\hat{R}(\\rho_{\\overline{K+1}})$. Now the system throughput is constrained by the minimum rate $\\hat{R}(\\rho_{\\bar{1}})=\\hat{R}(\\rho_{\\bar{2}})$. Following the same procedure for the preceding node, the system achieves a rate of $\\hat{R}(\\rho_{\\bar{1}})=\\hat{R}(\\rho_{\\bar{2}})=\\hat{R}^(\\rho_{\\bar{3}})\\leq\\ldots\\leq\\hat{R}(\\rho_{\\overline{K+1}})$. By continuously repeating and applying this same process to all the subsequent nodes, the system will achieve a rate where $\\hat{R}^{\\star}(\\rho^{\\star}_{\\bar{1}})=\\hat{R}^{\\star}(\\rho^{\\star}_{\\bar{2}})=\\ldots=\\hat{R}^{\\star}(\\rho^{\\star}_{\\overline{K+1}})$ as shown in Fig. ~\\ref{figproofmin}(d). From our solution, we are able to acquire closed-form solutions for our PS ratio at each of the $K+1$ nodes. These optimal PS ratios achieve the maximized set system SNR (i.e., $\\{\\hat{\\gamma}\\}^{K+1}_{k=1}=\\hat{\\gamma}^{\\star}$), this implies that, each node attains the maximized system rate established as $R^{\\star}=\\log_{2}(1+\\hat{\\gamma}^{\\star})$. $\\blacksquare$\n\\section{Number of Relay Nodes Estimation}\n\\label{app3}\nFrom (~\\ref{eqgeneraleq}), we have the estimated source power as \n\\begin{equation}\n\\label{app11}\n\\begin{aligned}\nE_{0}&\\quad\\approx \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}_{k}\\beta_{k}}{\\mathop{\\mathbb{E}}[\\Gamma_{k}]}=  \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}_{k}\\beta_{k}}{\\mathop{\\mathbb{E}}\\Big[\\prod^{k}_{j=1}C_{j}\\Big(\\frac{d_{j}}{d_{0}}\\Big)^{-\\alpha_{j}}\\beta_{j}\\vert h_{j}\\vert^{2}\\Big]}\\\\&\\quad = \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}_{k}\\beta_{k}}{\\prod^{k}_{j=1}C_{j}\\Big(\\frac{d_{j}}{d_{0}}\\Big)^{-\\alpha_{j}}\\beta_{j}\\mathop{\\mathbb{E}}\\Big[\\prod^{k}_{j=1}\\vert h_{j}\\vert^{2}\\Big]}\\\\&\\quad = \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}_{k}\\beta_{k}}{\\prod^{k}_{j=1}C_{j}\\Big(\\frac{d_{j}}{d_{0}}\\Big)^{-\\alpha_{j}}\\beta_{j}\\prod^{k}_{j=1}\\mathop{\\mathbb{E}}\\big[\\vert h_{j}\\vert^{2}\\big]}.\n\\end{aligned}\n\\end{equation}\nAssuming the sensor network is a homogeneous sensor network, let $\\beta_{1}=\\beta_{2}=\\ldots=\\beta_{K}$, $\\beta_{K+1}=1$, $\\sigma_{1}=\\sigma_{2}=\\ldots=\\sigma_{K}=\\sigma_{K+1}$, and $d_{1}=d_{2}=\\ldots=d_{K}=d_{K+1}$, then, \n\\begin{equation}\n\\label{app12}\n\\begin{aligned}\nE_{0}&\\quad\\approx \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\mathop{C^{k}\\Big(\\frac{d}{d_{0}}\\Big)^{-k\\alpha_{j}}\\beta^{k}\\prod^{k}_{j=1}\\mathbb{E}}\\big[\\vert h_{j}\\vert^{2}\\big]}.\n\\end{aligned}\n\\end{equation}\nSince $\\vert h_{j}\\vert^{2}$ is a random variable with an exponential distribution, hence, $\\mathop{\\mathbb{E}}\\big[\\vert h_{j}\\vert^{2}\\big]=1\/2=0.5$, because $h_{j} \\sim \\mathcal{CN}(0,1)$. This expectation solution is acquired from the fact that, the expectation of an exponential distribution acquired from a the square of Rayleigh distribution is the variance of the Rayleigh distribution divided by $2$, hence, we have $1\/2$. Therefore the optimal source power becomes\n\\begin{equation}\n\\label{app12a}\n\\begin{aligned}\nE_{0}&\\quad\\approx \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{C^{k}\\Big(\\frac{d}{d_{0}}\\Big)^{-k\\alpha_{j}}\\beta^{k}0.5^{k}}\\\\&\\quad= \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Big[ 0.5 C\\Big(\\frac{d}{d_{0}}\\Big)^{-\\alpha}\\beta\\Big]^{k}}.\n\\end{aligned}\n\\end{equation}\nLet $\\Gamma =0.5 C\\Big(\\frac{d}{d_{0}}\\Big)^{-\\alpha}\\beta$, then, the approximated source power becomes, \n\\begin{equation}\n\\label{app12b}\n\\begin{aligned}\nE_{0}\\approx \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{k}}.\n\\end{aligned}\n\\end{equation}\nUsing the approximation $E_{0}\\approx \\sum^{N}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{k}}$, we present the stepwise derivation of the estimated maximum number of DF-SWIPT relay nodes supported by a given source power. By expanding the (~\\ref{app12b}), we have, \n\\begin{equation}\n\\label{app13}\n\\begin{aligned}\nE_{0}&\\quad\\approx \\sum^{K+1}_{k=1}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{k}}= \\sum^{K+1}_{k=0}\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{k}}-\\bar{\\gamma}\\sigma^{2}\\beta\\\\&\\quad=\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{0}}+\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{1}}+\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{2}}+\\ldots+\\frac{\\bar{\\gamma}\\sigma^{2}\\beta}{\\Gamma^{K+1}}-\\bar{\\gamma}\\sigma^{2}\\beta.\n\\end{aligned}\n\\end{equation}\nIt is obvious from (~\\ref{app11}) that the $E_{0}$ equation is similar to the geometric progression sum equation, that is,\n\\begin{equation}\n\\label{app14}\n\\begin{aligned}\nE_{0}&\\quad\\approx \\alpha_{0}r^{0}+\\alpha_{0}r^{1}+\\alpha_{0}r^{2}+\\ldots+\\alpha_{0}r^{K+1}-\\alpha_{0}\\\\&\\quad = \\alpha_{0}\\Big(\\frac{1-r^{K+1}}{1-r}\\Big)-\\alpha_{0}=\\alpha_{0}\\Big(\\frac{1-r^{K+1}}{1-r}-1\\Big),\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha_{0}=\\bar{\\gamma}\\sigma^{2}\\beta$ and $r=\\frac{1}{\\Gamma}$ ~\\cite{Bird14,Stroud13}. By making $K+1$ the subject, we have,\n\\begin{equation}\n\\label{app15}\nE_{0}\\approx\\alpha_{0}\\Big(\\frac{1-r^{K+1}}{1-r}-1\\Big),\n\\end{equation}\n\\begin{equation}\n\\label{app16}\n1-\\Big(\\frac{E_{0}}{\\alpha_{0}}+1\\Big)\\Big(1-r\\Big)\\approx r^{K+1},\n\\end{equation}\n\\begin{equation}\n\\label{app17}\n\\ln\\Big[1-\\Big(\\frac{E_{0}}{\\alpha_{0}}+1\\Big)\\Big(1-r\\Big)\\Big]\\approx(K+1)\\ln r,\n\\end{equation}\n\\begin{equation}\n\\label{app18}\nK+1\\approx\\frac{\\ln\\Big[1-\\Big(\\frac{E_{0}}{\\alpha_{0}}+1\\Big)\\Big(1-r\\Big)\\Big]}{\\ln r}.\n\\end{equation}\nThe above solution is acquired from the geometric progression sum equation is for when $0 <r < 1$, that is, $\\Gamma > 1$. Following similar steps used above, when $r > 1$, that is, $\\Gamma < 1$, the solution becomes,\n\\begin{equation}\n\\label{app19}\nK+1\\approx\\frac{\\ln\\Big[1+\\Big(\\frac{E_{0}}{\\alpha_{0}}+1\\Big)\\Big(r-1\\Big)\\Big]}{\\ln r}.\n\\end{equation}\nNow, we replace $\\alpha_{0}$ and $r$ by their defined values to acquire the approximation of $K$ DF-SWIPT nodes as\n\\begin{equation}\n\\label{appnodenumb}\nK\\approx\\begin{cases}\n\\frac{\\ln\\Big[1-\\Big(\\frac{E_{0}}{\\bar{\\gamma}\\sigma^{2}\\beta}+1\\Big)\\Big(1-\\frac{1}{\\Gamma}\\Big)\\Big]}{-\\ln \\Gamma}-1,\\text{ }\\text{ }0<r<1,\\Gamma > 1 \\\\\n\\frac{\\ln\\Big[1+\\Big(\\frac{E_{0}}{\\bar{\\gamma}\\sigma^{2}\\beta}+1\\Big)\\Big(\\frac{1}{\\Gamma}-1\\Big)\\Big]}{-\\ln \\Gamma}-1,\\text{ }\\text{ }r>1,\\Gamma <1.\n\\end{cases}\n\\end{equation}\nIt can be observed that equation ($~\\ref{appnodenumb}$) is always positive and depends on the value of $\\Gamma$. $\\blacksquare$\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\\bibliographystyle{IEEEtr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe Weyl anomaly in quantum field theory has a long history (see e.g. \\cite{Duff:1993wm} for a historical overview). It states that the Weyl transformation of the metric in the curved space-time may not be a symmetry of the system even though the quantum field theory under consideration has the conformal symmetry in the flat space-time. Indeed, in most conformal field theories, the Weyl anomaly is non-vanishing, and we say that the Weyl symmetry is quantum mechanically broken in curved space-time.\n\nThe Weyl anomaly has played many important roles in theoretical physics.\nThe existence of the Weyl anomaly gives a constraint on the expectation values of energy-momentum tensor in curved background \\cite{Page:1982fm}, and may be related to the nature of Hawking radiation \\cite{Robinson:2005pd}\\cite{Iso:2006wa}\\cite{Kawai:2017txu}\\cite{Kawai:2014afa}.  The fact that vanishing of the Weyl anomaly happens only in a limited class of theories dictates the number of space-time dimensions in critical string theory. \nMore recently, we find that the universal terms in the entanglement entropy of conformal field theories are given by the coefficient of the Weyl anomaly, suggesting a deep relation between geometry and information \\cite{Ryu:2006ef}.\n\nWhat we would like to study in this paper is to find a way to cancel the Weyl anomaly from the other source, e.g. from the position dependent coupling constant.\\footnote{Sorry for the oxymoron. It is no longer constant. Probably it is Dirac who openly advocated this idea in the early days \\cite{Dirac:1938mt}.} Once we put a quantum field theory on a curved manifold, it is natural to assume that coupling constants are position dependent. The position dependent coupling constants then provide an extra contribution to the Weyl anomaly so that we may attempt to cancel the entire Weyl anomaly on the curved manifold. We would like to find under which condition such a cancellation is possible.\n\nThe similar idea of cancelling more general anomalies have been implicitly assumed in many places. For example, if we try to introduce background gauge fields for chiral current operators in the curved background (e.g. in the context of supersymmetric localization), then they may be mutually inconsistent due to the 't Hooft anomaly. One way to avoid this is to cancel the anomaly of the background gauge field from the space-time curvature and vice versa. Similarly, if preserving the Weyl anomaly is the critical issue (e.g. if we try to gauge it), our new way of doing it may be another option to be considered.\n\nThe organization of the paper is as follows. In section 2, we study the cancellation of the Weyl anomaly from position dependent coupling constant in two dimensional conformal field theories. In section 3, we study it in three dimensions and in section 4, we study it in four dimensions. We supplement the holographic viewpoint in section 5 and conclude with some discussions in section 6.\n\n\\section{Two dimensions}\nLet us consider a two-dimensional conformal field theory with an exactly marginal deformation denoted by $g$. For instance, we may take the Gaussian $c=1$ boson with the compactification radius as the exactly marginal deformation here.\nWe put the theory on a curved background with the metric $g_{\\mu\\nu}(x)$ and then vary the coupling constant $g(x)$ over the manifold: schematically we consider the action $S = S_0 + \\int d^2x\\sqrt{g} g(x) O(x)$. Even though the theory is conformal invariant in the Euclidean space $g_{\\mu\\nu} = \\delta_{\\mu\\nu}$ with $g(x) = g$, it is not necessarily so after turning on the background metric and position dependent coupling constant.\\footnote{We define the scale transformation  by the change of the difference of the coordinate as in \\cite{Osborn:1991gm} rather than the coordinate itself \\cite{Dong:2012ua}.} \nThis obstruction is known as the Weyl anomaly under the infinitesimal Weyl rescaling: $\\delta g_{\\mu\\nu}(x) = 2\\delta \\sigma g_{\\mu\\nu}$. \n\nIn terms of the free energy functional $e^{-F[g_{\\mu\\nu}(x),g(x)]} = \\int \\mathcal{D} \\Phi e^{-S[\\Phi]}$, the Weyl anomaly for a two-dimensional conformal field theory is given by (e.g. see  \\cite{Osborn:1991gm})\\footnote{In this paper, we always assume that the conformal field theories under consideration preserve the CP symmetry.}\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^2x\\sqrt{g} \\delta \\sigma(x) (cR -\\frac{1}{2} \\partial^\\mu g(x) \\partial_\\mu g(x)) \\  \\label{twoa}\n\\end{align}\nin a certain renormalization scheme so that the exactly marginal deformation has a flat line metric. Otherwise, we can always redefine the coupling constant or the renormalization scheme so that it is flat. It is clear that when $g(x) = g$, the only way to cancel the Weyl anomaly is to require $c=0$, which is typically what we demand in critical string theory.\n\nHowever, we see that this is not the only available option. Now, given a positive curvature $R(x)\\ge 0$, we may try to cancel the curvature term in the Weyl anomaly \\eqref{twoa} against the second term originating from the position dependent coupling constant by solving the equation\n\\begin{align}\nc R = \\frac{1}{2} \\partial^\\mu g(x) \\partial_\\mu g(x) \\ . \\label{twoc}\n\\end{align}\nThis is possible for positive curvature $R(x) \\ge 0$ (assuming $c>0$ in unitary conformal field theories). \n\n\nFor example, if we take the Fubini-Study metric on the sphere with the complex coordinate $z$ and $\\bar{z}$:\n\\begin{align}\nds^2 = \\frac{dzd\\bar{z}}{(1+|z|^2)^2} \\ , \n\\end{align}\nthe solution of \\eqref{twoc} is \n\\begin{align}\ng(x) = \\sqrt{c} \\cdot \\mathrm{arctan}(|z|)  + \\mathrm{const} \\ .\n\\end{align}\nIn this way, one can cancel the Weyl anomaly on the sphere by introducing the position dependent coupling constant. Note, however, that the position dependence of the coupling constant reduces the symmetry of the sphere from $SO(3)$ down to $SO(2)$. The idea here is we gained extra ``Weyl symmetry\" at the sacrifice of the rotational symmetry.\\footnote{This is not necessarily a bad idea: for examle, in the supersymmetric localization, we often do not keep the full isometry of the sphere but only the $U(1)$ subgroup of it.}\n\nThe above cancellation works both for infinitesimal generic Weyl transformation $\\delta g_{\\mu\\nu}(x) = 2\\delta \\sigma(x) g_{\\mu\\nu}(x)$ or finite but constant Weyl transformation $g_{\\mu\\nu}(x) \\to e^{2\\bar{\\sigma}} g_{\\mu\\nu}(x)$, where $\\bar{\\sigma}$ is a finite constant. The latter is because the equation to be solved in \\eqref{twoc} trivially scales under the constant Weyl transformation, so once it is solved then it is also solved after finite but constant Weyl transformation.\nFor finite generic Weyl transformation, however, the cancellation may not persist. The point is that the curvature term in the Weyl anomaly is non-trivially transforms under the Weyl transformation:\n\\begin{align}\nR \\to e^{-2\\sigma(x)} (R -2D^2 \\sigma) \\ , \n\\end{align}\nwhere $D^2$ is the Laplacian, \nwhile the Weyl transformation of the second term from the position dependent coupling constant $\\partial^\\mu g(x) \\partial_\\mu g(x)$ is trivial:\n\\begin{align}\n\\partial^\\mu g(x) \\partial_\\mu g(x) \\to e^{-2\\sigma(x)} \\partial^\\mu g(x) \\partial_\\mu g(x)\n\\end{align}\nThus, even though one may solve the cancellation condition for a given $g_{\\mu\\nu}(x)$ with a certain position dependent coupling constant $g(x)$, the cancellation does not persist for the Weyl transformed geometry. \n\n Nevertheless, we realize that the cancellation is still intact if we restrict\\footnote{A similar restriction on the Weyl transformation has been studied in \\cite{Edery:2014nha}.} ourselves to the harmonic Weyl transformation, which satisfies $D^2 \\sigma = 0$. Thus, we may construct a quantum field theory which is exactly invariant under the harmonic Weyl transformation by cancelling the Weyl anomaly from the position dependent coupling constant.\n\n\nMore generically, one may consider theories with several exactly marginal deformations. The Weyl anomaly has the generalized form\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^2x\\sqrt{g} \\delta\\sigma(x) (cR - \\chi_{ij}(g) \\partial^\\mu g^i(x) \\partial_\\mu g^j(x)) + \\partial_\\mu \\delta\\sigma(x) w_i(g) \\partial^\\mu g^i(x)  \\ , \\label{twog}\n\\end{align}\nwhere $\\chi_{ij}(g)$ and $w_i(g)$ may depend on the exactly marginal deformations $g^i(x)$. \nFor a constant Weyl transformation, the condition for the cancellation is essentially the same as before since the last term in \\eqref{twog} drops out.\n\nFor infinitesimal generic Weyl transformation, however, we have to think about the cancellation of the third term proportional to $\\partial_\\mu \\delta \\sigma(x)$. We did not talk about it in the single coupling case because we were able to remove it from the local counterterm $\\int d^2x\\sqrt{g} b(g) R$, but we have to discuss it now with several coupling constants when it has the non-trivial curvature $\\partial_i w_j - \\partial_j w_i$.\nWhile the Wess-Zumino consistency condition does not say anything about the (non-)existence of this term  \\cite{Osborn:1991gm}, the recent analysis in \\cite{Gomis:2015yaa} tells that on the conformal manifold spanned by the exactly marginal deformations, the curvature is trivial (i.e.  $\\partial_i w_j - \\partial_j w_i = 0$) and can be removed by the local counterterm $\\int d^2x\\sqrt{g} b(g^i) R$,\n so we actually do not have to worry about its cancellation. The non-existence of the curvature $\\partial_i w_j -\\partial_j w_i$ is related to the gradientness of the beta functions and it may have a deep implication in renormalization group flows  \\cite{Osborn:1991gm}\\cite{Friedan:2009ik}\\cite{Gukov:2016tnp}.\n\n\n\n\\section{Three dimensions}\nThere is no curvature dependent Weyl anomaly in three dimensions. The position dependent exactly marginal deformations do not introduce the additional Weyl anomaly either under the assumption of the CP symmetry \\cite{Nakayama:2013wda}. Thus there is no interesting scenario we can imagine in three dimensions.\n\n\\section{Four dimensions}\nLet us consider a four-dimensional conformal field theory with an exactly marginal deformation denoted by $g$. We put the theory on a curved background with the metric $g_{\\mu\\nu}(x)$ and then vary the coupling constant over the manifold $g(x)$. \nIn a certain renormalizaiton scheme, the first order Weyl transformation (i.e. the Weyl anomaly) is given by \\cite{Osborn:1991gm} (See also \\cite{Nakayama:2013is}\\cite{Jack:2013sha}.)\n\\begin{align}\n\\delta F_{\\sigma}  = \\int d^4x \\sqrt{g} \\delta\\sigma(x) &\\left( c(g) \\mathrm{Weyl}^2 - a \\mathrm{Euler} + (D^2 g D^2g - 2G_{\\mu\\nu} \\partial^\\mu g \\partial^\\nu g -\\frac{R}{3}\\partial^\\mu g\\partial_\\mu g)  \\right. \\cr\n& \\left. + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \\right) \\ .  \\label{foura}\n\\end{align}\nHere $G_{\\mu\\nu} = R_{\\mu\\nu} - \\frac{R g_{\\mu\\nu}}{2}$ is the Einstein tensor and $D^\\mu$ is the covariant derivative. In addition, we have introduced $\\mathrm{Weyl}^2 = R_{\\mu\\nu\\rho\\sigma}^2 -2R_{\\mu\\nu}^2 + \\frac{1}{3}R^2$ and $\\mathrm{Euler} = R_{\\mu\\nu\\rho\\sigma}^2 - 4R_{\\mu\\nu}^2 + R^2$. \nIn principle $c(g)$ can depend on $g$, but the only such theories known are constructed in somewhat artificial holographic realization \\cite{Nakayama:2017oye}.\n\n \nTo simplify the analysis, let us focus on the regime in which the last quartic term in \\eqref{foura} i.e. $\\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g$ can be neglected (e.g. in the small coupling regime). \nNeglecting the quartic term, we try to solve the equation\n\\begin{align}\n-c\\mathrm{Weyl}^2 + a \\mathrm{Euler} = (D^2 g D^2g - 2G_{\\mu\\nu} \\partial^\\mu g \\partial^\\nu g -\\frac{R}{3}\\partial^\\mu g\\partial_\\mu g) \\ . \\label{fourc}\n\\end{align}\nIn particular, suppose that the metric $g_{\\mu\\nu}(x)$ is Ricci flat. Then the equation \\eqref{fourc} becomes\n\\begin{align}\n\\sqrt{(a-c) R_{\\mu\\nu\\rho\\sigma}^2} = D^2 g \\ , \n\\end{align}\nwhich may be solved by using Green's function for the Laplacian \n\\begin{align}\ng(x) = \\int d^4x' G(x,x') \\sqrt{(a-c) R_{\\mu\\nu\\rho\\sigma}^2(x')}\n\\end{align}\nwhen the manifold is non-compact (otherwise the regular solution does not exist).\n\n\n\nAs in two-dimensions, the above argument works both for infinitesimal Weyl transformation or finite but constant Weyl transformation.\nFor finite generic Weyl transformation, one may define the analogue of harmonic Weyl transformation. For this purpose, it is more convenient to choose a different renormalization scheme so that the Weyl anomaly takes the form (e.g. \\cite{Gomis:2015yaa})\n\\begin{align}\n\\delta F_{\\sigma}  = \\int d^4x \\sqrt{g} \\delta\\sigma(x) &\\left( (c(g)-a) \\mathrm{Weyl}^2 - 4a Q   + g \\Delta_4 g \\right. \\cr\n& \\left. + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \\right) \\ , \\label{alt} \n\\end{align}\nwhere $\\Delta_4$ is the Fradkin-Tseytlin-Riegert-Paneitz conformal operator \\cite{Fradkin:1982xc}\\cite{Fradkin:1981jc}\\cite{Riegert:1984kt}\\cite{PA}\n\\begin{align}\n\\Delta_4 = (D^2)^2 + 2G_{\\mu\\nu}D^\\mu D^\\nu + \\frac{1}{3}(D^\\mu R) D_\\mu + \\frac{1}{3}R D^2, \n\\end{align}\nwhich is Weyl covariant $\\Delta_4 \\to e^{-4\\sigma} \\Delta_4$, \nand $Q$ is what is called the Q-curvature \\cite{Q}:\n\\begin{align}\nQ = \\frac{-1}{6} D^2 R -\\frac{1}{2}R^{\\mu\\nu}R_{\\mu\\nu} + \\frac{1}{6}R^2 \\ \n\\end{align}\nwhich has a nice mathematical property under the Weyl transformation\n\\begin{align}\nQ \\to e^{-4\\sigma} (Q + \\Delta_4 \\sigma)\n\\end{align}\n\nThe advantage of this rewriting or a choice of the particular local counterterm is as follows. Suppose we cancelled the Weyl anomaly at a particular background by demanding\n\\begin{align}\n0 = (c(g)-a) \\mathrm{Weyl}^2 - 4a Q   + g \\Delta_4 g + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \n\\end{align}\nThen, we are still able to cancel the Weyl anomaly on the Weyl transformed manifold whenever the Weyl rescaling is annihilated by the Fradkin-Tseytlin-Riegert-Paneitz operator:\n\\begin{align}\n\\Delta_4 \\sigma = 0 \\ . \\label{ann}\n\\end{align}\nThis is because all the terms in \\eqref{alt} except for the Q-curvature transform covariantly under the finite Weyl transformation.\nIf the Weyl scaling factor satisfies \\eqref{ann}, the cancellation of the Weyl anomaly therefore persists even for finite Weyl transformation. This is precisely analogous to the special role of harmonic Weyl transformation in two dimensions.\n\n\nLet us move on to the most generic cases with multiple coupling constants. The Weyl transformation is given by\n\\begin{align}\n\\delta F_{\\sigma}  = \\int d^4x \\sqrt{g} \\delta \\sigma(x) &( c(g) \\mathrm{Weyl}^2 - a \\mathrm{Euler} +  \\chi_{ij}(g) (D^2 g^i D^2g^j - 2G_{\\mu\\nu} \\partial^\\mu g^i \\partial^\\nu g^j -\\frac{R}{3}\\partial^\\mu g^i \\partial_\\nu g^j)  \\cr\n& \\left. + \\chi_{ijkl}(g) \\partial_\\mu g^i \\partial^\\mu g^j \\partial_\\nu g^k \\partial^\\nu g^l \\right)  \\cr\n&+ \\partial_\\mu \\delta\\sigma G^{\\mu\\nu} w_i(g) \\partial_\\nu g^j \\ . \\label{fourm}\n\\end{align}\nFor finite but constant Weyl transformation, we only have to cancel the first two lines in \\eqref{fourm}, which is  essentially equivalent to what we have done in the above. On the other hand, for infinitesimal but generic Weyl transformation, we have to cancel the third line as well, which requires either $\\partial_i w_j - \\partial_j w_i =0$ or $G_{\\mu\\nu} = 0$ in the background.\n\nTo conclude the analysis, we would like to mention the other obstructions to the Weyl transformation if the dimension two operator $O(x)$ exist in the theory. If this is the case, there is a further operator Weyl anomaly such as \n\\begin{align}\n\\int d^4x \\sqrt{g} \\delta\\sigma(x) (\\eta(g) R O(x) + \\epsilon(g) \\partial_\\mu g \\partial^\\mu g O + \\tau(g)  D^2 O + \\delta(g) D^2 g O) + \\partial_\\mu \\delta \\sigma(x) \\theta(g) \\partial^\\mu g O \\ .\n\\end{align}\nIt has been shown that such Weyl anomaly can be removed when $g(x) = g$ \\cite{Jack:2013sha} (see also \\cite{Nakayama:2013wda}\\cite{Farnsworth:2017tbz} for similar analysis), but with the space-time dependent coupling, we need the extra cancellation to get the consistent picture. Schematically, the Wess-Zumino consistency condition demands $\\eta = 0$ and  one can always remove $\\theta$ and $\\tau$ by local counterterms. Then we need to cancel $\\epsilon$ term against the $\\delta$ term. Since the existence of dimension two operator is non-generic, we will not pursue the cancellation in further details.\n\n\n\\section{Holographic models}\nWe revisit the cancellation mechanism we have studied in previous sections from  the holographic perspective. For definiteness we consider the case of four dimensional conformal field theories with the five dimensional bulk. Let us study the Einstein gravity coupled with a scalar field $\\phi$ given by the minimal action\n\\begin{align}\nS = \\int d^5x \\sqrt{g} \\left( R + \\Lambda + \\frac{1}{2}\\partial^M \\phi \\partial_M \\phi  \\right) .\n\\end{align}\n\nIn the AdS\/CFT correspondence, we compute the on-shell action for a given boundary condition at $\\rho= \\epsilon$ (i.e. $\\phi_{(0)}(x)$ and $g_{(0)\\mu\\nu}(x)$ below) with the expansion \n\\begin{align}\n\\phi &= \\phi_{(0)}(x) + \\rho \\phi_{(1)}(x) + \\rho^2 \\phi_{(2)}(x) + \\cdots \\cr\ng_{\\mu\\nu} &= g_{(0)\\mu\\nu}(x) + \\rho g_{(1)\\mu\\nu}(x) + \\rho^2 g_{(2) \\mu\\nu}(x) + \\cdots\n\\end{align}\nin the Graham-Fefferman gauge \n\\begin{align}\nds^2 = G_{MN} dx^M dx^N = \\frac{d\\rho^2}{\\rho^2} +  \\frac{g_{\\mu\\nu} dx^\\mu dx^\\nu}{\\rho}  \\ .\n\\end{align}\n\nThe resulting on-shell action is generically divergent in the limit $\\epsilon \\to 0$ from the $\\rho$ integration as $\\int_{\\epsilon} d\\rho \\rho^{-1} S_{\\log} = \\log \\epsilon S_{\\log}$, leading to the holographic Weyl anomaly \\cite{Henningson:1998gx}. Explicitly \\cite{Nojiri:1998dh}, we have\n\\begin{align}\nS = \\log \\epsilon \\int d^4x &\\left( \\frac{1}{8} R_{\\mu\\nu (0)}^2 -\\frac{1}{24} R_{(0)}^2  + \\frac{1}{4} (D^2 \\phi_{(0)})^2 \\right. \\cr \n& \\left.  - \\frac{1}{2} R_{(0)}^{\\mu\\nu} \\partial_\\mu \\phi \\partial_\\nu \\phi + \\frac{1}{6} R_{(0)} \\partial^\\mu \\phi_{(0)} \\partial_\\nu \\phi_{(0)} + \\frac{1}{3}(\\partial_\\mu \\phi_{(0)} \\partial^\\mu\\phi_{(0)})^2  \\right) \\ \n\\end{align}\nwhich is exactly what we had in section 4 for the constant Weyl transformation.\nThus cancelling the Weyl anomaly from the position dependent coupling constant corresponds to the choice of the boundary values of $\\phi(x)$ such that the on-shell gravity action is finite without the logarithmic divergence. The choice of such boundary conditions make the AdS\/CFT correlation functions better behaved, so classifying such supergravity background may be of theoretical interest.\n\n\\section{Discussions}\nIn this paper, we have studied a novel way to cancel the Weyl anomaly from the position dependent coupling constant. Here we would like to mention further possibilities to cancel the Weyl anomaly.\n\nFirst of all, if the theory under consideration possesses a conserved current $J_\\mu$, one may introduce the background field strength by the coupling $\\int d^4x \\sqrt{g} A^\\mu J_\\mu$. This gives another contribution to the Weyl anomaly as $\\int d^4x \\delta\\sigma(x) b_0 F_{\\mu\\nu}(x) F^{\\mu\\nu}(x)$, where $b_0$ is the coefficient of the one-loop beta function determined from the current two-point function (which is positive in unitary conformal field theories) and $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$. Then we may try to cancel the Weyl anomaly from this contribution.\nActually, simultaneous use of the gauge field and the position dependent coupling constant may not be a good idea because of the existence of the vector beta functions \\cite{Nakayama:2013ssa}. Again we have to think about the cancellation of the extra operator Weyl anomaly such as $\\int d^4x \\delta \\sigma \\rho(g) \\partial^\\mu g J_\\mu$. To avoid the appearance of the vector beta functions, we may only introduce the position dependent coupling constant which is neutral under the symmetry generated by $J_\\mu$.\n\nIt could have been extremely interesting if we were able to find a novel class of Weyl gauging without demanding $c=0$ in two dimensions, and $c=a=0$ in four dimensions.\\footnote{See e.g. \\cite{Fradkin:1983tg} for a possibility in the context of supersymmetric Weyl gravity.} \nCurrently, the closest way to do this is to demand all the non-trivial Weyl anomalies vanish, say $a=0$ in four dimensions, and then try to cancel the $c \\mathrm{Weyl}^2$ term against the space-time dependent coupling constants. Here, we should further employ the non-unitariness of the model (since $a=0$ from the beginning suggests it must be so) to obtain the cancellation in the Weyl anomalies. This is because unitarity demands the positivity of the both terms and the cancellation only happens by using the non-unitary property. Whether this is better than just demanding $c=a=0$ is yet to be seen in the context of quantum Weyl gravity in which we would like to gauge the Weyl symmetry exactly.\\footnote{Note, however, that the introduction of the position dependent coupling constant modifies the conservation of the energy-momentum tensor, and the consistecy with the dynamical gravity must be discussed more carefully. We stress that in the main part of this paper, we have focused on the fixed gravitational background, so there is no inconsistency. The author would like to thank S.~Deser for the discussions.}\n\nIn two dimensions, we did not obtain any new possibilities to gauge the entire Weyl symmetry than demanding $c=0$ from the beginning. We are still able to gauge the harmonic Weyl symmetry, but the physical interest in such gauging (e.g. whether it defines new class of quantum gravity in two dimensions) should be discussed more in detail.\n\nFinally, we point out that there is an alternative option. Once we know how to solve $g(x)$ to cancel the Weyl anomaly in a given metric $g_{\\mu\\nu}(x)$, one may introduce the extra transformation on $g(x)$ so that the Weyl anomaly is always cancelled (irrespective of the obstructions we have discussed above). This possibility requires further investigation if such transformation can be defined systematically and then whether such a generalized notion of the Weyl transformation is useful or not.\n\n\n\\section*{Acknowledgements}\nThis work is in part supported by JSPS KAKENHI Grant Number 17K14301. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Effects of the code improvements on the wind}\nIn this section, we evaluate the effect on the wind of the three improvements that we presented to the treatment of the radiation field: (i) radial dependence of $f_\\text{\\tiny UV}$, (ii) improved radiation transport, and (iii) relativistic corrections.\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/comparisons.pdf}\n    \\caption{Effects of our improvements to the treatment of the radiation field on the trajectories of gas blobs. All simulations have been done with $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in}=50R_\\mathrm{g}$.}\n    \\label{fig:comparisons}\n\\end{figure*}\nThe impact of each improvement is shown in \\autoref{fig:comparisons}. In the first panel (blue), we calculate the optical depth $\\tau_\\text{\\tiny UV}$ from the centre, rather than attenuating each individual light ray coming from the disc. We also take a constant $f_\\text{\\tiny UV}=0.85$ and we do not include relativistic corrections. Calculating $\\tau_\\text{\\tiny UV}$ from the centre overestimates the attenuation of the UV radiation field in the outer parts of the wind, since it assumes that the radiation is crossing all the inner gas. This produces a failed wind in the outer regions of the disc (which is on too small a scale to be seen in \\autoref{fig:comparisons}), since the inner gas is optically thick to the UV radiation. In the next panel (green), we calculate $\\tau_\\text{\\tiny UV}$ by attenuating each individual light ray as explained in \\autoref{sec:radiation_transport}, consequently, most of the wind escapes from the disc since the UV attenuation is less strong, with the average wind velocity being slightly lower because we are now averaging over the slower trajectories in the outer part of the wind. The next improvement we evaluate is the inclusion of the radial dependence of $f_\\text{\\tiny UV}$ (third panel, orange). The change in wind geometry can be explained by considering the distribution of the UV emissivity, which is skewed towards the centre of the disc thus pushing the outer wind along the equator. This also leads to a more powerful wind, due to the increase in UV luminosity at small radii, where the fastest and most massive streamlines originate. Lastly, the effect of including relativistic corrections is shown in the 4th panel (red). As expected, the velocity of the wind is considerably lower, and hence also its kinetic luminosity. Furthermore, the wind flows at a slightly higher polar angle, as the vertical component of the radiation force gets weaker where the wind has a high vertical velocity. \n\\section{Conclusions and future work}\n\nIn this paper, we have presented \\textsc{Qwind3}, a model that builds upon the non-hydrodynamical code \\textsc{Qwind} first introduced in \\cite{risaliti_non-hydrodynamical_2010} to model UV line-driven winds in the context of AGNs. In this new version, we generalise the CAK formalism for stellar winds to AGNs, which we use to calculate the initial density and velocity with which the wind is launched at the surface of the accretion disc. We have highlighted the importance of correctly accounting for the fraction of luminosity emitted in the UV by each disc annulus, which in other numerical codes is assumed to be a constant over the whole disk. We have also introduced an algorithm to do ray-tracing calculations throughout the wind, allowing us to compute the radiation transfer of the system taking into account the full geometry of the wind and the spatial distribution of the X-ray and UV radiation sources. Furthermore, we have also included special relativistic corrections to the calculation of the radiation flux, which are important for ensuring that the wind does not achieve superluminal speeds. We note two important assumptions that still remain: the simplified dependence of the X-ray opacity on the ionisation parameter and the X-ray luminosity fraction, which we here assume is constant at $f_X=0.15$, independent of black hole mass and mass accretion rate. We will address these issues in the next {\\sc Qwind} release. \n\nWe have used the new code to explore under what conditions UV line-driven winds are successfully launched from accretion discs, studying how the normalised mass loss rate, kinetic luminosity, momentum outflow rate, and final velocity change as a function of black hole mass, mass accretion rate, and initial wind launching radius. We find that winds can carry a mass loss rate up to 30\\% of the mass accretion rate,\nso this can have a moderate impact on the mass accretion rate at radii below the wind launching radius. The next {\\sc Qwind} release will address this by reducing the mass accretion rate through the disc, and hence reducing the UV flux produced at small radii, to make the model self-consistent (see also \\citealt{nomura_line-driven_2020}).\nThe current code produces winds where the \nkinetic power and especially the momentum outflow rate can \nbe comparable to that of the radiation power at high $\\dot{m}$, and even exceed it which is unphysical. This shows that the effect of line blanketing must be important in reducing the UV flux driving the wind below that predicted by electron scattering opacity alone. We will address this in the next {\\sc Qwind} release, but \noverall, the wind clearly carries sufficient power to meet the criteria for an efficient feedback mechanism in galaxy formation \\citep{hopkins_stellar_2016}.\n\nOur results here show that the outflow velocity is mildly relativistic across a broad parameter space, with velocity $0.1-0.3$~c even excluding the unphysically efficient winds. Thus UV line-driven winds can reach UFO velocities even when special relativistic corrections (radiation drag) are included. This contrasts with \\cite{luminari_speed_2021}, who conclude that UV line-driving is not capable of generating such high velocity winds. \nWe caution that the two codes make different assumptions about the initial conditions and ray tracing (both of which \\textsc{Qwind} does more accurately and self-consistently) as well as the opacity (which their code does better), so it is premature to rule out UV line-driving in favour of other mechanisms such as magnetic driving \\citep{blandford_hydromagnetic_1982, fukumura_magnetic_2017} as the origin of UFOs until these factors are all incorporated together. We will explore this more fully in the next \\textsc{Qwind} release.\n\nThe normalised wind mass loss rate and normalised kinetic luminosity vary substantially as a function of black hole mass and accretion rate, the latter being the most significant factor. The ratio of wind kinetic power to mass accretion rate scales steeply with $\\dot{m}$, in contrast to the constant ratio normally assumed in the AGN feedback models currently implemented in cosmological simulations of galaxy formation. Implementing this new more physically-based AGN feedback prescription in simulations will therefore change how galaxies are predicted to evolve across cosmic time.\n\n\n\\section{Calculating the gas trajectories}\n\\label{sec:gas_trajectories}\n\\subsection{Initial radii of gas trajectories}\n\nThe first thing to consider when calculating the gas trajectories is the initial location of the gas blobs. The innermost initial position of the trajectories is taken as a free parameter $R_\\mathrm{in}$, and the outermost initial position, $R_\\mathrm{out}$, is assumed to be the self-gravity radius of the disc, where the disc is expected to end \\citep{laor_massive_1989}, which for our reference BH corresponds to 1580 $R_g$. We initialise the first trajectory at $R_\\mathrm{in}$, the next trajectory starts at $R=R_\\mathrm{in} + \\Delta R$, where $\\Delta R$ is the distance between adjacent trajectories. We determine $\\Delta R$ by considering two quantities: (i) the change in optical depth between two adjacent trajectories along the base of the wind and (ii) the mass loss rate along a trajectory starting at $R$ and at a distance $\\Delta R$ to the next one. Regarding (i), the change in optical depth $\\Delta\\tau$ between two trajectories initially separated by $\\Delta R$ is given by\n\\begin{equation}\n    \\Delta\\tau = \\int_R^{R+\\Delta R_1} n(R')\\, \\sigma_{\\text{\\tiny T}} \\,\\mathrm{d} R'.\n\\end{equation}\nWe consider\n\\begin{equation}\n    \\Delta\\tau = \\begin{cases} \n    0.05 & \\mathrm{ if } \\;\\tau(R) < 5,\\\\\n    0.5 & \\mathrm{ if } \\;\\tau(R) < 10,\\\\\n    5 & \\mathrm{ if } \\;\\tau(R) < 100,\\\\\n    20 & \\mathrm{ if } \\;\\tau(R) > 100,\\\\\n    \\end{cases}\n\\end{equation}\nwhere $\\tau(R) = \\sum_i \\Delta\\tau_i$. This guarantees that the spacing between trajectories resolves the transition from optically thin to optically thick for both the UV and X-ray radiation. The quantity (ii) is given by\n\\begin{equation}\n    \\Delta \\dot M_\\mathrm{wind} = \\int_R^{R+\\Delta R_2}2\\pi R'\\rho(R')v(R') \\mathrm{d} R',\n\\end{equation}\nwhere we consider $\\Delta \\dot M_\\mathrm{wind}=0.01\\dot M$, so that no streamline represents more than $1\\%$ of the accreted mass rate. The $\\Delta R$ step is thus given by \n\\begin{equation}\n    \\Delta R = \\min (\\Delta R_1, \\Delta R_2)\n\\end{equation}\nand we repeat this process until $R=R_\\mathrm{out}$\n\n\\subsection{Solving the equation of motion}\n\nThe equation of motion of the wind trajectories is the same as in \\citetalias{quera-bofarull_qwind_2020}, and we solve it analogously by using the Sundials IDA integrator \\citep{hindmarsh_sundials_2005}. The wind trajectories are calculated until they either exceed a distance from the centre of $10^4 R_g$, fall back to the disc, or self intersect. To detect when a trajectory self intersects we use the algorithm detailed in Appendix \\ref{app:intersections}.\n\nBy considering the full wind structure for the radiation ray tracing, we run into an added difficulty: the interdependence of the equation of motion with the density field of the wind. To circumvent this, we adopt an iterative procedure in which the density field of the previous iteration is used to compute the optical depth factors for the current iteration. We first start assuming that the disc's atmosphere is void, with a vacuum density of $n_\\text{vac} = 10^2$ cm$^{-3}$. Under no shielding of the X-ray radiation, all the trajectories fall back to the disc in a parabolic motion. After this first iteration, we calculate the density field of the resulting failed wind and use it to calculate the wind trajectories again, only that this time the disc atmosphere is not void. We keep iterating until the mass loss rate and kinetic luminosity do not significantly change between iterations. It is convenient to average the density field in logarithmic space between iterations, that is, the density field considered for iteration $k$, $n_k$, is given by\n\\begin{equation}\n    \\log_{10}(n_{k}(R, z)) = \\frac{1}{2}\\, \\left(\\log_{10}(n_{k-1}(R,z)) + \\log_{10}(n_{k-2}(R,z))\\right).\n\\end{equation}\nWe do not take the average for the first two iterations. The number of iterations required depends on the initial radius of the wind, but we typically find convergence after $\\sim 20$ iterations, although we run several more to ensure that the standard deviation of the density field between iterations is small. The normalised mass loss rate and kinetic luminosity for our fiducial model at each iteration are shown in \\autoref{fig:iterations}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/iterations.pdf}\n    \\caption{Normalised mass loss rate and kinetic luminosity for each iteration for our fiducial case.}\n    \\label{fig:iterations}\n\\end{figure}\n\nFor a given iteration, solving the equation of motion for the different gas trajectories is an embarrassingly parallel problem, and so our code's performance scales very well upon using multiple CPUs, allowing us to quickly run multiple iterations, and scan the relevant parameter spaces. The computational cost of running one iteration is $\\sim 5$ CPU hours, so one is able to obtain a fully defined wind simulation after $\\sim 100$ CPU hours.\n\n\\section{Critical point derivation}\n\\label{app:initial_conditions}\n\nIn this appendix we aim to give a detailed and self-contained derivation of the initial conditions for launching the wind from the surface of the accretion disk. Let us first start with the 1D wind equation (now including gas pressure forces),\n\\begin{equation}\n    \\label{eq:ic_1d}\n    \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho (a_\\mathrm{rad}^z - a_\\mathrm{grav}^z) - \\frac{\\partial P}{\\partial z}.\n\\end{equation}\nThe left-hand side of the equation can be expanded to\n\\begin{equation}\n    \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho \\frac{\\partial v_z}{\\partial t} + \\rho v_z \\frac{\\partial v_z}{\\partial z} ,\n\\end{equation}\nwhere we note that $\\partial v_z \/ \\partial t$ = 0, since we are only interested in steady solutions that do not depend explicitly on time. For simplicity, we drop the partial derivative notation going forward, since only derivatives along the $z$ direction are involved. We assume that the wind is isothermal, with an equation of state given by\n\\begin{equation}\n    P = c_s^2 \\rho\n\\end{equation}\nwhere $c_s$ is the isothermal sound speed, which is assumed constant throughout the wind. Using the mass conservation equation (\\autoref{eq:mass_conservation}), $\\dot M = A\\, \\rho\\, v_z$, where $A = 2\\pi r \\Delta r$, we can write\n\\begin{equation}\n    \\frac{\\mathrm{d} P}{\\mathrm{d} z} =c_s^2\\frac{\\mathrm{d} \\rho}{\\mathrm{d} z} = -\\rho c_s^2 \\left(\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A} + \\frac{\\mathrm{d} v_z \/ \\mathrm{d} z}{v_z}\\right),\n\\end{equation}\nso we can rewrite \\autoref{eq:ic_1d} as\n\\begin{equation}\n     v_z \\frac{\\mathrm{d} v_z}{\\mathrm{d} z}\\left(1-\\frac{c_s^2}{v_z^2}\\right) = a_\\mathrm{rad}^z - a_\\mathrm{grav}^z + c_s^2\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A}.\n\\end{equation}\nWe now focus on the radiation force term, which is the sum of the electron scattering and line-driving components,\n\\begin{equation}\n    a_\\mathrm{rad}^z = a_\\mathrm{rad}^{\\mathrm{es}, z} + \\mathcal M \\, a_\\mathrm{rad}^{\\mathrm{es}, z}.\n\\end{equation}\nWhen the wind is rapidly accelerating, as it is the case at low heights, the force multiplier is well approximated by its simpler form (see section 2.2.3 of \\citetalias{quera-bofarull_qwind_2020}),\n\\begin{equation}\n    \\label{eq:fm_simple}\n    \\mathcal M(t) = k \\, t^{-\\alpha},\n\\end{equation}\nwhere $\\alpha$ is fixed to $0.6$, $k$ depends on the ionisation level of the gas (in the \\citetalias{stevens_x-ray_1990} parameterisation), and $t = \\kappa_{\\text{\\tiny e}} \\, \\rho \\, v_\\text{th} \\left | \\mathrm{d} v\/\\mathrm{d} z \\right | ^{-1}$. As discussed in section 3.1.3 of \\citetalias{quera-bofarull_qwind_2020}, the thermal velocity, $v_\\mathrm{th}$ (\\autoref{eq:thermal_velocity}), is computed at a fixed temperature of $T=2.5 \\times 10^4$ K. Once again making use of the mass conservation equation we can write\n\\begin{equation}\n    \\mathcal M = k\\, t^{-\\alpha} = \\frac{k}{(\\kappa_{\\text{\\tiny e}}\\, v_\\text{th}\\, \\rho)^\\alpha}\\left |\\frac{dv}{dz}\\right|^\\alpha = \\frac{k}{\\left(\\kappa_{\\text{\\tiny e}} v_\\text{th} \\, \\right)^\\alpha}\\left(\\frac{A}{\\dot M} v_z \\left|\\frac{dv_z}{dz}\\right|\\right)^\\alpha\n\\end{equation}\nand so the full equation to solve with all derivatives explicit is\n\\begin{equation}\n    \\begin{split}\n        v_z \\frac{\\mathrm{d} v_z}{\\mathrm{d} z}\\left( 1- \\frac{c_s^2}{v_z^2}\\right) &= a_\\mathrm{rad}^{\\mathrm{es}, z} (z) -a_\\mathrm{grav}^z (z) \\; \\\\\n                                                                      &+ a_\\mathrm{rad}^{\\mathrm{es}, z}(z)\\frac{k}{\\left(\\kappa_{\\text{\\tiny e}} v_\\text{th} \\, \\right)^\\alpha}\\left(\\frac{A(z)}{\\dot M} v_z \\left|\\frac{dv_z}{dz}\\right|\\right)^\\alpha \\\\\n                                                                      &+ c_s^2\\frac{\\mathrm{d} A(z) \/ \\mathrm{d} z}{A(z)}.\n    \\end{split}\n\\end{equation}\nWe assume that $\\mathrm{d} v_z \/ \\mathrm{d} z >0$ always, so the absolute value can be dropped. It is convenient to introduce new variables to simplify this last equation. The first change we introduce is $W=v_z^2\/2$, so that $v_z \\, \\mathrm{d} v_z \/ \\mathrm{d} z = \\mathrm{d} W \/ \\mathrm{d} z$. Next, we aim to make the equation dimensionless, so we consider $R$ as the characteristic length, $B_0 = GM_\\mathrm{BH}\/R^2$ as the characteristic gravitational acceleration value, $W_0=GM_\\mathrm{BH}\/R = B_0 R$ as the characteristic $W$ value, $A_0 = 2\\pi R \\Delta R$ as the characteristic $A$ value, and finally we take the characteristic radiation force value to be \n\\begin{equation}\n\\label{eq:gamma_0}\n    \\gamma_0 =  \\frac{k}{(\\kappa_{\\text{\\tiny e}} v_\\text{th})^\\alpha} \\; a_\\mathrm{rad, 0}^{\\mathrm{es}, z},\n\\end{equation}\nwhere $a_\\mathrm{rad, 0}^{\\mathrm{es}, z}$ is  \\autoref{eq:radiation_acceleration_approx} without considering the $\\tau_\\text{\\tiny UV}$ factor, since we do not include attenuation in this derivation. With all this taken into account, we can write\n\\begin{equation}\n    \\label{eq:half_way}\n    \\begin{split}\n        B_0 \\frac{dw}{dx} \\left(1 - \\frac{s}{w}\\right) & = -a_\\mathrm{grav}^z + a_\\mathrm{rad}^{\\mathrm{es}, z} \\left(1+ \\frac{k}{(\\kappa_{\\text{\\tiny e}} v_\\text{th} )^\\alpha}\\left(\\frac{A}{\\dot M} B_0\\frac{dw}{dx}\\right)^\\alpha\\right) \\\\\n                                                       & + c_s^2\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A},\n\\end{split}\n\\end{equation}\nwhere we have defined $x=z\/R$, $w=W\/W_0$, and $s=c_s^2 \/ (2 W_0)$. We also define $a=A\/A_0$, and $\\varepsilon = \\dot M \/ \\dot M_0$ with\n\\begin{equation}\n    \\dot M_0 = \\alpha (1-\\alpha)^{(1-\\alpha) \/ \\alpha} \\frac{(\\gamma_0 A_0)^{1 \/ \\alpha}}{(B_0 A_0)^{(1-\\alpha) \/ \\alpha}}.\n    \\label{eq:Mdot0_def}\n\\end{equation}\nThis last definition may seem a bit arbitrary, but it is taken such that $\\varepsilon = 1$ corresponds to the classical \\citetalias{castor_radiation-driven_1975} $\\dot M$ value for O-stars. \\autoref{eq:half_way} then becomes\n\\begin{equation}\n    \\begin{split}\n        \\frac{\\mathrm{d} w}{\\mathrm{d} x} \\left(1-\\frac{s}{w}\\right) &= \\frac{(-a_\\mathrm{grav}^z + a_\\mathrm{rad}^{\\mathrm{es}, z})}{B_0} + \\frac{1}{\\alpha^\\alpha(1-\\alpha)^{1-\\alpha}}\\frac{a_\\mathrm{rad}^{\\mathrm{es}, z}}{a_\\mathrm{rad,0}^{\\mathrm{es}, z}} \\left(\\frac{a}{\\varepsilon}\\frac{\\mathrm{d} w}{\\mathrm{d} x}\\right)^\\alpha \\\\\n    &+ \\frac{4sx}{a},\n    \\end{split}\n\\end{equation}\nwhere we have used\n\\begin{equation}\n    a = \\frac{A}{A_0} = \\frac{2\\pi r \\Delta r}{2\\pi r_0 \\Delta r_0} = \\frac{r^2}{r_0^2},\n\\end{equation}\nso that\n\\begin{equation}\n    \\frac{c_s^2}{B_0}\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A} = \\frac{4sx}{a}.\n\\end{equation}\nWe note that we have assumed that the area $A$ changes with $z$ like the 2D solution, despite the fact that we are considering here a 1D wind. This small correction guarantees that we find critical-point like solutions for all initial radii, since it guarantees that the \\citetalias{castor_radiation-driven_1975} conditions for the existence of a critical point are satisfied.\nFinally, we define\n\\begin{equation}\n\\label{eq:nozzle_f}\n    f = \\frac{1}{\\alpha^\\alpha (1-\\alpha)^{1-\\alpha}}\\frac{a_\\mathrm{rad}^{\\mathrm{es}, z}}{a_\\mathrm{rad, 0}^{\\mathrm{es}, z}},\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:nozzle_h}\n    h = \\frac{(a_\\mathrm{grav}^z - a_\\mathrm{rad}^{\\mathrm{es}, z})}{B_0} - 4sxa,\n\\end{equation}\nso that\n\\begin{equation}\n    \\frac{\\mathrm{d} w}{\\mathrm{d} x} \\left(1-\\frac{s}{w}\\right) = -h(x) + f(x) \\left(\\frac{a}{\\varepsilon}\\frac{\\mathrm{d} w}{\\mathrm{d} x}\\right)^\\alpha,\n\\end{equation}\nwhich is the dimensionless wind equation. The choice of sign in $h(x)$ is to make the further steps clearer.\nIt is useful to interpret this equation as an algebraic equation for $w'=\\mathrm{d} w \/ \\mathrm{d} x$,\n\\begin{equation}\n    F(x,w,w') =  w' \\left(1-\\frac{s}{w}\\right) + h(x) - f(x) \\left(\\frac{a}{\\varepsilon}w'\\right)^\\alpha = 0.\n    \\label{eq:CAK_F_wprime}\n\\end{equation}\nThis has the same general form as Equation~26 in \\citetalias{castor_radiation-driven_1975}, which applies to a spherical wind from a star, but the functions $h(x)$ and $f(x)$ are different. We note that $f(x)>0$ and $0<\\alpha<1$, but $h(x)$ can have either sign. (Also note that we have chosen the opposite sign for $h(x)$ to CAK, for later convenience.)\nGiven values of $x$ and $\\varepsilon$, we can distinguish 5 different regions of solutions for $w'$ (we follow the original enumeration of regions by \\citetalias{castor_radiation-driven_1975}):\n\n\\begin{itemize}\n    \\item Subsonic stage ($w<s$): We distinguish two cases\n        \\begin{itemize}\n            \\item Region V: If $h(x)<0$, then all the terms in Equation~\\ref{eq:CAK_F_wprime} are negative and there is no solution for $w'$.\n            \\item Region I: If $h(x)>0$, $F(w'=0)$ is positive and as $w'$ increases, $F$ goes to negative values crossing $F=0$ once, so there is one solution for $w'$.\n        \\end{itemize}\n    \\item Supersonic stage ($w >s$): We have three regions\n        \\begin{itemize}\n            \\item Region III: If $h(x)<0$, $F(w'=0)$ is negative and as $w'$ increases, $F$ goes to positive values crossing $F=0$ once, so there is one solution for $w'$.\n            \\item If $h(x) > 0$, $F(w'=0)$ is positive, then $F$ initially decreases until its minimum, $w_\\mathrm{min}'$, and then increases again. If $F(w_\\mathrm{min}') >0$, we have no solution  for $w'$ and if $F(w_\\mathrm{min}') \\leq 0$ we have two solutions, which are the same if $F(w_\\mathrm{min}') = 0$.\n                The minimum can be found by solving\n                \\begin{equation}\n                    \\frac{\\partial F}{\\partial w'} = 0,\n                \\end{equation}\n                which gives\n                \\begin{equation}\n                \\label{eq:w_min}\n                    w_\\mathrm{min}' = \\left( \\frac{1-s\/w}{\\alpha f (a\/\\varepsilon)^\\alpha}\\right)^\\frac{1}{\\alpha -1},\n                \\end{equation}\n                so we have\n                \\begin{itemize}\n                    \\item Region IV: If $F(w_\\mathrm{min}') > 0$, there is no solution.\n                    \\item Region II: If $F(w_\\mathrm{min}') \\leq 0$, there are two solutions, one with $w' < w_\\mathrm{min}'$ and the other with $w' > w_\\mathrm{min}'$\n                \\end{itemize}\n        \\end{itemize}\n\\end{itemize}\n\nWe assume that the wind starts subsonic ($w<s$), so it has to start in Region~I, since that is the only subsonic region with a solution. We note that $h(x\\to\\infty) < 0$, since for large $x$ both the gravitational and radiation force are small. Assuming that the wind ends as supersonic (w>s) and extends to $x\\to\\infty$, this means that the wind must end at Region~III. However, because $h(x)>0$ in Region~I and $h(x)<0$ in Region~III, these two regions must be connected by Region~II in between. At the boundary between Regions~I and II, $w=s$ and $h(x)>0$, while at the boundary between Regions~II and III, $w>s$ and $h(x)=0$.\n\nConsidering first the boundary between Regions~I and II, setting $w=s$ in \\autoref{eq:CAK_F_wprime} gives\n\\begin{equation}\n    h - f \\left(\\frac{a}{\\varepsilon} w'\\right)^\\alpha = 0,\n\\end{equation}\nwith $h>0$. Considering \\autoref{eq:w_min} for $w_\\mathrm{min}'$ as $w \\to s^{+}$, we find that the wind solution at this boundary must lie on the branch with $w' < w_\\mathrm{min}'$.\n\nConsidering next the boundary between Regions~II and III, setting $h=0$ in \\autoref{eq:CAK_F_wprime} (neglecting the $w'=0$ solution) gives\n\\begin{equation}\n    1-\\frac{s}{w} - f \\left(\\frac{a}{\\varepsilon}\\right)^\\alpha w'^{\\alpha-1} = 0,\n\\end{equation}\nAgain using \\autoref{eq:w_min} and $w>s$, we find that the wind solution at this boundary must lie on the branch with $w' > w_\\mathrm{min}'$.\n\n\nHowever, we assume that $w'$ must be continuous throughout the wind, which would not be the case if the two branches of region II were not connected. Therefore, both branches of region II must coincide at some point, so the condition\n\\begin{equation}\n    F(w_\\mathrm{min}') = 0\n\\end{equation}\nmust hold at that point, with $w' = w_\\mathrm{min}'$ for both branches. This point is in fact the critical point of the solution, and $\\partial F\/\\partial w' = 0$ there. Upon substitution of $w_\\mathrm{min}'$, this condition is equivalent to\n\\begin{equation}\n    \\label{eq:nozzle_function_sonic}\n    \\varepsilon \\, \\left(1 - \\frac{s}{w} \\right) = \\alpha (1-\\alpha)^\\frac{1-\\alpha}{\\alpha} \\, \\frac{f^{1\/\\alpha} a}{h^\\frac{1-\\alpha}{\\alpha}}.\n\\end{equation}\nThe right-hand side of the equation is usually referred to as the Nozzle function $\\mathcal N$,\n\\begin{equation}\n\\label{eq:nozzle_function}\n    \\mathcal N(x) = \\alpha (1-\\alpha)^\\frac{1-\\alpha}{\\alpha} \\frac{f^{1\/\\alpha}a}{h^\\frac{1-\\alpha}{\\alpha}}.\n\\end{equation}\nWe define $x=x_c$ to be the position of the critical point. We now assume that at the critical point the wind is highly supersonic ($w \\gg s$). We verify this assumption in \\autoref{subsection:verify_critical_point}. Then $1-s\/w \\approx 1$, and so the normalised mass loss rate is given by\n\\begin{equation}\n    \\varepsilon = \\mathcal N(x_c).\n    \\label{eq:eps_n_xc}\n\\end{equation}\n\nWe now show that the location of the critical point $x_c$ is at the minimum of $\\mathcal N(x)$. Let us consider the total derivative of $F$ with respect to $x$ taken along a wind solution. Since $F=0$ at all points along the solution,\n\\begin{equation}\n    \\frac{\\mathrm{d} F}{\\mathrm{d} x} = \\frac{\\partial F}{\\partial x} + \\frac{\\partial F}{\\partial w}w' + \\frac{\\partial F}{\\partial w'}\\frac{\\mathrm{d} w'}{\\mathrm{d} x} = 0.\n\\end{equation}\nBecause of our assumption $w\\gg s$, we have $\\frac{\\partial F}{\\partial w} = 0$, and, at the critical point, $\\frac{\\partial F}{\\partial w'} = 0$, so that\n\\begin{equation}\n    \\left .\\frac{\\mathrm{d} F}{\\mathrm{d} x}\\right |_{w'_\\mathrm{min}, x_c} = \\left .\\frac{\\partial F}{\\partial x}\\right |_{w'_\\mathrm{min}, x_c} = 0,\n\\end{equation}\nwhich gives (a prime denotes $\\mathrm{d} \/ \\mathrm{d} x$)\n\\begin{equation}\n\\label{eq:dF_dx}\n    h' - \\left(\\frac{w'_\\mathrm{min}}{\\varepsilon}\\right)^\\alpha \n    \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( f a^{\\alpha} \\right) = 0.\n\\end{equation}\nNow using \\autoref{eq:w_min} for $w'_\\mathrm{min}$ along with \\autoref{eq:eps_n_xc} and \\autoref{eq:nozzle_function}, we obtain $w'_\\mathrm{min} = \\frac{\\alpha}{(1-\\alpha)} h$. Substituting in \\autoref{eq:dF_dx} and using \\autoref{eq:eps_n_xc} and \\autoref{eq:nozzle_function} again, we obtain \n\\begin{equation}\n(1-\\alpha) \\frac{h'}{h} -  \\frac{1}{f a^{\\alpha}} \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( f a^{\\alpha} \\right) = 0.\n\\end{equation}\nComparing to \\autoref{eq:nozzle_function}, we see that this is equivalent to the condition $\\mathrm{d} \\mathcal{N}\/\\mathrm{d} x =0$. Therefore the critical point happens at an extremum of $\\mathcal{N}(x)$, but since the nozzle function is not upper bounded, the extremum has to be a minimum. We have therefore shown that the critical point $x_c$ corresponds to the minimum of the nozzle function $\\mathcal{N}(x)$.\n\n\n\nIn \\autoref{fig:nozzle_function}, we plot the nozzle function for $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, $k=0.03$, and $R=20R_\\mathrm{g}$ and we highlight the position of the critical point. The value of the nozzle function at the critical point determines the mass-loss rate along the streamline, through $\\dot M = \\varepsilon \\dot M_0$, so that the surface mass loss rate is given by\n\\begin{equation}\n    \\dot \\Sigma = \\frac{\\dot M}{A}.\n\\end{equation}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/nozzle_function.pdf}\n    \\caption{Nozzle function for $\\protectM_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$, $\\protect\\dot m =0.5$, $k=0.03$, and $R=20 R_\\mathrm{g}$}  \n    \\label{fig:nozzle_function}\n\\end{figure}{}\n\nIt is interesting to point out that the position of the critical point and the value of $\\epsilon$ do not depend on the chosen value of $k$ in the force multiplier parametrisation (\\autoref{eq:fm_simple}), however, \nthe mass loss rate does depend on $k$ through the value of $\\dot{M}_0$.\nIn the \\citetalias{stevens_x-ray_1990} parametrisation, $k$ is a function of the ionisation state of the gas, $k=k(\\xi)$, so the mass loss rate directly depends on the ionisation conditions at the critical point location. Since this would make our results dependent on the modelling of the vertical structure of the disc, which is out of scope for the purpose of this work, we assume that the gas is always ionised when it reaches the critical point, so we take $k=0.03$, corresponding to the minimum value of $k$ in \\citetalias{stevens_x-ray_1990}. Similarly, we set the initial height of the wind to $z=0$ to avoid dependencies on the disc vertical structure.\n\n\n\\subsection{Scaling of the initial conditions with BH properties}\n\\label{subsection:ic_scaling}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{figures\/nozzle_scaling.pdf}\n    \\caption{Value of the nozzle function at the critical point ($\\varepsilon = \\mathcal N(x_c)$) as a function of radius for varying $M_\\mathrm{BH}$ (left panel) with $\\dot m =0.5$ and varying $\\dot m$ (right panel) with $M_\\mathrm{BH}=10^8 \\mathrm{M}_\\odot$}\n    \\label{fig:nozzle_scaling}\n\\end{figure}\n\nSince we explore the BH parameter space in \\autoref{sec:results_bh}, it is useful to assess how the initial number density and velocity of the wind scale with $M_\\mathrm{BH}$ and $\\dot m$. To this end, we need to determine how the values of $\\varepsilon$ and $\\dot \\Sigma_0$ change with $M_\\mathrm{BH}$ and $\\dot m$. For this analysis, we ignore the dependence of $f_\\text{\\tiny UV}(R)$ on $M$ and $\\dot m$. We also have $v_\\text{th}(R) \\propto T^{1\/2} \\propto (M_\\mathrm{BH} \\dot{M}\/R^3)^{1\/8}$ (using \\autoref{eq:radiation_flux}). Accounting for the fact that the disk size scales as $R \\propto R_\\mathrm{g} \\propto M_\\mathrm{BH}$, this gives $v_0 = v_\\text{th} \\propto (\\dot{m}\/M_\\mathrm{BH})^{1\/8}$, where $v_0$ is the initial velocity. This is a weak dependence, so we ignore it here. Looking at \\autoref{eq:gamma_0}, and using \\autoref{eq:radiation_acceleration_approx}, we get $\\gamma_0 \\propto M_\\mathrm{BH} \\dot{M}\/R^3 \\propto \\dot m \/ M_\\mathrm{BH}$. Similarly, $B_0 \\propto M_\\mathrm{BH}\/R^2 \\propto 1 \/ M_\\mathrm{BH}$. Using \\autoref{eq:Mdot0_def}, we then have $\\dot \\Sigma_0 \\propto \\dot{M}_0\/A_0 \\propto \\gamma_0^{1\/\\alpha}\/B_0^{(1-\\alpha)\/\\alpha} \\propto \\dot m^{1\/\\alpha} \/ M_\\mathrm{BH}$.\n\nThe scaling of $\\varepsilon = \\mathcal{N}(x_c)$ is a bit more complicated, since it depends on the exact position of the critical point for each value of $M_\\mathrm{BH}$, $\\dot m$, and $R$. In \\autoref{fig:nozzle_scaling}, we plot the values of $\\varepsilon$ as a function of radius for varying $M_\\mathrm{BH}$ (left panel) and $\\dot m$ (right panel), ignoring the dependence of $f_\\text{\\tiny UV}$ with $M_\\mathrm{BH}$ and $\\dot m$ (we set $f_\\text{\\tiny UV} = 1$). We note that $\\varepsilon$ does not scale with $M_\\mathrm{BH}$, and changes very little with $\\dot m$. Including the dependence of $f_\\text{\\tiny UV}$ with $M_\\mathrm{BH}$ and $\\dot m$, effectively reduces the value of $\\varepsilon$ at the radii where $f_\\text{\\tiny UV}$ is small, but it does not change substantially in the radii that we would expect to launch an escaping wind. We can then conclude that $\\varepsilon$ has a very weak scaling with $M_\\mathrm{BH}$ and $\\dot m$ so that\n\\begin{equation}\n    \\label{eq:sigma_scaling}\n    \\dot\\Sigma \\propto \\dot\\Sigma_0 \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}},\n\\end{equation}\nwhich is the same result that \\cite{pereyra_steady_2006} found in applying the \\citetalias{castor_radiation-driven_1975} formalism to cataclysmic variables.\n\n\\subsection{Verification of the critical point conditions}\n\\label{subsection:verify_critical_point}\n\nFor our fiducial case (\\autoref{sec:results}), we plot the critical point location compared to the wind trajectories in \\autoref{fig:verify_critical_point}. All escaping trajectories are vertical at the critical point, so our treatment of the wind as a 1D flow for the initial conditions derivation is justified. Nonetheless, we emphasise again that this treatment does not hold for the inner failed wind. The wind is highly supersonic (~$10^3$ times the sound speed) at the critical point, as shown in the bottom panel of \\autoref{fig:verify_critical_point}, so our assumption $w\\gg s$ is validated.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/verify_critical_point.pdf}\n    \\caption{Results for fiducial case $\\protectM_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$ and $\\protect\\dot m =0.5$. Top panel: wind trajectories compared to the critical point position, plotted on a linear scale. Bottom panel: Velocity at the critical point plotted on a log scale}\n    \\label{fig:verify_critical_point}\n\\end{figure}\n\\section{Initial conditions}\n\\label{sec:initial_conditions}\n\nAs initial conditions, we determine the density and velocity at the base of the wind following the \\citetalias{castor_radiation-driven_1975} formalism. Let us consider a wind originating from the top of an accretion disc. At low heights, a gas blob is mostly irradiated by the local region of the disc that is just below it. Since this local disc area can be considered to be at a uniform temperature, the direction of the radiation force is mostly upwards, and thus the wind flows initially vertically and can be considered a 1D wind. The corresponding equation describing the vertical motion is\n\n\\begin{equation}\n    \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho\\, (a_\\mathrm{grav}^z  + a_\\mathrm{rad}^z) - \\frac{\\partial P}{\\partial z},\n\\end{equation}\nEven though the \\textsc{Qwind} model does not include hydrodynamic forces when solving the 2D trajectories of gas parcels, we do include the force term due to gas pressure here, since it is necessary for deriving critical point like solutions (see Appendix \\ref{app:initial_conditions}). \n\nThe study of 1D line-driven winds was pioneered by \\citetalias{castor_radiation-driven_1975}, who defined a framework to find steady state solutions of the 1D wind equation. Their methodology can be extended to any particular geometry of the gravitational and radiation fields, in particular, \\cite{pereyra_steady_2006} (hereafter \\citetalias{pereyra_steady_2006}) apply the \\citetalias{castor_radiation-driven_1975} formalism to the study of cataclysmic variables (CVs). We here aim to further extend this approach to our case, by using the \\citetalias{castor_radiation-driven_1975} formalism to calculate the properties of the 1D wind solutions from an accretion disk as initial conditions for the global 2D wind solution.\n\nThe core result of the \\citetalias{castor_radiation-driven_1975} approach is that if a steady state solution of the 1D wind equation satisfies the following conditions:\n\n\\begin{itemize}\n    \\item the velocity increases monotonically with height,\n    \\item the wind starts subsonic,\n    \\item the wind extends towards arbitrarily large heights,\n    \\item the wind becomes supersonic at some height,\n    \\item the velocity gradient is a continuous function of position,\n\\end{itemize}\nthen the wind must pass through a special point called the critical point $z_c$, which can be derived without solving the wind differential equation, and thus the global properties of the wind such as its mass loss rate can be determined without resolving the full wind trajectory. To keep the main text concise, we refer the reader to Appendix \\ref{app:initial_conditions} for a detailed derivation.\n\nThe previously specified conditions for the existence of a critical point solution may not be satisfied for all of the wind trajectories that we aim to simulate. For instance, a wind trajectory that starts in an upward direction and falls back to the disc because it failed to achieve the escape velocity does not have a velocity that increases monotonically with height. Furthermore, eventually the wind trajectory is no longer vertical, and the 1D approach breaks down. Having considered these possibilities, and only for the purpose of deriving the initial conditions of the wind, we assume that these conditions hold, so that we can derive the wind mass loss rate at the critical point, which we in turn use to determine the initial conditions of the wind. The full 2D solution of the wind may then not satisfy these conditions. \n\nThe location of the critical point as a function of radius is plotted in \\autoref{fig:critical_points}, where we also plot the height of the disc,\n\\begin{equation}\n    z_\\mathrm{h}(R) = \\frac{\\kappa_{\\text{\\tiny e}} \\mu_e \\mathcal F(R) R^3}{GM_\\mathrm{BH} c},\n\\end{equation}\ndefined as the point of equality between the vertical gravitational and radiation force. Overall, we notice that the critical point height increases slowly with radius, except for a bump at $R\\sim 20R_\\mathrm{g}$ which is caused by the UV fraction dependence with radius (see \\autoref{fig:uv_fractions}). For radii $R \\lesssim 50\\, R_\\mathrm{g}$ the critical point height is comparable to the disc radius, so our approximation that streamlines are vertical at that point may not be applicable. We assess the validity of this assumption in \\autoref{subsection:verify_critical_point}. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/critical_points.pdf}\n    \\caption{Position of the critical point $z_c$ as a function of radius, for $M_\\mathrm{BH} = 10^8 \\mathrm{M}_\\odot$ and $\\dot m =0.5$.}  \n    \\label{fig:critical_points}\n\\end{figure}{}\n\nWe assume that the wind originates from the disc surface with an initial velocity $v_0$ equal to the thermal velocity (or isothermal sound speed) at the local disc temperature, \n\\begin{equation}\n    \\label{eq:thermal_velocity}\n    v_\\text{th}(R) = \\sqrt{\\frac{k_\\mathrm{B} T(R)}{\\mu \\, m_p}}.\n\\end{equation}\nGiven that the critical point is close to the disc surface, and that the wind is supersonic at the critical point (see Appendix \\ref{app:initial_conditions}), this is a good starting point. Since mass conservation holds, the initial number density of the wind can then be calculated as\n\\begin{equation}\n    n_0 (R) = \\frac{\\dot \\Sigma (R)}{v_\\text{th}(T(R)) \\, \\mu \\, m_\\mathrm{p}},\n\\end{equation}\nwhere $\\dot \\Sigma(R)$ is the mass loss rate per unit area at the critical point. In \\autoref{fig:initial_conditions}, we plot the initial number density and velocity for $M_\\mathrm{BH} =10^8\\, \\mathrm{M}_\\odot$, $\\dot m =0.5$. We note that the initial velocity stays relatively constant, only varying by a factor of $\\sim 3$ across the radius range. However, the initial number density varies by more than 5 orders of magnitude, showing that the assumption of a constant density at the base of the wind used in the previous versions of the model was poor. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/initial_conditions.pdf}\n    \\caption{Initial conditions at the base of the wind as a function of radius. The original \\textsc{Qwind} code assumed \n    constant initial density and velocity across the disk, and these were free parameters. We now calculate these from first principles. The left panel shows that the initial density changes by a factor of $10^6$, very different to the original assumption, while the right panel (note change in y axis scale) shows that the initial velocity changes by less than a factor 10. }  \n    \\label{fig:initial_conditions}\n\\end{figure}{}\n\n\\subsection{Comparison to other models}\n\nAs a partial test of this new section of the code, we can compare our findings with \\cite{nomura_modeling_2013} (hereafter \\citetalias{nomura_modeling_2013}), in which the authors also use the sonic velocity as the initial velocity of the wind, and derive the initial density profile by assuming the same functional form for the mass loss rate as in \\citetalias{castor_radiation-driven_1975}, but using the AGN $M_\\mathrm{BH}$ and $\\dot m$ instead. We note that the use of the CAK formula directly for accretion discs may not be appropriate because the geometry of the system is very different from stellar winds. Additionally, in \\citetalias{nomura_modeling_2013} the dependence of $f_\\text{\\tiny UV}$ on the disc radius is not considered. Hence we first hardwire $f_\\text{\\tiny UV}$ for the comparison. This comparison is shown in the upper panels of \\autoref{fig:nomura_comparison} for different $M_\\mathrm{BH}$ (left, all at $\\dot{m}=0.5$), and (right) for $M_\\mathrm{BH}$ fixed at $10^8\\,\\mathrm{M}_\\odot$ with different $\\dot{m}$. It is clear that the initial density now derived in \\textsc{Qwind3} (dashed lines) has a steeper decrease with radius than in \\citetalias{nomura_modeling_2013}. This is probably due to their use of the direct CAK formula, which assumes a spherical geometry rather than an accretion disc. However, the inferred densities are within an order of magnitude of each other, and both approaches give a linear scaling of the initial number density profile with $M_\\mathrm{BH}$, but \\textsc{Qwind3} gives an almost quadratic scaling with $\\dot m$, compared to a linear one for \\citetalias{nomura_modeling_2013} (see \\autoref{subsection:ic_scaling}).\n\nThe lower panels of \\autoref{fig:nomura_comparison} show instead the comparison of the \\textsc{Qwind} models using the self consistent $f_\\text{\\tiny UV}$ with its radial dependence (solid lines) instead of assuming $f_\\text{\\tiny UV}=1$ (dashed lines). There is a very strong drop in the initial density at radii where $f_\\text{\\tiny UV}$ drops (see Fig. \\ref{fig:uv_fractions}). This shows the importance of including the self-consistent calculation of $f_\\text{\\tiny UV}$ in \\textsc{Qwind3}. \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/nomura_comparison.pdf}\n    \\caption{Initial density as a function of radius. In the top panels we compare with the results of \\protect\\citetalias{nomura_modeling_2013} (solid lines), the only other diskwind code which used UV line driving to calculate the initial density. That code assumes constant $f_{UV}$, so we fix $f_\\text{\\tiny UV}=1$ in  \\textsc{Qwind} to compare results (dashed lines). In the bottom panels we compare the \n    \\textsc{Qwind} results with $f_\\text{\\tiny UV}=1$ (dashed lines) with the full \\textsc{Qwind} results for $f_\\text{\\tiny UV}(R_\\mathrm{d})$. Left and right panels show  results for fixed $\\dot m = 0.5$, and vary $\\protectM_\\mathrm{BH} \\in (10^7, 10^8, 10^9) \\, \\mathrm{M}_\\odot$. Right panels: We fix $M_\\mathrm{BH} = 10^8\\,\\mathrm{M}_\\odot$ and vary $\\protect \\dot m \\in(0.05, 0.1, 0.5)$.}\n    \\label{fig:nomura_comparison}\n\\end{figure}{}\n\n\n\n\n\n\\section{Introduction}\n\nAGN feedback is a very important process in shaping the growth of galaxies, but the prescriptions for it that are included in current cosmological simulations are generally highly simplified and \nnot based on any deeper understanding of the physical processes involved. Jets from AGN are poorly understood, but some types of AGN winds can be calculated ab initio from the fundamental parameters of black hole mass, mass accretion rate and spin. Observations show the existence of ultra-fast outflows (UFOs) in AGN, likely originating from the accretion disc close to the central supermassive black hole (BH). These outflows can reach velocities of $v$ $\\sim (0.03-0.3)\\,c$ \\citep{weymann_comparisons_1991, pounds_evidence_2003, pounds_high-velocity_2003, reeves_compton-thick_2009, crenshaw_feedback_2012, tombesi_evidence_2010, fiore_agn_2017}. UV line driving is a mechanism which is especially likely to be present in luminous AGN, with their accretion disc spectrum peaking in the UV, where there are multiple strong atomic transitions in low ionisation material. These transitions absorb the photon momentum, producing the strong winds seen from similar temperature material in O star photospheres \\citep{howarth_stellar_1989}, which were first extensively studied by \\cite{castor_radiation-driven_1975} (hereafter  \\citetalias{castor_radiation-driven_1975}) and \\cite{abbott_theory_1980}. In the context of AGN, the study of UV line-driven winds started with analytical studies \\citep{murray_accretion_1995}, and continued with the use of radiation-hydrodynamic simulations \\citep{proga_radiation-driven_1998, nomura_radiation_2016}. However, the computational complexity of the radiation-hydrodynamics codes prevents us from efficiently exploring the input parameter space. Even more importantly, the complexity of these codes can obscure the effect of some of the underlying assumptions, e.g. the lack of scattered emission, on setting the radiation environment \\citep{higginbottom_line-driven_2014}, or the effect of wind mass loss on the net accretion rate and hence the disc emission \\citep{nomura_line-driven_2020}. \n\nTo circumvent this, we build on the pioneering approach of \\textsc{Qwind} \\citep{risaliti_non-hydrodynamical_2010} in developing a non-hydrodynamic code. This calculates ballistic trajectories, ignoring pressure forces (which should be negligible in a supersonic flow) but including gravity and radiation forces, to obtain the streamlines, making the computer code much faster, and simpler, so that it can be used to explore the parameter space much more fully. In \\cite{quera-bofarull_qwind_2020} (hereafter \\citetalias{quera-bofarull_qwind_2020}) we released a modern version of this code (\\textsc{Qwind2}), but this was still based on some underlying, arbitrary parameter choices, including for the launching of the wind from the accretion disk, and used simplified radiation transport. \nHere, we aim to significantly improve the predictive power of the \\textsc{Qwind} code. We present a model to derive the initial conditions of the wind, a radiative transfer algorithm that takes into account the wind geometry and density structure, and we include special relativistic corrections to our calculations of the radiation force on the wind. We then use this new model, which we refer as \\textsc{Qwind3}, to study the dependence of the mass loss rate and kinetic power of the wind on the BH mass and mass accretion rate. These results can form the basis for a physical prescription for AGN feedback that can be used in cosmological simulations to explore the coeval growth of galaxies and their central black holes across cosmic time. \n\n\\section*{Acknowledgements}\n\nAQB acknowledges the support of STFC studentship (ST\/P006744\/1) and the JSPS London Pre\/Postdoctoral Fellowship for Foreign Researchers. CD and CGL acknowledge support from STFC consolidated grant ST\/T000244\/1. CD acknowledges support for vists to Japan from Kavli Institute for the Physics\nand Mathematics of the Universe (IPMU) funding from the National\nScience Foundation (No. NSF PHY17-48958).\nThis work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST\/K00042X\/1, ST\/P002293\/1, ST\/R002371\/1 and ST\/S002502\/1, Durham University and STFC operations grant ST\/R000832\/1. DiRAC is part of the National e-Infrastructure. This work was supported in part by JSPS Grant-in-Aid for Scientific Research (A) JP21H04488 (KO), Scientific Research (C) JP18K03710 (KO), Early-Career Scientists JP20K14525 (MN). This work was also supported by MEXT as \"Program for Promoting Researches on the Supercomputer Fugaku\" (Toward a unified view of the universe: from large scale structures to planets, JPMXP1020200109) (KO), and by Joint Institute for Computational Fundamental Science (JICFuS, KO). This paper made use of the Matplotlib \\citep{hunter_matplotlib_2007} and SciencePlots \\citep{garrett_scienceplots_2021} software packages.\n\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Review of Qwind}\n\nThe \\textsc{Qwind} code of Q20 is based on the approach of \\cite{risaliti_non-hydrodynamical_2010}, which calculates ballistic trajectories of gas blobs launched from the accretion disc. These blobs are subject to two forces: the gravitational pull of the BH, and the outwards pushing radiation force, which can be decomposed  into an X-ray and a UV component, with the later being dominant. On the one hand, the X-ray photons couple to the accretion disc material via bound-free transitions with outer electrons, ionising the gas. On the other hand, the UV opacity is greatly enhanced when the material is not over-ionised, since the UV photons can then excite electrons to higher states while transferring their momentum to the gas in the process. If sufficient momentum is transferred from the radiation field to the gas, the latter may eventually reach the gravitational escape velocity, creating an outgoing flow. This mechanism for creating a wind is known as UV line-driving. The conditions under which the wind can escape depend on the density and velocity structure of the flow, where part of the material can be shielded from the X-ray radiation while being illuminated by the UV emitting part of the accretion disc. \n\n\\subsection{Radiation force}\n\nUsing a cylindrical coordinate system $(R, \\phi, z)$, with $r^2 = R^2 + z^2$, let us consider a BH of mass $M_\\mathrm{BH}$, located at $r=0$, accreting mass at a rate $\\dot M$, and an accretion disc located at the $z=0$ plane. We use the gravitational radius, $R_g = G M_\\mathrm{BH} \/ c^2$, as our natural unit of length. The total luminosity of the system is related to the accreted mass through\n\\begin{equation}\n    L_\\text{bol} = \\eta \\dot M c^2\n\\end{equation}\nwhere $\\eta$ is the radiation efficiency. We set $\\eta = 0.057$ throughout this work, as we only consider non-rotating BHs \\citep{thorne_disk-accretion_1974}. We frequently refer to  the Eddington fraction $\\dot m = \\dot M \/ \\dot M_\\text{Edd}$, where $\\dot M_\\text{Edd}$ is the mass accretion rate corresponding to the Eddington luminosity,\n\\begin{equation}\n    \\dot M_\\text{Edd} = \\frac{L_\\text{Edd}}{\\eta c^2} = \\frac{4\\pi G M_\\mathrm{BH}}{\\eta c \\kappa_{\\text{\\tiny e}}},\n\\end{equation}\nwhere $\\kappa_{\\text{\\tiny e}}$ is the electron scattering opacity, related to the electron scattering cross section $\\sigma_{\\text{\\tiny T}}$ through\n\\begin{equation}\n    \\kappa_{\\text{\\tiny e}} = \\frac{\\sigma_{\\text{\\tiny T}}}{m_p \\, \\mu_e},\n\\end{equation}\nwhere $m_p$ is the proton mass, and $\\mu_e$ is the mean molecular weight per electron. We set $\\mu_e = 1.17$ corresponding to a fully ionised gas with solar chemical abundance \\citep{asplund_chemical_2009}.\n\nThe emitted UV radiated power per unit area by a disc patch located at $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$ is given by \\citep{shakura_black_1973}\n\\begin{equation}\n    \\label{eq:radiation_flux}\n    \\mathcal F_{\\rm UV} = \\frac{3 G  M_\\mathrm{BH} \\dot M}{8\\pi R_\\mathrm{d}^3} f_\\text{\\tiny UV}(R_\\mathrm{d}) f_\\text{\\tiny NT}(R_\\mathrm{d}, R_\\mathrm{isco}),\n\\end{equation}\nwhere $f_\\text{\\tiny NT}$ are the Novikov-Thorne relativistic factors \\citep{novikov_astrophysics_1973}, and $f_\\text{\\tiny UV}$ is the fraction of power in the UV band, which we consider to be (200--3200) \\AA. (The total power radiated per unit area $\\mathcal F$ is given by setting $f_\\text{\\tiny UV}=1$ in the above equation.) The force per unit mass exerted on a gas blob at a position $(R, 0, z)$ due to electron scattering is (see \\citetalias{quera-bofarull_qwind_2020})\n\\begin{equation}\n    \\label{eq:radiation_acceleration}\n        \\bmath{a}_\\mathrm{rad}^\\mathrm{es}(R,z) = \\mathcal C \\,z\\int\\int\\frac{f_\\text{\\tiny UV} f_\\text{\\tiny NT}}{R_\\mathrm{d}^2 \\, \\Delta^4} e^{-\\tau_\\text{\\tiny UV}} \\begin{pmatrix}R-R_\\mathrm{d}\\cos\\phi_\\mathrm{d}\\\\ -R_\\mathrm{d} \\sin \\phi_\\mathrm{d}\\\\ z \\end{pmatrix} \\, \\mathrm{d} R_\\mathrm{d} \\mathrm{d} \\phi_\\mathrm{d},\n\\end{equation}\nwhere \n\\begin{equation}\n    \\mathcal C = \\frac{3 G M_\\mathrm{BH} \\dot M \\kappa_{\\text{\\tiny e}}}{8 \\pi^2 c},\n\\end{equation}\n$\\Delta^2 = R^2 + z^2 - 2 R R_\\mathrm{d} \\cos\\phi_\\mathrm{d}$, and $\\tau_\\text{\\tiny UV}$ is the UV optical depth measured from the disc patch to the gas blob. We note that it is enough to consider the case $\\phi=0$ due to the axisymmetry of the system, and, furthermore, the $\\phi$ component of the force vanishes upon integration. The total radiation force can be greatly amplified by the contribution from the line opacity, which we parameterise as $\\kappa_\\mathrm{line} = \\mathcal{M} \\,\\kappa_{\\text{\\tiny e}}$, such that the total radiation opacity is $(1 + \\mathcal{M}) \\kappa_{\\text{\\tiny e}}$, implying that\n\\begin{equation}\n    \\bmath{a}_\\mathrm{rad} = (1 + \\mathcal M) \\; \\bmath{a}_\\mathrm{rad}^\\mathrm{es}.\n\\end{equation}\nThe parameter $\\mathcal M$ is known as the force multiplier, and we use the same parametrisation as \\citetalias{quera-bofarull_qwind_2020} \\citep{stevens_x-ray_1990} (hereafter \\citetalias{stevens_x-ray_1990}). A limitation of our assumed parametrisation is that we do not take into account the dependence of the force multiplier on the particular spectral energy distribution (SED) of the accretion disc \\citep{dannen_photoionization_2019}. Furthermore, the force multiplier is also expected to depend on the metallicity of the gas \\citep{nomura_radiation_2021}. A self consistent treatment of the force multiplier with relation to the accretion disc and its chemical composition is left to future work. \n\nIt is useful to consider that, close to the disc's surface, the radiation force is well approximated by considering the radiation force produced by an infinite plane at a temperature equal to the local disc temperature,\n\\begin{equation}\n    \\label{eq:radiation_acceleration_approx}\n    \\bmath{a}_\\mathrm{rad, 0}^\\mathrm{es}(R) = \\frac{3 GM_\\mathrm{BH} \\dot M \\kappa_{\\text{\\tiny e}}}{8\\pi^2 R^3 c} f_\\text{\\tiny UV} f_\\text{\\tiny NT}\\, e^{-\\tau_\\text{\\tiny UV}}\\; \\bmath{\\hat{z}},\n\\end{equation}\nwhere $\\tau_\\text{\\tiny UV}$ is calculated along a vertical path. The radiation force is then vertical and almost constant at small heights \n($z \\lesssim 0.1 R_\\mathrm{g}$).\nTo speed up calculations and minimise numerical errors, we use this expression when $z < 0.01 R_\\mathrm{g}$.\n\n\n\\subsection{Equations of motion}\n\nThe equations of motion of the gas blob trajectories are\n\n\\begin{equation}\n\\label{eq:trajectory_ode}\n    \\begin{split}\n        &\\frac{\\mathrm{d} R}{\\mathrm{d} t} &=& \\; v_R,\\\\\n       \n        &\\frac{\\mathrm{d} z}{\\mathrm{d} t} &= &\\; v_z,\\\\\n        &\\frac{\\mathrm{d} v_R}{\\mathrm{d} t} &= &\\; a^\\mathrm{grav}_R + a^\\mathrm{rad}_R + \\frac{\\ell^2}{R^3},\\\\\n       \n        &\\frac{\\mathrm{d} v_z}{\\mathrm{d} t} &= &\\; a^\\mathrm{grav}_z + a^\\mathrm{rad}_z,\n    \\end{split}\n\\end{equation}\nwhere $\\ell$ is the specific angular momentum (assumed constant for a given blob), and $\\bmath{a}_\\mathrm{grav}$ is the gravitational acceleration,\n\\begin{equation}\n\\label{eq:gravity}\n\\bmath{a}_\\mathrm{grav}\\,(R,z) = -\\frac{GM_\\mathrm{BH}}{r^2}\\,\\begin{pmatrix}R\/r \\\\ 0 \\\\ z \/ r\\end{pmatrix}.\n\\end{equation}\nWe assume that initially the gas blobs are in circular orbits around the BH, so that $\\ell = \\sqrt{G M_\\mathrm{BH} R_0}$, where $R_0$ is the launch radius, and thus the azimuthal velocity component at any point is $v_\\phi = \\ell \/ r$.\n\nAs in \\citetalias{quera-bofarull_qwind_2020}, we ignore contributions from gas pressure (except when calculating the launch velocity and density) as these are negligible compared to the radiation force, especially since we focus our study on the supersonic region of the wind. We assume that the distance between two nearby trajectories at any point, $\\Delta r$, is proportional to the distance to the centre $\\Delta r \\propto r$, so that the surface mass loss rate along a particular streamline is $\\dot \\Sigma = \\dot M_\\mathrm{streamline} \/ (2 \\pi \\, r_0 \\, \\Delta r_0)$. This both captures the fact that streamlines are mostly vertical close to the disc, and diverge in a cone-like shape at large distances. The gas blob satisfies the approximate mass conservation equation along its trajectory  (\\citetalias{quera-bofarull_qwind_2020}),\n\\begin{equation}\n    \\label{eq:mass_conservation}\n    \\dot M_\\mathrm{streamline} = 2 \\, \\pi \\, r \\, \\Delta r \\, \\rho \\, v,\n\\end{equation}\nwhere $\\rho$ is the density of the wind, related to the number density $n$ through $\\rho = \\mu \\, m_p \\, n$, and $\\Delta r = (r \/ r_0) \\Delta R_0$. We set the mean molecular weight $\\mu$ to $\\mu = 0.61$, corresponding to a fully ionised gas with solar abundance \\citep{asplund_chemical_2009}. We use \\autoref{eq:mass_conservation} to determine the gas density at each point along a trajectory.\n\n\\subsection{Improvements to \\textsc{Qwind}}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, a series of assumptions are made to facilitate the numerical solution of the presented equations of motion. Furthermore, the initial conditions of the wind are left as free parameters to explore, limiting the predictive power of the model. In this work, we present a series of important improvements to the \\textsc{Qwind} code. \nFirstly we calculate $f_\\text{\\tiny UV}$ from the disc spectrum rather than have this as a free parameter (\\autoref{sec:fuv}).\nSecondly, we derive the initial conditions of the wind in \\autoref{sec:initial_conditions}, based on the methodology introduced in \\citetalias{castor_radiation-driven_1975}, and further developed in \\cite{abbott_theory_1982, pereyra_steady_2004, pereyra_further_2005, pereyra_steady_2006}. This removes the wind's initial velocity and density as degrees of freedom of the system. Thirdly, we vastly enhance the treatment of the radiative transfer in the code, reconstructing the wind density and velocity field from the calculated gas trajectories. This allows us to individually trace the light rays coming from the accretion disc and the central X-ray source, correctly accounting for their attenuation. This is explained in detail in \\autoref{sec:radiation_transport}. Lastly, we include the relativistic corrections from \\cite{luminari_importance_2020} in the calculation of the radiation force, solving the issue of superluminal winds, and we later compare our findings to \\cite{luminari_speed_2021}.\nReaders interested only in the results should skip to Section \\ref{sec:results}. \n\nThe improvement in the modelling of the physical processes comes at the expense of added computational cost. We have ported the \\textsc{Qwind} code to the Julia programming language \\citep{bezanson_julia_2017}, which is an excellent framework for scientific computing given its state of the art performance, and ease of use. The new code is made available to the community under the GPLv3 license on GitHub\\footnote{https:\/\/github.com\/arnauqb\/Qwind.jl}.\n\n\\section{Updates to the radiation transport}\n\\label{sec:radiation_transport}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, the radiation transfer is treated in a very simple way. The disc atmosphere (i.e. the wind) is assumed to have constant density, and so the line of sight absorption does not take into consideration the full geometry and density structure of the wind (see section 2 of \\citetalias{quera-bofarull_qwind_2020}). Furthermore, the UV optical depth is measured from the centre of the disc, and assumed to be the same for radiation from all disc patches, regardless of the position and angle relative to the gas parcel. In this section, we improve \\textsc{Qwind}'s radiative transfer model, by reconstructing the wind density from the gas blob trajectories, thus accounting fully for the wind geometry. The disc is assumed to be flat and thin, with constant height $\\bar z_\\mathrm{h} = 0$, thus we do not model the effect of the disc itself on the radiation transfer. To illustrate the improvements, we consider our fiducial model with $M_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$, and $\\dot m=0.5$, and present the ray tracing engine of the code in an arbitrary wind solution. We discuss particular physical implications and results in \\autoref{sec:results}.\n\n\\begin{figure}\n  \\includestandalone[width=\\columnwidth]{diagram}\n  \\caption{Diagram of the disc-wind geometry. The blue line corresponds to the light path an X-ray photon takes, while the violet line corresponds to an example of a UV light ray from the accretion disc.}\n  \\label{fig:geometry_diagram}\n\\end{figure}\n\n\\subsection{Constructing the density interpolation grid}\n\\label{sec:interp_grid}\n\nGiven a collection of trajectories, we aim to obtain the wind density field at every point in space. The first step is to delimit where the wind is spatially located by computing the concave hull that contains all the points of all wind trajectories. We use the algorithm described in \\cite{moreira_concave_2007} and implemented in \\cite{stagner_lstagnerconcavehulljl_2021}. The resulting concave set is illustrated in \\autoref{fig:wind_hull}. Outside the concave hull, the density is set to the vacuum density which is defined to be $n_\\text{vac} = 10^2$ cm$^{-3}$. \nSince the density varies by orders of magnitude within the wind, we compute the density at a point by linearly interpolating $\\log_{10} n$ in logarithmic space ($\\log R$ - $\\log z$) from the simulated wind trajectories. We use the interpolation algorithm \\textsc{LinearNDInterpolator} from \\textsc{SciPy} \\citep{virtanen_scipy_2020}. The resulting density map is shown in \\autoref{fig:density_grid}. We note that using a concave hull envelope is important, since the interpolation algorithm we use restricts the interpolation space to the convex hull of the input points, which, due to non-convexity of the wind geometry, would otherwise lead us to overestimate the obscuration in certain regions.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/wind_hull.pdf}\n    \\caption{Gas parcel trajectories encapsulated by the concave hull containing the points. Left panel on linear scale and right panel on logarithmic scale. Note that the non-convexity of the wind prevents us from using a simpler convex hull envelope.}\n    \\label{fig:wind_hull}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/density_grid.pdf}\n    \\caption{Interpolated density grid from the simulated wind positions and densities. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n    \\label{fig:density_grid}\n\\end{figure}\n\nOnce we have built the interpolator, we construct a rectilinear grid with the interpolated values. This allows us to implement an efficient ray tracing algorithm to compute the UV and X-ray optical depths. The vertical coordinates of the grid nodes are logarithmically spaced from $10^{-6}$ $R_g$ to the wind's maximum height. The horizontal coordinates are taken at the initial positions of the wind's trajectories, plus an additional range logarithmically spaced from the initial position of the last streamline to the highest simulated $R$ coordinate value.\n\n\n\\subsection{Ray tracing}\n\nTo compute the optical depth along different lines of sight, we need to calculate an integral along a straight path starting at a disc point $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$ to a point $(R, \\phi, z)$. Due to the axisymmetry of the system, the radiation acceleration is independent of $\\phi$ so we can set $\\phi = 0$. We can parametrise the curve in the Cartesian coordinate system with a single parameter $t\\in [0,1]$, (see \\autoref{fig:geometry_diagram}),\n\\begin{equation}\n    \\begin{split}\n        x(t) &= R_\\mathrm{d} \\cos\\phi_\\mathrm{d} \\, (1-t) + t \\, R , \\\\\n        y(t) &= R_\\mathrm{d} \\sin\\phi_\\mathrm{d} \\, (1-t),\\\\\n        z(t) &= t \\, z\\\\\n    \\end{split}\n\\end{equation}\nso that the cylindrical radius varies along the path as\n\\begin{equation}\n    R_t^2(t) = R_\\mathrm{d}^2(1-t)^2 + t^2 R^2 + 2R_\\mathrm{d} R \\cos\\phi_\\mathrm{d} \\,t (1-t) ,\n\\end{equation}\nand $\\phi(t) = \\arctan{(y(t) \/ x(t))}$. In this parametrisation, $t=0$ points to the disc plane point, and $t=1$ to the illuminated wind element. The integral to compute is thus\n\\begin{equation}\n    \\tau = \\Delta \\, \\int_0^1 \\sigma(t)\\; n(t)\\, \\mathrm{d} t,\n\\end{equation}\nwhere $\\Delta$ is the total path length defined earlier.\nGiven a rectilinear density grid in the R-z plane, we compute the intersections of the light ray with the grid lines, $\\{ R_i, z_i\\}$, such that we can discretise the integral as,\n\\begin{equation}\n    \\tau \\approx \\sum_i \\sigma(R_i, z_i) \\, n(R_i, z_i)\\Delta d_i,\n\\end{equation}\nwhere $\\Delta d_i$ is the 3D distance between the $i$-th intersection point and the $(i-1)$-th,\n\\begin{equation}\n    \\Delta d_i = \\sqrt{R_i^2 + R_{i-1}^2 + (z_i - z_{i-1})^2 -2 R_i R_{i-1} \\cos(\\phi_i-\\phi_{i-1})},\n\\end{equation}\nTo find the intersections, we need to calculate whether the light ray crosses an $R_i$ grid line or a $z_i$ grid line. We start at the initial point $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$, and compute the path parameter $t_R$ to hit the next $R$ grid line $R_i$ by solving the second degree equation $R(t_R) = R_i$, and similarly for the next $z_i$ line, $t_z = z_i \/ z$. The next intersection is thus given by the values of $R(t_m)$ and $z(t_m)$ where $t_m = \\min(t_R, t_z)$. Geometrically, the projection of the straight path onto the $(R-z)$ grid is in general a parabola as we can see in \\autoref{fig:tikz:ray_tracing}. \n\n\\begin{figure}\n  \\includestandalone[width=\\columnwidth]{ray_tracing_diagram}\n  \\caption{Projections of two typical light rays onto the $R-z$ interpolation grid. The X-ray radiation (path with blue dots) is assumed to come from the centre of the grid $(0,0)$, so the projection of the light curve onto the $R-z$ grid is always a straight line. However, for the UV case (purple dots), the light ray can originate from any $\\phi_\\mathrm{d}$, so the path on the interpolation grid is, in general, a parabola.}\n  \\label{fig:tikz:ray_tracing}\n\\end{figure}\n\n\\subsection{X-ray optical depth}\n\nThe X-ray opacity depends on the ionisation level of the gas and is assumed to have the same functional form as \\citetalias{quera-bofarull_qwind_2020},\n\\begin{equation}\n    \\label{eq:xray_opacity}\n    \\sigma_\\text{\\tiny X}(\\xi) = \\begin{cases}\\sigma_{\\text{\\tiny T}} & \\text{ if } \\xi > 10^5 \\text{erg cm s}^{-1}\\\\ 100 \\sigma_{\\text{\\tiny T}} & \\text{ if }\\xi \\leq 10^5 \\text{erg cm s}^{-1}\\end{cases},\n\\end{equation}\nwhere $\\sigma_{\\text{\\tiny T}}$ is the Thomson scattering cross section, $\\xi$ is the ionisation parameter,\n\\begin{equation}\n    \\xi = \\frac{4\\pi F_\\text{\\tiny X}}{n},\n\\end{equation}\nand $F_\\text{\\tiny X}$ is the X-ray radiation flux, $F_\\text{\\tiny X} = L_\\text{\\tiny X} \\exp(-\\tau_\\text{\\tiny X}) \/ (4\\pi r^2)$. We notice that to compute the value of $\\tau_\\text{\\tiny X}$ we need to solve an implicit equation, since the optical depth depends on the ionisation state of the gas which in turn depends on the optical depth. One thus needs to compute the distance $d_\\text{\\tiny X}$ at which the ionisation parameter drops below $\\xi_0 = 10^5 \\text{ erg cm s}^{-1}$,\n\\begin{equation}\n    \\xi_0 - \\frac{L_\\text{\\tiny X}}{n d_\\text{\\tiny X}^2} \\exp{(-\\tau_\\text{\\tiny X})} = 0.\n\\end{equation}\nThis equation needs to be tested for each grid cell along the line of sight as it depends on the local density value $n$. Therefore for a cell at a distance $d_{0_i}$ from the centre, density $n_i$, and intersection length $\\Delta d_i$ with the light ray (see \\autoref{fig:tikz:ray_tracing}), the contribution $\\Delta \\tau_\\text{\\tiny X}$ to the optical depth is\n\\begin{equation}\n    \\Delta \\tau_\\text{\\tiny X} = \\sigma_{\\text{\\tiny T}} \\, n_i \\cdot \\left[\\max(0, d_\\text{\\tiny X}-d_{0_i}) + 100 \\cdot \\max(0, \\Delta d_i - (d_\\text{\\tiny X} - d_{0_i}))\\right],\n\\end{equation}\nwhere $d_\\text{\\tiny X}$ is calculated from\n\\begin{equation}\n    \\xi_0 - \\frac{L_\\text{\\tiny X}}{n_i d_\\text{\\tiny X}^2} \\exp{\\left(-\\tau_{0_i} - \\sigma_{\\text{\\tiny T}} \\cdot n_i \\cdot (d_\\text{\\tiny X} - d_{0_i})\\right)} = 0,\n\\end{equation}\nwhere $\\tau_{0_i}$ is the accumulated optical depth from the centre to the current position. We solve the equation numerically using the bisection method. In \\autoref{fig:xray_grid} we plot the X-ray optical depth grid for our example wind. We observe that there is a region where $\\tau_\\text{\\tiny X} \\gg 1$ at very low heights. This shadow is caused by the shielding from the inner wind, which makes the ionisation parameter drop below $\\xi_0$, substantially increasing the X-ray opacity. As we will later discuss in the results section, the shadow defines the acceleration region of the wind, where the force multiplier is very high.\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/xray_tau_grid.pdf}\n    \\caption{X-ray optical depth as a function of position, measured from $R=0$ and $z=0$. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n    \\label{fig:xray_grid}\n\\end{figure}\n\n\\subsection{UV optical depth}\n\nThe UV opacity calculation is significantly simpler than that for the X-rays, since we  assume that the line shift due to the Doppler effect in an accelerating wind is sufficient to always reveal fresh, unabsorbed continuum, so that the opacity is constant at the Thomson (electron scattering) value, $\\sigma(R_i, z_i) = \\sigma_{\\text{\\tiny T}}$. \nIn \\autoref{fig:uv_grid}, we plot the UV optical depth as a function of $R$ and $z$ for light rays originating at the disc position $R_\\mathrm{d} = 500$, $\\phi_\\mathrm{d}=0$. Nevertheless, there are many more sight-lines to consider as the UV emission is distributed over the disc, making this ray tracing calculation the highest contributor to the computational cost of the model. The total UV flux and its resultant direction at any given position in the wind have to be calculated as the sum over each disc element (see \\autoref{eq:radiation_flux}) where now $\\tau_\\text{\\tiny UV} = \\tau_\\text{\\tiny UV}(R_d, \\phi_d, R, z)$. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/uv_tau_grid.pdf}\n    \\caption{UV optical depth as a function of position, measured from $R=500R_\\mathrm{g}$ and $z=0$. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n    \\label{fig:uv_grid}\n\\end{figure}\n\nAs already noted in \\citetalias{quera-bofarull_qwind_2020}, the integral in \\autoref{eq:radiation_acceleration} is challenging to calculate numerically, and in this case the computational cost is further increased by the refined UV ray tracing. In spite of that, by using the adaptive integration scheme presented in \\cite{berntsen_adaptive_1991} and implemented in \\cite{johnson_juliamathcubaturejl_2021}, the integration typically converges after $\\mathcal O(10^4)$ integrand evaluations, which results in a computation time of a few milliseconds, keeping the simulation tractable. At low heights, $z\\sim 0$, the trajectory solver requires several evaluations of the radiation force to correctly compute the adaptive time-step, which makes the computation particularly slow. Fortunately, the approximation in \\autoref{eq:radiation_acceleration_approx} comes in very handy at reducing the overall computational cost at low heights.\n\n\n\\section{Special relativity effects}\n\\label{sec:relativistic}\nWhen the gas trajectory approaches the speed of light, one should consider special relativistic effects such as relativistic beaming and Doppler shifting \\citep[see eg][chapter 4]{rybicki_radiative_1986}. The importance of taking these effects into account is highlighted in \\cite{luminari_importance_2020}. We include a correction to the radiation flux seen by the gas (\\autoref{eq:radiation_flux}),\n\\begin{equation}\n    \\mathrm{d} F_\\text{relativistic}= \\Psi(R_d, \\phi_d, R, z, v_R, v_z) \\; \\mathrm{d} F,\n\\end{equation}\nwhere $v_R$ and $v_z$ are the radial and vertical velocity components of the gas at the position $(R, 0, z)$. We ignore the contribution from the angular velocity component, $v_{\\phi}$ for simplicity, as its inclusion would break our assumption that angular momentum is conserved along a gas blob trajectory. The correction $\\Psi$ is given by \\citep{luminari_speed_2021},\n\\begin{equation}\n    \\Psi = \\frac{1}{\\gamma^4 (1+\\beta\\cos\\theta)^4},\n\\end{equation}\nwhere $\\gamma$ is the Lorentz factor, $\\beta = \\sqrt{v_R^2 + v_z^2} \/ c$, and $\\theta$ is the angle between the incoming light ray and the gas trajectory,\n\\begin{equation}\n    \\cos\\theta = \\frac{(R - R_d\\cos\\phi_d) v_R + z v_z}{\\beta \\Delta}.\n\\end{equation}\nIntuitively, when the incoming light ray is parallel to the gas trajectory, $\\cos\\theta=1$, so the correction reduces to $\\Psi = \\left(\\frac{1-\\beta}{1+\\beta}\\right)^2$, which is 0 when $\\beta=1$ and 1 when $\\beta=0$, as expected. \n\nIt is worth noting that this is a local correction which needs to be integrated along all the UV sight-lines (see \\autoref{eq:radiation_acceleration}). Nonetheless, the computational cost of calculating the radiation force is heavily dominated by ray tracing and the corresponding UV optical depth calculation, so this relativistic correction does not significantly increase the computation time. \n\nThe X-ray flux, which determines the ionisation state of the gas, is also likewise corrected for these special relativistic effects, but there is only one such sight-line to integrate along for any position in the wind, as the X-ray source is assumed to be point-like. \n\n\n\\section{Results}\n\\label{sec:results}\n\nHere, we evaluate the dependence of the wind properties on the initial wind radius, BH mass, and mass accretion rate. We also study the impact of the relativistic corrections on the wind velocity and structure. All of the parameters that are not varied are specified in \\autoref{table:fixed_parameters}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ c c c }\n Parameter & Value \\\\ \n \\hline\\hline \n $f_\\mathrm{\\tiny X}$ & 0.15 \\\\\n $z_0$ & 0 \\\\\n $R_\\text{out}\/ R_\\mathrm{g}$ & 1580 \\\\\n $\\mu$ & 0.61 \\\\  \n $\\mu_e$ & 1.17 \\\\  \n $\\alpha$ & 0.6  \\\\\n $k_\\text{ic}$ & 0.03 \\\\\n\\end{tabular}\n\\caption{\\label{table:fixed_parameters} Fixed parameters for the results section. Note that $k_\\text{ic}$ refers to the value of $k$ used to compute the initial conditions of the wind (\\autoref{eq:fm_simple}), but we use the SK90 parametrisation $k=k(\\xi)$ elsewhere.}\n\\end{table}\n\n\\subsection{The fiducial case}\n\nTo gain some intuition about the structure of the wind trajectories solutions, we first have a close look at our fiducial simulation with $M_\\mathrm{BH} = 10^8\\, \\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in} = 50\\,R_\\mathrm{g}$. We run the simulation iterating 50 times through the density field, to make sure that our density grid has converged (see \\autoref{sec:interp_grid}). \n\nIn \\autoref{fig:failed_wind}, we plot the wind streamline shapes, zooming in on the innermost region where we also show the ionisation state of the gas. \\autoref{fig:initial_conditions} shows that the initial density should be $\\sim 3\\times 10^{12}$~cm$^{-3}$. Hence the initial ionisation parameter at $R_\\mathrm{in}$ is $\\xi=f_\\text{\\tiny X}\\, 0.5\\, L_\\mathrm{Edd}\/(n_\\mathrm{in} R_\\mathrm{in}^2)\\sim 800$ so the base of the wind is already in the regime where the X-ray opacity is high. \nThe wind starts, but the drop in density as the material accelerates means it reaches a high ionisation parameter where the force multiplier is low before it reaches escape velocity. Hence the material falls back to the disc as a failed wind region. \n\nThe failed wind region has a size characterised by $\\tau_x \\lesssim 5$, acting as a shield to the outer wind from the central X-ray source. The X-ray obscuration is especially large in the failed wind shadow, due to the jump in X-ray opacity at $\\xi_0 = 10^5$ erg cm s$^{-1}$ (see contour shown by turquoise line in the left panel of \\autoref{fig:failed_wind}), but its opening angle may be small (\\autoref{fig:xray_grid}). The shadow region defines the acceleration region of the wind, where the force multiplier is greatly enhanced and the wind gets almost all of its acceleration. Eventually this acceleration is enough that the material reaches the escape velocity before it emerges from the shadow, and is overionised by the X-ray radiation. The left panel of \nFig. \\ref{fig:failed_wind} shows these first escaping streamlines (blue) which are close to $R_\\mathrm{in}$. \n\n\\autoref{fig:ufo_pros} shows the resulting wind parameters at \na distance $r=5000 \\, R_\\mathrm{g}$. We plot the wind column density, and density-weighted mean velocity and ionisation parameter as a function of the polar angle $\\theta = \\arctan(R\/z)$. The column is almost constant at $N_H\\sim 2\\times 10^{23}$~cm$^{-2}$ (optical depth of $\\sim 0.1$ to electron scattering) across the range $25^\\circ<\\theta<85^\\circ$. For $\\theta > 85^\\circ$, the sight-line intercepts the inner failed wind and the column density increases to $N_H\\sim 10^{24}$~cm$^{-2}$. The\ntypical wind velocity at this point is  $\\simeq (0.1-0.4)\\,c$\nbut it is always very ionised ($\\xi > 10^5$ erg cm s $^{-1}$) at these large distances. This is too ionised to allow even H- and He-like iron to give visible atomic features in this high velocity gas, although these species may exist at smaller radii where the material is denser. We will explore the observational impact of this in a future work, specifically assessing whether UV line-driving can be the origin of the ultra-fast outflows seen in some AGN\n(see also \\citealt{mizumoto_uv_2021}). \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/failed_wind.pdf}\n    \\caption{Simulated wind trajectories for our fiducial system ($M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in} = 50 R_\\mathrm{g}$) zooming in on the failed wind region, where we also colour-plot the ionisation parameter $\\xi$, showing the contour at $\\xi = 10^5$ erg cm s$^{-1}$ as the light turquoise line. We plot the failed trajectories in green and the escaping trajectories in blue.}\n    \\label{fig:failed_wind}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/ufo_pros.pdf}\n    \\caption{Wind properties for a system with $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, $R_\\mathrm{in}=50R_\\mathrm{g}$ measured along a sight-line at angle $\\theta$ at a distance $r=5000R_\\mathrm{g}$ from the centre. First panel: column density. Second panel: outward mean velocity weighted by density. Third panel: mean ionisation parameter weighted by density. Fourth panel: kinetic luminosity per unit angle. Fifth panel: wind momentum rate per unit angle.}\n    \\label{fig:ufo_pros}\n\\end{figure}\n\nThe escaping wind carries a mass loss rate of $\\dot M_\\mathrm{wind} \\simeq 0.26$ $\\mathrm{M}_\\odot \/$ yr, corresponding to $\\dot M_\\mathrm{wind} \/ \\dot M \\simeq 11\\%$ of the mass accretion rate, and a kinetic luminosity of $L_\\mathrm{kin} \\approx 7 \\times 10^{44}$ erg \/ s, which is equal to 10\\% of the bolometric luminosity. As the two bottom panels of \\autoref{fig:ufo_pros} show, most of the energy and momentum of the wind is located at small polar angles ($\\sim 20^\\circ$), which is consistent with the initial density profile since the innermost streamlines carry the largest amount of mass.\n\n\\subsection{Dependence on the initial radius \\texorpdfstring{$R_\\mathrm{in}$}{Rin}}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/rin_scan.pdf}\n    \\caption{Mass loss rate (first panel), kinetic luminosity (second panel), momentum loss rate (third panel) and average velocity (fourth panel) for different values of $R_\\mathrm{in}$, at a fixed $M_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$ and $\\dot m =0.5$. The mass loss rate is normalised to the system's mass accretion rate, while we normalise the luminosity to the bolometric luminosity. The average velocity is taken as \\protect$v_\\mathrm{r} = \\sqrt{2 L_\\mathrm{kin} \/ \\dot M_\\mathrm{wind}}$.}\n    \\label{fig:rin_scan}\n\\end{figure*}\n\nAs we already mentioned in \\autoref{sec:gas_trajectories}, the initial radius of the innermost trajectory ($R_\\mathrm{in}$) is left as a free parameter to explore. This parameter is likely dependent on the structure of the accretion flow, which we do not aim to model here. As we increase $R_\\mathrm{in}$, the amount of mass that can potentially be lifted from the disc decreases, both because of the reduction in the extent of the launching region and the decrease in initial density with radius (\\autoref{fig:initial_conditions}). Furthermore, increasing $R_\\mathrm{in}$ also narrows the failed wind shadow, since it reduces its subtended angle, thus reducing the accelerating region of the wind. We therefore expect the wind to flow at higher polar angles and smaller velocities when increasing $R_\\mathrm{in}$. In \\autoref{fig:rin_scan}, we plot the predicted normalised mass loss rate, kinetic luminosity, momentum loss rate, and average velocity of the wind as a function of $R_\\mathrm{in}$. As we expected, both the mass loss rate and the kinetic luminosity decrease with $R_\\mathrm{in}$, with the latter decreasing much faster. This difference in scaling is not surprising, since the fastest part of the wind originates from the innermost part of the disc, where most of the UV radiation is emitted. To further illustrate this, we plot the average velocity of the wind for each simulation in the rightmost panel of \\autoref{fig:rin_scan}, observing that the maximum velocity decreases with initial radius. We note that the wind successfully escapes for $R_\\mathrm{in} \\gtrsim 175 R_\\mathrm{g}$, which is a consequence of the initial number density profile (\\autoref{fig:initial_conditions}) sharply declining after $R \\gtrsim 100 R_\\mathrm{g}$. There is a physically interesting situation happening at $R_\\mathrm{in} \\sim 12R_\\mathrm{g}$, where the average velocity of the wind drops. This is due to the failed wind being located where most of the UV emission is, thus making the wind optically thick to UV radiation (with respect to electron scattering opacity). Lastly, we note that the wind consistently reaches velocities $\\gtrapprox 0.3$ $c$ for $R_\\mathrm{in} \\lesssim 50\\, R_\\mathrm{g}$.\n\n\n\\subsection{Dependence on BH mass and mass accretion rate}\n\\label{sec:results_bh}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/bh_scan.pdf}\n    \\caption{Wind mass loss rate normalised by the mass accretion rate (first panel column), kinetic luminosity normalised by bolometric luminosity (second panel column), momentum loss rate normalised by $L_\\mathrm{bol} \/ c$ (third panel column), and average velocity (fourth panel column) as functions of the Eddington fraction $\\dot m$ for different $M_\\mathrm{BH}$.The average velocity is taken as \\protect $v_\\mathrm{r} = \\sqrt{2 L_\\mathrm{kin} \/ \\dot M_\\mathrm{wind}}$.}\n    \\label{fig:bh_scan}\n\\end{figure*}\n\nWe now investigate the dependence of the wind mass loss rate, kinetic luminosity, momentum loss rate, and average velocity on $M_\\mathrm{BH}$, $\\dot m$, and $R_\\mathrm{in}$. We scan the BH parameter range for $M_\\mathrm{BH}\\in(10^6 - 10^{10})$, $\\dot m \\in(0.01-0.5)$, and $R_\\mathrm{in} = (10, 25, 50)R_\\mathrm{g}$. We fix $f_\\mathrm{X} = 0.15$ and recognise that keeping it constant throughout the parameter scan is a limitation, since in reality it depends on $M_\\mathrm{BH}$ and $\\dot m$.  The results are shown in \\autoref{fig:bh_scan}, where all of the quantities have been normalised to their characteristic scales. The red dashed line denotes the limit when quantities become unphysical since the wind is carrying more mass, energy, or momentum than the disc can provide. The first thing to note is that we do not obtain any wind for $\\dot m \\lesssim 0.06$, regardless of $M_\\mathrm{BH}$. This is initially surprising, as the force multiplier is of order $1000$ for cool material, apparently allowing a wind to escape for $\\dot{m}>0.001$. However, the initial density drops as $\\dot{m}^{1\/\\alpha}$ (see \\autoref{subsection:ic_scaling}) so the X-ray shielding drops dramatically, strongly suppressing the wind. Overall, the weakest winds are seen from the highest ($M_\\mathrm{BH}=10^{10}\\,\\mathrm{M}_\\odot$) and lowest ($M_\\mathrm{BH}=10^6\\,\\mathrm{M}_\\odot$) black hole masses. This can be explained by the behaviour of the UV fraction (\\autoref{fig:uv_fractions}). For the $M_\\mathrm{BH}=10^6\\,\\mathrm{M}_\\odot$ case, the UV bright disc annuli are located at large radii, where the disc luminosity is lower; for the $M=10^{10}\\mathrm{M}_\\odot$ case, $f_\\text{\\tiny UV}$ is only high at very small radii, and overall small in the wind launching region. Furthermore, the high disc temperatures expected for the lowest mass systems ($M_\\mathrm{BH} = 10^6\\,\\mathrm{M}_\\odot$), especially at high $\\dot{m}$, mean that the disc contributes to the ionising X-ray flux. This effect is not considered in our work here, but makes it likely that even our rather weak UV line-driven wind is an overestimate for these systems.\n\nFor the values of $M_\\mathrm{BH}$ where the wind is robustly generated across the rest of the parameter space, $M_\\mathrm{BH} \\in (10^7, 10^8, 10^9)\\mathrm{M}_\\odot$, we find a weak dependence of the normalised wind properties on $M_\\mathrm{BH}$. This is expected, since the initial density profile scales with $M_\\mathrm{BH}$ and $\\dot m$ as (\\autoref{subsection:ic_scaling})\n\\begin{equation}\n    n_0 \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}},\n\\end{equation}\nwhere we ignore the dependence of $f_\\text{\\tiny UV}$ on $M_\\mathrm{BH}$. The initial wind velocity $v_0 = v_\\text{th} \\propto T^{1\/2}$, hence\n\\begin{equation}\n    v_0 \\propto \\left( \\frac{\\dot m}{M_\\mathrm{BH}} \\right)^{1\/8} \n\\end{equation}\n(see \\autoref{eq:radiation_flux}.)\nThis  then implies\n\\begin{equation}\n    \\dot M_\\mathrm{wind} \\propto n_0 \\; v_0 \\; R^2 \\propto \\left(\\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}}\\right) \\left(\\frac{\\dot m}{M_\\mathrm{BH}}\\right)^{1\/8}\\left( M_\\mathrm{BH}^2\\right)\n\\end{equation}\nSince scaling due to the dependence on $v_0$ is particularly weak (it depends on the 1\/8-th power), we choose to ignore it so that we can write\n\\begin{equation}\n    \\frac{\\dot M_\\mathrm{wind}}{\\dot M_\\mathrm{acc}} \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}} \\, M_\\mathrm{BH}}{\\dot m M_\\mathrm{BH}}\\propto \\dot m^{\\frac{1}{\\alpha} -1},\n\\end{equation}\nwhere we have used the fact that $\\dot M_\\mathrm{acc} = \\dot m \\dot M_\\mathrm{Edd} \\propto \\dot m M_\\mathrm{BH}$. Similarly,\n\\begin{equation}\n    \\frac{L_\\mathrm{kin}}{L_\\mathrm{bol}} \\propto \\frac{\\dot M_\\mathrm{wind} v_f^2}{\\dot M_\\mathrm{acc}} = \\dot m^{1 + \\frac{1}{\\alpha}},\n\\end{equation}\nwhere $v_f$ is the final wind velocity and we have assumed $v_f \\propto \\dot m$. This last assumption is verified (for $0.1 \\lesssim \\dot m \\lesssim 0.5$) in the rightmost panel of \\autoref{fig:bh_scan}. Consequently, ignoring the scaling of $f_\\text{\\tiny UV}$, both the normalised mass loss rate and normalised kinetic luminosity are independent of $M_\\mathrm{BH}$. For our value of $\\alpha=0.6$, we find $L_\\mathrm{kin} \\propto \\dot m^{2.7} L_\\mathrm{bol}$. This scaling is significantly different from the one often assumed in models of AGN feedback used in cosmological simulations of galaxy formation, where the energy injection rate is assumed to be proportional to the mass accretion rate and hence to the bolometric luminosity \\citep{schaye_eagle_2015, weinberger_simulating_2017, dave_simba_2019}. However, it is consistent with the results found in the hydrodynamical simulations of \\cite{nomura_line-driven_2017} and with current observational constraints \\citep{gofford_suzaku_2015, chartas_multiphase_2021}.\n\nFor the $R_\\mathrm{in} = 10 R_\\mathrm{g}$ case, many parameter configurations of $M_\\mathrm{BH}$ and $\\dot m$ give rise to winds that are unphysical, since they carry more momentum than the radiation field. This is caused by us not considering the impact that the wind mass loss would have on the disc SED, and an underestimation of the UV opacity in our ray tracing calculation of the UV radiation field, in which we only include the Thomson opacity. We also observe that for the lower and upper ends of our $M_\\mathrm{BH}$ range the existence of a wind for different $\\dot m$ values depends on the value of $R_\\mathrm{in}$. This is a consequence of a complex dependence of the failed wind shadow on the initial density and $R_\\mathrm{in}$.\n\nWe now investigate the geometry of the wind: where it originates and at which angles it flows outwards. Despite the strong dependence of the kinetic luminosity on $\\dot m$, the wind launching region does not vary significantly with $\\dot m$, as is shown in \\autoref{fig:bh_scan_radii}, where we plot the average launch radius, weighted by mass loss rate or kinetic luminosity. The exception is the case $R_\\mathrm{in} =10R_\\mathrm{g}$, where the lower inner density at $R_\\mathrm{in}=10R_\\mathrm{g}$ (see \\autoref{fig:initial_conditions}), and its decrease with $\\dot m$ produce a larger failed wind region for $\\dot m \\lesssim 0.15$. The dependence of the mass weighted average radius with $M_\\mathrm{BH}$ is a consequence of the dependence of $f_\\text{\\tiny UV}$ on $M_\\mathrm{BH}$. Larger $M_\\mathrm{BH}$ black holes have the $f_\\text{\\tiny UV}$ peak closer in, so the wind carries more mass at smaller radii.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/bh_scan_avg_radius.pdf}\n    \\caption{Average launch radius weighted by the trajectories' mass loss rate (left panels) and kinetic luminosity (right panels) as a function of $M_\\mathrm{BH}$ and $\\dot m$ for the 3 values of $R_\\mathrm{in}$.}\n    \\label{fig:bh_scan_radii}\n\\end{figure}\n\nFinally, we plot the wind opening angle for the scanned parameter space in \\autoref{fig:wind_angle}. We again find a small dependence on $M_\\mathrm{BH}$, except for near the boundaries of our $M_\\mathrm{BH}$ range, where the angle is quite sensitive to $\\dot m$. The wind flows closer to the equator as we decrease $\\dot m$, hence whether the wind is more polar or equatorial depends on the mass accretion rate of the system. This can be explained by considering that lower disc luminosities do not push the wind as strongly from below, and thus the wind escapes flowing closer to the equator.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/wind_angle.pdf}\n    \\caption{Wind opening angle, measured as the smallest polar angle, $\\theta_\\mathrm{min}$, of all escaping streamlines as a function of $M_\\mathrm{BH}$ and $\\dot m$, for the 3 studied $R_\\mathrm{in}$ values.}\n    \\label{fig:wind_angle}\n\\end{figure}\n\n\\subsection{Can UV line-driven winds be UFOs?}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, wind trajectories could achieve arbitrarily large velocities, often surpassing the speed of light, due to the neglect of relativistic effects. With the introduction of relativistic corrections (\\autoref{sec:relativistic}), the wind is always sub-luminal, as we show in \\autoref{fig:rin_scan} and \\autoref{fig:bh_scan}. Nonetheless, throughout the simulated parameter space, outflows consistently achieve speeds of $(0.1 - 0.8)$ $c$, scaling approximately as $\\dot{m}$\nwith little dependence on \n$M_\\mathrm{BH}$. If we limit ourselves to the simulations that conserve the overall momentum and energy of the system, then the simulated wind still achieve speeds in the range $(0.1 - 0.4)$c. This implies that UV line-driving is a feasible mechanism to produce UFOs even when relativistic corrections are included. The final velocity of the wind depends on how much a gas blob can be accelerated while it is shadowed from the X-ray radiation. In \\autoref{fig:velocity_dependence}, we plot the velocity profile for a trajectory starting at $R=100R_\\mathrm{g}$ in our fiducial simulation density grid for different values of the initial velocity. We find that the final velocity of the trajectory is independent of the initial velocity and that the wind is able to drastically accelerate (up to 6 orders of magnitude in velocity) over a very small distance ($\\lesssim 1 R_\\mathrm{g}$), which suggests that line-driving can be very effective even when the X-ray shadowed region is very small.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/velocity_dependence.pdf}\n    \\caption{Velocity as a function of $z$ for a trajectory starting at $R=100R_\\mathrm{g}$ for our fiducial model. Different colours correspond to different initial velocities. }\n    \\label{fig:velocity_dependence}\n\\end{figure}\n\nWe thus find that UV line-driving is sufficient to reproduce the range of observed UFO velocities, as opposed to the findings of \\cite{luminari_speed_2021}. \nWe note that their code assumes an initial density (similar to \\citetalias{quera-bofarull_qwind_2020}) rather than calculating it from first principles, and does not include the full ray tracing of both UV and X-rays that are considered here. On the other hand, our treatment of the force multiplier is simplified compared to their calculation, where they use the radiative transfer code XSTAR \\citep{kallman_photoionization_2001} to compute the radiation flux absorbed by the wind. Nonetheless, our results show that \na small shaded region can produce a very fast wind (see \n\\autoref{fig:velocity_dependence}), \nand the range of $R_\\mathrm{in}$ that gives rise to velocities $\\geq 0.3 c$ is quite wide (see \\autoref{fig:bh_scan_radii}). It then seems\nquite likely that UV line-driven winds can indeed reach these velocities and hence be the origin of the majority of UFOs seen. \nThere are even higher velocities claimed for a few absorption features in the literature, but these are generally low signal-to-noise detections. \n\\section{Intersection of trajectories}\n\\label{app:intersections}\n\nOnce we have solved the equations of motion of the different gas blobs, it is common for the resulting trajectories to cross each other. Trajectories of gas elements computed using our ballistic model should not be confused with the streamlines of the actual wind fluid, since the latter cannot cross each other as it would imply the presence of singular points where the density and the velocity fields are not well defined.\n\nNonetheless, if we aim to construct a density and velocity field of the wind, we need to define the density and velocity at the crossing points. To circumvent this, once two trajectories intersect, we terminate the one that has the lowest momentum density at the intersection point.\n\nTo determine at which, if any, point two trajectories cross, we consider a trajectory as a collection of line segments $\\{s_i\\}$. Two trajectories $\\{s_i\\}$, and $\\{t_j\\}$ cross each other if it exists $i, j$ such that $s_i \\cap t_j \\neq 0$.\n\nSuppose the line segment $s_i$ is bounded by the points $\\bmath{p_1}$ and $\\bmath{p_2}$ such that $\\bmath{p_2} = \\bmath{p_1} + \\alpha' (\\bmath{p_2} - \\bmath{p_1})$ with $\\alpha' \\in [0,1]$. Similarly, $t_j$ is limited by $\\bmath{q_1}$ and $\\bmath{q_2}$ such that $\\bmath{q_2} = \\beta' (\\bmath{q_2} - \\bmath{q_1})$ with $\\beta' \\in [0,1]$. The condition that $s_i$ intersects $t_j$ is equivalent to finding $\\alpha$, $\\beta$ $\\in [0,1] \\times [0,1]$ such that\n\\begin{equation}\n    \\bmath{p_1} + \\alpha' (\\bmath{p_2} - \\bmath{p_1}) = \\bmath{q_1} + \\beta' (\\bmath{q_2} - \\bmath{q_1}),\n\\end{equation}\nwhich corresponds to the linear system $\\mathbfss{A}\\bmath{x} = \\bmath{b}$ with\n\\begin{equation}\n    \\mathbfss{A} = \\left(\\bmath{p_2}-\\bmath{p_1}, \\bmath{q_1}-\\bmath{q_2}\\right),\n\\end{equation}\n$\\bmath{x} = \\left(\\alpha', \\beta'\\right)^\\intercal$, and $\\bmath{b} = \\bmath{q_1} - \\bmath{p_1}$. The segments intersect if this linear system is determined with solution inside the unit square.\n\n\\section{Radial dependence of \\texorpdfstring{$f_\\text{\\tiny UV}$}{fuv}}\n\\label{sec:fuv}\n\nWe first address the validity of assuming a constant emitted UV fraction with radius. We can calculate its radial dependence using\n\\begin{equation}\n    f_\\text{\\tiny UV}(R_\\mathrm{d}) = \\frac{\\int_{E_1}^{E_2} B(E, T(R_\\mathrm{d}))\\, \\mathrm{d} E}{\\int_{0}^{\\infty} B(E, T(R_\\mathrm{d}))\\, \\mathrm{d} E},\n\\end{equation}\nwhere $B(E,T)$ is the blackbody spectral radiance, $E_1 = 0.0038$ keV and $E_2=0.06$ keV (the standard definition of the UV transition band: (3200-200) \\AA), and $T^4(R_\\mathrm{d})=\\mathcal{F} \/ \\sigma_\\text{\\tiny SB}$ (where $\\mathcal F$ is defined in \\autoref{eq:radiation_flux}). In \\autoref{fig:uv_fractions}, we plot the UV fraction as a function of radius for different $M_\\mathrm{BH}$ and $\\dot m$. The disc temperature is related to the total flux (given by setting $f_\\text{\\tiny UV}=1$ in \\autoref{eq:radiation_flux}) so $T^4\\propto (M\\dot{M}f_\\text{\\tiny NT}\/R^3)\\propto \\dot{m}\/M (R\/R_\\mathrm{g})^3$.\nThus the disc temperature increases with decreasing $R\/R_\\mathrm{g}$, and for the fiducial case of $M_\\mathrm{BH}=10^8 \\mathrm{M}_\\odot$ this leads to the majority of the disc emission in the UV coming from $R\/R_\\mathrm{g}\\le 100$ (left panel of \\autoref{fig:uv_fractions}: orange line).\nHowever, the increase in disc temperature at a given $R\/R_\\mathrm{g}$ for decreasing mass means that the same $\\dot{m}$ for $M_\\mathrm{BH}=10^6 \\mathrm{M}_\\odot$ gives a UV flux which peaks at $R\/R_\\mathrm{g}>100$, as the inner regions are too hot to emit within the defined UV bandpass (left panel of \\autoref{fig:uv_fractions}: blue line). Conversely, for the highest BH masses of $M_\\mathrm{BH}=10^{10}\\mathrm{M}_\\odot$ the disc is so cool that it emits UV only very close to the innermost stable circular orbit (left panel of \\autoref{fig:uv_fractions}: purple line). The universal upturn at $R=10R_\\mathrm{g}$ is caused by the the temperature sharply decreasing at the inner edge of the accretion disc due to the viscuous torque dropping to zero there.\n\nSimilarly, the right panel of \\autoref{fig:uv_fractions} shows the impact of changing $\\dot{m}$ for the fiducial mass of $10^8\\mathrm{M}_\\odot$. The dashed orange line shows the case $\\dot{m}=0.5$, as before, \nand the disc temperature decreases with decreasing $\\dot{m}$ to $0.1$ (dashed green line) and $0.05$ (dashed blue line), reducing the radial extent of the UV-emitting zone. \n\nWe note that the assumption used in \\citetalias{quera-bofarull_qwind_2020} (and many other UV line driven disc wind codes) of $f_\\text{\\tiny UV}$ being a constant value is poor overall, highlighting the importance of including the radial dependence of the UV flux.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/uv_fractions.pdf}\n    \\caption{UV fraction as a function of disc radius. Left panel: dependence on $M_\\mathrm{BH}$ for fixed $\\dot m =0.5$. Right panel: dependence on $\\dot m$ for fixed $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$.}\n    \\label{fig:uv_fractions}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCryptocurrencies are notoriously known to attract financial speculators who often seek to multiply their potential monetary upside and financial gains through leverage. Leverage is realized by borrowing assets to perform trades~---~commonly referred to as margin trading. It is apparent that margin trading, speculating with borrowed assets in general, is an incredibly risky endeavor. Yet, the borrowing and lending markets on blockchains are thriving and have reached a collective~$39.88$B USD of {\\em total value locked (TVL)} at the time of writing\\footnote{\\url{https:\/\/defipulse.com\/}}.\n\nLoans on a blockchain typically operate as follows. Lenders with a surplus of money provide assets to a lending smart contract. Borrowers then provide a security deposit, known as collateral, to borrow cryptocurrency. Because the lending and borrowing on blockchains lacks compulsory means on defaults,\nthe amount of debt borrowers can take on is typically inferior to the security deposit in value~---~resulting in \\emph{over-collateralized loans}. Over-collateralized loans are interesting from a financial perspective, as they enable borrowers to take on leverage.\n\nIf the collateral value decreases under a specific threshold (e.g., below $150$\\% of the debt value~\\cite{makerdao}), the associated debt can be recovered through three means: \\emph{(1)} a loan can be made available for liquidation by the smart contract. Liquidators then pay back the debt in exchange for receiving the collateral at a discount (i.e., \\emph{liquidation spread}), or the collateral is liquidated through an auction. \\emph{(2)} Debt can also be rescued by ``topping up'' the collateral, such that the loan is sufficiently collateralized. \\emph{(3)} Finally, the borrower can repay parts of their debt. While users can repay their debts manually, this appears impractical for the average user, as it requires infrastructure to constantly monitor the blockchain, collateral price, and transaction fee fluctuations. For example, even professional liquidation bots from MakerDAO failed to monitor and act upon price variations during blockchain congestion~\\cite{maker-fail}. \n\nIn this paper we make the following contributions. \n\n\\begin{enumerate}\n    \\item {\\bf Liquidation Models and Insights:} We provide the first longitudinal study of the four major lending platforms MakerDAO, Aave, Compound, and dYdX, capturing collectively over $85$\\% of the borrowing\/lending market on the Ethereum blockchain. By focusing on the protocol's liquidation mechanisms, we systematize their liquidation designs. MakerDAO, for instance, follows an auction-based liquidation process, while Aave, Compound, and dYdX operate under a fixed spread liquidation model.\n    \n    \\item {\\bf Data Analytics:} We provide on-chain data analytics covering the entire existence of the four protocols ($2$ years). Our findings show how the accumulative liquidation proceeds amount to~\\empirical{$347.62$M}~USD\\xspace, we identify~\\empirical{$1,600$}\\xspace unique liquidator addresses and~\\empirical{$21,792$}\\xspace liquidation events, of which~\\empirical{$265$}\\xspace} \\newcommand{\\empirical{$\\numprint{467.44}$K}~USD\\xspace}{\\empirical{$12.20$k}~USD\\xspace auction liquidations are not profitable for the liquidators. We show how~\\empirical{$72.85\\%$}\\xspace of the liquidations pay an above average transaction fee, indicating competitive behavior. We find the existence of bad debts, the borrowing positions that do not financially incentivize borrowers to perform a close. Notably, Aave V2 has accumulated up to~$87.4$K~USD of bad debts by the end of April,~2021.\n   \n    We quantify how sensitive debt behaves to collateral price declines and find that, for example, a~$43$\\% reduction of the ETH price (analogous to the ETH price decline on the~13th of March,~2020) would engender liquidatable collateral volume of~$1.07$B USD on MakerDAO.\n    \\item {\\bf Objective Liquidation Mechanism Comparison:} We provide a methodology to compare quantitatively whether a liquidation mechanism favors a borrower or a liquidator. We find evidence that fixed spread liquidation mechanisms favor liquidators over borrowers. That is, because existing DeFi systems are parameterized to allow more collateral than necessary to be liquidated.\n    \n    \\item {\\bf Optimal Fixed Spread Liquidation Strategy:} We propose an optimal fixed spread liquidation strategy. This strategy allows liquidators to lift the restrictions of the close factor (the upper limit of repaid debts in a single liquidation, cf.\\ Section~\\ref{sec:terminology}) within two successive liquidations. We provide a case study of a past liquidation transaction and show that the optimal strategy could have increased the liquidation profit by \\empirical{$53.96$K}~USD\\xspace (\\empirical{$1.36\\%$}\\xspace), validated through concrete execution on the real blockchain state. This optimal strategy can further aggravate the loss of borrowers.\n\\end{enumerate}\n\nThe remainder of the paper is organized as follows. Section~\\ref{sec:background} outlines the background on blockchain and lending, while we systematize existing liquidation mechanisms in Section~\\ref{sec:existing-protocols}. Section~\\ref{sec:insights} provides liquidation data insights from empirical data.\nWe discuss how to objectively compare liquidation mechanisms and the optimal liquidation strategy in Section~\\ref{sec:better-liquidation}.\nWe outline related work in Section~\\ref{sec:related-work} and conclude the paper in Section~\\ref{sec:conclusion}.\n\n\\section{Lending on the Blockchain}\\label{sec:background}\nWe proceed by outlining the required background on blockchain and DeFi for the remainder of the paper.\n\n\\subsection{Blockchain \\& Smart Contract}\nBlockchains are distributed ledgers that enable peers to transact without the need to entrust third-party intermediaries. There exist two categories of blockchains: \\textit{(i)} permissionless blockchains, where any entity is able to join and leave without permission; \\textit{(ii)} permissioned blockchains, which are typically composed of a group of authenticated participants. In this work, we only focus on permissionless blockchains, on top of which DeFi is built.\n\nAt its core, a blockchain is a hash-linked chain of blocks operating over a peer-to-peer (P2P) network~\\cite{bonneau2015sok}. A block is a timestamped data structure aggregating transactions, which record, e.g., asset transfers. To transfer assets, users need to broadcast digitally signed transactions through the P2P network. The so-called miners then collect transactions, pack transactions into blocks, and append blocks to the blockchain. The whole network follows a consensus protocol (e.g., Nakamoto consensus~\\cite{bitcoin}) allowing honest participants to agree on a consistent version of the blockchain. Transactions waiting to be confirmed on-chain are stored in the so-called mempool\\footnote{Note that there is no universal mempool across all network participants. Every node maintains its own mempool depending on the received transactions.}. We refer the reader to~\\cite{bonneau2015sok,bano2019sok} for a more thorough background on blockchains.\n\nSome blockchains, for example, Ethereum~\\cite{wood2014ethereum}, offer generic computation capabilities through smart contracts.\nIn essence, an Ethereum smart contract is an account controlled by an immutable program (i.e., bytecode). One can trigger the execution of the bytecode by sending a transaction, which contains the executing parameters specified by the transaction sender, to the smart contract account. The EVM, a quasi Turing-complete state machine~\\cite{atzei2017survey}, provides the runtime environment to the contract execution. Solidity~\\cite{dannen2017introducing}, which can be compiled into bytecode, is to date the most prevalent high-level language for implementing Ethereum smart contracts. Smart contracts are widely used to create cryptocurrencies (also known as tokens) on Ethereum in addition to the native coin ETH. Notably, WETH is a one-to-one equivalent token of ETH.\n\nTo submit a transaction on-chain, a user is required to pay a transaction fee. On Ethereum, the transaction fee equals the product of the gas (i.e., an integer measuring the computation complexity of a transaction) and the gas price (i.e., the amount of ETH that the transaction sender is willing to pay for a single unit of gas). Due to the limited space of an Ethereum block (i.e., the total amount of gas consumed in on block), a financially rational miner may include the transactions with the highest gas prices from the mempool into the next block. The blockchain network congests when the mempool grows faster than the transaction inclusion speed due to, for example, traders place substantial orders in a market collapse. Under such circumstances, users have to increase gas prices or wait longer than average to confirm their transactions.\n\n\\subsection{Decentralized Finance (DeFi)}\nSmart contracts allow, not only the creation of tokens, but further the construction of sophisticated on-chain financial systems, namely Decentralized Finance (DeFi). In DeFi, any entity can design a financial protocol, implement in smart contracts, and deploy on-chain. Compared to traditional finance, DeFi presents promising peculiarities, e.g., non-custody and public verifiability~\\cite{qin2021cefi}. Although most DeFi protocols are mirrored services from traditional finance (e.g., exchanges), a proper redesign appears to be necessary considering the special settlement mechanisms of the underlying blockchains. For instance, due to the limited computation capacity, a limit order book with a matching engine, which has been adopted in centralized exchanges for decades, is, however,  inefficient on blockchains. This leads to the invention of the automated market maker, where traders, instead trading against other traders, only need to interact with a pool of assets reserved in a smart contract. Since the rise of DeFi, we have observed numerous such innovative DeFi design, most of which, however, have not been thoroughly studied. As a result, the risks and threats that DeFi users are exposed to are still unclear, necessitating empirical research to provide objective insights.\n\nAt the time of writing, Ethereum is the dominating permissionless blockchain hosting DeFi. The DeFi ecosystem on Ethereum reached a TVL of over $80$B~USD\\footnote{In comparison, the Binance Smart Chain (BSC), ranked the second in terms of TVL at the time of writing, reaches~$20$B USD (cf.\\ \\url{https:\/\/debank.com\/ranking\/locked_value}). We omit BSC in this work because BSC starts to grow from early 2021, which has not accumulated sufficient data.}, with more than $50\\%$ contributed by lending protocols. Lending and borrowing is a popular way to realize a leverage (amplifying the profit) in DeFi. A typical use case is outlined as follows. A trader collateralizes $\\numprint{5000}$~USDT (a USD-pegged stablecoin, cf.\\ Section~\\ref{sec:stablecoinbackground}) to borrow~$1$ ETH, when the ETH\/USDT price is $\\numprint[ETH]{1} = \\numprint[USDT]{3000}$. The borrower then sells the borrowed $\\numprint[ETH]{1}$ for $\\numprint[USDT]{3000}$. If the ETH price declines to, for example, $\\numprint[ETH]{1} = \\numprint[USDT]{2000}$, the trader can purchase~$1$ ETH with $\\numprint[USDT]{2000}$, repay the debt, redeem the collateral, and finally realize a profit of $\\numprint[USDT]{1000}$. The trader at the same time bears the liquidation risk if the ETH price increases and the USDT collateral is insufficient to back the $\\numprint[ETH]{1}$ debt. In a liquidation, a liquidator repays the ETH debt for the trader and acquires the USDT collateral. The acquired collateral exceeds the rapid debt in value incurring a loss to the trader. Such repayment-acquisition liquidation mechanisms are adopted by most DeFi lending platforms. However, the incentives, risks (e.g., to what extend borrowers have lost in liquidation events), and stabilities of these protocols have not been thoroughly studied, which motivates this work.\n\nWe outline the details of the liquidation mechanisms in Section~\\ref{sec:existing-protocols}. In the following, we introduce the essential components of DeFi that are relevant to lending protocols.\n\n\\subsubsection{Price Oracle}\nBecause lending protocols aim to liquidate collateralized assets upon collateral price declines, the lending smart contract is required to know the price of the collateral asset. Prices can either be provided through an on-chain oracle, such as smart contract based exchanges (e.g., Uniswap~\\cite{uniswap2018}), or via an off-chain oracle (such as Chainlink~\\cite{arijuel2017chainlink}). On-chain price oracles are known to be vulnerable to manipulation~\\cite{qin2020attacking}.\n\n\\subsubsection{Flash Loan} The atomicity of blockchain transactions (executions in a transaction collectively succeed or fail) enables flash loans. A flash loan represents a loan that is taken and repaid within a single transaction~\\cite{qin2020attacking,allen2020design}. A borrower is allowed to borrow up to all the available assets from a flash loan pool and execute arbitrary logic with the capital within a transaction. If the loan plus the required interests are not repaid, the whole transaction is reverted without incurring any state change on the underlying blockchain (i.e., the flash loan never happened). Flash loans are shown to be widely used in liquidations~\\cite{qin2020attacking}.\n\n\\subsubsection{Stablecoin}\\label{sec:stablecoinbackground}\nStablecoins are a class of cryptocurrencies designed to provide low price volatility~\\cite{clark2020demystifying}. The price of a stablecoin is generally pegged to some reference point (e.g., USD). The typical stablecoin mechanisms are reserve of the pegged asset (e.g., USDT and USDC), loans (e.g., DAI), dual coin, and algorithmic supply adjustments~\\cite{moin2020sok}.\n\n\\subsection{Terminology}\\label{sec:terminology}\nWe adhere to the following terminologies in this paper.\n\\begin{description}\n\\item[Loan\/Debt:] A borrower, secured by a collateral deposit, temporarily takes capital from a lender. The collateral is the insurance of the lender against defaults.\n\\item[Interest Rate:] A loan is repaid by repaying the lent amount, plus a periodic percentage of the loan amount. The interest rate can be governed by the scarcity\/surplus of the available asset supply within the lending smart contract.\n\\item[Over\/Under-collateralization:] Blockchain based loans are typically over-collateralized, i.e., the borrower has to provide collateral assets of higher total value than the granted loan. A loan is under-collateralized when the value of the collateral is inferior to the debt.\n\\item[Position:] In this work, the collateral and debts are collectively referred to as a position. A position may consist of multiple-cryptocurrency collaterals and debts.\n\\item[Liquidation:] In the event of a negative price fluctuation of the debt collateral (i.e., a move below the liquidation threshold), a position can be liquidated. In permissionless blockchains, anyone can repay the debt and claim the collateral.\n\\item[Liquidation Threshold ($\\mathbf{LT}$):] Is the percentage at which the collateral value is counted towards the borrowing capacity (cf. Equation~\\ref{eq:borrowing-capacity}).\n\\item[Liquidation Spread ($\\mathbf{LS}$):] Is the bonus, or discount, that a liquidator can collect when liquidating collateral (cf.\\ Equation~\\ref{eq:ls}). This spread incentivises liquidators to act promptly once a loan crosses the liquidation threshold.\n\n\\begin{equation}\\label{eq:ls}\n\\begin{aligned}\n    &Value\\ of\\ Collateral\\ to\\ Claim \\\\&= Value\\ of\\ Debt\\ to\\ Repay\\times(1+\\mathbf{LS})\n\\end{aligned}\n\\end{equation}\n\\item[Close Factor ($\\mathbf{CF}$):] Is the maximum proportion of the debt that is allowed to be repaid in a single liquidation.\n\\item[Collateralization Ratio ($\\mathsf{CR}$):] Is the ratio between the total value of collateral and debt (cf.\\ Equation~\\ref{eq:collateralizationratio}) where $i$ represents the index of collateral or debt if the borrower owns collateral or owes debt in multiple cryptocurrencies.\n\n\\begin{equation}\\label{eq:collateralizationratio}\n    \\mathsf{CR}=\\frac{\\sum{Value\\ of\\ Collateral_i}}{\\sum{Value\\ of\\ Debt_i}}\n\\end{equation}\nA debt is under-collateralized if $\\mathsf{CR}<1$, otherwise the debt is over-collateralized.\n\\item[Borrowing Capacity ($\\mathsf{BC}$):] Refers to the total value that a borrower is allowed to request from a lending pool, given its collateral amount.\nFor each collateral asset $i$ of a borrower, its borrowing capacity is defined in Equation~\\ref{eq:borrowing-capacity}.\n\n\\begin{equation}\\label{eq:borrowing-capacity}\n    \\mathsf{BC} = \\sum{Value\\ of\\ Collateral_i\\times \\mathbf{LT}_i}\n\\end{equation}\n\\item[Health Factor ($\\mathsf{HF}$):] The health factor measures the collateralization status of a position, defined as the ratio of the borrowing capacity over the outstanding debts (cf.\\ Equation~\\ref{eq:health-factor}).\n\n\\begin{equation}\\label{eq:health-factor}\n\\mathsf{HF}=\\frac{\\mathsf{BC}}{\\sum{Value\\ of\\ Debt_i}}\n\\end{equation}\nIf $\\mathsf{HF}<1$, the collateral becomes eligible for liquidation.\n\\end{description}\n\n\n\n\n\n\n\\section{Systematization of Lending and Liquidation Protocols}\\label{sec:existing-protocols}\nIn this section, we systematize how the current borrowing mechanisms and their specific liquidation processes operate. \n\\subsection{Borrowing and Lending System Model}\\label{sec:borrowingandlendingsystemmodel}\n\n\\begin{figure}[tb!]\n     \\centering\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\includegraphics[width=0.9\\columnwidth]{figures\/high_level_system_diagrams.pdf}\n    \\caption{High-level system diagram of a lending pool system.\n    }\n    \\label{fig:liquidation_system_diagrams}\n\\end{figure}\n\nThe following aims to summarize the different actors engaging in borrowing and lending on a blockchain (cf.\\ Figure~\\ref{fig:liquidation_system_diagrams}).\n\nA \\textbf{lender} is an actor with surplus capital who would like to earn interest payments on its capital by lending funds to a third party, e.g., a borrower.\n\nA \\textbf{borrower} provides collateral to borrow assets from a lender. The borrower is liable to pay regular interest fees to the lender (typically measured as percentage of the loan). As lending in blockchains is typically performed without KYC, the borrower has to collateralize a value that is greater than the borrowed loan. In pure lending\/borrowing platforms such as Aave, Compound and dYdX, the collateral from borrowers is also lent out as loans. Borrowers hence automatically act as lenders.\n\nWhile lending could be performed directly on a peer-to-peer basis, blockchain-based lending protocols often introduce a \\textbf{lending pool} governed by a smart contract. A pool can hold several cryptocurrencies, and users can interact with the pool to deposit or withdraw assets according to the rules defined by the smart contract code.\n\nA \\textbf{liquidator} observes the blockchain for unhealthy positions (i.e., the health factor is below~$1$) to be liquidated. Liquidators typically operate bots, i.e., automated tools which perform a blockchain lookup, price observation, and liquidation attempt, if deemed profitable. Liquidators are engaging in a competitive environment, where other liquidators may try to front-run each other~\\cite{daian2019flash}. Notably, an atomic liquidation (e.g., a fixed spread liquidation) is settled in one blockchain transaction, while non-atomic liquidation mechanisms (e.g., auctions) generally require liquidators to interact with the lending pool in multiple transactions.\n\n\n\\subsection{Systematization of Liquidation Mechanisms}\\label{sec:liquidationmechanisms}\nWe observe that existing lending platforms mainly adopt two distinct liquidation mechanisms. One mechanism is based on a non-atomic \\emph{English auction}~\\cite{krishna2009auction} process, and the other follows an atomic fixed spread strategy. We formalize the two existing dominating mechanisms as follows.\n\n\n\\subsubsection{Auction Liquidation}\nAn auction based liquidation mechanism follows the subsequent methodology:\n\\begin{enumerate}\n    \\item A loan becomes eligible for liquidation (i.e., the health factor drops below $1$).\n    \\item A liquidator starts the auction process (which can last several hours).\n    \\item Interested liquidators provide their bids (e.g., the highest bid receives the loan collateral).\n    \\item The auction ends according to the rules set forth in the auction contract.\n\\end{enumerate}\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/makerdao.pdf}\n    \\caption{Two-phase liquidation auction of MakerDAO. Any actor can invoke the public function \\emph{bite} to initiate the collateral auction and invoke the public function \\emph{deal} to finalize the liquidation after the auction terminates.}\n    \\label{fig:maker-dao-auction}\n\\end{figure}\n\n\n\\paragraph{MakerDAO tend-dent auction} Specifically, MakerDAO employs a two-phase auction process, which we term \\emph{tend-dent auction} (cf.\\ Figure~\\ref{fig:maker-dao-auction}). \nWhen a position with $D$ value of debt and $C$ value of collateral is eligible for liquidation, i.e., $\\mathsf{HF}<1$, a liquidator is able to initiate the tend-dent auction. A liquidator is required to bid at a higher price than the last bid in the auction. The two-phase workflow is described as follows.\n\\begin{description}[leftmargin=10pt]\n    \\item[Tend:] In the \\emph{tend} phase, liquidators compete by bidding to repay parts of the debt in exchange for the entire collateral. We denote the amount of debt committed to repay in each bid by $d_i$, s.t.~$d_i \\leq D$ and $d_i>d_{i-1}$. If the auction terminates in the tend phase, the winning bidder receives all the collateral (i.e., $C$). When $d_i$ reaches $D$, the auction moves into the \\emph{dent} phase.\n    \\item[Dent:] In the dent phase, liquidators compete by bidding to accept decreasing amounts of collateral in exchange for the full debt (i.e., $D$) they will end up repaying. We denote the amount of collateral committed in each bid by $c_i$, s.t.~$c_i \\leq C$ and $c_i<c_{i-1}$. The winning bidder repays the full debt and receives the partial collateral (denoted by $c_{\\text{win}}$). The remaining collateral (i.e., $C-c_{\\text{win}}$) is returned to the position owner (i.e., the borrower).\n\\end{description}\n\nThe auction terminates when any of the following two conditions is satisfied. Note that the auction can terminate in the tend phase.\n\\begin{enumerate}\n    \\item \\textbf{Auction Length Condition:} the configurable auction length (e.g., $6$ hours) has passed since the initiation of the auction.\n    \\item \\textbf{Bid Duration Condition:} the configurable bid duration (e.g., $5$ hours) has passed since the last bid.\n\\end{enumerate}\n\nAfter the termination of an auction, the winning liquidator is allowed to finalize the liquidation to claim the proposed collateral.\n\n\\subsubsection{Fixed Spread Liquidation}\nInstead of allowing multiple liquidators to bid over a time-frame, a liquidatable loan can be instantly liquidated with a pre-determined discount (cf.\\ Figure~\\ref{fig:liquidation_system_diagrams}). Aave, for instance, allows liquidators to purchase the loan collateral at up to a 15\\% discount of the current market price. This discount, or liquidation spread, is known upfront, and the liquidators can hence locally decide whether to engage in a liquidation opportunity.\nFollowing a fixed spread model avoids hour-long liquidation auctions, which cost time and transaction fees. Liquidators, moreover, can choose to liquidate collateral with the use of atomic flash loans~\\cite{qin2020attacking}. While flash loans increase the transaction costs of the liquidators, they reduce the currency exposure risk of holding the assets required for liquidation.\n\n\\paragraph{Fixed Spread Liquidation Example}\nIn the following, we provide an example of a fixed spread liquidation:\n\\begin{enumerate}\n    \\item \\textbf{Currency values:} We assume an initial price of $\\numprint{3500}$ USD\/ETH.\n    \\item \\textbf{Collateral Deposit:} A user deposits $3$ ETH, and hence has $\\numprint[USD]{10500}$ worth of collateral. If we assume a liquidation threshold ($\\mathbf{LT}$, cf.\\ Section~\\ref{sec:terminology}) of $0.8$, the resulting borrowing capacity of the user is $\\mathsf{BC} = \\numprint[USD]{10500}\\times\\mathbf{LT} = \\numprint[USD]{8400}$.\n    \\item \\textbf{Borrowing:} In the next step the user borrows, for instance, $\\numprint[USDC]{8400}$ worth $\\numprint[USD]{8400}$.\n    \\item \\textbf{ETH price decline:} We now assume that the ETH value declines to $\\numprint[USD\/ETH]{3300}$, which means that the collateral value declines to $\\numprint[USD]{9900}$ with $\\mathsf{BC}=\\numprint[USD]{7920}$. The price oracle updates the ETH price on the lending smart contract. The health factor of the loan now drops to $\\mathsf{HF}=\\frac{\\numprint[USD]{7920}}{\\numprint[USD]{8400}}\\approx 0.94 < 1$ and thus the collateral is available for liquidation.\n    \\item \\textbf{Liquidation:} A liquidator submits a liquidation transaction to repay $50$\\% (close factor $\\mathbf{CF}$, cf.\\ Section~\\ref{sec:terminology}) of the debt, i.e., $\\numprint[USDC]{4200}$. In return, the liquidator is allowed to purchase collateral at the price of $\\frac{\\numprint[USD\/ETH]{3300}}{1+\\mathbf{LS}}=\\numprint[USD\/ETH]{3000}$ (we assume that the liquidation spread $\\mathbf{LS}$ is $10\\%$, cf.\\ Section~\\ref{sec:terminology}).\n    In this liquidation, the liquidator receives $\\frac{\\numprint[USD]{4200}}{\\numprint[USD\/ETH]{3000}}\\times\\numprint{3300}$ USD\/ETH $=\\numprint[USD]{4620}$ worth of ETH and realizes a profit of $\\numprint[USD]{420}$.\n    \n   \n\\end{enumerate}\n\n\\subsection{Studied Lending Protocols}\nWithin this work we focus on the biggest lending protocols, measured by total value locked, notably MakerDAO ($12.49$B~USD), Aave ($11.20$B~USD), Compound ($10.15$B~USD), and dYdX ($247.6$M~USD).\n\nAave is a pool-based lending and borrowing protocol~\\cite{aave}. Lenders deposit assets into a pool governed by open-source smart contracts, and borrowers can then take loans out of this pool. The interest rate of an Aave pool is decided algorithmically by the smart contract and depends on the available funds within the lending pool. The more users borrow an asset, the higher its interest rate rises. A lending pool can consist of several cryptocurrency assets, for instance ETH, DAI, and USDC.\nIn Aave, when the health factor drops below~$1$, any liquidator can call the public pool function \\emph{liquidationCall}, by repaying parts or all of the outstanding debt, while profiting from the liquidation spread. Aave specifies that only a maximum of $50\\%$ of the debt can be liquidated within one \\emph{liquidationCall} execution (referred to as a close factor). The liquidation spread on Aave ranges from 5\\% to 15\\%, depending on the considered markets. Aave bases its pricing feed on the external Chainlink oracle~\\cite{arijuel2017chainlink}. Aave was upgraded to a newer version in December 2020 while the core protocols remained nearly unchanged. In this work, we distinguish the two versions with Aave V1 and Aave V2.\n\nCompound~\\cite{compoundfinance} launched before Aave and operates in a similar fashion. Users deposit assets and earn interests based on the amount of interests paid by borrowers. When a borrower exceeds the borrowing capacity, at most 50\\% of outstanding debt can be repaid at once by a liquidator (same as the close factor in Aave). The liquidation may continue until the collateral guarantees a health factor superior to one. The liquidator in exchange receives the collateral at the current market price minus the liquidation spread.\n\ndYdX~\\cite{dydx} is divided into two sub-protocols, one for trading, borrowing, lending and one that also supports futures markets. Similar to Aave and Compound, dYdX operates at a fixed spread of $5\\%$ for the \\wrapletters{WETH\/USDC}, \\wrapletters{WETH\/DAI} and \\wrapletters{USDC\/DAI} markets, at the time of writing. dYdX's close factor is $100$\\%, allowing the liquidators to liquidate the entire collateral within one liquidation.\n\nContrary to the aforementioned borrowing\/lending protocols, MakerDAO provides a decentralized stablecoin called DAI that is pegged to the US dollar, while still functioning financially similar to a borrowing\/lending platform. A user can collateralize at least, e.g., $150$\\% of a crypto asset (for instance ETH) to mint $100$\\% DAI. Creating DAI opens a so-called \\emph{collateralized debt position (CDP)}, which can be liquidated if the collateralized value drops below the fixed collateralization ratio. MakerDAO adopts an auction-based liquidation mechanism (cf.\\ Section~\\ref{sec:liquidationmechanisms})\n\n\\section{Liquidation Insights}\\label{sec:insights}\nBy observing the publicly readable Ethereum blockchain we compiled the following insights on liquidation events.\n\\subsection{Measurement Setup}\n\\begin{figure}[b]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/setup_overview.pdf}\n    \\caption{Measurement setup overview.}\n    \\label{fig:setup-overview}\n\\end{figure}\nOur measurement setup is outlined in Figure~\\ref{fig:setup-overview}. We gather our data by crawling blockchain events (e.g., liquidation events) and reading blockchain states (e.g., oracle prices) from an Ethereum full archive node, on an AMD Ryzen Threadripper 3990X with $64$ cores, $256$ GB of RAM and $2\\times8$ TB NVMe SSD in Raid $0$ configuration. An Ethereum archive node stores not only the blockchain data but also the chain state at every historical block, which supports efficient historical state query (e.g., the borrowing position debt amount at a specific block). At the time of writing, an Ethereum archive node requires over $4$ TB disk space.\n\nThe Ethereum events are essentially EVM logs, which are also recorded on-chain. An event is indexed by its signature, a 256-bit hash, and the contract address emitting this event. We hence can filter the liquidation events emitted from the studied lending pools.\n\nWe also build our own custom Ethereum client based on the golang-based geth client\\footnote{\\url{https:\/\/github.com\/ethereum\/go-ethereum}} to execute transactions on a specific block (the block state is downloaded from the archive node) when necessary. For instance, in Section~\\ref{sec:optimalstrategy}, we validate our optimal fixed spread liquidation strategy through concrete executions on past blockchain states.\n\n\\subsection{Overall Statistics}\n\\begin{figure}[tb!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/accumulative_liquidation_line.pdf}\n    \\caption{From the inception of each considered lending protocol, we plot the accumulative collateral sold through liquidation (April,~2019~to April,~2021).}\n    \\label{fig:accumulative_liquidation_stacked}\n\\end{figure}\n\n\\begin{figure*}[tb]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/monthly_liquidation_profit.pdf}\n    \\caption{Monthly accumulated liquidator profit. We observe an outlier for MakerDAO in March,~2020, because the MakerDAO liquidation bots were faulty due to an excessive price decline of ETH. The outlier for Compound in November,~2020~is caused by an irregular price reported by a price oracle.}\n    \\label{fig:monthly-liquidation-profit}\n\\end{figure*}\n\nIn total, we observe \\empirical{$21,792$}\\xspace successful liquidations from the inception\\footnote{The inception blocks of Aave, Compound, dYdX and MakerDAO are~\\block{9241022}, \\block{7710733}, \\block{7575711} and \\block{8040587} respectively.} of the four platforms to block \\empirical{$11,363,269$}\\xspace, the last block in the month of April,~2021. We normalize the values of different cryptocurrencies to USD according to the prices given by the platforms' on-chain price oracles at the block when the liquidation is settled. We crawl this data on-chain, and do not rely on an external price oracle, or API.\n\nIn Figure~\\ref{fig:accumulative_liquidation_stacked}, we present the accumulative collateral sold through liquidation in terms of USD. The overall liquidated collateral on the four platforms Aave, Compound, dYdX, and MakerDAO accumulates to a total of~\\empirical{$347.62$M}~USD\\xspace. We notice an increase on Compound in November,~2020. This is caused by an irregular DAI price provided by the Compound price oracle, which triggers a large volume of cryptocurrencies to be liquidated~\\cite{OracleEx18:online}. Another remarkable boost on Compound in February,~2021 is caused by the drastic fluctuations in prices of cryptocurrencies~\\cite{Over117M34:online}.\n\n\n\\subsection{Incentives and Participation}\nIn the following, we measure how liquidators are incentivized to engage in liquidations and elaborate on the status quo of liquidator participation.\n\n\\subsubsection{Liquidator Profit \\& Loss}\\label{sec:profitandloss}\nTo measure the profit of each liquidation event, we assume that the purchased collateral is immediately sold by the liquidator at the price given by the price oracle. The total profit by the \\empirical{$21,792$}\\xspace liquidations sums up to a total of \\empirical{$33.53$M}~USD\\xspace. To better understand the temporal evolution of liquidation profits, we show the monthly collective profit yielded from each platform in Figure~\\ref{fig:monthly-liquidation-profit}.\n\nMakerDAO notably shows an outlier in March,~2020, when the MarkerDAO monthly profit reached~\\empirical{$13.13$M}~USD\\xspace. This outlier is due to the Ethereum network congestion caused by the~$43$\\% ETH price market collapse on the~13th March,~2020~\\cite{maker-fail}. The liquidation bots were not acting accordingly, which caused the liquidation transactions to not be swiftly included in the blockchain. This delay allowed other capable liquidators to manually win the auctions at a negligible cost.\n\nIn November,~2020, an irregular DAI price provided by the Compound price oracle~\\cite{OracleEx18:online} allowed liquidators to profit in total~\\empirical{$8.38$M}~USD\\xspace. We also observe that Compound contributes a strong liquidation profit of \\empirical{$9.61$M}~USD\\xspace in February,~2021, which, however, does not seem related to any bot failure or oracle irregularity.\n\nTo study the number of liquidators, we assume that each unique Ethereum address represents one liquidator. We then identify a total of \\empirical{$1,600$}\\xspace unique liquidators. On average the liquidators yield a profit of \\empirical{$20.96$k}~USD\\xspace each. We show the number of liquidators and their average profit on the four considered platforms in Table~\\ref{tab:liquidator_profit}. Remarkably, the most active liquidator performs \\empirical{$1,705$}\\xspace liquidations alone, which yield a total profit of \\empirical{$712.46$k}~USD\\xspace. The most profitable liquidator generates \\empirical{$5.16$M}~USD\\xspace in only \\empirical{$76$}\\xspace liquidations.\n\n\\begin{table}[bt!]\n    \\centering\n    \\caption{Number of the liquidations and liquidators on Aave, Compound, dYdX, MakerDAO and their average profit. We measure the number of liquidators based on their unique Ethereum address. We notice that some liquidators operate on multiple lending markets.\n    }\n    \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{lrrr}\n    \\toprule\n    \\bf Platform & \\bf Liquidations  & \\bf Liquidators & \\bf Average Profit \\\\\n    \\midrule\n    Aave V1 & \\empirical{$3,069$}\\xspace & \\empirical{$461$}\\xspace  & \\empirical{$9.58$k}~USD\\xspace \\\\\n    Aave V2 & \\empirical{$\\numprint{1039}$}\\xspace & \\empirical{$\\numprint{125}$}\\xspace  & \\empirical{$\\numprint{43.12}$K}~USD\\xspace \\\\\n    Compound & \\empirical{$5,499$}\\xspace & \\empirical{$594$}\\xspace  & \\empirical{$20.60$k}~USD\\xspace \\\\\n    dYdX & \\empirical{$7,463$}\\xspace & \\empirical{$543$}\\xspace & \\empirical{$6.49$k}~USD\\xspace \\\\\n    MakerDAO & \\empirical{$5,761$}\\xspace & \\empirical{$107$}\\xspace  & \\empirical{$124.80$k}~USD\\xspace \\\\\n    \\midrule\n    Total& \\empirical{$21,792$}\\xspace & \\empirical{$1,600$}\\xspace &  \\empirical{$20.96$k}~USD\\xspace \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n    \\label{tab:liquidator_profit}\n\\end{table}\n\nWe also discover~\\empirical{$265$}\\xspace} \\newcommand{\\empirical{$\\numprint{467.44}$K}~USD\\xspace}{\\empirical{$12.20$k}~USD\\xspace MakerDAO liquidations that are not profitable and incur a total loss of \\empirical{$\\numprint{467.44}$K}~USD\\xspace. After manually inspecting those non-profitable liquidation transactions, we can confirm that the liquidation losses are caused by collateral price fluctuations during the auctions.\n\n\\begin{figure*}[tb!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/liquidation_gas_prices.pdf}\n    \\caption{Gas prices paid by liquidators. We also report the 6000-blocks (1 day) moving average of the block median measured on-chain. Interestingly, several liquidations are below the average gas price. We notice a gas price spike in March,~2020~because of a collapse of ETH price~\\cite{maker-fail}. There is an uptrend of gas price since May,~2020~due to the growing popularity of DeFi.}\n    \\label{fig:liquidation-gas_prices}\n\\end{figure*}\n\n\\subsubsection{Fixed Spread Liquidations}\\label{sec:fixedspreadparticipation}\nWe observe \\empirical{$3,069$}\\xspace, \\empirical{$\\numprint{1039}$}\\xspace, \\empirical{$5,499$}\\xspace and \\empirical{$7,463$}\\xspace settled liquidations on Aave~V1, Aave~V2, Compound and dYdX respectively.\n\\paragraph{Liquidator Participation} In Figure~\\ref{fig:liquidation-gas_prices}, we show the gas price of every fixed spread liquidation transaction along with the average gas price. Note that we show the 6000-block moving average of the block gas price medians in the figure to smooth the curve for readability. The data in Figure~\\ref{fig:liquidation-gas_prices} shows that many liquidators pay significant gas fees (the y-axis is a log scale). We find that~\\empirical{$72.85\\%$}\\xspace of the liquidations pay an above average transaction fee, and hence allows the conclusion that liquidation events are competitive.\n\n\\subsubsection{Auction Liquidations}\\label{sec:auctionparticipation}\n\nOut of the recorded~\\empirical{$5,761$}\\xspace MakerDAO liquidations,~\\empirical{$3,286$}\\xspace auctions terminate in the tend phase and the other~\\empirical{$2,475$}\\xspace auctions terminate in the dent phase. The average number of bidders participating in a liquidation is only~\\empirical{$1.71$}\\xspace.\nWe notice that~\\empirical{$2.21\\pm1.58$}\\xspace bids (\\empirical{$1.46\\pm0.89$}\\xspace tend bids and~\\empirical{$0.75\\pm1.29$}\\xspace dent bids), are placed per auction.\n\n\\paragraph{Duration} We define the duration of a MakerDAO liquidation auction, as the time difference between the auction initiation and finalization. To capture time, we resort to the block timestamps. We visualize the duration of the MakerDAO liquidations in Figure~\\ref{fig:makerdao-auction-duration}. On average, an liquidation lasts for~\\empirical{$1.44\\pm6.33$}~hours\\xspace (mean$\\pm$standard deviation).\nThere are~\\empirical{$4,173$}\\xspace auctions terminating within one hour. We observe that few liquidations last longer than intended, which can be explained by that fact that the respective liquidators did not finalize the auction and hence didn't claim the liquidation proceeds. For example, the longest auction lasts for~\\empirical{$346.67$}~hours\\xspace, while its last bid is placed~\\empirical{$344.60$}~hours\\xspace prior to the termination.\n\n\\begin{figure}[tb!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/makerdao_auction_duration.pdf}\n    \\caption{Duration of the MakerDAO liquidations. Interestingly, auctions last longer than they're configured to. We can observe a change in system auction parameters after~the 13th March,~2020~event (the MakerDAO liquidation incident).}\n    \\label{fig:makerdao-auction-duration}\n\\end{figure}\n\n\\paragraph{Bid Intervals} To better understand the bidding process of the various liquidators, we study the number of bids, and their respective intervals. We note that the first bid is placed on average~\\empirical{$3.20\\pm25.69$}~minutes\\xspace after the auction initiation. Given that an auction can last several hours, it appears that most bidders engage early in the auction process. We observe that \\empirical{$3,549$}\\xspace auctions terminate with more than one bid placed. It also appears that bids come close together, as the average interval between bids is \\empirical{$24.05\\pm73.05$}~minutes\\xspace.\n\n\\subsection{Risks}\nWe proceed to discuss the risks that the participants of a lending pool (i.e., borrowers, lenders and liquidators) bear in liquidations.\n\\subsubsection{The Problem of Over-Liquidation}\\label{sec:overliquidation}\nWe observe that the liquidation mechanisms of Aave and Compound grant a liquidator the right to liquidate up to~$50$\\% of the collateral (i.e., the close factor) once a debt becomes liquidatable. dYdX even allows a liquidator to purchase~$100$\\% of the collateral at the fixed spread discount.\n\n\nSuch design decision favor the liquidators over the borrowers, as a debt can likely be rescued by selling less than $50$\\% of its value. Choosing an appropriate close factor is challenging, because the liquidation mechanism should minimize the number of liquidation events and overall transactions due to the limited transaction throughput of blockchains.\n\nAuction mechanisms do not specify a close factor and hence offer a more granular method to liquidate collateral. In return, auction liquidators are exposed to the risk of loss due to the price fluctuations of the collateral during the liquidation (cf.\\ Appendix~\\ref{app:postliquidationprice}). The MakerDAO tend-dent auction, is to the best of our knowledge, the only auction mechanism that is widely adopted in blockchain liquidations. Yet, the liquidation bots must remain robust and vigilant under blockchain congestion, otherwise the borrowers may endure excessive losses~\\cite{maker-fail}. Some alternative auction mechanisms (e.g., Vickrey auction~\\cite{ausubel2006lovely} and Dutch reverse auctions~\\cite{horlen2005reverse}) may have potential in mitigating the over-liquidation problem and could prove more resilient to network congestion. We leave an objective comparison of different auction designs for future work.\n\n\n\\subsubsection{Bad Debts}\n\\label{sec:bad_debts}\n\nWe define a borrowing position as a bad debt if it is financially rationale for neither the borrowers nor the lending platform to close the position. In the following we introduce two types of bad debts.\n\n\\begin{enumerate}\n\\item \\textbf{Type I bad debt (Under-collateralized position):} If the collateral value falls below the value of the debt, then either the borrower or the lending platform will suffer a loss if the corresponding position is closed. Type I bad debt is typically caused by overdue liquidations. An overdue liquidation can, e.g., occur when the collateral\/debt asset suffers a severe price fluctuation or the blockchain network is congested.\n\\item \\textbf{Type II bad debt (Excessive Transaction Fees):} When an over-collateralized position is closed, the borrower will regain the excess asset used for over-collateralization. However, if the value of the excess asset cannot cover the transaction fee, then there is no incentive for the borrower to repay and close this position.\n\\end{enumerate}\n\nBecause the value of the debt plus transaction fee is superior to the value of the bad debt collateral, repaying bad debt is not a financially rational endeavor for a borrower. The accumulation of bad debt reduces the total liquidity in a lending protocol, which necessarily leads to higher interest rates for borrowers. If the lending pool maintains exclusively bad debts, lenders will not be able to withdraw funds.\n\nIn the following we quantitatively measure the amount of bad debts present in existing lending protocols. To that end, we first have to assume a somewhat random cost which a borrower would need to bear, when repaying debt. For the sake of the example here, we choose a cost of~$100$ USD to repay the debt, and consider the blockchain state at block~\\block{12344944} (30th~Apr~2021). Given this cost, we have identified in total~$351$\/$\\numprint{3525}$ Type I\/II bad debts (cf.\\ Table~\\ref{tab:bad_debt}). Remarkably, the liquidity of Aave V2 is reduced by~$87.4$K~USD due to existing bad debts. It is worth mentioning that dYdX does not have any Type I bad debt at block~\\block{12344944}. This is, because dYdX apparently uses an external insurance fund, to write off bad debts of Type I.\n\n\\begin{table}[bt!]\n    \\centering\n    \\caption{Statistics of Type I\/II bad debts on Aave, Compound and dYdX at block~\\block{12344944} (30th~Apr~2021). For instance, if it costs~$100$ USD for a borrower to repay its debt, then~$2,550$ ($32.0\\%$) of the lending position are classified as Type II bad debts on Compound, which causes 14.3K USD collateral value to be locked.}\n    \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{l|c|cc}\n    \\toprule\n            & Type I                                                                     & \\multicolumn{2}{c}{Type II}                                                                                                                               \\\\\n    Transaction fee & -                                                                          & $\\leq$ 10 USD                                                              & $\\leq$ 100 USD                                                               \\\\\n    \\midrule\n    Aave V2         & \\begin{tabular}[c]{@{}l@{}}28 (0.5\\%)\\\\ \\numprint{25379} USD collateral\\end{tabular}  & \\begin{tabular}[c]{@{}l@{}}102 (1.9\\%)\\\\ \\numprint{4793} USD collateral\\end{tabular}  & \\begin{tabular}[c]{@{}l@{}}255 (4.7\\%)\\\\ \\numprint{62017} USD collateral\\end{tabular}   \\\\\\midrule\n    Compound        & \\begin{tabular}[c]{@{}l@{}}333 (4.2\\%)\\\\ \\numprint{27473} USD collateral\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}\\numprint{1681} (21.1\\%)\\\\ 675 USD collateral\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}\\numprint{2550} (32.0\\%)\\\\ \\numprint{14399} USD collateral\\end{tabular} \\\\\\midrule\n    dYdX            & -                                                                          & \\begin{tabular}[c]{@{}l@{}}411 (36.3\\%)\\\\ \\numprint{1287} USD collateral\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}720 (63.5\\%)\\\\ \\numprint{18019} USD collateral\\end{tabular}  \\\\\n    \\bottomrule\n    \\end{tabular}\n    }\n    \\label{tab:bad_debt}\n\\end{table}\n\n\\begin{table}[bt]\n \\centering\n    \\caption{Statistics of unprofitable liquidation opportunities on Aave, Compound and dYdX at block~\\block{12344944} (30th~Apr~2021). For instance, Aave configures a $50\\%$ liquidation threshold with at most $15\\%$ liquidation spread. Based on our measurement, at least $59.1\\%$ of the Aave liquidation opportunities are not profitable if the liquidation process costs~$100$.}\n    \\resizebox{\\columnwidth}{!}{%\n        \\begin{tabular}{llll}\n        \\toprule\n        Transaction Fee &                            & $\\leq$ 10 USD       & $\\leq$ 100 USD      \\\\\n        \\midrule\n        Aave V2         & $\\mathbf{LT}=50\\%, \\mathbf{LS} \\leq 15\\%$        & \\begin{tabular}[c]{@{}l@{}}$\\geq$ 6 (27.2\\%)\\\\ \\numprint{398} USD collateral\\end{tabular}  & \\begin{tabular}[c]{@{}l@{}}$\\geq$ 13 (59.1\\%)\\\\ \\numprint{3404} USD collateral\\end{tabular}  \\\\\n        Compound        & $\\mathbf{LT}=50\\%, \\mathbf{LS} = 8\\%$            & \\begin{tabular}[c]{@{}l@{}}325 (14.8\\%)\\\\ \\numprint{34025} USD collateral\\end{tabular}         & \\begin{tabular}[c]{@{}l@{}}350 (15.9\\%)\\\\ \\numprint{125722} USD collateral\\end{tabular}  \\\\\n        dYdX            & $\\mathbf{LT}=100\\%, \\mathbf{LS} = 5\\%$           & -                   & -            \\\\\n        \\bottomrule\n        \\end{tabular}\n    }\n    \\label{tab:unprofitable_liquidation}\n\\end{table}\n\n\n\n\n\n\n\n\\subsubsection{Unprofitable Liquidations}\\label{sec:unprofitable_liquidation}\n\nWe define a liquidation opportunity as unprofitable if the bonus collected by the liquidator cannot cover the transaction fee. Unprofitable liquidations imply that the lending position's health cannot be restored in time, which necessarily leads to an accumulation of Type I bad debt. We measure the number of unprofitable liquidation opportunities in Table~\\ref{tab:unprofitable_liquidation}. Given the average transaction fee at the time of writing, we choose a transaction cost of~$100$ USD for a single liquidation. Remarkably, by block~\\block{12344944} (30th~Apr~2021), we have identified~$350$ unprofitable liquidation opportunities on Compound, which corresponds to \\numprint{125722} USD worth of collateral. Rational liquidators will not attempt to liquidate unprofitable lending positions. Therefore, these positions will inevitably become Type I bad debts, if their health factor continues to fall.\n\n\\subsubsection{Flash Loan Usages}\\label{sec:flash_loan_usages}\nFixed spread liquidations can be conducted in one transaction alone, which reduces the complexity and risk for the liquidators. For instance, liquidators do not need to hold any assets ready to repay debt for a liquidation. Instead, the liquidators can resort to flash loan pools, lend the required capital to repay debt, and pay back the flash loan interests by the end of the liquidation call. A typical flow of a flash loan liquidation works as follows:\n\\begin{enumerate}\n    \\item The liquidator borrows a flash loan in currency $X$ to repay the debt.\n    \\item The liquidator repays the borrower's debt with a flash loan and receives collateral in currency $Y$ at a premium.\n    \\item The liquidator exchanges parts of the purchased collateral in exchange for currency $X$.\n    \\item To conclude the flash loan, the liquidator repays the flash loan together with flash loan interest. The remaining profit lies with the liquidator. If the liquidation is not profitable, the flash loan would not succeed.\n\\end{enumerate}\n\nTo understand to what degree liquidators engage in flash loans, we study the flash loan pools of Aave and dYdX (which also act as lending pools). We therefore filter the relevant events in the liquidation transactions that apply to flash loans. For our observation window, we observe a total of~\\empirical{$404$}\\xspace flash loans, that are borrowed for liquidations. The accumulative flash loan amount lent sums up to~\\empirical{$144.54$}M~USD\\xspace. We summarize further details in Table~\\ref{tab:flash_loan_usages}. In the table, we include the flash loans that are borrowed before and repaid after liquidation. We notice that in terms of accumulative amounts, dYdX flash loans are more popular than Aave, likely due to the low interest rate of dYdX flash loans.\n\n\\begin{table}[tb!]\n\\centering\n\\caption{Flash loan usages for liquidations.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}cccr@{}}\n\\toprule\n\\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Liquidation\\\\ Platform\\end{tabular}}} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Flash Loan\\\\ Platform\\end{tabular}}} & \\textbf{Flash Loans} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Accumulative\\\\ Amount\\end{tabular}}} \\\\ \\midrule\nAave V1 & dYdX & \\empirical{$264$}\\xspace & \\empirical{$74.65$M}~USD\\xspace \\\\ \\midrule\n\n\\multirow{2}{*}{Aave V2} & Aave V2 & \\empirical{$\\numprint{61}$}\\xspace & \\empirical{$\\numprint{1.27}$M}~USD\\xspace \\\\ \\cmidrule(l){2-4} \n & dYdX & \\empirical{$\\numprint{97}$}\\xspace & \\empirical{$\\numprint{317.29}$M}~USD\\xspace \\\\ \\midrule\n\n\\multirow{2}{*}{Compound} & Aave V1 & \\empirical{$113$}\\xspace & \\empirical{$32.49$k}~USD\\xspace \\\\ \\cmidrule(l){2-4} \n & dYdX & \\empirical{$27$}\\xspace & \\empirical{$69.86$M}~USD\\xspace \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{tab:flash_loan_usages}\n\\end{table}\n\n\\subsection{Instabilities}\nIn this section, we discuss the lending platform instabilities due to cryptocurrency price fluctuations.\n\\subsubsection{Liquidation Sensitivity}\\label{sec:sensitivity}\nTo understand how the lending platforms respond to price declines of different currencies, we quantify the liquidation sensitivity, i.e., the amount of collateral that would be liquidated, if the price of the collateral would decline by up to $100$\\%. We again capture Aave V2, Compound, MakerDAO, and dYdX in the snapshot state at block~\\empirical{$11,363,269$}\\xspace\\footnote{We ignore Aave V1 in the sensitivity measurement because the majority of the liquidity of Aave V1 had been migrated to Aave V2 at block~\\empirical{$11,363,269$}\\xspace.} to provide an exhaustive understanding of borrower risk profiles.\n\n\\newcommand\\mycommfont[1]{\\ttfamily\\textcolor{blue}{#1}}\n\\SetCommentSty{mycommfont}\n\\SetAlFnt{\\small}\n\\SetAlCapFnt{\\small}\n\\SetAlCapNameFnt{\\small}\n\\begin{algorithm}[t]\n\\SetAlgoLined\n\\SetKwProg{Fn}{Function}{:}{end}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\SetKwFunction{Collateral}{Col}\\SetKwFunction{Debt}{Debt}\n\n\\Input{Target currency $\\mathfrak{C}$; Price decline percentage $d\\%$; The set of borrowers $\\{\\mathcal{B}_i\\}$\\;}\n\\Output{Liquidatable collateral $\\mathsf{LC}$\\;}\n\\BlankLine\n\\Fn{\\Collateral{$\\mathcal{B}$, $c$}}{\n    \\KwRet{The value of collateral in currency $c$ owned by $\\mathcal{B}$\\;}\n}\n\\BlankLine\n\\Fn{\\Debt{$\\mathcal{B}$, $c$}}{\n\\KwRet{The value of debt in currency $c$ owed by $\\mathcal{B}$\\;}}\n\\BlankLine\n$\\mathsf{LC} \\leftarrow 0$\\;\n\\ForEach{$\\mathcal{B}\\in\\{\\mathcal{B}_i\\}$}{\n\\If{$\\mathcal{B}$ owns collateral in currency $\\mathfrak{C}$}{\n\n\\tcp*[h]{The collateral value of $\\mathcal{B}$}\n\n\\tcp*[h]{after the price decline}\n\n$\\mathsf{C}\\leftarrow\\sum_c$\\Collateral{$\\mathcal{B}$, $c$}$-$\\Collateral{$\\mathcal{B}$, $\\mathfrak{C}$}$\\times d\\%$\\;\n\n\\tcp*[h]{The borrowing capacity of $\\mathcal{B}$}\n\n\\tcp*[h]{after the price decline}\n\n$\\mathsf{BC}\\leftarrow\\sum_c$\\Collateral{$\\mathcal{B}$, $c$}$\\times \\mathbf{LT}_c - $\\Collateral{$\\mathcal{B}$, $\\mathfrak{C}$}$\\times \\mathbf{LT}_\\mathfrak{C}\\times d\\%$\\;\n\n\\tcp*[h]{The debt value of $\\mathcal{B}$} \n\n\\tcp*[h]{after the price decline}\n\n$\\mathsf{D}\\leftarrow\\sum_c$\\Debt{$\\mathcal{B}$, $c$}\\;\n\n\\If{$\\mathcal{B}$ owes debt in currency $\\mathfrak{C}$}{\n$\\mathsf{D}\\leftarrow\\mathsf{D}-$\\Debt{$\\mathcal{B}$, $\\mathfrak{C}$}$\\times d\\%$\\;\n}\n\\If{$\\mathsf{BC}<\\mathsf{D}$}{\n$\\mathsf{LC} \\leftarrow \\mathsf{LC} + \\mathsf{C}$\\;\n}\n}\n}\n\\KwRet{$\\mathsf{LC}$\\;}\n\\caption{Sensitivity measurement algorithm.}\n\\label{alg:liquidationsensitivity}\n\\end{algorithm}\n\n\\begin{figure*}[tb]\n     \\centering\n     \\begin{subfigure}[b]{0.495\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/aavev2_sensitivity.pdf}\n         \\caption{Aave V2}\n         \\label{fig:aave-liquidation-sensitivity}\n     \\end{subfigure}\n   \n     \\begin{subfigure}[b]{0.495\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/compound_sensitivity.pdf}\n         \\caption{Compound}\n         \\label{fig:compound-liquidation-sensitivity}\n     \\end{subfigure}\n   \n     \\begin{subfigure}[b]{0.495\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/dydx_sensitivity.pdf}\n         \\caption{dYdX}\n         \\label{fig:dydx-liquidation-sensitivity}\n     \\end{subfigure}\n   \n     \\begin{subfigure}[b]{0.495\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/maker_sensitivity.pdf}\n         \\caption{MakerDAO}\n         \\label{fig:maker-liquidation-sensitivity}\n     \\end{subfigure}\n     \\caption{Liquidation sensitivity to price decline in four lending platforms. Liquidation sensitivity denotes the amount of collateral that would be liquidated, if the price of the collateral would decline by up to $100$\\%. We find that all of the four lending platforms are sensitive to the price decline of ETH. Although Aave V2 and Compound follow similar liquidation mechanisms and have similar TVL, Aave V2 is more stable to price declines.}\n     \\label{fig:liquidation-sensitivity}\n\\end{figure*}\n\nWe detail how we measure the liquidation sensitivity in Algorithm~\\ref{alg:liquidationsensitivity}. Specifically, we examine whether each debt becomes liquidatable because of the price decline of the given cryptocurrency. Note that when counting the liquidatable collateral value, we consider the value decrease due to the price decline. We then present the sensitivity results in Figure~\\ref{fig:liquidation-sensitivity}. We find that all of the four lending platforms are sensitive to the price decline of ETH. For example, an immediate $43$\\% decline of the ETH price (analogous to the ETH price decline on the~13th of March,~2020), would result in up to~$1.07$B~USD collateral to become liquidatable on MakerDAO. To our surprise, although Aave V2 and Compound follow similar liquidation mechanisms and have similar TVL, Aave V2 is more stable to price declines in terms of liquidatable collateral. By manually inspecting, we find that this is because Aave V2 users prefer adopting a multiple-cryptocurreny collateral. Hence, the positions in Aave V2 are less likely to become liquidatable due to the price decline of a single cryptocurrency.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Stability of Stablecoins}\\label{sec:stabilityofstablecoins}\nWe observe that certain borrowers collateralize one stablecoin and borrow another stablecoin. Through such strategy, a borrower reduces the likelihood of liquidations, because the prices of stablecoins are deemed stable (cf.\\ Section~\\ref{sec:stablecoinbackground}). To measure the stability of this stablecoin borrowing strategy, we collect the prices of three popular stablecoins, DAI, USDC, and USDT, reported by the price oracle Chainlink~\\cite{arijuel2017chainlink}, from block~\\block{9976964} (May-01-2020) to~\\block{12344944} (Apr-30-2021). We find that the price differences among the three stablecoins is within~$5\\%$ in~$99.97\\%$ of the measured ~$\\numprint{2367981}$ blocks ($1$~year). This indicates that the aforementioned stablecoin borrowing strategy could have been stable for most of the time in 2020. However, we remark that liquidation risks still remain. The maximum price difference we detect is $11.1\\%$ between USDC and DAI at block~\\block{10578280}.\n\n\\subsection{Remarks}\nOur measurements and analysis provide the following insights on DeFi liquidation mechanisms.\n\\begin{enumerate}\n    \\item Existing liquidation mechanisms generate remarkable financial rewards for liquidators (cf.\\ Section~\\ref{sec:profitandloss}). Liquidators are well incentivized to actively perform liquidations. This is confirmed by the severe gas price competition among liquidation transactions (cf.\\ Section~\\ref{sec:fixedspreadparticipation}) and short bid intervals in auction liquidations (cf.\\ Section~\\ref{sec:auctionparticipation}). However, the fixed spread liquidation allows liquidators to over-liquidate a borrowing position, which incurs unnecessary losses to the borrowers (cf.\\ Section~\\ref{sec:overliquidation}).\n    \\item Excessive transaction fees necessarily lead to unprofitable liquidation opportunities and Type II bad debt. Overdue liquidations increase the likelihood of Type I bad debt (cf.\\ Section~\\ref{sec:bad_debts} and Section~\\ref{sec:unprofitable_liquidation}).\n    \\item Fixed spread liquidators can use flash loans to eliminate the risk of holding a specific asset (cf.\\ Section~\\ref{sec:flash_loan_usages}). Auction liquidators are exposed to the risk of price fluctuations during auctions, and may hence suffer a loss (cf.\\ Appendix~\\ref{app:postliquidationprice}).\n    \\item We show that at the time of writing, the studied lending platforms in this work (Aave V2, Compound, dYdX, and MakerDAO) are sensitive (i.e., the amount of the liquidatable collateral due to the price decline of a cryptocurrency) to the price decline of ETH (cf.\\ Section~\\ref{sec:sensitivity}).\n    \\item We evaluate the lending and borrowing practice when focussing exclusively on stablecoins. We show that such strategy mitigates the risk of liquidations for most of the time, while liquidations can still occur (cf.\\ Section~\\ref{sec:stabilityofstablecoins}).\n\\end{enumerate}\n\n\\section{Towards Better Liquidation Mechanisms}\\label{sec:better-liquidation}\nIn this section, we objectively compare the studied liquidation mechanisms and present an optimal fixed spread liquidation strategy that aggravates the loss of borrowers.\n\n\\subsection{Objectively Comparing Liquidation Mechanisms}\nDefining the optimality of a liquidation mechanism is challenging, because a liquidation process is a zero-sum game: the liquidator wins, what the borrower loses. As such, we can rather quantitatively reason about which liquidation mechanism is likely advantageous to a liquidator, or to a borrower.\n\nBased on our empirical data, we proceed by comparing the aforementioned liquidation mechanisms. We define the monthly \\emph{profit-volume ratio} as the ratio between the monthly accumulated liquidation profit and the monthly average collateral volume. To avoid that our results are biased by the asset price fluctuations of different cryptocurrencies, we only study the liquidations repaid in DAI and collateralized in ETH, which are available across all the studied lending platforms. We present the monthly profit-volume ratios of the four platforms from November,~2019~to April,~2021~in Figure~\\ref{fig:profit_volume_ratio}.\n\n\\begin{figure*}[tb!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/profit_volume_ratio_wide.pdf}\n    \\caption{Comparison of the monthly liquidation profit over the monthly average collateral volume for the DAI\/ETH lending markets. The lower the profit-volume ratio is, the better the liquidation protocol is for borrower.}\n    \\label{fig:profit_volume_ratio}\n\\end{figure*}\n\n\nOur results show that dYdX has a higher profit-volume ratio than the other four platforms, meaning that dYdX is in expectation favorable to liquidators and worse for borrowers. This observation matches the fact that dYdX does not set a close factor, which means that a debt can be fully liquidated once its health factor declines below $1$. We further observe that MakerDAO consistently shows a smaller ratio than Compound, except for the outlier in March,~2020~(cf.\\ Section~\\ref{sec:insights}). Surprisingly, while Aave follows the same liquidation mechanism of Compound, the profit-volume ratio of Aave, especially Aave V1, remains below Compound. We infer that this is because the number of DAI\/ETH liquidations events on Aave are rare (cf.\\ Table~\\ref{tab:DAI-ETH-liquidations}, Appendix~\\ref{app:daiethliquidations})~\u2013-~we hence believe that the Aave market is not sufficiently indicative to draw a representative conclusion. Overall our results suggest that the auction mechanism favors the borrowers more than a fixed spread liquidation with a close factor beyond $50\\%$.\n\n\n\n\n\n\n\n\n\\subsection{Optimal Fixed Spread Liquidation Strategy}\\label{sec:optimalstrategy}\nThe configuration of a close factor (cf.\\ Section~\\ref{sec:terminology}) restricts the profit of the liquidator (i.e., the loss of the borrower) in a single fixed spread liquidation. We denote the strategy of liquidating up to the close factor limit within a single liquidation as the \\emph{up-to-close-factor} strategy. Intuitively, a liquidator is rational to perform the up-to-close-factor strategy, because the profit is positively correlated to the liquidation amount. However, we find that a liquidator can lift the restriction of the close factor by performing two successive liquidations. The optimal strategy leverages the rule that a position remains liquidatable as long as it is in an unhealthy state, no matter this position has been liquidated or not previously.\n\nIn this optimal strategy, instead of pushing to close factor limit, the liquidator liquidates as much as possible but still keeps the position in an unhealthy state in the first liquidation. Then, in the second liquidation, the liquidator liquidates the remaining collateral up to the close factor. We detail the optimal fixed spread liquidation strategy in Algorithm~\\ref{alg:optimalfixedspreadliquidation}. \n\n\\begin{algorithm}[tb]\n\\SetAlgoLined\n\\SetKwProg{Fn}{Function}{:}{end}\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\\SetKwFunction{Liquidate}{Liquidate}\n\\SetKwFunction{IsLiquidatable}{Liquidatable}\n\n\\Input{A liquidatable position $\\mathcal{POS}=\\left \\langle C, D \\right \\rangle$, where $C$ represents the collateral value, while $D$ represents the debt value; Liquidation threshold $\\mathbf{LT}$; Liquidation spread $\\mathbf{LS}$; Close factor $\\mathbf{CF}$.}\n\\Output{Amount of debt to repay in the two optimal successive liquidations, $repay_1$ and $repay_2$.}\n\\BlankLine\n\\Fn{\\IsLiquidatable{$\\mathcal{POS}$}}{\n    \\KwRet{$\\frac{\\mathcal{POS}.C \\times \\mathbf{LT}}{\\mathcal{POS}.D} > 1$}\\;\n}\n\\BlankLine\n\\Fn{\\Liquidate{$\\mathcal{POS}$, $repay$}}{\n    $\\mathcal{POS}'\\leftarrow\\left \\langle C - repay\\times (1+\\mathbf{LS}) , D - repay \\right \\rangle$\\;\n    \\KwRet{$\\mathcal{POS}'$}\\;\n}\n\\BlankLine\n$repay_1 \\leftarrow \\operatorname{argmax}_r$ \\IsLiquidatable{\\Liquidate{$\\mathcal{POS}, r$}}\\;\n$\\mathcal{POS}'\\leftarrow$ \\Liquidate{$\\mathcal{POS}, repay_1$}\\;\n$repay_2 \\leftarrow \\mathcal{POS}'.D \\times \\mathbf{CF}$\\;\n\n\\caption{Optimal fixed spread liquidation strategy.}\n\\label{alg:optimalfixedspreadliquidation}\n\\end{algorithm}\n\n\\subsubsection{Optimality Analysis} Given a liquidatable borrowing position with $C$ collateral value and $D$ debt (cf.\\ Equation~\\ref{eq:position}), we proceed to analyze the profit of our optimal strategy.\n\n\\begin{equation}\\label{eq:position}\n    \\mathcal{POS} = \\left \\langle C, D \\right \\rangle\n\\end{equation}\n$\\mathbf{LT}$, $\\mathbf{LS}$, $\\mathbf{CR}$ denote the liquidation threshold, liquidation spread and close factor respectively (cf.\\ Section~\\ref{sec:terminology}).\n\nFollowing Algorithm~\\ref{alg:optimalfixedspreadliquidation}, the repaid debt amounts in the two successive liquidations are given in Equation~\\ref{eq:repay1} and~\\ref{eq:repay2}. Note that the $D-\\mathbf{LT}\\cdot C > 0$ because $\\mathcal{POS}$ is liquidatable (i.e., the debt is greater than the borrowing capacity). We show in Appendix~\\ref{app:reasonalconfiguration} that a reasonable fixed spread liquidation configuration satisfies $1-\\mathbf{LT}(1+\\mathbf{LS})>0$.\n\n\\begin{equation}\\label{eq:repay1}\n\\begin{aligned}\n    repay_1 &= \\operatorname{argmax}_r \\frac{\\mathbf{LT}(C-r(1+\\mathbf{LS}))}{D-r}\\geq 1\\\\\n    &=\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{eq:repay2}\n    repay_2 = \\mathbf{CF}\\left(D-repay_1\\right)= \\mathbf{CF}\\left(D-\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\\right)\n\\end{equation}\nThe overall profit of the two liquidations is shown in Equation~\\ref{eq:overallprofit}.\n\\begin{equation}\\label{eq:overallprofit}\n\\begin{aligned}\n    profit_{o} &= (repay_1 + repay_2)\\times \\mathbf{LS}\\\\\n    &=\\mathbf{LS}\\cdot\\mathbf{CF}\\cdot D+\\mathbf{LS}(1-\\mathbf{CF})\\left(\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\\right)\n\\end{aligned}\n\\end{equation}\n\nIf the liquidator instead chooses to perform the up-to-close-factor strategy, the repay amount is $\\mathbf{CF}\\cdot D$ and the profit hence is $profit_{c}=\\mathbf{LS}\\cdot\\mathbf{CF}\\cdot D$. Therefore, the optimal strategy can yield more profit than the up-to-close-factor strategy. The increase rate of the liquidation profit is shown in Equation~\\ref{eq:increaserate}.\n\n\\begin{equation}\\label{eq:increaserate}\n    \\Delta R_{profit} = \\frac{profit_o-profit_c}{profit_c}=\\frac{\\mathbf{CF}}{1-\\mathbf{CF}}\\cdot\\frac{1-\\mathbf{LT}\\cdot\\mathsf{CR}}{1-\\mathbf{LT}(1+\\mathbf{LS})}\n\\end{equation}\nwhere $\\mathsf{CR}=\\frac{C}{D}$ is the collateralization ratio (cf.\\ Section~\\ref{sec:terminology}). We notice that the optimal strategy is more effective when $\\mathsf{CR}$ is low. \n\n\\subsubsection{Case Study}\nIn the following, we study the most profitable fixed spread liquidation transaction (\\empirical{$4.04$M}~USD\\xspace)\\footnote{Transaction hash: \\etherscantx{0x53e09adb77d1e3ea593c933a85bd4472371e03da12e3fec853b5bc7fac50f3e4}} we detect and showcase how the optimal fixed spread liquidation strategy increases the profit of a liquidator. In this Compound liquidation, the liquidator first performs an oracle price update\\footnote{Compound allows any entity to update the price oracle with authenticated messages signed by, for example, off-chain price sources.}, which renders a borrowing position liquidatable. The liquidator then liquidates the position within the same transaction. In Table~\\ref{tab:casestudystatus}, we present the status change of the position following the price update. Before the price update (block~\\block{11333036}), the position owns a total collateral of $135.07$M~USD (with a borrowing capacity of $101.30$M~USD), and owes a debt of~$101.18$M~USD. After the price of DAI increases from $1.08$ to $1.095299$~USD\/DAI, the total debt reaches $102.61$M~USD, while the borrowing capacity is only $102.55$M~USD. The health factor drops below~$1$, and hence the position becomes liquidatable.\n\n\\begin{table}[tb!]\n\\centering\n\\caption{Status of the borrowing position (\\account{0x909b443761bbD7fbB876Ecde71a37E1433f6af6f}). Note we ignore the tiny amount of collateral and debt in USDT that the borrower owns and owes. The liquidation thresholds (i.e., $\\mathbf{LT}$) of DAI and USDC are both~$0.75$.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}ccc|c|c@{}}\n\\toprule\n\\multirow{2}{*}{\\bf Token} & \\multirow{2}{*}{\\bf Collateral} & \\multirow{2}{*}{\\bf Debt} & \\multicolumn{2}{c}{\\bf Price (USD)} \\\\ \\cmidrule(l){4-5} \n&  &  & Block \\block{11333036} & After Price Update \\\\ \\midrule\nDAI & $108.51$M & $93.22$M & $1.08$ & $1.095299$ \\\\\nUSDC & $17.88$M & $506.64$k & $1$ & $1$ \\\\ \\midrule\\midrule\n\\multicolumn{3}{c|}{\\bf Total Collateral (USD)} & $135.07$M & $136.73$M \\\\\n\\multicolumn{3}{c|}{\\bf Borrowing Capacity (USD)} & $101.30$M & $102.55$M \\\\\n\\multicolumn{3}{c|}{\\bf Total Debt (USD)} & $101.18$M & $102.61$M \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\label{tab:casestudystatus}\n\\end{table}\n\n\\begin{table}[tb!]\n\\centering\n\\caption{The depiction of liquidation strategies. At the time of the original liquidation, the price of DAI is $1.095299$~USD\/DAI. The close factor is $50\\%$. The optimal strategy is the most profitable liquidation mechanism for the liquidator.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}c|l|l@{}}\n\\toprule\n\\textbf{Original liquidation} & \\multicolumn{2}{l}{\\begin{tabular}[c]{@{}l@{}}Repay $46.14$M~USD\\\\ Receive $49.83$M~DAI\\\\ \\emph{Profit} $3.69$M~DAI\\end{tabular}} \\\\ \\midrule\\midrule\n\\textbf{Up-to-close-factor strategy} & \\multicolumn{2}{l}{\\begin{tabular}[c]{@{}l@{}}Repay $46.61$M~DAI\\\\ Receive $50.34$M~DAI\\\\ \\emph{Profit} $3.73$M~DAI\\end{tabular}} \\\\ \\midrule\\midrule\n\\multirow{4}{*}{\\textbf{Optimal strategy}} & Liquidation 1 & Liquidation 2 \\\\ \\cmidrule(l){2-3} \n & \\begin{tabular}[c]{@{}l@{}}Repay $296.61$K~DAI\\\\ Receive $320.34$K~DAI\\\\ \\emph{Profit} $23.73$K~DAI\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Repay $46.46$M~DAI\\\\ Receive $50.18$M~DAI\\\\ \\emph{Profit} $3.72$M~DAI\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\label{tab:liquidation-strategy-comparisons}\n\\end{table}\n\nTo evaluate the up-to-close-factor strategy and our optimal liquidation strategy, we implement the original liquidation and the two liquidation strategies\\footnote{We publish the smart contract code at \\url{https:\/\/anonymous.4open.science\/r\/An-Empirical-Study-of-DeFi-Liquidations-Anonymous\/CompoundLiquidationCaseStudy.sol}.} in Solidity v0.8.4\\footnote{\\url{https:\/\/docs.soliditylang.org\/en\/v0.8.4\/}}.\nWe execute them on the corresponding blockchain states\\footnote{We fork the Ethereum mainchain locally from block~\\block{11333036} and apply all the transactions executed prior to the original liquidation transaction in block~\\block{11333037}. We then execute the liquidation strategies to ensure that they are validated on the exact same state of the original liquidation.} and present the results in Table~\\ref{tab:liquidation-strategy-comparisons}. We find that the optimal strategy is superior to the up-to-close-factor strategy and can generate an additional profit of $49.26$K~DAI (\\empirical{$53.96$K}~USD\\xspace) compared to the original liquidation.\n\n\n\\subsubsection{Mitigation}\nThe aforementioned optimal strategy defeats the original intention of a close factor, which incurs undesirably additional losses to borrowers. A possible mitigation solution is that for every position only one liquidation is permitted within one block. Such a setting enforces a liquidator adopting the optimal strategy to settle the two liquidations in two blocks, which decreases the success probability.\n\nWe proceed to assume the existence of a mining liquidator with a mining power $\\alpha$. Given a liquidation opportunity, the up-to-close-factor strategy produces a profit of $profit_c$, while the optimal strategy yields $profit_{o_1}$ and $profit_{o_2}$ respectively in the two successive liquidations. We further assume that there is no ongoing consensus layer attack (e.g., double-spending), implying that a miner with an $\\alpha$ fraction mining power mines the next block is with a probability of $\\alpha$. We hence derive the the expected profit of the two strategies as shown in Equation~\\ref{eq:expected-profit-close} and~\\ref{eq:expected-profit-optimal}\\footnote{To ease understanding, we assume other competing liquidators adopt the up-to-close-factor strategy.}.\n\n\\begin{equation}\\label{eq:expected-profit-close}\n    \\mathbb{E}[\\text{up-to-close-factor}] = \\alpha\\cdot profit_c\n\\end{equation}\n\\begin{equation}\\label{eq:expected-profit-optimal}\n    \\mathbb{E}[\\text{optimal}] = \\alpha\\cdot profit_{o_1} + \\alpha^2\\cdot profit_{o_2}\n\\end{equation}\nThe liquidator is incentivized to perform the optimal strategy only when $\\mathbb{E}[\\text{optimal}] > \\mathbb{E}[\\text{up-to-close-factor}]$, leading to Equation~\\ref{eq:optimal-rational-condition}.\n\n\\begin{equation}\\label{eq:optimal-rational-condition}\n\\alpha > \\frac{profit_c-profit_{o_1}}{profit_{o_2}}\n\\end{equation}\nIntuitively, $profit_{o_1}$ is relatively small compared to $profit_{c}$ and $profit_{o_2}$ because the liquidator needs to keep the position unhealthy after the first liquidation. The expected profit in the second liquidation then should be sufficient to cover the opportunity cost in the first one, which is typically unattainable.\nInstantiating with our case study liquidation (cf.\\ Table~\\ref{tab:casestudystatus}), we show that a rational mining liquidator would attempt the optimal strategy in two consecutive blocks only if its mining power is over $99.68\\%$. Therefore, we conclude that the one liquidation in one block effectively reduces the expected profit of the optimal liquidation strategy, protecting borrowers from a further liquidation losses.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\\label{sec:related-work}\n\n\\point{Blockchains and DeFi}\nThere is a growing body of literature on blockchains and DeFi. Qin \\textit{et al.}~\\cite{qin2020attacking} study flash loan attacks and present an optimization approach to maximize the profit of DeFi attacks. Zhou \\textit{et al.}~\\cite{zhou2020high} analyze sandwich attacks in decentralized exchanges. Eskandari \\textit{et al.}~\\cite{eskandari2019sok} provide an overview of the blockchain front-running attacks. Daian \\textit{et al.}~\\cite{daian2019flash} investigate the front-running attacks in decentralized exchanges and propose the concept of \\emph{Miner Extractable Value} (MEV), a financial revenue miners can extract through transaction order manipulation. Qin \\textit{et al.}~\\cite{qin2021quantifying} quantify the extracted MEV on the Ethereum blockchain, including fixed spread liquidations, and present a generalized front-running algorithm, transaction replay. Zhou \\textit{et al.}~\\cite{zhou2021just} propose a framework called \\textsc{DeFiPoser} that allows to automatically create profit-generating transactions given the blockchain state.\n\n\\point{Blockchain Borrowing and Lending Markets}\nDarlin \\textit{et al.}~\\cite{darlin2020optimal} study the MakerDAO liquidation auctions. The authors optimize the costs for participating in the auctions and find that most auctions conclude at higher than optimal prices. The work appears real-world relevant, as it considers the transaction fees, conversion costs and cost of capital, yet it does not consider potential gas bidding contests by the end of MakerDAO auctions~\\cite{daian2019flash}. Kao \\textit{et al.}~\\cite{kao2020analysis} and ZenGo~\\cite{zengo-compound} are to our knowledge the first to have investigated Compound's liquidation mechanism (the third biggest lending protocol in terms of USD at the time of writing). Perez~\\textit{et al.}~\\cite{perez2020liquidations} follow up with a report that focuses on additional on-chain analytics of the Compound protocol. DragonFly Research provides a blog post~\\cite{medium-liquidation} about the liquidator profits on Compound, dYdX and MakerDAO. Minimizing financial deposit amounts in cryptoeconomic protocols, while maintaining the same level of security is studied in Balance~\\cite{harzbalance}.\n\n\\point{Liquidations in Traditional Finance} Liquidations are essential to traditional finance (TradFi) and are well studied in the related literature~\\cite{titman1984effect,shleifer1992liquidation,alderson1995liquidation,almgren1999value,reinhart2011liquidation}. We remark that liquidations in blockchain systems are fundamentally different from those in TradFi in terms of high-level designs and settlement mechanisms.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nDue to their significant volatility when compared to alternative financial vehicles cryptocurrencies are attracting speculators. Furthermore, because speculators seek to further their risk exposure, non-custodial lending and borrowing protocols on blockchains are thriving. The risks of borrowing, however, manifests themselves in the form of liquidation profits claimed by liquidators.\n\nIn this paper we study the lending platforms that capture $85$\\% of the blockchain lending market. We systematize the most prevalent liquidation mechanisms and find that many liquidations sell excessive amounts of borrower's collateral. In this work we provide extensive data analytics covering over $2$ years the prevalent $4$ lending protocols. We systematize their respective liquidation mechanisms and show that most liquidation systems are unfavorable to the borrowers. We finally show an optimal liquidation strategy which we have not yet observed in the wild.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA \\emph{hole} in a graph is an induced cycle of length at least four. A \\emph{proper coloring} of a graph is a function that assigns to each vertex a color with the constraint that two adjacent vertices are not given the same color. The \\emph{chromatic number} of $G$, denoted by $\\chi(G)$, is the smallest number of colors needed to color the graph properly. All the colorings considered in the sequel are proper, so we just call them colorings. \nThe size of the largest clique of $G$ is denoted $\\omega(G)$.\nWe obviously have $\\omega(G)\\leq \\chi(G)$, and one may wonder whether the equality holds. In fact, it does not hold in the general case, and the simplest counter-example is any \\emph{odd hole}, i.e. any hole of odd length, for which $\\omega(G)=2$ but $\\chi(G)=3$.  Graphs for which the equality $\\chi(G')=\\omega(G')$ holds for every induced subgraph $G'$ of $G$ are called \\emph{perfect}, and the Strong Perfect Graph Theorem \\cite{SPGT} proved that \na graph if perfect if and only if it is\n\\emph{Berge}, that is to say there is no odd hole in $G$ nor in its complement.\nIn order to get some upper bound on $\\chi(G)$, Gy\\'arf\\'as \\cite{G54} introduced the concept of $\\chi$-bounded class: a family $\\mathcal{G}$ of graphs is called $\\chi$\\emph{-bounded}  if there exists a function $f$ such that $\\chi(G')\\leq f(\\omega(G'))$  whenever $G'$ is an induced subgraph of $G\\in \\mathcal{G}$.\n\nThis notion has widely been studied since then, in particular in hereditary classes (\\emph{hereditary} means closed under taking induced subgraph). \nA classical result of Erd\\H{o}s \\cite{E40} asserts that there exist graphs with arbitrarily large \\emph{girth} (that is, the length of the shortest induced cycle) and arbitrarily large chromatic number. Thus forbidding only one induced subgraph $H$ may lead to a $\\chi$-bounded class only if $H$ is acyclic. It is conjectured that this condition is also sufficient \\cite{G32, S52}, but it is proved only if $H$ is a path, a star \\cite{G54} or a tree of radius two \\cite{TreeRadius2} (or three, with additional conditions \\cite{TreeRadius3}). Scott \\cite{ScottSubdivTree} also proved it for any tree $H$, provided that we forbid every induced subdivision of $H$, instead of just $H$ itself.\n\nConsequently, forbidding holes in order to get a $\\chi$-bounded class is conceivable only if we forbid infinitely many hole lengths. \nTwo parameters should be taken into account: first, the length of the holes, and secondly, the parity of their lengths. \nIn this respect, Gy\\'arf\\'as \\cite{G54} made a famous series of three  conjectures. The first one asserts the class of graphs with no odd hole is $\\chi$-bounded. The second one asserts that, for every $k$, the class of graphs with no hole of length at least $k$ is $\\chi$-bounded. The last one generalizes the first two conjectures and asserts that for every $k$, the class of graphs with no odd hole of length at least $k$ is $\\chi$-bounded. After several partial results \\cite{RS99, ScottCycles, CSSGyarfas}, the first and the second conjectures were recently solved by Chudnovsky, Scott and Seymour \\cite{SSOddHoles, CSSLongHoles}. Moreover, we learned while writing this article that Scott and Seymour claimed\\footnote{Their article is still in preparation.} to have proved a very general result implying the triangle-free case of the third conjecture (which would also imply the result of this paper): for every $k\\geq 0$, every triangle-free graph with large enough chromatic number admits a sequence of holes  of $k$ consecutive lengths.\n\n\nThe class of even-hole-free graphs has extensively been studied from a structural point of view. A decomposition theorem together with a recognition algorithm have been found by Conforti, Cornu\\'ejols, Kapoor and Vu\\v{s}kovi\\'c \\cite{EvenHole1, EvenHole2,ChangLu}. Reed conjectured \\cite{R58} that every even-hole-free graphs has a vertex whose neighborhood is the union of two cliques (called a \\emph{bisimplicial} vertex), which he and his co-authors proved \\cite{EvenHoleBisimplicial} a few years later. As a consequence, they obtained that every even-hole-free graph $G$ satisfies $\\chi(G)\\leq 2\\omega(G)-1$.\n\nForbidding $C_4$ is in fact a strong restriction since $C_4$ can also be seen as the complete bipartite graph $K_{2,2}$: K\\\"uhn and Osthus \\cite{KuhnOsthus} proved that for every graph $H$ and for every integer $s$, every graph of large average degree (with respect to $H$ and $s$) with no $K_{s,s}$ as a (non-necessarily induced) subgraph contains an induced subdivision of $H$, where each edge is subdivided at least once. This strong result implies that $\\chi$ is bounded in any class $\\mathcal{C}$ defined as graphs with no triangles, no induced $C_4$ and no cycles of length divisible by $k$, for any fixed integer $k$. Indeed, let $G\\in \\mathcal{C}$ be a minimal counter-example to $\\chi(G)\\leq t$ (with $t$ chosen large enough with respect to $k$), then it has large minimum degree. Moreover it has neither induced $C_4$ nor triangles, consequently it has no $C_4$ subgraphs. By K\\\"uhn and Osthus' theorem, there exists an induced subdivision $H$ of $K_\\ell$ for some well-chosen integer $\\ell$ depending on $k$. Consider $K_\\ell$ as an auxiliary graph where we color each edge with $c\\in\\{1,\\ldots, k \\}$ if this edge is subdivided $c$ times modulo $k$ in $H$. By Ramsey's theorem \\cite{Ramsey}, if $\\ell$ is large enough, then we can find a monochromatic clique $K$ of size $k$. Let $C_0$ be a Hamiltonian cycle through $K$ and call $C$ the corresponding cycle in the subdivided edges in $H$. Since $K$ was monochromatic in $K_\\ell$, the edges used in $C_0$ are subdivided the same number of times modulo $k$, consequently $C$ has length divisible by $k$. Moreover, it is an induced cycle since each edge is subdivided at least once in $H$.\n\n\nThis is why we are interested in finding a $\\chi$-boundedness result when every even hole except $C_4$ is forbidden, which was conjectured by Bruce Reed \\cite{ReedPrivate}. In this paper, we achieve a partial result by forbidding also triangles. This is a classical step towards $\\chi$-boundedness, and Thomass\\'e \\emph{et al.} \\cite{FPTBullFree} even asked whether this could always be sufficient, namely: does there exist a function $f$ such that for every class $\\mathcal{C}$ of graphs and any $G\\in \\mathcal{C}$, $\\chi(G)\\leq f(\\chi_T(G), \\omega(G))$, where $\\chi_T(G)$ denotes the maximum chromatic number of a triangle-free induced subgraph of $G$?\n\n\n\nThe result of this paper is closely related to the following recent one, by Bonamy, Charbit and Thomass\\'e, answering to a question by Kalai and Meshulam on the sum of Betti numbers of the stable set complex (see \\cite{Thomasse0mod3} for more details):\n\n\\begin{theorem}[\\cite{Thomasse0mod3}]\nThere exists a constant $c$ such that every graph $G$ with no induced cycle of length divisible by 3 satisfies $\\chi(G) < c$.\n\\end{theorem}\n\nIndeed, the so-called Parity Changing Path (to be defined below) is directly inspired by their Trinity Changing Path. The structure of the proofs also have several similarities.\n\n\n\n\n\\bigskip\n\n\\paragraph{Contribution} We prove the following theorem:\n\n\\begin{theorem} \\label{th: C chi borne}\nThere exists a constant $c$ such that every graph $G$ with no triangle and no induced cycle of even length at least 6  satisfies $\\chi(G) < c$.\n\\end{theorem}\n\nThe outline is to prove the result when the 5-hole is also forbidden (see Lemma \\ref{lem: sans C5} below), which should intuitively be easier, and then deduce the theorem for the general case.\n\n\n\nTo begin with, let us introduce and recall some notations:\nthe class under study, namely graphs with no triangle and no induced $C_{2k}$ with $k\\geq 3$ (meaning that every even hole is forbidden except $C_4$) will be called \\noindent {\\bf Definitions. }{$\\mathcal{C}_{3, 2k\\geq 6}$} for short. Moreover, we will consider in Subsection \\ref{sec: sans C5} the subclass \\noindent {\\bf Definitions. }{$\\mathcal{C}_{3, 5, 2k\\geq 6}$} of $\\mathcal{C}_{3, 2k\\geq 6}$ in which the 5-hole is also forbidden. For two subsets of vertices $A, B \\subseteq V$, $A$ \\noindent {\\bf Definitions. }{dominates} $B$ if $B\\subseteq N(A)$.\nA \\noindent {\\bf Definitions. }{major connected component} of $G$ is a connected component $C$ of $G$ for which $\\chi(C)=\\chi(G)$. Note that such a component always exists.  \nFor any induced path $P=x_1x_2\\cdots x_\\ell$ we say that $P$ is a path from its \\noindent {\\bf Definitions. }{origin} $x_1$ to its \\noindent {\\bf Definitions. }{end} $x_\\ell$ or an \\noindent {\\bf Definitions. }{$x_1x_\\ell$-path}. Its \\noindent {\\bf Definitions. }{interior} is $\\{x_2, \\ldots, x_{\\ell-1}\\}$ and its \\noindent {\\bf Definitions. }{length} is $\\ell-1$. \n\nMoreover, we use a rather common technique called a \\emph{levelling} \\cite{SSOddHoles, CSSGyarfas} :\ngiven a vertex $v$, the \\noindent {\\bf Definitions. }{$v$-levelling} is the partition $(N_0, N_1, \\ldots, N_k, \\ldots)$ of the vertices according to their distance to $v$: $N_k$ is the set of vertices at distance exactly $k$ from $v$ and is called the \\noindent {\\bf Definitions. }{$k$-th level}. In particular, $N_0=\\set{v}$ and $N_1=N(v)$. We need two more facts about levellings: \nif $x$ and $y$ are in the same part $N_k$ of a $v$-levelling, we call an \\noindent {\\bf Definitions. }{upper $x y$-path} any shortest path from $x$ to $y$ among those with interior in $N_0\\cup \\cdots \\cup N_{k-1}$. Observe that it always exists since there is an $xv$-path and a $vy$-path (but it may take shortcuts; in particular, it may be just one edge). Moreover, in any $v$-levelling, there exists $k$ such that $\\chi(N_k)\\geq \\chi(G)\/2$: \nindeed, if $t$ is the maximum of $\\chi(N_i)$ over all levels $N_i$, one can color $G$ using $2t$ colors by coloring $G[N_{i}]$ with the set of colors $\\{1, \\ldots, t\\}$ if $i$ is odd, and with the set of colors $\\{t+1, \\ldots , 2t\\}$ if $i$ is even. Such a level with chromatic number at least $\\chi(G)\/2$ is called a \\noindent {\\bf Definitions. }{colorful} level.\n\n\n\nLet us now introduce the main tool of the proof, called \\noindent {\\bf Definitions. }{Parity Changing Path} (\\noindent {\\bf Definitions. }{PCP} for short) which, as already mentioned, is inspired by the \\emph{Trinity Changing Path (TCP)} appearing in \\cite{Thomasse0mod3}: intuitively (see Figure \\ref{fig: PCP} for an unformal diagram), a PCP is an induced sequence of induced subgraphs and paths $(G_1, P_1, \\ldots, G_\\ell, P_\\ell, H)$ such that each block $G_i$ can be crossed by two possible paths of different parities, and the last block $H$ typically is a stock of big chromatic number.\nFormally, a \\noindent {\\bf Definitions. }{PCP of order $\\ell$} in $G$ is a sequence of induced subgraphs $G_1, \\ldots, G_{\\ell}, H$ (called \\noindent {\\bf Definitions. }{blocks}; the $G_i$ are the \\noindent {\\bf Definitions. }{regular} blocks) and induced paths $P_1, \\ldots , P_{\\ell}$ such that the origin of $P_i$ is some vertex $y_i$ in $G_i$, and the end of $P_i$ is some vertex $x_{i+1}$ of $G_{i+1}$ (or of $H$ if $i=\\ell$). Apart from these special vertices which belong to exactly two subgraphs of the PCP, the blocks and paths $G_1, \\ldots, G_\\ell, H, P_1, \\ldots, P_\\ell$ composing the PCP are pairwise disjoint. The only possible edges have both endpoints belonging to the same block or path.\nWe also have one extra vertex $x_1\\in G_1$ called the \\noindent {\\bf Definitions. }{origin} of the PCP. \nMoreover in each block $G_i$, there exists one induced $x_i y_i$-path of odd length, and one induced $x_i y_i$-path of even length (these paths are not required to be disjoint one from each other). In particular $x_i \\neq y_i$ and $x_i y_i$ is not an edge. For technical reasons that will appear later, we also require that $H$ is connected, every $G_i$ has chromatic number at most 4 and every $P_i$ has length at least 2.  Finally the chromatic number of $H$ is called the \\noindent {\\bf Definitions. }{leftovers}. \n\nIn fact in Subsection \\ref{sec: Yes C4 avec C5}, we need a slightly stronger definition of PCP: a \\noindent {\\bf Definitions. }{strong PCP} is a PCP for which every $G_i$ contains an induced $C_5$.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=1]{fig\/PCP}\n\\caption{An informal diagram for a PCP of order 3. Grey curved lines stand for the even and odd length $x_iy_i$-paths.}\n\\label{fig: PCP}\n\\end{figure}\n\n\\bigskip\n\nWe first bound the chromatic number in $\\mathcal{C}_{3, 5, 2k\\geq 6}$ (see Lemma \\ref{lem: sans C5} below), which is easier because we forbid one more cycle length, and then deduce the theorem for $\\mathcal{C}_{3, 2k\\geq 6}$. The proofs for $\\mathcal{C}_{3, 2k\\geq 6}$ and $\\mathcal{C}_{3, 5, 2k\\geq 6}$ follow the same outline, which we informally describe here:\n\n\\begin{enumerate}\n\\item If $\\chi(G)$ is large enough, then for every vertex $v$ we can grow a PCP whose origin is $v$ and whose leftovers are large (Lemmas \\ref{lem: debut PCP sans C5}, \\ref{lem: existence PCP sans C5} and then Lemma \\ref{lem: debut PCP}).\n\\item Using (i), if $\\chi(G)$ is large enough and $(N_0, N_1, \\ldots)$ is the $v$-levelling,\nwe can grow a \\noindent {\\bf Definitions. }{rooted} PCP:\nit is a PCP in a level $N_k$, which has a \\noindent {\\bf Definitions. }{root}, \\emph{i.e.} a vertex in the previous level $N_{k-1}$ whose unique neighbor in the PCP is the origin (Lemma \\ref{lem: rooted PCP sans C5} and then Lemma \\ref{lem: rooted PCP}).\n\\item Given a rooted PCP in a level $N_k$, if a vertex $x\\in N_{k-1}$ has a neighbor in the last block $H$, then it has a neighbor in every regular block $G_i$ (Lemma \\ref{obs: sommets riches sans C5}).\n\\item Given a rooted PCP of order $\\ell$ in a level $N_k$ and a stable set $S$ in $N_{k-1}$, the set of neighbors of $S$ inside $N_k$ can not have a big chromatic number. Consequently, the \\noindent {\\bf Definitions. }{active lift} of the PCP, defined as $N(G_\\ell)\\cap N_{k-1}$, has high chromatic number (Lemmas \\ref{lem: shadow stable bornee sans C5}, \\ref{lem: active lift gros si neighb stable gros} and \\ref{lem: shadow borne bornee sans C5} and then Lemmas \\ref{lem: active lift gros si neighb stable gros}, \\ref{lem: shadow stable bornee}, \\ref{lem: shadow borne bornee}).\n\\item The final proofs put everything together: consider a graph of $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ (resp. $\\mathcal{C}_{3, 2k\\geq 6}$) with chromatic number large enough. Then pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level. By (ii), grow inside $N_k$ a rooted PCP $P$. Then by (iv), get an active lift $A$ of $P$ inside $N_{k-1}$ with big chromatic number. Grow a rooted PCP $P'$ inside $A$, and get an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number big enough to find an edge $xy$ (resp. a 5-hole $C$) in $A'$ . Then ``clean\" $P'$ in order to get a stable set $S$ inside the last regular block of $P'$, dominating this edge (resp. hole). Now find an even hole of length $\\geq 6$ in $\\set{x,y}\\cup S\\cup P$ (resp. $C\\cup S\\cup P$), a contradiction.\n\\end{enumerate}\n\n\n\\section{Forbidding 5-holes}\n\\label{sec: sans C5}\n\nThis section is devoted to the proof of the following lemma :\n\n\\begin{lemma} \\label{lem: sans C5}\nThere exists a constant $c'$ such that every graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ satisfies $\\chi(G)< c'$.\n\\end{lemma}\n\n\nWe follow the outline described above. Let us start with step (i):\n\n\\begin{lemma} \\label{lem: debut PCP sans C5}\nLet $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be a connected graph and $v$ be any vertex of $G$. For every $\\delta$ such that $\\chi(G)\\geq \\delta \\geq 18$, there exists a PCP of order 1 with origin $v$ and leftovers at least $h(\\delta)=\\delta\/2 -8$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is illustrated on Figure \\ref{fig: build PCP overview}. Let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level.\nLet $N'_k$ be a major connected component of $G[N_k]$, so $\\chi(N'_k)\\geq \\delta\/2$.\n Let $xy$ be an edge of $N'_k$, and $x'$ (resp. $y'$) be a neighbor of $x$ (resp. $y$) in $N_{k-1}$.\nLet  $Z' = N(\\{x',y',x,y\\})\\cap N'_k$ and $Z=Z' \\setminus \\{x,y\\}$. \n Let $z\\in Z$ be a vertex having a neighbor $z_1$ in a major connected component $M_1$ of $N'_k \\setminus Z'$. Observe that $N'_k \\setminus Z'$ is not empty since  $\\chi(Z')\\leq 6$ (the neighborhood of any vertex is a stable set).\n The goal is now to find two $vz$-paths $P$ and $P'$ of different parities with interior in $G[N_0 \\cup \\ldots \\cup \\{ x', y' \\} \\cup \\{x,y\\}]$. Then we can set $G_1=G[P\\cup P']$, $P_1=G[\\{z,z_1\\}]$ and $H=G[M_1]$ as parts of the wanted PCP. In practice, we need to be a little more careful to ensure the condition on the length of $P_1$ and the non-adjacency between $z$ and $H$, which is described after finding such a $P$ and a $P'$.\n\nLet $P_0$ (resp. $P'_0$) be a $vx'$-path (resp. $vy'$-path) of length $k-1$ (with exactly one vertex in each level).\nBy definition of $Z$, $z$ is connected to $\\{x',y',x,y\\}$.\n\\begin{enumerate}\n\\item (see Figure \\ref{fig: build PCP case 1}) If $z$ is connected to $x$ or $y$, say $x$, then $z$ is connected neither to $x'$ nor to $y$, otherwise it creates a triangle. We add the path $x'xz$ to $P_0$ to form $P$. Similarly, we add either the edge $y'z$ if it exists, or else the path $y' y x z$ to $P'_0$ to form $P'$. Observe that $P'$ is indeed an induced path since there is no triangle.\n\\item (see Figure \\ref{fig: build PCP case 2}) Otherwise, $z$ is connected neither to $x$ nor to $y$, thus $z$ is connected to exactly one of $x'$ and $y'$, since otherwise it would either create a triangle $x', y', z$ or a 5-hole $z x' x y y'$, so say $zx'\\in E$ and $zy'\\notin E$. We add the edge $x'z$ to $P_0$ to form $P$. We add the path $y' x' z$ if $y'x'\\in E$, otherwise add the path $y' y x x' z$ to $P'_0$ to form $P'$. Observe that this is an induced path since $G$ has no triangle and no 5-hole.\n\\end{enumerate}\nNow come the fine tuning.\nChoose in fact $z_1\\in M_1\\cap N(z)$ so that $z_1$ is connected to a major connected component $M_2$ of $M_1 \\setminus N(z)$. Choose $z_2$ a neighbor of $z_1$ in $M_2$ such that $z_2$ is connected to a major connected component $M_3$ of $M_2\\setminus N(z_1)$. We redefine $H=G[\\{z_2 \\cup M_3\\}]$ and $P_1=G[\\{z,z_1,z_2\\}]$. Then $P_1$ is a path of length 2, $G_1$ is colorable with 4 colors as the union of two induced paths, and $H$ is connected. Moreover $H$ has chromatic number at least $\\chi(N'_k)-\\chi(Z')-\\chi(N(z))-\\chi(N(z_1))$. Since the neighborhood of any vertex is a stable set, $\\chi(Z')\\leq 6$ and $\\chi(N(z)), \\chi(N(z_1))\\leq 1$. Thus $\\chi(H)\\geq \\delta\/2- 8$.\n\n\\end{proof}\n\n\\begin{figure}\n\\center\n\n\\subfigure[Overview of the situation\\label{fig: build PCP overview}]{\\includegraphics[scale=1, page=1]{fig\/buildPCP}}\n\\subfigure[Case (i)\\label{fig: build PCP case 1}]{\\includegraphics[scale=1, page=2]{fig\/buildPCP}}\n\n\\subfigure[Case (ii)\\label{fig: build PCP case 2}]{\\includegraphics[scale=1, page=3]{fig\/buildPCP}}\n\\caption{Illustrations for the proof of Lemma \\ref{lem: debut PCP sans C5}. Dashed edges stand for non-edges, and grey edges stand for edges that may or may not exist.}\n\\label{fig: build PCP}\n\\end{figure}\n\nWe can iterate the previous process to grow some longer PCP. In the following, for a function $f$ and an integer $k$, $f^{(k)}$ denotes the $k$-th iterate of $f$, that is to say that $f^{(k)}(x)=\\underbrace{(f \\circ \\ldots \\circ f)}_{k \\text{ times}} (x)$.\n\n\\begin{lemma}\n\\label{lem: existence PCP sans C5}\nLet $h(x)=x\/2-8$ be the function defined in Lemma \\ref{lem: debut PCP sans C5}. For every positive integers $\\ell, \\delta\\in \\mathbb{Z}_+$, if $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ is connected and satisfies $\\chi(G)\\geq \\delta$ and $h^{(\\ell-1)}(\\delta)\\geq 18$, then from any vertex $x_1$ of $G$, one can grow a PCP of order $\\ell$ with leftovers at least $h^{(\\ell)}(\\delta)$.\n\\end{lemma}\n\n\\begin{proof}\nWe prove the result by induction on $\\ell$. For $\\ell=1$, the result follows directly from Lemma \\ref{lem: debut PCP sans C5}. Now suppose it is true for $\\ell-1$, and let $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be such that $\\chi(G)\\geq \\delta$ and $h^{(\\ell-1)}(\\delta)\\geq 18$. Then $\\delta\\geq h^{(\\ell-1)}(\\delta)\\geq 18$, so we can apply Lemma \\ref{lem: debut PCP sans C5} to get a PCP of order 1 and leftovers at least $h(\\delta)$ from any vertex $x_1$. Let $x_2$ be the common vertex between the last block $H$ and the first path $P_1$ of the PCP (as in the definition). Now apply the induction hypothesis to $H$, knowing that $H$ is connected, $\\chi(H)\\geq h(\\delta)=\\delta'$ and $h^{(\\ell-2)}(\\delta')\\geq 18$. Then we obtain a PCP of order $\\ell-1$ with origin $x_2$ and leftovers at least $h^{(\\ell-2)}(\\delta')$, which finishes the proof by gluing the two PCP together.\n\\end{proof}\n\n\n\nNow we grow the PCP in a level $N_k$ of high chromatic number, and we want the PCP to be rooted (\\emph{i.e.} there \nexists a root $u'\\in N_{k-1}$ that is adjacent to the origin $u$ of the PCP, but to no other vertex of the PCP). This is step (ii).\n\n\n\\begin{lemma}\n\\label{lem: rooted PCP sans C5}\nLet $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be a connected graph, $v\\in V(G)$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $h$ be the function defined in Lemma \\ref{lem: debut PCP sans C5}. For every $k, \\delta$ such that $\\chi(N_k)\\geq \\delta+1$ and $h^{(\\ell-1)}(\\delta)\\geq 18$, there exists a rooted PCP of order $\\ell$ in $N_k$ with leftovers at least $h^{(\\ell)}(\\delta)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $N'_k$ be a major connected component of $N_k$ and $u\\in N'_k$. Consider a neighbor $u'$ of $u$ in $N_{k-1}$. Since there is no triangle, $N'_k\\setminus N(u')$ still has big chromatic number (at least $\\delta$), and let $N''_k$ be a major connected component of $N'_k\\setminus N(u')$. Let $z$ be a vertex of $N(u)\\cap N'_k$ having a neighbor in $N''_k$. Then we apply Lemma \\ref{lem: existence PCP sans C5} in $\\{z\\}\\cup N''_k$ to grow a PCP of order $\\ell$ from $z$ with leftovers at least $h^{(\\ell)}(\\delta)$. Now $u'$ has an only neighbor $z$ on the PCP, which is the origin.\n\\end{proof} \n \nLet us observe the properties of such a rooted PCP. We start with step (iii):\n\n\\begin{lemma} \\label{obs: sommets riches sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP $(G_1, P_1, \\ldots, G_\\ell, P_\\ell, H)$ of order $\\ell$ in a level $N_k$ for some $k$. If $x'\\in N_{k-1}$ has a neighbor $x$ in $H$, then $x$ has a neighbor in every $G_i$ for $1\\leq i \\leq \\ell$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $u$ be the origin of the PCP and $u'$ its root. Since $x'\\neq u'$ by definition of the root, there exists an upper $x'u'$-path $P_{up}$ of length at least one. \nConsider a $ux$-path $P$ inside the PCP. Let $v_1, \\ldots, v_r$ be the neighbors of $x'$ on this path, different from $x$ (if any), in this order (from $u$ to $x$).\nNow we can show that any regular block $G_i$ contains at least one $v_j$: suppose not for some index $i$, let $j$ be the greatest index such that $v_j$ is \\emph{before} $x_i$, \n\\emph{i.e.} $v_j\\in G_1\\cup P_1 \\cup \\cdots \\cup G_{i-1} \\cup P_{i-1}$.\n\nIf such an index does not exist (\\emph{i.e.} all the $v_j$ are after $G_i$), then there is an odd and an even path from $u$ to $v_1$ of length at least 3 by definition of a regular block,\nand this path does not contain any neighbor of $x'$. Close them to build two induced cycles by going through \n$x'$, $P_{up}$ and $u'$: one of them is an even cycle, and its length is at least 6.\n\nIf $j=r$ (\\emph{i.e.} all the $v_j$ are before $G_i$), then we can use the same argument with a path of well-chosen parity from $v_r$ to $x$, crossing $G_i$. \n\n\nOtherwise, there is an odd and an even path in the PCP between $v_j$ and $v_{j+1}$, crossing $G_i$, and its length is at least 4 because $x_i$ and $y_i$ are at distance at least 2 one from each other. We can close the even path path by going back and forth to $x$: this gives an even hole of length at least 6.\n\\end{proof}\n\nNote that, in the lemma above, $G$ is taken in $\\mathcal{C}_{3, 2k\\geq 6}$ and not in $\\mathcal{C}_{3, 5, 2k\\geq 6}$. In particular, we will use Lemma \\ref{obs: sommets riches sans C5} in the next section as well. Let us now continue with step (iv):\n \n\\begin{lemma} \\label{lem: shadow stable bornee sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. \nLet $S\\subseteq N_{k-1}$ be a stable set. Then $\\chi(N(S)\\cap N_k)\\leq 52$.\n\n\\end{lemma}\n\n\\begin{proof}\nLet $\\delta=\\chi(N(S)\\cap N_{k-1})-1$. Suppose by contradiction that $\\delta\\geq 52 $, then $h(\\delta)\\geq 18$ hence by Lemma \\ref{lem: rooted PCP sans C5}, we can grow a rooted PCP of order 2 inside $N(S)\\cap N_k$. Let $u$ be the origin of the PCP and $u'$ its root. Observe in particular that $S$ dominates $G_2$. Let $xy$ be an edge of $G_2$, and let $x'$ (resp. $y'$) be a neighbor of $x$ (resp. $y$) in $S$. \nBy Lemma \\ref{obs: sommets riches sans C5}, both $x'$ and $y'$ have a neighbor in $G_1$.\nThis gives an $x'y'$-path $P_{down}$ with interior in $G_1$. In order not to create an even hole nor a 5-hole by closing it with $x' x y y'$, we can ensure that $P_{down}$ is an even path of length at least 4. Moreover, there exists an upper $x'y'$-path $P_{up}$. Then either the hole formed by the concatenation of $P_{up}$ and $x' x y y'$, or the one formed by the concatenation of $P_{up}$ and $P_{down}$ is an even hole of length $\\geq 6$, a contradiction.\n\\end{proof}\n\nThe previous lemma allows us to prove that one can \\emph{lift} the PCP up into $N_{k-1}$ to get a subset of vertices with high chromatic number. We state a lemma that will be reused in the next section:\n\n\n\n\\begin{lemma}\n\\label{lem: active lift gros si neighb stable gros}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP of order $\\ell$ in a level $N_k$ (for some $k\\geq 2$) with leftovers at least $\\delta$. Let $A=N(G_\\ell)\\cap N_{k-1}$ (called the active lift of the PCP).\nSuppose that for every stable set $S\\subseteq A$, we have $\\chi(N(S)\\cap N_{k})\\leq \\gamma$,\n then $\\chi(A)\\geq {\\delta}\/{\\gamma}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $r=\\chi(A)$, suppose by contradiction that $r<\\delta\/\\gamma$ and decompose $A$ into $r$ stable sets $S_1, \\ldots, S_r$. Then $N(A)\\cap N_k$ is the (non-necessarily disjoint) union of $r$ sets $N(S_1)\\cap N_k, \\ldots, N(S_r)\\cap N_k$, and each of them has chromatic number at most $\\gamma$ by assumption. Consequently $\\chi(N(A)\\cap N_k)\\leq r \\gamma < \\delta$ and hence $\\chi(H\\setminus N(A)) \\geq \\chi(H) - \\chi(N(A)\\cap N_k) \\geq 1$.\nLet $x$ be any vertex of $H\\setminus N(A)$ and $x'$ be a neighbor of $x$ in $N_{k-1}$. By construction, $x'\\notin A$ so $x'$ has no neighbor in $G_\\ell$. This is a contradiction with Lemma \\ref{obs: sommets riches sans C5}.\n\\end{proof}\n\nBy Lemmas \\ref{lem: shadow stable bornee sans C5} and \\ref{lem: active lift gros si neighb stable gros} with $\\gamma=52$, we can directly deduce the following:\n\n\\begin{lemma} \\label{lem: shadow borne bornee sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP of order $\\ell$ in a level $N_k$ (for some $k\\geq 2$) with leftovers at least $\\delta$. Let $A=N(G_\\ell)\\cap N_{k-1}$ be the active lift of the PCP, then we have $\\chi(A)\\geq g(\\delta)={\\delta}\/{52}$.\n\\end{lemma}\n\n\n\n\nWe can now finish the proof, this is step (v). Recall that a sketch was provided, and it may help to understand the following proof.\n\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem: sans C5}] \nLet $c'$ be a constant big enough so that \n$$g\\left(h^{(2)}\\left(g\\left(h^{(2)}\\left(\\frac{c'}{2}-1\\right)\\right)-1\\right)\\right)\\geq 5 \\ .$$\nSuppose that $\\chi(G)\\geq c'$. Pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level, so $\\chi(N_k)\\geq \\chi(G)\/2\\geq c_1+1$ where $c_1=c'\/2-1$. By Lemma \\ref{lem: rooted PCP sans C5}, grow a rooted PCP  $P=(G_1, P_1, G_2, P_2, H)$ inside $N_k$ of order 2 with leftovers at least $c_2=h^{(2)}(c_1)$. Then apply Lemma \\ref{lem: shadow borne bornee sans C5} and get an active lift $A$ of $P$ inside $N_{k-1}$ with chromatic number at least $c_3=g(c_2)$. Since $h(c_3 -1)\\geq 18$, apply again Lemma \\ref{lem: rooted PCP sans C5} to get a rooted PCP $P'=(G'_1, P'1, G'_2, P'_2, H')$ of order 2 inside $N_{k-1}$ with leftovers at least $c_4=h^{(2)}(c_3 -1)$. Now apply Lemma \\ref{lem: shadow borne bornee sans C5} to get an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number at least $c_5=g(c_4)$.\n\nBecause of the chromatic restriction in the definition of the PCP, one can color $G'_2$ with 4 colors. Moreover, $G'_2$ dominates $A'$ by definition. Thus there exists a stable set $S\\subseteq G'_2$ such that $\\chi(N(S)\\cap A')\\geq c_6=c_5\/4$ (since $A'$ is the union of the $N(S')\\cap A'$ for the four stable sets $S'$ that partition $G_2'$).\n\nNow $c_6>1$ so there is an edge $xy$ inside $N(S) \\cap A'$. Call $x'$ (resp. $y'$) a vertex of $S$ dominating $x$ (resp. $y$). Both $x'$ and $y'$ have a neighbor in $G_2$ by definition of $A$ and, by Lemma \\ref{obs: sommets riches sans C5}, both $x'$ and $y'$ also have a neighbor in $G_1$.  This gives an $x'y'$-path $P_1$ (resp. $P_2$) with interior in $G_1$ (resp. $G_2$). Due to the path $x'x y y'$ of length 3, $P_1$ and $P_2$ must be even paths of length at least 4. Thus the concatenation of $P_1$ and $P_2$ is an even hole of length at least 6, a contradiction.\n\\end{proof}\n\n\\section{General case}\n\\label{sec: Yes C4 avec C5}\n\nThis section aims at proving Theorem \\ref{th: C chi borne}, using the result of the previous section. As already mentioned, we follow the same outline, except that we now need the existence of a $C_5$ several times. Let us start by a technical lemma to find both an even and an odd path out of a 5-hole and its dominating set:\n\n\n\n\\begin{lemma}\n\\label{lem: C5 chemin pair chemin impair}\nLet $G$ be a triangle-free graph inducing a 5-hole $C$. Let $S\\subseteq V(G)$ be a minimal dominating set of $C$, assumed to be disjoint from $C$. If we delete the edges with both endpoints in $S$, then for every vertex $t\\in S$, there exists a vertex $t'\\in S$ such that \none can find an induced $tt'$-path of length 4 and an induced $tt'$-path of length 3 or 5, both with interior in $C$.\n\\end{lemma}\n\n\\begin{proof}\nLet $t\\in S$, call $v_1$ a neighbor of $t$ on the cycle and number the others vertices of $C$ with $v_2, \\ldots , v_5$ (following the adjacency on the cycle). Since $G$ is triangle-free, $t$ can not be adjacent to both $v_3$ and $v_4$, so up to relabeling the cycle in the other direction we assume that $t$ is not adjacent to $v_3$. Let $t'\\in S$ be a vertex dominating $v_3$. Then $tv_1v_2v_3t'$ is an induced path of length 4 between $t$ and $t'$. Moreover, $tv_1v_5v_4t'$ is a (non-necessarily induced) path of length 5 between $t$ and $t'$. If this path is not induced, the only possible chords are $tv_4$ and $t'v_5$ since $G$ is triangle-free, which in any case gives an induced $tt'$-path of length 3.\n\\end{proof}\n\n\n\n\nRecall that in this section, we are interesting in strong PCP, \\emph{i.e.} PCP, all regular blocks $G_i$ of which contain an induced $C_5$. We start with step (i):\n\n\n\\begin{lemma} \\label{lem: debut PCP}\nLet $c'$ be the constant of Lemma \\ref{lem: sans C5}, let $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ and $v$ be any vertex of $G$. For every $\\delta \\in \\mathbb{N}$ such that $\\chi(G)\\geq \\delta \\geq 2c'$, there exists a strong PCP of order 1 with origin $v$ and leftovers at least $f(\\delta)=\\delta\/2-15$.\n\n\\end{lemma}\n\n\\begin{proof}\n\n\nLet $(N_0, N_1, \\ldots)$ be the $v$-levelling, $N_k$ be a colorful level and let $N'_k$ be a major connected component of $G[N_k]$, so $\\chi(N'_k)\\geq c'$. Using Lemma \\ref{lem: sans C5}, there exists a 5-hole $C$ in $G[N'_k]$. Consider a minimum dominating set $D$ of $C$ inside $N_{k-1}$. \n\n\nFrom now on, the proof is very similar to the one of Lemma \\ref{lem: debut PCP sans C5}. Similarly, we define $Z'=N(D\\cup C)\\cap N'_k$ and $Z=Z' \\setminus C$. Let $z\\in Z$ be a vertex having a neighbor $z_1$ in a major connected component $M_1$ in $N'_k \\setminus Z'$. The goal is now to find two $vz$-paths $P$ and $P'$ of different parity with interior in $N_0 \\cup \\ldots \\cup D \\cup C$, then we can set $G_1=G[P\\cup P'\\cup C]$, $P_1=G[\\{z,z_1\\}]$ and $H=G[M_1]$ as parts of the wanted PCP. In practice, we need to be a little more careful to ensure the condition on the length of $P_1$ and the non-adjacency between $z$ and $H$.\n\n\nLet us now find those two paths $P$ and $P'$. By definition of $Z$, $z$ also has a neighbor in $D$ or in $C$. \n\\begin{enumerate}\n\\item If $z$ has a neighbor $x\\in C$, let $y\\in C$ be a vertex adjacent to $x$ on the hole. Let $x'$ and $y'$ be respectively a neighbor of $x$ and a neighbor of $y$ in $D$. Observe that $z$ is connected neither to $x'$ nor to $y$, otherwise it creates a triangle. We grow $P$ by starting from an induced path of length $k-1$ from $v$ to $x'$ and then add the path $x'x z$. Similarly, we grow $P'$ by starting from an induced path of length $k-1$ from $v$ to $y'$, and then add the edge $y'z$ if it exists, or else the path $y' y x z$. Observe that $P'$ is indeed an induced path since there is no triangle.\n\\item If $z$ has no neighbor in $C$, then it has at least one neighbor $x'$ in $D$. Apply Lemma \\ref{lem: C5 chemin pair chemin impair} to get a vertex $y'\\in D$ such that there exists an $x'y'$-path of length 3 or 5, and another one of length 4, both with interior in $C$. Observe that \n$x'$ and $y'$ cannot have a common neighbor $u$ in $N_{k-2}\\cup \\{z\\}$, otherwise there would be either a triangle $x', u, y'$ (if $x'y'\\in E$), or a $C_6$ using the \\mbox{$x'y'$-path} of length 4 with interior in $C$. Now we grow $P$ by starting from an induced path of length $k-1$ from $v$ to $x'$, and add the edge $x'z$. We grow $P'$ by starting from an induced path of length $k-1$ from $v$ to $y'$, and then add the edge $x'y'$ if it exists, otherwise add the $x'y'$-path of length 3 or 5 with interior in $C$, and then finish with the edge $x'z$.\n\\end{enumerate}\n\n\nNow come the fine tuning.\nChoose in fact $z_1\\in M_1\\cap N(z)$ so that $z_1$ is connected to a major connected component $M_2$ of $M_1 \\setminus N(z)$. Choose $z_2$ a neighbor of $z_1$ in $M_2$ such that $z_2$ is connected to a major connected component $M_3$ of $M_2\\setminus N(z_1)$. We redefine $H=G[\\{z_2 \\cup M_3\\}]$ and $P_1=G[\\{z,z_1,z_2\\}]$. Then $P_1$ is a path of length 2, $H$ is connected and $G_1$ is colorable with 4 colors (it is easily 7-colorable as the union of a 5-hole and two paths; a careful case analysis shows that it is 4-colorable). Moreover $H$ has chromatic number at least $\\chi(N'_k)-\\chi(Z')-\\chi(N(z))-\\chi(N(z_1))$. Since the neighborhood of any vertex is a stable set, $\\chi(Z')\\leq |D|+|C| +\\chi(C)\\leq 13$ and $\\chi(N(z)), \\chi(N(z_1))\\leq 1$. Thus $\\chi(H)\\geq \\delta\/2- 15$.\n\n\\end{proof}\n\nWe go on with step (ii): find a strong rooted PCP.\nThe following lemma is proved in the same way as Lemma \\ref{lem: rooted PCP sans C5} by replacing the use of Lemma \\ref{lem: existence PCP sans C5} by Lemma \\ref{lem: debut PCP}, so we omit the proof here.\n\n\\begin{lemma}\n\\label{lem: rooted PCP}\nLet $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ be a connected graph, $f$ be the function defined in Lemma \\ref{lem: debut PCP}, $v$ be a vertex of $G$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. For every $k, \\delta$ such that $\\chi(N_k)\\geq \\delta+1\\geq 2c'+1$, there exists a strong rooted PCP of order 1 in $N_k$ with leftovers at least $f(\\delta)$.\n\\end{lemma}\n\nStep (iii) is proved by Lemma \\ref{obs: sommets riches sans C5} from the previous section, and was valid not only for $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ but also for $G\\in \\mathcal{C}_{3, 2k\\geq 6}$. So we continue with step (iv):\n\n\\begin{lemma} \\label{lem: shadow stable bornee}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $(N_0, N_1, \\ldots)$ be the $v$-levelling. \nLet $S$ be a stable set inside $N_{k-1}$. Then $\\chi(N(S)\\cap N_k)\\leq 2c'$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose by contradiction that $\\chi(N(S)\\cap N_k)\\geq 2c'+1$. By Lemma \\ref{lem: rooted PCP}, we can grow in $N(S)\\cap N_k$ a rooted PCP of order 1, and in particular $S$ dominates $G_1$.\nBy definition of a strong PCP, there is a 5-hole $C$ in $G_1$. Since $S$ is a dominating set of $C$, we can apply Lemma \\ref{lem: C5 chemin pair chemin impair} to get two vertices $t, t'\\in S$ such that one can find both an even and an odd $tt'$-path with interior in $C$ and length at least 3. Then any upper $tt'$-path close a hole of even length $\\geq 6$.\n\\end{proof}\n\nIn fact, as in previous section, we can directly deduce from Lemmas \\ref{lem: active lift gros si neighb stable gros} and \\ref{lem: shadow stable bornee} that one can lift the PCP up into $N_{k-1}$ to get a subset of vertices with high chromatic number:\n\n\n\n\\begin{lemma} \\label{lem: shadow borne bornee}\nLet $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $v\\in V(G)$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a strong rooted PCP of order $1$ in a level $N_k$ (for some $k\\geq 2$) with leftovers $\\delta$. Let $A=N(G_1)\\cap N_{k-1}$ be the active lift of the PCP. If $\\delta \\geq 2c'$, then $\\chi(A)\\geq \\varphi(\\delta)=\\frac{\\delta}{2c'}$.\n\\end{lemma}\n\n\n\n\nWe are now ready to finish the proof, this is step (v). Recall that a sketch was given and may be useful to have a less technical overview of the proof.\n\n\\begin{proof}[Proof of Theorem \\ref{th: C chi borne}] \nLet $c$ be a constant such that \n$$\\varphi\\left(f\\left(\\varphi\\left(f\\left(\\frac{c}{2} -1\\right)\\right)-1\\right)\\right)\\geq 4c' \\ .$$ \nSuppose that $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ has chromatic number $\\chi(G)\\geq c$. Then pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level, consequently $\\chi(N_k)\\geq c_1+1=c\/2$.\nApply Lemma \\ref{lem: rooted PCP} and grow inside $N_k$ a strong rooted PCP $P=(G_1, P_1, H)$ of order 1 with leftovers at least $c_2=f(c_1)$. Then apply Lemma~\\ref{lem: shadow borne bornee} and get an active lift $A=N(G_1)$ of $P$ inside $N_{k-1}$ with chromatic number at least $c_3=\\varphi(c_2)$. By Lemma \\ref{lem: rooted PCP}, we can obtain a strong rooted PCP $P'=(G'_1, P'_1, H')$ inside $A$ with leftovers at least $c_4=f(c_3-1)$, and by Lemma~\\ref{lem: shadow borne bornee} we obtain an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number at least $c_5=\\varphi(c_4)$. Because of the chromatic restriction in the definition of the PCP, one can color $G'_1$ with 4 colors. Moreover, $G'_1$ dominates $A'$ by definition. Thus there exists a stable set $S\\subseteq P'$ such that $\\chi(N(S)\\cap A')\\geq c_6=c_5\/4$.\nNow $c_6\\geq c'$ thus Lemma \\ref{lem: sans C5} proves the existence of a 5-hole $C$ inside $N(S) \\cap A'$. Let us give an overview of the situation: we have a 5-hole $C$ inside $N_{k-2}$, dominated by a stable set $S$ inside $N_{k-1}$, and every pair of vertices $t,t'$ of $S$ can be linked by a $tt'$-path $P_{down}$ with interior in $G_1\\subseteq N_k$. Lemma \\ref{lem: C5 chemin pair chemin impair} gives the existence of two vertices $t, t'\\in S$ linked by both an odd path and an even path of length $\\geq 3$ with interior in $C$. Closing one of these paths with $P_{down}$ gives an induced even hole of length $\\geq 6$, a contradiction.\n\\end{proof}\n\n\n\n\\section*{Acknowledgment}\n\nThe author would like to sincerely thank St\\'ephan \\textsc{Thomass\\'e} for bringing this problem to her knowledge and for useful discussions about Trinity\/Parity Changing Paths.\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGiven a spin manifold $M$ of dimension $n$, we consider the universal spinor bundle $\\Sigma M$. This is the bundle whose sections consist of pairs of metrics $g \\in \\Gamma(\\odot^2_+ T^*M)$ and spinor fields $\\varphi \\in \\Gamma(\\Sigma_g M)$.\nWe denote by $\\mathcal{N}$ the set\n$$\\{ (g, \\varphi) \\in \\Gamma( \\Sigma M) : |\\varphi| = 1\\}$$\nand define the spinorial energy functional\n$$\\mathcal{E}: \\mathcal{N} \\to \\mathbb R$$\n$$\\mathcal{E}(g, \\varphi) = \\frac{1}{2}\\int_M |\\nabla^g \\varphi|^2 \\operatorname{vol}_g.$$\nIf the dimension of $M$ is at least three, the only critical points of $\\mathcal{E}$ are absolute minimizers. This implies $\\nabla^g \\varphi = 0$ for a critical point $(g, \\varphi)$, i.e. $\\varphi$ is a parallel spinor with respect to the metric $g$. Existence of parallel spinors is a strong constraint on the metric $g$. Indeed, such a metric is necessarily Ricci flat and of special holonomy. Conversely, Ricci flat manifolds with special holonomy admit a parallel spinor.  Given that manifolds with such metrics are difficult to construct, it is natural to consider the negative gradient flow\n$$\\partial_t \\Phi_t = Q(\\Phi_t)$$\nof $\\mathcal{E}$ to find such a metric. Here $Q: \\mathcal{N} \\to T \\mathcal{N}$ is the negative gradient of $\\mathcal{E}$ with respect to the natural $L^2$ metric on $\\mathcal{N}$. It turns out that $Q$ is weakly elliptic and has negative symbol.\nThe spinorial energy and the associated negative gradient flow, called spinor flow, were first examined in \\cite{Ammann2015}. There, short time existence of this flow on closed manifolds was established.\nFrom here on we assume $M$ to be a closed manifold and $\\dim M = n \\geq 3$.\nWe will prove that critical points of $\\mathcal{E}$, i.e. pairs of Ricci flat metrics and parallel spinor fields, are stable with respect to the spinor flow, that is:\n\\begin{thm}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a critical point of $\\mathcal{E}$ and suppose $\\bar{g}$ has no Killing fields. Then there exists a $C^{\\infty}$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi$ smoothly converges to a critical point. In any $C^k$ norm the speed of convergence is exponential.\n\\end{thm}\nA Ricci flat manifold is up to a finite covering a product of irreducible Ricci-flat manifolds and a flat torus. Hence the condition on the Killing fields can also be read as saying that $\\bar{g}$ has no torus factor. This can for instance be ruled out by the topological condition that the fundamental group of $M$ be finite.\nThe strategy of the proof will be roughly as follows: first, we establish a \u0141ojasiewicz-Simon type inequality for the spinorial energy. This inequality implies exponential decay of the energy along the flow. We will then show that this implies convergence to a critical point. The inequality depends in its optimal form on the fact that the critical set of $\\mathcal{E}$ is smooth. This was shown in \\cite{Ammann2015b}.\n\nWe will also consider the stability of volume constrained critical points. A section $\\Phi \\in \\mathcal{N}$ is a {\\em volume constrained critical point}, if\n$$\\frac{d}{dt}\\Bigr|_{t=0} \\mathcal{E}(\\Phi_t) = 0$$\nfor all volume preserving variations $\\Phi_t$ of $\\Phi$. Such a critical point evolves under the spinor flow by rescaling. A {\\em volume constrained minimizer} $\\Phi = (g, \\varphi)$ is a volume constrained critical point, such that for any $\\Psi \\in \\mathcal{N}$ close to $\\Phi$ we have $\\mathcal{E}(\\Phi) \\leq \\mathcal{E}(\\Psi)$, provided the metrics induced by $\\Phi$ and $\\Psi$ have equal volume.\n\\begin{thm}\nLet $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ be a volume constrained minimizer of $\\mathcal{E}$. Suppose that the critical set near $\\bar{\\Phi}$ is a manifold and suppose $\\bar{g}$ has no Killing fields. Then there exists a $C^{\\infty}$ neighborhood $U$ of $\\bar{\\Phi}$, such that the volume normalized spinor flow converges smoothly to a volume constrained minimizer, if the initial condition is in $U$. The convergence speed in $C^k$ is exponential. \n\\end{thm}\nThe strategy for the proof is essentially the same as in the case of critical points. However, here both the condition on Killing fields and the assumption that the critical set near $\\bar{\\Phi}$ is a manifold are strong restrictions. Indeed, suppose $(g, \\varphi)$ is such that\n$$\\nabla^g_X \\varphi = \\lambda X \\cdot \\varphi \\text{ for all } X \\in \\Gamma(TM)$$\nwith $\\lambda \\in \\mathbb R$.\nThen $(g,\\varphi)$ is a volume constrained critical point. The spinor $\\varphi$ is called a Killing spinor.\nIf $g$ carries a Killing spinor, then the cone $((0,\\infty)\\times M, dr^2 + r^2g)$ carries a parallel spinor.\nA large class of metrics with Killing spinors is supplied by Sasaki--Einstein manifolds.\nSince the Reeb vector field is a Killing vector field, all Sasaki--Einstein manifolds carry Killing fields.\nFurthermore, the moduli space of Sasaki--Einstein manifolds is not known to be smooth in general.\nWhat's more, in contrast to the space of parallel spinors, whose dimension is locally constant under Ricci flat deformations of the metric, the dimension of the space of killing spinors can jump under Einstein deformations of the metric. Indeed, a 3-Sasakian manifold admits three linearly independent Killing spinors. Van Coevering found that a toric 3-Sasakian manifold has Einstein deformations $g_t$, such that the space of Killing spinors is two-dimensional for any $t\\neq 0$.\n\nSince the spinor flow is a generalization of the heat flow for $G_2$-structures introduced in \\cite{Weiss2012}, our result is a generalization of the stability result proven there. However, the arguments of our proof are closer in spirit to the proofs in \\cite{Haslhofer2011}, \\cite{Haslhofer2014}, \\cite{Krncke2014}, where stability of Ricci-flat and Einstein metrics with respect to the Ricci flow is shown.\n\n\\section*{Acknowledgements}\nThe author thanks Hartmut Wei\u00df for posing the problem and numerous discussions related to it.\n\n\n\\section{The universal spinor bundle and the spinor flow}\nFor convenience and completeness, we recall the precise definitions of the spinor flow as well as results on short time existence of the flow. Details may be found in \\cite{Ammann2015}. \nWe defined the spinor energy to be a functional on sections of the universal spinor bundle. We will now construct this universal spinor bundle.\nBefore we do this, let us first recall the ordinary spinor bundle on a spin manifold. \nThe orientation preserving component of the general linear group $\\operatorname{GL}_+(n)$ has fundamental group $\\mathbb Z_2$ and hence there exists a universal double covering group $\\widetilde{\\operatorname{GL}}_+(n)$ together with a covering map $\\xi: \\widetilde{\\operatorname{GL}}_+(n) \\to \\operatorname{GL}_+(n)$, which is also a homomorphism.\nLet $M$ be a spin manifold of dimension $n$. By this we mean a manifold $M$ and a $\\widetilde{\\operatorname{GL}}_+(n)$ principal bundle $\\tilde{P}$ which covers the $\\operatorname{GL}_+(n)$ frame bundle $P$, $\\pi: \\tilde{P} \\to P$, so that for any $g \\in \\widetilde{\\operatorname{GL}}_+(n), p \\in \\tilde{P}$ we have\n$$\\pi(p \\cdot g) = \\pi(p) \\cdot \\xi(g).$$\nNow let $g$ be a metric on $M$. The metric induces a reduction of the structure group of $P$ to the oriented orthonormal frame bundle $P_{\\operatorname{SO}(n)}$. The group $\\operatorname{Spin}(n) = \\xi^{-1}(\\operatorname{SO}(n))$ is called the spin group. Thus the structure group of $\\tilde{P}$ reduces to $\\operatorname{Spin}(n)$ and we call this bundle $P_{\\operatorname{Spin}(n)}$, which double covers $P_{\\operatorname{SO}(n)}$.\nNow we define the (complex) spinor bundle as the associated vector bundle\n$$\\Sigma_g M = P_{\\operatorname{Spin}(n)} \\times_{\\Delta_n} \\Sigma_n$$\nwhere $\\Delta_n : \\operatorname{Spin}(n) \\to \\operatorname{End}(\\Sigma_n)$, $\\Sigma_n = \\mathbb C^{2^{[n\/2]}}$, is the standard complex spin representation. Up to scaling, there exists one $\\operatorname{Spin}(n)$ invariant Hermitian product on $\\Sigma_n$. This turns $\\Sigma_g M$ into a Hermitian bundle.\nThe universal spinor bundle gives us a way to compare spinors over different metrics.\nRecalling that\n$$\\faktor{\\operatorname{GL}_+(n)}{\\operatorname{SO}(n)} \\cong \\odot^2_+ \\mathbb R^n,$$\nwe conclude\n$$\\odot^2_+ T^*M = P \\times_{\\operatorname{GL}_+(n)} \\faktor{\\operatorname{GL}_+(n)}{\\operatorname{SO}(n)} = \\tilde{P} \\times_{\\widetilde{\\operatorname{GL}}_+(n)} \\faktor{\\widetilde{\\operatorname{GL}}_+(n)}{\\operatorname{Spin}(n)} = \\faktor{\\tilde{P}}{\\operatorname{Spin}(n)}.$$\nWe define\n$$\\Sigma M = \\tilde{P} \\times_{\\Delta_n} \\Sigma_n.$$\nThis is a vector bundle over $\\odot^2_+ T^*M$, i.e. we have the structure of two nested fibrations:\n$$\\Sigma M \\xrightarrow{\\pi_{\\Sigma}} \\odot^2_+ T^*M \\xrightarrow{\\pi_{\\mathcal{M}}} M.$$\nGiven a metric $g$ we can identify $\\pi_{\\Sigma}^{-1}(g)$ and $\\Sigma_g M$. Using this identification any element $\\Phi \\in\\Sigma M$ can be considered as a pair of a metric $g_{\\Phi} = \\pi_{\\Sigma}(\\Phi)$ and a spinor $\\varphi_{\\Phi} \\in \\Sigma_{g_{\\Phi}} M$.\nAs above we also get a Hermitian inner product $h$ on $\\Sigma M$.\nWe denote by $\\langle \\cdot, \\cdot \\rangle = \\operatorname{Re} h$ the real part of $h$ and $| \\cdot |$ the associated norm.\nNow the definition\n$$\\mathcal{N} = \\{\\Phi \\in \\Gamma(\\Sigma M) : |\\Phi| = 1\\}$$\nfrom the introduction is fully explained. To make sense of the gradient of $\\mathcal{E}$ we need to compute the tangent spaces of $\\mathcal{N}$. For this we need to compare spinors in different fibers $\\Sigma_{g_1} M$ and $\\Sigma_{g_2} M$. This can be done using the Bourgignon--Gauduchon connection.\n\nSuppose we have a vector space $V$ and two inner products $\\langle \\cdot, \\cdot \\rangle_1$ and $\\langle \\cdot, \\cdot \\rangle_2$. Then there exists a unique endomorphism $A^2_1 : V \\to V$, such that\n$$\\langle v, w \\rangle_2 = \\langle A_1^2 v, w \\rangle_1 \\text{ for all } v,w \\in V.$$\nDenote by $B_1^2$ the square root of $A_1^2$. The operator $B_1^2$ maps orthonormal bases of $(V, \\langle \\cdot, \\cdot \\rangle_1)$ to orthonormal bases of $(V, \\langle \\cdot, \\cdot \\rangle_2)$. Since no choices are involved and $B_1^2$ depends smoothly on the inner products, we can transfer this construction to Riemannian manifolds $(M,g_i)$, $i=1,2$, and consequently obtain a smooth principal bundle isomorphism\n$$P^{g_2}_{\\operatorname{SO}(n)} \\to P^{g_1}_{\\operatorname{SO}(n)}.$$\nThis map lifts to the spinor bundle and hence induces a isomorphism $\\hat{B}^{g_2}_{g_1}:\\Sigma_{g_2} M \\to \\Sigma_{g_1} M$.\nSince the metric on $\\Sigma_n$ is $\\operatorname{Spin}(n)$-invariant, this is an isometry.\nNotice that the restriction of $\\hat{B}_{g_1}^{g_2}$ to a fibre over a point $x \\in M$ only depends on the scalar products $g_1(x), g_2(x)$ on $T_xM$.\nWe now have a canonical isometry between two spinor bundles over the same manifold with two distinct metrics. From this we can derive a horizontal distribution\n$$\\mathcal{H}_{\\Phi} = \\left\\{ \\frac{d}{dt}\\Bigr|_{t=0} \\hat{B}^{g}_{g_t} \\varphi \\; \\Big| \\; g: (-\\epsilon, \\epsilon) \\to \\odot^2_+ T_x^*M, g(0) = g \\right\\} \\subset T_{\\Phi} \\Sigma M $$\nwhere $\\Phi = (g, \\varphi) \\in \\Sigma M_x$. By construction, $\\mathcal{H}_{\\Phi} \\cong \\odot^2 T_x^*M$.\nThis distribution yields a splitting of the tangent bundle\n$$T_{\\Phi} \\Sigma M_x = \\mathcal{H}_{\\Phi} \\oplus \\Sigma_g M  \\cong \\odot^2 T_x^*M \\oplus \\Sigma_g M.$$\nTurning to sections of the universal spinor bundle, this implies that for $\\Phi = (g, \\varphi) \\in \\Gamma (\\Sigma M)$ we have the splitting\n$$T_{\\Phi} \\Gamma(\\Sigma M) = \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma( \\Sigma_g M)$$\nand if $\\Phi \\in \\mathcal{N}$\n$$T_{\\Phi} \\mathcal{N} = \\Gamma(\\odot^2 T^* M) \\oplus \\Phi^{\\perp},$$\nwhere\n$$\\Phi^{\\perp} = \\{ \\psi \\in \\Gamma(\\Sigma_g M) : \\langle \\varphi, \\psi\\rangle \\equiv 0 \\}.$$\nNow we define for $\\Phi \\in \\Gamma (\\Sigma M)$ and $\\Psi_1, \\Psi_2 \\in T_{\\Phi} \\Gamma( \\Sigma M)$\n$$\\left( \\Psi_1, \\Psi_2 \\right)_{L^2} = \\int_M g(h_1, h_2) \\operatorname{vol}_g + \\int_M \\langle \\psi_1, \\psi_2 \\rangle_{\\Sigma_g M} \\operatorname{vol}_g$$\nwhere $(h_i, \\psi_i) \\in \\Gamma(\\odot^2_+ T^*M) \\oplus \\Gamma (\\Sigma_g M)$ are the sections corresponding to $\\Psi_i$ according to the isomorphisms above. From now on we will use these isomorphisms implicitly.\nNow the negative gradient\n$$Q: \\mathcal{N} \\to T\\mathcal{N}$$\nis defined by the property\n$$\\left(Q(\\Phi), \\Psi \\right) = - \\frac{d}{dt}\\Bigr|_{t=0} \\mathcal{E}(B^g_{g+th} (\\varphi + t \\psi)),$$\nwhere $\\Phi = (g, \\varphi) \\in \\mathcal{N}$ and $\\Psi = (h, \\psi) \\in T_{\\Phi} \\mathcal{N}$.\n\n\\section{Diffeomorphism invariance, the gauged spinor flow and a slice theorem}\nWe denote by $\\operatorname{Diff}_s(M)$ the group of spin diffeomorphisms, i.e. the orientation preserving diffeomorphisms of $M$, which lift to $\\tilde{P}$. To be more precise, by a lift of a orientation preserving diffeomorphism $f: M \\to M$ to $\\tilde{P}$, we mean a lift of the map\n$$ P_x \\ni [e_1, ..., e_n] \\mapsto [Df e_1,..., Df e_n] \\in P_{f(x)}$$\ninduced by $f$ on the oriented frame bundle $P$ to the topological spin bundle $\\tilde{P}$.\nSince $\\tilde{P}$ is a $\\mathbb Z_2$ bundle over $P$, there is a choice of lift and the group of lifts of spin diffeomorphisms $\\widehat{\\operatorname{Diff}}_s(M)$ fits into an exact sequence\n$$0 \\to \\mathbb Z_2 \\to \\widehat{\\operatorname{Diff}}_s(M) \\to \\operatorname{Diff}_s(M) \\to 0.$$\nThe group $\\widehat{\\operatorname{Diff}}_s(M)$ acts on $\\Gamma( \\Sigma M)$ in the following way. Let $F \\in \\widehat{\\operatorname{Diff}}_s(M)$, $\\Phi = (g, \\varphi) \\in \\Gamma(\\Sigma M)$. The map $F : \\tilde{P} \\to \\tilde{P}$ is a lift of a diffeomorphism $f: M \\to M$. Restricting $\\tilde{P}$ to $P_{\\operatorname{Spin}(n)}^g$ we obtain an isomorphism\n$$F : P_{\\operatorname{Spin}(n)}^g \\to P^{f_* g}_{\\operatorname{Spin}(n)}.$$\nThen we define locally\n$$F_*\\varphi = [F \\circ b \\circ f^{-1}, \\varphi \\circ f^{-1}] \\in \\Gamma(\\Sigma_{f_* g} M),$$\nif $\\varphi = [b, \\varphi]$, $b$ a local section of $P_{\\operatorname{Spin}(n)}^g$, $\\varphi$ a $\\Sigma_n$ field. The push forward preserves the metric in the following sense:\n$$|F_* \\varphi|_{f_* g} (x)  = |\\varphi|_g (f^{-1}(x)).$$\nIn particular $F_*$ preserves $\\mathcal{N}$. Moreover, we have\n$$\\mathcal{E}(F_* \\Phi) = \\mathcal{E}(\\Phi),$$\n$$Q(F_* \\Phi) = F_* Q(\\Phi).$$\nIn particular, the spinor flow is not strongly parabolic, since $Q$ is invariant under an infinite dimensional group. This invariance is reflected on the infinitesimal level by the following Bianchi-type identity\n$$\\lambda_{g,\\varphi} Q(g,\\varphi) = 0$$\nwhere\n$$\\lambda_{g, \\varphi} : \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M) \\to \\Gamma(TM)$$\nis defined as the formal adjoint of\n$$\\lambda_{g,\\varphi}^* : \\Gamma(TM) \\to \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$$\n$$X \\mapsto \\left(2 \\delta^*_g X^{\\flat}, \\nabla^g_X \\varphi - \\frac{1}{4} dX^{\\flat} \\cdot \\varphi\\right) = \\left( \\mathcal{L}_X g, \\tilde{\\mathcal{L}}_X \\varphi\\right) =: \\tilde{\\mathcal{L}}_X \\Phi.$$\nIndeed, the tangent space of the orbit $\\widehat{\\operatorname{Diff}}_s(M).(g,\\varphi)$ is the image of $\\lambda_{g,\\varphi}^*$.\nAt a critical point $(g,\\varphi) \\in \\Gamma( \\Sigma M)$, we get the following exact sequence\n$$0 \\to \\Gamma(TM) \\xrightarrow{\\lambda_{g, \\varphi}^*} \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\varphi^{\\perp}) \\xrightarrow{L_{g,\\varphi}} \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\varphi^{\\perp}) \\xrightarrow{\\lambda_{g, \\varphi}} \\Gamma(TM) \\to  0,$$\nwhere $L_{g,\\varphi} = DQ (g,\\varphi)$. It turns out that with\n$$X_{\\bar{g}} : \\Gamma(\\odot^2 T^*M) \\to \\Gamma(TM)$$\n$$g \\mapsto -2 (\\delta_{\\bar{g}} g)^{\\sharp}$$\nand $\\bar{g}$ any given metric the operator\n$$\\tilde{Q}_{\\bar{g}}(\\Phi) = Q(\\Phi) + \\lambda^*_{g,\\varphi}(X_{\\bar{g}}(\\Phi))$$\nis strongly parabolic for any $\\Phi = (\\bar{g}, \\varphi) \\in \\mathcal{N}_{\\bar{g}}$ and hence the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t)$$\nexists for short time. We call this flow {\\em gauged spinor flow} or {\\em spinor-DeTurck flow}.\nMoreover, the spinor flow and the gauged spinor flow differ only by a family of diffeomorphisms, i.e.\nif $\\Phi_t = (g_t, \\varphi_t)$ is a solution of the spinor flow and $\\tilde{\\Phi}_t = (\\tilde{g}_t, \\tilde{\\varphi}_t)$ is a solution of the gauged spinor flow with $\\Phi_0 = \\tilde{\\Phi}_0$, then there exists a family $F_t \\in \\widehat{\\operatorname{Diff}}_s(M)$, induced by $f_t \\in \\operatorname{Diff}(M)$, such that\n$$\\tilde{\\Phi}_t = F_{t*} \\Phi_t.$$\nThis family obeys the partial differential equation\n$$\\partial_t f_t = P_{g_t, \\bar{g}}(f_t)$$\nwith initial condition $f_0 = \\operatorname{id}_M$, where\n$$P_{g,\\bar{g}} : \\mathcal{C}^{\\infty}(M,M) \\to T \\mathcal{C}^{\\infty}(M,M)$$\n$$f \\mapsto -df (X_{f^* \\bar{g}} (g)).$$\nFor future reference we note that the linearization of $P_{\\bar{g},\\bar{g}}$ at $\\operatorname{id}_M$ is given by\n$$\\Gamma(TM) \\ni X \\mapsto -4 (\\delta_{\\bar{g}} \\delta^*_{\\bar{g}} X^{\\flat})^{\\sharp} \\in \\Gamma(TM).$$\nBecause\n$$T_{\\Phi} \\Gamma(\\Sigma M) = \\ker \\lambda_{\\Phi} \\oplus \\operatorname{im} \\lambda_{\\Phi}^* = \\ker \\lambda_{\\Phi} \\oplus T_{\\Phi} \\widehat{\\operatorname{Diff}}_s(M).\\Phi,$$\nwe can consider $\\ker \\lambda_{\\Phi}$ to be an infinitesimal slice to the diffeomorphism action.\n Indeed, we will prove that, in a weak sense, $\\ker \\lambda_{g,\\varphi}$ parametrizes a slice in a simple way. To see this we first need a parametrization of $\\Gamma(\\Sigma M)$ by the set $T_{(g,\\varphi)} \\Gamma(\\Sigma M) = \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$ near $(g,\\varphi)$. This will be frequently useful and throughout the rest of the article $\\Xi = \\Xi_{g,\\varphi}$ denotes this parametrization.\nWe define\n$$\\Xi_{g,\\varphi}: (U_g \\subset \\Gamma(\\odot^2 T^*M)) \\times \\Gamma(\\Sigma_g M) \\to \\Gamma(\\Sigma M)$$\n$$(h, \\psi) \\mapsto (g+h, \\hat{B}^g_{g+h}(\\varphi + \\psi))$$\nand its inverse\n$$\\Xi^{-1}: \\Gamma(\\Sigma M) \\to U_g \\times \\Gamma(\\Sigma_g M)$$\n$$(g', \\varphi') \\mapsto (g'-g, \\hat{B}^{g'}_g (\\varphi') - \\varphi).$$\nHere $U_g = \\{h \\in \\Gamma(\\odot^2 T^*M) : g+h \\text{ is a metric}\\}$.\nIn terms of this parametrization we can formulate the following slice theorem:\n\\begin{prop}\nLet $\\Phi = (g,\\varphi) \\in \\Gamma(\\Sigma M)$ and assume $g$ has no Killing fields. Then there exists a $C^{k+1,\\alpha}$ neighborhood $U$ of $\\Phi$, such that for any $\\tilde{\\Phi} \\in U$, there exists a $C^{k+2,\\alpha}$ diffeomorphism $f: M\\to M$, such that\n$$\\lambda_{\\Phi}(\\Xi^{-1}(F^* \\tilde{\\Phi})) = 0.$$\n\\end{prop}\n\\begin{proof}\nWe base the proof on \\cite{Viaclovsky13}, theorem 3.6.\nConsider the map\n$$G: \\Gamma^{k+1, \\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M) \\times \\Gamma^{k+2, \\alpha}(TM) \\to \\Gamma^{k,\\alpha}(TM)$$\n$$((h, \\psi), X) \\mapsto \\lambda_{\\Phi}(\\phi^{X*}_1(g+h, B^g_{g+h}(\\varphi + \\psi)))$$\nThen the derivative of $G$ at $((0,0), 0)$ in $X$ direction is given by\n$$\\frac{d}{dt}\\Bigr|_{t=0}  \\lambda_{\\Phi}(\\phi^{tV*}_1 \\Phi) = \\lambda_{\\Phi}(\\tilde{\\mathcal{L}}_V \\Phi) = \\lambda_{\\Phi} \\lambda_{\\Phi}^* V$$\nfor $V \\in \\Gamma^{k+2, \\alpha}(TM)$. Since $g$ posesses no Killing fields, $\\lambda_{\\Phi} \\lambda_{\\Phi}^*$ is injective, because in the first component $\\lambda_{\\Phi} \\lambda_{\\Phi}^* X$ is just $\\delta_g \\delta_g^* X^{\\flat}$. Additionally, $\\lambda_{\\Phi} \\lambda_{\\Phi}^*$ is an elliptic operator. It is selfadjoint and hence it must also be surjective. Thus we may apply the implicit function theorem and we find that there exists a neighborhood $U \\subset \\Gamma^{k+1, \\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M)$ of $(0,0)$ and a map $H: U \\to \\Gamma^{k+2, \\alpha}(TM)$, such that $G((h,\\psi), H(h,\\psi)) = 0$.\nNow let $\\tilde{\\Phi} \\in \\Xi(U)$.\nThen denote by $f$ the time-$1$ map of the vector field $H(\\Xi^{-1}(\\tilde{\\Phi}))$. Then\n$$\\lambda_{\\Phi}(F^*(\\Xi^{-1}(\\tilde{\\Phi}))) = 0$$\nby construction.\nThe statement then follows, because\n$$F^*(\\Xi^{-1}(\\tilde{\\Phi})) = \\Xi^{-1}(F^*(\\tilde{\\Phi})).$$\n\\end{proof}\n\n\n\\section{Volume normalized spinor flow}\nVolume constrained critical points evolve by rescaling under the spinor flow. We expect similar behavior near such a point. To address convergence questions in this situation, it is thus useful to rescale the solutions to a fixed volume. In this section, we introduce the volume normalized spinor flow and describe its evolution equation.\nLet $\\Phi_t = (g_t, \\phi_t)$ be a solution to the spinor flow. We denote by $\\mu(t)$ the normalizing factor $\\left(\\int_M \\operatorname{vol}_{g_t}\\right)^{-2\/n}$. Then $\\int_M \\operatorname{vol}_{\\mu(t) g_t} = 1$.\nNow let $\\tilde{\\Phi}(t) = (\\tilde{g}(t), \\tilde{\\varphi}(t))$, where\n$$\\tilde{g}(t) = \\mu (\\tau(t)) g_{\\tau(t)},$$\n$$\\tilde{\\varphi}(t) =  \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\varphi_{\\tau(t)}),$$\nwhere $\\tau: I \\subset \\mathbb R \\to J \\subset \\mathbb R$ is some time reparametrization.\nThen we have\n$$\\partial_t \\tilde{g}_t = \\dot{\\mu}(\\tau(t)) \\tau'(t) g_{\\tau(t)} + \\mu(\\tau(t)) \\dot{g}_{\\tau(t)} \\tau'(t),$$\n$$\\partial_t \\tilde{\\varphi}_t = \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\dot{\\varphi}_{\\tau(t)}) \\tau'(t).$$\nSolving a separable ordinary differential equation, we can arrange $\\tau'(t) \\mu(\\tau(t)) = 1$.\nWe call $\\tilde{\\Phi}_t$ with this choice of time rescaling the {\\em volume normalized spinor flow}.\nFor any $h\\in \\Gamma(\\odot^2 T^*M)$, we denote by $\\mathring{h}$ the tensor \n$$h - \\frac{\\int_M \\operatorname{tr}_g h \\operatorname{vol}_g}{n \\int_M \\operatorname{vol}_g} g.$$\nSince $\\tilde{g}_t$ has constant volume $1$, it follows that $\\int_M \\partial_t g_t \\operatorname{vol}_{g_t} = 0$. \nThus we have\n$$\\partial_t \\tilde{g}_t = \\mathring{Q}_1(g_{\\tau(t)}, \\varphi_{\\tau(t)}).$$\nBy corollary 4.5 in \\cite{Ammann2015}, we moreover have $Q_1(c^2 g, \\hat{B}^g_{c^2 g}(\\varphi)) = Q_1(g, \\varphi)$, which implies\n$$\\partial_t \\tilde{g}_t = \\mathring{Q}_1(g_{\\tau(t)}, \\varphi_{\\tau(t)}) = \\mathring{Q}_1\\left(\\mu (\\tau(t)) g_{\\tau(t)}, \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\varphi_{\\tau(t)})\\right) = \\mathring{Q}_1(\\tilde{\\Phi}_t).$$\nAgain by corollary 4.5 in op. cit., we have $Q_2(c^2 g, \\hat{B}^g_{c^2 g} (\\varphi)) = c^{-2} \\hat{B}^g_{c^2 g} Q_2(g, \\varphi)$. Thus\n$$\\partial_t \\tilde{\\varphi}_t = \\mu(t)^{-1} \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (Q_2(g_{\\tau(t)}, \\varphi_{\\tau(t)})) = Q_2(\\tilde{\\Phi}_t).$$\nWe define\n$$\\mathring{Q}(\\Phi) = (\\mathring{Q}_1(\\Phi), Q_2(\\Phi))$$\nand can rewrite the evolution of $\\tilde{\\Phi}_t$ as\n$$\\partial_t \\tilde{\\Phi}_t = \\mathring{Q}(\\tilde{\\Phi}_t).$$\nSince $\\mathring{Q}$ is the negative gradient of $\\mathcal{E}$ restricted to the set\n$$\\mathcal{N}^1 = \\left\\{ \\Phi = (g, \\varphi) \\in \\mathcal{N} : \\int_M \\operatorname{vol}_g = 1\\right\\},$$\nwe conclude that the volume normalized spinor flow coincides with the negative gradient flow of $\\mathcal{E}$ restricted to $\\mathcal{N}^1$.\n\n\\section{Analytical setup}\nIn the following proof of stability we will analyze three flows: the spinor flow, the gauged spinor flow and the mapping flow. Each of these flows is defined on an infinite dimensional manifold rather than a vector space and we feel it is appropiate to clarify our analytic setup, so that we can proceed in a somewhat more formal manner later on without bypassing rigor altogeher. \n\nThe set of unit spinors $\\mathcal{N}$ forms a Fr\u00e9chet manifold with the $\\mathcal{C}^{\\infty}$ topology. We will however never use this topology directly. Instead, we will typically restrict to a chart and work with the Sobolev or $C^{k,\\alpha}$ topologies. We do this as follows.\nFix $\\Phi_0 = (g_0, \\varphi_0) \\in \\Gamma(\\Sigma M)$. We already constructed the chart \n$$\\Xi^{-1}_{\\Phi_0} : U \\subset \\Gamma(\\Sigma M) \\to V \\subset \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_{g_0} M).$$\nThe metric $g_0$ then induces the usual $H^k$ and $C^{k,\\alpha}$ norms on $\\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$ and we simply pull them back via the chart.\nLocally we can now consider the spinor energy $\\mathcal{E}$ as a map $V \\to \\mathbb R$ and $Q$ as a map $V \\to V$.\nWhenever we use a $C^k$ or $H^s$ norm we implicitly use this construction. In particular, when we write $\\|\\Phi - \\Phi_0\\|_{X}$ for a fixed $\\Phi_0$ and a nearby $\\Phi$, we mean  $\\|\\Xi^{-1}_{\\Phi_0} (\\Phi)\\|_{X}$, where $X$ is one of the discussed Banach spaces.\n\nFor the mapping flow we proceed in a similar manner. Note first that for $f_0 \\in \\mathcal{C}^{\\infty}(M,M)$, there is a local chart around $f_0$ given by\n$$U \\subset \\mathcal{C}^{\\infty}(M,M) \\to V \\subset \\Gamma(f_0^* TM)$$\n$$f \\mapsto (x \\mapsto (\\exp_{f_0(x)})^{-1}(f(x))),$$\nwhere $\\exp$ is the exponential map of some Riemannian metric on $g$ and $V$ is a neighborhood of the $0$ section in $TM$, such that $exp_x$ is a diffeomorphism from $V_x = T_x M \\cap V$ to $\\exp (V_x)$ for every $x \\in M$. \nThen we define\n$$U = \\{f: M\\to M \\Big| (f_0,f)(M) \\subset \\exp(V)\\}.$$\nWe can define appropiate norms in the standard manner using some Riemannian metric on $M$, for example\n$$\\left(X,Y\\right)_{L^2} = \\int_M g_{f_0(p)}(X(p), Y(p)) \\operatorname{vol}_g$$\nfor $X,Y \\in \\Gamma(f_0^* TM)$.\n\nFor future reference we also quote a standard parabolic estimate and prove an interior estimate following from this.\n\\begin{thm}\nSuppose $A_t$ is an elliptic differential operator of order $m$, uniformly elliptic in $t$, with $\\mathcal{C}^{\\infty}$ coefficients in $x$ and $t$.\nThen for any $s \\in \\mathbb R$ and $T > 0$, there exists $C > 0$ such that\n$$\\|u_t\\|_{H^s}^2 + \\int_0^T \\|u_{t'}\\|_{H^{s+m'}}^2 dt' \\leq C \\left( \\|u_0\\|_{H^s}^2 + \\int_0^T \\|\\partial_t u_{t'} - A_{t'} u_{t'}\\|^2_{H^{s-m'}} dt' \\right)$$\nfor any $t \\in [0, T]$ and $u \\in C^1([0,T], H^s) \\cap C^0([0,T], H^{s+m'})$, where $m' = m\/2$.\n\\end{thm}\nFor a proof, see 6.5.2 in \\cite{Chazarain1982}.\nWe will need the following estimate for solutions, derived from this inequality:\n\\begin{cor}\n\\label{PE}\nFor any $\\delta > 0$ and any \n $A_t$ as above, there exists $C, \\tilde{C}>0$, such that for any $u_t$ a solution of\n$$\\partial_t u_t = A_t u_t,$$\nwe have\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{H^r}^2d \\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{H^{s}}^2 d\\tau,$$\nas well as\n$$\\|u_{t}\\|_{H^r}^2 \\leq \\tilde{C} \\int_0^T \\|u_{\\tau}\\|_{H^{s}}^2 d\\tau$$\nfor any $r,s \\in \\mathbb R$ and any $t \\in [\\delta, T]$.\n\\end{cor}\n\\begin{proof}\nFor $r < s$ the inequality is trivial.\nFor $r > s$ the claim follows inductively from\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|^2_{H^{s+m'}} d\\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{H^{s-m'}}^2 d\\tau.$$\nFor this consider $f:[0,T] \\to [0,1]$ smooth such that $f(0) = 0, f(\\delta) = 1$.\nThen\n$$\\partial_t (f(t) u_t) - A_t u_t = (\\partial_t f(t)) u_t.$$\nHence the above estimate yields\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{s+m'}^2 d\\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{s-m'}^2 d\\tau,$$\nwhere $C = \\max |\\partial_t f|$.\n\nWe have shown that\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{H^r}^2 d\\tau \\leq \\tilde{C} \\int_0^T \\|u_{\\tau}\\|_{H^s}^2 d\\tau.$$\nSince $\\partial_t u_t = A_t u_t$ and $u_t$ is a differential operator of order $m$ this implies\n$$\\int_{\\delta}^T \\|\\partial_{\\tau} u_{\\tau}\\|^2_{H^{r-m}} d\\tau \\leq  \\tilde{C} \\int_0^T \\|u_{\\tau}\\|^2_{H^s} d\\tau$$\nand hence by the Sobolev embedding $W^{1,2}([a,b]; H^{l+1}, H^l) \\hookrightarrow C^0([a,b]; H^l)$ (cf. \\cite{Cherrier12}, Theorem 1.7.4 and (1.7.62))  we conclude\n$$\\|u_t\\|_{H^{r-m}} \\leq \\hat{C} \\int_0^T \\|u_{\\tau}\\|^2_{H^s} d\\tau.$$\n(Here\n$$W^{1,2}([a,b]; H^{l+1}, H^l) = L^2([a,b]; H^{l+1}) \\cap \\{u: [a,b] \\to H^l : \\partial_t u \\in L^2([a,b]; H^l) \\}$$\nwith the obvious norm.)\n\\end{proof}\n\n\n\\section{The \u0141ojasiewicz inequality and gradient estimates}\nThe \u0141ojasiewicz inequality relates the norm of the gradient of a differentiable function to its value near a critical point in a way that allows us to show convergence of the gradient flow. There are two situations when \u0141ojasiewicz inequalities are known to hold. The optimal situation is when the function is a Morse function or less restrictively a Morse--Bott function. Then we have\n$$|f(x) - f(x_0)| \\leq C \\|\\operatorname{grad} f(x)\\|^2$$\nfor $x_0$ a critical point of $f$ and some constant $C > 0$. This can be easily seen by applying the Morse--Bott lemma: near a critical manifold we may write a Morse--Bott function as\n$$f(x_1, ..., x_n) = c + x_1^2 + ... + x_r^2 - x_{r+1}^2 - ... - x_s^2,$$\nwhere $(x_1, ..., x_n)$ are coordinates with $x_0$ at the origin and critical manifold $\\{x_{s+1} = ... = x_{n} = 0\\}$.\nBecause in a small neighborhood the Riemannian metric is very close to being Euclidean, we get the inequality\n$$|f(x) - c| \\leq C |\\operatorname{grad} f(x)|^2$$\nfor some $C > 0$.\nThe other case is that $f$ is analytic. Then there exists $\\theta \\in (1,2)$, such that\n$$|f(x) - f(x_0)| \\leq \\|\\operatorname{grad} f(x)\\|^{\\theta}.$$\nWe will make use of both versions. The inequality for analytic functions is a difficult theorem in the theory of semianalytic sets, due to \u0141ojasiewicz. The first version will be employed to demonstrate stability of parallel spinors, since there we know $\\mathcal{E}$ to be Morse--Bott. For volume constrained critical points we do not know this and instead use the weaker inequality for analytic functions.\nBoth inequalities are known in this general form only for functions on finite dimensional domains. We will spend most of the rest of the section justifying these inequalities for the spinor energy functional.\n\n\\begin{prop}[Optimal \u0141ojasiewicz inequality for parallel spinors]\n\\label{LIP}\nLet $\\bar{\\Phi}$ be a critical point of $\\mathcal{E}$. (Hence $\\bar{\\Phi}$ is an absolute minimiser with $\\mathcal{E}(\\bar{\\Phi}) = 0$.) Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$ and some constant $C>0$, such that for any $\\Phi \\in U$ we have\n$$\\mathcal{E}(\\Phi) \\leq C \\|Q(\\Phi)\\|_{L^2}^2.$$\n\\end{prop}\n\n\\begin{prop}[\u0141ojasiewicz inequality for volume constrained critical points]\n\\label{LIVC}\nLet $\\bar{\\Phi} = (\\bar{g},\\bar{\\varphi})$ be a volume constrained critical point of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$ and some constant $\\theta \\in (1,2)$, such that for any $\\Phi=(g,\\varphi)$ with $\\int_M \\operatorname{vol}_g = \\int_M \\operatorname{vol}_{\\bar{g}}$ we have\n$$|\\mathcal{E}(\\Phi) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\|\\mathring{Q}(\\Phi)\\|_{L^2}^{\\theta}.$$\nIf the set of volume constrained critical points near $\\bar{\\Phi}$ is a manifold, this can be improved to\n$$|\\mathcal{E}(\\Phi) - \\mathcal{E}(\\bar{\\Phi})| \\leq C \\|\\mathring{Q}(\\Phi)\\|_{L^2}^2.$$\n\\end{prop}\nThe proofs of both propositions rely on the following infinite-dimensional form of the \u0141ojasiewicz inequality, due to Colding and Minicozzi II, see \\cite{Colding2013}.\n\\begin{thm}\n\\label{CML}\n\\begin{enumerate}\n\\item Suppose $E \\subset L^2$ is a closed subspace, $U$ is an open neighborhood of $0$ in $C^{2, \\beta} \\cap E$.\n\\item Suppose $G: U \\to \\mathbb R$ is an analytic function or that there is a neighborhood $V$ of $0$, such that $\\{x \\in V : \\operatorname{grad} G (x) = 0\\}$ is a finite dimensional submanifold.\n\\item Suppose the gradient $\\operatorname{grad} G : U \\to C^{\\beta} \\cap E$ is $C^1$, $\\operatorname{grad} G (0) = 0$ and\n$$\\|\\operatorname{grad} G(x) - \\operatorname{grad} G(y) \\|_{L^2} \\leq C \\|x - y\\|_{H^2}$$\n\\item $L = D \\operatorname{grad} G (0)$ is symmetric, bounded from $C^{2, \\beta} \\cap E$ to $C^{\\beta} \\cap E$ and from $H^2 \\cap E$ to $L^2 \\cap E$ and Fredholm from $C^{2, \\beta} \\cap E$ to $C^{\\beta} \\cap E$.\n\\end{enumerate}\nThen there exists $\\theta \\in (1,2)$ so that for all $x\\in E$ sufficiently small\n$$|G(x) - G(0)| \\leq \\|\\operatorname{grad} G(x)\\|_{L^2}^\\theta$$\nIf there is a neighborhood $V$ of $0$, such that $\\{x \\in V : \\operatorname{grad} G (x) = 0\\}$ is a finite dimensional submanifold, we get the stronger inequality\n$$|G(x) - G(0)| \\leq C \\|\\operatorname{grad} G(x)\\|_{L^2}^2$$\nfor some $C>0$.\n\\end{thm}\n{\\em Remark.} Colding and Minicozzi II prove this for $G$ analytic. The alternative condition we give is essentially that $G$ is Morse--Bott at $0$. The proof in that case is the same except that when the finite dimensional \u0141ojasiewicz inequality is used, we instead invoke the stronger inequality for Morse--Bott functions.\n\nSince this theorem requires the linearisation of the gradient to be Fredholm we will be working on a slice of the spin diffeomorphism group. \n\\begin{lemma}\nLet $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ be a critical point. Let $\\iota: \\ker \\lambda_{\\bar{\\Phi}} \\to \\Gamma(\\Sigma M)$ be the inclusion. $f = \\mathcal{E} \\circ \\Xi_{\\bar{\\Phi}} \\circ \\iota$ fulfills the conditions of theorem \\ref{CML}.\nIn particular we have\n$$|f(x)| \\leq C \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\n\\end{lemma}\n\\begin{proof}\nWe equip $\\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\Sigma_g M)$ with the $L^2$ metric induced by $\\bar{g}$, and similarly we define the $C^{2,\\alpha}$ norm in terms of $\\bar{g}$.\nThen clearly $\\mathcal{E} \\circ \\Xi_{\\bar{\\Phi}} \\circ \\iota$ is a smooth function and by \\cite{Ammann2015b} its critical set is smooth, thus the second condition in theorem \\ref{CML} is fulfilled.\nMoreover $0$ corresponds to $\\bar{\\Phi}$ and hence is a critical point, i.e. $\\operatorname{grad} f(0) = 0$.\nThe gradient of $f$ can be considered as a nonlinear second order differential operator. In fact, it is a smooth map \n$$\\operatorname{grad} f: \\Gamma^{2,\\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M) \\to \\Gamma^{\\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M).$$\nOn any bounded $C^{2,\\alpha}$ neighborhood $U$ of $0$ we have\n$$\\|\\operatorname{grad} f(x) - \\operatorname{grad} f(y)\\|_{L^2} \\leq C \\|x - y\\|_{H^2}.$$\nThis is a simple consequence of the fact that $Q(g,\\varphi)$ can be locally represented as a polynomial expression\nin the coordinate expressions of $g$ and $\\varphi$ and their first and second derivatives. In a bounded $C^{2,\\alpha}$ neighborhood we then estimate terms as needed to get an expression which is bounded by $\\|(g,\\varphi)\\|_{H^2}$.\nThis concludes the argument for conditions 1,2 and 3.\n\nSince $DQ(\\bar{\\Phi})$ is symmetric (by \\cite{Ammann2015}), so is $L$. Since $L$ is a linear second order differential operator, it induces continuous maps $C^{2,\\alpha} \\to C^{\\alpha}$ and $H^2 \\to L^2$.\nIt remains to be shown that $L$ is Fredholm.\nTo see this, remember that we have a splitting\n$$T_{\\bar{\\Phi}} \\mathcal{N} = \\ker \\lambda_{\\bar{\\Phi}} \\oplus \\operatorname{im} \\lambda^*_{\\bar{\\Phi}}.$$\nWith respect to these operators, we know the two identities\n$$DQ(\\bar{\\Phi}) \\circ \\lambda^*_{\\bar{\\Phi}} = 0 \\text{ and } \\lambda_{\\bar{\\Phi}} \\circ DQ(\\bar{\\Phi}) = 0,$$\nboth of which reflect diffeomorphism invariance of $Q$.\nMoreover, we introduced the perturbed gradient $\\tilde{Q}_{\\bar{\\Phi}}$, which we know is strongly elliptic and thus its linearization is Fredholm. Its linearization is also symmetric.\nThus we conclude that $DQ(\\bar{\\Phi})$ has the form\n$$\\bordermatrix{\n                      & \\ker \\lambda_{\\bar{\\Phi}} & \\operatorname{im} \\lambda_{\\bar{\\Phi}}^* \\cr\n\\ker \\lambda_{\\bar{\\Phi}} & P                   & 0                      \\cr\n\\operatorname{im} \\lambda_{\\bar{\\Phi}}^* & 0                   & 0                      \\cr\n},$$\nwhereas $D\\tilde{Q}_{\\bar{\\Phi}}(\\bar{\\Phi})$ has the form \n$$\\bordermatrix{\n                      & \\ker \\lambda_{\\bar{\\Phi}} & \\operatorname{im} \\lambda_{\\bar{\\Phi}}^* \\cr\n\\ker \\lambda_{\\bar{\\Phi}} & P                   & 0                      \\cr\n\\operatorname{im} \\lambda_{\\bar{\\Phi}}^* & 0                   & R                      \\cr\n}.$$\nSince $D\\tilde{Q}_{\\bar{\\Phi}}(\\bar{\\Phi})$ is Fredholm, so is $P = \\pi \\circ DQ(\\Phi) \\circ \\iota$, where $\\pi: T_{\\bar{\\Phi}} \\mathcal{N} \\to \\ker \\lambda_{\\bar{\\Phi}}$ denotes the orthogonal projection.\nWe compute\n$$D\\operatorname{grad} f (0) = D(\\Xi \\circ \\iota)(0)^* DQ(\\Xi \\circ \\iota(x)) = \\pi \\circ D\\Xi(0)^* DQ(\\bar{\\Phi}).$$\nSince the domain is restricted to $\\ker \\lambda_{\\bar{\\Phi}}$ and $D\\Xi(0) = \\operatorname{id}$, we conclude that\n$$D \\operatorname{grad} f(0) = P,$$\nand hence $L = D\\operatorname{grad} f(0)$ is Fredholm as required. Thus we have checked all conditions in theorem \\ref{CML}, and the inequality holds.\n\\end{proof}\n\n\\begin{proof}[Proof of proposition \\ref{LIP}]\nWhat remains to be shown is that the inequality\n$$|f(x)| \\leq \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\nimplies the inequality\n$$|\\mathcal{E}(\\Phi)| \\leq C \\|Q(\\Phi)\\|^2_{L^2}.$$\nFirst, by the slice theorem there exists a $C^{k+1,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ there exists a diffeomorphism $f: M\\to M$, such that\n$$\\lambda_{\\bar{\\Phi}}(\\Xi^{-1}(F_* \\Phi)) = 0.$$\nSince\n$$\\mathcal{E}(F_* \\Phi) = \\mathcal{E}(\\Phi), \\quad F_* Q(\\Phi) = Q(F_* \\Phi)$$\nand since the $L^2$ metric is diffeomorphism invariant, we can assume that $\\Phi$ lies in the slice, i.e. $\\lambda_{\\bar{\\Phi}}(\\Xi^{-1}(\\Phi)) = 0$. Then we have\n$$|\\mathcal{E}(\\Phi)| = f(\\Xi^{-1}(\\Phi)) \\leq \\|\\operatorname{grad} f(\\Xi^{-1}(\\Phi))\\|_{L^2}^2.$$\nHence we must show\n$$\\|\\operatorname{grad} f (\\Xi^{-1}(\\Phi))\\|_{L^2}^2 \\leq \\|Q(\\Phi)\\|_{L^2}.$$\nFirst note that the metric on $\\ker \\lambda_{\\bar{\\Phi}}$ is the metric induced by $\\bar{\\Phi}$. By making the neighborhood smaller if necessary, we can assume that all $L^2$ metrics in that neighborhood are uniformly equivalent. We have\n$$\\operatorname{grad} f(\\Xi^{-1}(\\Phi)) = D(\\Xi \\circ \\iota) (\\Xi^{-1}(\\Phi))^* Q(\\Phi).$$\nSince $D(\\Xi \\circ \\iota)$ is clearly Lipschitz, we obtain our estimate. This concludes the proof of the \u0141ojasiewicz inequality in this case.\n\\end{proof}\n\\begin{proof}[Proof of proposition \\ref{LIVC}]\nFor the purposes of the following discussion, read the spaces of smooth mappings as the spaces of $C^{2,\\alpha}$ mappings, so that they are Banach spaces or Banach manifolds.\nBy the analytic regular value theorem, we can find an analytic parametrization of\n$$\\left\\{ g \\in \\Gamma(\\odot^2_+ T^*M) : \\int_M \\operatorname{vol}_g = 1\\right\\}$$\nby\n$$\\left\\{ h \\in \\Gamma(\\odot^2_+ T^* M) : \\int_M \\operatorname{tr}_g h \\operatorname{vol}_g = 0\\right\\}.$$\n(For a treatment of the implicit function theorem in the analytic category on Banach spaces, take for example \\cite{Hajek14}, theorem 174.)\nWe combine this parametrization with $\\Xi_{g,\\varphi}$ to obtain an analytic parametrization\n$$\\Psi: U \\subset V_0 \\to \\mathcal{N}^1$$\nwhere\n$$V_0 = \\left\\{(h, \\psi) \\in \\ker \\lambda_{g,\\varphi} : \\int_M \\operatorname{tr}_g h \\operatorname{vol}_g = 0\\right\\}$$\nand\n$$\\mathcal{N}^1 = \\left\\{\\Phi = (g,\\varphi) \\in \\mathcal{N} : \\int_M \\operatorname{vol}_g = 1\\right\\}.$$\nDefine $f = \\mathcal{E} \\circ \\iota \\circ \\Psi$, with $\\iota: \\mathcal{N}^1 \\to \\mathcal{N}$ the inclusion.\nThen $f$ fulfills the conditions of theorem \\ref{CML}, which can be shown as in the previous lemma. \nApplying the theorem, we thus obtain\n$$|f(x) - f(0)| \\leq \\|\\operatorname{grad} f (x)\\|_{L^2}^{\\theta},$$\nwhere $\\theta \\in (1,2)$. If the critical set is a manifold near $\\bar{\\Phi}$, we use the optimal version theorem of theorem \\ref{CML} and obtain\n$$|f(x) - f(0)| \\leq C \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\nfor some $C > 0$.\n What remains to be shown is\n$$\\|\\operatorname{grad} f(\\Psi^{-1}(\\Phi))\\|_{L^2} \\leq C \\|\\mathring{Q}(\\Phi)\\|_{L^2}.$$\nAs in the previous proposition, we compute\n$$\\operatorname{grad} f(\\Psi^{-1}(\\Phi)) = (D\\Psi )^*(\\Psi^{-1}(\\Phi)) \\operatorname{grad} (\\mathcal{E} \\circ \\iota)(\\Phi).$$\nThen the claim follows, since, on the one hand, $D\\Psi$ is Lipschitz by the regular value theorem, and on the other hand\n$$\\operatorname{grad}(\\mathcal{E} \\circ \\iota)(\\Phi) = D\\iota(\\Phi)^* \\operatorname{grad} \\mathcal{E}(\\Phi) = D\\iota(\\Phi)^* Q(\\Phi).$$\nSince $D\\iota(\\Phi)^* : T_{\\Phi} \\mathcal{N} \\to T_{\\Phi} \\mathcal{N}^1$ is the orthogonal projection,\nthis implies\n$$\\operatorname{grad}(\\mathcal{E} \\circ \\iota)(\\Phi) = \\mathring{Q}(\\Phi).$$\n\\end{proof}\n\n\n\\begin{thm}[Energy decay]\n\\label{L2cv}\nSuppose $M$ is a compact manifold.\n\\begin{enumerate}\n\\item Suppose $\\bar{\\Phi}$ is a critical point of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neigborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ the following inequalities hold\n$$\\mathcal{E}(\\Phi_t) \\leq C e^{-\\alpha t},$$\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|^2_{L^2} dt \\leq C e^{-\\alpha T},$$\nand\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C e^{-\\alpha T},$$\nwhere $C, \\alpha>0$ and $\\Phi_t$ is the solution of\n$$\\partial_t \\Phi_t = Q(\\Phi_t), \\Phi_0 = \\Phi.$$\n\\item Suppose $\\bar{\\Phi}$ is a volume constrained minimizer of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ it holds\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\frac{C}{1+T^{\\beta}},$$\n$$\\int_T^{\\infty} \\|\\mathring{Q}(\\Phi_t)\\|^2_{L^2} dt \\leq \\frac{C}{1 + T^{\\beta}}$$\nand\n$$\\int_T^{\\infty} \\|\\mathring{Q}(\\Phi_t)\\|_{L^2} dt \\leq \\frac{C}{1+T^{\\gamma}},$$\nfor some $C, \\beta > 1$. If the set of volume constrained critical sets is a manifold near $\\bar{\\Phi}$, we can instead choose exponential bounds as in the first case. Here we assume $\\Phi_t$ is the volume normalized spinor flow with initial condition $\\Phi_0 = \\Phi \\in U$.\n\\end{enumerate}\nThe integrals are to be read as the integral from $T$ to the maximal time of existence in the neighborhood $U$. The constants $C, \\alpha, \\beta$ only depend on the constants $C$ and $\\theta$ in the \u0141ojasiewicz inequalities.\n\\end{thm}\n{\\em Remark. } The constants $\\beta$ and $\\gamma$ can be computed from the constant $\\theta$ in the \u0141ojasiewicz inequality as $\\beta = \\frac{\\theta }{2 - \\theta}$ and $\\gamma = \\frac{\\theta - 1}{2 - \\theta}$. As $\\theta$ tends to $2$, $\\beta$ tends to infinity, i.e. the convergence rate improves. As $\\theta$ tends to $1$, $\\beta$ tends to $1$, i.e. the convergence rate gets worse. Likewise, $\\gamma$ tends to $\\infty$ if $\\theta$ tends to $2$, but $\\gamma$ tends to $0$ as $\\theta$ tends to $1$. \n\\begin{proof}\nFirst we note that\n$$\\frac{d}{dt} \\mathcal{E}(\\Phi_t) =  -\\|Q(\\Phi_t)\\|_{L^2}^2$$\nimplies that the integral of the gradient over all future time is controlled by the energy at a fixed time.\nNow applying the optimal \u0141ojasiewicz inequality, we obtain \n$$\\frac{d}{dt} \\mathcal{E}(\\Phi_t) \\leq -\\frac{1}{C} \\mathcal{E}(\\Phi_t).$$\nIntegrating this differential inequality, we obtain\n$$\\mathcal{E}(\\Phi_t) \\leq \\mathcal{E}(\\Phi_0) e^{-(1\/C) t}.$$\nChoosing the neighborhood so that $\\mathcal{E}$ is bounded, we obtain the desired inequality.\n\nFor the second case consider\n$$\\frac{d}{dt} |\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| = -\\|Q(\\Phi_t)\\|_{L^2}^2 \\leq -|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})|^{2\/\\theta}.$$\nIntegrating this differential inequality, we obtain\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\left(\\frac{2}{\\theta} - 1\\right) \\frac{1}{(C + t)^{\\beta}}$$\nwhere $\\beta = \\frac{1}{2\/\\theta - 1}$ and $C = |\\mathcal{E}(\\Phi_0) - \\mathcal{E}(\\bar{\\Phi})|^{1-2\/\\theta}$.\nBy continuity of $\\mathcal{E}$ we can find a lower bound for $C$ on a small neighborhood, and using this lower bound we obtain the desired inequality. The bound for the integral of $\\|\\mathring{Q}(\\Phi_t)\\|_{L^2}$ follows as above.\n\nFor the estimates of $\\int_T^{\\infty} \\|Q(\\Phi_t)\\| dt$, notice that the \u0141ojasiewicz inequality implies $\\mathcal{E}(\\Phi)^{-1\/\\theta} \\geq C \\|Q(\\Phi)\\|^{-1}$. (Here we actually have $\\theta = 2$. The case of volume constrained minimizers is analogous with $\\theta \\neq 2$ in general.)\nThis implies\n\\begin{eqnarray*}\n-\\frac{d}{dt}\\mathcal{E}(\\Phi_t)^{1-1\/\\theta} & = & (1-1\/\\theta) \\mathcal{E}(\\Phi_t)^{-1\/\\theta} \\|Q(\\Phi_t)\\|^2\\\\\n& \\geq & C \\|Q(\\Phi_t)\\| \n\\end{eqnarray*}\nHence\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C \\mathcal{E}(\\Phi_T)^{1-1\/\\theta}.$$\nPlugging in the estimate for $\\mathcal{E}(\\Phi_T)$ then gives the desired result.\n\\end{proof}\n\n\\section{Mapping flow estimates}\nSuppose $\\Phi_t$ solves\n$$\\partial_t \\Phi_t = Q(\\Phi_t).$$\nIn the previous section we proved a strong estimate of the gradient along the flow in the $L^2$ norm, provided $\\Phi_t$ is near a critical point. We would now like to improve this to an estimate in some higher regularity norm. Since the gradient $Q_t = Q(\\Phi_t)$ satisfies the linear parabolic equation\n$$\\partial_t Q_t = DQ(\\Phi_t) Q_t,$$\nthis is reasonable by parabolic regularity. Unfortunately, this equation is only weakly parabolic and hence we can not directly apply parabolic regularity. However, we recall that $\\tilde{\\Phi}_t = F_{t*} \\Phi_t$ obeys the strongly parabolic equation\n$$\\partial_t \\tilde{\\Phi}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$$\nif $f_t$ satisfies the mapping flow equation\n$$\\partial_t f_t = P_{g_t, g_0}(f_t), f_0 = \\operatorname{id}_M.$$\nThe gauged gradient $\\tilde{Q}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$ satisfies the linear strongly parabolic equation\n$$\\partial_t \\tilde{Q}_t = D\\tilde{Q}(\\tilde{\\Phi}_t) \\tilde{Q}_t.$$\nParabolic regularity applies to $\\tilde{Q}_t$, but we have no estimate of $\\tilde{Q}_t$! To obtain such an estimate, we will now show how to control $\\partial_t f_t$ along the mapping flow. In the next section, we will combine this estimate with the gradient estimate of the previous section to obtain an estimate of $\\tilde{Q}_t$.\n\n\\begin{lemma}\n\\label{MFE}\nLet $\\tilde{g} \\in \\Gamma(\\odot^2_+ T^* M)$ and $k > \\frac{n}{2} + 2$. Suppose $\\tilde{g}$ has no Killing fields.\nThen there exists a $H^k$ neighborhood $U \\times V$ of $(\\operatorname{id}_M, \\tilde{g})$ and constants $C, \\lambda > 0$, such that for a solution $f_t$ and a metric $g_t \\in V$, $g_t$ once differentiable in time, of an initial value problem\n$$f_0 = \\operatorname{id}_M$$\n$$\\dot{f_t} = P_{g_t, \\tilde{g}}(f_t)$$\nwe have\n$$\\int_{t_1}^{t_2} \\|P_{g_t, \\tilde{g}}\\|_{H^{-2}} dt \\leq C \\left( \\int_0^{t_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda (t - t_1)} dt + \\int_{t_1}^{t_2} \\|\\dot{g}_t\\|_{L^2} dt  + e^{-\\lambda t_1}\\right)$$\nfor some $C, \\lambda > 0$, provided the flow exists until time $t_2$ in the neighborhood $U \\times V$.\n\\end{lemma}\n\\begin{proof}\nAs computed in \\cite{Ammann2015},\n$$DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X = -4 (\\delta_{\\tilde{g}} \\delta_{\\tilde{g}}^* X^{\\flat})^{\\sharp}.$$\nA computation of the symbol then shows that this operator is strongly elliptic. \nFurthermore, this formula implies\n$$\\left(DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X, X\\right)_{L^2} = -4 \\left(\\delta^*_{\\tilde{g}} X^{\\flat}, \\delta^*_{\\tilde{g}} X^{\\flat} \\right) = -4 \\left(\\mathcal{L}_X \\tilde{g}, \\mathcal{L}_X \\tilde{g} \\right)_{L^2}.$$\nSince we assume $\\tilde{g}$ has no Killing fields, this implies $DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M)$ is strictly negative definite, i.e. there exists $\\mu > 0$, such that\n$$\\left(DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X, X\\right)_{L^2} \\leq -\\mu \\left(X,X\\right)_{L^2}.$$\nSince the coefficients of the operator $P_{g_1, g_2}(f)$ are continuous in $f$ and the first derivatives of $g_1$ and $g_2$ and recalling that by the Sobolev embedding theorem $H^k$ continuously embeds in $C^2$, we conclude that there is a $H^k$ neighborhood $U$ of $\\tilde{g}$, a neighborhood $V$ of $\\operatorname{id}_M$ and a constant $0 < \\lambda < \\mu$, such that $DP_{g, \\tilde{g}}(f)$ is strongly elliptic and strictly negative definite with a constant $\\lambda$.\n\nSince $L = DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M)$ is strictly negative definite, it induces an invertible operator from $H^{s+2} \\to H^s$. We have, up to equivalence,\n$$\\|f\\|_{H^{-2}} = \\|L^{-1}f\\|_{L^2}.$$\nThis implies, in particular, that $DP_{g, \\tilde{g}}(f)$ is also strictly negative definite with respect to the Sobolev inner product $\\langle \\cdot, \\cdot \\rangle_{H^{-2}}$.\n\nWe will now derive a differential inequality for $\\|\\dot{f}_t\\|_{H^{-2}}^2$, where\n$$\\dot{f}_t = P_{g_t, \\tilde{g}}(f_t).$$\nFor brevity, we let $P_{g_t, \\tilde{g}}(f_t) = P_{g_t}(f_t)$.\nIn what follows, we tacitly assume $g_t \\in U$, $f_t \\in V$ for all $t$, as per the statement of the lemma. \nWe calculate\n\\begin{eqnarray*}\n\\frac{1}{2} \\frac{d}{dt} \\langle P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}} & = &  \\langle \\frac{d}{dt} P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}\\\\\n& = & \\langle P_{\\dot{g_t}}(f_t) + DP_{g_t}(f_t) \\dot{f}_t, P_{g_t}(f_t) \\rangle_{H^{-2}} \\\\\n& = & \\langle P_{\\dot{g_t}}(f_t), P_{g_t}(f_t) \\rangle + \\langle DP_{g_t}(f_t) P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}.\n\\end{eqnarray*}\nThe map\n$$g \\mapsto P_g(f) = 2 df(\\delta_{f^*\\tilde{g}} g),$$\nis a linear first order differential operator with bounds dependent on $\\|f\\|_{C^1}$ and $\\|\\tilde{g}\\|_{C^1}$. As such we can estimate, using that bound and the Cauchy-Schwarz inequality\n$$|\\langle P_{\\dot{g_t}}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}| \\leq \\|P_{\\dot{g_t}} (f_t)\\|_{H^{-2}} \\|P_{g_t}(f_t)\\|_{H^{-2}}  \\leq  C \\|\\dot{g}_t\\|_{L^2} \\|P_{g_t}(f_t)\\|_{H^{-2}}.$$\nThen we obtain for\n$$a(t) = \\langle P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}$$\nthe inequality\n$$\\frac{1}{2} \\dot{a}(t) \\leq C \\|\\dot{g}_t\\|_{L^2} \\sqrt{a(t)} - \\lambda a(t).$$\nLet $b(t) = \\sqrt{a(t)}$. The function $b$ then satisfies the following differential inequality\n$$\\dot{b}(t) \\leq -\\lambda b(t) + \\|g_t\\|_{L^2}.$$\nDefine\n$$\\beta(t) = e^{-\\lambda t} \\left( b(0) + \\int_0^t e^{\\lambda s} \\|\\dot{g}_s\\|_{L^2} ds \\right).$$\nThen we have\n$$\\dot{\\beta}(t) = -\\lambda \\beta(t) + \\|\\dot{g}_t\\|_{L^2}.$$\nWe deduce\n$$\\frac{d}{dt} (b-\\beta) \\leq -\\lambda (b-\\beta),$$\nand since $b(0) = \\beta(0)$, $b(t) \\leq \\beta(t)$ follows.\nTo obtain the claim of the lemma, we will now estimate the integral of $\\beta(t)$. For brevity, we denote $\\gamma(t) = \\|\\dot{g}_t\\|_{L^2}$. Define $\\chi(s,t) = 1$ if $0 \\leq s \\leq t$ and $\\chi(s,t) = 0$ otherwise. Then we calculate\n\\begin{eqnarray*}\n\\int_{t_1}^{t_2} e^{-\\lambda t} \\int_0^t e^{\\lambda s} \\gamma(s) ds dt\n& = & \\int_{t_1}^{t_2} \\int_0^t e^{\\lambda(s-t)} \\gamma(s) ds dt \\\\\n& = & \\int_{t_1}^{t_2} \\int_0^{t_2} \\chi(s,t) e^{\\lambda(s-t)} \\gamma(s) ds dt\\\\\n& = & \\int_0^{t_2} \\gamma(s) \\int_{t_1}^{t_2} \\chi(s,t) e^{\\lambda(s-t)} dt ds\\\\\n& = & \\int_0^{t_2} \\gamma(s) \\int_{\\max\\{s, t_1\\}}^{t_2} e^{\\lambda(s-t)} dt ds\\\\\n& = & \\int_0^{t_1} \\gamma(s) \\int_{t_1}^{t_2} e^{\\lambda(s-t)} dt ds + \\int_{s}^{t_2} \\gamma(s) \\int_{t_1}^{t_2} e^{\\lambda (s-t)} dt ds\\\\\n& \\leq & \\lambda^{-1} \\left( \\int_0^{t_1} e^{\\lambda (s-t_1)}\\gamma(s) ds + \\int_{t_1}^{t_2} \\gamma(s) ds \\right)   \n\\end{eqnarray*}\nThe integral of the term $b(0) e^{-\\lambda t}$ is\n$$\\int_{t_1}^{t_2} b(0) e^{-\\lambda t} dt = \\lambda^{-1} b(0) \\left( e^{-\\lambda t_1} - e^{-\\lambda t_2} \\right).$$\nThus\n\\begin{eqnarray*}\n  \\int_{t_1}^{t_2} \\beta(t) dt & \\leq & \\lambda^{-1} \\left( b(0) e^{-\\lambda t_1} +  \\int_0^{t_1} e^{\\lambda (s-t_1)}\\gamma(s) ds + \\int_{t_1}^{t_2} \\gamma(s) ds \\right) \n\\end{eqnarray*}\nand the claim of the lemma follows.\n\\end{proof}\n\n\\section{Smooth convergence of the flow}\nNow everything is in place to prove stability of the spinor flow. We obtain slightly sharper theorems than in the introduction:\n\\begin{thm}\n\\label{stabP}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a critical point of $\\mathcal{E}$, such that $\\bar{g}$ has no Killing fields. Then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that any solution of the negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is exponential.\n\\end{thm}\n\\begin{thm}\n\\label{stabVC}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a volume constrained minimizer of $\\mathcal{E}$ and suppose the set of critical points is a manifold near $\\bar{\\Phi}$. Suppose furthermore, that $\\bar{g}$ has no Killing fields and $k > \\frac{n}{2} + 5$. Then there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the volume constrained negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is exponential.\n\nIf the critical set is not a manifold, but $\\theta$ in proposition \\ref{LIVC} can be chosen to be larger than $3\/2$, then there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the volume constrained negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is $O(T^{-\\kappa})$, $\\kappa = \\frac{2\\theta -3}{2-\\theta} >0$.\n\\end{thm}\n\nWe will reduce the proof of these theorems to the following two lemmas:\n\\begin{lemma}[Existence near critical points]\n\\label{UniformExistence}\nLet $\\bar{\\Phi}$ be a critical point of $\\mathcal{E}$ and let $T, \\epsilon > 0, k > \\frac{n}{2} + 2$.\nThen there exists $\\delta > 0$, such that for any $\\Phi$ with $\\|\\Phi - \\bar{\\Phi}\\|_{H^k} < \\delta$, the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t), \\Phi_0 = \\Phi$$\nexists until time $T$ and $\\|\\Phi_T - \\bar{\\Phi}\\|_{H^k} < \\epsilon$. The same result holds for volume constrained critical points and the volume constrained flow. \n\\end{lemma}\nThe proof is analogous to the proof of corollary 8.6 in \\cite{Weiss2012}\n\\begin{lemma}[Decay of the gradient in a Sobolev norm]\n\\label{GradientDecay}\nSuppose $\\bar{\\Phi}$ is a critical point of $\\mathcal{E}$. Then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, a neighborhood $V$ of $\\operatorname{id}_M$ in $\\operatorname{Diff}(M)$, constants $C, \\alpha > 0$, such that for $\\Phi \\in U$ the gauged spinor flow $\\tilde{\\Phi}_t$ with initial condition $\\Phi$ fulfills the following estimate \n\\begin{equation}\n\\label{GEa}\n\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq C e^{-\\alpha T}\n\\end{equation}\nas long as $\\Phi_t$ and $f_t$ remain in the neighborhoods $U$ and $V$ respectively.\n\nAnalogously, if $\\bar{\\Phi}$ is a volume constrained critical point of $\\mathcal{E}$ and the critical set near $\\bar{\\Phi}$ is a manifold, then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, a neighborhood $V$ of $\\operatorname{id}_M$ in $\\operatorname{Diff}(M)$, constants $C, \\alpha > 0$, such that for $\\Phi \\in U$ the volume normalized gauged spinor flow $\\tilde{\\Phi}_t$ with initial condition $\\Phi$ fulfills the following estimate \n\\begin{equation}\n\\label{GEb}\n\\|\\mathring{\\tilde{Q}}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq C e^{-\\alpha T}\n\\end{equation}\nas long as $\\Phi_t$ and $f_t$ remain in the neighborhoods $U$ and $V$ respectively. If the critical set is not a manifold we instead find $C, \\beta > 0$, such that\n\\begin{equation}\n\\label{GEc}\n\\|\\mathring{\\tilde{Q}}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq \\frac{C}{1+ T^\\beta}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}[Proof of the lemma]\nWe start with the first case.\nWe will show this estimate by combining the gradient estimate from the \u0141ojasiewicz inequality and the estimate of the mapping flow. This will give us an estimate of the time integral of $\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s}$ for $s=-3$, which we will then improve via parabolic regularity.\nWe consider the spinor flow\n$$\\partial_t \\Phi_t = Q(\\Phi_t), \\Phi_0 = \\Phi,$$\nthe gauged spinor flow\n$$\\partial_t \\tilde{\\Phi}_t = \\tilde{Q}(\\tilde{\\Phi}_t), \\tilde{\\Phi}_0 = \\Phi$$\nand the mapping flow\n$$\\partial_t f_t = P_{g_t, \\bar{g}}(f), f_0 = \\operatorname{id}_M.$$\nThen we have that\n$$\\tilde{\\Phi}_t = F_t^* \\Phi_t$$\nand hence\n\\begin{eqnarray*}\n\\tilde{Q}(\\tilde{\\Phi}_t) & = & \\partial_t (F_t^* \\Phi_t) \\\\\n& = & F_t^* \\tilde{\\mathcal{L}}_{X_t} \\Phi_t + F_t^* \\dot{\\Phi}_t\n\\end{eqnarray*}\nwhere $X_t = \\frac{d}{dt} f_t$ and $\\tilde{\\mathcal{L}}$ is the spinorial Lie derivative.\n\nMultiplication of Sobolev functions $H^k \\times H^s \\to H^s$ for negative $s$ and positive $k$ is continous, if $k > -s$ and $k > n\/2$, where $n$ is the dimension of the manifold, see theorem 2 (i), sect. 4.4.3 in \\cite{Runst96}. In particular, our choice of $k$ allows any $s \\geq -3$.\n\nWe will use this to estimate $\\tilde{\\mathcal{L}}_{X_t} \\Phi_t$ in the $H^s$ norm.\nRecall that \n$$\\tilde{\\mathcal{L}}_X \\Phi = (\\mathcal{L}_X g, \\tilde{\\mathcal{L}}_X \\varphi) = (2 \\delta_g^* X^{\\flat}, \\nabla_X^g \\varphi - \\frac{1}{4} dX^{\\flat} \\cdot \\varphi).$$\nIn local coordinates we have\n$$\\mathcal{L}_X g = p_1(g_{jk}, \\partial_l g_{mn}, X^i) +  p_2(g_{ij}, \\partial_k X^l)$$\nfor some polynomials $p_1, p_2$, which are linear in the partial derivative terms and the $X^i$ terms. Likewise we have\n$$\\tilde{\\mathcal{L}}_X \\varphi = q_1(X^i, \\partial_j\\varphi^{\\alpha}) + q_2(g_ij, \\partial_l g_{mn}, X^k, \\varphi^{\\alpha})$$\nfor polynomials $q_1, q_2$, linear in the partial derivative terms and the $X^i$ terms.\nFrom this follows, using the multiplication theorem above and the fact that $H^{k-1}$ is a Banach algebra (since it embeds into $C^2$),\n\\begin{eqnarray*}\n\\|\\tilde{\\mathcal{L}}_X \\Phi\\|_{H^s} & \\leq & C \\left( \\|DX\\|_{H^s} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k-1}}^d +  \\|X\\|_{H^s} \\sum_{d=1}^r \\|D\\Phi\\|_{H^{k-1}}^d \\right)\\\\\n& \\leq & C \\left( \\|X\\|_{H^{s+1}} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k-1}}^d + \\|X\\|_{H^s} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k}}^d \\right)\\\\\n& \\leq & C \\left(\\|X\\|_{H^{s+1}} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k}}^d\\right)\n\\end{eqnarray*}\nfor $k > -s + n\/2 + 2$, where $r$ is the maximal degree of the polynomials $p_1, p_2, q_1, q_2$.\nSince we will choose $s=-3$ and $k > n\/2 + 5$, this will be the case.\n\nFurthermore, given a diffeomorphism $f: M \\to M$ and a lift to the topological spin structure $F: \\tilde{P} \\to \\tilde{P}$, we have\n$$F^* \\Phi = \\Phi \\circ F,$$\nwhere we view $\\Phi$ as an equivariant map $\\Phi: \\tilde{P} \\to \\left(\\widetilde{\\operatorname{GL}_n^+} \\times \\Sigma_n\\right)\/\\operatorname{Spin}(n)$.\nUsing the transformation rule, we can derive an estimate\n$$\\|u \\circ f\\|_{W^{k,p}(M)} \\leq \\nu(\\|f\\|_{C^{\\max \\{k, 1\\}}}) \\|u\\|_{W^{k,p}(M)}$$\nfor the integral Sobolev spaces.\nFor real $s$, we conclude the following inequality by interpolation and duality\n$$\\|F^* \\Phi\\|_{H^s} \\leq \\tilde{\\nu} (\\|F\\|_{C^{\\lceil|s|\\rceil}}) \\|\\Phi\\|_{H^s},$$\nwhere $\\nu, \\tilde{\\nu}: [0, \\infty) \\to [0, \\infty)$ are continuous functions.\n \nIn conclusion we obtain\n\\begin{eqnarray*}\n\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s} & = & \\|F_t^* \\tilde{\\mathcal{L}}_{X_t} \\Phi_t + F_t^* \\dot{\\Phi}_t\\|_{H^s} \\\\\n& \\leq & C \\nu(\\|F_t\\|_{C^{\\lceil|s|\\rceil}}) ( \\|X_t\\|_{H^{s+1}} \\|\\Phi_t\\|_{H^k} +  \\|\\dot{\\Phi}_t\\|_{H^s} )\n\\end{eqnarray*}\nWe will assume both $f_t$ and $\\Phi_t$ to remain in a bounded $H^k$ neighborhood, thus we can estimate their norms by a constant, hence we obtain\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s} \\leq C (\\|\\dot{f}_t\\|_{H^{s+1}} + \\|\\dot{\\Phi}_t\\|_{H^s}).$$\n\nIt remains to choose a neighborhood of $\\bar{\\Phi}$ so that we can also estimate the terms $\\|\\dot{f}_t\\|_{H^{s+1}}$ and $\\|\\dot{\\Phi}_t\\|_{H^s}$.\n\nBy theorem \\ref{L2cv} there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ it holds\n$$\\int_T^{T_{\\max}} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C e^{-\\alpha T}.$$\n\nChoose a neighborhood $U\\times V_m$ of $(\\operatorname{id}_M, \\bar{g})$ such that we have the mapping flow estimate \\ref{MFE}.\nChoose a neighborhood $V_s$ of $\\bar{\\Phi}$, such that we have the $L^2$ estimate of the gradient along the spinor flow as in theorem \\ref{L2cv}. We may assume that $\\pi_{\\Sigma}(V_s) = V_m$.\nFurthermore, we choose the neighborhoods to be bounded in $H^k$.\n\nNow choose $\\Phi \\in V_s$ as initial condition for the spinor and the spinor-DeTurck flow. As above we denote these flows by $\\Phi_t$ and $\\tilde{\\Phi}_t$ respectively and by $f_t$ we mean the associated mapping flow.\nWe will now estimate the integral of the $H^{-3}$ norm of $\\tilde{Q}(\\tilde{\\Phi}_t)$.\nRecall that we have \n$$\\int_{T_1}^{T_2} \\|\\dot{\\Phi}_t\\|_{L^2} dt \\leq C e^{-\\alpha T_1}$$\nfrom theorem \\ref{L2cv}. For $\\dot{f_t}$ we get the estimate\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt \\leq C \\left( \\int_0^{T_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt + \\int_{T_1}^{T_2} \\|\\dot{g}_t\\|_{L^2} dt + e^{-\\lambda T_1} \\right).$$\nThe second term can be bounded by $C e^{-\\alpha T_1}$ by the previous estimate, since $\\|\\dot{g}_t\\|_{L^2} \\leq \\|\\dot{\\Phi}_t\\|_{L^2}$. The first term we decompose into\n$$\\int_0^{T_1\/2} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt < C e^{-\\lambda T_1\/2}$$\nand\n$$\\int_{T_1\/2}^{T_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt  < C e^{-\\alpha T_1\/2}$$\nagain using the estimate for $\\|\\dot{g}_t\\|$.\nThus\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt < C e^{-\\mu T}$$\nfor some $C>0, \\mu >0$. We will use the same constants in the estimate of $\\dot{g}_t$.\nPutting these estimates together we obtain\n\\begin{eqnarray*}\n\\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt & \\leq & C \\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} + \\|\\dot{\\Phi}_t\\|_{H^{-3}} dt\\\\\n& \\leq & C e^{-\\mu T_1}\n\\end{eqnarray*}\nSince $\\tilde{Q}$ is a continuous map from $H^k$ to $H^{k-2}$, because $H^k$ embeds into $C^3$, and $\\tilde{\\Phi}_t$ is in a bounded $H^k$ neighborhood, we obtain that $\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} \\leq \\tilde{C}$.\nHence we may estimate\n$$\\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}}^2 dt \\leq \\tilde{C} \\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt \\leq C \\tilde{C} e^{-\\mu T_1}.$$\n\nSince $\\tilde{Q}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$ fulfills the linear strongly parabolic equation\n$$\\partial_t \\tilde{Q}_t = D\\tilde{Q}(\\tilde{\\Phi}_t) \\tilde{Q}_t,$$\nwe may now apply the parabolic estimate \\ref{PE}  to obtain\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_{T + \\delta})\\|_{H^k} \\leq C e^{-\\mu T} = \\tilde{C} e^{-\\mu (T+\\delta)}.$$\n(Since $\\tilde{\\Phi}_t$ remains in a bounded neighborhood of $\\bar{\\Phi}$, the parabolic inequality for $D\\tilde{Q}(\\tilde{\\Phi}_t)$ can be chosen independent of $\\tilde{\\Phi}_t$.\nIn particular $\\delta$ can be chosen independently of $T$ and $\\Phi$, hence the estimate gets worse by a constant factor $e^{\\mu \\delta}$.)\n\nThe argument for the estimate (\\ref{GEb}) is identical and for the estimate (\\ref{GEc}) the argument runs in parallel until we apply the gradient estimate. Then we get the following estimate:\n$$\\int_{T_1}^{T_2} \\|\\dot{\\Phi}_t\\|_{L^2} dt \\leq \\frac{C}{1 + T^{\\beta}}.$$\nSimilarly as above, we can estimate\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt \\leq \\frac{C}{1 + T^{\\beta}}.$$\nThus\n$$\\int_{T_1}^{T_2} \\|\\tilde{\\mathring{Q}}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt \\leq \\frac{C}{1 + T^{\\beta}}$$\nand hence\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq \\frac{C}{1 + T^{\\beta}}$$\nas claimed.\n\\end{proof}\n\n\n\\begin{proof}[Proof of theorem \\ref{stabP}]\nIn the following $B_\\rho$ denotes the ball of radius $\\rho$ around $\\bar{\\Phi}$ with respect to the $H^k$ norm, and in this proof ``flow'' always refers to the gauged spinor flow. \nUsing lemmas \\ref{UniformExistence} and \\ref{GradientDecay}, choose $0 < \\gamma < \\delta < \\epsilon$ and $T$, such that\n\\begin{enumerate}[label=(\\roman*)]\n\\item The estimate from lemma \\ref{GradientDecay} holds on $B_{\\epsilon}$.\n\\item For any $\\Phi \\in B_{\\delta}$ the flow exists until time $1$ and stays in $B_{\\epsilon}$\n\\item $\\int_T^{\\infty} C e^{-\\alpha t} dt < \\frac{\\delta}{3}$, where $C$ and $\\alpha$ as in lemma \\ref{GradientDecay}\n\\item For any $\\Phi \\in B_{\\gamma}$ the flow exists until time $T$ and remains in $B_{\\delta\/3}$.\n\\end{enumerate}\nNow let $\\Phi \\in B_{\\gamma}$.\nThen denote by $\\Phi_t$ the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t), \\Phi_0 = \\Phi.$$\nDenote by $\\hat{T} \\in (0, \\infty]$ the maximal time, such that the flow with initial condition $\\Phi$ exists in $B_{\\delta}$. The condition on $B_{\\delta}$ ensures that $\\Phi_{\\hat{T}}$ exists and $\\|\\Phi_{\\hat{T}} - \\bar{\\Phi}\\|_{H^k} = \\delta$.\nOn the other hand,\n\\begin{eqnarray*}\n\\|\\bar{\\Phi} - \\Phi_{\\hat{T}}\\|_{H^k} & \\leq & \\|\\bar{\\Phi} - \\Phi_T\\|_{H^k} + \\|\\Phi_T - \\Phi_{\\hat{T}}\\|_{H^k} \\\\\n& \\leq & \\frac{\\delta}{3} + \\int_T^{\\hat{T}} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\\\\n& \\leq & \\frac{\\delta}{3} + \\int_T^{\\hat{T}} C e^{-\\alpha t} dt \\\\\n& \\leq & \\frac{2}{3} \\delta\n\\end{eqnarray*}\nThis is a contradiction and we conclude $\\hat{T} = \\infty$.\nAdditionally, \n$$\\int_T^{\\infty} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\leq \\frac{\\delta}{3},$$\nand we conclude that the limit\n$$\\Phi_{\\infty} = \\Phi_T + \\int_T^{\\infty} \\tilde{Q}(\\Phi_t) dt$$\nexists in $H^k$ and\n$$\\|\\Phi_{\\infty} - \\Phi_t\\|_{H^k} \\leq \\int_t^{\\infty} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\leq C e^{-\\alpha t}.$$\nSince \n$$\\lim_{t\\to \\infty} \\mathcal{E}(\\Phi_t) = 0,$$\n$\\Phi_{\\infty}$ is a critical point.\nWe have shown that the gauged spinor flow converges for $\\Phi \\in B_{\\gamma}$ to a critical point in $B_{\\delta}$.\nGiven that the mapping flow is a strongly parabolic equation, the velocity along the flow solves a linear strongly parabolic equation and we can apply the parabolic regularity estimate and the mapping flow estimate to obtain that the mapping flow converges exponentially in any $H^k$ norm. Since the spinor flow is given by $(F_t^{-1})^* \\Phi_t$, the spinor flow also converges exponentially.\n\\end{proof}\n\\begin{proof}[Proof of theorem \\ref{stabVC}]\nWhen the critical set is a manifold, the proof is entirely analogous to the previous proof. If the critical set is not a manifold, we have the weaker estimate\n$$\\|\\tilde{Q}(\\Phi_t)\\|_{H^k} \\leq \\frac{C}{1 + T^{\\gamma}}.$$\nThe exponent $\\gamma$ can be computed from $\\theta$ in the \u0141ojasiewicz inequality as $\\gamma = \\frac{\\theta - 1}{2 - \\theta}$. Hence if $\\theta > 3\/2$, $\\gamma > 1$. In that case we find\n$$\\int_T^{\\infty} \\frac{C}{1 + t^{\\gamma}} dt \\leq C\\frac{1}{T^{\\gamma-1}} \\xrightarrow{T \\to \\infty} 0$$\nand we can show existence and convergence of the flow as in the previous proof.\nWe define\n$$\\Phi_{\\infty} = \\Phi_T + \\int_T^{\\infty} \\tilde{Q}(\\Phi_t) dt$$\nand using that\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\frac{C}{1+ T^{\\beta}}$$\nwe obtain\n$$\\mathcal{E}(\\Phi_{\\infty}) = \\lim_{t\\to \\infty} \\mathcal{E}(\\Phi_t) = \\mathcal{E}(\\bar{\\Phi})$$\nand hence $\\Phi_{\\infty}$ is also a local minimum, and in particular a critical point of $\\mathcal{E}|_{\\mathcal{N}^1}$.\nThe speed of convergence is then given by $\\frac{1}{T^{\\gamma-1}}$.\n\\end{proof}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIdentifying line outages is crucial to understanding current system conditions and preventing cascading outages.\nHistorically, line outages have been identified based on breaker status indications; however, sometimes the reported line statuses are incorrect. \nFailure to update the system models can cause inaccurate state estimation and threaten system reliability~\\cite{NERC_EMSoutage_2017}, especially for systems that are under stress. \nFor example, in the 2011 San Diego blackout, operators could not detect overloaded lines because of an incorrect network model~\\cite{OutageReport2011,Chen2015Quickest}.\n\nPhasor measurement units (PMUs), because of their unique characteristics (namely, global synchronization and high reporting rate), have been installed on many modern power grids to provide better visibility of grid behavior.\nIn particular, there have been many reported efforts to use PMUs to improve topology models (in particular, to detect one or more line outages) using various approaches (\\cite{Tate_line_2008,Chen2015Quickest,Tate_double_2009,Zhu_sparse_2012,Emami_external_2013}).\nMethods in~\\cite{Tate_line_2008} and~\\cite{Tate_double_2009} are based on hypothesis testing for single-line and double-line outage cases, while reference~\\cite{Zhu_sparse_2012} proposes an overcomplete representation and formulates the problem in terms of sparse vector estimation.\nAlternative approaches include integer programming (\\cite{Emami_external_2013}) and identification based on the small-signal linearized power grid model (\\cite{Chen2015Quickest}).\nMost of the existing works use PMU voltage angle measurements rather than both magnitude and angle data since they rely on dc power flow models, which only considers angles.\nPMU-based disturbance detection methods have been developed and used by utilities and operators.\nLine switching events are identified as one of the applications ~\\cite{NASPI_using}.\nIn fact, several applications are being developed and tested (e.g., by ISO New England for external system transmission element tripping and PJM for detection and triangulation of large disturbances).\nMeanwhile, Operador Nacional do Sistema El\u00e9trico (ONS) in Brazil deployed a WAMS system which identifies transmission line tripping based on angle disturbance of nearby PMUs.\nTo the best of our knowledge, details of these applications are not publicly available for comparison.\nIn this research, we extended the single line outage identification algorithm in~\\cite{Tate_line_2008} by utilizing voltage phasor measurements from PMUs.\nFirst, the algorithm computes expected voltage phasors for all possible outage scenarios by solving ac power flows.\nNext the pre- and post-outage voltage phasor difference can be calculated (hereinafter called expected values).\nBy comparing the expected and the observed values, the hypothetical outage scenario that is closest to the observation will be identified. \nUnlike prior approaches, which have relied on the relatively inaccurate dc power flow model, the proposed method uses the full ac power flow model to identify outages.\nIdeally, system responses after different outages are distinct enough to be correctly identified.\nHowever, due to measurement uncertainty, if two outages lead to similar (but not identical) responses, they may be confused and thus misidentified.\nIn some cases, misidentification may lead to wrong operation that aggravates the situation especially when the system is under stressful conditions.\nFor example, misoperation (tripping heavily loaded but not faulted lines) played a significant role in the 2003 US-Canada blackout and the 2015 Turkish blackout~\\cite{abdullah_distance_2018}.\nIn such cases, a more conservative result (inconclusive) sometimes is better than an incorrect result (i.e., no identification is better than an erroneous identification).\nThis practical problem has not been addressed in previous research.\nIn order to acknowledge measurement uncertainty and improve identification accuracy, a rejection filtering technique is introduced.\nWith the rejection filter in place, instead of definitive identification results, the events are now labeled as conclusive or inconclusive.\nIn summary, a two-stage framework for single line outage identification is proposed.\n\nExtensive tests have been conducted to show the relationships between identification results and different conditions (including filtering methods, threshold values, and PMU placements) on the IEEE 30-bus system.\nAdditionally, results using the Ontario network are presented.\nSection~\\ref{sec: Algorithm} gives the first stage of the algorithm for single line outage identification.\nSection~\\ref{sec:filtering} discusses the rejection filtering algorithm (the second stage). \nCase studies are presented in Section~\\ref{sec:case study}.\nLastly, concluding remarks and future work are presented in Section~\\ref{sec: conclusions and future work}.\n\n\\section{Stage 1: Main Identification Algorithm}\n\\label{sec: Algorithm}\nIn this section, the main identification algorithm will be presented. \nThe assumptions we make include: \\text{(1)} all measurements are taken from a system in quasi-steady state, and \\text{(2)} the ac power flow model is available, since steady-state models are available in modern energy management systems for power flow calculations, such as contingency analysis~\\cite{wu_power_2005}.\n\nInspired by~\\cite{Tate_line_2008}, the algorithm is based on hypothesis testing using voltage phasor measurements.\nBy comparing the simulated voltage changes due to each hypothetical line outage to the observed, the case that is closest to the observations is identified as the outage source.\nOutages are assumed to be equally likely in all lines (that would not lead to islands).\n\\Cref{Eqn: criterion} to \\Cref{Eqn: V obs} describe the identification algorithm. \n\\begin{align}\n\\label{Eqn: criterion}\nl^{\\ast}&=\\text{arg} \\underset{l \\in \\mathcal{L}} {\\: \\min} \\:E(l)\\\\\n\\label{Eqn: E def}\nE(l)&=\\|\\Delta\\boldsymbol{\\bar{V}_{exp,l}}-\\Delta\\boldsymbol{\\bar{V}_{obs}}\\|\\\\\n\\label{Eqn: V exp}\n\\Delta\\boldsymbol{\\bar{V}_{exp,l}}&=(\\boldsymbol{\\bar{V}_{exp,l}}-\\boldsymbol{\\bar{V}^{pre}})\\\\\n\\label{Eqn: V obs}\n\\Delta\\boldsymbol{\\bar{V}_{obs}}&=\\boldsymbol{\\bar{V}_{obs}^{post}}-\\boldsymbol{\\bar{V}_{obs}^{pre}}\n\\end{align}\nAn error measure $E(l)$ given a hypothetical outage in line $l$ (where ${l \\in \\mathcal{L}}$) is defined as (\\ref{Eqn: E def}) to quantify the difference between the expected voltage phasor change and its observed counterpart.\nSimilar to the list commonly used in online contingency analysis~\\cite{savulescu2014real}, the set $\\mathcal{L}$ represents all lines to be checked for outage occurrence, the cardinality of which (denoted by ${L}$) is the number of lines to be checked.\nThe most likely line ${l^{\\ast}}$ that leads to the smallest ${{E}(l)}$ (\\ref{Eqn: criterion}) is identified as the cause of outage.\nThe observed voltage change after an event ($\\Delta\\boldsymbol{\\bar{V}_{obs}}$) is defined as the difference between the observed pre-event voltage ($\\boldsymbol{\\bar{V}_{obs}^{pre}}$) and the post-event voltage ($\\boldsymbol{\\bar{V}_{obs}^{post}}$).\nAnalogously, the expected voltage change (${\\Delta \\boldsymbol{\\bar{V}_{exp,l}}}$) is defined based on the pre-event voltage ($\\boldsymbol{\\bar{V}^{pre}}$, computed from state estimator) and the expected post-event voltage (${\\boldsymbol{\\bar{V}_{exp,l}}}$, computed for all potential line outages ${l \\in \\mathcal{L}}$ through power flow calculations).\n\nNote that $\\boldsymbol{\\bar{V}^{pre}}$ is used instead of the pre-outage voltage measured by PMUs because (\\ref{Eqn: V exp}) can then be updated constantly and does not require detection of an outage.\nIn contrast, (\\ref{Eqn: criterion}), (\\ref{Eqn: E def}), (\\ref{Eqn: V obs}) and (\\ref{Eqn: rank}) are only updated when an outage is detected.\nIn our study, the ac power flow method, in particular, the Newton method, is used to solve for ${\\boldsymbol{\\bar{V}_{exp,l}}}$.\nThis is the most computationally expensive step.\nFor example, one successful ac power flow solution takes around 0.1 s for the Ontario system (with 3488 buses, introduced in Section \\ref{sec: line out Ontario}) using MATPOWER~\\cite{Zimmerman_matpower_2011}.\nBy comparison, the rest of the outage identification only takes $0.0075$ s due to highly efficient sorting algorithms.  \nAll identification algorithms were implemented in MATLAB on a computer with an Intel Xeon E5-1607 processor and 8 GB RAM.  \nThe results can likely be improved using better computation resources and simple parallelism \\cite{Jun_1995,roberge_parallel_2017}.\nHowever, simulation of system responses due to hypothetical outages is commonly used in contingency analysis, which reduces the additional computation burden associated with the proposed algorithm.\nNote that unlike phase angles that are typically used in previous works, all voltage vectors in our algorithm are phasors (with both magnitude and angle) with a length of $P$, where $P$ is the number of PMUs in a system.\nEmpirically, the algorithm is based on the following ranking of the error measure ${{E}(l)}$.\n\\setlength{\\belowdisplayskip}{3pt}\n\\setlength{\\abovedisplayskip}{3pt}\n\\setlength{\\belowdisplayshortskip}{3pt}\n\\setlength{\\abovedisplayshortskip}{3pt}\n\\begin{align}\n\\label{Eqn: rank}\n0\\leq E(l^{\\ast}=l_{r_1})\\leq E(l_{r_2}) \\leq \\ldots \\leq E(l_{r_{L}})\n\\end{align}\nThe subscript of $l_{r_i}$ means this line occupies the $i^{\\text{th}}$ place in the ranking among $L$ potential candidates.\nThe line outage ($l^{\\ast}$) that leads to the smallest error (i.e., $1^{\\text{st}}$ in the ranking) with respect to the observation is identified as the cause. \nGiven PMU measurements due to the actual outage of line $l_a$, if  ${l^{\\ast} = l_a}$, then the outage is successfully identified.\n\n\\section{Stage 2: Rejection Filtering Algorithms}\n\\label{sec:filtering}\nWhen measurements are ideal without uncertainty, almost all outage cases should be identified correctly even with a small number of PMUs installed in the system. \nTwo exceptions would be series lines with zero injection at shared buses and identical parallel lines. \nGiven measurements with uncertainties, more outage cases (not the aforementioned situations) are likely to be misidentified.\nMeasurement uncertainty may stem from measurement errors (e.g., instrument transformers, the A\/D converters, and the communication cables~\\cite{zhu_enhanced_2006}) or random errors (e.g. unknown fluctuations that can not be captured deterministically in principle).\nIn our research, we use independent Gaussian distributions for voltage phasor magnitude and angle to model measurement uncertainty.\n\nWhether a case will be misidentified is determined by the system as well as the measurement uncertainty model.\nA simple example for a 4-bus system (\\cite{Zimmerman_matpower_2011,grainger_power_1994}) is used to demonstrate the impact of uncertainty on identification results.\nThe system consists of 4 buses and 4 branches with a PMU installed on bus 2 and 1 (the reference bus).\nAssuming there is no measurement error on the reference bus, the ranking in (\\ref{Eqn: rank}) solely depends on measurements at bus 2.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{delta_phasor_misranks.pdf}\n\t\\caption[Impact of measurement uncertainty on line outage identification in a 4-bus system]{Demonstration of measurement uncertainty impact on identification, ${\\Delta\\boldsymbol{\\bar{V}_{exp,l}},l\\in \\{1,2\\}}$ and ${\\Delta\\boldsymbol{\\bar{V}_{obs,1}}}$ in 4-bus system, ${\\sigma_{v}=0.002\/\\sqrt{3}}$ pu and ${\\sigma_{\\theta}=0.01\/\\sqrt{3}}$ degrees}\n\t\\label{fig_4gs}\n\\end{figure}\nFig.~\\ref{fig_4gs} shows simulated responses after the outage of line 1 (pink crosses) considering uncertainty, along with the ideal response due to the outage of line 1 (blue cross) and line 2 (blue star).\nThe pink crosses represents 1000 random realizations corresponding to possible measurements after the line 1 outage, generated based on independent distributions.\nSpecifically, we assume the magnitude and the angle uncertainty follow a zero-mean Gaussian distribution with standard deviation ${\\sigma_{v}}$ and ${\\sigma_{\\theta}}$ respectively.\nThe main identification algorithm is represented by the solid line that partitions the graph into two regions.\nAny point in the left region is closer to ${\\Delta\\boldsymbol{\\bar{V}_{exp,1}}}$ (represented by the blue cross), which means if an observation falls in this region, line 1 will be correctly identified.\nHowever, due to measurement uncertainty, the observations may fall to the right region and be closer to ${\\Delta\\boldsymbol{\\bar{V}_{exp,2}}}$ (represented by the blue star). \nIn such cases, the event will be misidentified.\n\nIf we set up a rejection filter (represented by the band delineated by two dashed lines on each side of the original identification solid line) so that all cases in the right region are labeled as inconclusive, then they will not be misidentified.\nThey are in fact misidentified-filtered cases (those would have been misidentified but filtered out to be inconclusive).\nHowever, depending on how we define the filter threshold (band position and width), it is likely that some cases in the left region also fall within the band and become inconclusive (i.e., correct-filtered cases).\n\nThere may be different ways to set up the filter threshold. \nIn this study, the threshold is determined by $\\Delta E$, where ${\\Delta E = E(l_{r_2}) - E (l_{r_1})}$. \nNamely the difference between the first two most promising candidates is used to set up the threshold.\n\\begin{equation}\n\\Delta E^{(r)}_{l_a} = E^{(r)}_{l_a}(l_{r_2}) - E^{(r)}_{l_a} (l_{r_1}) < \\epsilon,\n\\end{equation}\nwhere ${E^{(r)}_{l_a}(l)}$ denote the error measure in (\\ref{Eqn: E def}) but for the actual outage ${l_a}$ specifically.\nDue to randomness, multiple runs are conducted based on multiple, independent uncertainty distributions.\nThe superscript $r$ indicates the error measure is computed based on the $r^{th}$ random realization.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{four_types_graffle_random.pdf}\n\t\\caption{Categories of line identification results (30-bus system with 20 PMUs)}\n\t\\label{fig_results_illustrate_1}\n\\end{figure}\nA visual representation of this process is in \\cref{fig_results_illustrate_1}.\nThe area of each circle represents the number of cases in each category.\nThe red inner circle represents misidentified cases which correspond to the cases in the right region in Fig.~\\ref{fig_4gs} (i.e., misidentified).\nIf the rejection filter is defined as an ellipse which will not be coincident with the inner circle, then by covering the inner circle (thus eliminating misidentified), some correct cases will become inconclusive (correct filtered) as well.  \nThe filter threshold defines the sizes of ellipses.\nAs will be shown in the case studies, as the threshold is increased, the misidentified cases will decrease to zero while the inconclusive cases will go up as a result of increased number of correct-filtered cases.\nThis corresponds the bottom right circle in Fig.~\\ref{fig_results_illustrate_1}. \nIn the course of minimizing the misidentified cases, we probably sacrifice some cases which would have been identified correctly. \nThe goal is to minimize misidentified cases without introducing a significant number of correct-filtered cases.\n\n\n\\section{Case Studies}\n\\label{sec:case study}\nA comprehensive comparison of different techniques and the choices of filter thresholds are presented in this section. \nThe first experiment is the comparison of the ac and the dc approach using the IEEE 30-bus test system. \nThen the results using rejection filtering are presented.\nLastly, the impact of the rejection filter threshold is discussed. \nThe identification algorithm is further tested using a 3488-bus model of the Ontario power system.\nIn all tests, we assume the measurement uncertainty in the magnitude and angle observed follows a zero mean Gaussian distribution~\\cite{chakhchoukh_pmu_2014} with standard deviation ${0.002\/\\sqrt{3}}$ pu and ${0.01\/\\sqrt{3}}$ degrees respectively.\n1000 different placements are generated randomly for each possible ${P\\in\\{2,\\ldots,B\\}}$ where ${B}$ is the total number of buses in a system.\nIf the total number of placements is less then 1000, all placements are considered.\nFor comparison, 100 randomly generated measurements for each PMU are shared among all placements.\n\n\\subsection{IEEE 30-Bus System}\n\\begin{figure}[hbtp]\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 15 0,width=\\linewidth]{ac_dc_pcorrect.pdf}\n\t\t\\caption{correctly identified, no filter}\n\t\t\\label{fig_ac_dc_pcorrect}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 15 0,width=\\linewidth]{ac2_set2.pdf}\n\t\t\\caption{ac, $\\Delta E$}\n\t\t\\label{fig_ac2_set2}\n\t\\end{subfigure}\n\t\\caption[Comparison of dc and ac approaches (${E_{r_2}}$ and $\\Delta E$) for line outage identification]{Comparison of dc and ac approaches (with or without $\\Delta E$) for line outage identification (The threshold $\\epsilon$ is chosen so that the mean misidentification rate is driven under $0.00015\\%$)}\n\t\\label{fig: Comparison of DC and 2 AC}\n\\end{figure}\nFig.~\\ref{fig_ac_dc_pcorrect} shows the correctly identified rate of ac and dc approaches without filtering techniques for the IEEE 30-bus system.\nAs expected, this figure shows that the ac approach is needed to achieve high identification accuracy.\nFor example, even in the best-case scenario of complete PMU coverage, the dc accuracy is only 70.15\\%, whereas the ac accuracy is 97.4\\%.  \nFig.~\\ref{fig_ac2_set2} shows\nperformance of the rejection filter based on four types of identification results: correct, misidentified, correct-filtered and misidentified-filtered.\nThe results in Fig. \\ref{fig: Comparison of DC and 2 AC} are generated using thresholds so that the misidentification rate is driven to nearly zero (specifically the mean misidentification rate is driven under $0.00015\\%$ over all random realizations, all placements and all outages).\nIf we compare Fig.~\\ref{fig_ac_dc_pcorrect} to \\ref{fig_ac2_set2}, the drop of correctly identified rate is because of increased inconclusive cases.\nFor example, for the ac approach, with 30 PMUs the accuracy was 97.4\\%.\nIt drops to around $83\\%$ when the $\\Delta E$ filter is used.\n\n\\subsection{Results of Different Rejection Filter Thresholds}\nThis section discusses the impact of the rejection filter $\\epsilon$ on the identification results. \nTo isolate impact of the threshold, the results should be generated using different PMU placements.\nAt the beginning of the section, we mentioned 1000 different placements are generated randomly for each possible ${P\\in\\{2,\\ldots,B\\}}$ where ${B=30}$.\nAdditionally, 100 randomly generated measurements for each PMU are shared among all placements.\nThe number of lines in the 30-bus system to be checked for outages is 38.\nIf we consider all these possible combinations, $1.102\\times10^8$ cases are needed for just one threshold level.\nSince enumeration over all possible scenarios takes significant amount of time, we focus on the behavior with a varying range of $\\epsilon$ (from 0 to 0.0045) and four specific levels of PMU coverage (10\\%, 20\\%, 50\\% and 100\\%).\nThe measurement uncertainty levels remain the same as in the previous section.\n\n\n\\begin{figure}[!ht]\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus10per.pdf}\n\t\t\\caption{10\\% coverage}\n\t\n\t\t\\label{fig_diff_threshold_30bus_10per}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus20per.pdf}\n\t\t\\caption{20\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_20per}\n\t\\end{subfigure}\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus50per.pdf}\n\t\t\\caption{50\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_50per}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 0 0, width=0.83\\linewidth]{30bus100per.pdf}\n\t\t\\caption{100\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_100per}\n\t\\end{subfigure}\n\t\\caption[Identification results vs $\\epsilon$ (30-bus system, $\\Delta E$ filter, four coverage levels)]{Identification results vs $\\epsilon$ (30-bus system, $\\Delta E$ filter with different coverage (legends are the same for 4 figures))}\n\t\\label{fig_ac_thresholds_complete}\n\\end{figure}\nThe trade-off between correct and misidentified cases as a result of different $\\epsilon$ is illustrated by Fig.~\\ref{fig_ac_thresholds_complete}.\nThe horizontal lines in each figure represents the correctly identified\/misidentified level without filtering.\nFor example, without any rejection filter, with $10\\%$ coverage, the misidentified rate is about $45\\%$ while the correctly identified rate is $55\\%$.\nThis corresponds to a red inner circle with an area of $45\\%$ of the entire circle in Fig.~\\ref{fig_results_illustrate_1}.\nBy increasing the threshold, more misidentified cases (in red) will be filtered out and become misidentified-filtered (in pink), whilst correctly identified cases (in green) become correct-filtered (in yellow).\n\nFig.~\\ref{fig_ac_thresholds_complete} indicates the misidentified percentage (in red) is very sensitive to the threshold when $\\epsilon < 1\\times10^{-3} $ (particularly when the coverage is low).\nIn such cases, the misidentified rate drops rapidly as $\\epsilon$ increases from zero.\nIt also decays faster than the correctly identified rate (by comparing red to green).   \nFor example, with 10\\% coverage (Fig.~\\ref{fig_diff_threshold_30bus_10per}), by increasing the threshold from 0 to $1\\times10^{-3}$, the misidentified rate drops more than 40\\% while the correctly identified rate only drops about 30\\%.\nHowever, when the misidentified rate is low (around zero), further elimination of misidentified cases leads to significant loss of correctly identified cases. \nFor example, by increasing the threshold from $1\\times10^{-3}$ to $2\\times10^{-3}$ in Fig.~\\ref{fig_diff_threshold_30bus_10per}, the green area decreases significantly (about $10\\%$) compared to the trivial gain in eliminating red cases. \nThe results also indicate that, ultimately, higher coverage is needed to achieve better identification accuracy.\nAs the number of PMUs goes up, the ratio between correctly identified and correct-filtered cases also increases (i.e., green versus yellow).\nThis means fewer correctly identified cases will be sacrificed when the number of PMUs is high. \n\nDepending on the acceptable level of the misidentification rate or the correct-filtered rate, operators can decide on an empirical value for the threshold.\nIf misidentification is considered to be worse than a lower correct rate (e.g. compared to false alarms, the operator would rather accept more inconclusive cases), then the threshold can be set accordingly to eliminate misidentified cases.\nOtherwise, a very small non-zero level (e.g. $0.0015\\%$) can be chosen instead, with minimal impact on the correctly identified cases.\n\n\\subsection{Results of the Ontario Power System}\n\\label{sec: line out Ontario}\nTo evaluate performance on a more realistic system, results were also obtained using a model of the Ontario power system, consisting of 3488 buses, 864 generators, 1290 loads, 2242 branches and 1697 transformers.\nAmong the original branches, 1762 single-line outage cases are to be checked for occurrence without introducing islands.\nPMUs are assumed to be placed on the actual PMU locations and, alternatively, a set of high voltage buses in Ontario.\n\nFirst, tests were conducted using 26 bus voltages based on the current PMU locations in the Ontario network \\cite{curtis_north_2011}.\nTo evaluate the performance of the line outage detection algorithms for realistic, future PMU deployments, we also considered cases where all buses above a certain voltage level are monitored by PMUs.\nIn particular, we consider two cases: monitoring all buses with nominal voltages greater than or equal to 230 kV (an additional 53 buses) or 220 kV (an additional 842 buses).\n\\begin{figure}[hbtp]\n\t\\centering\n\t\\begin{subfigure}[b]{0.5\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim=0 0 20 20, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_mis.pdf}\n\t\t\\caption{26 PMUs}\n\t\t\\label{fig_ieso_26PMU}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.5\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim=0 0 20 0, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_79_mis.pdf}\n\t\t\\caption{79 PMUs (230 kV+)}\n\t\t\\label{fig_ieso_79PMU}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\includegraphics[trim=0 0 20 0, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_868_mis.pdf}\n\t\t\\caption{868 PMUs (220 kV+)}\n\t\t\\label{fig_ieso_868PMU}\n\t\\end{subfigure}\n\t\\caption{Identification results vs $\\epsilon$ (the Ontario system, $\\Delta E$ filter, three coverage levels)}\n\t\\label{fig_ieso_complete}\n\\end{figure}\nFig.~\\ref{fig_ieso_complete} shows the identification results using different thresholds.\nConsistent with the previous results, the observations about impact of filter threshold and PMU coverage are still valid for the Ontario system.\nWhen the coverage goes up, the threshold required to eliminate misidentified cases also decreases, which results in not only a higher correctly identified rate but also higher ratio of correct versus correct-filtered.\nThe horizontal line in each figure represents the separation of correctly identified (above the line) and misidentified (below the line) without rejection filter.\nThe accuracy given by the initial 26 PMUs is low (22.7\\% correct versus 77.3\\% misidentified) due to the very limited number of PMUs and the large number of outage scenarios.\nHowever, as the coverage increases by adding PMUs at high voltage buses, the results have been improved significantly.\nWhen 220 kV+ buses are monitored (868 PMUs), the correctly identified rate without filtering rises up to more than 70\\%.\nThis indicates monitoring high voltage buses is very beneficial and should be considered for future PMU deployments.\n\n\\section{Conclusions and Future Work}\n\\label{sec: conclusions and future work}\nThe current proposed methods have been implemented and tested on several systems of different scales.\nThe results show that the proposed identification algorithm using the ac power flow model achieves better identification accuracy compared to the dc approach. \nAdditionally, relatively high identification accuracy is achieved with a small number of PMUs.\nBy using rejection filtering techniques, the misidentified rate can be further reduced, which is crucial for online utilization of the event detection.\nThe results also demonstrate there are significant benefits of having a higher PMU coverage.\nLow number of PMUs makes it difficult to identify the outage on the Ontario system, but future PMU deployments (e.g., with all high voltage buses monitored) exhibit good performance. \n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMixed action approaches have been studied by several groups,\nsuch as Domain-wall fermion (DWF) valence on Asqtad fermion sea~\\cite{Edwards:2005ym}, overlap valence on DWF sea~\\cite{Allton:2006nu}, overlap valence on \nclover sea~\\cite{Durr:2007ez}, and overlap valence\non twisted-mass fermion sea~\\cite{Cichy:2009dy}. In view of the fact that it is numerically intensive to simulate chiral fermions\n(DWF or overlap), it is deemed practical to use the cheaper fermion formulation for generating gauge configurations and\nthe more expensive fermion discretization for the valence as an expedient approach toward dynamical QCD simulations with chiral fermions.\nMany current algebra relations depend only on the chiral properties of the\nvalence sector. The mixed action theory with different fermions for the valence and the sea is a generalization of the\npartially quenched theory with different sea and valence quark masses.\nThe mixed action partially quenched chiral perturbation theory (MAPQ$\\chi$PT) has been developed for\nGinsparg-Wilson fermions on Wilson sea~\\cite{Bar:2003mh} and staggered sea~\\cite{Bar:2005tu}, and has been worked out for\nmany hadronic quantities to next-to-leading order (NLO), such as pseudoscalar masses and decay \nconstants~\\cite{Bar:2003mh,Bar:2005tu,Chen:2007ug},\nisovector scalar $a_0$ correlator~\\cite{Golterman:2005xa,Prelovsek:2005rf,Aubin:2008wk,WalkerLoud:2008bp},\nheavy-light decay constants~\\cite{Aubin:2005aq}, and baryon masses~\\cite{Tiburzi:2005is, WalkerLoud:2008bp}.\n\nIn the mixed action chiral perturbation theory with chiral valence fermions, it is shown~\\cite{Bar:2003mh} that to\nNLO there is no $\\mathcal{O}(a^2)$ correction to the valence-valence meson mass due to\nthe chiral symmetry of the valence fermion. Furthermore, both the chiral Lagrangian and the chiral extrapolation\nformulas for hadron properties to the one-loop level (except $\\theta$-dependent quantities) are independent of the\nsea fermion formulation~\\cite{Chen:2006wf}. The LO mixed-action chiral Lagrangian involves only one more term with\n$\\mathcal{O}(a^2)$ discretization dependence which is characterized by a low energy constant $\\Delta_{mix}$.\nThe LO pseudoscalar meson masses for overlap valence and DWF sea are given as\n\\begin{eqnarray}   \\label{eq:dd}\nm_{vv'}^2 &=& B_{ov}(m_v +m_{v'}), \\nonumber \\\\\nm_{vs}^2 &=& B_{ov}m_v + B_{dw}(m_s + m_{res}) + a^2\\Delta_{mix}, \\nonumber \\\\\nm_{ss'}^2 &=& B_{dw}(m_s + m_{s'} + 2 m_{res}),\n\\end{eqnarray}\nwhere $m_{vv'}\/m_{ss'}$ is the mass of the pseudoscalar meson made up of valence\/sea quark and antiquark.\n$m_{vs}$ is the mass of the mixed valence and sea pseudoscalar meson. Up to numerical accuracy, there is no\nresidual mass for the valence overlap fermion. The DWF sea has a residual mass $m_{res}$ which vanishes as\n$L_S \\rightarrow \\infty$. $\\Delta_{mix}$ enters in the mixed meson mass $m_{vs}$ as an\n$\\mathcal{O}(a^2)$ error which vanishes in the continuum limit. We should note that, unlike the partially\nquenched case, even when the quark masses in the valence and sea match, the unitarity is still violated due\nto the use of mixed actions. The degree of unitarity violation at finite lattice spacing depends on the size of $\\Delta_{mix}$.\n\n$\\Delta_{mix}$ has been calculated for DWF\nvalence and Asqtad fermion sea which gives\n$\\Delta_{mix} = 0.249(6)\\,{\\rm GeV}^4$ \\cite{Orginos:2007tw} at $a=0.125\\,{\\rm fm}$, $0.211(16)\\,{\\rm GeV}^4$ at $a=0.12\\,{\\rm fm}$ \\cite{Aubin:2008wk}, and $0.173(36)\\,{\\rm GeV}^4$ at $a=0.09\\,{\\rm fm}$ \\cite{Aubin:2008wk}. It is also calculated for overlap valence and clover sea which yields $\\Delta_{mix} = 0.35(14)\/0.55(23)\\,{\\rm GeV}^4$ at $m_{\\pi} = 190\/300\\,{\\rm MeV}$ and $a=0.09\\,{\\rm fm}$~\\cite{Durr:2007ef}.~This means that for a valence pion of $300\\,{\\rm MeV}$, $\\Delta_{mix}$ produces, at $a=0.12\\,{\\rm fm}$, a shift of $\\sim$ $110-240\\,{\\rm MeV}$ and,\nat $a =0.09\\,{\\rm fm}$, a shift of $\\sim$ $55-153\\,{\\rm MeV}$ for these cases, which are substantial portions of\nthe valence pion mass. \n\nWe have used valence overlap fermions on the 2+1-flavor DWF sea to study hadron \nspectroscopy~\\cite{Li:2010pw,Dong:2009wk,Mathur:2010ed,Gong:2011nr}.  In this work, we calculate \n$\\Delta_{mix}$ for such a mixed action approach which is needed for chiral extrapolation\nin MAPQ$\\chi$PT.\n\n\n\\section{Calculation Details}\nThe $\\Delta_{mix}$ parameter is calculated on four ensembles of the $2+1$-flavor domain wall fermion gauge configurations \\cite{Allton:2008pn, Mawhinney:2009jy}. Two different lattice spacings were used to study the dependence on the cutoff. In addition, we also used multiple sea masses, for the $32^3 \\times 64$ lattices, to study the sea quark mass dependence of $\\Delta_{mix}$.   Details of the ensembles are listed in Table \\ref{tab:ensembles}.\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nLattice Size& $a^{-1}(\\,{\\rm GeV})$& $a m_l$& $ m_{ss}(\\,{\\rm MeV})$\\\\ \n\\hline\n\\hline\n$24^3 \\times 64$ & 1.73(3) & 0.005 & 329(1)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.004 & 298(1)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.006 & 350(2)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.008 & 399(1)\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Details of the DWF ensembles used in this work.}\n\\label{tab:ensembles}\n\\end{table}\n\nThere are different parameterization schemes \\cite{Orginos:2007tw, Aubin:2008wk} one can use to relate $\\Delta_{mix}$ to the quark and pseudoscalar masses from Eq.~(\\ref{eq:dd}). In this paper we choose to parameterize $\\Delta_{mix}$ as \n \\begin{equation}\n\\label{eq:dmix}\nm_{vs}^2 -\\frac{1}{2}m_{ss}^2 = B_{ov} m_v + a^2\\Delta_{mix}.\n\\end{equation}\nThis is similar to the parameterization used in \\cite{Aubin:2008wk}. The quantity $\\delta m^2 \\equiv m_{vs}^2 -\\frac{1}{2}m_{ss}^2$, has a linear behavior in $m_v$ and can be calculated directly by computing only the pseudoscalar masses $m_{vs}$, and $m_{ss}$.  We see that in the regime where Eq.~(\\ref{eq:dmix}) is valid, $\\Delta_{mix}$ is equivalent to $\\delta m^2$ in the limit $ m_v \\rightarrow 0$.\n\nIn this work we used an overlap operator with a HYP-smeared kernel. This was shown to have better numerical properties~\\cite{Li:2010pw}. We calculated the masses $m_{vs}$ and $m_{ss}$ using 50 configurations for each ensemble.  Recall from Section I that $m_{ss}$ requires the propagators for the sea quark mass which were computed with DWF. $m_{vs}$ needs the propagators for both overlap and DWF. The DWF propagators were made available by LHPC.  The overlap propagators were computed using a polynomial approximation \\cite{Alexandru:2011sc} to the matrix sign function. They  were used to compute $14-16$ values of $m_{vs}$. A multi-shifted version of the conjugate gradient algorithm \\cite{Jegerlehner:1996pm}  was implemented for the overlap propagator calculation to compute all masses at once. To accelerate the inversions for the overlap propagators we employed a deflation technique which has been seen to speed up the calculation significantly~\\cite{Li:2010pw}.  \n\nWe fitted correlators using single state exponential fits to extract the pion masses. The fitting windows were adjusted to get a reasonable $\\chi^2\/dof$ and are the same for all the masses in each ensemble. For the $24^3 \\times 64$ ensemble the details are presented in Table \\ref{tab:pfit1}.  For one of the sea quark mass in the  $32^3 \\times 64$ ensemble, the results are presented in Table \\ref{tab:pfit2}. \n\\begin{table}[htdp]\n\\begin{center}\n\\caption{Pion mass fitting details for the $24^3 \\times 64$ lattice. $m_{ss}  \\simeq 329\\,{\\rm MeV}$.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$a m_{v}$& $m_{vs}(\\,{\\rm MeV})$  & $\\chi^2_{vs}\/dof$ & $m_{vv}(\\,{\\rm MeV})$  &$\\chi^2_{vv}\/dof$ \\\\ \n\\hline\n\\hline\n0.0014& 274(3) & 1.0 &122(4) & 1.8 \\\\ \n0.0027& 278(2) & 1.1 &154(2) & 1.4 \\\\ \n0.0046& 287(2) & 1.2 &188(2) & 1.6 \\\\ \n0.0081& 305(2) & 1.2 &242(2) & 1.6 \\\\ \n0.0102& 315(2) & 1.2 &270(2) & 1.4 \\\\ \n0.0135& 331(2) & 1.1 &308(2) & 1.3 \\\\ \n0.0153& 339(1) & 1.1 &327(2) & 1.3 \\\\ \n0.0160& 342(1) & 1.1 &334(2) & 1.3 \\\\ \n0.0172& 347(1) & 1.1 &346(2) & 1.3 \\\\ \n0.0243& 378(1) & 1.2 &409(1) & 1.6 \\\\ \n0.0290& 397(1) & 1.2 &445(1) & 1.7 \\\\ \n0.0365& 426(1) & 1.2 &498(1) & 1.5 \\\\ \n0.0434& 451(1) & 1.2 &542(1) & 1.3 \\\\ \n0.0489& 471(1) & 1.2 &576(1) & 1.2 \\\\ \n0.0670& 531(1) & 1.0 &677(1) & 1.3 \\\\ \n0.0710& 543(1) & 1.0 &698(1) & 1.4 \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\label{tab:pfit1}\n\\end{center}\n\\end{table}\n\\begin{table}[htdp]\n\\begin{center}\n\\caption{Pion mass fitting details for the $32^3 \\times 64$ lattice for $a m_l = 0.004$.  $m_{ss}  \\simeq 298\\,{\\rm MeV}$.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$a m_{v}$& $m_{vs}(\\,{\\rm MeV})$  & $ \\chi^2_{vs}\/dof$ & $m_{vv}(\\,{\\rm MeV})$  &$\\chi^2_{vv}\/dof$ \\\\ \n\\hline\n\\hline\n0.0007& 237(3)&1.0& 126(10)&1.5\\\\\n0.0015& 238(2)&1.0& 136(7)& 1.2\\\\\n0.0025& 246(2)&1.0& 160(4)& 1.2\\\\\n0.0035& 254(2)&1.0& 184(3)& 1.2\\\\\n0.0046& 264(2)&1.0& 209(3)& 1.2\\\\\n0.0059& 274(2)&1.0& 235(3)& 1.2\\\\\n0.0068& 282(2)&1.0& 253(3)& 1.3\\\\\n0.0076& 289(2)&1.0& 268(3)& 1.4\\\\\n0.0089& 299(2)&1.0& 288(3)& 1.6\\\\\n0.0112& 317(2)&1.1& 324(4)& 1.0\\\\\n0.0129& 329(2)&1.1& 337(5)& 1.0\\\\\n0.0152& 344(2)&1.2& 365(5)& 1.3\\\\\n0.0180& 362(2)&1.3& 396(6)& 1.5\\\\\n0.0240& 399(3)&1.6& 454(9)& 1.9\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{tab:pfit2}\n\\end{center}\n\\end{table}%\n\n\n\\section{Fitting strategies and Results}\n\n\\begin{figure*}[t]\n\\includegraphics[width= 3.4in]{mvvsq24_2.pdf}\n\\includegraphics[width= 3.4in]{mvvsq32_2.pdf}\n\\caption{$(a m_{vv})^2\/2(a m_v)$ as a function of $a m_v$ for\nthe $24^3 \\times 64$ ensemble (left) and \nthe three $32^3 \\times 64$ ensembles (right).}\n\\label{fitregion}\n\\end{figure*}\n\nExtracting $\\Delta_{mix}$ requires fits to squares of multiple pion propagators with different valence quark masses; the resulting \nmasses will be correlated since they all come from a single ensemble. \nTo compute the covariance matrix we would need to perform an augmented $\\chi^2$-fit involving the pion propagators for all valence quark masses simultaneously.  This is not numerically stable due to the large number of parameters in the model.  We thus have to use an alternative procedure to take into account these correlations. To gauge the systematic errors introduced by our choice of fitting method we will use multiple fitting procedures. In this section we will describe three different fitting strategies. These methods differ in the way we account for correlations among the different valence masses.\n \nIn method I we follow the standard jackknife philosophy by defining a $\\Delta_{mix}$ estimator directly in terms of the raw pion propagators; the error is then determined by the variance over the jackknife ensemble. In method II we use the jackknife procedure to estimate the $\\delta m^2$ covariance matrix and then compute $\\Delta_{mix}$ using a standard correlated fit~\\cite{Luscher:2010ae}. These two procedures are used for both the $24^3 \\times 64$ and $32^3 \\times 64$ ensembles. However, since there are three sea quark masses for the $32^3 \\times 64$ volume, we can perform an {\\em uncorrelated} fit using $\\delta m^2$ computed on the three independent ensembles---this is our method III. From the different methods we can obtain a systematic uncertainty for our results.   \n\nIn each method we performed a binned jackknife analysis of $\\delta m^2 =m_{vs}^2 -\\frac{1}{2}m_{ss}^2$. Because we performed four inversions with four different point sources per configuration we binned in units of four and eight. This allowed us to drop one or two whole configurations in each jackknife subsample. We find the uncertainties in both cases to be of comparable size, indicating that autocorrelation is negligible.  \n\n\nThe first step in our fitting is to determine a range of quark masses where the tree-level relation between $m^2_{vv}$ and $m_v$ holds so that we can use Eq.~(\\ref{eq:dd}). Below this range, one expects to see chiral logs including partially quenched logs and other non-linear $m_v$ dependence from the NLO in $\\chi$PT. Above this range, tree-level $\\chi$PT is not expected to be valid.  We do this by plotting $(a m_{vv})^2 \/2(a m_v)$ as a function of $a m_v$ as shown in Fig. \\ref{fitregion}. We choose the fitting range in the region where the ratio $(a m_{vv})^2\/2 (a m_v)$ is fairly flat; these ranges are tabulated in Table \\ref{dmix:fitrange} and are used in all three fitting methods.  We note that in the range we are fitting, $m_{\\pi}L > 4$ for both $m_{vs}$ and $m_{vv}$ so that the volume dependence is expected to be small.\n\n\\begin{table}[b]\n\\begin{center}\n \\begin{tabular}{|c|c|c|}\n  \\hline\n   Lattice&$a m_l$& $a m_v $ fit range\\\\\n   \\hline\n   \\hline\n   $24^364$&0.005&0.0243 - 0.0489 \\\\\n   $32^364$&0.004&0.0112 - 0.0240\\\\\n   $32^364$&0.006&0.0112 - 0.0240 \\\\\n   $32^364$&0.008&0.0112 - 0.0240\\\\\n   \\hline\n   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Range of quark masses, $a m_v$, used in the fitting procedures to extract $\\Delta_{mix}$ via Eq.~(\\ref{eq:dd}).}\n\\label{dmix:fitrange}\n\\end{table}\n\n\\begin{figure*}[t]\n\\includegraphics[width= 3.4in]{dmix_24.pdf}\n\\includegraphics[width= 3.4in]{all3.pdf}\n\\caption{Extracting $\\Delta_{mix}$ from a linear extrapolation of $\\delta m^2$ for the $24^3 \\times 64$ ensemble (left) \nand the three $32^3 \\times 64$ ensembles (right). For the $32^3 \\times 64$ plot the inset shows the intercept of the \nfit and the error bars of the extracted values of $\\Delta_{mix}$.}\n\\label{dmixfit}\n\\end{figure*}\n\n\\subsection{Method I:  weighted averaging}\nIn this method we used a weighted linear fit to extract the value of $\\Delta_{mix}$ for each bin.  The weights, $\\sigma_{\\delta m^2}^2$, are given by \n\\begin{equation}\n\\sigma_{\\delta m^2}^2 = 4 m_{vs}^2\\sigma_{ m_{vs}}^2 + m_{ss}^2 \\sigma_{m_{ss}}^2,\n\\label{dmix:err1}\n\\end{equation}\nwhere $\\sigma_{m_{vs}}$, and  $\\sigma_{m_{ss}}$ are the uncertainties of the mixed and DWF pion masses respectively. Eq.~(\\ref{dmix:err1}) was derived using the standard error propagation formula neglecting correlation between $m_{vs}$ and $m_{ss}$. The cross-correlations will be accounted for by the external jackknife procedure. By using weighted fitting, less importance is given to data points with larger uncertainties.  After fitting each bin we then have a jackknife ensemble \\{$\\Delta_{mix}$\\}. We use square brackets, [~], to indicate a particular jackknife sample and angle brackets, $\\langle ~\\rangle$, to denote jackknife averages.  For the case of $\\Delta_{mix}$, its jackknife average and the corresponding uncertainty is given by\n\\begin{eqnarray}\n\\langle \\Delta_{mix} \\rangle &=& \\frac{1}{N}\\sum_{k=1}^{N}\\left[ \\Delta_{mix}\\right]_k \\nonumber \\\\\n\\sigma_{\\Delta_{mix}} &=& \\sqrt{(N-1)\\left(\\langle \\Delta_{mix}^2\\rangle-\\langle \\Delta_{mix}\\rangle^2 \\right)}\\nonumber. \\\\\n\\end{eqnarray}\n\nThe extracted values of $\\Delta_{mix}$, using this method, are presented in Table \\ref{dmix:method1}. We also list the corresponding values of $B_{ov}$.  Figure \\ref{dmixfit} shows $a^2 \\delta m^2$ as a function of $a m_{v}$ and the corresponding linear fit.\n\n\\begin{table}[b]\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|c|}\n \\hline\n   \\multirow{2}{*}{Lattice}&\\multirow{2}{*}{$a m_l$}& \\multicolumn{2}{|c|}{$\\Delta_{mix}(\\,{\\rm GeV}^4)$}& \\multicolumn{2}{|c|}{$B_{ov}$(\\,{\\rm GeV})}\\\\\n   &&\\multicolumn{1}{c}{I}&II&\\multicolumn{1}{c}{I}&II\\\\\n   \\hline\n   \\hline\n   $24^364$&0.005&0.032(6)  & 0.028(5)  &1.85(2)  &1.88(2) \\\\\n   $32^364$&0.004&0.040(14)& 0.025(9)  &1.88(10)&1.95(6)\\\\\n   $32^364$&0.006&0.054(9)  & 0.050(8)  &1.81(5)  &1.81(5)\\\\\n   $32^364$&0.008&0.059(13)& 0.063(13)&1.74(6)  &1.73(6) \\\\\n   \\hline\n   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Extracted values of $\\Delta_{mix}$ and $B_{ov}$ using fitting methods I and II.}\n\\label{dmix:method1}\n\\end{table}\n\n\n\n \n  \n  \n  \n  \n  \n  \n  \n  \n \n  \n  \n\n\\subsection{Method II: covariance matrix}\nIn our second method we perform a correlated valence-quark mass fit to the function $\\delta m^2  = B_{ov} m_v + a^2 \\Delta_{mix}$. The central values are taken to be the jackknife averages,\n\\begin{equation}\n\\langle(\\delta m^2)_i \\rangle = \\frac{1}{N}\\sum_{k=1}^N\\left [(\\delta m^2)_i\\right]_k \\,.\n\\end{equation}\nWe minimize the correlated $\\chi^2$-function,\n\\begin{eqnarray*}\n\\chi^2& =& \\sum_{i,j} \\left(\\langle(\\delta m^2)_i \\rangle - (\\delta m^2)_i\\right)C_{i,j}^{-1} \\nonumber\\\\\n&\\times&\\left(\\ \\langle(\\delta m^2)_j \\rangle -(\\delta m^2)_j\\right) \\,, \\nonumber \\\\\n\\end{eqnarray*}\nwith the covariance matrix, $C_{i,j}$, given by the jackknife estimate\n\\begin{eqnarray}\nC_{ij} &=& \\frac{(N-1) } {N} \\sum_{k=1}^N \\left(\\left[(\\delta m^2)_i\\right]_k - \\langle(\\delta m^2)_i \\rangle \\right) \\nonumber \\\\\n&\\times& \\left(\\left[(\\delta m^2)_j\\right]_k - \\langle (\\delta m^2)_j \\rangle\\right) \\,. \\nonumber \\\\\n\\label{covmat}\n\\end{eqnarray}\nThe subscripts $i$ and $j$ index the valence-quark mass. The factor $(N-1)\/N$ differs from the usual definition of the covariance matrix to account for the correlation of the jackknife samples~\\cite{Luscher:2010ae}. The fit uncertainties are obtained by constructing the standard Hessian matrix. Results of this method are tabulated in Table \\ref{dmix:method1}.\n  \n   \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\\subsection{Method III: uncorrelated fitting}\nThe methods described in the two previous sections closely parallel the method performed by \\cite{Aubin:2008wk} in which for each lattice ensemble a string of partially quenched meson masses are calculated; from them we extract $\\Delta_{mix}$.  In this section we perform a fit on the three $32^3 \\times 64$ ensembles based on the value of $\\delta m^2$ measured at the point where the valence and sea quark masses match \\footnote{As mentioned in \\cite{Aubin:2008wk} one can never get rid of partial quenching effects even in the case where the pseudoscalar mesons of both the DWF and overlap actions match. This is because discretization errors are different for the two actions.}.  There are no cross-correlations among the masses since the ensembles are independent. \n\nThe value of $\\delta m^2$ is computed for the pion mass that most closely satisfies the condition $m_{ss} \\approx m_{vv}$. Because it is not  easy to match a priori the pseudoscalar masses and it is expensive to regenerate overlap propagators with different masses, we performed interpolation among the existing data points to obtain a better approximation of where $m_{ss}$ and $m_{vv}$ match.\n\nWe do an uncorrelated $\\chi^2$-fit with the error bars obtained by a jackknife procedure. Figure \\ref{dmix:32method3} shows the data points and the fit results. For this case we find $\\Delta_{mix} = 0.042(24)\\,{\\rm GeV}^4$, and $B_{ov} = 1.82(16)\\,{\\rm GeV}$. \n\n\\begin{figure}[t]\n\\includegraphics[width= 3.4in]{3264_full3.pdf} \n\\caption{Determining $\\Delta_{mix}$ by performing the linear extrapolation as in  Fig.~\\ref{dmixfit}.  Each point corresponds to one of the $32^3 \\times 64$ lattices where $m_{ss} \\approx m_{vv}$. A magnified view of the extrapolation in the neighborhood of the chiral limit, along with the extrapolated value, is shown in the inset.}\n\\label{dmix:32method3}\n\\end{figure}\n\n\\section{Discussion}\n\n\nAs a first step, we look at the lattice spacing dependence. We compare the two ensembles with $m_{ss}$ close to $300\\,{\\rm MeV}$.  \nThese correspond to $a m_l = 0.005$ and $a m_l = 0.004$ with $a = 0.114\\,{\\rm fm}$ and $a=0.085\\,{\\rm fm}$ respectively.\nWe average the central values and the errors from the two fitting methods, separately for each lattice spacing. \nWe find at $a = 0.114\\,{\\rm fm}$ $\\Delta_{mix} = 0.030(6)\\,{\\rm GeV}^4$ and for $a = 0.085\\,{\\rm fm}$ $\\Delta_{mix} = 0.033(12)\\,{\\rm GeV}^4$. \nWe see that the lattice spacing dependence is smaller than our errors; this indicates that $\\Delta_{mix}$ is capturing \nthe dominant lattice artifact for the parameters used in this study.  \n\nTo discuss the sea quark dependence of $\\Delta_{mix}$ we plot in Fig.~\\ref{dmix:32method4} the results of \nmethods~I, II, and III for the ensembles with $a = 0.085\\,{\\rm fm}$.  We note that the results are consistent within \ntwo sigma. This is consistent with LO MAPQ$\\chi$PT in Eq.~(\\ref{eq:dd}) where $\\Delta_{mix}$ is a low energy \nconstant, independent of the valence and sea masses.\n\nWe now combine the $\\Delta_{mix}$ values extracted from the two lightest sea mass ensembles to produce our final result.\nWe use these ensembles because the LO MAPQ$\\chi$PT is expected to describe the data better at lower sea quark masses.\nFor each of the fitting methods we combine the results of the $a = 0.114\\,{\\rm fm}$ and $a=0.085\\,{\\rm fm}$ ensembles.\nSince the two ensembles are statistically independent, it is straightforward to combine these results: for method~I we get \n$\\Delta_{mix} = 0.033(6)\\,{\\rm GeV}^4$ and for method~II we get $\\Delta_{mix} = 0.027(5)\\,{\\rm GeV}^4$. We can now use the values\ndetermined using these two methods to estimate the systematic fitting errors. \nWe quote the final result with two uncertainties, the first statistical and the second associated with fitting systematics.\nThe central value and the statistical error are taken to be the average of the results from methods~I and II. \nThe systematic error is the standard deviation of the results from these two methods. \nWe get $\\Delta_{mix}=0.030(6)(5)\\,{\\rm GeV}^4$.\n\n\n\\begin{figure}[t]\n\\includegraphics[width= 3.4in]{3264_vs_chiral.pdf} \n\\caption{$\\Delta_{mix}$ for $ a= 0.085\\,{\\rm fm}$. The empty symbols indicate the results of method I and the full symbols are from method II. The continuous line and the shaded region are the results of the chiral extrapolation in method III.}\n\\label{dmix:32method4}\n\\end{figure}\n\n\n\n\n\nTable \\ref{dmix:others} lists calculated values of $\\Delta_{mix}$ for\npion masses close to $300\\,{\\rm MeV}$ using different mixed actions.  We\nnotice that our values of $\\Delta_{mix}$ are  significantly smaller\nthan overlap on clover or DWF on Asqtad for comparable lattice spacings and pion masses. \nFor both $m_{vv}$ and $m_{ss}$ at $300\\,{\\rm MeV}$, our calculated $\\Delta_{mix}$ will\nshift the pion mass up by $10$ and $16\\,{\\rm MeV}$ for the $32^3 \\times 64$ lattice at\n$a = 0.085\\,{\\rm fm}$ and $24^3 \\times 64$ lattice at $a = 0.114\\,{\\rm fm}$, respectively.\nThese are substantially smaller than the corresponding $55-153\\,{\\rm MeV}$ and\n$110-240\\,{\\rm MeV}$ shifts that we mentioned in Sec. I.\nThe value of $\\Delta_{mix}$ is significantly smaller in our case probably because \nthe sea and valence fermion actions are similar.  Both of them are approximations of \nthe matrix sign function, but with different kernels.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n~~~~Mixed Action& Ref. & $a\\,{\\rm fm}$&$\\Delta_{mix}(\\,{\\rm GeV}^4)$\\\\\n\\hline\n\\hline\nDWF on staggered& \\cite{Orginos:2007tw}&0.125& 0.249(6)\\\\\nDWF on staggered&\\cite{Aubin:2008wk}& 0.12& 0.211(16)\\\\\nDWF on staggered& \\cite{Aubin:2008wk}& 0.09& 0.173(36)\\\\\noverlap on clover& \\cite{Durr:2007ef}&0.09&0.55(23)\\\\\noverlap on DWF    & this work & 0.114 & 0.030(6)\\\\\noverlap on DWF    & this work & 0.085& 0.033(12)\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{$\\Delta_{mix}$ values for DWF valence quarks on staggered sea quarks for pion mass at $300\\,{\\rm MeV}$.}\n\\label{dmix:others}\n\\end{table}\n\nThe previous calculation \\cite{Li:2010pw} of $\\Delta_{mix}$, for\noverlap on the DWF sea, has roughly the same value but the sign is\nnegative. That calculation measured $\\Delta_{mix}$ by examining the\nstates that wrap around the time boundary.  This indirect \nmethod is less reliable and the errors are large. \n\n\\section{Conclusion}\nWe calculated the additive mixed action pseudoscalar meson mass parameter,\n$\\Delta_{mix}$, for the case of valence overlap fermions on a DWF sea.\n$\\Delta_{mix}$ is significant because it enters\ninto mixed action partially quenched chiral perturbation theory (MAPQ$\\chi$PT) for\nchiral extrapolation of many low energy observables.\n\nTwo different lattice spacings were used to examine the cut-off behavior.\nFor a pion mass close to $300\\,{\\rm MeV}$ we find $\\Delta_{mix}=0.030(6)\\,{\\rm GeV}^4$ at $a =0.114\\,{\\rm fm}$ and \n$\\Delta_{mix}=0.033(12)\\,{\\rm GeV}^4$ at $a =0.085\\,{\\rm fm}$.  They are the same within errors. \nOur calculated $\\Delta_{mix}$ will\nshift the pion mass up by $10$ and $16\\,{\\rm MeV}$ for the $32^3 \\times 64$ lattice at\n$a = 0.085\\,{\\rm fm}$ and $24^3 \\times 64$ lattice at $a = 0.114\\,{\\rm fm}$, respectively.\nWe studied the sea quark mass dependence of $\\Delta_{mix}$ at $a = 0.085\\,{\\rm fm}$ and\nwe find that they agree within two sigma. Combining the results extracted from\nthe ensembles with the lightest sea quarks, we get $\\Delta_{mix}=0.030(6)(5)\\,{\\rm GeV}^4$,\nwhere the first error is statistical and the second is the systematic error\nassociated with the fitting method.\n \nWhen compared to previous mixed action studies, DWF on staggered or overlap on clover, the values of $\\Delta_{mix}$ of overlap on  DWF\nare almost an order of magnitude smaller. This is most likely due to the fact that the sea and valence fermions\nare similar. \n\n\\begin{acknowledgements}\nWe would like to thank J. Negele, A. Pochinsky, M. Engelhardt, and\nLHPC for generously making available the DWF propagators for all the\nensembles used in this paper. This work is supported in part by\nU.S. Department of Energy grants DE-FG05-84ER40154,\nDE-FG02-95ER40907, GW IMPACT collaboration, DE-FG02-05ER41368, and DST-SR\/S2\/RJN-19\/2007, India.\n\\end{acknowledgements}\n\n\\newpage\n\\bibliographystyle{jhep-3}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nThe compensation of quantum numbers plays a key role in our understanding of the\nfragmentation process whereby partons transform into observable hadrons. \nConsequently, baryon production in hadronic \\ensuremath{{\\rm e}^+{\\rm e}^-}\\ annihilation final states\nprovides data very well suited to test \nphenomenological fragmentation models. In particular, the study of di-lambda \npairs allows a subtle testing of model predictions because of the relatively \nlarge rates and the necessity to compensate two quantum numbers:  \nbaryon number and strangeness.  \n\nFragmentation models such as \\jt \\cite{jt} and \\hw \\cite{hw} are based on\na chainlike production of hadrons with local compensation of quantum numbers.\nIn \\jt, particle production is implemented via string fragmentation. Baryons (B)\nare formed when a diquark pair is contained in the string \n(see diagram a below), thus resulting in a strong baryon-antibaryon \ncorrelation. This correlation can be softened by the ``popcorn effect'' when \nan additional meson (M) is produced between the baryon pair as shown in \nthe diagrams b and c below.\nIn contrast, \\hw\\ describes fragmentation via the formation of clusters and \ntheir subsequent decay. Baryons are produced by the isotropic cluster decay \ninto a baryon pair, which can result in stronger correlations than those \npredicted by \\jt .\n\n\\vspace*{-2.5cm}\n\\begin{center}\n  \\resizebox{\\textwidth}{!}{\n    \\includegraphics{pr251_diagr.eps}\n    }\n\\end{center}\n\\vspace{-2.5cm}\n\nDi-lambda production in multihadronic \\ensuremath{{\\rm Z}^0}\\ decays has been studied over the \npast years by experiments at {\\sc Petra}, {\\sc Pep} and \\lep \n\\cite{petra,pep,aleph_corr,delphi_corr,opal_corr}. These experiments \nreport short-range correlations as observed in the distributions of the \nrapidities $y$ or rapidity differences \\ensuremath{|\\Delta y|}\\ of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs.\nThe rapidity of a particle is defined as\n$y=\\frac{1}{2}\\ln\\left(\\frac{E+p_{\\scriptscriptstyle\\parallel}}{E-\np_{\\scriptscriptstyle\\parallel}}\\right)$, \nwhere $E$ is the energy of the particle and $p_{\\scriptscriptstyle\\parallel}$ \nthe longitudinal momentum with respect to the thrust axis. Rapidity differences \nare Lorentz-invariant under boosts along the event axis.\nThese correlations are compared to predictions of \\jt\\ and \\hw . \nSatisfactory agreement is found with the predictions of \\jt\\ when $\\rho$, the \n``popcorn parameter''\\footnote{The value of $\\rho$ can be set in \\jt\\ with the \nparameter PARJ(5): $\\rho=\\frac{BM\\bar{B}\\rule{0cm}{2.5ex}}{B\\bar{B}+BM\\bar{B}} = \n\\frac{\\rm PARJ(5)}{0.5+{\\rm PARJ(5)}}$.},\nis set to the default value, $\\rho=0.5$. However, the tune of other parameters \nmodeling baryon production significantly influences the predictions \n\\cite{james_early}. \n\\hw\\ on the other hand predicts correlations much larger than those \nexperimentally observed.\n\nThe full data sample of 4.3~million hadronic \\ensuremath{{\\rm Z}^0}\\ decays collected with \nthe {\\sc Opal}\\ detector at \\lep\\ in the region of the \\ensuremath{{\\rm Z}^0}\\ peak is used in this \ninvestigation. It supplements the earlier {\\sc Opal}\\ \nwork\\cite{opal_corr} by increased statistics and a more robust technique  \nto remove the background contributions from the \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ \nsamples in order to obtain a correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ sample which is as clean as \npossible. \nThe \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations are investigated mainly via rapidity differences. \nThey are compared to the earlier \\lep\\ results and to the predictions of  \n\\jt\\ and \\hw. The predictions of the recent \\jt\\ modification \n{\\sc Mops}\\ (MOdified Popcorn Scenarium)~\\cite{mops} are also considered. \nCorrelated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs are also studied in 2-jet events in which models can\nbe tested with improved sensitivity (compared to the full data sample)\nwhen rapidity differences are investigated. \nFinally, we study correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs within the same and within different\njets. \n\nSection~\\ref{sec:procedure} gives a short description of the {\\sc Opal}\\ detector \nand\npresents the selection of the \\l\\ events\\footnote \n{For simplicity \\l\\ refers to both \\l\\ and \\ensuremath{\\bar{\\Lambda}}.}\nin the total sample and also in 2-jet and 3-jet events, for both experimental \nand simulated data.\nIn section~\\ref{sec:method} the separation of the  \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and  \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ samples \nfrom the background and the determination of the rates of correlated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs as a function of the rapidity differences \\ensuremath{|\\Delta y|}\\ are discussed. \nSection~\\ref{sec:results} contains the measured rates with their errors and a \ncomparison to earlier results as well as the presentation of the differential \ndistributions as a function of \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*}, where \\ensuremath{\\theta^*}\\  is the angle \nbetween the thrust axis and the \\l\\ momentum calculated in the rest frame of \nthe di-lambda pair.\nIn section~\\ref{sec:comparison} the models are tested using the production \nrates of \\l\\ pairs as well as the \\ensuremath{\\cos \\theta^*}\\ and \\ensuremath{|\\Delta y|}\\ spectra of correlated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs. \nThe range of di-lambda correlations is investigated in section~\\ref{sec:jets}\nby the assignment of the \\l's to the jets. \nConclusions are drawn in section~\\ref{sec:summary}.\n\\section{Experimental Procedure}\n\\label{sec:procedure}\n\\subsection{The OPAL Detector}\n\\label{subs:detec}\nA detailed description of the {\\sc Opal}\\ detector can be found in \nRef.~\\cite{detector}. \nOf most relevance for the present analysis is the tracking system and \nthe electromagnetic calorimeter. \nThe tracking system consists of a silicon microvertex detector, an inner\nvertex gas chamber, a large-volume jet chamber and specialized chambers at\nthe outer radius of the jet chamber which improve the measurements in\nthe $z$ direction ($z$-chambers)\\footnote\n{The coordinate system is defined so that $z$~is the coordinate parallel \nto the e$^-$ beam axis, $r$~is the coordinate normal to the beam axis, \n$\\phi$~is the azimuthal angle around the beam axis, and $\\theta$~is the \npolar angle \\mbox{with respect to~$z$.}}. The tracking system covers the region\n$|\\cos\\theta|<0.95$ and is located within a solenoidal magnet coil with\nan axial field of~0.435~T. \nThe tracking detectors provide momentum measurements of charged\nparticles, and particle identification from measurements of the\nionization energy loss,  d$E$\/d$x$.\nElectromagnetic energy is measured by a lead-glass calorimeter \nlocated outside the magnet coil, which covers $|\\cos\\theta|<0.98$.\n\\subsection{Data Samples}\n\\label{subs:data}\nThe analysis is based on hadronic \\ensuremath{{\\rm Z}^0}\\ decays collected around the \n\\ensuremath{{\\rm Z}^0}\\ peak from 1990 to 1995 (total \\lep\\ 1 statistics). The hadronic events \nwere selected with the standard {\\sc Opal}\\ procedure~\\cite{tkmh} based on the \nnumber and quality of the measured tracks and the electromagnetic clusters and \non the amount of visible energy in the event. In addition, events with the \nthrust axis close to the beam direction were rejected by requiring \n$|\\cos \\theta_{\\rm {thrust}}|< 0.9$, where $\\theta_{\\rm thrust}$ is the polar \nangle of the thrust axis. With the additional requirement that the jet chamber\nand the z-chambers were fully operational, a total of 3.895 million hadronic \nevents remained for further analysis, with an efficiency of ($98.4 \\pm 0.4$)\\%. \nThe remaining background processes, \nsuch as $\\ensuremath{{\\rm e}^+{\\rm e}^-} \\rightarrow \\tau^+ \\tau^-$ and two photon events, were estimated \nto be at negligible level (0.1\\% or less).\n\nAfter the \\l-selection which will be described below, the selection of 2- and \n3-jet events was performed. Charged tracks and electromagnetic \nclusters not associated with any track were grouped into jets using the \n\\ensuremath{{\\rm k}_{\\perp}}\\ recombination algorithm \\cite{durham} with a cut value\n$y_{\\rm cut}=0.005$. In addition to the standard selection criteria, the\nenergy of the clusters and the momenta of the charged tracks had to be less \nthan 60~GeV\/$c$.\nTo improve the quality of the jets it was finally required that there be at \nleast two charged particles per jet (in addition to the possible tracks from \n\\l\\ decays) and that the minimum energy per jet was $5$~GeV.\nThe cuts on the quality of jets were chosen to be this loose to keep the \nkinematic range as large as possible for comparison with fragmentation models. \nIn total, samples of 1.7 million 2-jet events and 1.4 million 3-jet events \nwere available for further analysis corresponding to $45\\%$ and $36\\%$,\nrespectively, of the entire data set.\n\\subsection{Monte Carlo Event Samples} \n\\label{subs:mc} \nMonte Carlo hadronic events with a full simulation of the OPAL detector \n\\cite{gopal} and including initial-state photon radiation were used \n(a) for evaluation of detector acceptance and resolution \nand (b) for studying the efficiency of the di-lambda reconstruction as a \nfunction of the rapidity differences.  \nIn total, seven million simulated events were available, of which four million\nwere generated by \\jt~7.4 with fragmentation parameters  \ndescribed in~\\cite{jt7.4}, and three million were generated by \\jt~7.3  with  \nfragmentation parameters described in~\\cite{jt7.3}.\nThe two \\jt\\ versions differ in the particle decay tables and heavy meson  \nresonances. \nThere are also some differences in the simulation of baryon production between \nthe two samples.\nTheir small influence on the efficiency correction to the experimental \ndata is accounted for in the systematic errors (see \nsection~\\ref{subs:syserr}). \n\nFor comparison with the experimental results, the Monte Carlo models \n\\jt~7.4 and \\hw~5.9\\cite{hw5.8}\\footnote \n{The fragmentation parameters of \\hw~5.9 were identical to those used in  \n our tuned version of \\hw~5.8\\cite{hw5.8} with the exception of the maximum\n cluster mass ({\\tt CLMAX}) which was set to 3.75 GeV in order to improve the  \n description of the mean charged particle multiplicity in inclusive  \n hadronic \\ensuremath{{\\rm Z}^0}\\ decays.} \nwere used. Both models give a good description of global  \nevent shapes and many inclusive particle production rates, but differ in  \ntheir description of the perturbative phase and their implementation of the  \nhadronization mechanism. \n\nTracks and clusters are selected in the Monte Carlo events, which \ninclude detector simulation, in the same way as for the data, and the \nresulting four-vectors of particles are referred to as being at the \n`detector level'.  Alternatively, for testing the model predictions, \nMonte Carlo samples without \ninitial-state photon radiation nor detector simulation are used, with \nall charged and neutral particles with mean lifetimes greater than \n$3\\times10^{-10}$~s treated as stable.  The four-vectors of the \nresulting particles are referred to as being at `generator level'. \n\\subsection{\\l\\ Reconstruction } \n\\label{subs:lambda} \nNeutral strange \\l\\ baryons were reconstructed in their decay  \nchannel \\l ~$\\rightarrow \\pi^-$p as described in \\cite{opal_sp}. \nBriefly, tracks of opposite charge were paired and regarded as a secondary  \nvertex candidate if the track pair intersection in the plane  \nperpendicular to the beam axis satisfied the criteria of a neutral two-body  \ndecay with a decay length of at least 1 cm.\n \nEach candidate track pair was refitted with the \nconstraint that the tracks originated from a common vertex, and \nbackground from photon conversions was suppressed. Information from  \nd$E$\/d$x$ measurements was used as in \\cite{opal_sp} to help identify the \n$\\pi$ and p for further background suppression, primarily due to \n\\ensuremath{{\\rm K}^{0}_{\\rm S}} $\\rightarrow \\pi^+ \\pi^-$. Two sets of cuts, called `method 1' and \n`method 2' are described in \\cite{opal_sp} for \\l\\ identification.   \nFor the present analysis, \\l\\ candidates were reconstructed using method~1,  \nwhich is optimized to have good mass and momentum resolution. \n\n\\begin{sloppypar} \nBy these means a narrow \\l\\ mass peak above a small background has been \nobtained. The selection of di-lambda candidates with both invariant masses in  \nthe range \\mbox{1.1057~GeV\/$c^2<$ $m_{\\pi p}<$ 1.1257~GeV\/$c^2$} (region A in \nfigure~\\ref{fig:m2dim}) retains most of the \\l\\ signal for further analysis. \n\\end{sloppypar}\n\\section{Selection of Correlated \\l-pairs}\n\\label{sec:method}\n\\subsection{Method}\n\\label{subs:mgeneral}\nEvents with more than one \\l\\ candidate that had passed the above selection \ncriteria were considered and all possible pair combinations of the \\l\\ and \n\\ensuremath{\\bar{\\Lambda}}\\ baryons within an event were formed. This resulted in pairs of \\ensuremath{\\Lambda\\bar{\\Lambda}}, \n\\l\\l\\ and \\ensuremath{\\bar{\\Lambda}}\\lb. Combinations were rejected if the pair had a track in \ncommon. The remaining pairs are henceforth referred to as \\l-pair candidates. \n\nThe three types of baryon pairs can be grouped into two classes: pairs with \ndifferent baryon numbers \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and pairs with equal baryon numbers \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}. \nOnly in \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs can the baryon and flavor quantum numbers be compensated \nby correlated production. \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs can never be produced in correlation\nand hence they will occur only in events with more than one baryon-antibaryon \npair (B$\\bar{\\rm B}$). In such events uncorrelated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs from \ndifferent (B$\\bar{\\rm B}$) pairs are also possible. The number of uncorrelated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs corresponds to the number of pairs with same baryon number. \nHence, the number of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs can be derived via \n\\begin{equation}\nN_{\\ensuremath{\\Lambda\\bar{\\Lambda}}}^{\\rm corr.} = N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}} - (N_{\\ll} + N_{\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}})\\: .\n\\end{equation}\nAt this stage 9479 \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and 4217 (\\l\\l+\\ensuremath{\\bar{\\Lambda}}\\lb) pair candidates are \nselected.\n\\subsection{Background Subtraction and Efficiency Correction}\n\\label{subs:backgr} \nDue to the small statistical errors it is necessary to keep systematic \nuncertainties as low as possible in this analysis. \nThe correct subtraction of {\\mbox{non-\\l\\ }} background from the pairs is \ntherefore of  particular importance. This background consists mainly of other \nlong-lived particles with similar decay topologies (namely \\ensuremath{{\\rm K}^{0}_{\\rm S}} $\\rightarrow\n\\pi^+ \\pi^-$) and random \ntrack  combinations. An important contribution to the contamination is the \nso-called correlated background from \\l candidates that have been reconstructed \nwith one false decay track.   \nThey are more numerous in pairs with opposite baryon number because the \nnumber of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs is far higher than the number of \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs.  \nFor this reason the background has to be estimated in the two samples  \nseparately. \nBackground pairs occur when either one or both \\l-candidates are fake.\nIn the two-dimensional mass plane in figure~\\ref{fig:m2dim}, pairs with one \nfake \\l\\ form horizontal and vertical bands of background, while pairs with \ntwo fake candidates are uniformly distributed in the region above the lower \nmass bounds. \n\nThe background was subtracted using a two-dimensional sideband method. The  \nbackground in the signal region A was measured from two mass windows \n(sidebands) of the same size (regions B$_1$ and B$_2$) placed \nin the two bands of background.  \nIn this way the background with two fake candidates is counted twice. The latter\nwas determined from region C outside the bands.\nHence, the signal is obtained with the subtraction:  \n\\begin{center} \n   Signal = $N_{\\rm A} - (N_{\\rm B_1} + N_{\\rm B_2} - N_{\\rm C})$ .   \n\\end{center}\nWe optimized the position of the sidebands with a MC test investigating the\ndeviations between the background-corrected sample and the true-\\l\\ sample.  \nThe stability of this method was tested in the experimental data by shifting \nthe position of the sidebands by one half of the band size from the optimized \nposition. The fluctuations were of the same size as the deviations found in \nthe MC. \n\nFinally the background-corrected \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ signal distributions were \ncorrected for detector acceptance and reconstruction efficiency as functions \nof \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} . The average efficiency in the total hadronic sample \nwas found to be $\\approx 2\\%$, varying between 1.3\\% and 2.5\\% over the \n\\ensuremath{|\\Delta y|}\/\\ensuremath{\\cos \\theta^*}\\ range.\n\\section{Experimental Results}\n\\label{sec:results}\nIn an earlier OPAL paper~\\cite{opal_corr} based on the 1990 and 1991 data \nsamples we already investigated the production dynamics of baryon-antibaryon \npairs. \nIn this section, we present the rates and differential distributions of \n\\l~pairs using the full 1990 to 1995 LEP~1 data in three samples: \nthe entire set of multihadronic events, the 2-jet and the 3-jet events. \n\\subsection{Pair Production Rates}\n\\label{subs:rates}\nThe resulting rates for \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs in all hadronic events, \ndetermined as sum over all corrected \\ensuremath{|\\Delta y|}\\ bins, are \ngiven in table~\\ref{tab:rates}. The rates for the correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs \nare derived according to equation (1) from the difference of the opposite \nand same baryon number pairs. Compared to the results from other \\lep\\ \nexperiments and to the previous {\\sc Opal}\\ publication, good agreement is found. \n\nThe di-lambda rates in 2- and 3-jet events are listed in  \ntable~\\ref{tab:2_3jrates}.\nIn 3-jet events, due to the higher color charge of the gluons, the average \npair multiplicity is higher. \n\\subsection{Differential Distributions}\n\\label{subs:distributions}\nWe studied the correlations in the differential \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ spectra using the \nobservables \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*}\\ as they are particularly sensitive for  \ncomparison with Monte Carlo models.\nThe differential distributions are shown in figure~\\ref{fig:distrib}.\nThe short-range correlations show up as a peak in the region \\ensuremath{|\\Delta y|}\\ $\\le$ 2.0.\n\nWhen investigating \\ensuremath{|\\Delta y|}\\ distributions, we will restrict ourselves to \n2-jet events.\nThis is due to the fact that in 3-jet events many particle momenta have large \nangles to the thrust axis, resulting in smaller longitudinal momenta and \nsmaller rapidity differences, independent of correlations. As a result the \\ensuremath{|\\Delta y|}\\ \ndistribution is broader and less steep in 2-jet events than in the 3-jet or the \ntotal sample (see figure~\\ref{fig:distrib}a). Consequently, also the range of \nvariations is larger in 2-jet events and yields a higher sensitivity in the \ncomparison with model predictions.\n\\subsection{Systematic Errors}\n\\label{subs:syserr}\nThe systematic error is found to be largely independent of \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} ,  \nand in the subsequent discussion of the differential distributions of \nthe correlated pairs, only normalized distributions are considered. These are \nlargely insensitive to effects of systematic uncertainties. Consequently, the \nsystmatic errors discussed below are mainly relevant for the total rates.\n\nFor the determination of the experimental uncertainties we considered the \nfollowing sources of systematic effects:\n\\begin{itemize}\n\\item Uncertainties due to the subtraction of background via the \n  sidebands. These were estimated using simulated events by applying the \n  analysis to the fully detector simulated MC and comparing the rate from the \n  background corrected sample to the true number.\n\\item Efficiency uncertainties. These were estimated from the difference of \n  the results when the efficiency\n  correction was done using both \\jt\\ versions 7.3 \n  and 7.4 samples in combination and using them separately.\n\\item The statistical error of the efficiency due to the limited sample size\n  of the simulated events at detector level.\n\\item Uncertainties in the modelling of the cut variables used for the \\l\\\n  selection. This error is taken from a former analysis \n  \\cite{opal_sp} where it was determined very precisely for single \\l 's. \n  The error given there is doubled for the \\l\\ pairs in the present \n  analysis.\n\\end{itemize}\n\nThese effects contribute to the total systematic error as shown in\ntable~\\ref{tab:syserr}, where the relative systematic errors from the \ndifferent sources are compared to the total systematic as well as to the \nstatistical error. Statistical and total systematic errors contribute about \nequally.\n\\section{Comparison with Fragmentation Models}\n\\label{sec:comparison}\n\nWe start the discussion with the numbers and distributions of the models with\nOPAL default tunes that optimize the general performance of the models and the \nagreement with the measured single particle rates. \n\n\\subsection{Pair Production Rates}\n\\label{subs:comp_rates}\n\nWe investigate the di-lambda rates first in the total hadronic data sample\ncomparing the measured rates to the predictions of the models \\jt~7.4, {\\sc Mops}\\ \nand \\hw~5.9 (see table~\\ref{tab:rates}). None of the models gives a perfect \ndescription of the data but \\hw\\ clearly exhibits the largest disagreement. \n\nThe comparison of the di-lambda rates in 2- and 3-jet events is given in \ntable~\\ref{tab:2_3jrates}. The higher multiplicity in 3-jet events compared to \n2-jet events is qualitatively well described by all three models. \nHowever, only \\jt\\ yields a prediction compatible with the measured numbers.\nIn the 2-jet event sample the agreement is excellent.\nIn the 3-jet sample all the measured rates exceed the \\jt\\ predictions. \nThis can be compared to the observation that \\l\\ rates in gluon jets are too low\nin \\jt~\\cite{opal_jets} .\n\n\\subsection{Differential Distributions}\n\\label{subs:comp_distributions}\n\nTo further investigate the nature of the \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations we \ncompare the differential distributions of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs with the \npredictions of the various models. We use the variables \\ensuremath{\\cos \\theta^*}\\ and \\ensuremath{|\\Delta y|}\\ \nand test their sensitivity to distinguish between the different fragmentation \nmodels and baryon production mechanisms. All distributions are of the type \n$ \\frac{1}{N} \\frac{dN}{d(\\ensuremath{|\\Delta y|})}$, $N$ being the total number of entries. \nThis has the advantage that they are independent of the total rates and \nthat the systematic errors mostly cancel out, since they are nearly independent \nof both \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} .\n\nThe angle \\ensuremath{\\theta^*}\\ is particularly suited to distinguish between string and \ncluster fragmentation.  The mostly isotropic cluster decay (\\hw) results in a \nrelatively flat \\ensuremath{\\cos \\theta^*}\\ distribution  whereas string fragmentation produces \nthe correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ system predominantly close to the thrust axis, i.e., with \n$\\ensuremath{\\cos \\theta^*} \\approx 1$. These predictions are compared to the measurement \nin figure~\\ref{fig:comp_theta}. The data show a distribution that is strongly \npeaked towards \\ensuremath{\\cos \\theta^*} = 1 and therefore clearly rule out the \\hw\\ cluster \nmodel. \nThe predictions of {\\sc Mops}\\ agree somewhat better with the experimental \ndistribution but they also fail to model the forward peak correctly. \nOnly \\jt\\ yields a good description of the data.\n\nOn the other hand, especially in 2-jet events, the rapidity difference \\ensuremath{|\\Delta y|}\\ is \nmore sensitive to show differences in the strength of the correlations.\nThe experimental data and model predictions are compared in \nfigure~\\ref{fig:compall_2j}.\nAgain \\jt\\ gives the best, albeit not completely satisfactory, description of \nthe measured distribution. \\hw\\ generates correlations which are far too \nstrong .\nThe {\\sc Mops}\\ model with its built-in facility to allow for several ``popcorn \nmesons'' should yield weaker correlations than \\jt ; however, in contrast to \nthis naive expectation it produces a narrower \\ensuremath{|\\Delta y|}\\ distribution, i.e. \nstronger correlations. \nWe see the following possible reasons for this: first of all, and different \nfrom \\jt , a new kinematic property is built into {\\sc Mops}: the \nlow-$\\Gamma$-suppression\\cite{mops}. This suppresses popcorn fluctuations \nat early times in the color field, resulting in very strong correlations. \nSecondly, it appears that the strength of the correlations is influenced more\nby the rate of baryon production via the popcorn mechanism than by the actual \nnumber of intermediate mesons produced.  \nAs \\jt\\ and {\\sc Mops}\\ are tuned to show the same mean number of popcorn mesons \ninstead of popcorn systems, {\\sc Mops}\\ has fewer popcorn systems and therefore \nstronger correlations.\n\n\\subsection{Tuning of Models}\n\\label{subs:tuning}\n\nIn an earlier {\\sc Opal}\\ analysis of strange baryons\\cite{james_early} it was \nobserved that the agreement between experimental data and \\jt\\ predictions\ncan be improved by adjusting some of the diquark parameters: \nimproving the predicted shape of the \\ensuremath{|\\Delta y|}\\ distribution was possible by varying \nthe popcorn parameter, $\\rho$=PARJ(5), that influences the frequency of \npopcorn production and hence the correlation strength. It was found that \n$\\rho$ acts on both the shape of the rapidity spectrum and the production \nrates.\nTwo other parameters were used to correct for this change of predicted \nmultiplicities:\nthe ratio of the strange to non-strange diquarks over strange to \nnon-strange quarks, (us:ud\/s:d) = PARJ(3), and the ratio of spin-1 to spin-0 \ndiquarks, $(1\/3 \\cdot [{\\rm qq}]_1\/[{\\rm qq}]_0)$=PARJ(4). \nThese last parameters affect mainly the rates and leave the spectra nearly\nunmodified.\nWhen attempting to improve the predictions of the {\\sc Mops}\\ model in the same \nmanner, the most direct correspondence to $\\rho$ in \\jt\\ is the \n{\\sc Mops}\\ parameter PARJ(8)=$\\beta({\\rm u})$, the transverse mass of an \nintermediate u-quark. The higher the transverse mass of the intermediate \nsystem (with several quark pairs possible), the lower the probability to \nproduce this popcorn system and the stronger the correlations. \nTherefore, in both \\jt\\ and {\\sc Mops}\\ we tried first to improve the \nagreement with the data distributions by tuning \nthe parameters that influence \nthe correlation strength (figure~\\ref{fig:compjt_tunes} for \\jt .) \nIn \\jt , the popcorn probability $\\rho$ was varied from 0\\%-90\\%, while in \n{\\sc Mops}\\ the transverse mass of a u-quark, $\\beta({\\rm u})$=PARJ(8), was altered \nbetween 0.2 and 1.0~GeV$^{-1}$. \nAll other parameters remained at the {\\sc Opal}\\ default values.\nAs can be seen from table~\\ref{tab:jtrates_tune}, for the $\\rho$ parameter,\nthese variations affect not only the shape of the \\ensuremath{|\\Delta y|}\\ distribution but \nalso the di-lambda rates, as expected.\n\nThe predictions with the different popcorn parameter values in \\jt\\ are \ncompared to the data in figure~\\ref{fig:compjt_tunes}a.\nOnly the results from parameter settings above the default value \nof $\\rho = 0.5$ are shown, since lower values give a poorer agreement with the \ndata. \nThe best agreement \nis found in the range $0.6<\\rho<0.8$.\nPopcorn values within this range also yield good agreement between data and \npredictions for the \\ensuremath{\\cos \\theta^*}\\ distribution.\nHowever, when the influence of the popcorn parameter on the predicted\ndi-lambda rates is also considered (table~\\ref{tab:jtrates_tune})  \nuse has to be  made of the other two \\jt\\ parameters that affect the strange \nbaryon production in order to tune the rates back to values corresponding to\nthe measurement. It can be seen from table~\\ref{tab:jtrates_tune} and \nfigure~\\ref{fig:compjt_tunes}b that such\na tune clearly produces a better agreement with the rates (also for single \nparticle production) while it does not change the spectra of rapidity \ndifferences significantly. Using the results of other {\\sc Opal}\\ analyses,\nit can also be seen that the tune does not change the strange meson (\\ensuremath{{\\rm K}^{0}_{\\rm S}})\nrate, nor does it affect the non-strange baryon (p) rate significantly.\nThe known problems~\\cite{james_early} in modeling the decuplet baryon rates \nare also seen here.\nNo further attempt has been made to globally optimize the parameter set, \nhowever. \n\nThe tune of parameter PARJ(8) in {\\sc Mops}\\ did not result in an improvement. \nAlthough the value of the parameter was varied in a comparatively wide \nrange, the effect on the \\ensuremath{|\\Delta y|}\\ distribution was almost imperceptible. \nPARJ(8) clearly is not suited to adjust the {\\sc Mops}\\ model to the data. \nTherefore, we tested another parameter of the model using the relative \ndifference between the fragmentation function $f(z)$ for baryons and mesons, \nthe parameter PARJ(45). Again, the variation did not notably change the shape \nof the distribution. \nThis relatively poor performance of the {\\sc Mops}\\ Monte Carlo in describing \nthe \\ensuremath{|\\Delta y|}\\ dependent \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations seems to be connected to the known \nshortcomings of the model in describing  $p_{\\perp}$-related \ndistributions~\\cite{mops}.\n\\section{Di-lambdas in Jets}\n\\label{sec:jets}\nAfter studying the strength of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations in the \\ensuremath{|\\Delta y|}\\ spectra, we will \nnow present results on the range of the correlations by assigning \nboth partners from a correlated pair to the reconstructed jets in an event. \nFor short-range correlations both partners are expected \nwithin the same jet whereas long-range correlations (which can be obtained by \nthe production of baryons from the primary quarks) should result in an \nassignment to different jets. \nWe use the following two classifications for the assignment study: \nboth partners within the same jet, and each partner in a different jet. \n\nDue to the fact that it is impossible to map 2-jet events at detector level \nto 2-jet events at generator level, we do not attempt to apply efficiency \ncorrections but compare our uncorrected results with the \\jt\\ predictions at \ndetector level.\nWe count the number of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs in \neach sample and obtain the number of correlated pairs again from the \nrelation \n\\mbox{$N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}}^{\\rm correlated} = N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}} - (N_{\\ll} + N_{\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}})$}.\nThe amount of background in like- and unlike-sign pairs approximately cancels \nout in this subtraction as long as the contribution from the correlated \nbackground (see section 3.2) can be neglected. \nThe numbers of pairs obtained from the same jet and from different jets are \nlisted in table~\\ref{tab:injets} \nfor both 2- and 3-jet events.\nThe major part of the correlated pairs is reconstructed \nwithin the \nsame jet (about 96\\% in 2-jet events, 81\\% in 3-jet events) whereas only a \nvery small fraction is found in different jets. These experimental numbers are \nin excellent agreement with the \\jt\\ predictions at detector level and support\nthe assumption of short-range compensation of baryon number \nand strangeness in the fragmentation process.\n\\section{Summary}\n\\label{sec:summary}\n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations have been studied in 4.3 million multihadronic \\ensuremath{{\\rm Z}^0}\\\ndecays, with the correlated sample obtained from the difference: \n\\ensuremath{\\Lambda\\bar{\\Lambda}_{\\rm corr}} =\\ensuremath{\\Lambda\\bar{\\Lambda}}--(\\ll+\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}).\nThe analysis has been performed in terms of \\ensuremath{\\cos \\theta^*}\\ and rapidity differences \n\\ensuremath{|\\Delta y|} . As the rapidity is defined with respect to the event (thrust) \naxis, the sensitivity of the analysis is seen to be higher in 2-jet events.\nTherefore three data samples have been analyzed: the \nentire hadronic event sample, 2-jet events ($45\\%$), and 3-jet events ($36\\%$).\nThe experimental findings have been used to study the baryon production\nmechanism implemented in various phenomenological fragmentation models.\n\nThe following results have been obtained:\n\\begin{itemize}\n\\item  In the full data set, the measured production rates of \\ensuremath{\\Lambda\\bar{\\Lambda}}, \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ \n  and, consequently, \\ensuremath{\\Lambda\\bar{\\Lambda}_{\\rm corr}}\\ are in good agreement with a previous {\\sc Opal}\\ \n  measurement and results from \\aleph\\ and {\\sc Delphi} , but show significantly \n  smaller errors. \n\\item The \\ensuremath{\\cos \\theta^*}\\ distribution of correlated \\l-pairs is well suited to \n  distinguish between isotropic cluster and non-isotropic string decay, and \n  clearly favours the latter, implemented in \\jt . The predictions of the \n  isotropic cluster model \\hw\\ are ruled out by the data: they do not describe \n  the features of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ production. \n\\item  The rapidity difference \\ensuremath{|\\Delta y|}\\ is used to study the strength of correlated \n  di-lambda production. The measured distribution \n  exhibits strong local correlations.\n\\item  Satisfactory reproduction of the experimental results is obtained\n  with the predictions of the string fragmentation model \\jt . Improved\n  agreement can be found by tuning some of the default parameters used \n  by {\\sc Opal} . After adjusting the popcorn parameter, to improve the description \n  of the \\ensuremath{|\\Delta y|}\\ spectrum, other parameters, fixing the fraction of diquarks with \n  strangeness and spin1, have to be modified to readjust the predicted rates \n  to the experimental ones. This procedure does not affect the previously \n  optimized \\ensuremath{|\\Delta y|}\\ distribution. \n  The \\hw\\ model cannot describe the measured \\ensuremath{|\\Delta y|}\\ spectra, and the predictions \n  of {\\sc Mops} , a recently published modification of the \\jt\\ model, also fail to \n  reproduce the experimental data, even after some parameter tuning. \n\\item  In the 2-jet and 3-jet event samples it is found that correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ \n  pairs are produced predominantly within the same jet, supporting the \n  assumption of a short-range compensation of quantum numbers.\n  Again, the \\jt\\ predictions are in good agreement with the experimental \n  results. \n\\end{itemize} \nIn conclusion, the analysis of correlated di-lambda pairs proves to be a\nvery effective tool to test fragmentation models. \\jt\\ is the only candidate \nmodel studied, describing the data successfully. \n\n\\bigskip \\bigskip \\bigskip \\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{INTRODUCTION}\n\nThe connection between endomorphisms of factors and families of isometric operators has been explored most notably by Arveson \\cite{arveson}, among others \\cite{brenken1,brenken2,laca,longo}. In \\cite{laca}, Laca determines that given a normal $*$-endomorphism $\\alpha$ of $B(H)$ there exists an $n\\in\\mathbb{N}\\cup\\{\\infty\\}$ and $*$-representation $\\pi:\\mathcal{E}_n\\to B(H)$, where $\\mathcal{E}_n$ denotes the Toeplitz algebra for $n$ orthogonal isometries $v_1,...,v_n$, such that\n\\begin{equation*}\\alpha(T)=\\sum_{i=1}^n\\pi(v_i)T\\pi(v_i)^*\\end{equation*}\nfor each $T\\in B(H)$. The $n$ value is unique but the representation $\\pi$ may differ by automorphisms of $\\mathcal{E}_n$ which arise specifically from unitary transformations of the Hilbert space $\\ell^2(\\{v_1,...,v_n\\})\\subseteq\\mathcal{E}_n$ \\cite[Proposition 2.2]{laca}. \n\nIn \\cite{brenken1} and \\cite{brenken2} Brenken extends the connection by determining that, for a von Neumann algebra which can be decomposed into a direct sum of Type I factors, a certain class of $*$-endomorphisms correspond to representations of certain $C^*$-algebras associated with (possibly infinite) matrices which arise as the adjacency matrices for directed graphs. The $*$-endomorphisms studied by Brenken are, however, required to either be unital \\cite{brenken1} or satisfy a number of restrictive conditions \\cite[pp 25]{brenken2}.\n\nOur results extend the work of Brenken by eliminating the conditions imposed on the $*$-endomorphisms. The fundamental difference in our approach is that we will investigate representations of the Toeplitz algebra of a $C^*$-correspondence, whereas Brenken concerned himself with the so-called relative Cuntz-Pimnser algebras. We recover Brenken's results as a special case when the endomorphisms are assumed unital. Our primary result in this line is Theorem \\ref{theorem3}.\n\nOur results also continue the spirit of Laca's investigations which allow for equivalencies between $*$-endomorphisms to be encoded as transformations of an underlying linear object: a Hilbert space in the case of Laca and a $C^*$-correspondence in the present work. Our primary results along these lines are Theorems \\ref{theorem1} and \\ref{theorem2}.\n\n\\section{PRELIMINARIES}\nFirst we will establish our terminology and notation.\n\n\\begin{defi} A \\emph{graph} is a tuple $E=(E^0,E^1,r,s)$ consisting of a \\emph{vertex set} $E^0$, an \\emph{edge set} $E^1$, and \\emph{range} and \\emph{source} maps $r,s:E^1\\to E^0$.\n\\end{defi}\n\nWe will only consider graphs where $E^0$ and $E^1$ are at most countable.\n\n\\begin{defi} Let $A$ be a $C^*$-algebra. A set $X$ is a \\emph{$C^*$-correspondence over $A$} provided that it is a right Hilbert $A$-module and there is a $*$-homomorphism $\\phi:A\\to L(X)$, where $L(X)$ denotes the space of adjointable $A$-module homomorphisms from $X$ to itself. \n\\end{defi} \n\nGiven a $C^*$-correspondence $X$ over $A$, we will denote the $A$-valued inner product of $x,y\\in X$ by ``$\\langle x,y\\rangle_A$\" (perhaps omitting the $A$) and the right action of $a\\in A$ on $x\\in X$ will be written as ``$x\\cdot a$\". The map $\\phi:A\\to L(X)$ may sometimes be written as $\\phi_X$ for clarity.\n\nOur primary objects of study will be certain $C^*$-correspondences which arise from graphs. The following constructions are due originally to Fowler and Raeburn \\cite[Example 1.2]{fowlerraeburn}, although we will adopt the modern convention for the roles of $r$ and $s$. \n\n\\begin{defi} Given a graph $E$, the \\emph{graph correspondence} $X(E)$ is the set of all functions $x:E^1\\to\\mathbb{C}$ for which $\\hat{x}(v):=\\sum_{e\\in s^{-1}(v)}|x(e)|^2$ extends to a function $\\hat{x}\\in C_0(E^0)$. We give $X(E)$ the structure of a $C^*$-correspondence over $C_0(E^0)$ as follows:\n\\begin{align*}x\\cdot a&:e\\mapsto x(e)a(s(e)),\\\\\n\\phi(a)x&:e\\mapsto a(r(e))x(e),\\\\\n\\langle x,y\\rangle&:v\\mapsto\\sum_{e\\in s^{-1}(v)}\\overline{x(e)}y(e).\\end{align*}\nwhich is to say that $a\\in C_0(E^0)$ acts on the right of $X(E)$ as multiplication by $a\\circ s$ and acts on the left as multiplication by $a\\circ r$.\n\\end{defi}\n\nThe sets $\\{\\delta_e:e\\in E^1\\}$ and $\\{\\delta_v:v\\in E^0\\}$ are dense in $X(E)$ and $C_0(E^0)$, respectively, in the appropriate senses. For $e\\in E^1$ and $v\\in E^0$ we have the following useful relations: $\\langle \\delta_e,\\delta_e\\rangle=\\delta_{s(e)}$, $\\delta_e\\cdot\\delta_v=\\delta_e$ if $v=s(e)$ and is $0$ otherwise, and $\\phi(\\delta_v)\\delta_e=\\delta_e$ if $v=r(e)$ and is $0$ otherwise.\n\n\n\\begin{defi}\\label{toeplitzrepdef} Given a $C^*$-correspondence $X$ over $A$ and given another $C^*$-algebra $B$, a \\emph{Toeplitz representation} of $X$ in $B$ is a pair $(\\sigma,\\pi)$ consisting of a linear map $\\sigma:X\\to B$ and a $*$-homomorphism $\\pi:A\\to B$ such that for all $x,y\\in X$ and $a\\in A$\n\t\\begin{enumerate}\n\t\\item $\\sigma(x\\cdot a)=\\sigma(x)\\pi(a)$,\n\t\\item $\\sigma(\\phi(a)x)=\\pi(a)\\sigma(x)$, and\n\t\\item $\\pi(\\langle x,y\\rangle)=\\sigma(x)^*\\sigma(y).$\n\t\\end{enumerate}\n\\end{defi}\n\nFor a graph correspondence $X(E)$, a Toeplitz representation $(\\sigma,\\pi)$ is determined entirely by the values $\\{\\sigma(\\delta_e):e\\in E^1\\}$ and $\\{\\pi(\\delta_v):v\\in E^0\\}$. Property (iii) of a Toeplitz representation guarantees that $\\sigma(\\delta_e)$ is a partial isometry with source projection $\\pi(\\delta_{s(e)})$.\n\n\\begin{defi}\\cite[Proposition 1.3]{fowlerraeburn} Given a $C^*$-correspondence $X$ over $A$, the \\emph{Toeplitz algebra of $X$}, denoted $\\mathcal{T}_X$, is the $C^*$-algebra which is universal in the following sense: there exists a Toeplitz representation $(\\sigma_u,\\pi_u)$ of $X$ in $\\mathcal{T}_X$ such that if $(\\sigma,\\pi)$ is another Toeplitz representation of $X$ in a $C^*$-algebra $B$ then there exists a unique $*$-homomorphism $\\rho_{\\sigma,\\pi}:\\mathcal{T}_X\\to B$ such that $\\sigma=\\rho_{\\sigma,\\pi}\\circ\\sigma_u$ and $\\pi=\\rho_{\\sigma,\\pi}\\circ\\pi_u$.\n\\end{defi}\n\nThat $\\mathcal{T}_X$ exists was proven by Pimnser in \\cite{pimsner}.\n\nGiven a graph $E$ we may consider the Toeplitz algebra of its graph correspondence, cumbersomely denoted $\\mathcal{T}_{X(E)}$. Unless there is danger of confusion, we will abuse notation and make no distinction between elements of $X(E)$ and $C_0(E^0)$ and their images in $\\mathcal{T}_{X(E)}$ under the universal maps $\\sigma_u$ and $\\pi_u$. \n\nIf $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ is a $*$-representation then, for each $e\\in E^1$, $\\tau(\\delta_e)$ is a partial isometry with source projection $\\tau(\\delta_{s(e)})$ and range projection contained in $\\tau(\\delta_{r(e)})$.\n\nIf $E$ is the graph with but a single vertex and $n$ edges then $X(E)$ is a Hilbert space of dimension $n$ and $\\mathcal{T}_{X(E)}$ is isomorphic to the classical Toeplitz algebra $\\mathcal{E}_n$. In this case the elements $\\{\\delta_e:e\\in E^1\\}$ are precisely the generating isometries of $\\mathcal{E}_n$. The space $X(E)$ plays a significant role in the analysis of endomorphisms of $B(H)$ in \\cite{laca}, and it is for this reason that we are considering the generalized Toeplitz algebras $\\mathcal{T}_{X(E)}$ in our investigations.\n\n\\section{COHERENT UNITARY EQUIVALENCE}\n\nTwo graphs $E$ and $F$ are isomorphic if there are two bijections $\\psi^0:E^0\\to F^0$ and $\\psi^1:E^1\\to F^1$ for which $r_F\\circ\\psi^1=\\psi^0\\circ r_E$ and $s_F\\circ\\psi^1=\\psi^0\\circ s_E$. In order to encode such an isomorphism at the level of the graph correspondences $X(E)$ and $X(F)$, we offer the following novel definition.\n\n\\begin{defi} Let $X$ and $Y$ be $C^*$-correspondences over $A$ and $B$, respectively. A \\emph{coherent unitary equivalence} between $X$ and $Y$ is a pair $(U,\\alpha)$ consisting of a bijective linear map $U:X\\to Y$ and a $*$-isomorphism $\\alpha:A\\to B$ for which \n\t\\begin{enumerate}\n\t\\item $U(x\\cdot a)=(Ux)\\cdot \\alpha(a)$ for all $x\\in X$ and $a\\in A$,\n\t\\item $U(\\phi_X(a)x)=\\phi_Y(\\alpha(a))Ux$ for all $x\\in X$ and $a\\in A$, and\n\t\\item $\\langle Ux,y\\rangle_Y=\\alpha(\\langle x,U^{-1}y\\rangle_X)$ for all $x\\in X$ and $y\\in Y$.\n\t\\end{enumerate}\nRoutine calculations will verify that coherent unitary equivalence is an equivalence relation.\n\\end{defi}\n\n\\begin{proposition} If $E$ and $F$ are isomorphic graphs then $X(E)$ and $X(F)$ are coherently unitarily equivalent.\n\\end{proposition} \n\\begin{proof} We'll assume $(\\psi^0,\\psi^1)$ to be an isomorphism from $F$ to $E$.\n\nFor $a\\in C_0(E^0)$, $\\alpha(a):=a\\circ\\psi^0$ clearly defines a $*$-isomorphism $\\alpha:C_0(E^0)\\to C_0(F^0)$. For $x\\in X(E)$ define $Ux:=x\\circ\\psi^1$.\nFor $v\\in F^1$ we have\n\\begin{equation*}\\sum_{e\\in s_F^{-1}(v)}|Ux(e)|^2=\\sum_{e\\in s_F^{-1}(v)}|x(\\psi^1(e))|^2=\\sum_{f\\in s_E^{-1}(\\psi^0(v))}|x(f)|^2\\end{equation*}\n(using the fact that if $s_E(e)=v$ then $s_F(\\psi^1(e))=\\psi^0(v)$) and so $\\widehat{Ux}(v)=\\hat{x}(\\psi^1(v))$. As $\\hat{x}\\in C_0(E^0)$ it follows immediately that $\\widehat{Ux}\\in C_0(F^0)$, i.e. $Ux\\in X(F)$. Identical arguments show that $U^{-1}y:=y\\circ(\\psi^1)^{-1}$ is a map from $X(F)$ to $X(E)$ which is a two-sided inverse for $U$. Hence $U:X(E)\\to X(F)$ is a bijection which is naturally linear.\n\nGiven $x\\in X(E)$, $a\\in C_0(E^0)$, and $e\\in E^1$ we have\n\\begin{align*}\nU(x\\cdot a)&=(x(a\\circ s_E))\\circ\\psi^1=(x\\circ\\psi^1)(a\\circ s_E\\circ\\psi^1)=(Ux)(a\\circ\\psi^0\\circ s_F)=Ux\\cdot\\alpha(a)\\\\\nU(\\phi_E(a)x)&=((a\\circ r_E)x)\\circ\\psi^1=(a\\circ r_E\\circ\\psi^1)(x\\circ\\psi^1)=(a\\circ \\psi^0\\circ r_F)(Ux)=\\phi_F(\\alpha(a))Ux\n\\end{align*}\nand, given $v\\in F^0$,\n\\begin{align*}\n\\langle Ux,y\\rangle(v)&=\\sum_{e\\in s_F^{-1}(v)}\\overline{Ux(e)}y(e)=\\sum_{e\\in s_F^{-1}(v)}\\!\\!\\overline{x(\\psi^1(e))}y(e)=\\!\\!\\!\\sum_{f\\in s_E^{-1}(\\psi^0(v))}\\hspace{-.5cm}\\overline{x(f)}y((\\psi^1)^{-1}(f))\\\\\n&=\\sum_{f\\in s_E^{-1}(\\psi^0(v))}\\overline{x(f)}U^{-1}y(f)=\\langle x,U^{-1}y\\rangle(\\psi^0(v))= \\alpha(\\langle x,U^{-1}y\\rangle)(v)\n\\end{align*}\n(the first inner product is that of $X(F)$ and the later two are that of $X(E)$). Thus the pair of $U$ and $\\alpha$ satisfies the definition of a coherent unitary equivalence.\n\\end{proof}\n\nNot every coherent unitary equivalence comes from a graph isomorphism in the sense of the preceding Proposition. As a simple example, consider the graph $E$ with but a single vertex and two edges. In this case $C_0(E^0)=\\mathbb{C}$ and $X(E)=\\mathbb{C}^2$. Hence any unitary $U\\in M_2(\\mathbb{C})$ forms (with the identify on $C_0(E^0)$) a coherent unitary equivalence. However, the only such equivalences arising from graph isomorphisms would be those of the two permutation matrices.\n\n\\begin{proposition}\\label{isotoeplitz} If there is a coherent unitary equivalence between $X$ and $Y$ then $\\mathcal{T}_X$ and $\\mathcal{T}_Y$ are $*$-isomorphic. \n\\end{proposition}\n\\begin{proof} Let $A$ and $B$ be the coefficient $C^*$-algebras for $X$ and $Y$, respectively.\nSuppose that $(U,\\alpha)$ is a coherent unitary equivalence between $X$ and $Y$ and let $(\\sigma,\\pi)$ be a Toeplitz representation of $Y$. For $x\\in X$ and $a\\in A$\n\\begin{align*}\n\\sigma(U(x\\cdot a))&=\\sigma(Ux\\alpha(a))=\\sigma(Ux)\\pi(\\alpha(a))\\\\\n\\sigma(U(\\phi_X(a)x))&=\\sigma(\\alpha(a)Ux)=\\pi(\\alpha(a))\\sigma(Ux)\n\\end{align*}\nand for $x_1,x_2\\in X$\n\\begin{equation*}\\pi\\circ\\alpha(\\langle x_1,x_2\\rangle_A)=\\pi(\\langle Ux_1,Ux_2\\rangle_B)=\\sigma(Ux_1)^*\\sigma(Ux_2).\\end{equation*}\nHence $(\\sigma\\circ U,\\pi\\circ \\alpha)$ is a Toeplitz representation of $X$.\n\nIn particular, $(\\sigma_Y\\circ U,\\pi_B\\circ\\alpha)$ is a Toeplitz representation of $X$ where $(\\sigma_Y,\\pi_B)$ is the universal Toeplitz representation of $Y$ in $\\mathcal{T}_Y$. By the universal property of $\\mathcal{T}_X$, there is a $*$-homomorphism $\\theta:\\mathcal{T}_X\\to\\mathcal{T}_Y$ such that $\\theta\\circ\\sigma_X=\\sigma_Y\\circ U$ and $\\theta\\circ\\pi_A=\\pi_B\\circ \\alpha$, where $(\\sigma_X,\\pi_A)$ is the universal representation of $X$ in $\\mathcal{T}_X$.\n\nSimilarly $(\\sigma_X\\circ U^{-1},\\pi_A\\circ \\alpha^{-1})$ is a Toeplitz representation of $Y$ and induces a $*$-homomorphism $\\theta':\\mathcal{T}_Y\\to \\mathcal{T}_X$ for which $\\theta'\\circ\\sigma_Y=\\sigma_X\\circ U^{-1}$ and $\\theta'\\circ\\pi_B=\\pi_A\\circ \\alpha^{-1}$. Thus\n$$\\sigma_Y=\\sigma_Y\\circ U\\circ U^{-1}=\\theta\\circ\\sigma_X\\circ U^{-1}=\\theta\\circ\\theta'\\circ\\sigma_Y$$\nand similarly $\\pi_B=\\theta\\circ\\theta'\\circ\\pi_B$. Since the identity $id$ on $\\mathcal{T}_Y$ also has the property that $\\pi_B=id\\circ\\pi_B$ and $\\sigma_Y=id\\circ\\sigma_Y$, it follows by the universal property of $\\mathcal{T}_Y$ that $\\theta\\circ\\theta'=id$. Identical reasoning verifies that $\\theta'\\circ\\theta$ is the identity on $\\mathcal{T}_X$. Thus $\\theta$ is our desired $*$-isomorphism.\n\\end{proof}\n\nGoing forward we will be exclusively interested in Toeplitz algebras associated to graph correspondences, and so offer the following corollary.\n\\begin{corollary}\\label{autosfromcoherent} Let $E$ and $F$ be graphs. If $(U,\\alpha)$ is a coherent unitary equivalence between $X(E)$ and $X(F)$ then there is a $*$-isomorphism $\\Gamma_{U,\\alpha}:\\mathcal{T}_{X(E)}\\to\\mathcal{T}_{X(F)}$ for which $\\Gamma_{U,\\alpha}(\\delta_e)=U\\delta_{e}$ and $\\Gamma_{U,\\alpha}(\\delta_v)=\\alpha(\\delta_v)$ for all $e\\in E^1$ and $v\\in E^0$.\n\\end{corollary}\nThis is immediately seen from the proof of the previous Proposition if we recall that we identify $X(E)$ and $X(F)$ with their images in $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$, respectively, under the appropriate universal maps. This also implies that if $E$ and $F$ are isomorphic graphs then $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$ are $*$-isomorphic, which is unsurprising.\n\n\\section{ENDOMORPHISMS FROM GRAPHS}\n\nThroughout this section we will let $E=(E^0,E^1,r,s)$ be a given graph whose vertex and edge sets are at most countable. All $*$-representations will be assumed non-degenerate.\n\n\\begin{proposition} Given a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$, the assignments\n\\begin{equation*}Ad_\\tau(w):=\\sum_{e\\in E^1}\\tau(\\delta_e)w\\tau(\\delta_e)^*\\end{equation*}\n(the sum is taken as a SOT limit) define a $*$-endomorphism $Ad_\\tau$ of the von Neumann algebra $W=\\{\\tau(\\delta_v):v\\in E^0\\}'$ (this notation will denote the relative commutant in $B(H)$).\n\\end{proposition}\n\\begin{proof} First, notice that for $e\\in E^1$ and $w\\in W$ the term $\\tau(\\delta_e)w\\tau(\\delta_e)^*$ has its support projection contained in $\\tau(\\delta_e^*\\delta_e)$. Since the partial isometries $\\tau(\\delta_e)$ have mutually orthogonal ranges, it follows that for every $h\\in H$, $\\tau(\\delta_e)w\\tau(\\delta_e)^*h$ is nonzero for at most one $e\\in E^1$. Thus the sum converges in the SOT.\n\nCertainly $Ad_\\tau$ is linear and has $Ad_\\tau(w^*)=Ad_\\tau(w)^*$ for each $w\\in W$. Given $w_1,w_2\\in W$ we find that\n\\begin{align*}Ad_\\tau(w_1)Ad_\\tau(w_2)&=\\bigg(\\sum_{e\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_e)^*\\bigg)\\bigg(\\sum_{f\\in E^1}\\tau(\\delta_f)w_2\\tau(\\delta_f)^*\\bigg)\\\\\n\t&= \\sum_{e,f\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_e)^*\\tau(\\delta_f)w_2\\tau(\\delta_f)^*\\\\\n\t&= \\sum_{e\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_{s(e)})w_2\\tau(\\delta_e)^*\\\\\n\t&=\\sum_{e\\in E^1}\\tau(\\delta_e)\\tau(\\delta_{s(e)})w_1w_2\\tau(\\delta_e)^*\\\\\n\t&=\\sum_{e\\in E^1}\\tau(\\delta_e)w_1w_2\\tau(\\delta_e)^*\\\\\n\t&=Ad_\\tau(w_1w_2)\n\t\\end{align*}\nand so $Ad_\\tau$ is multiplicative. Note that any potential issues with SOT-convergence of the product are circumvented by $E^1$ being at most countable. All that remains is to verify that $Ad_\\tau(w)\\in W$ for each $w\\in W$. To that end we first note that $\\delta_e^*\\delta_v=\\delta_e^*$ if $v=r(e)$ and is zero otherwise. By taking adjoints, $\\delta_v\\delta_e=\\delta_e$ if $v=r(e)$ and is zero otherwise. Thus, given $w\\in W$ and $v\\in E^0$ we find\n\\begin{equation*}Ad_\\tau(w)\\tau(\\delta_v)=\\sum_{e\\in r^{-1}(v)}\\tau(\\delta_e)w\\tau(\\delta_e)^*=\\tau(\\delta_v)Ad_\\tau(w)\\end{equation*}\nand so $Ad_\\tau(w)$ commutes with each $\\tau(\\delta_v)$.\n\\end{proof}\n\nWe note that the von Neumann algebra $W=\\{\\tau(\\delta_v):v\\in E^0\\}$, because the $\\tau(\\delta_v)$ are a family of mutually orthogonal projections, is precisely equal to $\\displaystyle{\\bigoplus_{v\\in E^0} \\tau(\\delta_v)B(H)\\tau(\\delta_v)}$ which is a sum of Type I factors. \n\nThe following is a construction which we believe to be folklore, but our use of it is motivated by observations made by Muhly and Solel \\cite{muhlysolel}. Given a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ let $W=\\{\\tau(\\delta_v):v\\in E^0\\}'$. The space\n\\begin{equation*}\\mathcal{I}_\\tau:=\\left\\{T\\in B(H):Ad_\\tau(w)T=Tw\\ ,\\ w\\in W \\right\\}\\end{equation*}\nis a $C^*$-correspondence over $W'$. The left and right actions of $W'$ are simply multiplication within $B(H)$ and the $W'$-valued inner product is defined by $\\langle T,S\\rangle_{W'}:=T^*S$.\n\nBecause our endomorphism is of the form $Ad_\\tau$,  we can say more: for $w\\in W$ and $e\\in E^1$\n\\begin{equation*}Ad_\\tau(w)\\tau(\\delta_e)=\\sum_{f\\in E^1}\\!\\!\\tau(\\delta_f)w\\tau(\\delta_f)^*\\tau(\\delta_e)=\\tau(\\delta_e)w\\tau(\\delta_{s(e)})=\\tau(\\delta_e)\\tau(\\delta_{s(e)})w=\\tau(\\delta_e)w\\end{equation*}\nand so $\\tau(\\delta_e)\\in\\mathcal{I}_\\tau$ for each $e\\in E^1$. As $\\tau(\\delta_v)\\in W'$ for each $v\\in E^0$ we finally have $\\tau(X(E))\\subseteq \\mathcal{I}_\\tau$.\n\n\\begin{theorem}\\label{theorem1} Suppose that $E$ and $F$ are graphs and $\\tau_1:\\mathcal{T}_{X(E)}\\to B(H)$ and $\\tau_2:\\mathcal{T}_{X(F)}\\to B(H)$ are two faithful $*$-representations. If $Ad_{\\tau_1}=Ad_{\\tau_2}$ on $W=\\{\\tau_1(\\delta_v):v\\in E^0\\}'=\\{\\tau_2(\\delta_v):v\\in F^0\\}'$ then there is a coherent unitary equivalence $(U,\\alpha)$ between $X(E)$ and $X(F)$ such that $\\tau_2=\\tau_1\\circ\\Gamma_{U,\\alpha}$. \n\\end{theorem}\nHere $\\Gamma_{U,\\alpha}$ is the $*$-isomorphism from $\\mathcal{T}_{X(E)}$ to $\\mathcal{T}_{X(F)}$ arising from $(U,\\alpha)$ as given in Corollary \\ref{autosfromcoherent}.\n\\begin{proof} \nSince $\\{\\tau_1(\\delta_v):v\\in E^0\\}$ and $\\{\\tau_2(\\delta_v):v\\in F^0\\}$ are sets of orthogonal projections with the same commutant they are in fact equal, and in particular $E^0$ and $F^0$ are of the same cardinality. To ease notation we'll denote these projections by $P_v$, $v\\in E^0$, (with no assumption that $P_v=\\tau_1(\\delta_v)$ or similar) hence\n\\begin{equation*}\\{P_v:v\\in E^0\\}=\\{\\tau_1(\\delta_v):v\\in E^0\\}=\\{\\tau_2(\\delta_v):v\\in F^0\\}.\\end{equation*}\n\nAs $Ad_{\\tau_1}=Ad_{\\tau_2}$ we have that $\\mathcal{I}_{\\tau_1}=\\mathcal{I}_{\\tau_2}$ and we'll call this module simply $\\mathcal{I}$.\n\nAs $\\tau_1(\\delta_e)\\in\\mathcal{I}$ for each $e\\in E^1$ we have\n\\begin{equation*}\\tau_1(\\delta_e)=\\tau_1(\\delta_e)I=Ad_{\\tau_2}(I)\\tau_1(\\delta_e)=\\sum_{f\\in F^1}\\tau_2(\\delta_f)\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\end{equation*}\nhence $\\tau_1(\\delta_e)$ is in the $W'$-submodule of $\\mathcal{I}$ generated by $\\tau_2(X(E))$. Similarly, for each $f\\in F^1$, $\\tau_2(\\delta_f)$ is in the $W'$-submodule generated by $\\tau_1(X(E))$. Thus $\\tau_1(X(E))$ and $\\tau_2(X(E))$ generate the same $W'$-submodule of $\\mathcal{I}$.\n\nGiven $e\\in E^1$ and $f\\in F^1$ we have seen that \n\\begin{equation*}\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\in W'=\\{P_v:v\\in E^0\\}''=\\ell^\\infty(\\{P_v:v\\in E^0\\}).\\end{equation*}\nNotice however that $\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\tau_1(\\delta_v)=0$ unless $v=s(e)$ and hence $\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)$ is a multiple of $\\tau_1(\\delta_{s(e)})$ only, i.e. is an element of $C_0(\\{P_v:v\\in E^0\\})$. Since before we obtained $\\tau_1(\\delta_e)=\\sum_{f\\in F^1}\\tau_2(\\delta_f)\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)$ for all $e\\in E^1$, it now follows that $\\tau_1(X(E))$ and $\\tau_2(X(E))$ generate the same correspondence over $C_0(\\{P_v:v\\in E^0\\})$. It is important to note that this correspondence has three different actions of $C_0(\\{P_v:v\\in E^0\\})$: the ones inherited through $\\tau_1$ and $\\tau_2$ and simple operator multiplication in $B(H)$.\n \nFinally we have that $\\tau_1(C_0(E^0))=\\tau_2(C_0(F^0))$ and $\\tau_1(X(E))=\\tau_2(X(F))$ as sets and so, because both representations are faithful by hypothesis, $\\tau_2^{-1}\\circ\\tau_1$ is a well-defined bijection between $X(E)$ and $X(F)$ and between $C_0(E^0)$ and $C_0(F^0)$. Denote by $U$ and $\\alpha$ the restrictions of $\\tau_2^{-1}\\circ\\tau_1$ to $X(E)$ and to $C_0(E^0)$, respectively.\n\nGiven $x\\in X(E)$, $y\\in X(F)$, and $a\\in C_0(E^0)$ we have\n\\begin{align*}U(xa)&=\\tau_2^{-1}\\circ\\tau_1(xa)=\\tau_2^{-1}\\circ\\tau_1(x)\\tau_2^{-1}\\circ\\tau_1(a)=(Ux)\\alpha(a),\\\\\nU(\\phi(a)x)&=\\tau_2^{-1}\\circ\\tau_1(\\phi(a)x)=\\tau_2^{-1}\\circ\\tau_1(a)\\tau_2^{-1}\\circ\\tau_1(x)=\\alpha(a)Ux,\\\\\n\\langle Ux,y\\rangle&=[\\tau_2^{-1}\\circ\\tau_1(x)]^*y=\\tau_2^{-1}\\circ\\tau_1(x^*\\tau_1^{-1}\\circ\\tau_2(y))=\\alpha(\\langle x,\\tau_1^{-1}\\circ\\tau_2(y)\\rangle)=\\alpha(\\langle x,U^{-1}y\\rangle).\\end{align*}\nand so $(U,\\alpha)$ is a coherent unitary equivalence between $X(E)$ and itself.\n\nIt follows from Corollary \\ref{autosfromcoherent} that $(U,\\alpha)$ induces a $*$-isomorphism $\\Gamma_{U,\\alpha}:\\mathcal{T}_{X(E)}\\to\\mathcal{T}_{X(F)}$ and, by construction, $\\tau_2\\circ \\Gamma_{U,\\alpha}=\\tau_1$.\n\\end{proof}\n\nNotice we have shown that if $\\tau_1:\\mathcal{T}_{X(E)}\\to B(H)$ and $\\tau_2:\\mathcal{T}_{X(F)}\\to B(H)$ generate identical $*$-endomorphims $Ad_{\\tau_1}$ and $Ad_{\\tau_2}$ then $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$ are $*$-isomorphic as $C^*$-algebras. Hence in subsequent results we will concern ourselves only with two representations $\\tau_1$, $\\tau_2$ of the same Toeplitz algebra $\\mathcal{T}_{X(E)}$.\n\n\nOur result is a generalization of Laca's \\cite[Proposition 2.2]{laca}. When $E$ is the graph with a single vertex and $n\\in\\mathbb{N}\\cup\\{ \\infty\\}$ edges we have already seen that $\\mathcal{T}_{X(E)}=\\mathcal{E}_n$. If $\\tau_1$ and $\\tau_2$ are faithful and nondegenerate then $W=B(H)$. The map $\\alpha$ is the identity on $C_0(E^0)=\\mathbb{C}$ and $U$ is a unitary operator on the Hilbert space $X(E)=\\ell^2(\\{v_1,...,v_n\\})$. Hence $\\Gamma_{U,\\alpha}$ is a $*$-automorphism of $\\mathcal{E}_n$ which fixes the Hilbert space $X(E)$ and implements the equivalence between the two representations.\n\nWe will conclude this section with a discussion of conjugacy conditions for endomorphisms of the type we've been examining. Recall that two endomorphisms $\\alpha$ and $\\beta$ are said to be \\emph{conjugate} if there is an automorphism $\\gamma$ such that $\\alpha\\circ\\gamma=\\gamma\\circ\\beta$.\n\n\\begin{lemma}\\label{spatialdiagonal} If $P_1,P_2,...\\in B(H)$ is an at most countable family of orthogonal projections and $\\gamma$ is a $*$-automorphism of $W=\\{P_1,P_2,...\\}'$ then there exists a unitary $U\\in B(H)$ such that $\\gamma(w)=UwU^*$ for all $w\\in W$.\n\\end{lemma}\n\\begin{proof} Note that for each $n$, $\\gamma$ restricts to a $*$-isomorphism $\\gamma_n$ between $P_nB(H)P_n=B(P_nH)$ and $\\gamma(P_n)B(H)\\gamma(P_n)=B(\\gamma(P_n)H)$. Such isomorphisms are always spatial and so there are unitaries $U_n:B(P_nH)\\to B(\\gamma(P_n)H)$ such that $\\gamma_n(w)=U_nwU_n^*$. It is then immediate that $U:=U_1\\oplus U_2\\oplus...$ is a unitary in $B(H)$ and $UwU^*=\\gamma(w)$ for each $w\\in W$.\n\\end{proof}\n\n\\begin{theorem}\\label{theorem2} Suppose that $\\tau_1,\\tau_2:\\mathcal{T}_{X(E)}\\to B(H)$ are two faithful $*$-representations such that $Ad_{\\tau_1}$ and $Ad_{\\tau_2}$ are conjugate $*$-endomorphisms of $W=\\{\\tau_1(\\delta_v):v\\in E^0\\}'=\\{\\tau_2(\\delta_v):v\\in E^0\\}'$. Then there is a coherent unitary equivalence $(U,\\alpha)$ between $X(E)$ and itself such that $\\tau_2$ and $\\tau_1\\circ\\Gamma_{U,\\alpha}$ are unitarily equivalent $*$-representations.\n\\end{theorem}\n\n\\begin{proof} Let $\\gamma$ be an $*$-automorphism of $W$ such that $Ad_{\\tau_1}\\circ\\gamma=\\gamma\\circ Ad_{\\tau_2}$ and let $V\\in B(H)$ be the unitary for which $\\gamma(w)=VwV^*$ according the Lemma \\ref{spatialdiagonal}. Then $Ad_{\\tau_2}(w)=V^*Ad_{\\tau_1}(VwV^*)V$ for all $w\\in W$. Define $\\kappa:\\mathcal{T}_{X(E)}\\to B(H)$ by $\\kappa(t):=V\\tau_1(t)V^*$ and note that $\\kappa$ is a $*$-representation of $\\mathcal{T}_{X(E)}$ such that \n\\begin{equation*}Ad_{\\kappa}(w)=\\sum_{e\\in E^1}\\kappa(\\delta_e)w\\kappa(\\delta_e)^*=\\sum_{e\\in E^1}V\\tau_1(\\delta_e)V^*wV\\tau_1(\\delta_e)^*V^*=VAd_{\\tau_1}(V^*wV)V^*\\end{equation*}\nand so $Ad_{\\kappa}=Ad_{\\tau_2}$ on $W$. Applying Theorem \\ref{theorem1} we obtain a coherent unitary equivalence $(U,\\alpha)$ inducing the $*$-automorphism $\\Gamma_{U,\\alpha}$ of $\\mathcal{T}_{X(E)}$ such that $\\tau_2=\\kappa\\circ\\Gamma_{U,\\alpha}$. As now $\\tau_2(t)=V[\\tau_1\\circ\\Gamma_{U,\\alpha}(t)]V^*$ for each $t\\in \\mathcal{T}_{X(E)}$, we have that $\\tau_2$ and $\\tau_1\\circ\\Gamma_{U,\\alpha}$ are unitarily equivalent, as desired.\n\\end{proof}\n\n\\section{GRAPHS FROM ENDOMORPHISMS}\n\nIn this final section we will demonstrate that all $*$-endomorphisms of von Neumann algebras which are sums of Type I factors are obtained in the natural way from representations of Toeplitz algebras for graph correspondences. Our result is a significant generalization of \\cite[Theorem 3.9]{brenken2} which places technical restrictions on the endomorphisms. In the case of unital endomorphisms our results are comparable.\n\n\\begin{theorem}\\label{theorem3} Let $W=\\bigoplus W_i\\subseteq B(H)$ be a countable sum of Type I factors. Let $P_1,P_2,...\\in B(H)$ be projections such that $W_i=P_iB(H)P_i$. If $\\alpha$ is a normal $*$-endomorphism of $W$ then there exists a graph $E$ and $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ such that $\\alpha=Ad_\\tau$.\n\\end{theorem}\n\\begin{proof} Without loss of generality we may assume that $\\sum P_i=I$. If this were not the case we may define $P_0=(\\sum P_i)^\\perp$ and $W_0=P_0B(H)P_0$ and extend $\\alpha$ to a normal $*$-endomorphism of $W\\oplus W_0$ with $\\alpha|_{W_0}=id$. \n\nFor $i>0$ define $H_i=P_iH$. For $i,j>0$ and $x\\in W$ define $\\alpha_{ij}(x)=P_j\\alpha(P_ix)$. Then $\\alpha_{ij}$ restricts to a $*$-homomorphism between $B(H_i)=P_iB(H)P_i=W_i$ and $B(H_j)=P_jB(H)P_j=W_j$ as seen by\n\\begin{equation*}P_j\\alpha(P_ix)P_j\\alpha(P_iy)=P_j\\left(P_j\\alpha(P_ix)\\right)\\alpha(P_iy)=P_j\\alpha(P_ixP_iy)=P_j\\alpha(P_ixy).\\end{equation*}\n\nSo $\\alpha_{ij}$ is a $*$-homomorphism between two Type I factors, and thus by \\cite[Proposition 2.1]{arveson} if $\\alpha_{ij}$ is nonzero there exists $n_{ij}\\in\\mathbb{N}\\cup\\{\\infty\\}$ and isometries $V_k^{(ij)}\\in B(H_i,H_j)$, $k=1,...,n_{ij}$ such that $\\alpha_{ij}|_{B(H_i)}(T)=\\sum_{k=1}^{n_{ij}}V_k^{(ij)}TV_k^{(ij)*}$.\nWe will identify the $V_k^{(ij)}$ with their associated partial isometries in $B(H)$, so that $V_k^{(ij)*}V_k^{(ij)}=P_i$ and $V_k^{(ij)}V_k^{(ij)*}\\leq P_j$.\n\nSet $E^0:=\\{P_1,P_2...,\\}$ and $E^1:=\\bigcup_{i,j}\\{V_k^{(ij)}:k=1,...,n_{ij}\\}$. Define maps $r,s:E^1\\to E^0$ by $r(V_k^{(ij)})=P_j$ and $s(V_k^{(ij)})=P_i$. Then $E=(E^0,E^1,r,s)$ is a graph. It is trivial to see that the identity maps on $E^1$ and $E^0$ extend to a Toeplitz representation of $X(E)$ which in turn induces a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$.\n\nFinally, we have that for each $x\\in W$\n\\begin{equation*}\\alpha(x)=\\sum_{i,j>0} P_j\\alpha(P_ix)=\\sum_{i,j>0}\\alpha_{ij}(x)=\\sum_{i,j>0}\\sum_{k=1}^{n_{ij}} V_k^{(ij)}xV_k^{(ij)*}=\\sum_{f\\in E^1}\\tau(\\delta_f)x\\tau(\\delta_f)^*\\end{equation*}\nas desired.\n\\end{proof}\nAs a consequence, the possible $*$-endomorphisms of a given von Neumann algebra $W=\\{P_1,P_2,...\\}'$ coincide, up to conjugacy, with the coherent unitary equivalence classes of $C^*$-correspondences over $C_0(\\{P_1,P_2,...\\})$. \n\n\\subsection{Cuntz-Pimsner Algebras} To discuss the special case of unital $*$-endomorphisms we will need to briefly sketch the defining features and universal properties of the so-called Cuntz-Pimsner algebras. The relationship of these Cuntz-Pimnser algebras to our Toeplitz algebras is analogous to the relationship between the classical Toeplitz algebras $\\mathcal{E}_n$ and classical Cuntz algebras $\\O_n$. The Cuntz-Pimsner algebras were originally defined in \\cite{pimsner} though our treatment will take its cues from \\cite{fowlerraeburnmuhly} and \\cite{fowlerraeburn}.\n\nRecalling that $X(E)$ is a $C^*$-correspondence over $C_0(E^0)$, notice that $I=C_0(\\{v\\in\\ E^0:r^{-1}(v)\\text{ is finite }\\})$ is a closed, two-sided ideal in $C_0(E^0)$. As noted in \\cite[Proposition 4.4]{fowlerraeburn} (and recalling our modern reversal of $r$ and $s$ from the presentation in that work), elements of $I$ are precisely those elements of $C_0(E^0)$ whose left action on $X(E)$ is compact. Since our graphs have at most countable vertices and edges, $X(E)$ is countably generated as a $C^*$-correspondence over $C_0(E^0)$. It follows, \\cite[Remark 3.9 following Definition 3.8]{pimsner} that the \\emph{Cuntz-Pimnser algebra} of $X(E)$ is the $C^*$-algebra $\\O_{X(E)}$ which is universal for Toeplitz representations $(\\sigma,\\pi)$ (as in Definition \\ref{toeplitzrepdef}) which additionally satisfy:\n\\begin{enumerate}\n\\item[(iv)] $\\displaystyle{\\pi(a)=\\sum_{f\\in E^1}\\sigma(\\phi(a)\\delta_f)\\sigma(\\delta_f)^*}$ for all $a\\in I$.\n\\end{enumerate}\nor, equivalently  (see \\cite[Example 1.5]{fowlerraeburnmuhly}),\n\\begin{enumerate}\n\\item[(iv)$'$] $\\displaystyle{\\pi(\\delta_v)=\\sum_{f\\in r^{-1}(v)}\\sigma(\\delta_f)\\sigma(\\delta_f)^*}$ for all $v\\in E^0$ with $|r^{-1}(v)|<\\infty$.\n\\end{enumerate}\nRepresentations satisfying (i)-(iv)$'$ are often termed ``coisometric\" Toeplitz representations. With this characterization of $\\O_{X(E)}$ we may recognize that Theorem \\ref{theorem3} has more to say when the endomorphism is unital.\n\n\\begin{cor} Let $W=\\bigoplus W_i\\subseteq B(H)$ be a countable sum of Type I factors. Let $P_1,P_2,...\\in B(H)$ be projections such that $W_i=P_iB(H)P_i$. If $\\alpha$ is a normal, unital $*$-endomorphism of $W$ then there exists a graph $E$ and $*$-representation $\\tau:\\O_{X(E)}\\to B(H)$ such that $\\alpha=Ad_\\tau$.\n\\end{cor}\n\\begin{proof} We define the partial isometries $V_k^{(ij)}$ and graph $E$ in the same manner as in the proof of Theorem \\ref{theorem3}. Since $\\alpha$ is unital we have\n\n$$P_n=P_n\\alpha(I)=\\sum_{i,j>0}\\sum_{k=1}^{n_{ij}} P_nV_k^{(ij)}V_k^{(ij)*}=\\sum_{i>0}\\sum_{k=1}^{n_{in}} V_k^{(in)}V_k^{(in)*}$$\nfor every $P_n$. Recalling that the representation of $(\\sigma,\\pi)$ $E$ on $B(H)$ is the pair of identity maps, this becomes\n$$\\pi(\\delta_v)=\\sum_{f\\in E^1}\\pi(\\delta_v)\\sigma(\\delta_f)\\sigma(\\delta_f)^*=\\sum_{f\\in r^{-1}(v)}\\sigma(\\delta_f)\\sigma(\\delta_f)^*$$\nfor all $v\\in E^0$. In particular this holds for all edges $v$ with $r^{-1}(v)$ finite (i.e.\\ all $P_j$ such that $\\{V_k^{(ij)}: i>0, k=1,...,n_{ij}\\}$ is finite). Thus condition (iv)$'$ is satisfied and the identity maps on $E^0$ and $E^1$ induce a representation of $\\O_{X(E)}$. The fact that $\\alpha=Ad_\\tau$ follows as before.\n\\end{proof}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nSequential execution of multi-robot coordinated behaviors can be employed to solve real-world complex missions. However, sequences of behaviors can be executed only if the robots meet all required communication constraints in finite time. In this paper, we described a distributed framework for the sequential composition of coordinated behaviors designed on finite-time convergence control barrier functions. The resulting composition framework is formulated in the form of a quadratic program, which is solved locally by individual robots. \\mymod{Although the focus of this paper is on coordinated motion, the application of the proposed framework is relevant to other form of autonomous collaborations where the robots need to satisfy prescribed pair-wise proximity requirements that change over time}. Finally, a large-scale multi-task scenario, denoted ``Securing a Building\" mission is proposed as an ideal environment for testing multi-robot techniques.\n\n\\appendices\n\\input{sections\/researchOpportunities}\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Behaviors Dynamic} \\label{sec:appendixB}\n\\paragraph{Rendezvous}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} x_j - x_i\n\\end{equation}\n\n\\paragraph{Scatter}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} x_i - x_j\n\\end{equation}\n\n\n\\paragraph{Formation control}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i)\n\\end{equation}\n\nwhere $\\theta_{ij}$ must be feasible.\n\n\\paragraph{Leader-follower}\n\\begin{align}\n\tu_f &= \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\\\\n\tu_l &= \\sum_{j \\in \\mathcal{N}_i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \n\\end{align}\n\n\\paragraph{Leader formation}\n\\begin{align}\n\tu_f &= \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\\\\n\tu_l &= \\sum_{j \\in \\mathcal{N}_i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \n\\end{align}\nwhere $\\theta_{ij}$ must be feasible.\n\n\\paragraph{Cyclic pursuit}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} R(\\theta)\\,(x_j - x_i) \n\\end{equation}\nwhere $\\mathcal{G}$ must be $C_N$ (cycle graph).\n\n\\paragraph{Lattice}\n\\begin{equation}\nu_i = \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta^2)(x_j - x_i)\n\\end{equation}\nwhere $\\theta \\simeq 0.8$ max sensing radius. \n\n\\paragraph{Coverage}\n\\begin{equation}\nu_i = c_i - x_i\n\\end{equation}\nwhere $c_i$ centroid of Voronoi tessellation.\n\n\\paragraph{Swarming}\n\\begin{equation}\nu_i = \\sum_{j \\in \\mathcal{N}_i} \\dots (x_j - x_i)\n\\end{equation}\n\n\\section{Securing a Building as Benchmark Scenario} \\label{sec:appendixA}\nTesting the performance of techniques and algorithms for the control of multi-robot systems in real-world scenarios is a challenging task. \\mymod{This is particularly true when addressing novel approaches, as the focus on specific aspects of the problem might obscure all-around performance assessments.} To this end, thanks to its modularity, the {\\it Securing a Building} mission is an ideal testing framework. In this section, we suggest a number of selected research topics, for which this mission could serve as a testing framework when aiming to evaluate performance of new techniques. \\mymod{This appendix is by no mean proposed as a complete list of subjects relevant to multi-robot systems but rather as a discussion to stimulate application of the {\\it Securing a Building} as a versatile, real-world testing scenario.}\n\n\\paragraph{Team Assembly} Considerable efforts have been devoted to the development of team composition techniques for heterogeneous robots~\\cite{prorok2016formalizing},~\\cite{koes2005heterogeneous}. Based on the skill set required to solve a particular task, e.g., certain actuation, sensing, locomotion, or communication capabilities, the question is to find a recruitment rule that produces a team capable of delivering the best performance. For instance, in the RESCUE phase, robots capable of opening doors may be required for the {\\it maneuvering} agents, while agility and communication capabilities might be preferred during the FIND phase.\n\n\\paragraph{Communication} In the context of autonomous networked systems, central roles are played by the flow of information between agents, and the infrastructure required for it~\\cite{gupta2016survey}. A number of questions can be posed in relation to the distribution of agents over a domain, given the constraints of communication systems, such as limited range, power requirements, and privacy of the information.\n\n\\paragraph{Unknown Environment} The amount of prior knowledge about the environment plays an important role in the definition of both low-level robot controllers and high-level mission plans. The performance of distributed solutions to the localization and mapping problems~\\cite{forster2013collaborative} can be tested on the {\\it Securing a Building}. Aspect of interest include balancing between exploitation and exploration of the environment applied, for instance, to the building exploration planning.\n\n\\paragraph{Resilience} Failure of the mission can be attributed to factors such as damaged components, sensing errors, communication dropouts, delays, control disturbances, reduction of functionalities due to adversarial attacks, etc. A number of different research thrusts focus on the problem of detecting and responding to faults and malicious attacks in multi-agent and cyber-physical systems~\\cite{pasqualetti2011consensus,pierpaoli2018fault,fawzi2014secure}.\n\n\n\\section{\\mymod{Distributed Composition of Behaviors}}\\label{sec:multAgImp}\nThe composition framework discussed in the previous section reduces to the team-wise minimum norm controller~(\\ref{eq:minQP1}), which is not directly solvable by individual robots. \\mymod{In addition to this, a centralized supervisor is needed in order to synchronize behavior transitions.} In this section, we \\mymod{formulate a decentralized solution to problem~\\ref{pr:problem} which can be implemented by the robots using only information from their neighbors. Furthermore, we also include those} additional constraints necessary for the safe operations of the robots, e.g., inter-agent collisions and obstacles avoidance~\\cite{wang2016multi}. The formulation is derived following the approach described in~\\cite{squires2019composition}, which we adapt here to our framework.\n\n\\subsection{Distributed Finite-Time Convergence Control Barrier Functions}\nThe limitation in solving problem~(\\ref{eq:minQP1}) in a distributed fashion is represented by the fact that knowledge of dynamics, input $\\hat{u}$, and state $x$ for the entire team need to be available. In addition, solution of~(\\ref{eq:minQP1}), provides the control inputs for the entire team, which are unnecessary to the individual robots.\n\nIn order to develop the correct decentralized formulation of~(\\ref{eq:minQP1}), we first define a decomposition of the dynamics~(\\ref{eq:ensembleDynamics}). We denote by $\\mathcal{D}_i\\subset$ \\mymod{$\\mathbb{R}^d$} and $U_i \\subset$ \\mymod{$\\mathbb{R}^m$} configuration space and set of feasible controls for agent $i$ respectively. In addition, by denoting with $\\bar{f},\\,\\bar{g}: \\mathcal{D}_i \\mapsto$ \\mymod{$\\mathbb{R}^d$} the node-level terms of the control affine dynamics of agent $i$, the ensemble dynamics can be written as:\n\\begin{equation} \\label{eq:decoup_dyn}\n\t\\dot{x} = \\bar{f}(x_i)\\otimes {\\bf 1}_n + (\\bar{g}(x_i) \\otimes I_n)\\,\\begin{bmatrix} u_1 \\\\ \\vdots \\\\ u_n \\end{bmatrix},\n\\end{equation}\nwhere $u_i \\in U_i$ is the $i^{\\text{th}}$ robot's control input, $\\otimes$ is  the Kronecker product, and ${\\bf 1}_n$ and $I_n$ are vector of ones and identity matrix of size $n$ respectively.\n\nLet's consider two sequential behaviors $\\mathcal{B}_{k-1}$ and $\\mathcal{B}_{k}$. Upon completion of $\\mathcal{B}_{k-1}$, for all edges $(i,j)\\in E_k$, robots' configuration should satisfy \n\\begin{equation} \\label{eq:dec_const_edge}\n\t\\dot{h}_{ij}^c(x_i,x_j) + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c(x_i,x_j)) \\geq 0.\n\\end{equation}\nFrom the $i^{\\text{th}}$ robot's point of view, the set of constraints that need to be satisfied in order to execute the new behavior are\n\\begin{equation} \\label{eq:dec_const_agenti}\n\t\\dot{h}^c_{ij}(x_i,x_j) + \\bar{\\alpha}_{\\rho,\\gamma}(h^c_{ij}(x_i,x_j)) \\geq 0 \\quad \\forall j\\in \\mathcal{N}_{k}^i,\n\\end{equation}\nwhere we recall that $\\mathcal{N}_{k}^i$ is the set of neighbors to robot $i$ required by behavior $\\mathcal{B}_k$. However, since constraint~(\\ref{eq:dec_const_agenti}) appears exactly twice across the team of robots, it can be relaxed by considering the admissible set of control inputs\n\\begin{equation} \\label{eq:admGraphControl}\nK_{k}^{c,i} = \\bigcap_{j\\in\\mathcal{N}_k^i} K_{k,ij}^{c,i}\n\\end{equation}\nwith\n\\begin{equation}\nK_{k,ij}^{c,i} = \\{u_i\\in U_i \\,| \\,L_{\\bar{f}}h_{ij}^c + L_{\\bar{g}}h_{ij}^cu_i + \\frac{\\bar{\\alpha} _{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 \\},\n\\end{equation}\nwhere dependence from $x_i$ and $x_j$ is omitted for clarity.\n\n\\begin{theorem} \\label{thm:fcbfControl_dist}\n\tDenoting with $x_0 = [x_{0,1}^T,\\dots,x_{0,n}^T]^T$ the initial state of a multi-agent system with dynamics described as in~(\\ref{eq:decoup_dyn}), any controller $\\mathcal{U}_i:\\mathcal{D}_i^{|\\mathcal{N}_k^i|} \\mapsto U_i$ such that $\\mathcal{U}_i(x_{0}) \\in K_{k}^{c,i}$ for all $x_{0} \\in \\mathcal{D}_i^{|\\mathcal{N}_k^i|}$, will drive the ensemble state to $\\mathcal{C}^c_k $ within time\n\\begin{equation}\n\tT_k = \\max_{\\substack{(i,j) \\in E_k \\\\ \\text{s.t.} \\,\\, h^c_{ij}(x_{0,i},x_{0,j})<0}} \\left\\{ \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_{0,i},x_{0,j}) |^{1-\\rho} \\right\\}.\n\\end{equation}\n\\end{theorem}\n\n\n\n\\begin{IEEEproof}\nFrom Theorem~\\ref{thm:FCBF}, agents $i$ and $j$, with $(i,j)\\in E_k$, will satisfy $h_{ij}^c \\geq 0$ in finite time if\n\\begin{equation} \\label{eq:connect_const}\n\\dot{h}_{ij}^c + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0. \n\\end{equation}\n\nConsidering the node level dynamics in~(\\ref{eq:decoup_dyn}), the constraint~(\\ref{eq:connect_const}) reduces to\n\n\\begin{equation} \\label{eq:const2agents}\n\\begin{aligned}\n&\\frac{\\partial h_{ij}^c}{\\partial x_i} \\left( \\bar{f} + \\bar{g}u_i \\right) \\, + \\, \\frac{\\partial h_{ij}^c}{\\partial x_j}\\left(\\bar{f} + \\bar{g}u_j\\right) + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0 \\\\\n&2\\,L_{\\bar{f}}h_{ij}^c + L_{\\bar{g}}h_{ij}^c\\,u_i + L_{\\bar{g}}h_{ij}^c u_j + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0\n\\end{aligned}\n\\end{equation}\nwhich will be satisfied if both agents $i$ and $j$ satisfy the constraint \n\\begin{equation} \\label{eq:dec_const}\n\t\\dot{h}_{ij}(x_i,x_j) + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}(x_i,x_j))}{2} \\geq 0.\n\\end{equation}\nIn addition, as discussed in Theorem~\\ref{thm:fcbfControl},  constraint~(\\ref{eq:const2agents}) will still be satisfied at time\n\n\\begin{equation}\n\tT_{ij} \\leq \\frac{1}{\\gamma(1-\\rho)} | h_{ij}^c(x_{0,i},x_{0,j}) |^{1-\\rho}.\n\\end{equation}  \n\nThe same argument can be repeated for all pairs $(i,j) \\in E_k$, and condition $\\mathcal{G}_k \\subseteq \\mathcal{G}(t)$ will be satisfied within time\n\n\\begin{equation}\n\tT_k = \\max_{\\substack{(i,j) \\in E_k \\\\ \\text{s.t.} \\,\\, h^c_{ij}(x_{0,i},x_{0,j})<0}} \\left\\{ T_{ij} \\right\\}.\n\\end{equation}\t\n\\end{IEEEproof}\n\nApplying the same design principle described in Section~\\ref{sec:ftcontrolBF}, the minimally invasive control action can be computed by each robot as\n\\begin{equation}\\label{eq:minQP1_dist}\nu^*_i = \\argmin_{u_i \\in U_i}  \\| \\hat{u}_{k-1,i} - u_i \\|^2 \\\\\n\\end{equation}\nsubject to\n\\begin{equation}\\label{eq:constrTransition_dist}\nL_{\\bar{f}}\\,h_{ij}^c + L_{\\bar{g}}\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 , \\quad \\forall j\\in  \\mathcal{N}_{k-1}^i \\cup \\mathcal{N}_{k}^i.\n\\end{equation}\nSimilarly to constraint~(\\ref{eq:constrTransition}), once all edges in $E_k$ are available, constraint~(\\ref{eq:constrTransition_dist}) is substituted with\n\\begin{equation}\\label{eq:constrExecution_dist}\nL_{\\bar{f}}\\,h_{ij}^c + L_{\\bar{g}}\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 , \\quad \\forall j\\in\\mathcal{N}_{k}^i,\n\\end{equation}\nwhich remains active until $\\mathcal{B}_k$ is completed.\n\nWe note that, in order for agent $i$ to respect~(\\ref{eq:constrExecution_dist}), the only external information needed is the state of all current neighbors, i.e. $x_j$ for all $j\\in\\mathcal{N}^i_k$. On the other side, in order to respect (\\ref{eq:constrTransition_dist}), robots need to have access to the state of the future neighbors. This requirement can be satisfied through a state estimation scheme (e.g. EKF~\\cite{williams2015observability}), which in turn requires knowledge of robots' dynamics (known for homogeneous teams) or network localization techniques~\\cite{aspnes2006theory}.\n\\mymod{\n\\begin{remark}\nThe ability of each robot to have access to an estimate of their future neighbors' state does not eliminate the necessity of establishing neighborhood relationships. In fact, a certain proximity structure between robots might be required by desired controllers' performance that cannot be met through state estimations, or by collaboration tasks that require physical interaction between the robots, e.g. collaborative manipulation~\\cite{culbertson2018decentralized}, sharing of resources~\\cite{ramachandran2019resilience}.\n\\end{remark}}\n\n\\subsection{Additional Constraints}\nIn addition to the proximity constraints discussed above, additional limitations might be imposed on the robots' configuration by the mission and the environment. For illustrative purposes, we consider inter-robots collisions and obstacle avoidance. Following the approach described in~\\cite{wang2016multi}, we encode each pair-wise separation condition through the following barrier certificate\n\\begin{equation}\n\th_{ij}^a(x) = \\| x_i-x_j \\|^2 - D_a^2 \n\\end{equation}\nand the minimum separation $D_a$ between the robots is satisfied if $h_{ij}^a(x) \\geq 0$, for all physical neighbors $j\\in \\mathcal{N}^i(t)$.\n\nSimilarly, avoidance of fixed obstacles can be introduce by considering $M$ ellipsoidal regions of the domain, described by centers $o=[o_1^T,\\dots,o_M^T]^T$. For every agent-obstacle pair $(i,m)$ we define a pairwise barrier function as\n\\begin{align}\nh_{im}^o(x) &=(x_i-o_m)^T\\,P_m\\,(x_i-o_m) - 1 \\\\\nP_m &= \\begin{bmatrix}\n\ta_m & 0 \\\\ 0 & b_m \n\\end{bmatrix}\n\\quad a_m,b_m > 0.\n\\end{align}\nThe object avoidance constraints are satisfied if $h_{im}^o(x)\\geq 0$, for all $i\\in V$ and $m \\in \\{1,\\dots,M \\}=\\mathcal{I}_M$.\n\nCollecting all the constraints, we expand the problem formulation in~(\\ref{eq:minQP1}) to\n\n\\begin{equation} \\label{eq:minQP2}\n\\begin{aligned}\n&\\qquad u_i^* = \\arg \\min_{u_i \\in U_i}  \\| \\hat{u}_{k-1,i} - u_i \\|^2 &\\\\\n&L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0, & \\forall j\\in \\mathcal{N}_{k}^i \\\\\n&L_f\\,h_{ij}^s + L_g\\,h_{ij}^s\\,u_i + \\alpha(h_{ij}^{s}) \\geq 0, &\\forall j\\in \\mathcal{N}^i(t)\\\\\n&L_f\\,h_{im}^o + L_g\\,h_{im}^o\\,u_i + \\alpha(h_{ij}^{s}) \\geq 0, &\\forall m \\in \\mathcal{I}_M\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$ is a locally Lipschitz extended class-$\\mathcal{K}$ function and the first constraint is replaced by~(\\ref{eq:constrTransition_dist}) during transitions. In conclusion, \\mymod{if there exists a set of control inputs $u = [u_1,\\dots,u_N]$ that simultaneously satisfies all constraints in~(\\ref{eq:minQP2}), for all behaviors $k=1,\\dots,M$, Problem~\\ref{pr:problem} will be solved by the robots.}\n\n\\subsection{\\mymod{Decentralized Behaviors Sequencing}}\n\\mymod{For the correct execution of the behaviors sequencing, each robot should start assembling a new graph only after all other robots have completed the current behavior. Similarly, a new behavior should start once all robots satisfy the neighbors' requirements for it. Now, we describe a decentralized strategy that allows execution of these two transitions without the need of a supervisor, nor synchronization between the robots.}\n\n\\mymod{With reference to Fig.~\\ref{fig:automata}, at any given time, each robot's mode of operation is described by a binary variable $\\alpha_i$ that describes whether robot $i$ is assembling the graph for an upcoming behavior ($\\alpha_i=1$) or executing a behavior ($\\alpha_i=0$). Without loss of generality, assume robots' initial configuration satisfies the communication requirements for the first behavior, which is then executed ($\\alpha_i = 0$). Once all robots have completed the first behavior, they start assembling the graph required by the following one ($\\alpha_i = 1$), while minimally perturbing the behavior just concluded. Once the new graph is satisfied $\\mathcal{G}_2 \\subseteq \\mathcal{G}(t)$, robots start behavior $\\mathcal{B}_2$ and exit from assembly mode ($\\alpha_i = 0$). This process repeats, until no successive behavior exists.} \n\n\\mymod{A correct execution of this process requires robot to agree on when to perform transitions $\\alpha_i = 0 \\rightarrow 1$ and $\\alpha_i = 1 \\rightarrow 0$. To this end, we take inspiration from the consensus-based algorithm described in~\\cite{wagenpfeil2009distributed} and we note that this choice is not central to the contribution of this paper. For each robot, we define a binary variable available only to robot $i$, $s_{t,i} \\in \\{0,1\\}$ that denotes whether robot $i$ itself has completed its current task $s_{t,i} = 1$ ($s_{t,i} = 0$ if robot has not completed its current task). In addition, we introduce a variable $\\sigma_i \\in \\mathbb{R}_+$, shared among neighbors, continuously updated through the following consensus-based process\n\\begin{equation} \\label{eq:sync}\n    \\sigma_i^+ = s_{t,i} \\frac{1}{|\\mathcal{N}_i(t)|+1}\\left(\n    \\sum_{j \\in \\mathcal{N}_i(t)} \\sigma_j + 1 \\right),\n\\end{equation}\nwhere $\\sigma_i^+$ represent the variable's value after the update. Owing to the diffusion of $\\sigma_1,\\dots,\\sigma_N$ throughout the network, we can interpret $\\sigma_i$'s as local measures of the team-wise completion of a task. As proved in~\\cite{wagenpfeil2009distributed}, if $s_{t,i}=1$ for all $i=1,\\dots N$ (i.e., all robots are capable to complete the current behavior), by following~(\\ref{eq:sync}), $\\lim_{t \\rightarrow \\infty} \\sigma_i = 1$, for all $i=1,\\dots N$. Therefore, robot $i$ starts assembling a new communication graph once the value of $\\sigma_i$ is close enough to $1$ (see~\\cite{wagenpfeil2009distributed} for a discussion on how to choose the switching threshold). A similar process is used for the transition $\\alpha_i=1 \\rightarrow 0$, where we replace $s_{t,i}$ and $\\sigma_i$ with $s_{a,i}$ and $\\eta_i$ respectively. The distributed sequencing procedure is summarized in Algorithm~\\ref{alg:dis_sequence}.}\n\n\n\\begin{algorithm}[h]\n\\caption{Distributed composition of behaviors.} \\label{alg:dis_sequence}\n$\\pi \\gets \\{\\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}$ \\tcc*[r]{initialize behaviors}\n$k = 1$\\;\n$\\alpha_i \\gets 0$\\;\n\\While{$k < M+1$}{\n\\tcc*[h]{Aggregate data from neighbors} \\\\\n\\For{$j \\in \\mathcal{N}_i(t)$}\n{\n$\\{X_i, \\Sigma_i, H_i\\} \\gets \\{X_i, \\Sigma_i, H_i\\} \\cup \\{ x_j , \\sigma_j , \\eta_j \\}$ \\;\n}\n\\tcc*[h]{Compute nominal control} \\\\\n$\\hat{u}_i \\gets \\mathcal{U}_k(x_i,X_i)$\\; \n\\tcc*[h]{Compute team-wise completion states} \\\\\n\n\\eIf{$\\alpha_i == 0$}{\n\\lIf{{\\tt task complete}}\n{$s_e \\gets 1$}\n\\lElse{$s_e \\gets 0$} \n$\\sigma_i:=s_e\\,\\frac{1}{|\\mathcal{N}_k^i|+1}(\\sum_{j \\in \\mathcal{N}_i(t)} \\sigma_j + 1)$\\;\n\\If{$\\sigma_i > \\bar{\\sigma}$}{\n$\\alpha_i\\gets 1$\\; $k \\gets k+1$ \\;\n}\n}{\n\\lIf{{\\tt assembly complete}}\n{$s_a \\gets 1$}\n\\lElse{$s_a \\gets 0$} \n$\\eta:=s_a\\,\\frac{1}{|\\mathcal{N}_k^i|+1}(\\sum_{j \\in \\mathcal{N}_i(t)} \\eta_j + 1)$\\;\n\\If{$\\eta>\\bar{\\eta}$}{\n$\\alpha_i\\gets 0$\\; $s_e\\gets 0$ \\;}\n}\n\\tcc*[h]{Solve FCBF QP} \\\\\n$u_i \\gets QP(\\hat{u}_i,X_i,x_i)$\n}\n\\end{algorithm}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{graphics\/automata.pdf}\n    \\caption{Representation of the distributed sequencing framework and information flow. At all times, each robot's state is in either {\\tt behavior execution} ($\\alpha_i=0$) or {\\tt graph assembly} ($\\alpha_i=0$) modes. Switching between the two modes is triggered by the variables $\\sigma_i$ and $\\eta_i$ whose values is continuously) updated through~(\\ref{eq:sync}). When a switching between {\\tt graph assembly} and {\\tt behavior execution} occurs, a new behavior is started.}\n    \\label{fig:automata}\n\\end{figure}\n\n\\subsection{Applications}\n\\begin{figure*}[t]\n\\centerline{ \n\\subcaptionbox{\\label{fig2:a}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T0_task2_fixed}}~\n\\subcaptionbox{\\label{fig2:b}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T1_task2_fixed}}~\n\\subcaptionbox{\\label{fig2:c}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T2_task23_fixed}}\n} \n\\vspace{0.25cm}\n\\centerline{ \n\\subcaptionbox{\\label{fig2:d}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T3_task3_fixed}}~\n\\subcaptionbox{\\label{fig2:e}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T4_task3_fixed}}~\n\\subcaptionbox{\\label{fig2:f}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T5_task3_fixed}}\n}\n\\caption{Overhead screen-shots from experiments on the Robotarium. Five robots execute two behaviors in sequence (cyclic-pursuit and formation). In figure, green patches represent robots that have completed their task, black rings represent robots that have all neighbors needed for the following task, and green lines represent edges that are available in the current communication graph. From (a) to (b) robots complete the first behavior. During second behavior, additional edges $(2,5)$ and $(3,5)$ are required (red dashed line represent missing edges). From (c) to (d), robots $2,3,5$ reduce their distance below the communication threshold. After the new graph is complete (d), robots initiate the second behavior (e) and complete it (f).\n\\label{fig:5RobotsExp}}\n\\end{figure*}\n\n\\mymod{We implemented the distributed sequencing framework on the Robotarium~\\cite{pickem2017robotarium}, on a team of $5$ differential drive robots. For this example, controllers are designed assuming a single integrator model, i.e. $\\bar{f}(x_i)=[0,0]^T$ and $\\bar{g}(x_i)=I_2$. In this example, robots execute a transition between two behaviors, where $\\mathcal{B}_1$ is a cyclic-pursuit behavior and $\\mathcal{B}_2$ is a formation assembly with leader. Cyclic-pursuit behavior is obtained through the following controller:\n\\begin{equation*}\n\t\\hat{u}_i = \\sum_{j \\in \\mathcal{N}_1^i} R(\\phi)\\,(x_j - x_i) \\quad \\forall \\, i=1,\\dots,5,\n\\end{equation*}\nwhere $R(\\phi)\\in SO(2)$ is the rotation matrix of angle $\\phi$, which is related to the desired cycle radius. Importantly, for this behavior to work, the communication graph $\\mathcal{G}_1$ must be a cycle graph. Considering robot $1$ as leader, the formation control behavior can be achieved with\n\\begin{align*}\n\\hat{u}_1 &= \\sum_{j \\in \\mathcal{N}_2^i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \\\\\n\t\\hat{u}_i &= \\sum_{j \\in \\mathcal{N}_2^i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\quad i = 2,\\dots 5\n\\end{align*}\nwhere $\\theta_{ij} \\in \\mathbb{R}_+$ is the desired inter-robot distance, $x_g \\in \\mathcal{D}$ is the leader's goal, and $\\gamma_g \\in \\mathbb{R}_+$ the corresponding proportional gain. In the case of formation control, it is known that the Euclidean embedding of $\\mathcal{G}_2$ must be a rigid framework (see for instance~\\cite{mesbahi2010graph} and references therein). With reference to Fig.~\\ref{fig:5RobotsExp}, robots initially execute $\\mathcal{B}_1$ for a certain amount of time (a). Once completed (b) (green patches represent robots that have completed their current behavior), robots start assembling $\\mathcal{G}_2$ (c), after which $\\mathcal{B}_2$ is executed until $\\| \\hat{u}_i \\|$ is below a pre-defined threshold (d-f).}\n\n\\mymod{In Fig.~\\ref{fig:transition} we can observe the value of the two consensus variables $\\sigma_i$ and $\\eta_i$ for all robots during the behavior transition. Background colors represent the time intervals during which the two behaviors were executed, while the darker region in the middle corresponds to the assembly of the new graph. We observe the assembly and task variables $\\eta_i$ and $\\sigma_i$ approaching the value $1$ simultaneously for all robots, thus triggering a synchronized start of the successive phase.}\n\n\\mymod{The robustness of our technique was tested by simulating uniformly distributed delays between the robots. Results for this case are shown in Fig.~\\ref{fig:transition_delay} where we observe that although convergence of $\\eta_i$ and $\\sigma_i$ is no longer monotonic, robots still reach agreement on when to switch to the successive phase.}\n\n\\mymod{Finally, in order to show the benefits of the minimally invasive approach, we compare it with an alternative technique inspired by~\\cite{twu2010graph}, where, upon collective completion of a behavior, robots execute rendezvous until the communication graph required by the successive behavior is assembled. As shown by the simulation results for a sequence of $7$ behaviors (Fig.~\\ref{fig:energyComp}), the mean of the input's norm when considering our framework (red solid line) is always lower than the one obtained using the rendezvous as {\\it glue} behavior. Importantly, since transitions between behaviors occur faster in the minimally invasive case, the lower control effort cannot be attributed to a more {\\it relaxed} choice of controller gains.}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = \\columnwidth]{graphics\/transition.png}\n    \\caption{Task and assembly consensus variables $\\sigma_i$ and $\\eta_i$ for $i=1,\\dots,N$ during a transition between two behaviors.}\n    \\label{fig:transition}\n\\end{figure}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = \\columnwidth]{graphics\/transition_delay.png}\n    \\caption{Task and assembly consensus variables $\\sigma_i$ and $\\eta_i$ for $i=1,\\dots,N$ during a transition between two behaviors with communication delays.}\n    \\label{fig:transition_delay}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[trim ={2cm 0 0 0}, width=1.05\\columnwidth]{graphics\/energyComparison.png}\n    \\caption{Control input comparison between the minimally invasive sequencing framework proposed in this paper (red) and a sequencing based on rendezvous as {\\it gluing} behavior (blue). Solid lines represent the mean of the control input across all robots, while shaded regions represent the interval between minimum and maximum control input.}\n    \\label{fig:energyComp}\n\\end{figure}\n\\section{Problem Formulation}\n\\label{sec:problem}\nWe denote the state of a team of $n$ homogeneous mobile robots operating in a $d$-dimensional and connected domain $\\mathcal{D}$ as $x(t) = [x_1(t)^T,\\dots,x_n (t)^T]^T \\in \\mathcal{D} \\subset$ $\\mathbb{R}^{dn}$ where $x_i(t)\\in\\mathbb{R}^d$ is the position of robot $i$ at time $t$. As part of the coordinated nature of the behaviors being performed by the robots, each robot executes a control protocol which depends on the state of the subset of robots with which it interacts. We assume robots can communicate if the distance between them is less or equal to a sensing threshold $\\Delta\\in\\mathbb{R}_{>0}$. Thus, the list of possible interactions between agents are described by a time-varying, undirected, proximity graph $\\mathcal{G}(t)=(V,E(t))$, where $V=\\{1,\\dots,n \\}$ is the set of nodes representing the robots and $E(t)$ is the set of interacting pairs at time $t$, where \n\\begin{equation}\nE(t) = \\{ (i,j) \\in V \\times V\\, | \\, \\| x_i(t)-x_j(t)\\| \\leq \\Delta \\}.\t\n\\end{equation}\nFor each robot $i=1,\\dots,n$, we denote the set of available neighbors at time $t$ as $\\mathcal{N}_i(t) = \\{ j \\in V\\, | \\, (i,j) \\in E(t) \\}$, which depends on the position of the robots at time $t$.\n\nThe ensemble dynamics of the multi-agent system is described by\n\\begin{equation}\n\t\\dot{x} = f(x) + g(x)\\,u\n\t\\label{eq:ensembleDynamics}\n\\end{equation}\nwhere $f$ and $g$ are continuous locally Lipschitz continuous functions and $u = [u_1^T,\\dots,u_n^T]^T \\in U \\subset \\mathbb{R}^m$ is the vector of inputs, which depends on the particular behavior being executed. At all times, the control input $u$ in~(\\ref{eq:ensembleDynamics}) is given by a controller $\\mathcal{U}$, which can be defined as a state feedback law $\\mathcal{U}: \\mathcal{D} \\mapsto U$ or by a combination of both external parameters and state feedback law $\\mathcal{U}: \\mathcal{D} \\times \\Theta  \\mapsto U$, where $\\Theta$ is a space of parameters appropriate for the behavior. For instance, the controller corresponding to a {\\it weighted consensus} belongs to the first case. On the other side, a leader-follower protocol where followers maintain prescribed inter-agent distances is described by a controller that depends on both state feedback (followers' control) and exogenous parameters (leader's goal) (see Section~\\ref{sec:multAgImp} for examples).\n\nWe represent a {\\it mission} by an ordered sequence of $M$ coordinated behaviors\n\\begin{equation}\n\t\\pi = \\{ \\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}.\n\\end{equation}\nThe $k^{\\text{th}}$ behavior in $\\pi$ is defined by the pair\n\\begin{equation}\\label{eq:tuple}\n\t\\mathcal{B}_k = \\{ \\mathcal{U}_k,\\, \\mathcal{G}_k\\},\n\\end{equation}\nwhere $\\mathcal{U}_k$ represents the coordinated controller described above and $\\mathcal{G}_k$ is the interaction graph required by behavior $\\mathcal{B}_k$ to function properly. We assume the list of behaviors $\\pi$ to be fixed and available to all robots. \\mymod{We will use the term {\\it behavior} to refer to a generalized multi-robot controller in the form~(\\ref{eq:tuple}) and to {\\it task} as the objective of the controller.}\n\nAs discussed in Section~\\ref{sec:intro}, each behavior requires a certain interaction structure between the robots (i.e., pairs of robots that need to be neighbors). With reference to~(\\ref{eq:tuple}), we describe an interaction structure via the graph $\\mathcal{G}_k=(V,E_k)$. Thus, denoting by $t_k^\\vdash$ and $t_k^\\dashv$ the starting and ending times for behavior $k$, the robots' configuration needs to satisfy $\\mathcal{G}_k \\subseteq \\mathcal{G}(t)$ for all $t \\in [t_k^\\vdash, t_k^\\dashv]$. In other words, as shown in Fig.~\\ref{fig:beh_seq}, the interaction structure required by each behavior needs to be a spanning graph of the graph induced by the state of the agents during the interval of time the behavior is executed. \\mymod{At this point, given a list of behaviors constituting the mission $\\pi$ and the corresponding multi-robot controllers, we want to design a procedure that enables robots to assemble and maintain the communication graph required by each behavior.}\n\n\n\\mymod{\n\\begin{problem} \\label{pr:problem}\nGiven an ordered sequence of coordinated behaviors $\\pi= \\{ \\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}$, where each $\\mathcal{B}_k = \\{ \\mathcal{U}_k,\\, \\mathcal{G}_k\\}$ can be completed by the robots in finite-time, design a feedback control policy to compose the behaviors such that\n\\begin{equation}\n\\mathcal{G}(t) \\supseteq\n    \\begin{cases}\n    \\mathcal{G}_k \\, &t \\in [t_k^\\vdash , t_k^\\dashv] \\\\\n    \\mathcal{G}_k \\cup \\mathcal{G}_{k+1} \\, &t \\in (t_k^\\dashv,t_{k+1}^\\vdash)\n\\end{cases} \\quad \\forall \\, k=1,\\dots,M-1.\n\\end{equation}\n\\end{problem}\n}\n\\section{Finite-Time Barrier Functions} \\label{sec:fcbf}\nIn this section we review the general definition of Finite-time Convergence Control Barrier Function (FCBF) which was first introduced in~\\cite{li2018formally} and inspired by the finite-time stability analysis for autonomous system introduced in~\\cite{bhat2000finite}. Given a dynamical system operating in an open set $\\mathcal{D} \\subseteq \\mathbb{R}^n$ and a set $\\mathcal{C}\\subset\\mathcal{D}$, barrier functions~\\cite{xu2015robustness} are Lyapunov-like functions that guarantee forward invariance of $\\mathcal{C}$ with respect to the state of the system. In other words, if an appropriate barrier function exists, it can be used to show that if the state of a system is in $\\mathcal{C}$ at some time, it will be in $\\mathcal{C}$ thereafter. The concept of barrier functions was extended to Zeroing Control Barrier Functions (ZCBF) in~\\cite{xu2015robustness}, where asymptotic convergence of the state to the set $\\mathcal{C}$ was discussed. Thus, provided that an appropriate ZCBF exists, if the state of the system is not in $\\mathcal{C}$ at some initial time, it will asymptotically converge to $\\mathcal{C}$. \n\nAs discussed in the introduction, before execution of a coordinated behavior, robots need to satisfy certain spatial configurations imposed by the behavior itself. Importantly asymptotic convergence to the correct configuration is not sufficient. In fact, if we consider $\\mathcal{C}$ as the joint set of all initial configurations required for a particular behavior, the state must strictly belong to $\\mathcal{C}$ for the behavior to work properly. Following this observation, the need for a finite-time convergence extension of the previous concepts becomes clear. In particular, denoting the state of the system as $x(t)\\in\\mathcal{D}$, we are interested in verifying the following conditions:\n\\begin{itemize}\n\t\\item if $x(t_0)\\in\\mathcal{C}$, then $x(t)\\in\\mathcal{C}$ for all $t > t_0$\n\t\\item if $x(t_0) \\notin \\mathcal{C}$, then $x(t)\\in\\mathcal{C}$ for some $t_0<t<\\infty$.\n\\end{itemize}\nIn order to do this, we encode the set $\\mathcal{C} \\subset \\mathcal{D} \\subseteq \\mathbb{R}^n$, through the superzero-level set of a continuously differentiable function $h: \\mathcal{D} \\rightarrow \\mathbb{R}$, i.e.,\n\\begin{equation}\n\t\\mathcal{C} = \\{ x\\in \\mathcal{D} \\, | \\, h(x)\\geq 0 \\}.\n\t\\label{eq:sefdef}\n\\end{equation}\n\\begin{definition}\nWe introduce the following class-$\\mathcal{K}$ function\n\\begin{equation}\n\t\\bar{\\alpha}_{\\rho,\\gamma}(h(x)) = \\gamma \\cdot \\,\\text{sign}(h(x))\\, \\cdot |h(x)|^\\rho,\t\\label{eq:alpha}\n\\end{equation}\nwith $\\rho \\in [0,1)$ and $\\gamma>0$, which is continuous everywhere and locally Lipschitz everywhere except at the origin~\\cite{bhat2000finite}.\n\\end{definition}\n\n\\begin{definition}~\\cite{li2018formally} For a dynamical system \n\\begin{equation} \\label{eq:affinesystem}\n\t\\dot{x} = f(x)+g(x)u\n\\end{equation} with $x\\in\\mathcal{D}$, $u \\in U \\subset \\mathbb{R}^m$, and for a set $\\mathcal{C}$ induced by $h$, if there exists a function $\\bar{\\alpha}_ {\\rho,\\gamma}(h(x))$ of the form~(\\ref{eq:alpha}) such that\n\t\\begin{equation}\n\t\t\\sup_{u \\in U}\\bigg\\{ L_fh(x) + L_gh(x)u + \\bar{\\alpha} _{\\rho,\\gamma}(h(x)) \\bigg\\} \\geq 0 \\quad \\forall x\\in\\mathcal{D},\n\t\\end{equation}\nthen, the function $h$ is a {\\it Finite-time Convergence Barrier Function }(FCBF) defined on $\\mathcal{D}$.\n\\end{definition}\n\nFollowing from the definition above, we define the set of admissible control inputs as\n\\begin{equation} \\label{eq:admisU}\n\tK(x) = \\{ u \\in U \\, | \\, L_fh(x) + L_gh(x)u + \\bar{\\alpha} _{\\rho,\\gamma}(h(x)) \\geq 0\\}.\n\\end{equation}\n\n\\begin{theorem} \\label{thm:FCBF} \\cite{li2018formally}~Given a set $\\mathcal{C}\\subset \\mathbb{R}^n$, any Lipschitz continuous controller $\\mathcal{U}: \\mathcal{D} \\mapsto U$ such that \n\\begin{equation} \\label{eq:fcbf_controllers}\n\t\\mathcal{U}(x) \\in K(x) \\qquad \\forall x\\in\\mathcal{D},\n\\end{equation}\nrenders $\\mathcal{C}$ forward invariant for the system~(\\ref{eq:affinesystem}). Moreover, given an initial state $x_0 \\in \\mathcal{D} \\backslash \\mathcal{C}$, the same controller $\\mathcal{U}$ results in $x(T)\\in\\mathcal{C}$, where\n\t\\begin{equation}\n\t\tT \\leq \\frac{1}{\\gamma(1-\\rho)}|h(x_0)|^{1-\\rho}.\n\t\\end{equation}\n\\end{theorem}\n\n\\iffalse\n\\begin{IEEEproof}\n\tLet's consider the following Lyapunov function $V(x)=\\max\\{0,-h(x)\\}$. It can be verified that\n\\begin{align}\n\t\tV(x) &> 0 \\qquad  x\\in\\mathcal{D}\\backslash \\mathcal{C} \\\\\n\t\tV(x) &= 0 \\qquad  x\\in\\mathcal{C}\n\\end{align}\t\n\nIn addition, since\n\\begin{equation}\n\t\\frac{\\partial V(x)}{\\partial h(x)}=\n\t\\begin{cases}\n\t-1 &\\qquad x\\in\\mathcal{D}\\backslash \\mathcal{C} \\\\\n\t 0 &\\qquad  x\\in\\mathcal{C}\n\t\\end{cases}\n\\end{equation} \t\nit follows that $\\dot{V}(x(t)) \\leq -\\gamma\\,V^\\rho(x(t))$, for all $t$. \n\nConsider $x_0 = x(t_0) \\in \\mathcal{C}$. Since $V(x_0)=0$ and $\\dot{V}(t)=0$, we have $x(t)\\in\\mathcal{C}$ for all $t>t_0$, from which forward invariance of $\\mathcal{C}$ under $\\mathcal{U}$ follows. Now, consider $x_0 \\in \\mathcal{D}\\backslash\\mathcal{C}$. As shown in~\\cite{bhat2000finite}, the dynamics $\\dot{h} _1(t) = -\\gamma\\,\\text{sign}(h_1(t))\\, |h_1(t)|^\\rho$, with $\\rho \\in [0,1)$ and $\\gamma>0$ drives $h_1$ to the origin, and the minimum time $T_1$ for which $h_1$ reaches $0$ (denoted finite-settling time) is\n\\begin{equation}\n\tT_1 = \\frac{1}{\\gamma(1-\\rho)}|h_1(t_0)|^{1-\\rho}.\n\\end{equation} \nBy applying the comparison lemma~\\cite{pachpatte1997inequalities}, if $h(0) \\geq h_1(0)$ and $\\dot{h}(t) \\geq \\dot{h}_1(t)$, then $h(t) \\geq h_1(t)$ for all $t \\geq 0$ and consequently under the effect of $\\mathcal{U}$ we have\n\\begin{equation}\n\th(x(T)) = 0, \\quad T \\leq T_1.\n\\end{equation}\n\\end{IEEEproof}\n\\fi\n\nIn conclusion, by selecting a controller that verifies condition~(\\ref{eq:fcbf_controllers}), both forward invariance and finite-time convergence to the desired set are guaranteed. \n\n\n\\section{Composition of Coordinated Behaviors}\n\\label{sec:seq_framework}\n\nIn addition to the list $\\pi$, transitions between behaviors need to be synchronized, i.e., for each behavior $\\mathcal{B}_k$, $k=1,\\dots, M$, robots must 1) start assembling $\\mathcal{G}_{k+1}$ only after all robots have completed $\\mathcal{B}_k$ and 2) start executing $\\mathcal{B}_{k+1}$ only after condition $\\mathcal{G}_{k+1} \\subseteq \\mathcal{G}(t)$ is satisfied. We assume the existence of a discrete counter $\\sigma \\in [1,\\dots,M]$ which indicates the active behavior and a binary signal\n\\begin{equation}\n\\eta(\\sigma) = \\begin{cases}\n\t1 \\quad \\text{if} \\quad \\mathcal{G}_k \\subseteq \\mathcal{G}(t) \\\\\n\t0 \\quad \\text{o.w.}\n\\end{cases}\n\\end{equation} \nwhich describes whether the interaction structure required by behavior $\\mathcal{B}_\\sigma$ is available. In this section, we assume both signals to be controlled by a supervisor and made available to the robots at all times, e.g., through a dedicated static communication network. \\mymod{In the next section, we discuss the extension to a fully distributed framework.}\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\columnwidth]{graphics\/figure_Sequence.pdf}\n  \\caption{Schematic representation of the behaviors sequencing framework. Behavior $\\mathcal{B}_k$ is executed during the blue portion of the timeline and $\\mathcal{B}_{k+1}$ is executed during the orange portion. Sequential execution of behaviors requires each agent to reach a spatial configuration such that the desired graph is a spanning graph of the communication graph, i.e., $\\mathcal{G}_k \\subseteq \\mathcal{G}(t_k^\\vdash)$ and $\\mathcal{G}_{k+1} \\subseteq \\mathcal{G}(t_{k+1}^\\vdash)$ respectively. \\label{fig:beh_seq}}\n\\end{figure}\n\nFollowing from the communication modality assumed for the robots, communication constraints can be expressed in terms of relative distance between the robots. In other words, behavior $\\mathcal{B}_k$ can be correctly executed if, for all $t\\in[t_k^\\vdash,t_k^\\dashv]$, all the distances between pairs in $E_k$ are below the proximity threshold $\\Delta$. To this end, a convenient pair-wise connectivity FCBF can be defined as\n\\begin{equation}\n\th_{ij}^c(x) = \\Delta^2 - \\| x_i - x_j \\|^2,\n\t\\label{eq:commBarriers}\n\\end{equation}\nand we note that if $\\|x_i-x_j\\| \\leq \\Delta$, then $h_{ij}^c(x)\\geq 0$.\nIn addition, the edge-level and ensemble-level connectivity constraint sets for behavior $\\mathcal{B}_k$ are\n\\begin{align}\n\\mathcal{C}_{ij}^c &= \\{ x \\in \\mathcal{D} \\,|\\, h_{ij}^c(x) \\geq 0 \\} \\\\\n\\mathcal{C}^c_k &= \\{ x \\in \\mathcal{D} \\,|\\, h_{ij}^c(x) \\geq 0, \\, \\forall (i,j)\\in  E_k\\}.\n\\end{align}\n\nFollowing the definition given in~(\\ref{eq:admisU}), the admissible set of control inputs that guarantees finite-time convergence to $\\mathcal{C}^c_k$ is:\t\n\\begin{multline} \\label{eq:admGraphinput}\n\tK_k^c (x) = \\{ u \\in U \\, | \\, \\dot{h}_{ij}^c(x) + \\bar{\\alpha} _{\\rho,\\gamma}(h_{ij}^c(x)) \\geq 0 , \\\\ \\forall (i,j)\\in E_k \\}\n\\end{multline}\n\n\\begin{theorem} \\label{thm:fcbfControl}\n\tDenoting with $x_0$ the initial state of the system with dynamics~(\\ref{eq:ensembleDynamics}), any controller $\\mathcal{U}:\\mathcal{D} \\mapsto U$ such that $\\mathcal{U}(x_0) \\in K_k^c(x_0)$ for all $x_o \\in \\mathcal{D}$, will drive the system to $\\mathcal{C}^c_k $ within time\n\\begin{equation} \\label{eq:fcTime}\n\tT_k = \\max_{ (i,j) \\in E_k |  h^c_{ij}(x_0)<0} \\left\\{ \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_0) |^{1-\\rho} \\right\\}.\\end{equation}\n\\end{theorem}\n\n\\begin{IEEEproof} \n\tConsider all pairs of agents $i$ and $j$, such that $(i,j) \\in E_k$. If $h_{ij}^c(x_0) \\geq 0$, i.e., agents $i$ and $j$ are within communication distance, the forward invariance property of $\\mathcal{U}$, guarantees that $i$ and $j$ will stay connected. In this case, the state will reach $\\mathcal{C}_{ij}^c$, within time $T_{ij}=0$. On the other side, consider $h_{ij}^c(x_0)<0$. Any $\\mathcal{U}(x_0) \\in K_k^c(x_0)$ satisfies the finite-time convergence barrier certificates, and because of Theorem~\\ref{thm:FCBF}, if $x_0 \\notin \\mathcal{C}_{ij}^c$, then $x(T_{ij})\\in \\mathcal{C}_{ij}^c $, with\n\\begin{equation}\nT_{ij} \\leq  \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_0) |^{1-\\rho}.\n\\end{equation}   \nSince every communication constraint $\\mathcal{C}_{ij}^c $ will be reached within time $T_{ij}$, the total time required to drive $x(t)$ to $\\mathcal{C}_k^c$ is upper bounded by\n\\begin{equation}\n\tT_k = \\max_{(i,j) \\in E_k |  h^c_{ij}(x_0)<0} T_{ij}.\n\\end{equation}\t\n\\end{IEEEproof} \nWhen selecting control inputs from set~(\\ref{eq:admGraphinput}), the system~(\\ref{eq:ensembleDynamics}) will satisfy requirements for behavior $\\mathcal{B}_k$ in finite time.\n\n\\subsection{Finite-Time Convergence Control Barrier Functions}\n\\label{sec:ftcontrolBF}\nOnce behavior $\\mathcal{B}_{k-1}$ is completed, robots are required to converge to the set $\\mathcal{C}^c_{k}$ before behavior $\\mathcal{B}_{k}$ can start. Under the lead of the external supervisor, the change of behavior is communicated to the robots through the signal $\\sigma$, which transitions from $k-1$ to $k$ once $\\mathcal{B}_{k-1}$ is completed. Now, although finite-time convergence to $\\mathcal{C}^c_{k}$ can be achieved by selecting any control input in $K_{k}^c(x)$, we seek to minimally perturb the execution of the behavior just concluded, namely $\\mathcal{B}_{k-1}$. This can be accomplished by solving a problem similar to the one proposed in~\\cite{ames2014control}, which we adapt to our framework. Denoting with $\\hat{u}_{k}=\\mathcal{U}_{k}(x)$ the nominal control input from behavior $\\mathcal{B}_{k}$, during transition between $\\mathcal{B}_{k-1}$ and $\\mathcal{B}_{k}$ the actual control input to the robots $u^*$ is defined as\n\\begin{equation}\\label{eq:minQP1}\nu^* = \\argmin_{u \\in U}  \\| \\hat{u}_{k-1} - u \\|^2 \\\\\n\\end{equation}\nsubject to\n\\begin{equation} \\label{eq:constrTransition}\n   L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0, \n\\end{equation}\nfor all $(i,j) \\in  E_{k-1} \\cup E_{k}$. Once all required edges $E_{k}$ are established (i.e., $\\eta=1$), edges in $E_{k-1}$ are no longer necessary. At this point, under the effect of the controller $\\mathcal{U}_{k}$, the list of constraints in~(\\ref{eq:constrTransition}) is substituted with\n\\begin{equation} \\label{eq:constrExecution}\n   L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0,\n\\end{equation}\nfor all $(i,j)\\in  E_{k}$. Since the cost function is convex and the inequality constraints~(\\ref{eq:constrTransition}) and~(\\ref{eq:constrExecution}) are control affine, the problem can be solved in real-time. In conclusion, because of the finite-time convergence and forward invariance properties of the above formulation, \\mymod{if $\\mathcal{B}_{k-1}$ can be completed and a solution to~(\\ref{eq:minQP1}-\\ref{eq:constrTransition}) (or (\\ref{eq:minQP1}-\\ref{eq:constrExecution})) exists}, robots will converge to the configuration required by $\\mathcal{B}_{k}$, and maintain it throughout its execution.\n\\mymod{\n\\begin{remark}\nThe solution of~(\\ref{eq:minQP1}-\\ref{eq:constrTransition}) (or (\\ref{eq:minQP1}-\\ref{eq:constrExecution})) is contingent upon the existence of a control input capable to solve all constraints. In other words, $K_k^c(x) \\cap K_{k+1}^c(x)$ (or $K_k^c(x)$) should not be empty for all times. For this, it is necessary that a robot's configuration that satisfies all constraints of the problem exists. However, this is not sufficient as the progress towards the desired configuration might be obstructed by constraints on the actuators or deadlock configurations. Although we do not address this directly, it is possible to mitigate feasibility issues by considering, for example, constraints relaxation, sum of squares barrier functions, or pre-defined back-up controllers (see~\\cite{ames2019control} and references therein). \\label{rmk:feasibility}\n\\end{remark}}\n\n\n\n\\subsection{Initial Constraints}\nIn addition to the communication constraints considered above, certain missions might require additional conditions to be met before each behavior can start. For example, during the exploration tasks it might be desirable for one robot to always stay within range of communication with a human-operator, or to maintain a minimum distance from an unsafe area. Assuming $\\mathcal{B}_k$ requires a number of distinct $s_k$ of such constraints, we encode the entire set of initial conditions through a list of barrier functions $h_\\ell^s(x)$, with $\\ell = 1,\\dots,s_k $:\n\\begin{equation} \\label{eq:initialSet}\n\\mathcal{C}^{s}_k = \\{ x \\in \\mathcal{D} \\,|\\, h_\\ell^s(x) \\geq 0, \\, \\forall \\ell = 1,\\dots,s_k \\}.\n\\end{equation}\nFollowing this definition, we define a set of admissible control inputs similar to the one in~(\\ref{eq:admGraphinput}) that will drive the state of the system to the desired set within finite time:\n\\begin{multline} \\label{eq:admInitCond}\n\tK_k^s (x) = \\{ u\\in U \\, | \\, \\dot{h}_\\ell^s(x) + \\bar{\\alpha} _{\\rho,\\gamma}(h_\\ell^s(x)) \\geq 0, \\\\\n\t \\forall \\ell = 1,\\dots,s_k \\}.\n\\end{multline}\n\nThe set of controls satisfying both communication and initial conditions constraints can thus be obtained by intersection of set~(\\ref{eq:admInitCond}) and~(\\ref{eq:admGraphinput}):\n\\begin{equation}\n\tK_k(x) = K_k^c (x) \\bigcap K_k^s (x).\n\\end{equation}\n\nWe note that the results in Theorem~\\ref{thm:fcbfControl} and the formulation of minimally invasive controller in~(\\ref{eq:minQP1}) still holds valid by considering the set $K_k(x)$ instead of $K_k^c (x)$ as the set of admissible control inputs.\n\\section{Case Study: Securing a Building} \\label{sec:securing}\nThe objective of this section is to describe the {\\it Securing a Building} mission, which will be used as testing scenario for the composition framework. We describe now the main structure and objective of the mission, while we deconstruct it into coordinated behaviors in the next subsection.\n\n\\subsection{Mission Overview}\nIn the Securing a Building mission, a group of robots are deployed in an urban environment to identify an unknown target building and rescue a subject located inside. Based on \\cite{FieldManual}, we\ndecompose this mission into the following 4 phases:\n\nFIND - First, the robots are tasked with identifying the target building by means of surveillance of the perimeters of all the buildings. For efficient exploration, robots can be broken into sub-teams. Each team reports collected information at the base after each building has been investigated. Once the target building has been identified, the robots reunite and prepare for the next phase.\n\nISOLATE - The robots isolate the target building by patrolling its perimeter. To achieve this, the robots are divided into two subgroups - the {\\it security agents} responsible for boundary protection and the {\\it maneuvering agents} tasked with entering the building.\n\nRESCUE - During the rescue phase, the security agents keep patrolling around the building. In the meanwhile, the maneuvering agents enter the building, clear the rooms, and seize positions as they maneuver through the building to find the subject to be rescued. Once the subject has been located, the robots transport it to the safe zone.\n\nFOLLOW-THROUGH - As the interior of the building is being cleared, individual robots are left inside as beacons, while the remaining robots from the maneuvering agents leave the building, gather on the outside with the security agents, and report back to the base station.\n\nA number of arguments support the choice of the Securing a Building mission as an ideal scenario for testing multi-robot techniques and algorithms. First, the requirement of spatially diverse functionalities that cannot be provided by single robots naturally requires the use of multi-robot systems. Second, the final goal of the mission, namely rescuing the subjects of interest, reflect the fact that general real-world missions cannot be accomplished with single controllers. Lastly, thanks to its modularity, techniques focusing on specific aspects of the mission can be integrated and tested without influencing the overall structure of the mission (see the Appendix for details).\n\n\\subsection{Securing a Building Through Composition of Behaviors}\n\n\\begin{figure*}[t]\n\\centerline{ \\includegraphics[width=1.9\\columnwidth]{graphics\/missionChart1.pdf}}\n\\caption{Mission design chart showing how coordinated behaviors are composed together to tackle the Securing a Building mission. The four bold titles are the mission phases and the large boxes below them indicate specific agent roles\nand associated behaviors. The arrows in the chart indicate the transitions between different behaviors. We note that he choice of controllers that produces the behaviors in the chart is not unique.\\label{fig:missionChart}}\n\\end{figure*}\n\nWe deconstruct the Securing a Building mission through ordered sequences of coordinated behaviors. The process is summarized in Fig.~\\ref{fig:missionChart}. We refer to behaviors in terms of their main objectives, acknowledging that different implementations can be used to achieve the same results. We highlight these behaviors in parenthesis.\n\n\\paragraph{FIND} Robots initially coordinate with the operator at the base station ({\\it rendezvous}). After that, robots are divided into different search teams, each assigned with a list of buildings to investigate ({\\it task allocation}). Subsequently, all the teams investigate their own lists of buildings. First team of robots travels to the vicinity of a building ({\\it leader-follower}), then start to survey the exterior of the building ({\\it perimeter patrol}), and return to the base ({\\it leader-follower}). This process repeats until the target building is discovered.\n\n\\paragraph{ISOLATE} Robots gather near the base ({\\it rendezvous}), then are divided into {\\it security} and {\\it maneuvering} agents ({\\it task allocation}). After traveling from the base to the vicinity of the target building ({\\it go-to-goal}), security agents protect the building's perimeter ({\\it cyclic pursuit}), until the end of the RESCUE phase. Meanwhile, the maneuvering agents locate the building's entrance, by following its perimeter ({\\it perimeter patrol}). Once the entrance has been found, the maneuvering agents gather at the entrance ({\\it rendezvous}) and create a formation ({\\it formation control}) before entering.\n\n\\paragraph{RESCUE} The maneuvering agents enter the building in formation ({\\it formation control}) and cover the interior area ({\\it area coverage}). Once the location of the subject to rescue is identified, the robots form a circular closure around the subject ({\\it cyclic pursuit}). Then, the robots transport the subject to the safety zone, while maintaining the circular closure around the subject ({\\it containment control}).\n\n\\paragraph{FOLLOW-THROUGH} Maneuvering agents spread ({\\it scatter}) over the interior of the building. To signify that the area has been cleared, few robots are left inside the building as beacons ({\\it persistent coverage}). The rest of the maneuvering agents and the security agents reunite outside the building ({\\it rendezvous}). At last, they return to the base ({\\it leader-follower}).\n\n\n\\subsection{Results}\nWe tested the behavior composition framework described in Section~\\ref{sec:multAgImp} on the Securing a Building mission, which was executed on the Robotarium~\\cite{pickem2017robotarium}. In Fig.~\\ref{fig:experiment}, we display selected snapshots of the mission obtained by a camera mounted on the ceiling. In the experiment, $8$ differential-drive robots, indexed $1,\\dots,8$ are deployed in a simulated urban environment composed of $6$ buildings, blue polygons indexed $1,\\dots,6$. In this experiment, we simulate a maximum sensor range $\\Delta = 0.5$m. Because of the different spatial scales between FIND\/ISOLATE phases and RESCUE\/FOLLOW-THROUGH phases, the entire mission is divided in two parts. In the first part (Fig.~\\ref{fig2:a} to Fig.~\\ref{fig2:d}) the experiment is performed at a {\\it neighborhood}-level scale. The remaining two phases are executed in a zoomed-in environment, which focuses on the one building of interest (Fig.~\\ref{fig2:d} to Fig.~\\ref{fig2:f}).\n\nDuring FIND phase (Fig.~\\ref{fig2:a} and~\\ref{fig2:b}), two groups of robots $\\text{\\sc team}1:\\{1,2,3,4\\}$ and $\\text{\\sc team}2:\\{5,6,7,8\\}$ investigates preassigned lists of buildings, leaving some agents near the base station (the purple filled dot in the top right corner) if destination building cannot be reached without breaking the connectivity constraints. The red polygon in Fig.~\\ref{fig2:b} and~\\ref{fig2:c} is the target building after being identified by $\\text{\\sc team}1$. During the ISOLATE phase (Fig.~\\ref{fig2:c}), maneuvering agents look for the entrance, while the security agents secure the outer perimeter. \n\nDuring the RESCUE phase (Fig.~\\ref{fig2:d} to~\\ref{fig2:e}), the agents inside the building, i.e. $\\text{\\sc team}1$, localize the target (red dot) using Voronoi coverage (Fig.~\\ref{fig2:d}) and escort it to the safe area (red circle) as shown in (Fig.~\\ref{fig2:e}). Finally, robots $1$ and $2$ are left as beacon inside the building, while remaining robots return to the base (Fig.~\\ref{fig2:f}).\n\n\\begin{figure*}[t]\n\\centerline{ \n\\subcaptionbox{\\label{fig2:a}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation1}}~\n\\subcaptionbox{\\label{fig2:b}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation2}}~\n\\subcaptionbox{\\label{fig2:c}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation4}}\n} \\vspace{0.25cm}\n\\centerline{ \n\\subcaptionbox{\\label{fig2:d}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation7}}~\n\\subcaptionbox{\\label{fig2:e}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation9}}~\n\\subcaptionbox{\\label{fig2:f}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation11}}\n}\n\\caption{Overhead screen-shots from experiments on the Robotarium. A team of eight robots is divided in $\\text{\\sc team}1:\\{1,2,3,4\\}$ and $\\text{\\sc team}2:\\{5,6,7,8\\}$. Because of the different spatial scales between FIND\/ISOLATE phases and RESCUE\/FOLLOW-THROUGH phases the mission is executed on two different environments. Each team is assigned with a list of three buildings to inspect sequentially. FIND: (a) perimeter patrol of buildings $2$ and $5$; (b) building $4$ is identified as the target building, while $\\text{\\sc team}1$ waits for $\\text{\\sc team}2$ to return to base. ISOLATE: (c) $\\text{\\sc team}2$ secures perimeter of building, while $\\text{\\sc team}1$ inspects exterior of building, searching for the entrance. RESCUE: after entering the building, $\\text{\\sc team}1$ performs domain coverage of the building until target (red dot) is identified (d); after this, (e) robots escort target to safe location (red circle). FOLLOW-THROUGH: finally, two robots are left as beacons inside the building while all remaining robots return to base (f).   \n\\label{fig:experiment}}\n\n\\end{figure*}\n\\section{Related Work} \nThe problem of partitioning complex objectives into simpler tasks can be solved by sequentially composing {\\it primitives}, e.g.,~\\cite{cassandras2009introduction}, or by blending them simultaneously in a {\\it hierarchical} fashion. An example of hierarchical composition for single robot motion control is navigation between points while avoiding obstacles, e.g.,~\\cite{arkin1998behavior}. \\mymod{In general, the problem of controlling a system by composing different modes of operation pertains to hybrid systems and {\\it multi-modal} control domains~\\cite{koutsoukos2000supervisory}.}\n\nBecause of the complexity emerging from the composition of distinct controllers, guarantees on the safety and correctness of the final results need to be established~\\cite{kress2018synthesis}. Provable correct composition of control policies is investigated in the formal methods literature. Recently, compositional strategies inspired from formal methods have been used for the development of control strategies for multi-robot systems~\\cite{srinivasan2018control,garg2019control,meyer2019hierarchical,chen2018verifiable}. In particular, the authors of~\\cite{srinivasan2018control} use tools from linear temporal logic (LTL) for the specification of behaviors to be executed by the system. The solution is based on a sequence of constrained reachability problems, each consisting of a target set to be reached in finite time and a safety set within which the system must stay at all times. \\mymod{A related approach is developed in~\\cite{garg2019control}, where the problem of prescribed-time convergence to spatio-temporal specifications is formulated using control barrier functions.} The authors in~\\cite{meyer2019hierarchical} discuss a hierarchical decomposition method for controller synthesis given LTL specifications.\n\nIn the context of controllers composition for multi-robot systems, in~\\cite{belta2007symbolic} the authors use symbolic methods in order to generate high-level instructions from form of {\\it human-like} language. The authors of~\\cite{klavins2000formalism} introduce a framework for the composition of controllers in robotic systems using Petri Nets. In~\\cite{marino2009behavioral}, behaviors from the Null-Space-Behaviors framework are combined in order to solve ad-hoc tasks, such as perimeter patrol. A supervisor, represented as a finite state automata, selects high-level behaviors by assembling low-level behaviors. In~\\cite{nagavalli2017automated}, a revised version of the $A^*$ algorithm is used to generate an optimal path of behaviors, such that the overall cost of the mission is minimized. Similarly, in~\\cite{vukosavljev2019hierarchically} motion planning for a team of quadcopters is solved by defining higher level motion primitives obtained by a spatial partition of the environment. However, none of these approaches specifically address the problem of correct composition between primitives, which is the focus of this paper.\n\nAs discussed in the previous section, coordination between agents is possible only if particular interactions exist between the robots. In multi-robot systems, interaction requirements are commonly investigated in terms of connectivity maintenance, i.e., a certain graph or node-connectivity needs to be guaranteed at all times. Methods employed in the solution to this problem include edge weight functions~\\cite{ji2007distributed}, control rules based on estimate of algebraic connectivity~\\cite{sabattini2013distributed}, hybrid control~\\cite{zavlanos2009hybrid}, passivity~ \\cite{igarashi2009passivity}, and barrier functions~\\cite{wang2016multi}. If connectivity between agents needs to be guaranteed in non-nominal circumstances, resilient solutions must be in place as well, e.g.,\n\\cite{ramachandran2019resilience}, \\cite{panerati2019robust}, and~\\cite{varadharajan2019unbroken}. Notably, a technique based on graph process specifications for the sequential composition of different multi-agent controllers is discussed in~\\cite{twu2010graph}. Similar to our work, the authors in~\\cite{twu2010graph} bridge the gap between composition of controllers and the topology requirements by encoding requisites for each controller in terms of graphs.\nHowever, while in \\cite{twu2010graph} {\\it incompatible} controllers are combined through the introduction of a {\\it bridging} controller, in our approach controllers are minimally modified by the robots in order to satisfy upcoming requirements. Our approach significantly reduces the complexity of the composition process, \\mymod{minimizes the energy spent by the robots to switch between behaviors}, and can accommodate additional constraints, such as inter-robot collisions and obstacles avoidance.\n\n\\section{Introduction} \\label{sec:intro}\n\\IEEEPARstart{A}{s} our understanding of how to structure control and coordination protocols for teams of robots increases, a number of application domains have been identified, such as entertainment~\\cite{ackerman2014flying}\\cite{du2018fast}, surveillance~\\cite{santos2018coverage}~\\cite{shishika2018local}, manipulation~\\cite{han2018hybrid}, and search-and-rescue~\\cite{suarez2011survey}. Along with a decrease in the production and manufacturing costs associated with the platforms themselves, these applications have been enabled by a number of theoretical results that have emerged at the intersection of different disciplines such as robotics, controls, computer science, and graph theory~\\cite{zelazo2018graph}. \n\nFrom a motion controls perspective, one notable requirement is given by the need to define actions capable to solve team-wise objectives on the basis of locally available information. For instance, different extensions of the consensus equation have been used to arrive at locally defined controllers with provable, global convergence properties~\\cite{cortes2017coordinated}. In this way, it is possible to construct coordinated controllers for the solution of motion control problems, such as rendezvous~\\cite{lin2003multi}~\\cite{ren2005coordination}, cyclic pursuit~\\cite{ramirez2009cyclic}, formation control~\\cite{lawton2003decentralized}~\\cite{buckley2017infinitesimally}, coverage~\\cite{cortes2004coverage}~\\cite{santos2018coverage}, leader-based control~\\cite{mesbahi2010graph}, and flocking~\\cite{tanner2007flocking}. Particular instantiations of some of these behaviors are shown in Fig.~\\ref{fig:coordBehExamples} on a group of six simulated differential drive robots.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[trim={2cm 2cm 1cm 2cm},width=0.35\\columnwidth]{graphics\/redezvous.pdf}~\n\t\t\\includegraphics[trim={3cm 2cm 2.5cm 2cm},width=0.32\\columnwidth]{graphics\/cyclicPurs.pdf}~\n\t\t\\includegraphics[trim={2cm 2cm 2cm 2cm},width=0.32\\columnwidth]{graphics\/leadFollow.pdf}\n\t\t\\caption{Simulation of three distributed multi-agent behaviors on a group of differential drive robots. From the left: rendezvous, cyclic-pursuit, and leader-follower. Solid lines indicate the past trajectories of the robots. \\label{fig:coordBehExamples}}\n\t\\end{center}\n\\end{figure}\n\nFor the correct execution of the controllers mentioned, a sufficiently rich set of information needs to be available to the robots. Representing the flow of information between the robots through {\\it graphs}, with vertices and edges being respectively the robots and the pair-wise ability of sharing information, those conditions can be encoded in terms of particular graphs that need to exist between the robots. For example, rendezvous requires a spanning out-branching tree~\\cite{mesbahi2010graph}, cyclic-pursuit requires a cyclic graph~\\cite{ramirez2009cyclic}, formation control a rigid graph~\\cite{mesbahi2010graph}, and a Delaunay graph is required for most of coverage control problems~\\cite{cortes2004coverage}. \n\nEven though the coordinated behaviors mentioned above can address a number of different tasks, they have limited utility in the context of real-world missions, which can rarely be represented as single tasks. However, the utility of these behaviors can be greatly expanded if they are sequenced together, which is the primary consideration in this paper. But, for a construction like this to work, it is necessary that the required information is available to the robots as they transition from one behavior to the next.\n\nAs such, the problem of composing different behaviors, can be recast in terms of the ability of the robots to establish the interactions needed at each stage of a mission. In particular, when the communication between agents depends on their relative configurations (e.g. relative distance or orientation), realizing a certain communication structure directly affects the configuration of the system, which in turn, affects the execution of the mission itself. In order to overcome this coupling, we separate the problem of generating a sequence of behaviors that corresponds to the solution of a mission objective (e.g.~\\cite{nagavalli2017automated}) from their composition. In this work we focus on the problem of designing a composition framework given a sequence of coordinated behaviors. \\mymod{Although the focus of this paper is confined to motion control tasks, our framework is applicable to other forms of autonomous collaboration where desired interaction structures between the robots are required by the mission, e.g., sharing of resources in heterogeneous teams~\\cite{ramachandran2019resilience} or coordinated manipulation~\\cite{culbertson2018decentralized}}.\n\nThe contribution of this paper is twofold. Firstly, extending the results in~\\cite{li2018formally}, we propose a fully decentralized framework for composing a given sequence of multi-robot coordinated behaviors. Secondly, responding to the lack of established large-scale scenarios for the testing of multi-robot techniques, we propose a scenario called {\\it Securing a Building}, which is rich and complex enough to capture many challenges and objectives of real-world implementations. \\mymod{The significance of our framework is demonstrated through implementation of the Securing a Building scenario on a team of mobile robots.}\n\nThe remaining of this paper is organized as follows. In Section~\\ref{sec:fcbf} we review the definition of finite-time convergence barrier functions, while in Section~\\ref{sec:problem} we present a centralized multi-robot composition framework, which is extended to a \\mymod{fully decentralized} formulation in Section~\\ref{sec:multAgImp}. In Section~\\ref{sec:securing}, we describe the {\\it Securing a Building} case study and its implementation. Finally, motivated by the lack of well-established scenarios for testing and comparing multi-agent robotics techniques, \\mymod{in Appendix~\\ref{sec:appendixA} we discuss supportive arguments for considering the Securing a Building as a multi-robot benchmark scenario.}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction \\label{sec:introduction}}\n\n\nThe bismuth chalcogenides \\ch{Bi2Ch3} (\\ch{Ch} = \\ch{S}, \\ch{Se}, or \\ch{Te}) of the layered tetradymite structure are an interesting class of highly two dimensional narrow bandgap semiconductors.\nStrong spin-orbit coupling inverts the energy ordering of their bands, making them bulk \\glspl{ti} characterized by a single Dirac cone at the Brillouin zone centre and a topologically protected metallic surface state~\\cite{2009-Hsieh-N-460-1101, 2009-Zhang-NP-5-438}.\nThis has augmented longstanding interest in their thermoelectric properties with significant efforts (theoretical and experimental) to understand their electronic properties in detail~\\cite{2013-Cava-JMCC-1-3176, 2017-Heremans-NRM-2-17049}.\nConsisting of weakly interacting \\ch{Ch-Bi-Ch-Bi-Ch} atomic \\glspl{ql} (see \\latin{e.g.}, Figure~1 in Ref.~\\onlinecite{2019-McFadden-PRB-99-125201}), they can also accommodate intercalant species such as \\ch{Li^{+}} in the \\gls{vdw} gap between \\glspl{ql}~\\cite{1988-Paraskevopoulos-MSEB-1-147, 1989-Julien-SSI-36-113, 2010-Bludska-JSSC-183-2813}, similar to the layered \\glspl{tmd}~\\cite{1978-Whittingham-PSSC-12-41, 1987-Friend-AP-36-1}.\nAlthough their bandgaps are \\SI{\\sim 150}{\\milli\\electronvolt}, doping by intrinsic defects, such as \\ch{Ch} vacancies, yields crystals that are far from insulating.\nTo increase the contrast in conductivity between the bulk and the metallic \\gls{tss}, the crystals are often compensated extrinsically.\nFor example, \\ch{Ca} substitution for \\ch{Bi} suppresses the self-doped $n$-type conductivity in \\ch{Bi2Se3}~\\cite{2009-Hor-PRB-79-195208, 2009-Hsieh-N-460-1101}.\nDoping can also be used to modulate their magnetic properties, yielding magnetic \\glspl{ti} where the \\gls{tss} is gapped~\\cite{2010-Chen-S-329-659}, for example, by substitution of \\ch{Bi} with a paramagnetic transition metal~\\cite{2010-Hor-PRB-81-195203, 2019-Tokura-NRP-1-126}.\n\n\nThe intriguing electronic properties of the tetradymite \\glspl{ti} have predominantly been investigated using surface sensitive probes in real and reciprocal space (\\latin{e.g.}, \\gls{sts} and \\gls{arpes}), as well as other bulk methods.\nIn complement to these studies, \\gls{nmr} offers the ability to probe their electronic ground state and low-energy excitations through the hyperfine coupling of the nuclear spin probe to the surrounding electrons.\nSuch a local probe is especially useful when disorder masks sharp reciprocal space features, as is the case in \\ch{Bi2Ch3}.\nThe availability of a useful \\gls{nmr} nucleus, however, is usually determined by elemental composition and natural (or enriched) isotopic abundance, as well as the specific nuclear properties such as the gyromagnetic ratio $\\gamma$ and, for spin $> 1\/2$, the nuclear electric quadrupole moment $Q$.\nWhile the \\ch{Bi2Ch3} family naturally contain several \\gls{nmr} nuclei~\\cite{2013-Nisson-PRB-87-195202, 2014-Koumoulis-AFM-24-1519, 2016-Levin-JPCC-120-25196}, they are either low-abundance or have a large $Q$.\nAs an alternative, here we use an ion-implanted \\gls{nmr} probe at ultratrace concentrations, with detection based on the asymmetric property of radioactive $\\beta$-decay, known as \\gls{bnmr}~\\cite{2015-MacFarlane-SSNMR-68-1}.\n\n\nA key feature of ion-implanted \\gls{bnmr} is the depth resolution\nafforded by control of the incident beam energy~\\cite{2014-Morris-HI-225-173, 2015-MacFarlane-SSNMR-68-1},\nwhich dictates the stopping distribution of the implanted \\gls{nmr} probes.\nAt \\si{\\kilo\\electronvolt} energies,\nthe depth can be varied on the nanometer length-scale and,\nat the lowest accessible energies,\nit may be able to sense the \\gls{tss} in the tetradymite \\glspl{ti},\nwhich is likely confined to depths \\SI{< 1}{\\nano\\meter} below the surface.\nHere one expects Korringa relaxation and Knight shifts from the \\gls{tss} electrons,\nmodified by the phase space restrictions imposed by their chirality.\nThe magnitude of each will depend on both the \\gls{tss} carrier density and\nthe strength of the coupling to the implanted nuclei.\nWhile the motivation to study the \\gls{tss} is strong,\nhere we report the ``bulk'' response of an implanted \\ch{^{8}Li} probe\nin two doped \\ch{Bi2Ch3} \\glspl{ti}.\nThis is an essential step toward detecting the\n\\gls{tss}~\\cite{2014-MacFarlane-PRB-90-214422, 2019-McFadden-PRB-99-125201},\nbut it also demonstrates the sensitivity of the implanted \\ch{^{8}Li} to\nthe carriers in such heavily compensated narrow gap semiconductors.\n\n\nUsing \\gls{bnmr}, we study two single crystals of doped \\ch{Bi2Ch3} --- compensated \\gls{bsc} and magnetic \\gls{btm} --- each with a beam of highly polarized \\ch{^{8}Li^{+}}.\nIn many respects, \\gls{bnmr} is closely related to \\gls{musr}, but the radioactive lifetime is much longer, making the frequency range of dynamics it is sensitive to more comparable to conventional \\gls{nmr}.\nIn addition to purely electronic phenomena, in solids containing mobile species, \\gls{nmr} is also well known for its sensitivity to low frequency diffusive fluctuations~\\cite{1948-Bloembergen-PR-73-679, 1982-Kanert-PR-91-183, 1994-MullerWarmuth-PIR-17-339}, as are often encountered in intercalation compounds.\nAt ion-implantation energies sufficient to probe the bulk of \\gls{bsc} and \\gls{btm}, we find evidence for ionic mobility of \\ch{^{8}Li^{+}} above \\SI{\\sim 200}{\\kelvin}, likely due to \\gls{2d} diffusion in the \\gls{vdw} gap.\nAt low temperature, we find Korringa relaxation and a small temperature dependent negative Knight shift in \\gls{bsc}, allowing a detailed comparison with \\ch{^{8}Li} in the structurally similar \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nIn \\gls{btm}, the effects of the \\ch{Mn} moments predominate, but remarkably the signal can be followed through the magnetic transition.\nAt low temperature, we find a prominent critical peak in the relaxation that is suppressed in a high applied field, and a broad, intense resonance that is strongly shifted.\nThis detailed characterization of the \\ch{^{8}Li} \\gls{nmr} response is an important step towards using depth-resolved \\gls{bnmr} to study the low-energy properties of the chiral \\gls{tss}.\n\n\n\\section{Experiment \\label{sec:experiment}}\n\n\nDoped \\gls{ti} single crystals \\gls{bsc} and \\gls{btm} with nominal stoichiometries \\ch{Bi_{1.99}Ca_{0.01}Se_{3}} and \\ch{Bi_{1.9}Mn_{0.1}Te_{3}} were grown as described in Refs.~\\onlinecite{2009-Hor-PRB-79-195208, 2010-Hor-PRB-81-195203} and magnetically characterized using a Quantum Design \\gls{mpms}.\nIn the \\gls{btm}, a ferromagnetic transition was identified at $T_{C} \\approx \\SI{13}{\\kelvin}$, consistent with similar \\ch{Mn} concentrations~\\cite{2010-Hor-PRB-81-195203, 2013-Watson-NJP-15-103016, 2016-Zimmermann-PRB-94-125205, 2019-Vaknin-PRB-99-220404}.\nIn contrast, the susceptibility of the \\gls{bsc} crystal was too weak to measure accurately,\nbut the data show no evidence for a Curie tail at low-$T$ that could originate from dilute paramagnetic defects.\n\n\n\\gls{bnmr} experiments were performed at TRIUMF's \\gls{isac} facility in Vancouver, Canada.\nDetailed accounts of the technique can be found in Refs.~\\onlinecite{2015-MacFarlane-SSNMR-68-1, 2019-McFadden-PRB-99-125201}.\nA low-energy highly polarized beam of \\ch{^{8}Li^{+}} was implanted into the samples mounted in one of two dedicated spectrometers~\\cite{2014-Morris-HI-225-173, 2015-MacFarlane-SSNMR-68-1}.\nPrior to mounting, the crystals were cleaved in air and affixed to sapphire plates using \\ch{Ag} paint (SPI Supplies, West Chester, PA).\nThe approximate crystal dimensions were \\SI{7.8 x 2.5 x 0.5}{\\milli\\meter} (\\gls{bsc}) and \\SI{5.3 x 4.8 x 0.5}{\\milli\\meter} (\\gls{btm}).\nWith the crystals attached, the plates were then clamped to an aluminum holder threaded into an \\gls{uhv} helium coldfinger cryostat.\nThe incident \\ch{^{8}Li^{+}} ion beam had a typical flux of \\SI{\\sim e6}{ions\\per\\second} over a beam spot \\SI{\\sim 2}{\\milli\\metre} in diameter.\nAt the implantation energies $E$ used here (between \\SIrange{1}{25}{\\kilo\\electronvolt}), \\ch{^{8}Li^{+}} stopping profiles were simulated for \\num{e5} ions using the \\gls{srim} Monte Carlo code (see \\Cref{sec:implantation})~\\cite{srim}.\nFor $E > \\SI{1}{\\kilo\\electronvolt}$, a negligible fraction of the \\ch{^{8}Li^{+}} stop near enough to the\ncrystal surface to sense the \\gls{tss}.\nMost of the data is taken at \\SI{20}{\\kilo\\electronvolt}, where the implantation depth is \\SI{\\sim 100}{\\nano\\meter}, and the results thus reflect the bulk behavior.\n\n\nThe probe nucleus \\ch{^{8}Li} has nuclear spin $I=2$, gyromagnetic ratio $\\gamma \/ 2 \\pi = \\SI{6.3016}{\\mega\\hertz\\per\\tesla}$, nuclear electric quadrupole moment $Q = \\SI[retain-explicit-plus]{+32.6}{\\milli\\barn}$, and radioactive lifetime $\\tau_{\\beta} = \\SI{1.21}{\\second}$.\nThe nuclear spin is polarized in-flight by collinear optical pumping with circularly polarized light~\\cite{2014-Levy-HI-225-165}, yielding a polarization of \\SI{\\sim 70}{\\percent}~\\cite{2014-MacFarlane-JPCS-551-012059}.\nIn each measurement, we alternate the sense of circular polarization\n(left and right) of the pumping light,\nproducing either ``positive'' or ``negative'' helicity in the \\ch{^{8}Li} beam\n(i.e., its nuclear spin polarization is aligned or counter-aligned with the beam).\nData were collected separately for each helicity, which are usually combined,\nbut in some cases, are considered independently to reveal ``helicity-resolved'' properties.\nWhile helicity-resolved spectra are useful to elucidate details of resonance lines\n(see \\Cref{sec:helicities}),\ncombined spectra are helpful to remove detection systematics\n(see \\latin{e.g.},~\\cite{1983-Ackermann-TCP-31-291, 2015-MacFarlane-SSNMR-68-1}).\nThe \\ch{^{8}Li} polarization was monitored after implantation through the anisotropic radioactive $\\beta$-decay, similar to \\gls{musr}.\nSpecifically, the experimental asymmetry $A$ (proportional to the average longitudinal spin-polarization) was measured by combining the rates in two opposed scintillation counters~\\cite{1983-Ackermann-TCP-31-291, 2015-MacFarlane-SSNMR-68-1}.\nThe proportionality constant depends on the experimental geometry and the details of the $\\beta$-decay (here, on the order of \\num{\\sim 0.1}).\n\n\n\\Gls{slr} measurements were performed by monitoring the transient decay of spin-polarization both during and following a pulse of beam lasting several seconds.\nDuring the pulse, the polarization approaches a steady-state value, while after the pulse, it relaxes to essentially zero.\nAt the edge of the pulse, there is a discontinuity in the slope, characteristic of \\gls{bnmr} \\gls{slr} spectra (see \\latin{e.g.}, \\Cref{fig:bsc-slr-spectra}).\nNote that unlike conventional \\gls{nmr}, no \\gls{rf} field is required for the \\gls{slr} measurements.\nAs a result, it is generally more expedient to measure \\gls{slr} than the resonance;\nhowever, there is no spectral resolution of the relaxation, which represents the \\gls{slr} of all the \\ch{^{8}Li}.\nThe temperature dependence of the \\gls{slr} rate was measured at several applied magnetic fields $B_{0}$:\n\\SI{6.55}{\\tesla} parallel to the \\ch{Bi2Ch3} trigonal $c$-axis;\nand at lower fields \\SI{\\leq 20}{\\milli\\tesla} perpendicular to the $c$-axis.\nA typical \\gls{slr} measurement took \\SI{\\sim 20}{\\minute}.\n\n\nResonances were acquired using a \\gls{dc} \\ch{^{8}Li^{+}} beam and a \\gls{cw} transverse \\gls{rf} magnetic field $B_{1}$.\nIn this measurement mode, the \\gls{rf} frequency is stepped slowly through the \\ch{^{8}Li} Larmor frequency\n\\begin{equation*} \\label{eq:larmor}\n   \\omega_{0} = 2 \\pi \\nu_{0} = \\gamma B_{0}\n\\end{equation*}\nand the spin of any on-resonance \\ch{^{8}Li} is rapidly precessed, resulting in a loss in the average time-integrated $\\beta$-decay asymmetry.\nThe resonance amplitudes are determined by several factors:\nthe baseline asymmetry (\\latin{i.e.}, the time integral of the \\gls{slr});\nthe \\gls{rf} amplitude $B_{1}$;\nthe presence of slow, spectral \\ch{^{8}Li} dynamics occurring up to the second timescale (see \\latin{e.g.}, Ref.~\\onlinecite{2017-McFadden-CM-29-10187});\nand, for quadrupole satellite transitions, the relative populations of the magnetic sublevels are somewhat different than conventional pulsed \\gls{nmr}~\\cite{1990-Slichter-PMR, 2015-MacFarlane-SSNMR-68-1, 2019-McFadden-PRB-99-125201}.\nResonances were recorded over a temperature range of \\SIrange{4}{315}{\\kelvin} at both high and low magnetic fields.\nAt high field, the resonance frequency was calibrated against its position in a single crystal \\ch{MgO} at \\SI{300}{\\kelvin}, with the superconducting solenoid persistent.\nA single spectrum typically took \\SI{\\sim 30}{\\minute} to acquire.\n\n\n\\section{Results and analysis \\label{sec:results}}\n\n\n\\subsection{\\ch{Bi2Se3:Ca} \\label{sec:results:bsc}}\n\n\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\gls{bsc}, at both high and low magnetic field, are shown in \\Cref{fig:bsc-slr-spectra}.\nTo aid comparison, $A(t)$ has been normalized by its initial value $A_{0}$ determined from fits described below.\nClearly, the \\gls{slr} is strongly temperature and field dependent.\nAt low field, the \\gls{slr} is very much faster, due to additional relaxation from fluctuations of the host lattice nuclear spins~\\cite{2009-Hossain-PRB-79-144518}.\nThe temperature dependence of the relaxation is non-monotonic, indicating that some of the low frequency fluctuations at $\\omega_{0}$ are frozen out at low temperature.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-slr-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} \\acrlong{slr} spectra in \\ch{Ca} doped \\ch{Bi2Se3} at high and low magnetic field.\n]{ \\label{fig:bsc-slr-spectra}\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\ch{Ca} doped \\ch{Bi2Se3} at high (left) and low (right) magnetic field with \\ch{^{8}Li^{+}} implanted at \\SI{20}{\\kilo\\electronvolt}.\nThe shaded region indicates the duration of the \\ch{^{8}Li^{+}} beam pulse.\nThe relaxation is strongly field dependent, increasing at lower fields, and it increases non-monotonically with increasing temperature.\nThe solid black lines show fits to a stretched exponential described in the text.\nThe initial asymmetry $A_{0}$ from the fits is used to normalize the data which are binned by a factor of \\num{20} for clarity.\n}\n\\end{figure}\n\n\nThe relaxation is non-exponential at \\emph{all} temperatures \\emph{and} fields, so the data were fit with the phenomenological stretched exponential.\nThis approach was also used for \\ch{^{8}Li} in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201} and in conventional \\gls{nmr} of related materials~\\cite{2013-Nisson-PRB-87-195202, 2014-Koumoulis-AFM-24-1519, 2016-Levin-JPCC-120-25196}.\nExplicitly, for a \\ch{^{8}Li^{+}} implanted at time $t^{\\prime}$, the spin polarization at time $t > t^{\\prime}$ follows:\n\\begin{equation} \\label{eq:strexp}\n   R \\left ( t, t^{\\prime} \\right ) = \\exp \\left \\{ - \\left [ \\lambda \\left ( t-t^{\\prime} \\right ) \\right ]^{\\beta} \\right \\},\n\\end{equation}\nwhere $\\lambda \\equiv 1\/T_{1}$ is the \\gls{slr} rate and $0 < \\beta \\leq 1$ is the stretching exponent.\nThis is the simplest model that fits the data well with the minimal number of free parameters, for the entire \\ch{Bi2Ch3} tetradymite family of \\glspl{ti}.\n\n\nUsing \\Cref{eq:strexp} convoluted with the beam pulse, \\gls{slr} spectra in \\gls{bsc}, grouped by magnetic field $B_{0}$ and implantation energy $E$, were fit simultaneously with a shared common initial asymmetry $A_{0}(B_{0}, E)$.\nNote that the statistical uncertainties in the data are strongly time-dependent (see \\latin{e.g.}, \\Cref{fig:bsc-slr-spectra}), which must be accounted for in the analysis.\nUsing custom C++ code incorporating the MINUIT minimization routines~\\cite{1975-James-CPC-10-343} implemented within the ROOT data analysis framework~\\cite{1997-Brun-NIMA-389-81}, we find the global least-squares fit for each dataset.\nThe fit quality is good ($\\tilde{\\chi}_{\\mathrm{global}}^{2} \\approx 1.02$) and a subset of the results are shown in \\Cref{fig:bsc-slr-spectra} as solid black lines.\nThe large values of $A_{0}$ extracted from the fits (\\SI{\\sim 10}{\\percent} for $B_{0} = \\SI{6.55}{\\tesla}$ and \\SI{\\sim 15}{\\percent} for $B_{0} = \\SI{15}{\\milli\\tesla}$) are consistent with the full beam polarization, with no missing fraction.\nThe fit parameters are plotted in \\Cref{fig:bsc-slr-fits}, showing agreement with the qualitative observations above.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-slr-fits.pdf}\n\\caption[\nTemperature and field dependence of the \\ch{^{8}Li} \\acrlong{slr} rate $1\/T_{1}$ and stretching exponent $\\beta$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n]{ \\label{fig:bsc-slr-fits}\nTemperature and field dependence of the \\ch{^{8}Li} \\gls{slr} rate $1\/T_{1}$ and stretching exponent $\\beta$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n$\\beta$ is nearly independent of temperature and field at \\num{\\sim 0.6} (dotted line), except at low field around the large $1\/T_{1}$ peak seen in the bottom panel.\nThe solid and dashed black lines are global fits to \\Cref{eq:rlx}, consisting of a linear $T$-dependence with a non-zero intercept and two \\gls{slr} rate peaks, labelled with index $i$.\nIndependent of the choice of $J_{n}$ used in the analysis, the model captures all the main features of the data.\nThe $T$-linear contribution to $1\/T_{1}$ is shown as the dotted black line.\n}\n\\end{figure}\n\n\nWe now consider a model for the temperature and field dependence of $1\/T_{1}$.\nWe interpret the local maxima in $1\/T_{1}$ in \\Cref{fig:bsc-slr-fits} as \\gls{bpp} peaks~\\cite{1948-Bloembergen-PR-73-679}, caused by a fluctuating field coupled to the \\ch{^{8}Li} nuclear spin with a characteristic rate that sweeps through $\\omega_{0}$ at the peak temperature~\\cite{1948-Bloembergen-PR-73-679, 1979-Richards-TCP-15-141, 1988-Beckmann-PR-171-85}.\nPotential sources of the fluctuations are discussed below.\nThe rate peaks are superposed on a smooth background that is approximately linear, reminiscent of Korringa relaxation in metals~\\cite{1950-Korringa-P-16-601, 1990-Slichter-PMR}.\nThis is surprising, since \\gls{bsc} is a semiconductor, but it is similar to \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nWe discuss this point further in \\Cref{sec:discussion:electronic}.\n\n\nFrom this, we adopt the following model for the total \\gls{slr} rate:\n\\begin{equation} \\label{eq:rlx}\n   1\/T_{1} = a + b T + \\sum_{i} c_{i} \\left ( J_{1,i} + 4J_{2,i} \\right ) .\n\\end{equation}\nIn \\Cref{eq:rlx}, the first two terms account for the $T$-linear contribution with a finite intercept $a$, while the remaining terms describe the $i^{\\mathrm{th}}$ $1\/T_{1}$ peak in terms of a coupling constant $c_{i}$ (proportional to the mean-squared transverse fluctuating field) and the $n$-quantum \\gls{nmr} spectral density functions $J_{n,i}$~\\cite{1988-Beckmann-PR-171-85}.\nIn general, $J_{n,i}$ is frequency dependent and peaked at a temperature where the fluctuation rate matches $\\sim n \\omega_{0}$.\nWhile the precise form of $J_{n,i}$ is not known \\latin{a priori}, the simplest expression, obtained for isotropic \\gls{3d} fluctuations, has a Debye (Lorentzian) form~\\cite{1948-Bloembergen-PR-73-679, 1988-Beckmann-PR-171-85}:\n\\begin{equation} \\label{eq:j3d}\n   J_{n}^{\\mathrm{3D}} = \\frac{\\tau_{c}}{1 + \\left (n \\omega_{0} \\tau_{c} \\right )^{2}} ,\n\\end{equation}\nwhere $\\tau_{c}$ is the (exponential) correlation time of the fluctuations.\nAlternatively, when the fluctuations are \\gls{2d} in character, as might be anticipated for such a layered crystal,\n$J_{n}$ may be described by the empirical expression~\\cite{1979-Richards-TCP-15-141, 1994-Kuchler-SSI-70-434}:\n\\begin{equation} \\label{eq:j2d}\n   J_{n}^{\\mathrm{2D}} = \\tau_{c} \\ln \\left ( 1 + \\left ( n \\omega_{0} \\tau_{c} \\right )^{-2} \\right ).\n\\end{equation}\nFor both \\Cref{eq:j3d,eq:j2d}, we assume that $\\tau_{c}$ is thermally activated, following an Arrhenius dependence:\n\\begin{equation} \\label{eq:arrhenius}\n   \\tau_{c}^{-1} = \\tau_{0}^{-1} \\exp \\left ( - \\frac{ E_{A} }{ k_{B} T } \\right ),\n\\end{equation}\nwhere $E_{A}$ is the activation energy, $\\tau_{0}$ is a prefactor, $k_{B}$ is the Boltzmann constant.\nIf the fluctuations are due to \\ch{^{8}Li^{+}} hopping, $\\tau_{c}^{-1}$ is the site-to-site hop rate.\n\n\nUsing the above expressions, we fit the $1\/T_{1}$ data using a global procedure wherein the kinetic parameters (\\latin{i.e.}, $E_{A, i}$ and $\\tau_{0, i}^{-1}$) are shared at all the different $\\omega_{0}$.\nThis was necessary to fit the data at \\SI{6.55}{\\tesla} where the relaxation is very slow.\nFor comparison, we applied this procedure using both $J_{n}^{\\mathrm{3D}}$ and $J_{n}^{\\mathrm{2D}}$ and the fit results are shown in \\Cref{fig:bsc-slr-fits} as solid ($J_{n}^{\\mathrm{3D}}$) and dashed ($J_{n}^{\\mathrm{2D}}$) lines, clearly capturing the main features of the data.\nThe analysis distinguishes two processes, $i = 1, 2$ in \\Cref{eq:rlx}:\none $(i = 1)$ that onsets at lower temperature with a shallow Arrhenius slope of \\SI{\\sim 0.1}{\\electronvolt}\nthat yields the weaker peaks in $1\/T_1$ at both fields;\nand a higher barrier process $(i = 2)$ with an $E_{A}$ of \\SI{\\sim 0.4}{\\electronvolt} that\nyields the more prominent peak in the low field relaxation,\nwhile the corresponding high field peak must lie above the accessible temperature range.\nThe resulting fit parameters are given in \\Cref{tab:bsc-slr-fits}.\nWe discuss the results in \\Cref{sec:discussion:dynamics}.\n\n\n\\begin{table}\n\\centering\n\\caption[\nArrhenius parameters obtained from the analysis of the temperature dependence of $1\/T_{1}$ in \\ch{Ca} doped \\ch{Bi2Se3}.\n]{ \\label{tab:bsc-slr-fits}\nArrhenius parameters in \\Cref{eq:arrhenius} obtained from the analysis of the temperature dependence of $1\/T_{1}$ in \\ch{Ca} doped \\ch{Bi2Se3} shown in \\Cref{fig:bsc-slr-fits}.\nThe two processes giving rise to the rate peaks are labelled with index $i$.\nGood agreement is found between the $E_{A}$s determined using the spectral density functions $J_{n}$ for \\gls{2d} and \\gls{3d} fluctuations [\\Cref{eq:j2d,eq:j3d}].\n}\n\\begin{tabular}{c S S S S}\n\\toprule\n& \\multicolumn{2}{c}{$i = 1$} & \\multicolumn{2}{c}{$i = 2$} \\\\\n$J_{n}$ & {$\\tau_{0}^{-1}$ (\\SI{e10}{\\per\\second})} & {$E_{A}$ (\\si{\\electronvolt})} & {$\\tau_{0}^{-1}$ (\\SI{e14}{\\per\\second})} & {$E_{A}$ (\\si{\\electronvolt})} \\\\\n\\midrule\n3D & 8.4 \\pm 2.7 & 0.113 \\pm 0.005 & 7 \\pm 5 & 0.395 \\pm 0.015 \\\\\n2D & 9 \\pm 3 & 0.106 \\pm 0.005 & 110 \\pm 90 & 0.430 \\pm 0.016 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nWe now turn to the \\ch{^{8}Li} resonances, with typical spectra shown in \\Cref{fig:bsc-1f-spectra-lf}.\nAs anticipated for a non-cubic crystal, the spectrum is quadrupole split, confirmed unambiguously by the helicity-resolved spectra (see \\Cref{fig:bsc-1f-spectra-helicities} in \\Cref{sec:helicities}).\nThis splitting, on the order of a few \\si{\\kilo\\hertz}~\\footnote{Note that the large $A_{0}$ down to low field precludes \\ch{^{8}Li^{+}} sites with very large \\glspl{efg}.},\nis determined by the \\gls{efg} and is a signature of the crystallographic \\ch{^{8}Li} site.\nBesides this, an unsplit component is also apparent, very close to (within \\SI{\\sim 100}{\\hertz}) the centre-of-mass of the four satellites.\nAt low temperature, the ``central'' and split components are nearly equal, but as the temperature is raised, the unsplit line grows to dominate the spectrum, accompanied by a slight narrowing.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-1f-spectra-lf.pdf}\n\\caption[\n\\ch{^{8}Li} resonance spectra in \\acrlong{bsc} at low magnetic field.\n]{ \\label{fig:bsc-1f-spectra-lf}\n\\ch{^{8}Li} resonance spectra in \\gls{bsc} at low magnetic field.\nThe vertical scale is the same for all spectra;\nthey have been normalized to account for changes in intensity due to \\gls{slr}~\\cite{2009-Hossain-PB-404-914}, with their baselines (shown as dashed grey lines) shifted to match the temperature.\nThe spectra consist of a small and nearly temperature-independent quadrupole split pattern, centred about an unsplit Lorentzian line, whose amplitude grows above \\SI{\\sim 150}{\\kelvin}.\nNote the quadrupole pattern of an integer spin nucleus like \\ch{^{8}Li}, has no central satellite (main line).\nThe solid black lines are fits to a sum of this Lorentzian plus and $2I = 4$ quadrupole satellites (see text).\n}\n\\end{figure}\n\n\nThe scale of the quadrupole splitting is determined by the product of the principal component of the \\gls{efg} tensor $eq$ with the nuclear electric quadrupole moment $eQ$.\nWe quantify this with a conventional definition of the quadrupole frequency (for $I=2$)~\\cite{1957-Cohen-SSP-5-321}:\n\\begin{equation*}\n   \\nu_{q} = \\frac{e^{2} q Q}{8 h}.\n\\end{equation*}\nIn high field, a first order perturbation treatment of the quadrupole interaction is sufficient to obtain accurate satellite positions.\nHowever, at low field, where $\\nu_{q} \/ \\nu_{0} \\approx \\SI{6}{\\percent}$, second order terms are required~\\cite{1957-Cohen-SSP-5-321, 1990-Taulelle-NASC-393}.\nBased on the change in satellite splittings by a factor \\num{2} in going from $B_{0} \\parallel c$ to $B_{0} \\perp c$, we assume the asymmetry parameter of the \\gls{efg} $\\eta = \\num{0}$ (\\latin{i.e.}, the \\gls{efg} is axially symmetric).\nThis is reasonable based on likely interstitial sites for \\ch{^{8}Li^{+}}~\\cite{2019-McFadden-PRB-99-125201}.\nPairs of helicity-resolved spectra were fit with $\\nu_{0}$ and $\\nu_{q}$ as shared free parameters, in addition to linewidths and amplitudes.\nAs the difference between the frequency of the unsplit line and the center of the quadrupole split pattern was too small to measure accurately,\nthe fits were additionally constrained to have the same central frequency $\\nu_{0}$.\nThis is identical to the approach used for \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nA subset of the results (after recombining the two helicities) are shown in \\Cref{fig:bsc-1f-spectra-lf} as solid black lines.\n\n\nThe main result is the strong temperature dependence of the resonance amplitude shown in \\Cref{fig:bsc-1f-fits-amp}.\nWhile the satellite amplitudes are nearly temperature independent, the central component increases substantially above \\SI{150}{\\kelvin}, tending to plateau above the $1\/T_{1}$ peak.\nThe other parameters are quite insensitive to temperature.\nTypical linewidths (\\latin{i.e.}, \\acrlong{fwhm}) are \\SI{\\sim 2.2}{\\kilo\\hertz} for the satellites and \\SI{\\sim 3.8}{\\kilo\\hertz} for the central component.\nThe quadrupole frequency $\\nu_{q} \\approx \\SI{5.5}{\\kilo\\hertz}$ varies weakly, increasing slightly as temperature is lowered.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{bsc-1f-fits-amp.pdf}\n\\caption[\nThe resonance amplitude as a function of temperature in \\ch{Ca} doped \\ch{Bi2Se3} at \\SI{15}{\\milli\\tesla}.\n]{ \\label{fig:bsc-1f-fits-amp}\nThe resonance amplitude as a function of temperature in \\ch{Ca} doped \\ch{Bi2Se3} at \\SI{15}{\\milli\\tesla}.\nWhile the amplitude of the satellite lines are nearly temperature independent, the central component increases substantially above \\SI{150}{\\kelvin}, plateauing on the high-$T$ side of the $1\/T_{1}$ maximum (grey band).\nThe solid line is a guide, while the dashed line indicates the estimated saturation value for the Lorentizan component.\n}\n\\end{figure}\n\n\nWe also measured resonances at room and base temperature in high field (\\SI{6.55}{\\tesla}) where \\ch{^{8}Li} is sensitive to small magnetic shifts.\nFrom the fits, we use $\\nu_{0}$ to calculate the raw relative frequency shift $\\delta$ in parts per million (\\si{\\ppm}) using:\n\\begin{equation} \\label{eq:shift}\n   \\delta = 10^{6} \\left ( \\frac{\\nu_{0} - \\nu_{\\ch{MgO}} }{ \\nu_{\\ch{MgO}} } \\right ) ,\n\\end{equation}\nwhere $\\nu_{\\ch{MgO}}$ is the reference frequency position in \\ch{MgO} at \\SI{300}{\\kelvin}.\nThe shifts are small:\n\\SI[retain-explicit-plus]{+12 \\pm 2}{\\ppm} at room temperature and \\SI{-17 \\pm 3}{\\ppm} at \\SI{5}{\\kelvin}, the latter considerably smaller in magnitude than in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nBecause \\ch{^{8}Li} \\gls{nmr} shifts are generally so small, it is essential to account for the demagnetization field of the sample itself.\nFrom $\\delta$, the corrected shift $K$ is obtained by the \\gls{cgs} expression~\\cite{2008-Xu-JMR-191-47}:\n\\begin{equation}\n   K = \\delta + 4 \\pi \\left ( N - \\frac{1}{3} \\right ) \\chi_{v}\n\\end{equation}\nwhere $N$ is the dimensionless demagnetization factor that depends only on the shape of the sample and $\\chi_{v}$ is the dimensionless (volume) susceptibility.\nFor a thin film, $N$ is \\num{1}~\\cite{2008-Xu-JMR-191-47}, but for the thin platelet crystals used here, we estimate $N$ is on the order of \\num{\\sim 0.8}, treating them as oblate ellipsoids~\\cite{1945-Osborn-PR-67-351}.\nFor the susceptibility, we take the average of literature values reported for pure \\ch{Bi2Se3}~\\cite{1958-Matyas-CJP-8-309, 2003-Kulbachinskii-PB-329-1251, 2015-Pafinlov-JPCM-27-456002, 2016-Chong-JAC-686-245}, giving $\\chi_{v}^{\\mathrm{CGS}} \\approx \\SI{-2.4e-6}{\\emu\\per\\centi\\meter\\cubed}$.\nNote that we have excluded several reports~\\cite{2008-Janicek-PB-403-3553, 2012-Young-PRB-86-075137, 2014-Zhao-NM-13-580} whose results disagree by an order of magnitude from those predicted by Pascal's constants~\\cite{2008-Bain-JCE-85-532}.\nApplying the correction for \\gls{bsc} yields $K$s of \\SI[retain-explicit-plus]{-2 \\pm 2}{\\ppm} and \\SI{-31 \\pm 3}{\\ppm} at room and base temperature, respectively.\nWe discuss this below in \\Cref{sec:discussion:electronic}.\n\n\n\\subsection{\\ch{Bi2Te3:Mn} \\label{sec:results:btm}}\n\n\nTypical \\ch{^{8}Li} \\gls{slr} spectra at high and low field in the magnetically doped \\gls{btm} are shown in \\Cref{fig:btm-slr-spectra}.\nIn contrast to nonmagnetic \\gls{bsc}, the relaxation at high field is fast, typical of paramagnets with unpaired electron spins~\\cite{2015-MacFarlane-PRB-92-064409, 2016-Cortie-PRL-116-106103,2011-Song-PRB-84-054414}.\nThe fast high field rate produces a much less pronounced field dependence.\nAt low field, the \\gls{slr} rate is peaked at low temperature.\nThe relaxation is also non-exponential and fits well using \\Cref{eq:strexp}, with a stretching exponent systematically smaller than in the nonmagnetic \\gls{bsc} or \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nWe analyzed the data with the same global approach, obtaining good quality fits ($\\tilde{\\chi}_{\\mathrm{global}}^{2} \\approx 1.01$) demonstrated by the solid black lines in \\Cref{fig:btm-slr-spectra}.\nThe shared values of $A_{0}$ from the fits are large (\\SI{\\sim 10}{\\percent} for $B_{0} = \\SI{6.55}{\\tesla}$ and \\SI{\\sim 15}{\\percent} for $B_{0} = \\SI{20}{\\milli\\tesla}$), consistent with the full beam polarization, implying that there is remarkably no magnetic wipeout from very fast relaxation~\\cite{2015-MacFarlane-PRB-92-064409}, even at low field.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-slr-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} \\acrlong{slr} spectra in \\ch{Mn} doped \\ch{Bi2Te3} at high and low magnetic field.\n]{ \\label{fig:btm-slr-spectra}\nTypical \\ch{^{8}Li} \\gls{slr} spectra in \\ch{Mn} doped \\ch{Bi2Te3} at high (left) and low (right) magnetic field for \\ch{^{8}Li^{+}} implantation energies of \\SI{20}{\\kilo\\electronvolt} and \\SI{8}{\\kilo\\electronvolt}, respectively.\nThe shaded region denotes the duration of the \\ch{^{8}Li^{+}} beam pulse.\nThe \\gls{slr} is substantial and orders of magnitude faster than in \\gls{bsc} (see \\Cref{fig:bsc-slr-spectra}) at high field.\nThe field dependence to the \\gls{slr} is much weaker than in the nonmagnetic tetradymites.\nThe solid black lines are fits to a stretched exponential convoluted with the \\ch{^{8}Li^{+}} beam pulse as described in the text.\nThe initial asymmetry $A_{0}$ from the fit is used to normalize the spectra.\nThe high and low field spectra have been binned for by factors of \\num{20} and \\num{5}, respectively.\n}\n\\end{figure}\n\n\nThe fit parameters are shown in \\Cref{fig:btm-slr-fits}.\nAt all temperatures, especially at high field, the \\gls{slr} rate $1\/T_{1}$ is orders of magnitude larger than in the nonmagnetic analogs.\nNo clear $1\/T_{1}$ \\gls{bpp} peaks can be identified between \\SIrange[range-units=single,range-phrase=--]{100}{300}{\\kelvin};\nhowever, in the low field data, a critical divergence is evident at the magnetometric transition at about \\SI{13}{\\kelvin}.\nIn high field, this feature is largely washed out, with a remnant peak near \\SI{50}{\\kelvin}.\nAbove \\SI{200}{\\kelvin}, the \\gls{slr} rate increases very rapidly and is well-described by $1\/T_{1} \\propto \\exp [ - E_{A} \/ ( k_{B} T )]$,\nwith $E_{A} \\approx \\SI{0.2}{\\electronvolt}$ at both fields.\nWe discuss this below in \\Cref{sec:discussion:dynamics}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-slr-fits.pdf}\n\\caption[\nTemperature dependence of the \\ch{^{8}Li} \\acrlong{slr} rate $1\/T_{1}$ in \\ch{Mn} doped \\ch{Bi2Te3} at high and low field.\n]{ \\label{fig:btm-slr-fits}\nTemperature dependence of the \\ch{^{8}Li} \\gls{slr} rate $1\/T_{1}$ in \\ch{Mn} doped \\ch{Bi2Te3} at high and low field.\nAt low field, $1\/T_{1}$ shows a critical peak at the ferromagnetic transition at $T_{C} \\approx \\SI{13}{\\kelvin}$, as the \\ch{Mn} spin fluctuations freeze out.\nAbove \\SI{200}{\\kelvin}, the \\gls{slr} rate increases exponentially in manner nearly independent of applied field.\nThe solid grey lines are drawn to guide the eye.\n}\n\\end{figure}\n\n\nIn contrast to \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, the resonance in \\gls{btm} consists of a single broad Lorentzian with none of the resolved fine structure (see \\Cref{fig:btm-1f-spectra}).\nSurprisingly, the very broad line has significant intensity, dwarfing the quadrupole pattern in \\gls{bsc} in both width and amplitude.\nIn addition, there is a large negative shift at \\SI{10}{\\kelvin}.\nAt room temperature, the line is somewhat narrower, and the shift is reduced in magnitude.\nQuantitative results from Lorentzian fits are summarized in \\Cref{tab:btm-1f-fits}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{btm-1f-spectra.pdf}\n\\caption[\nTypical \\ch{^{8}Li} resonances in \\ch{Mn} doped \\ch{Bi2Te3} and \\ch{Ca} doped \\ch{Bi2Se3} at high magnetic field.\n]{ \\label{fig:btm-1f-spectra}\nTypical \\ch{^{8}Li} resonances in \\ch{Mn} doped \\ch{Bi2Te3} and \\ch{Ca} doped \\ch{Bi2Se3} at high magnetic field.\nThe vertical scale is the same for both spectra;\nthey have been normalized to account for changes in intensity and baseline~\\cite{2009-Hossain-PB-404-914}.\nThe lineshape in the magnetic \\gls{btm} is well-described by a broad Lorentzian (solid black line) with no quadrupolar splitting.\nA large negative shift is also apparent for \\gls{btm} with respect to the reference frequency in \\ch{MgO} (vertical dashed line).\nThe dotted vertical line indicates the expected resonance position due to demagnetization, revealing a large positive hyperfine field (\\SI{\\sim 38}{\\gauss}) at \\SI{5}{\\kelvin} in the magnetic state.\n}\n\\end{figure}\n\n\n\\begin{table}\n\\centering\n\\caption[\nResults from the analysis of the \\ch{^{8}Li} resonance in \\acrlong{btm} at high and low temperature with $B_{0} = \\SI{6.55}{\\tesla} \\parallel (001)$.\n]{ \\label{tab:btm-1f-fits}\nResults from the analysis of the \\ch{^{8}Li} resonance in \\gls{btm} at high and low temperature with $B_{0} = \\SI{6.55}{\\tesla} \\parallel (001)$.\nThe (bulk) magnetization $M$ measured with $\\SI{1.0}{\\tesla} \\parallel (001)$ is included for comparison.\n}\n\\begin{tabular}{S S S S[retain-explicit-plus] S}\n\\toprule\n{$T$ (\\si{\\kelvin})} & {$\\tilde{A}$ (\\si{\\percent})} & {\\acrshort{fwhm} (\\si{\\kilo\\hertz})} & {$\\delta$ (\\si{\\ppm})} & {$M$ (\\si{\\emu\\per\\centi\\meter\\cubed})} \\\\\n\\midrule\n294 & 33 \\pm 5     & 16.2 \\pm 1.2 &  +10 \\pm 9 &  0.076 \\\\\n 10 & 14.4 \\pm 0.8 & 41.7 \\pm 1.6 & -206 \\pm 12 & 7.698 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Discussion \\label{sec:discussion}}\n\n\nThe \\ch{^{8}Li} \\gls{nmr} properties of nonmagnetic \\gls{bsc} are quite similar to previous measurements in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nThe resonance spectra show a similar splitting ($\\nu_q$ is about \\SI{25}{\\percent} smaller in \\gls{bsc}), indicating a similar site for \\ch{^{8}Li}.\nThe resemblance of the spectra extends to the detailed temperature dependence, including the growth of the unsplit line approaching room temperature.\nSurprisingly, the \\gls{bsc} spectra are better resolved than \\gls{bts}, implying a higher degree of order, despite the \\ch{Ca} doping.\nThis is also evident in the \\gls{slr}, with a stretching exponent $\\beta$ closer to unity in \\gls{bsc} than in \\gls{bts}.\nThis likely reflects additional disorder in \\gls{bts} from \\ch{Bi\/Te} anti-site defects~\\cite{2012-Scanlon-AM-24-2154} that are much more prevalent than for \\ch{Bi\/Se}, due to the difference in radii and electronegativity.\nThe sharp quadrupolar pattern indicates a well-defined crystallographic \\ch{^{8}Li^{+}} site, and the corresponding small \\gls{efg} suggests it is in the \\gls{vdw} gap.\n\\Gls{dft} calculations of the \\gls{efg} may enable a precise site assignment.\nThe high field \\gls{slr} is also similar to \\gls{bts}:\nit is slow and near the lower limit measurable due to the finite \\ch{^{8}Li} lifetime $\\tau_\\beta$ and comparable to the \\gls{vdw} metal \\ch{NbSe2}~\\cite{2006-Wang-PB-374-239}, where the carrier concentration is much higher, but significantly slower than the \\gls{ti} alloy \\ch{Bi_{1-x}Sb_{x}}~\\cite{2014-MacFarlane-PRB-90-214422}.\nThe low field enhancement of $1\/T_1$ is also similar, so it cannot be essentially related to the dilute \\ch{^{125}Te} moments that are absent in \\gls{bsc}, but probably determined primarily by the \\SI{100}{\\percent} abundant \\ch{^{209}Bi}.\nFrom such a detailed similarity, it is clear that a quantitative comparison with \\gls{bts} and other \\gls{vdw} materials will be useful.\n\n\nWith these similarities in mind, the rest of the discussion is organized as follows: in \\Cref{sec:discussion:dynamics},\nwe consider evidence of mobility of the \\ch{^{8}Li^{+}} ion;\nin \\Cref{sec:discussion:electronic}, electronic effects at low temperature in \\gls{bsc};\nand the magnetic properties of \\gls{btm} in \\Cref{sec:discussion:magnetic}.\n\n\n\\subsection{Dynamics of the \\ch{Li^{+}} ion \\label{sec:discussion:dynamics}}\n\nIn \\gls{bts}, we considered if the evolution of the spectrum with temperature (similar to \\Cref{fig:bsc-1f-spectra-lf}) was the result of a site change transition from a meta-stable quadrupolar \\ch{^{8}Li^{+}} site to a lower energy site with very small \\gls{efg} at higher temperature~\\cite{2019-McFadden-PRB-99-125201}, similar to elemental \\ch{Nb}~\\cite{2009-Parolin-PRB-80-174109}.\nWe now consider an alternative explanation; namely, dynamic averaging of the quadrupolar interaction due to \\ch{^{8}Li^{+}} motion.\nExamples of this are found in conventional \\ch{^{7}Li} \\gls{nmr}, where, unlike \\ch{^{8}Li}, the $I = 3\/2$ quadrupole spectrum has a main line (the $m = \\pm 1\/2$ satellite) overlapping the averaged resonance~\\cite{2012-Galven-CM-24-3335, 2012-Indris-JPCC-116-14243}.\nDynamic averaging is suggested by the onset near, but below, the \\gls{slr} rate peak (see \\Cref{fig:bsc-1f-fits-amp}).\nHowever, for hopping between equivalent interstitial sites (probably the quasi-octahedral Wyckoff $3b$ site in the \\gls{vdw} gap --- see Figure~12 in Ref.~\\onlinecite{2019-McFadden-PRB-99-125201}),\none does not expect that the \\gls{efg} will average to a value near zero (required to explain the unsplit line).\nA point charge estimate reveals the quasi-tetrahedral (Wyckoff $6c$) site, thought to be the saddle point in the potential for \\ch{Li^{+}} between adjacent $3b$ sites~\\cite{2016-Gosalvez-PRB-93-075429}, has an \\gls{efg} of opposite sign to the $3b$ site.\nIf $6c$ is instead a shallow minimum, the \\ch{^{8}Li^{+}} residence time there may be long enough that the \\gls{efg} averages to near zero.\nIn the fast motion limit at higher temperatures, one would then expect the quadrupole splitting to re-emerge when the residence time in the $6c$ ``transition'' site becomes much shorter~\\cite{2012-Indris-JPCC-116-14243}.\n\n\nWe now examine the two kinetic processes causing the $1\/T_{1}$ peaks in \\gls{bsc}.\nIt is surprising to find two distinct thermally activated processes sweeping through the \\gls{nmr} frequency, especially since only a single process was found in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}.\nFirst, we consider the weaker feature, the low temperature $(i=1)$ peaks.\nIn layered materials, small intercalates can undergo highly localized motion at relatively low temperatures below the onset of free diffusion~\\cite{1982-Kanert-PR-91-183, 1994-MullerWarmuth-PIR-17-339}.\nSuch local motion may be the source of the \\gls{slr} rate peak, but it is quite ineffective at narrowing the resonance, consistent with the absence of any lineshape changes in the vicinity of the $i=1$ peaks.\nCaged local motion is usually characterized by a small activation barrier, comparable to the \\SI{\\sim 0.1}{\\electronvolt} observed here.\nSimilar phenomena have been observed at low temperature, for example, in neutron activated \\ch{^{8}Li} \\gls{bnmr} of \\ch{Li} intercalated graphite, \\ch{LiC12}~\\cite{1989-Schirmer-SM-34-589, 1995-Schirmer-ZNA-50-643}.\nIt is not clear why such motion would be absent in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, which has a larger \\gls{vdw} gap than \\gls{bsc} (\\SI{2.698}{\\angstrom} vs.\\ \\SI{2.568}{\\angstrom}).\nAlternatively, this feature in the relaxation may have an electronic origin,\nperhaps related to the emergent low-$T$ magnetism in \\ch{MoTe2} observed\nby \\gls{musr}~\\cite{2018-Guguchia-SA-4-eaat3672} and \\ch{^{8}Li} \\gls{bnmr}~\\cite{Krieger-tbp}.\n\n\nIn contrast, the \\gls{slr} rate peak above \\SI{200}{\\kelvin} $(i = 2)$ is almost certainly due to \\gls{efg} fluctuations caused by stochastic \\ch{^{8}Li^{+}} motion.\nFrom the data, we cannot conclude that this is long-range diffusion, but the room temperature \\ch{Li^{+}} intercalability of \\ch{Bi2Se3}~\\cite{1988-Paraskevopoulos-MSEB-1-147, 1989-Julien-SSI-36-113, 2010-Bludska-JSSC-183-2813} suggests it is.\nIts barrier, on the order of \\SI{\\sim 0.4}{\\electronvolt}, is comparable to other \\gls{vdw} gap layered ion conductors, but it is about twice as high as in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, possibly a result of the \\ch{Se} (rather than \\ch{Te}) bounded \\gls{vdw} gap, which provides less space between neighbouring \\glspl{ql}.\n\n\nWe now consider the Arrhenius law prefactors $\\tau_{0}^{-1}$, that, for ionic diffusion, are typically in the range \\SIrange[range-phrase=--,range-units=single]{e12}{e14}{\\per\\second}.\nFor the low-$T$ process ($i = 1$), independent of the form of $J_{n}$ (see \\Cref{tab:bsc-slr-fits}), $\\tau_{0}^{-1} \\approx \\SI{9e10}{\\per\\second}$ is unusually low.\nIn contrast, for the high-$T$ ($i = 2$) process, it is much larger and depends strongly on $J_{n}$.\nFor \\gls{3d} diffusion, it is in the expected range, while the \\gls{2d} model yields an extremely large value, \\SI{\\sim e16}{\\per\\second},\nin the realm of prefactor anomalies~\\cite{1983-Villa-SSI-9-1421} and opposite to the small value expected for\nlow dimensional diffusion~\\cite{1978-Richards-SSC-25-1019}.\nSimilar behaviour was observed recently in \\ch{^{7}Li} \\gls{nmr} of \\ch{LiC6}~\\cite{2013-Langer-PRB-88-094304},\nwhere surprisingly, $J_{n}^{\\mathrm{2D}}$ was concluded to be less appropriate than $J_{n}^{\\mathrm{3D}}$,\nsuggesting that \\ch{Li} motion in the \\gls{vdw} gap is not as ideally \\gls{2d} as might be expected.\nIn \\gls{bsc}, the anomaly may be related to local dynamics that onset at lower $T$, imparting some \\gls{3d} character to the motion.\n\n\nGiven the evidence for long-range \\ch{Li^{+}} motion in \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, the absence of a relaxation peak in \\gls{btm} may seem unexpected.\nBoth \\ch{Ca^{2+}} and \\ch{Mn^{2+}} dopants (substitutional for \\ch{Bi^{3+}}) have an effective $-1$ charge yielding an attractive trapping potential for the positive interstitial \\ch{^{8}Li^{+}}, but the \\ch{Mn} concentration is an order of magnitude larger.\nThe high trap density in \\gls{btm} will suppress \\ch{Li^{+}} mobility.\nThe exponential increase in $1\/T_{1}$ above \\SI{200}{\\kelvin} may be the onset of a diffusive \\gls{bpp} peak, but, in this case, one does not expect it to be so similar between the two very different magnetic fields.\nThis may reflect a trade-off between the increase in $\\omega_{0}$ that shifts the peak to higher temperature, slowing the relaxation on its low-$T$ flank, and the increased polarization of the \\ch{Mn} moments by the field that amplifies local magnetic inhomogeneities.\nA motional origin for this increase is consistent with the apparent $E_{A} \\sim \\SI{0.2}{\\electronvolt}$, similar to \\ch{^{8}Li^{+}} in \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201}, which also has a \\ch{Te} bounded \\gls{vdw} gap of similar size to \\gls{btm} (\\SI{2.620}{\\angstrom}).\nHowever, it may have a different explanation, see below in \\Cref{sec:discussion:magnetic}.\n\n\n\\subsection{Electronic effects at low temperature \\label{sec:discussion:electronic}}\n\n\nBismuth chalcogenide (\\ch{Bi2Ch3}) crystals exhibit substantial bulk conductivity, despite a narrow gap in the \\gls{3d} band structure, making it difficult to distinguish effects of the metallic \\gls{tss}.\nThis is due to native defects (\\latin{e.g.}, \\ch{Ch} vacancies) that are difficult or impossible to avoid~\\cite{2013-Cava-JMCC-1-3176}.\nExtrinsic dopants, such as substitutional \\ch{Ca\/Bi}, can be used to compensate the spontaneous $n$-type doping.\nBrahlek \\latin{et al.}\\ have argued~\\cite{2015-Brahlek-SSC-215-54} that, even for the most insulating compensated samples, the carrier densities far exceed the Mott criterion, making them heavily doped semiconductors in the metallic regime.\nIn this case, we expect metallic \\gls{nmr} characteristics~\\cite{1978-Holcomb-SUSSP-19-251}, namely a magnetic Knight shift $K$, proportional to the carrier spin susceptibility $\\chi_{s}$~\\footnote{%\nThe \\gls{nmr} shift $K$ is comprised of several terms including both the spin and orbital response of all the surrounding electrons. In metals, where the local carrier density is substantial,\nthe Fermi contact coupling with the electron spins often dominates, as we have assumed here.%\n}.\nIn the simplest (isotropic) case,\n\\begin{equation} \\label{eq:knight-shift}\n   K = A \\chi_{s},\n\\end{equation}\nwhere $A$ is the hyperfine coupling constant, which is accompanied by a \\gls{slr} rate following the Korringa law~\\cite{1950-Korringa-P-16-601, 1990-Slichter-PMR},\n\\begin{equation} \\label{eq:korringa-rate}\n   \\frac{1}{T_{1}} = 4 \\pi \\hbar A^{2} \\gamma_{n}^{2} \\left ( \\frac{\\chi_{s}}{g^{*}\\mu_{B}} \\right )^{2} k_{B} T.\n\\end{equation}\nHere, $\\gamma_{n}$ is the nuclear gyromagnetic ratio, $g^{*}$ is the carrier $g$-factor, and $\\mu_{B}$ is the Bohr magneton.\nCombining \\Cref{eq:knight-shift,eq:korringa-rate}, we obtain the Korringa product, which is independent of the value of $A$,\n\\begin{equation} \\label{eq:korringa-product}\n   T_{1}TK^{2} = \\frac{\\hbar (g^{*} \\mu_B)^{2}}{4 \\pi k_{B} \\gamma_{n}} = S(g^{*}).\n\\end{equation}\nFor \\ch{^{8}Li},\n\\begin{equation*}\n   S(g^{*}) \\approx 1.20 \\times 10^{-5} \\left ( \\frac{ g^{*} }{g_0} \\right )^{2}~\\si{\\second\\kelvin},\n\\end{equation*}\nwhere, unlike in metals, we have allowed for an effective $g$-factor that may be far from its free electron value\n$g_0 \\approx 2$~\\cite{1972-Look-PRB-5-3406}.\nIndeed, recent \\gls{epr} measurements in \\ch{Bi2Se3} find $g^{*} \\approx 30$~\\cite{2016-Wolos-PRB-93-155114}.\n\n\nAccording to Ref.~\\onlinecite{2015-Brahlek-SSC-215-54}, \\gls{bts} and \\gls{bsc} lie on opposite sides of the Ioffe-Regel limit, where the carrier mean free path is equal to its Fermi wavelength, with \\gls{bsc} having a higher carrier density and mobility.\nA comparative Korringa analysis could test this assertion and, to this end, using \\Cref{eq:korringa-product} we define the dimensionless Korringa ratio as\n\\begin{equation} \\label{eq:korringa-constant}\n   \\mathscr{K} = \\frac{T_{1}TK^2}{S(g^{*})}.\n\\end{equation}\nBelow the Ioffe-Regel limit, the autocorrelation function of the local hyperfine field at the nucleus, due to the carriers (that determines $T_{1}$) becomes limited by the diffusive transport correlation time.\nThis has been shown to enhance the Korringa rate (\\latin{i.e.}, shortening $T_{1}$)~\\cite{1971-Warren-PRB-3-3708, 1983-Gotze-ZPB-54-49}.\nFrom this, one expects $\\mathscr{K}$ would be smaller in \\gls{bts} than in \\gls{bsc}.\n\n\nThere are, however, significant difficulties in determining the experimental $\\mathscr{K}$.\nFirst, the Korringa slope depends on magnetic field.\nAt low fields, this is due to coupling with the host nuclear spins, a phenomenon that is quenched in high fields where the \\ch{^{8}Li} \\gls{nmr} has no spectral overlap with the \\gls{nmr} of host nuclei.\nFor example, in simple metals, we find the expected field-independent Korringa slope at high fields in the Tesla range~\\cite{2015-MacFarlane-SSNMR-68-1}.\nIn contrast, in \\gls{bts}, the slope decreases substantially with increasing field, even at high fields~\\cite{2019-McFadden-PRB-99-125201}.\nWe suggested this could be the result of magnetic carrier freeze-out.\nWhile we do not have comparably extensive data in \\gls{bsc}, we can compare the slope at the same field, \\SI{6.55}{\\tesla} (see \\Cref{tab:korringa}).\nHere, in both materials, the relaxation is extremely slow, exhibiting no curvature in the \\gls{slr} during the \\ch{^{8}Li} lifetime\n(\\Cref{fig:bsc-slr-spectra}), so the uncertainties in $1\/T_{1}T$ are likely underestimates.\nThe larger slope is, however, consistent with a higher carrier density $n$ in \\gls{bsc}.\nThe Korringa slopes should scale~\\cite{1972-Look-PRB-5-3406} as $n^{2\/3}$.\nUsing $n \\sim \\SI{1e19}{\\per\\centi\\meter\\cubed}$ in \\gls{bsc}~\\cite{2009-Hor-PRB-79-195208} and \\SI{\\sim 2e17}{\\per\\centi\\meter\\cubed} in \\gls{bts}~\\cite{2011-Jia-PRB-84-235206}, the slopes should differ by a factor of \\num{\\sim 14}, while experimentally the ratio is \\num{\\sim 5}.\n\n\nThe next difficulty is accurately quantifying the shift $K$, which is quite small with a relatively large demagnetization correction.\nExperimentally, the zero of shift, defined by the calibration in \\ch{MgO}, differs from the true zero (where $\\chi_{s} = 0$) by the difference in chemical (orbital) shifts between \\ch{MgO} and the chalcogenide.\nHowever, because \\ch{Li} chemical shifts are universally very small, this should not be a large difference, perhaps a few \\si{\\ppm}.\nThe negative low temperature shift is also somewhat surprising.\nThe hyperfine coupling $A$ for \\ch{Li} is usually determined by a slight hybridization of the vacant $2s$ orbital with the host conduction band.\nAs the $s$ orbital has density at the nucleus, the resulting coupling is usually positive, with the $d$ band metals \\ch{Pd} and \\ch{Pt} being exceptional~\\cite{2015-MacFarlane-SSNMR-68-1}.\nFor a positive $A$, the sign of $K$ is determined by the sign of $g^{*}$, which has not yet been\nconclusively measured in either \\gls{bsc} or \\gls{bts}.\nA more serious concern is that $K$ depends on temperature (in contrast to simple metals) and, at least in \\gls{bts}, also on applied field~\\cite{2019-McFadden-PRB-99-125201}.\nTo avoid ambiguity from the field dependence, we similarly restrict comparison to the same field, \\SI{6.55}{\\tesla}.\nA similarly temperature dependent shift (for the \\ch{^{207}Pb} \\gls{nmr}) was found in the narrow band semiconductor \\ch{PbTe}~\\cite{1973-Hewes-PRB-7-5195}, where it was explained by the temperature dependence of the Fermi level $E_{F}$~\\cite{1970-Senturia-PRB-1-4045}.\nAt low-$T$ in the heavily $p$-type \\ch{PbTe}, $E_{F}$ occurs in the valence (or a nearby impurity) band, but with increasing temperature, moves upward into the gap, causing a reduction in $|K|$.\nWith this in mind, we assume the low temperature shift is the most relevant for a Korringa comparison.\nWithout a measured $g^{*}$ in \\gls{bts}, we simply assume it is the same as \\gls{bsc}, and use the $g^{*}_{\\parallel}$ from \\gls{epr}~\\cite{2016-Wolos-PRB-93-155114} to calculate the values of $\\mathscr{K}$ in \\Cref{tab:korringa}.\n\n\n\\begin{table}\n\\centering\n\\caption[\nKorringa analysis of \\acrlong{bsc} and \\acrlong{bts} at \\SI{6.55}{\\tesla} and low temperature.\n]{ \\label{tab:korringa}\nKorringa analysis of \\gls{bsc} and \\gls{bts}~\\cite{2019-McFadden-PRB-99-125201} at \\SI{6.55}{\\tesla} and low temperature.\nTo calculate $\\mathscr{K}$, we take $S(g^{*})$ in \\Cref{eq:korringa-constant} to be \\SI{2.69e-3}{\\second\\kelvin}, using the $g_{\\parallel}^{*}$ from \\gls{epr}~\\cite{2016-Wolos-PRB-93-155114}.\n}\n\\begin{tabular}{l S S S}\n\\toprule\n    & {$1\/(T_{1}T)$ (\\SI{e-6}{\\per\\second\\per\\kelvin})} & {$K$ (\\si{\\ppm})} & {$\\mathscr{K}$} \\\\\n\\midrule\n\\acrlong{bsc} & 9.5  \\pm 0.8  &  -31 \\pm 3 &   0.038 \\pm 0.008 \\\\\n\\acrlong{bts} & 1.79 \\pm 0.07 & -115 \\pm 3 &   2.78  \\pm 0.18  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nThe values of $\\mathscr{K}$ are just opposite to the expectation of faster relaxation for diffusive \\gls{bts} compared to metallic\n\\gls{bsc}~\\cite{2015-Brahlek-SSC-215-54}.\nElectronic correlations can, however, significantly alter the Korringa ratio to an extent that depends on disorder~\\cite{1994-Shastry-PRL-72-1933}.\nThere should be no significant correlations in the broad bulk bands of the chalcogenides, but in narrow impurity bands, this is certainly a possibility.\nWe note that $\\mathscr{K}$ is also less than \\num{1} in \\ch{PbTe}~\\cite{1973-Alexander-JN-1-251}, similar to \\gls{bsc}.\nAt this stage, without more data and a better understanding of the considerations mentioned above, it is premature to draw further conclusions.\n\n\n\\subsection{Magnetism in \\acrlong{btm} \\label{sec:discussion:magnetic}}\n\n\nIn the \\ch{Mn} doped \\ch{Bi2Te3} at low field, the relaxation from magnetic \\ch{Mn^{2+}} becomes faster as the spin fluctuations slow down on cooling towards $T_{C}$.\nIn particular, the low-$T$ increase in $1\/T_{1}$ occurs near where correlations among the \\ch{Mn} spins become evident in \\gls{epr}~\\cite{2016-Zimmermann-PRB-94-125205, 2017-Talanov-AMR-48-143}.\nIn remarkable contrast to ferromagnetic \\ch{EuO}~\\cite{2015-MacFarlane-PRB-92-064409}, the signal is not wiped out in the vicinity of $T_{C}$, but $1\/T_{1}$ does become very fast.\nThis is likely a consequence of a relatively small hyperfine coupling consistent with a \\ch{Li} site in the \\gls{vdw} gap.\n\n\nHigh applied field slows the \\ch{Mn} spins more continuously starting from a higher temperature, suppressing the critical peak and reducing $1\/T_{1}$ significantly, a well-known phenomenon in \\gls{nmr} and \\gls{musr} at magnetic transitions (see \\latin{e.g.}, Ref.~\\onlinecite{2001-Heffner-PRB-63-094408}).\nThis also explains the small critical peak in the \\ch{^{8}Li} \\gls{slr} in the dilute magnetic semiconductor, \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}.\nAs in \\ch{GaAs}, \\ch{Mn} in \\ch{Bi2Te3} is both a magnetic and electronic dopant.\nAt this concentration, \\gls{btm} is $p$-type with a metallic carrier density of \\SI{\\sim 7e19}{\\per\\centi\\meter\\cubed}~\\cite{2010-Hor-PRB-81-195203, 2016-Zimmermann-PRB-94-125205}.\nHowever, the difference in scale of $1\/T_{1}$ at high field between \\Cref{fig:btm-slr-fits,fig:bsc-slr-fits} shows that the \\ch{Mn} spins completely dominate the carrier relaxation.\n\n\nIt is also remarkable that the resonance is so clear in the magnetic state, in contrast to \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}.\nThe difference is not the linewidth, but rather the enhanced amplitude.\nThis may be due to slow \\ch{^{8}Li} spectral dynamics occurring on the timescale of $\\tau_{\\beta}$ that effectively enhance the amplitude, \nfor example, slow fluctuations of the ordered \\ch{Mn} moments, not far below $T_{C}$.\nSimilar behavior was found in rutile \\ch{TiO2} at low temperature, where it was attributed to field fluctuations due to a nearby electron polaron~\\cite{2017-McFadden-CM-29-10187}.\nEnhancement of the \\gls{rf} field at nuclei in a ferromagnet~\\cite{1959-Gossard-PRL-3-164} may also play a role.\n\n\nAbove \\SI{200}{\\kelvin}, the activated increase in $1\/T_{1}$ may indicate the onset of diffusive \\ch{^{8}Li^{+}} motion, similar to the nonmagnetic analogs, with the additional effect that the local magnetic field from the \\ch{Mn} spins is also modulated by \\ch{^{8}Li} hopping (not just the \\gls{efg}), similar to the ordered magnetic \\gls{vdw} layered \\ch{CrSe2}~\\cite{2020-Ticknor-RSCA-10-8190}.\nHowever, it may instead mark the onset of thermal excitation of electrons across the bandgap that is narrowed by \\ch{Mn} doping\\cite{2010-Hor-PRB-81-195203}.\nThermally excited carriers may also explain the increase in $1\/T_1$ at comparable temperatures in \\ch{Ga_{1-x}Mn_{x}As}~\\cite{2011-Song-PRB-84-054414}, a very different medium for \\ch{Li^+} diffusion.\nThermally increased carrier density would strengthen interaction between the \\ch{Mn} moments, extend their effects via \\gls{rkky} polarization, and increase $1\/T_{1}$.\nMeasurements at higher temperatures may be able to discriminate these possibilities, but it is likely that both processes will contribute.\n\n\nThe stretched \\gls{slr} and field-dependent $1\/T_{1}$\nare characteristic of the \\gls{nmr} response in a disordered glassy magnetic state, consistent with\nthe random Mn\/Bi site disorder. The magnetic properties of \\gls{btm}~\\cite{2010-Hor-PRB-81-195203, 2013-Watson-NJP-15-103016, 2016-Zimmermann-PRB-94-125205, 2019-Vaknin-PRB-99-220404} are similar to the dilute (ferro)magnetic semiconductors that include a uniform magnetization of the carriers. In this context, our data provides a \\emph{local} characterization of the inhomogeneous magnetic state of \\gls{btm} that will be useful in developing a detailed microscopic understanding of its magnetism.\nHaving established the effects of \\ch{Mn} magnetism in the bulk, it would be interesting to use lower implantation energies to study how they may be altered in the surface region by coupling to the \\gls{tss}.\n\n\n\\section{Conclusion \\label{sec:conclusion}}\n\n\nUsing implanted \\ch{^{8}Li} \\gls{bnmr}, we have studied the electronic and magnetic properties of the doped \\glspl{ti} \\gls{bsc} and \\gls{btm}.\nFrom \\gls{slr} measurements, we find evidence at temperatures above \\SI{\\sim 200}{\\kelvin} for site-to-site hopping of isolated \\ch{Li^{+}} with an Arrhenius activation energy of \\SI{\\sim 0.4}{\\electronvolt} in \\gls{bsc}.\nAt lower temperature the electronic properties dominate, giving rise to Korringa-like relaxation and negative Knight shifts, similar to isostructural \\gls{bts}.\nA quantitative comparison reveals Korringa ratios opposite to expectations across the Ioffe-Regel limit.\nIn \\gls{btm}, the magnetism from dilute \\ch{Mn} moments dominates all other spin interactions, but the \\gls{bnmr} signal remains measurable through the magnetic transition at $T_{C}$, where a critical peak in the \\gls{slr} rate is observed.\nThe \\SI{\\sim 0.2}{\\electronvolt} activation energy from the high temperature increase in the \\gls{slr} may be related to \\ch{Li} mobility or to thermal carrier excitations.\n\n\nWith these new results, a more complete picture of the implanted \\ch{^{8}Li} \\gls{nmr} probe of the tetradymite \\ch{Bi} chalcogenides (and other \\gls{vdw} chalcogenides) is beginning to emerge.\nAt high temperatures, isolated \\ch{^{8}Li^{+}} has a tendency to mobilize, providing unique access to the kinetic parameters governing \\ch{Li^{+}} diffusion in the ultra-dilute limit.\nAt low temperature, \\ch{^{8}Li} is sensitive to the local metallic and magnetic properties of the host.\nWith the bulk \\gls{nmr} response now established in \\ch{Bi2Ch3} \\glspl{ti}, the prospect of directly probing the chiral \\gls{tss} with the depth resolution provided by \\gls{bnmr} remains promising.\n\n\n\\begin{acknowledgments}\nWe thank:\nR.\\ Abasalti, D.\\ J.\\ Arseneau, S.\\ Daviel, B.\\ Hitti, K.\\ Olchanski, and D.\\ Vyas for their excellent technical support;\nM.\\ H.\\ Dehn, T.\\ J.\\ Parolin, O.\\ Ofer, Z.\\ Salman, Q.\\ Song, and D.\\ Wang for assistance with early measurements;\nas well as D.\\ E.\\ MacLaughlin, S.\\ D.\\ Senturia, and A.\\ Wolos for useful discussions.\nThis work was supported by NSERC Discovery grants to R.F.K.\\ and W.A.M.\nAdditionally, R.M.L.M.\\ and A.C.\\ acknowledge support from their NSERC CREATE IsoSiM Fellowships.\nThe crystal growth at Princeton University was supported by the ARO-sponsored MURI on topological insulators, grant number W911NF1210461.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Parameters of the electron number density\nprofiles.}\\label{table_neCC} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{lcccccc}\n    \\hline\n   \n    Cluster &  $n_{e0}$ & $\\beta$ & $\\theta_{c1}$ & $\\theta_{c2}$ & $f$ \\\\\n      & (10$^{-2}$cm$^{-3}$) &  & (arcsec) & (arcsec) &  \\\\\n    \\hline\n    \\hline\n    A1413 & $3.89\\pm0.54$ & $0.535\\pm0.016$ & $6.7\\pm1.4$ & $40.1\\pm4.1$ & $0.760\\pm0.020$ \\\\\n    A1689  & $4.15\\pm0.31$ & $0.871\\pm0.040$ & $21.6\\pm1.0$ & $104.5\\pm5.3$ & $0.870\\pm0.010$ \\\\\n    A1835  & $11.3\\pm0.4$ & $0.802\\pm0.015$ & $9.3\\pm0.2$ & $63.8\\pm1.6$ & $0.940\\pm0.001$ \\\\\n    A2204  & $20.4\\pm1.1$ & $0.716\\pm0.028$ & $7.5\\pm0.3$ & $67.6\\pm1.9$ & $0.959\\pm0.004$ \\\\\n    A2261  & $4.07\\pm0.59$ & $0.631\\pm0.024$ & $10.2\\pm1.8$ & $39.1\\pm5.9$ & $0.760\\pm0.050$ \\\\\n    MS1358.4+6245  & $9.63\\pm0.79$ & $0.676\\pm0.017$ & $3.3\\pm0.2$ & $37.0\\pm1.8$ & $0.934\\pm0.003$ \\\\\n    RXJ1347.5-1145  & $28.5\\pm1.4$ & $0.632\\pm0.009$ & $4.0\\pm0.2$ & $23.3\\pm1.6$ & $0.942\\pm0.004$ \\\\\n    ZW3146  & $16.9\\pm0.3$ & $0.669\\pm0.005$ & $4.4\\pm0.1$ & $25.8\\pm0.6$ & $0.882\\pm0.004$ \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nClusters of galaxies are the largest gravitationally-bound objects\narising thus far from the process of hierarchical structure\nformation (\\cite{Voit05}). As the most recent and most massive\nobjects of the Universe, clusters are excellent probes for studying\nits formation and evolution. The observed state of gas within a\ncluster is determined by a combination of shock heating during\naccretion, radiative cooling, and thermal feedback produced by the\ncooling itself, so the density and temperature of the ICM represent\nthe full thermal history of clusters' formation. To better\nunderstand the physics of ICM, it is necessary to have sufficient\nknowledge of the gas density and temperature distributions. Though\nclusters are the ideal target objects for X-ray observations of the\nhot ICM, millimeter and sub-millimeter measurements provide\nindependent and complementary tools for studying the same ICM by\nexploiting the Sunyaev Zel'dovich (SZ) effect (\\cite{Sunyaev72}).\n\nThe SZ effect is the Comptonization of the cosmic microwave\nbackground (CMB) photons, coming from the last scattering surface,\nby the hot electrons population of the ICM. The photon energy\nvariation, which is caused by the scattering process, can be\nexpressed as CMB temperature variations\n\n\\begin{equation}\\label{eq_SZ}\n\\Delta T_{SZ}=yT_{CMB}f(x)(1+\\delta_n(x,\\theta_e))+\\Delta T_{kin}\n\\end{equation}\nwhere\n\n\\begin{equation}\\label{eq_y}\ny=\\int\\theta_ed\\tau_e=\\int\\left(\\frac{k_BT_e}{m_ec^2}\\right)\\sigma_Tn_edl\\propto\\int\nP_edl\n\\end{equation}\nrepresents the comptonization parameter, $x=(h\\nu)\/(k_BT_{CMB})$ the\ndimensionless frequency, $h$ and $k_B$ are respectively the Planck\nand Boltzmann constants, $T_{CMB}$, $m_e$ and $\\sigma_T$, the CMB\ntemperature at $z=0$, the electron mass at rest and the Thomson\ncross section, $\\theta_e$ represents the dimensionless thermal\nenergy of the ICM, $\\tau_e$ is the electron optical depth. The\nparameters $n_e$, $T_e$, and $P_e$ are the electron number density,\ntemperature, and pressure of the ICM,\n$\\delta_n(x,\\theta_e)=f_n(x)\\theta_e^n\/f(x)$ is the relativistic\ncorrection term that accounts for the thermal energy of the\nelectrons involved in the scattering processes, where\n$f(x)=x[(e^x+1)\/(e^x-1)-4]$ is a dimensionless quantity that\ndescribes the spectral signature of the effect, and the subscript\n$n$ indicates the maximum order of the relativistic correction\n($n=4$ in this work, \\cite{Nozawa95}). The last term of Eq.\n\\ref{eq_SZ} is the kinematic component of the SZ effect, which\ncontains the contribution from the bulk motion of the electron\npopulation with respect to the last scattering surface reference\nframe. For the purpose of this paper, this term is omitted, assuming\nthat it is disentangled from the thermal component by\nmulti-frequency observations, together with the signal from the\nprimary CMB emission.\n\nThe SZ effect is redshift independent and, for this reason, it is\npossible to detect distant clusters without any existing X-ray or\noptical observations. This is the case of the ongoing ground-based\nexperiments such as SPT (\\cite{Ruhl04}), ACT (\\cite{Kosowsky03}),\nand the all sky survey like Planck (\\cite{Planck11a}) or the\nupgraded MITO (\\cite{DePetris07}) and OLIMPO (\\cite{Masi08}) with\nnew spectroscopic capabilities and the proposed 30-m diameter C-CAT\n(\\cite{Sebring06}). However, some assumptions on cluster physics\nstill have to be made in order to directly extract cluster\nobservables.\n\nEstimates of cluster's total mass can be derived by SZ observations\nwhen X-ray or lensing measurements are available or by empirically\ncalibrated scaling relations linking the SZ flux to the total mass\n(e.g. \\cite{Vikhlinin09, Arnaud10, Planck11b, Comis11}). Total mass\ncan also be determined by SZ observations alone when applying\nthermal energy constraints (\\cite{Mroczkowski11}).\n\nTo accurately reproduce the gas inside the cluster, an ICM universal\nmodel is mandatory (e.g. \\cite{Nagai07, Arnaud10}). In this paper we\nconfirm that the simple isothermal \\textit{beta}-model is clearly an\ninappropriate cluster representation for total mass recovery by SZ\nobservations, particularly in the presence of relaxed cool core (CC)\nclusters. These objects show a well studied peaked density profile\nwith a temperature decrement in the core region (\\cite{Jones84}). In\nthe local universe this class of clusters is observationally a\nsignificant percentage of the total cluster population\n(\\cite{Eckert11}). Even if the X-ray estimated CC fraction is biased\nby selection effects in flux-limited samples, recently a 35\\% of\nclusters have picked up in the SZ high signal-to-noise ratio Planck\nearly cluster data-set (\\cite{Planck11c}) are CC clusters. Large\nscatter in mass estimates of CC clusters has been highlighted\npreviously using numerical simulations by Hallman et al. (2006) and\nHallman et al. (2007).\n\nWe investigate the bias on the mass in a limited sample of eight\nnearby ($0.1<z<0.5$) and high-mass ($M>10^{14}$ $M_{\\odot}$) CC\nclusters observed by Chandra. The SZ maps of these clusters, which\nare expressed in thermodynamic temperature units and convolved with\nseveral instrumental beams, are dealt with by applying the\nisothermal \\textit{beta}-model. The total mass is derived in three\ndifferent ways: by assuming hydrostatic equilibrium and a fixed gas\nfraction and by applying a self similar scaling relation. To focus\nonly on the consequences of the employed ICM model, in our analysis\nwe neglect all the contaminants present in the sky by assuming in\nthis way the best situation to recover cluster total mass.\n\nIn Sect. \\ref{sec_ne_Te}, we discuss the electron number density\nradial profile and follow self-similar studies to characterize a\nuniversal electron temperature radial profile of a limited sample of\neight CC clusters observed by Chandra. In Sect. \\ref{sec_MCMC} we\ngenerate maps of the SZ effect in thermodynamic temperature. In\nSect. \\ref{sec_total_mass} we evaluate cluster total mass under\ndifferent sets of assumptions. The bias on the recovered mass is\ndescribed in Sect. \\ref{sec_MB}, which discusses the main\ncontributions. Conclusions are summarized in Sect.\n\\ref{sec_conclusions}.\n\n\\section{Electron number density and temperature\nprofiles}\\label{sec_ne_Te}\n\n\\begin{table*}\n\\begin{minipage}[hbpt]{\\textwidth}\n\\caption{CC galaxy clusters properties used in the analysis to\ngenerate a universal $T_e$ profile for this class of clusters.}\n\\label{table_te} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{l c c c c c c}\n    \\hline\n   \n    Cluster & z & $D_A$ & $r_{500}$\\footnote{\\cite{Morandi07}} & $\\theta_{500}$ & $T_X$\\footnote{Temperature scale  calculated by a weighted mean of Bonamente et al. (2006) data.}& $T_{e0}$\\footnote{Electron temperature of the cluster obtained by fitting the isothermal \\textit{beta}-model to X-ray data (\\cite{Bonamente06}).}\\\\\n    name &  & (Gpc) & (kpc) & (arcsec) & (keV)& (keV)\\\\\n    \\hline\n    \\hline\n    $A1413$ & 0.142 & 0.52 & $1195\\pm232$ & $321\\pm62$ & $6.6\\pm0.6$& $7.3\\pm0.2$\\\\\n    $A1689$ & 0.183 & 0.63 & $1402\\pm260$ & $377\\pm70$ & $8.7\\pm0.9$& $10.0\\pm0.3$\\\\\n    $A1835$ & 0.252 & 0.81 & $1439\\pm414$ & $387\\pm111$ & $10.5\\pm1.0$& $8.4\\pm0.2$\\\\\n    $A2204$ & 0.152 & 0.55 & $1796\\pm320$ & $483\\pm86$ & $11.3\\pm1.8$& $6.5\\pm0.2$\\\\\n    $A2261$ & 0.224 & 0.74 & $1201\\pm168$ & $323\\pm45$ & $7.0\\pm1.0$& $7.2\\pm0.4$\\\\\n    $MS1358.4+6245$ & 0.327 & 0.97 & $1633\\pm885$ & $439\\pm238$ & $8.4\\pm1.1$& $8.3\\pm0.6$\\\\\n    $RXJ1347.5-1145$ & 0.451 & 1.19 & $1734\\pm170$ & $466\\pm46$ & $14.8\\pm1.5$& $13.5\\pm0.5$\\\\\n    $ZW3146$ & 0.291 & 0.90 & $1804\\pm344$ & $485\\pm92$ & $8.7\\pm0.4$& $6.6\\pm0.1$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nA general parametric model for the cluster atmosphere must be\ndefined to forecast the shape of cluster SZ signals in matched\nfilter techniques for detecting clusters in blind surveys. ACT has\ndetected new clusters assuming a two-dimensional Gaussian profile as\nfilter (\\cite{Sehgal10}), while SPT has detected a projected\nspherical \\textit{beta} profile (\\cite{Vanderlinde10}) and Planck a\nuniversal pressure profile (\\cite{Melin11}).\n\nThe approach for determining the total mass cluster can be\ndifferent. High-quality X-ray data allows an accurate modeling of\ncluster morphology, but in the case of low angular resolution and\/or\nlow signal-to-noise ratio a simple isothermal \\textit{beta}-model is\nstill applied (e.g. \\cite{Marriage10, Sayers11}).\n\nIn this work we analyze this model (\\cite{Cavaliere78}), which is\nbased on the very general assumption that the electron temperature\n$T_e$ is constant along the whole considered cluster radial\nextension and that the electron number density follows a spherical\ndistribution as\n\n\\begin{equation}\\label{eq_beta}\nn_{e,ISO}(r)=n_{e0}{\\left(1+\\frac{r^2}{r_c^2}\\right)}^{-\\frac{3}{2}\\beta},\n\\end{equation}\nwhere $n_{e0}$ is the central electron number density, $r_c$ the\ncore radius, and $\\beta$ the power law index. The subscript ISO\nindicates, hereafter, the isothermal \\textit{beta}-model. The proved\ninadequacy of this model is compensated for by the advantage of\nextracting a simple analytic expression for the $y$ parameter along\nthe off-axis angular separation, $\\theta$,\n\n\\begin{equation}\\label{eq_ybeta}\ny_{ISO}(\\theta)=y_0{\\left(1+\\frac{\\theta^2}{\\theta_c^2}\\right)}^{\\frac{1}{2}-\\frac{3}{2}\\beta}\n\\end{equation}\nwhere\n\n\\begin{equation}\\label{eq_y0}\ny_0=n_{e0}\\frac{k_BT_e}{m_ec^2}\\sigma_T\nr_c\\sqrt{\\pi}\\frac{\\Gamma\\left(\\frac{3}{2}\\beta-\\frac{1}{2}\\right)}{\\Gamma\\left(\\frac{3}{2}\\beta\\right)},\n\\end{equation}\nand $\\theta_c=r_c\/D_A$, with $D_A$ the angular diameter distance,\nwhich has been calculated for each cluster by using\n\n\\begin{equation}\nD_A=\\frac{c}{H_0(1+z)}\\int^z_0\\frac{dz'}{E(z')},\n\\end{equation}\nwhere $H_0$ is the Hubble constant and\n$E(z)={\\left[\\Omega_{M}(1+z)^3+\\Omega_{\\Lambda}\\right]}^{1\/2}$. We\nadopt a $\\Lambda CDM$ cosmology with $H_0=70$ km\/s\/Mpc,\n$\\Omega_M=0.3$, and $\\Omega_{\\Lambda}=0.7$.\n\nIt is easy to find clusters that are not relaxed or that display\nstructures that are very difficult to describe with this model.\nTherefore, it is realistic to assume that many newly discovered\nclusters in blind SZ surveys exhibit such significant deviations as\nwell. To explore a particular ICM gas morphology, we focus on CC\nclusters. We started analyzing a small sample of eight objects\nextracted from the Chandra dataset investigated in Bonamente et al.\n(2006).\n\nThe study of a central region, commonly known as the core region,\nhas been challenged by high-resolution numerical simulations\n(\\cite{Navarro95, Borgani04, Kay04, Nagai07, Henning09}). These\nworks lead to an agreement on whether there is a cooling core in the\nvery central denser gas region ($r<0.1r_{500}$) of some clusters, as\nwell as a slower decline in the temperature at large radii\n($r>0.2r_{500}$). As usual we refer to $r_{500}$ as the radius of\nthe cluster that defines a volume with mean density $500$ times the\ncritical density $\\rho_{crit}$ at cluster redshift. The choice of\n$r_{500}$ is motivated by simulation results from Evrard et al.\n(1996) showing the gas within this radius relaxed and in hydrostatic\nequilibrium.\n\nMoreover, many observational studies in X-rays have shown that the\nX-ray surface brightness, hence the underlying density, cannot be\nrepresented correctly by a \\textit{beta} profile. A second component\nshould be added or a peaked central part introduced in order to\nproperly fit the observation. The observed deprojected density\nprofiles are peaked for CC systems and flatter for morphologically\ndisturbed clusters.\n\nThe $n_e$ cluster profile can be described by a double\n\\textit{beta}-model profile\n\n\\begin{eqnarray}\\label{eq_2beta}\nn_{e,CC}(r)&=&n_{e0}\\left[f{\\left(1+\\frac{r^2}{r_{c1}^2}\\right)}^{-\\frac{3}{2}\\beta}+(1-f)\n{\\left(1+\\frac{r^2}{r_{c2}^2}\\right)}^{-\\frac{3}{2}\\beta}\\right],\\nonumber\\\\\n\\end{eqnarray}\nwhere the parameters' data have been taken from Bonamente et al.\n(2006) and adapted to this work to have a symmetric standard\ndeviation (D'Agostini, 2003).\n\nThis distribution is a generalization of a double\n\\textit{beta}-model profile of the electron number density,\ndeveloped by La Roque et al. (2005), but instead using the same\n$\\beta$ parameter for both the central region and the outskirts, as\nin Bonamente et al. (2006). The $r_{c1}=\\theta_{c1}\/D_A$ and\n$r_{c2}=\\theta_{c2}\/D_A$ are the core radii of the inner and outer\ndistributions, and $f$ is a parameter defined between 0 and 1 that\nrepresents how the core region dominates the outer region. These\nparameters, together with $n_{e0}$, are taken from Bonamente et al.\n(2006) and summarized in Table \\ref{table_neCC} for the selected\nclusters.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Contefg1.eps}\\\\\n  \\caption{\\itshape Radial electron temperature profiles of a Chandra-selected sample of CC galaxy clusters (\\cite{Bonamente06}) and the best\n  fit (solid line) of the electron temperature profile with the 1$\\sigma$ error\n  (dotted lines) proposed in this work. Temperatures and radii\n  are expressed in terms of $T_X$ and $r_{500}$, respectively, which\n  are used as scale quantities throughout this work.}\\label{fig_fitTe}\n\\end{figure}\n\nWe describe the temperature profile as\n\n\\begin{eqnarray}\\label{eq_Te_general}\n\\frac{T_{e,CC}(r)}{T_X}=\\left\\{\n\\begin{array}{rl}\nA_1{\\left(\\frac{r}{r_{500}}\\right)}^{m_1} & \\mbox{  for } \\frac{r}{r_{500}}<0.1 \\\\\nA_2{\\left(\\frac{r}{r_{500}}\\right)}^{m_2} & \\mbox{  for }\n\\frac{r}{r_{500}}>0.2\n\\end{array}\n\\right.\n\\end{eqnarray}\nwhere $A_{1,2}$ are determined by fixing the position at which the\ntwo power laws, described by the $m_1$ and $m_2$ parameters,\nintersect each other, as explained in Appendix \\ref{sec_appendix}.\n\nThe search for a universal temperature profile in the cluster halo\nregion has been a target of several works based on observations\n(e.g. \\cite{Markevitch98, DeGrandi02, Zhang04, Vikhlinin05,\nVikhlinin06, Sanderson06, Zhang07, Pratt07, Zhang08}) and\nhydrodynamical simulations (e.g., \\cite{Loken02, Borgani04, Kay04,\nPiffaretti08}).\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Parameters of the single \\textit{beta}-model, estimated\nusing the MCMC procedure described in the text, as best fit of the\nCC cluster maps.}\\label{table_neISO} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llccccc}\n    \\hline\n   \n     Cluster & FWHM fov & $\\Delta T_{SZ0}$ & $\\beta$ & $\\theta_c$  \\\\\n     & (arcmin) & ($\\mu$K) &   & (arcsec)  \\\\\n    \\hline\n    \\hline\n   $Cl_{A1413}$ & 1& $-273\\pm9$ & $0.731\\pm0.006$ &  $66.1\\pm3.0$   \\\\\n    & 4.5& $-224\\pm14$ & $0.743\\pm0.009$ &  $81.7\\pm7.3$   \\\\\n    & 7& $-195\\pm11$ & $0.758\\pm0.011$ &  $96.4\\pm8.0$   \\\\\n    \\hline\n    $Cl_{A1689}$& 1& $-554\\pm8$ & $0.979\\pm0.006$ &  $80.2\\pm1.5$   \\\\\n    & 4.5& $-430\\pm12$ & $0.994\\pm0.004$ &  $93.7\\pm2.2$   \\\\\n    & 7& $-592\\pm84$ & $0.909\\pm0.018$ &  $61.4\\pm8.3$   \\\\\n    \\hline\n    $Cl_{A1835}$& 1& $-709\\pm10$ & $0.951\\pm0.006$ &  $56.8\\pm1.1$   \\\\\n    & 4.5& $-468\\pm29$ & $0.980\\pm0.013$ &  $76.0\\pm5.0$   \\\\\n    & 7& $-818\\pm88$ & $0.870\\pm0.010$ &  $36.0\\pm3.6$   \\\\\n    \\hline\n    $Cl_{A2204}$& 1& $-727\\pm8$ & $0.848\\pm0.003$ &  $68.4\\pm0.9$   \\\\\n    & 4.5& $-551\\pm18$ & $0.865\\pm0.007$ &  $86.0\\pm3.6$   \\\\\n    & 7& $-514\\pm39$ & $0.859\\pm0.017$ &  $86.9\\pm9.0$   \\\\\n    \\hline\n    $Cl_{A2261}$ & 1& $-399\\pm10$ & $0.830\\pm0.006$ &  $52.3\\pm1.7$   \\\\\n    & 4.5& $-273\\pm17$ & $0.849\\pm0.008$ &  $71.6\\pm4.6$   \\\\\n    & 7& $-323\\pm32$ & $0.816\\pm0.013$ &  $55.5\\pm6.4$   \\\\\n    \\hline\n    $Cl_{MS1358.4+6245}$& 1& $-304\\pm12$ & $0.857\\pm0.011$ &  $49.6\\pm2.7$   \\\\\n    & 4.5& $-245\\pm28$ & $0.856\\pm0.013$ &  $56.0\\pm6.1$   \\\\\n    & 7& $-1246\\pm116$ & $0.753\\pm0.009$ &  $8.7\\pm1.2$   \\\\\n    \\hline\n    $Cl_{RXJ1347.5-1145}$& 1& $-1683\\pm15$ & $0.835\\pm0.002$ &  $37.7\\pm0.5$   \\\\\n    & 4.5& $-893\\pm30$ & $0.860\\pm0.005$ &  $62.7\\pm2.5$   \\\\\n    & 7& $-739\\pm41$ & $0.873\\pm0.011$ &  $73.1\\pm5.1$   \\\\\n    \\hline\n    $Cl_{ZW3146}$& 1& $-645\\pm13$ & $0.839\\pm0.005$ &  $42.4\\pm1.1$   \\\\\n    & 4.5& $-404\\pm38$ & $0.857\\pm0.013$ &  $61.3\\pm6.7$   \\\\\n    & 7 & $-585\\pm77$ & $0.801\\pm0.011$ &  $36.5\\pm5.2$   \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe profile, proposed in this paper specifically for CC clusters,\nfollows both the central drop and the outer decline of the gas\ntemperature. The function is formalized in the\n$\\log\\left(r\/r_{500}\\right)-\\log\\left(T_e(r)\/T_X\\right)$ plane on\nwhich the power laws of Eq. \\ref{eq_Te_general} become linear\nfunctions (see Appendix \\ref{sec_appendix} for a complete\ntreatment). To fix the parameters $A_1$ and $A_2$ of the radial\nelectron temperature profile and to confirm the power laws indices\n$m_1$ and $m_2$, we fit a co-adding of Chandra electron temperature\nnormalized to $T_X$ data, of the CC clusters selection with the\nproposed function. $T_X$ represents the average temperature value in\nthe range $(0.1\\div1.0)r_{500}$, and it is used to scale each\ncluster, in order to fit the universal temperature function to the\nmeasured profiles. In Table \\ref{table_te} we report the cluster\nredshift, $z$, and the angular diameter distance, $D_A$. The scale\nradius $r_{500}$ and temperature $T_X$ are also collected. The\nchosen radial range, which is used to calculate the scale\ntemperature $T_X$, corresponds to a cut in the central region\n($r<0.1r_{500}$). Obviously this value cannot be compared easily\nwith results of other works because it strictly depends on the data\nradial extension. In fact, an important source of bias is the\ntemperature definition (\\cite{Vikhlinin06}). Here, we use the\nspectroscopic temperature $T_X$. In Figure \\ref{fig_fitTe} the\ntemperature data of our cluster sample with the best fit are\nplotted. By following Appendix \\ref{sec_appendix}, the resulting\n$T_e$ profile parameters are $A_1=2.41\\pm0.14$, $A_2=0.55\\pm0.10$,\n$m_1=0.38\\pm0.02$ and $m_2=-0.29\\pm0.11$, which univocally define\nthe $T_e(r)$ function. We note that, even if the power law that\ndescribes the outskirts of the cluster temperature distribution\nsuffers larger uncertainties, $m_1$ and $m_2$ are both compatible,\nwithin one standard deviation, with estimates available in the\nliterature. For example, Zhang et al. (2008) find $m_1=0.38\\pm0.04$,\nfor $r<0.2r_{500}$, in agreement with Sanderson et al. (2006), who\nproposes $m_1=0.4$, for $r<0.1r_{500}$. For radii larger then\n$0.2r_{500}$, Zhang et al. (2008) fitted a selected sample of data\nfrom XMM-Newton finding structurally similar behavior to ours with\n$m_2=-0.28\\pm0.19$.\n\n\\section{Pipeline of cluster simulation}\\label{sec_MCMC}\n\nThe analysis reported in this paper can be summarized in the\nfollowing steps:\n\\begin{itemize}\n    \\item construction of an SZ signal distortion profile $\\Delta T_{SZ}\n    (\\theta)$, using the ICM information coming from existing X-ray\n    observations;\n    \\item convolution of the cluster SZ map with several instrumental beam profiles;\n    \\item extraction of the ICM parameters as in the assumptions of the isothermal \\textit{beta}-model;\n    \\item estimation of the cluster total mass $M_{tot}$;\n    \\item calculation of the bias on cluster total mass due to\n    the incorrect description of the ICM.\n\\end{itemize}\n\nIn the first step, we generate angular profiles of the SZ signal\ndistortion $\\Delta T_{SZ} (\\theta)$, assuming an observing frequency\nof 150 GHz. The electron number density $n_e$ and temperature $T_e$\nprofiles are constructed using the equations presented in Sect.\n\\ref{sec_ne_Te}, which we assume to be a good representation of a\ncool core ICM. The angular profile of the comptonization parameter\nis then evaluated by projecting the electron pressure profile on the\nplane orthogonal to the cluster line of sight. The SZ signal is\nobtained using Eq. \\ref{eq_SZ}.\n\nThe $\\Delta T_{SZ} (\\theta)$ profiles are then convolved by\nconsidering three different instrumental beam profiles modeled as a\nGaussian, corresponding to large single dishes (SPT or ACT) with\n$FWHM=1$ arcmin, medium size telescopes (MITO or OLIMPO) with\n$FWHM=4.5$ arcmin, and small apertures (Planck) with $FWHM=7$\narcmin.\n\nThe errors associated to the convolved $\\Delta T_{SZ}(\\theta)$\nprofiles are treated as only due to instrumental noise. An\noptimistic choice of the sensitivity, for all the observatories, is\n6 $\\mu$K\/beam, corresponding to the Planck channel at 143 GHz\n(\\cite{Planck11c}) assuming the necessary integration time on source\nfor the other experiments. Contaminants are not included in the\nstudy since we wish to assess our ability to extract the mass of the\nclusters under ideal conditions.\n\nTo simulate the missing knowledge of X-ray observational results, we\nignore cluster morphology and assume the most general model for it:\nan isothermal \\textit{beta}-model, that expressed in temperature is\n\n\\begin{equation}\\label{eq_DeltaT_beta}\n\\Delta T_{SZ}=\\Delta\nT_{SZ0}{\\left(1+\\frac{\\theta^2}{\\theta_c^2}\\right)}^{\\frac{1}{2}-\\frac{3}{2}\\beta}.\n\\end{equation}\n\nWe apply the Metropolis Hastings (M-H) algorithm Monte Carlo Markov\nChain (MCMC) procedure to fit this equation (after a convolution\nwith the corresponding instrumental beam) on the simulated profiles\nto extract the parameters $\\Delta T_{SZ0}$, $\\beta$ and $\\theta_c$.\nFor each cluster, at a fixed field of view (\\textit{fov}), we\nanalyze the accepted set of parameters derived by the MCMC\nprocedure. The degeneracy among the extracted \\emph{beta}-model\nparameters affects their uncertainties.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\columnwidth]{Contefg2.eps}\\\\\n  \\caption{\\itshape Radial profiles of the electron temperature, $T_e$ (top),\n  number density, $n_e$ (middle) and  pressure, $P_e$ (bottom) for\n  the cluster ZW3146. The red dashed curve describes the CC template\n  while the black solid curve represents the ISO model (the shadowed regions define\n  1$\\sigma$ uncertainties), with parameters extracted by MCMC analysis\n  considering $\\Delta T_{SZ}(\\theta)$ profiles convolved with a beam\n  of 7 arcmin (FWHM).}\\label{fig_ne_Te_pe_profiles1}\n\\end{figure}\n\nWe obtain the \\textit{beta}-model parameter set that is most\nconsistent with the $\\Delta T_{SZ}(\\theta)$ profile, given the\nassumed instrumental noise and beam sizes. All the parameters\nresulting from the MCMC analysis are collected in Table\n\\ref{table_neISO} for each cluster and for each \\textit{fov}. Figure\n\\ref{fig_ne_Te_pe_profiles1} shows the electron temperature (top),\nnumber density (middle), and pressure (bottom) profiles for only the\ncluster ZW3146, as an example, of both the original CC ICM and the\nrecovered ISO model. The errors associated to the curves account for\nthe 1$\\sigma$ uncertainties on the parameters.\n\n\\section{Cluster total mass estimation}\\label{sec_total_mass}\n\nWe want to stress the consequences of the assumptions on the ICM\nphysics when we miss X-ray information. A quantity, such as the\ntotal mass $M_{tot}$, can be biased by a different physical state of\nthe ICM (i.e. mergers or cooling flow mechanisms). In particular we\nestimate the mass for both the ICM discussed templates ($CC$ and\n$ISO$), by using the following different approaches:\n\n\\begin{itemize}\n    \\item hydrostatic equilibrium assumption for the cluster\n    gas (hydrostatic equilibrium, HE);\n    \\item gas fraction independence of cluster physical state (fixed gas fraction, FGF);\n    \\item $M_{tot}-Y$ scaling relation (scaling law,\n    SL), as derived in the standard self-similar collapse scenario.\n\\end{itemize}\n\nThe masses are calculated, in particular, within a fixed integration\nradius $r_{int}$ (aperture radius), which we arbitrarily choose\nequal to the $r_{500}$ values as reported in Morandi et al. (2007,\nsee Table \\ref{table_te}). This choice is motivated by the need to\nfix an aperture radius within which to estimate integrated\nquantities. We point out that $r_{int}$ does not always correspond\nto the same overdensity, due to the different assumed ICM templates.\nIt is clear that, hereafter, results associated to the clusters\nsimulated in this work cannot be considered as describing the true\nICM physics of the observed objects. We choose, however, to maintain\nthe link with the ``native'' cluster in the name ($NAME \\rightarrow\nCl_{NAME}$).\n\n\\subsection{Hydrostatic equilibrium}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Total cluster masses calculated considering the HE and FGF\napproaches.} \\label{table_mass HE_FGF} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llcc}\n    \\hline\n   \n    Cluster & ICM template & $M_{tot,HE}$ & $M_{tot,FGF}$\\footnote{derived by $M_{gas}$ assuming\n$f_{gas}=0.1$}  \\\\\n     & (FWHM fov)  & (10$^{14}$ $M_{\\odot}$)& (10$^{14}$ $M_{\\odot}$)  \\\\\n    \\hline\n    \\hline\n    A1413 & CC   &$3.85\\pm0.77$& $8.73\\pm2.03$\\\\\n    $Cl_{A1413}$& ISO (1')   &$6.59\\pm0.60$&$6.21\\pm0.35$\\\\\n    & ISO (4.5')   &$6.67\\pm0.59$&$6.02\\pm0.59$\\\\\n    & ISO (7')   &$6.66\\pm0.58$&$5.93\\pm0.89$\\\\\n    \\hline\n    A1689 & CC  & $8.96\\pm1.97$&$12.72\\pm2.33$\\\\\n    $Cl_{A1689}$& ISO (1')  &$13.56\\pm1.41$& $9.88\\pm3.12$\\\\\n    & ISO (4.5')   &$13.61\\pm1.33$&$9.14\\pm4.08$\\\\\n    & ISO (7')   &$12.66\\pm1.33$&$8.86\\pm2.01$\\\\\n    \\hline\n    A1835 & CC  & $10.23\\pm2.30$&$12.72\\pm1.18$\\\\\n    $Cl_{A1835}$& ISO (1') &$16.21\\pm1.61$& $10.08\\pm0.33$\\\\\n    & ISO (4.5')   &$16.32\\pm1.55$&$9.13\\pm0.94$\\\\\n    & ISO (7')   &$15.08\\pm1.40$&$7.99\\pm1.44$\\\\\n    \\hline\n    A2204 & CC  & $12.91\\pm3.30$&$14.84\\pm2.96$\\\\\n    $Cl_{A2204}$& ISO (1') &$19.88\\pm3.27$&$10.13\\pm0.24$\\\\\n    & ISO (4.5')   &$20.33\\pm3.05$&$10.87\\pm0.67$\\\\\n    & ISO (7')   &$19.89\\pm2.86$&$10.32\\pm1.54$\\\\\n    \\hline\n    A2261 & CC  & $4.79\\pm1.11$&$10.55\\pm3.48$\\\\\n    $Cl_{A2261}$& ISO (1')  &$7.88\\pm1.09$&$7.80\\pm3.50$\\\\\n     & ISO (4.5')   &$7.84\\pm1.18$&$7.24\\pm1.22$\\\\\n    & ISO (7')   &$7.81\\pm1.08$&$6.94\\pm1.09$\\\\\n   \\hline\n    MS1358.4+6245 & CC  & $8.09\\pm1.75$&$10.23\\pm1.65$\\\\\n    $Cl_{MS1358.4+6245}$& ISO (1')   &$13.27\\pm1.77$&$7.74\\pm0.63$\\\\\n    & ISO (4.5')   &$13.17\\pm1.88$&$7.27\\pm1.22$\\\\\n    & ISO (7')   &$11.90\\pm1.51$&$6.12\\pm1.14$\\\\\n    \\hline\n    RXJ1347.5-1145 & CC  & $14.66\\pm3.02$&$32.34\\pm4.05$\\\\\n    $Cl_{RXJ1347.5-1145}$& ISO (1')  &$24.45\\pm2.49$& $25.04\\pm0.48$\\\\\n    & ISO (4.5')   &$24.47\\pm2.54$&$22.08\\pm1.30$\\\\\n    & ISO (7')   &$24.66\\pm2.49$&$21.07\\pm1.84$\\\\\n    \\hline\n    ZW3146 & CC  & $9.06\\pm1.64$&$17.88\\pm1.05$\\\\\n    $Cl_{ZW3146}$& ISO (1')  &$15.05\\pm0.60$& $13.77\\pm0.58$\\\\\n    & ISO (4.5')   &$15.14\\pm0.67$&$12.72\\pm1.97$\\\\\n    & ISO (7')   &$14.44\\pm0.64$&$12.03\\pm2.61$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe first approach, HE, assumes a spherical symmetry for the\ncluster, so that the hydrostatic equilibrium equation can be written\n(\\cite{Sarazin88}) as\n\n\\begin{equation}\\label{eq_hyd_eq}\n\\frac{dP_{gas}(r)}{dr}=-\\rho_{gas}(r)G\\frac{M_{tot}(<r)}{r^2}\n\\end{equation}\nwhere $M_{tot}(<r)$ is the total cluster mass within radius $r$ and\nunder the ideal gas assumption, $P_{gas}=(\\rho_{gas}k_BT_{gas})\/(\\mu\nm_p)$, with $\\rho_{gas}=\\mu_e m_p n_e$, $\\mu$ and $\\mu_e$ are the\ntotal and electron mean molecular weights (i.e. the mean particle\nmass per electron in units of the proton mass $m_p$), $G$ is the\ngravitational constant and $T_{gas}=T_i=T_e$, the ion and electron\ntemperatures respectively, because the system is assumed to be in\nthermal equilibrium. To adapt this equation to the SZ physics, we\nsubstitute the gas pressure, which accounts for both electrons and\nions, with the simple electron pressure $P_e$, which causes the SZ\neffect, so we have\n\n\\begin{equation}\\label{eq_hyd_eq}\n\\frac{dP_e(r)}{dr}=-\\mu m_p G\\frac{M_{tot}(<r)}{r^2}n_e(r).\n\\end{equation}\nThe total mass can be derived as\n\n\\begin{equation}\\label{eq_hyd_eq_Mtot}\nM_{tot}(<r)=-\\frac{kT_e(r)}{G\\mu\nm_p}r\\left[\\frac{\\partial\\ln(n_e(r))}{\\partial\\ln(r)}+\\frac{\\partial\\ln(kT_e(r))}{\\partial\\ln(r)}\\right],\n\\end{equation}\nwhich reduces, in the simple case of the isothermal\n\\textit{beta}-model, to\n\n\\begin{equation}\\label{eq_hyd_eq_Mtot_iso}\nM_{tot}(<r)=\\frac{3\\beta kT_e}{G\\mu m_p}\\frac{r^3}{r^2_c+r^2}.\n\\end{equation}\nThe previous equations yield $M_{tot,HE,CC}$ and $M_{tot,HE,ISO}$,\nrespectively. If we also calculate the cluster gas mass by\n\n\\begin{equation}\\label{eq_Mgas}\nM_{gas}=\\mu_em_p\\int n_edV,\n\\end{equation}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{Contefg3.eps}\\\\\n  \\caption{\\itshape Radial profiles of the cumulative total (top) and\n  gas (middle) cluster masses and the gas fraction (bottom) for the cluster $Cl_{ZW3146}$. The\n  plots refer to the two ICM templates (CC and ISO) discussed\n  in the text, under the hydrostatic hypothesis. The ISO model parameters have been extracted by MCMC analysis\n  considering $\\Delta T_{SZ}(\\theta)$ profiles convolved with a beam\n  of 7 arcmin (FWHM).}\\label{fig_Mtot_Mgas_log}\n\\end{figure}\nit is easy to estimate the gas fractions, $f_{gas,HE,CC}$, and\n$f_{gas,HE,ISO}$. The plots in Fig. \\ref{fig_Mtot_Mgas_log} refer,\nstill as an example, to the cluster $Cl_{ZW3146}$ and show the\nradial profiles of the cumulative cluster total and gas masses for\nboth the ICM templates and the gas fraction. The $1\\sigma$ lines are\nderived by considering the uncertainties of the ICM parameters.\n\n\\subsection{Fixed gas fraction}\n\nThe second approach, FGF, assumes that the gas fraction, estimated\nat $r_{int}$, is independent of the cluster dynamical state, as\ndeduced by simulations (e.g. \\cite{Rasia06, Lau09}) and as recently\nderived by XMM-Newton and Subaru observations (\\cite{Zhang10}).\nUnder this hypothesis, we can derive the cluster total masses\ndirectly by gas masses as $M_{tot,FGF}=M_{gas}\/f_{gas}$.\n\nBecause the integrated quantities are calculated by referring to a\nfixed radius $r_{int}$, defined in the previous section, it is\nevident that, for each cluster and for each considered ICM model, we\nare dealing with different overdensities. We calculate these\noverdensities by\n\n\\begin{equation}\\label{eq_delta}\n\\Delta_{int}=\\frac{2G}{H_0^2E(z)^2r_{int}^3}M_{tot}(r_{int}).\n\\end{equation}\n\nThe results obtained for the HE and FGF approaches are collected in\nTable \\ref{table_mass HE_FGF}. In order to derive all the quantities\ncorresponding to the ISO template from the parameters obtained with\nthe MCMC procedure, we need to convert the $\\Delta T_{SZ0}$ values\nto central electron number densities. This is possible using\nequation\n\n\\begin{equation}\\label{eq_n_e0}\nn_{e0}=\\frac{\\Delta T_{SZ0}}{T_{CMB}f(x)}\\frac{m_ec^2}{\\sigma_T\n\\theta_cD_A\\sqrt{\\pi}}\\frac{\\Gamma(\\frac{3}{2}\\beta)}{\\Gamma(\\frac{3}{2}\\beta-\\frac{1}{2})}\\frac{1}{T_{e0}},\n\\end{equation}\nwhich is derived by expressing Eq. \\ref{eq_y0} in terms of $\\Delta\nT_{SZ}$. Of course this expression introduces a degeneracy between\ntemperature and density, possibly producing the same $\\Delta\nT_{SZ0}$ value. For this reason the masses presented in this work\nhave been obtained by fixing the electron temperature to the $T_X$\nvalues (Table \\ref{table_te}) and assuming $f_{gas}=0.1$.\n\n\\subsection{$M_{tot}-Y$ scaling relation}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Total mass as derived by applying the $M_{tot}-Y$ scaling\nlaw as in Eq. \\ref{eq_scaling_M-Ycyl}.} \\label{table_SL} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{llcccc}\n    \\hline\n   \n    Cluster &ICM template& $D_A^2Y$ &$f_{gas}$ & $\\Delta_{int}$& $M_{tot,SL}$ \\\\\n    name & (FWHM fov)& (kpc$^2$) & & & (10$^{14}$ $M_{\\odot}$) \\\\\n    \\hline\n    \\hline\n    A1413 & CC& $100\\pm1$ & $0.23\\pm0.05$ & $345\\pm69$& $1.94\\pm0.23$ \\\\\n    $Cl_{A1413}$& ISO (1')& $95\\pm5$ & $0.10\\pm0.01$ & $591\\pm53$& $2.81\\pm0.18$ \\\\\n     & ISO (4.5')& $92\\pm12$ & $0.09\\pm0.01$ & $598\\pm53$& $2.86\\pm0.30$ \\\\\n     & ISO (7')& $90\\pm9$ & $0.09\\pm0.01$ & $597\\pm52$& $2.81\\pm0.26$ \\\\\n    \\hline\n    A1689 & CC& $177\\pm2$ & $0.15\\pm0.03$ & $477\\pm105$& $3.34\\pm0.38$ \\\\\n     $Cl_{A1689}$&ISO (1')& $152\\pm6$ & $0.07\\pm0.01$ & $722\\pm75$& $4.15\\pm0.23$ \\\\\n     &ISO (4.5')& $142\\pm6$ & $0.07\\pm0.01$ & $724\\pm71$& $4.17\\pm0.23$ \\\\\n     &ISO (7')& $139\\pm32$ & $0.07\\pm0.02$ & $674\\pm71$& $4.16\\pm0.82$ \\\\\n    \\hline\n    A1835 & CC& $263\\pm2$ & $0.13\\pm0.03$ & $468\\pm105$& $4.48\\pm0.36$ \\\\\n     $Cl_{A1835}$&ISO (1')& $186\\pm7$ & $0.06\\pm0.01$ & $741\\pm74$& $5.05\\pm0.25$ \\\\\n      &ISO (4.5')& $171\\pm17$ & $0.06\\pm0.01$ & $746\\pm71$& $5.12\\pm0.51$ \\\\\n      &ISO (7')& $152\\pm28$ & $0.05\\pm0.01$ & $689\\pm64$& $5.00\\pm0.85$ \\\\\n    \\hline\n     A2204 & CC& $271\\pm2$ & $0.12\\pm0.04$ & $337\\pm86$& $5.28\\pm0.72$ \\\\\n     $Cl_{A2204}$&ISO (1')& $241\\pm6$ & $0.06\\pm0.01$ & $520\\pm86$& $6.78\\pm0.47$ \\\\\n     &ISO (4.5')& $231\\pm15$ & $0.05\\pm0.01$ & $531\\pm80$& $6.89\\pm0.59$ \\\\\n      &ISO (7')& $222\\pm31$ & $0.05\\pm0.01$ & $520\\pm75$& $6.88\\pm0.99$ \\\\\n    \\hline\n    A2261 & CC& $134\\pm2$ & $0.22\\pm0.06$ & $388\\pm90$& $2.31\\pm0.38$ \\\\\n     $Cl_{A2261}$&ISO (1')& $110\\pm5$ & $0.10\\pm0.02$ & $639\\pm89$& $2.87\\pm0.19$ \\\\\n    &ISO (4.5')& $104\\pm9$ & $0.09\\pm0.02$ & $635\\pm95$& $2.93\\pm0.29$ \\\\\n     &ISO (7')& $99\\pm16$ & $0.09\\pm0.02$ & $633\\pm88$& $2.93\\pm0.48$ \\\\\n    \\hline\n    MS1358.4+6245& CC& $166\\pm4$ & $0.13\\pm0.03$ & $233\\pm50$& $3.82\\pm0.35$ \\\\\n   $Cl_{MS1358.4+6245}$&ISO (1')& $127\\pm11$ & $0.06\\pm0.01$ & $381\\pm51$& $4.67\\pm0.45$ \\\\\n   &ISO (4.5')& $113\\pm21$ & $0.06\\pm0.01$ & $379\\pm54$& $4.56\\pm0.76$ \\\\\n   &ISO (7')& $96\\pm18$ & $0.05\\pm0.01$ & $342\\pm43$& $4.36\\pm0.76$ \\\\\n     \\hline\n   RXJ1347.5-1145 & CC& $1165\\pm4$ & $0.23\\pm0.03$ & $305\\pm63$& $8.09\\pm0.52$ \\\\\n      $Cl_{RXJ1347.5-1145}$&ISO (1')& $719\\pm14$ & $0.10\\pm0.01$ & $509\\pm52$& $8.68\\pm0.40$ \\\\\n      &ISO (4.5')& $654\\pm41$ & $0.09\\pm0.01$ & $509\\pm52$& $8.86\\pm0.60$ \\\\\n      &ISO (7')& $649\\pm66$ & $0.09\\pm0.01$ & $513\\pm52$& $9.11\\pm0.80$ \\\\\n     \\hline\n   ZW3146 & CC& $312\\pm4$ & $0.20\\pm0.04$ & $201\\pm37$& $4.42\\pm0.31$ \\\\\n     $Cl_{ZW3146}$&ISO (1')& $224\\pm11$ & $0.09\\pm0.01$ & $334\\pm13$& $5.21\\pm0.23$ \\\\\n     &ISO (4.5')& $208\\pm38$ & $0.08\\pm0.01$ & $336\\pm15$& $5.31\\pm0.81$ \\\\\n     &ISO (7')& $205\\pm43$& $0.08\\pm0.02$ & $321\\pm14$ & $5.41\\pm0.11$ \\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe total cluster mass can be inferred by applying a self-similar\nrelation that links it to the integrated comptonization parameter.\nThe $M_{tot}-Y$ scaling law is usually calibrated for a fixed\noverdensity value, while in this work we are dealing with masses\ncalculated at different overdensities. For this reason in the\nscaling law, we make the dependence on overdensity explicit. Simple\nconsiderations, based on the assumption that cluster evolution is\ncompletely determined by gravitational processes, lead to an easy\nscaling relation that connects the total mass of a cluster of\ngalaxies, $M_{tot}$, to its temperature, $T_e$, considering an\nisothermal structure for the ICM. Following Kravtsov et al. (2006)\nand Bryan \\& Norman (1998) and keeping the overdensity dependence\nexplicit, we have\n\n\\begin{equation}\\label{eq_scaling_M-T}\nk_BT_e=\\mu m_p{\\left[\\frac{27}{16}\\Delta_{int}\nG^2H_0^2E(z)^2\\right]}^{1\/3}M_{tot}^{2\/3}.\n\\end{equation}\n\nTo connect a spherically integrated quantity (e.g. cluster total\nmass) to one derived by a cylindrical integration (e.g. integrated\nSZ flux), we consider the spherical analogous to the SZ flux,\n$Y_{S}$. This quantity is directly proportional to the cluster total\nmass as\n\n\\begin{equation}\\label{eq_scaling_M-Y-T}\nY_{S}=\\frac{k_B\\sigma_T}{m_ec^2}\\int\nn_eT_edV=\\frac{k_B\\sigma_T}{m_ec^2}T_{mw}\n\\frac{f_{gas}M_{tot}}{\\mu_e m_p}\n\\end{equation}\nwhere $M_{tot}$ is the cluster total mass inside a sphere with\nradius equal to $r_{int}$, and $T_{mw}$ is the gas mass-weighted\nmean temperature.\n\nBy combining the previous equations, we get\n\n\\begin{equation}\\label{eq_scaling_M-Ysph}\nM_{tot}=0.248{\\left[f_{gas}^3E(z)^2\\Delta_{int}\\right]}^{-1\/5}Y_{S}^{3\/5}\\mbox{$10^{14}$\n$M_{\\odot}$}.\n\\end{equation}\nThis means that, by estimating $Y_{S}$, we can easily infer the\ncorresponding  total cluster mass. Furthermore, in order to derive\nthe integrated SZ flux, we need to solve the equation\n\n\\begin{equation}\nY=2\\pi\\int^{\\theta_{int}}_0 y(\\theta)\\theta d\\theta,\n\\end{equation}\nwhere $\\theta_{int}=r_{int}\/D_A$ and $y(\\theta)$ is extracted by the\nsimulated SZ temperature decrement profiles (in the CC case) or\ncalculated directly from Eq. \\ref{eq_ybeta}, by considering the\nparameters as derived from the MCMC analysis in Sect. \\ref{sec_MCMC}\n(in the ISO case). The two defined integrated Comptonization\nparameters ($Y_S$ and $Y$) are connected by the dimensionless\nquantity $C=D_A^2Y\/Y_{S}$.\n\nWe averaged the $C$ factors, assuming different cylindric depths\nequal to 2, 5, and 10 $r_{int}$, obtaining $C_2=1.52\\pm0.07$,\n$C_5=1.85\\pm0.14$, and $C_{10}=2.00\\pm0.19$, respectively,\nconverging to the value in Bonamente et al. (2008).\n\nThe scaling law can be rewritten as\n\n\\begin{equation}\\label{eq_scaling_M-Ycyl}\nM_{tot}=0.248{\\left[f_{gas}^3E(z)^2\nC_{\\infty}^3\\Delta_{int}\\right]}^{-1\/5}(D_A^2Y)^{3\/5}\\mbox{$10^{14}$\n$M_{\\odot}$},\n\\end{equation}\nwhich can directly give an estimation of cluster total mass\n$M_{tot,SL}$. All the quantities included in the scaling law are\nlisted in Table \\ref{table_SL}\n\n\\section{Bias on the total mass}\\label{sec_MB}\n\n\\begin{table*}\n\\begin{minipage}[htbp]{\\textwidth}\n\\caption{Biases on the cluster total mass estimate for three\ndifferent approaches as derived by three different fovs.}\n\\label{table_bias} \\centering\n\\renewcommand{\\footnoterule}{} \n  \\begin{tabular}{l l c c c}\n    \\hline\n   \n    Cluster& FWHM fov & $MB_{HE}$ & $MB_{FGF}$ & $MB_{SL}$ \\\\\n    name& (arcmin) & & &\\\\\n    \\hline\n    \\hline\n    $Cl_{A1413}$& 1 & $0.75\\pm0.31$ & $-0.29\\pm0.19$ & $0.58\\pm0.32$\\\\\n    & 4.5 & $0.79\\pm0.35$ & $-0.30\\pm0.15$ & $0.59\\pm0.34$\\\\\n    & 7 & $0.81\\pm0.31$ & $-0.33\\pm0.17$ & $0.61\\pm0.31$\\\\\n    \\hline\n    $Cl_{A1689}$& 1 & $0.56\\pm0.31$ & $-0.24\\pm0.14$ & $0.43\\pm0.24$\\\\\n    & 4.5 & $0.59\\pm0.33$ & $-0.30\\pm0.15$ & $0.45\\pm0.28$\\\\\n    & 7 & $0.50\\pm0.30$ & $-0.35\\pm0.20$ & $0.55\\pm0.45$\\\\\n    \\hline\n    $Cl_{A1835}$& 1 & $0.63\\pm0.31$ & $-0.26\\pm0.07$ & $0.44\\pm0.16$\\\\\n    & 4.5 & $0.64\\pm0.28$ & $-0.32\\pm0.09$ & $0.47\\pm0.21$\\\\\n    & 7 & $0.51\\pm0.32$ & $-0.41\\pm0.11$ & $0.45\\pm0.29$\\\\\n    \\hline\n    $Cl_{A2204}$& 1 & $0.59\\pm0.32$ & $-0.25\\pm0.15$ & $0.45\\pm0.27$\\\\\n    & 4.5 & $0.62\\pm0.30$ & $-0.29\\pm0.15$ & $0.50\\pm0.27$\\\\\n    & 7 & $0.63\\pm0.30$ & $-0.31\\pm0.18$ & $0.47\\pm0.33$\\\\\n    \\hline\n    $Cl_{A2261}$&1  & $0.75\\pm0.30$ & $-0.23\\pm0.25$ & $0.52\\pm0.36$\\\\\n    & 4.5 & $0.74\\pm0.33$ & $-0.28\\pm0.29$ & $0.53\\pm0.40$\\\\\n    & 7 & $0.69\\pm0.32$ & $-0.30\\pm0.25$ & $0.54\\pm0.40$\\\\\n    \\hline\n    $Cl_{MS1358.4+6245}$& 1 & $0.71\\pm0.31$ & $-0.28\\pm0.12$ & $0.50\\pm0.21$\\\\\n    & 4.5 & $0.69\\pm0.34$ & $-0.32\\pm0.17$ & $0.55\\pm0.34$\\\\\n    & 7 & $0.52\\pm0.29$ & $-0.43\\pm0.14$ & $0.47\\pm0.32$\\\\\n    \\hline\n    $Cl_{RXJ1347.5-1145}$& 1 & $0.79\\pm0.32$ & $-0.26\\pm0.08$ & $0.51\\pm0.17$\\\\\n    & 4.5 & $0.77\\pm0.30$ & $-0.35\\pm0.08$ & $0.56\\pm0.19$\\\\\n    & 7 & $0.82\\pm0.38$ & $-0.38\\pm0.10$ & $0.61\\pm0.26$\\\\\n    \\hline\n    $Cl_{ZW3146}$& 1 & $0.66\\pm0.32$ & $-0.28\\pm0.06$ & $0.47\\pm0.16$\\\\\n    & 4.5 & $0.67\\pm0.32$ & $-0.33\\pm0.12$ & $0.53\\pm0.27$\\\\\n    & 7 & $0.62\\pm0.33$ & $-0.40\\pm0.14$ & $0.57\\pm0.31$\\\\\n    \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table*}\n\nThe results clearly show that a general interpretation of cluster\nphysics, as assumed in a pure self-similar scaling law like\n$M_{tot}-Y$, can infer a wrong estimate of cluster total mass. To\nemphasize  and to quantify this point, we define a mass bias as\n\n\\begin{equation}\\label{eq_MB}\nMB=\\frac{M_{tot,ISO}-M_{tot,CC}}{M_{tot,CC}}.\n\\end{equation}\nThis quantifies the difference between the SZ derived mass, as\nresults from the isothermal \\textit{beta}-model assumption (ISO) and\nthe X-ray derived mass (CC), which we deduce from the HE, FGF, and\nSL approaches. We notice no significant bias dependence on the beam\nsize used to convolve the SZ signals, under the assumption of the\npresence of instrumental noise alone. Therefore, all plots refers to\nresults obtained with a \\textit{fov} of 7 arcmin FWHM, considering\n$T_X$ values as in Table \\ref{table_te}. We calculate the mass bias\nfor all the analyzed approaches.\n\nSince the $\\Delta T_{SZ0}$ parameter does not give us unique\ninformation on the physics of the ICM, we have to study different\npairs of the parameters $n_{e0}$ and $T_x$. Thus, we select values\nof $T_x$ that describe a reliable range of electron temperatures\n(from 5 keV to 15 keV) and calculate the corresponding $n_{e0}$. The\nmass biases are plotted in Figure \\ref{fig_bias}a, where the three\ncases (HE, FGF, and SL) are shown all together in the plots\ncorresponding to clusters $Cl_{A1689}$, $Cl_{A2204}$, and\n$Cl_{RXJ1347.5-1145}$, representing a wide span in the electron\ntemperature values. Table \\ref{table_bias} lists the mass biases, as\nestimated at the X-ray derived electron temperature ($T_X$ in Table\n\\ref{table_te}).\n\nA check of the goodness of the procedure was done by simulating the\nobservation of a cluster having an isothermal \\textit{beta}-model\n$\\Delta T_{SZ}(\\theta)$ profile instead of a CC cluster. For this\nvalidation procedure, named \\textit{Test}, we used the same\nanalytical expression of the SZ signal to extract the parameters.\nThe assumed electron temperature is the one obtained by Bonamente et\nal. (2006) and reported in the last column in Table \\ref{table_te}.\nIn Figure \\ref{fig_bias}b we represent the cluster total mass bias\nwith these assumptions. While the bias is always zero in the HE\napproach, for the FGF and SL cases, as expected, we notice a mass\nbias dependence on $T_e$. Due to the degeneracy between the electron\ntemperature and number density, the mass bias varies with $n_e$. In\nall cases, it is worth noting that these biases nullify for electron\ntemperatures equal to $T_{e0}$ values, thus proving that the method\nis not affected by systematics.\n\nThe degeneracy between the ICM parameters, resulting from yielding\nthe same SZ signal, produces different trends on FGF and SL biases\nwith electron temperature. For increasing values of electron\ntemperature it underestimates the $M_{gas}$. Considering the FGF\napproach, this implies an underestimation of the total mass, too.\nFor the SL method this would instead produce an increasing trend in\nthe mass bias with temperature because an $M_{gas}$ underestimation\ncorresponds to a decreasing gas fraction.\n\nThe net result is a mass bias that is always different from zero for\nthe HE approach, as well as for the FGF and SL ones, unless it is\nfor an electron temperature suitable to correctly modeling the CC\ncluster as an isothermal one but generally different from the true\nICM temperature (see the crossovers in mass biases in Figure\n\\ref{fig_bias}).\n\nThe mass bias reflects a combination of different contributions\n(\\cite{Piffaretti08}). In particular for this work, the bias can be\nreduced to an \\textit{integration bias} and a \\textit{modeling\nbias}, which are estimated and discussed separately. The\n\\textit{integration bias} refers to the mass bias due to the\ndifferent integration ranges along the line of sight. An infinite\nintegration range is considered for the analytic $\\Delta T_{SZ}$\nexpression, while a finite range is assumed in the numerical\nprojection method, which is limited by the missing knowledge of the\nelectron pressure profile outside the region defined by the data.\nThe \\textit{integration bias} that can be evaluated by still\napplying the whole procedure to an ISO template (Figure\n\\ref{fig_bias}c). The \\textit{modeling bias} reflects the dependence\nof the bias on the assumed model for the ICM. In order to highlight\nit and to cancel the \\textit{integration bias}, we apply the\nprocedure to the CC cluster, recovered as an isothermal one. In this\ncase both the cluster signals (CC mock data and ISO recovered data)\nare obtained by projecting the three-dimensional electron pressure\nprofile. For disentangling the \\textit{integration bias}, the ISO\nprofile is also integrated over a limited range along the line of\nsight. The HE bias ranges between $50\\%$ and $80\\%$, always implying\na mass overestimation that is independent of the electron\ntemperature.\n\n\\begin{figure*}\n\\centering\n  \\begin{flushleft}\\hspace{2.5cm} \\Large{(a)} \\hspace{4.6cm} \\Large{(b)} \\hspace{3.0cm} \\Large{(c)}  \\hspace{3.0cm} \\Large{(d)}\\\\\\end{flushleft}\n  \\includegraphics[width=\\textwidth]{Contefg4.eps}\n  \\caption{\\itshape Total mass biases estimated\n  with different approaches, HE (black), FGF (red), SL (blue), for three representative\n  clusters of our sample (from top to bottom: $Cl_{A1689}$,\n  $Cl_{A2204}$, and\n  $Cl_{RXJ1347.5-1145}$). The ISO parameters used for the bias\n  estimation are associated to SZ signals convolved with a beam\n  of 7 arcmin (FWHM). In panel (a) the total mass bias is shown, while in\n  panel (b) the reliability of bias estimation has been tested. The two\n  bias contributions are disentangled: integration (c) and modeling (d)\n  biases. Assuming no prior for $T_e$, the mass biases are plotted\n  versus a reliable range of electron temperatures (5-15 keV).}\\label{fig_bias}\n\\end{figure*}\n\n\\section{Conclusions}\\label{sec_conclusions}\n\nWe studied the bias that affects the estimate of cluster total mass\nby SZ observations when an isothermal \\textit{beta}-model is assumed\nto describe the ICM physical properties, specifically in the case of\nCC clusters when X-ray and\/or lensing information is missing.\n\nIt is well known that, rather than the central Comptonization\nparameter $y_0$, affected by the choice of cluster profile modeling,\nan integrated quantity, like the parameter $Y$, appears to be a more\nrobust mass proxy. Nevertheless, we have shown that CC clusters can\ngenerate observed $y$ maps in which the ICM morphology could still\nbe substantially hidden, even for the current most sensitive\nexperiments operating from the largest available mm\/submm\ntelescopes.\n\nWhile a general assumption of cluster morphology is efficient at\ndetecting them in blind SZ maps, the possible mismatch with the\nactual cluster profile results in a mass bias. In fact simple ICM\nmodels, like the isothermal \\textit{beta}-model, applied to SZ\nobservations can wrongly estimate cluster total mass in the presence\nof peculiar ICM dynamics as in CC clusters, which are studied in the\ncurrent analysis, and mergers.\n\nWe analyzed the mass bias as derived in a limited sample of eight CC\nclusters observed by Chandra, both nearby ($0.1<z<0.5$) and with\nhigh mass ($M>10^{14}$ $M_{\\odot}$). Under the assumption of an\nisothermal \\textit{beta}-model, the cluster total mass was derived\napplying three different approaches: the hydrostatic equilibrium\nequation, a fixed gas fraction, and a self-similar $M_{tot}-Y$\nrelation.\n\nAssuming we had no information from X-ray observations, we reported\nthe bias on the derived total mass as dependent on electron gas\ntemperature. Only in the case of hydrostatic equilibrium does this\nbias appear almost constant for the considered clusters in the range\nof 50-80 \\%. Incidentally, we notice that an electron temperature\nvalue exists for which the FGF and SL mass biases vanish. This could\nbe the only case in which a simple isothermal \\textit{beta}-model\naccurately reproduces the mass of CC clusters.\n\nThe large biases on total cluster mass recovery in CC clusters\nrepresent another reason to definitely discard the isothermal\n\\textit{beta}-model for this purpose and to firmly support more\nsophisticated models, with universal pressure profiles (e.g.\n\\cite{Arnaud10}). This is already employed for modeling cluster\natmospheres in almost all the present blind-survey data reduction\n(SPT and Planck), and it is planned in the next future for ACT\nobservations.\n\n\\begin{acknowledgements}\nPart of this work has been supported by funding from Ateneo\n2009-C26A09FTJ7. We thank S. Borgani and the anonymous referee for\ntheir useful comments and suggestions.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe {\\it mapping class group} of a compact connected, possibly with boundary, surface  \n$S$, $\\mod_S$, is defined as \nthe group of isotopy\nclasses of homeomorphisms $S\\rightarrow S$. The algebraic investigation\nof that group goes as far back as Dehn and Nielsen, who first showed that the mapping class\ngroup of an orientable surface is embedded into the group of outer automorphisms of the\nfundamental group of $S$, ${\\rm Out}(\\pi_1(S))$. This embedding was later extended to a wider \nclass of 2-orbifolds. Maclachlan and Harvey \\cite{mh} proved that $\\mod_S\\le {\\rm Out}(\\pi_1(S))$ \nfor a surface $S$ obtained from a compact surface of genus $k$ by deleting $s$ points,\n$t$ discs and $r$ marked points. Recently, Fujiwara \\cite{fujiwara} extended the above \nembedding to the case of hyperbolic 2-orbifolds (orientable or non-orientable) with finite volume. \n\nGrossman \\cite{Gr} was the first to show that if $S_k$ is a compact\norientable surface of genus $k$, then $\\mod_{S_k}$\nis residually finite. Recently, Allenby, Kim and Tang \\cite{akt} extended the result\nof Grossman for non-orientable closed surfaces. Ivanov \\cite{ivanov} gave a geometric proof\nof the result of Grossman which seems that can easily be extended to the non-orientable \ncase. The proofs of both results in \\cite{Gr} and \\cite{akt} are using\ncombinatorial group theory arguments. One of the key ingredients of both results\nis the fact that the group of conjugating automorphisms of the fundamental group of the surface\ncoincides with the group of its inner automorphisms. If $G$ is a group, then an automorphism \n$f$ of $G$ is called a {\\it conjugating automorphism} if  $f(g)$ is a conjugate of $g$ for \nevery $g\\in G$. The conjugating automorphisms of a group $G$ form a  subgroup of the group \nof  automorphisms of $G$, ${\\rm Aut}(G)$, which\nwe denote by ${\\rm Conj}(G)$. Moreover, ${\\rm Conj}(G)$ is normal in ${\\rm Aut}(G)$ and \ncontains the subgroup of inner automorphisms of $G$, ${\\rm Inn}(G)$. \n\n\nIn the present note we investigate the residual finiteness of ${\\rm Out}(G)$ for hyperbolic\ngroups $G$. Our study is, as well, based on the investigation of the group of conjugating \nautomorphisms of $G$.\nIn fact, using the powerful geometric ideas developed by Paulin \\cite{Pau} and \nBestvina \\cite{bestvina} in the context of ultralimits of metric spaces, \nwe show that in a hyperbolic group $G$, the quotient group ${\\rm Conj}(G)\/{\\rm Inn}(G)$ is\nalways finite. The main theorem shows that every virtually torsion-free subgroup of the\nouter automorphism group of a conjugacy separable, hyperbolic group is residually\nfinite.\n\nAs a result, we obtain the residual finiteness of the outer automorphism group \nof finitely generated Fuchsian groups and of free-by-finite groups. Consequently the \nmapping class groups of hyperbolic 2-orbifolds with finite volume are \nresidually finite. Hence, we retrieve the results of Grossman and of\nAllenby, Kim and Tang as special cases of our corollaries.\n\nIn an addendum at the end of the paper we show how to generalize the main\nresults for relatively hyperbolic groups. \n\n\n\\section{Main Results}\n\nThe next lemma is what really proved in ~\\cite[Theorem 1]{Gr}. We\nreproduce here the proof for the reader's convenience.\n\n\\begin{lem}\\label{lem:Gr} Let $G$ be a finitely generated, conjugacy separable\ngroup. Then the quotient group ${\\rm Aut}(G)\/{\\rm Conj}(G)$ is residually\nfinite.\n\\end{lem}\n\n\\begin{proof}\nSince $G$ is finitely generated, we can find a family\n$(K_{i})_{i\\in I}$ of characteristics subgroups of finite index in\n$G$ such that for each normal subgroup $N$ of finite index in $G$\nthere is a subgroup $K_{i}$ in the above family, contained in $N$. \nEvery automorphism $f$ of $G$ induces an\nautomorphism $f_{i}$ of $G\/K_{i}$ which acts as a permutation on the\nconjugacy classes of $G\/K_{i}$. Therefore, for each $i\\in I$ we have a\nhomomorphism $\\pi_{i}:{\\rm Aut}(G)\\rightarrow S_{i}$, where $S_{i}$ denotes the\npermutation group on the set of conjugacy classes of the finite group\n$G\/K_{i}$. Since each conjugating automorphism preserves conjugacy\nclasses, the group ${\\rm Conj}(G)$ is contained in the intersection\n$\\bigcap _{i\\in I}$ Ker$(\\pi_{i})$ of the kernels of the $\\pi_{i}$'s.\nNow let $f\\in \\bigcap _{i\\in I}$ Ker$(\\pi_{i})$. Then for every $g\\in G$, \nthe elements $f(g)$ and $g$ are conjugate in each quotient\n$G\/K_{i}$. In particular, the elements $f(g)$ and $g$ are\nconjugate in each finite quotient of $G$ and thus they are\nconjugate in $G$, by the conjugacy separability of $G$. This means\nthat $f$ is a conjugating automorphism of $G$ and hence\n${\\rm Conj}(G)=\\bigcap_{i\\in I}$ Ker$(\\pi_{i})$.\n\\end{proof}\n\nThe following material on ultralimits of metric spaces can be\nfound in detail in the paper of Kapovich and Leeb \\cite{kl}.\n\nA $\\textsl{filter}$ on the set of natural numbers $\\mathbb{N}$ is\na non-empty set $\\omega$ of subsets of $\\mathbb{N}$ with the\nfollowing properties:\n\n\\begin{enumerate}\n\\item The empty set is not contained in $\\omega$.\n\\item $\\omega$ is closed under finite intersections.\n\\item If $A\\in \\omega$ and $A\\subseteq B$, then $B\\in \\omega$.\n\\end{enumerate}\n\nA filter $\\omega$ is called $\\textsl{ultrafilter}$ if it is\nmaximal. An ultrafilter is called \\textsl{non-principal} if it\ncontains the complements of the finite subsets of $\\mathbb{N}$.\nLet $\\omega$ be an ultrafilter on $\\mathbb{N}$ and\n$a:\\mathbb{N}\\rightarrow [0,\\infty)$ a sequence of non-negative real\nnumbers. Then there is a unique point $l$, which we denote by\n$\\textrm{lim}_{\\omega}a_{i}$, in the one-point compactification\n$[0,\\infty]$ of $[0,\\infty)$ such that for each neighborhood $U$\nof $l$, the inverse image $a^{-1}(U)$ of $U$ under $a$ is\ncontained in $\\omega$.\n\nLet $(X_{i},d_{i},x_{i}^{0})_{i\\in \\mathbb{N}}$ be a sequence of\nbased metric spaces and let $\\omega$ be an ultrafilter on\n$\\mathbb{N}$. On the subspace $Y$ of $\\prod_{i\\in\n\\mathbb{N}}X_{i}$ consisting of all sequences $(x_{i})$ for which\n$\\textrm{lim}_{\\omega}d_{i}(x_{i},x_{i}^{0})<\\infty$, we define a\npseudo-metric $d_{\\omega}$ by\n$d_{\\omega}\\big((x_{i}),(y_{i})\\big)=\\textrm{lim}_{\\omega}\\big(d_{i}(x_{i},y_{i})\\big)$.\nThe \\textsl{based ultralimit}\n$(X_{\\omega},d_{\\omega},x_{\\omega})$, where\n$x_{\\omega}=(x_{i}^{0})_{i\\in \\mathbb{N}}$, is the associated\nmetric space.\n\n\\bigskip\n\nLet $X$ be a geodesic metric space and $\\delta$ a non-negative real number.\nThe space $X$ is called {\\it  $\\delta$-hyperbolic\\\/} if for every triangle $\\Delta\\subset X$\nwith geodesic sides, each side is contained in the $\\delta$-neighbourhood of the\nunion of the two other sides.\n\nLet $G$ be a finitely generated group and let $X=X(G,S)$ be the Cayley graph of\n$G$ with respect to a finite generating set $S$ for $G$, closed under inverses.\nThe group $G$ is called {\\it $\\delta$-hyperbolic\\\/} if $X$ is a $\\delta$-hyperbolic\nspace with respect to the word metric. \n\n\nIn the next lemma we use various results concerning actions of groups on $\\mathbb R$-trees.\nFor details, we refer the reader \nto the paper of Morgan and Shalen \\cite{ms}.\n\nThe key point of Lemma \\ref{lem:finite} is the following statement.\nAssume that an action of a hyperbolic group $G$ on a real tree $Y$ is obtained \nas a limit of a sequence of actions of $G$ on its Cayley graph $X$, where each \naction in the sequence is the natural action of $G$ on $X$ twisted by an \nautomorphism of $G$. If the conjugacy class of an element $g$ in $G$ is periodic \nunder these automorphisms, then $g$ is elliptic when acting on the limit tree $Y$. \n\n\nAs it was pointed out by the referee, the above statement seems\nto be well known to the experts (see \\cite{bestvina2}). Nonetheless, the\nauthors failed to track down a reference for its proof.  So, a proof of it,\nis included in the lemma for the reader's convenience and completeness. \n\n\\begin{lem}\\label{lem:finite} Let $G$ be a hyperbolic group. Then the group \n${\\rm Inn}(G)$ of inner automorphisms of $G$ is of\nfinite index in ${\\rm Conj}(G)$.\n\\end{lem}\n\n\\begin{proof} Suppose to the contrary that ${\\rm Inn}(G)$ is of infinite\nindex in ${\\rm Conj}(G)$. Fix an infinite sequence\n$f_{1},f_{2},\\dots,f_{n},\\dots$ of conjugating automorphisms of\n$G$ representing pairwise distinct cosets of ${\\rm Inn}(G)$ in ${\\rm Conj}(G)$. We\napply the method of Bestvina and Paulin, using ultrafilters, to\nconstruct an $\\mathbb{R}$-tree on which $G$ acts by isometries.\n\nLet $X=X(G,S)$ be the Cayley graph of $G$ with respect to a finite\ngenerating set $S$ closed under inverses. Then $X$ with the associated\nword metric $d$ is a $\\delta$-hyperbolic metric space for some $\\delta\\ge 0$. \nEach $f_{i}$ gives an\naction $\\rho_{i}$ by isometries of $G$ on $X$ by\n$\\rho_i(g,x)=f_{i}(g)x$. The outer automorphism group of a\nvirtually cyclic group is finite. Thus, we may assume that $G$ is\nnon-elementary. In that case the action of $G$ on the boundary\n$\\partial X$ of $X$ is non-trivial and Lemma 2.1 in ~\\cite{Pau}\napplies to any action $\\rho_{i}$, yielding a sequence $x_{i}^{0}$\nof elements of $X$ such that\n\\begin{equation} \\label{eq:min} \\underset{g\\in\nS}{\\textrm{max}}\\,d\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)\\leq\n\\underset{g\\in S}{\\textrm{max}}\\,d\\big(x,f_{i}(g)x\\big)\n\\end{equation}\nfor all $x$ in $X$. Let $\\lambda_{i}=\\underset{g\\in\nS}{\\textrm{max}}\\,d\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)$. Then\n\\begin{equation} \\label{eq:trineq}\nd\\big(x_{i}^{0},f_{i}(g)x_{i}^{0}\\big)\\leq \\lambda_{i} \\|g\\|\n\\end{equation}\nfor all $g$ in $G$, where $\\|g\\|$ denotes the word-length of $g$\nwith respect to $S$. If the sequence $(\\lambda_{i})$ contains a\nbounded subsequence, then the argument in ~\\cite[Case 1, p.\n338]{Pau} shows that there are indices $i$ and $j$ with $i\\neq j$\nsuch that the automorphisms $f_{i}$ and $f_{j}$ differ by an inner automorphism\nof $G$ and consequently they give rise to the same coset of ${\\rm Inn}(G)$, \ncontradicting the choice of the $f_{i}$. It\nfollows that\n$\\underset{i\\rightarrow\\infty}{\\textrm{lim}}\\lambda_{i}=\\infty$.\nWe consider the sequence $(X_{i},d_{i},x_{i}^{0})$ of based metric\nspaces, where $X_{i}=X$ and $d_{i}=\\frac{d}{\\lambda_{i}}$. Note\nthat the space $(X_{i},d_{i},x_{i}^{0})$ is\n$\\frac{\\delta}{\\lambda_{i}}$-hyperbolic for all $i$.\nFor any non-principal\nultrafilter $\\omega$ on $\\mathbb{N}$, let\n$(X_{\\omega},d_{\\omega},x_{\\omega})$ be the corresponding based\nultralimit. The fact that the distance $d_{\\omega}(x,y)$ of two\npoints $x=(x_{i})$ and $y=(y_{i})$ of $X_{\\omega}$ is approximated\nby the distances $d_{i}(x_{i},y_{i})$ for infinitely many $i$,\nimplies that $X_{\\omega}$ is a $0$-hyperbolic space (i.e., an\n$\\mathbb{R}$-tree), since $\\underset{i\\rightarrow\n\\infty}{\\textrm{lim}}\\frac{\\delta}{\\lambda_{i}}=0$. The action of\n$G$ on $X_{\\omega}$ is given by\n$g\\cdot(x_{i})=\\big(f_{i}(g)x_{i}\\big)$. Inequality\n~(\\ref{eq:trineq}) ensures that the action is well-defined. We will\nshow that the action of $G$ on $X_{\\omega}$ is trivial (i.e. there\nis a global fixed point).\n\nLet $g$ be an element of $G$ which acts as a hyperbolic isometry\non $X_{\\omega}$, and let $\\t_{\\omega}(g)$ denote its translation\nlength. Fix $x=(x_{i})\\in X_{\\omega}$ such that $x$ lies on the axis of $g$. Then\n$$\\t_{\\omega}(g)=d_{\\omega}(gx,x)=\\left(\\textrm{lim}_{\\omega}\nd_i(f_i(g)x_i,x_i)=\\right)\n\\displaystyle\\frac{d_{\\omega}(g^{n}x,x)}{n}$$ for all positive integers $n$.\nIn particular,\n$d_{\\omega}(gx,x)=\\displaystyle\\frac{d_{\\omega}(g^{2}x,x)}{2}$ and thus\n\\begin{equation}\\label{eq:2}\n\\textrm{lim}_{\\omega}\\Big(2d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)-d_{i}\n\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\Big)\n=0.\n\\end{equation}\nFor each $i$, we fix an element $y_{i}$ of $X$ on which the\ndisplacement function of $f_{i}(g)$ attains its minimum\n$\\t(f_{i}(g))$, i.e.\n$\\t(f_{i}(g))=d\\big(f_{i}(g)y_{i},y_{i}\\big)=\\textrm{inf\\,}\\{d\\big(f_{i}(g)y,y\\big)|\\,y\\in\nX\\}$.\\footnote{The reader should not confuse this with the algebraic translation length of \nthe elements of a group $G$.} \nSince $X$ is a $\\delta$-hyperbolic space, there is a non-negative\nconstant $K(\\delta)$ depending only on $\\delta$ such that\n\\begin{equation}\\label{ineq:a}\nd\\big(f_{i}(g)x_{i},x_{i}\\big)\\geq\n2d(x_{i},y_{i})+\\t(f_{i}(g))-K(\\delta).\n\\end{equation}\nThen,\n\\begin{equation} \\label{ineq:b}\n\\begin{array}{ccl} A & = &\n2d\\big(f_{i}(g)x_{i},x_{i}\\big)-d\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\\\\n & \\geq &\n 4d(x_{i},y_{i})+2\\t(f_{i}(g))-2K(\\delta)-d\\big(f_{i}(g)^{2}x_{i},x_{i}\\big)\\\\\n & \\geq & 4d(x_{i},y_{i})+2\\t(f_{i}(g))-2K(\\delta)-2d(x_{i},y_{i})-2\\t(f_{i}(g))\\\\\n & = & 2d(x_{i},y_{i})-2K(\\delta)\\,,\n \\end{array}\\end{equation}\nwhere the second inequality follows from the triangle one. Working\nin a similar way, we see that\n\\begin{equation}\n2d(x_{i},y_{i})-K(\\delta)\\leq\nd\\big(f_{i}(g)x_{i},x_{i}\\big)-\\t(f_{i}(g))\\leq 2d(x_{i},y_{i})\n\\end{equation}\nand hence we can say that \n\\begin{equation}\\label{ineq:d}\n\\big|d\\big(f_{i}(g)x_{i},x_{i}\\big)-\\t(f_{i}(g))\\big|\\leq\n2d(x_{i},y_{i})+K(\\delta).\n\\end{equation}\nConsequently,\n\\[\\begin{array}{ccl}\n\\Big|\\t_{\\omega}(g)-\\frac{\\t(f_{i}(g))}{\\lambda_{i}}\\Big| & \\leq &\n\\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+\n\\Big|d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)-\n\\frac{\\t(f_{i}(g))}{\\lambda_{i}}\\Big| \\\\\n & \\leq &\n \\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+2\\displaystyle\\frac{d(x_{i},y_{i})}\n{\\lambda_{i}}+\\frac{K(\\delta)}{\\lambda_{i}}\\\\\n & \\leq &\n \\Big|\\t_{\\omega}(g)-d_{i}\\big(f_{i}(g)x_{i},x_{i}\\big)\\Big|+\\displaystyle\\frac{|A|}{\\lambda_{i}}+\n 3\\frac{K(\\delta)}{\\lambda_{i}}\\,,\\\\\n\\end{array}\\]\nwhere the second inequality follows from ~(\\ref{ineq:d}) and the third\none from ~(\\ref{ineq:b}). The $\\omega$-limit of each term in\nthe right-hand side of the above inequality is $0$ (for the second term\nsee ~(\\ref{eq:2})). Therefore\n\\[\\t_{\\omega}(g)=\\textrm{lim}_{\\omega}\\frac{\\t(f_{i}(g))}{\\lambda_{i}}=0\\,,\\]\nsince $\\t(f_{i}(g))=\\t(g)$ for all $i$ ($f_{i}$ being a conjugating\nautomorphism). Hence, each element $g$ of the finitely generated\ngroup $G$ acts as an elliptic isometry on $X_{\\omega}$. This means\nthat the action of $G$ on $X_{\\omega}$ has a global fixed point, say $z=(z_i)$. \nIt follows that for every $\\varepsilon>0$ and every\nfinite subset $F$ of $G$, there is an $\\Omega\\in \\omega$ such that\n\\[d_{i}\\big(z_{i},f_{i}(g)z_{i}\\big)=\\frac{d(z_{i},f_{i}(g)z_{i})}{\\lambda_{i}}<\n\\varepsilon,\\] for all $i\\in \\Omega$ and $g\\in F$. This\ncontradicts to the minimality of $\\lambda_{i}$ (see ~(\\ref{eq:min})).\n\\end{proof}\n\nWe should mention here that this is the best possible result in that generality.\nIndeed, as shown by Burnside \\cite{burnside} and subsequently by several other\nauthors (see also Sah \\cite{sah}), there are finite groups that posses non-trivial \nouter conjugating automorphisms\n(known also as outer, class preserving automorphisms). Therefore one can \neasily construct free-by-finite groups with outer conjugating automorphisms by\nconsidering the direct product of the above mentioned finite groups by free groups.  \n\nWe are now able to show our main theorem.\n\n\\begin{thm}\\label{theorem}\nLet $G$ be a conjugacy separable, hyperbolic group. Then each\nvirtually torsion-free subgroup of the outer automorphism group\n${\\rm Out}(G)$ of $G$ is residually finite.\n\\end{thm}\n\\begin{proof} It suffices to show that each torsion-free subgroup $H$\nof ${\\rm Out}(G)$ is residually finite. We consider the following short\nexact sequence\n\\[1 \\rightarrow {\\rm Conj}(G)\\big\/{\\rm Inn}(G)\\stackrel{i}{\\rightarrow} {\\rm Aut}(G)\\big\/{\\rm Inn}(G)\\stackrel{\\pi}{\\rightarrow} \n{\\rm Aut}(G)\\big\/{\\rm Conj}(G) \\rightarrow 1\\,.\\]\n By Lemma ~\\ref{lem:finite} the first term is finite. This implies that the \nrestriction of $\\pi$ on $H$\nis a monomorphism. It follows that $H$ is residually finite being\nisomorphic to a subgroup of ${\\rm Aut}(G)\\big\/{\\rm Conj}(G)$, which is\nresidually finite by Lemma ~\\ref{lem:Gr}.\n\\end{proof}\n\n\nIn view of the above theorem, given a group $G$ it is natural to\nseek conditions under which the outer automorphism group ${\\rm Out}(G)$\nof $G$ is virtually torsion-free. Recently, Guirardel and Levitt ~\\cite{gl} have shown that the\nouter automorphism group of a hyperbolic group $G$ is virtually\ntorsion-free, provided that $G$ is virtually torsion-free. Thus, by\nTheorem ~\\ref{theorem}, the outer automorphism group of a\nconjugacy separable hyperbolic group is residually finite, since \na residually finite hyperbolic group is virtually torsion-free. \n\nThe next lemma is more or less\nknown (see ~\\cite{gl,Mc}).\n\n\n\\begin{lem}\\label{lem:vir} Let $G$ be a finitely generated group containing a\nnormal subgroup $N$ of finite index whose center is trivial. If\n${\\rm Out}(N)$ is virtually torsion-free, then so is ${\\rm Out}(G)$.\n\\end{lem}\n\n\\begin{proof} Let ${\\rm Aut}_N(G)$ denote the subgroup of ${\\rm Aut}(G)$\nconsisting of those automorphisms of $G$ which fix $N$ and induce\nthe identity on $G\/N$. The restriction map $\\phi: {\\rm Aut}_N(G) \\rightarrow {\\rm Aut}(N)$ is an\ninjection. Indeed, let $g\\in G$ and $f\\in {\\rm Aut}_N(G)$. Then\n$f(g)=gh_{g}$ for some $h_{g}\\in N$. Suppose now that $f$ is in\nthe kernel of the restriction map. Then for each $h$ in $N$ we\nhave $ghg^{-1}=f(ghg^{-1})=gh_{g}hh_{g}^{-1}g^{-1}$. This implies\nthat $h_{g}$ is in the center of $N$, which is trivial, and\ntherefore $f$ is the identity. From the injectivity of $\\phi$ we\nget $\\phi\\big({\\rm Inn}_N(G)\\big)={\\rm Inn}(N)$, where ${\\rm Inn}_N(G)$ denotes\nthe (normal) subgroup of ${\\rm Inn}(G)$ consisting of all inner\nautomorphisms of $G$ induced by elements of $N$. We conclude that\nthe quotient group ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ embeds into ${\\rm Out}(N)$. In\nparticular, ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is virtually torsion-free.\n\nNow the restriction of the natural projection $\\pi:{\\rm Aut}(G) \\rightarrow\n{\\rm Out}(G)$ to ${\\rm Aut}_N(G)$ induces a map $\\tilde{\\pi}:{\\rm Aut}_N(G)\/{\\rm Inn}_N(G) \n\\rightarrow {\\rm Out}(G)$. The kernel of $\\tilde{\\pi}$ (which is equal to\n${\\rm Inn}(G)\/{\\rm Inn}_N(G)$) is finite, since $N$ is of finite index in\n$G$, while its image has finite index in ${\\rm Out}(G)$, since ${\\rm Aut}_N(G)$ \nis of finite index in ${\\rm Aut}(G)$, by ~\\cite[Lemma 1]{Mc}. It\nfollows that each finite-index, torsion-free subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ \nembeds as a subgroup of finite index in ${\\rm Out}(G)$,\nwhich proves the lemma.\n\\end{proof}\n\n\\begin{cor}\\label{cor:free-by-finite}\nThe outer automorphism group of a finitely generated,\nfree-by-finite group is residually finite.\n\\end{cor}\n\\begin{proof} It is well-known that every finitely generated,\nfree-by-finite group $G$ is hyperbolic. Furthermore, $G$ is\nconjugacy separable by ~\\cite{Dy}. Therefore Theorem\n~\\ref{theorem} applies to $G$. On the other hand, it is also known\nthat the outer automorphism group of a free group is virtually\ntorsion-free. If $G$ is virtually infinite cyclic, then ${\\rm Out}(G)$\nis finite. In the case where $G$ is not virtually cyclic, the\ncenter of a free subgroup of finite index in $G$ is trivial, and\nthe result follows from Lemma ~\\ref{lem:vir}.\n\\end{proof}\n\nRecall that a Fuchsian group is a discrete subgroup of the group of \nisometries of the hyperbolic plane $\\mathbb{H}^{2}$.\n\n\\begin{cor} The outer automorphism group of a finitely generated,\nFuchsian group is residually finite.\n\\end{cor}\n\n\\begin{proof} Every finitely generated Fuchsian group $G$ contains  \na normal torsion-free subgroup of finite \nindex, say $N$, which is either a free group or the fundamental group of an\norientable surface group of genus $g\\ge 2$. So $N$ is hyperbolic since it is either free or \nquasi-isometric to $\\H^2$. Moreover $N$ is conjugacy separable \\cite[Theorem 3.3]{stebe} and \nits outer automorphism group ${\\rm Out}(N)$ is virtually torsion-free and so $N$ \nsatisfies the hypotheses of\nTheorem \\ref{theorem}. Hence, ${\\rm Out}(N)$ is residually finite.\n\nIf $N$ is cyclic then $G$ is virtually cyclic and so ${\\rm Out}(G)$ is finite. In all other\ncases, $N$ is a normal subgroup of finite index in $G$ with trivial centre (since it is\na non-cyclic torsion-free hyperbolic group) and so from the proof of \nLemma \\ref{lem:vir} we have that ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is a subgroup of ${\\rm Out}(N)$. \nHence, every subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ is residually finite. But, again from the proof of \nLemma \\ref{lem:vir}, there is a torsion-free subgroup of ${\\rm Aut}_N(G)\/{\\rm Inn}_N(G)$ \nthat  embeds as a finite index subgroup \nin ${\\rm Out}(G)$. Therefore, ${\\rm Out}(G)$ is residually finite.\n\\end{proof}\n\n\nThe corollary below generalizes the results of Grossman \\cite{Gr} and of \nAllenby, Kim and Tang \\cite{akt}. Notice that for the exceptional cases of the Torus and\nthe Klein bottle it is easily verified that the result still holds.\n\\begin{cor}\nThe mapping class group $\\mod_S$ of a hyperbolic 2-orbifold $S$ with finite volume\nis residually finite.\n\\end{cor}\n\n\\begin{proof}\nThe fundamental group of a hyperbolic 2-orbifold with finite volume is a\nfinitely generated Fuchsian group. Hence, the proof is an immediate consequence \nof the results of Fujiwara \\cite{fujiwara}, the\nabove corollaries and the fact that subgroups of residually finite groups\nare residually finite. \n\\end{proof}\n\n\n\\subsubsection*{Addendum. Added, October 8, 2005.} Only recently the authors found out that their \nmain result can be\ngeneralized to relatively hyperbolic groups by using \\cite[Theorem\n1.1]{bs}.\n\nRelatively hyperbolic groups were introduced by\nGromov in \\cite{gromov}, in order to generalize notions such as\nthe fundamental group of a complete, non-compact, finite volume \nhyperbolic manifold and to give a hyperbolic version of small \ncancellation theory over free groups by adopting the geometric language \nof manifolds with cusps. \n\nThis notion has been \ndeveloped by several authors and, in particular, various \ncharacterizations of relatively hyperbolic groups\nhave been given (see \\cite{bow,osin} and \\cite{ds} \nand references therein).\n\nWe recall here one of Bowditch's equivalent definitions. A finitely\ngenerated group $G$ is {\\it hyperbolic relative to a\nfamily of finitely generated subgroups\\\/} $\\mathcal{G}$ if $G$ admits a \nproper, discontinuous and isometric action on a proper, hyperbolic \npath metric space $X$ such that $G$ acts on the ideal boundary of \n$X$ as a geometrically\nfinite convergence group and the elements of $\\mathcal G$\nare the  maximal parabolic subgroups of $G$.\n\n We should mention here that Farb\n\\cite{farb} introduced a weaker notion of relative hyperbolicity \nfor groups using constructions on the Cayley graph of the groups.\n\n\n\n\n\nExcept of the fundamental groups of hyperbolic manifolds\nof finite volume, another interesting family of relatively \nhyperbolic groups are the fundamental groups of graphs of finitely generated\ngroups with finite edge groups which are hyperbolic relative to the \nfamily of vertex groups (since their action on the Bass-Serre tree \nsatisfies definition 2 in \\cite{bow}).\n\nThe key lemma of the paper, Lemma 2.2, generalizes to\nrelatively hyperbolic groups.\n\n\\setcounter{section}{2} \\setcounter{thm}{1}\n\n\\begin{lem}\\hskip -.2cm{$\\mathbf '$}\nLet $G$ be a relatively hyperbolic group. Then the group\nInn$(G)$ of inner automorphisms of $G$ is of finite\nindex in Conj$(G)$.\n\\end{lem}\n\n\\noindent {\\it Sketch of proof.} In this case the Cayley graph\nof $G$ is replaced by the $\\d$-hyperbolic metric space $X$ on\nwhich $G$ acts by isometries.\n\nLet $\\l_i=\\inf\\limits_{x\\in X}\\max\\limits_{s\\in S} d(x,f_i(s)x)$\nwhere $S$ is a fixed finite generating set $G$ and let $x_i^0\\in X$\nsuch that $\\max\\limits_{s\\in S}d(x_i^0,f_i(s)x_i^0)\\le \\l_i+\\frac 1i$.\n\nBy the proof of Theorem 1.1 in \\cite{bs}, the sequence $\\l_i$\nconverges to infinity. Hence for every non-primitive ultrafilter\n$\\omega$ on $\\mathbb N$ the based ultralimit $(X_{\\omega},d_{\\omega},x_{w}^0)$ of the\nsequence of based metric spaces $(X,\\frac{d}{\\l_i},x_i^0)$ is an\n$\\mathbb R$-tree. Moreover, there is an induced non-trivial isometric\n$G$-action on $(X_{\\omega},d_{\\omega},x_{w}^0)$, given by $g\\cdot\nx_i=f_i(g)x_i$.\n\nFollowing step-by-step the proof of Lemma 2.2 we arrive again at the\ncontradiction that the action has a global fixed point.\n\nThe reader should be careful in the following.\n\\begin{enumerate}\n\\item The elements $y_i$ of $X$ are chosen such that\n$$\\t(f_i(g))\\le d(f_i(g)y_i,y_i) \\le \\t(f_i(g))+\\frac 1i.$$\n\\item The inequalities (5), (6) and (7) become\n$$ A\\ge 2d(x_i,y_i)-2K(\\d) -\\frac 2i \\eqno (5') $$\n$$ 2d(x_i,y_i)-K(\\d) \\le d(f_i(g)x_i,x_i)-\\t(f_i(g))\\le 2d(x_i,y_i)+\\frac 1i \\eqno (6')$$\n$$ \\textrm{and} \\quad |d(f_i(g)x_i,x_i)-\\t(f_i(g))| \\le 2d(x_i,y_i)+K(\\d)+\\frac 1i, \\eqno (7')$$\n\\noindent respectively.\n\n\\item Finally, the last inequality becomes\n$$\\left|\\t_{\\omega}(g) - \\frac{\\t(f_i(g))}{\\l_i}\\right|\\le\n\\left| \\t_{\\omega}(g)- d_i(f_i(g)x_i,x_i)\\right|+\\frac{|A|}{\\l_i}+3\\frac{K(\\d)}{\\l_i}+\\frac{3}{i\\l_i}.$$\n\\end{enumerate}\n\\hfill$\\Box$\n\nConsequently, Theorem 2.3 generalizes as follows.\n\n\\setcounter{thm}{2}\n\\begin{thm}\\hskip -.2cm{$\\mathbf '$}\nLet $G$ be a conjugacy separable, relatively hyperbolic group.\nThen each virtually torsion-free subgroup of the outer\nautomorphism group {\\rm Out}$(G)$ of $G$ is residually finite.\n\\end{thm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe Thomas-Ehrman Level Displacement formalism (TELD)~\\cite{bib:ehr51,bib:tho52} is an established technique for calculating the level displacement between mirror pairs. It \nis found to be particularly useful in situations where a reaction proceeds via a proton resonant state in a proton-rich nucleus.\nLargely, this usefulness derives from the fact that such states are above the particle decay threshold, usually resulting in proton partial\nwidths too narrow to be measured experimentally. Thus, by appealling to the charge symmetry of the nuclear force, one may make use of\nrelatively abundant spectroscopic data of analogue states in the mirror nucleus to determine the properties of the astrophysically interesting states. \nExamples in the literature are the $^{21}$Ne-$^{21}$Na~\\cite{bib:mar57}, $^{20}$F-$^{20}$Na~\\cite{bib:lan86}, \n$^{18}$O-$^{18}$Ne~\\cite{bib:wie88}, $^{22}$Ne-$^{22}$Mg~\\cite{bib:rui03} and $^{46}$Ti-$^{46}$Cr~\\cite{bib:hor02,bib:he07} mirror nuclear pairs.\nHowever, a survey of the literature finds inconsistency in the definition of critical parameters, leading to errors in the calculations. \nIn the present work a complete and consistent TELD formalism is presented and made available for wider use. \n\n\\section{The Wave Function}\nHere we reproduce and expand upon the original work of Thomas~\\cite{bib:tho52}, using, for consistency, exactly the same terminology. The channel radius\nof two interacting bodies is defined as $a_c=1.44\\times(A_{1c}^{1\/3}+A_{2c}^{1\/3})$ fm, with $A_{1c}$ and $A_{2c}$ being the mass numbers of the bodies\nof the pair; the reduced mass is $M_c= A_{1c} A_{2c}\/(A_{1c}+A_{2c})$; the energy of relative motion is $\\epsilon_c$, which may be positive or negative.\nThe subscript $c$ is used to describe all of the features of the channel, unless it is necessary to distinguish the positive-energy ($\\epsilon_{c+}>0$)\nfrom the negative-energy ($\\epsilon_{c-}<0$) channels in which case the symbols $c+$ and $c-$ are used, respectively.\n\nFor external wave functions, a radial factor (Equ. 1 of ref~\\cite{bib:tho52}) may be written that satisfies the wave equation\n\\begin{equation}\n\\overline{F}_c^{\\prime\\prime} + (2M_c\/\\hbar^2)(\\epsilon_c - \\mho_c)\\overline{F}_c = 0,\n\\label{eq:1}\n\\end{equation}\nwhere a prime signifies differentiation with respect to $r$ (in the following descriptions, all the derivatives are with respect to $r$ unless stated\notherwise). The interaction potential may be written\n\\begin{equation}\n\\mho_c = Z_{1c}Z_{2c}e^2r_c^{-1} + (\\hbar^2\/2M_c)\\ell(\\ell+1)r_c^{-2},\n\\label{eq:2}\n\\end{equation}\nwhere the nuclear potential term disappears in the external region. In the notation of Yost, Wheeler, and Breit~\\cite{bib:yos36}, the positive-energy\nsolution, which is regular at the origin, is designated by ${F(kr)}$ and has the asymptotic form for large $r$,\n\\begin{equation}\nF_{c+} \\thicksim \\sin(x-\\frac{1}{2}\\ell \\pi -\\eta \\mathrm{ln}2x+ \\sigma).\n\\label{eq:3}\n\\end{equation}\nLikewise, there is a solution which is linearly independent of $F$ and irregular at the origin which is conveniently taken with the asymptotic form\nfor large $r$,\n\\begin{equation}\nG_{c+} \\thicksim \\cos(x-\\frac{1}{2}\\ell \\pi -\\eta \\mathrm{ln}2x+ \\sigma).\n\\label{eq:4}\n\\end{equation}\nThe quantities entering Equ.~\\ref{eq:3} and~\\ref{eq:4} are\n\\begin{eqnarray}\nx_{c\\pm} & = & kr \\nonumber \\\\\nk_{c\\pm} & = & p\/\\hbar =(2M_c|\\epsilon|\/\\hbar^2)^{1\/2} \\nonumber,\n\\end{eqnarray}\nwith Sommerfeld parameter\n\\begin{eqnarray}\n\\eta_{c\\pm}=M_cZ_{1c}Z_{2c}e^2\/\\hbar^2k =Z_{1c}Z_{2c}e^2\/\\hbar v \\nonumber,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\sigma_{c+}=\\mathrm{arg}\\Gamma(1+\\ell+i\\eta) \\nonumber.\n\\end{eqnarray}\nIt is worth noting that $x$ is replaced with $\\rho$ in some formulations.\n\nThe general solution of this equation, $\\overline{F}(r)$, is a linear combination of $F$ and $G$. The Wronskian relation for these two particular\nsolutions, which directly follows from Equ.~\\ref{eq:1}-~\\ref{eq:4} is\n\\begin{equation}\nF^\\prime G-G^\\prime F=k_{c+}.\n\\label{eq:5}\n\\end{equation}\nExtensive tables~\\cite{bib:blo50} and several computer codes~\\cite{bib:bar74,bib:bar82,bib:sea02} have been developed for evaluating $F$ and $G$ and\ntheir derivatives when $\\eta>0$.\n\nFor the $c-$ channels, only the solution to Equ.~\\ref{eq:1}, vanishing at large distances from the origin, can occur; it is the\nWhittaker function~\\cite{bib:whi35,bib:mag48},\n\\begin{widetext}\n\\begin{eqnarray}\nW_{-\\eta,\\ell +\\frac{1}{2}}(2x_{c-})=\n\\frac{e^{-x-\\eta \\mathrm{ln}2x}}{\\Gamma (1+\\ell+\\eta)}\\int_0^\\infty t^{\\ell+\\eta}e^{-t}\\left (1+\\frac{t}{2x} \\right) ^{\\ell-\\eta}dt.\n\\label{eq:8}\n\\end{eqnarray}\n\\end{widetext}\nWhittaker function and its derivative may be accurately calculated using the {\\tt whittaker\\_w}~\\cite{bib:nob04} computer code.\nHowever, it is useful to note that if there is no Coulomb interaction in a $c-$ channel, one has from Equ.~\\ref{eq:8} for $s$, $p$, $d$, and $f$\norbitals the simpler relations\n\\begin{eqnarray}\n& & W_{0,\\frac{1}{2}}(2x)=e^{-x} \\label{eq:9} \\\\\n& & W_{0,\\frac{3}{2}}(2x)=(1+x^{-1})e^{-x} \\label{eq:10} \\\\\n& & W_{0,\\frac{5}{2}}(2x)=(1+3x^{-1}+3x^{-2})e^{-x} \\label{eq:11} \\\\\n& & W_{0,\\frac{7}{2}}(2x)=(1+6x^{-1}+15x^{-2}+15x^{-3})e^{-x} \\label{eq:12}\n\\end{eqnarray}\nwhich can be used for checking the results from a more complicated code.\n\nIn discussing conditions at the nuclear surface, one needs to evaluate the real and imaginary parts of the logarithmic derivatives,\n$g_c=E^\\prime\/E$, and these are \\cite{bib:tho52},\n\\begin{eqnarray}\n& & g_{c+}^{Re}=(FF^\\prime+GG^\\prime)(F^2+G^2)^{-1} \\label{eq:13} \\\\ \n& & g_{c+}^{Im}=k(F^2+G^2)^{-1} \\label{eq:14} \\\\\n& & g_{c-}^{Re}=W^\\prime W^{-1} \\label{eq:15} \\\\\n& & g_{c-}^{Im}=0 \\label{eq:16} \n\\end{eqnarray}\nwhere $g_c=g^{Re}+\\mathrm{i}g^{Im}$ and $r_c=a_c$. Although the simple WKB approximation \\cite{bib:tho52,bib:mar57} can perform well in calculating the\nlogarithmic derivatives of the Coulomb and Whittaker functions in specified regions, modern computer codes perform essentially exact calculations and are preferred.\nFor example, the difference between the WKB approximation and the exact evaluation of $g_{c-}$, performed using the code {\\tt whittaker\\_w}~\\cite{bib:nob04},\nin the region 0.1$<x<$5.0, 0.1$<\\eta<$10.0 (roughly corresponding to $Z_{1c}Z_{2c}\\leq$20 and bound nucleon\nenergy 0.1$<|E_b|<$10.0 MeV), can be as large as 3\\% ($\\ell$=1),\n1\\% ($\\ell$=2) and 0.6\\% ($\\ell$=3), respectively, and becomes\nsmaller as $\\eta$ increases. In the case that $\\ell$=0, the differences\ncan be as much as 10\\% when $x<$0.1 and $\\eta<$0.9,\nor $x<$0.2 and $\\eta<$0.5, or $x<$0.5 and $\\eta<$0.3. \nIn addition, it should be noted that a frequently used subroutine, {\\tt COULFG}~\\cite{bib:bar82}, was unable to reproduce the results\n($F$, $G$ and their derivatives) of the subroutine {\\tt RCWFN} near values of $\\eta=1.70x+5.13$ unless the\n{\\tt ACCUR} variable was set to be less than $10^{-17}$ (of course the smaller the safer, {\\em e.g.}, $10^{-30}$\nif possible). \n\n\\section{Calculation of level displacements}\n\nUnder the assumption of charge symmetry of nuclear forces, the $nn$ and $pp$ nuclear interactions are identical. The difference in the excitation energy\nof levels in mirror nuclei is therefore due to differences in the Coulomb, electromagnetic spin-orbit, and mass energies. As suggested\nby~\\cite{bib:tho52}, one can evaluate this by considering $H_p-H_n \\equiv V$. Furthermore, in the one-level approximation, irrespective of the\nboundary conditions, in the internal region the proton and neutron wavefunctions are the same to within a multiplicative constant.\nUnder these assumptions, one obtains (Equ. 30a in ref.~\\cite{bib:tho52})\n\\begin{equation}\nE_{n}-E_{p}=-\\langle V \\rangle_\\tau + \\Delta_{\\lambda n} - \\Delta_{\\lambda p},\n\\label{eq:18}\n\\end{equation}\nwhere there is a boundary condition that satisfies $\\overline{b}_{nc}=\\overline{b}_{pc}$ (defined in Equ. 24c of~\\cite{bib:tho52}).\nHere, $\\langle V \\rangle_\\tau$ is the mean value of $V$ in the internal region, and $E_n$ and $E_p$ are the Eigenvalues satisfying\n$H_{n(p)}\\Psi_{n(p)}=E_{n(p)}\\Psi_{n(p)}$ of the nucleus with the odd neutron or proton, respectively.\nThe difference of the level displacements of the neutron and proton states (see Equ. 30b of~\\cite{bib:hor02}),\n\\begin{equation}\n\\Delta_{\\lambda}=\\Delta_{\\lambda n} - \\Delta_{\\lambda p} = -\\sum_{c\\pm}\\gamma_c^2(g_{nc}^{Re}-g_{pc}^{Re}),\n\\label{eq:19}\n\\end{equation}\nis referred to as the boundary condition level displacement.\nThe energy difference of corresponding levels of mirror nuclei, following from Equ.~\\ref{eq:18} and ~\\ref{eq:19}, can be written as\n\\begin{equation}\n(E_{n}^{\\ast}-E_{n}^{g.s.})-(E_{p}^{\\ast}-E_{p}^{g.s.})=\\Delta_{\\lambda}^{\\ast}-\\Delta_{\\lambda}^{g.s.},\n\\label{eq:20}\n\\end{equation}\nwhere it is assumed that the quantity $\\langle V \\rangle_\\tau$ is the same in the excited state as in the ground state. The scripts $\\ast$ and {\\em g.s.}\ndenote the corresponding quantities are being evaluated with respect to the excited state and the ground state, respectively. Assuming that the level\ndisplacements of the two ground states are the same simplifies this relation further,\n\n\\begin{equation}\nE^\\ast(n)-E^\\ast(p)=\\Delta_{\\lambda}^{\\ast},\n\\label{eq:21}\n\\end{equation}\nwhere $E^\\ast(n)$ and $E^\\ast(p)$ are the corresponding excitation energies in the mirror nuclei.\nTherefore, the observed energy difference between mirror nuclear states is due to different level displacements,\n$\\Delta_{\\lambda n}$ and $\\Delta_{\\lambda p}$, in the two nuclei.\n\n\\begin{table*}\n\\caption{\\label{table1} Various definitions of $\\Gamma_{\\lambda c}$,$\\gamma_{\\lambda c}^2$, $\\theta_{\\lambda c}^2$ and\n$\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60} of a level $\\lambda$ in the literature.}\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n \n   & $\\Gamma_{\\lambda c}$ & $\\gamma_{\\lambda c}^2$ & $\\theta_{\\lambda c}^2$ & $\\theta_{sp}^2$ \\\\\n  \\hline\n  Thomas~\\cite{bib:tho52}          & 2$P_c(\\gamma_{\\lambda c}^2\/a_c)$ & $\\frac{\\hbar^2}{2M_c a_c}\\theta_{\\lambda c}^2$    & $a_c\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$                    &  \\\\\n  Lane and Thomas~\\cite{bib:lan58,bib:lan60} & 2$P_c\\gamma_{\\lambda c}^2$       & $\\frac{\\hbar^2}{M_c a_c^2}\\theta_{\\lambda c}^2$   & C$^2$S$\\theta_{sp}^2$ & $\\frac{a_c}{2}\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$ \\\\\n  French~\\cite{bib:fre60}          & 2$P_c\\gamma_{\\lambda c}^2$       & $\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{\\lambda c}^2$ & C$^2$S$\\theta_{sp}^2$ & $\\frac{a_c}{3}\\times\\frac{u^2(a_c)}{\\int_{0}^{a_c}u^2(r)dr}$ \\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\nWe find that the definitions used by various authors of partial width $\\Gamma_{\\lambda c}$, reduced width $\\gamma_{\\lambda c}^2$, dimensionless reduced\nwidth $\\theta_{\\lambda c}^2$ and dimensionless single-particle reduced width $\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60}, of a level $\\lambda$, are not\nconsistent, as shown in Table~\\ref{table1}. Following the previous work, the definitions of French~\\cite{bib:fre60} are adopted in the present work,\nthough it is possible to achieve consistency using the alternative definitions~\\cite{bib:tho52,bib:lan58}.\nOne should pay attention that the definition of $\\gamma_{\\lambda c}^2$~\\cite{bib:fre60} is different from that of~\\cite{bib:tho52} by a factor of $3\/a_c$.\nThus, for the positive-energy channel,\n\\begin{equation}\n-\\gamma_{c+}^2g_{c+}^{Re}=-\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{c+}^2P_c(FF_{x}^\\prime+GG_{x}^\\prime)\n\\label{eq:22}\n\\end{equation}\nwith the Coulomb penetrability $P_c=x\/(F^2+G^2)$; and for the negative-energy channel,\n\\begin{equation}\n-\\gamma_{c-}^2g_{c-}^{Re}=-\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_{c-}^2 x \\frac{W_{x}^\\prime}{W}.\n\\label{eq:23}\n\\end{equation}\nWhere $F_x^\\prime$,$G_x^\\prime$ and $W_x^\\prime$ represent differentiation with respect to $x$. Equ.~\\ref{eq:22} \\&~\\ref{eq:23} were defined as\n$\\Delta_b$ and $\\Delta_r$ in previous literature by assuming $\\overline{b}_{nc}=\\overline{b}_{pc}=0$ (see Fig.~\\ref{prcfig1}).\n\nThe reasonable assumption that the reduced widths, $\\gamma_{c}^2$, are the same for the mirror levels leads to an\nassumption of $\\theta_{c}^2=\\theta_{p}^2=\\theta_{n}^2$. Thus, the excitation-energy displacements of mirror nuclei can be expressed as\n\\begin{equation}\n\\Delta_{\\lambda}^{\\ast}=\\frac{3\\hbar^2}{2M_c a_c^2}\\theta_c^2 \\left\\{\\left[P_c(FF_{x}^\\prime +GG_{x}^\\prime)\\right]_{\\mid E=E_r}- \\left(x\\frac{W_{x}^\\prime}{W}\\right)_{\\mid E=E_b} \\right\\},\n\\label{eq:24}\n\\end{equation}\nwhere $E_r$ [=$E^\\ast(p)-S_p^p$] and $E_b$ [=$|E^\\ast(n)-S_n^n|$] are the energies relative to the respective nucleon thresholds ($S_p^p$ and $S_n^n$;\nthe superscripts $p$ and $n$ refer to the odd-proton (or proton-rich) and odd-neutron (or neutron-rich) nuclei, respectively; the subscripts $p$ and $n$\ndenote the corresponding nucleon separation energies). A pictorial representation of the physical meanings of the parameters described here are illustrated  \nin Fig.~\\ref{prcfig1} which shows the case of the 6.424~MeV state in $^{46}$Ti and its analogue state in $^{46}$Cr~\\cite{bib:hjj07}. It is clear from the \nfigure that the level displacement may equally well be written as,\n\\begin{equation}\nE^\\ast(n)-E^\\ast(p)=[E^\\ast(n)-S_p^p]-[E^\\ast(p)-S_p^p]=E_b^\\prime-E_r,\n\\label{eq:25}\n\\end{equation}\nwhere the quantity $E_b^\\prime$ is different from that of $E_b$ defined above. It appears that confusion \nover these definitions has, in part, been the source of errors in the past. For example, in several previous \nworks~\\cite{bib:lan86,bib:wie88,bib:hor02,bib:rui03} the level displacement has been written as \n$E^\\ast(n)-E^\\ast(p)=E_b-E_r$ in contrast to the correct expression of $E^\\ast(n)-E^\\ast(p)=E^\\prime_b-E_r$ \nwhich follows from Marion {\\em et al.}~\\cite{bib:mar57}. However, the pressent work validates that the correct\nexpression was used in the calculations~\\cite{bib:gor06}.\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[width=7cm]{mirror.eps}\n\\end{center}\n\\caption{\\label{prcfig1} Definitions and calculation results of relevant quantities in case of the $^{46}$Ti-$^{46}$Cr mirror pair. The calculated values\n(in units of MeV) are with respect to the analogue states at 6.424 MeV (in $^{46}$Ti) and 6.246 MeV (in $^{46}$Cr)~\\cite{bib:hjj07}.}\n\\end{figure}\n\nThe final ingredient which is needed to allow calculation of the level shift is the dimensionless reduced width, $\\theta_c^2$. \nThis can be calculated according to the relation $\\theta_c^2$=C$^2$S$\\theta_{sp}^2$~\\cite{bib:lan60,bib:fre60}, \nwhere the dimensionless single-particle reduced width $\\theta_{sp}^2$ has already been\ndetermined~\\cite{bib:ili97,bib:bar98}, and the factor C$^2$S is calculated via a shell-model code such as {\\tt OXBASH}~\\cite{bib:bro92}. \n\n\\section{Online resources}\nThe algorithms developed here have been made available online~\\cite{bib:heweb}.\nThe Visual Basic tool requires the {\\tt TELD\\_VB} and {\\tt TELD} executables be in the same directory. Opening the {\\tt TELD} \napplication provides a window in which one defines the parameters of the mirror states to be considered \n($A$, $Z$, $S_n$, $S_p$ and $E_x(n)$). One also could change values for the channel radius, the orbital angular momentum of the\nsingle particle and spectroscopic factor for the state. \nActivation of the ``calc\" button then performs the TELD formalism and returns the resulting excitation of the \nproton-rich analogue state and the proton decay width for this state. This last parameter is calculated using the relation\n\\begin{equation}\n\\Gamma_p=\\frac{3\\hbar^2P_{c}\\theta_p^2}{M_c a_c^2}.\n\\label{eq:29}\n\\end{equation}\nIn addition, a Fortran source code, which allows users to make changes whenever required, is also encloded in the TELD.rar package.\nFigure~\\ref{VBfig} shows a screenshot from the {\\tt TELD} program with all relevant parameters for the case of the \n6.424~MeV state in $^{46}$Ti and its analogue state in $^{46}$Cr~\\cite{bib:hjj07}. \n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[width=7cm]{vbfig.eps}\n\\end{center}\n\\caption{\\label{VBfig} Screenshot from the {\\tt TELD} program for the case of the 6.424~MeV state in $^{46}$Ti and its analogue state \nin $^{46}$Cr~\\cite{bib:hjj07}.} \n\\end{figure}\n\n\n\\section{Summary}\nA complete and consistent Thomas-Ehrman Level Displacement formalism has been presented and made available on the World Wide Web. With this, \nif one has knowledge (or makes a reasonable assumption) of the quantity $\\theta_c^2$ and spectroscopic factor $S$, one may estimate the\nlocation of the mirror to a known excited state. Alternatively, experimental measurement of the mirror level displacement\nprovides a route to determining $\\theta_c^2$ (or $S$ factor).\n\n\\begin{center}\n\\textbf{Acknowledgments}\n\\end{center}\n\nThe authors acknowledge support from the Engineering and Physical Sciences Research Council, UK.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent proposals to perform quantum computation in nuclear spin systems\nusing nuclear magnetic resonance (NMR)\ntechniques~\\cite{Gershenfeld97,Cory97x,Cory97b,Chuang97e}, coupled logic\noperations are performed using spin-spin couplings that occur naturally in\nmolecular systems.\nWhile this is straightforward for small systems with a few spins,\ngeneralization to complex molecular structures has been challenging.  \nThe complication is caused by the many spin-spin couplings which occur along\nwith the desired one.\nThis fundamental task to turn off spurious evolution is so difficult that,\ncoercing a complex system to {\\em do nothing}~\\cite{Jozsa98} -- ceasing all\nevolution -- can be just as difficult as making it do something\ncomputationally useful.\n\nThe task of turning off all couplings is known in the art of NMR as {\\em\ndecoupling}; doing this for all but a select subset of couplings is known as\n{\\em selective recoupling}.  \nA common method to perform these tasks is to interrupt the free evolution by\ncarefully chosen pulses.  These pulses are single spin-1\/2 (qubit) operations\nthat transform the hamiltonian in the time between pulses in such a manner\nthat unwanted evolutions in consecutive time intervals cancel out each other.\n\nPulse sequences which perform selective recoupling are generally difficult to\nfind for a large system.  Each pulse simultaneously affects many coupling\nterms in the hamiltonian.  To turn off all but one of the coupling terms,\nthese pulses have to satisfy many simultaneous requirements.\nIngenious sequences have been found for usual NMR\napplications~\\cite{Ernst94,Slichter,Mansfield} but they do not address the\nproblems relevant to quantum computation.  In usual NMR applications, the\nstructure of the spin systems is not known a-priori.  Therefore, pulse\nsequences are designed to address all the spins together rather than\nindividual spins.\nQuantum computation brings new requirements, and initial\nefforts~\\cite{Linden98} have been made to develop pulse sequences to satisfy\nthese needs; however, to-date, schemes have necessitated resources (such as\ntotal number of pulses applied) exponential in the number of spins being\ncontrolled.\n\nIn this paper, we present an {\\em efficient} scheme to perform selective\nrecoupling.\nIn contrast to the situation with traditional NMR, this scheme addresses the\nproblem relevant to NMR quantum computation, in which the molecular structure\nis assumed to be well-known, and spins are individually addressible.\nThe method is related to a class of well-known matrices called {\\em Hadamard\nmatrices}.  \nWe derive from any $n \\times n$ Hadamard matrix a pulse scheme that decouples\n$n$ spins using $n$ time intervals and ${\\cal O}(n^2)$ pulses.\nThis decoupling scheme can easily be modified  \n(i) to remove Zeeman evolution and \n(ii) to implement selective recoupling.  \nThis completes the construction for $n$ spins whenever $n \\times n$ Hadamard\nmatrices exist. \nWhen $n \\times n$ Hadamard matrices do not exist, a scheme for $n$ spins can\nstill be constructed using larger existing Hadamard matrices.  \nIn doing so, an extra amount of effort is required, but this can be bounded\nusing existence properties of Hadamard matrices.\nAltogether, the scheme requires $cn$ time intervals and less than \n$cn^2$ pulses, where $c \\approx 1$ with upper bound $c \\leq 2$.\nOur method applies whenever the spins couple pairwise and have very\ndifferent Zeeman frequencies, such as in heteronuclear spin systems.\n\nThe paper is structured as follows.  \nIn Section~\\ref{sec:statement}, we review relevant concepts in NMR quantum\ncomputing and re-state the problem precisely. \nIn Section~\\ref{sec:scheme}, we first motivate the construction of the\ndecoupling scheme with examples, and then describe the general construction\nrelated to Hadamard matrices.  Important properties of Hadamard matrices \nare summarized.  Modifications of the decoupling scheme to perform selective\nrecoupling are described.\nWe conclude with some general remarks and discussions of various properties \nand limitations of the scheme.  \n\n\\section{NMR quantum computing and the statement of the problem} \n\\label{sec:statement}\n\nIn this section, we describe the NMR system and describe how a\nuniversal set of (non-fault tolerant)\noperations~\\cite{Div95,Barenco95,Deutsch95}, namely, the single qubit\noperations and the controlled-NOT gate, can be realized using basic NMR\nprimitives.\n\nWe shall consider a physical system which consists of a solution of identical\nmolecules.  Each molecule has $n$ non-magnetically equivalent nuclear spins\nwhich serve as qubits.\nA static magnetic field is applied externally along the $+\\hat{z}$ direction.\nThis magnetic field splits the energy levels of the spin states aligned with\nand against it.  This is described in the hamiltonian by the Zeeman terms,\nwhich, in the energy eigenbasis, are given by\n\\begin{equation}\n\t{\\cal H}_{\\rm Z} \n\t= -\\frac{1}{2} \\sum_i \\hbar \\omega_i \\sigma_z^{(i)} \n\\,,\n\\label{eq:ham:zeeman}\n\\end{equation}\nwhere $i$ is the spin index, $\\omega_i\/2\\pi$ is the {\\em Zeeman\nfrequency} for the $i$-th spin, and $\\sigma_z^{(i)}$ is the Pauli\nmatrix operating on the $i$-th spin.  \nThe convention $\\hbar = 1$ is used for the rest of the paper.\nThe spins have very different Zeeman frequencies, a situation loosely termed \nas ``heteronuclear'' in this paper.\n\nNuclear spins can interact via the dipolar coupling~\\cite{Slichter,Abragam}.\nThis is given by the hamiltonian\n\\begin{equation}\n\t{\\cal H}_{\\rm d} = \\sum_{i<j} g_{ij}^{\\rm d}  \\left[\\rule{0pt}{2.4ex}  \n\t\t\\vec{\\sigma}^{(i)} \\cdot \\vec{\\sigma}^{(j)} \n\t\t- 3 (\\hat{r}_{ij} \\cdot \\vec{\\sigma}^{(i)})  \n\t\t\\otimes (\\hat{r}_{ij} \\cdot \\vec{\\sigma}^{(j)})  \\right] \n\\label{eq:dipolarfull}\n\\,,\n\\end{equation}\nwhere $\\hat{r}_{ij}$ denotes the unit displacement vector from the $i$-th to\nthe $j$-th spin, and $g_{ij}^{\\rm d}$ denotes the coupling constant between\nthem.\nSpin-spin coupling can also be mediated by coupling to electrons.  This\nindirect coupling has a tensor part and a scalar part.  The tensor part is\nusually of the same form as ${\\cal H}_{\\rm d}$.  The scalar part is given by\nthe hamiltonian \n\\begin{equation} \n\t{\\cal H}_{\\rm s} = \\sum_{i<j} g_{ij}^{\\rm s} \n\t\t\t\\vec{\\sigma}^{(i)} \\cdot \\vec{\\sigma}^{(j)} \n\\label{eq:scalar}\n\\,.  \n\\end{equation} \nIf the molecules tumble fast and isotropically, dipolar coupling and\nindirect tensor coupling will be averaged away; otherwise, the physics \ncan be more complicated.  \nHowever, in the presence of a strong external magnetic field, only the {\\em\nsecular} part (terms that commute with ${\\cal H}_{\\rm Z}$) is\nimportant~\\cite{Slichter,Abragam}.\nFor a heteronuclear system, the resulting coupling becomes \n\\begin{equation} \n\t{\\cal H}_{\\rm c} = \\sum_{i<j} g_{ij}^{\\rm c} \\sigma_z^{(i)} \\otimes\n\t\\sigma_z^{(j)}\n\\label{eq:dipolar}\n\\,, \n\\end{equation} \nindependent of the original form of coupling.  \n\nSingle qubit operations are performed by applying {\\em pulsed} radio frequency\n(RF) magnetic fields along some directions $\\hat{\\eta}$ perpendicular to the\nstatic field.\nTo address the $i$-th spin, the frequency of the RF field is chosen to\napproximate $\\omega_i\/2\\pi$.\nWhen the $\\omega_i$'s are very different, very short pulses can be used, so\nthat during the pulses, all other evolutions are negligible except for the\nrotation operator $e^{-i \\frac{\\theta}{2} \\vec{\\sigma}^{(i)} \\cdot\n\\hat{\\eta}}$ where $\\theta$ is proportional to the pulse duration and the\npower.  The Lie group of all single qubit operations can be generated by\nrotations about $\\hat{x}$ and $\\hat{y}$.\nOur scheme uses only rotations of $\\theta = \\pi$ along $\\hat{x}$, which\nimplement $\\sigma_x$ (up to an irrelevant overall phase) on the spins being\naddressed.  We denote this operation by $X$, superscripted by the spin index\nwhenever appropriate.\n\nCoupled operations such as controlled-phase-shift or controlled-NOT acting on\nthe $i$-th and the $j$-th spins can be performed given the primitive, \n\\begin{equation} \n      \tZ\\!\\!Z_{ij} = \n\te^{-i \\frac{\\pi}{4} \\sigma_z^{(i)} \\otimes \\sigma_z^{(j)}} \n\\label{eq:zzij}\n\\,.\n\\end{equation} \nFor instance, a controlled-NOT from the $i$-th to the $j$-th spin\ncan be implemented by\n\\begin{equation}\n\te^{-i \\frac{\\pi}{4} \\sigma_y^{(i)}} \n\te^{i \\frac{\\pi}{4} \\sigma_x^{(i)}} \n\te^{i \\frac{\\pi}{4} \\sigma_y^{(i)}} \n\te^{-i \\frac{\\pi}{4} \\sigma_x^{(j)}} \n\te^{i \\frac{\\pi}{4} \\sigma_y^{(j)}}\n\tZ\\!\\!Z_{ij} \n\te^{-i \\frac{\\pi}{4} \\sigma_y^{(j)}}  \n\\,.\n\\end{equation}\n\nThe ultimate goal is to be able to efficiently realize arbitrary quantum\noperations on an $n$ spin system with arbitrary couplings.  In this paper, we\nconsider a more limited objective, which can now be stated precisely, using\nthe definitions of Eq.(\\ref{eq:ham:zeeman}), Eq.(\\ref{eq:dipolar}), and\nEq.(\\ref{eq:zzij}):\n\\begin{quote} \nGiven a heteronuclear system of $n$ spins with free evolution $e^{-i ({\\cal\nH}_{\\rm Z} + {\\cal H}_{\\rm c}) t}$, controlled using typical RF pulses, how\ncan $Z\\!\\!Z_{ij}$ be implemented efficiently?\n\\end{quote}\n\n\\section{Construction of the scheme} \n\\label{sec:scheme} \n\nWe will first construct a decoupling scheme to remove the entire coupling term\n${\\cal H}_{\\rm c}$ in the total evolution.  The scheme is derived from\nHadamard matrices, which will be reviewed.\nMethods to remove ${\\cal H}_{\\rm Z}$ and to implement selective recoupling\nwill be described afterwards.\n\n\\subsection{Construction of the decoupling scheme} \n\\label{sec:examples} \n\nTo construct the decoupling scheme, we only consider ${\\cal H}_{\\rm c}$ in\nthe evolution.  Effects of ${\\cal H}_{\\rm Z}$ can be included later since all\nmatrix exponents commute.\n\nTo motivate the general construction, we analyze the simplest example of\ndecoupling two spins.  The evolution operator for an arbitrary duration $t$ is\ngiven by $\\tau = e^{-i g_{12}^{\\rm c} t \\sigma_z^{(1)} \\otimes\n\\sigma_z^{(2)}}$.  The sequence of events, $\\tau X^{(2)} \\tau X^{(2)}$, known\nas {\\em refocusing} in NMR, will first couple and then decouple the spins.\nThis can be described mathematically as:\n\\begin{eqnarray}\n\t&   & \\tau (X^{(2)} \\tau  X^{(2)}) \n\\label{eq:refocus}\n\\\\\n\t& = & e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t    (\\sigma_x^{(2)} \n\t      e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t     \\sigma_x^{(2)})  \n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t\te^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_x^{(2)} \n\t\t\\sigma_z^{(2)} \\sigma_x^{(2)}}\n\\label{eq:third}\n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\te^{-i \\theta \\sigma_z^{(1)} \\otimes (-\\sigma_z^{(2)})}\n\\label{eq:fourth}\n\\\\\n\t& = &  e^{-i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\t\te^{i \\theta \\sigma_z^{(1)} \\otimes \\sigma_z^{(2)}} \n\\\\\n\t& = &  I \n\\end{eqnarray}\nwhere $\\theta = g_{12}^{\\rm c} t$.  Eq.(\\ref{eq:third}) is obtained using\nTaylor series expansion of the matrix exponents and using $\\sigma_x^2 = I$,\nand Eq.(\\ref{eq:fourth}) is obtained using anticommutivity of $\\sigma_x$ and\n$\\sigma_z$.\n\nThe essential features of this decoupling procedure are:\n\\\\ \\noindent {\\bf 1.} The pair of $X^{(2)}$ negates the sign of\n$\\sigma_z^{(2)}$ in the evolution between the pulses.\n\\\\ \\noindent {\\bf 2.} The $X$ pulses make the signs of the $\\sigma_z$\nmatrices of the two spins disagree for exactly half of the time.\n\\\\ \\noindent {\\bf 3.} Since the coupling is bilinear in the $\\sigma_z$\nmatrices, it is unchanged (negated) when the signs of the $\\sigma_z$\nmatrices agree (disagree).  The $X$ pulses therefore negate the coupling for\nexactly half of the time.\n\\\\ \\noindent {\\bf 4.} Since the matrix exponents commute, negating the\ncoupling for exactly half of the time suffices for the evolution to be \ncancelled out. \n\nThe most crucial point leading to decoupling is that, the signs of the\n$\\sigma_z$ matrices of the coupled spins, controlled by pairs of $X$ pulses,\ndisagree for half of the time.\n\nIn general, we consider pulse sequences which concatenate equal-time intervals\nand use $X$ pulses to control the signs of the $\\sigma_z$ of each spin.\nThe essential information on the signs can be represented by \na ``sign matrix'' defined as follows.\nThe ``sign matrix'' of a pulse scheme for $n$-spins with $m$ time intervals is\nthe $n \\times m$ matrix with the $(i,a)$ entry being the {\\em sign} of\n$\\sigma_z^{(i)}$ in the $a$-th time interval.\nThese sign matrices have one-to-one correspondence with our restricted class\nof pulse sequences.\nWe denote any sign matrix for $n$ spins by $S_n$. \nFor example, the sequence in Eq.(\\ref{eq:refocus}) can be represented by \nthe sign matrix\n\\begin{equation}\n\tS_2 = \\left[ \\begin{array}{cc}\n\t{+}&{+}\n\\\\\t{+}&{-} \n\t\\end{array} \\right] \n\\label{eq:s2}\n\\,. \n\\end{equation} \nThe general construction of decoupling scheme is now reduced to finding \nsign matrices such that every two rows disagree in half of the entries.  \n\nAs a second example, we construct a decoupling scheme for four spins.  We\nfirst find a correct sign matrix following the previous observations, and\nthen derive the corresponding pulse sequence.  For example, a possible \nsign matrix is given by \n\\begin{equation} \n\tS_4 = \\left[ \\begin{array}{cccc}\n\t{+}&{+}&{+}&{+}\n\\\\\t{+}&{+}&{-}&{-}\n\\\\\t{+}&{-}&{-}&{+}\n\\\\\t{+}&{-}&{+}&{-}\n\t\\end{array} \\right] \n\\label{eq:s4}\n\\,, \n\\end{equation} \nin which any two rows disagree in exactly two entries.  \n$S_4$ can be converted to a pulse scheme by converting each column to a time\ninterval before and after which $X$ pulses are applied to spins (rows) given\nby $-$'s.  No pulses are applied to spins (rows) with $+$'s.  The resulting\nsequence,\n\\begin{eqnarray}\n\t\\tau \n\t(X^{(3)} X^{(4)} \\tau X^{(3)} X^{(4)})  \n\t(X^{(2)} X^{(3)} \\tau X^{(2)} X^{(3)})  \\times \n\\nonumber\n\\\\\n\t\\hfill (X^{(2)} X^{(4)} \\tau X^{(2)} X^{(4)})  \n\\label{eq:dec4}\n\\,, \n\\end{eqnarray}\nis the identity by construction and this can also be verified directly.  \nNote that ${\\cal H}_{\\rm c}$ in $\\tau = e^{-i {\\cal H}_{\\rm c} t}$ now denotes\nthe sum of six possible coupling terms for four spins.\nNote also Eq.(\\ref{eq:dec4}) is written in such a way that it corresponds\nvisually to the sign matrix, though the evolutions are actually in reverse\ntime order relative to $S_4$.  However, such ordering is irrelevant for\ncommuting evolutions.  Since $X^{(i)} X^{(i)} = I$, Eq.(\\ref{eq:dec4}) can be\nsimplified to\n\\begin{equation}\n\t\\tau \n\t(X^{(3)} X^{(4)} \\tau X^{(4)})  \n\t(X^{(2)} \\tau X^{(3)})  \n\t(X^{(4)} \\tau X^{(2)} X^{(4)})  \n\\label{eq:dec4short}\n\\,.\n\\end{equation}\nThis simplified pulse sequence can also be obtained directly from\nEq.(\\ref{eq:s4}) by converting columns to time intervals and inserting\n$X^{(i)}$ between intervals whenever the $i$-th row changes sign or \nwhenever a $-$ sign reaches either end of the row.  Pulse sequences for\nEq.(\\ref{eq:dec4}) and Eq.(\\ref{eq:dec4short}) are shown in\nFig.~\\ref{fig:pulseseq}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=s24.eps,width=3.3in}}\n\\vspace*{2ex}\n\\caption{(a) Pulse sequence corresponding to Eq.(\\ref{eq:dec4}).  From $S_4$,\neach ``$-$'' sign in the $i$-th row and $a$-th column translates to two $X$\npulses at $\\omega_i$ before and after the $a$-th time interval.\n(b) Pulse sequence obtained from simplifying (a).  This corresponds to\nEq.(\\ref{eq:dec4short}), and can be constructed directly from $S_4$ by\ntranslating each change of sign in the $i$-th row to an $X$ pulse at\n$\\omega_i$.  A ``$-$'' sign at the end of the row also gives rise to an $X$\npulse at end of the last time interval.}\n\\label{fig:pulseseq}\n\\end{center}\n\\end{figure}\n\nThe above scheme can be generalized to decouple $n$ spins with $m$ time\nintervals as follows:\n\\begin{quote} \nConstruct the $n \\times m$ sign matrix $S_n$, with entries $+$ or $-$, such\nthat {\\em any} two rows disagree in exacly half of the entries.  For\neach $-$ sign \nin the $i$-th row and the $a$-th column, apply $X^{(i)}$ before and after the\n$a$-th time interval.\n\\end{quote}\nBecause of the pulses, the sign of the $\\sigma_z$ matrix for each spin in\neach time interval is as given by the sign matrix.  The $\\sigma_z$ matrices\nof any two spins therefore have opposite signs for half of the time, so\nthat their coupling is negated for exactly half of the time, and the \nevolution is always cancelled.\n\nFor $n$ spins, $n \\times m$ sign matrices which correspond to decoupling\nschemes do not necessarily exist for arbitrary $m$, but they always exist for\nlarge and special values of $m$.  A possible structure is:\n\\begin{eqnarray}\n\\nonumber\n\tS_n = \\left[ \\begin{array}{cccccccccccc}\n\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}\n\\\\\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}&{-}&{\\cdots}&{+}\n\\\\\t{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }\n\\\\\t{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}&{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}\n\\\\\t{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ }&{ }&{\\cdots}&{ } \n\\\\\t{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}&{+}&{\\cdots}&{-}\n\t\\end{array} \\right] \n\\,, \n\\end{eqnarray} \nin which intervals are bifurcated when rows (spins) are added.  Such\nbifurcation takes place whenever it is impossible to add an extra row that is\northogonal to all the existing ones (``depletion'').  If such depletion occurs\nfrequently, the sign matrix will have exponential number of columns, and\ndecoupling will take exponential number of steps as $n$ increases.\nThe challenge is to find correct sign matrices with subexponential number of\ncolumns. \nThis difficult problem turns out to have solutions given by the Hadamard \nmatrices, which will be decribed next.  \n\n\\subsection{Hadamard matrices}\n\\label{sec:Hadamard}\n\nHadamard matrices have applications in many areas such as the construction of\ndesigns, error correcting codes and Hadamard\ntransformations~\\cite{CRC96,Sloanehp,vanLint92,MacWilliams77}.\n\nA Hadamard matrix of order $n$, denoted by $H(n)$, is an $n \\times n$ matrix\nwith entries $\\pm 1$, such that\n\\begin{equation}\n\tH(n)H(n)^{T} = n I\n\\label{eq:ortho}\n\\,.\n\\end{equation}\nThe rows are pairwise orthogonal, therefore any two rows agree in exactly half\nof the entries.  Likewise columns are pariwise orthogonal.  We abbreviate\n``$\\pm 1$'' as ``$\\pm$''.  $S_2$ and $S_4$ in Eqs.(\\ref{eq:s2}) and\n(\\ref{eq:s4}) are simple examples of $H(2)$ and $H(4)$.  An example of $H(12)$\nis given by\n\\begin{eqnarray}\n\\nonumber\n\tH(12) = \\left[ \\begin{array}{cccccccccccc}\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\\end{array} \\right] \n\\,.\n\\end{eqnarray} \n\nThe following is a list of useful facts about Hadamard matrices (details and \nproofs omitted):\n\n\\begin{enumerate} \n\\item {\\em Equivalence}~~Permutations or negations of rows or columns of\nHadamard matrices leave the orthogonality condition invariant.  Two Hadamard\nmatrices are equivalent if one can be transformed to the other by a series of\nsuch operations.  Each Hadamard matrix is equivalent to a {\\em normalized}\none, which has only $+$'s in the first row and column.  For instance, $H(12)$\nshown previously can be {\\em normalized} by negating the 7-${\\rm th}$ row\nand column.\n\n\\item {\\em Necessary conditions}~~$H(n)$ exists only for $n = 1$, $n = 2$ or\n$n \\equiv 0 \\bmod 4$.  This is obvious if the matrix is normalized, and the\ncolumns are permuted so that the first three rows become: \n\\begin{eqnarray}\n\\nonumber\n\t\\left[ \\begin{array}{cccccccccccc}\n\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}& {+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}\n\\\\\t{+}&{\\cdots}&{+}&{+}&{\\cdots}&{+}& {-}&{\\cdots}&{-}&{-}&{\\cdots}&{-}\n\\\\\t{+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}& {+}&{\\cdots}&{+}&{-}&{\\cdots}&{-}\n\\\\\t{\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}&   {\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}\n\t&{\\cdot}&{\\cdot}&{\\cdot}&{\\cdot}\n\t\\end{array} \\right] \n\\,.\n\\end{eqnarray} \n\n\\item {\\em Hadamard's conjecture}~\\cite{Hadamard93}~~$H(n)$ exists for every\n$n \\equiv 0 \\bmod 4$.  This famous conjecture is verified for all $n < 428$.  \n\n\\item {\\em Sylvester's construction}~\\cite{Sylvester67}~~If $H(n)$ and $H(m)$ \nexist, then $H(nm)$ can be constructed as $H(n) \\otimes H(m)$. \nIn particular, $H(2^r)$ can be constructed as $H(2)^{\\otimes r}$, which is\nproportional to the matrix representation of the Hadamard transformation\nfor $r$ qubits.\n\n\\item {\\em Paley's construction}~\\cite{Paley33}~~Let $q$ be an odd prime\npower.  If $q \\equiv 3 \\bmod 4$, then $H(q+1)$ exists; if $q \\equiv 1 \\bmod 4$,\nthen $H(2(q+1))$ exists.  \n\n\\item {\\em Numerical facts}~\\cite{CRC96}~~\nFor an arbitrary integer $n$, let $\\underline{n}$ and $\\overline{n}$ be the\nlargest and smallest integers that satisfy $\\underline{n} < n \\leq\n\\overline{n}$ with {\\em known} $H(\\overline{n})$ and $H(\\underline{n})$.  We\ndefine the ``gap'', $\\delta_n$, to be $\\overline{n} - \\underline{n}$ (see\nFig.~\\ref{fig:gap}).\nFor $n\\leq 1000$, $H(n)$ is known for every possible order except for 6 cases,\nand the maximum gap is 8.  For $n \\leq 10000$, $H(n)$ is unknown for 192\npossible orders and the maximum gap is 32.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=gap.eps,width=1.4in}}\n\\vspace*{2ex}\n\\caption{The gap $\\delta_n$ between two existing orders of Hadamard matrices.}\n\\label{fig:gap}\n\\end{center}\n\\end{figure}\n\n\\end{enumerate}\n\nThe importance of the full connection to Hadamard matrices will become \nclear after we construct the scheme for an arbitrary number of spins \nin the next section.  \n\n\\subsection{General Construction of Scheme} \n\nIn this section, a decoupling scheme for an arbitrary number of spins is\nconstructed.  Variations of the scheme to remove Zeeman evolution and to\nimplement selective recoupling are also constructed.  The requirements of the\nscheme are discussed.\n\nRecall that $\\overline{n}$ is the minimum integer no smaller than $n$ with\nknown $H(\\overline{n})$, and $m$ is the number of columns in a sign matrix.\nFor any variations of the scheme for $n$ spins, $m_n$ is defined to be the\nminimum number of columns in any valid sign matrix and it represents the\nnumber of time intervals needed for the intended operation.\n\nTo construct a decoupling scheme for $n$ spins when $H(n)$ does not\nnecessarily exist, any $H(\\overline{n})$ with $\\overline{n}-n$ rows omitted\ncan be used as the sign matrix since subsets of rows of $H(\\overline{n})$ are\nstill pairwise orthogonal.\nIn other words, any $n \\times \\overline{n}$ submatrix of $H(\\overline{n})$ is\na valid sign matrix for decoupling $n$ spins, and $m_n = \\overline{n}$.\n\nTo remove both ${\\cal H}_{\\rm Z}$ and ${\\cal H}_{\\rm c}$, we use the fact that\nthe Zeeman term for the $i$-th spin is linear in $\\sigma_z^{(i)}$, and\nnegating $\\sigma_z^{(i)}$ for half of the time results in no net Zeeman\nevolution for the $i$-th spin.  Therefore, Zeeman evolution for all spins can\nbe removed if the sign matrix has identically zero row sum.\nSuch a sign matrix can be constructed by starting with a {\\em normalized}\n$H(\\overline{n})$ and excluding the first row of $H(\\overline{n})$ in the sign\nmatrix.  Since a normalized $H(\\overline{n})$ has only $+$'s in the first row,\nall other rows have zero row sums by orthogonality.\nSuch construction is possible unless $n = \\overline{n}$, in which case \nconstruction should start with $H(\\overline{n+1})$.  Therefore, $m_n =\n\\overline{n}$ if $n < \\overline{n}$ and $m_n = \\overline{n+1}$ otherwise.\n\nTo implement selective recoupling between the $i$-th and the $j$-th spins, the\nsign matrix should have equal $i$-th and $j$-th rows but any other two rows\nshould be orthogonal.\nThe coupling term $g_{ij}^{\\rm c} \\sigma_z^{(i)} \\otimes \\sigma_{\\rm z}^{(j)}$\nnever changes sign and that coupling is implemented, while all other couplings\nare removed.\nThe sign matrix can be obtained from a normalized $H(\\overline{n})$ by\nreplacing the $1$-st row with the $j$-th row, and replacing the $j$-th row\nwith the $i$-th row.  This scheme also removes ${\\cal H}_{\\rm Z}$ and $m_n =\n\\overline{n}$.\nTo implement $Z\\!\\!Z_{ij}$, the duration of each interval $t$ is chosen\nto satisfy $g_{ij}^{\\rm c} \\overline{n} t = \\pi\/4$.  Note that the total time\nused to implement $Z\\!\\!Z_{ij}$ is the shortest possible, since the coupling\nis always ``on''.\n\nThe scheme requires $m_n$ time intervals.  Consequently, it requires at\nmost $n m_n$ pulses, since $XX=I$ and the $X$ pulses are only used in pairs.  \nThe remaining question is: how does $m_n$ depend on $n$?\nFor simplicity, we consider $\\overline{n}$ in place of $m_n$. \nIf Hadamard matrices exist and can be constructed for all orders,\n$\\overline{n} = n$.\nHowever, some Hadamard matrices are missing, either because no construction\nmethods are known or they simply cannot exist.\nDue to missing Hadamard matrices, $\\overline{n} = cn$ where $c \\geq 1$.  \nWe argue in the following that the scheme is still very efficient.  \nFirst of all, we prove $c < 2$.  For each $n$, there exists $r$ such that \n$2^{r-1} \\!  \\leq n < 2^r$.  Since $H(2^r)$ exists by Sylvester's\nconstruction, $cn = \\overline{n} \\leq 2^r \\!\\! < 2n$.\nWe now show that $c$ is close to the ideal value 1 in most cases, due to the\nexistence of Hadamard matrices of orders other than powers of 2.\nThis is why the full connection to Hadamard matrices is important.  \nIn Fig.~\\ref{fig:c}, $c$ as a function of $n$ is plotted for $n \\leq 10000$.  \nWithin this technologically relevant range of $n$, $c$ deviates significantly\nfrom 1 only for a few exceptional values of $n$ when $n$ is small.\nOne arrives at the same conclusion by considering the gap $\\delta_n =\n\\overline{n} - \\underline{n} > \\overline{n} - n$, which bounds the extra\nnumber of intervals needed in the scheme caused by missing Hadamard matrices.\nFor $n \\leq 10000$, $\\delta_n\/n \\ll 1$ except for the few exceptional values\nof $n$ as a numerical fact.\nFor completeness, we present arguments for $c \\approx 1$ for {\\em arbitrarily\nlarge} $n$ in Appendix~\\ref{sec:largen}.  \nThis is based on Paley's construction and the prime number theorem. \nFinally, if Hadamard's conjecture is proven, $\\delta_n \\leq 3$ $\\forall n$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\mbox{\\psfig{file=cvsnsmall.eps,width=1.7in}\\psfig{file=cvsn.eps,width=1.74in}}\n\\vspace*{2ex}\n\\caption{Plots of $c$ vs $n$, where $cn = \\overline{n} = m_n$ is the minimun\nnumber of time intervals required to perform decoupling or selective\nrecoupling for an $n$-spin system.  $c$ for $n \\leq 100$ and $101 \\geq n \\leq\n10000$ are plotted separately.}\n\\label{fig:c}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion} \n\nWe reduce the problem of decoupling and selective recoupling in heteronuclear\nspin systems to finding sign matrices which is further reduced to finding\nHadamard matrices.\nWhile the most difficult task of constructing Hadamard matrices is not\ndiscussed in this paper, solutions already exist in the literature. \nEven more important is that the connection to Hadamard matrices results in\nvery efficient schemes.\n\nSome properties of the scheme are as follows.  \nFirst of all, the scheme is optimal in the following sense.  \nThe rows of Hadamard matrices and their negations form the codewords of first\norder Reed-Muller codes, which are {\\em perfect\ncodes}~\\cite{vanLint92,MacWilliams77}.\nIt follows that, for each Hadamard matrix, it is impossible to add an extra\nrow which is orthogonal to all the existing ones.\nTherefore, for a given $n$, $m_n = \\overline{n}$ is in fact the minimum number\nof time intervals necessary for decoupling or recoupling, if one restricts \nto the class of pulse sequences considered.\nSecond, the scheme applies for arbitrary duration of the time intervals.  \nThis is a consequence of the commutivity of all the terms in the hamiltonian,\nwhich in turn comes from the large separations of the Zeeman frequencies\ncompared to the coupling constants.  \nSpin systems can be chosen to satisfy this condition.\nFinally, disjoint pairs of spins can couple in parallel. \n\nWe outline possible simplifications of the scheme for systems with restricted\nrange of coupling.\nFor example, a linear spin system with $n$ spins but only $k$-nearest\nneighbor coupling can be decoupled by a scheme for $k$ spins only.  \nThe $i$-th row of the $n \\times \\overline{k}$ sign matrix can be chosen to be\nthe $r$-th row of $H(\\overline{k})$, where $i \\equiv r \\bmod k$.\nSelective recoupling can be implemented using a decoupling scheme for $k+1$\nspins.  The sign matrix is constructed as in decoupling using\n$H(\\overline{k+1})$ but the rows for the spins to be coupled are chosen to be\nthe $k+1$-th row different from all existing rows~\\cite{kplus1}.\nThis method involving periodic boundary conditions generalizes to other\nspatial structures.  The size of the scheme depends on $k$ and the exact \nspatial structure but not on $n$.\n\nThe scheme has several limitations.  \nFirst of all, it only applies to systems in which spins can be individually\naddressed by short pulses and coupling has the simplified form given by\nEq.(\\ref{eq:dipolar}).\nThese conditions are essential to the simplicity of the scheme.  They can all\nbe satisfied if the Zeeman frequencies have large separations.\nSecond, generalizations to include couplings of higher order than bilinear\nremain to be developed.\nFurthermore, in practice, RF pulses are inexact and have finite\ndurations, leading to imperfect transformations and residual errors.\n\nThe present discussion is only one example of a more general issue, that the\nnaturally occuring hamiltonian in a system does not directly give rise to\nconvenient quantum logic gates or other computations such as simulation of\nquantum systems~\\cite{Terhal98}.\nEfficient conversion of the given system hamiltonian to a useful form is\nnecessary and is an important challenge for future research.\n\n\\section{Acknowledgments}\n\nThis work was supported by DARPA under contract DAAG55-97-1-0341 and Nippon\nTelegraph and Telephone Corporation (NTT).  D.L. acknowledges support of an\nIBM Co-operative Fellowship.  We thank Hoi-Fung Chau, Kai-Man Tsang, Hoi-Kwong\nLo, Alex Pines, Xinlan Zhou and Lieven Vandersypen for helpful comments.\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nA key element in the development of machine learning methods is the exploitation of the underlying structure of the data through appropriate architectures.\nFor example, convolutional neural networks make use of local, translation-invariant correlations in image-like data and compile them into characteristic features \\cite{Hinton, Ciresan, ILSVRC2012, He2015, deOliveira:2015xxd, Aurisano:2016jvx, Komiske:2016rsd, Kasieczka:2017nvn, Erdmann:2017str, Shilon:2018xlp, Delaquis:2018zqi, Adams:2018bvi}.\n\nIn particle physics, characteristic features are used to identify particles, to separate signal from background processes, or to perform calibrations.\nThese features are usually expressed in manually engineered variables based on physics reasoning.\nAs input for machine learning methods in particle physics, these so-called high-level variables were in many cases superior to the direct use of particle four momenta, often referred to as low-level variables.\nRecently, several working groups have shown that deep neural networks can perform better if they are trained with both specifically constructed variables and low-level variables \\cite{Baldi2014:exotics, Baldi2015:higgs, Adam2015:kaggle, Guest2016:jetflavor, Baldi:2016fql, Shimmin:2017PhRvD, Louppe:2017ipp, Datta:2017rhs, Erdmann:ttH2017, Butter:2017cot, deOliveira:2017pjk, Stoye:2017, Sirunyan:2018mvw}.\nThis observation suggests that the networks extract additional useful information from the training data.\n\nIn this paper, we attempt to autonomize the process of finding suitable variables that describe the main characteristics of a particle physics task.\nThrough a training process, a neural network should learn to construct these variables from raw particle information.\nThis requires an architecture tailored towards the structure of particle collision events.\nWe will present such an architecture here and investigate its impact on the separation power for signal and background processes in comparison to domain-unspecific deep neural networks.\nFurthermore, we will uncover characteristic features which are identified by the network as particularly suitable.\n\nParticle collisions at high energies produce many particles which are often short-lived.\nThese short-lived particles decay into particles of the final state, sometimes through a cascade of multiple decays.\nIntermediate particles can be reconstructed by using energy-momentum conservation when a parent particle decays into its daughter particles.\nBy assigning to each particle a four-vector defined by the particle energy and its momentum vector, the sum of the four-vectors of the daughter particles gives the four-vector of the parent particle.\nFor low-energy particle collisions, bubble chamber images of parents and their daughter particles were recorded in textbook quality.\nEvidently, here the decay angular distributions of the daughter particles are distorted by the movement of the parent particle, but can be recovered in the rest frame of the parent particle after Lorentz transformation.\n\nThe same principles of particle cascades apply to high-energy particle collisions at colliders.\nHere, the particles of interest are, for example, top quarks or Higgs bosons, which remain invisible in the detector due to their short lifetimes, but can be reconstructed from their decay products.\nExploiting the properties of such parent particles and their daughter particles in appropriate rest frames is a key to the search for high-level variables characterizing a physics process.\n\nFor the autonomous search for sensitive variables, we propose a two-stage network, composed of a so-called Lorentz Boost Network (LBN) followed by an application-specific deep neural network (NN).\nThe LBN takes only the four-vectors of the final-state particles as input.\nIn the LBN there are two ways of combining particles, one to create composite particles, the other to form appropriate rest frames.\nUsing Lorentz transformations, the composite particles are then boosted into the rest frames.\nThus, the decay characteristics of a parent particle can be exploited directly.\n\nFinally, characteristic features are derived from the boosted composite particles: masses, angles, etc.\nThe second network stage (NN) then takes these variables as input to solve a specific problem, e.g. the separation of signal and background processes.\nWhile the first stage constitutes a novel network architecture, the latter network is interchangeable and can be adapted depending on the analysis task.\n\nThis paper is structured as follows.\nFirst, we explain the network architecture in detail.\nSecond, we present the simulated dataset we use to investigate the performance of our architecture in comparison to typical deep neural networks.\nThereafter, we review the particles, rest frames, and characteristic variables created by the network to gain insight into what is learned in the training process, before finally presenting our conclusions.\n\n\n\\section{Network architecture}\n\\label{sec:architecture}\n\nIn this section we explain the structural concept on which the Lorentz Boost Network (LBN) is based and introduce the network architecture.\n\nThe measured final state of a collision event is typically rather complex owing to the high energies of the particles involved.\nIn principle, all available information is encoded in the particles' four-vectors, but the comprehensive extraction of relevant properties poses a significant challenge.\nTo this end, physicists engineer high-level variables to decipher and factorize the probability distributions inherent to the underlying physics processes.\nThese high-level variables are often fed into machine learning algorithms to efficiently combine their descriptive power in the context of a specific research question.\nHowever, the consistent result of \\cite{Baldi2014:exotics, Baldi2015:higgs, Adam2015:kaggle, Guest2016:jetflavor, Baldi:2016fql, Shimmin:2017PhRvD, Louppe:2017ipp, Datta:2017rhs, Erdmann:ttH2017, Butter:2017cot, deOliveira:2017pjk, Stoye:2017, Sirunyan:2018mvw} is that the combination of both low- and high-level variables tends to provide superior performance.\nThis observation suggests that low-level variables potentially contain additional, useful information that is absent in hand-crafted high-level variables.\n\nThe aim of the LBN is, given only low-level variables as input, to autonomously determine a comprehensive set of physics-motivated variables that maximizes the relevant information for solving the physics task in the subsequent neural network application.\n\\Fig{fig:lbn_arch} shows the proposed two-stage network architecture (LBN+NN) in detail.\nThe first stage is the LBN and constitutes the novel contribution.\nIt consists of several parts, namely the combination of input four-vectors to particles and rest frames, subsequent Lorentz transformations, and the extraction of suitable high-level variables.\nThe second stage can be some form of deep neural network (NN) with an objective function depending on the specific research question.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{lbn_architecture.pdf}\n    \\caption{\n        The two-stage deep neural network architecture consists of the Lorentz Boost Network (LBN) and a subsequent deep neural network (NN).\n        In the LBN, the input four-vectors ($E, p_x, p_y, p_z$) are combined in two independent ways before each of the combined particles is boosted into its particular rest frame, which is formed from a different particle combination.\n        The boosted particles are characterized by variables which can be, e.g., invariant masses, transverse momenta, pseudorapidities, and angular distances between them.\n        These features serve as input to the second network designed to accomplish a particular analysis task.\n    }\n    \\label{fig:lbn_arch}\n\\end{figure}\n\nThe LBN combines $N$ input four-vectors, consisting of energies $E$ and momentum components $p_x, p_y, p_z$, to create $M$ particles and $M$ corresponding rest frames according to weighted sums using trainable weights.\nThrough Lorentz transformation, each combined particle is boosted into its dedicated rest frame.\nAfter that, a generic set of features is extracted from the properties of these $M$ boosted particles.\n\nExamples of variables that can be reconstructed with this structure include spin-dependent angular distributions, such as those observed during the decay of a top quark with subsequent leptonic decay of the W boson.\nBy boosting the charged lepton into the rest frame of the W boson, its decay angular distribution can be investigated.\nIn more sophisticated scenarios, the LBN is also capable of accessing further properties that rely on the characteristics of two different rest frames.\nAn example is a variable usually referred to as $\\cos(\\theta^*)$, which is defined by the angular difference between the directions of the charged lepton in the W boson's rest frame, and the W boson in the rest frame of the top quark.\nThe procedure of how this variable can be reconstructed in the LBN is depicted in \\Fig{fig:lbn_example}.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{lbn_theta_star.pdf}\n    \\caption{\n        Example of a possible feature engineering in top quark decays addressing the angular distance of the direction of the W boson in the top rest system and the direction of the lepton in the W boson rest system, commonly referred to as $\\cos(\\theta^*)$.\n    }\n    \\label{fig:lbn_example}\n\\end{figure}\n\nThe number $N$ of incoming particles is to be chosen according to the research question.\nThe number of matching combinations $M$ is a hyperparameter to be adjusted.\nIn this paper we introduce a specific version of the LBN which constructs $M$ particle combinations, and each combination will have its own suitable rest system.\nOther variants are conceivable and will be mentioned below.\n\nIn the following paragraphs we describe the network architecture in detail.\n\n\\subsection*{Combinations}\n\\label{sec:architecture:combinations}\n\nThe purpose of the first LBN layer is to construct arbitrary particles and suitable rest frames for subsequent boosting.\nThe construction is realized via linear combinations of $N$ input four-vectors,\n\\begin{equation}\n    X =\n    \\begin{bmatrix}\n        E_1    & p_{x,1} & p_{y,1} & p_{z,1}\\\\\n        E_2    & p_{x,2} & p_{y,2} & p_{z,2}\\\\\n        \\vdots & \\vdots  & \\vdots  & \\vdots\\\\\n        E_N    & p_{x,N} & p_{y,N} & p_{z,N}\n    \\end{bmatrix},\n\\end{equation}\nto a number of $M$ particles and rest frames, which are represented by four-vectors accordingly.\nHere, $M$ is a hyperparameter of the LBN and its choice is related to the respective physics application.\nThe coefficients $W$ of all linear combinations $C$,\n\\begin{equation}\n    C_{m} = \\sum_{n=1}^N W_{mn} \\cdot X_{n}\n\\end{equation}\nwith $m \\in \\left[1, M\\right]$, are free parameters and subject to optimization within the scope of the training process.\nIn the following, combined particles and rest frames are referred to as $C^P$ and $C^R$, respectively.\nTaking both into consideration, this amounts to a total of $2 \\cdot N \\cdot M$ degrees of freedom in the LBN.\nIn order to prevent the construction of objects which would lead to unphysical implications when applying Lorentz transformations, i.e., four-vectors not fulfilling $E > m > 0$, all parameters $W_{mn}$ are restricted to positive values.\nWe initialize the weights randomly according to a half-normal distribution with mean $0$ and standard deviation $1 \/ M$.\nIt should be noted that, in order to maintain essential physical properties and relations between input four-vectors, feature normalization is not applied at this point.\n\n\n\\subsection*{Lorentz transformation}\n\\label{sec:architecture:boost}\n\nThe boosting layer performs a Lorentz transformation of the combined particles into their associated rest frames.\nThe generic transformation of a four-vector $q$ is defined as $q^* = \\Lambda \\cdot q$ with the boost matrix\n\\begin{align}\n    \\Lambda &=\n    \\begin{bmatrix}\n        \\gamma           & -\\gamma\\beta n_x       & -\\gamma\\beta n_y       & -\\gamma\\beta n_z\\\\\n        -\\gamma\\beta n_x & 1 + (\\gamma - 1) n_x^2 & (\\gamma - 1) n_x n_y   & (\\gamma - 1) n_x n_z\\\\\n        -\\gamma\\beta n_y & (\\gamma - 1) n_y n_x   & 1 + (\\gamma - 1) n_y^2 & (\\gamma - 1) n_y n_z\\\\\n        -\\gamma\\beta n_z & (\\gamma - 1) n_z n_x   & (\\gamma - 1) n_z n_y   & 1 + (\\gamma - 1) n_z^2\n    \\end{bmatrix}.\n    \\label{eqn:lambda}\n\\end{align}\nThe relativistic parameters $\\gamma = E \/ m$, $\\vec{\\beta} = \\vec{p} \/ E$, and $\\vec{n} = \\vec{\\beta} \/ \\beta$ are to be derived per rest-frame four-vector $C^R_m$.\n\nThe technical implementation within deep learning algorithms requires a vectorized representation of the Lorentz transformation.\nTo this end, we rewrite the boost matrix in \\Eqn{eqn:lambda} as\n\\begin{align}\n    \\Lambda &= I \\, + \\, (U \\oplus \\gamma) \\, \\odot \\, ((U \\oplus 1) \\cdot \\beta \\, - \\, U) \\, \\odot \\, (e \\cdot e^T)\\\\[2mm]\n    \\text{with} \\quad U &=\n    \\begin{bmatrix}\n        -1^{1 \\times 1} & 0^{1 \\times 3}\\\\\n        0^{3 \\times 1}  & -1^{3 \\times 3}\n    \\end{bmatrix},\n    \\quad e =\n    \\begin{bmatrix}\n        1^{1 \\times 1}\\\\\n        -\\vec{n}^{3 \\times 1}\n    \\end{bmatrix},\n\\end{align}\nand the $4 \\times 4$ unit matrix $I$.\nThe operators $\\oplus$ and $\\odot$ denote elementwise addition and multiplication, respectively.\nThis notation also allows for the extension by additional dimensions to account for the number of combinations $M$ and an arbitrary batch size.\nAs a result, all boosted four-vectors $B$ are efficiently computed by a single, broadcasted matrix multiplication, $B = \\Lambda^R \\cdot C^P$.\n\nWhile the description above focuses on a pairwise mapping approach, i.e., particle $C^P_m$ is boosted into rest frame $C^R_m$, other configurations are conceivable:\n\\begin{itemize}\n    \\item\n    The input four-vectors are combined to build $M$ particles and $K$ rest frames.\n    Each combined particle is transformed into all rest frames, resulting in $K \\cdot M$ boosted four-vectors.\n\n    \\item\n    The input four-vectors are combined only to build $M$ particles, which simultaneously serve as rest frames.\n    Each combined particle is transformed into all rest frames derived from the other particles, resulting in $M^2 - M$ boosted four-vectors.\n\\end{itemize}\nThe specific advantages of these configurations can, in general, depend on aspects of the respective physics application.\nResults of these variants will be the target of future investigations.\n\n\n\\subsection*{Feature extraction}\n\\label{sec:architecture:features}\n\nFollowing the boosting layer, features are extracted from the Lorentz transformed four-vectors, which can then be utilized in a subsequent neural network.\nFor the projection of $M \\times 4$ particle properties into $F \\times 1$ features, we employ a distinct yet customizable set of generic mappings.\nThe autonomy of the network is not about finding entirely new features, but rather about factorizing probability densities in the most effective way possible to answer a scientific question.\nTherefore, we let the LBN network work autonomously to find suitable particle combinations and rest frames which enable this factorization, but then use established particle characterizations.\n\nWe differentiate between two types of generic mappings:\n\\begin{enumerate}\n    \\item\n    Single features are extracted per boosted four-vector.\n    Besides the vector elements ($E$, $p_x$, $p_y$, $p_z$) themselves, derived features such as mass, transverse and longitudinal momentum, pseudorapidity, and azimuth can be derived.\n\n    \\item\n    Pairwise features are extracted for all pairs of boosted four-vectors.\n    Examples are the cosine of their spatial angular difference, their distance in the $\\eta-\\phi$ plane, or their distance in Minkowski space.\n    In contrast to single features, pairwise features introduce close connections among boosted four-vectors and, by means of backpropagation, between trainable combinations of particles and rest frames.\n\\end{enumerate}\nThe set of extracted features is concatenated to a single output vector.\nProvided that the employed batch size is sufficiently large, batch normalization with floating averages adjusted during training can be performed after this layer \\cite{batch_norm}.\n\n\n\\subsection*{Subsequent problem-specific application}\n\\label{sec:architecture:application}\n\nIn the previous sections, we described the LBN as shown in the left box in \\Fig{fig:lbn_arch}, namely how input vectors are combined and boosted into dedicated rest frames, followed by how features of these transformed four-vectors are compiled.\nThese features are intended to maximize the information content to be exploited in a subsequent, problem-specific deep neural network NN as shown in the right box in \\Fig{fig:lbn_arch}.\n\nThe objective function of the NN defines the type of learning process.\nWeight updates are performed as usual through backpropagation.\nThese updates apply to the weights of the subsequent network as well as to the trainable weights of the combination layer in the LBN.\n\nIn the following, we evaluate how well the autonomous feature engineering of the compound LBN+NN network operates.\nWe compare its performance with only low-level information to that of typical, fully-connected deep neural networks (DNNs).\nWe alternatively supply the DNNs with low-level information, sophisticated high-level variables (see \\Apx{sec:appendix}), and the combination of both.\n\n\n\\section{Simulated datasets}\n\\label{sec:simulations}\n\nThe Pythia $8.2.26$ program package \\cite{Pythia} was used to simulate $\\ttH$ and $\\ttbb$ events.\nExamples of corresponding Feynman diagrams are shown in \\Fig{fig:feynman}.\nThe matrix elements contain angular correlations of the decay products from heavy resonances.\nBeam conditions correspond to LHC proton-proton collisions at $\\sqrt{s} = 13$\\,TeV.\nOnly the dominant gluon-gluon process is enabled and Higgs boson decays into bottom quark pairs are favored.\nHadronization is performed with the Lund string fragmentation model.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{feynman_ttH.pdf}\n        \\caption{}\n        \\label{fig:feynman_ttH}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{feynman_ttbb.pdf}\n        \\caption{}\n        \\label{fig:feynman_ttbb}\n    \\end{subfigure}\n    \\caption{Example Feynman diagrams of a) $\\ttH$ and b) $\\ttbb$ processes.}\n    \\label{fig:feynman}\n\\end{figure}\n\nTo analyze a typical final state as observed in an LHC detector, we use the DELPHES package \\cite{deFavereau:2013fsa}.\nDELPHES provides a modular framework designed to parameterize the simulation of a multi-purpose detector.\nIn our study, we chose to simulate the CMS detector response.\nAll important effects such as pile-up, deflection of charged particles in magnetic fields, electromagnetic and hadronic calorimetry, and muon detection systems are covered.\nThe simulated output consists of muons, electrons, tau leptons, photons, jets, and missing transverse momentum from a particle flow algorithm.\nAs the neutrino leaves the detector without interacting, we identify its transverse momentum components from the measurement of missing transverse energy, and we reconstruct its longitudinal momentum by constraining the mass of the leptonically decaying W boson to $80.4$\\,GeV.\n\nFollowing the detector simulation, an event selection with the following criteria was carried out:\nEvents must have at least six jets clustered with the anti-$k_T$ algorithm, implemented in the FastJet package \\cite{Cacciari:2011ma}, with a radius parameter of $\\Delta R = 0.4$.\nA jet is accepted if its transverse momentum fulfills $p_t > 25$\\,GeV and its absolute pseudorapidity is within $|\\eta| < 2.4$.\nFurthermore, exactly one electron or muon with $p_t > 20$\\,GeV and $|\\eta| < 2.1$ is required.\nEvents with further leptons are rejected to focus solely on semi-leptonic decays of the $\\ttbar$ system.\nFinally, a matching is performed between the final-state partons of the generator and the jets of the detector simulation with the maximum distance $\\Delta R = 0.3$, whereby the matching must be unambiguously successful for all partons.\nFor $\\ttH$ events, the combined selection efficiency was measured as $2.3$\\,\\%.\n\nFor $\\ttbb$ processes, further measures are taken to identify the two additional bottom quark jets.\nThe definition is an adaption of \\cite{Sirunyan:2018mvw}.\nAt generator level, the anti-$k_T$ algorithm with a radius parameter of $\\Delta R = 0.4$ is used to cluster all stable final-state particles.\nIf a generator jet is found to originate from one of the top quarks or W boson decays, or if its transverse momentum is below a threshold of $20$\\,GeV, it is excluded from further considerations.\nFor each remaining jet, we then count the number of contained bottom hadrons using the available hadronization and decay history.\nAn event is a $\\ttbb$ candidate if at least two generator jets contain one or more distinct bottom hadrons.\nFinally, a matching is performed between the four quarks resulting from the $\\ttbar$ decay and the two identified generator jets on the one hand, and the selected, reconstructed jets on the other hand.\nSimilar to $\\ttH$, a $\\ttbb$ event is accepted if all six generator-level objects are unambiguously matched to a selected jet.\nThis leaves a fraction of $0.02$\\,\\% of the generated $\\ttbb$ events.\n\nA total of $10^6$ events remain, consisting of an evenly distributed number of $\\ttH$ and $\\ttbb$ events, which are then divided into training and validation datasets at a ratio of $80:20$.\n\nFor further analysis, a distinct naming scheme for reconstructed jets is introduced that is inspired by the semi-leptonic decay characteristics of the $\\ttbar$ system.\nThe two light quark jets of the hadronically decaying W boson are named $\\qi$ and $\\qii$, where the index $1$ refers to the jet with the greater transverse momentum.\nThe bottom jet that is associated to this W boson within a top quark decay is referred to as $\\bhad$.\nAccordingly, the bottom quark jet related to the leptonically decaying W boson is referred to as $\\blep$.\nThe remaining jets are named $\\bi$ and $\\bii$, and transparently refer to the decay products of the Higgs boson for $\\ttH$, or to the additional b quark jets for $\\ttbb$ events.\n\nIn \\Fig{fig:dataset} we show, by way of example, distributions of the generated $\\ttH$ and $\\ttbb$ events.\nIn \\Fig{fig:dataset-a} we compare the transverse momentum of the jet $\\bi$, and in \\Fig{fig:dataset-b} the largest difference in pseudorapidity between a jet and the charged lepton.\nIn \\Fig{fig:dataset-c} we show the invariant mass of the closest jet pair, and in \\Fig{fig:dataset-d} the event sphericity.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{dataset_bj1_pt.pdf}\n        \\caption{}\n        \\label{fig:dataset-a}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{dataset_jet_lep_max_abs_deta.pdf}\n        \\caption{}\n        \\label{fig:dataset-b}\n    \\end{subfigure}\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{dataset_jet_closest_pair_mass.pdf}\n        \\caption{}\n        \\label{fig:dataset-c}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{dataset_sphericity.pdf}\n        \\caption{}\n        \\label{fig:dataset-d}\n    \\end{subfigure}\n    \\caption{\n        Exemplary comparisons of kinematic distributions of $\\ttH$ and $\\ttbb$ events; a)~transverse momentum of the jet $\\bi$, b)~largest difference in pseudorapidity of a jet to the charged lepton, c)~invariant mass of the closest jet pair, d)~event sphericity.\n    }\n    \\label{fig:dataset}\n\\end{figure}\n\n\n\\section{Benchmarks of network performance}\n\\label{sec:performance}\n\nAs a benchmark, the LBN is utilized in the classification task of distinguishing a signal process from a background process.\nIn this example, we use the production of a top quark pair in association with a Higgs boson decaying into bottom quarks ($\\ttH$) as a signal process, and top quark pairs produced with $2$ additional bottom jets ($\\ttbb$) as a background process.\nIn both processes, the final state consists of eight four-vectors, four of which represent bottom quark jets, two describe light quark jets, one denotes a charged lepton, and one a neutrino.\n\nWithin the benchmark, the LBN competes against other frequently used deep neural network setups, which we refer to as DNN here.\nFor a meaningful comparison, we perform extensive searches for the optimal hyperparameters in each setup.\nAs explained above, the LBN consists of only very few parameters to be trained in addition to those of the following neural network (LBN+NN).\nStill, by varying the number $M$ of particle combinations to be constructed from the eight input four-vectors, and by varying the number $F$ and types of generated features, we performed a total of $346$ training runs of the LBN+NN setup, and a similar amount for the competing DNN.\nThe best performing architectures are listed in \\Apx{sec:appendix}.\n\nThe LBN network exclusively receives the four-vectors of the eight particles introduced above, which shall be denoted by 'LBN low' in the following.\nFor the networks marked with DNN we use three variants of inputs.\nIn the first variant, the DNN receives only the four-vectors of the eight particles ('DNN-low').\nFor the second variant, physics knowledge is applied in the form of $26$ sophisticated high-level variables which are inspired by \\cite{Sirunyan:2018mvw} and incorporate comprehensive event information ('DNN-high').\nA list of these variables, such as the event sphericity \\cite{event_shape_variables} and Fox-Wolfram moments \\cite{fox_wolfram_moments}, can be found in the \\Apx{sec:appendix}.\nIn the third variant, we provide the DNNs with both low-level and high-level variables as input ('DNN-combined').\n\nIn the first set of our benchmark tests, the task of input particle identification is bypassed by using the generator information in order to exclusively measure the performance of the LBN.\nJets associated to certain quarks through matching are consistently placed at the same position of the $N$ input four-vectors.\nWe quantify how well the event classification of the networks performs with the integral of the receiver operating characteristic curve (ROC AUC).\n\n\\Fig{fig:perf_comparison_a} shows the performance of all training runs involved in the hyperparameter search.\nThe best training is marked with a horizontal line, while the distribution of results is denoted by the orange and blue shapes which manifestly depend on the structure of the scanned hyperparameter space.\nFor the LBN, the training runs achieve stable results since there are only minor variations in the resulting ROC AUC.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{training_distributions_best.pdf}\n        \\caption{}\n        \\label{fig:perf_comparison_a}\n    \\end{subfigure}%\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{training_distributions_reco.pdf}\n        \\caption{}\n        \\label{fig:perf_comparison_b}\n    \\end{subfigure}\n    \\caption{\n        Benchmarks of the LBN performance in comparison to typical deep neural network architectures (DNNs) with three variants of input variables;\n        low=four-vectors, high=physics-motivated variables, combined=low+high.\n        Ordering of the input four-vectors is done by a) generator information and b) jet transverse momenta.\n    }\n    \\label{fig:perf_comparison}\n\\end{figure}\n\nAlso shown are the training runs of the three input variants for the DNNs.\nIf the eight input four-vectors are ordered according to the generator information, the 'DNN low' setup already achieves results equivalent to the combination of low-level and high-level variables ('DNN combined').\nHowever, both setups are unable to match the 'LBN low' result.\nNote that high-level variables are mostly independent of the input ordering by construction, and hence contain reduced information compared to the ordered four-vectors.\nConsequently, the weaker performance of the 'DNN high' is well understood.\n\nIn the second set of our benchmark test, we disregard generator information and instead sort the six jets according to their transverse momenta, followed by the charged lepton and the neutrino.\nThis order can be easily established in measured data.\n\\Fig{fig:perf_comparison_b} shows that this alternative order consistently reduces the ROC AUC score of all networks.\n\n\nThe training runs of the LBN again show stable results with few exceptions and achieve the best performance in separating events from signal and background processes.\nFor the DNNs, the above-mentioned hierarchy emerges, i.e., the best results are achieved using the combination of high- and low-level variables, followed by only high-level variables with reasonably good results,\nwhereas the DNN exhibits the weakest performance with only low-level information.\n\nOverall, the performance of the compound LBN+NN structure based on the four-vector inputs is quite convincing in this demanding benchmark test.\n\n\n\\section{Visualizations of network predictions}\n\\label{sec:insights}\n\nSeveral components of the LBN architecture have a direct physics interpretation.\nTheir investigation can provide insights into what the network is learning.\nIn the following, we investigate only the best network trained on the generator-ordered input in order to establish which of the quark and lepton four-vectors are combined.\n\nThe number of particle combinations to create is a hyperparameter of the architecture.\nFor the generator ordering, we obtain the best results for $13$ particles.\nThis matches well with our intuition since the Feynman diagram also contains $13$ particles in the cascade.\nIn total we extract $F = 143$ features in the LBN that serve as input to the subsequent deep neural network NN (\\Fig{fig:lbn_arch}).\nFor details refer to the \\Apx{sec:appendix}.\n\nIn our particular setup, each combined particle has a separate corresponding rest frame.\nAs a consequence of demanding the second network to solve a research question, the decisions about which four-vectors are combined for the composite particle and which for its rest frame are strongly correlated.\nA correlation also exists between all $13$ systems of combined particles and their rest frames from the pairwise angular distances exploited by the LBN as additional properties.\n\nWe start by looking at the weights used to combine the input four-vectors to form the particle combinations and rest frames.\n\n\n\\subsection*{Weight matrices of particle and rest-frame combinations}\n\n\\Fig{fig:weight_gen_particles_normed} shows the weight matrices of the LBN.\nEach column of the two matrices for combined particles (red) and rest frames (green) is normalized so that the weight of each input four-vector (bottom quark jet $\\bi$ of the Higgs, etc.) is shown as a percentage.\nThe color code reflects the sum of the particle weights for the Higgs boson (or $b\\bar{b}$ system for $\\ttbb$) and the two top quarks, respectively.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{1\\textwidth}\n        \\centering\n        \\includegraphics[width=0.65\\textwidth]{weight_particles.pdf}\n        \\caption{}\n        \\label{fig:weight_gen_particles_normed-a}\n    \\end{subfigure}\n    \\begin{subfigure}{1\\textwidth}\n        \\centering\n        \\includegraphics[width=0.65\\textwidth]{weight_restframes.pdf}\n        \\caption{}\n        \\label{fig:weight_gen_particles_normed-b}\n    \\end{subfigure}\n    \\caption{\n        Particle combinations (red) and their rest frames (green).\n        The numbers represent the relative weights of the input particles for each combination $i$.\n        The color code reflects the sum of the particle weights for the Higgs boson ($\\ttH$), the $b\\bar{b}$ system ($\\ttbb$), or the two top quarks.\n        For better clarity of the presentation, the zero weights are kept white.\n    }\n    \\label{fig:weight_gen_particles_normed}\n\\end{figure}\n\nNote that combined particles and rest frames are complementary.\nExtreme examples of this can be seen in columns $2$ and $6$, in which one of the two bottom quark jets from the Higgs boson decay was selected as the rest frame and a broad combination of all other four-vector vectors was formed whose properties are exploited for the event classification.\n\nConversely, in column $0$ a Higgs-like combination is transformed into the rest frame of all particles, to which the hadronic top quark makes the main contribution.\nSimilarly, in columns $1$ and $7$, top quark-like combinations are formed and also transformed into rest frames formed by many four-vectors.\nThese types of combinations allow for the Lorentz boost of the center-of-mass system of the scattering process to be nearly compensated.\n\nIt is striking that four-vector combinations forming a top quark are often selected.\nIn \\Fig{fig:correlations}, we quantitatively assess the $13$ possible combinations of the combined particles (red) and the $13$ rest frames (green) in order to determine which combinations are typically formed between the eight input four-vectors.\n\\begin{figure}[htbp]\n    \\centering\n    \\begin{subfigure}{.49\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{correlations_particles.pdf}\n        \\caption{}\n        \\label{fig:correlations-a}\n    \\end{subfigure}\n    \\begin{subfigure}{.49\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{correlations_restframes.pdf}\n        \\caption{}\n        \\label{fig:correlations-b}\n    \\end{subfigure}\n    \\caption{\n        Correlations of quarks and leptons for particle combinations (red) and for rest frames (green).\n        Both the numbers and the color code represent the correlation strength.\n    }\n    \\label{fig:correlations}\n\\end{figure}\n\nIn order to build rest frames (green), the LBN network recognizes combinations leading to the top quarks and treats the bottom quark jets from the Higgs individually.\nThis appears to be an appropriate choice as the top quark pair is the same in $\\ttbb$ and $\\ttH$ events while the two bottom quarks typically originate from gluon splitting or the Higgs decay, respectively.\nFor the hadronic top quark, the W boson ($75$\\,\\%) is often accounted for as well.\nWith the leptonic top, the LBN network considers bottom quark jet and lepton ($91$\\,\\%).\nThe lepton and neutrino are related to form the W ($68$\\,\\%).\n\nTo form the particle combinations (red), the LBN network builds the light quark jets to the W boson ($78$\\,\\%) and the hadronic top quark.\nIn the leptonic top quark, the LBN combines bottom quark jets and leptons ($61$\\,\\%).\nSometimes the lepton and the neutrino are also correlated to build the W boson ($22$\\,\\%).\nThe LBN rarely combines the Higgs boson ($-4$\\,\\%).\n\nThe positive and negative correlations show that the LBN forms physics-motivated particle combinations from the training data.\nIn the next step, we investigate the usefulness of these combinations for the separation of $\\ttH$ and $\\ttbb$ events.\n\n\\subsection*{Distributions resulting from combined particle properties}\n\nThe performance of particle combinations and their transformations into reference frames in the LBN is evaluated through the extracted features which support the classification of signal and background events.\nHere, we present the distributions of different variables calculated from the combined particles.\n\n\\Fig{fig:m_gen_ttH_a} shows the invariant mass $m$ distributions of all $13$ combined particles for the $\\ttH$ signal dataset.\n\\Fig{fig:m_gen_ttH_b} shows the mass distributions for the $\\ttbb$ background dataset.\nThe difference between the two histograms is shown in Figure~\\ref{fig:m_gen_ttH_c}, where combined particle $0$ of the first column shows the largest difference between signal and background.\n\\Fig{fig:weight_gen_particles_normed} shows that this combination $0$ approximates a Higgs-like configuration (red).\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{feature_m_gen_ttH.pdf}\n        \\caption{}\n        \\label{fig:m_gen_ttH_a}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{feature_m_gen_ttbb.pdf}\n        \\caption{}\n        \\label{fig:m_gen_ttH_b}\n    \\end{subfigure}\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{feature_m_gen_diff.pdf}\n        \\caption{}\n        \\label{fig:m_gen_ttH_c}\n    \\end{subfigure}%\n   \n   \n   \n    \\caption{\n        Masses of combined particles for a) $\\ttH$, b) $\\ttbb$, and c) the differences between $\\ttH$ and $\\ttbb$.\n    }\n    \\label{fig:m_gen_ttH}\n\\end{figure}\n\nFor this Higgs-like configuration (combined particle $0$), we show mass distributions in \\Fig{fig:slices_a}.\nFor $\\ttH$, the invariant mass exhibits a narrow distribution around $m = 22$ a.u. reflecting the Higgs-like combined particle, while the $\\ttbb$ background is widely distributed.\nNote that because of the weighted four-vector combinations, arbitrary units are used, so that the Higgs boson mass is not stated in GeV.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_m_gen_0.pdf}\n        \\caption{}\n        \\label{fig:slices_a}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_m_gen_5.pdf}\n        \\caption{}\n        \\label{fig:slices_b}\n    \\end{subfigure}\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_pt_gen_0.pdf}\n        \\caption{}\n        \\label{fig:slices_c}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_pt_gen_5.pdf}\n        \\caption{}\n        \\label{fig:slices_d}\n    \\end{subfigure}\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_eta_gen_0.pdf}\n        \\caption{}\n        \\label{fig:slices_e}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_eta_gen_5.pdf}\n        \\caption{}\n        \\label{fig:slices_f}\n    \\end{subfigure}\n    \\caption{\n        Example comparisons of kinematic distributions for $\\ttH$ and $\\ttbb$ processes.\n        Figures a) and b) show the masses of the combined particles $0$ and $5$.\n        Figures c)-f) also present their transverse momenta and pseudorapidities in the rest frames.\n    }\n    \\label{fig:slices}\n\\end{figure}\n\nAnalogously, we determine the transverse momentum $p_t$ and pseudorapidity $\\eta$ distributions of the $13$ particles for signal and background events, and then compute their difference to visualize distinctions between $\\ttH$ and $\\ttbb$.\nWhile the invariant mass distribution can also be formed without Lorentz transformation, both the $p_t$ and $\\eta$ distributions are determined in the respective rest frames.\nClear differences in the distributions are apparent in \\Fig{fig:gen_ttH_ttbb_a} for the transverse momenta of the combined particles $11, 8, 5, 10$ (in descending order of importance), while the combined particles $5, 9, 12, 10$ are most prominent for the pseudorapidities (\\Fig{fig:gen_ttH_ttbb_b}).\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{feature_pt_gen_diff.pdf}\n        \\caption{}\n        \\label{fig:gen_ttH_ttbb_a}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.9\\textwidth]{feature_eta_gen_diff.pdf}\n        \\caption{}\n        \\label{fig:gen_ttH_ttbb_b}\n    \\end{subfigure}\n    \\caption{\n        Differences between $t\\ttH$ and $\\ttbb$ processes for combined particles in the rest frames of particle combinations, a) transverse momenta and b) pseudorapidities.\n    }\n    \\label{fig:gen_ttH_ttbb}\n\\end{figure}\n\nAs a selection of the many possible distributions, in \\Fig{fig:slices} we show the invariant mass $m$, the transverse momentum $p_t$, and the pseudorapidity $\\eta$ for two combined particles.\n\\Figs{fig:slices_a}, \\ref{fig:slices_c}, and \\ref{fig:slices_e} present the distributions of the combined Higgs-like particle $0$.\nIn addition to the prominent difference in the mass distributions, for $\\ttH$ events its $p_t$ is smaller while its direction is more central in $\\eta$ compared to $\\ttbb$ events.\n\nFurthermore, in \\Figs{fig:slices_b}, \\ref{fig:slices_d}, and \\ref{fig:slices_f} we show combined particle $5$ because of its separation power in pseudorapidity and in transverse momentum (\\Fig{fig:gen_ttH_ttbb}).\nCombination $5$ is essentially determined by the bottom quark jet $\\bi$ boosted into a rest frame of all other four-vectors (\\Fig{fig:weight_gen_particles_normed}).\nHere, the distribution of mass $m$ for $\\ttH$ events is smaller, $p_t$ is larger and more defined, and $\\eta$ is more central than for $\\ttbb$ events.\n\nIn the example of the characteristic angular distribution of the charged lepton in top quark decays (\\Fig{fig:lbn_example}, $\\cos{\\theta^*}$), two different reference systems are combined to determine the opening angle $\\theta^*$ between the W boson in the top quark rest frame and the lepton in the W boson rest frame.\nAs mentioned above, the LBN is capable of calculating these complex angular correlations involving four-vectors evaluated in different rest frames.\n\nAs an example, we select distributions for angular distances of combined particles $2$ and $5$ (\\Fig{fig:weight_gen_particles_normed}).\nParticle $2$ is a combination of all four-vectors boosted into the rest frame of the bottom quark jet $\\bii$, whereas particle $5$ corresponds to the bottom quark jet $\\bi$ boosted into a combination of all four-vectors.\n\\Fig{fig:cosangle_a} shows that, for $\\ttbb$ background events, combined particles $2$ and $5$ predominantly propagate back-to-back, while the $\\ttH$ signal distribution scatters broadly around $90$ deg.\n\\begin{figure}[h!tbp]\n    \\centering\n    \\begin{subfigure}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_pair_cos_gen_2_5.pdf}\n        \\caption{}\n        \\label{fig:cosangle_a}\n    \\end{subfigure}%\n    \\begin{subfigure}{.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{sliced_pair_cos_gen_2_6.pdf}\n        \\caption{}\n        \\label{fig:cosangle_b}\n    \\end{subfigure}\n    \\caption{\n        Cosine of the opening angles between combined particle $2$ ($\\ttbar + \\bi$) in its rest frame $\\bii$ and a) combined particle $5$ ($\\bi$) in its rest frame ($\\ttbar + \\bi + \\bii$), and b) combined particle $6$ ($\\ttbar + \\bii$) in its rest frame ($\\bi$).\n    }\n    \\label{fig:cosangle}\n\\end{figure}\n\nThe opposite holds true for the angles between combined particles $2$ and $6$ (\\Fig{fig:weight_gen_particles_normed}).\nHere, the bottom quark jet $\\bi$, or alternatively $\\bii$, serves as a rest frame for characterizing combinations of both top quarks together with the other bottom quark jet.\n\\Fig{fig:cosangle_b} shows that, for $\\ttbb$, combined particles $2$ and $6$ preferably propagate in the same direction, whereas for $\\ttH$ signal events this is less pronounced.\n\nOverall, it can be concluded that the LBN network, when given only low-level variables as input, seems to autonomously build and transform particle combinations leading to extracted features that are suitable for the task of separating signal and background events.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThe various physics processes from which the events of high-energy particle collisions originate lead to complex particle final states.\nOne reason for the complexity is the high energy that leads to Lorentz boosts of the particles and their decay products.\nFor an analysis task such as the identification of processes, we reconstruct aspects of the probability distributions from which the particles were generated.\nHow well probability distributions of individual physical processes are reflected is compiled in a series of characteristic variables.\nTo accomplish this, both the Minkowski metric for the correct calculation of energy-momentum conservation or invariant masses and the Lorentz transformation into the rest frames of parent particles are necessary.\nSo far, such characteristic variables have been engineered by physicists.\n\nIn this paper, we presented a general two-stage neural network architecture.\nThe first stage, the novel Lorentz Boost Network, contains at its core an efficient and fully vectorized implementation for performing Lorentz transformations.\nThe aim of this stage is to autonomously generate variables suitable for the characterization of collision events when using only particle four-vectors.\nFor this purpose, the input four-vectors are combined separately into composite particles and rest frames.\nThe composite particles are boosted into corresponding rest frames and a generic set of variables is extracted from the boosted particles.\nThe second stage of the network then uses these autonomously generated variables to solve a particular physics question.\nBoth the weights used to create the combinations of input four-vectors and the parameters of the second stage are learned together in a supervised training process.\n\nTo assess the performance of the LBN, we investigated a benchmark task of distinguishing $\\ttH$ and $\\ttbb$ processes.\nWe demonstrated the improved separation power of our model compared to domain-unspecific deep neural networks where the latter even used sophisticated high-level variables as input in addition to the four-vectors.\nFurthermore, we developed visualizations to gain insight into the training process and discovered that the LBN learns to identify physics-motivated particle combinations from the data used for training.\nExamples are top-quark-like combinations or the approximate compensation of the Lorentz boost of the center-of-mass system of the scattering process.\n\nThe LBN is a multipurpose method that uses Lorentz transformations to exploit and uncover structures in particle collision events.\nIt is part of an ongoing comparison of methods to identify boosted top quark jets and has already been successfully applied at the IML workshop challenge at CERN \\cite{iml2018}.\nThe source code of our implementation in TensorFlow \\cite{tensorflow} is publically available under BSD license \\cite{lbncode}.\n\n\\begin{appendix}\n\n\\section*{Appendix}\n\\label{sec:appendix}\n\n\\subsection*{Network parameters}\n\nThe network parameters of the best performing networks are illustrated in Table~\\ref{tab:network-parameters}.\n\\begin{table}[h!tbp]\n    \\centering\n    \\caption{\n        All deep neural networks use a fully connected architecture with $n_\\text{layers}$ and $n_\\text{nodes}$.\n        ELU is used as the activation function.\n        The Adam optimizer \\cite{kingma2014} is employed with decay parameters $\\beta_1 = 0.9$, $\\beta_2 = 0.999$, and a learning rate of $10^{-4}$.\n        $L_2$ normalization is applied with a factor of $10^{-4}$.\n        Batch normalization between layers was utilized in every configuration.\n        The generic mappings in LBN feature extraction layer create $E, p_T, \\eta, \\phi, m$, and pairwise $\\cos(\\varphi)$.\n    }\n    \\label{tab:network-parameters}\n    \\begin{tabular}{c|cc|cc|cc|cc}\n        \\toprule\n        Network & \\multicolumn{2}{c|}{\\textbf{LBN+NN}} & \\multicolumn{6}{c}{\\textbf{DNN}}\\\\\n        Variables & \\multicolumn{2}{c|}{\\textbf{low-level}} & \\multicolumn{2}{c|}{\\textbf{low-level}} & \\multicolumn{2}{c|}{\\textbf{high-level}} & \\multicolumn{2}{c}{\\textbf{combined}} \\\\\n        Input Ordering & \\textbf{gen.} & $\\mathbf{p_T}$ & \\textbf{gen.} & $\\mathbf{p_T}$ & \\textbf{gen.} & $\\mathbf{p_T}$ & \\textbf{gen.} & $\\mathbf{p_T}$\\\\\n        \\midrule\n        $M_\\text{part.,rest fr.}$ & $13$ & $16$ & \\verb|-| & \\verb|-| & \\verb|-| & \\verb|-| & \\verb|-| & \\verb|-|\\\\\n        $n_\\text{layers}$ & $8$ & $8$ & $8$ & $8$ & $4$ & $4$ & $8$ & $6$\\\\\n        $n_\\text{nodes}$ & $1024$ & $1024$ & $1024$ & $1024$ & $512$ & $512$ & $1024$ & $1024$\\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\n\\subsection*{High-level variables}\n\nThe high-level variables employed in the training of the DNN benchmark comparison are inspired by \\cite{Sirunyan:2018mvw}:\n\\begin{itemize}\n    \\item\n    The event shape variables sphericity, transverse sphericity, aplanarity, centrality \\cite{event_shape_variables}.\n\n    \\item\n    The first five Fox-Wolfram moments \\cite{fox_wolfram_moments}.\n\n    \\item\n    The cosine of spatial angular difference $\\theta^*$ between the charged lepton in the W boson rest frame and the W boson direction when boosted into the rest frame of its corresponding top quark.\n    In the hadronic branch, the down-type quark is used owing to its increased spin analyzing power \\cite{spin_analyzing_power}.\n\n    \\item\n    The minimum, maximum and average of the distance in pseudorapidity $\\eta$ and $\\phi$ phase space $\\Delta R$ of jet pairs.\n\n    \\item\n    The minimum, maximum and average $|\\Delta\\eta|$ of jet pairs.\n\n    \\item\n    The minimum and maximum of the distance in $\\Delta R$ of jet-lepton pairs.\n\n    \\item\n    The minimum, maximum and average $|\\Delta\\eta|$ of jet-lepton pairs.\n\n    \\item\n    The sum of the transverse momenta of all jets.\n\n    \\item\n    The transverse momentum and the mass of the jet pair with the smallest $\\Delta R$.\n\n    \\item\n    The transverse momentum and the mass of the jet pair whose combined mass is closest to the Higgs boson mass m$_H = 125$\\,GeV \\cite{CMS2012:higgs}.\n\\end{itemize}\n\n\\end{appendix}\n\n\n\\acknowledgments\nWe wish to thank Jonas Glombitza for his valuable comments on the manuscript.\nWe would also like to thank Jean-Roch Vlimant, Benjamin Fischer, Dennis Noll, and David Schmidt for a variety of fruitful discussions.\nThis work is supported by the Ministry of Innovation, Science and Research of the State of North Rhine-Westphalia, and by the Federal Ministry of Education and Research (BMBF).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nScattering amplitudes and cross sections simplify in infrared kinematic limits enabling insight into their all orders perturbative structure.\nOne of the most powerful approaches to studying the all orders structure is the use of renormalization group (RG) techniques. Depending on the particular nature of the kinematic limit considered, these RG equations often describe evolution not just in the standard virtuality scale, $\\mu$, but also in other additional physical scales. A common example in gauge theories are rapidity evolution equations, which allow for the resummation of infrared logarithms associated with hierarchical scales in rapidity. Classic examples include the Sudakov form factor \\cite{Collins:1989bt},  the Collins-Soper equation \\cite{Collins:1981uk,Collins:1981va,Collins:1984kg} describing the $p_T$ spectrum for color singlet boson production in hadron colliders, the BFKL evolution equations describing the Regge limit  \\cite{Kuraev:1977fs,Balitsky:1978ic,Lipatov:1985uk},  and the rapidity renormalization group \\cite{Chiu:2011qc,Chiu:2012ir} describing more general event shape observables.\n\nThere has recently been significant effort towards understanding subleading power corrections to infrared limits  \\cite{Manohar:2002fd,Beneke:2002ph,Pirjol:2002km,Beneke:2002ni,Bauer:2003mga,Hill:2004if,Lee:2004ja,Dokshitzer:2005bf,Trott:2005vw,Laenen:2008ux,Laenen:2008gt,Paz:2009ut,Benzke:2010js,Laenen:2010uz,Freedman:2013vya,Freedman:2014uta,Bonocore:2014wua,Larkoski:2014bxa,Bonocore:2015esa,Bonocore:2016awd,Kolodrubetz:2016uim,Moult:2016fqy,Boughezal:2016zws,DelDuca:2017twk,Balitsky:2017flc,Moult:2017jsg,Goerke:2017lei,Balitsky:2017gis,Beneke:2017ztn,Feige:2017zci,Moult:2017rpl,Chang:2017atu,Beneke:2018gvs,Beneke:2018rbh,Moult:2018jjd,Ebert:2018lzn,Ebert:2018gsn,Bhattacharya:2018vph,Boughezal:2018mvf,vanBeekveld:2019prq,vanBeekveld:2019cks,Bahjat-Abbas:2019fqa,Beneke:2019kgv,Boughezal:2019ggi,Moult:2019mog,Beneke:2019mua,Cieri:2019tfv,Moult:2019uhz,Beneke:2019oqx} with the ultimate goal of achieving a systematic expansion, much like for problems where their exists a local operator product expansion (OPE). Using Soft Collinear Effective Theory (SCET) \\cite{Bauer:2000ew, Bauer:2000yr, Bauer:2001ct, Bauer:2001yt} subleading power infrared logarithms were resummed to all orders using RG techniques for a particular class of event shape observables where only virtuality evolution is required. This was achieved both in pure Yang-Mills theory \\cite{Moult:2018jjd}, and including quarks in $\\mathcal{N}=1$ QCD \\cite{Moult:2019uhz}, and a conjecture for the result including quarks in QCD was presented in \\cite{Moult:2019uhz}. Subleading power infrared logarithms have also been resummed for color singlet production at kinematic threshold, when only soft real radiation is present \\cite{Beneke:2018gvs,Beneke:2019mua,Bahjat-Abbas:2019fqa}.\n\nIn this paper we build on the recent advances in understanding the structure of subleading power renormalization group equations in SCET, and consider for the first time the resummation of subleading power rapidity logarithms. Using renormalization group consistency arguments, we derive a class of subleading power rapidity evolution equations. These equations involve mixing into new class of operators, which play a crucial role in the renormalization group equations. We call these operators ``rapidity identity operators\", and we derive their renormalization group properties, and solve the associated RG equations. \n\nWe apply our evolution equations to derive the all orders structure of the power suppressed leading logarithms for the energy-energy correlator (EEC) event shape in $\\mathcal{N}=4$ super-Yang-Mills (SYM) theory in the back-to-back (double light cone) limit. Denoting these power suppressed contributions by $\\text{EEC}^{(2)}$, we find the remarkably simple formula\n\\begin{align}\n\\boxed{\\text{EEC}^{(2)}=-\\sqrt{2a_s}~D\\left[ \\sqrt{\\frac{\\Gamma^{\\mathrm{cusp}}}{2}} \\log(1-z) \\right]}\\,,\n\\end{align}\nwhere $D(x)=1\/2\\sqrt{\\pi}e^{-x^2}\\mathrm{erfi}(x)$ is Dawson's integral, $a_s=\\alpha_s\/(4\\pi)C_A$, and $\\Gamma^{\\mathrm{cusp}}$ is the cusp anomalous dimension \\cite{Korchemsky:1987wg}. This result provides insight into new all orders structures appearing in subleading power infrared limits. Since this extends the classic Sudakov exponential \\cite{Sudakov:1954sw}, we will refer to this functional form as ``Dawson's Sudakov\". The particular example of the EEC observable was chosen in this paper, since its exact structure for generic angles is known to $\\mathcal{O}(\\alpha_s^3)$ due to the remarkable calculation of \\cite{Henn:2019gkr}, and we find perfect agreement with the expansion of their results in the back-to-back limit to this order, providing a strong check of our techniques. While we focus on the EEC in $\\mathcal{N}=4$, this observable has an identical resummation structure to $p_T$ resummation, and therefore  the techniques we have developed apply more generally, both to the EEC in QCD, and to the $p_T$ distribution of color singlet bosons at hadron colliders.\n\nAn outline of this paper is as follows. In \\Sec{sec:EEC_b2b} we review the known structure of the EEC observable in the back-to-back limit, and relate it to the case of the $p_T$ spectrum of color singlet bosons, which is perhaps more familiar to the resummation community. In \\Sec{sec:FO} we perform a fixed order calculation of the EEC at subleading power in SCET, which allows us to understand the structure of the subleading power rapidity divergences, and provides the boundary data for our RG approach. In \\Sec{sec:RRG_NLP} we study the structure of subleading power rapidity evolution equations, introduce the rapidity identity operators, and analytically solve their associated evolution equations. In \\Sec{sec:N4} we apply these evolution equations to the particular case of the EEC in $\\mathcal{N}=4$ SYM to derive the subleading power leading logarithmic series, and we comment on some of the interesting features of the result. We also compare our result with the fixed order calculation of \\cite{Henn:2019gkr} expanded in the back-to-back limit, finding perfect agreement.  We conclude in \\Sec{sec:conc}, and discuss many directions for improving our understanding of the infrared properties of gauge theories at subleading powers. \n\n\\section{The Energy-Energy Correlator in the Back-to-Back Limit}\\label{sec:EEC_b2b}\n\nIn this section we introduce the EEC observable, and review its structure in the back-to-back limit at leading power. We then discuss the resummation of the associated infrared logarithms using the rapidity renormalization group approach. This will allow us to introduce our notation, before proceeding to subleading power. \n\nAn additional goal of this section is to make clear the relation between the EEC in the back-to-back limit and more standard $p_T$ resummation, which may be more familiar to the resummation community. This should also make clear that the techniques we develop are directly applicable to the case of $p_T$ resummation, although we leave a complete treatment of $p_T$ resummation to a future publication due to complexities related to the treatment of the initial state hadrons. Some other recent work towards understanding subleading power factorization for $p_T$ can be found in \\cite{Balitsky:2017flc,Balitsky:2017gis}.\n\n\nFor a color singlet source, the EEC is defined as \\cite{Basham:1978bw}\n\\begin{align}\\label{eq:EEC_intro}\n\\text{EEC}(\\chi)=\\sum\\limits_{a,b} \\int d\\sigma_{V\\to a+b+X} \\frac{2 E_a E_b}{Q^2 \\sigma_{\\text{tot}}}   \\delta(\\cos(\\theta_{ab}) - \\cos(\\chi))\\,,\n\\end{align}\nwhere the sum is over all pairs of final state particles, $E_a$ are the energies of the particles, and $\\theta_{ab}$ are the angles between pairs of particles. Energy correlators are a theoretically nice class of event shape observable, since they can be directly expressed in terms of energy flow operators \\cite{Hofman:2008ar,Belitsky:2013xxa,Belitsky:2013bja,Belitsky:2013ofa}\n\\begin{align}\n\\mathcal{E}(\\vec n) =\\int\\limits_0^\\infty dt \\lim_{r\\to \\infty} r^2 n^i T_{0i}(t,r \\vec n)\\,.\n\\end{align}\nIn particular, the EEC is given by the four-point Wightman correlator\n\\begin{align}\n\\frac{1}{\\sigma_{\\rm tot}} \\frac{d\\sigma}{dz}=\\frac{\\langle \\mathcal{O} \\mathcal{E}(\\vec n_1)  \\mathcal{E}(\\vec n_2) \\mathcal{O}^\\dagger \\rangle }{\\langle \\mathcal{O}  \\mathcal{O}^\\dagger \\rangle}\\,,\n\\end{align}\nwhere we have introduced the convenient variable\n\\begin{align}\nz=\\frac{1-\\cos {\\chi}}{2}\\,,\n\\end{align}\nand $\\mathcal{O}$ is a source operator that creates the excitation.\n\nThere has been significant recent progress in understanding the EEC, both in QCD, as well as in conformal field theories. In QCD, the EEC has been computed for arbitrary angles at next-to-leading order (NLO) analytically \\cite{Dixon:2018qgp,Luo:2019nig} and at NNLO numerically \\cite{DelDuca:2016csb,DelDuca:2016ily}. In $\\mathcal{N}=4$ it has been computed for arbitrary angles to NNLO \\cite{Belitsky:2013ofa,Henn:2019gkr}. There has also been significant progress in understanding the limits of the EEC, namely the collinear ($z\\to 0$) limit \\cite{Dixon:2019uzg,Kravchuk:2018htv,Kologlu:2019bco,Kologlu:2019mfz,Korchemsky:2019nzm}, and the back-to-back ($z\\to 1$) limit \\cite{deFlorian:2004mp,Moult:2018jzp,Korchemsky:2019nzm,Gao:2019ojf}. Here we will focus on the EEC in the back-to-back limit, where it exhibits Sudakov double logarithms. As we will explain shortly, these double logarithms are directly related to those appearing for transverse momentum resummation. In this section we follow closely the factorization derived in \\cite{Moult:2018jzp} using the rapidity renormalization group \\cite{Chiu:2012ir,Chiu:2011qc}. An alternative approach to studying this limit directly from the four point correlator was given in \\cite{Korchemsky:2019nzm}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{figures\/back_to_back}\n\\end{center}\n\\caption{The kinematics of the EEC in the back-to-back limit. Wide angle soft radiation recoils the two jets in a manner identical to the case of $p_T$ for color singlet boson production in hadronic collisions. This provides a precise relation between the factorization formulas in the two cases. (Figure from \\cite{Moult:2018jzp}.)  }\n\\label{fig:schematic_factorization}\n\\end{figure}\n\nThe back-to-back limit corresponds to the region of phase space where there are two collimated jets accompanied by low energy soft radiation, which recoils the directions of the jets so that they are not exactly back-to-back. This configuration is illustrated in \\Fig{fig:schematic_factorization}. A simple exercise shows that the angle between two partons correlated by the EEC is related to the transverse momentum of these particles within the jets and the transverse momentum of the soft radiation that recoils these jets, by\n\\begin{align}\n\\label{eq:z_in_back_to_back}\n1-z=\\frac{1}{Q^2}  \\left|  \\frac{\\vec k_{\\perp,i}^h}{x_i}+\\frac{\\vec k_{\\perp,j}^h}{x_j}-\\vec k_{\\perp,s}^h \\right|^2 +\\mathcal{O}(1-z)\\,.\n\\end{align}\nHere $x_i=2E_i\/Q$, where $Q$ is the center of mass energy.\nThis relation makes clear the connection between the EEC in the back-to-back limit and transverse momentum resummation, which we will shortly extend to subleading powers.\n\nTo describe the EEC in this limit, we use an effective field theory description of the soft and collinear dynamics, where the relevant modes have the scalings\n\\begin{align}\\label{eq:pc_modes}\np_s\\sim Q(\\lambda, \\lambda, \\lambda)\\,, \\qquad p_c \\sim Q(\\lambda^2, 1, \\lambda) \\,, \\qquad p_{\\bar c} \\sim Q(1,\\lambda^2,  \\lambda)\\,.\n\\end{align}\nHere $\\lambda$ is a power counting parameter and is defined as\n\\begin{align}\n\\lambda\\sim \\sqrt{1-z}\\,.\n\\end{align}\nThis scaling defines what is referred to as an SCET$_{\\rm II}$~ theory \\cite{Bauer:2002aj}. Crucially, unlike in SCET$_{\\rm I}$, the soft and collinear modes in SCET$_{\\rm II}$~ have the same virtualities, but different rapidities. This factorization into soft and collinear modes therefore introduces divergences as $k^+\/k^-\\to \\infty$ or $k^+\/k^-\\to 0$  \\cite{Collins:1992tv,Manohar:2006nz,Collins:2008ht,Chiu:2012ir,Vladimirov:2017ksc}, which are referred to as rapidity divergences. To regulate these divergences, one must introduce a regulator that breaks boost invariance, allowing the soft and collinear modes to be distinguished. Once such a regulator is introduced, renormalization group evolution equations can be derived to resum the associated rapidity logarithms \\cite{Chiu:2012ir,Chiu:2011qc}.\n\nUsing the effective field theory, one can systematically expand the cross section for either transverse momentum, or for the EEC in powers of the observable. For the EEC, we write\n\\begin{align}\\label{eq:xsec_expand}\n\\frac{\\mathrm{d} \\sigma}{\\mathrm{d} z}\n&= \\frac{\\mathrm{d}\\sigma^{(0)}}{\\mathrm{d} z} + \\frac{\\mathrm{d}\\sigma^{(2)}}{\\mathrm{d} z}+ \\frac{\\mathrm{d}\\sigma^{(4)}}{\\mathrm{d} z} + \\dotsb{\\nonumber} \\\\\n&=\\text{EEC}^{(0)}+\\text{EEC}^{(2)}+\\text{EEC}^{(4)}\\,,\n\\end{align}\nwhere we will occasionally use the second notation, since it is more compact.\nHere\n\\begin{align}\n\\frac{\\mathrm{d}\\sigma^{(0)}}{\\mathrm{d} z}\n&\\sim \\delta(1-z)+ \\biggl[\\frac{ \\ord{1} }{1-z}\\biggr]_+\n\\,, \n\\end{align}\nis referred to as the leading power cross section, and\ndescribes all terms scaling like $\\mathcal{O}((1-z)^{-1})$ modulo logarithms. All the other terms in the expansion of the cross section are suppressed by explicit powers of $(1-z)$\n\\begin{align}\\label{eq:scaling_lam2}\n\\frac{\\mathrm{d}\\sigma^{(2k)}}{\\mathrm{d} z} &\\sim \\ord{(1-z)^{k-1}}\n\\,.\\end{align}\nThe focus of this paper will be on deriving the structure of the leading logarithms in $\\mathrm{d}\\sigma^{(2)}\/\\mathrm{d} z$, which is also referred to as the next-to-leading power (NLP) cross section. \n\n\nFor the leading power cross section, $d\\sigma^{(0)}\/dz$, one can derive a factorization formula describing in  a factorized manner the contributions of the soft and collinear modes to the EEC in the $z\\to 1$ limit \\cite{Moult:2018jzp}\n\\begin{equation}\\label{eq:fact_final}\n\\hspace{-0.35cm}\\frac{d\\sigma^{(0)}}{dz}= \\frac{1}{2}  \\int d^2 \\vec k_\\perp \\int \\frac{d^2 \\vec b_\\perp}{(2 \\pi)^2} e^{-i \\vec b_\\perp \\cdot \\vec k_\\perp} H(Q,\\mu)  J^q_{\\text{EEC}} (\\vec b_\\perp,\\mu,\\nu) J^{\\bar q}_{\\text{EEC}} (\\vec b_\\perp,\\mu,\\nu) S_{\\text{EEC}}(\\vec b_\\perp,\\mu,\\nu)  \\delta \\left( 1-z- \\frac{\\vec k_\\perp^2}{Q^2} \\right )\\,,\n\\end{equation}\nin terms of a hard function, $H$, jet functions, $J$, and a soft function, $S$. This factorization is nearly equivalent to the factorization for the $p_T$ for color singlet boson production (This factorization formula was originally derived in  \\cite{Collins:1981uk,Collins:1981va,Collins:1984kg}, and was derived in terms of the rapidity renormalization group used here in \\cite{Chiu:2012ir,Chiu:2011qc}), \n\\begin{align}\\label{eq:pt_fact}\n\\frac{1}{\\sigma} \\frac{d^3 \\sigma^{(0)}}{d^2 \\vec p_T dY dQ^2} = H(Q,\\mu) \\int \\frac{d^2 \\vec b_\\perp}{(2\\pi)^2} e^{i\\vec b_\\perp \\cdot \\vec p_T} \\left[  B \\times B  \\right] (\\vec b_\\perp, \\mu, \\nu) S_\\perp(\\vec b_\\perp, \\mu, \\nu)\\,,\n\\end{align}\nup to the fact that the jet functions are moved to the initial state, where they are referred to as beam functions \\cite{Stewart:2009yx}. Apart from our intrinsic interest in understanding the kinematic limits of the EEC, this relation between the EEC and $p_T$ is one of our primary motivations for studying the EEC. Lessons derived from the EEC can be directly applied to understanding the structure of subleading power logarithms for $p_T$, which is a phenomenologically important observable at the LHC, for example, for precision studies of the Higgs boson. Here we briefly discuss the objects appearing in the factorization formula, both to emphasize the close connections between the EEC and $p_T$, as well as to introduce the general structure of the $\\mu$ and $\\nu$ rapidity evolution equations. \n\n\nThe hard functions, $H(Q,\\mu)$, appearing in the factorization formulas for the EEC and $p_T$ are identical. They describe hard virtual corrections, and satisfy a multiplicative renormalization group equation (RGE) in $\\mu$\n\\begin{align}\n\\mu \\frac{d}{d\\mu} H(Q,\\mu) =2 \\left[\\Gamma^{\\mathrm{cusp}}(\\alpha_s) \\ln\\frac{Q^2}{\\mu^2} +  \\gamma^H(\\alpha_s) \\right] H(Q,\\mu)\\,.\n\\end{align}\nThey are independent of rapidity.  The soft functions appearing in both $p_T$ and the EEC can be proven to be identical \\cite{Moult:2018jzp}. They are matrix elements of Wilson lines, which for quarks and gluons are defined as\n\\begin{align}\n S_q(\\vec p_T) &= \\frac{1}{N_c} \\big\\langle 0 \\big| \\mathrm{Tr} \\bigl\\{\n   \\mathrm{T} \\big[S^\\dagger_{\\bn} S_n\\big]\n   \\delta^{(2)}(\\vec p_T-\\mathcal{P}_\\perp)\\overline{\\mathrm{T}} \\big[S^\\dagger_{n} S_{\\bn} \\big]\n   \\bigr\\{ \\big| 0 \\big\\rangle\n\\,,{\\nonumber}\\\\\n S(\\vec p_T) &= \\frac{1}{N_c^2 - 1} \\big\\langle 0 \\big| \\mathrm{Tr} \\bigl\\{\n   \\mathrm{T} \\big[\\mathcal{S}^\\dagger_{\\bn} \\mathcal{S}_n\\big]\n  \\delta^{(2)}(\\vec p_T-\\mathcal{P}_\\perp) \\overline{\\mathrm{T}} \\big[\\mathcal{S}^\\dagger_{n} \\mathcal{S}_{\\bn} \\big]\n   \\bigr\\} \\big| 0 \\big\\rangle\n\\,.\\end{align}\nHere $\\mathrm{T}$ and $\\overline{\\mathrm{T}}$ denote time and anti-time ordering, and $S_n$ and $\\mathcal{S}_n$ denote Wilson lines in the fundamental and adjoint representations, respectively. Explicitly, \n\\begin{align}\n S_n(x) &= \\mathrm{P} \\exp\\biggl[ \\mathrm{i} g \\int_{-\\infty}^0 \\mathrm{d} s\\, n \\cdot A(x+sn)\\biggr]\\,,\n\\end{align}\nand similarly for the adjoint Wilson lines. These soft functions satisfy the $\\mu$ and $\\nu$ RGEs\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S(\\vec p_T)&=\\int d\\vec q_T \\gamma^S_\\nu(p_T-q_T) S(\\vec q_T)\\,, {\\nonumber} \\\\\n\\mu \\frac{d}{d\\mu}S(\\vec p_T)&=\\gamma_\\mu^S S(\\vec p_T)\\,,\n\\end{align}\nwith the anomalous dimensions\n\\begin{align}\n\\gamma_\\mu^S &=4\\Gamma^{\\mathrm{cusp}}(\\alpha_s)\\log \\left( \\frac{\\mu}{\\nu}\\right)\\,,{\\nonumber} \\\\\n\\gamma_\\nu^S &=2\\Gamma^{\\mathrm{cusp}} (\\alpha_s)\\mathcal{L}_0\\left( \\vec p_T,\\mu \\right)\\,.\n\\end{align}\nHere the color representation is implicit in the cusp anomalous dimension, and $\\mathcal{L}_0$ is a standard plus function (see e.g. \\cite{Ebert:2016gcn} for a detailed discussion of two-dimensional plus distributions, and for a definition of the conventions that are used here). Since we will ultimately be interested in $\\mathcal{N}=4$ where all particles are in the same representation, we will drop the quark and gluon labels on the soft functions.\n\nThe only difference between $p_T$ and the EEC lies in the collinear sector, namely whether one uses beam functions or jet functions. The jet functions for the EEC were recently computed to NNLO  \\cite{Luo:2019bmw,Luo:2019hmp}. Since in this paper we will be focused on resummation at LL, we can always choose to run all functions to the jet scale, and thereby avoid a discussion of the collinear sector for simplicity. The structure of the power corrections to the beam (jet) functions, and their matching to the parton distributions (fragmentation functions) is interesting, and will be presented in future work, since it is important for a complete understanding of $p_T$ at subleading powers.\n\nThese renormalization group evolution equations in both $\\mu$ and $\\nu$ allow for a derivation of the all orders structure of logarithms in the $z\\to 1$ limit, at leading order in the $(1-z)$ expansion. Performing the renormalization group evolution, one finds for a non-conformal field theory \\cite{Moult:2018jzp} (i.e. allowing for a running coupling)\n\\begin{align}\n  \\label{eq:resformula}\n\\frac{d\\sigma^{(0)}}{dz} = &\\; \\frac{1}{4} \\int\\limits_0^\\infty db\\, b\n  J_0(bQ\\sqrt{1-z})H(Q,\\mu_h) j^q_{\\text{EEC}}(b,b_0\/b,Q) j^{\\bar\n  q}_{\\text{EEC}}(b,b_0\/b,Q) S_{\\text{EEC}}( b,\\mu_s, \\nu_s) \n{\\nonumber}\\\\\n&\\; \\cdot\n  \\left(\\frac{Q^2}{\\nu_s^2}\\right)^{\\gamma^r_{\\text{EEC}}(\\alpha_s(b_0\/b))}\n  \\exp \\left[ \\int\\limits_{\\mu_s^2}^{\\mu_h^2}\n  \\frac{d\\bar{\\mu}^2}{\\bar{\\mu}^2} \\Gamma^{\\mathrm{cusp}}(\\alpha_s(\\bar \\mu)) \\ln\n  \\frac{b^2\\bar{\\mu}^2}{b_0^2} \\right.\n{\\nonumber}\\\\\n&\\;\n\\left. +\n  \\int\\limits_{\\mu_h^2}^{b_0^2\/b^2}\\frac{d\\bar{\\mu}^2}{\\bar{\\mu}^2}\n  \\left(\\Gamma^{\\mathrm{cusp}}(\\alpha_s(\\bar \\mu)) \\ln\\frac{b^2 Q^2}{b_0^2} +\n   \\gamma^H (\\alpha_s(\\bar \\mu)) \\right) -\n  \\int\\limits_{\\mu_s^2}^{b_0^2\/b^2}\\frac{d\\bar{\\mu}^2}{\\bar{\\mu}^2}\n   \\gamma^s_{\\text{EEC}} (\\alpha_s(\\bar \\mu))  \\right]\\,.\n\\end{align}\nFor the particular case of a conformal theory, this expression simplifies considerably, both due to the fact that the coupling doesn't run, and also since in a conformal field theory there is an equivalence between the rapidity anomalous dimension and the soft anomalous dimension  \\cite{Vladimirov:2016dll,Vladimirov:2017ksc}. Combining this equivalence with the relations for the soft anomalous dimension derived in \\cite{Dixon:2008gr} (see also \\cite{Falcioni:2019nxk} for recent work on relations between different soft functions), we have\n\\begin{align}\n\\gamma^r= -\\mathcal{G}_0+2B\\,,\n\\end{align}\nwhere $B$ is the virtual anomalous dimension (the coefficient of $\\delta(1-x)$ in the DGLAP kernel), and $\\mathcal{G}_0$ is the collinear anomalous dimension. We then find\n\\begin{align}\n  \\label{eq:resformula_N4}\n\\frac{d\\sigma^{(0)}}{dz} = &\\; \\frac{1}{4} \\int\\limits_0^\\infty db\\, b\n  J_0(bQ\\sqrt{1-z})H(Q,\\mu_h) j^q_{\\text{EEC}}(b,b_0\/b,Q) j^{\\bar\n  q}_{\\text{EEC}}(b,b_0\/b,Q) S_{\\text{EEC}}( b,\\mu_s, \\nu_s) \n{\\nonumber}\\\\\n&\\exp \\left[ \\Gamma^{\\mathrm{cusp}} \\log^2\\left( \\frac{b^2 Q^2}{b_0^2} \\right)  +2B\\log\\left( \\frac{b^2 Q^2}{b_0^2} \\right)  \\right]\\,.\n\\end{align}\nBoth the cusp anomalous dimension, as well as the $B$ anomalous dimension are known from integrability \\cite{Eden:2006rx,Beisert:2006ez,Freyhult:2007pz,Freyhult:2009my,Fioravanti:2009xt}.  It is interesting that only the two anomalous dimensions that are known from integrability appear in the final result. The collinear anomalous dimension, which drops out of the final result in a conformal theory, is known to four loops in $\\mathcal{N}=4$ \\cite{Dixon:2017nat}.\n\n\n\nThis formula describes the leading power asymptotics of the EEC in the $z\\to 1$ limit to all orders in the coupling (Indeed, in $\\mathcal{N}=4$, it should also apply at finite coupling). The goal of this paper will be to start to understand the all orders structure of the subleading power corrections to this formula in $(1-z)$. While we do not have a complete factorization formula or all orders understanding, we will be able to deduce much of the structure from general consistency and symmetry arguments. Ultimately, we would like to be able to classify the operators that appear in the description of the subleading powers in this limit, and understand their renormalization group evolution. This paper represents a first step in this direction. \n\n\n\nWe conclude this section by noting that the result of \\Eq{eq:resformula_N4} can also be derived in a conformal field theory by directly considering the structure of the four point correlator in the double light cone limit \\cite{Korchemsky:2019nzm} (see also \\cite{Alday:2013cwa}), and using the duality between the correlator and a Wilson loop \\cite{Alday:2010zy}, as well as known results for the structure constants \\cite{Eden:2012rr}. It would be interesting to understand systematically the OPE of the correlator in this limit and the operators that appear from this perspective. There has been some study of the double light cone limit in \\cite{Alday:2015ota}. It would be interesting to understand it in more detail and develop a systematic OPE, much like exists in the collinear limit \\cite{Alday:2010ku,Basso:2014jfa,Basso:2007wd,Basso:2010in,Basso:2015uxa,Basso:2013vsa,Basso:2013aha,Basso:2014koa,Basso:2014nra}. It would also be interesting if the recently introduced light ray OPE \\cite{Kravchuk:2018htv,Kologlu:2019bco,Kologlu:2019mfz} can provide insight into this limit. However, we leave these directions to future work.\n\n\n\\section{Fixed Order Calculation of the EEC at Subleading Power}\\label{sec:FO}\n\nHaving discussed the structure of the EEC in the back-to-back limit, as well as the factorization theorem describing its leading power asymptotics, in this section we begin our study of the subleading corrections in powers of $(1-z)$ by performing a fixed order calculation. This is important both for understanding the structure of the subleading power rapidity divergences for the EEC, and for providing the boundary conditions for the renormalization group studied later in the paper. We will perform this calculation both in QCD, as well as in $\\mathcal{N}=4$. We follow closely the calculation for the power corrections for $p_T$ presented in \\cite{Ebert:2018gsn}. In \\Sec{sec:intuition} we summarize some of the intuition derived from this calculation, which provides significant insight into the structure of the subleading power renormalization group evolution, which we will then study in more detail in \\Sec{sec:RRG_NLP}.\n\n\n\\subsection{Leading Order Calculation in $\\mathcal{N}=4$ SYM and QCD}\\label{sec:FO2}\n\n\nIn this section, we perform the leading order (LO) calculation of the EEC at NLP in both QCD and $\\mathcal{N}=4$. This section is rather technical, and assumes some familiarity with fixed order calculations at subleading power in SCET (see e.g. \\cite{Moult:2016fqy,Moult:2017jsg,Ebert:2018lzn,Ebert:2018gsn} for more detailed discussions).\n\nThe EEC observable can be written as\n\\begin{align}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &= \\sum_{a,b} \\int \\mathrm{d} \\Phi_{V \\to a+b+X} \\left|\\mathcal{M}_{V \\to a+b+X}\\right|^2 \\frac{E_a E_b}{q_V^2}\\delta\\left(y-\\cos^2 \\frac{\\theta_{ab}}{2}\\right) \\,,\n\\end{align}\nwhere we have used $y \\equiv 1-z$, so that $y \\to 0$ in the back-to-back limit. To perform the calculation it is convenient to write the observable definition in a boost invariant manner\n\\begin{align}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &= \\frac{1}{2 (1-y) q_V^2}\\sum_{a,b} \\int \\mathrm{d} \\Phi_{V \\to a+b+X} \\left|\\mathcal{M}_{V \\to a+b+X}\\right|^2\\, p_a\\cdot p_b\\,\\delta\\left[y - \\left(1 -\\frac{q_V^2p_a\\cdot p_b}{2 p_a\\cdot q_V p_b \\cdot q_V}\\right)\\right] \\,,\n\\end{align}\nwhere $q_V^\\mu$ is the momentum of the vector boson. This definition is convenient since if we are correlating a particular pair of particles $\\{a,b\\}$, we can boost the system such that the particles being correlated are back-to-back and the vector boson (or source) recoils against the unmeasured radiation in the perpendicular direction.\n\n\nGiven this setup, we begin by considering a single perturbative emission, which is sufficient for the LO calculation. We will denote by $p_a^\\mu$ and $p^\\mu_b$ the momenta of the particles we are correlating, and $k^\\mu$ the momentum of the unmeasured radiation. This translates to the following choice of kinematics\\footnote{Note that here $p_{a,b}^\\mu$ and $k^\\mu$ are outgoing while $q_V^\\mu$ is incoming.}\n\\begin{align}\\label{eq:kinematic}\n\tp_a^\\mu &= (q^-_V - k^-) \\frac{n^\\mu}{2}\\,,&& k^\\mu = k^- \\frac{n^\\mu}{2} + k^+ \\frac{\\bn^\\mu}{2} + k_\\perp^\\mu\\,,{\\nonumber}\\\\\n\tp_b^\\mu &= (q^+_V - k^+) \\frac{\\bn^\\mu}{2}\\,,&& q_V^\\mu = q_V^- \\frac{n^\\mu}{2} + q_V^+ \\frac{\\bn^\\mu}{2} + k_\\perp^\\mu\\,.\n\\end{align}\nThe measurement of the EEC observable then takes the form of the following constraint\n\\begin{align}\n\ty = 1 -  \\frac{(q^-_V - k^-)(q^+_V - k^+) q_V^2}{q_V^+ (q^-_V - k^-) q_V^- (q^+_V - k^+)} = \\frac{q^+_V q^-_V - q_V^2}{q^+_V q^-_V} = \\frac{{\\vec{k}_\\perp^{\\,2}}}{q_V^2 - {\\vec{k}_\\perp^{\\,2}}}\\,,\n\\end{align}\nwhich we can rewrite  as a constraint on ${\\vec{k}_\\perp^{\\,2}}$ \n\\begin{equation}\n\t\\delta\\left(y - \\frac{{\\vec{k}_\\perp^{\\,2}}}{q_V^2 - {\\vec{k}_\\perp^{\\,2}}} \\right) = \\delta\\left({\\vec{k}_\\perp^{\\,2}} - q_V^2\\frac{y}{1-y}\\right) \\frac{q_V^2}{(1-y)^2}\\,.\n\\end{equation}\nThis extends the relation between the EEC and $p_T$ to subleading powers.\n\n\\subsection*{Master Formula}\n\nUsing this result for the measurement function, the cross section for the EEC with one emission reads\n\\begin{align}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &= \\frac{1}{2 (1-y)^3}\\sum_{a,b}\\int \\mathrm{d} \\Phi |\\mathcal{M}_{V \\to q\\bar{q} g}|^2\\, p_a \\cdot p_b\\,\\delta\\left({\\vec{k}_\\perp^{\\,2}} - q_V^2\\frac{y}{1-y}\\right){\\nonumber}\\,.\n\\end{align}\nFor our particular choice of frame where $p_a^\\mu$ has no $\\perp$ component, we have \\cite{Fleming:2007qr}\n\\begin{align}\n\t \\int \\frac{\\mathrm{d}^{d-2} p_{a\\perp}}{(2\\pi)^{d-2}} &=  \\sum_n d^2 p_{\\perp,n} \\int \\frac{\\mathrm{d}^{d-2} p_{a\\perp}}{(2\\pi)^{d-2}}\\delta^{(d-2)}(p^\\mu_{a\\perp}) = \\frac{|\\vec{p}_a|^{d-2}}{(2\\pi)^{d-2}} \\int \\mathrm{d} \\Omega_{d-2}\\,.\n\\end{align}\nUsing\n\\begin{align}\\label{eq:paPS}\n\t\\int {\\mathrm{d}\\!\\!\\!{}^-}^d p_a \\delta_+(p_a^2) &=  \\frac{1}{2} \\int \\frac{\\mathrm{d} p_a^-}{(2\\pi)} \\frac{\\mathrm{d} p_a^+}{(2\\pi)} (2\\pi) \\theta(p_a^+ + p_a^-)\\delta(p_a^+ p_a^- ) |\\vec{p}_a|^2 \\int  \\frac{ \\mathrm{d}\\Omega_{d-2}}{(2\\pi)^{d-2}} {\\nonumber}\\\\\n\t&= \\frac{1}{2(2\\pi)^{d-1}}  \\int \\frac{\\mathrm{d} p_a^-}{p_a^-} \\theta(p_a^-) |p_a^-\/2|^2\\int \\mathrm{d} \\Omega_{d-2} = C \\times \\int \\mathrm{d} p_a^- \\,p_a^-\\,,   \n\\end{align}\nwe can write the cross section with a single emission as\n\\begin{align}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{\\text{EEC}}}}{\\mathrm{d}  y} &\\sim \\frac{1}{2 (1-y)^3}\\sum_{a,b} \\int {\\mathrm{d}\\!\\!\\!{}^-}^d k \\delta_+(k^2) |\\mathcal{M}_{V \\to q\\bar{q} g}|^2\\, (p_a \\cdot p_b)^2\\,\\delta\\left({\\vec{k}_\\perp^{\\,2}} - q_V^2\\frac{y}{1-y}\\right){\\nonumber}\\,\\\\\n\t&\\sim \\frac{1}{2 (1-y)^3}\\sum_{a,b} \\int_0^{q^-} \\frac{\\mathrm{d} k^-}{k^-}  \\int \\mathrm{d} {\\vec{k}_\\perp^{\\,2}} \\frac{\\mu^{2\\epsilon}}{{\\vec{k}_\\perp^{\\,2\\epsilon}}} |\\mathcal{M}_{V \\to q\\bar{q} g}|^2\\, (p_a \\cdot p_b)^2\\,\\delta\\left({\\vec{k}_\\perp^{\\,2}} - q_V^2\\frac{y}{1-y}\\right){\\nonumber}\\,\\\\\n\t&\\sim \\frac{1}{2 (1-y)^3}\\left(\\frac{\\mu^{2}}{q_V^2}\\frac{1-y}{y}\\right)^\\epsilon \\sum_{a,b} \\int_0^{q^-} \\frac{\\mathrm{d} k^-}{k^-} |\\mathcal{M}_{V \\to q\\bar{q} g}|^2(k^-,y)\\, (p_a \\cdot p_b)^2 (k^-,y)\\,,\n\\end{align}\nwhere everything is a function of $k^-$, $y$ and $q_V^2$. Up to this point, we have not expanded anything, but we have enforced the measurement, momentum conservation and the choice of frame.  We can now express $k^-$ as a dimensionless fraction $x$ via\n\\begin{equation}\\label{eq:xdefinition}\n\tx = \\frac{k^-}{q^-}\\,,\n\\end{equation} \nand we can write all Mandelstam invariants in terms of $x$ and $y$\n\\begin{align}\n\tk^-&= x q^- \\,,\\qquad {\\vec{k}_\\perp^{\\,2}} = q_V^2\\frac{y}{1-y}\\,,\\qquad k^+ = \\frac{{\\vec{k}_\\perp^{\\,2}}}{k^-} = \\frac{q_V^2}{q^-}\\frac{y}{1-y} \\frac{1}{x}\\,,{\\nonumber} \\\\\n\tq^2_V &\\equiv q^+ q^- - {\\vec{k}_\\perp^{\\,2}} = q_V^+ q_V^- - q_V^2\\frac{y}{1-y} \\quad\\implies\\quad q_V^2 = q^+_V q^-_V(1-y)\\,, {\\nonumber} \\\\\n\ts_{ab}(x,y) &= 2p_a \\cdot p_b = (q_V^- - k^-)(q_V^+ - k^+) = q_V^+q_V^- (1-x)\\left(1 - \\frac{y}{x}\\right) {\\nonumber} \\\\\n\t&= q_V^2 \\frac{(1-x)}{(1-y)}\\left(1 - \\frac{y}{x}\\right) \\,,{\\nonumber}\\\\\n\ts_{ak}(x,y) &= 2p_a \\cdot k = (q_V^- - k^-)k^+ = q_V^2\\frac{y}{(1-y)}\\frac{(1-x)}{x}\\,,{\\nonumber}\\\\\n\ts_{bk}(x,y) &= 2p_b \\cdot k = (q_V^+ - k^+)k^- = q_V^2 \\frac{x}{(1-y)}\\left(1 - \\frac{y}{x}\\right)\\,.\n\\end{align}\nWith these expressions one can easily check that\n\\begin{equation}\n\ts_{ab}+ s_{ak}+ s_{kb} = q_V^2\\,,\n\\end{equation}\nwith no power corrections.\nWe can now use the expressions for $s_{ab}$ to arrive at\n\\begin{align}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &= \\frac{1}{(1-y)^5}\\left(\\frac{\\mu^{2}}{q_V^2}\\frac{1-y}{y}\\right)^\\epsilon \\sum_{a,b} \\int_0^{1} \\frac{\\mathrm{d} x}{x} (1-x)^2  \\left(1 - \\frac{y}{x}\\right)^2|\\mathcal{M}_{V \\to q\\bar{q} g}|^2(x,y)\\,.\n\\end{align}\nExpanding in the collinear limit is now the same as expanding in $y$, and we find up to NLP\n\\begin{align}\\label{eq:collinearMasterUnreg}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &=\\left(\\frac{\\mu^{2}}{q_V^2 y}\\right)^\\epsilon \\sum_{a,b} \\int_0^{1} \\frac{\\mathrm{d} x}{x} (1-x)^2  \\left[A^{(0)}(x)+A^{(2)}(x) + y A^{(0)}(x)\\left( 5-\\frac{2}{x} -\\epsilon \\right) \\right]\\,.\n\\end{align}\nHere $A^{(0)}(x)$ and $A^{(2)}(x)$ are the expansion of the squared matrix elements in the collinear limits,\n\\begin{align}\n|\\mathcal{M}_{V \\to q\\bar{q} g}|^2(x,y)=A^{(0)}(x)+A^{(2)}(x) + y A^{(0)}(x)+\\cdots.\n\\end{align}\n\n\\subsection*{Rapidity Regulator}\n\nThe expression for the EEC in \\eq{collinearMasterUnreg} is divergent as $x \\to 0$. This is a rapidity divergence and must be regulated with a rapidity regulator. Here we present the result using pure rapidity regularization \\cite{Ebert:2018gsn}, which greatly simplifies the calculation, particularly for the constant (the non-logarithmically enhanced term). We have also computed the logarithm using the more standard $\\eta$-regulator \\cite{Chiu:2012ir,Chiu:2011qc}, and find an identical result.\n\n\nWe take as a regulator the rapidity in the $n$-collinear sector, normalized by the rapidity of the color singlet\\footnote{In the rest frame of the decaying boson this would be 1. Here we use a slightly boosted frame, so adding this factor is necessary to guarantee that the result is independent of the frame.}\n\\begin{equation}\n\te^{-Y_V}e^Y_n = \\frac{q_V^+}{q_V^-}\\frac{p_1^- + k^-}{p_1^+ + k^+} = \\frac{q_V^+}{k^+} = \\frac{q_V^+ q_V^- x}{{\\vec{k}_\\perp^{\\,2}}} = \\frac{q_V^+ q_V^- x (1-y)}{q_V^2 y } =  \\frac{x}{y}\\,.\n\\end{equation}\nThe rapidity regulated result is then given by\n\\begin{align}\\label{eq:collinearMasterReg}\n\t\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}}{\\mathrm{d}  y} &=\\left(\\frac{\\mu^{2}}{q_V^2 y}\\right)^\\epsilon \\frac{y^\\eta}{\\upsilon^\\eta} \\sum_{a,b} \\int_0^{1} \\frac{\\mathrm{d} x}{x^{1+\\eta}} (1-x)^2  \\left[A^{(2)}(x) + y A^{(0)}(x)\\left( 5-\\frac{2}{x} -\\epsilon \\right) \\right]\\,,\n\\end{align}\nwhich can be straightforwardly integrated.\n\n\\subsection*{Results}\n\nPlugging in the expression for $A^{(0)}$ and $A^{(2)}$ in QCD and $\\mathcal{N}=4$ gives the result for the EEC  up to NLP\n\\begin{align}\\label{eq:results}\n\t\\frac{1}{\\sigma_0}\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}^\\text{QCD}}{\\mathrm{d}  y} &= -\\frac{1}{y}\\left(\\log y + \\frac{3}{4}\\right) - 5\\log y-\\frac{9}{4} +\\mathcal{O}(y)\\,, {\\nonumber} \\\\\n\t\\frac{1}{\\sigma_0}\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}^{\\mathcal{N}=4}}{\\mathrm{d}  y} &=- \\frac{\\log y}{y}- 2\\log y +\\mathcal{O}(y)\\,.\n\\end{align}\nThis agrees with the expansion of the full angle result for $\\mathcal{N}=4$ in \\cite{Belitsky:2013ofa,Belitsky:2013xxa,Belitsky:2013bja}, as well as the classic QCD result \\cite{Basham:1978bw} for both the logarithm and the constant. This agreement (in particular for the constant) illustrates that we have the correct effective field theory setup, and that we understand the regularization of rapidity divergences at subleading power. This is further supported by our calculation of the subleading power corrections for the case of $p_T$ in \\cite{Ebert:2018gsn}, which was performed using the same formalism. While this expansion is an inefficient way to compute these subleading terms, which can much more easily be obtained by performing the full calculation and expanding the result, the ability to systematically compute the terms at each order in the power expansion will allow us to perform an all orders resummation by deriving renormalization group evolution equations in rapidity.\n\n\n\\subsection{Physical Intuition for Subleading Power Rapidity Divergences}\\label{sec:intuition}\n\nIn this section we wish to summarize some of the general lessons learned from the above fixed order calculation (as well as from the calculation of the fixed order subleading rapidity logarithms for $p_T$ in \\cite{Ebert:2018gsn}), which provides significant clues into the structure of the subleading power rapidity renormalization group.  Since we have shown above that the description of the EEC in the back-to-back limit is ultimately formulated in the EFT in terms of transverse momentum, here we will phrase all the functions in terms of transverse momentum, however, these can straightforwardly be converted back to derive results for the EEC.\nWe will also phrase this discussion in terms of the $\\eta$ regulator \\cite{Chiu:2012ir,Chiu:2011qc}, due to the fact that it is more familiar for most readers.\n\nThe first important observation from the fixed order calculation is that at NLP there are no purely virtual corrections at lowest order in $\\alpha_s$ (such a correction would appear as $y\\delta(y)=0$). The lowest order result for the EEC in $\\mathcal{N}=4$, which we use as an example due to its simpler structure, is given by\n\\begin{align}\\label{eq:N4_NLP_fixedorder}\n\t\\frac{1}{\\sigma_0}\\frac{\\mathrm{d} \\sigma_{\\text{EEC}}^{\\mathcal{N}=4}}{\\mathrm{d}  (1-z)} &=  - 2\\log(1-z) +\\mathcal{O}(1-z)\\,.\n\\end{align}\nHere the single logarithm comes from real soft or collinear emissions. Since the soft and collinear sectors lie at the same virtuality, this guarantees that at subleading power the lowest order logarithm must be a rapidity logarithm. This can be made explicit by writing down a general ansatz for the one-loop result. This approach was first introduced in the SCET$_{\\rm I}$~ case in \\cite{Moult:2016fqy}.  Here we will phrase it in terms of the underlying $p_T$ dependence, since this is perhaps more familiar to the resummation community. The general form of the one-loop result for the NLP corrections to $p_T$ can be written as\n\\begin{align}\n\\frac{d\\sigma^{(2)}}{dp_T^2}=\\left( \\frac{\\mu^2}{p_T^2}  \\right)^\\epsilon   \\left( \\frac{\\nu}{p_T}  \\right)^\\eta  \\left[  \\frac{s_\\epsilon}{\\epsilon}+\\frac{s_\\eta}{\\eta}  \\right] +\\left( \\frac{\\mu^2}{p_T^2}  \\right)^\\epsilon   \\left( \\frac{\\nu}{Q}  \\right)^\\eta  \\left[  \\frac{c_\\epsilon}{\\epsilon}+\\frac{c_\\eta}{\\eta}  \\right] \\,.\n\\end{align}\nExpanding this, we find\n\\begin{align}\n\\frac{d\\sigma^{(2)}}{dp_T^2}=\\left( \\frac{c_\\epsilon+s_\\epsilon}{\\epsilon}  \\right)   +     \\left( \\frac{c_\\eta+s_\\eta}{\\eta}  \\right) +2(c_\\epsilon+s_\\epsilon)  \\log\\frac{\\mu}{p_T} +s_\\eta \\log \\frac{\\nu}{p_T} +c_\\eta \\log\\frac{\\nu}{Q}\\,.\n\\end{align}\nDemanding that there are no $1\/\\epsilon$ or $1\/\\eta$ poles in the final answer imposes the conditions \n\\begin{align}\nc_\\epsilon=-s_\\epsilon\\,, \\qquad c_\\eta=-s_\\eta\\,,\n\\end{align}\nand shows that the lowest order logarithm appearing in the cross section is a rapidity logarithm.\n\nThis simple observation provides considerable insight into the structure of the subleading power renormalization group evolution equations. In \\cite{Moult:2018jjd} it was shown that the way that a single logarithm at the first non-vanishing order can be generated is through renormalization group mixing. In the particular case studied in \\cite{Moult:2018jjd} there was only a virtuality renormalization group, and therefore the single logarithm was generated by the $\\mu$-RGE. However, here we see that for observables that have both a $\\mu$ and $\\nu$ RGE, this mixing will always occur in the $\\nu$ RGE, since the lowest order logarithm is always a rapidity logarithm. \n\nOur fixed order calculation also provides insight into the structure of the cancellation of rapidity divergences at subleading power.  Although we have not written down a complete set of SCET$_{\\rm II}$~operators, we briefly comment on the physical intuition for the cancellation of rapidity anomalous dimensions, which will then determine how renormalization group consistency appears in the effective theory. We can consider the case of the NLP correction from a soft quark, since this will provide the clearest picture of the differences between rapidity divergences and virtuality divergences. At lowest order, the soft function for the emission of a soft quark is given by\n\\begin{align}\\label{eq:quark_soft}\nS^{(2)}_q&= \\fd{3cm}{figures\/soft_quark_scetii_low}{\\nonumber} \\\\\n&\\propto \\mu^{2\\epsilon} \\nu^\\eta \\int d\\hspace*{-0.08em}\\bar{}\\hspace*{0.1em}^d k |k^+-k^-|^{-\\eta} \\delta^+ (k^2) \\frac{(p_\\perp^2)^{-\\epsilon}}{\\Gamma(1-\\epsilon) \\pi^\\epsilon} \\delta^{d-2} (p_\\perp-\\hat \\mathcal{P}_\\perp) \\\\\n&=\\frac{1}{\\eta} + \\log{\\frac{\\nu}{p_\\perp}} +\\cdots\\,.\n\\end{align}\nThis soft function is $\\epsilon$ finite, but exhibits the expected $\\eta$ divergence. This divergence cannot be absorbed into the renormalization of this soft function, since it starts at $\\alpha_s$, and therefore we must find the class of operators that this soft function mixes into. We will derive the structure of these operators in \\Sec{sec:RRG_NLP}.\n\nWhile the fact that this operator must mix into a new operator is familiar from other studies at NLP, what is different is that the integrand for the soft function is ``1\".  This implies that an equal part of the divergence comes from when the quark goes to infinite rapidity in either direction. This has an interesting interpretation, which can guide the physical intuition for how the cancellation of rapidity divergences occurs. As the soft quark goes collinear in the direction of the  collinear quark, the rapidity divergence must be cancelled by a collinear rapidity divergence from a subleading power jet function describing two collinear quarks. One the other hand, as the soft quark goes collinear with the gluon, it must be cancelled by a subleading power jet function involving a collinear quark and gluon field. In pictures, the two limits are shown as\n\\begin{align}\n\\fd{3cm}{figures\/qq_op_low.pdf}\\xleftarrow[]{\\eta\\to -\\infty} \\fd{3cm}{figures\/soft_quark_low.pdf}  \\xrightarrow[]{\\eta\\to \\infty} \\fd{3cm}{figures\/qg_collinear_low.pdf}\\,.\n\\end{align}\nThis is of course exactly what happens at leading power, however, at leading power the jet functions are identical in all limits, giving a much simpler structure. The structure for the cancellation of the rapidity divergences at subleading power implies that renormalization group consistent subsets of operators will appear in triplets, as opposed to doublets, as was observed in \\cite{Moult:2018jjd} for the $\\mu$-RGE. The factorization formula for a single pair of triplets will take the (extremely) schematic form\n\\begin{align}\n\\frac{d\\sigma^{(2)}}{dz}&=H_1 J_{\\bar n}^{(2)} J_n^{(0)} S^{(0)} \n+H_2 J_{\\bar n}^{(0)} J_n^{(2)} S^{(0)}\n+H_3 J_{\\bar n}^{(0)} J_n^{(0)} S^{(2)}\\,,\n\\end{align}\nwhere the jet and soft functions with superscript $(2)$ denote power suppressed functions. In terms of pictures, we have\n\\begin{align}\\label{eq:schematic_factorization}\n\\frac{d\\sigma^{(2)}}{dz}&={\\nonumber} \\\\\n&\\underbrace{  \\left| \\fd{1.5cm}{figures\/LP_hardfunc_low.pdf}~ \\right|^2  \\cdot   \\int dr_2^+ dr_3^+ \\fd{3cm}{figures\/soft_quark_collinear_piece_purjet_low.pdf}\\otimes \\fd{3cm}{figures\/1quark_jetfunction_low} \\otimes  \\fd{3cm}{figures\/soft_quark_diagram_wilsonframe_low.pdf}   }_{\\text{Soft Quark Correction}}{\\nonumber} \\\\\n +&\\underbrace{ \\int d \\omega_1 d \\omega_2 \\left| \\fd{1.5cm}{figures\/NLP_hard_2quark_low}~ \\right|^2\\otimes\\fd{3cm}{figures\/2quark_jetfunc_low}\\otimes \\fd{3cm}{figures\/1gluon_jetfunc_low}\\otimes \\fd{3cm}{figures\/eikonal_factor_gluon_low.pdf} }_{\\text{Collinear Quark Correction 1}} {\\nonumber} \\\\\n +&\\underbrace{ \\int d \\omega_1 d \\omega_2 \\left| \\fd{1.5cm}{figures\/NLP_1quark1gluon_hard_low}~\\right|^2  \\otimes~\\fd{3cm}{figures\/1quark_jetfunction_low}\\otimes  \\fd{3cm}{figures\/1quark1gluon_jetfunc_low}\\otimes \\fd{3cm}{figures\/eikonal_factor_gluon_low.pdf}}_{\\text{Collinear Quark Correction 2}}\\,,\n\\end{align}\nwhich perhaps makes more clear schematically how the cancellation between $\\nu$ anomalous dimension occurs between the soft and collinear sectors of the theory. A similar triplet exists from the corrections associated with soft gluon emission. Each of the power suppressed functions in \\Eq{eq:schematic_factorization} will mix in rapidity, as was illustrated for the soft function in \\Eq{eq:quark_soft}. To perform the renormalizaton, we must therefore identify which operators are being mixed into in each case.\n\nIt is interesting to compare this to the structure of the subleading power factorization for SCET$_{\\rm I}$~ event shapes, which is described in \\cite{Moult:2018jjd,Moult:2019uhz}. There subleading power jet functions involving two quark fields, and those involving a quark and gluon field are in separately renormalization group consistent pairings. Therefore, we find that the SCET$_{\\rm II}$~ case gives rise to a much tighter structure in the EFT, since it links multiple hard scattering operators. Note that this also suggests that the issue of endpoint divergences is harder to avoid, although this is a topic that we leave for future study.\n\nIn conclusion, we have learned a number of general lessons about the subleading power $\\nu$-RG from our perturbative calculation. In particular, we have seen that subleading power corrections to the EEC (and more generally any observable that involves both rapidity and virtuality renormalization group flows) involve a mixing in rapidity at the first non-trivial order into a new class of operators. In \\Sec{sec:RRG_NLP} we will derive the structure of these new operators by studying the consistency of the structure of the renormalization group.\n\n\\section{Rapidity Renormalization Group at Subleading Power}\\label{sec:RRG_NLP}\n\nIn the previous section we have shown that for the EEC, and more generally for subleading corrections to observables with both rapidity and virtuality scales, the subleading power rapidity RGE will always involve mixing into additional operators. The goal of this section will be to derive the structure of the operators that arise from this mixing, as well as their renormalization group properties. As in our study of thrust at subleading powers \\cite{Moult:2018jjd}, our approach to gain a handle on the structure of the RG equations at subleading power will be to use an illustrative example of subleading power jet and soft functions whose renormalization group structure can be obtained from the known leading power RG equations. Once the particular form of the operators is derived, they can then be applied in other situations. \n\nWe will find that for the case of rapidity divergences considered here, the use of an illustrative example is more subtle than for the $\\mu$ RGE due to the appearance of additional divergences that appear. We will also find that due to this, there are multiple (two distinct) operators that can be mixed into. Therefore, our analysis in \\Sec{sec:example} should be viewed as providing motivation for the type of operators that appear, although in the initial form that they arise, they will involve unregularized integrals. With this motivation for their structure, we are then able to use the commutativity of the $\\mu$ and $\\nu$ RGEs in \\Sec{sec:consistency} to fix the RG properties of these operators and provide regularized definitions. We then solve the associated evolution equations for the newly introduced operators in \\Sec{sec:solution}.\n\n\\subsection{An Illustrative Example}\\label{sec:example}\n\nWe will begin by considering the LP soft function for $p_T$, defined as\n\\begin{align}\n S(\\vec p_T) &= \\frac{1}{N_c^2 - 1} \\big\\langle 0 \\big| \\mathrm{Tr} \\bigl\\{\n   \\mathrm{T} \\big[\\mathcal{S}^\\dagger_{\\bn} \\mathcal{S}_n\\big]\n  \\delta^{(2)}(\\vec p_T-\\mathcal{P}_\\perp) \\overline{\\mathrm{T}} \\big[\\mathcal{S}^\\dagger_{n} \\mathcal{S}_{\\bn} \\big]\n   \\bigr\\} \\big| 0 \\big\\rangle\n\\,.\\end{align}\nThis soft function satisfies the $\\mu$ and $\\nu$ RGEs\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S(\\vec p_T)&=\\int d\\vec q_T \\gamma^S_\\nu(p_T-q_T) S(\\vec q_T)\\,, {\\nonumber} \\\\\n\\mu \\frac{d}{d\\mu}S(\\vec p_T)&=\\gamma_\\mu^S S(\\vec p_T)\\,,\n\\end{align}\nwith the anomalous dimensions\n\\begin{align}\n\\gamma_\\mu^S &=4\\Gamma^{\\mathrm{cusp}}(\\alpha_s)\\log \\left( \\frac{\\mu}{\\nu}\\right)\\,,{\\nonumber} \\\\\n\\gamma_\\nu^S &=2\\Gamma^{\\mathrm{cusp}} (\\alpha_s)\\mathcal{L}_0\\left( \\vec p_T,\\mu \\right)\\,.\n\\end{align}\nCrucially, the $\\mu$ anomalous dimension is multiplicative, while the $\\nu$ anomalous dimension is a convolution in $p_T$ space. It is this fact that will ultimately lead to the subleading power rapidity renormalization group having a more interesting structure. \n\nA simple trick to understand the structure of subleading power RG equations that was first used in \\cite{Moult:2018jjd} is to consider jet and soft functions obtained by multiplying the leading power jet and soft functions by a kinematic invariant. In the present case, we \ncan consider the subleading power soft function defined by\n\\begin{align}\nS_{p_T^2}^{(2)}(\\vec p_T)=\\vec p_T^2 S(\\vec p_T)\\,.\n\\end{align}\nThe superscript $(2)$ indicates that this function is power suppressed due to the explicit factor of $p_T^2$, and the subscript $p_T^2$ is meant to identify the nature of this power suppression. The structure of this function is known to all orders, since it is inherited from the known structure of the leading power soft function. However, understanding how this structure is manifested in the renormalization group structure of $S_{p_T^2}^{(2)}$ is non-trivial, and will reveal new operators.\n\n\n\nFirst, we note that the $\\mu$ RGE for this subleading power soft function is identical to the RGE of the leading power soft function, since the $\\mu$ RGE is multiplicative. We therefore have\n\\begin{align}\n\\mu \\frac{d}{d\\mu}S_{p_T^2}^{(2)}(\\vec p_T)=\\gamma^\\mu_S   S_{p_T^2}^{(2)}(\\vec p_T)\\,.\n\\end{align}\nHowever, we find a more interesting behavior for the $\\nu$ RGE, due to the fact that it is a convolution in $p_T$. Multiplying both sides of the LP RGE by $\\vec p_T^2$, and using the identity\n\\begin{align}\\label{eq:pt_expand}\n\\vec p_T^2 =(\\vec p_T-\\vec q_T)^2 +\\vec q_T^2 +2(\\vec p_T- \\vec q_T) \\cdot \\vec q_T\\,,\n\\end{align}\nwe arrive at the equation\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S_{p_T^2}^{(2)}(\\vec p_T)&=\\int d\\vec q_T  (\\vec p_T-\\vec q_T)^2  \\gamma_S(p_T-q_T) S(q_T)  \\\\\n&+\\int d\\vec q_T  \\gamma_S(p_T-q_T) \\left[ 2(\\vec p_T- \\vec q_T) \\cdot \\vec q_T S(q_T)\\right] + \\int d\\vec q_T \\gamma_S(p_T-q_T) \\left[ \\vec q_T^2 S(\\vec q_T)\\right]\\,. {\\nonumber}\n\\end{align}\nNote that we must arrange \\Eq{eq:pt_expand} in this form, so that it is a kernel in $\\vec p_T- \\vec q_T$ multiplying a function of $q_T$.\nSimplifying this result, we find that we can write it  as\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S_{p_T^2}^{(2)}(\\vec p_T)&=2\\Gamma^{\\mathrm{cusp}} {\\mathbb{I}}_S \\\\\n&+\\int d\\vec q_T  \\gamma_S(p_T-q_T) 2(\\vec p_T- \\vec q_T) \\cdot   \\vec S^{(1)}(q_T) +\\int d\\vec q_T \\gamma_S(p_T-q_T) S_{p_T^2}^{(2)}(\\vec q_T)\\,.{\\nonumber}\n\\end{align}\nHere we see a renormalization group mixing with two power suppressed functions. The first is \n\\begin{align}\n{\\mathbb{I}}_S\\propto\\int d^2\\vec q_T~   S(\\vec q_T)\\,,\n\\end{align}\nwhich we will refer to as the ``rapidity identity operator\", and will play a crucial role in our subsequent analysis. As written, the integral over $q_T$ goes to infinity, and therefore this expression is ill-defined, and will require regularization. For this reason we have also been glib about what this operator depends on. In the next section we will present a way of deriving its renormalization group properties, as well as a regularized definition. The goal of this section is merely to illustrate that the subleading power soft function mixes into a new operator which is loosely related to a moment of the leading power soft function. The second operator arising in the mixing is \n\\begin{align}\n\\vec S^{(1)}(\\vec p_T)=\\vec p_T S^{(0)}(\\vec p_T)\\,,\n\\end{align}\nwhich is a vector soft function which scales like $\\mathcal{O}(\\lambda)$. This function can only appear in a factorization formula if it is dotted into a vector jet function, or some other vector quantity. \n\n\nWhile we believe that it would be extremely interesting to study in more detail the complete structure of this illustrative example, and we will return to this in future work, here we focus only on the elements of these equations that are required at leading logarithmic accuracy. We note that\n\\begin{align}\n{\\mathbb{I}}_S=1+\\mathcal{O}(\\alpha_s)\\,, \\qquad S^{(1)}(\\vec p_T)=0+\\mathcal{O}(\\alpha_s)\\,.\n\\end{align}\nTherefore, in the leading logarithmic series, $S_{p_T^2}^{(2)}$ mixes into ${\\mathbb{I}}_S$, and we can ignore $S^{(1)}$. We can therefore simplify our equation to\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S_{p_T^2}^{(2)}(\\vec p_T,\\mu,\\nu)&=2\\Gamma^{\\mathrm{cusp}} {\\mathbb{I}}_S  +\\int d\\vec q_T \\gamma_S(p_T-q_T) S_{p_T^2}^{(2)}(\\vec q_T,\\mu,\\nu)\\,.\n\\end{align}\nWe see that what is occurring is that the power suppressed soft function is mixing with the rapidity identity operator. This provides a renormalization group derivation of the perturbative calculations in \\Sec{sec:FO}, where one generates a rapidity divergence and associated logarithm at the lowest non-trivial order. This is simply a perturbative description of the mixing into ${\\mathbb{I}}_S$. Now, with this general structure in mind, we would like to understand the properties of this rapidity identity operator.\n\n\n\nWe emphasize again that we have performed only a cursory study of this illustrative example so as to be able to illustrate the renormalization group mixing in rapidity, and to identify the structure of the rapidity identity operator. It would be particularly interesting to study the complete structure of this illustrative example to all logarithmic orders, and in particular to better understand the structure of the convolutions in $p_T$. However, since the focus of this paper is on deriving the leading logarithmic series for the EEC, we will leave this to future work.\n\n\n\n\n\\subsection{Rapidity Identity Operators}\\label{sec:consistency}\n\n\n\nIn the previous section, we found that the subleading power rapidity renormalization group involves a mixing into the rapidity identity operator, which is loosely related to the first moment of the leading power soft function\n\\begin{align}\\label{eq:identity_unregularized}\n{\\mathbb{I}}_S\\propto\\int d^2\\vec q_T~   S(\\vec q_T)\\,.\n\\end{align}\nThe goal of this section will be to understand how to make sense of this operator, since it is ill-defined as currently written. This is a crucial difference as compared with the case of thrust considered in \\cite{Moult:2018jjd}. There a similar first moment operator appears, defined as \\cite{Moult:2018jjd}\n\\begin{align}\\label{eq:theta_soft_first}\nS^{(2)}_{g,\\theta}(k,\\mu)&= \\frac{1}{(N_c^2-1)} {\\rm tr} \\langle 0 | \\mathcal{Y}^T_{\\bar n} (0)\\mathcal{Y}_n(0) \\theta(k-\\hat \\mathcal{T}) \\mathcal{Y}_n^T(0) \\mathcal{Y}_{\\bar n}(0) |0\\rangle\\,.\n\\end{align}\nHowever, there the first moment is a finite integral, and does not introduce new divergences, allowing all the properties of this operator to be immediately deduced from those of the leading power operator. For the case of $p_T$ considered here, additional arguments must be used to fully fix the structure of the rapidity identity operator.\n\n\nThe $\\mu^2$ dependence of the rapidity identity operator can be derived using the commutativity of the RG \\cite{Chiu:2012ir,Chiu:2011qc}, namely that\n\\begin{align}\n\\left[  \\frac{d}{d\\mu}, \\frac{d}{d\\nu}  \\right]=0\\,,\n\\end{align}\nwhich ensures the path independence of the $\\mu$ and $\\nu$ rapidity evolution. Here we consider a general subleading power soft operator $S^{(2)}$ (not necessarily $S_{p_T^2}^{(2)}$). If we assume that this operator mixes into a single identity type operator, then we obtain\n\\begin{align}\n\\mu \\frac{d}{d\\mu} \\left[  \\nu \\frac{d}{d\\nu} S^{(2)}  \\right]&=\\mu \\frac{d}{d\\mu} \\left[ \\gamma_{\\delta{\\mathbb{I}} }  {\\mathbb{I}}_S+\\gamma_\\nu S^{(2)}  \\right]\\,,{\\nonumber} \\\\\n\\nu \\frac{d}{d\\nu} \\left[  \\mu \\frac{d}{d\\mu} S^{(2)}  \\right]&= \\nu\\frac{d}{d\\nu} \\left[  \\gamma^S_\\mu S^{(2)} \\right]\\,.\n\\end{align}\nHere, we have used $\\gamma_{\\delta{\\mathbb{I}} }$ to denote the mixing anomalous dimension.\nPerforming the next differentiation, we  then obtain the equality\n\\begin{align}\n\\gamma_{\\delta{\\mathbb{I}} } \\mu \\frac{d}{d\\mu} {\\mathbb{I}}_S +\\left[  \\mu \\frac{d}{d\\mu} \\gamma_\\nu \\right]S^{(2)} +\\gamma_\\nu \\gamma_\\mu^S S^{(2)} =\\gamma_\\mu^S \\gamma_{\\delta{\\mathbb{I}} }  {\\mathbb{I}}_S +\\left[ \\nu \\frac{d}{d\\nu} \\right]S^{(2)} +\\gamma_\\nu \\gamma_\\mu^S S^{(2)}\\,.\n\\end{align}\nUsing the fact that commutativity is satisfied for the leading power anomalous dimensions, we arrive at\n\\begin{align}\n\\mu \\frac{d}{d\\mu}{\\mathbb{I}}_S = \\gamma_\\mu^S {\\mathbb{I}}_S\\,.\n\\end{align}\nThis fixes the $\\mu$ anomalous dimension of the rapidity identity operator. \n\n\nTo fix the $\\nu$ anomalous dimension, we now apply commutativity to ${\\mathbb{I}}_S$ itself, and use the fact that we know the $\\mu$ anomalous dimension. We then have\n\\begin{align}\n\\left[  \\frac{d}{d\\mu}, \\frac{d}{d\\nu}  \\right]  {\\mathbb{I}}_S =0\\,,\n\\end{align}\nwhich gives the equation (at lowest order in $\\alpha_s$, which is sufficient for LL)\n\\begin{align}\n\\mu \\frac{d}{d\\mu} \\left( \\nu \\frac{d}{d\\nu}  \\right) {\\mathbb{I}}_S=-4 \\Gamma^{\\mathrm{cusp}}\\,,\n\\end{align}\nwhich can then be solved for\n\\begin{align}\n\\nu \\frac{d}{d\\nu} {\\mathbb{I}}_S= -4 \\Gamma^{\\mathrm{cusp}} \\log \\left( \\frac{\\mu^2}{\\Lambda^2}  \\right)\\,,\n\\end{align}\nwhere $\\Lambda^2$ is an as yet to be determined scale. The only available scales at leading logarithmic accuracy are $\\Lambda^2=p_T^2, \\mu^2, \\nu^2$. The cases $\\Lambda^2=p_T^2, \\nu^2$ both give the same behavior for the $\\nu$ RGE on the hyperbola $\\mu=p_T$, and therefore we will not treat them separately. It would be interesting to explore in more detail  the differences between these RGEs.\n\n\nWe therefore find two distinct identity operators with two different rapidity anomalous dimensions\n\\begin{align}\n\\nu \\frac{d}{d\\nu}{\\mathbb{I}}^\\nu_S(\\mu, \\nu) &= -\\gamma_\\mu^S {\\mathbb{I}}^\\nu_S(\\mu, \\nu)\\,, {\\nonumber} \\\\\n\\nu \\frac{d}{d\\nu}{\\mathbb{I}}^{p_T^2}_S(p_T^2,\\mu, \\nu) &= 2 \\Gamma^{\\mathrm{cusp}} \\log \\left( \\frac{p_T^2}{\\mu^2}  \\right) {\\mathbb{I}}^{p_T^2}_S(p_T^2,\\mu, \\nu)\\,.\n\\end{align}\nHere we have again used the superscript $p_T^2$ to indicate the identity function that the $S_{p_T^2}^{(2)}$ function mixes into, as we will argue shortly, and the superscript $\\nu$ to indicate the identity function that has a non-trivial $\\nu$ RGE on the $\\mu=p_T$ hyperbola.\n\nWhile this argument allows us to derive the renormalization group properties of these operators, which is sufficient for the purposes of this paper, it is also interesting to give explicit example of functions that realize this behavior. This is easy to do by defining regularizations of the integral in \\Eq{eq:identity_unregularized}. \nFunctions which give the behavior of the different identity operators at LL are\n\\begin{align}\n{\\mathbb{I}}^\\nu_S(\\mu, \\nu)&=\\int_0^{\\nu^2} d^2\\vec q_T~   S(\\vec q_T)\\,, \\\\\n{\\mathbb{I}}_S^{p_T^2}(p_T^2,\\mu, \\nu)&=\\int_0^{p_T^2} d^2\\vec q_T~   S(\\vec q_T)\\,.\n\\end{align}\nThe first function is a function of only $\\mu^2\/\\nu^2$, while the second also depends on $p_T^2$. These two functions have different properties under boosts, and therefore do not themselves mix. This provides leading logarithmic definitions of the rapidity identity operators. It would be extremely interesting to understand how to extend these definitions beyond leading logarithm, however, we leave this to future work.\n\n\nFor the particular soft function considered in our illustrative example, $S_{p_T^2}^{(2)}=p_T^2 S^{(0)}$, one can use the knowledge of the two loop soft function to show that it is the operator ${\\mathbb{I}}^{p_T^2}_S(p_T^2,\\mu, \\nu)$ that is being mixed into. This can also be argued directly by symmetry grounds: at the scale $\\mu=p_T$, the leading power soft function does not flow in $\\nu$ at leading logarithmic accuracy. This is due to its boost invariance. This property is not broken by multiplying by $p_T^2$, and therefore must also be a property of the counterterm operator that is being mixed with. This identifies the operator ${\\mathbb{I}}^{p_T^2}_S(p_T^2,\\mu, \\nu)$. We can therefore make more precise the equation earlier for the RG of this function\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S_{p_T^2}^{(2)}(\\vec p_T,\\mu,\\nu)&=2\\Gamma^{\\mathrm{cusp}} {\\mathbb{I}}^{p_T^2}_S(\\vec p_T,\\mu,\\nu)  +\\int d\\vec q_T \\gamma_S(p_T-q_T) S_{p_T^2}^{(2)}(\\vec q_T,\\mu,\\nu)\\,.\n\\end{align}\nFor subleading power soft functions with explicit fields inserted, this argument no longer holds, and one can mix into the other operator.\n\nIt is also interesting to arrive at these conclusions for the structure of the renormalization group by manipulation of the renormalization group equations. This approach is ultimately ill-defined due to the lack of convergence of the integrals, but it gives the same results as derived from the commutativity of the RG, and provides additional insight into the origin of this behavior.\nRecall that the leading power renormalization group evolution equations are\n\\begin{align}\n\\nu \\frac{d}{d\\nu}S(\\vec p_T)&=\\int d\\vec q_T \\gamma^S_\\nu(p_T-q_T) S(q_T)\\,, {\\nonumber} \\\\\n\\mu \\frac{d}{d\\mu}S(\\vec p_T)&=\\gamma_\\mu^S S(\\vec p_T)\\,,\n\\end{align}\nwith the anomalous dimensions\n\\begin{align}\n\\gamma_\\mu^S &=4\\Gamma^{\\mathrm{cusp}}(\\alpha_s)\\log \\left( \\frac{\\mu}{\\nu}\\right)\\,,{\\nonumber} \\\\\n\\gamma_\\nu^S &=2\\Gamma^{\\mathrm{cusp}} (\\alpha_s)\\mathcal{L}_0\\left( \\vec p_T,\\mu \\right)\\,.\n\\end{align}\nSince the $\\mu$ anomalous dimension is multiplicative, this should not be changed if we integrate over $q_T$. In other words, we have\n\\begin{align}\n\\int d^2 p_T \\left[ \\mu \\frac{d}{d\\mu}S(\\vec p_T)=\\gamma^\\mu_S S(\\vec p_T) \\right] \\implies \\mu \\frac{d}{d\\mu} {\\mathbb{I}}_S=  \\gamma^\\mu_S {\\mathbb{I}}_S\\,,\n\\end{align}\nwhich immediately leads to the fact that ${\\mathbb{I}}_S$ is multiplicatively renormalized in $\\mu$ with the same anomalous dimension as the leading power soft function.  For the $\\nu$ renormalization group equation, we have\n\\begin{align}\n\\nu \\frac{d}{d\\nu} {\\mathbb{I}}_S &=\\int d^2 \\vec q_T \\nu \\frac{d}{d\\nu} S(\\vec q_T){\\nonumber} \\\\\n&=\\int d^2 \\vec q_T \\left[   \\int d^2 \\vec p_T \\gamma_S(\\vec q_T-\\vec p_T) S(\\vec p_T)  \\right]\\,.\n\\end{align}\nNow, performing the shift $\\vec q_T\\to \\vec q_T +\\vec p_T$, we obtain\n\\begin{align}\n\\nu \\frac{d}{d\\nu} {\\mathbb{I}}_S &=\\left[ \\int d^2 \\vec q_T \\gamma_S(\\vec q_T)  \\right]{\\mathbb{I}}_S\\,,\n\\end{align}\nwhich is again multiplicative. The expression in square brackets is not well defined, and must be fixed by some regularization, as was shown above. In this case, this shift argument may no longer be valid. However, this exercise is merely meant to illustrate another perspective on why the rapidity identity operator should satisfy a multiplicative $\\nu$ RGE, and the argument presented earlier in this section should be taken as primary.\n\n\nTherefore, in summary, we have shown that there are non-trivial rapidity identity operators that arise at subleading power, and we have identified the renormalization group properties of these operators at leading logarithmic accuracy. The first rapidity identity operator does not depend on the observable, and its anomalous dimensions are given by\n\\begin{align}\\label{eq:rap_identity_summary}\n\\mu \\frac{d}{d\\mu} {\\mathbb{I}}^\\nu_S \\left(  \\frac{\\mu^2}{\\nu^2} \\right) &= -2\\Gamma^{\\mathrm{cusp}}\\log\\left(  \\frac{\\nu^2}{\\mu^2} \\right)~ {\\mathbb{I}}^\\nu_S\\left(  \\frac{\\mu^2}{\\nu^2} \\right)\\,,  {\\nonumber} \\\\\n\\nu \\frac{d}{d\\nu} {\\mathbb{I}}^\\nu_S\\left(  \\frac{\\mu^2}{\\nu^2} \\right) &= 2\\Gamma^{\\mathrm{cusp}} \\log\\left(  \\frac{\\nu^2}{\\mu^2} \\right)~ {\\mathbb{I}}^\\nu_S\\left(  \\frac{\\mu^2}{\\nu^2} \\right)\\,.\n\\end{align}\nThe second rapidity identity operator depends on the observable, and its anomalous dimensions are given by\n\\begin{align}\\label{eq:rap_identity2_summary}\n\\mu \\frac{d}{d\\mu} {\\mathbb{I}}^{p_T^2}_S \\left( p_T^2,\\mu, \\nu \\right) &= -2\\Gamma^{\\mathrm{cusp}}\\log\\left(  \\frac{\\nu^2}{\\mu^2} \\right)~ {\\mathbb{I}}^{p_T^2}_S\\left( p_T^2,\\mu, \\nu \\right)\\,, {\\nonumber} \\\\\n\\nu \\frac{d}{d\\nu} {\\mathbb{I}}^{p_T^2}_S\\left( p_T^2,\\mu, \\nu \\right) &= 2\\Gamma^{\\mathrm{cusp}}\\log\\left(  \\frac{p_T^2}{\\mu^2} \\right)~ {\\mathbb{I}}^{p_T^2}_S\\left( p_T^2,\\mu, \\nu \\right)\\,.\n\\end{align}\nHere we have written the anomalous dimension in terms of the cusp anomalous dimension \\cite{Korchemsky:1987wg}, which is given by $\\Gamma^{\\mathrm{cusp}}=4(\\alpha_s\/4\\pi)C_A+\\mathcal{O}(\\alpha_s^2)$.\nWe expect these functions to appear ubiquitously in subleading power rapidity renormalization, and we therefore believe their identification is an important first step towards an understanding of the structure of subleading power rapidity logarithms.\n\nOne also has rapidity identity operators in the jet\/beam sectors, that are defined analogously to the soft operators. Their anomalous dimensions are fixed to be identical to the soft rapidity identity operators (up to a sign) by RG consistency. For the particular case of the EEC, we can avoid them by always running to the jet scale. They are interesting for the case of $p_T$ resummation where one must consider their matching onto the parton distribution functions (PDFs). We will present a more detail discussion of these structures in future work.\n\nIt is also interesting to note that the subleading power Regge limit for massive scattering amplitudes has recently been studied in $\\mathcal{N}=4$ SYM in \\cite{Bruser:2018jnc}. Their solution also involves an interesting operator mixing. Since there are connections between the Regge limit and the EEC at leading power due to conformal transformations, it would be interesting to understand if these persist at subleading power.\n\n\\subsection{Analytic Solution of Renormalization Group Evolution Equations}\\label{sec:solution}\n\nIn this section we provide an analytic solution to the renormalization group evolution equations of the subleading power soft functions when mixing with either type of identity operator. Here we will consider only the case of fixed coupling, since our current application will be to $\\mathcal{N}=4$ (which is conformal), and the reader who is interested in extending these results to running coupling can consult  \\cite{Moult:2018jjd}. We will also only study the renormalization group flow in $\\nu$ at the scale $\\mu=p_T$ to simplify the analysis. This will be sufficient for our applications, since we can always run the hard function down to the scale $\\mu=p_T$. We will consider separately the two different types of identity operators, since we know that due to their different properties under boosts, they themselves cannot mix to generate a more complicated RG structure.\n\n\\subsection*{Identity Operator ${\\mathbb{I}}^{p_T^2}_S \\left( p_T^2,\\mu, \\nu \\right)$:}\n\nWe first consider the case of mixing with the operator ${\\mathbb{I}}_S \\left( p_T^2,\\mu, \\nu \\right)$, which gives rise to a simple $\\nu$ RGE at the scale $\\mu=p_T$. In this case, we get the RGE\n\\begin{align}\n\\nu \\frac{d}{d\\nu}\\left(\\begin{array}{c} S^{(2)} \\\\  {\\mathbb{I}}^{p_T^2}_S \\end{array} \\right) &=  \\left( \\begin{array}{cc}0&\\gamma_{\\delta{\\mathbb{I}} }\\,  \\\\  0 & 0   \\end{array} \\right) \\left(\\begin{array}{c} S^{(2)} \\\\  {\\mathbb{I}}^{p_T^2}_S \\end{array} \\right) \\,, \n\\end{align}\nwith the boundary conditions\n\\begin{align}\n{\\mathbb{I}}^{p_T^2}_S(\\mu=p_T, \\nu=p_T)=1\\,, \\qquad S^{(2)}(\\mu=p_T,\\nu=p_T)=0\\,.\n\\end{align}\nNote that the specific choice of $\\mu=p_T$ is important for achieving this simple form of the RG since it eliminates the need to consider the diagonal terms in the mixing matrix. More generally, these would be required, but for LL resummation as considered in this paper, the particular path considered here suffices, as explained in more detail in \\Sec{sec:N4_LL}.\n\nThis RGE is trivial to solve, and generates a single logarithm from the mixing\n\\begin{align}\nS^{(2)}(\\mu=p_T, \\nu=Q)=\n \\gamma_{\\delta{\\mathbb{I}} }  \\log(p_T\/Q)   {\\mathbb{I}}^{p_T^2}_S(\\mu=p_T, \\nu=p_T) \\,.\n\\end{align}\nNo additional logarithms are generated.  Since this is the case that applies to the soft function $S_{p_T^2}^{(2)}=p_T^2 S^{(0)}$, one can easily check using the known form of the two-loop soft function that there is indeed only a single logarithm at any $\\nu$ scale for $\\mu=p_T$. \n\nIn an actual factorization formula, this single rapidity logarithm is then dressed by a Sudakov coming from running the hard function to the scale $\\mu=p_T$, giving rise to a result of the form\n\\begin{align}\n\\gamma_{\\delta{\\mathbb{I}} }  \\log(p_T\/Q)   \\exp \\left[ -2 a_s \\log^2\\left( \\frac{ p_T^2}{Q^2} \\right)  \\right]  H(Q) {\\mathbb{I}}^{p_T^2}_S(\\mu=p_T, \\nu=p_T)\\,.\n\\end{align}\n This provides quite an interesting structure, namely a single logarithm arising from $\\nu$ evolution, which is then dressed by double logarithms from $\\mu$ evolution. The $\\nu$ and $\\mu$ RGEs therefore completely factorize at leading logarithmic order. An identical structure was observed for the case of thrust in \\cite{Moult:2018jjd}, however, there both the single logarithm and the tower of double logarithms arise from the $\\mu$ RGE. \n\n\\subsection*{Identity Operator ${\\mathbb{I}}^\\nu_S \\left(\\mu, \\nu \\right)$:}\n\nMixing with the rapidity identity operator ${\\mathbb{I}}^\\nu_S \\left(\\mu, \\nu \\right)$ gives rise to a more non-trivial rapidity flow at the scale $\\mu=p_T$.\nIn this case, we get the RGE\n\\begin{align}\n\\nu \\frac{d}{d\\nu}\\left(\\begin{array}{c} S^{(2)} \\\\  {\\mathbb{I}}^\\nu_S \\end{array} \\right) &=  \\left( \\begin{array}{cc}0&\\gamma_{\\delta{\\mathbb{I}} }\\,  \\\\  0 & \\gamma^\\mu_S   \\end{array} \\right) \\left(\\begin{array}{c} S^{(2)} \\\\  {\\mathbb{I}}^\\nu_S \\end{array} \\right) \\,, \n\\end{align}\nwith the boundary conditions\n\\begin{align}\n{\\mathbb{I}}^\\nu_S(\\mu=p_T, \\nu=p_T)=1\\,, \\qquad S^{(2)}(\\mu=p_T,\\nu=p_T)=0\\,.\n\\end{align}\nThis RGE is is a specific case of the general RGE solved in \\cite{Moult:2018jjd}. However, here we can solve it in two steps by first solving for the identity operator, and then plugging it in to the solution for the soft function. This will provide some insight into the structure of the final solution.\n\nThe solution of\n\\begin{align}\n\\nu \\frac{d}{d\\nu}  {\\mathbb{I}}^\\nu_S =\\gamma^\\mu_S  {\\mathbb{I}}^\\nu_S \\equiv \\tilde  \\gamma \\log\\left(  \\frac{\\nu^2}{p_T^2} \\right)  {\\mathbb{I}}^\\nu_S\\,,\n\\end{align}\nis easily found to be\n\\begin{align}\n {\\mathbb{I}}^\\nu_S=\\exp\\left( \\frac{\\tilde \\gamma}{4} \\log^2\\left(  \\frac{\\nu^2}{p_T^2} \\right) \\right){\\mathbb{I}}^\\nu_S(\\mu=p_T, \\nu=p_T)\\,.\n\\end{align}\nThe original soft function then satisfies the inhomogeneous equation\n\\begin{align}\n\\nu \\frac{d}{d\\nu} S^{(2)}=\\gamma_{\\delta{\\mathbb{I}} }  \\exp\\left( \\frac{\\tilde \\gamma}{4} \\log^2\\left(  \\frac{\\nu^2}{p_T^2} \\right) \\right){\\mathbb{I}}_S^\\nu(\\mu=p_T, \\nu=p_T)\\,.\n\\end{align}\nWe can integrate this equation up to the scale $\\nu=Q$ to find\n\\begin{align}\nS^{(2)}(\\mu=p_T, \\nu=Q)=\n - \\frac{\\sqrt{\\pi} \\gamma_{\\delta{\\mathbb{I}} }}{ \\sqrt{\\tilde \\gamma}}  \\mathrm{erfi}\\left[ \\sqrt{\\tilde \\gamma}  \\log(p_T\/Q)  \\right] {\\mathbb{I}}^\\nu_S(\\mu=p_T, \\nu=p_T) \\,, \n\\end{align}\nwhere $\\mathrm{erfi}$ is the imaginary error function, defined as \n\\begin{align}\n\\mathrm{erfi}(z)=-i \\mathrm{erf}(iz)\\,.\n\\end{align}\nThe fact that the solution is not simply a Sudakov is quite interesting, and shows that the subleading power logarithms have a more interesting structure than at leading power. We expect that this structure will be quite common in the study of subleading power corrections to observables with rapidity evolution.\nThe $\\mathrm{erfi}$ function satisfies the identify\n\\begin{align}\n\\frac{d}{dz}\\mathrm{erfi}(z) = \\frac{2}{\\sqrt{\\pi}} e^{z^2}\\,.\n\\end{align}\nIt is perhaps quite intuitive that an integral of a Sudakov appears, since the rapidity identity operator is the integral of the leading power soft function. However, this is a new structure that has not previously appeared in subleading power calculations. It is amusing to note that a similar structure also appears in the calculation of Sudakov safe observables \\cite{Larkoski:2015lea} due to the integration over a resummed result.\n\n\\section{The Energy-Energy Correlator in $\\mathcal{N}=4$ SYM}\\label{sec:N4}\n\nIn this section we use our subleading power rapidity renormalization group to derive the leading logarithmic series at NLP for a physical observable, namely the EEC in $\\mathcal{N}=4$ SYM. We then compare our predictions with the recent calculation of \\cite{Henn:2019gkr} to $\\mathcal{O}(\\alpha_s^3)$ finding perfect agreement.\n\n\\subsection{Leading Logarithmic Resummation at Subleading Power}\\label{sec:N4_LL}\n\nIn performing the resummation of subleading power logarithms for the EEC, we must clearly state several assumptions that are made, which we hope can be better understood in future work. Nevertheless, we believe that the fact that our result agrees with the calculation of  \\cite{Henn:2019gkr} provides strong support for these assumptions. The goal of this paper has been to understand how far one can get in understanding the subleading power rapidity renormalization group using only symmetries and consistency arguments.  Using this approach, we found that at leading logarithmic order, there are two distinct identity operators with different renormalization group properties. To derive the structure of the resummed result for the EEC in the back-to-back limit, our approach will therefore be to match a linear combination of these two solutions to the know expansion of the EEC. To fix both coefficients requires two inputs, which we take to be the $\\alpha_s$ and $\\alpha_s^2$ leading logarithms. This then completely fixes our result, which can then be used to predict the coefficient of the $\\alpha_s^3$ leading logarithm, for which we will find agreement with the calculation of \\cite{Henn:2019gkr}. This approach should simple be viewed as a shortcut to a complete operator based analysis, which enables us to explore the general structure of the rapidity evolution equations at NLP, and show that they predict non-trivial behavior of the NLP series for a physical observable.  A more complete operator based analysis will be presented in a future paper.\n\nSecondly, we must also assume that there exists a consistent factorization at subleading powers that does not have endpoint divergences. The presence of endpoint divergences in the factorization formula would violate our derivations based on the consistency of the RG. At this stage, both for the standard $\\mu$ renormalization group at subleading power, and for the subleading power rapidity renormalization group considered here, this is still ultimately an assumption. In general, endpoint divergences are known to appear generically at next-to-leading logarithm, but have also been shown to appear even at LL in certain cases when fields with different color representations are involved (e.g. both quarks and gluons) \\cite{Moult:2019uhz}. However, since in $\\mathcal{N}=4$ all fields are in the same representation, we work under the assumption that there are no endpoint divergences at leading logarithmic accuracy. We will see that this assumption is strongly supported by the fact that we are able to exactly reproduce to $\\mathcal{O}(\\alpha_s^3)$ the highly non-trivial series that arises from the exact calculation of \\cite{Henn:2019gkr}. However, we acknowledge that before our techniques can be more widely applied, it will be important to understand when endpoint divergences do occur in the rapidity renormalization group, and how they can be resolved. \n\nTherefore, working under the assumption of convergent convolutions for the subleading power factorization, all anomalous dimensions are fixed by symmetries, as described above, and the resummation of the subleading power logarithms is now a simple application of the renormalization group evolution equations derived in \\Sec{sec:RRG_NLP}. To perform the resummation, one must resum logarithms in both $\\mu$ and $\\nu$. We choose to perform this resummation using the following evolution path\n\\begin{itemize}\n\\item Run the hard functions from $\\mu=Q$ to $\\mu=p_T$.\n\\item Run the soft functions in rapidity from $\\nu=p_T$ to $\\nu=Q$.\n\\end{itemize}\nThis path is the most convenient, since it avoids the need to perform any resummation of the rapidity anomalous dimensions \\cite{Chiu:2012ir,Chiu:2011qc}. To run the hard function from $\\mu=Q$ down to the soft scale, we use the evolution equations\n\\begin{align}\n\\mu \\frac{d}{d\\mu} H = \\gamma_H H\\,, \\qquad \\gamma_H =-8 a_s \\log \\left( \\frac{ \\mu^2}{Q^2} \\right)\\,,\n\\end{align}\nHere and throughout this section, we will use $a_s=\\alpha_s\/(4\\pi)C_A$ to simplify the notation.\nThe renormalization group equation for the hard function has the simple solution\n\\begin{align}\nH(p_T)=H(Q) \\exp \\left[ -2 a_s \\log^2\\left( \\frac{ p_T^2}{Q^2} \\right)  \\right]\\,.\n\\end{align}\nFor the soft function evolution, we use the results derived in \\Sec{sec:RRG_NLP} for the evolution of the two different types of rapidity identity operators. Since these rapidity identity functions cannot themselves mix due to different boost properties, the result at LL order is necessarily a linear combination of the two. For the first type of mixing, we have\n\\begin{align}\nS_1^{(2)}(\\mu=p_T, \\nu=Q)=\n \\gamma_{\\delta{\\mathbb{I}},1 }  \\log(p_T\/Q)   {\\mathbb{I}}^{p_T^2}_S(\\mu=p_T, \\nu=p_T) \\,,\n\\end{align}\nwhere $ \\gamma_{\\delta{\\mathbb{I}},1 }$ is the anomalous dimension for mixing into the  $ {\\mathbb{I}}^{p_T^2}_S$ soft function, and for the second type, we have\n\\begin{align}\nS_2^{(2)}(\\mu=p_T, \\nu=Q)=\n - \\frac{\\sqrt{\\pi} \\gamma_{\\delta{\\mathbb{I}},2 }}{ \\sqrt{\\tilde \\gamma}}  \\mathrm{erfi}\\left[ \\sqrt{\\tilde \\gamma}  \\log(p_T\/Q)  \\right]  {\\mathbb{I}}_S(\\mu=p_T, \\nu=p_T)\\,,\n\\end{align}\nwith $\\tilde \\gamma =8 a_s$, and where $ \\gamma_{\\delta{\\mathbb{I}},2 }$ is the anomalous dimension for mixing into the  $ {\\mathbb{I}}^{\\nu}_S$ soft function.\nOur general prediction is then a linear combination of these two \n\\begin{align}\n\\text{EEC}^{(2)}=\\gamma_{\\delta{\\mathbb{I}},1 }  \\log(p_T\/Q)   \\exp \\left[ -2 a_s \\log^2\\left( \\frac{ p_T^2}{Q^2} \\right)  \\right]   - \\frac{\\sqrt{\\pi} \\gamma_{\\delta{\\mathbb{I}},2 }}{ \\sqrt{\\tilde \\gamma}}  \\mathrm{erfi}\\left[ \\sqrt{\\tilde \\gamma}  \\log(p_T\/Q)  \\right]\\,.\n\\end{align}\nMatching to the $a_s$ and $a_s^2$ coefficients from expanding the result of \\cite{Belitsky:2013ofa,Belitsky:2013xxa,Belitsky:2013bja}(these coefficients are given in \\Sec{sec:N4_expand}), we find that $\\gamma_{\\delta{\\mathbb{I}},1 }=0$. We therefore find the simple result for the NLP leading logarithmic series to all orders in $a_s$ in $\\mathcal{N}=4$ SYM theory\n\\begin{align}\n\\text{EEC}^{(2)}=-\\frac{\\sqrt{\\pi}a_s}{ \\sqrt{2a_s}}  \\mathrm{erfi}\\left[ \\sqrt{2 a_s} \\log(1-z)  \\right]  \\exp\\left[ -2 a_s \\log(1-z)^2  \\right]\\,.\n\\end{align}\nInterestingly, for the particular case of $\\mathcal{N}=4$, we find that the result only involves the operator ${\\mathbb{I}}^\\nu_S \\left(\\mu, \\nu \\right)$.\nThis result takes an interesting form, going beyond the simple Sudakov exponential \\cite{Sudakov:1954sw} for the leading logarithms at leading power. We believe that this structure will appear somewhat generically in subleading power rapidity resummation.  It is interesting to note that up to the prefactor this particular structure is in fact a well known special function, called Dawson's integral, which is defined as\n\\begin{align}\nD(x)=\\frac{1}{2}\\sqrt{\\pi}e^{-x^2}\\mathrm{erfi}(x)\\,.\n\\end{align}\nWe can therefore write our answer for the NLP leading logarithmic series in the simple form\n\\begin{align}\n\\text{EEC}^{(2)}=-\\sqrt{2a_s}~D\\left[ \\sqrt{2 a_s} \\log(1-z) \\right]\\,.\n\\end{align}\nSince the anomalous dimension is fixed by renormalization group consistency to be the cusp anomalous dimension (see \\Eq{eq:rap_identity_summary}), we can rewrite this as\n\\begin{align}\n\\boxed{\\text{EEC}^{(2)}=-\\sqrt{2a_s}~D\\left[ \\sqrt{\\frac{\\Gamma^{\\mathrm{cusp}}}{2}} \\log(1-z) \\right]\\,.}\n\\end{align}\nThis expression is a primary result of this paper. We will refer to this functional form as ``Dawson's Sudakov\". We find it pleasing that despite the somewhat non-trivial functional structure, the double logarithmic asymptotics of the EEC in the back-to-back limit are still driven by the cusp anomalous dimension \\cite{Korchemsky:1987wg} much like at leading power (see \\Eq{eq:resformula_N4}), and as is expected physically. It would be extremely interesting to understand if the subleading power logarithms at next-to-leading power are driven by the collinear anomalous dimension, as is the case at leading power. We note that from \\Eq{eq:N4_NLP_fixedorder} there is no constant term at NLP (it can easily be checked that this is true at any power), and therefore it is plausible that there is a simple generalization of this formula that also incorporates the next-to-leading logarithms.\n\nIn \\Fig{fig:plot}, we compare the standard Sudakov with Dawson's Sudakov. Note that while $\\mathrm{erfi}\\left[ \\sqrt{2 a_s} \\log(1-z)  \\right]$ diverges as $z\\to 1$, this is overcome by the suppression from the Sudakov exponential. However, the behavior of the $\\mathrm{erfi}$ leads to a more enhanced behavior of the distribution as $z\\to 1$, as compared to a standard Sudakov.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{figures\/sudakovs}\n\\end{center}\n\\caption{A comparison of the standard Sudakov exponential which describes the all orders exponentiation of the leading power logarithms for the EEC with the subleading power Sudakov involving the imaginary error function ($\\mathrm{erfi}$) derived in this section, which we refer to as Dawson's Sudakov.  Dawson's Sudakov exhibits a more peaked structure as $z\\to 1$. A value of $\\alpha_s=0.12$ was chosen to plot the numerical results.}\n\\label{fig:plot}\n\\end{figure}\n\nIt is also interesting to consider the Taylor expansion of our result in $a_s$. We find that it can be written as a remarkably simple series in terms of the double factorial\n\\begin{align}\n\\text{EEC}^{(2)}\n&=\\sum\\limits_{n=0}^{\\infty} \\frac{(-1)^{n+1}}{(2n+1)!!} a_s^{n+1} \\log((1-z)^2)^{2n+1}\\,,\n\\end{align}\n(here we have chosen to move factors of $2$ into the definition of the logarithm, but they can equally well be moved into the definition of $a_s$)\nwhere we recall that the double factorial is defined as\n\\begin{align}\nn!!=n(n-2)(n-4)\\cdots .\n\\end{align}\nExplicitly, the first few terms in the expansion are\n\\begin{align}\\label{eq:EEC_RG_expand}\n\\text{EEC}^{(2)}&=-2 a_s \\log(1-z) +\\frac{8}{3}a_s^2 \\log^3(1-z)-\\frac{32}{15}a_s^3 \\log^5(1-z) +\\frac{128}{105}a_s^4 \\log^7(1-z) {\\nonumber} \\\\\n&-\\frac{512}{945}a_s^5 \\log^9(1-z)+\\frac{2048}{10395}a_s^6 \\log^{11}(1-z)-\\frac{8192}{135135}a_s^7 \\log^{13}(1-z)+\\cdots\\,.\n\\end{align}\nThe presence of the double factorial as compared with the single factorial at leading power generates a more interesting series of rational coefficients.  In \\cite{Dixon:2019uzg} the logarithms in the collinear limit of the EEC at each logarithmic order were written as simple infinite sums of factorials multiplied by polynomials in logarithms. It would be very interesting to understand if the subleading logarithms at NLP in the back-to-back limit can also be written as generalized double factorial sums.\n\nIt will be important to understand if this structure persists in QCD. The results in QCD are only known to $a_s^2$ \\cite{Dixon:2018qgp,Luo:2019nig}, therefore, without performing a more complete operator analysis, these results can be used to fix our prediction as a linear combination of our two RG solutions, but do not enable a non-trivial check, which is particularly important due to the possible presence of endpoint divergences. While we will perform an operator based analysis in a future publication, we find that it interesting to conjecture a result. Based on our intuition from the case of the thrust observable \\cite{Moult:2019uhz}, we expect that we are most likely to avoid endpoint divergences for the case of the EEC in Higgs decays to gluons in pure Yang-Mills. Under the assumption that there are no endpoint divergences, we can fix an ansatz in terms of our two RG solutions to be\n\\begin{align}\n\\left.\\text{EEC}^{(2)}\\right|_{\\text{Yang-Mills}}=2 a_s \\log(1-z)   \\exp \\left[ -2 a_s \\log^2\\left(1-z \\right)  \\right]   -4\\sqrt{2a_s}~D\\left[ \\sqrt{\\frac{\\Gamma^{\\mathrm{cusp}}}{2}} \\log(1-z) \\right]\\,,\n\\end{align}\nwhich takes a slightly more complicated form than its $\\mathcal{N}=4$ counterpart, since it involves both types of rapidity identity operators. Unfortunately, unlike for the case of the EEC in $\\mathcal{N}=4$, there is no $a_s^3$ result to which we are able to compare this result. \nIt will interesting to verify or disprove this conjecture using a complete operator analysis, which we leave to future work. This will provide insight into the presence (or lack of) endpoint divergences in this particular case.\n\nIt would also be extremely interesting to understand how to derive this result directly from the four point correlator, or from the light ray OPE \\cite{Kravchuk:2018htv,Kologlu:2019bco,Kologlu:2019mfz}. From the point of view of the four point correlator, the back-to-back limit corresponds to the so called double light cone limit. This has been used to study the EEC in \\cite{Korchemsky:2019nzm}, has been studied in gauge theories in \\cite{Alday:2010zy,Alday:2013cwa}, and has been studied in more general conformal field theories in \\cite{Alday:2015ota}. \n\n\\subsection{Comparison with Direct Fixed Order Calculation to $\\mathcal{O}(\\alpha_s^3)$}\\label{sec:N4_expand}\n\nAs mentioned earlier, we have chosen to perform the resummation for the EEC in $\\mathcal{N}=4$ since we can directly compare with the remarkable calculation of the EEC for arbitrary angles using the triple discontinuity of the four point correlator \\cite{Henn:2019gkr} (The result to $\\mathcal{O}(\\alpha_s^2)$ in $\\mathcal{N}=4$ was calculated in \\cite{Belitsky:2013ofa,Belitsky:2013xxa,Belitsky:2013bja}), and exploiting the large amount of information known about its structure, see e.g. \\cite{Eden:2011we,Eden:2012tu,Drummond:2013nda,Korchemsky:2015ssa,Bourjaily:2016evz}. This calculation of the EEC provides extremely valuable data for understanding the structure of kinematic limits at subleading power, both for the particular case considered here, and beyond.\n\nAlthough we will be interested in the expansion in the $z\\to 1$ limit, it is interesting to understand what parts of the full angle result the back-to-back limit is sensitive to at subleading powers, so we briefly review the structure of the result of \\cite{Henn:2019gkr}.\nThe result of \\cite{Henn:2019gkr} is written as\n \\begin{align}\\label{defF}\n F(\\zeta) \\equiv 4 \\zeta^2 (1-\\zeta) {\\text{EEC}} (\\zeta) \\,,\n \\end{align}\nwhere at NNLO, \n \\begin{align}  \\label{resultFatNNLO}\nF_{\\rm NNLO} (\\zeta) &= f_{\\rm HPL} (\\zeta) +   \\int_{0}^1 d \\bar z  \\int_{0}^{\\bar z } dt \\, \\frac{\\zeta-1}{t(\\zeta-\\bar z)+(1-\\zeta)\\bar z} \\nt \n& \\times \\left[  R_1 ( z, \\bar z )  P_1 (z, \\bar z ) + \n  R_2 ( z, \\bar z )  P_2 (z, \\bar z ) \\right] \\,,\n\\end{align} \nwith\n \\begin{align} \n  R_1 = \\frac{z \\bar z }{ 1 - z - \\bar z}\\, , \\quad  R_2 = \\frac{z^2 \\bar z }{ (1-z)^2  ( 1 - z \\bar z)}\\,.\n \\end{align}\n Here $P_1$ and $P_2$ are weight three HPLS in $z$ and $\\bar z$. These two fold integrals are believed to be elliptic, and so we will refer to them as elliptic contributions. The term $f_{\\rm HPL} (\\zeta)$ is expressed in terms of harmonic polylogarithms. The leading power asymptotics in the back-to-back limit are described entirely by $ f_{\\rm HPL}$. At NLP, we require also the elliptic contributions, showing that subleading powers probe more of the structure of the result.  This in turn makes the agreement with our result derived from the RG more non-trivial.\n \n\nThe expansion of $f_{\\rm HPL} (\\zeta)$ can be performed straightforwardly, and produces only $\\zeta$ values, $\\log^n(2)$, and $\\text{Li}_n(1\/2)$. On the other hand, the expansion of the elliptic contribution is more non-trivial, and leads to a more complicated set of constants. To compute the result, we expanded under the integral sign, and integrated using HyperInt \\cite{Panzer:2014caa}. This produced polylogarithms of sixth roots of unity up to weight 5. These were reduced using results from  \\cite{Henn:2015sem} to a basis of constants. The final result involves several non-zeta valued constants which were guessed using hints for the classes of numbers that should appear in the answer from \\cite{Broadhurst:1998rz,Fleischer:1999mp,Davydychev:2000na} and reconstructed using the PSLQ algorithm. We found that the elliptic piece could be expressed as \n\\begin{align}\n\\frac{\\text{Elliptic}_{\\text{NLP}}}{2}&=2\\frac{L^5}{5!}+\\frac{1}{2}\\zeta_2 \\frac{L^3}{3!}  +\\frac{3}{4}\\zeta_3 \\frac{L^2}{2!}+ \\left(-\\frac{67}{32}+\\frac{3}{4}\\zeta_2+\\frac{9}{4}\\zeta_3 -\\frac{7}{4} \\zeta_4\\right) L{\\nonumber} \\\\\n&+\\frac{85}{32}-\\frac{49}{16}\\zeta_2+\\frac{87}{8}\\zeta_3 +\\frac{37}{4}\\zeta_4 +\\frac{3}{4}\\zeta_2\\zeta_3+\\frac{5}{2}\\zeta_5-\\frac{611}{108}\\zeta_4 \\sqrt{3} \\pi+6\\sqrt{3}I_{2,3}\\,.\n\\end{align}\nHere $I_{2,3}$ is a higher weight Clausen function \\cite{Broadhurst:1998rz} \n\\begin{align}\nI_{2,3}=\\sum\\limits_{m>n>0}\\frac{\\sin\\left(\\frac{\\pi(m-n)}{3}\\right)}{m^{b-a}n^{2a}}\\,.\n\\end{align}\nWhile the leading power result has uniform trascendental weight when expanded in the back-to-back region, this is no longer true at subleading power.  \n\n\nCollecting all the terms from both the elliptic and HPL contributions, we find (in our normalization) the following expression for the leading power suppressed logarithms up to $\\mathcal{O}(a_s)^3$\n\\begin{align}\n &\\text{EEC}^{(2)}=-2a_s \\log(1-z) {\\nonumber} \\\\\n &+a_s^2 \\left[\\frac{8}{3} \\log^3(1-z) +3 \\log^2(1-z) +(4+16\\zeta_2)\\log(1-z)+(-12-2\\zeta_2+36 \\zeta_2\\log(2)+5\\zeta_3) \\right]{\\nonumber} \\\\\n &+a_s^3 \\left[-\\frac{32}{15} \\log^5(1-z)-\\frac{16}{3}\\log^4(1-z)-\\left(  \\frac{8}{3}+24 \\zeta_2 \\right)\\log^3(1-z) +(4-36\\zeta_2-50\\zeta_3)\\log^2(1-z) \\right. {\\nonumber} \\\\\n &\\left.  -\\left(\\frac{131}{2}+4\\zeta_2+372 \\zeta_4+12\\zeta_3   \\right) \\log(1-z) + \\text{const}  \\right]\\,,\n\\end{align}\nwhere\n\\begin{align}\n\\text{const}&=-\\frac{3061}{2}\\zeta_5  -96\\zeta_2 \\zeta_3 -\\frac{4888}{27}\\zeta_4 \\pi \\sqrt{3} +192 \\sqrt{3} I_{2,3}+1482 \\zeta_4 \\log(2) -256 \\zeta_2 \\log^3(2) {\\nonumber} \\\\\n&-\\frac{64}{5}\\log^5(2)\n+1536 \\text{Li}_5\\left( \\frac{1}{2} \\right) -544 \\zeta_4 +192 \\zeta_2 \\log^2(2)+16 \\log^4(2)+384 \\text{Li}_4\\left( \\frac{1}{2} \\right) {\\nonumber} \\\\\n& -288 \\zeta_2 \\log(2)+158 \\zeta_3 +55 \\zeta_2 +\\frac{533}{2}\\,.\n\\end{align}\nExtracting out the leading logarithmic series\n\\begin{align}\n\\left. \\text{EEC}^{(2)}\\right|_{\\text{LL}}=-2a_s \\log(1-z) +\\frac{8}{3}a_s^2 \\log^3(1-z) -\\frac{32}{15}a_s^3 \\log^5(1-z)\\,,\n\\end{align}\nwe find that this result agrees exactly with the result derived from the subleading power renormalization group given in \\Eq{eq:EEC_RG_expand}! Note that due to the matching of our RG predictions to the fixed order results, as was explained above, it is really only the $a_s^3$ coefficient that is a prediction. However, this agreement is highly non-trivial, since it probes both the elliptic and polylogarithmic sectors of the full result, and therefore we believe that it provides strong support that our subleading power renormalization group evolution equation is correct. The next-to-leading logarithms also have relatively simple rational coefficients, \n\\begin{align}\n\\left. \\text{EEC}^{(2)}\\right|_{\\text{NLL}}=3a_s^2 \\log^2(1-z) -\\frac{16}{3}a_s^3 \\log^4(1-z)\\,,\n\\end{align}\nand provide data for understanding the structure of subleading power resummation beyond the leading logarithm. It would be interesting to derive them directly from the renormalization group approach.\n\nIt would be interesting to better understand the structure of the numbers appearing in the expansion of the EEC both in the collinear and back-to-back limits, and the functions appearing in the full angle result. This could ultimately allow the result to be bootstrapped from an understanding of these limits, in a similar manner to the hexagon bootstrap for $N=4$ SYM amplitudes \\cite{Dixon:2011pw,Dixon:2014iba,Caron-Huot:2016owq,Dixon:2016nkn,Caron-Huot:2019vjl}. However, the presence of elliptic functions makes this seem like a daunting task, unless more information is known about their structure.\n\n\n\\section{Conclusions}\\label{sec:conc}\n\nIn this paper we have shown how to resum subleading power rapidity logarithms using the rapidity renormalization group, and have taken a first step towards a systematic understanding of subleading power corrections to observables exhibiting hierarchies in rapidity scales. Much like for the virtuality renormalization group at subleading power, the rapidity renormalization group at subleading power involves a non-trivial mixing structure. Using the consistency of the RG equations combined with symmetry arguments, we were able to identify the operators that arise in the mixing,  which we termed ``rapidity identity operators\", and we derived their anomalous dimensions. We believe that these operators will play an important role in any future studies of the rapidity renormalization group at subleading power, and are the key to understanding its structure. \n\nTo illustrate our formalism, we resummed the subleading power logarithms appearing in the back-to-back limit of the EEC in $\\mathcal{N}=4$ SYM. This particular observable was chosen since the full analytic result is known to $\\mathcal{O}(\\alpha_s^3)$ from the remarkable calculation of \\cite{Henn:2019gkr}. We found perfect agreement between our result derived using the renormalization group, and the expansion in the back-to-back limit of the calculation of \\cite{Henn:2019gkr}, which provides an extremely strong check on our results. The analytic form of the resummed subleading power logarithms takes an interesting, but extremely simple form, being expressed in terms of Dawson's integral with argument related to the cusp anomalous dimension. We called this structure ``Dawson's Sudakov\". We expect this structure to be generic at subleading power for rapidity dependent observables, much like the Sudakov exponential is at leading power. \n\nSince this represents the first resummation of subleading power rapidity logarithms, there are many directions to extend our results, as well as to better understand the structures that we have introduced in this paper. First, although we have arrived at the structure of the leading logarithms using symmetry and consistency arguments, it would be interesting to use a complete basis of SCET$_{\\rm II}$ ~operators to derive the operator structure of all the subleading power jet and soft functions in SCET$_{\\rm II}$, and perturbatively compute their anomalous dimensions. Second, to go beyond LL, it will be necessary to better understand the structure of the momentum convolutions appearing in the subleading power factorization. We expect that at subleading power this is best done in momentum space using the formalism of \\cite{Ebert:2016gcn}. Finally, we also expect that away from $\\mathcal{N}=4$ SYM theory, divergent convolutions will appear even at LL order, as occurs in SCET$_{\\rm I}$, and so it will be important to understand when these occur, and how they can be overcome.\n\nOn the more formal side, it will be interesting to understand how to extract the subleading power logarithms directly from the four point correlation function, following the approach of  \\cite{Korchemsky:2019nzm}, or using the light ray OPE \\cite{Kravchuk:2018htv,Kologlu:2019bco,Kologlu:2019mfz}. While there have been some studies of the double light cone limit in conformal field theories \\cite{Alday:2010zy,Alday:2015ota}, further studies of this limit in conformal field theories could provide insight into behavior of phenomenological interest in QCD, and the EEC provides an example that is of both formal and phenomenological interest.\n\nIt will also be important to apply our formalism to observables of direct phenomenological interest, such as the EEC in QCD, the color singlet $p_T$ distribution in hadron colliders, or to the study of power suppressed logarithms appearing in the Regge limit, which can also be formulated in terms of the rapidity renormalization group \\cite{Rothstein:2016bsq,Moult:2017xpp}.  Our work represents the first step in extending recent successes in understanding the structure of subleading power infrared logarithms to subleading power rapidity logarithms, and we hope that this will allow for a much wider set of phenomenologically important applications.\n\n\\begin{acknowledgments}\nWe thank Martin Beneke, Sebastian Jaskiewicz, Robert Szafron, Iain Stewart, Frank Tackmann, Johannes Michel, Markus Ebert, David Simmons-Duffin, Cyuan-Han Chang, Lance Dixon, Johannes Henn, and HuaXing Zhu for useful discussions. We thank Vladimir Smirnov for providing us with reduction tables for polylogarithms of roots of unity. This work was supported in part by the Office of Nuclear Physics of the U.S.\nDepartment of Energy under Contract No. DE-SC0011090, and by the Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-76SF00515. This research received\nfunding from the European Research Council (ERC) under the European\nUnion's Horizon 2020 research and innovation programme\n(grant agreement No. 725110), ``Novel structures in scattering amplitudes\".\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\n\nIn \\cite{C}, Cohn introduced  the  notion of Schreier domain.  A domain $D$ is said to be a {\\em Schreier domain} if $(1)$ $D$ is integrally closed and $(2)$  whenever $I, J_1, J_2$ are principal ideals of $D$ and $I\\supseteq J_1J_2$, then  $I = I_1I_2$ for some principal ideals $I_1,I_2$ of $D$ with $I_i \\supseteq J_i$ for $i = 1,2$.  The study of Schreier domains was continued in \\cite{MR} and \\cite{Zps}. In  \\cite{Zps}, a domain was called  a {\\em pre-Schreier domain} if it satisfies  condition $(2)$ above.\nSubsequently, extensions of the ``(pre)-Schreier domain'' concept were studied in \\cite{DM}, \\cite{ADZ}, \\cite{DK}, \\cite{AD} and \\cite{ADE}.\n\nIn \\cite{ADE}, we studied a class of domains that satisfies a Schreier-like condition for all ideals. More precisely, a domain $D$ was called  a {\\em sharp domain } if whenever $I \\supseteq AB$ with $I$, $A$, $B$  nonzero ideals of $D$, there exist  ideals $A'\\supseteq A$ and $B'\\supseteq B$ such that $I=A'B'$. We recall  several results from \\cite{ADE}.\nIf the domain $D$ is  Noetherian or Krull, then $D$ is sharp if and only if $D$ is a  Dedekind domain \\cite[Corollaries 2 and 12]{ADE}. A sharp domain is  pseudo-Dedekind; in particular, a sharp domain is a completely integrally closed GGCD domain \\cite[Proposition 4]{ADE}.\nRecall (cf. \\cite{Z} and \\cite{AK}) that a domain $D$ is called a {\\em pseudo-Dedekind domain} (the name used in \\cite{Z}  was {\\em generalized Dedekind domain}) if the $v$-closure of each nonzero ideal of $D$ is invertible. Also, recall from \\cite{AA} that a  domain $D$ is called a {\\em generalized GCD domain (GGCD domain)} if  the $v$-closure of each nonzero finitely generated ideal of $D$ is invertible. The definition of the $v$-closure is recalled below.\nA valuation domain is sharp if and only if the value group  of $D$ is  a complete subgroup of the reals \\cite[Proposition 6]{ADE}.\nThe localizations of a  sharp domain at the maximal ideals are valuation domains with value group a complete subgroup of the reals; in par\\-ti\\-cu\\-lar, a sharp domain is a Pr\\\"ufer domain of dimension at most one  \\cite[Theorem 11]{ADE}.\nThe converse is true for the domains of finite character  \\cite[Theorem 15]{ADE}, but not true in general \\cite[Example 13]{ADE} (recall that a {\\em domain of finite character} is a domain whose nonzero elements are contained in only finitely many maximal ideals).  A countable sharp domain is a Dedekind domain \\cite[Corollary 17]{ADE}.\n\n\n\n\n\n\n\n\n\n\n\n\nThe purpose of this paper is to study the ``sharp domain''  concept in the  star operation setting. To facilitate the reading of the  paper, we first review some basic facts about $*$-operations. Let $D$ be a domain with quotient field $K$ and let $F(D)$ denote the set of nonzero fractional ideals of $D$.\nA function       $A\\mapsto A^*: F(D) \\rightarrow F(D)$ is called a {\\em star operation} on\n$D$ if $*$ satisfies the following three conditions for all $0\n\\neq a \\in K$ and all $I,J \\in F(D)$:\n\n$(1)$ $D^{*} = D$ and\n$(aI)^{*}=aI^{*}$,\n\n$(2)$ $I \\subseteq I^{*}$ and if $ I\n\\subseteq J$, then $I^{*} \\subseteq J^{*}$,\n\n$(3)$\n$(I^{*})^{*} = I^{*}.$\n\n\\noindent An ideal $I \\in F(D)$ is called a $*$-ideal if $I = I^{*}.$\nFor all $I,J\\in F(D)$, we have\n$(IJ)^{*}=(I^{*}J)^{*}=(I^{*}J^{*})^{*}$.\nThese equations define the so-called {\\em $*$-multiplication.}\nIf $\\{I_\\alpha\\}$ is a subset of $F(D)$ such that $\\cap I_\\alpha\\neq 0$, then\n$\\cap I_\\alpha^*$ is a $*$-ideal.\nAlso, if $\\{I_\\alpha\\}$ is a subset of $F(D)$ such that $\\sum I_\\alpha$ is a fractional ideal, then\n$(\\sum I_\\alpha)^*=(\\sum I_\\alpha^*)^*$.\nThe star operation $*$ is said to be  {\\em stable}  if  $(I \\cap J)^{*} = I^{*}\\cap J^{*}$ for all $I,J\\in F(D)$.\nIf $*$ is a star operation, the function $*_f: F(D) \\rightarrow F(D)$ given by $I^{*_f} = \\cup_H H^{*}$, where $H$ ranges over all\nnonzero finitely generated subideals of $I$, is also a star operation. The star operation $*$ is said to be of\n{\\em finite character} if $*=*_f$. Clearly $(*_f)_f=*_f$.\nDenote by $Max_*(D)$  the set of maximal $*$-ideals,  that is, ideals maximal among proper integral\n$*$-ideals of $D$. Every maximal $*$-ideal is a prime ideal.\nThe {\\em $*$-dimension} of $D$ is\n$sup \\{ n\\mid 0\\subset P_1\\subset \\cdots \\subset P_n,$  $P_i$  prime $*$-ideal of $D\\}$.\nAssume that $*$ is a star operation of finite character. Then every proper $*$-ideal is contained in some maximal $*$-ideal, and\nthe map $I\\mapsto I^{\\tilde{*}} = \\cap _{P \\in Max_*(D)}ID_P$ for all $I\\in F(D)$ is a stable star operation of finite character, cf. \\cite[Theorems 2.4, 2.5 and 2.9]{AC}.\nMoreover, $*$ is stable if and only if $*=\\tilde{*}$, cf. \\cite[Corollary 4.2]{A}.\nA $*$-ideal $I$ is of {\\em finite type} if\n$I=(a_1,...,a_n)^{*}$ for some $a_1,...,a_n\\in I.$\nA {\\em Mori domain} is a domain whose $t$-ideals are of finite type (see \\cite{B}). By \\cite{HZ}, an integral domain is said to be a {\\em TV domain} if every $t$-ideal is a $v$-ideal. A Mori domain is a TV domain.\n\nA fractional ideal $I \\in F(D)$ is said to be {\\em $*$-invertible} if\n$(II^{-1})^{*} = D$, where $I^{-1}=(D:I)=\\{ x \\in K \\mid xI \\subseteq D\\}$.\nIf $*$ is of finite character, then $I$ is $*$-invertible if and only if\n$II^{-1}$ is not contained in any maximal $*$-ideal of $D$; in this case $I^*=(a_1,...,a_n)^*$ for some $a_1,...,a_n\\in I$.\nLet $*_1,*_2$ be star operations on $D$. We write $*_1\\leq *_2$, if $I^{*_1}\\subseteq I^{*_2}$ for all\n$I\\in F(D)$. In this case we get $(I^{*_1})^{*_2}=I^{*_2}=(I^{*_2})^{*_1}$ and every\n$*_1$-invertible ideal is $*_2$-invertible.\nSome well-known star operations are: the {\\em $d$-operation} (given by $I\\mapsto I$),\nthe {\\em $v$-operation} (given  by $I\\mapsto I_v = (I^{-1})^{-1}$) and   the {\\em $t$-operation} (defined by $t=v_f$).\nThe {\\em $w$-operation} is the star operation given by $I \\mapsto I_w= \\{x\\in K \\mid xH \\subseteq\nI$ for some finitely\ngenerated ideal $H$ of $D$ with $H^{-1} =D\\}$. The $w$-operation is a stable star operation of finite character.\nFor   an integrally closed domain $D$, the {\\em $b$-operation} on $D$ is the star operation  defined by $I\\mapsto I_b=\\cap _{V} IV$  where $V$ runs in the set of all valuation overrings of $D$ (see \\cite[Page 398]{G}). For every $I\\in F(D)$, we have\n$I\\subseteq I_w \\subseteq I_t \\subseteq I_v$. It is known that $Max_w(D)=Max_t(D)$, cf.\n\\cite[Corollaries 2.13 and 2.17]{AC} and $I_w = \\cap _{M \\in Max_t(D)}ID_M$, cf. \\cite[Corollary 2.10]{AC}.\nConsequently, a nonzero fractional ideal is $w$-invertible if and only if it is $t$-invertible. Recall \\cite{EFP} that an integral domain $D$ is said to be {\\em $*$-Dedekind} if every nonzero fractional ideal of $D$ is $*$-invertible. A domain $D$ is called a Prufer {\\em $*$-multiplication domain (P$*$MD)} if every nonzero finitely generated ideal of $D$ is $*_f$-invertible (see \\cite{FJS}).\nFor the general theory of star operations we refer the reader to \\cite[Sections 32 and 34]{G}.\\\\\n\n\nWe introduce the key concept of this paper.\n\n\\begin{definition}\\label{1}\nLet $*$ be a star operation on $D$. We say that a domain $D$ is a {\\em  $*$-sharp domain} if whenever $I$, $A$, $B$  are nonzero ideals of $D$ with $I \\supseteq AB$, there exist nonzero ideals $H$ and $J$ such that $I^{*}=(HJ)^{*}$, $H^{*}\\supseteq A$ and $J^{*}\\supseteq B$.\n\\end{definition}\n\n\nThe $d$-sharp domains are just the sharp domains studied in \\cite{ADE}. If $*_1 \\leq *_2$ are star operations and $D$ is $*_1$-sharp, then $D$ is $*_2$-sharp (Proposition \\ref{81}). In particular, if $*$ is a star operation, then  every sharp domain is $*$-sharp and every $*$-sharp domain is $v$-sharp. A $t$-sharp domain is $v$-sharp but the converse is not true in general (Remark \\ref{121}).\n\nIn Section 2, we study the $*$-sharp domains in general.\nIn this new context, we generalize most of the results obtained in \\cite{ADE}.\nFor  $*\\in \\{d,b,w,t\\}$, every fraction ring of a $*$-sharp domain is $*$-sharp (Proposition \\ref{81}).\nEvery $*$-Dedekind domain is $*$-sharp. In particular, every Krull domain is  $t$-sharp (Proposition \\ref{3}).\nLet $D$ be a domain and  $*$  a finite character  stable star operation such that $D$ is $*$-sharp. Then $D$ is a P$*$MD of $*$-dimension $\\leq 1$; moreover   $D_M$ is a  valuation domain with value group a complete subgroup of the reals, for each  $M\\in Max_*(D)$ (Proposition \\ref{77}).\nThe converse is true for domains  whose  nonzero elements are contained in only finitely many $*$-maximal ideals (Proposition \\ref{200}).\nIf  $*$ is a star operation on $D$ such that $D$ is a $*$-sharp domain, then $I_v$ is $*$-invertible for each nonzero ideal $I$\n(Proposition \\ref{3a}).\nIf $*$ is a finite character stable star operation  on $D$ such that $D$ is a $*$-sharp TV domain, then $D$ is $*$-Dedekind (Corollary \\ref{111}).\nA domain $D$ is $v$-sharp if and only if $D$ is completely integrally closed (Corollary \\ref{100}). In particular, every $*$-sharp domain is completely integrally closed.\nIf  $*$ is a stable star operation on $D$ such that $D$ is a $*$-sharp domain, then every finitely generated nonzero ideal of $D$ is $*$-invertible (Proposition \\ref{102}).\nIf $D$ is a countable domain and $*$ a finite character  stable star operation on $D$  such that $D$ is $*$-sharp, then $D$ is a $*$-Dedekind domain (Corollary \\ref{845}).\n\nIn Section 3, we study the $t$-sharp domains. We obtain the following results.\nEvery $t$-sharp domain $D$ is a PVMD with $t$-dimension $\\leq 1$ and $D_M$ is a  valuation domain with value group a complete subgroup of the reals, for each maximal $t$-ideal $M$ of $D$ (Proposition  \\ref{11}). A domain is $t$-sharp if and only if it is  $w$-sharp  (Proposition \\ref{103}).\nA domain  $D$ is a Krull domain if and only if $D$  is a $t$-sharp TV domain (Corollary \\ref{400}).\nIf $D$ is a countable $t$-sharp domain, then $D$ is a Krull domain\n(Corollary \\ref{300}).\nA domain $D$ is $t$-sharp if and only if $D[X]$ is $t$-sharp\n(Proposition \\ref{331}) if and only if $D[X]_{N_v}$ is sharp (Proposition \\ref{133}). Here  $N_v$ denotes the multiplicative subset of $D[X]$ consisting of all  $f\\in D[X]-\\{0\\}$ with $c(f)_v=D$, where $c(f)$ is the ideal generated by the coefficients of $f$.\nLet $D$ be a $t$-sharp domain. Then $N'_v=\\{ f\\in D[[X]]-\\{0\\}\\mid c(f)_v=D\\}$ is  a multiplicative set,\n$D[[X]]_{N'_v}$ is a sharp domain and every ideal of $D[[X]]_{N'_v}$ is extended from $D$ (Proposition \\ref{1024}). Moreover, $D[[X]]_{N'_v}$ is a faithfully flat $D[X]_{N_v}$-module\nand there is a one-to-one correspondence between the ideals of $D[X]_{N_v}$ and the  ideals of $D[[X]]_{N'_v}$\n(Corollary \\ref{307}).\n\n\nThroughout this paper all rings are (commutative unitary) integral domains. Any unexplained material is standard, as in \\cite{G}, \\cite{H}.\n\n\\section{$*$-sharp domains.}\n\n\n\n\nIn this section we study the $*$-sharp domains  for an arbitrary star operation $*$ (see Definition \\ref{1}).  We obtain $*$-operation analogues for most of the results  in \\cite{ADE}.\n\n\n\\begin{proposition} \\label{4}\nLet $D$ be a domain, $S\\subseteq D$ a multiplicative set and $*$ (resp. $\\sharp$)  star operations  on $D$ (resp. $D_S$) such that\n$I^*\\subseteq (ID_S)^\\sharp$  for each nonzero ideal $I$ of $D$.\nIf $D$ is  $*$-sharp, then the fraction ring $D_S$  is  $\\sharp$-sharp.\n\\end{proposition}\n\\begin{proof}\nNote that the condition $I^*\\subseteq (ID_S)^\\sharp$ in the hypothesis is equivalent to $(I^*D_S)^\\sharp=(ID_S)^\\sharp$.\nLet $I,A,B$ be nonzero ideals  of $D$ such that $ID_S\\supseteq ABD_S$. Then $C=ID_S \\cap D \\supseteq  AB$. As $D$ is $*$-sharp, we have $C^*=(HJ)^*$ with $H,J$ ideals of $D$ such that  $H^*\\supseteq A$ and $J^*\\supseteq B$.  Since $(WD_S)^\\sharp=(W^*D_S)^\\sharp$ for every nonzero ideal $W$,\nwe get $(ID_S)^\\sharp=(C^*D_S)^\\sharp=((HJ)^*D_S)^\\sharp=(HJD_S)^\\sharp$,  $(HD_S)^\\sharp=(H^*D_S)^\\sharp\\supseteq AD_S$ and  $(JD_S)^\\sharp\\supseteq BD_S$.\n\\end{proof}\n\n\n\n\n\\begin{proposition} \\label{81}\nLet $D$ be a domain, $*_1 \\leq  *_2$   star operations and $D$ and $S\\subseteq D$ a multiplicative set.\n\n$(a)$  If $D$ is $*_1$-sharp, then D is $*_2$-sharp.\n\n$(b)$ If $*\\in \\{d,t,w,b\\}$ and $D$ is  $*$-sharp (with $D$ integrally closed if $*=b$), then  $D_S$  is  $*$-sharp.\n\\end{proposition}\n\\begin{proof}\n$(a)$. Apply Proposition \\ref{4} for $S=\\{1\\}$, $*=*_1$ and $\\sharp=*_2$.\n$(b)$. By Proposition \\ref{4}, it suffices to show that $I^*\\subseteq (ID_S)^*$  for each nonzero ideal $I$ of $D$. This is clear for $*=d$ and true for $*=t$, cf. \\cite[Lemma 3.4]{Kg}. Assume that $x\\in I_w$. Then $xH\\subseteq I$ for some finitely generated nonzero ideal $H$ of $D$ such that $H_v=D$. Hence $(HD_S)_v=D_S$ (cf. \\cite[Lemma 3.4]{Kg}) and $xHD_S\\subseteq ID_S$, thus $x\\in (ID_S)_w$. Assume that $D$ integrally closed. If $V$ is a valuation overring of $D_S$, then  $V$ is an overring of $D$, so $I_b\\subseteq IV$. Thus $I_b\\subseteq (ID_S)_b$.\n\\end{proof}\n\nIn \\cite[Theorem 11]{ADE}, it was shown that a sharp domain is a Prufer domain of dimension at most $1$. We extend this result.\n\n\\begin{proposition} \\label{77}\nLet $D$ be a domain and $*$ a finite character  stable star operation on $D$  such that $D$ is $*$-sharp.  Then    $D_M$ is a  valuation domain with value group a complete subgroup of the reals, for each  $M\\in Max_*(D)$. In particular, $D$ is a P$*$MD of $*$-dimension $\\leq 1$.\n\\end{proposition}\n\\begin{proof}\nLet $M$ be a maximal $*$-ideal. If  $I$ is a nonzero ideal  of $D$, then $I^*D_M=ID_M$, cf. \\cite[Corollary 4.2]{A}. By   Proposition \\ref{4}, applied for $S=D-M$, $*=*$ and $\\sharp=d$, we get that $D_M$ is a sharp domain. Apply \\cite[Theorem 11]{ADE}. The ``in particular'' assertion is clear.\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{3}\nLet $D$ be a domain and $*$ a star operation on $D$. If $D$ is $*$-Dedekind, then $D$ is $*$-sharp. In particular, every Krull domain is   $t$-sharp.\n\\end{proposition}\n\\begin{proof}\nLet $I,A,B$ be nonzero ideals of $D$ such that $I \\supseteq AB$. Set $H=I+A$ and $J=IH^{-1}$. Note that $J\\subseteq D$ and $A\\subseteq H$. Since $(HH^{-1})^*=D$, we get $I^*=(HJ)^*$. From $BH=B(A+I) \\subseteq I$, we get  $B \\subseteq (BHH^{-1})^* \\subseteq (IH^{-1})^*=J^*$. For the ``in particular statement'', recall that  the $t$-Dedekind domains are the Krull domains, cf. \\cite[Theorem 3.6]{Kg1}.\n\\end{proof}\n\n\n\n\n\\begin{proposition}\\label{3a}\nLet $D$ be a domain and $*$ a star operation on $D$ such that $D$ is $*$-sharp. Then $I_v$ is $*$-invertible for each nonzero ideal $I$.\n\\end{proposition}\n\\begin{proof}\nLet $I$ be a nonzero ideal of $D$ and $x  \\in I-\\{0\\}$. Then $I(xI^{-1}) \\subseteq xD$. Since $D$ is $*$-sharp, there exist $H,J$ ideals of $D$ such that  $H^* \\supseteq I$,  $J^* \\supseteq xI^{-1}$ and  $xD=(HJ)^*$. Hence $H$ is $*$-invertible and we get\n$H^{-1}=(x^{-1}J)^*\\supseteq (xx^{-1}I^{-1})^*=I^{-1}$, so $H_v\\subseteq I_v$. The opposite inclusion follows from  $H^* \\supseteq I$. Thus $I_v=H_v$ is $*$-invertible, because $H^*=H_v$ since $H$ is $*$-invertible.\n\\end{proof}\n\nNext, we extend  \\cite[Corollary 12]{ADE} to the star operation setting.\n\n\n\\begin{corollary}\\label{111}\nLet $D$ be a domain and $*$ a finite character  stable star operation on $D$  such that $D$ is $*$-sharp. If $D$ is a TV domain (e.g. a Mori domain), then $D$ is $*$-Dedekind.\n\\end{corollary}\n\\begin{proof}\nBy Proposition \\ref{77}, $D$ is a P$*$MD , so $*=t$, cf. \\cite[Proposition 3.15]{FJS}. As $D$ is a TV domain, we get $*=t=v$. By Proposition \\ref{3a}, $D$ is $*$-Dedekind.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{100}\nA domain $D$ is $v$-sharp if and only if $D$ is completely integrally closed. In particular, any  $*$-sharp domain  is  completely integrally closed.\n\\end{corollary}\n\\begin{proof}\nBy Propositions \\ref{3} and \\ref{3a} (for $*=v$),  $D$ is $v$-sharp if and only if $D$ is $v$-Dedekind. By \\cite[Theorem 34.3]{G} or \\cite[Proposition 3.4]{F}, a domain is $v$-Dedekind if and only if it is  completely integrally closed. For the ``in particular'' assertion, apply  Proposition \\ref{81} taking into account that $*\\leq v$ for each star operation $*$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{remark}\\label{121}\n$(a)$ There exist  a completely integrally closed domain $A$  having some fraction ring which is not completely integrally closed (for instance the ring of entire functions, cf. \\cite[Exercises 16 and 21, page 147]{G}). Thus the $v$-sharp property does not localize, cf. Corollary \\ref{100}. Note that $A$ cannot be $t$-sharp because the $t$-sharp property localizes, cf.\nProposition \\ref{81}.\n$(b)$ Let $D$ be a completely integrally closed domain which is not a PVMD (such a domain is constructed in  \\cite{D}). By  Corollary \\ref{100} and Proposition  \\ref{11}, such a domain is $v$-sharp but not $t$-sharp.\n$(c)$ Let $D$ be a Krul domain of dimension $\\geq 2$ (e.g. $\\mathbb{Z}[X]$).  By  Proposition \\ref{3} and \\cite[Theorem 11]{ADE}, $D$ is  $t$-sharp but not sharp.\n\n\n\\end{remark}\n\nIn the next lemma we recall two well-known facts.\n\n\n\n\\begin{lemma} \\label{71}\nLet $D$ be a domain, $*$ a  star operation on $D$ and $I,J,H\\in F(D)$.\n\n$(a)$ If $(I+J)^*=D$, then $(I\\cap J)^*=(IJ)^*$.\n\n$(b)$ If $I$ is $*$-invertible, then $(I(J\\cap H))^*=(IJ\\cap IH)^*$.\n\\end{lemma}\n\\begin{proof}\n$(a)$ Clearly,  $(IJ)^*\\subseteq (I\\cap J)^*$. Conversely, since  $(I+J)^*=D$, we have $(I\\cap J)^*=((I\\cap J)(I+J))^*\\subseteq (IJ)^*$, thus $(I\\cap J)^*=(IJ)^*$.\n$(b)$ Clearly, $(I(J\\cap H))^*\\subseteq (IJ\\cap IH)^*$. Conversely, because $I$ is $*$-invertible, we have $(IJ\\cap IH)^*=(II^{-1}(IJ\\cap IH))^*\\subseteq (I(I^{-1}IJ\\cap I^{-1}IH))^*\\subseteq (I(J\\cap H))^*$.\n\\end{proof}\n\nThe next result generalizes \\cite[Proposition 10]{ADE}.\n\n\\begin{proposition} \\label{211p}\nLet $D$ be a domain and $*$ a stable star operation on $D$  such that $D$ is $*$-sharp.  If $I,J$ are nonzero ideals of $D$ such that $(I+J)_v=D$, then $(I_v+J_v)^*=D$.\n\\end{proposition}\n\\begin{proof}\nLet $K$ be the quotient field of $D$. Changing $I$ by $I_v$ and $J$ by $J_v$, we may assume that $I,J$ are $*$-invertible $v$-ideals, cf. Proposition \\ref{3a}.\nSince $(I+J)^2 \\subseteq I^2+J$ and $D$ is $*$-sharp, there exist two nonzero ideals $A$, $B$ such that $(I^2+J)^*=(AB)^*$ and $I+J\\subseteq A^* \\cap B^*$. We {\\em claim} that $(I^2+J)^*:I=(I+J)^*$. To prove the claim, we perform the following step-by-step computation. First,\n$(I^2+J)^*:I=((I^2+J)^*:_KI)\\cap D=((I^2+J)I^{-1})^*\\cap D=(I+JI^{-1})^*\\cap D$ because $I$ is $*$-invertible. As $*$ is stable, we get $(I+JI^{-1})^*\\cap D=((I+JI^{-1})\\cap D)^*=(I+(JI^{-1}\\cap D))^*$ by modular distributivity.\nSince $I$ is $*$-invertible, we get\n$(I+(JI^{-1}\\cap D))^*=(I+I^{-1}(J\\cap I))^*$, cf. Lemma \\ref{71}.\nUsing the fact that $I$ is $*$-invertible (hence $v$-invertible) and Lemma \\ref{71}, we derive that $(I+I^{-1}(J\\cap I))^* \\subseteq (I+I^{-1}(IJ)_v)^*\\subseteq  (I+(II^{-1}J)_v)^*=(I+J_v)^*=(I+J)^*$. Putting all these facts together, we get $(I^2+J)^*:I\\subseteq (I+J)^*$ and the other inclusion is clear. So the claim is proved.\nFrom $(I^2+J)^*=(AB)^*$, we get $A^*\\subseteq (I^2+J)^*:B^*\\subseteq (I^2+J)^*:I=(I+J)^*$, so $A^*=(I+J)^*$. Similarly, we get $B^*=(I+J)^*$, hence $(I^2+J)^*=((I+J)^2)^*$.\nIt follows that $J^*\\subseteq (I^2+J)^*=((I+J)^2)^*\\subseteq  (J^2+I)^*$. So $J^*=J^*\\cap (J^2+I)^*=(J\\cap (J^2+I))^*=(J^2+(J\\cap I))^*$ where we have used the fact that $*$ is stable and the modular distributivity. By Lemma \\ref{71}, we have $I\\cap J\\subseteq (IJ)_v$, so we get $J^*=(J^2+(J\\cap I))^*\\subseteq (J^2+(IJ)_v)^*$. Since $J$ is $*$-invertible, we have $D=(JJ^{-1})^*\\subseteq ((J^2+(IJ)_v)J^{-1})^* \\subseteq (J+I_v)^*=(J+I)^*$.\nThus $(I+J)^*=D$.\n\\end{proof}\n\nNote that from Proposition \\ref{211p} we can recover easily  \\cite[Proposition 10]{ADE}. Next, we give another extension of \\cite[Theorem 11]{ADE} (besides Proposition \\ref{77}).\n\n\n\\begin{proposition} \\label{102}\nLet $D$ be a domain and $*$ a stable star operation on $D$  such that $D$ is $*$-sharp. Then every finitely generated nonzero ideal of $D$ is $*$-invertible.\n\\end{proposition}\n\\begin{proof}\nLet $x,y\\in D-\\{0\\}$. By Proposition \\ref{3a}, the ideal $I=(xD+yD)_v$ is $*$-invertible (hence $v$-invertible), so $(xI^{-1}+yI^{-1})_v=D$. By Proposition \\ref{211p} we get $(xI^{-1}+yI^{-1})^*=D$, hence $I=((xI^{-1}+yI^{-1})I)^*=(xD+yD)^*$ because $I$ is $*$-invertible. Thus every two-generated nonzero ideal of $D$ is $*$-invertible. Now  the proof of \\cite[Proposition 22.2]{G} can be easily adapted to show that every finitely generated nonzero ideal of $D$ is $*$-invertible.\n\\end{proof}\n\n\\begin{remark}\nUnder the assumptions  of Proposition \\ref{102}, it does not follow that $D$ is a P$*$MD. Indeed, let $D$ be a completely integrally closed domain which is not a PVMD (such a domain is constructed in  \\cite{D}).  The $v$-operation on $D$ is stable (cf. \\cite[Theorem 2.8]{ACl}) and   $D$ is $v$-sharp (cf. Corollary \\ref{100}).\n\\end{remark}\n\n\nLet $D$ be a domain with quotient field $K$.\nAccording to \\cite{AZ}, a family   $\\mathcal{F}$ of nonzero prime ideals of $D$ is called    {\\em independent of finite character family (IFC family)},\nif $(1)$ $D=\\cap_{P\\in \\mathcal{F}}D_P$, $(2)$ every nonzero $x\\in D$ belongs to only finitely many members of\n$\\mathcal{F}$ and $(3)$ every nonzero prime ideal of $D$ is contained in at most one member of $\\mathcal{F}$. The follwing result extends \\cite[Theorem 15]{ADE}.\n\n\n\n\n\n\n\\begin{proposition}\\label{200}\nLet $D$ be a domain and $*$ a finite character star operation on $D$. Assume that\n\n$(a)$ every $x\\in D-\\{0\\}$ is contained in only finitely many maximal $*$-ideals, and\n\n$(b)$ for every $M\\in Max_*(D)$, $D_M$ is a  valuation domain with value group a complete subgroup of the reals.\n\\\\ Then $D$ is a $\\tilde{*}$-sharp domain and hence $*$-sharp.\n\\end{proposition}\n\\begin{proof}\nBy \\cite{Gr}, $D=\\cap_M D_M$ where $M$ runs in the set of maximal $*$-ideals. Since each $D_M$ is a  valuation domain with value group a complete subgroup of the reals, every $M$ has height one.\nIt follows that $Max_*(D)$  is an IFC family. Consider the $\\tilde{*}$ operation, i.e. $I\\mapsto I^{\\tilde{*}}=\\cap_M ID_M$. We show that $D$ is $\\tilde{*}$-sharp.\nLet $I,A,B$ be nonzero ideals  of $D$ such that $I\\supseteq AB$.\nLet $P_1$,...,$P_n$ the maximal $*$-ideals of $D$ containing $AB$.\nSince $D_{P_i}$ is sharp, there exist $H_i$, $J_i$ ideals of $D_{P_i}$ such that $ID_{P_i}=H_iJ_i$, $H_i\\supseteq AD_{P_i}$ and\n$J_i\\supseteq BD_{P_i}$ for all $i$ between $1$ and $n$.\nSet $H'_i=H_i\\cap D$, $J'_i=J_i\\cap D$, $i=1,...,n$,\n$H=H'_1\\cdots H'_n$ and $J=J'_1\\cdots J'_n$.\nBy \\cite[Lemma 2.3]{AZ}, $P_i$ is the only element of $Max_*(D)$ containing $H'_i$ (resp. $J'_i$), thus it can be checked that $ID_P=(HJ)D_P$, $HD_P\\supseteq AD_P$ and $JD_P\\supseteq BD_P$ for each $P\\in Max_*(D)$. So, we have $I^{\\tilde{*}}=(HJ)^{\\tilde{*}}$, $H^{\\tilde{*}}\\supseteq A$ and $J^{\\tilde{*}}\\supseteq B$. Consequently, $D$ is $\\tilde{*}$-sharp. By Proposition \\ref{81}, $D$ is  $*$-sharp because  $\\tilde{*}\\leq *$, cf. \\cite[Theorem 2.4]{AC}.\n\\end{proof}\n\n\\begin{proposition}\\label{822}\nLet $D$ be a contable domain and $*$ a finite character star operation on $D$ such that $D$ is a P$*$MD and  $I_v$ is $*$-invertible for each nonzero ideal $I$ of $D$. Then  every nonzero element of $D$ is contained in only finitely many maximal $*$-ideals.\n\\end{proposition}\n\\begin{proof}\nDeny. By \\cite[Corollary 5]{DZ},  there exists a nonzero element $z$ and an infinite family $(I_n)_{n\\geq 1}$ of $*$-invertible proper ideals containing $z$ which are mutually $*$-comaximal (that is, $(I_m+I_n)^*=D$ for every $m\\neq n$). For each nonempty set  $\\Lambda$ of natural numbers, consider the $v$-ideal $I_\\Lambda=\\cap_{n\\in \\Lambda} I_n$ (note that $z\\in I_\\Lambda$). By hypothesis, $I_\\Lambda$ is $*$-invertible. We claim that $I_\\Lambda\\neq I_{\\Lambda'}$ whenever $\\Lambda$, $\\Lambda'$ are distinct nonempty sets of natural numbers.\nDeny. Then there exists a nonempty set  of natural numbers $\\Gamma$ and some $k\\notin \\Gamma$ such that $I_k\\supseteq I_\\Gamma$. Consider the ideal  $H=(I_k^{-1} I_\\Gamma)^*\\supseteq I_\\Gamma$. If $n\\in \\Gamma$, then $I_n \\supseteq I_\\Gamma =(I_kH)^*$, so $I_n \\supseteq H$, because $(I_n+I_k)^*=D$. It follows that $I_\\Gamma\\supseteq H$, so $I_\\Gamma = H=(I_k^{-1} I_\\Gamma)^*$. Since $I_\\Gamma$ is $*$-invertible, we get $I_k=D$, a contradiction. Thus the claim is proved. But then it follows that $\\{I_\\Lambda\\mid \\emptyset\\neq \\Lambda\\subseteq \\mathbb{N}\\}$ is an uncountable set of $*$-invertible ideals. This leads to a contradiction, because $D$ being countable, it has countably many $*$-ideals of finite type.\n\\end{proof}\n\n\n\\begin{corollary}\\label{845}\nLet $D$ be a countable domain and $*$ a finite character  stable star operation on $D$  such that $D$ is $*$-sharp.\nThen $D$ is a $*$-Dedekind domain.\n\\end{corollary}\n\\begin{proof}\nWe may assume that $D$ is not a field.\nBy Proposition \\ref{77}, $D$ is a P$*$MD. Now Propositions \\ref{3a} and \\ref{822} show that every nonzero element of $D$ is contained in only finitely many maximal $*$-ideals. Let $M$ be a maximal $*$-ideal of $D$. By Proposition \\ref{77}, $D_M$ is a countable valuation domain with value group $\\mathbb{Z}$ or $\\mathbb{R}$, so $D_M$ is a DVR. Thus $D$ is a $*$-Dedekind domain, cf. \\cite[Theorem 4.11]{EFP}.\n\\end{proof}\n\n\n\\section{$t$-sharp domains.}\n\n\nThe $t$-operation is a very useful  tool in multiplicative ideal theory. In this section we give some results  which are specific for  the $t$-sharp domains.\n\n\n\n\\begin{proposition}\\label{11}\nLet $D$ be a $t$-sharp domain. Then $D$ is a PVMD of $t$-dimension $\\leq 1$ and $D_M$ is a  valuation domain with value group a complete subgroup of the reals for each maximal $t$-ideal $M$ of $D$.\n\\end{proposition}\n\\begin{proof}\nLet $I$ be finitely generated nonzero ideal of $D$. Then $I_v=I_t$, so Proposition \\ref{3a} shows that $I_t$ is $t$-invertible. Thus $D$ is a PVMD.\nLet $M$ be a maximal $t$-ideal  of $D$.\nBy  part $(b)$ of Proposition \\ref{81},  $D_M$ is a  $t$-sharp valuation domain, so $D_M$ is sharp since for valuation domains $t=d$. Now apply \\cite[Proposition 6]{ADE}.\n\\end{proof}\n\n\n\n\n\n\\begin{proposition}\\label{103}\nLet $D$ be a domain. Then $D$ is $t$-sharp if and only if $D$ is $w$-sharp.\n\\end{proposition}\n\\begin{proof}\nIf $D$ is $w$-sharp, then $D$ is $t$-sharp (cf. Proposition \\ref{81}) because $w\\leq t$. Conversely, assume that $D$ is $t$-sharp. By Proposition \\ref{11}, $D$ is a PVMD. But in a PVMD the $w$-operation coincides with the $t$-operation (cf. \\cite[Theorem 3.5]{Kg}),  so $D$ is also $w$-sharp.\n\\end{proof}\n\nCombining  Corollary \\ref{111} and Proposition \\ref{103}, we get\n\n\\begin{corollary}\\label{400}\nA domain  $D$ is a Krull domain if and only if $D$  is a $t$-sharp TV domain.\n\\end{corollary}\n\n\n\n\\begin{corollary}\\label{300}\nIf $D$ is a countable $t$-sharp domain, then $D$ is a Krull domain.\n\\end{corollary}\n\\begin{proof}\nAssume that $D$ is a countable $t$-sharp domain.\nBy Proposition \\ref{103},  $D$ is $w$-sharp. Moreover the $w$-operation is stable and of finite character, cf. \\cite[Corollary 2.11]{AC}. By Corollary \\ref{845}, $D$ is a $t$-Dedekind domain, that is a Krull domain.\n\\end{proof}\n\n\n\n\n\n\nBy \\cite{Zac}, a domain $D$ is called  a {\\em pre-Krull domain} if $I_v$ is $t$-invertible for each nonzero ideal $I$ of $D$ (see also \\cite{L} where a pre-Krull domain is called a $(t,v)$-Dedekind domain).\n\n\n\n\n\n\\begin{proposition} \\label{3x}\nA domain $D$ is  $t$-sharp   if and only if\n\n$(a)$ $D$ is pre-Krull, and\n\n$(b)$ for all nonzero ideals  $I$,$A$,$B$   of $D$ such that $I_v=D$ and $I\\supseteq AB$, there exist nonzero ideals $H$ and $J$ such that $I_t=(HJ)_t$, $H_t\\supseteq A$ and $J_t\\supseteq B$.\n\\end{proposition}\n\\begin{proof}\nThe implication $(\\Rightarrow)$ follows from Proposition \\ref{3a}. Conversely, assume that $(a)$ and $b$ hold.\nLet $I$, $A$ and $B$ be nonzero ideals of $D$ such that $I\\supseteq AB$.\nThen $I_v \\supseteq A_vB_v$ and $I_v$, $A_v$, $B_v$ are $t$-invertible ideals, cf. $(a)$.\nSince $D$ is a pre-Krull domain,  $D$ is a  PVMD and hence a $t$-Schreier domain cf. \\cite[Corollary 6]{DZ1}.\nSo there exist  $t$-invertible ideals $H$ and $J$ such that $I_v=(HJ)_t$, $H_t\\supseteq  A_v$ and $J_t\\supseteq  B_v$.\nWe have $(II^{-1})_t = (IH^{-1}J^{-1})_t  \\supseteq (AH^{-1})(BJ^{-1})$. Set $M=(II^{-1})_t$ and note that  $AH^{-1}$ and $BJ^{-1}$ are integral ideals.\nSince $I_v$ is $t$-invertible, $M_v=(I_vI^{-1})_v=D$.\nBy $(b)$, there exist  nonzero ideals $N$ and $P$ such that $M_t=(NP)_t$, $N_t\\supseteq AH^{-1}$ and $P_t\\supseteq BJ^{-1}$.\nSumming up, we get\n$I_t=(I_vM)_t=(HJNP)_t=((HN)(JP))_t$, $(HN)_t\\supseteq (AHH^{-1})_t=A_t$ and $(JP)_t\\supseteq (BJJ^{-1})_t=B_t$.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{332}  If D is an integrally closed domain and $I$ a nonzero ideal of $D[X]$ such that $I_v=D[X]$, then $I\\cap D \\neq 0$.\n\\end{lemma}\n\\begin{proof} Assume that  $I \\cap D = 0$. By \\cite[Theorem 2.1]{AKZ}, there exist $f\\in D[X]-\\{0\\}$ and $a\\in D-\\{0\\}$ such that $J:=(a\/f)I \\subseteq D[X]$ and $J\\cap D \\neq 0$.  We get  $(a\/f)D[X]=(a\/f)I_v \\subseteq D[X]$, hence  $a\/f \\in D[X]$ and thus $a\/f\\in D$, because $a\\in D-\\{0\\}$. We get $J\\subseteq I$  which is a contradiction because $J\\cap D  \\neq 0$   and   $I \\cap D = 0$.\n\\end{proof}\n\n\\begin{proposition} \\label{331}\nA domain $D$ is $t$-sharp if and only if $D[X]$ $t$-sharp.\n\\end{proposition}\n\\begin{proof} $(\\Rightarrow).$ Set $\\Omega=D[X]$. Let $I$,$A$,$B$ be nonzero ideals of $\\Omega$ such that  $I\\supseteq AB$.\nBy Proposition \\ref{3x}, $D$ is pre-Krull, so $D[X]$ is pre-Krull, cf. \\cite[Theorem 3.3]{L}. Applying Proposition \\ref{3x} for $\\Omega$, we can assume that $I_v=\\Omega$. Changing $A$ by $I+A$ and $B$ by $I+B$, we may assume that $A,B \\supseteq I$. By Lemma \\ref{332} we get $I\\cap D\\neq 0$, so $A\\cap D\\neq 0$ and $B\\cap D\\neq 0$. By \\cite[Theorem 3.2]{AKZ}, we have $I_t=I'_t\\Omega$, $A_t=A'_t\\Omega$ and $B_t=B'_t\\Omega$ for some nonzero ideals $I',A'$ and $B'$ of D. From $I\\supseteq AB$, we get $I'_t\\Omega=I_t\\supseteq A_tB_t=(A'_tB'_t)\\Omega$, hence $I'_t\\supseteq A'B'$.\nAs $D$ is $t$-sharp, there exist nonzero ideals $H$ and $J$ of $D$ such that $I'_t=(HJ)_t$, $H_t\\supseteq  A'$ and $J_t\\supseteq  B'$.\nHence $I_t=(HJ\\Omega)_t$, $(H\\Omega)_t\\supseteq  A$ and $(J\\Omega)_t\\supseteq  B$. $(\\Leftarrow).$ Let $I$,$A$,$B$ be nonzero ideals of $D$ such that  $I\\supseteq AB$. As $D[X]$ $t$-sharp, there exist nonzero ideals $H$ and $J$ of $\\Omega$ such that $(I\\Omega)_t=(HJ)_t$, $H_t\\supseteq  A$ and $J_t\\supseteq  B$.\nSince  $H_t\\supseteq A$ and $A\\neq 0$, we derive that $H_t\\cap D\\neq 0$,  hence $H_t=(M\\Omega)_t$ for some nonzero ideal $M$ of $D$, cf. \\cite[Theorem 3.2]{AKZ}. Similarly, $J_t=(N\\Omega)_t$ for some nonzero ideal $N$ of $D$. Combining the relations above, we get $I_t=(MN)_t$, $M_t\\supseteq  A$ and $N_t\\supseteq  B$.\n\\end{proof}\n\n\\begin{remark}\nNotice that we do not have a ``$d$-analogue'' of  Proposition \\ref{331} because  a sharp domain  has dimension $\\leq 1$ (see  \\cite[Theorem 11]{ADE}). But remark that we do  have a ``$v$-analogue'' of  Proposition \\ref{331}. Indeed, a domain $D$ is $v$-sharp  if and only if $D$ is completely integrally closed\n(cf. Corollary \\ref{100}) and  $D$ is completely integrally closed if and only if so is $D[X]$. Similarly, $D$ is $v$-sharp  if and only if the power series ring $D[[X]]$ is $v$-sharp.\n\\end{remark}\n\nDenote by $N_v$ the multiplicative set of $D[X]$ consisting of all nonzero polynomials $a_0+a_1X+\\cdots +a_nX^n$ such that $(a_0,a_1,...,a_n)_v=D$. The ring $D[X]_{N_v}$ was studied in \\cite{Kg}.\n\n\n\n\n\n\\begin{proposition}\\label{133}\nA domain $D$ is $t$-sharp if and only if $D[X]_{N_v}$ is sharp.\n\\end{proposition}\n\\begin{proof}\nIf $D$ is $t$-sharp, then $D$ is a PVMD, cf. Proposition  \\ref{11}.\nIf $D[X]_{N_v}$ is sharp, then $D[X]_{N_v}$ is a Prufer domain (cf. \\cite[Theorem 11]{ADE}) hence $D$ is a PVMD, cf. \\cite[Theorem 3.7]{Kg}. So we may assume from the beginning that $D$ is a PVMD. Note that the $t$-sharp  property of $D$ is in fact a property of the ordered monoid of all integral $t$-ideals of $D$ under the $t$-multiplication. Similarly, the sharp  property of $D[X]_{N_v}$ is  a property of the ordered monoid of all integral ideals of $D$ under the usual multiplication. Since $D$ is a PVMD, these two monoids are isomorphic (cf. \\cite[Theorem 3.14]{Kg}), so the proof is complete.\n\\end{proof}\n\nWe end our paper with a (partial) power series analogue of Proposition \\ref{133}. A lemma is in order.\n\n\n\\begin{lemma}\\label{101}\nLet $D\\subseteq E$ be a domain extension and every ideal of $E$ is extended from $D$. If $D$ is sharp then $E$ is also sharp.\n\\end{lemma}\n\\begin{proof}\nLet $I,A,B$ be nonzero ideals  of $D$ such that $IE\\supseteq ABE$. Then $C=IE \\cap D \\supseteq  AB$. As $D$ is sharp, we have $C=HJ$ with $H,J$ ideals of $D$ such that  $H\\supseteq A$ and $J\\supseteq B$.  We get $IE=CE=HJE$,  $HE\\supseteq AE$ and  $J\\supseteq BE$.\n\\end{proof}\n\nLet $D$ be a $t$-sharp domain which is not a field. \nBy Proposition \\ref{11}, $D$ is PVMD with $t$-dimension one. Hence \\cite[Proposition 3.3]{AK2} shows that $c(fg)_t=(c(f)c(g))_t$ (thus $c(fg)_v=(c(f)c(g))_v$)\nfor every $f,g\\in D[[X]]-\\{0\\}$,  where $c(f)$ is the ideal generated by the coefficients of $f$.\nThen $N'_v=\\{ f\\in D[[X]]-\\{0\\}\\mid c(f)_v=D\\}$\nis a multiplicative subset of the power series ring $D[[X]]$.\n The fraction ring $D[[X]]_{N'_v}$ was studied in \\cite{AK2} and \\cite{L}. Note that $D\\subseteq D[X]_{N_v}\\subseteq D[[X]]_{N'_v}$, where $N_v=\\{ f\\in D[X]-\\{0\\}\\mid c(f)_v=D\\}$.\n\n\n\n\\begin{proposition}\\label{1024}\nIf  $D$ is a $t$-sharp domain, then $D[[X]]_{N'_v}$ is sharp and every ideal of $D[[X]]_{N'_v}$ is extended from $D$.\n\\end{proposition}\n\\begin{proof} \nWe may assume that $D$ is not a field. \nBy Proposition \\ref{3x}, $D$ is a pre-Krull domain (alias $(t,v)$-Dedekind domain). As seen in the paragraph preceding this proposition, $c(fg)_v=(c(f)c(g))_v$\nfor every $f,g\\in D[[X]]-\\{0\\}$. By  \\cite[Theorem 4.3]{L}\n it follows that every ideal of $D[[X]]_{N'_v}$ is extended from $D$, then, a fortiori, from $D[X]_{N_v}$.\n By Proposition \\ref{133} it follows that $D[X]_{N_v}$ is a sharp domain, hence so is $D[[X]]_{N'_v}$, cf. Lemma \\ref{101}. \n\\end{proof}\n\n\\begin{corollary}\\label{307}\nLet  $D$   be a  $t$-sharp domain. Then  $D[[X]]_{N'_v}$ is a faithfully flat $D[X]_{N_v}$-module and the extension map \n$I\\mapsto ID[[X]]_{N'_v}$ \n is a bijection from the set of  ideals of $D[X]_{N_v}$ to the set of  ideals of $D[[X]]_{N'_v}$.\n\\end{corollary}\n\\begin{proof}\nSet $E=D[X]_{N_v}$ and $F=D[[X]]_{N'_v}$.\nBy the proof of Proposition \\ref{133} it follows that $E$ is a Prufer domain. Hence \n$F$ is a flat $E$-module because over a Prufer domain every torsion-free module is flat. \nWe show that every proper ideal of $E$ extends to a proper ideal of $F$. \nLet $J$ be a proper nonzero ideal of $E$. By \\cite[Theorem 3.14]{Kg}, $J=IE$ for some ideal $I$ of $D$ such that $I_t\\neq D$. Assume that $JF=F$. Then $IF=JF=F$, so $ID[[X]]$ contains some power series $f$ with $c(f)_v=D$. Write $f=a_1f_1+...+a_nf_n$ with $a_i \\in I$ and $f_i \\in D[[X]]$. Then $D=c(f)_v \\subseteq  (a_1,...,a_n)_v\\subseteq I_t$, so $I_t=D$, a contradiction. As $F$ is a flat $E$-module and every proper ideal of $E$ extends to a proper ideal of $F$, it follows that $F$ is a faithfully flat $E$-module, cf. \\cite[Exercise 16, page 45]{AM}. In particular, $HF\\cap E=H$ for each ideal $H$ of $E$. \nBy Proposition \\ref{1024},  every ideal of $F$ is extended from $D$ and hence from $E$ because $D\\subseteq  E\\subseteq  F$.\nCombining these two facts, it follows that the extension map  $I\\mapsto IF$ is a bijection from the set of  ideals of $E$ to the set of  ideals of $F$.\n\\end{proof}\n \n\n\n\n\n\n\n{\\bf Acknowledgements.} The first author was partially supported by an HEC (Higher Education Commission, Pakistan)  grant. The second author gratefully acknowledges the warm\nhospitality of the Abdus Salam School of Mathematical Sciences GCU Lahore during his visits in 2006-2011. The third author was supported by UEFISCDI, project number 83\/2010, PNII-RU code TE\\_46\/2010, program Human Resources.\n\\\\[7mm]\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:1}}\n\nIn this note, we describe a~generalization of the Carath\\'eodory\nform of the calculus of variations in \\emph{second-order }and, for\nspecific Lagrangians, in\\emph{ higher-order} field theory. Our approach\nis based on a geometric relationship between the Poincar\\'e\\textendash Cartan\nand Carath\\'eodory forms,\\emph{ }and analysis of the corresponding\nglobal properties. In \\cite{CramSaun2}, Crampin and Saunders obtained\nthe Carath\\'eodory form for second-order Lagrangians as a~certain\nprojection onto a~sphere bundle. Here, we confirm this result by\nmeans of a different, straightforward method which furthermore allows\nhigher-order generalization. It is a~standard fact in the global\nvariational field theory that the local expressions,\n\\begin{equation}\n\\Theta_{\\lambda}=\\mathscr{L}\\omega_{0}+\\sum_{k=0}^{r-1}\\left(\\sum_{l=0}^{r-1-k}(-1)^{l}d_{p_{1}}\\ldots d_{p_{l}}\\frac{\\partial\\mathscr{L}}{\\partial y_{j_{1}\\ldots j_{k}p_{1}\\ldots p_{l}i}^{\\sigma}}\\right)\\omega_{j_{1}\\ldots j_{k}}^{\\sigma}\\wedge\\omega_{i},\\label{eq:PoiCar}\n\\end{equation}\nwhich generalize the well-known Poincar\\'e\\textendash Cartan form\nof the calculus of variations, define, in general, differential form\n$\\Theta_{\\lambda}$ globally for Lagrangians $\\lambda=\\mathscr{L}\\omega_{0}$\nof order $r=1$ and $r=2$ only; see Krupka \\cite{Krupka-Lepage}\n($\\Theta_{\\lambda}$ is known as the principal component of a~Lepage\nequivalent of Lagrangian $\\lambda$), and Hor\\'ak and Kol\\'a\\v{r}\n\\cite{HorakKolar} (for higher-order Poincar\\'e\\textendash Cartan\nmorphisms). We show that if $\\Theta_{\\lambda}$ is globally defined\ndifferential form for a~\\emph{class} of Lagrangians of order $r\\geq3$,\nthen a~higher-order Carath\\'eodory equivalent for Lagrangians belonging\nto this class naturally arises by means of geometric operations acting\non $\\Theta_{\\lambda}$. To this purpose, for order $r=3$ we analyze\nconditions, which describe the obstructions for globally defined principal\ncomponents of Lepage equivalents \\eqref{eq:PoiCar} (or, higher\\textendash order\nPoincar\\'e\\textendash Cartan forms). \n\nThe above-mentioned differential forms are examples of \\emph{Lepage\nforms}; for a~comprehensive exposition and original references see\nKrupka \\cite{Handbook,Krupka-Book}\\emph{. }Similarly as the well-known\nCartan form describes analytical mechanics in a~coordinate-independent\nway, in variational field theory (or, calculus of variations for multiple-integral\nproblems) this role is played by Lepage forms, in general. These objects\ndefine the same variational functional as it is prescribed by a~given\nLagrangian and, moreover, variational properties (as variations, extremals,\nor Noether's type invariance) of the corresponding functional are\nglobally characterized in terms of geometric operations (such as the\nexterior derivative and the Lie derivative) acting on integrands -\nthe Lepage equivalents of a~Lagrangian.\n\nA concrete application of our result in second-order field theory\nincludes the Carath\\'eodory equivalent of the Hilbert Lagrangian\nin general relativity, which we determine and it will be further studied\nin future works.\n\nBasic underlying structures, well adapted to this paper, can be found\nin Voln\\'a and Urban \\cite{Volna}. If $(U,\\varphi)$, $\\varphi=(x^{i})$,\nis a chart on smooth manifold $X$, we set\n\\begin{equation*}\n{\\color{black}{\\color{black}\\omega_{0}}=dx^{1}\\wedge\\ldots\\wedge dx^{n},}\\qquad{\\color{black}\\omega_{j}}{\\color{black}=i_{\\partial\/\\partial x^{j}}\\omega_{0}=\\frac{1}{(n-1)!}\\varepsilon_{ji_{2}\\ldots i_{n}}dx^{i_{2}}\\wedge\\ldots\\wedge dx^{i_{n}},\n\\end{equation*}\nwhere $\\varepsilon_{i_{1}i_{2}\\ldots i_{n}}$ is the Levi-Civita permutation\nsymbol. If $\\pi:Y\\rightarrow X$ is a~fibered manifold and $W$ an\nopen subset of $Y$, then there exists a~unique morphism $h:\\Omega^{r}W\\rightarrow\\Omega^{r+1}W$\nof exterior algebras of differential forms such that for any fibered\nchart $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$, where $V\\subset W$,\nand any differentiable function $f:W^{r}\\rightarrow\\mathbb{R}$, where\n$W^{r}=(\\pi^{r,0})^{-1}(W)$ and $\\pi^{r,s}:J^{r}Y\\rightarrow J^{s}Y$\nthe jet bundle projection, \n\\[\nhf=f\\circ\\pi^{r+1,r},\\quad\\quad hdf=(d_{i}f)dx^{i},\n\\]\nwhere\n\\begin{equation}\nd_{i}=\\frac{\\partial}{\\partial x^{i}}+\\sum_{j_{1}\\leq\\ldots\\leq j_{k}}\\frac{\\partial}{\\partial y_{j_{1}\\ldots j_{k}}^{\\sigma}}y_{j_{1}\\ldots j_{k}i}^{\\sigma}\\label{eq:FormalDerivative}\n\\end{equation}\nis the $i$-th formal derivative operator associated with $(V,\\psi)$.\nA~differential form $q$-form $\\rho\\in\\Omega_{q}^{r}W$ satisfying\n$h\\rho=0$ is called \\emph{contact}, and $\\rho$ is generated by contact\n$1$-forms\n\\begin{equation*}\n\\omega_{j_{1}\\ldots j_{k}}^{\\sigma}=dy_{j_{1}\\ldots j_{k}}^{\\sigma}-y_{j_{1}\\ldots j_{k}s}^{\\sigma}dx^{s},\\qquad0\\leq k\\leq r-1\n\\end{equation*}\nThroughout, we use the standard geometric concepts: the exterior derivative\n$d$, the contraction $i_{\\Xi}\\rho$ and the Lie derivative $\\partial_{\\Xi}\\rho$\nof a differential form $\\rho$ with respect to a vector field $\\Xi$,\nand the pull-back operation $*$ acting on differential forms.\n\n\\section{Lepage equivalents in first- and second-order field theory}\n\nBy a~\\textit{Lagrangian} $\\lambda$ for a~fibered manifold $\\pi:Y\\rightarrow X$\nof order $r$ we mean an element of the submodule $\\Omega_{n,X}^{r}W$\nof $\\pi^{r}$-horizontal $n$-forms in the module of $n$-forms $\\Omega_{n}^{r}W$,\ndefined on an open subset $W^{r}$ of the $r$-th jet prolongation\n$J^{r}Y$. In a~fibered chart $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$,\nwhere $V\\subset W$, Lagrangian $\\lambda\\in\\Omega_{n,X}^{r}W$ has\nan expression\n\\begin{equation}\n\\lambda=\\mathscr{L}\\omega_{0},\\label{eq:Lagrangian}\n\\end{equation}\nwhere $\\omega_{0}=dx^{1}\\wedge dx^{2}\\wedge\\ldots\\wedge dx^{n}$ is\nthe (local) volume element, and $\\mathscr{L}:V^{r}\\rightarrow\\mathbb{R}$\nis the \\textit{Lagrange function} associated to $\\lambda$ and $(V,\\psi)$.\n\nAn $n$-form $\\rho\\in\\Omega_{n}^{s}W$ is called a\\textit{~Lepage\nequivalent} of $\\lambda\\in\\Omega_{n,X}^{r}W$, if the following two\nconditions are satisfied: \n\n(i) $(\\pi^{q,s+1})^{*}h\\rho=(\\pi^{q,r})^{*}\\lambda$ (i.e. $\\rho$\nis \\textit{equivalent }with $\\lambda$), and \n\n(ii) $hi_{\\xi}d\\rho=0$ for arbitrary $\\pi^{s,0}$-vertical vector\nfield $\\xi$ on $W^{s}$ (i.e. $\\rho$ is a\\textit{~Lepage form}). \n\nThe following theorem describe the structure of the Lepage equivalent\nof a~Lagrangian (see \\cite{Krupka-Lepage,Handbook}).\n\\begin{thm}\n\\label{Thm:LepageEquiv}Let $\\lambda\\in\\Omega_{n,X}^{r}W$ be a~Lagrangian\nof order $r$ for $Y$, locally expressed by \\eqref{eq:Lagrangian}\nwith respect to a~fibered chart $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$.\nAn $n$-form $\\rho\\in\\Omega_{n}^{s}W$ is a~Lepage equivalent of\n$\\lambda$ if and only if\n\\begin{equation}\n(\\pi^{s+1,s})^{*}\\rho=\\Theta_{\\lambda}+d\\mu+\\eta,\\label{eq:LepDecomp}\n\\end{equation}\nwhere $n$-form $\\Theta_{\\lambda}$ is defined on $V^{2r-1}$ by \\emph{\\eqref{eq:PoiCar},}\n$\\mu$ is a~contact $(n-1)$-form, and an $n$-form $\\eta$ has the\norder of contactness $\\geq2$.\n\\end{thm}\n\n$\\Theta_{\\lambda}$ is called the \\emph{principal component} of the\nLepage form $\\rho$ with respect to fibered chart $(V,\\psi)$. In\ngeneral, decomposition \\eqref{eq:LepDecomp} is \\emph{not} uniquely\ndetermined with respect to contact forms $\\mu$, $\\eta$, and the\nprincipal component $\\Theta_{\\lambda}$ need \\emph{not} define a~global\nform on $W^{2r-1}$. Nevertheless, the Lepage equivalent $\\rho$ satisfying\n\\eqref{eq:LepDecomp} is globally defined on $W^{s}$; moreover $E_{\\lambda}=p_{1}d\\rho$\nis a~globally defined $(n+1)$-form on $W^{2r}$, called the \\emph{Euler\\textendash Lagrange\nform} associated to $\\lambda$.\n\nWe recall the known examples of Lepage equivalents of first- and second-order\nLagrangians, determined by means of additional requirements.\n\\begin{lem}\n\\textbf{\\textup{\\label{Lem:PrincipalLepEq}(Principal Lepage form)}}\n\\emph{(a)} For every Lagrangian $\\lambda$ of order $r=1$, there\nexists a unique Lepage equivalent $\\Theta_{\\lambda}$ of $\\lambda$\non $W^{1}$, which is $\\pi^{1,0}$-horizontal and has the order of\ncontactness $\\leq1$. In a fibered chart $(V,\\psi)$, $\\Theta{}_{\\lambda}$\nhas an expression \n\\begin{equation}\n\\Theta_{\\lambda}=\\mathscr{L}\\omega_{0}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega^{\\sigma}\\wedge\\omega_{j}.\\label{eq:Poincare-Cartan}\n\\end{equation}\n\n\\emph{(b)} For every Lagrangian $\\lambda$ of order $r=2$, there\nexists a unique Lepage equivalent $\\Theta_{\\lambda}$ of $\\lambda$\non $W^{3}$, which is $\\pi^{3,1}$-horizontal and has the order of\ncontactness $\\leq1$. In a fibered chart $(V,\\psi)$, $\\Theta{}_{\\lambda}$\nhas an expression \n\n\\begin{equation}\n\\Theta_{\\lambda}=\\mathscr{L}\\omega_{0}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}\\right)\\omega^{\\sigma}\\wedge\\omega_{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\wedge\\omega_{j}.\\label{eq:Poincare-Cartan-2ndOrder}\n\\end{equation}\n\\end{lem}\n\nFor $r=1$ and $r=2$, the principal component $\\Theta_{\\lambda}$\n\\eqref{eq:PoiCar} is a~\\emph{globally defined} Lepage equivalent\nof $\\lambda$. We point out that for $r\\geq3$ this is \\emph{not}\ntrue (see \\cite{HorakKolar,Krupka-Lepage}). \\eqref{eq:Poincare-Cartan}\nis the well-known \\emph{Poincar\\'e-Cartan form} (cf. Garc\\'ia \\cite{Garcia}),\nand it is generealized for second-order Lagrangians by globally defined\n\\emph{principal Lepage equivalent} \\eqref{eq:Poincare-Cartan-2ndOrder}\non $W^{3}\\subset J^{3}Y$.\n\\begin{lem}\n\\textbf{\\textup{\\label{Lem:Fundamental}(Fundamental Lepage form)}}\nLet $\\lambda\\in\\Omega_{n,X}^{1}W$ be a~Lagrangian of order 1 for\n$Y$, locally expressed by \\eqref{eq:Lagrangian}. There exists a~unique\nLepage equivalent $Z_{\\lambda}\\in\\Omega_{n}^{1}W$ of $\\lambda$,\nwhich satisfies $Z_{h\\rho}=(\\pi^{1,0})^{*}\\rho$ for any $n$-form\n$\\rho\\in\\Omega_{n}^{0}W$ on $W$ such that $h\\rho=\\lambda$. With\nrespect to a fibered chart $(V,\\psi)$, $Z_{\\lambda}$ has an expression\n\\begin{align}\nZ_{\\lambda} & =\\mathscr{L}\\omega_{0}+\\sum_{k=1}^{n}\\frac{1}{(n-k)!}\\frac{1}{(k!)^{2}}\\frac{\\partial^{k}\\mathscr{L}}{\\partial y_{j_{1}}^{\\sigma_{1}}\\ldots\\partial y_{j_{k}}^{\\sigma_{k}}}\\varepsilon_{j_{1}\\ldots j_{k}i_{k+1}\\ldots i_{n}}\\label{eq:Fundamental}\\\\\n & \\quad\\cdot\\omega^{\\sigma_{1}}\\land\\ldots\\wedge\\omega^{\\sigma_{k}}\\wedge dx^{i_{k+1}}\\wedge\\ldots\\wedge dx^{i_{n}}.\\nonumber \n\\end{align}\n\\end{lem}\n\n$Z_{\\lambda}$ \\eqref{eq:Fundamental} is known as the \\emph{fundamental\nLepage form} \\cite{Krupka-Fund.Lep.eq.}, \\cite{Betounes}), and it\nis characterized by the equivalence: $Z_{\\lambda}$ is closed if and\nonly if $\\lambda$ is trivial (i.e. the Euler\\textendash Lagrange\nexpressions associated with $\\lambda$ vanish identically). Recently,\nthe form \\eqref{eq:Fundamental} was studied for variational problems\nfor submanifolds in \\cite{UrbBra}, and applied for studying symmetries\nand conservation laws in \\cite{Javier}.\n\\begin{lem}\n\\textbf{\\textup{\\label{Lem:Caratheodory}(Carath\\'eodory form) }}Let\n$\\lambda\\in\\Omega_{n,X}^{1}W$ be a~non-vanishing Lagrangian of order\n1 for $Y$ \\eqref{eq:Lagrangian}. Then a~differential $n$-form\n$\\Lambda_{\\lambda}\\in\\Omega_{n}^{1}W$, locally expressed as\n\\begin{align}\n\\Lambda_{\\lambda} & =\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega^{\\sigma}\\right)\\label{eq:CaratheodoryForm}\\\\\n & =\\frac{1}{\\mathscr{L}^{n-1}}\\left(\\mathscr{L}dx^{1}+\\frac{\\partial\\mathscr{L}}{\\partial y_{1}^{\\sigma_{1}}}\\omega^{\\sigma_{1}}\\right)\\wedge\\ldots\\wedge\\left(\\mathscr{L}dx^{n}+\\frac{\\partial\\mathscr{L}}{\\partial y_{n}^{\\sigma_{n}}}\\omega^{\\sigma_{n}}\\right),\\nonumber \n\\end{align}\nis a Lepage equivalent of $\\lambda$.\n\\end{lem}\n\n$\\Lambda_{\\lambda}$ \\eqref{eq:Fundamental} is the well-known \\emph{Carath\\'eodory\nform} (cf. \\cite{Caratheodory}), associated to Lagrangian $\\lambda\\in\\Omega_{n,X}^{1}W$,\nwhich is nowhere zero. $\\Lambda_{\\lambda}$ is uniquely characterized\nby the following properties: $\\Lambda_{\\lambda}$ is (i) a~Lepage\nequivalent of $\\lambda$, (ii) decomposable, (iii) $\\pi^{1,0}$-horizontal\n(i.e. semi-basic with respect to projection $\\pi^{1,0}$).\n\n\\section{The Carath\\'eodory form: second-order generalization}\n\nLet $\\lambda\\in\\Omega_{n,X}^{1}W$ be a~\\emph{non-vanishing,} \\emph{first-order}\nLagrangian on $W^{1}\\subset J^{1}Y$. In the next lemma, we describe\na~new observation, showing that the Carath\\'eodory form $\\Lambda_{\\lambda}$\n\\eqref{eq:CaratheodoryForm} arises from the Poincar\\'e-Cartan form\n$\\Theta_{\\lambda}$ \\eqref{eq:Poincare-Cartan} by means of contraction\noperations on differential forms with respect to the formal derivative\nvector fields $d_{i}$ \\eqref{eq:FormalDerivative}.\n\\begin{lem}\n\\label{lem:Car-PC}The Carath\\'eodory form $\\Lambda_{\\lambda}$ \\eqref{eq:CaratheodoryForm}\nand the Poincar\\'e-Cartan form $\\Theta_{\\lambda}$ \\eqref{eq:Poincare-Cartan}\nsatisfy\n\\begin{align*}\n\\Lambda_{\\lambda} & =\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nFrom the decomposable structure of $\\Lambda_{\\lambda}$, we see that\nwhat is needed to show is the formula\n\\begin{equation*}\ni_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}=(-1)^{n-j}\\left(\\mathscr{L}dx^{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega^{\\sigma}\\right\n\\end{equation*}\nfor every $j$, $1\\leq j\\leq n$. Since $dx^{k}\\wedge\\omega_{j}=\\delta_{j}^{k}\\omega_{0}$,\nthe Poincar\\'e-Cartan form is expressible as\n\\begin{align*}\n\\Theta_{\\lambda} & =\\mathscr{L}\\omega_{0}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega^{\\sigma}\\wedge\\omega_{j}=\\mathscr{L}\\omega_{0}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}dy^{\\sigma}\\wedge\\omega_{j}-\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}y_{k}^{\\sigma}dx^{k}\\wedge\\omega_{j}\\\\\n & =\\left(\\mathscr{L}-\\frac{\\partial\\mathscr{L}}{\\partial y_{1}^{\\sigma}}y_{1}^{\\sigma}-\\frac{\\partial\\mathscr{L}}{\\partial y_{2}^{\\sigma}}y_{2}^{\\sigma}-\\ldots-\\frac{\\partial\\mathscr{L}}{\\partial y_{n}^{\\sigma}}y_{n}^{\\sigma}\\right)\\omega_{0}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}dy^{\\sigma}\\wedge\\omega_{j}.\n\\end{align*}\nApplying the contraction operations to $\\Theta_{\\lambda}$, we obtain\nby means of a straightforward computation for every $j$,\n\\begin{align*}\n & i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}=\\left(\\mathscr{L}-\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}y_{j}^{\\sigma}-\\ldots-\\frac{\\partial\\mathscr{L}}{\\partial y_{n}^{\\sigma}}y_{n}^{\\sigma}\\right)dx^{j}\\wedge\\ldots\\wedge dx^{n}\\\\\n & \\quad\\quad+(-1)^{j-1}\\sum_{k=j}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}dy^{\\sigma}\\wedge i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad\\quad+\\sum_{l=1}^{j-1}(-1)^{l-1}y_{l}^{\\sigma}i_{d_{j-1}}\\ldots i_{d_{l+1}}i_{d_{l-1}}\\ldots i_{d_{1}}\\left(\\sum_{k=j}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}\\omega_{k}\\right),\n\\end{align*}\nand\n\\begin{align*}\n & i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\\\\\n & \\quad=-\\left(\\mathscr{L}-\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}y_{j}^{\\sigma}-\\sum_{k=j+2}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}y_{k}^{\\sigma}\\right)dx^{j}\\wedge dx^{j+2}\\wedge\\ldots\\wedge dx^{n}\\\\\n & \\quad+(-1)^{j}\\sum_{k=j}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}dy^{\\sigma}\\wedge i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad+\\sum_{l=1}^{j-1}(-1)^{l-1}y_{l}^{\\sigma}i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{l+1}}i_{d_{l-1}}\\ldots i_{d_{1}}\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega_{j}+\\sum_{k=j+2}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}\\omega_{k}\\right)\\\\\n & \\quad+(-1)^{j-1}y_{j+1}^{\\sigma}i_{d_{j-1}}\\ldots i_{d_{1}}\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega_{j}+\\sum_{k=j+2}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}\\omega_{k}\\right).\n\\end{align*}\nFollowing the inductive structure of the preceding expressions, we\nget after the next $n-j-1$ steps,\n\\begin{align*}\n & i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\\\\\n & \\quad=(-1)^{n-j}\\left(\\mathscr{L}-\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}y_{j}^{\\sigma}\\right)dx^{j}+(-1)^{n-j}\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}dy^{\\sigma}-(-1)^{n-j}\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\sum_{k\\neq j}y_{k}^{\\sigma}dx^{k}\\\\\n & \\quad=(-1)^{n-j}\\left(\\mathscr{L}dx^{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}\\omega^{\\sigma}\\right),\n\\end{align*}\nas required.\n\\end{proof}\nAn intrinsic nature of Lemma \\ref{lem:Car-PC} indicates a~possible\nextension of the Carath\\'eodory form \\eqref{eq:CaratheodoryForm}\nfor higher-order variational problems. We put\n\\begin{equation}\n\\Lambda_{\\lambda}=\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda},\\label{eq:2nd-Caratheodory}\n\\end{equation}\nwhere $\\Theta_{\\lambda}$ in \\eqref{eq:2nd-Caratheodory} denotes\nthe principal Lepage equivalent \\eqref{eq:Poincare-Cartan-2ndOrder}\nof a~\\emph{second-order} Lagrangian $\\lambda$, and verify that formula\n\\eqref{eq:2nd-Caratheodory} defines a~global form.\n\\begin{thm}\n\\label{thm:Main}Let $\\lambda\\in\\Omega_{n,X}^{2}W$ be a~non-vanishing\nsecond-order Lagrangian on $W^{2}\\subset J^{2}Y$. Then $\\Lambda_{\\lambda}$\nsatisfies:\n\n\\emph{(a)} Formula \\eqref{eq:2nd-Caratheodory} defines an $n$-form\non $W^{3}\\subset J^{3}Y$. \n\n\\emph{(b)} If $\\lambda\\in\\Omega_{n,X}^{2}W$ has an expression \\eqref{eq:Lagrangian}\nwith respect to a~fibered chart $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$,\nsuch that $V\\subset W$, then $\\Lambda_{\\lambda}$ \\eqref{eq:2nd-Caratheodory}\nis expressed by\n\\begin{align}\n\\Lambda_{\\lambda} & =\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right).\\label{eq:2ndCaratheodoryExpression}\n\\end{align}\n\n\\emph{(c)} $\\Lambda_{\\lambda}\\in\\Omega_{n}^{3}W$ \\eqref{eq:2nd-Caratheodory}\nassociated to a~second-order Lagrangian $\\lambda\\in\\Omega_{n,X}^{2}W$\nis a~Lepage equivalent of $\\lambda$, which is decomposable and $\\pi^{3,1}$-horizontal.\n\\end{thm}\n\n\\begin{proof}\n1. Suppose $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$, and $(\\bar{V},\\bar{\\psi})$,\n$\\bar{\\psi}=(\\bar{x}^{i},\\bar{y}^{\\sigma})$, are two overlapping\nfibered charts on $W$. For $\\lambda\\in\\Omega_{n,X}^{2}W$, the corresponding\nchart expressions $\\lambda=\\mathscr{L}\\omega_{0}$ and $\\lambda=\\bar{\\mathscr{L}}\\bar{\\omega}_{0}$\nsatisfy\n\\begin{equation}\n\\mathscr{L}=\\left(\\bar{\\mathscr{L}}\\circ\\bar{\\psi}^{-1}\\circ\\psi\\right)\\det\\frac{\\partial\\bar{x}^{i}}{\\partial x^{j}}.\\label{eq:LagrangianTransform}\n\\end{equation}\nSince the push-forward vector field $\\bar{d}_{k}$,\n\\[\n\\bar{d}_{k}=\\frac{\\partial}{\\partial\\bar{x}^{k}}+\\bar{y}_{k}^{\\sigma}\\frac{\\partial}{\\partial\\bar{y}^{\\sigma}}+\\bar{y}_{kl}^{\\sigma}\\frac{\\partial}{\\partial\\bar{y}_{l}^{\\sigma}},\n\\]\nof vector field $(\\partial x^{i}\/\\partial\\bar{x}^{k})d_{i}$ with\nrespect to the chart transformation $\\bar{\\psi}^{-1}\\circ\\psi$ satisfies\n\\[\n(\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\left(i_{\\bar{d}_{k}}\\Theta_{\\lambda}\\right)=i_{\\frac{\\partial x^{i}}{\\partial\\bar{x}^{k}}d_{i}}\\left((\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\right)=\\frac{\\partial x^{i}}{\\partial\\bar{x}^{k}}i_{d_{i}}\\left((\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\right),\n\\]\nand $\\Theta_{\\lambda}$ \\eqref{eq:Poincare-Cartan-2ndOrder} is globally\ndefined, we get\n\\begin{align*}\n & (\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\frac{1}{\\mathscr{\\bar{L}}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{\\bar{d}_{n}}\\ldots i_{\\bar{d}_{j+1}}i_{\\bar{d}_{j-1}}\\ldots i_{\\bar{d}_{1}}\\Theta_{\\lambda}\\\\\n & \\quad=\\frac{1}{\\left(\\bar{\\mathscr{L}}\\circ\\bar{\\psi}^{-1}\\circ\\psi\\right)^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}\\frac{\\partial x^{i_{1}}}{\\partial\\bar{x}^{1}}\\ldots\\frac{\\partial x^{i_{j-1}}}{\\partial\\bar{x}^{j-1}}\\frac{\\partial x^{i_{j+1}}}{\\partial\\bar{x}^{j+1}}\\ldots\\frac{\\partial x^{i_{n}}}{\\partial\\bar{x}^{n}}\\\\\n & \\quad\\quad\\cdot i_{d_{i_{n}}}\\ldots i_{d_{i_{j+1}}}i_{d_{i_{j-1}}}\\ldots i_{d_{i_{1}}}\\left((\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\right)\\\\\n & \\quad=\\frac{1}{\\left(\\bar{\\mathscr{L}}\\circ\\bar{\\psi}^{-1}\\circ\\psi\\right)^{n-1}}\\left(\\frac{\\partial x^{i_{1}}}{\\partial\\bar{x}^{1}}\\ldots\\frac{\\partial x^{i_{n}}}{\\partial\\bar{x}^{n}}\\varepsilon_{i_{1}\\ldots i_{n}}\\right)^{n}\\\\\n & \\quad\\quad\\cdot\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}(\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\\\\n & \\quad=\\frac{1}{\\left((\\bar{\\mathscr{L}}\\circ\\bar{\\psi}^{-1}\\circ\\psi)\\det\\frac{\\partial\\bar{x}^{i}}{\\partial x^{j}}\\right)^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}(\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\\\\n & \\quad=\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}(\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\Theta_{\\lambda}\\\\\n & \\quad=\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda},\n\\end{align*}\nas required.\n\n2. Analogously to the proof of Lemma \\ref{lem:Car-PC}, we find a~chart\nexpression of $1$-form\n\\[\ni_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda},\n\\]\nwhere $\\Theta_{\\lambda}$ is the principal Lepage equivalent \\eqref{eq:Poincare-Cartan-2ndOrder}.\nUsing $dx^{k}\\wedge\\omega_{j}=\\delta_{j}^{k}\\omega_{0}$, we have\n\\begin{align*}\n\\Theta_{\\lambda} & =\\left(\\mathscr{L}-\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}\\right)y_{j}^{\\sigma}-\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}y_{ij}^{\\sigma}\\right)\\omega_{0}\\\\\n & +\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}\\right)dy^{\\sigma}\\wedge\\omega_{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}dy_{i}^{\\sigma}\\wedge\\omega_{j}.\n\\end{align*}\nThen\n\\begin{align*}\n & i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\\\\\n & \\quad=\\left(\\mathscr{L}-\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{k}^{\\sigma}-\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{ik}^{\\sigma}\\right)dx^{j}\\wedge\\ldots\\wedge dx^{n}\\\\\n & \\quad+\\sum_{l=1}^{j-1}(-1)^{l-1}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{l}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{il}^{\\sigma}\\right)i_{d_{j-1}}\\ldots i_{d_{l+1}}i_{d_{l-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad+(-1)^{j-1}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)dy^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}dy_{i}^{\\sigma}\\right)\\wedge\\left(i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\right)\\\\\n & \\quad=\\left(\\mathscr{L}-\\sum_{k=j}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{k}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{ik}^{\\sigma}\\right)\\right)dx^{j}\\wedge\\ldots\\wedge dx^{n}\\\\\n & \\quad+\\sum_{l=1}^{j-1}(-1)^{l-1}\\sum_{k=j}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{l}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{il}^{\\sigma}\\right)i_{d_{j-1}}\\ldots i_{d_{l+1}}i_{d_{l-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad+(-1)^{j-1}\\sum_{k=j}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)dy^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}dy_{i}^{\\sigma}\\right)\\wedge\\left(i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\right),\n\\end{align*}\nand\n\\begin{align*}\n & i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\\\\\n & \\quad=\\Biggl(-\\mathscr{L}+\\sum_{\\begin{array}{c}\nk=j\\\\\nk\\neq j+1\n\\end{array}}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{k}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{ik}^{\\sigma}\\right)\\Biggr)dx^{j}\\wedge\\bigwedge_{l=j+2}^{n}dx^{l}\\\\\n & \\quad+\\sum_{l=1}^{j-1}(-1)^{l-1}\\sum_{\\begin{array}{c}\nk=j\\\\\nk\\neq j+1\n\\end{array}}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{l}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{il}^{\\sigma}\\right)\\\\\n & \\quad\\quad i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{l+1}}i_{d_{l-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad+(-1)^{j-1}\\sum_{\\begin{array}{c}\nk=j\\\\\nk\\neq j+1\n\\end{array}}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)y_{j+1}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}y_{i,j+1}^{\\sigma}\\right)i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\\\\n & \\quad+(-1)^{j}\\sum_{k=j}^{n}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pk}^{\\sigma}}\\right)dy^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}dy_{i}^{\\sigma}\\right)\\wedge\\left(i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\omega_{k}\\right).\n\\end{align*}\nAfter another $n-j-1$ steps we obtain\n\\begin{align*}\n & i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda}\\\\\n & =(-1)^{n-j}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right).\n\\end{align*}\n\n3. From \\eqref{eq:2ndCaratheodoryExpression} it is evident that $\\Lambda_{\\lambda}$\n\\eqref{eq:2nd-Caratheodory} is decomposable, $\\pi^{3,1}$-horizontal,\nand obeys $h\\Lambda_{\\lambda}=\\lambda$. It is sufficient to verify\nthat $\\Lambda_{\\lambda}$ is a~Lepage form, that is $hi_{\\xi}d\\Lambda_{\\lambda}=0$\nfor arbitrary $\\pi^{3,0}$-vertical vector field $\\xi$ on $W^{3}\\subset J^{3}Y$.\nThis follows, however, by means of a~straightforward computation\nusing chart expression \\eqref{eq:2ndCaratheodoryExpression}. Indeed,\nwe have\n\\begin{align*}\n & d\\Lambda_{\\lambda}=(1-n)\\frac{1}{\\mathscr{L}^{n}}d\\mathscr{L}\\wedge\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad+\\frac{1}{\\mathscr{L}^{n-1}}\\sum_{k=1}^{n}(-1)^{k-1}d\\left(\\mathscr{L}dx^{k}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad\\wedge\\bigwedge_{j\\neq k}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right),\n\\end{align*}\nand the contraction of $d\\Lambda_{\\lambda}$ with respect to $\\pi^{3,0}$-vertical\nvector field $\\xi$ reads\n\\begin{align*}\n & i_{\\xi}d\\Lambda_{\\lambda}=(1-n)\\frac{1}{\\mathscr{L}^{n}}i_{\\xi}d\\mathscr{L}\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad-(1-n)\\frac{1}{\\mathscr{L}^{n}}d\\mathscr{L}\\wedge\\sum_{l=1}^{n}(-1)^{l-1}\\frac{\\partial\\mathscr{L}}{\\partial y_{jl}^{\\sigma}}\\xi_{j}^{\\sigma}\\\\\n & \\quad\\quad\\bigwedge_{j\\neq l}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad+\\frac{1}{\\mathscr{L}^{n-1}}\\sum_{k=1}^{n}(-1)^{k-1}\\left(i_{\\xi}d\\mathscr{L}dx^{k}+i_{\\xi}d\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\omega^{\\sigma}\\right.\\\\\n & \\quad\\left.-\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\xi_{j}^{\\sigma}dx^{j}+i_{\\xi}d\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\omega_{i}^{\\sigma}-\\xi_{i}^{\\sigma}d\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}-\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\xi_{is}^{\\sigma}dx^{s}\\right)\\\\\n & \\quad\\quad\\wedge\\bigwedge_{j\\neq k}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad+\\frac{1}{\\mathscr{L}^{n-1}}\\sum_{k=1}^{n}(-1)^{k-1}d\\left(\\mathscr{L}dx^{k}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad\\quad\\wedge\\sum_{l<k}(-1)^{l-1}\\frac{\\partial\\mathscr{L}}{\\partial y_{il}^{\\sigma}}\\xi_{i}^{\\sigma}\\bigwedge_{j\\neq k,l}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\n\\end{align*}\n\\begin{align*}\n & \\quad+\\frac{1}{\\mathscr{L}^{n-1}}\\sum_{k=1}^{n}(-1)^{k-1}d\\left(\\mathscr{L}dx^{k}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\\\\n & \\quad\\quad\\wedge\\sum_{l>k}(-1)^{l}\\frac{\\partial\\mathscr{L}}{\\partial y_{il}^{\\sigma}}\\xi_{i}^{\\sigma}\\bigwedge_{j\\neq k,l}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right).\n\\end{align*}\nHence the horizontal part of $i_{\\xi}d\\Lambda_{\\lambda}$ satisfies\n\\begin{align*}\nhi_{\\xi}d\\Lambda_{\\lambda} & =(1-n)i_{\\xi}d\\mathscr{L}\\omega_{0}-(1-n)\\frac{1}{\\mathscr{L}}\\sum_{l=1}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{jl}^{\\sigma}}\\xi_{j}^{\\sigma}d_{l}\\mathscr{L}\\omega_{0}+ni_{\\xi}d\\mathscr{L}\\omega_{0}\\\\\n & -\\sum_{k=1}^{n}\\left(\\xi_{i}^{\\sigma}d_{k}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\xi_{ik}^{\\sigma}\\right)\\omega_{0}-\\sum_{k=1}^{n}\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{k}^{\\sigma}}-d_{i}\\frac{\\partial\\mathscr{L}}{\\partial y_{ik}^{\\sigma}}\\right)\\xi_{k}^{\\sigma}\\omega_{0}\\\\\n & +\\frac{1}{\\mathscr{L}}\\sum_{k=1}^{n}(-1)^{k-1}d_{s}\\mathscr{L}dx^{s}\\wedge dx^{k}\\wedge\\sum_{l<k}(-1)^{l-1}\\frac{\\partial\\mathscr{L}}{\\partial y_{il}^{\\sigma}}\\xi_{i}^{\\sigma}\\bigwedge_{j\\neq k,l}dx^{j}\\\\\n & +\\frac{1}{\\mathscr{L}}\\sum_{k=1}^{n}(-1)^{k-1}d_{s}\\mathscr{L}dx^{s}\\wedge dx^{k}\\wedge\\sum_{l>k}(-1)^{l}\\frac{\\partial\\mathscr{L}}{\\partial y_{il}^{\\sigma}}\\xi_{i}^{\\sigma}\\bigwedge_{j\\neq k,l}dx^{j}\\\\\n & =-(1-n)\\frac{1}{\\mathscr{L}}\\sum_{l=1}^{n}\\frac{\\partial\\mathscr{L}}{\\partial y_{jl}^{\\sigma}}\\xi_{j}^{\\sigma}d_{l}\\mathscr{L}\\omega_{0}-\\frac{1}{\\mathscr{L}}\\sum_{k=1}^{n}\\sum_{l\\neq k}\\frac{\\partial\\mathscr{L}}{\\partial y_{il}^{\\sigma}}\\xi_{i}^{\\sigma}d_{s}\\mathscr{L}dx^{s}\\wedge\\omega_{l}\\\\\n & =0,\n\\end{align*}\nwhere the identity $dx^{k}\\wedge\\omega_{l}=\\delta_{l}^{k}\\omega_{0}$\nis applied.\n\\end{proof}\nLepage equivalent $\\Lambda_{\\lambda}$ \\eqref{eq:2nd-Caratheodory}\nis said to be the \\emph{Carath\\'eodory form} associated to $\\lambda\\in\\Omega_{n,X}^{2}W$.\n\n\\section{The Carath\\'{e}odory form and principal Lepage equivalents in higher-order\ntheory}\n\nWe point out that in the proof of Theorem \\ref{thm:Main}, (a), the\nchart independence of formula \\eqref{eq:2nd-Caratheodory} is based\non principal Lepage equivalent $\\Theta_{\\lambda}$ \\eqref{eq:Poincare-Cartan-2ndOrder}\nof a~second-order Lagrangian, which is defined \\emph{globally}. Since\nfor a Lagrangian of order $r\\geq3$, principal components of Lepage\nequivalents are, in general, \\emph{local} expressions (see the Introduction),\nwe are allowed to apply the definition \\eqref{eq:2nd-Caratheodory}\nfor such class of Lagrangians of order $r$ over a~fibered manifold\nwhich assure invariance of local expressions $\\Theta_{\\lambda}$ \\eqref{eq:PoiCar}.\n\nConsider now a~\\emph{third-order} Lagrangian $\\lambda\\in\\Omega_{n,X}^{3}W$.\nThen the principal component $\\Theta_{\\lambda}$ of a~Lepage equivalent\nof $\\lambda$ reads\n\\begin{align}\n\\Theta_{\\lambda} & =\\mathscr{L}\\omega_{0}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}+d_{p}d_{q}\\frac{\\partial\\mathscr{L}}{\\partial y_{pqj}^{\\sigma}}\\right)\\omega^{\\sigma}\\wedge\\omega_{j}\\label{eq:PrinComp3}\\\\\n & +\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{kj}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{kpj}^{\\sigma}}\\right)\\omega_{k}^{\\sigma}\\wedge\\omega_{j}+\\frac{\\partial\\mathscr{L}}{\\partial y_{klj}^{\\sigma}}\\omega_{kl}^{\\sigma}\\wedge\\omega_{j}.\\nonumber \n\\end{align}\nIn the following lemma we describe conditions for invariance of \\eqref{eq:PrinComp3}.\n\\begin{lem}\n\\label{lem:3rdCond}The following two conditions are equivalent:\n\n\\emph{(a)} $\\Theta_{\\lambda}$ satisfies\n\\[\n(\\bar{\\psi}^{-1}\\circ\\psi)^{*}\\bar{\\Theta}_{\\lambda}=\\Theta_{\\lambda}.\n\\]\n\\emph{(b)} For arbitrary two overlapping fibered charts on $Y$, $(V,\\psi)$,\n$\\psi=(x^{i},y^{\\sigma})$, and $(\\bar{V},\\bar{\\psi})$, $\\bar{\\psi}=(\\bar{x}^{i},\\bar{y}^{\\sigma})$,\n\\begin{equation}\nd_{k}\\left(\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{l_{1}l_{2}k}^{\\tau}}\\frac{\\partial x^{s}}{\\partial\\bar{x}^{p}}-\\frac{\\partial\\mathscr{L}}{\\partial y_{l_{1}l_{2}s}^{\\tau}}\\frac{\\partial x^{k}}{\\partial\\bar{x}^{p}}\\right)\\frac{\\partial^{2}\\bar{x}^{p}}{\\partial x^{l_{1}}\\partial x^{l_{2}}}\\frac{\\partial y^{\\tau}}{\\partial\\bar{y}^{\\sigma}}\\right)=0.\\label{eq:Obstruction-3rd}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nEquivalence conditions (a) and (b) follows from the chart transformation\n\\begin{align*}\n & \\bar{x}^{k}=\\bar{x}^{k}(x^{s}),\\quad\\bar{y}^{\\sigma}=\\bar{y}^{\\sigma}(x^{s},y^{\\nu}),\\\\\n & \\bar{y}_{k}^{\\sigma}=\\frac{d}{d\\bar{x}^{k}}(\\bar{y}^{\\sigma})=\\frac{\\partial x^{s}}{\\partial\\bar{x}^{k}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial x^{s}}+\\frac{\\partial x^{s}}{\\partial\\bar{x}^{k}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial y^{\\nu}}y_{s}^{\\nu},\\\\\n & \\bar{y}_{ij}^{\\sigma}=\\frac{d}{d\\bar{x}^{j}}(\\bar{y}_{i}^{\\sigma})=\\frac{\\partial x^{t}}{\\partial\\bar{x}^{j}}\\frac{d}{dx^{t}}\\left(\\frac{\\partial x^{s}}{\\partial\\bar{x}^{i}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial x^{s}}+\\frac{\\partial x^{s}}{\\partial\\bar{x}^{i}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial y^{\\nu}}y_{s}^{\\nu}\\right),\\\\\n & \\bar{y}_{ijk}^{\\sigma}=\\frac{d}{d\\bar{x}^{k}}(\\bar{y}_{ij}^{\\sigma})=\\frac{\\partial x^{r}}{\\partial\\bar{x}^{k}}\\frac{d}{dx^{r}}\\left(\\frac{\\partial x^{t}}{\\partial\\bar{x}^{j}}\\frac{d}{dx^{t}}\\left(\\frac{\\partial x^{s}}{\\partial\\bar{x}^{i}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial x^{s}}+\\frac{\\partial x^{s}}{\\partial\\bar{x}^{i}}\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial y^{\\nu}}y_{s}^{\\nu}\\right)\\right),\n\\end{align*}\napplied to $\\Theta_{\\lambda}$, where Lagrange function $\\mathscr{L}$\nis transformed by \\eqref{eq:LagrangianTransform} and the following\nidentities are employed,\n\\begin{align*}\n & \\bar{\\omega}_{j}=\\frac{\\partial x^{s}}{\\partial\\bar{x}^{j}}\\det\\left(\\frac{\\partial\\bar{x}^{p}}{\\partial x^{q}}\\right)\\omega_{s},\\\\\n & \\bar{\\omega}^{\\sigma}=\\frac{\\partial\\bar{y}^{\\sigma}}{\\partial y^{\\nu}}\\omega^{\\nu},\\quad\\bar{\\omega}_{i}^{\\sigma}=\\frac{\\partial\\bar{y}_{i}^{\\sigma}}{\\partial y^{\\nu}}\\omega^{\\nu}+\\frac{\\partial\\bar{y}_{i}^{\\sigma}}{\\partial y_{p}^{\\nu}}\\omega_{p}^{\\nu},\\quad\\bar{\\omega}_{ij}^{\\sigma}=\\frac{\\partial\\bar{y}_{ij}^{\\sigma}}{\\partial y^{\\nu}}\\omega^{\\nu}+\\frac{\\partial\\bar{y}_{ij}^{\\sigma}}{\\partial y_{p}^{\\nu}}\\omega_{p}^{\\nu}+\\frac{\\partial\\bar{y}_{ij}^{\\sigma}}{\\partial y_{pq}^{\\nu}}\\omega_{pq}^{\\nu}.\n\\end{align*}\n\\end{proof}\n\\begin{thm}\n\\label{thm:Main-3rd}Suppose that a~fibered manifold $\\pi:Y\\rightarrow X$\nand a~non-vanishing third-order Lagrangian $\\lambda\\in\\Omega_{n,X}^{3}W$\nsatisfy condition \\emph{\\eqref{eq:Obstruction-3rd}}. Then $\\Lambda_{\\lambda}$\n\\emph{\\eqref{eq:2nd-Caratheodory}}, where $\\Theta_{\\lambda}$ is\ngiven by \\emph{\\eqref{eq:PrinComp3},} defines a~differential $n$-form\non $W^{5}\\subset J^{5}Y$, which is a~Lepage equivalent of $\\lambda$,\ndecomposable and $\\pi^{5,2}$-horizontal. In a~fibered chart $(V,\\psi)$,\n$\\psi=(x^{i},y^{\\sigma})$, on W, $\\Lambda_{\\lambda}$ has an expression\n\n\\begin{align}\n\\Lambda_{\\lambda} & =\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{pj}^{\\sigma}}+d_{p}d_{q}\\frac{\\partial\\mathscr{L}}{\\partial y_{pqj}^{\\sigma}}\\right)\\omega^{\\sigma}\\right.\\label{eq:3rdCaratheodoryExpression}\\\\\n & \\qquad\\qquad\\qquad+\\left.\\left(\\frac{\\partial\\mathscr{L}}{\\partial y_{ij}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{L}}{\\partial y_{ipj}^{\\sigma}}\\right)\\omega_{i}^{\\sigma}+\\frac{\\partial\\mathscr{L}}{\\partial y_{ikj}^{\\sigma}}\\omega_{ik}^{\\sigma}\\right)\\nonumber \n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nThis is an immediate consequence of Lemma \\ref{lem:3rdCond} and the\nprocedure given by Lemma \\ref{lem:Car-PC} and Theorem \\ref{thm:Main}.\n\\end{proof}\n\\begin{rem}\nNote that according to Lemma \\ref{lem:3rdCond}, characterizing obstructions\nfor the principal component $\\Theta_{\\lambda}$ \\eqref{eq:PrinComp3}\nto be a~global form, the Carath\\'eodory form \\eqref{eq:3rdCaratheodoryExpression}\nis well-defined for third-order Lagrangians on fibered manifolds,\nwhich satisfy condition \\eqref{eq:Obstruction-3rd}. Trivial cases\nwhen \\eqref{eq:Obstruction-3rd} holds identically include namely\n(i) Lagrangians \\emph{independent} of variables $y_{ijk}^{\\sigma}$,\nand (ii) fibered manifolds with bases endowed by smooth structure\nwith \\emph{linear} chart transformations. An example of (ii) are fibered\nmanifolds over two-dimensional open \\emph{M\\\"{o}bius strip} (for\ndetails see \\cite{UV}).\n\\end{rem}\n\nSuppose that a~pair $(\\lambda,\\pi)$, where $\\pi:Y\\rightarrow X$\nis a~fibered manifold and $\\lambda\\in\\Omega_{n,X}^{r}W$ is a Lagrangian\non $W^{r}\\subset J^{r}Y$, induces \\emph{invariant} principal component\n$\\Theta_{\\lambda}$ \\eqref{eq:PoiCar} of a~Lepage equivalent of\n$\\lambda$, with respect to fibered chart transformations on $W$.\nWe call $n$-form $\\Lambda_{\\lambda}$ on $W^{2r-1}$,\n\\begin{equation}\n\\Lambda_{\\lambda}=\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}(-1)^{n-j}i_{d_{n}}\\ldots i_{d_{j+1}}i_{d_{j-1}}\\ldots i_{d_{1}}\\Theta_{\\lambda},\\label{eq:Caratheodory-rth}\n\\end{equation}\nwhere $\\Theta_{\\lambda}$ is given by \\eqref{eq:PoiCar}, the \\emph{Carath\\'eodory\nform} associated to Lagrangian $\\lambda\\in\\Omega_{n,X}^{r}W$.\n\nIn a~fibered chart $(V,\\psi)$, $\\psi=(x^{i},y^{\\sigma})$, on W,\n$\\Lambda_{\\lambda}$ has an expression\n\n\\begin{align}\n\\Lambda_{\\lambda} & =\\frac{1}{\\mathscr{L}^{n-1}}\\bigwedge_{j=1}^{n}\\left(\\mathscr{L}dx^{j}+\\sum_{s=0}^{r-1}\\left(\\sum_{k=0}^{r-1-s}(-1)^{k}d_{p_{1}}\\ldots d_{p_{k}}\\frac{\\partial\\mathscr{L}}{\\partial y_{i_{1}\\ldots i_{s}p_{1}\\ldots p_{k}j}^{\\sigma}}\\right)\\omega_{i_{1}\\ldots i_{s}}^{\\sigma}\\right).\\label{eq:HOCaratheodoryExpression}\n\\end{align}\n\n\n\\section{Example: The Carath\\'eodory equivalent of the Hilbert Lagrangian}\n\nConsider a~fibered manifold $\\mathrm{Met}X$ of metric fields over\n$n$-dimensional manifold $X$ (see \\cite{Volna} for geometry of\n$\\mathrm{Met}X$). In a~chart $(U,\\varphi)$, $\\varphi=(x^{i})$,\non $X$, section $g:X\\supset U\\rightarrow\\mathrm{Met}X$ is expressed\nby $g=g_{ij}dx^{i}\\otimes dx^{j}$, where $g_{ij}$ is symmetric and\nregular at every point $x\\in U$. An induced fibered chart on second\njet prolongation $J^{2}\\mathrm{Met}X$ reads $(V,\\psi)$, $\\psi=(x^{i},g_{jk},g_{jk,l},g_{jk,lm})$.\n\nThe \\emph{Hilbert Lagrangian }is an odd-base $n$-form defined on\n$J^{2}\\mathrm{Met}X$ by \n\\begin{equation}\n\\lambda=\\mathscr{R}\\omega_{0},\\label{eq:Hilbert}\n\\end{equation}\nwhere $\\mathscr{R}=R\\sqrt{|\\det(g_{ij})|}$, $R=R(g_{ij},g_{ij,k},g_{ij,kl})$\nis the \\emph{scalar curvature} on $J^{2}\\mathrm{Met}X$, and $\\mu=\\sqrt{|\\det(g_{ij})|}\\omega_{0}$\nis the \\emph{Riemann volume element}.\n\nThe principal Lepage equivalent of $\\lambda$ \\eqref{eq:Hilbert}\n(cf. formula \\eqref{eq:Poincare-Cartan-2ndOrder}), reads\n\\begin{equation}\n\\Theta_{\\lambda}=\\mathscr{R}\\omega_{0}+\\left(\\left(\\frac{\\partial\\mathscr{R}}{\\partial y_{j}^{\\sigma}}-d_{p}\\frac{\\partial\\mathscr{R}}{\\partial y_{pj}^{\\sigma}}\\right)\\omega^{\\sigma}+\\frac{\\partial\\mathscr{R}}{\\partial y_{ij}^{\\sigma}}\\omega_{i}^{\\sigma}\\right)\\wedge\\omega_{j},\\label{eq:FLE-Hilbert}\n\\end{equation}\nand it is a~globally defined $n$-form on $J^{1}\\mathrm{Met}X$.\n\\eqref{eq:FLE-Hilbert} was used for analysis of structure of Einstein\nequations as a~system of \\emph{first-order} partial differential\nequations (see \\cite{KruStep}). Another Lepage equivalent for a~second-order\nLagrangian in field theory which could be studied in this context\nis given by \\eqref{eq:2ndCaratheodoryExpression},\n\\begin{align*}\n & \\Lambda_{\\lambda}=\\frac{1}{\\mathscr{R}^{n-1}}\\bigwedge_{k=1}^{n}\\left(\\mathscr{R}dx^{k}+\\left(\\frac{\\partial\\mathscr{\\mathscr{R}}}{\\partial g_{ij,k}}-d_{l}\\frac{\\partial\\mathscr{\\mathscr{R}}}{\\partial g_{ij,kl}}\\right)\\omega_{ij}+\\frac{\\partial\\mathscr{\\mathscr{R}}}{\\partial g_{ij,kl}}\\omega_{ij,l}\\right),\n\\end{align*}\nwhere $\\omega_{ij}=dg_{ij}-g_{ij,s}dx^{s}$, $\\omega_{ij,l}=dg_{ij,l}-g_{ij,ls}dx^{s}$.\nUsing a~chart expression of the scalar curvature, we obtain \n\\begin{align*}\n & \\Lambda_{\\lambda}=\\frac{1}{\\mathscr{R}^{n-1}}\\bigwedge_{k=1}^{n}\\left(\\mathscr{R}dx^{k}+\\frac{1}{2}\\sqrt{g}\\left(g^{qp}g^{si}g^{jk}-2g^{sq}g^{pi}g^{jk}+g^{pi}g^{qj}g^{sk}\\right)g_{pq,s}\\omega_{ij}\\right.\\\\\n & \\qquad\\qquad\\qquad\\qquad+\\sqrt{g}\\left(g^{il}g^{kj}-g^{kl}g^{ji}\\right)\\omega_{ij,l}\\biggr).\n\\end{align*}\nThis is the Carath\\'eodory equivalent of the Hilbert Lagrangian.\n\n\\section*{Acknowledgements}\n\nThis work has been completed thanks to the financial support provided\nto the VSB-Technical University of Ostrava by the Czech Ministry of\nEducation, Youth and Sports from the budget for the conceptual development\nof science, research, and innovations for the year 2020, Project No.\nIP2300031.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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Leave 0.25~inches of space\nbefore the heading and 0.15~inches after the heading.\n\nSimilarly, subsection headings should be numbered, flush left, and set\nin 10~pt bold type with the content words capitalized. Leave\n0.2~inches of space before the heading and 0.13~inches afterward.\n\nFinally, subsubsection headings should be numbered, flush left, and\nset in 10~pt small caps with the content words capitalized. Leave\n0.18~inches of space before the heading and 0.1~inches after the\nheading.\n\nPlease use no more than three levels of headings.\n\n\\subsubsection{Paragraphs and Footnotes}\n\nWithin each section or subsection, you should further partition the\npaper into paragraphs. Do not indent the first line of a given\nparagraph, but insert a blank line between succeeding ones.\n\nYou can use footnotes\\footnote{Footnotes\nshould be complete sentences.} to provide readers with additional\ninformation about a topic without interrupting the flow of the paper.\nIndicate footnotes with a number in the text where the point is most\nrelevant. Place the footnote in 9~point type at the bottom of the\ncolumn in which it appears. Precede the first footnote in a column\nwith a horizontal rule of 0.8~inches.\\footnote{Multiple footnotes can\nappear in each column, in the same order as they appear in the text,\nbut spread them across columns and pages if possible.}\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{icml_numpapers}}\n\\caption{Historical locations and number of accepted papers for International\nMachine Learning Conferences (ICML 1993 -- ICML 2008) and International\nWorkshops on Machine Learning (ML 1988 -- ML 1992). At the time this figure was\nproduced, the number of accepted papers for ICML 2008 was unknown and instead\nestimated.}\n\\label{icml-historical}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\\subsection{Figures}\n\nYou may want to include figures in the paper to illustrate\nyour approach and results. Such artwork should be centered,\nlegible, and separated from the text. Lines should be dark and at\nleast 0.5~points thick for purposes of reproduction, and text should\nnot appear on a gray background.\n\nLabel all distinct components of each figure. If the figure takes the\nform of a graph, then give a name for each axis and include a legend\nthat briefly describes each curve. Do not include a title inside the\nfigure; instead, the caption should serve this function.\n\nNumber figures sequentially, placing the figure number and caption\n\\emph{after} the graphics, with at least 0.1~inches of space before\nthe caption and 0.1~inches after it, as in\nFigure~\\ref{icml-historical}. The figure caption should be set in\n9~point type and centered unless it runs two or more lines, in which\ncase it should be flush left. You may float figures to the top or\nbottom of a column, and you may set wide figures across both columns\n(use the environment \\texttt{figure*} in \\LaTeX). Always place\ntwo-column figures at the top or bottom of the page.\n\n\\subsection{Algorithms}\n\nIf you are using \\LaTeX, please use the ``algorithm'' and ``algorithmic''\nenvironments to format pseudocode. These require\nthe corresponding stylefiles, algorithm.sty and\nalgorithmic.sty, which are supplied with this package.\nAlgorithm~\\ref{alg:example} shows an example.\n\n\\begin{algorithm}[tb]\n   \\caption{Bubble Sort}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} data $x_i$, size $m$\n   \\REPEAT\n   \\STATE Initialize $noChange = true$.\n   \\FOR{$i=1$ {\\bfseries to} $m-1$}\n   \\IF{$x_i > x_{i+1}$}\n   \\STATE Swap $x_i$ and $x_{i+1}$\n   \\STATE $noChange = false$\n   \\ENDIF\n   \\ENDFOR\n   \\UNTIL{$noChange$ is $true$}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Tables}\n\nYou may also want to include tables that summarize material. Like\nfigures, these should be centered, legible, and numbered consecutively.\nHowever, place the title \\emph{above} the table with at least\n0.1~inches of space before the title and the same after it, as in\nTable~\\ref{sample-table}. The table title should be set in 9~point\ntype and centered unless it runs two or more lines, in which case it\nshould be flush left.\n\n\n\\begin{table}[t]\n\\caption{Classification accuracies for naive Bayes and flexible\nBayes on various data sets.}\n\\label{sample-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcccr}\n\\toprule\nData set & Naive & Flexible & Better? \\\\\n\\midrule\nBreast    & 95.9$\\pm$ 0.2& 96.7$\\pm$ 0.2& $\\surd$ \\\\\nCleveland & 83.3$\\pm$ 0.6& 80.0$\\pm$ 0.6& $\\times$\\\\\nGlass2    & 61.9$\\pm$ 1.4& 83.8$\\pm$ 0.7& $\\surd$ \\\\\nCredit    & 74.8$\\pm$ 0.5& 78.3$\\pm$ 0.6&         \\\\\nHorse     & 73.3$\\pm$ 0.9& 69.7$\\pm$ 1.0& $\\times$\\\\\nMeta      & 67.1$\\pm$ 0.6& 76.5$\\pm$ 0.5& $\\surd$ \\\\\nPima      & 75.1$\\pm$ 0.6& 73.9$\\pm$ 0.5&         \\\\\nVehicle   & 44.9$\\pm$ 0.6& 61.5$\\pm$ 0.4& $\\surd$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nTables contain textual material, whereas figures contain graphical material.\nSpecify the contents of each row and column in the table's topmost\nrow. Again, you may float tables to a column's top or bottom, and set\nwide tables across both columns. Place two-column tables at the\ntop or bottom of the page.\n\n\\subsection{Citations and References}\n\nPlease use APA reference format regardless of your formatter\nor word processor. If you rely on the \\LaTeX\\\/ bibliographic\nfacility, use \\texttt{natbib.sty} and \\texttt{icml2020.bst}\nincluded in the style-file package to obtain this format.\n\nCitations within the text should include the authors' last names and\nyear. If the authors' names are included in the sentence, place only\nthe year in parentheses, for example when referencing Arthur Samuel's\npioneering work \\yrcite{Samuel59}. Otherwise place the entire\nreference in parentheses with the authors and year separated by a\ncomma \\cite{Samuel59}. List multiple references separated by\nsemicolons \\cite{kearns89,Samuel59,mitchell80}. Use the `et~al.'\nconstruct only for citations with three or more authors or after\nlisting all authors to a publication in an earlier reference \\cite{MachineLearningI}.\n\nAuthors should cite their own work in the third person\nin the initial version of their paper submitted for blind review.\nPlease refer to Section~\\ref{author info} for detailed instructions on how to\ncite your own papers.\n\nUse an unnumbered first-level section heading for the references, and use a\nhanging indent style, with the first line of the reference flush against the\nleft margin and subsequent lines indented by 10 points. The references at the\nend of this document give examples for journal articles \\cite{Samuel59},\nconference publications \\cite{langley00}, book chapters \\cite{Newell81}, books\n\\cite{DudaHart2nd}, edited volumes \\cite{MachineLearningI}, technical reports\n\\cite{mitchell80}, and dissertations \\cite{kearns89}.\n\nAlphabetize references by the surnames of the first authors, with\nsingle author entries preceding multiple author entries. Order\nreferences for the same authors by year of publication, with the\nearliest first. Make sure that each reference includes all relevant\ninformation (e.g., page numbers).\n\nPlease put some effort into making references complete, presentable, and\nconsistent. If using bibtex, please protect capital letters of names and\nabbreviations in titles, for example, use \\{B\\}ayesian or \\{L\\}ipschitz\nin your .bib file.\n\n\\section*{Software and Data}\n\nIf a paper is accepted, we strongly encourage the publication of software and data with the\ncamera-ready version of the paper whenever appropriate. This can be\ndone by including a URL in the camera-ready copy. However, \\textbf{do not}\ninclude URLs that reveal your institution or identity in your\nsubmission for review. Instead, provide an anonymous URL or upload\nthe material as ``Supplementary Material'' into the CMT reviewing\nsystem. Note that reviewers are not required to look at this material\nwhen writing their review.\n\n\\section*{Acknowledgements}\n\n\\textbf{Do not} include acknowledgements in the initial version of\nthe paper submitted for blind review.\n\nIf a paper is accepted, the final camera-ready version can (and\nprobably should) include acknowledgements. In this case, please\nplace such acknowledgements in an unnumbered section at the\nend of the paper. Typically, this will include thanks to reviewers\nwho gave useful comments, to colleagues who contributed to the ideas,\nand to funding agencies and corporate sponsors that provided financial\nsupport.\n\n\n\\nocite{langley00}\n\n\n\\section{Introduction}\n    With the availability of large collections of unlabeled data, recent work has led to significant advances in self-supervised learning. In particular, contrastive methods have been tremendously successful in learning representations for visual and sequential data \\citep{logeswaran2018efficient,wu2018unsupervised,oord2018representation,henaff2020data,tian2019contrastive,hjelm2018learning,bachman2019learning,he2019momentum,chen2020simple,schneider2019wav2vec,Baevski2020vqwav2vec,baevski2020wav2vec,ravanelli2020multi}.     %\n    While a number of explanations have been provided as to why contrastive learning leads to such informative representations, existing theoretical predictions and empirical observations appear to be at odds with each other~\\citep{tian2019contrastive,bachman2019learning,wu2020importance,saunshi2019theoretical}. \n    \n    In a nutshell, contrastive methods aim to learn representations where related samples are aligned (positive pairs, e.g. augmentations of the same image), while unrelated samples are separated (negative pairs)~\\citep{chen2020simple}.\n    Intuitively, this leads to invariance to irrelevant details or transformations (by decreasing the distance between positive pairs), while preserving a sufficient amount of information about the input for solving downstream tasks (by increasing the distance between negative pairs)~\\citep{tian2020makes}.\n    This intuition has recently been made more precise by \\cite{wang2020understanding}, showing that a commonly used contrastive loss from the InfoNCE family~\\citep{Gutmann12JMLR, oord2018representation, chen2020simple} asymptotically converges to a sum of two losses: an \\emph{alignment} loss that pulls together the representations of positive pairs, and a \\emph{uniformity} loss that maximizes the entropy of the learned latent distribution.\n    \n    \\begin{figure*}\n        \\centering\n        \\includegraphics[width=0.8\\textwidth]{figures\/figure_1_reduced.pdf}\n        \\caption{We analyze the setup of contrastive learning, in which a feature encoder $f$ is trained with the InfoNCE objective \\citep{Gutmann12JMLR, oord2018representation, chen2020simple} using positive samples (green) and negative samples (orange). We assume the observations are generated by an (unknown) injective generative model $g$ that maps unobservable latent variables from a hypersphere to observations in another manifold. Under these assumptions, the feature encoder $f$ implictly learns to invert the ground-truth generative process $g$ up to linear transformations, i.e., $f = \\mathbf{A} g^{-1}$ with an orthogonal matrix $\\mathbf{A}$, if $f$ minimizes the InfoNCE objective.}\n        \\label{fig:header_figure}\n    \\end{figure*}\n    \n    We show that an encoder learned with a contrastive loss from the InfoNCE family can recover the true generative factors of variation (up to rotations) if the process that generated the data fulfills a few weak statistical assumptions. This theory bridges the gap between contrastive learning, nonlinear independent component analysis (ICA) and generative modeling (see Fig.~\\ref{fig:header_figure}).\n    Our theory reveals implicit assumptions encoded in the InfoNCE objective about the generative process underlying the data. If these assumptions are violated, we show a principled way of deriving alternative contrastive objectives based on assumptions regarding the positive pair distribution.\n    We verify our theoretical findings with controlled experiments, providing evidence that our theory holds true in practice, even if the assumptions on the ground-truth generative model are partially violated. %\n    \n    To the best of our knowledge, our work is the first to analyze under what circumstances representation learning methods used in practice provably represent the data in terms of its underlying factors of variation. Our theoretical and empirical results suggest that the success of contrastive learning in many practical applications is due to an implicit and approximate inversion of the data generating process, which explains why the learned representations are useful in a wide range of downstream tasks.\n    \n    In summary, our contributions are:\n    \\begin{itemize}\n        \\item We establish a theoretical connection between the InfoNCE family of objectives, which is commonly used in self-supervised learning, and nonlinear ICA. We show that training with InfoNCE inverts the data generating process if certain statistical assumptions on the data generating process hold.\n        \\item We empirically verify our predictions when the assumed theoretical conditions are fulfilled. In addition, we show successful inversion of the data generating process even if these theoretical assumptions are partially violated. %\n        \\item We build on top of the CLEVR rendering pipeline~\\citep{johnson2017clevr} to generate a more visually complex disentanglement benchmark, called \\emph{3DIdent}, that contains hallmarks of natural environments (shadows, different lighting conditions, a 3D object, etc.). We demonstrate that a contrastive loss derived from our theoretical framework can identify the ground-truth factors of such complex, high-resolution images.\n    \\end{itemize}\n    \n\\section{Related Work}\n\\paragraph{Contrastive Learning}\n    Despite the success of contrastive learning (CL), our understanding of the learned representations remains limited, as existing theoretical explanations yield partially contradictory predictions. One way to theoretically motivate CL is to refer to the InfoMax principle \\citep{linsker1988self}, which corresponds to maximizing the mutual information (MI) between different views \\citep{oord2018representation, bachman2019learning, hjelm2018learning, chen2020simple, tian2020makes}. However, as optimizing a tighter bound on the MI can produce worse representations \\citep{tschannen2019mutual}, it is not clear how accurate this motivation describes the behavior of CL.\n    \n    Another approach aims to explain the success by introducing latent classes \\citep{saunshi2019theoretical}. While this theory has some appeal, there exists a gap between empirical observations and its predictions, e.g. the prediction that an excessive number of negative samples decreases performance does not corroborate with empirical results~\\citep{wu2018unsupervised,tian2019contrastive,he2019momentum,chen2020simple}. However, recent work has suggested some empirical evidence for said theoretical prediction, namely, issues with the commonly used sampling strategy for negative samples, and have proposed ways to mitigate said issues as well~\\citep{robinson2020contrastive, chuang2020debiased}.   \n    \n    \n    More recently, the behavior of CL has been analyzed from the perspective of \\emph{alignment} and \\emph{uniformity} properties of representations, demonstrating that these two properties are correlated with downstream performance~\\citep{wang2020understanding}.\n    We build on these results to make a connection to cross-entropy minimization from which we can derive identifiability results.%\n\n\\paragraph{Nonlinear ICA}\n    Independent Components Analysis (ICA) attempts to find the underlying sources for multidimensional data. In the nonlinear case, said sources correspond to a well-defined nonlinear generative model $g$, which is assumed to be invertible (i.e., injective)~\\citep{Hyvabook,Jutten10}. In other words, nonlinear ICA solves a demixing problem:\n    Given observed data $\\mathbf{x} = g(\\mathbf{z})$, it aims to find a model $f$ that equals the inverse generative model $g^{-1}$, which allows for the original sources $\\mathbf{z}$ to be recovered.\n    \n    \\citet{hyvarinen2018nonlinear} show that the nonlinear demixing problem can be solved as long as the independent components are conditionally mutually independent with respect to some auxiliary variable. The authors further provide practical estimation methods for solving the nonlinear ICA problem~\\citep{hyvarinen2016unsupervised,hyvarinen2017nonlinear}, similar in spirit to noise contrastive estimation (NCE; \\citealp{Gutmann12JMLR}). Recent work has generalized this contribution to VAEs~\\citep{khemakhem2020variational,locatello2020weakly,klindt2020slowvae}, as well as (invertible-by-construction) energy-based models~\\citep{khemakhem2020ice}. We here extend this line of work to more general feed-forward networks trained using InfoNCE~\\citep{oord2018representation}.\n    \n    In a similar vein, \\citet{roeder2020linear} build on the work of \\citet{hyvarinen2018nonlinear} to show that for a model family which includes InfoNCE, distribution matching implies parameter matching. In contrast, we associate the learned latent representation with the ground-truth generative factors, showing under what conditions the data generating process is inverted, and thus, the true latent factors are recovered.\n    \n\\section{Theory}\n    \n    We will show a connection between contrastive learning and identifiability in the form of nonlinear ICA. For this, we introduce a feature encoder $f$ that maps observations $\\mathbf{x}$ to representations.\n    We consider the widely used \\emph{InfoNCE} loss, which often assumes $L^2$ normalized representations \\citep{wu2018unsupervised, he2020momentum, tian2019contrastive, bachman2019learning,chen2020simple},\n    \\begin{align} \\label{eq:contrastive_loss}\n        &\\mathcal{L}_\\mathsf{contr}(f; \\tau, M) \\quad := \\\\ \n        &\\underset{\\substack{\n            ({\\bf{x}}, {\\bf{\\tilde x}}) \\sim p_\\mathsf{pos} \\\\\n            \\{\\mathbf{x}^-_i\\}_{i=1}^M \\overset{\\text{i.i.d.}}{\\sim} p_\\mathsf{data}\n        }}{\\mathbb{E}} \\left[\\, {- \\log \\frac{e^{f(\\mathbf{x})^{\\mathsf{T}} f({\\bf{\\tilde x}}) \/ \\tau }}{e^{f(\\mathbf{x})^{\\mathsf{T}} f({\\bf{\\tilde x}}) \/ \\tau } + \\sum\\limits_{i=1}^M e^{f(\\mathbf{x}^-_i)^{\\mathsf{T}} f({\\bf{\\tilde x}}) \/ \\tau }}}\\,\\right]. \\nonumber\n    \\end{align}\n    Here $M\\in\\mathbb{Z}_+$ is a fixed number of negative samples, $p_{\\rm{data}}$ is the distribution of all observations and $p_{\\rm{pos}}$ is the distribution of positive pairs.\n    This loss was motivated by the InfoMax principle \\citep{linsker1988self}, and has been shown\n    to be effective by many recent representation learning methods \\citep{logeswaran2018efficient,wu2018unsupervised,tian2019contrastive,he2019momentum,hjelm2018learning,bachman2019learning,chen2020simple,baevski2020wav2vec}. Our theoretical results also hold for a loss function whose denominator only consists of the second summand across the negative samples (e.g., the SimCLR loss \\citep{chen2020simple}). %\n    \n    In the spirit of existing literature on nonlinear ICA \\cite{hyvarinen1999nonlinear, harmeling2003kernel,sprekeler2014extension,hyvarinen2016unsupervised,hyvarinen2017nonlinear, Gutmann12JMLR, hyvarinen2018nonlinear, khemakhem2020variational}, we assume that the observations $\\mathbf{x} \\in \\mathcal{X}$ are generated by an invertible (i.e., injective) generative process $g: \\mathcal{Z} \\to \\mathcal{X}$, where $\\mathcal{X} \\subseteq \\mathbb{R}^K$ is the space of observations and $\\mathcal{Z} \\subseteq \\mathbb{R}^N$ with $N\\leq K$ denotes the space of latent factors. Influenced by the commonly used feature normalization in InfoNCE, we further assume that $\\mathcal{Z}$ is the unit hypersphere $\\mathbb{S}^{N-1}$ (see Appx.~\\ref{apx:gt_assumptions}).\n    Additionally, we assume that the ground-truth marginal distribution of the latents of the generative process is uniform and that the conditional distribution (under which positive pairs have high density) is a von Mises-Fisher (vMF) distribution:\n    \\begin{align} \\label{eq:vmf_conditional}\n        p({\\bf{z}}) &= |\\mathcal{Z}|^{-1}, \\quad\\quad p({\\bf{z}}|{\\tilde{\\mathbf{z}}}) = C_p^{-1} e^{\\kappa {\\bf{z}}^\\top {\\tilde{\\mathbf{z}}}} \\quad \\text{with} \\\\ C_p :&= \\int e^{\\kappa {\\bf{z}}^\\top {\\tilde{\\mathbf{z}}}} \\,\\d{\\tilde{\\mathbf{z}}} = \\text{const.}, \\quad \\mathbf{x} = g({\\bf{z}}), \\quad {\\tilde{\\mathbf{x}}} = g({\\tilde{\\mathbf{z}}}). \\nonumber\n    \\end{align}\n    \n    Given these assumptions, we will show that if $f$ minimizes the contrastive loss $\\mathcal{L}_\\mathsf{contr}$, then $f$ solves the demixing problem, i.e., inverts $g$ up to orthogonal linear transformations. %\n    \n    Our theoretical approach consists of three steps:\n    (1) We demonstrate that $\\mathcal{L}_\\mathsf{contr}$ can be interpreted as the cross-entropy between the (conditional) ground-truth and inferred latent distribution. %\n    (2) Next, we show that encoders minimizing $\\mathcal{L}_\\mathsf{contr}$ maintain distance, i.e., two latent vectors with distance $\\alpha$ in the ground-truth generative model are mapped to points with the same distance $\\alpha$ in the inferred representation. %\n    (3) Finally, we leverage distance preservation to show that minimizers of $\\mathcal{L}_\\mathsf{contr}$  invert the generative process up to orthogonal transformations.\n    Detailed proofs are given in Appx.~\\ref{apx:proofs}.\n    \n    Additionally, we will present similar results for general convex bodies in $\\mathbb{R^N}$ and more general similarity measures, see Sec.~\\ref{sec:extension_rn}. For this, the detailed proofs are given in Appx.~\\ref{apx:rn_extension}. %\n    \n\\subsection{Contrastive learning is related to cross-entropy minimization}\n    From the perspective of nonlinear ICA, we are interested in understanding how the representations $f({\\bf{x}})$ which minimize the contrastive loss $\\mathcal{L}_\\mathsf{contr}$ (defined in Eq.~\\eqref{eq:contrastive_loss}) are related to the ground-truth source signals ${\\bf{z}}$. To study this relationship, we focus on the map $h = f\\circ g$ between the recovered source signals $h({\\bf{z}})$ and the true source signals ${\\bf{z}}$. Note that this is merely for mathematical convenience; it does not necessitate knowledge regarding neither $g$ nor the ground-truth factors during learning (beyond the assumptions stated in the theorems).\n    \n    A core insight is a connection between the contrastive loss and the cross-entropy between the ground-truth latent distribution and a certain model distribution. For this, we expand the theoretical results obtained by \\citet{wang2020understanding}:\n    \\vspace{\\topsep}\n    \\begin{customtheorem}{\\ref*{thm:extended_asym_inf_negatives_CE}}[$\\mathcal{L}_\\mathsf{contr}$ converges to the cross-entropy between latent distributions] \\label{thm:asym_inf_negatives_CE}\n        If the ground-truth marginal distribution $p$ is uniform, then for fixed $\\tau > 0$, as the number of negative samples $M \\rightarrow \\infty$, the (normalized) contrastive loss converges to\n        \\begin{equations}\n            \\lim_{M \\rightarrow \\infty} \\mathcal{L}_\\mathsf{contr}(f; \\tau, M) - \\log M + \\log |\\mathcal{Z}| = \\\\ \\expectunder{{\\bf{z}} \\sim p({\\bf{z}})}{H(p(\\cdot | {\\bf{z}}), q_h(\\cdot | {\\bf{z}}))}\n            \\label{eq:contrastive_loss_CE_limit}\n        \\end{equations}\n        where $H$ is the cross-entropy between the ground-truth conditional distribution $p$ over positive pairs and a conditional distribution $q_{\\rm{h}}$ parameterized by the model $f$,\n        \\begin{equations} \\label{eq:qhjoint}\n            q_{\\rm{h}}({\\tilde{\\mathbf{z}}}|{\\bf{z}}) &= C_h(\\mathbf{z})^{-1} e^{h({\\tilde{\\mathbf{z}}})\\T h(\\mathbf{z}) \/\\tau} \\\\ \\text{with} \\quad C_h(\\mathbf{z}) :&= \\int e^{h({\\tilde{\\mathbf{z}}})\\T h(\\mathbf{z}) \/\\tau} \\,\\d{\\tilde{\\mathbf{z}}},\n        \\end{equations}\n        where $C_h({\\bf{z}})\\in\\mathbb{R}^{+}$ is the partition function of $q_{\\rm{h}}$ (see Appx.~\\ref{apx:model_assumptions}).\n    \\end{customtheorem}\n    \n    Next, we show that the minimizers $h^{*}$ of the cross-entropy~(\\ref{eq:qhjoint}) are isometries in the sense that $\\kappa {\\bf{z}}^\\top{\\tilde{\\mathbf{z}}} = h^{*}({\\bf{z}})^\\top h^{*}({\\tilde{\\mathbf{z}}})$ for all ${\\bf{z}}$ and ${\\tilde{\\mathbf{z}}}$. In other words, they preserve the dot product between ${\\bf{z}}$ and ${\\tilde{\\mathbf{z}}}$. %\n    \\vspace{\\topsep}\n    \\begin{customproposition}{\\ref*{prop:extended_correct_model_ce_isometry}}[Minimizers of the cross-entropy maintain the dot product] \\label{prop:correct_model_ce_isometry}\n        Let $\\mathcal{Z} = \\mathbb{S}^{N-1}$, $\\tau > 0$ and consider the ground-truth conditional distribution of the form $p({\\tilde{\\mathbf{z}}} | {\\bf{z}}) = C_p^{-1} \\exp(\\kappa {\\tilde{\\mathbf{z}}}^\\top \\mathbf{z})$. Let $h$ map onto a hypersphere with radius $\\sqrt{\\tau \\kappa}$.\\footnote{Note that in practice this can be implemented as a learnable rescaling operation as the last operation of the network $f$.} Consider the conditional distribution $q_h$ parameterized by the model, as defined above in Theorem~\\ref{thm:asym_inf_negatives_CE}, where the hypothesis class for $h$ (and thus $f$) is assumed to be sufficiently flexible such that $p({\\tilde{\\mathbf{z}}} | \\mathbf{z})$ and $q_{\\rm{h}}({\\tilde{\\mathbf{z}}}|\\mathbf{z})$ can match.\n        If $h$ is a minimizer of the cross-entropy $\\mathbb{E}_{p({\\tilde{\\mathbf{z}}} | \\mathbf{z})}[- \\log q_{\\rm{h}}({\\tilde{\\mathbf{z}}} | \\mathbf{z})]$, then $p({\\tilde{\\mathbf{z}}}|\\mathbf{z}) = q_{\\rm{h}}({\\tilde{\\mathbf{z}}} | \\mathbf{z})$ and $\\forall {\\bf{z}}, {\\tilde{\\mathbf{z}}}: \\kappa {\\bf{z}}^\\top{\\tilde{\\mathbf{z}}} = h({\\bf{z}})^\\top h({\\tilde{\\mathbf{z}}})$.\n    \\end{customproposition}\n\n\n\\subsection{Contrastive learning identifies ground-truth factors on the hypersphere}\n    From the strong geometric property of isometry, we can now deduce a key property of the minimizers $h^*$: %\n    \\vspace{\\topsep}\n    \\begin{customproposition}{\\ref*{prop:extended_mazurulamspheres}}[Extension of the Mazur-Ulam theorem to hyperspheres and the dot product]\n        \\label{prop:mazurulamspheres}\n        Let $\\mathcal{Z} = \\mathbb{S}^{N-1}$. If $h: \\mathcal{Z} \\to \\mathcal{Z}$ maintains the dot product up to a constant factor, i.e., $\\forall {\\bf{z}}, {\\tilde{\\mathbf{z}}} \\in \\mathcal{Z}: \\kappa {\\bf{z}}^\\top {\\tilde{\\mathbf{z}}} = h({\\bf{z}})^\\top h({\\tilde{\\mathbf{z}}})$, then $h$ is an orthogonal linear transformation.\n    \\end{customproposition}\n\n    In the last step, we combine the previous propositions to derive our main result: the minimizers of the contrastive loss $\\mathcal{L}_\\mathsf{contr}$ solve the demixing problem of nonlinear ICA up to linear transformations, i.e., they identify the original sources ${\\bf{z}}$ for observations $g({\\bf{z}})$ up to orthogonal linear transformations. For a hyperspherical space $\\mathcal{Z}$ these correspond to combinations of permutations, rotations and sign flips.\n    \\vspace{\\topsep}\n    \\begin{customtheorem}{\\ref*{thm:extended_ident_matching}}\\label{thm:ident_matching}\n        Let $\\mathcal{Z} = \\mathbb{S}^{N-1}$, the ground-truth marginal be uniform, and the conditional a vMF distribution (cf. Eq.~\\ref{eq:vmf_conditional}). Let the mixing function $g$ be differentiable and injective. If the assumed form of $q_{\\rm{h}}$, as defined above, matches that of $p$, i.e., both are based on the same metric, and if $f$ is differentiable and minimizes the CL loss as defined in Eq.~\\eqref{eq:contrastive_loss}, then for fixed $\\tau > 0$ and $M\\to\\infty$, $h = f \\circ g$ is linear, i.e., $f$ recovers the latent sources up to orthogonal linear transformations.\n    \\end{customtheorem}\n    Note that we do not assume knowledge of the ground-truth generative model $g$; we only make assumptions about the conditional and marginal distribution of the latents.\n    On real data, it is unlikely that the assumed model distribution $q_{\\rm{h}}$ can exactly match the ground-truth conditional. We do, however, \n    provide empirical evidence that $h$ is still an affine transformation even if there is a severe mismatch, see Sec.~\\ref{sec:experiments}.\n    \n    \n\\subsection{Contrastive learning identifies ground-truth factors on convex bodies in \\texorpdfstring{$\\mathbb{R}^N$}{RN}} \\label{sec:extension_rn}\n    While the previous theoretical results require $\\mathcal{Z}$ to be a hypersphere, we will now show a similar theorem for the more general case of $\\mathcal{Z}$ being a convex body in $\\mathbb{R}^N$. Note that the hyperrectangle $[a_1, b_1] \\times \\ldots \\times [a_N, b_N]$ is an example of such a convex body.\n    \n    We follow a similar three step proof strategy as for the hyperspherical case before:\n    (1) We begin again by showing that a properly chosen contrastive loss on convex bodies corresponds to the cross-entropy between the ground-truth conditional and a distribution parametrized by the encoder. For this step, we additionally extend the results of \\citet{wang2020understanding} to this latent space and loss function.\n    (2) Next, we derive that minimizers of the loss function are isometries of the latent space. Importantly, we do not limit ourselves to a specific metric, thus the result is applicable to a family of contrastive objectives.\n    (3) Finally, we show that these minimizers must be affine transformations.\n    For a special family of conditional distributions (rotationally asymmetric generalized normal distributions~\\citep{subbotin1923law}), we can further narrow the class of solutions to permutations and sign-flips. %\n    For the detailed proofs, see Appx.~\\ref{apx:rn_extension}. \n    \n    As earlier, we assume that the ground-truth marginal distribution of the latents is uniform. However, we now assume that the conditional distribution is exponential:\n    \\begin{equations} \\label{eq:rn_conditional}\n        p({\\bf{z}}) &= |\\mathcal{Z}|^{-1}, \\quad\\quad p({\\bf{z}}|{\\tilde{\\mathbf{z}}}) = C_p^{-1} e^{- \\delta({\\bf{z}}, {\\tilde{\\mathbf{z}}})} \\quad \\text{with} \\\\ C_p({\\bf{z}}) :&= \\int e^{-\\delta({\\bf{z}}, {\\tilde{\\mathbf{z}}})} \\,\\d{\\tilde{\\mathbf{z}}}, \\quad \\mathbf{x} = g({\\bf{z}}), \\quad {\\tilde{\\mathbf{x}}} = g({\\tilde{\\mathbf{z}}}),\n    \\end{equations}\n    where $\\delta$ is a metric induced by a norm (see Appx.~\\ref{apx:rn_gt_assumptions}).\n    \n    To reflect the differences between this conditional distribution and the one assumed for the hyperspherical case, we need to introduce an adjusted version of the contrastive loss in \\eqref{eq:contrastive_loss}:  \n    \\begin{definition}[$\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$ objective] \\label{def:delta_contrastive_loss}\n        Let $\\delta: \\mathcal{Z} \\times \\mathcal{Z} \\to \\mathbb{R}$ be a metric on $\\mathcal{Z}$. We define the general InfoNCE loss, which uses $\\delta$ as a similarity measure, as\n        \n        \\begin{align} \\label{eq:delta_contrastive_loss}\n            &\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}(f; \\tau, M) \\quad :=\\\\\n            &\\underset{\\substack{\n                ({\\bf{x}}, {\\bf{\\tilde x}}) \\sim p_\\mathsf{pos} \\\\\n                \\{\\mathbf{x}^-_i\\}_{i=1}^M \\overset{\\text{i.i.d.}}{\\sim} p_\\mathsf{data}\n            }}{\\mathbb{E}} \\hspace{-1em}\\Bigg[\n            {- \\log \\frac{e^{-\\delta(f(\\mathbf{x}), f({\\bf{\\tilde x}})) \/ \\tau }}{e^{\\text{--}\\delta(f(\\mathbf{x}), f({\\bf{\\tilde x}})) \/ \\tau } \\hspace{-.3em}+\\hspace{-.3em} \\sum\\limits_{i=1}^M e^{\\text{--}\\delta(f(\\mathbf{x}^\\text{--}_i), f({\\bf{\\tilde x}})) \/ \\tau }}}\\,\\Bigg]. \\nonumber\n        \\end{align}\n    \\end{definition}\n    Note that this is a generalization of the InfoNCE criterion in Eq.~(\\ref{eq:contrastive_loss}). In contrast to the objective above, the representations are no longer assumed to be $L^2$ normalized, and the dot-product is replaced with a more general similarity measure $\\delta$.\n    \n    Analogous to the previously demonstrated case for the hypersphere, for convex bodies $\\mathcal{Z}$, minimizers of the adjusted $\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$ objective solve the demixing problem of nonlinear ICA up to invertible linear transformations:\n    \\begin{customtheorem}{\\ref*{thm:extended_rn_linear_identifiable}} \\label{thm:rn_linear_identifiable}\n        Let $\\mathcal{Z}$ be a convex body in $\\mathbb{R}^N$, $h = f\\circ g:\\mathcal{Z}\\to\\mathcal{Z}$, and $\\delta$ be a metric or a semi-metric (cf. Lemma~\\ref{lem:semimetric} in Appx.~\\ref{apx:rn_ce_min_identifiability}), induced by a norm. Further, let the ground-truth marginal distribution be uniform and the conditional distribution be as Eq.~\\eqref{eq:rn_conditional}. Let the mixing function $g$ be differentiable and injective. If the assumed form of $q_{\\rm{h}}$ matches that of $p$, i.e., \n        \\begin{equations}\n            q_{\\rm{h}}({\\tilde{\\mathbf{z}}}|{\\bf{z}}) &= C_q^{-1}(\\mathbf{z})e^{-\\delta(h({\\tilde{\\mathbf{z}}}), h(\\mathbf{z}))\/\\tau}\\quad \\\\ \\text{with} \\quad C_q(\\mathbf{z}) :&= \\int e^{-\\delta(h({\\tilde{\\mathbf{z}}}), h(\\mathbf{z}))\/\\tau} \\,\\d{\\tilde{\\mathbf{z}}},\n        \\end{equations}\n        and if $f$ is differentiable and minimizes the $\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$ objective in Eq.~\\eqref{eq:delta_contrastive_loss} for $M \\to \\infty$, we find that $h = f \\circ g$ is invertible and affine, i.e., we recover the latent sources up to affine transformations.\n    \\end{customtheorem}\n    Note that the model distribution $q_{\\rm{h}}$, which is implicitly described by the choice of the objective, must be of the same form as the ground-truth distribution $p$, i.e., both must be based on the same metric. Thus, identifying different ground-truth conditional distributions requires different contrastive $\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$ objectives.\n    This result can be seen as a generalized version of Theorem~\\ref{thm:ident_matching}, as it is valid for any convex body $\\mathcal{Z} \\subseteq \\mathbb{R}^N$, allowing for a larger variety of conditional distributions.\n    \n    Finally, under the mild restriction that the ground-truth conditional distribution is based on an $L^p$ similarity measure for $p \\geq1, p \\neq 2$, $h$ identifies the ground-truth generative factors up to generalized permutations. A generalized permutation matrix $\\bf{A}$ is a combination of a permutation and element-wise sign-flips, i.e., $\\forall {\\bf{z}}: (\\bf{A}{\\bf{z}})_i = \\alpha_i {\\bf{z}}_{\\sigma(i)}$ with $\\alpha_i = \\pm 1$ and $\\sigma$ being a permutation.\n    \\begin{customtheorem}{\\ref*{thm:extended_rn_permutation_identifiable}} \\label{thm:rn_permutation_identifiable}\n        Let $\\mathcal{Z}$ be a convex body in $\\mathbb{R}^N$, $h: \\mathcal{Z} \\to \\mathcal{Z}$, and $\\delta$ be an $L^\\alpha$ metric or semi-metric (cf. Lemma~\\ref{lem:semimetric} in Appx.~\\ref{apx:rn_ce_min_identifiability}) for $\\alpha \\geq 1, \\alpha \\neq 2$. Further, let the ground-truth marginal distribution be uniform and the conditional distribution be as Eq.~\\eqref{eq:rn_conditional}, and let the mixing function $g$ be differentiable and invertible. If the assumed form of $q_{\\rm{h}}(\\cdot|{\\bf{z}})$ matches that of $p(\\cdot|{\\bf{z}})$, i.e., both use the same metric $\\delta$ up to a constant scaling factor, and if $f$ is differentiable and minimizes the $\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$ objective in Eq.~\\eqref{eq:delta_contrastive_loss} for $M \\to \\infty$, we find that $h = f \\circ g$ is a composition of input independent permutations, sign flips and rescaling.\n    \\end{customtheorem}\n\n\n\\section{Experiments} \\label{sec:experiments}\n\n\\subsection{Validation of theoretical claim} \\label{sec:toy_experiments}\n    We validate our theoretical claims under both perfectly matching and violated conditions regarding the ground-truth marginal and conditional distributions. We consider source signals of dimensionality $N=10$, and sample pairs of source signals in two steps: First, we sample from the marginal $p({\\bf{z}})$. For this, we consider both uniform distributions which match our assumptions and non-uniform distributions (e.g., a normal distribution) which violate them. Second, we generate the positive pair by sampling from a conditional distribution $p({\\tilde{\\mathbf{z}}} | {\\bf{z}})$.\n    Here, we consider matches with our assumptions on the conditional distribution (von Mises-Fisher for $\\mathcal{Z} = \\mathbb{S}^{N-1}$) as well as violations (e.g. normal, Laplace or generalized normal distribution for $\\mathcal{Z} = \\mathbb{S}^{N-1}$). Further, we consider spaces beyond the hypersphere, such as the bounded box (which is a convex body) and the unbounded $\\mathbb{R}^N$.\n    \n    We generate the observations with a multi-layer perceptron (MLP), following previous work~\\citep{hyvarinen2016unsupervised,hyvarinen2017nonlinear}.\n    Specifically, we use three hidden layers with leaky ReLU units and random weights; to ensure that the MLP $g$ is invertible, we control the condition number of the weight matrices.\n    For our feature encoder $f$, we also use an MLP with leaky ReLU units, where the assumed space is denoted by the normalization, or lack thereof, of the encoding. Namely, for the hypersphere (denoted as \\emph{Sphere}) and the hyperrectangle (denoted as \\emph{Box}) we apply an $L^2$ and $L^\\infty$ normalization, respectively. For flexibility in practice, we parameterize the normalization magnitude of the \\emph{Box}, including it as part of the encoder's learnable parameters. On the hypersphere we optimize $\\mathcal{L}_\\mathsf{contr}$ and on the hyperrectangle as well as the unbounded space we optimize $\\mathcal{L}_{\\delta\\text{-}\\mathsf{contr}}$. For further details, see Appx.~\\ref{apx:experiment_details}.  %\n    \n    To test for identifiability up to affine transformations, we fit a linear regression between the ground-truth and recovered sources and report the coefficient of determination ($R^2$). To test for identifiability up to generalized permutations, we leverage the mean correlation coefficient (MCC), as used in previous work~\\citep{hyvarinen2016unsupervised,hyvarinen2017nonlinear}. For further details, see Appx.~\\ref{apx:experiment_details}.\n    \n    \\begin{table*}[t]\n        \\centering\n        \\caption{Identifiability up to affine transformations. Mean $\\pm$ standard deviation over $5$ random seeds. Note that only the first row corresponds to a setting that matches (\\cmark) our theoretical assumptions, while the others show results for violated assumptions (\\xmark; see column \\emph{M.}). Note that the identity score only depends on the ground-truth space and the marginal distribution defined for the generative process, while the supervised score additionally depends on the space assumed by the model. \n        }\n        \\resizebox{\\textwidth}{!}{%\n        \\begin{tabular}{ccccccccc}\n            \\toprule\n            \\multicolumn{3}{c}{Generative process $g$} & \\multicolumn{3}{c}{Model $f$} & \\multicolumn{3}{c}{$R^2$ Score [\\%]} \\\\\n            Space & $p(\\cdot)$ & $p(\\cdot|\\cdot)$ & Space & $q_{\\rm{h}}(\\cdot|\\cdot)$ & M. & Identity & Supervised & Unsupervised \\\\\n            \\midrule\n            Sphere & Uniform & vMF($\\kappa{=}1$) & Sphere & vMF($\\kappa{=}1$) & \\cmark & $66.98 \\pm 2.79$ & $99.71  \\pm 0.05$ & $99.42 \\pm 0.05$ \\\\\n            Sphere & Uniform & vMF($\\kappa{=}10$) & Sphere & vMF($\\kappa{=}1$) & \\xmark & \\dittotikz & \\dittotikz & $99.86 \\pm 0.01$ \\\\\n            Sphere & Uniform & Laplace($\\lambda{=}0.05$) & Sphere & vMF($\\kappa{=}1$) & \\xmark & \\dittotikz & \\dittotikz & $99.91 \\pm 0.01$ \\\\\n            Sphere & Uniform & Normal($\\sigma{=}0.05$) & Sphere & vMF($\\kappa{=}1$) & \\xmark & \\dittotikz & \\dittotikz & $99.86 \\pm 0.00$\\\\\n            \\midrule\n            Box & Uniform & Normal($\\sigma{=}0.05$) & Unbounded & Normal & \\xmark & $67.93 \\pm 7.40$ & $99.78 \\pm 0.06$ & $99.60 \\pm 0.02$ \\\\\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Unbounded & Normal & \\xmark & \\dittotikz & \\dittotikz & $99.64 \\pm 0.02$ \\\\\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Unbounded & GenNorm($\\beta{=}3$) & \\xmark & \\dittotikz & \\dittotikz & $99.70 \\pm 0.02$\\\\\n            Box & Uniform & Normal($\\sigma{=}0.05$) & Unbounded & GenNorm($\\beta{=}3$) & \\xmark & \\dittotikz & \\dittotikz & $99.69 \\pm 0.02$\\\\\n            \\midrule\n            Sphere & Normal($\\sigma{=}1$) & Laplace($\\lambda{=}0.05$) & Sphere & vMF($\\kappa{=}1$) & \\xmark & $63.37 \\pm 2.41$ & $99.70 \\pm 0.07$ & $99.02 \\pm 0.01$ \\\\\n            Sphere & Normal($\\sigma{=}1$) & Normal($\\sigma{=}0.05$) & Sphere & vMF($\\kappa{=}1$) & \\xmark & \\dittotikz & \\dittotikz & $99.02 \\pm 0.02$ \\\\\n            \\midrule          Unbounded & Laplace($\\lambda{=}1$) & Normal($\\sigma{=}1$) & Unbounded & Normal & \\xmark & $62.49 \\pm 1.65$ & $99.65 \\pm 0.04$ & $98.13 \\pm 0.14$ \\\\ Unbounded & Normal($\\sigma{=}1$) & Normal($\\sigma{=}1$) & Unbounded & Normal & \\xmark & $63.57 \\pm 2.30$ & $99.61 \\pm 0.17$ & $98.76 \\pm 0.03$ \\\\\n            \\bottomrule\n        \\end{tabular}}\n        \\label{tab:results_linear}\n    \\end{table*}\n    \n    \\begin{table*}[t]\n        \\centering\n        \\caption{Identifiability up to generalized permutations, averaged over $5$ runs. \n        Note that while Theorem~\\ref{thm:extended_rn_permutation_identifiable} requires the model latent space to be a convex body and $p(\\cdot|\\cdot)=q_{\\rm{h}}(\\cdot|\\cdot)$, we find that empirically either is sufficient.\n        The results are grouped in four blocks corresponding to different types and degrees of violation of assumptions of our theory showing identifiability up to permutations: (1) no violation, violation of the assumptions on either the (2) space or (3) the conditional distribution, or (4) both.\n        }\n        \\resizebox{\\textwidth}{!}{%\n        \\begin{tabular}{ccccccccc}\n            \\toprule \\multicolumn{3}{c}{Generative process $g$} & \\multicolumn{3}{c}{Model $f$} & \\multicolumn{3}{c}{MCC Score [\\%]} \\\\\n            Space & $p(\\cdot)$ & $p(\\cdot|\\cdot)$ & Space & $q_{\\rm{h}}(\\cdot|\\cdot)$ & M. & Identity & Supervised & Unsupervised \\\\\n            \\midrule\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Box & Laplace & \\cmark & $46.55 \\pm 1.34$ & $99.93 \\pm 0.03$ & $98.62 \\pm 0.05$ \\\\\n            Box & Uniform & GenNorm($\\beta{=}3$; $\\lambda{=}0.05$) & Box & GenNorm($\\beta{=}3$) & \\cmark & \\dittotikz & \\dittotikz & $99.90 \\pm 0.06$ \\\\\n            \\midrule\n            Box & Uniform & Normal($\\sigma{=}0.05$) & Box & Normal & \\xmark & \\dittotikz & \\dittotikz & $99.77 \\pm 0.01$ \\\\\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Box & Normal & \\xmark & \\dittotikz & \\dittotikz & $99.76 \\pm 0.02$ \\\\\n            Box & Uniform & GenNorm($\\beta{=}3$; $\\lambda{=}0.05$) & Box & Laplace & \\xmark & \\dittotikz & \\dittotikz & $98.80 \\pm 0.02$ \\\\\n            \\midrule\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Unbounded & Laplace & \\xmark & \\dittotikz & $99.97 \\pm 0.03$ & $98.57 \\pm 0.02$ \\\\\n            Box & Uniform & GenNorm($\\beta{=}3$; $\\lambda{=}0.05$) & Unbounded & GenNorm($\\beta{=}3$) & \\xmark & \\dittotikz & \\dittotikz & $99.85 \\pm 0.01$ \\\\\n            \\midrule\n            Box & Uniform & Normal($\\sigma{=}0.05$) & Unbounded & Normal & \\xmark & \\dittotikz & \\dittotikz & $58.26 \\pm 3.00$ \\\\\n            Box & Uniform & Laplace($\\lambda{=}0.05$) & Unbounded & Normal & \\xmark & \\dittotikz & \\dittotikz & $59.67 \\pm 2.33$ \\\\\n            Box & Uniform & Normal($\\sigma{=}0.05$) & Unbounded & GenNorm($\\beta{=}3$) & \\xmark & \\dittotikz & \\dittotikz & $43.80 \\pm 2.15$ \\\\\n            \\bottomrule\n        \\end{tabular}}\n        \\label{tab:perm_results}\n    \\end{table*}\n    \n    We evaluate both identifiability metrics for three different model types.\n    First, we ensure that the problem requires nonlinear demixing by considering the identity function for model $f$, which amounts to scoring the observations against the sources (\\textbf{Identity Model}).\n    Second, we ensure that the problem is solvable within our model class by training our model $f$ with supervision, minimizing the mean-squared error between $f(g({\\bf{z}}))$ and ${\\bf{z}}$ (\\textbf{Supervised Model}). Third, we fit our model without supervision using a contrastive loss (\\textbf{Unsupervised Model}).\n      \n    Tables~\\ref{tab:results_linear} and~\\ref{tab:perm_results} show results evaluating identifiability up to affine transformations and generalized permutations, respectively. When assumptions match (see column M.), CL recovers a score close to the empirical upper bound.\n    Mismatches in assumptions on the marginal and conditional do not lead to a significant drop in performance with respect to affine identifiability, but do for permutation identifiability compared to the empirical upper bound.\n    In many practical scenarios, we use the learned representations to solve a downstream task, thus, identifiability up to affine transformations is often sufficient.\n    However, for applications where identification of the individual generative factors is desirable, some knowledge of the underlying generative process is required to choose an appropriate loss function and feature normalization.\n    Interestingly, we find that for convex bodies, we obtain identifiability up to permutation even in the case of a normal conditional, which likely is due to the axis-aligned box geometry of the latent domain.\n    Finally, note that the drop in performance for identifiability up to permutations in the last group of Tab.~\\ref{tab:perm_results} is a natural consequence of either the ground-truth or the assumed conditional being rotationally symmetric, e.g., a normal distribution, in an unbounded space. Here, rotated versions of the latent space are indistinguishable and, thus, the model cannot align the axes of the reconstruction with that of the ground-truth latent space, resulting in a lower score.\n    \n    To zoom in on how violations of the uniform marginal assumption influence the identifiability achieved by a model in practice, we perform an ablation on the marginal distribution by interpolating between the theoretically assumed uniform distribution and highly locally concentrated distributions.\n    In particular, we consider two cases: (1) a sphere ($\\mathcal{S}^9$) with a vMF marginal around its north pole for different concentration parameters $\\kappa$; (2) a box ($[0,1]^{10}$) with a normal marginal around the box's center for different standard deviations $\\sigma$.\n    For both cases, Fig.~\\ref{fig:uniformity_violation} shows the $R^2$ score as a function of the concentration $\\kappa$ and $1\/\\sigma^2$ respectively (black). As a reference, the concentration of the used conditional distribution is highlighted as a dashed line.\n    In addition, we also display the probability mass (0--100\\%) that needs to be moved for converting the used marginal distribution (i.e., vMF or normal) into the assumed uniform marginal distribution (blue) as an intuitive measure of the mismatch (i.e., $\\frac{1}{2}\\int |p(\\mathbf{z})\\mathrm{-}p_{\\mathrm{uni}}|\\, \\mathrm{d}\\mathbf{z}$).\n    While, we observe significant robustness to mismatch, in both cases, we see performance drop drastically once the marginal distribution is more concentrated than the conditional distribution of positive pairs. In such scenarios, positive pairs are indistinguishable from negative pairs. \n\n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=.9\\linewidth]{figures\/uniformity_violation_sigma_kappa_ablation.pdf}\n        \\vspace*{-0.5cm}\n        \\caption{Varying degrees of violation of the uniformity assumption for the marginal distribution. The figure shows the $R^2$ score measuring identifiability up to linear transformations (black) as well as the difference between the used marginal and assumed uniform distribution in terms of probability mass (blue) as a function of the marginal's concentration. The black dotted line indicates the concentration of the used conditional distribution.}\n        \\label{fig:uniformity_violation}\n    \\end{figure}\n    \n\\subsection{Extensions to image data}\n    Previous studies have demonstrated that representation learning using constrastive learning scales well to complex natural image data \\citep{chen2020simple, chen2020big, henaff2020data}.\n    Unfortunately, the true generative factors of natural images are inaccessible, thus we cannot evaluate identifiability scores.\n\n    We consider two alternatives.\n    First, we evaluate on the recently proposed benchmark \\textit{KITTI Masks}~\\citep{klindt2020slowvae}, which is composed of segmentation masks of natural videos.\n    Second, we contribute a novel benchmark (\\textit{3DIdent}; cf. Fig.~\\ref{fig:3dident_examples}) which features aspects of natural scenes, e.g. a complex 3D object and different lighting conditions, while still providing access to the continuous ground-truth factors. For further details, see Appx.~\\ref{apx:3dident_comparison}. \\textit{3DIdent} is available at \\href{https:\/\/zenodo.org\/record\/4502485\/}{zenodo.org\/record\/4502485}.\n\n\\subsubsection{KITTI Masks} \\label{sec:kitti_experiments}\n    KITTI Masks~\\citep{klindt2020slowvae} is composed of pedestrian segmentation masks extracted from an autonomous driving vision benchmark KITTI-MOTS~\\citep{geiger2012are}, with natural shapes and continuous natural transitions. We compare to SlowVAE~\\citep{klindt2020slowvae}, the state-of-the-art on the considered dataset. In our experiments, we use the same training hyperparameters (for details see Appx.~\\ref{apx:experiment_details}) and (encoder) architecture as \\citet{klindt2020slowvae}. The positive pairs consist of nearby frames with a time separation $\\overline{\\Delta t}$.\n    \n    \\begin{table}\n        \\centering\n        \\caption{\\textbf{KITTI Masks}. Mean $\\pm$ standard deviation over 10 random seeds. $\\overline{\\Delta t}$ indicates the average temporal distance of frames used.}\n        \\label{table:MOTSComp}\n        \\small\n        \\begin{tabular}{clll}\n            \\toprule\n                    & Model & Model Space & MCC [\\%] \\\\\n            \\midrule\n            \\parbox[t]{16mm}{\\multirow{5}{*}{$\\overline{\\Delta t}=0.05s$}} & SlowVAE &  Unbounded & 66.1 $\\pm$ 4.5 \\\\\n            & Laplace &  Unbounded & 77.1 $\\pm$ 1.0 \\\\\n            &  Laplace &  Box & 74.1 $\\pm$ 4.4 \\\\\n            &  Normal &  Unbounded & 58.3 $\\pm$ 5.4 \\\\\n            &  Normal &  Box & 59.9 $\\pm$ 5.5 \\\\\n             \\midrule\n           \\parbox[t]{16mm}{\\multirow{5}{*}{$\\overline{\\Delta t}=0.15s$}} &  SlowVAE & Unbounded & 79.6 $\\pm$ 5.8 \\\\\n            &  Laplace &  Unbounded & 79.4 $\\pm$ 1.9 \\\\\n            &  Laplace &  Box & 80.9 $\\pm$ 3.8 \\\\\n            &  Normal &  Unbounded & 60.2 $\\pm$ 8.7 \\\\\n            &  Normal &  Box & 68.4 $\\pm$ 6.7 \\\\\n            \\bottomrule\n        \\end{tabular}\n    \\end{table}\n    \n    As argued and shown in \\citet{klindt2020slowvae}, the transitions in the ground-truth latents between nearby frames is sparse. Unsurprisingly then, Table~\\ref{table:MOTSComp} shows that assuming a Laplace conditional as opposed to a normal conditional in the contrastive loss leads to better identification of the underlying factors of variation. %\n    SlowVAE also assumes a Laplace conditional~\\citep{klindt2020slowvae} but appears to struggle if the frames of a positive pair are too similar ($\\overline{\\Delta t}=0.05s$).\n    This degradation in performance is likely due to the limited expressiveness of the decoder deployed in SlowVAE.\n    \n    \n    \n    \n\\subsubsection{3DIdent} \\label{sec:3dident_experiments}\n    \n\\paragraph{Dataset description}\n    \\begin{figure*}[htb]\n        \\centering\n        \\includegraphics[width=\\textwidth]{figures\/3ddis_samples_reduced.pdf}\n        \\caption{\\textbf{3DIdent}. Influence of the latent factors ${\\bf{z}}$ on the renderings ${\\bf{x}}$. Each column corresponds to a traversal in one of the ten latent dimensions while the other dimensions are kept fixed. %\n        }\n        \\label{fig:3dident_examples}\n    \\end{figure*}\n    \n    We build on \\citep{johnson2017clevr} and use the Blender rendering engine \\citep{blender} to create visually complex 3D images (see Fig.~\\ref{fig:3dident_examples}). Each image in the dataset shows a colored 3D object which is located and rotated above a colored ground in a 3D space. Additionally, each scene contains a colored spotlight focused on the object and located on a half-circle around the scene. The observations are encoded with an RGB color space, and the spatial resolution is $224\\times224$ pixels.\n\n    The images are rendered based on a $10$-dimensional latent, where: (1) three dimensions describe the XYZ position, (2) three dimensions describe the rotation of the object in Euler angles, (3) two dimensions describe the color of the object and the ground of the scene, respectively, and (4) two dimensions describe the position and color of the spotlight. We use the HSV color space to describe the color of the object and the ground with only one latent each by having the latent factor control the hue value. For more details on the dataset see Sec.~\\ref{apx:3dident_details}.\n    \n    The dataset contains $250\\,000$ observation-latent pairs where the latents are uniformly sampled from the hyperrectangle $\\mathcal{Z}$. To sample positive pairs $({\\bf{z}}, {\\tilde{\\mathbf{z}}})$ we first sample a value ${\\tilde{\\mathbf{z}}}'$ from the data conditional $p({\\tilde{\\mathbf{z}}}'|{\\bf{z}})$, and then use nearest-neighbor matching\\footnote{We used an Inverted File Index (IVF) with Hierarchical Navigable Small World (HNSW) graph exploration for fast indexing.} implemented by FAISS \\citep{JDH17} to find the latent ${\\tilde{\\mathbf{z}}}$ closest to ${\\tilde{\\mathbf{z}}}'$ (in $L^2$ distance) for which there exists an image rendering. In addition, unlike previous work~\\citep{locatello2018challenging}, we create a hold-out test set with $25\\,000$ distinct observation-latent pairs. \n\n\\paragraph{Experiments and Results}\n    \\begin{table*}[htb]\n        \\vspace{-0.05cm}\n        \\centering\n        \\caption{Identifiability up to affine transformations on the test set of 3DIdent. Mean $\\pm$ standard deviation over 3 random seeds. As earlier, only the first row corresponds to a setting that matches the theoretical assumptions for linear identifiability; the others show distinct violations. Supervised training with unbounded space achieves scores of $R^2=(98.67 \\pm 0.03)$\\% and $\\text{MCC}=(99.33 \\pm 0.01)$\\%. The last row refers to using the image augmentations suggested by \\citet{chen2020simple} to generate positive image pairs.\n        For performance on the training set, see Appx.~Table~\\ref{tab:3dident_results_train}.\n        }\n        \\small\n        \\begin{tabular}{ccccccc}\n            \\toprule\n            Dataset & \\multicolumn{3}{c}{Model $f$} & Identity [\\%] & \\multicolumn{2}{c}{Unsupervised [\\%]} \\\\\n            $p(\\cdot|\\cdot)$ & Space & $q_{\\rm{h}}(\\cdot|\\cdot)$ & M. & $R^2$ & $R^2$ & MCC \\\\\n            \\midrule\n            \n            Normal & Box & Normal & \\cmark & $5.25 \\pm 1.20$ & $96.73 \\pm 0.10$ & $98.31 \\pm 0.04$\\\\            \n\n            Normal & Unbounded & Normal & \\xmark & \\dittotikz & $96.43 \\pm 0.03$ & $54.94 \\pm 0.02$\\\\\n\n            Laplace & Box & Normal & \\xmark & \\dittotikz & $96.87 \\pm 0.08$ & $98.38 \\pm 0.03$\\\\\n            \n            Normal & Sphere & vMF & \\xmark & \\dittotikz & $65.74 \\pm 0.01$ & $42.44 \\pm 3.27$\\\\\n            \n            Augm. & Sphere & vMF & \\xmark & \\dittotikz & $45.51 \\pm 1.43$ & $46.34 \\pm 1.59$\\\\\n\n            \\bottomrule\n        \\end{tabular}\n        \\label{tab:3dident_results_test}\n    \\end{table*}\n    \n    We train a convolutional feature encoder $f$ composed of a ResNet18 architecture~\\citep{he2015deep} and an additional fully-connected layer, with a LeakyReLU nonlinearity as the hidden activation. For more details, see Appx.~\\ref{apx:experiment_details}. Following the same methodology as in Sec.~\\ref{sec:toy_experiments}, i) depending on the assumed space, the output of the feature encoder is normalized accordingly and ii) in addition to the CL models, we also train a supervised model to serve as an upper bound on performance. We consider normal and Laplace distributions for positive pairs. Note, that due to the finite dataset size we only sample from an approximation of these distributions.\n    \n    As in Tables~\\ref{tab:results_linear} and~\\ref{tab:perm_results}, the results in Table~\\ref{tab:3dident_results_test} demonstrate that CL reaches scores close to the topline (supervised) performance, and mismatches between the assumed and ground-truth conditional distribution do not harm the performance significantly. However, if the hypothesis class of the encoder is too restrictive to model the ground-truth conditional distribution, we observe a clear drop in performance, i.e., mapping a box onto a sphere. Note, that this corresponds to the InfoNCE objective for $L^2$-normalized representations, commonly used for self-supervised representation learning~\\citep{wu2018unsupervised, he2020momentum, tian2019contrastive, bachman2019learning,chen2020simple}.\n    Finally, the last result shows that leveraging image augmentations~\\citep{chen2020simple} as opposed to sampling from a specified conditional distribution of positive pairs $p(\\cdot|\\cdot)$ results in a performance drop. For details on the experiment, see Appx. Sec.~\\ref{apx:experiment_details}.  We explain this with the greater mismatch between the conditional distribution assumed by the model and the conditional distribution induced by the augmentations. \n    In all, we demonstrate validation of our theoretical claims even for generative processes with higher visual complexity than those considered in Sec.~ \\ref{sec:toy_experiments}.\n    \n    \n\\section{Conclusion}\n    We showed that objectives belonging to the InfoNCE family, the basis for a number of state-of-the-art techniques in self-supervised representation learning, can uncover the true generative factors of variation underlying the observational data. To succeed, these objectives implicitly encode a few weak assumptions about the statistical nature of the underlying generative factors. While these assumptions will likely not be exactly matched in practice, we showed empirically that the underlying factors of variation are identified even if theoretical assumptions are severely violated.\n    \n    Our theoretical and empirical results suggest that the representations found with contrastive learning implicitly (and approximately) invert the generative process of the data. This could explain why the learned representations are so useful in many downstream tasks. It is known that a decisive aspect of contrastive learning is the right choice of augmentations that form a positive pair. We hope that our framework might prove useful for clarifying the ways in which certain augmentations affect the learned representations, and for finding improved augmentation schemes.\n \n    Furthermore, our work opens avenues for constructing more effective contrastive losses. As we demonstrate, imposing a contrastive loss informed by characteristics of the latent space can considerably facilitate inferring the correct semantic descriptors, and thus boost performance in downstream tasks. \n    While our framework already allows for a variety of \\emph{conditional} distributions, it is an interesting open question how to adapt it to \\emph{marginal} distributions beyond the uniform implicitly encoded in InfoNCE. Also, future work may extend our theoretical framework by incorporating additional assumptions about our visual world, such as compositionality, hierarchy or objectness. \n    Accounting for such inductive biases holds enormous promise in forming the basis for the next generation of self-supervised learning algorithms. \n    \n    Taken together, we lay a strong theoretical foundation for not only understanding but extending the success of state-of-the-art self-supervised learning techniques.\n\n\\subsection*{Author contributions}\n    The project was initiated by WB.\n    RSZ, SS and WB jointly derived the theory. RSZ and YS implemented and executed the experiments. The 3DIdent dataset was created by RSZ with feedback from SS, YS, WB and MB.\n    RSZ, YS, SS and WB contributed to the final version of the manuscript.\n\n\\subsection*{Acknowledgements}\n    We thank\n    Muhammad Waleed Gondal,\n    Ivan Ustyuzhaninov,\n    David Klindt,\n    Lukas Schott\n    and Luisa Eck\n    for helpful discussions.\n    We thank Bozidar Antic, Shubham Krishna and Jugoslav Stojcheski for ideas regarding the design of 3DIdent.\n    We thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting RSZ, YS and StS.\n    StS acknowledges his membership in the European Laboratory for Learning and Intelligent Systems (ELLIS) PhD program.\n    We acknowledge support from the German Federal Ministry of Education and Research (BMBF) through the Competence Center for Machine Learning (TUE.AI, FKZ 01IS18039A) and the Bernstein Computational Neuroscience Program T\\\"ubingen (FKZ: 01GQ1002). WB acknowledges support via his Emmy Noether Research Group funded by the German Science Foundation (DFG) under grant no. BR 6382\/1-1 as well as support by Open Philantropy and the Good Ventures Foundation. MB and WB acknowledge funding from the MICrONS program of the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior\/Interior Business Center (DoI\/IBC) contract number D16PC00003.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn the survey of the Galactic plane carried out by the \\h experiment, many sources emitting at TeV energies remain unidentified to date  \\citep[e.g.][]{ah08}.\nMost of the sources are extended beyond the point spread function (PSF) of the  \\h experiment ($\\sim 0.06^{\\circ}$ for the analysis presented in this paper). The largest number of conclusive \nidentifications so far can be attributed to pulsar wind nebulae (PWNe) as presented in e.g. \\citet{gallant08}. \nRecently, a new radio SNR, catalogued as \\radio, was discovered by \\citet{tl08} to be in spatial coincidence with \\src,  one of the\n unidentified sources presented in \\citet{ah08}. The diameter of the radio shell is nearly 0.5$\\degr$ \nwhich allows, given the brightness of the source and the \\h angular resolution of $\\sim$0.06$\\degr$, for a morphological comparison of the \\g-ray source with the\n shell observed in radio. \nMoreover, at least up to the current date, no radio pulsar or X-ray PWN candidate was found that might alternatively explain the TeV emission. \nThis situation should be compared to other \\g-ray sources like HESS\\,J1813$-$178 or HESS\\,J1640$-$465 \\citep{aharonian06}, which are also in spatial coincidence with radio SNR shells. However, for these latter sources a morphological identification with the radio shells is not possible, \nand the emission can plausibly originate from a PWN seen in X-rays as discussed in\n\\citet{funk07b} for HESS\\,J1640$-$465 and in \\citet{funk07,gotthelf09} for   HESS\\,J1813$-$178.\n\n\nObservations  of the north-eastern part of  \\src with the X-ray satellites \\x, Chandra, and Suzaku have confirmed an X-ray counterpart\n found in archival ROSAT data \\citep[presented in ][]{ah08,tl08}.\nAn X-ray shell partly matching\nthe radio morphology was found and the spectral analysis has revealed that the X-ray emission is of \nsynchrotron origin,  indicating  that the shock\nwave of the SNR has accelerated electrons up to   TeV  energies \\citep{ap09,tl10}.\nA compact (unresolved) X-ray source XMMU J173203.3$-$344518  \\citep{halpern10} was observed towards the geometrical center of the remnant\nand has spectral properties reminiscent of central compact objects (CCOs) found in several other supernova shells  \\citep[e.g.][]{ps04}.\nA search for pulsations using the EPIC PN cameras onboard \\x shows only marginal evidence of a 1\\,s period \\citep{halpern10}.\n\nGiven the recent discovery of \\radio, little is known about its age and distance. \\cite{tl08} suggested a distance of 3.2 $\\pm$ 0.8 kpc  \nassuming that the SNR is at the same distance as the H{\\sc ii} region G353.42-0.37.\n\nAdditional \\h observations, carried out since the discovery paper of \\src \\citep{ah08}, allow  to investigate the compatibility of the\nTeV source with the radio shell SNR \\radio.  The observations and the data analysis are described\nin Sect. \\ref{sect:data} and  the morphological and spectral results in Sect. \\ref{sect:results}. \nThe multi-wavelength counterparts of \\src are described in Sect.  \\ref{sect:multi} and a general discussion is presented in Sect. \\ref{sect:discussion}.\n\n\n\n\n\\section{\\h observations and analysis methods }\n\\label{sect:data}\n\n\n\\h  is an array of four identical imaging atmospheric Cherenkov telescopes (IACTs)\nlocated in the Khomas Highland of Namibia 1800 m above sea level   \\citep{bernlohr03}.\nThe survey of the Galactic plane by the \\h collaboration has led to the discovery of \nthe \\g-ray source HESS J1731-347, presented as an unidentified extended source in \\citet{ah08}.\nIn this first data set,  14 hours of observation time were available. \nAdditional dedicated observations were carried out in July 2007 and \nin July and August 2009 with zenith angles ranging from 9$\\degr$ to 42$\\degr$, the mean angle being 16.5$\\degr$. \nThe total \\h observation time for this target is 59 hours after data quality cuts.\n\nThe  data set\nwas analyzed using the \\textit{Model analysis} \\citep{dr09} which exploits the\nfull pixel information by comparing the recorded shower images with a pre-calculated shower model\nusing log-likelihood minimization. In comparison with conventional analysis techniques, no cleaning  or parametrization of the image shape is required and the full camera information is used.\nThis method leads to a more accurate reconstruction and better background suppression\nthan more conventional techniques and thus to an improved sensitivity.\n\nSpectral and spatial analyses were carried out using a minimum image intensity of 60 photoelectrons (p.e.) resulting in an energy threshold of 240 GeV and an angular resolution of  0.06$^{\\circ}$ (68\\% containment radius).\n All results presented were cross-checked with a multivariate analysis \\citep{ohm09} using an independent calibration and gamma\/hadron separation,\n  which yielded consistent results. Unless otherwise quoted, the error bars in the following section are given at 1$\\sigma$. \\\\\n \n\\section{TeV \\g-rays analysis results}\n\\label{sect:results}\n\n\n\nThe \\h  excess map of the region of \\src is shown in Fig.~\\ref{fig:excess} smoothed with a Gaussian of $\\sigma$=0.04$^{\\circ}$. \nFor the background estimation in the image and in the morphology studies, the \\textit{ring background} method presented in \\citet{berge07} was used.\nBecause of the larger data set and the more sensitive reconstruction technique, \nthe presented image is much more detailed than the one shown in the\ndiscovery paper \\citep{ah08}. \nThis reveals a complex region composed of a large and bright structure (\\srca), detected\nat 22$\\sigma$, with a suggestive shell-like morphology.\n\nA smaller and fainter structure named \\object{\\srcb} (detected at 8$\\sigma$) is also observed, the \nproperties of which are presented separately in Sect. \\ref{sect:srcb}.\n\n\n\\subsection{TeV energy morphology}\n\n\\begin{figure}\n   \\centering\n\n   \\includegraphics[bb= 89 200 465 565, clip,width=7.7cm]{16425f1.ps}  \n   \\includegraphics[bb= 465 114 516 581, clip,width=0.812cm]{16425f1.ps}\n   \n      \\vspace{-0.6cm}\n\n    \n   \\caption{TeV \\g-ray excess map ($1.5\\degr \\times1.5\\degr$) of the \\src region smoothed with a Gaussian width $\\sigma$=0.04$^{\\circ}$.\n    The average \\h PSF for the dataset is shown in the inset. The regions used for the spectral analysis of \\srca and \\srcb are respectively represented by the large and small dashed circles.\n    The position of the central compact object detected in X-rays is shown with a white cross. The linear scale is in units of excess counts per smoothing Gaussian width. The transition between blue and red in the color scale is at the level of 4$\\sigma$.}\n   \\label{fig:excess}\n\\end{figure}\n\nTo further test    the hypothesis of a shell morphology for \\srca and its association with the radio SNR, radial and azimuthal profiles in radio and \\g-rays were extracted centered on the\nposition of the CCO ($\\alpha_{\\mathrm{J2000}}=$17h32m03s,  $\\delta_{\\mathrm{J2000}}=-34\\degr45'18''$), also coincident\nwith the geometrical center of the radio SNR. The profiles were derived from the uncorrelated \\g-ray excess map\nand corrected for the field of view (FoV) acceptance.\n  \nFor the  \\g-ray radial profile, the position angles\\footnote{Position angle 0$\\degr$ corresponds to North and 90$\\degr$ to East.} from 270$\\degr$ to 310$\\degr$ were excluded, to avoid contamination from \\srcb, and the resulting radial profile was compared with a sphere and a shell model. \nThe first model is a uniformly emitting sphere of adjustable radius, projected on the sky and then folded with  the PSF derived\nfor this analysis ($r_{\\rm 68\\%}=0.06\\degr$).\nThe shell model consists of a uniformly emitting shell of variable outer radius and thickness (defined as $r_{\\rm outer} - r_{\\rm inner}$) projected on the sky and then folded with the same PSF. \n\nIn the morphological test, the best fit statistically favors the shell model and the sphere model  is \nruled out at 3.9 $\\sigma$ ($\\chi^{2}$\/dof = \\chishell  and \\chisphere   \nfor the shell and sphere models respectively).  \nIn the case of a shell model, the best fit  radius is $0.27^{\\circ}  \\pm \\,0.02^{\\circ}$ and the emission is compatible with  a thin, spatially\nunresolved, shell with an upper limit thickness of 0.12$^{\\circ}$ (90\\% confidence level). \n \nTo compare the TeV morphology with the shell seen in radio, \n the radio continuum map from the ATCA southern Galactic plane survey (SGPS)  \\citep{hg06} was \nsmoothed to match the \\h spatial resolution and a radial profile was extracted (excluding point sources).\nThe radio profile was then scaled  by a normalization factor\ncalculated as the ratio of the total number of excess  $\\gamma$-rays over the total radio flux  on the whole remnant. \nThe resulting profiles, presented in Fig.~\\ref{fig:radprof}, show an extended emission in \n\\g-rays similar to that seen in radio. \n\n\nIn contrast with \\rxj which is brighter in the North-West and SN 1006 that exhibits a bipolar morphology, \nthe  azimuthal profile of \\srca (see Fig.~\\ref{fig:azprof}) integrated   for $r \\leqslant 0.3 \\degr$   shows no significant deviation from a flat profile ($\\chi^{2}$\/dof = 8.8 \/ 9).\n\n\n\\begin{figure}\n   \\centering\n      \\includegraphics[width=\\columnwidth]{16425f2.ps}\n            \\vspace{-0.6cm}\n   \\caption{The \\g-ray excess and radio radial profiles are shown with  green crosses and red squares respectively. \nThe best fits  to the \\g-ray data of a sphere and a shell model are overlaid. Both radial profiles \nare centered on the compact central object ($\\alpha_{\\mathrm{J2000}}=$17h32m03s,  $\\delta_{\\mathrm{J2000}}=-34\\degr45'18''$). }\n   \\label{fig:radprof}\n\\end{figure}\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=\\columnwidth]{16425f3.ps}\n      \\vspace{-0.6cm}\n   \\caption{Normalized azimuthal \\g-ray excess profile restricted to radius r $\\leq0.3\\degr$ and using the same center as in Fig.~\\ref{fig:radprof}. The brightness distribution is compatible with a flat profile.}\n   \\label{fig:azprof}\n\\end{figure}\n\n\n\\subsection{Spectral results}\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=\\columnwidth]{16425f4.ps}\n   \\caption{Differential energy spectra of \\srca (filled circles) and \\srcb (open circles). The normalization for the second source has been divided by \n   10 for graphical purposes.  Events were binned to reach a significance of at least 2$\\sigma$. \n   The best fit power-law models along with the residuals for \\srca are also shown.\n   The grey bands correspond to the range of the power-law fit, taking into account statistical errors.}\n   \\label{fig:spec}\n\\end{figure}\n\n\n\nThe energy spectrum of the SNR was obtained by means of a forward-folding maximum likelihood fit \\citep{piron01}\nfrom a circular region of 0.3$\\degr$ centered \non the CCO, illustrated by the large dashed circle ($r=0.3^{\\circ}$) in Fig.~\\ref{fig:excess}, chosen to fully enclose\nthe emission of the remnant. \nThe background is estimated using the \\textit{multiple reflected-regions}\ntechnique where background events are selected from regions of the\nsame size and shape as the source region and at equal angular distance from the\nobservation position \\citep{berge07}. \nThe resulting spectrum, shown in Fig.~\\ref{fig:spec},   is well described by  a power-law model (equivalent $\\chi^{2}$\/dof = 27.7 \/ 35) defined as\n${\\rm d}N\/{\\rm d}E = N_{\\rm 0} (E\/E_{\\rm 0})^{-\\Gamma} $ where $E_{\\rm 0}$ is the decorrelation energy (energy at which the correlation between the slope and the normalization vanishes).\n The best fit parameters, listed in Table \\ref{tab:param}, \nresult in an integrated 1-10 TeV energy flux of (\\fluxa $\\pm \\, 1.38_{\\rm syst}) \\times 10^{-12}$ erg cm$^{-2}$s$^{-1}$.  The flux measured here is lower than what has been derived initially in\n \\citet{ah08} : $(16.2 \\pm  3.6_{\\rm stat} \\pm \\, 3.2_{\\rm syst}) \\times 10^{-12}$ erg cm$^{-2}$s$^{-1}$ in the same energy band.\n  However, the region of extraction in the discovery paper was much larger ($r=0.6^{\\circ}$ versus $r=0.3^{\\circ}$\n in this paper),  including \\srcb and possibly some surrounding diffuse emission. \n A cross-check to derive the flux from the SNR only using the same data set as used in \\citet{ah08} and following the original analysis method \n gave results consistent with the complete data set presented here thus confirming that the flux difference was mainly due to the choice of the integration region.\n A power-law model with an exponential cutoff was also tested which did not improve the quality of the fit (equivalent  $\\chi^{2}$\/dof = 24.0 \/ 34). \n\n\n\n\\begin{figure*}[!t]\n   \\centering\n   \\begin{tabular}{ccc}\n\\hspace{-0.6cm}\n { \\includegraphics[bb= 43 137 570 650 , clip,width=6.23cm]{16425f5.ps}  } & \\hspace{-6mm}\n { \\includegraphics[bb= 40 137 565 641 , clip,width=6.20cm]{16425f6.ps} }  & \\hspace{-6mm}\n { \\includegraphics[bb= 43 137 570 648 , clip,width=6.23cm]{16425f7.ps} } \n \n   \\end{tabular}\n\n\\vspace{-3mm}\n\n   \\caption{Multi-wavelength view of the  \\src region. The radio and the \\g-ray image show the same field\n    of view  while the X-ray image is zoomed in order to show the details of the shell structure. The significance contours at 4, 6 and 8 $\\sigma$ obtained with an\n   integration radius of 0.06$\\degr$ and the Galactic plane (white dashed line) are overlaid in the three panels.  \n     \\textit{Left} : ATCA radio map at 1.4 GHz from the south Galactic plane survey (SGPS)\n     in units of Jy\/beam with a beam of 100''. \n   The HII regions G353.42-0.37 (left) and G353.381-0.114 (right) are marked with arrows. \\textit{Middle} :   \\x observation of a sub-region of the SNR, in the 0.5-4.5 keV energy band, using MOS instruments with units in ph\/cm$^{2}$\/s\/arcmin$^{2}$.  The position of the source XMMU J173203.3$-$344518, \n   which is likely to be the CCO of the SNR is shown by the red arrow. \\textit{Right} : \n   TeV \\g-ray excess map of \\src  smoothed with a Gaussian with $\\sigma$=0.04$^{\\circ}$.\n    The region used to derive the radio flux and the spectral parameters in X- and \\g-rays for the SED is also shown. }\n\n   \\label{fig:4contrib}\n\\end{figure*}\n\n\n\n\\subsection{\\srcb}\n\\label{sect:srcb}\n\n\nA \\g-ray excess of TeV emission was found  at the best fit position \n$\\alpha_{\\mathrm{J2000}}=$17h29m35s,  $\\delta_{\\mathrm{J2000}}=-34\\degr32'22''$ with a statistical error of 0.035\\degr and the source was therefore\nlabeled \\srcb.\n\nThe source is extended beyond the size of the PSF  (Gaussian width $\\sigma=0.12\\degr \\pm 0.03\\degr$)\n and the region used to derive \nthe spectral parameters is shown by the small dashed circle ($r=0.14\\degr$) in Fig.~\\ref{fig:excess}. The spectrum obtained is well modeled by a power-law model (see Fig.~\\ref{fig:spec}) and the best fit parameters are listed in Table \\ref{tab:param}. The integrated flux in the 1-10 TeV energy band is  (\\fluxb $\\pm \\,   0.18_{\\rm syst}) \\times 10^{-12}$ erg cm$^{-2}$s$^{-1}$.\n\n\n\n\\begin{table*}\n\\begin{minipage}[t]{\\textwidth}\n\\centering\n\\caption{Best fit spectral parameters obtained for different extraction regions in \\src. The model used is a power-law of the form ${\\rm d}N\/{\\rm d}E = N_{\\rm 0} (E\/E_{\\rm 0})^{-\\Gamma} $. The systematic errors are conservatively  estimated  to be $\\pm$ 0.2 on the photon index and 20\\% on the flux.   }\n\\label{tab:param}\n\n\\renewcommand{\\footnoterule}{} \n\\begin{tabular}{l c c c c  }    \n\\hline\n\nRegion & Photon index $\\Gamma$ & Decorrelation energy $E_{\\rm 0}$& Normalization $N_{\\rm 0}$ & 1-10 TeV integrated flux \\\\\n           &                          & TeV            &   $10^{-12}$ cm$^{-2}$s$^{-1}$ TeV$^{-1}$  &  $10^{-12}$ erg cm$^{-2}$s$^{-1}$ \\\\\n\\hline\n\n   \\srca & \\slopea &  0.783 & \\norma & \\fluxa  \\\\\n   sub-region of  \\srca \\footnote{A spectral analysis corresponding to the FoV of the \\x  data (see Fig.~\\ref{fig:4contrib}, center) has been carried out in order to build a SED.}  &   \\slopesub &  0.780 &\\normsub & \\fluxsub  \\\\\n      \\srcb &   \\slopeb & 0.861 &\\normb & \\fluxb  \\\\\n\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{table*}\n\n\n\\section{Multi-wavelength counterparts}\n\\label{sect:multi}\n\n\n\n\nOne of the interesting characteristics  of \\srca is that  non-thermal emission is clearly identified in \nradio, X-rays and at  TeV energies.\nIn X-rays however,  the access to the spectral properties  is limited to a subregion of the SNR as \nthe coverage with the \\x, Chandra and Suzaku  satellites is only partial, and the statistics in the ROSAT All Sky Survey data are too low.\nIn order to study the spectral energy distribution (SED) of the source, \n the radio flux and the TeV spectral properties were extracted only from the region observed in X-rays (see region definition in Fig.~\\ref{fig:4contrib}, right).  \nThe multi-wavelength counterparts of \\srcb are discussed later in Sect. \\ref{sect:robin}. \n\n\n\\subsection{Radio Continuum}\n\\label{sect:radio}\n\n\nThe shell  observed in radio is spatially coincident with the \\g-ray shell and has a similar extent (see radial profile in Fig.~\\ref{fig:radprof}).\nThe flux obtained (excluding point sources)  from the SGPS ATCA data in the region observed by the \\x pointing is  0.8 $\\pm$ 0.3 Jy at 1420 MHz.\nThe total radio flux for the SNR measured by  \\citet{tl08} is of 2.2 $\\pm$ 0.9 Jy.\nThe compact HII region (G353.42-0.37) located to the West of the remnant at a distance of  3.2 $\\pm$ 0.8 kpc \\citep{tl08} is indicated \nin Fig.~\\ref{fig:4contrib} (left).\n\n  \n\n\\subsection{X-rays}\n\\label{sect:xray}\n\n\nIn order to derive spectral information from the X-ray emission from the remnant, the \\x pointing obtained as a follow up of the HESS source \n(ObsId: 0405680201 ; PI: G. P\\\"uhlhofer) was analyzed. \nTo clean the proton flare contamination during the observation, a histogram of the 10-12 keV count rates of each \ncamera was built. A Gaussian fit was then performed in order to remove time intervals where the count rates were beyond 3 $\\sigma$ from the mean value \\citep{pa02}. The remaining exposure time after flare screening is 22 ks out of the 25 ks of observation for MOS and 15 ks for PN. For the image generation, the instrumental background was derived from the compilation of blank sky observations by \\citet{cr07}\n and renormalized in the 10-12 keV band over the whole  FoV. The image resulting from the combination of the\n  two MOS instruments  is presented in Fig.~\\ref{fig:4contrib} (middle). \nFor this mosaic, the data from the PN instrument were not used because of  straylight contamination to the\nNorth-East (photons  singly reflected by the mirrors) from a bright X-ray source located outside the FoV. \nThis results in some spurious arc features near the border of the FoV in the North-East.\n\n\nThe X-ray emission is characterized by extended emission which is concentrated in arc-like features, similar to broken shell seen from many shell-type SNRs. \nSome of the arcs partly coincide with the radio and \\g-ray shell (see Fig.~\\ref{fig:4contrib}). Some of the structures could hint at an additional, smaller shell, \nbut might also come from irregular SNR expansion in an inhomogeneous and\/or dense medium \\citep{blondin01}.  \nA double-shell structure is also observed in \\rxj in  X-rays \\citep{lazendic04,cd04,acero09}. \n\nThe spectral analysis of the diffuse X-ray emission was carried out using the  Extended Source Analysis Software\n(ESAS\\footnote{http:\/\/xmm2.esac.esa.int\/external\/xmm\\_sw\\_cal\/background\/\nepic\\_esas.shtml})\n provided in the \\x Science Analysis System (SAS v9.0) to model the particle and instrumental backgrounds. \nThe error bars in this section are quoted at 90\\% level confidence.\nFor this analysis, the three instruments PN+MOS1+MOS2 were used and the regions were selected to avoid the straylight features.\n\nThe spectrum derived from the region covered by the FoV of \\x  that is used for the SED  is shown in Fig.~\\ref{fig:xrayspec}. \nThe emission is well represented by an absorbed power-law model and no emission lines were found \\citep[see also][]{tl10}. \nThe best fit parameters obtained from  a joint fit of MOS1, MOS2 and PN spectra  are \n$N_{\\rm H}=(1.08 \\pm 0.02) \\times 10^{22}$ cm$^{-2}$,  a spectral index $\\Gamma= 2.28  \\pm 0.03$ and a normalization at 1 keV  \n$N_{\\rm 0}= (1.37  \\pm 0.05) \\times 10^{-2}$ cm$^{-2}$  s$^{-1}$ keV$^{-1}$.\nA search for spatial spectral variations  of the diffuse emission revealed that the power-law index is in most locations in the \nrange $\\Gamma = 2.1-2.5$. Under the assumption  of a pure power-law hypothesis, the  absorption column significantly increases towards the Galactic plane from \n$N_{\\mathrm{H}} = (0.93  \\pm  0.05) \\times 10^{22}$ cm$^{-2}$ in the South-East region to \n$N_{\\mathrm{H}} = (2.23   \\pm 0.21) \\times 10^{22}$ cm$^{-2}$  in the North-West region (see Fig.~\\ref{fig:xraynh} ; left).\nThe errors on the absorption column shown in Fig.~\\ref{fig:xraynh} (left) are ranging from 5\\% to 12\\%.\n\n  \n The bright point source XMMU J173203.3$-$344518 lies at the geometrical center of the radio and \\g-ray shell.  \nMarginal evidence for a pulsation at a period of 1 s for the pulsar candidate and a faint nebula (radius of 30'') \nwhose spectral properties are compatible with a dust-scattered halo have been reported by \\citet{halpern10}. \nAs no optical or \nIR counterpart of the point source have been  detected and as the X-ray spectrum is well described by a blackbody emission model \nwith $kT\\sim0.5$ keV,  the object is\na good candidate to be  the CCO of the SNR \\citep{ap09,halpern10,tl10}. \n \n\n\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[bb= 70 25 590 705,clip,width=6.4cm,angle=-90]{16425f8.ps}\n   \\vspace{-0.2cm}\n\n   \\caption{X-ray spectrum, using the PN camera, extracted from the sub-region of \\srca shown in Fig.~\\ref{fig:4contrib} (right). \n   The non-thermal emission from the SNR  is  described by an absorbed power-law model (dashed line). \n   The local astrophysical background, fitted to an off region outside the SNR, is modeled by two components (dotted lines).\n    The low energy component is an APEC model (astrophysical plasma emission code, see http:\/\/hea-www.harvard.edu\/APEC)\n  representing the background from the Local Bubble and\n     the high-energy component is an absorbed power-law representing the hard X-ray background  (unresolved AGNs, cataclysmic variables, etc). \n     The residuals of the total model (SNR+local astrophysical background) are shown in the lower panel and the $\\chi^{2}$\/ndof  is 1921\/1569.}\n   \\label{fig:xrayspec}\n\\end{figure}\n\n\n\n\\begin{figure*}\n   \\centering\n    \\begin{tabular}{cc}\n    \n   {\\includegraphics[bb=-80 28 504 519,clip,width=7.8cm]{16425f9.ps} } & \\hspace{1.cm}\n   {\\includegraphics[bb=45 156 570 620,clip,width=7.6cm]{16425f10.ps} } \\\\   \\vspace{0.5cm}\n   \n \\hspace{-2.1cm}   {\\includegraphics[bb=-120 0 440 27,clip,width=7.5cm]{16425f9.ps}}      &   \n  \\hspace{-1.5cm} {\\includegraphics[bb=-120 0 440 27,clip,width=7.5cm]{16425f9.ps}} \n   \n      \\vspace{-0.7cm}\n\n   \\end{tabular}\n\n\n\n   \\caption{ \\textit{Left } : X-ray absorption map derived from a spectral fit to \\x data assuming an absorbed power-law model. A significant increase of $N_{\\rm H}$ towards the Galactic plane is observed. \\textit{ Right } : Absorption column map derived from atomic and molecular hydrogen when integrating over radial velocities from 0 km\/s to  $-$25 km\/s (see Sect. \\ref{sect:CO}  for more details). The Galactic plane is represented by the white dashed line. In both panels, the \\x field of view is represented by a dashed circle and the X-ray contours\n   obtained from Fig.~\\ref{fig:4contrib} (center)  are overlaid. }\n   \\label{fig:xraynh}\n\\end{figure*}\n\n\n\\subsection{$^{12}$CO (J=1-0) and HI}\n\\label{sect:CO}\n\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[bb=70 360 585 1040,clip,width=\\columnwidth]{16425f11.ps}\n       \\vspace{-0.5cm}\n     \\caption{ \n      \\textit{Top} : Cumulative absorbing column density (solid line) as a function of radial velocity at the position of highest X-ray absorption (see Sect. \\ref{sect:CO}).\n       The relative contributions from the atomic and molecular hydrogen are represented by the dashed and dash-dotted lines respectively.\n   \\textit{Middle} : Rotation curve towards the same direction as derived from the model of Galactic rotation of  \\citet{hou09}.   \n    \\textit{Bottom} :  $^{12}$CO (dashed line) and HI (dash-dotted line) spectra obtained the region highest X-ray absorption. \n}\n   \\label{fig:COspec}\n\\end{figure}\n\nThe comparison of the  absorption along the line of sight derived from X-ray data,  $^{12}$CO and HI observations can be used to constrain the distance to the SNR.\nThe velocity spectra of the $^{12}$CO emission \\citep[using data from the CfA survey, ][]{dame01} and the HI emission \n\\citep[using data from the SGPS survey, ][]{hg06}  derived from the region of highest X-ray \nabsorption ($\\alpha_{\\mathrm{J2000}}=$17h31m43s,  $\\delta_{\\mathrm{J2000}}=-34\\degr34'58''$) are shown in Fig.~\\ref{fig:COspec} (bottom).\n\nIn order to derive a lower limit on the integration distance required to match the $N_{\\rm H}$ derived from X-rays, all the material is\nassumed to be at the near distance allowed by the Galactic rotation curve. \nUnder this hypothesis,  the cumulative absorption column derived from the atomic and molecular  hydrogen shown in Fig.~\\ref{fig:COspec} (top) \nis similar to the one observed in X-rays, $N_{\\mathrm{H}} = (2.23   \\pm 0.21) \\times 10^{22}$ cm$^{-2}$, when integrating\n up to a radial velocity relative to the \\textit{local standard of rest} (LSR) of $-$25 km\/s.\n The CO-to-H$_{2}$ mass conversion factor and the HI brightness temperature to column density used are respectively of\n$1.8\\times10^{20}$ cm$^{-2}$ K$^{-1}$ km$^{-1}$ s \\citep{dame01} and $1.82\\times10^{18}$ cm$^{-2}$ K$^{-1}$ km$^{-1}$ s \\citep{dickey90}.\n \n When integrating up to the same velocity, the  map of $N_{\\rm H}$ derived from the atomic and molecular hydrogen shown in Fig.~\\ref{fig:xraynh} \n(right) exhibits an increase of absorption towards the Galactic plane similar to that in the X-ray absorption map  in Fig.~\\ref{fig:xraynh} (left).\nThe peak of $^{12}$CO emission at a LSR velocity of $-$18 km\/s is thus in the foreground of the SNR and is likely to be the cause of the gradient of absorption seen in X-rays.\nUsing the circular Galactic rotation model of \\cite{hou09} with a distance to the Galactic center of 8.0 kpc, the \nnearest distance corresponding to the LSR velocity of $-$18 km\/s is \\dist kpc thus setting a lower limit for the distance of the remnant.\n\n\n\\begin{figure}\n   \\centering\n\\hspace{-0.3cm}   \\includegraphics[bb=56 210 525 585,clip,width=9.3cm]{16425f12.ps}\n\\hspace{-0.43cm}    \\includegraphics[bb=56 210 525 585,clip,width=9.3cm]{16425f13.ps}\n   \\caption{$^{12}$CO map of the vicinity of \\src integrated from LSR velocity $-$13 km\/s to $-$25 km\/s (top) and from $-$75 km\/s to $-$87 km\/s (bottom)\n    respectively corresponding to the intervals of the second and third $^{12}$CO peak shown in Fig.~\\ref{fig:COspec}.\n   The HESS significance contours from Fig.~\\ref{fig:4contrib} together with the Galactic plane and the \n   Fermi 95\\% position confidence level contours presented in the 1 year catalog are overlaid.\n   The linear scale is in units of K km\/s. }\n   \\label{fig:COslice}\n\\end{figure}\n\n\n\n\\subsection{GeV \\g-rays}\n\\label{sect:fermi}\n\n\n\nIn the Fermi-LAT first year catalog \\citep{abdo10cat} the source 1FGL J1729.1$-$3452c is found in the neighborhood of \\src as shown in Fig.~\\ref{fig:COslice}. \nThe Fermi source has an analysis flag that indicates that the source position  moved beyond its 95\\% error ellipse when changing the model \nof diffuse emission.\nThe $^{12}$CO map in Fig.~\\ref{fig:COslice} shows that the Fermi source is located near a small scale gas clump that could be not well represented in the  \ndiffuse emission model. The position of the source presented in the catalog is to be used with caution and is therefore possibly not incompatible with \\srcb.\n\nThe Fermi source has a photon spectral slope of 2.26$\\pm$0.08 and shows neither indication for spectral curvature nor time variability on a time scale of months \n(the catalog does not address shorter or longer time variations). This source is the closest Fermi detection\nnear the newly discovered SNR and the flux derived in the Fermi catalog is used as an upper limit in the SED of the SNR in Fig.~\\ref{fig:SED}.\n\n\n\n\\subsection{Multi-wavelength counterparts for \\srcb}\n\\label{sect:robin}\n\n\nAt radio wavelengths, the \\g-ray contours of \\srcb lie near the HII region G353.381-0.114. Using HI radio recombination line data, the LSR velocity\ncorresponding to this source is either $-$54 km\/s or $-$82 km\/s \\citep{caswell87}.\n In the latter case this HII region could be associated with the molecular cloud observed around\nvelocities of $\\sim$ $-$80 km\/s (see Fig.~\\ref{fig:COslice}). \nAt X-ray energies, no archival dedicated observations were found, and no emission is detected in the ROSAT all sky survey, probably due to the high \nabsorption in the line of sight. As discussed in the previous section, a Fermi source is found to lie close to \\srcb. \n\n\n\n\\section{Discussion}\n\\label{sect:discussion}\n\n\nThe newly discovered SNR \\srca is in several ways comparable to \\rxj and \\vela. Those objects are X-ray synchrotron emitters and exhibit\n no thermal emission lines. A CCO is also found within those three SNRs indicating a core collapse SN.\nMoreover at a distance of \\dist kpc (see Sect. \\ref{sect:CO}),\nthe TeV luminosity of \\srca in the 1-30 TeV energy band is 1.07$\\times (d\/\\dist \\, \\rm{kpc})^{2} \\, 10^{34}$ erg s$^{-1}$ which is similar to the luminosity of \\rxj \n(the brightest TeV shell SNR detected until now), of 0.81$\\times 10^{34}$ erg s$^{-1}$ using a distance of 1 kpc \\citep{fm03,cd04,mt05} \nand slightly higher than that of  \\vela with 0.65$\\times 10^{34}$ erg s$^{-1}$ at a distance of 0.75 kpc \\citep{katsuda08}.\nA difference with \\rxj is that the flat \\g-ray  azimuthal profile of \\srca (see Fig.~\\ref{fig:azprof}) suggests that the remnant is evolving in a relatively uniform ambient medium and \nthat it is not in interaction with the cloud (shown in Fig.~\\ref{fig:COslice}, top) used to derive a lower limit to the distance of the SNR. \nThis significantly differs from the case of \\rxj which exhibits much brighter \\g-ray emission in the North-West where the shock is thought to interact with denser material.\n\nThe distance used for the luminosity is derived from the absorption in the foreground and provides only a lower limit of \\dist kpc.\nHowever, as it is believed that supernova explosions are more likely to  occur in the spiral arms of the Galaxy  where the density of massive stars \n(i.e. SNR progenitors) is higher \\citep{russeil03,hou09}, it is likely that \\srca could be located within\nthe  Scutum-Crux or  Norma arms, which cross the line of sight at $l=353.5^{\\circ}$ at $\\sim$3.0 and $\\sim$4.5 kpc respectively \\citep{hou09}.\nThe next arm in the  same line of sight is the Sagittarius arm lying at a distance of 12 kpc. This latter possibility for the location\nof the SNR would lead to a much higher \\g-ray luminosity,  an order of magnitude higher than \\rxj. Also at such a distance, the physical size of the \nremnant would exceed 50 pc, substantially larger than other TeV shell SNRs whose physical size is $\\lesssim$ 15 pc. \nAs a result it is reasonable to believe that the real distance to the SNR should not be much larger than the derived lower limit of \\dist kpc. \n\nThe radio flux and the X- and \\g-ray spectra derived in Sect. \\ref{sect:multi} from the sub-region of \\srca that is covered by the FoV of \\x  were combined\nin the SED presented in Fig.~\\ref{fig:SED}. The X-ray data were corrected for the interstellar absorption with \n$N_{\\rm H}=1.08 \\times 10^{22}$ cm$^{-2}$.\n To model the SED, a simple one-zone stationary model \\citep[presented in ][]{acero10} was used.  \nIn this model, the spectrum of electrons and protons is represented by a power-law of slope $s$ with exponential cutoffs at energies\n$E_{\\rm c,e}$ and $E_{\\rm c,p}$ for the electrons and protons respectively.   \nFor the modeling of the object, it is assumed that the measured multi-wavelength emission from the sub-region of \\srca is \nentirely coming from the SNR located at a distance of \\dist kpc. \nAs this distance is only a lower limit, the total energy of accelerated particles ($W_{\\rm e}$ and $W_{\\rm p}$) \nin the SNR should also be viewed as lower limits.\n\nIn the pure leptonic scenario, the slope of the electrons is constrained by the radio and the X-ray synchrotron emission between 1.9 and 2.1 and\nthe strength of the magnetic field required to reproduce the ratio of observed synchrotron and IC emission \nlies between 20 and 30 $\\mu$G  for $15 \\leq E_{\\rm c,e} \\leq 25$ TeV. Although the relative ratio of radio, X- and TeV fluxes can be fairly well\nreproduced by this leptonic scenario, the model is inadequate to account for the X-ray and the  \\g-ray spectral slope \nas illustrated in Fig.~\\ref{fig:SED} (top). The corresponding parameters for the latter model are summarized in Table \\ref{tab:SED}.\n\nThis limitation no longer occurs in a scenario where the TeV emission is dominated by hadronic\nprocesses as the X- and \\g-ray emission are now independent and both spectral slopes can be reproduced as shown in Fig.~\\ref{fig:SED} (bottom).\nMoreover, the strength of the magnetic field can be increased  as it is no longer fixed by the X\/\\g ratio. \nIn order to reproduce the observed TeV flux, the total energy in high-energy protons ($E \\, \\geq$ 1 GeV) assuming a spectral slope of 2.0 \nis $W_{\\rm p} = 2 \\times 10^{50}$ ($n$\/1 cm$^{-3})^{-1}$ ($d$\/\\dist kpc)$^{2}$ erg. \nIt should be noted that this energy content only represents a sub-region of the SNR accounting for $\\sim$ 1\/3 of the total TeV flux\n(see Table \\ref{tab:param}) implying that the total energy transferred to accelerated protons in the whole SNR is \n a substantial fraction of the energy available in the remnant for $n \\sim$ 1 cm$^{-3}$.\nFor this energetic reason, gas densities much below this value appear incompatible with the hadronic emission scenario.\n\n\n\nAlthough it is not possible to measure the density of the ambient medium surrounding the SNR as no X-ray thermal emission is detected,\nan upper limit on the density can be derived. \nIn order to do so, a thermal component, whose normalization is fixed for a given density using the method presented in \\citet{ab07} (Sect. 3.1), is\n added to the X-ray spectrum. \n The shocked ambient medium is assumed to be in a non equilibrium ionization state with an ionization timescale parameter\n$\\tau=10^{9}$ cm$^{-3}$ s and an electron plasma\ntemperature $kT_{\\rm e}$=1 keV.  Such values are commonly observed in other young SNRs \nfor which the X-ray emission of the shocked ambient medium has been studied as in e.g. RCW86 \\citep[see][]{vink06}.\nFor the given parameters, the derived upper limit (90\\% confidence level) on the ambient medium density is $10^{-2}$ cm$^{-3}$.\nIn the case of a lower temperature ($kT_{\\rm e}$=0.15 keV),  an upper limit of 1 cm$^{-3}$ is reached.\n\nFor a density of 1 cm$^{-3}$  the corresponding shock speed and age of the SNR\nwould be  $\\sim$410 km\/s and 14000 yrs in order to match a physical radius of \n$R_{\\rm shock}$=15 pc (0.27$^{\\circ}$ at \\dist kpc), for  a  SN  explosion of $E_{\\rm SN}=1 \\times 10^{51}$ erg with a mass  of ejecta of 5 $M_\\odot$ \nusing equations from \\citet{tm99}.\nHowever, this shock speed is an order of magnitude lower than what has been measured in other bright synchrotron emitting SNRs like SN 1006 \n\\citep[$V_{\\rm sh}$=5000 $\\pm$ 400 km\/s at a distance of 2.2 kpc ; ][]{katsuda09}, RCW 86 \\citep[$V_{\\rm sh}$=6000 $\\pm$ 3000  km\/s ; ][]{helder09}, \nCasA \\citep[$V_{\\rm sh}$=4900  km\/s ; ][]{patnaude09} or Tycho \\citep[$V_{\\rm sh} = 3000 \\pm 1000 $ km\/s at a distance of 2.3 kpc ; ][]{katsuda10}. \nAs a rough estimate, the required density  to reproduce a canonical shock speed of 3000 km\/s  using the aforementioned SN parameters\nis of the order of 0.01 cm$^{-3}$  (compatible with the upper limit derived from the lack of thermal X-ray emission in the previous paragraph) for a corresponding age of $\\sim$ 2500 yrs. \n\nTo summarize, the presented static one-zone model suffers from limitations in both scenarios. In the leptonic case,\nthe model allows to estimate  the average B-field ($\\sim 25 \\, \\mu$G) and the total energy in accelerated electrons present in the shell of the SNR but\n fails to reproduce the observed X-ray and \\g-ray spectral slope. In the hadronic model, the high medium density required to \nreproduce the observed TeV flux is hardly compatible with the hydrodynamics of the SNR.\nMore detailed models using  non-linear diffuse shock acceleration theory have been developed \n\\citep[e.g.][]{zirakashvili10,ellison10} and would  provide more accurate predictions than the simple model presented here.\nIt should be noted that the considered model does not take into account evolution related to radiative cooling\nwhich  could yield a steeper gamma-ray spectrum, in better agreement with the data.  \nAlso, the presented scenarios do not cover possible non-homogeneous surroundings such as wind bubble blown by the progenitor.\nSuch detailed spectral and evolutionary modeling depends on many poorly known parameters and is therefore beyond the scope of the\npresent discussion.\n\n\nConcerning the source \\srcb, detected in the vicinity of the SNR, the presented multi-wavelength data\ndo not provide a clear understanding of the nature of the object. \nThe closest structures located near the \\g-ray emission are \nthe  HII region G353.381-0.114 (seen in the radio in Fig.~\\ref{fig:4contrib}, left) and\na molecular gas clump observed in $^{12}$CO (see Fig.~\\ref{fig:COslice}, bottom) when integrating around a LSR velocity\n of $-$80 km\/s  (corresponding to near and far kinematic distances of $\\sim$6 and $\\sim$10 kpc respectively). \nIf the \\g-ray source \\srcb is associated with those gas structures, it would therefore not be associated with the SNR HESS J1731-347 thought to lie at a closer distance.\n \n\n\n\n\\begin{table}\n\\centering\n\\caption{List of the parameters used for the spectral energy distribution modeling presented in Fig. \\ref{fig:SED}. \nThe spectral slope are  fixed at 2.0 for the electron and the proton distribution. The density of the ambient medium was set to 1 cm$^{-3}$ in the case of the \nhadronic model.}\n\\label{tab:SED}\n\n\\renewcommand{\\footnoterule}{} \n\\begin{tabular}{l c c c c c }    \n\\hline\n\nModel & $E_{\\rm c,e}$ & $E_{\\rm c,p}$ & $W_{\\rm e}$ & $W_{\\rm p}$ & $B$ \\\\\n           &          TeV                &  TeV            &   $10^{47}$ erg  & $10^{50}$ erg & $\\mu$G \\\\\n\\hline\n\nleptonic   &        18     &       $-$         &      1.1      &       $-$       &      25    \\\\\nhadronic   &       16      &        100        &       0.25     &       2.0       &    50     \\\\\n\n\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[bb= 142 55 575 707, clip,width=6cm,angle=-90]{16425f14.ps} \\\\\n   \\includegraphics[bb= 142 55 575 707 , clip,width=6cm,angle=-90]{16425f15.ps} \n      \\vspace{-0.2cm}\n\n   \\caption{Broadband SED for the sub-region of \\srca that is observed in X-rays (see Fig.~\\ref{fig:4contrib}, right, for region definition).\n    A purely leptonic (top) and a hadronic (bottom) scenario  are shown and the corresponding parameters for both models are\n     presented in Table \\ref{tab:SED}. The infrared (IR) seed photons energy density and temperature were \n     respectively set to 1 eV cm$^{-3}$ and 40 K following the model from \\citet{porter08} for a galactocentric radius\n     of 4 kpc. \n     The flux from the nearby Fermi source 1FGL J1729.1$-$3452c is represented and used as an upper limit. }\n   \\label{fig:SED}\n\\end{figure}\n\n\n\\section{Conclusion}\n\n\nThe newly discovered SNR \\srca exhibits a significant shell morphology spatially resolved by \\h,  similar to the one observed in radio.\nTogether with \\rxj, \\vela and SN 1006, \\srca is now the fourth\\footnote{With the current statistics, the shell morphology of RCW86 is not statistically significant \\citep{aharonian09}.} TeV \\g-ray source to join this small but growing class. \nA lower limit to the distance of the SNR of \\dist kpc was obtained by comparing the absorption derived from the X-rays and from HI and $^{12}$CO observations.\n\nThe multi-wavelength emission from \nthe SNR, detected in radio, X-rays and  \\g-rays, was combined in an SED to investigate the origin of the \\g-ray emission assuming that the broadband emission\nstems from the same region (one-zone model). While the measured fluxes can\nbe accounted for in a purely leptonic model with a magnetic field of the order of 25 $\\mu$G, this simple model fails to reproduce the spectral shape of\n the X- and  \\g-ray emission.\nA second model that assumes that the TeV emission is produced by hadronic processes is able to reproduce the spectral slopes in X- and \\g-rays at the cost\nof requiring that a large fraction of the kinetic energy of the explosion must be transferred to the accelerated protons and a high ambient medium density \nof $n\\sim$1  cm$^{-3}$ for $d \\geq$ \\dist kpc.\nMoreover  for such a density, the corresponding shock speed of the SNR would be an order of magnitude lower than in other SNRs exhibiting bright \nsynchrotron emission.\n\n\n\n\n\n\n\\begin{acknowledgements}\nThe support of the Namibian authorities and of the University of Namibia\nin facilitating the construction and operation of HESS is gratefully\nacknowledged, as is the support by the German Ministry for Education and\nResearch (BMBF), the Max Planck Society, the French Ministry for Research,\nthe CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the\nCNRS, the U.K. Science and Technology Facilities Council (STFC),\nthe IPNP of the Charles University, the Polish Ministry of Science and \nHigher Education, the South African Department of\nScience and Technology and National Research Foundation, and by the\nUniversity of Namibia. We appreciate the excellent work of the technical\nsupport staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris,\nSaclay, and in Namibia in the construction and operation of the\nequipment. Article based on observations obtained with XMM-Newton, \nan ESA science mission with instruments and contributions directly \nfunded by ESA Member States and NASA. \n\\end{acknowledgements}\n\\vspace{-0.4cm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplemental Material}\nSec.~I provides additional information on the experimental setup, on the reliability of the measurements yielding negative values for $Q_D$ and on the absence of ions within the laser excitation volume. Within the theoretical description based on the DCS, after a brief recall of the mean-field Sec.~II describes the construction of a basis of\nnon-symmetric states the construction of the bath of the non-symmetric states and their excitation by the laser. Sec.~III discusses the vdW parameters determining the Rydberg interactions.\n\n\\section{Set-up}\n\\indent  Experiments are performed using a two-photon excitation scheme for Rydberg excitations in a small cloud of around $10^5$ $^{87}$Rb atoms trapped in a MOT of radius around 30~$\\mu{}$m and densities between $1\\times10^{10}$ and $5\\times10^{10}$ atoms\/cm$^{3}$ determined to within a factor of $2$ from fluorescence images (for details see \\cite{Viteau2010}). The sizes of the excitation beams (around 200~$\\mu{}$m) are chosen such as to obtain intense beams that are approximately uniform across the atomic sample. Single atom two-photon Rabi frequencies of around 100 kHz are achieved. The excitation lasers are pulsed for up to $20\\,\\mathrm{\\mu s}$, after which the Rydberg atoms are field ionized and the ions are detected by a channeltron. For the two-photon excitation, a laser beam at 421~nm is detuned around 0.5--1~GHz from the $\\mathrm{5S_{1\/2}(F=2)}\\rightarrow\\mathrm{6P_{3\/2}(F'=3)}$ transition, and an infra-red beam of wavelength 1013--1015~nm provides the second step to the Rybderg state $\\mathrm{6P_{3\/2}(F'=3)}\\rightarrow\\mathrm{nS\/nD}$. Both of these beams are controlled using acousto-optic modulators (AOMs) with a rise-time of around 80~ns. Both lasers are stabilised to a scanning Fabry-Perot cavity to prevent long-term frequency drift. We estimate that the two-photon linewidth on the timescale of the excitation pulses is around 300~kHz. In order to achieve reliable statistics, the excitation sequence is repeated up to a few hundred times with the repetition rate limited to $10$ Hz in order to avoid a buildup of charge on the quartz cell surrounded by the external field plates \\cite{Viteau2011}. The complete acquisition procedure takes around 30~s.\nThe detection efficiency for ions in our experiment is around $40\\%$.\\\\\n\\indent In order to verify that the observed sub-Poissonian statistics for resonant excitation is actually due to collective Rydberg excitations and not an artefact of the limitations of the experimental apparatus, we performed a series of additional experiments. Most importantly, we performed an experiment that interpolated between Poissonian and sub-Poissonian statistics for a fixed average number of Rydberg excitations, thus ruling out a spurious effect due to a saturation of the channeltron. To that end, the volume of the MOT was varied using a combination of magnetic field gradients, trapping beam sizes and laser powers chosen such as to keep the number of excited Rydberg atoms constant. As expected, for a small cloud the Rydberg dynamics was in the highly collective regime and a negative $Q_D$ was measured. As the volume increased with the number of Rydberg excitations kept fixed, the excitation dynamics tended to a Poissonian distribution with $Q\\approx 0$. Moreover, we performed experiments in which the first step laser was tuned to resonance with the $\\mathrm{5S_{1\/2}(F=2)}\\rightarrow\\mathrm{6P_{3\/2}(F'=3)}$ transition so that atoms were directly ionized through the absorption of two photons at 421 nm. Up until counting saturation, this process was found to obey Poissonian statistics with $Q_{D}=0$.\\\\\n\\indent The creation of spurious ions, that is, those not created by field ionization of Rydberg atoms, is a delicate issue in studies of the dipole blockade. The Stark shift due to a single ion is larger than the van-der-Waals interaction between two atoms in the investigated $D_{5\/2}$ state. Thus the creation of one ion in the sample can lead to a blockade due to Coulomb interactions which suppresses the excitation of other Rydberg atoms and thus mimicks the dipole blockade. A possible cause for the presence of spurious ions is the absorption of two 421~nm photons leading to direct ionization from the intermediate 6$^2$P$_{3\/2}$ state \\cite{Viteau2011}. By detuning the blue laser 1~GHz from this state excitation the probability of creating an ion is $3\\times 10^{-7}$ per $2\\,\\mathrm{\\mu s}$ pulse. For the low atoms numbers in our experiment this equates to around $3\\times 10^{-2}$ per shot and thus is not a problem. Also, black-body radiation limits the lifetime of Rydberg atoms, for the 71$D_{5\/2}$ state the effective lifetime is $\\sim$150$~\\mu{}$s~\\cite{Beterov2009}; however, the excitation and ionization is performed within $<10~\\mu$s and thus should have negligible impact on the counting statistics. Finally, at the typical atomic densities used in the experiment the decay process is not affected by super-radiance. Any loss from the initially prepared state will, therefore, follow Poissonian statistics.\n\n\\section{Theory}\n\\label{theory}\n\\subsection{Mean field}\nIn a many-body system with interactions, the mean-field theory replaces all interatomic interactions with an average or effective interaction, reducing the many-body problem to an effective one-body problem. Within this approach the Rydberg state energy is shifted by the vdW interaction characterized by the parameter $W_{0}$ given by a sum over all the  Rydberg pairs inside the atomic cloud, see \\cite{Tong2004}.  For instance, in a medium with uniform atomic density $n$ and for a $C_6$ interaction depending as $r^{-6}$ on the interatomic distance $r$,  $W_0$ is approximatively  $\\frac{C_{6}}{<r>^6}\\approx C_{6}\\left( \\frac{4\\pi }{3}\\right)^{2}n^2$.\n\n\n\\subsection{vdW basis in the cooperative approach}\nThe vdW interaction couples the fully symmetrical DCSs to the remaining\nDCSs within the fixed $N$ subspace  having\nthe large $C_{N_0}^{N}$ degeneracy. The most appropriate orthonormal basis within that subspace is composed of the symmetrical DCSs $ \\left\\vert N\\right\\rangle_s$  and the superposition of non-symmetrical states $\\left\\{ \\left\\vert N,q \\right\\rangle_{ns} \\right\\}$, with $1\\le q \\le C_{N_0}^{N}-1$, for which  the vdW interaction matrix is diagonal.   To obtain the correct number of states within the $C_{N_0}^{N}$ ensemble, we first remove from the $\\left\\{ \\left\\vert N,q\\right\\rangle \\right\\}$ set one state labelled  $\\left\\vert N,p\\right\\rangle $, assuming $W_{pp}^{N}=W_{ss}^{N}$. Considering\nthe state\n\\begin{equation}\n\\left\\vert N,{\\widetilde q}\\right\\rangle =\\left\\vert N,\n\\right\\rangle +\\frac{1}{\\sqrt{C_{N_0}^{N}}-1} \\left\\vert N,p\\right\\rangle,\n\\end{equation}\nfor  $1\\le q \\le C_{N_0}^{N}-1$. Then the basis is constructed by subtracting for each $\\vert N,{\\widetilde q}\\rangle $ state  its projection on the symmetrical\n$\\left\\vert N\\right\\rangle _{s}$ state. We obtain the $\\left\\{ \\left\\vert N,q \\right\\rangle_{ns} \\right\\} $  states\ndefined a\n\\begin{equation}\n\\left\\vert N,q\\right\\rangle _{ns}=\\left[ \\left\\vert N,{\\widetilde q}\n\\right\\rangle -_{s}\\left\\langle N\\mid N,{\\widetilde q}\\right\\rangle \\left\\vert\nN\\right\\rangle _{s}\\right] .\n\\end{equation}\nIt is easy to verify that such a basis $\\left\\{ \\left\\vert N\\right\\rangle\n_{s},\\left\\{ \\left\\vert N,q\\right\\rangle _{ns}\\right\\} \\right\\} $ is\northonormal. The van der Waals coupling is diagonal with eigenvalue $W_{qq}^{N}$ for the restricted basis $\\left\\{ \\left\\vert N,q\\right\\rangle _{ns}\\right\\} $; these eigenvalues are not modified by the above basis changes. The coupling between the states $\\left\\vert N\\right\\rangle\n_{s}$\\ and $\\left\\vert N,q\\right\\rangle _{ns}$ i\n\\begin{equation}\nW_{sq}^{N}=_{s}\\left\\langle N\\right\\vert W\\left\\vert N,q\\right\\rangle _{ns}\n\\frac{1}{\\sqrt{C_{N}^{N}}}\\left[ W_{qq}^{N}-W_{ss}^{N}\\right] .\n\\label{Wenergies}\n\\end{equation}\n\\indent The above construction of the basis is very general, provided we know\nthe basis which diagonalizes the coupling. The only approximation is the\nexistence of the state  $\\left\\vert N,p\\right\\rangle $. For\na large number of atoms, such an approximation is reasonable because it is\npossible to identify a state $\\left\\vert N,\\widetilde{p}\\right\\rangle $ such that $W_{{\\widetilde p}{\\widetilde p}}^{N}\\approx W_{ss}^{N}$.\n\n\\subsection{Enlarged wavefunction evolution}\nBy writing the enlarged atomic wave-function as\n\\begin{equation}\n\\left\\vert \\Psi \\left( t \\right) \\right\\rangle =\\sum_{N=0}^{N_0} a_{N}\\left( t\\right) \\left\\vert N\\right\\rangle_s + \\sum_{q} b^q_{N }\\left( t\\right) \\left\\vert\nN,q \\right\\rangle_{ns},\n\\end{equation}  the $a_N$ time evolution of\n Eqs. (2) and (3) in the main text  is completed by terms describing the  ($b^q_N, a_N$) vdW coupling. As a first approximation, we neglect the laser excitation of the amplitudes $b^q_N$ because of their large degeneracy and the resulting\nweak occupation for each of them. Thus eliminating the equations for $b^q_N$, we obtain the following integro-differential equations:\n\\begin{eqnarray}\n{\\dot a}_{N} &=&-i(W_{ss}^N-N\\delta)a_{N}-i\\frac{\\Omega  }{2}\\left[\\sqrt{\\left( N_0-N\\right) \\left(\nN+1\\right) }a_{N+1}+\\sqrt{N\\left( N_0-N+1\\right)}a_{N-1}\\right] \\nonumber \\\\\n&+&\\sum_q\\int_{0}^{t} \\left[ -\\frac{d^{2\n}{d\\tau ^{2}}-2i\\frac{d}{d\\tau }W_{ss}^N+\\left(W_{ss}^N\\right)^{2}\\right]   \\frac{e^ { i\\left(W_{qq}^N-N\\delta\\right)\\tau}}{\\sqrt{C_{N_0}^N}} a_{N}\\left(\nt-\\tau \\right) d\\tau.\n\\label{differointegral}\n\\end{eqnarray}\n\n\\begin{figure}[htp]\n\\includegraphics[width=0.75\\linewidth]{rydberg_Fig5.eps}\n\\caption{Real and imaginary part of $f^s_{N}\\left( t \\right) $ vs the delay time $\\tau$ for typical parameters investigated in our experiment.  }\n\\label{correlationfunction}\n\\end{figure}\n\nThe coupling $b_N \\to a_N$ appearing in Eq. \\eqref{differointegral} is described through the correlation function $f_N(\\tau)$  as \\begin{equation}\n{\\dot a_{N}}=-\\int_{0}^{t}f_N(\\tau) e^{iN\\delta \\tau} a_N\\left(t-\\tau \\right) d\\tau,\n\\end{equation} with  the correlation function given by\n\\begin{equation}\nf_N\\left( \\tau \\right) =\\sum_q\\frac{1}{C^N_{N_0}}\\left[ \\left[ \n\\frac{d^{2}}{d\\tau ^{2}}-2i\\frac{d}{d\\tau }W^N_{ss}+(W^N_{ss})^{2}\\right]e^\n{ -iW^N_{qq}\\tau} \\right],\n\\end{equation\nwhere $W^N_{sq}$ was eliminated using Eq. \\eqref{Wenergies}. With only two Rydberg excited atoms  within the volume $V$, $f_N$ becomes\n\\begin{equation}\nf_N\\left( \\tau \\right) =\\frac{W_0^{2}}{4V}\\int_{2V\/N}^{V}\\left[\n\\frac{1}{v^{4}}-\\frac{N}{v^{2}V^{2}}+\\frac{N^{2}}{4V^{4}}\\right]\ne^{ -i\\frac{2W_0}{v^{2}}\\tau} dv.\n\\end{equation}\nThe above integral  has an analytical solution. The solution for the correlation function is composed of a long-memory component\n$f^l_{N}\\left( \\tau \\right) $ and a short-memory one, $f^s_{N}\\left( \\tau \\right) $. The real and imaginary parts of $f^s_{N}\\left( \\tau \\right) $ are represented in  Fig.~\\ref{correlationfunction}. The $f^s_{N}\\left( \\tau \\right) $ contribution treated in the Markov approximation leads to the following imaginary term:\n\\begin{equation}\n-\\int_{0}^{t}f^s_{N}\\left( \\tau \\right) \\exp{\\left(i\\delta j \\tau\\right)}a_{N}\\left( t-\\tau \\right) d\\tau\n=iW^N_{ss}a_{N}\\left( t\\right).\n\\end{equation\nThis term exactly compensates exactly the blockade energy term introduced in Eq.~\\eqref{differointegral} above for the symmetric DCS theory and, as a consequence, produces the symmetrical resonance excitation profiles reported in the Fig. 1 of the main text.\\\\\n\\indent The long-memory part can be treated exactly, but a good approximation, up to a few tens of Rydberg excitations, is\n\\begin{equation}\nf^l_{N}\\left( \\tau \\right) =(W^N_{ss})^{2}.\n\\end{equation}\nUsing the above short-memory and long-memory parts,\nthe vdW coupling can be described by representing the\nensemble of the non-symmetrical levels with $j$ Rydberg excitation in terms of a single\nlevel $\\ c_j$ defined as the superposition all asymmetric DCS's of  the fixed $N$ subspace. Within this simplified scheme for each number $N$ of Rydberg excitations the  evolution  is described by the following system of two equations, one for $a_N$ and one for a single level $\\ c_N$, reported also in the text:\n\\begin{eqnarray}\n{\\dot a_{N}}&=&-iN\\delta  a_{N}\n-i\\frac{\\Omega  }{2}\\left[\\sqrt{\\left( N_0-N\\right) \\left( N+1\\right)}\na_{j+1}+\\sqrt{N\\left( N_0-N+1\\right)}a_{N-1}\\right] -iW_{ss}^Nc_{N}, \\label{equationA} \\\\\n{\\dot c}_{N} &=&-iN\\delta c_{N}-iW_{ss}^Na_{N}.\n\\label{equationC}\n\\end{eqnarray}\n\\subsection{Laser excitation of the non-symmetric bath}\nThe last step of this theoretical analysis is the introduction of the laser\nexcitation into evolution of $c_N$.\nWe treat it perturbatively, with a limitation on the temporal evolution imposed by the  spectral linewidth $\\Delta \\omega$ of the excitation laser.  The final result is a modification of Eq. \\eqref{equationC} into \\begin{equation}\n{\\dot c}_{N} =-\\left(\\frac{\\Gamma_N}{2}+iN\\delta\\right) c_{N}-iW_{ss}^Na_{N} +\\frac{\\Gamma_N}{2} \\left|c_{N-1}\\right|,\n\\label{FinalEquation}\n\\end{equation}\nwith all coherence terms $c_Nc^*_{N-1}$ set to zero.  $\\Gamma_N$ depends on the laser linewidth $\\Delta \\omega$ as\n\\begin{equation}\n\\Gamma_N=\\frac{\\Omega^2(N_0-2N)}{4}\\left[\\frac{\\Delta \\omega\/2}{\\left(\\Delta \\omega\/2\\right)^2+\\left(\\delta-NW_{ss}\\right)^2}+\\frac{\\Delta \\omega\/2}{\\left(\\Delta \\omega\/2\\right)^2+\\left(\\delta+NW_{ss}\\right)^2}\\right].\n\\end{equation}\n\n\\section{Rydberg interaction parameters}\n\\label{parameters}\nThe numerical results rely on the precise values for the vdW coupling strengths. In a first approximation, we can consider that, in zero field, two Rydberg atoms  experience a ${C_6}\/{r^6}$ interaction, $C_6$ being  calculated on the basis of  perturbation theory of \\cite{SInger2005}. This model fails to describe the interaction accurately, first at small interatomic distances and secondly because of the Zeeman degeneracy of the two-atom Rydberg state. While at large interatomic distances, the two Rydberg states are dipole coupled via one additional state, at very short interatomic distances a large number of states is coupled, and the overall coupling is reduced. In our calculation of the interaction energy $W_0$, we introduce an appropriate cut-off radius (roughly twice the atomic size) because the excitation of two Rydberg atoms having a distance smaller than the cut-off radius is very unlikely. At intermediate interatomic distances  the perturbative treatment of \\cite{SInger2005} is not fully valid because of crossing in the molecular levels. Then a full diagonalization of the Hamiltonian leads to a more complex dependence of the interatomic coupling on the interatomic distance,  ${C_6}\/{r^6}$ at large distance and ${C_3}\/{r^3}$, i.e. a dipole-dipole interaction, at smaller ones. Finally, Walker and Saffman~\\cite{WalkerSaffman2008} have shown the Rydberg interaction is strongly dependent of the relative orientation of magnetic momentum of the colliding atoms of the pair. For the of $D_{\\frac{5}{2}}$ states this leads to a variety of 21 possible interaction curves whose strengths vary by more than two orders of magnitude. The average presented in~\\cite{WalkerSaffman2008} allowed us to obtain an effective dipole-dipole interaction, i.e., a single interaction curve for the  calculation of the parameter $W_0$. The variation in the interaction curves is reflected in the uncertainty in our determination of the coupling strength for the theoretical curves in Fig. 1 of the main text.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe developing field of superconducting spintronics\ncomprises a plenty of fascinating phenomena that may complement\nnonsuperconducting spintronics devices~\\cite{rf:Zutic}.\nMesoscopic hybrid structures consisting of superconducting and magnetic materials have attracted\nconsiderable attention as devices with novel functionalities~\\cite{rf:Buzdin1}.\nOne of most interesting effects is the formation of $\\pi$-states in\nsuperconductor\/ferromagnetic-metal\/superconductor (S\/FM\/S) Josephson junctions~\\cite{rf:Bulaevskii}.\nUnder appropriate conditions a ferromagnet can become a $\\pi$-phase shifter, providing \nthe phase difference $\\phi=\\pi$ between two superconductors in the ground state\nin contrast to $\\phi=0$ in ordinary Josephson junctions.\nRecently a quiet qubit based on S\/FM\/S $\\pi$-junction~\\cite{rf:Ioffe} has been\nsuggested as a promising device to realize quantum computation because\nthe spontaneously generated two-level system in this structure is robust against\ndecoherence due to external fluctuations.\nHowever, low energy quasiparticle excitations in a FM provide strong dissipation~\\cite{rf:Schon}.\nTherefore Josephson $\\pi$ junctions with a nonmetallic interlayers are highly\ndesired for qubit applications~\\cite{rf:Kawabata1}.\nMoreover, from the fundamental view point, the Josephson transport through a $ferromagnetic$ $insulator$ (FI) has been studied based on phenomenological models~\\cite{rf:SFIS} and not yet been explored explicitly.\n\nIn this Letter, we study theoretically the Josephson effect\nin superconductor\/ferromagnetic-insulator\/superconductor (S\/FI\/S)\njunctions using the tight-binding model.\nWe show that the ground state in such structures alternates between the 0- and $\\pi$-states\nwhen the thickness of a FI ($L_F$) is increasing by a single atomic layer.\nThis remarkable effect originates from the characteristic band\nstructure of a FI. Quasiparticles in the electron and hole branches\nacquire different phase shifts while propagating across a FI.\nWe will show that the phase difference is exactly $\\pi  L_F$ due to the band structure of a FI,\nthus providing the atomic-scale 0-$\\pi$ transition.\nThis mechanism is in striking contrast to the proximity induced  $0$-$\\pi$ transition in conventional S\/FM\/S junctions.\nOn the basis of the obtained results, we predict a stable $\\pi$-state in a\nJosephson junction based on high-$T_c$ superconductors with a La$_2$BaCuO$_5$ barrier, \nwhere electric current flows along the $c$ axis\nof cuprates. \n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\end{center}\n\\caption{(color online) (a)\nThe Josephson junction with a ferromagnetic-insulator (FI) barrier\non the tight-binding lattice.\nThe magnetic moment in FI is chosen along the $z$ axis in spin space.\nThe band structure of a FI (b) and of standard band insulator (c) in the\nBogoliubov-de Gennes picture. The dispersion for a hole with spin $\\sigma$ is\nobtained as a mirror image of the dispersion for an electron with spin\n$\\sigma$ with respect to Fermi energy.\n }\n\\label{fig1}\n\\end{figure}\n\n\n\nLet us first consider an S\/FI\/S junction in the two-dimensional tight-binding model as shown\nin Fig.~1(a). The vector $\\boldsymbol{r}=j{\\boldsymbol{x}}+m{\\boldsymbol{y}}$ points\nto a lattice site, where ${\\boldsymbol{x}}$ and ${\\boldsymbol{y}}$ are unit vectors\nin the $x$ and $y$ directions, respectively.\nThe thickness of a FI layer is $L_F$.\nIn the $y$ direction, we apply the hard wall boundary condition for the number of\nlattice sites being $W$.\nElectronic states in a superconductor are described by the\nmean-field Hamiltonian,\n$\n{\\cal H} =(1\/2)\n\\sum_{\\boldsymbol{r},\\boldsymbol{r}^{\\prime }  \\in \\text{S}    }%\n( \\tilde{c}_{\\boldsymbol{r}}^{\\dagger }\\;\\hat{h}_{\\boldsymbol{r},%\n\\boldsymbol{r}^{\\prime }}\\;\\tilde{c}_{\\boldsymbol{r}^{\\prime }}^{{}}-%\n\\overline{\\tilde{c}_{\\boldsymbol{r}}}\\;\\hat{h}_{\\boldsymbol{r},\\boldsymbol{r}%\n^{\\prime }}^{\\ast }\\;\\overline{\\tilde{c}_{\\boldsymbol{r}^{\\prime }}^{\\dagger\n}})\n+(1\/2)\n\\sum_{\\boldsymbol{r}\\in \\text{S}} ( \\tilde{c}_{%\n\\boldsymbol{r}}^{\\dagger }\\;\\hat{\\Delta}\\;\\overline{\\tilde{c}_{\\boldsymbol{r}%\n}^{\\dagger }}-\\overline{\\tilde{c}_{\\boldsymbol{r}}}\\;\\hat{\\Delta}^{\\ast }\\;%\n\\tilde{c}_{\\boldsymbol{r}} )\n$\nwith\n$\\overline{\\tilde{c}}_{\\boldsymbol{r}}=\\left( c_{\\boldsymbol{r}%\n,\\uparrow },c_{\\boldsymbol{r},\\downarrow }\\right) $,\n where\n$ c_{\\boldsymbol{r} ,\\sigma }^{\\dagger }$ ($c_{\\boldsymbol{r},\\sigma }^{{}}$)\n is the creation (annihilation) operator of an electron at $\\boldsymbol{r}$ with spin $\\sigma\n(= \\uparrow $ or $\\downarrow $ ), $\\overline{\\tilde{c}}$ means the\ntranspose of $\\tilde{c}$,  and $\\hat{\\sigma}_{0}$ is $2\\times 2$ unit matrix.\nWe introduce the hopping integral $t$ among nearest neighbor sites\nand measure the length in the units of the lattice constant $a$.\nIn superconductors, the Hamiltonian leads\n$\n\\hat{h}_{\\boldsymbol{r},\\boldsymbol{r}'}= [ -t \\delta _{|\\boldsymbol{r}-\\boldsymbol{r}'|,1}\n+\n(-\\mu_s+4t)\\delta_{\\boldsymbol{r},\\boldsymbol{r}'} ] \\hat{\\sigma}_{0}\n$,\nthe chemical potential $\\mu_s$ is measured from the band bottom\nand $\\hat{\\Delta}=i\\Delta \\hat{\\sigma}_{2}$, where $\\Delta$ is the amplitude\nof the pair potential in the $s$-wave symmetry and $\\hat{\\sigma}_{j}$ for $j=1-3$\nare Pauli matrices. We describe a FI by\n$\n\\hat{h}_{\\boldsymbol{r},\\boldsymbol{r}'}= -t \\delta_{|\\boldsymbol{r}-\\boldsymbol{r}'|,1}\n\\hat{\\sigma}_{0}\n- (g\/2+4t) \\delta_{\\boldsymbol{r},\\boldsymbol{r}'}\\hat{\\sigma}_3\n$,\nwhere $g$ corresponds to a gap of a FI as shown in Fig.~1(b).\n\nThe Hamiltonian is diagonalized by the Bogoliubov transformation.\nThe Andreev bound state consists of subgap states whose\nwave functions decay far from the junction interface.\nIn what follows, we focus on the subspace for\nspin-$\\uparrow$ electron and spin-$\\downarrow$ hole\n[the dispersions shown by solid curves in Fig.~1(b)].\nIn superconductors, the wave function of a bound state is given by\n\\begin{align}\n\\Psi_L(\\boldsymbol{r})=&\\Phi_L \\left[\n \\left(\\begin{array}{c} u \\\\ v \\end{array} \\right)\n A e^{-ikj} +\n \\left(\\begin{array}{c} v \\\\ u \\end{array} \\right)B e^{ik^*j}\\right] \\chi_l(m),\\\\\n\\Psi_R(\\boldsymbol{r})=&\\Phi_R \\left[\n \\left(\\begin{array}{c} u \\\\ v \\end{array} \\right)\n C e^{ikj} +\n \\left(\\begin{array}{c} v \\\\ u \\end{array} \\right)D e^{-ik^*j}\\right] \\chi_l(m),\n\\end{align}\nwhere $\\nu=L$ ($R$) indicates a superconductor in the left (right) hand side,\n$\\phi_\\nu$ is the phase of a superconductor,\n$\\Phi_\\nu=\\mathrm{diag} \\left(  e^{i\\phi_\\nu\/2} , e^{-i\\phi_\\nu\/2} \\right)$,\n$u(v)=[ ( 1+(-) \\sqrt{E^2-\\Delta^2}\/E )\/ 2 ]^{1\/2}$, and $A, B, C$ and $D$\nare amplitudes of the wave function for an outgoing quasiparticle.\nThe wave function in the $y$ direction is $\\chi_l(m)=\\sqrt{2\/W}\\sin[l m\\pi \/(W+1)]$,\nwhere $l$ indicates a transport channel.\nThe energy $E$ is measured from the Fermi energy\nand $k= \\cos^{-1} [2-\\mu_s\/2t  - \\cos \\{ l \\pi \/ (W+1) \\} + i \\sqrt{\\Delta^2 - E^2} \/E]$ is the complex wave number.\nThese wave functions decay as $e^{-(j-L_F)\/\\xi_0}$ for $j > L_F$\nand $e^{j\/\\xi_0}$ for $j<0$ with $\\xi_0$ being the coherence length.\nIn a FI, the wave function is given by\n\\begin{align}\n\\Psi_{FI}(\\boldsymbol{r})=&\n\\left[ \\left(\\begin{array}{c} f_L e^{-iq_ej} \\\\ g_L e^{-iq_hj}\\end{array} \\right)\n+\\left(\\begin{array}{c} f_R e^{iq_ej} \\\\ g_R e^{iq_hj}\\end{array} \\right)\\right] \\chi_l(m),\\\\\n q_e=&\\pi + i\\beta_\\uparrow, \\label{wn1}\\\\\n   q_h=&0+i\\beta_\\downarrow,\\label{wn2}\n\\end{align}\nwhere $\\cosh\\beta_\\uparrow= 1 + E\/2t+g\/4t + \\cos [ l \\pi \/ (W+1) ]- \\cos [ W \\pi \/ (W+1) ]$,\n $\\cosh \\beta_\\downarrow= 1 + E\/2t+g\/4t - \\cos [ l \\pi \/ (W+1) ]- \\cos [  \\pi \/ (W+1) ]$, and\n$f_L, f_R, g_L$ and $g_R$ are amplitudes of wave function in a FI.\nThe Andreev levels $\\varepsilon_{n,l} (\\phi=\\phi_L-\\phi_R)$ [$n=1, \\cdots, 4$]\ncan be calculated from boundary conditions\n$\\Psi_L(\\lambda,m)=\\Psi_{FI}(\\lambda,m)$ and\n$\\Psi_R(L_F+\\lambda,m)=\\Psi_{FI}(L_F+\\lambda,m)$ for $\\lambda =0$ and 1.\nThe Josephson current is related to $\\varepsilon_{n,l}$ via\n$ I_J (\\phi) =(2e \/ \\hbar) \\sum_{n,l}  \\left[ \\partial \\varepsilon_{n,l}\n(\\phi)\/ \\partial \\phi \\right]  f  \\left( \\varepsilon_{n,l} (\\phi) \\right),\n$\nwhere $ f\\left( \\varepsilon  \\right)$ is the Fermi-Dirac distribution function.\nThe Josephson critical current  $I_C$ is defined by $I_C = I_J(\\pi\/2)$.\n\n\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig2.eps}\n\\end{center}\n\\caption{(color online)\n(a) The Andreev levels $\\varepsilon_{i} \\equiv \\varepsilon_{i,1}$ are plotted as functions of $\\phi$ for an one-dimensional S\/FI\/S junction with $W=1$.\n Left  (right)  panel shows the results for odd $L_F$ (even $L_F$) with\n$g=0.5t$.\n(b) Josephson critical current $I_C$ at $T=0.01 T_c$ as a function of the thickness $L_F$ for\n$W=1$ and $g=0.5 t$. The large red (small blue) circles indicate the $\\pi$(0) junction.\nIn the inset, the 0-$\\pi$ phase diagram on the $g$-$L_F$ plane is shown for\na two-dimensional  S\/FI\/S junction with $W=10$ at $T=0.01 T_c$, where the \nred and blue regimes correspond to the $\\pi$- and 0-states, respectively.}\n\\label{fig2}\n\\end{figure}\n\nIn Fig.~\\ref{fig2}(a), we first show the Andreev levels $\\varepsilon_{n,1} \\equiv  \\varepsilon_{n}$ for odd $L_F$ ($=3$ and 5) and even  $L_F$ ($=4$ and 6) with $W=1$,\n$\\mu_s=2t$, and $\\Delta=0.01t$.\nThe results show that the ground state for odd $L_F$ is at $\\phi=\\pi$, whereas that for even $L_F$ is at $\\phi=0$.\nThis atomic-scale 0-$\\pi$ transition persists  even if we increase $L_F$ and $W$.\nIn Fig.~2(b), we show the Josephson critical current as a function of $L_F$ for $W=1$.\nTemperature $T$ is set to be $0.01T_c  \\ll T_c$, where $T_c$ is the transition temperature of a superconductor.\nThe $\\pi$(0)-state is always more stable than the 0($\\pi$)-state when the thickness of FI is an odd(even) integer.\nThe reason is as follows.\nAt low temperatures, only the Andreev levels below the Fermi energy i.e.,\n$\\varepsilon_{1}$ and\n$\\varepsilon_{2}$, contribute to $I_C$ [see Fig. 2(a)].\nIn the odd (even) $L_F$ cases, the $\\pi$- (0-) state is stable because of\n$  \\partial_\\phi \\varepsilon_1   |_{\\phi=\\pi\/2}> (<) 0 $,\n$ \\partial_\\phi \\varepsilon_2   |_{\\phi=\\pi\/2}< (>) 0 $, and\n$ |  \\partial_\\phi \\varepsilon_2   |_{\\phi=\\pi\/2} |>  |\\partial_\\phi \\varepsilon_1   |_{\\phi=\\pi\/2} |$.\nThe atomic-scale 0-$\\pi$ transition is  insensitive to $W$ and material parameters such as $\\mu_s$, $g$, and $\\Delta$.\nAs an example, inset of Fig.~2(b) shows the phase diagram on the $g$-$L_F$ plane for $W=10$.\n\n\n\n\n\n\n\n\n\n\nThe mechanism of the 0-$\\pi$ transition in a FI is very different from that\nin a FM. The key feature is expressed by the wave number of a quasiparticle\nin a FI as shown in Eqs.~(\\ref{wn1}) and (\\ref{wn2}),\nwhere $q_e$ and $q_h$ are the wave numbers for an electron spin-$\\uparrow$\nand a hole spin-$\\downarrow$, respectively.\nThe real parts of $q_e$ and $q_h$ reflect the wave number at the $q$ points, where energy\nis closest to the Fermi energy, and differ by $\\pi$ from each other .\n As shown in Fig. 1(b), the real part of $q_e$ is\n$\\pi$ because the top of the electron band is located at $q=\\pi$.\nOn the other hand, the real part of $q_h$  is 0 because the top of the hole\nband is at $q=0$. This is the origin of the difference between $q_e$ and $q_h$ which\naccounts the atomic-scale 0-$\\pi$ transition.\nWhen we consider a usual band insulator as shown in Fig. 1(c),\nwe always obtain $q_e=q_h$ and their real parts equal $\\pi$\nbecause both the top of the electron band and the bottom of the hole band are located at $q=\\pi$.\nAs a consequence, 0-state is always stable in usual band insulators.\nThus we conclude that the characteristic band structure of a FI is the origin\nof atomic-scale 0-$\\pi$ transitions.\nThese features basically remain unchanged\neven when we consider Josephson junctions in higher dimensions. In such junctions, however, the appearance of \n0-$\\pi$ transitions depends on relative directions between the\ncurrent and the crystalline axis. We will address this issue below.\nIt should be emphasized that peculiar results presented above cannot be described by the standard quasiclassical Green's function method~\\cite{rf:Fogelstrom} where band structure structure far from the Fermi energy is is ignored.\n\n\n\nLet us reconsider atomic-scale 0-$\\pi$ transitions from a different view point of quasiparticle\ntransmission coefficient.\nIn the high barrier limit ($g \\gg t$), the Josephson critical current is perturbatively given by\n$I_C \\propto T_\\downarrow^* T_\\uparrow$~\\cite{rf:Kawabata1}.\nHere $T_{\\uparrow(\\downarrow)}$ is a transmission coefficient of a FI for\nan electron with spin-$\\uparrow$ (-$\\downarrow$).\nBy using the transfer-matrix method~\\cite{rf:Datta}, $T_\\sigma$ for\none-dimensional junctions can be obtained analytically\n$\n  T_\\sigma \\approx\n\\alpha_{L_F}\\left(  \\rho_\\sigma t  \/  g \\right)^{L_F}.\n$\nHere $\\rho_{\\uparrow(\\downarrow)} = -(+)1$ and $\\alpha_{L_F}$ is a spin-independent complex constant.\nWe immediately find $T_\\uparrow\/T_\\downarrow = (-1)^{L_F}$.\nThus the relative phase\nof $T_\\sigma$ between spin-$\\uparrow$ and spin-$\\downarrow$\n is $\\pi$ (0) for the odd (even) number of $L_F$.\nAs a consequence, the sign of $I_C \\propto  (-1)^{L_F} $ becomes negative for odd $L_F$ and positive for even $L_F$.\nIn other words, a FI acts as a $\\pi$-phase-shifter for the spin-$\\uparrow$ electron for odd $L_F$.\n\n\n\nThe transfer-matrix method in real space also enables us to extend the calculations to\nanother magnetic materials.\nUp to now, we have considered uniform magnetic moment in FI, which can be schematically expressed by\nS\/$\\uparrow_1 \\uparrow_2 \\cdots \\uparrow_{L_F}$\/S or S\/$\\downarrow_1 \\downarrow_2\n\\cdots \\downarrow_{L_F}$\/S. The arrows $\\uparrow_j$ and\n$\\downarrow_j$ indicate the $z$-axis magnetization at $j$.\nWe can extend the above simple analysis to the arbitrary\nmagnetization configuration, $e.g.,$ a\nrandom alignment described by S\/$\\downarrow_1 \\uparrow_2 \\downarrow_ 3 \\cdots\n\\uparrow_{L_F-2} \\uparrow_{L_F-1} \\downarrow_{L_F}$\/S.\nIn such junctions, we find\n$I_C \\sim \\prod_{i=1,L_F} T_{i, \\mathrm{\\downarrow}}^* T_{i, \\mathrm{\\uparrow}}\n =  \\prod_{i=1,L_F} (-1)=(-1)^{L_F}$,\nwhere $T_{i,\\sigma}$ is the transmission coefficient of an FI layer at $i$.\nTherefore we obtain a noticeable result, $i.e.$,\nthe sign of\n $I_C$\nis independent of magnetization configurations and is negative (positive) for odd (even) $L_F$.\nThe appearance of the atomic-scale 0-$\\pi$ transition has been also predicted in\nS\/antiferromagnetic-interlayer\/S junctions~\\cite{rf:Barash}.\nIn their theory, however, the antiferromagnetic configuration\nis found to be essential for the atomic-scale transition.\nOn the other hand, we conclude that the magnetization symmetry is not necessary and that\nthe $\\pi$-phase difference between $T_\\uparrow$ and $T_\\downarrow$ is an essential feature\nfor the atomic-scale transition.\nTherefore our analysis provides more general view for the physics of the atomic scale 0-$\\pi$ transition.\n\n\n\n\n\n\n\nFinally, we show the possibility of the atomic-scale 0-$\\pi$ transition in a three-dimensional\njunction using realistic materials.\nHere we focus on La$_2$BaCuO$_5$ (LBCO)~\\cite{rf:Mizuno} which is an important representative FI in spintronics.\nAccording to a first-principle band calculation~\\cite{rf:LBCO}, the bottom of the minority spin band is at the $\\Gamma$ point\nwhose wave number is $(k_a, k_b, k_c) = (0,0,0)$, where $k_j$ for $j=a,b,$ and $c$ is the\nwave number along $j$ axis (see Fig.~6 in Ref.~\\onlinecite{rf:LBCO}).\nThe mirror image of the minority spin band with respect to the Fermi energy corresponds to\nthe hole band with minority spin in the Bogoliubov-de Gennes picture.\nThus the top of the minority spin hole band is at the $\\Gamma$ point.\nOn the other hand, top of the majority spin band is at the $Z$ point with\n$(k_a, k_b, k_c) = (0,0,\\pi)$.\nThus we can predict that the $\\pi$ state would be possible if one fabricates a Josephson\njunction along $c$ axis as shown in Fig.~4(a). Note that it is impossible to\nrealize the $\\pi$-state if current flows in the $ab$-plane.\nThis is because wave numbers in $ab$-plane at the bottom of the minority spin band and\nthose at the top of the majority spin band are given by the same wave number\n$(k_a, k_b) = (0,0)$~\\cite{rf:LBCO}.\n\n\n\n\nFrom the perspectives of the S\/FI interface matching and the high-temperature device-operation,\nthe usage of high-$T_c$ cuprate superconductors (HTSC), e.g.,\nYBa${}_2$Cu${}_3$O${}_{7-\\delta}$ and La${}_{2-x}$Sr${}_x$CuO${}_4$ (LSCO) is desirable.\nRecent development of the pulsed laser deposition technique enables layer-by-layer\nepitaxial-growth of oxide superlattices~\\cite{rf:Prellier}.\nIn order to show the possibility of $\\pi$-coupling in such realistic HTSC junctions, we have calculated the $c$-axis Josephson critical current $I_C$ based on a three-dimensional tight-binding model with $L_a$ and $L_b$ being the numbers of lattice sites in $a$ and $b$ directions [Fig. 3 (a)].\nIn the calculation we have taken into account the $d$-wave order-parameter symmetry in HTSC, i.e., $\\Delta= \\Delta_d (\\cos k_x a- \\cos k_y a) \/ 2$. The tight-binding parameters $t$ and $g$  have been determined  by fitting to the first-principle band structure calculations along the line from $\\Gamma$ to $Z$ point~\\cite{rf:LBCO}.\nFigure 3 (c) shows the thickness $L_F$ dependence of $I_C$ at $T=0.01 T_c$ for a LSCO\/LBCO\/LSCO junction with $g\/t=20$, $\\Delta_d\/t=0.6$, and $L_a=L_b=100$.\nAs expected, the atomic-scale  0-$\\pi$ transitions can be realized in such oxide-based $c$-axis stack junctions.\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=8.8cm]{fig3.eps}\n\\end{center}\n\\caption{\n(color online) (a) Schematic picture of a $c$-axis stack high-$T_c$\nsuperconductor\/FI\/high-$T_c$ superconductor Josephson junction and (b) a $d$-wave ring to detect the $\\pi$-junction behavior experimentally.\n(c) The Josephson critical current $I_C$ as a function of the FI thickness $L_F$ at $T=0.01 T_c$ for a $c$-axis stack LSCO\/LBCO\/LSCO junction with $g\/t=20$, $\\Delta_d\/t=0.6$, and $L_a=L_b=100$.\nThe large red (small blue) circles indicate the $\\pi$(0) junction.}\n\\label{fig3}\n\\end{figure}\n\nThe experimental detection of the $\\pi$-junction is possible by using a superconducting ring\nwhich contains two Josephson junctions as shown in Fig.~\\ref{fig3}(b).\nWhen both junctions are in 0- (or $\\pi$-) state at the same time,\nthe critical current of the ring reaches its maximum at zero\nexternal magnetic flux. On the other hand, the critical current reaches its minimum\nat zero magnetic flux when the 0 state is stable in one junction and $\\pi$ is stable\nin the other~\\cite{rf:Sigrist}.\n\n\nFrom the  view point of qubit applications, it is important to note that the harmful influence  of  midgap Andreev resonant states~\\cite{rf:Tanaka,rf:KawabataMQT1} and nodalquasiparticles due to the $d$-wave symmetry on the macroscopic quantum dynamics  in $c$-axis HTSC junctions~\\cite{rf:Kleiner} is found to be weak, both theoretically~\\cite{rf:KawabataMQT2} and experimentally~\\cite{rf:MQTexperiment}.\nTherefore we conclude that  HTSC\/LBCO\/HTSC\n$\\pi$-junctions would be promising candidates as basic-elements for  quiet qubits.\n\n\n\n\nIn summary, we have studied the Josephson effect in S\/FI\/S junctions based on the tight-binding model.\nWe predict the formation of the atomic-scale 0-$\\pi$ transitions in such junctions.\nThis result is insensitive to the material parameters such as the gap $g$ of the FI and the superconducting gap $\\Delta$, indicating that it is a robust and universal phenomenon.\nOur findings suggest the way of realizing  {\\it ideal quiet qubits} which possess both the quietness and the weak quasiparticle-dissipation nature.\n\n\n\nWe  would like to thank J. Arts, A. Brinkman, M. Fogelstr\\\"om, T. Kato, P. J. Kelly, T. L\\\"ofwander, T. Nagahama, F. Nori, J. Pfeiffer, A. S. Vasenko, and M. Weides for useful discussions.\nThis work was  supported by CREST-JST, a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (Grant No. 19710085),  and  NanoNed (Grant TCS.7029).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{P}{hotonic}-crystal (PhC) waveguides are made by creating a line-defect in a PhC slab. The defect introduces guided modes in the photonic band-gap of the crystal and allows for an efficient propagation of optical signals. In particular, a major advantage of PhC line-defect waveguides (LDWGs), as compared to conventional waveguides, is the possibility to exploit slow-light (SL) propagation \\cite{Baba2008}. In the SL region of the waveguide dispersion relation, the group velocity is greatly reduced and ideally tends to zero at the band edge. As a consequence, in the SL region, a Bloch wave propagating within an active PhC waveguide experiences an effective gain per unit length which is greatly enhanced with respect to the modal gain coefficient of a conventional waveguide \\cite{ek2014a}. This gain-enhancement allows for the realization of shorter devices and makes PhC waveguides ideal candidates for high-density photonic integrated circuits. PhC lasers have attracted large research as efficient light-sources in on-chip and chip-to-chip interconnections. They allow for scaling the active volume while maintaining a high cavity Q-factor, thus exhibiting low threshold current and operating energy \\cite{MatsuoReview}. Different types of PhC lasers exist. A typical PhC laser consists of a so called $\\mathrm{LN}$ cavity \\cite{Okano2010}, that is formed by only omitting $N$ holes in a PhC slab. Initially, due to heat accumulating in the active region, room-temperature continuous-wave (RT-CW) operation was difficult to achieve in this type of PhC laser. In the last years, significant progress has been made and RT-CW operation has been demonstrated under optical pumping by using ultra-small cavities with embedded quantum dots (QDs) \\cite{Nomura2006}. A major breakthrough, addressing the issues of high-speed direct modulation and thermal stability, has been the introduction of the so called lambda-scale-embedded active region PhC laser (LEAP laser), working under electrical \\cite{Matsuo2012} pumping conditions. In this laser, the cavity is made of a buried heterostructure (BH) embedded in a PhC LDWG. Furthermore, differently from PhC lasers based on $\\mathrm{LN}$ cavities \\cite{Xue2015,Xue2016}, in LEAP lasers light can be emitted into in-plane waveguides. The output waveguide can be either shifted with respect to the active region \\cite{MatsuoNatPhot2013,MatsuoJSTQE2013} or placed in-line with it \\cite{Shinya}. Therefore, LEAP lasers are promising sources for photonic integrated circuits \\cite{MatsuoReview}.\n\\IEEEpubidadjcol\n\nRigorous approaches to analyze PhC devices, such as FDTD \\cite{Okano2010,Okano2009} or RCWA \\cite{LalanneRCWA2005,LalanneRCWA2007,SongRCWA}, are time- and memory-consuming and cannot always provide an intuitive understanding of the physical phenomena into play. The aim of this paper is to present an alternative and simple approach for analyzing active PhC LDWGs and lasers based on this type of waveguides. Our approach is based on coupled-mode theory (CMT) \\cite{Yariv1973,Marcuse_Book1974}, which has proved to be an effective tool to study and design lasers with periodic gain and\/or refractive index perturbation, such as edge-emitting DFB lasers \\cite{KogelnikShank}. More recently, CMT has also been used to analyze both passive \\cite{Patterson2010,MichaelisPR} and active \\cite{Chen2015} PhC waveguides. By following a simple CMT approach, we show that a weak perturbation of the gain and\/or refractive index of a PhC waveguide causes a strong coupling between the forward- and backward-propagating Bloch modes of the unperturbed waveguide, with coupling coefficients strongly increasing as we move towards the band edge. We start from the formulation already presented in \\cite{Chen2015}, where the Bloch modes of a passive, reference waveguide were used as a basis for the field expansion in a finite-length, active section; the presence of gain in this section was treated as a weak perturbation to the passive structure, coupling the otherwise independent, counter-propagating Bloch modes. However, the system of coupled propagation equations was solved numerically in \\cite{Chen2015}, thus not providing important insight on the coupling mechanism. In this work, we push further the formulation of \\cite{Chen2015} to generally take into account both a real and imaginary refractive index perturbation. Avoiding the numerical solution, we derive a simple, closed-form expression for the unit cell transmission matrix of an active PhC waveguide. Therefore, consistently with the rigorous, non-perturbative approach of \\cite{Gric2012}, we show that, in the presence of gain, the group index does not diverge at the band edge and that the maximum attainable SL gain-enhancement is limited by the gain itself. \n\nIn this work, our coupled-Bloch-mode approach is then applied to analyze the threshold condition of various types of PhC lasers. As a first example, we analyze a laser cavity made up of different sections of both passive and active PhC LDWGs. This laser is conceptually similar to the one characterized in \\cite{Shinya}. Consistenly with \\cite{Shinya}, we show that three different operating regimes can be identified, which can be explained on the basis of the interplay between the distributed feedback in the active region and the passive mirrors. This reveals the great impact of coherent distributed feedback effects in these PhC lasers. As a second example, we analyze a typical PhC laser based on a $\\mathrm{LN}$ cavity, such as that of \\cite{Xue2016}. In this case, the laser cavity is an active PhC LDWG bounded on either side by classical PhC mirrors. As a last example, we examine a structure simply consisting of an active section, with zero back reflections from the interfaces with the passive output waveguides. Interestingly, in this type of structure the distributed feedback, caused entirely by the gain perturbation in the active region, is enough to allow lasing with reasonable threshold gain.\n\nThe paper is organized as follows: in Section II, we present our model and discuss the peculiar characteristics of the self- and cross-coupling coefficients of an active PhC LDWG. In Section III, we present the numerical results applied to both active PhC waveguides and lasers. In Section IV, we finally draw the conclusions.                  \n\n \n\\section{Numerical model}\n\\begin{figure}[ht]\n\t\\centering\\includegraphics[width=\\linewidth]{Supercell}\n\t\\caption{\\label{fig:Supercell} Reference PhC waveguide supercell (a). \\uline{Cross section of the reference waveguide  supercell at either the input or output plane. The black lines represent QWs or QDs layers} (b). Finite-length, perturbed section in red and reference waveguide in grey (c).}\n\\end{figure}\n\\begin{figure}[ht]\n\t\\centering\\includegraphics[width=0.75\\linewidth]{BandStructure_ns3171}\n\t\\caption{\\label{fig:BandStructure_ns3171} Projected band structure of the reference PhC waveguide TE-like modes. The membrane refractive index is $n_s = 3.171$, the lattice constant $a = \\mathrm{438 \\,nm}$ \\uline{and the thickness $h = \\mathrm{250 \\,nm}$; the other parameters are those of the PhC LDWG on which the lasers in \\cite{Xue2016} are based. The membrane is assumed to be suspended in air.} The fundamental guided mode is even with respect to $x$ (red, solid line), the other mode is odd (red, dashed line). \\uline{The blue lines represent a subset of the continuum of photonic crystal slab modes found by MPB.}}\n\\end{figure}\nWe consider an active, PhC LDWG of finite length bounded on either side by semi-infinite, passive PhC waveguides, as illustrated in Fig.~\\ref{fig:Supercell}(c). The passive sections have the same lattice constant and membrane thickness as the active section. \\uline{In this context, for \\textit{membrane} we simply mean a thin layer of semiconductor, suspended in a low refractive index medium and periodically patterned with holes \\cite{DeRossi2009}, as shown in cross-section in Fig.~\\ref{fig:Supercell}(b)}. \\uline{The \\textit{material}} gain \\uline{$g$} in the active section may be provided by layers of quantum wells (QWs) or QDs embedded within the PhC membrane, \\uline{shown as black lines in Fig.~\\ref{fig:Supercell}(b)}. For simplicity, the entire membrane is assumed to contain active material, as in the case of optically pumped waveguides \\cite{ek2014a} and lasers based on $\\mathrm{LN}$ cavities \\cite{Xue2015,Xue2016}, which lack a structure for lateral carrier confinement. However, we note that the coupled-Bloch-mode approach is also applicable to structures with a buried active region, such as LEAP lasers. We consider the gain of the active section and the variation of its refractive index with respect to that of the passive sections as a weak perturbation.\n\\begin{figure*}[ht]\n\n\t\\begin{subfigure}[b]{0.33\\linewidth}\n\t\t\\centering\\includegraphics[width=\\linewidth]{Gamma11}\n\t\t\\caption{\\label{fig:ConfFacta}}\n\t\\end{subfigure}%\n\n\t\\begin{subfigure}[b]{0.33\\linewidth}\n\t\t\\centering\\includegraphics[width=\\linewidth]{Gamma12ABS}\n\t\t\\caption{\\label{fig:ConfFactb}}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.33\\linewidth}\n\t\t\\centering\\includegraphics[width=\\linewidth]{Gamma12ANGLE}\n\t\t\\caption{\\label{fig:ConfFactc}}\n\t\\end{subfigure}%\n\n\t\\caption{\\label{fig:ConfFact} \\sout{confinement factors} \\uline{Spatial dependence of} $\\Gamma_{11}$ and $\\Gamma_{12}$ of the reference waveguide at various $k_z$ values. The membrane refractive index is $n_s = 3.171$; the other parameters are those of the PhC LDWG on which the lasers in \\cite{Xue2016} are based. (\\ref{fig:ConfFacta}) $\\Gamma_{11}$. (\\ref{fig:ConfFactb}) Magnitude of $\\Gamma_{12}$. (\\ref{fig:ConfFactc}) Phase of $\\Gamma_{12}$.}\n\\end{figure*}\nThe forward- (+) and backward-propagating \\mbox{(-)} Bloch modes of the reference waveguide in the frequency-domain are denoted by $\\mathbf{E}_{0,\\pm}(\\mathbf{r},\\omega) = \\mathbf{e}_{0,\\pm}(\\mathbf{r},\\omega) e^{\\pm ik_z(\\omega)z}$, where $\\mathbf{e}_{0,\\pm}(x,y,z,\\omega) = \\mathbf{e}_{0,\\pm}(x,y,z+a,\\omega)$, with $k_z$ being the Bloch wave number along the length of the waveguide ($z$) and $a$ the PhC lattice constant. In the active section, the complex refractive index variation compared to the reference waveguide is $\\Delta n_s +i\\Delta n_i$, where $\\Delta n_s$ is the variation of refractive index and $\\Delta n_i$ reflects the \\uline{\\textit{modal}} gain coefficient $g_0$, with $\\Delta n_i = -(c\/\\omega) g_0\/2$. \\uline{Here, $g_0$ is given by $\\Gamma_y g$, where $\\Gamma_y$ is the optical confinement factor of the considered electric field guided mode along the y-direction and within the QWs or QDs layers.} This polarization perturbation couples to each other in the active waveguide the otherwise independent Bloch modes of the reference waveguide. Therefore, in the limit of a weak perturbation, the electric field in the active section can be expanded in the basis of the Bloch modes of the reference waveguide, with slowly-varying amplitudes $\\psi_{\\pm}(z,\\omega)$ caused by the coupling. The electric field in the active section thus reads as $\\mathbf{E}(\\mathbf{r},\\omega) = \\psi_{+}(z,\\omega)\\mathbf{E}_{0,+}(\\mathbf{r},\\omega) + \\psi_{-}(z,\\omega)\\mathbf{E}_{0,-}(\\mathbf{r},\\omega)$. At a given $\\omega$, by neglecting nonlinear effects, two coupled differential equations in $\\psi_{\\pm}$ are derived \\cite{Chen2015}:\n\\begin{equation}\n\\label{eq:CoupledBlochModeEquations}\n\\begin{aligned}\n\\partial_z\\psi_+(z) & = i\\kappa_{11}(z)\\psi_+(z) + i\\kappa_{12}(z) e^{-2ik_z z} \\psi_-(z)\\\\\n-\\partial_z\\psi_-(z) & = i\\kappa_{21}(z) e^{+2ik_z z}\\psi_+(z) + i\\kappa_{11}(z)\\psi_-(z)  \n\\end{aligned}\n\\end{equation} \nThe self- and cross-coupling coefficients are written as $\\kappa_{11;12;21}(z,\\omega) \\simeq [\\Delta n_s(\\omega)+i\\Delta n_i(\\omega)](\\omega\/c)[n_{g,0}(\\omega)\/n_s]\\Gamma_{11;12;21}(z,\\omega)$, where $n_{g,0}$ and $n_s$ are the group index and the membrane material refractive index of the reference waveguide. $\\Gamma_{11;12;21} (z,\\omega)$ are given by\n\\begin{equation}\n\\label{eq:ConfFact}\n\\begin{aligned}\n\\Gamma_{11}(z,\\omega) & = \\frac{a\\int_{S} \\epsilon_0 n_s^2|\\textbf{e}_{0}(\\textbf{r},\\omega)|^2 F(\\textbf{r})dS}{\\int_{V} \\left[\\epsilon_0 n_b^2(\\textbf{r}) |\\textbf{e}_0(\\textbf{r},\\omega)|^2\\right] dV}\\\\ \\Gamma_{12}(z,\\omega) & = \\frac{a\\int_{S} \\epsilon_0 n_s^2\\left[\\textbf{e}_{0,-}(\\textbf{r},\\omega) \\cdot \\textbf{e}^*_{0,+}(\\textbf{r},\\omega)\\right] F(\\textbf{r})dS}{\\int_{V} \\left[\\epsilon_0 n_b^2(\\textbf{r}) |\\textbf{e}_0(\\textbf{r},\\omega)|^2\\right] dV}\n\\end{aligned}\n\\end{equation}\nwith $\\Gamma_{21} = \\Gamma_{12}^*$. Here, $V$ is the volume of the supercell (see Fig.~\\ref{fig:Supercell}(a)), $S$ the transverse section at position $z$ and $n_b (\\mathbf{r}) = n_s (=n_{\\mathrm{holes}})$ in the membrane (holes) is the reference waveguide background refractive index. The spatial distribution of the complex perturbation is taken into account by $F(\\mathbf{r})$, which is equal to 1 (0) in the membrane (holes) of the active section. \\uline{$\\Gamma_{11}$ is the ratio between the electric field energy in the gain region of a supercell and that stored in the whole supercell, by assuming the modal gain $g_0$ to be uniform through the semiconductor slab.} \n\\begin{figure}[ht]\n\t\\centering\\includegraphics[width=\\linewidth]{Gamma11_12Harmonics}\n\t\\caption{\\label{fig:Gamma11_12Harmonics} Magnitude of the Fourier components of the self-coupling $\\kappa_{11}(z)$ (a) and cross-coupling coefficient $\\kappa_{12}(z)$ (b) at various $k_z$ values.}\n\\end{figure}\n\\uline{We note that the self-coupling coefficient is caused by the fact that the modes used as a basis for the electric field see, in the perturbed waveguide, an average complex refractive index profile which is different from that of the reference waveguide where the modes are defined}. \\sout{We note that,} Due to the $z-$periodicity of $\\mathbf{e}_{0,\\pm}$ and $F(\\mathbf{r})$, \\sout{confinement factors} $\\Gamma_{11}$ and $\\Gamma_{12}$ and, therefore, the coupling coefficients are periodic with $z$. As an example, we consider a reference waveguide with refractive index $n_s = 3.171$; the other parameters are those of the PhC waveguide on which the lasers in \\cite{Xue2016} are based, with $a = 438\\mathrm{\\,nm}$, \\uline{$n_{\\mathrm{holes}} = 1$ and membrane thickness $h = 250\\mathrm{\\,nm}$. The membrane is assumed to be suspended in air, consistently, for instance, with \\cite{Xue2015,Xue2016} or \\cite{MatsuoNatPhot2013,MatsuoJSTQE2013}, which investigate air-bridge structures}. Band structure and TE-like Bloch modes of the reference waveguide are computed by the plane wave eigensolver MIT Photonic-Bands (MPB) \\cite{MPBpaper} (Fig.~\\ref{fig:BandStructure_ns3171}). \\uline{The blue lines represent a subset of the continuum of photonic crystal slab modes found by MPB \\cite{skorobogatiy_yang_2008}. These modes are confined to the slab along the y-direction, but are delocalized in the x- and z-direction}. We apply the coupled-Bloch-mode approach to the fundamental guided mode (red, solid line), \\uline{which is confined both along the x- and y-direction}. The magnitude and phase of $\\Gamma_{11}$ and $\\Gamma_{12}$ over a unit cell are displayed in Fig.~\\ref{fig:ConfFact} at various $k_z$ values. Given the spatial distribution of the perturbation (that is $F(\\textbf{r})$), they only depend on the reference waveguide geometry and Bloch modes and not on the magnitude of the perturbation. For uniformly pumped membranes, \\sout{confinement factors} $\\Gamma_{11}$ and $\\Gamma_{12}$ are intrinsic parameters of the reference waveguide and they determine how strong the cross-coupling is with respect to self-coupling. The coupling coefficients are indeed proportional to \\sout{confinement factors} $\\Gamma_{11}$ and $\\Gamma_{12}$ through $n_{g,0}$ and the complex refractive index perturbation \\sout{and they are periodic in $z$ with the same periodicity of $\\Gamma_{11,12,21}$}. Therefore, they can be expanded in a Fourier series as $\\kappa_{11;12}(z,\\omega) = \\sum_q \\kappa_{11,q;12,q}(\\omega) \\exp(+iq2\\pi z\/a)$. Since the phase of $\\Gamma_{12}$ is approximately linear with $z$ with a slope equal to $2\\pi\/a$, $\\kappa_{12,q=1}$ is proportional to $\\left\\langle |\\kappa_{12}(z)|\\right\\rangle$, which is comparable with $\\kappa_{11,q=0}$. This is illustrated in Fig.~\\ref{fig:Gamma11_12Harmonics}, showing the Fourier components of $\\kappa_{11}(z)$ and $\\kappa_{12}(z)$  in arbitrary units at various $k_z$ values. Interestingly, we find that the complex refractive index perturbation not only produces a self-coupling coefficient \\sout{(as expected in any standard waveguide)} \\uline{proportional to the average of the perturbation (i.e. the average of $\\Gamma_{11}(z)$ in Fig.~\\ref{fig:ConfFact}(a))}, but also a strong cross-coupling, whose magnitude is close to that of the self-coupling. This strong cross-coupling is possible thanks to the linear phase of $\\Gamma_{12}(z)$; if this linear phase component were not present, the cross-coupling would be negligible. By analyzing the contribution of the various field components ($\\hat{x}$, $\\hat{y}$ and $\\hat{z}$ component) to $\\Gamma_{12}(z)$, we have found that the linear phase variation is caused by a significant $\\hat{z}$ component of the TE-like guided mode, which is typically negligible in standard waveguides. Here the non-negligible $\\hat{z}$ component is due to the strong lateral (\\sout{$y-$} \\uline{$x-$}direction) confinement obtained by the PhC. \n\nHaving now more insight into the main properties of the coupling coefficients, we derive an analytical expression for the transmission matrix describing the field propagation over a unit cell. As illustrated in Fig.~\\ref{fig:Gamma11_12Harmonics}, all harmonics in the self-coupling coefficient, other than $q=0$, are negligible; similarly, the harmonic with $q = 1 (=-1)$ is the dominant one in the cross-coupling coefficient $\\kappa_{12}$ ($\\kappa_{21}$). This is true over a wide frequency range and especially close to the band edge. Therefore, by only retaining the dominant harmonics, Eqs.~(\\ref{eq:CoupledBlochModeEquations}) are turned into \n\\begin{equation}\n\\label{eq:CoupledBlochModeEquationsFourier}\n\\begin{aligned}\n\\partial_z\\psi_+ & \\simeq i\\kappa_{11,q=0}(\\omega)\\psi_+ + i\\kappa_{12,q=1}(\\omega) e^{+2i\\delta(\\omega) z} \\psi_-\\\\\n-\\partial_z\\psi_- & \\simeq i\\kappa_{21,q=-1}(\\omega) e^{-2i\\delta(\\omega) z}\\psi_+ + i\\kappa_{11,q=0}(\\omega)\\psi_-  \n\\end{aligned}\n\\end{equation}     \nwhere $\\delta(\\omega) = \\pi\/a - k_z(\\omega)$ is the detuning from the band edge. \\uline{By defining $b^{\\pm}(z,\\omega) = \\psi_{\\pm}(z,\\omega)\\exp\\{\\mp i\\delta(\\omega)z\\}$, the system of Eqs.~(\\ref{eq:CoupledBlochModeEquationsFourier}) is turned into a pair of partial differential equations with z-independent coefficients. This allows to analytically solve it over a unit cell as an initial value problem. That is, $b^{\\pm}(z_0 + a,\\omega)$ are computed by assuming $b^{\\pm}(z_0,\\omega)$ to be known, with the input coordinate $z_0$ of the unit cell conveniently chosen to be zero. Therefore, the unit cell transmission matrix in terms of $b^{\\pm}$ is obtained. However, we need the transmission matrix for $c^{\\pm}(z,\\omega)$, with $c^{\\pm}(z,\\omega) = \\psi_{\\pm}(z,\\omega)\\exp\\{\\pm ik_z(\\omega)z\\}$. Since $b^{\\pm}(z,\\omega) = c^{\\pm}(z,\\omega)\\exp\\{\\mp i(\\pi\/a)z\\}$, the unit cell transmission matrix in terms of $c^{\\pm}$ is obtained through the change of variables $b^{\\pm}(z_0 + a,\\omega) = c^{\\pm}(z_0 + a,\\omega)\\exp\\{\\mp i\\pi\\}$ and $b^{\\pm}(z_0,\\omega) = c^{\\pm}(z_0,\\omega)$. Therefore,} \\sout{By putting $\\psi_{\\pm} = c^\\pm e^{\\mp ik_z z}$ and solving the resulting equations over a unit cell}, we can relate $c^\\pm$ at the input ($N-1$) and output ($N$) of the generic $N_{\\mathrm{th}}$ cell by\n\\begin{equation}\n\\label{eq:cpm_Ta}\n\\left[\n\\begin{aligned}\n& c_N^+ (\\omega) \\\\\n& c_N^- (\\omega)\n\\end{aligned}\n\\right]\n=\n\\mathbf{T}_a (\\omega)\n\\left[\n\\begin{aligned}\n& c_{N-1}^+ (\\omega) \\\\\n& c_{N-1}^- (\\omega)\n\\end{aligned}\n\\right] \n\\end{equation}\n\\sout{$[c_N^+ (\\omega)   c_N^- (\\omega)]^T = \\mathbf{T}_a (\\omega) [c_{N-1}^+ (\\omega)   c_{N-1}^- (\\omega)]^T$,} where $\\mathbf{T}_a$ is the unit cell transmission matrix \\uline{in terms of $c^{\\pm}$} \\sout{and $T$ denotes the transpose operator}. The elements of $\\mathbf{T}_a$ are given by \n\\begin{equation}\n\\label{eq:Tmatrix}\n\\begin{aligned}\nT_{a,11;a,22} & = -\\cosh\\left(\\gamma a\\right) \\mp i\\left(\\tilde{\\delta}\/\\gamma\\right)\\sinh\\left(\\gamma a\\right) \\\\ \nT_{a,12;a,21} & = \\mp i\\left(\\kappa_{12,q=1;21,q=-1}\/\\gamma\\right)\\sinh\\left(\\gamma a\\right) \n\\end{aligned}\n\\end{equation}\nwith $\\tilde{\\delta}(\\omega) = \\kappa_{11,q=0}(\\omega) - \\delta(\\omega)$ and $\\gamma(\\omega) = \\sqrt{\\kappa_{12,q=1}(\\omega)\\kappa_{21,q=-1}(\\omega) - \\tilde{\\delta}^2(\\omega)}$. This approach allows to directly relate the eigenvalues of $\\mathbf{T}_a$ to the coupling coefficients and the detuning. In fact, the eigenvalues $\\lambda_{1,2}$ are readily obtained as $\\lambda_{1,2} = \\exp\\{\\pm\\mathrm{Re}\\left\\lbrace\\gamma a\\right\\rbrace\\} \\exp\\{\\pm i[\\mathrm{Im}\\left\\lbrace\\gamma a\\right\\rbrace +\\pi]\\}$. From here, we define $\\pm g_{\\mathrm{eff}} = \\pm2\\mathrm{Re}\\left\\lbrace\\gamma\\right\\rbrace$ as the effective gain per unit length, representing the gain experienced by the forward ($+$) and backward ($-$) Bloch modes of the \\textit{active} section while propagating in a unit cell. Similarly, $\\pm \\beta_{\\mathrm{eff}}a = \\pm[\\mathrm{Im}\\left\\lbrace\\gamma a\\right\\rbrace +\\pi]$ is the phase shift per cell of these Bloch modes. From this phase shift, we can then calculate the group index of the Bloch modes of the active waveguide as $n_{g}(\\omega) = \\mathrm{c}\\:d\\beta_{\\mathrm{eff}}\/d\\omega$. By applying Frobenius theorem \\cite{Frobenius}, the transmission matrix of an active waveguide of $N$ unit cells is computed as \n\\begin{equation}\n\\label{eq:FrobeniusTheorem}\n\\mathbf{T}^N_a = \\mathbf{M}\\mathbf{\\lambda_a}^N\\mathbf{M}^{-1} \n\\end{equation}\nwhere $\\mathbf{M}$ and $\\mathbf{\\lambda_a}$ are given by\n\\begin{equation}\n\\label{eq:M_Lambda}\n\\mathbf{M} = \n\\left[\n\\begin{aligned}\nu_{11} & \\quad u_{12} \\\\\nu_{21} & \\quad u_{22}\n\\end{aligned}\n\\right], \n\\quad\n\\mathbf{\\lambda_a} = \n\\left[\n\\begin{aligned}\n\\lambda_1 & \\quad 0 \\\\\n0 & \\quad \\lambda_2\n\\end{aligned}\n\\right] \n\\end{equation}\nwith $\\mathbf{u}_1 = \\left[u_{11} \\quad u_{21}\\right]^{T}$ and $\\mathbf{u}_2 = \\left[u_{12} \\quad u_{22}\\right]^{T}$ being the eigenvectors of $\\mathbf{T}_a$ \\uline{and $T$ denoting the transpose operator}. As well as being numerically efficient, this approach for computing $\\mathbf{T}^N_a$ allows for a useful physical interpretation: $\\mathbf{\\lambda_a}^N$ can be seen as the transmission matrix describing the propagation of the Bloch modes of the \\textit{active} section over the $N$ unit cells. $\\mathbf{M}^{-1}$ and $\\mathbf{M}$ can be interpreted as the transmission matrices of the interfaces between the active section and, respectively, the left and right passive waveguides; they physically account for the mismatch between the Bloch modes of the active and passive sections. \\uline{By applying the relationships between a transmission and a scattering matrix \\cite{Coldren}, the transmission matrix $\\mathbf{T}^N_a$ can be turned into the corresponding scattering matrix. The scattering parameters are given by}\n\\begin{equation}\n\\label{eq:Smatrix}\n\\begin{aligned}\nS_{11} & = \\frac{\\left(\\lambda_1^N - \\lambda_2^N\\right)T_{a,21}}{\\left(\\lambda_1^N - \\lambda_2^N\\right)T_{a,11} + \\left(\\lambda_2^{N+1} - \\lambda_1^{N+1}\\right)} \\\\ \nS_{12;21} & = \\frac{\\left(\\lambda_2 - \\lambda_1\\right)}{\\left(\\lambda_1^N - \\lambda_2^N\\right)T_{a,11} + \\left(\\lambda_2^{N+1} - \\lambda_1^{N+1}\\right)} \\\\\nS_{22} & = \\frac{\\left(\\lambda_2^N - \\lambda_1^N\\right)T_{a,12}}{\\left(\\lambda_1^N - \\lambda_2^N\\right)T_{a,11} + \\left(\\lambda_2^{N+1} - \\lambda_1^{N+1}\\right)}\n\\end{aligned}\n\\end{equation} \n\nAs outlined in \\cite{Chen2015}, the coupled-Bloch-mode approach breaks down at large values of $g_0$. Specifically, the larger $n_{g,0}$ becomes, the smaller is the value of the gain coefficient at which the model starts to fail. As a further limitation, here we also point out that the model is actually not applicable at the band edge of the reference waveguide, independently of the value of $g_0$. This is because the group velocity of the Bloch modes used as a basis for the field expansion is ideally equal to zero at the band edge of the reference waveguide. These considerations imply that, in principle, we could not analyze active PhC waveguides in the frequency range close to the critical point $k_z = \\pi\/a$. However, this limitation can be overcome with the following trick: to analyze an active PhC waveguide including the frequency range of the band edge and some frequencies in the stop-band, we start from a reference waveguide with index $n_s$ slightly \\textit{larger} than that of the waveguide to be studied. The waveguide of interest is then seen as having a real refractive index perturbation $\\Delta n_s<0$ with respect to the reference one; therefore, its dispersion relation is shifted to higher frequencies with respect to the band edge of reference waveguide, where the model is not applicable. This approach will be \\sout{illustrated} \\uline{applied} in section IIIA. \n\n\n\\section{Simulation results}\n\n\\subsection{Line-defect active waveguide}\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{CouplCoeff}\n\t\\caption{\\label{fig:CouplCoeff}Magnitude of self- ($\\kappa_{11,q=0}$) and cross-coupling coefficient ($\\kappa_{12,q=1}$) for $g_0 = 0$ (blue) and $g_0 = 50 \\,\\mathrm{cm^{-1}}$ (red). In both cases, $\\Delta n_s = -0.001$.}\n\\end{figure}\nFig.~\\ref{fig:CouplCoeff} shows the magnitude of the self- ($\\kappa_{11,q=0}$) and cross- coupling coefficient ($\\kappa_{12,q=1}$) for $g_0 = 0$ (blue) and $g_0 = 50 \\,\\mathrm{cm^{-1}}$ (red). In both cases, $\\Delta n_s = -0.001$. This figure proves that the cross-coupling coefficient is always comparable to the self-coupling coefficient. This is consistent with results also shown in \\cite{MichaelisPR}. \nHowever, as compared to \\cite{MichaelisPR}, we have clarified here the physical origin of this peculiar behaviour. We also observe that the coupling coefficients depend on the intensity of the perturbation, as expected, but also on frequency and they significantly increase as the frequency approaches the band edge as consequence of the SL effect. Therefore, Bloch modes at smaller frequency and\/or with higher gain of the active section experience stronger distributed feedback as compared to higher frequency Bloch modes and\/or lower active section gain.  \n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{gammaIM_MPBcomp_Updated}\n\t\\caption{\\label{fig:gammaIM_MPBcomp} Dispersion curve computed by MPB for $n_s = 3.171$ (black) and $n_s = 3.170$ (pink). Phase shift per cell of the Bloch modes of the perturbed waveguide with (light-blue, dotted) and without (light-blue, solid) cross-coupling; the reference waveguide has $n_s = 3.171$ and the real refractive index perturbation is $\\Delta n_s = -0.001$, with $g_0 = 0$. The dark-blue, dotted curve is for $\\Delta n_s = -0.002$, $g_0 = 0$.} \n\\end{figure}\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{gammaIM_ngPert}\n\t\\caption{\\label{fig:gammaIM_ngPert} Phase shift per cell of the Bloch modes of the active waveguide (a) and corresponding group index (b) at various gain values. The reference waveguide has $n_s = 3.171$ and the real refractive index perturbation is $\\Delta n_s = -0.001$.}\n\\end{figure}\nIn this section, we analyze how a generally complex refractive index perturbation impacts the dispersion relation of a PhC waveguide. To validate our model, we first study the case of a purely real refractive index perturbation, because the results obtained by our approach can be compared with MPB simulations. We consider a reference passive waveguide with  $n_s = 3.171$ and we report in black in Fig.~\\ref{fig:gammaIM_MPBcomp} the corresponding dispersion relation computed by MPB. We then consider a small, real refractive index perturbation $\\Delta n_s<0$ and we compare the dispersion relation calculated with our model (i.e. $\\beta_{\\mathrm{eff}}(\\omega)a$) with that calculated by MPB for a passive waveguide with refractive index $n_s+\\Delta n_s$. As $n_s$ decreases, the dispersion curve shifts to higher frequencies, meaning that the Bloch modes become evanescent at lower frequencies. The formation of this stopband for \\textit{Bloch} modes is correctly reproduced by our approach: the light-blue, dotted curve  in Fig.~\\ref{fig:gammaIM_MPBcomp} is the dispersion relation calculated by our approach for $\\Delta n_s = -0.001$ and it perfectly overlaps with that obtained by MBP for $n_s=3.170$. If the contribution of the cross-coupling coefficients in Eqs.~(\\ref{eq:CoupledBlochModeEquationsFourier}) is neglected, we find the dispersion relation shown by the light-blue, solid curve in Fig. \\ref{fig:gammaIM_MPBcomp}; in this case, the Bloch modes of the perturbed waveguide are not evanescent in the stop-band and the dispersion relation disagrees with the MPB prediction. These results validate the correctness of our approach and emphasize the role of the cross-coupling terms. Furthermore, the possibility to analyze the propagation of the Bloch-modes in a portion of the stop-band of the perturbed, passive waveguide will be exploited in the next section to study laser configurations similar to the LEAP lasers of \\cite{Matsuo2012,Matsuo2010}. \n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{gammaRE_EnhcFact}\n\t\\caption{\\label{fig:gammaRE_EnhcFact} Effective gain as a function of frequency at various gain values (a). Gain enhancement factor at $f\\simeq\\mathrm{189.388\\,THz}$ (b). The reference waveguide has $n_s = 3.171$ and the real refractive index perturbation is $\\Delta n_s = -0.001$.}\n\\end{figure}    \nAs a second example, we consider the impact of a gain  perturbation, assuming now $\\Delta n_s = -0.001$ and $g_0>0$. Fig.~\\ref{fig:gammaIM_ngPert}(a) shows how the dispersion relation  of the Bloch modes of the active waveguide is modified by the gain perturbation and Fig.~\\ref{fig:gammaIM_ngPert}(b) reports the corresponding group index at various gain values. For $g_0 = 0$, the group index diverges at $f\\simeq\\mathrm{189.388\\,THz}$, because this frequency corresponds to the band edge of the perturbed waveguide. At lower frequencies, the Bloch modes are evanescent and the group index is zero,  \\uline{because evanescent waves do not carry any active power}. As $g_0$ increases, the group index is gradually reduced in the passband, thus limiting the SL effect; on the contrary, it gradually increases in the stopband, where $\\beta_{\\mathrm{eff}}$ is now different from zero. This is consistent with results reported in \\cite{Gric2012}, which were obtained through a non-perturbative approach. The corresponding $g_{\\mathrm{eff}}$ versus frequency is shown in Fig.~\\ref{fig:gammaRE_EnhcFact}(a) at various  values of $g_0$; we also report in Fig.~\\ref{fig:gammaRE_EnhcFact}(b) the gain  enhancement factor $g_{\\mathrm{eff}}\/g_0$ at $f\\simeq\\mathrm{189.388\\,THz}$ obtained by our approach. Since cross-coupling is not negligible, the gain-induced distributed feedback becomes more and more important as the gain increases, thus causing the decrease of the group index; as a consequence, the gain enhancement factor, which is based on the SL effect, is reduced by increasing the pumping of the active region. This result is consistent with \\cite{Gric2012} and confirms that a fundamental limitation to the achievable SL enhancement of the gain is imposed by the gain itself. \n\n\\subsection {PhC Lasers}\n\nIn this section, we  apply the coupled-Bloch-mode approach to model the threshold characteristics of various types of PhC laser cavities. The threshold condition is found by calculating the complex loop-gain (LG) of the cavity.\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{TypeABCcavScheme}\n\t\\caption{\\label{fig:TypeABCcavScheme} Laser cavity made up of three sections, perturbed in refractive index (passive mirrors) and gain (active section) with respect to a reference, passive waveguide with $n_s = 3.171$. The  section to the left of the active region (broad-band mirror) is passive, with $\\Delta n_s = -0.002$ and length fixed to $L = 30a$; the active section has $\\Delta n_s = -0.001$, $g_0>0$ and $L = 10a$. The section to the right of the active region is also passive and acts as a buffer mirror, with $\\Delta n_s = -0.001$ (Type A), $\\Delta n_s = -0.0005$ (Type B) or $\\Delta n_s = -0.002$ (Type C) and variable length $L_{\\mathrm{buffer}}$. The output waveguide coincides with the reference waveguide. \\uline{The position of the reference plane to compute the complex loop-gain is denoted by $z_{\\mathrm{ref}}$}.}\n\\end{figure} \n\nAs a first example, we analyze the cavity shown in Fig.~\\ref{fig:TypeABCcavScheme}. The cavity is made up of three sections, perturbed in refractive index (passive mirrors) and gain (active section) with respect to a reference, passive waveguide with $n_s = 3.171$. Referring to Fig.~\\ref{fig:TypeABCcavScheme}, the  section to the left of the active region (rear broad-band mirror) is passive, with $\\Delta n_s = -0.002$ and length fixed to $L = 30a$; the active section has $\\Delta n_s = -0.001$, $g_0>0$ and $L = 10a$. The section to the right of the active region (front mirror) is also passive and acts as a buffer mirror, coupling the active section with the reference, output waveguide. Depending on the refractive index perturbation of this buffer region, the band edge of its dispersion relation shifts with respect to that of the active region; on the basis of the buffer refractive index perturbation, we have identified three different types of buffer, thus providing different back reflections to the active region. These will be denoted as Type A (moderate back reflection), Type B (low back reflection) and Type C (high back reflection). These laser cavities are conceptually similar to those in \\cite{Shinya}, where the relative shift of the dispersion relation of the passive sections with respect to the active section was obtained by changing the width of the corresponding waveguides \\cite{Notomi2001}. For this laser cavity, the LG is computed as the product between the left ($r_{\\mathrm{eq,L}}$) and right ($r_{\\mathrm{eq,R}}$) field reflectivity at the interface between the active section and the rear mirror (see Fig.~\\ref{fig:TypeABCcavScheme}). \\uline{Specifically, $r_{\\mathrm{eq,L}}$ is readily obtained by Eqs.~(\\ref{eq:Smatrix}) as the $S_{22}$ parameter of the broad-band mirror; $r_{\\mathrm{eq,R}}$ corresponds to the field reflectivity resulting from the cascade of the active section and buffer mirror scattering matrices, i.e.} \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:reqR}\nr_{\\mathrm{eq,R}} & = S_{\\mathrm{11,active}} + r_{\\mathrm{eq,2}} \\\\\nr_{\\mathrm{eq,2}} & = \\frac{S_{\\mathrm{12,active}}S_{\\mathrm{11,buffer}}S_{\\mathrm{21,active}}}{1-S_{\\mathrm{11,buffer}}S_{\\mathrm{22,active}}}\n\\end{aligned}\n\\end{equation}\nwith $S_{ij,\\mathrm{buffer}}$ and $S_{ij,\\mathrm{active}}$ denoting the buffer and active section scattering parameters computed by Eqs.~(\\ref{eq:Smatrix}). The longitudinal resonant modes are those satisfying  $\\angle \\mathrm{LG}=2m\\pi$; the threshold gain $g_{\\mathrm{0,th}}$ is the smallest $g_0$ value ensuring  $|\\mathrm{LG}|=1$ at the frequency of the longitudinal modes. \\sout{$r_{\\mathrm{eq,R,L}}$ are calculated from the transmission matrices of the active section and buffer mirror as computed by Eqs.~(\\ref{eq:FrobeniusTheorem})}. \n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{MirrorsSpectra}\n\t\\caption{\\label{fig:MirrorsSpectra} Broad-band mirror reflectivity; $\\Delta n_s = -0.002$ and $L=30a$ (green). Buffer mirror reflectivity for $L_{\\mathrm{buffer}}=15a$ and $\\Delta n_s = -0.002$ (type C, black), $\\Delta n_s = -0.001$ (type A, red) and $\\Delta n_s = -0.0005$ (type B, blue). The bullets indicate the position of the lasing mode, in the three different operating regimes, both on the broad-band and buffer mirror reflection spectrum.}\n\\end{figure}    \n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{gth_fTHzth_etaOut_TypeABC_DoubleAxes}\n\t\\caption{\\label{fig:gth_fTHzth_etaOut_TypeABC_DoubleAxes} Threshold gain (solid line) and lasing frequency (dashed line) as a function of the buffer mirror length (a) for type A (red), B (blue) and C (black). Output coupling efficiency (b). The active section has $\\Delta n_s = -0.001$ and $L_{\\mathrm{active}}=10a$.}\n\\end{figure}\nFig.~\\ref{fig:MirrorsSpectra} shows the reflectivity of the broad-band mirror (green) and that of the buffer mirror in the three different operating conditions: Type A (red), Type B (blue) and Type C (black). In all three cases, the buffer length is fixed to $L_{\\mathrm{buffer}} = 15a$. The bullets denote the position of the lasing mode (calculated in the following) in the three cases, with respect to both the broad-band and buffer mirror reflection spectrum. In Type C (black), the buffer refractive index perturbation coincides with that of the broad-band mirror. Therefore, at the lasing frequency, the reflectivity of both mirrors is high (because the lasing mode lies in the stop-band of the buffer mirror) and the laser threshold behaviour is mainly dominated by the mirrors feedback. In Type A, the buffer refractive index perturbation is $\\Delta n_s = -0.001$; the lasing frequency is slightly shifted with respect to Type C, but the buffer reflectivity is smaller. In Type B, the buffer refractive index perturbation is $\\Delta n_s = -0.0005$; as a result, the band edge of the buffer dispersion relation is in the stopband of the active section. The lasing frequency is practically the same as for type A, but the buffer reflectivity is even smaller. In Type A and B, we can expect the interplay of the feedback in both the active section and the mirrors to play a role in determining the laser threshold.\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{reflTermsMAG_ANGLE_TypeB_separate_WithThresholdGain}\n\t\\caption{\\label{fig:reflTermsMAG_ANGLE_TypeB_separate_WithThresholdGain}\\uline{Scattering terms of Eq.~(\\ref{eq:reqR}) for Type B configuration of Fig.~\\ref{fig:TypeABCcavScheme} evaluated at the lasing frequency and threshold gain. Magnitude of $S_{\\mathrm{11,active}}$ (blue, solid line), $r_{\\mathrm{eq,2}}$ (blue, dashed line) and $S_{\\mathrm{11,buffer}}$ (blue, dotted line); threshold gain (pink line) (a).  Phase of $S_{\\mathrm{11,active}}$ (blue, solid line) and $r_{\\mathrm{eq,2}}$ (blue, dashed line) (b).}}\n\\end{figure} \n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{S11mirror_Lbuf5a17a21a28a}\n\t\\caption{\\label{fig:S11mirror_Lbuf5a17a21a28a} Type B buffer mirror reflectivity for different buffer lengths; the bullets denote, for each length, the lasing mode position.}\n\\end{figure} \nFig.~\\ref{fig:gth_fTHzth_etaOut_TypeABC_DoubleAxes}(a) shows threshold gain (solid curve) and lasing frequency (dashed curve); Fig.~\\ref{fig:gth_fTHzth_etaOut_TypeABC_DoubleAxes}(b) reports the output coupling efficiency $\\eta_{\\mathrm{out}}$, calculated as power transmission through the buffer, as a function of the buffer mirror length. Type C configuration has the smallest threshold gain, which monotonically decreases with increasing buffer length. In this case, the lasing mode is in the stop-band of the buffer mirror, whose reflectivity rapidly increases with increasing number of unit cells; as a result, the power transmission through the buffer rapidly deteriorates with increasing buffer length. This situation is similar to the operating condition of typical LEAP lasers \\cite{MatsuoNatPhot2013,MatsuoJSTQE2013}. In Type A configuration, the distributed feedback in the active section provides a back reflection comparable with that of the buffer mirror. In this case, the threshold gain is larger than in type C, but the output coupling efficiency is high and does not significantly deteriorate with increasing buffer length. In Type B configuration, the lasing mode lies outside the stop-band of the buffer mirror; consequently, this configuration exhibits the largest threshold gain, but also the largest $\\eta_{\\mathrm{out}}$. In this case, the buffer mirror reflectivity is very low and lasing is mainly possible thanks to the active section distributed feedback. Nevertheless, the buffer also slightly contributes to the laser threshold behaviour. The interplay between buffer back reflection and distributed feedback in the active region can be understood by analyzing, with reference to Fig.~\\ref{fig:TypeABCcavScheme}, the expression for $r_{\\mathrm{eq,R}}$ \\uline{given by Eqs.~(\\ref{eq:reqR})}. \\sout{For the sake of convenience, we denote by $r_{\\mathrm{eq,2}}$ the second term on the RHS of Eq.~(\\ref{eq:reqR})}. Fig.~\\ref{fig:reflTermsMAG_ANGLE_TypeB_separate_WithThresholdGain} shows in blue for Type B the magnitude (a) and phase (b) of the scattering terms, evaluated at the lasing frequency and threshold gain, which contribute to $r_{\\mathrm{eq,R}}$; it also reports the corresponding threshold gain (pink line, (a)). Fig.~\\ref{fig:S11mirror_Lbuf5a17a21a28a} shows the buffer reflectivity for various buffer lengths; the bullets denote the position of the corresponding lasing mode.\nFor $L_{\\mathrm{buffer}}=5a$, the threshold gain is $g_{\\mathrm{0,th}}\\simeq \\mathrm{70\\, cm^{-1}}$ and the lasing frequency $f\\simeq\\mathrm{189.434\\, THz}$; at $L_{\\mathrm{buffer}}=21a$, the lasing frequency is practically the same, but the threshold gain is much larger. This is because the mode is close to the zero of the buffer reflection spectrum for $L_{\\mathrm{buffer}}=21a$; as a result, although the LG phase condition is satisfied at the same frequency for these two different buffer lengths, $|r_{\\mathrm{eq,R}}|$ is smaller in the second case, thus requiring a larger gain for achieving threshold. For $L_{\\mathrm{buffer}}=28a$, the lasing frequency is the same again, but the threshold gain practically coincides with that of the structure with $L_{\\mathrm{buffer}}=5a$. This is because the mode is now close to the top of the buffer reflection spectrum side-lobe, rather than to the zero (see Fig.~\\ref{fig:S11mirror_Lbuf5a17a21a28a}); furthermore, as for $L_{\\mathrm{buffer}}=5a$, $S_{\\mathrm{11,active}}$ and $r_{\\mathrm{eq,2}}$ are nearly in-phase (see Fig.~\\ref{fig:reflTermsMAG_ANGLE_TypeB_separate_WithThresholdGain}). For the case of $L_{\\mathrm{buffer}}=17a$, although $|r_{\\mathrm{eq,2}}|$ is maximum, the threshold gain is not minimum, but rather close to its maximum value. The reason is that $S_{\\mathrm{11,active}}$ and $r_{\\mathrm{eq,2}}$ are now nearly out-of-phase. \\uline{We note that $S_{\\mathrm{11,active}}$ and $r_{\\mathrm{eq,2}}$ are also nearly out-of-phase for the case $L_{\\mathrm{buffer}}=21a$; in this case, however, the magnitude of $r_{\\mathrm{eq,2}}$ is too small to significantly affect $r_{\\mathrm{eq,R}}$}. As a result of this complex interaction between the active section distributed feedback and the buffer back reflection, the Type B configuration exhibits an optimum number of buffer cells minimizing the threshold gain. These examples prove the great impact of coherent distributed feedback effects in PhC cavities like that in Fig.~\\ref{fig:TypeABCcavScheme}, which is similar to a LEAP laser with an in-line coupled waveguide.\n  \n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{HighZeroFacetReflcavScheme}\n\t\\caption{\\label{fig:HighZeroFacetReflcavScheme} Typical PhC laser based on a $\\mathrm{LN}$ cavity (a). PhC laser cavity consisting of an active section with zero back reflections \\uline{for Bloch modes} at the interfaces with the passive output waveguides (b). In both cases, the reference waveguide has $n_s = 3.171$ and the active section $\\Delta n_s = -0.001$. \\uline{The position of the reference plane to compute the complex loop-gain is denoted by $z_{\\mathrm{ref}}$}.}\n\\end{figure} \nAs a second example, we analyze the PhC lasers shown in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a). This configuration is similar to the optically pumped PhC laser  of \\cite{Xue2016}, where the cavity mirrors are classical PhC mirrors. Referring to Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a), $r_{\\mathrm{m,R}}$ is the reflectivity that the forward-propagating Bloch mode of the reference, passive waveguide (whose perturbation is accounted for, in the active section, by the coupled-Bloch-mode equations) undergoes when impinging on the right mirror; this mode is reflected back to the active section because it becomes evanescent within the mirror. A similar interpretation holds for $r_{\\mathrm{m,L}}$. This reflectivity cannot be computed by our approach, because the classical PhC mirror cannot be viewed as a weak perturbation to a reference waveguide (due to the high refractive index contrast between slab and air holes). However, it has been computed in \\cite{Lalanne2008} by modelling the cavity as an effective Fabry-Perot (FP) resonator and by fitting the Q-factor with that obtained through a RCWA approach \\cite{LalanneRCWA2007}. For the sake of simplicity and neglecting the impact of disorder, we assume a high, frequency-independent reflectivity $r_{\\mathrm{m,L}} = r_{\\mathrm{m,R}} = 0.98$, which represents a reasonable approximation \\cite{Xue2016,Lalanne2008,Rigal2017}.\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{rFacet098_fTHz_g0th}\n\t\\caption{\\label{fig:rFacet098_fTHz_g0th} Numerically computed resonant frequencies (a) and threshold gain ((b), solid curve) for the laser in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a). The threshold gain estimated by the SL-enhanced FP formula is also shown ((b), dotted curve). Each colour corresponds to a different longitudinal resonant mode.}\n\\end{figure}\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{rFacet098_resModeLocations}\n\t\\caption{\\label{fig:rFacet098_resModeLocations}. Resonant mode condition for the laser in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a). Each colour corresponds to a different longitudinal resonant mode. $\\beta_{\\mathrm{eff}}$ is evaluated at the numerically computed resonant frequencies and corresponding threshold gain.}\n\\end{figure}\nTo compute the LG, we choose the reference plane at the interface between the active section and the left mirror (see Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a)).\n\\uline{As a result, $r_{\\mathrm{eq,L}}$ is equal to $r_{\\mathrm{m,L}}$ and $r_{\\mathrm{eq,R}}$ is obtained from Eq.~(\\ref{eq:reqR}) by replacing $S_{\\mathrm{11,buffer}}$ with $r_{\\mathrm{m,R}}$}. The laser threshold behaviour is summarized in Fig.~\\ref{fig:rFacet098_fTHz_g0th}, showing the numerically computed resonant frequencies (a) and threshold gain ((b), solid curve); each colour corresponds to a different longitudinal resonant mode. In the existing literature, similar cavities have been studied in \\cite{Okano2010} through FDTD simulations. In \\cite{Okano2010}, it has been shown that the resonant modes of a passive $\\mathrm{LN}$ cavity correspond to the fullfillment of the condition\n\\begin{equation}\n\\label{eq:ResCond}\n\\left(0.5 - k_za\/2\\pi\\right) = m\/2N\n\\end{equation}\nwith $N = L\/a$ being the number of unit cells, $m$ the mode order and $k_z$ the dispersion relation of the passive PhC LDWG of the cavity. For this reason, we have evaluated the quantity $0.5 - \\beta_{\\mathrm{eff}}a\/2\\pi$ at the numerically computed resonant frequencies and corresponding threshold gain of Fig.~\\ref{fig:rFacet098_fTHz_g0th}; it is shown as the solid curve in Fig.~\\ref{fig:rFacet098_resModeLocations}. We also note that \nthe quantity $0.5 - \\beta_{\\mathrm{eff}}a\/2\\pi$ \\sout{exactly} \\uline{practically} coincides, at a given cavity length, with $m\/2N$ (dashed curve in Fig.~\\ref{fig:rFacet098_resModeLocations}). Since the required threshold gain is low, the gain-induced distributed feedback is negligible. As a consequence, the gain does not impact on the position of the resonant modes. As the cavity length increases, the modes move towards the SL region along the dispersion relation of the umpumped waveguide. This \\uline{effect} is consistent with experimental \\cite{Xue2016} and numerical \\cite{Cartar2017} trends and is \\sout{just a FP effect} \\uline{independent of the perturbation-induced distributed feedback}. In fact, by setting $\\Delta n_s=0$ and $\\kappa_{12,q=1} = \\kappa_{21,q=-1} = 0$ in Eqs.~(\\ref{eq:Tmatrix}), Eq.~\\ref{eq:ResCond} can be easily obtained. The threshold gain reported in Fig.~\\ref{fig:rFacet098_fTHz_g0th}(b) is compared with the expression $\\left[1\/\\left(S\\: L_{\\mathrm{active}}\\right)\\right]\\ln\\left[1\/\\left(r_{\\mathrm{m,L}}r_{\\mathrm{m,R}}\\right) \\right]$ (dotted curve in Fig.~\\ref{fig:rFacet098_fTHz_g0th}(b)) \\cite{Xue2016}, with $S = n_{g}\/n_s$ being the slow-down factor, evaluated at the resonant frequencies, and $L_{\\mathrm{active}}$ the cavity length; again, the group index is that of the umpumped waveguide. \\uline{This expression resembles that of a standard FP laser, but the threshold gain is scaled down by the slow-down factor}. Since the gain is low, $n_g$ is not reduced and the SL enhancement of the gain is not limited. On the basis of these considerations, we conclude that the laser with classical PhC mirrors modelled as high-reflectivity, frequency-independent reflectors behaves as a SL-enhanced FP laser \\uline{for Bloch modes}.\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{rFacet098Zero_g0th_fth_comparison_withMatlab}\n\t\\caption{\\label{fig:rFacet098Zero_g0th_fth_comparison_withMatlab} Threshold gain and corresponding resonant frequency for the laser in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a) (a) and Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(b) (b) as a function of cavity length. Each colour corresponds to a different longitudinal resonant mode. The cavity length ranges from $10a$ to $20a$.}\n\\end{figure} \n\\begin{figure}\n\t\\centering\\includegraphics[width=\\linewidth]{rFacet098Zero_GainMargin_NaEVEN_ODD}\n\t\\caption{\\label{fig:rFacet098Zero_GainMargin_NaEVEN_ODD} Difference between $\\mathrm{M_2}$ and $\\mathrm{M_1}$ threshold gain for the laser in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(a) (red) and (b) (green) as a function of the cavity length.}\n\\end{figure}\nAs a last example, we focus on the structure in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(b), which consists of an active section with zero back reflections \\uline{for \\textit{Bloch} modes} at the interfaces with the passive output waveguides. \\sout{We note that this condition can be practically implemented by making the active section refractive index (or width) sufficiently smaller than that of the output waveguides, such that the cavity resonant modes are in the propagation band of the output waveguides. This implies that, differently from the typical\nLEAP laser implementation, the passive sections do not\nact as mirrors}. \\uline{The practical implementation of this \\textit{matching} condition is outside the scope of this paper; various promising solutions are reported in the existing literature \\cite{DUTTAreview,Hugonin2007}.} We show for the first time, to the best of our knowledge, that this type of structure can ideed achieve lasing with reasonable threshold gain. \\sout{The output waveguides coincide with} The reference waveguide \\sout{having} has $n_s = 3.171$, while the active section has $\\Delta n_s = -0.001$ and $g_0>0$. \\uline{Similarly to the second example, here a given $\\Delta n_s$ is only considered in order to shift away the active section dispersion relation from the critical point $k_z = \\pi\/a$ and investigate the laser operation also at frequencies near the band edge (as explained in the end of section II).} To compute the LG, we choose a reference plane within the active region, at the interface between any two unit cells; $N_R$ unit cells are located on the right of the reference plane and $N_L$ on the left, with $N = N_L + N_R$ being the total number of unit cells in the whole active region. \\sout{Therefore, to obtain $r_{\\mathrm{eq,R}}$, the transmission matrix of an active section of $N_R$ unit cells is computed by Eqs.~(\\ref{eq:FrobeniusTheorem}) and turned into a scattering matrix; the $\\mathrm{S_{11}}$ parameter of this scattering matrix is $r_{\\mathrm{eq,R}}$. Similarly, $r_{\\mathrm{eq,L}}$ is computed as the $S_{22}$ parameter of the scattering matrix corresponding to the other $N_L$ cells.} \\uline{Therefore, $r_{\\mathrm{eq,R}}$ is given by the $S_{11}$ parameter of Eqs.~(\\ref{eq:Smatrix}) with $N = N_R$; similarly, $r_{\\mathrm{eq,L}}$ is computed as the $S_{22}$ parameter of Eqs.~(\\ref{eq:Smatrix}) with $N = N_L$.} The threshold gain and corresponding resonant frequencies of this laser are shown in Fig.~\\ref{fig:rFacet098Zero_g0th_fth_comparison_withMatlab}(b) and compared with those of the laser with classical PhC mirrors (Fig.~\\ref{fig:rFacet098Zero_g0th_fth_comparison_withMatlab}(a)). The cavity length ranges from $10a$ to $20a$ and each colour corresponds to a different longitudinal resonant mode. As a first remark, we note that the threshold gain is considerably larger for the laser with zero back reflections; however, the threshold gain of the lasing mode ($\\mathrm{M_1}$) turns out to be reasonable, being of the same order of magnitude as observed for the Type B configuration in Fig.~\\ref{fig:TypeABCcavScheme}. Secondly, the lasing mode frequency is essentially independent of the cavity length and it is located close to the band edge of the active section dispersion relation with $g_0 = 0$. On the contrary, the higher-order modes shift towards the SL region as the cavity length increases. This is somehow similar to what occurs in purely gain-coupled DFB lasers, whose lasing frequency is independent of the cavity length and exactly located at the Bragg frequency \\cite{KogelnikShank}. Finally, we note that the laser in Fig.~\\ref{fig:HighZeroFacetReflcavScheme}(b) exhibits a much better spectral selectivity as compared to a laser with classical PhC mirrors. This is illustrated in Fig.~\\ref{fig:rFacet098Zero_GainMargin_NaEVEN_ODD}, comparing the gain margin (defined as the difference between $\\mathrm{M_2}$ and $\\mathrm{M_1}$ threshold gain)  for the lasers in Fig.~\\ref{fig:HighZeroFacetReflcavScheme} as a function of cavity length.    \n\n\n\\section{Conclusions}\n\nBy starting from a set of two coupled-Bloch-mode equations \\cite{Chen2015}, we have derived a simple, closed-form expression for the unit cell transmission matrix of a PhC LDWG with a generally complex refractive index perturbation as compared to a reference waveguide. This allows for a simple and numerically efficient analysis of active PhC LDWGs and lasers based on this type of waveguide, such as LEAP lasers.\n\nIn particular, we have derived the expression of the coupling coefficients and explained that the magnitude of the cross-coupling is always comparable to that of self-coupling; this is due to the non-negligible longitudinal component of TE-like Bloch modes in PhC LDWGs. We have shown that our approach can correctly reproduce the formation of a stop-band for Bloch modes as a consequence of a purely real refractive index perturbation. We have further validated it by computing the group index and gain enhancement factor of an active PhC waveguide; consistently with the rigorous, non-perturbative approach of \\cite{Gric2012}, we have shown that the maximum attainable SL gain enhancement is limited by the gain itself.          \n\nWe have then applied our coupled-Bloch-mode approach to analyze the threshold condition of three types of PhC laser cavities. The first cavity is conceptually similar to that characterized in \\cite{Shinya}. Depending on the buffer refractive index perturbation, we have identified, consistently with \\cite{Shinya}, three different operating regimes, thus proving the great impact of coherent distributed feedback effects in this type of PhC cavity. The second cavity is the one characterized in \\cite{Xue2016}. By neglecting the impact of fabrication disorder and modelling the classical PhC mirrors as standard reflectors with a high, frequency-independent reflectivity, we have shown that the gain-induced distributed feedback is negligible in this type of cavity, which simply behaves as a SL-enhanced FP laser \\uline{for Bloch modes}. As a last example, we have analyzed a structure consisting of an active section bounded on either side by passive waveguides, which are assumed to \\sout{provide zero back reflections} \\uline{be matched for the reference waveguide Bloch modes}. This means that this configuration is different from the typical LEAP laser implementation. Interestingly, we have shown that this cavity can lase with reasonable threshold gain, with lasing only sustained by the active region distributed feedback. \n\nIn conclusion, we have presented an effective approach that will be useful to provide insights on the characteristics of PhC lasers and might be also extended to study the laser dynamics of these structures.   \n\n\\section{Acknowledgements}\n\nM.S. wishes to express his gratitude to Prof. I. Montrosset and Prof. R. Orta for fruitful discussions on CMT and Bloch modes. J. M\u00f8rk acknowledges support for the Villum Foundation through the NATEC Centre of Excellence (grant 8692). \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\n\nThe heating of the solar chromosphere and corona is mediated by magnetic fields, which guide Poynting flux generated by plasma motions in the convection zone into the outer atmosphere and provide a means to convert the transported energy into heat. \nThe conversion mechanisms can be broadly divided into dissipation of wave energy \\citep[e.g.,][]{2020SSRv..216..140V} and dissipation of electric currents induced by the slow evolution of the magnetic field \\citep[e.g.,][]{1988ApJ...330..474P}, but their relative contribution to the energy balance is still unclear. In the quiet-Sun (QS) chromosphere, heating by acoustic waves may \\citep{2020A&A...642A..52A} or may not \\citep{2021ApJ...920..125M} act as the dominant process, but active regions (ARs) definitely require other energy sources.\n\nRecent research has focused mostly on the corona \\citep[e.g.,][]{2011Natur.475..477M,2013Natur.493..501C,2015ApJ...811..106H,2015RSPTA.37340256K} but much less on the chromosphere, which requires more sophisticated models that only recently became sufficiently realistic to allow for quantitative comparisons with observations \\citep{2019ARA&A..57..189C}. Furthermore, it is not trivial to interpret spectral diagnostics formed under optically thick, nonlocal thermodynamic equilibrium (non-LTE) conditions, such as the resonance lines of \\ion{Mg}{II} and \\ion{Ca}{II}, which require detailed three-dimensional (3D) radiative transfer calculations including partial frequency redistribution for full accuracy \\citep[see a review by][]{2017SSRv..210..109D}. \nHowever, energy losses are much greater in the chromosphere than in the corona: canonical estimates of average (total) losses for QS and AR are 4\\,kW\\,m$^{-2}$ and  20\\,kW\\,m$^{-2}$ for the chromosphere but only 0.3\\,kW\\,m$^{-2}$ and $<$\\,10\\,kW\\,m$^{-2}$, respectively, for the corona \\citep{1977ARA&A..15..363W}. In the era of high-resolution solar physics the focus should shift to obtaining new detailed models that reproduce the observed fine structure rather than averaged observables \\citep{2019ARA&A..57..189C}. Yet studies quantifying spatially and temporally resolved energy losses are scarce. Recent estimates with a time resolution of 30\\,s and a spatial resolution of $\\sim$100\\,km indicate losses that can locally be as high as 160\\,kW\\,m$^{-2}$ in the chromosphere \\citep{2020arXiv201206229D}. This was attributed to magnetic reconnection but no attempt was made at reconciling simulations with observations.  \n\nIt has been well established that radiative cooling is stronger in regions where the magnetic field is more concentrated, but this relationship is not linear \\citep[e.g.,][]{1987A&A...180..241S,1999ApJ...515..812H,2018A&A...619A...5B}. Low resolution ($\\gtrsim$\\,$10^{\\prime\\prime}$) observations in the millimeter range show brightness enhancements associated with network and ARs \\citep[e.g.,][]{1991ApJ...383..443L,2009A&A...497..273L}, while high-resolution ($\\sim$\\,$0.1^{\\prime\\prime}$) optical observations show that the \\ion{Ca}{II} K brightness in ARs is dominated by an extended component that is associated with more space-filling, horizontal magnetic fields \\citep{2018A&A...612A..28L}. \n\nDissipation of electric currents induced by the magnetic field is a prime candidate for explaining this heating, but determining the electric current vector in the chromosphere is notoriously challenging. It has only been reported in sunspots using \\ion{Ca}{II} 8542\\,\\AA~observations \\citep{2005ApJ...633L..57S,2021A&A...652L...4L} and in pores using the \\ion{He}{I}\\,10830\\,\\AA~multiplet \\citep{2003Natur.425..692S}. While \\ion{Ca}{II} is partially sensitive to temperatures, the \\ion{He}{I} line is not, and thus cannot be used to establish a direct link between heating and electric currents in the atmosphere. \n\nObservational evidence for magnetic reconnection in the lower atmosphere comes from the association of small-scale optical or ultraviolet (UV) brightenings, such as Ellerman bombs, UV bursts, and jets near or above patches of opposite magnetic polarity in the photosphere \\citep[e.g.,][]{2013ApJ...774...32V,2017A&A...605A..49C,2019ApJ...887...56T,2020A&A...633A..58O}, while numerical simulations show how these events could be related to heating in current sheets \\citep[e.g.,][]{2015ApJ...799...79N,2017A&A...601A.122D,2019A&A...626A..33H,2020ApJ...891...52S}.\n\nHigh-resolution observations of the free-free millimeter (mm) continuum are now provided by the Atacama Large Millimeter\/sub-millimeter Array \\citep[ALMA,][]{2009IEEEP..97.1463W}. This radiation is optically thick in the chromopshere, and the opacity is dominated by electron-proton collisions \\citep[see review by][]{2016SSRv..200....1W}. The mm continuum provides strong temperature constraints in inversions of non-LTE lines \\citep{2018A&A...620A.124D,2020A&A...634A..56D}, which should allow us to revise estimates of radiative energy losses that can be used to benchmark numerical simulations. \n\nSmall-scale transient {\\it mm-bursts} in ARs have also been linked to magnetic reconnection events \\citep{2020A&A...643A..41D}, but the lack of chromospheric magnetic field measurements did not enable a definitive conclusion in this regard. Here, we conducted a follow-up study that combines ALMA Band\\,3 (100\\,GHz or 3\\,mm) data with optical spectropolarimetry obtained at the 1-m Swedish Solar Telescope \\citep[SST,][]{2003SPIE.4853..341S}, providing further evidence for heating in the upper chromosphere in an AR by current dissipation of up to $\\sim$5\\,kW\\,m$^{-2}$ at the current Band\\,3 spatial resolution ($\\sim$\\,1.2$^{\\prime\\prime}$). This is supported by a magnetohydrostatic extrapolation and quantitatively reproduced by a 3D radiative-magnetohydrodynamics (r-MHD) simulation.\n\n\\section{Observations}\n\\label{section:observations}\n\nWe obtained simultaneous interferometric brightness temperature, $T_{\\rm b}$, maps of the 3\\,mm continuum with ALMA and spectropolarimetric observations using the CRISP instrument \\citep{2008ApJ...689L..69S} at the SST in the Fe\\,I 6173\\,\\AA~and the Ca\\,II 8542\\,\\AA~lines (hereafter $\\lambda6173$ and $\\lambda8542$) of NOAA AR 12738 on April 13, 2019. We also use ultraviolet (UV) imaging provided by the Atmospheric Imaging Assembly \\citep[AIA,][]{2012SoPh..275...17L} and magnetogram data from the Helioseismic and Magnetic Imager \\citep[HMI,][]{2012SoPh..275..207S} on board the Solar Dynamics Observatory \\citep[SDO,][]{2012SoPh..275....3P}.\n\nThe ALMA data set consists of $T_{\\rm b}$ maps taken approximately between 18:20-18:55\\,UTC and 19:16-19:51\\,UTC at 2\\,s cadence with 140\\,s calibration intervals every 10\\,min. These data have been previously presented by \\citet{2020A&A...643A..41D} and we refer to their study for further details of the data reduction and calibration.\nIn this paper, we only use ALMA maps taken within the time span of the SST observations (see below). The field of view (FOV) has a diameter of $60^{\\prime\\prime}$ and the pixel scale is $0.3^{\\prime\\prime}$. The noise root-mean-square level is approximately 20\\,K. \n\nCoordination between ALMA in Chile and the SST in La Palma is challenging because of the difference in time zones. The SST\/CRISP observations started as soon as the seeing conditions improved in the late afternoon but it was technically unfeasible to prolong the campaign for a long period given the closeness to local sunset. Therefore, the CRISP spectropolarimetric data (full-Stokes) consist of a single line scan of sufficient quality in 17 wavelength positions in $\\lambda8542$ in the range of $\\pm700$\\,m\\AA~and 15 positions in $\\lambda6173$ within $\\pm 275$\\,m\\AA~from the line center taken between 18:48:36\\,UTC and 18:48:56\\,UTC. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{FigSpl1_v3.pdf}\n    \\caption{Extended view of NOAA AR 12738 provided by SDO. Panel A: SDO\/AIA\\,1700\\,\\AA~intensity in square-root scale; panel B: Composite of AIA 171\\,\\AA~(red), 193\\,\\AA~(green), and 211\\,\\AA~(blue) intensities (unsharpened). Solid circle and dashed square show the ALMA and SST fields, respectively.} \n    \\label{fig:Spl1}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{Fig1_6p_v7.pdf}\n    \\caption{Multiwavelength imaging of a solar active region. The leftmost panels show an extended view of the target, while the ones on the right show a closer look at the center. Panel A: AIA\\,304\\,\\AA~intensity in logarithmic scale (18:48:53\\,UTC); panel B: SST\/CRISP $\\lambda$8542\\,\\AA~core (18:48:41\\,UTC); panel C: ALMA brightness temperature at 3\\,mm (18:48:53\\,UTC); panel D: HMI LOS magnetogram clipped at +\/- 1\\,kG (white\/black, 18:49:30\\,UTC); panel E: Composite of total linear polarization (red) and total circular polarization (blue) in $\\lambda$6173 (18:48:53\\,UTC); panel F: TLP in $\\lambda$8542 (18:48:41\\,UTC). Solid circle and dashed square show the ALMA and SST fields. The cyan contours correspond to $T_{\\mathrm{b}}[3\\,\\mathrm{mm}] = 9$\\,kK.}\n    \\label{fig:fig1}\n\\end{figure*}\n\nThe SST data were reduced using the \\texttt{CRISPRED} pipeline \\citep{2015A&A...573A..40D}, which includes flat-field and dark correction, cross-talk correction, polarimetric calibration, image reconstruction through Multi-Object Multi-Frame Blind Deconvolution \\citep[MOMFBD,][]{2002SPIE.4792..146L,2005SoPh..228..191V}, and fringe removal using Fourier filtering. An additional fringe removal step using principal component analysis was necessary in order to remove large-scale patterns in Stokes Q and U \\citep{2020A&A...644A..43P}. \nThe absolute intensity and wavelength calibrations were performed using the solar atlas of \\citet{1984SoPh...90..205N} as reference. The pixel scale is $0.059^{\\prime\\prime}$. \nWe co-aligned the CRISP and HMI data by cross-correlating the 6173\\,\\AA~continuum images taken by both instruments. The calibrated SST and ALMA data sets were analyzed using inversion codes (Section\\,\\ref{section:inv}).\n\nThe UV and extreme-UV (EUV) images taken by AIA and the line-of-sight (LOS) magnetograms obtained by HMI were downloaded from the Virtual Solar Observatory using the \\texttt{vso\\_search} routine in SolarSoftWare (SSW) \\citep{1998SoPh..182..497F}; they were further processed using IDL tools in SSW and the \\texttt{SunPy} package \\citep{2015CS&D....8a4009S} as described in \\citet{2020A&A...643A..41D}.  \nThe LOS magnetograms were deconvolved using \\texttt{Enhance}\\footnote{\\url{https:\/\/github.com\/cdiazbas\/enhance}} \\citep{2018A&A...614A...5D}.\nIn addition, we used the full-vector magnetogram that was closest in time (taken at 18:48:01\\,UTC) to the CRISP scans for the magnetic field extrapolation (Section\\,\\ref{section:extrapol}). \nThe vector magnetogram has been processed with the \\texttt{SHARP} pipeline \\citep{2014SoPh..289.3549B} and it was obtained from the JSOC interface \\footnote{\\url{http:\/\/jsoc.stanford.edu\/HMI\/HARPS.html}}.\n\nFigure\\,\\ref{fig:Spl1} shows a context view of the AR provided by SDO. The SST and ALMA pointings were on a group of pores and an arch-filament system (AFS) southwest of a large sunspot close to disk center. The EUV composite image was generated using the \\texttt{make\\_lupton\\linebreak\\_rgb} function in \\texttt{Astropy} \\citep{astropy:2018,2004PASP..116..133L}. The focus of this paper is the area surrounding the pore at the center of the ALMA FOV, which is the location of the western footpoints of the AFS.\n\n\n\\section{Methods}\n\\label{section:methods}\n\n\\subsection{Data inversions}\n\\label{section:inv}\n\nWe ran two different inversion codes on the SST data for different purposes: \\texttt{PyMilne}\\footnote{\\url{https:\/\/github.com\/jaimedelacruz\/pyMilne}} \\citep{2019A&A...631A.153D}, a code based on the Milne-Eddington (ME) approximation for photospheric lines using analytic response functions \\citep{2007A&A...462.1137O}; and \\texttt{STiC}\\footnote{\\url{https:\/\/github.com\/jaimedelacruz\/stic}} \\citep{2019A&A...623A..74D}, a multi-atom, non-LTE inversion code based on the Rybicki-Hummer code \\citep[RH,][]{2001ApJ...557..389U} that is suitable for both photospheric and chromospheric lines. The former provides the mean magnetic field vector within the formation region of the lines in a large FOV in a quick manner, whereas the latter requires more computing power but it allows for a detailed investigation of the thermodynamic stratification of the plasma as function of logarithmic optical depth of the 500\\,nm continuum (here simply $\\log \\tau$), while taking radiative transfer effects into account \\citep[e.g.,][]{2017SSRv..210..109D}.\n\n\\subsubsection{Milne-Eddington inversions}\n\\label{section:ME}\n\nWe fitted the $\\lambda$6173 spectra taken by SST\/CRISP using \\texttt{PyMilne} in order to provide an input for the magnetic field extrapolation code (Section\\,\\ref{section:extrapol}). We took into account the spectral point-spread-function of the instrument and cavity errors \\citep{2015A&A...573A..40D}. The magnetic filing factor is assumed to be unity. The results are shown in the supplementary Fig.\\,\\ref{fig:ME}.\n\nUpper limits on the uncertainty in the parameters can be obtained from their spatial variation on scales shorter than those of typical photospheric features (e.g., granule size $\\sim$1$^{\\prime\\prime}$). Using 5$\\times$5\\,px boxes at five different locations in the magnetic areas of the region of interest (ROI; see Fig.\\,\\ref{fig:fig2}), we found average standard deviations of $\\delta\\norm{B}=27$\\,G, $\\delta v_{\\rm LOS}=0.07\\,\\rm km\\,s^{-1}$, $\\delta \\theta=2^{\\circ}$, and $\\delta \\phi=8^{\\circ}$, in the magnetic field strength, line-of-sight velocity, inclination angle, and azimuth angle, respectively.\n\n\\subsubsection{Non-LTE inversions}\n\\label{section:nlte}\n\n\\begin{figure*}[t]\n    \\centering\n    \\sidecaption\n    \\includegraphics[width=120mm]{Fig2_v9.pdf}\n    \\caption{ Non-LTE inversions of the spectral data and radiative losses in the upper chromosphere. Temperature, integrated radiative losses within the contribution function of the 3\\,mm continuum, longitudinal field strength, and transverse field strength at selected optical depths from the photosphere to the chromosphere are shown. The cyan contours correspond to $T_{\\mathrm{b}}[3\\,\\mathrm{mm}] = 9$ and 10\\,kK. The middle row shows example observed (markers) and fitted (solid lines) intensity of the $\\lambda6173$ and $\\lambda8542$ lines at three locations indicated by different markers overlaid on the other panels; the intensity is normalized by $\\bar{I_{0}}$ -- the mean intensity at the bluest sampled wavelength of each line.}\n    \\label{fig:fig2}\n\\end{figure*}\n\nWe also ran non-LTE inversions of the $\\lambda$6173 and $\\lambda$8542 lines along with the ALMA $T_{\\rm b}[\\rm 3\\,mm]$ map using the \\texttt{STiC} code similarly to the approach described in \\citet{2018A&A...620A.124D}.\nThe Band\\,3 $T_{\\rm b}$ maps were converted into intensity in c.g.s. units using the Planck function and linearly resampled to the pixel scale of the CRISP data. The inherent inversion uncertainties (see below) outweigh the interpolation uncertainties.\nWe note that there is a factor of ten difference in spatial resolution between the optical\/infrared and mm diagnostics. Multiresolution spectral data can be dealt with using linear operators and global inversion schemes \\citep[e.g.,][]{2019A&A...631A.153D} but this has not yet been implemented into \\texttt{STiC}. Therefore, we have decided to preserve the high-resolution information provided by the SST\/CRISP spectropolarimetry and oversample the ALMA maps. This approach worked well here because of the distinct formation heights of $\\lambda$8542 and the 1.25mm continuum. The \nhigh-resolution information in the $\\lambda$6173 and $\\lambda$8542 lines provides the temperature stratification  at high resolution in the lower atmosphere. Because the chromosphere at depths lower than $\\log \\tau \\sim-5$ is so unconstrained from inversions of those spectral lines \\citep{2018A&A...620A.124D}, \\texttt{STiC} will essentially provide a low-resolution upper chromosphere constrained by ALMA on top of the high-resolution lower atmosphere constrained by the spectral lines.\nWe restricted the inversions to a $\\sim$$16.5^{\\prime\\prime}$$\\times$$24.8^{\\prime\\prime}$ subfield that encloses the ROI in order to reduce the computational cost. This analysis step required a few million core-hours.\n\nWe treated each pixel independently (1.5-D approximation) by simultaneously solving the statistical equilibrium equation for non-LTE populations of a 4-level H atom, a 6-level Ca\\,II atom, and a 16-level Fe\\,I atom along with the charge conservation equation, while other atoms and molecules are treated in LTE. Compared to LTE, treating the hydrogen atom in non-LTE and correcting the electron densities using charge conservation provides more realistic values that directly impact the calculation of the opacity in the mm continuum \\citep[e.g.,][]{2020A&A...634A..56D}. \nWe find that treating the Fe\\,I atom in non-LTE leads to average differences in the temperature and magnetic field strength of the order of a few percent at $\\log \\tau=0$ (relative to LTE). Differences in the field strength can be up to $\\sim$\\,60\\% in some patches. \\citet{2020A&A...633A.157S} investigated this in detail for the \\ion{Fe}{I} 6301, 6302\\,\\AA~lines using a more complete atomic model, but this would further increase the computational cost of our inversions. It is not critical for our conclusions as we are mainly interested in estimating radiative losses in the chromosphere, however, these effects contribute to the uncertainties of the ME inversions and field extrapolations (Section\\,\\ref{section:extrapol}).\nWe then ran another spectral synthesis solving the statistical equilibrium equation for a 11-level Mg\\,II atom in order to obtain population densities and radiative rates for computing radiative energy losses (Section\\,\\ref{section:enlosses}). The calculations include PRD in the \\ion{Mg}{II} h and k lines. The gas pressure is obtained by integration of the hydrostatic equilibrium equation and the value at the top boundary is treated as a free parameter \\citep{2019A&A...623A..74D}. \n\nThe polarization signals are assumed to be dominated by the Zeeman effect and we do not take the Hanle effect into account. This is a good approximation since the ROI features magnetic fields that are stronger than 500\\,G \\citep{2021arXiv210604478C}.\n\n\\texttt{STiC} uses parameterization by nodes that are interpolated using B\u00e9zier splines in an optical depth grid. We used nine nodes in temperature, four nodes in line-of-sight velocity, and two nodes in microturbulence, parallel and transverse magnetic field, and magnetic azimuth angle.\n\nThe uncertainties were estimated using a Monte Carlo approach \\citep{1992nrca.book.....P} on selected pixels in the ROI (supplementary Fig.\\,\\ref{fig:STiC_spl}). After obtaining good fits to the SST and ALMA observations, we generated up to 100 different synthetic spectra adding a random component that is the sum in quadrature of white noise and flux calibration uncertainties. The latter amounts to less than 1\\% for the spectral lines \\citep{1984SoPh...90..205N} and about 5\\% for the 3\\,mm continua. The synthetic spectra were inverted using different randomly generated atmospheres as initial guesses. This provides probability distributions for each parameter from which we computed the 16th, 50th and 84th percentiles at every optical depth (supplementary Fig.\\,\\ref{fig:STiC_spl}). \nTypical uncertainties in temperature are of the order of $\\sim$50\\,K or lower within $\\log\\tau=[-4,0]$, but they increase at lower optical depths reaching $\\sim$600\\,K at $\\log\\tau=-6$; below these values, the response functions of the diagnostics are weak. The uncertainty in the magnetic field strength is $\\sim$10-20\\,G for the longitudinal (or LOS) component and $\\sim$10-70\\,G for the transverse component within $\\log\\tau=[-5,0]$. The sensitivity of the $\\lambda$6173 and $\\lambda$8542 lines to magnetic fields is weak at optical depths lower than $\\log\\tau=-5$.\n\n\n\\subsection{Magnetic field extrapolation}\n\\label{section:extrapol}\n\nWe performed a magnetic field extrapolation using a magnetohydrostatic (MHS) model based on the SST and HMI data. Since the SST\/CRISP FOV covers only part of the magnetic connectivity, we embedded a cutout of the FOV in the SHARP-processed (disambiguated) HMI vector magnetogram, which covers the entire AR (top panels in supplementary Fig.\\,\\ref{fig:extrapolSpl}). \nIn this way we make use of the high-resolution information provided by the SST data in the ROI and we take advantage of the more extended HMI FOV for context. This is necessary to interpret the interaction of small- and large-scale loop systems. The resulting magnetogram is in cylindrical-equal-area projection (CEA) coordinates.\n\nTo ensure consistency between the CRISP and HMI magnetograms, we disambiguated the azimuth angle provided by the ME inversion of the CRISP data by imposing an acute angle with the HMI\/SHARP magnetogram. Both magnetograms are derived from the same spectral line ($\\lambda6173$). The borders of the CRISP cutout were smoothed with a Gaussian kernel.  \nFinally, the maps of the three components of the magnetic field vector, $\\vec{B}$, were convolved with a median filter to reduce the impact of bad pixels and inversion noise in the extrapolation. \n\nThe MHS model takes into account the effect of plasma forces which cannot be ignored in the lower atmosphere. An optimization approach is used to numerically solve the equations \\citep{2019A&A...631A.162Z}:\n\\begin{eqnarray}\n    \\frac{1}{\\mu_{0}}(\\nabla \\times \\vec{B}) \\times \\vec{B} - \\nabla p + \\rho\\,\\vec{g} = 0,\\\\\n    \\nabla \\cdot \\vec{B} = 0,\n\\end{eqnarray}\nstarting from a nonlinear force-free field extrapolation as initial guesses \n\\citep{2004SoPh..219...87W}.\nHere, $\\mu_0$ is the magnetic permeability, $p$ is the gas pressure, $\\rho$ is the mass density, and $\\vec{g}$ is the gravitational acceleration. The optimization procedure aims to reach a static equilibrium state between the Lorentz force, the pressure gradient, and gravity.\n\nThe model uses the photospheric vector magnetogram (Section\\,\\ref{section:ME}) as the lower boundary condition. The plasma pressure at the lower boundary is calculated from:\n\\begin{equation}\n    p+\\frac{B^2_{z}}{2} = P_\\mathrm{ph},\n\\end{equation}\nwhere $P_\\mathrm{ph},$ is a typical photospheric pressure. Further details of the MHS method can be found in \\citet{2018ApJ...866..130Z,2020A&A...640A.103Z}.\nThe resulting model has a size of $2000\\times1800\\times256$ pixels with a uniform pixel scale of 40\\,km.\n\n\\subsection{3D radiative-magnetohydrodynamics simulation}\n\\label{section:simulation}\n\nThe simulation presented here was performed with the \\texttt{MURaM} code \\citep{2005A&A...429..335V,2017ApJ...834...10R}, which includes the following physics: single fluid MHD, 3D grey LTE radiative transfer, a tabulated LTE equation of state, Spitzer heat conduction, and optically thin radiative losses in the corona based on \\texttt{CHIANTI} \\citep{2012ApJ...744...99L}. The chromospheric and coronal parts of the simulation domain is heated by the Poynting flux generated through magnetoconvection in the photosphere and convection zone. Earlier simulations with this code have reproduced many observed chromospheric phenomena \\citep[e.g.,][]{2019A&A...631A..33B,2020LRSP...17....3L}.\n\nThe simulation domain has an extent of $40\\times 40 \\times 22$\\,Mm, spanning the vertical direction from $-8$\\,Mm to 14\\,Mm above the average $\\tau_{500\\,\\mathrm{nm}}$\\,$=$\\,1 height. The pixel size is 0.078\\,Mm. The run was initialized with a bipolar uniform magnetic field of 200\\,G \\citep{2011A&A...533A..86C}, which is added to the well-developed nonmagnetic convection simulation to form extended magnetic field concentrations at meso- to super-granular scales. This was done with the aim of reproducing the plasma dynamics of solar plage -- regions of moderate magnetic activity. The computational domain was then extended to include the upper solar atmosphere, and the magnetic field from the preexisting simulation was used for potential field extrapolation into the rest of the domain. The new simulation was then run until a relaxed state was achieved. An additional bipolar flux system was advected through the bottom boundary through an ellipsoidal region with major axes (a, b) = (3, 1) Mm and a field strength of 8000\\,G \\citep{2019NatAs...3..160C}, which emerged from the convection zone into the chromosphere directly beneath the preexisting filaments.  Once the flux reaches the photosphere, its field strength decreased to around 1.5\\,kG.\n\nSince the non-LTE spectral synthesis is quite computationally intensive and memory-demanding, here we analyzed one single simulation snapshot where we identified a region of enhanced $T_{\\rm b}[3\\rm\\,mm]$ at a later stage of the flux emergence $t=17$\\,min after the magnetic flux reached the photosphere. Anyhow, the lack of spectropolarimetric time series in $\\lambda6173$ and $\\lambda8542$ does not allow us to compare the observations and the unraveling of the simulated flux emergence. \n\nWe computed the emerging intensities in $\\lambda$8542 from the simulation in non-LTE 1.5D (full-Stokes) using \\texttt{STiC} and in 3D (intensity only) using \\texttt{Multi3D} \\citep{2009ASPC..415...87L}. The synthetic \\ion{Ca}{II} lines were convolved with the CRISP response function.\nThe 3\\,mm continuum intensities were computed assuming statistical equilibrium and charge conservation using \\texttt{STiC} \\citep[see also][]{2020A&A...634A..56D}.\nBesides the synthetic intensities, we also stored the opacities $\\alpha (\\nu, z)$ as a function of frequency, $\\nu$, and height, $z$ for each pixel. We investigated the formation of the 3\\,mm continuum in detail using contribution functions (CF) defined as\n\\begin{equation}\n    \\mathrm{CF}_{\\nu} (z)=\\alpha_{\\nu}(z)\\,S_{\\nu}(z)\\,e^{-\\tau_{\\nu}(z)},\n\\end{equation}\n\\noindent where $\\nu=100$\\,GHz, $S_{\\nu}$ is the source function, which is given by the Planck function, and $\\tau_{\\nu}$ is the optical depth. This quantity essentially quantifies the contribution from different heights of the atmosphere to the emerging intensities.\n\n\\subsection{Radiative energy losses}\n\\label{section:enlosses}\n\nBesides the radiative losses, $Q_{\\rm l}$, that can be directly retrieved from the \\texttt{MURaM} simulation output \\citep{2017ApJ...834...10R}, we also obtained $Q_{\\rm l}[\\rm STiC]$ from the non-LTE synthesis with \\texttt{STiC} by summing the contributions to the radiative cooling from the Ca\\,II H, K, infrared triplet, Mg\\,II h, k, UV triplet, H$\\alpha$, and Ly$\\alpha$ lines as follows:\n\\begin{equation}\n    Q_{\\rm l}[\\mathrm{STiC}] = h\\nu_{0}(n_{\\rm u}R_{\\rm ul} - n_{\\rm l}R_{\\rm lu}),\n\\end{equation}\n\\noindent where $h$ is the Planck constant, $\\nu_{0}$ is the frequency of the transition, $n_{\\rm u}$ and $n_{\\rm l}$ are the population densities of the upper and lower levels, and $R_{\\rm ul}$ and $R_{\\rm lu}$ are the radiative rates of the transitions\n\\citep[e.g.,][]{2020arXiv201206229D}. These chromospheric losses are more suitable for comparison with the losses that can be inferred from observations using data inversions (Section\\,\\ref{section:nlte}).\n\nTotal radiative losses were obtained by integration in the height range spanned by the CF of the 3\\,mm continuum using a threshold of 1\\% of the maximum in each pixel. This criterion was used both for the CFs obtained from the simulation and observations. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{extrapol_v4.png}\n    \\caption{Magnetohydrostatic extrapolation of the SST and HMI composite magnetogram and millimeter continuum brightness. The background shows a SST\/CRISP and HMI magnetogram composite image clipped at +\/-1\\,kG (white\/black). The magnetic field lines are computed from the MHS extrapolation and they are color-coded with the horizontal field strength. The pink shade shows regions where $T_{\\rm b}\\rm\\,[3\\,mm]>9$\\,kK.} \n    \\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{FigSpl5_v7.pdf}\n    \\caption{Simulated magnetograms and synthetic emission. Strength of the vertical (panel A) and horizontal (panel B) components of the magnetic field at $z=0$\\,Mm; panel B has been gamma-adjusted for display purposes. Panel C: Intensity in the core of $\\lambda 8542$; the range is capped for display purposes. Panel D and panel E: Continuum $T_{\\rm b}[\\rm 3\\,mm]$ at full resolution and convolved with a Gaussian kernel with full-width-at-half-maximum of 1.2$^{\\prime\\prime}$; the cyan contours show $T_{\\rm b}[\\rm 3\\,mm]=9$\\,kK. The dashed box delimits the area displayed in Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{B}, \\ref{fig:fig4}\\textcolor{blue}{C}. The lower right panels show selected $\\lambda 8542$ profiles (normalized intensity) in the flux emergence region. Vertical cuts through various parameters of the simulated atmosphere along the slices {\\it S3}, {\\it S4}, and {\\it S5} are displayed in the supplementary Fig.\\,\\ref{fig:simSlices}.}\n    \\label{fig:simextra0} \n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{merge_test9_v6.pdf}\n    \\caption{Magnetic topology, heating rates, and the formation of the millimeter continuum in a 3-D radiative-magnetohydrodynamics simulation. Panel A: Photospheric magnetogram clipped at +\/- 1\\,kG; the yellow shade shows where $T_{\\rm b}[3 \\mathrm{mm}]>9$\\,kK. The area inside the dotted box is displayed in panels B and C, which show $T_{\\rm b}[3 \\mathrm{mm}]$ and the sum of viscous and resistive heating integrated within the CF of the 3\\,mm continuum. The group of panels D show from the left to the right: Current density squared per mass unit (square-root scaling), logarithm of temperature, and CF$_{\\rm 3\\,mm}$ slices along the dotted paths {\\it S1} and {\\it S2} overlaid on panels B and C; the white dashed lines show where $\\tau_{\\rm 3\\,mm}=1$, and the dotted black lines show where CF is at 1\\% of the maximum in each column. } \n    \\label{fig:fig4}\n\\end{figure*}\n\n\n\\section{Results}\n\\label{section:results}\n\nAR 12738 showed ongoing magnetic flux emergence into a preexisting AFS. The ALMA map (Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{C}) reveals elongated patches of enhanced $T_{\\rm b}\\rm [3\\,mm]$ by $\\sim$3,000\\,K relative to QS values, which is indicative of local heating in the upper chromosphere. The ALMA time series (supplementary movie) shows recurring brightenings exhibiting several hundred kelvin $T_{\\rm b}\\rm [3\\,mm]$ variations, which last from a few tens of seconds to a few minutes with no clear periodicity. These hot spots are also fully visible in the AIA\\,304\\,\\AA~images (Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{A}) and partially in AIA\\,171\\,\\AA~(Fig.\\,\\ref{fig:Spl1}) -- but not in hotter channels. Correlation between Band\\,3 brightenings and coronal emission in ARs has been reported before \\citep{2020A&A...643A..41D,2021A&A...651A...6B}. \nThere are other bright regions in the FOV in the 304\\,\\AA~images that have no counterpart in the ALMA maps. This may imply varying relative differences in formation heights of both diagnostics at different locations.\n\nA comparison to the HMI LOS magnetogram (Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{D}) shows that this region is located between concentrations of magnetic field of opposite polarity, which are the footpoints of chromospheric loops that connect the two patches. The SST data indeed show short bright loop-like structure in the $\\lambda$8542 core (Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{B}) and significant total linear polarization signals ($\\mathrm{TLP}=\\int\\sqrt{Q^2+U^2}\/I\\,d\\lambda$) indicative of strong transverse magnetic field in the photosphere and chromosphere (Figs.\\,\\ref{fig:fig1}\\textcolor{blue}{E}, \\ref{fig:fig1}\\textcolor{blue}{F}). The composite image of other AIA EUV channels shows longer dark filaments overlying the small-scale loops (Fig.\\,\\ref{fig:Spl1}).\n\n\\subsection{Physical properties from inversions}\n\nThe non-LTE inversions provide a well-resolved temperature structure from the photosphere to the top of the chromosphere where the 3\\,mm continuum is formed. The sensitivity of the spectra to magnetic fields is lower, and the inversions provide the vector magnetic field in the photosphere and mid-chromosphere. \n\nFigure~\\ref{fig:fig2} shows the results of the \\texttt{STiC} inversions. We find increased temperatures in the chromosphere at least up to optical depth $\\log \\tau$\\,$\\sim$\\,$-6$ where $T$\\,$\\sim$\\,10,000\\,K, which is on the order of the observed $T_{\\rm b}\\rm\\,[3\\,mm]$. The warm locations feature $\\lambda$8542 profiles with central reversals or raised intensities in the wings and small Doppler shifts ($\\lesssim$\\,$4\\,\\rm km\\,s^{-1}$), which are well-reproduced by our models (supplementary Fig.\\,\\ref{fig:STiC_spl}). \n\nWe computed the radiative energy losses as a proxy for the heating rate, which cannot be directly observed. In Fig.\\,\\ref{fig:fig2} we also show integrated losses from the inferred atmosphere in the strongest lines of H\\,I, Ca\\,II, and Mg\\,II (Section\\,\\ref{section:enlosses}) in the height range spanned by the CF of the 3\\,mm continuum. This is done in geometrical height scale assuming hydrostatic equilibrium.\nEnergy losses range from 2.6 to 4.9\\,$\\rm kW\\,m^{-2}$ with a mean value of $\\sim$4\\,$\\rm kW\\,m^{-2}$ within the $T_{\\rm b}\\rm\\,[3\\,mm]=9$\\,kK contours, which is higher than previous estimates of $\\sim$2\\,$\\rm kW\\,m^{-2}$ in the upper chromosphere \n\\citep{1977ARA&A..15..363W}. \n\nThe longitudinal and transverse photospheric field in the ROI have a maximum strength of $\\lvert B_{\\rm ln}\\rvert$\\,$=$\\,$1890$\\,G and $\\lvert B_{\\rm tr}\\rvert$\\,$=$\\,$1380$\\,G and the uncertainty is of the order of 10\\,G. The chromospheric magnetic field is overall weaker, but the transverse component remains strong with an average (maximum) value of $\\lvert B_{\\rm tr}\\rvert\\sim480$\\,G (1060\\,G) with an uncertainty of $\\sim$70\\,G at $\\log \\tau=-5$ within the $T_{\\rm b}\\rm\\,[3\\,mm]=9$\\,kK contours. The transverse field traces the near-horizontal tops of the emerging loops and coincides with higher temperature regions. \n\n\n\\subsection{Magnetic topology}\n\nThe fidelity of the magnetic field derived from the inversions is high, but the vertical resolution is low and the FOV is small. Therefore, we complement our analysis with a MHS extrapolation based on a high-resolution magnetogram derived from the SST $\\lambda$6173 data (supplementary Fig.\\,\\ref{fig:ME}) embedded in a lower resolution magnetogram provided by HMI (Section\\,\\ref{section:extrapol}). \n\nFigure~\\ref{fig:fig3} was produced using the \\texttt{VAPOR} software\\footnote{\\url{www.vapor.ucar.edu}} \\citep{atmos10090488} and it shows a 3D rendering of the overlying field connecting the pore and the plage region, whose direction is the same as the AFS in the EUV images (c.f. Fig.\\,\\ref{fig:Spl1}).\n\nA comparison with Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{B} confirms that the field lines align with the direction of the bright $\\lambda8542$ fibrils inside the ALMA $T_{\\rm b}$ contours, in agreement with the picture drawn from the imaging and inversions. The horizontal magnetic field strength in the short loops derived from the extrapolation (as high as $\\sim$\\,800\\,G around $z$\\,$\\sim$\\,1\\,Mm) is consistent with the inversion results. \nPatches of high $T_{\\rm b}\\rm\\,[3\\,mm]$ coincide with the interaction region between the loop systems where current sheets (tangential discontinuities) must exist between them. \n\nWe computed the current density ($\\vec{j}=\\nabla \\times \\vec{B}\/\\mu_{0}$) from the extrapolation for completeness (supplementary Fig.\\,\\ref{fig:extrapolSpl}), but we note that spurious currents arise from the lack of smoothness in the magnetic field introduced by inversion noise in the magnetograms and the azimuth angle disambiguation, as well as limitations of the MHS extrapolation algorithm itself \\citep{2018ApJ...866..130Z}. We find current strands over a range of heights relevant to the formation of the 3\\,mm continuum ($\\sim$1-4\\,Mm, Fig.\\,\\ref{fig:fig4} and Fig.\\,\\ref{fig:simextra2}\\textcolor{blue}{C}) that is cospatial with both the short $\\lambda8542$ loops and the long, overlying ones seen in the AIA images, but we could not identify a single height that correlates strongly with $T_{\\rm b}\\rm[3\\,mm]$. However, we do expect spatial and temporal variations of the formation height of the mm continuum \\citep[e.g.,][]{2015A&A...575A..15L,2020ApJ...891L...8M}.\n\nWe attempted to correlate $T_{\\rm b}[\\rm 3\\,mm]$ with $j^{2}\/\\rho$ in the MHS extrapolation at the height where $\\tau_{\\rm 3\\,mm}=1$, obtained from the non-LTE inversions assuming hydrostatic equilibrium (typically $z$\\,$\\sim$\\,1.5-1.8\\,Mm) and taking into account the fact that the zero point of the MHS extrapolation is defined by the mean formation height of $\\lambda6173$; however, this turned out to be inconclusive. This comparison was also made in column mass scale. \nThis is partly due to the fact that the height scales obtained from these two methods are different, but we also expect the MHS extrapolation to underestimate the magnetic structure and field strength compared to $\\lambda8542$ spectropolarimetry \\citep{2021arXiv210902943V}, so the electric current values carry some uncertainty. Moreover, the height-integration effect of the CF of the 3\\,mm continuum may smear such correlation, as further discussed in Section\\,\\ref{section:formation}.\n\nTherefore, we cannot unambiguously link the observed millimeter emission to heating in current sheets based on these data alone. Nonetheless, the 3\\,mm continuum forms above $\\lambda8542$ \\citep{2018A&A...620A.124D}, but below the He\\,II\\,304\\,\\AA~line, which strongly suggests that the mm continuum originates in the the shear layer. \n\n\\subsection{Investigating the formation of the mm continuum}\n\\label{section:formation}\n\n\\begin{figure*}[t]\n    \\centering\n    \\sidecaption\n    \\includegraphics[width=120mm]{FigSpl7_v7.pdf}\n    \\caption{Heating rates and the formation of the millimeter continuum in the simulation. Panel A: Integrated $j^{2}\/\\rho$ weighted by the contribution function of the 3\\,mm continuum (CF$_{\\rm  3\\,mm}$). Panel B: Photospheric magnetogram (range $\\pm1$\\,kG) with 3-D rendering of magnetic field lines and CF$_{\\rm 3\\,mm}$ (blue shade) of the region indicated by the arrow in panel A. A threshold was applied to the CF values not to hide the field lines. Panel C: Height at which $\\tau=1$ at 3\\,mm; the errorbar overlaid on the colormap shows the median and the range between the 16th and 84th percentiles of the distribution. Panel D: Temperature (dashed lines), total heating rates per mass (yellow lines), and CF$_{\\rm 3\\,mm}$ (blue lines) as a function of height at two locations indicated in panels A and C. }\n    \\label{fig:simextra2} \n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\sidecaption\n    \\includegraphics[width=120mm]{FigSpl12_v4.pdf}\n    \\caption{Radiative losses in the simulation. Total radiative losses integrated within CF$_{\\rm 3 mm}$ from the \\texttt{MURaM} output (panel A) and non-LTE values calculated using \\texttt{STiC} (panel B) at full resolution; panels C and D show the corresponding quantities convolved with a Gaussian kernel with FWHM of 1.2$^{\\prime\\prime}$. Cyan contours show $T_{\\rm b}[\\rm 3\\,mm]=9$\\,kK. Top row panels are displayed in square-root scale.}\n    \\label{fig:simLosses}\n\\end{figure*}\n\nTo gain further insight into the magnetic flux emergence and how electric currents heat the atmosphere in that process, we ran a 3D r-MHD simulation using \\texttt{MURaM}, as explained in Section\\,\\ref{section:simulation}. Figure~\\ref{fig:simextra0} displays the strength of the vertical and horizontal components of the photospheric magnetic field vector along with the synthetic emission in the core of $\\lambda8542$ (corrected for Doppler shifts) and $T_{\\rm b}[\\rm 3\\,mm]$. In order to simulate the effect of the ALMA beam to first order, we convolved $T_{\\rm b}[\\rm 3\\,mm]$ at the simulation resolution (Fig.\\,\\ref{fig:simextra0}\\textcolor{blue}{D}) with a Gaussian kernel with full-width-at-half-maximum $\\rm FWHM$\\,$=$\\,1.2$^{\\prime\\prime}$ (Fig.\\,\\ref{fig:simextra0}\\textcolor{blue}{E}). In this paper we are primarily interested in the region inside the dashed box that encloses a patch of emerging flux and where we see some compact mm brightenings as well as bright fibrils. At the ALMA resolution the bright strands appear more blob-like as in the observations (Fig.\\,\\ref{fig:fig1}\\textcolor{blue}{C}). The simulation is also able to reproduce $\\lambda$8542 profiles qualitatively similar to the observations featuring central reversals and raised intensities in the wings. There are patches where the line core is in full emission, which we do not find in the single CRISP scan that we have.  Emission profiles are uncommon in the simulation, and they occur at sites of enhanced heating at the $\\tau=1$ layer of the line core (e.g., panels S4 in Fig.\\,\\ref{fig:simLosses}). We would need a longer $\\lambda$8542 time series to investigate whether these kinds of emission profiles can occur as in the simulation.\n\nFigure~\\ref{fig:fig4}\\textcolor{blue}{A} shows the footprint of the emerging bubble together with field lines highlighting the direction of the emerging loops (green), the overlying filaments (red), and the two sets of reconnected field lines (blue and orange). The synthetic 3\\,mm emission shows a bifurcated structure following the reconnected field lines in the interaction region \n(Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{B}). \n\nWe find enhanced total heating rates, defined as the sum of the viscous, $Q_{\\rm v}$, and resistive, $Q_{\\rm r}$, heating within CF$_{\\rm 3\\,mm}$ in thin strands in the interaction region (Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{C}). The mean (maximum) values at the location where $T_{\\rm b}\\rm\\,[3\\,mm]>10,000$~K are 12\\,kW\\,m$^{-2}$ (184\\,kW\\,m$^{-2}$).\nThis heating is caused by dissipation of currents and viscous dissipation of mass flows, both of which are driven by the interaction of the magnetic loop systems. This does not only directly cause Joule heating, but also drives flows through the Lorentz force that dissipate through viscosity. Our simulation employs numerical diffusive and resistive terms with an effective magnetic Prandtl number $P_\\mathrm{m} > 1$ so that viscous heating is the largest contributor. However, in the real chromosphere $P_\\mathrm{m} < 1$ and the heating is dominated by electric resistivity. The 3\\,mm continuum is optically thick where $P_\\mathrm{m} < 1$ (supplementary Fig.\\,\\ref{fig:simSlices2}).\n\nThe vertical cuts along the slices {\\it S1} and {\\it S2} (Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{D}) underscore that CF$_{\\rm 3\\,mm}$ peaks at locations of high $j^2\/\\rho$ in the chromosphere where there are magnetic field gradients, but the resulting $T_{\\rm b}\\rm\\,[3\\,mm]$ may reflect contributions from multiple strands along the LOS. In supplementary Fig.\\,\\ref{fig:simSlices} we also provide additional 2D slices of the total heating rates per mass unit, which show the same qualitative picture. \nThe CF$_{\\rm 3\\,mm}$ shows a loop-like structure following the shape of the chromosphere-transition region boundary where the gas is still partially ionized. Locations of larger $j^2\/\\rho$ at transition region and coronal temperatures have a negligible (3\\,mm) opacity so they do not contribute to $T_{\\rm b}\\rm\\,[3\\,mm]$. However, the heating rate in those strands can be very large, and hot pockets embedded in the 3\\,mm formation height range lead to the large peak values in Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{C}. \n\nThe simulated chromosphere is pervaded by multiple current sheets at different heights and shows a complicated thermal structure (Figs.\\,\\ref{fig:fig4}\\textcolor{blue}{D}), while the analysis of the opacity data shows that the formation height of the 3\\,mm continuum varies significantly across the flux emergence region (Fig.\\,\\ref{fig:simextra2}\\textcolor{blue}{C}). The CF of the 3\\,mm continuum peaks where the loop systems meet an angle (Fig.\\,\\ref{fig:simextra2}\\textcolor{blue}{B}) and the atmosphere is locally heated (see also the supplementary Fig.\\,\\ref{fig:simSlices}).  Figure~\\,\\ref{fig:simextra2}\\textcolor{blue}{A} shows integrated $j^{2}\/\\rho$ weighted by CF$_{\\rm 3\\,mm}$, which reveals the filamentary structure of the heating that is very similar to the synthetic emission itself (c.f. Fig.\\,\\ref{fig:simextra0}\\textcolor{blue}{D}).\n\nThe layer where optical depth is unity in the core of $\\lambda$8542 is typically located below that of the 3\\,mm continuum in the flux emergence region (supplementary Fig.\\,\\ref{fig:simSlices}). This suggests that the former traces the top of the low-lying fibrils, whereas the latter sees the heating at or above the $\\lambda$8542 canopy. \n\nFigure~\\,\\ref{fig:simLosses} shows a comparison between the total radiative losses $Q_{\\rm l}$ and $Q_{\\rm l}[\\rm STiC]$ in the simulated chromosphere. At the native resolution of the simulation, the losses show the same bifurcated structure in the interaction region (c.f. Fig.\\,\\ref{fig:fig4}\\textcolor{blue}{B}, \\ref{fig:fig4}\\textcolor{blue}{C}). Total $Q_{\\rm l}$ in the simulation reach above 100\\,$\\rm kW\\,m^{-2}$ in some pixels due to the inclusion of losses at transition-region and coronal temperatures in thin pockets within the formation range of the millimeter continuum (c.f. the supplementary Fig.\\,\\ref{fig:simSlices}). These are not included in the \\texttt{STiC} losses, hence, the lower values. Much of the filamentary structure is lost at the ALMA resolution, and the values of the radiative losses decrease significantly; the mean(standard deviation) is $\\sim$\\,$6(\\pm 2)\\,\\rm kW\\,m^{-2}$, where $T_{\\rm b}\\rm\\,[3\\,mm]$\\,>\\,9\\,kK, which is similar to the values derived from the non-LTE inversions of the SST and ALMA data (Fig.\\,\\ref{fig:fig2}). \n\n\\section{Discussion and conclusions}\n\\label{section:conclusions}\n\nIn this paper we present an analysis of joint SST and ALMA observations of an AR on the Sun using inversion methods, field extrapolations, and a numerical simulation. We find enhanced $T_{\\rm b}[\\rm 3\\,mm]$ and bright $\\lambda$8542 profiles between a pore and parasitic opposite polarity patch in the photosphere. This is also the location of short, low-lying chromospheric fibrils, where the magnetic field is more horizontal to the solar surface, underneath an overlying AFS seen in absorption in the AIA EUV channels.\n\nOur analysis validates dissipation in current sheets as, at least, a locally dominant source of atmospheric heating, which produces brightenings in chromospheric diagnostics within ARs. Integrated radiative losses in the strongest chromospheric lines obtained from the simulation can be an order of magnitude higher than the ones determined from the observations. However, degrading the former to the spatial resolution of ALMA yields a mean(standard deviation) value of $\\sim$\\,6\\,$(\\pm 2)\\,\\rm kW\\,m^{-2}$, which is consistent with the observed values. The heating occurs on spatial scales that are not resolved in the ALMA Band\\,3 data. We note that this is only an approximate way of comparing losses obtained from observations and simulations at different spatial resolutions. In principle we would need to investigate the impact of the lack of constraints in the inversions on the estimates of the radiative losses by inverting synthetic data from the simulation at different resolutions, but this is beyond the scope of this paper.\n\nOur estimates are much lower than the values reported in an observed magnetic reconnection event of up to $\\sim$\\,160$\\,\\rm kW\\,m^{-2}$ \\citep{2020arXiv201206229D}. This is partly due to differences in the integration method but, most importantly, that event was much stronger and showed strong flows and an associated surge, which we did not detect. However, our limited spectropolarimetric data only provide one time frame and do not allow us to investigate the dynamics in detail. The ALMA time series does show recurring elongated brightenings at that location prior to the SST campaign, which would be consistent with the bright strands that we see in the simulation.\n\nWe note that the simulation makes use of numerical diffusivity and viscosity terms that are larger than their physical values. This is common in numerical simulations of the type employed here, and experiments show that this has only a marginal effect on the total energy dissipation rate \n\\citep{1996JGR...10113445G,2017ApJ...834...10R}, \nwhich is mainly determined by the Poynting flux input in the photosphere. \nThe simulation was run with an effective numerical magnetic Prandtl number $P_\\mathrm{m}>1$; about two thirds of the energy dissipated in the chromosphere is through viscosity and one third by electric resistivity. The chromosphere has a $P_\\mathrm{m}$ that is significantly below unity, especially if ambipolar diffusion is taken into account \n\\citep{2012ApJ...753..161M}, which appears as an additional cross-field resistivity in the MHD approximation. \nWhile the majority of the dissipation of magnetic energy in the simulated chromosphere occurs through the Lorentz force, which drives flows that are then damped through viscosity, the solar chromosphere causes the majority of the energy to dissipate through currents \\citep[e.g.,][]{2017ApJ...834...10R,2019ApJ...879...57B}.\n\nAlthough our simulation lacks some chromospheric physics such as ion-neutral interactions and non-equilibrium ionization \\cite[e.g.,][]{2012ApJ...747...87K,2020A&A...633A..66N}, it reproduces well the observed magnetic configuration, infrared and radio spectra, and energetics remarkably. It also shows how the flux emergence leads to enhanced emission in the mm continuum, which is a good proxy for local chromospheric heating, and how the opacity may vary across the flux emergence region. However, the height-integration effect of the contribution function implies that the observed brightness temperatures may be a weighted average of contributions from several different layers along the LOS \\cite[see also][]{2020ApJ...891L...8M}, which complicates the interpretation of observations.\n\nThe MHS extrapolation reveals a textbook magnetic topology similar to previously proposed models of the interaction of emergent flux and the canopy \\cite[e.g.,][]{2003Natur.425..692S,2014LRSP...11....3C,2016ApJ...825...93O}, leading to the formation of current sheets between the two flux systems \\cite[e.g.,][]{2007ApJ...666..516G,2014ApJ...788L...2A}. If the loop interaction occurs at coronal heights this may lead to much higher temperatures ($\\gtrsim$\\,1\\,MK) and produce the recently discovered {\\it campfire} EUV signatures, as simulations suggest \\citep{2021A&A...656L...7C}.\n\nThese findings may play a role in explaining the solar cycle modulation of brightness temperatures in the millimeter range and their correlation with the sunspot number \\citep{2020ApJ...902..136G}, as well as the excess millimeter brightness in chromospheres of other stars \\citep{2015ApJ...809...47M,2015A&A...573L...4L,2017A&A...602L..10O}. Our analysis quantifies radiative losses, which can also be used to benchmark simulations of solar and stellar atmospheres. \n\n\\begin{acknowledgements}\n\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2018.1.01518.S. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof\u00edsica de Canarias. \nThe Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2017-00625). ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ. \nPart of the calculations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) at Link\\\"{o}ping University and the PDC Centre for High Performance Computing (PDC-HPC) at the Royal Institute of Technology in Stockholm. \nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (SUNMAG grant agreement 759548 and SOLARNET grant agreement 824135), the Knut and Alice Wallenberg Foundation, Swedish Research Council (2021-05613) and Swedish National Space Agency (2021-00116); this material is partly based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977; X.Z. is supported by the mobility program (M-0068) of the Sino-German Science Center. \n\nThis research has made use of \\texttt{Astropy} (\\url{https:\/\/astropy.org}) -- a community-developed core Python package for Astronomy \\citepads{astropy:2018} and \\texttt{SunPy} (\\url{https:\/\/sunpy.org}) -- an open-source and free community-developed solar data analysis Python package \\citep{2015CS&D....8a4009S}.\n\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nE-mail, or electronic mail, is one of the most popular forms of communication in the world, with\nover~3.9 billion active email users~\\cite{campaign_monitor}. As a side effect of this rapid growth, \nthe number of unwanted bulk email messages---i.e., spam messages---sent with commercial \nor malicious intent has also grown. According to~\\cite{campaign_monitor},\n60 billion spam emails will be sent each \nday for the next three years. \n\nWhile text-based spam filtering systems are in use by most, \nif not all, e-mail clients~\\cite{machine_learning_for_email_spam_filtering}, spammers \ncan embed messages in attached images to evade such systems---such messages are known as \nimage spam. Image spam detectors based on optical character recognition (OCR) have been \ndeployed to combat such e-mail. As a countermeasure, spammers can modify images \nso as to disrupt OCR based techniques~\\cite{image_spam_hunter}. \n\nIn recent years, deep learning models, such as multi-layer perceptrons and convolutional neural networks, \nhave been successfully applied to the image spam problem~\\cite{image_spam_hunter, \nimage_spam_analysis_and_detection, statistical_feature_extraction_for_classification, \nsupport_vector_machines_for_image_spam, deepimagespam, image_spam_filtering_using_conv, \nconvolutional_neural_networks_for_image_spam}. Note that these techniques do not rely\non OCR, but instead detect image spam directly, based on characteristics of the images.\n\nWith the recent development of perturbation methods, the possibility exists for spammers \nto utilize adversarial techniques to defeat image-based machine learning \ndetectors~\\cite{intriguing_properties}. To date, we are not aware of perturbation techniques\nhaving been used by image spammers, but it is highly likely that this will occur in the\nnear future.\n\nThe main contributions of our research are the following.\n\\begin{itemize}\n\\item We show that the universal perturbation adversarial attack is best suited for the \ntask of bypassing deep learning-based image spam filters.\n\\item We propose a new image transformation-based attack that utilizes the maximization \nof layer activations to produce spam images containing universal perturbations. This technique\nfocuses perturbations in the most salient regions, as well as concentrating natural features \nin the remaining regions.\n\\item We compare our proposed adversarial technique to existing attacks and find that our \napproach outperforms all others in terms of accuracy reduction, computation time per example, \nand perturbation magnitude.\n\\item We generate a large dataset containing both non-spam and adversarial spam images using \nour proposed attack. The authors will make this dataset available to researchers.\n\\end{itemize}\n\nThe remainder of this chapter is organized as follows. In Section~\\ref{sec:bac}, we provide\nan overview of relevant research and related work.\nIn Section~\\ref{sec:adv}, we evaluate adversarial attacks in the context of image spam, \nand in Section~\\ref{sec:univ}, we present our proposed attack.\nFinally, Section~\\ref{sec:con} concludes this chapter, where we have included\nsuggestions for future work.\n\n\n\\section{Background}\\label{sec:bac}\n\n\\subsection{Image Spam Filtering}\n\nThe initial defenses against image spam relied on\noptical character recognition (OCR). In such OCR-based systems, \ntext is extracted from an image, at which point\na traditional text-based spam filter can be used~\\cite{spam_assassin}. \nAs a reaction to OCR-based techniques,\nspammers introduced images with slight modifications, such as \noverlaying a light background of random artifacts on images, which are\nsufficient to render OCR ineffective. The rise of learning algorithms, however, \nhas enabled the creation of image spame filtering systems based directly on \nimage features. \n\nIn~2008, a filtering system using a global image feature-based probabilistic boosting \ntree was proposed, and achieved an~89.44\\%\\ detection rate with a false positive rate \nof~0.86\\%~\\cite{image_spam_hunter}. Two years later, an artificial neural network for \nimage classification was proposed~\\cite{statistical_feature_extraction_for_classification}. \nThese latter authors used were able to classify image spam\nwith~92.82\\%\\ accuracy based on color histograms, \nand~89.39\\%\\ accuracy based on image composition extraction. \n \nThe two image spam detection methods presented \nin~\\cite{image_spam_analysis_and_detection} rely on principal component analysis (PCA)\nand support vector machines (SVM). In addition, the authors \nof~\\cite{image_spam_analysis_and_detection} introduce a new dataset that their \nmethods cannot reliably detect. \nTwo years later, the authors of~\\cite{support_vector_machines_for_image_spam} improved \non the results in~\\cite{image_spam_analysis_and_detection} by training a linear SVM \non~38 image features, achieving~98\\%, \naccuracy in the best case. The authors also introduce a challenge dataset that is even \nmore challenging than the analogous dataset presented \nin~\\cite{image_spam_analysis_and_detection}. \n \nThe recent rise of deep learning, a subfield of machine learning, coupled with advances \nin computational speed has enabled the creation of filtering systems capable of considering \nnot only image features, but entire images at once. In particular, \nconvolutional neural networks (CNN) \nare well suited to computer vision tasks due to their powerful feature extraction capabilities. \n\nIn recent years, CNNs have been applied to the task of image spam detection.\nFor example, in~\\cite{image_spam_filtering_using_conv} a CNN is trained on \nan augmented dataset of spam images, achieving a~6\\%\\ improvement in accuracy,\nas compared to previous work. Similarly, the authors of~\\cite{deepimagespam} \nconsider a CNN, which achieved~91.7\\%\\ accuracy. \nIn~\\cite{convolutional_neural_networks_for_image_spam},\na CNN-based system is proposed, which achieves an accuracy of~99\\%\\ on \na real-world image spam dataset,~83\\%\\ accuracy  \non the challenge dataset in~\\cite{image_spam_analysis_and_detection}\n(an improvement over previous works), \nand~68\\%\\ on the challenge dataset in~\\cite{support_vector_machines_for_image_spam}. \n\nFrom the challenge datasets introduced in~\\cite{image_spam_analysis_and_detection} and~\\cite{support_vector_machines_for_image_spam}, we see that\nthe accuracy of machine learning-based filtering systems can be \nreduced significantly with appropriate modifications to spam images. In this research, \nwe show that the accuracy of \nsuch systems can be reduced far more by using the adversarial learning\napproach that we present below. \n\n\\subsection{Adversarial Learning}\n\nThe authors of~\\cite{intriguing_properties} found that by applying an imperceptible \nfilter to an image, a given neural network's prediction can be arbitrarily changed. \nThis filter can be generated from the optimization problem\n\\begin{equation}\\nonumber\n    \\begin{split}\n&    \\textbf{minimize } \\|r\\|_2 \\\\\n&\\textbf{subject to } f(x+r) = l \\text{ and } x+r \\in [0, 1]^m\n    \\end{split}\n\\end{equation}\nwhere $f$ is the classifier, $r$ is the minimizer, $l$ is the target label, and $m$ is the dimension of the image.  \nThe resulting modified images are said to be \\textit{adversarial examples}, and the attack presented in~\\cite{intriguing_properties} is known as the \\textit{L-BFGS Attack}. These adversarial examples generalize well to different network architectures and networks.\n \nMore recently, many advances have been made in both adversarial example generation and \ndetection. For example, in~\\cite{adversarial_examples_attacks_and_defenses} a taxonomy \nis proposed\nfor generation and detection methods, as well as a threat model. Based on this threat model, \nthe task of attacking neural network-based image spam detectors requires an attack that is \nfalse-negative (i.e. generative of positive samples misclassified as negative) and black-box \n(i.e. the attacker does not have access to the trained model). \nAttacks on image spam classifiers must satisfy these two criteria. \n\nAfter the introduction of the L-BFGS Attack, the authors of~\\cite{explaining_and_harnessing} \nbuilt on their work in~\\cite{intriguing_properties} by introducing \nthe \\textit{Fast Gradient Sign Method} (FGSM). This method uses the gradient of the \nloss function  with respect to a given input image to efficiently create a new image that \nmaximizes the loss, via backpropagation. This can be summarized with the expression\n$$\n    \\mbox{adv}_x = x + \\epsilon\\,\\mbox{sign}\\bigl(\\nabla_x J(\\theta, x, y)\\bigr)\n$$\nwhere $\\theta$ is the parameters of the model, $x$ is the input image, $y$ is the target label, \nand $J$ is the cost function used to train the model.\nThese authors also introduce the notion that adversarial examples result from linear \nbehavior in high-dimensional spaces. \n\nThe authors of~\\cite{towards_evaluating_the_robustness} introduce \\textit{C\\&W's Attack}, \na method designed to combat \\textit{defensive distillation}, which consists of training a \npair of models such that there is a low probability of successively attacking both models. \nC\\&W's Attack is a non-box constrained variant of the L-BFGS Attack that is more easily \noptimized and effective against both distilled and undistilled networks. \nThey formulate adversarial example generation as the optimization problem\n\\begin{equation}\\nonumber\n  \\begin{split}\n& \\textbf{minimize } D(x, x + \\delta) + c \\cdot f(x + \\delta) \\\\ \n& \\textbf{such that } x + \\delta \\in [0, 1]^n  \n\\end{split}  \n\\end{equation}\nwhere $x$ is the image, $D$ is one of the three distance metrics described below, and $c$ is a suitably chosen constraint (the authors choose $c$ with binary search).  \nThe authors also utilize three distance metrics for measuring perturbation: $L_0$ (the number of altered pixels), $L_2$ (the Euclidean distance), and $L_{\\infty}$ (the maximum change to any of the coordinates), and introduced three subvariants of their attack that aim to minimize each of these distance metrics. \n \nIt is important to note that the previously mentioned attacks require knowledge of the classifier's gradient and, as such, cannot be directly deployed in a black-box attack. In~\\cite{practical_black_box}, the authors propose using a surrogate model for adversarial example generation to enable the transferability of adversarial examples to attack black-box models. Differing from gradient-based methods, the authors of~\\cite{zoo} introduced a method, \n\\textit{Zeroth Order Optimization} (ZOO), which is inspired by\nthe work in~\\cite{towards_evaluating_the_robustness}.\nThe ZOO technique employs gradient estimation, with the most significant downside \nbeing that it is computationally expensive. \n\nThe paper~\\cite{deepfool} introduces the \\textit{DeepFool} attack, which aims to find the minimum\ndistance from the original input images to the decision boundary for adversarial examples. \nThey found that the minimal perturbation needed for an affine classifier is the distance to the separating affine hyperplane, which is expressed (for differentiable binary classifiers) as \n\\begin{equation}\\nonumber\n\\begin{split}\n& \\textbf{argmin}_{\\eta_i} \\|\\eta_i\\|_2 \\\\\n& \\textbf{such that } f(x_i) + \\nabla f(x_i)^T \\eta_i = 0  \n\\end{split}\n\\end{equation}\nwhere $i$ denotes the iteration, $\\eta$ is the perturbation, and $f$ is the classifier.\nIn comparison to FGSM, DeepFool minimizes the magnitude of the perturbation, instead of the number  of selected features. This would appear to be ideal for spammers, since it would tend to minimize the effect on an image.\n \nThe \\textit{universal perturbation} attack presented in~\\cite{universal} is also suited to \nthe task at hand. We believe that universal adversarial examples are most likely \nto be deployed by spammers against black-box models due to their simplicity \nand their transferability across architectures. Generating universal \nperturbations is an iterative process, as the goal is to find a vector $v$ that satisfies \n$$\n    \\|v\\|_p \\leq \\xi \\mbox{\\ \\ and\\ \\ }\n    \\mathbb{P}_{x\\sim \\mu} (\\hat{k}(x+v) \\neq \\hat{k}(x)) \\geq 1 - \\delta\n$$\nwhere $\\mu$ is a distribution of images, $\\hat{k}$ is a \nclassification function that outputs for each image $x$ and a label $\\hat{k}(x)$.\nThe results in~\\cite{universal} show that universal perturbations are misclassified with high probability, suggesting that the existence of such perturbations are correlated to certain regions of the decision boundary of a deep neural network.\n\nFinally, the authors of~\\cite{restoration_as_a_defense} propose input restoration with a \npreprocessing network to defend against adversarial attacks. \nThe authors' defense improved the classification precision of a CNN \nfrom~10.2\\%\\ to~81.8\\%, on average. These results outperform \nexisting input transformation-based defenses. \n\n\\section{Evaluating Adversarial Attacks}\\label{sec:adv}\n\n\\subsection{Experimental Design}\n\nThe two multi-layer perceptron and convolutional neural network architectures presented in~\\cite{convolutional_neural_networks_for_image_spam} are each trained on both of\nthe dataset presented in~\\cite{image_spam_hunter}, which henceforth will be referred to \nas the \\textit{ISH Dataset}, and the dataset presented \nin~\\cite{support_vector_machines_for_image_spam}, which henceforth will be referred to as \nthe \\textit{MD Dataset} (modified Dredze). We use TensorFlow~\\cite{tensorflow} to train\nour models---both architectures have been trained as they were presented in their \nrespective articles on each of the datasets. NumPy~\\cite{numpy} and \nOpenCV~\\cite{opencv} are used for numerical operations and image processing \ntasks, respectively. All computation are performed on a\nlaptop with~8GB ram, using Google Colaboratory's Pro GPU. \n\nThe ISH Dataset contains~928 spam images and~830 non-spam images, \nwhile the MD Dataset contains~810 spam images and~784 non-spam images;\nall images in both datasets are in \\textit{jpg} format. These datasets are summarized\nin Table~\\ref{tab:data}.\n\n\\begin{table}[!htb]\n\\caption{Image spam datasets}\\label{tab:data}\n\\centering\n\\begin{tabular}{ccc}\n\\midrule\\midrule\n\\textbf{Name} & \\textbf{Spam images} & \\textbf{Non-spam images} \\\\\n\\midrule\nISH dataset & \\zz928 & \\zz830 \\\\\nMD dataset & \\zz810 & \\zz784 \\\\\n\\midrule\nTotal & 1738 & 1613 \\\\\n\\midrule\\midrule\n\\end{tabular}\n\\end{table}\n\nDataset preprocessing for the networks \npresented in~\\cite{convolutional_neural_networks_for_image_spam} consist \nof downsizing each of the images such that their dimensions are~$32\\time 32\\times 3$, \napplying zero-parameter Canny edge detection~\\cite{canny} to a copy of the downsized \nimage, and concatenating the downsized image with the copy that had \nCanny edge detection applied. This process results in~$64\\time 32\\time 3$ images, \nwhich are used to train the two neural networks, one for the ISH dataset, and one for the MD dataset. \nThe four resulting models achieved accuracies within roughly~7\\%\\ of the accuracies \nreported in~\\cite{convolutional_neural_networks_for_image_spam}. \n\nTo enable the generation of adversarial examples, four larger models with an input size of 400x400 are also trained on \nthe original datasets. The first few layers of each of these models are simply used to \ndownscale input images such that the original architectures can be used after downscaling. \nThese four alternative models achieve accuracy roughly equivalent to the \noriginal models. The four adversarial attacks \n(FGSM, C\\&W's Attack, DeepFool, and Universal Perturbation) \nutilize these four alternative models to generate adversarial examples that can then be formatted \nas the original datasets to attack the original four models. This procedure attempts to exploit \nthe transferability of adversarial examples to similar architectures. \n\nThe IBM Adversarial Robustness Toolbox (ART)~\\cite{adversarial_robustness_toolbox} is \nused to implement C\\&W's Attack, DeepFool, \nand Universal Perturbations, while FGSM was implemented independently from scratch.\nAn attempt was made to optimize the parameters of each technique---the resulting\nparameters are summarized in Table~\\ref{tab:parms}. Note that\nfor the Universal Perturbation attack, FGSM was used as the base attack, as the IBM ART \nallows any adversarial attack to be used for computing universal perturbations.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n\\caption{Attack parameters}\\label{tab:parms}\n\\centering\n\\begin{tabular}{c|ll}\n\\midrule\\midrule\n\\textbf{Attack} & \\textbf{Description} & \\textbf{Value} \\\\\n\\midrule\nFGSM \n  & perturbation magnitude & 0.1 \\\\ \\midrule\n\\multirow{6}{*}{C\\&W's attack} \n  & target confidence & 0 \\\\ \n  & learning rate & 0.001 \\\\\n  & binary search steps & 20 \\\\ \n  & maximum iterations & 250 \\\\\n  & initial trade-off & 100 \\\\\n  & batch size & 1 \\\\ \\midrule\n\\multirow{4}{*}{DeepFool}\n  & max iterations & 500 \\\\ \n  & overshoot parameter & $10^{-6}$ \\\\ \n  & class gradients & 10 \\\\\n  & batch size & 1 \\\\ \\midrule\n\\multirow{4}{*}{Universal Perturbation}\n  & target accuracy & 0\\% \\\\ \n  & max iterations & 250 \\\\\n  & step size & 64 \\\\\n  & norm & $\\infty$ \\\\\n\\midrule\\midrule\n\\end{tabular}\n\\end{table}\n\nThe metrics used to evaluate each of the four attacks are the average accuracy, \narea under the curve (AUC) of the receiver operating characteristic (ROC) curve, average~$L_2$ perturbation measurement \n(Euclidean distance), and average computation time per example for each of the four models. \nScikit-learn~\\cite{scikit-learn} was used to generate the ROC curves for each attack. \n\nWe use~251 data points for accuracy \nand $L_2$ distances collected for the FGSM, DeepFool, and Universal Perturbation experiments, \nin accordance with the full size of the test dataset, which contains~251 examples for generating adversarial examples. However, only~$28$ data points were \ncollected from the C\\&W's Attack experiment due to \nthe large amount of time required to generate each data point (roughly five minutes per data point). \nThe technique that will be used as the basis of our proposed attack will be selected based \non the performance of each attack, as presented in the next section.\n\n\\subsection{Analysis}\n\n\nThe mean accuracy, computation time per example, and~$L_2$ distance were \nrecorded for each of the four models attacked by each of the attack methods. \nThis data was compiled into the tables discussed in this section. \n\nFrom Table~\\ref{tab:meanacc1} we see that for FGSM, the accuracy of the attacked models \nis shown to vary inconsistently while Figure~\\ref{fig:fgsml2} shows that the distribution of \nthe~$L_2$ distances of the generated adversarial examples skew right. \nBased on these results \nand corresponding density plots of the accuracy and~$L_2$ distance distributions, \nthe FGSM attack can be ruled out as a candidate due to poor accuracy. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mean accuracy per adversarial example}\n    \\label{tab:meanacc1}\n    \\begin{tabular}{c|cccc}\n     \\midrule\\midrule\n    \n      \\multirow{2}{*}{\\textbf{Model}}\n       & \\multirow{2}{*}{\\textbf{FGSM}}\n       & \\textbf{C\\&W's}\n       & \\multirow{2}{*}{\\textbf{DeepFool}}\n       & \\textbf{Universal}\\\\\n                             &                          &  \\textbf{Attack} &                              &  \\textbf{Perturbation}\\\\\n     \\midrule\n     MLP (ISH) & 95.2\\% & 89.2\\% & 98.8\\% & 98.7\\%\\\\\n     CNN (ISH) & 36.2\\% & 49.6\\% & 61.5\\% & 49.9\\%\\\\\n     MLP (MD) & 69.7\\% & 75.6\\% & 93.5\\% & 94.3\\%\\\\\n     CNN (MD) & 82.8\\% & 77.2\\% & 14.5\\% & \\zz8.4\\%\\\\\n     \\midrule\\midrule\n    \\end{tabular}\n\\end{table}\n\nThe mean~$L_2$ (Euclidean) distances of the adversarial examples\nare given in Table~\\ref{tab:meandist1}. The distribution of distances appears to be roughly equivalent across all attacks.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering  \n    \\caption{Mean $L_2$ (Euclidean) distance of adversarial examples from original images}\n    \\label{tab:meandist1}\n    \\begin{tabular}{c|cccc}\n     \\midrule\\midrule\n    \n      \\multirow{2}{*}{\\textbf{Model}}\n       & \\multirow{2}{*}{\\textbf{FGSM}}\n       & \\textbf{C\\&W's}\n       & \\multirow{2}{*}{\\textbf{DeepFool}}\n       & \\textbf{Universal}\\\\\n                             &                          &  \\textbf{Attack} &                              &  \\textbf{Perturbation}\\\\\n     \\midrule\n     MLP (ISH) & 11537.55 & 10321.77 & 11513.26 & 11483.72\\\\\n     CNN (ISH) & 11108.44 & 10924.14 & 11216.19 & 11416.58\\\\\n     MLP (MD) & \\zz8998.71 & \\zz9185.04 & \\zz9566.02 & \\zz9490.56\\\\\n     CNN (MD) & \\zz9144.49 & \\zz9009.91 & \\zz9128.99 & \\zz9381.15\\\\\n     \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=7.8cm]{images\/FGSMl2.png}\n\\caption{Density plot of~$L_2$ (Euclidean) distances (Fast Gradient Sign Method)}\n\\label{fig:fgsml2}\n\\centering\n\\end{figure}\n\nDeepFool can also be ruled as a candidate, as the attack has been seen to be \nonly marginally better than the FGSM attack in terms of performance, while also \nhaving a significantly higher average computation time per adversarial example. This\ncan be observed in Table~\\ref{tab:meancomp1}, where the computation time \nper example varies greatly. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mean computation time per adversarial example}\n    \\label{tab:meancomp1}\n    \\begin{tabular}{c|cccc}\n     \\midrule\\midrule\n    \n      \\multirow{2}{*}{\\textbf{Model}}\n       & \\multirow{2}{*}{\\textbf{FGSM}}\n       & \\textbf{C\\&W's}\n       & \\multirow{2}{*}{\\textbf{DeepFool}}\n       & \\textbf{Universal}\\\\\n                             &                          &  \\textbf{Attack} &                              &  \\textbf{Perturbation}\\\\\n     \\midrule\n     MLP (ISH) & 0.180 & 269.65 & 19.90 & 4.37\\\\\n     CNN (ISH) & 0.038 & 251.01 & \\zz4.75 & 2.87\\\\\n     MLP (MD) & 0.164 & 270.58 & 36.30 & 3.71\\\\\n     CNN (MD) & 0.165 & 244.47 & \\zz1.48 & 5.23\\\\\n     \\hline\n    \\end{tabular}\n\\end{table}\n\nIn contrast, C\\&W's Attack shows consistent performance in all three metrics at the cost of \nhigh computation time (roughly five minutes per adversarial example).\nThe consistency of this attack is ideal from a spammer's perspective, \nthough the trade-off is a relatively high computation time. In addition, the \nleft skew of this attack with respect to~$L_2$ distance, as presented in Figure~\\ref{fig:cwl2}, \nindicates that the perturbation made to spam images is much lower in comparison to the other attacks.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=8cm]{images\/CWl2.png}\n\\caption{Density plot of~$L_2$ (Euclidean) distances (C\\&W's Attack)}\n\\label{fig:cwl2}\n\\centering\n\\end{figure}\n\nThe Universal Perturbation attack is inconsistent in terms of accuracy, as shown \nin Table~\\ref{tab:meanacc1} where the mean accuracy across the four models is clearly \nshown to fluctuate wildly, but this \nis simply due to the fact that only one perturbation (albeit with varying success across architectures) is applied to all spam images, which is highly advantageous for spammers.\nThe generation and application of this perturbation to an image takes roughly four seconds,\nwhich would result in greater performance in a real-world spam setting in comparison \nto C\\&W's Attack. \n \nTo further compare C\\&W's Attack and the Universal Perturbation attack, \nthe ROC curves of the two are presented in Figure~\\ref{fig:cwroc_uproc}.\nThese ROC curves can be used to quantify the diagnostic ability of the \nmodels attacked by each method.\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/CWroc.png}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/UProc.png}\n  \\end{minipage}\n  \\caption{ROC curves of C\\&W's Attack and the Universal Perturbation when used to attack the four classifiers}\n  \\label{fig:cwroc_uproc}\n\\end{figure}\n\nThe ROC curve for C\\&W's attack is much noisier due to being generated from only~28 \ndata points. Taking this into consideration, it can be inferred that both C\\&W's Attack \nand the Universal Perturbation attack are able to reduce the areas under the ROC curve (AUC) \nof the attacked models to values close to~0.5. This suggests that both attacks are able to reduce the class separation capacity of attacked image spam classifiers to essentially random. \n\nTo analyze the differences in distribution of the accuracy and~$L_2$ distance data collected from \nthe trials conducted on C\\&W's Attack and the Universal Perturbation attack, the Mann-Whitney U Test was utilized via its implementation \nin SciPy~\\cite{scipy}. The Mann-Whitney U Test compares two populations---in \nthis case, the accuracy and~$L_2$ distance data from both attacks for each attacked model.\nThe null hypothesis (H0) for the test is that the probability is~50\\%\\ that a randomly drawn \nvalue from the first population will exceed a value from the second population. The result of each \ntest is a Mann-Whitney U Statistic (not relevant in our case) and a~$p$-value. We use\nthe~$p$-value to determine whether the difference between the data is statistically \nsignificant, where the standard threshold is~$p = 0.05$. The results of these tests are \ngiven in Table~\\ref{tab:mannwhitneyu}.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mann-Whitney U Test results comparing C\\&W's Attack and the Universal Perturbation attack}\n    \\label{tab:mannwhitneyu}\n    \\begin{tabular}{l|ll}\n     \\midrule\\midrule\n     \\multicolumn{1}{c|}{\\textbf{Model}}\n                 & \\multicolumn{1}{c}{\\textbf{Accuracy $p$-value}} \n                 & \\multicolumn{1}{c}{\\textbf{$L_2$ distance $p$-value}} \\\\\n     \\midrule\n     MLP ISH & 0.000 (H0 is rejected) & 0.034 (H0 is rejected)\\\\\n     CNN ISH & 0.384 (H0 is not rejected) & 0.098 (H0 is not rejected)\\\\\n     MLP MD & 0.000 (H0 is rejected) & 0.057 (H0 is not rejected)\\\\\n     CNN MD & 0.000 (H0 is rejected) & 0.016 (H0 is rejected)\\\\\n     \\midrule\\midrule\n    \\end{tabular}\n\\end{table}\n\nThe results  in Table~\\ref{tab:mannwhitneyu} imply that the performance of these two attacks (C\\&W's Attack and the Universal Perturbation attack)\nare nearly identical when attacking a CNN trained on the ISH dataset, \nas evidenced in the second row, where the null hypothesis is not rejected. \nHowever, the~$L_2$ distance measurement for spam images \nthat have had the universal perturbation applied should remain constant relative \nto the original spam image. Therefore, the results of these tests suggest that \nthe Universal Perturbation attack is able to achieve similar performance \nto C\\&W's Attack, in terms of perturbation magnitude, \nwith a much lower computation time per example in comparison to C\\&W's Attack. \n\nGiven the above evidence, the Universal Perturbation attack is the best choice for \nimage spam, as it is unrivaled in terms of potential performance in a real-world \nsetting. The key advantages of the Universal Perturbation attack include\nthat it generates a single perturbation to be applied to all spam images, \nand its relatively fast computation time per adversarial example. Therefore,\nUniversal Perturbation will be used as a basis for our image transformation \ntechnique, as discussed and analyzed in the remainder of this paper. A sample adversarial spam image generated with the Universal Perturbation attack is presented in Figure~\\ref{fig:sampleadv}.\n\n\n\\begin{figure}[!htb]\n  \\centering\n    \\includegraphics[width=8cm]{images\/newspamimage.png}\n    \\caption{Adversarial spam image generated with the Universal Perturbation attack}\n    \\label{fig:sampleadv}\n\\end{figure}\n\n\\section{Inceptionism-Augmented Universal Perturbations}\\label{sec:univ}\n\n\\subsection{Procedure}\n\nBased on the results and discussion above, a transformation that\nis applied to spam images prior to generating adversarial examples, since perturbations cannot be transformed after application, should \nmeet the following conditions.\n\\begin{itemize}\n    \\item Lower the misclassification rate\n    \\item Preserve adversarial effects after image resizing\n    \\item Make non-spam features more prominent, while retaining legibility\n\\end{itemize}\nGiven the above criteria, a reasonable approach would be to maximize the presence \nof \"natural features\" in a given spam image. That is, the features characteristic of non-spam images learned by classifiers should be maximized whilst retaining legibility. To accomplish this, the procedure for maximizing \nthe activation of a given output neuron (in this case, the non-spam output neuron), \nas introduced in~\\cite{deepdream}, dubbed \"DeepDream\", can be used to increase the number of natural features in all images from the non-spam subsets of the ISH and MD datasets.\nThis is accomplished by maximizing the activations of the convolutional layer \nwith the greatest number of parameters and output layer in the corresponding CNNs. \nThe resulting two sets of images that have had DeepDream applied (``dreamified'' images) are then grouped into batches of four images. The weighted average of the four images in each batch can then be taken to produce two \nprocessed non-spam datasets of images with high concentrations of natural features, as batches of greater than four images may result in high noise. Each of the images in the resulting two non-spam datasets \nare henceforth referred to as \\textit{natural perturbations}.\n\nTo preserve the adversarial effect that the universal perturbation introduces, \nthe Gradient-weighted Class Activation Mapping (Grad-CAM) technique \nintroduced in~\\cite{gradcam} \nis used to generate a class activation map for each spam \nimage in each dataset. The inverse of each such map is used with a natural perturbation \ngenerated from the same dataset to remove the regions of the \nnatural perturbation where the class activation map is highest. \nBy superimposing the resulting natural perturbations onto the corresponding spam images, \nthe regions where the universal perturbation is most effective are left intact while the \nregions of the spam images affected by the natural perturbations benefit by being\nmore non-spam-like. The presence of natural features in the resulting spam images \nshould also result in robustness against resizing prior to inference by a deep \nlearning-based image spam detection model, as the natural features should be still be somewhat preserved even after being shrunken. \n\nThe universal perturbation is then applied to each of the resulting spam images. \nThe result is that we potentially reduce a deep learning-based image spam \ndetector's accuracy due to the presence of a natural perturbation and a universal adversarial perturbation and retain some sort of adversarial effect in the case of resizing.\nThis procedure also allows for the retention of legible text within spam images. \n\n\\subsection{Implementation}\n\nTo generate our two sets of ``dreamified'' images, the CNN architecture presented in~\\cite{convolutional_neural_networks_for_image_spam} is trained on both \nthe ISH and MD datasets, with inverted labels to allow for the maximization of the \nactivations of the neurons corresponding to non-spam images, as the activations for spam images would be maximized if the labels weren't inverted. \nThese two models are trained with the TensorFlow Keras API, with the \nhyperparameters given in~\\cite{convolutional_neural_networks_for_image_spam}. \nFor each of the models, the convolutional layer with the highest number of parameters \nand the output layer were chosen as the layers in which the activation should be maximized \nvia gradient ascent, as the aforementioned convolutional layer is responsible for recognizing the most complex natural features. Each of the images from the non-spam subsets of the ISH and MD datasets were used for inference on the two CNN models. The CNN models use \nthe losses of the chosen layers to iteratively update the non-spam images with gradient \nascent so that the number of non-spam features is maximized. \nEach non-spam image \nis updated for~64 iterations with an update size of~$0.001$. The resulting ``dreamified'' \nimages are then grouped into batches of~4 and blended via evenly distributed weighted \naddition to produce a total of~392 grayscale images, each of size~$400\\times 400\\times 1$.\nThese~392 grayscale images are evenly split between the ISH dataset and MD datasets.\n \nTo utilize GradCAM, \nthe CNN architecture presented in~\\cite{convolutional_neural_networks_for_image_spam} \nis trained on both the ISH and MD datasets with normal labels. \nFor each image from the spam subsets of the ISH and MD datasets, \nGradCAM is used to generate a corresponding class activation map based \non the activations of the last convolutional layer in each of the two models.\nThis is accomplished by computing the gradient of the top predicted class with \nrespect to the output feature map of the last convolutional layer, \nusing the mean intensity of the gradient over specific feature map channels. \nOpenCV~\\cite{opencv} is then used to upscale each of the class activation maps \nto~$400\\times 400$, convert them to binary format, and invert the result to allow the class activation maps to be applied to the natural perturbations such that only the areas with highest activation will contain the natural perturbations. \nThe bitwise AND of each processed class activation map and a randomly selected \nnatural perturbation can then be used to generate two sets of processed \nnatural perturbations, which are superimposed on the corresponding spam images \nfrom each of the two spam subsets. This procedure results in two subsets \nof spam images with natural perturbations. \n\nLastly, the universal perturbation is generated and applied to all images within \nthe two spam image subsets that have had natural perturbations applied. For this\noperation, we use the IBM Adversarial Robustness Toolbox~\\cite{adversarial_robustness_toolbox}.\nThe hyperparameters for the Universal Perturbation attack remain the same \nas those given in Table~\\ref{tab:parms}, above.\n\n\\subsection{Performance Evaluation}\n\nThe mean accuracy, computation time per example, and~$L_2$ distance were recorded for \neach of the four models attacked using spam images with modified universal perturbations.\nThis is analogous to what was done during the attack selection process. \nThis data has been compiled into the tables discussed in this section.\n\nAs can be seen from the results in Table~\\ref{tab:meanacc2},\nthe proposed method for generating adversarial \nspam images is capable of lowering a learning-based model's accuracy \nto~23.7\\%. In addition, on average, our proposed technique is\nmuch more effective while being evenly distributed in terms of accuracy \non similar learning-based models. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mean accuracy of each model with spam images created by the proposed method}\n    \\label{tab:meanacc2}\n    \\resizebox{0.85\\textwidth}{!}{\n    \\begin{tabular}{c|cccc}\n     \\midrule\n     \\midrule\n      \\textbf{Images} & \\textbf{MLP (ISH)}  & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n     \\midrule\n     Modified spam images & 80.1\\% & 98.8\\% & 98.4\\% & 75.3\\% \\\\\n     Modified spam images with & \\multirow{2}{*}{72.2\\%} \n     \t\t\t\t\t\t& \\multirow{2}{*}{50.4\\%} \n\t\t\t\t\t\t& \\multirow{2}{*}{78.7\\%} \n\t\t\t\t\t\t& \\multirow{2}{*}{23.7\\%} \\\\\n     Universal Perturbations \\\\\n     \\midrule\\midrule\n    \\end{tabular}}\n\\end{table}\n\nFrom Table~\\ref{tab:meancomp2}, we see that\nin contrast to C\\&W's Attack, which on average takes 258.93 seconds per example, the time necessary to generate adversarial spam images \nwith natural perturbations is significantly lower and comparable to \nthat of the original Universal Perturbation attack.\nThis is another advantage of our proposed attack.\n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mean computation time per adversarial spam image (in seconds)}\n    \\label{tab:meancomp2}\n    \\begin{tabular}{cccc}\n     \\midrule\\midrule\n      \\textbf{MLP (ISH)}  & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n     \\midrule\n      5.46 & 5.15 & 5.87 & 4.80 \\\\\n     \\midrule\\midrule\n    \\end{tabular}\n\\end{table}\n\nThe mean~$L_2$ distances and the distribution of the~$L_2$ distances of the modified adversarial spam \nimages are given in Table~\\ref{tab:meandist2}. From Figure~\\ref{fig:lastl2}, we see that the distributions \nof these distances are, on average, not skewed, indicating that the natural perturbations have \nhad a slight negative effect on the spam image $L_2$ distances, as the distributions for the original Universal Perturbation attack were skewed to the left. \n\n\\begin{table}[!htb]\n\\advance\\tabcolsep by 4pt\n    \\centering\n    \\caption{Mean~$L_2$ (Euclidean) distance of modified adversarial spam images from original images}\n    \\label{tab:meandist2}\n    \\begin{tabular}{cccc}\n     \\midrule\\midrule\n      \\textbf{MLP (ISH)}  & \\textbf{CNN (ISH)} & \\textbf{MLP (MD)} & \\textbf{CNN (MD)} \\\\\n     \\midrule\n      11392.02 & 11309.40 & 9440.69 & 9628.61 \\\\\n     \\midrule\\midrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{figure}[!htb]\n  \\centering\n    \\includegraphics[width=8cm]{images\/lastl2.png}\n    \\caption{Density plot of~$L_2$ (Euclidean) distances of the modified adversarial spam images from the original images}\n    \\label{fig:lastl2}\n\\end{figure}\n\nThe ROC curves of the models attacked by the proposed method, \nwhich appear in Figure~\\ref{fig:lastroc}, are slightly worse in comparison to \nthat of the original Universal Perturbation attack, suggesting once more that the \nattack is capable of reducing the class separation capacity of attacked image spam \nclassifiers to essentially random.\n\n\\begin{figure}[!htb]\n  \\centering\n    \\includegraphics[width=8cm]{images\/lastroc.PNG}\n    \\caption{ROC curves of each of the four models attacked by the modified spam images generated with the proposed method}\n    \\label{fig:lastroc}\n\\end{figure}\n\n\\subsection{Proposed Dataset Analysis}\n\nFigure~\\ref{fig:spamimage2} contains an example of a modified adversarial spam images.\nFrom this image, we observe that\nthe proposed method was able to effectively utilize class activation maps \ngenerated with GradCAM to selectively apply a random natural perturbation \nto the spam image. As discussed in the previous section, this decreases \nclassification accuracy even prior to the application of a universal perturbation. \n\n\\begin{figure}[!htb]\n  \\centering\n    \\includegraphics[width=6cm]{images\/img213.jpg}\n    \\caption{Example of modified adversarial spam image generated with the proposed method}\n    \\label{fig:spamimage2}\n\\end{figure}\n\nTo fully evaluate the effect of the modified adversarial spam images from the two \nmodified datasets, two sets of class activation maps are generated from the spam \nsubsets of the two datasets using GradCAM and the corresponding CNN models.\nThese activation maps are then averaged to obtain two heatmaps \nfrom the class activation maps, as shown in Figures~\\ref{fig:origishheatmap1} \nand~\\ref{fig:origsvmheatmap1}. For comparison, the same process was applied to \nthe original datasets to obtain Figures~\\ref{fig:ishheatmap1} and~\\ref{fig:svmheatmap1}.\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}[b]{0.425\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/ishheatmap1.png}\n    \\caption{ISH spam data}\n    \\label{fig:ishheatmap1}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.425\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/svmheatmap1.png}\n    \\caption{MD spam data}\n    \\label{fig:svmheatmap1}\n  \\end{minipage}\n\\end{figure}\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}[b]{0.425\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/origishheatmap1.png}\n    \\caption{Modified ISH spam data}\n    \\label{fig:origishheatmap1}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.425\\textwidth}\n    \\includegraphics[width=\\textwidth]{images\/origsvmheatmap1.png}\n    \\caption{Modified MD spam data}\n    \\label{fig:origsvmheatmap1}\n  \\end{minipage}\n\\end{figure}\n\nAs can be seen in Figures~\\ref{fig:ishheatmap1} and~\\ref{fig:svmheatmap1}, \nthe activation regions for spam images from the original ISH and MD datasets \nare skewed towards the top and bottom. The narrow shape of these regions represent \nthe regions in spam images that generate the highest activations in \nthe neurons of the deep learning-based classifier. \nThe central region of the average class activation map for spam images from the \nMD dataset is much darker in comparison to that of spam images \nfrom the ISH dataset due to the superimposition of natural images \ndirectly onto spam features, \nas described in~\\cite{support_vector_machines_for_image_spam}. \n\nIn contrast, Figures~\\ref{fig:origishheatmap1} and~\\ref{fig:origsvmheatmap1} indicate \nthat the introduction of natural and universal adversarial perturbations \nare able to more evenly distribute the activation regions. This result shows \nthat the spam images from the modified datasets are much closer---in terms \nof natural features---to non-spam images. This also suggests that the proposed \nmethod outperforms the procedure used to generate the original MD dataset \nas outlined in~\\cite{support_vector_machines_for_image_spam}. \n\n\\section{Conclusion and Future Work}\\label{sec:con}\n\nModern deep learning-based image spam classifiers can accurately classify \nimage spam that has appeared to date in the wild. \nHowever, spammers are constantly creating new countermeasures to\ndefeat anti-spam technology. Consequently, the eventual use of adversarial examples \nto combat deep learning-based image spam filters is inevitable. \n\nIn this chapter, four adversarial attacks were selected based on specific restrictions \nand constraints of the image spam problem. These adversarial attacks \nwere evaluated on the CNN and MLP architectures introduced in~\\cite{convolutional_neural_networks_for_image_spam}. For training data,\nwe used the dataset presented in~\\cite{image_spam_hunter} \nand~\\cite{support_vector_machines_for_image_spam}. \nThe Fast Gradient Sign Method (FGSM) attack, C\\&W's Attack, DeepFool, and the \nUniversal Perturbation attack were all evaluated based on mean accuracy reduction, \nmean computation time per adversarial spam image, mean $L_2$ distance from the original spam \nimages, and ROC curves of the attacked classifiers. Through further statistical analysis, \nthe Universal Perturbation was chosen as a base for our proposed image transformation \nattack, due to its versatility and overall high performance \nin terms of accuracy reduction and computation time.\n\nTo maximize the number and intensity of natural features in an attack, the approach \nintroduced in~\\cite{deepdream} for maximizing activations of certain layers in \na deep neural network was used. This technique serves to generate sets of ``natural perturbations'' \nfrom the non-spam subsets of the image spam datasets. These natural perturbations \nwere then modified via the class activation maps of all spam images in both datasets. \nThe class activations were generated using GradCAM from the two convolutional \nneural networks trained on the ISH and MD datasets. These activation maps allow the regions in spam images recognized to contribute most to the spam classification to benefit from a universal adversarial perturbation. \n\nOur technique resulted in comparable---if not greater---accuracy reduction as\ncompared to C\\&W's Attack. In addition, our approach is \ncomputation much more efficient than C\\&W's Attack. \nFurthermore, the nature of our attack implies that the only potential \ncomputational bottleneck is generating the modified natural perturbations.\nThis aspect of the attack\nwould not be an issue in practice, unless a spammer generates \nvast numbers (i.e., in the millions) of modified \nadversarial spam images. \n\nA dataset of modified adversarial \nspam images has been generated by the authors by applying the proposed attack to the spam subsets of the ISH and MD datasets. This dataset will be made freely available to researchers.\n\nFuture work will include evaluating the ability of adversarial attack defense \nmethods. We will consider defensive distillation against adversarial spam images \ngenerated with our proposed attack. The goal of this research will be to develop defenses specifically designed for natural perturbation-augmented adversarial spam images. For example,\nthe subtraction of predicted adversarial perturbations is one path that we intend to pursue. \n\n\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction Definitions and Results}\nOriginally, the Chuang's inequality is a standard estimate in Nevanlinna theory but later on this inequality is used as a valuable tool in the study\nof value distribution of differential polynomials. For example, Recently, using this inequality, some sufficient conditions are obtained for which two differential polynomials sharing a small function satisfies the conclusions of Br\\\"{u}ck conjecture (\\cite{bb}, \\cite{bc1}, \\cite{bc2}, \\cite{bm}). \\par\nAt this point, we recall some notations and definitions to proceed further.\\par\nIt will be convenient to let $E$ denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence.\\par\nLet $f$ be a non constant meromorphic function in the open complex plane $\\mathbb{C}$. For any non-constant meromorphic function $f$, we denote by $S(r, f)$ any quantity satisfying $$S(r, f) = o(T(r, f))\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\ (r\\lra \\infty, r\\not\\in E).$$\nA meromorphic function $a(\\not\\equiv \\infty)$ is called a small function with respect to $f$ provided that $T(r,a)=S(r,f)$ as $r\\lra \\infty, r\\not\\in E$.\\par\nWe use $I$ to denote any set of infinite linear measure of $0<r<\\infty$.\\\\\nNow  we recall the following definition.\n\\begin{defi} (\\cite{4})Let $n_{0j},n_{1j},\\ldots,n_{kj}$ be non negative integers.\\\\\nThe expression $M_{j}[f]=(f)^{n_{0j}}(f^{(1)})^{n_{1j}}\\ldots(f^{(k)})^{n_{kj}}$ is called a differential monomial generated by $f$ of degree $d(M_{j})=\\sum\\limits_{i=0}^{k}n_{ij}$ and weight\n$\\Gamma_{M_{j}}=\\sum\\limits_{i=0}^{k}(i+1)n_{ij}$.\n\nThe sum $P[f]=\\sum\\limits_{j=1}^{t}b_{j}M_{j}[f]$ is called a differential polynomial generated by $f$ of degree $\\ol{d}(P)=max\\{d(M_{j}):1\\leq j\\leq t\\}$\nand weight $\\Gamma_{P}=max\\{\\Gamma_{M_{j}}:1\\leq j\\leq t\\}$, where $T(r,b_{j})=S(r,f)$ for $j=1,2,\\ldots,t$.\n\nThe numbers $\\underline{d}(P)=min\\{d(M_{j}):1\\leq j\\leq t\\}$ and k (the highest order of the derivative of $f$ in $P[f]$ are called respectively the lower degree and order of $P[f]$.\n\n$P[f]$ is said to be homogeneous if $\\ol{d}(P)$=$\\underline{d}(P)$.\n\n$P[f]$ is called a Linear Differential Polynomial generated by $f$ if $\\ol {d} (P)=1$. Otherwise $P[f]$ is called Non-linear Differential Polynomial. We also denote by $\\mu =max\\; \\{\\Gamma _{M_{j}}-d(M_{j}): 1\\leq j\\leq t\\}=max\\; \\{ n_{1j}+2n_{2j}+\\ldots+kn_{kj}: 1\\leq j\\leq t\\}$.\n\\end{defi}\nNow we are position to state the Chuang's inequality.\n\\begin{theoA} \\label{l2.4} (\\cite{3a}) Let $f$ be a meromorphic function and $P[f]$ be a differential polynomial. Then\n$$ m\\left(r,\\frac{P[f]}{f^{\\ol {d}(P)}}\\right)\\leq (\\ol {d}(P)-\\underline {d}(P)) m\\left(r,\\frac{1}{f}\\right)+S(r,f).$$\n\\end{theoA}\nIn this article, we give a short proof of the above inequality with some restriction.\n\\begin{theo} \\label{l2.5} Let $f$ be a meromorphic function and $P[f]$ be a differential polynomial. Then\n$$ m\\left(r,\\frac{P[f]}{f^{\\ol {d}(P)}}\\right)\\leq (\\ol {d}(P)-\\underline {d}(P)) m\\left(r,\\frac{1}{f}\\right)+S(r,f)~\\text{as}~ r \\to \\infty~\\text{and}~r \\not\\in E_{0},$$\nwhere $E_{0}$ is a set whose linear measure is not greater than $2$.\n\\end{theo}\n For this, we need to recall the lemma of logarithmic derivative.\n\\begin{lem}[\\emph{Lemma of Logarithmic Derivative}] (\\cite{1234})\nSuppose that $f(z)$ is a non constant meromorphic function in whole complex plane. Then\n$$m\\bigg(r,\\frac{f'}{f}\\bigg)=S(r,f)~\\text{as}~ r \\to \\infty~\\text{and}~r \\not\\in E_{0},$$\nwhere $E_{0}$ is a set whose linear measure is not greater than $2$.\n\\end{lem}\n\\begin{lem}\\label{ghgh}\nSuppose that $f(z)$ is a non constant meromorphic function in whole complex plane and $l$ is natural number. Then\n$$m\\bigg(r,\\frac{f^{(l)}}{f}\\bigg)=S(r,f)~\\text{as}~ r \\to \\infty~\\text{and}~r \\not\\in E_{0},$$\nwhere $E_{0}$ is a set whose linear measure is not greater than $2$.\n\\end{lem}\n\\section {Proof of Chuang's inequality}\n\\begin{proof} [Proof of theorem \\ref{l2.5}]\nSuppose that $P[f]=\\sum_{j=1}^{t}b_{j}M_{j}[f]$ be a differential polynomial generated by a non constant meromorphic function $f$. Further suppose that $m_{j}=d(M_{j})$ for $j=1,2,...,t$. Without loss of any generality, we can assume that $m_1\\leq m_2\\leq...\\leq m_t$.\\par We have to prove this inequality by induction on $t$.\\\\\nIf $t=1$, then in view of Lemma \\ref{ghgh} the inequality follows. Next we assume that the inequality holds for $t=l(\\geq2)$. Now we have to show that the inequality holds for $t=l+1$.\\\\\nAssume $$P[f]=\\sum\\limits_{j=1}^{l+1}b_{j}M_{j}[f]=Q[f]+bM[f],$$\nwhere $Q[f]=\\sum\\limits_{j=1}^{l}b_{j}M_{j}[f]$, $M[f]=M_{l+1}[f]$ and $b=b_{l+1}$.\\\\\nThen $m_1\\leq m_2\\leq...\\leq m_l\\leq m_{l+1}$ and by hypothesis $$m\\left(r,\\frac{Q[f]}{f^{\\ol {d}(Q)}}\\right)\\leq (\\ol {d}(Q)-\\underline {d}(Q)) m\\left(r,\\frac{1}{f}\\right)+S(r,f)~\\text{as}~ r \\to \\infty~\\text{and}~r \\not\\in E_{0},$$\nwhere $E_{0}$ is a set whose linear measure is not greater than $2$.\\par\nThus \\beas m\\left(r,\\frac{P[f]}{f^{\\ol {d}(P)}}\\right)&=&m\\left(r,\\frac{Q[f]+b M[f]}{f^{\\ol {d}(P)}}\\right)\\\\\n&\\leq& m\\left(r,\\frac{Q[f]}{f^{\\ol {d}(P)}}\\right)+m\\left(r,\\frac{ M[f]}{f^{\\ol {d}(P)}}\\right)+S(r,f)\\\\\n&\\leq& (\\ol {d}(Q)-\\underline {d}(Q)) m\\left(r,\\frac{1}{f}\\right)+(\\ol {d}(P)-\\ol {d}(Q)) m\\left(r,\\frac{1}{f}\\right)\\\\\n&+&(\\ol {d}(P)-d(M)) m\\left(r,\\frac{1}{f}\\right)+S(r,f)\\\\\n&\\leq& (\\ol {d}(P)-\\underline {d}(P)) m\\left(r,\\frac{1}{f}\\right)\\\\\n&+& (\\ol {d}(P)+\\underline {d}(P)-\\ol{d}(Q)-d(M)) m\\left(r,\\frac{1}{f}\\right)+S(r,f)\\\\\n&\\leq& (\\ol {d}(P)-\\underline {d}(P)) m\\left(r,\\frac{1}{f}\\right)+S(r,f)\n\\eeas\nas $r \\to \\infty~\\text{and}~r \\not\\in E_{0}$, where $E_{0}$ is a set whose linear measure is not greater than $2$, and\n$(\\ol {d}(P)+\\underline {d}(P)-\\ol{d}(Q)-d(M))=m_{l+1}+m_{1}-m_{l}-m_{l+1}\\leq 0.$\\\\\nThus by the principle of Mathematical Induction, the inequality follows.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSystems of ultra-cold atoms with time-periodic potential differences have recently\nstarted to be implemented experimentally both on the single-particle level~\\cite{KierigEtAl08} and for\nBose-Einstein condensates (BECs)~\\cite{LignierEtAl07,SiasEtAl08}. These experiments realise schemes of\ntunnelling-control devised in recent years~\\cite{GrossmannEtAl91,EckardtEtAl05}. Very recently, even the control of\nthe Mott-insulator transition via periodic shaking~\\cite{EckardtEtAl05b,CreffieldMonteiro06}\nhas been realised experimentally~\\cite{Zenesini08}. This would, roughly speaking, correspond to making\na glass of water freeze by turning on a loud-speaker. Proposals for shaking-induced tunnelling\ncontrol for ultra-cold atoms\nnot realised so far include schemes for entanglement\ngeneration~\\cite{TeichmannWeiss07,WeissTeichmann07,Creffield2007}. The tunnelling control\nbased on shaking the potentials uses concepts successfully implemented in the dressed-atom picture~\\cite{CohenTannoudjiReynaud77}.\n\nThe double-well potentials~\\cite{GatiOberthaler07} used for the single-particle experiments~\\cite{KierigEtAl08} would\npromise further exciting experimental results for BECs: a simple mean-field model of this system is known to be chaotic for\nnot too small\ninteractions~\\nocite{UtermannEtAl94}\\nocite{AbdullaevKraenkel00}\\nocite{GhoseEtAl01}\\nocite{LeeEtAl01}\\cite{UtermannEtAl94,GhoseEtAl01,LeeEtAl01,AbdullaevKraenkel00,LasPhys08}.\nFor BECs in a double-well potential with periodic shaking, we demonstrated numerically the\nemergence of ``Schr\\\"odinger-cat'' like states in phase-space \nif the parameters are chosen such that the mean-field system\ndisplays a coexistence of regular and chaotic dynamics~\\cite{WeissTeichmann08}.\nIn this paper,\nmore general entangled states are investigated. As shown by the authors of\nRef.~\\cite{PezzeSmerzi2007}, the quantum Fisher information~\\cite{BraunsteinEtAl96,BraunsteinCaves94} can be used to identify\nmulti-particle entanglement. This leads to a a greater variety of highly entangled states than\nwe investigated in Ref.~\\cite{WeissTeichmann08}.\n\nThis manuscript is organised as follows: Section~\\ref{sec:mod} introduces the models used to\ndescribe the BEC in a double well both on the mean-field (Gross-Pitaevskii) level and on the\n$N$-particle quantum level. Section~\\ref{sec:multi} explains the way entangled states are\ndetected in the numerics. Before concluding the paper, Sec.~\\ref{sec:results} shows the\nnumerical results for the emergence of entanglement.\n\n\n\\section{Model\\label{sec:mod}}\nBose-Einstein condensates in double-well potentials are often described via a model\noriginally developed in nuclear physics~\\cite{LipkinEtAl65}, a multi-particle Hamiltonian in two-mode\napproximation~\\cite{MilburnEtAl97}: \n\\begin{eqnarray}\n\\label{eq:H}\n\\hat{H} &=& -\\frac{\\hbar\\Omega}2\\left(\\hat{a}_1^{\\phantom\\dag}\\hat{a}_2^{\\dag}+\\hat{a}_1^{\\dag}\\hat{a}_2^{\\phantom\\dag} \\right) + \\hbar\\kappa\\left(\\hat{a}_1^{\\dag}\\hat{a}_1^{\\dag}\\hat{a}_1^{\\phantom\\dag}\\hat{a}_1^{\\phantom\\dag}+\\hat{a}_2^{\\dag}\\hat{a}_2^{\\dag}\\hat{a}_2^{\\phantom\\dag}\\hat{a}_2^{\\phantom\\dag}\\right)\\nonumber\\\\\n&+&\\hbar\\big(\\mu_0+\\mu_1\\sin(\\omega t)\\big)\\left(\\hat{a}_2^{\\dag}\\hat{a}_2^{\\phantom\\dag}-\\hat{a}_1^{\\dag}\\hat{a}_1^{\\phantom\\dag}\\right)\\;,\n\\end{eqnarray}\nwhere the operator $\\hat{a}^{(\\dag)}_j$ creates (annihilates) a boson in well~$j$; $\\hbar\\mu_0$ is\nthe tilt between well~1 and well~2 and $\\hbar\\mu_1$ is the driving amplitude. The interaction\nbetween a pair of particles in the same well is denoted by $2\\hbar\\kappa$.\nApplications of such Hamiltonians include\nmulti-particle entanglement~\\cite{MicheliEtAl03,MahmudEtAl03,TeichmannWeiss07,Creffield2007}, high precision\nmeasurements, many-body quantum coherence~\\cite{Lee06,LeeEtAl08} and spin\nsystems~\\cite{DusuelVidal05}.\n\nWithin the approximation on which the Gross-Pitaevskii equation is based, a multi-particle\nwave-function is characterised by two variables,  $\\theta$ and $\\phi$. The fraction of atoms\nfound in well~1 (well~2) is given by $\\cos^2[\\theta\/2]$ ($\\sin^2[\\theta\/2]$); the\nphase-factor between both wells is given by $\\exp(i\\phi)$. All atoms occupy this single\nparticle wave-function. One can also find corresponding $N$-particle wave-functions \nknown as ``atomic coherent states''~\\cite{MandelWolf95}. In an expansion in the Fock-basis $|\nn, N-n \\rangle$ with $n$ atoms in well~$1$ and  $N-n$ atoms in  well~$2$ these wave-functions read:\n\\begin{eqnarray}\n\\label{eq:atomic}\n\\left|\\theta,\\phi\\right>&=& \\sum_{n=0}^N \\genfrac{(}{)}{0pt}{}{N}{n}^{1\/2}\\cos^{n}(\\theta\/2)\n                 \\sin^{N-n}(\\theta\/2)\n                 \\nonumber\\\\\n                 &\\times& e^{i(N-n)\\phi}| n, N-n \\rangle\\;.\n\\end{eqnarray}\n\nThe Gross-Pitaevskii dynamics can be mapped to that  of a nonrigid\npendulum~\\cite{SmerziEtAl97}. Including the term  describing the  periodic shaking, the \nHamilton function is given by ($z=\\cos^2(\\theta\/2)-\\sin^2(\\theta\/2)$):\n\\begin{eqnarray}\nH_{\\rm mf}& = &\\frac{N\\kappa}{\\Omega}z^2-\\sqrt{1-z^2}\\cos(\\phi)\\nonumber\\\\\n&-&2z\\left(\\frac{\\mu_0}{\\Omega}+\n\\frac{\\mu_1}{\\Omega}\\sin\\left({\\textstyle\\frac{\\omega}{\\Omega}}\\tau\\right)\\right)\\;,\\quad \\tau =t\\Omega\\;.\n\\label{eq:mean}\n\\end{eqnarray}\nIn our case, $z\/2$ is \nthe experimentally measurable~\\cite{GatiOberthaler07} population imbalance which can\nbe used\nto characterise the mean-field dynamics. The dynamics for this system are known to become\nchaotic~\\cite{GuckenheimerHolmes83}. \n\n\n\\section{Multi-particle-entanglement \\& Quantum Fisher information\\label{sec:multi}}\n\nMulti-particle-entanglement~\\cite{Yukalov2003,Vaucher08,DunninghamEtAl05,PoulsenEtAl05,CiracEtAl98,MicheliEtAl03,MahmudEtAl03,Dounas-frazerEtAl07}\nis a hot topic of current research; to experimentally realise ``Schr\\\"odinger-cat'' like\nsuperpositions of BECs is still a challenge of fundamental research. For the periodically driven\ndouble-well potential, the relation between emergence of ``Schr\\\"odinger-cat'' like mesoscopic superpositions\nin phase-space and mean-field chaos has been discovered in Ref.~\\cite{WeissTeichmann08}. An ideal\nexample of such a mesoscopic superposition is shown\nin Fig.~\\ref{fig:cat}; the fidelity~\\cite{fidelity} of some of the highly entangled states found numerically\nwas well above 50\\%~\\cite{WeissTeichmann08}. In this manuscript, we employ the fact that the\nquantum Fisher information can be used to detect multi-particle entanglement~\\cite{PezzeSmerzi2007}.\n\\begin{figure}\n\\vspace{-1cm}\n\\includegraphics[angle=-90,width=\\linewidth]{cat_perfect.jpg}\n\\caption{\\label{fig:cat}(Colour online) An ideal ``Schr\\\"odinger-cat'' like state which is the superposition of\n  two atomic coherent states~(\\ref{eq:atomic}) with hardly any overlap. \nThe figure\nshows the projection \n$|\\langle \\psi_{\\rm cat}| \\theta, \\phi\\rangle|^2$\n of the\n``Schr\\\"odinger-cat'' like state $\\psi_{\\rm cat} = \\frac{1}{\\sqrt{2}} \\left( |z=-0.6, \\phi = 1.2 \\rangle + |z = 0.65, \\phi=-2.20 \\rangle\n\\right)$ onto the atomic\ncoherent states $|\\theta, \\phi \\rangle$ (e.q. \\ref{eq:atomic}) in dependece of\n$z$ and $\\phi$.\nWhile\n  Ref.~\\cite{WeissTeichmann08} concentrated on identifying such highly entangled state, the\n  present manuscript uses a different approach to search highly entangled states: the sufficient condition~(\\ref{eq:sufficient}) derived in\n  Ref.~\\cite{PezzeSmerzi2007} by using the quantum Fisher information~(\\ref{eq:fisher})\n  (Ref.~\\cite{PezzeSmerzi2007} and references therein).}\n\\end{figure}\n \nBefore defining the quantum Fisher information, we note that the creation and annihilation\noperators can be used to define\n\\begin{eqnarray}\n\\hat{J}_x =& &\\frac12\\left(\\hat{a}^{\\dag}_1\\hat{a}_2^{\\phantom{\\dag}}+\\hat{a}^{\\dag}_2\\hat{a}^{\\phantom{\\dag}}_1\\right)\\;,\\\\\n\\hat{J}_y =& -&\\frac i2\\left(\\hat{a}_1^{\\dag}\\hat{a}_2^{\\phantom{\\dag}}-\\hat{a}_2^{\\dag}\\hat{a}^{\\phantom{\\dag}}_1\\right)\\;,\\\\\n\\hat{J}_z =& &\\frac 12\\left(\\hat{a}_1^{\\dag}\\hat{a}_1^{\\phantom{\\dag}}-\\hat{a}_2^{\\dag}\\hat{a}_2^{\\phantom{\\dag}}\\right)\\;,\n\\end{eqnarray}\nwhich satisfy angular momentum commutation rules. Except for a factor of $N$, the operator $\\hat{J}_z$ is the operator\nof the (experimentally measurable~\\cite{GatiOberthaler07}) population imbalance.\n\nWhile the quantum Fisher information~$F_{\\rm Q}$ can be defined for statistical mixtures, it is\nparticularly simple for pure states~\\cite{PezzeSmerzi2007}:\n\\begin{equation}\nF_{\\rm Q} = 4(\\Delta \\hat{J}_n)^2\\;;\n\\end{equation}\nwhere $\\Delta \\hat{J}_n$ are the mean-square fluctuations of $\\hat{J}$ in direction $n$. In\nthe following we choose the $z$-direction, thus\n\\begin{equation}\n\\label{eq:fisher}\nF_{\\rm Q} = 4(\\Delta \\hat{J}_z)^2\\;.\n\\end{equation}\nFor $N$ particles, a sufficient condition for multi-particle entanglement is given\nby~\\cite{PezzeSmerzi2007}:\n\\begin{eqnarray}\n\\label{eq:sufficient}\nF_{\\rm ent}&>&1\\;,\\\\\nF_{\\rm ent}&\\equiv&\\frac{F_{\\rm Q}}N\\;.\n\\label{eq:flag}\n\\end{eqnarray}\nFor the ideal ``Schr\\\"odinger-cat'' state in real space,\n\\begin{equation}\n|\\psi_{\\rm NOON}\\rangle=\\frac1{\\sqrt{2}}\\left(|N,0\\rangle+|0,N\\rangle\\right)\\;,\n\\end{equation}\nthis entanglement flag reaches a value of $N$. Thus, while Eq.~(\\ref{eq:sufficient}) already\nindicates multi-particle entanglement, values of \n\\begin{equation}\n\\label{eq:highly}\nF_{\\rm ent}\\gg 1\n\\end{equation}\n demonstrate highly entangled\nstates.\n\n\n\n\\section{Results\\label{sec:results}}\nIn order to characterise whether or not the mean-field dynamics are chaotic, Poincar\\'e\nsurface of sections are particularly suited: for a set of initial conditions, the mean-field\ndynamics of the periodically driven system characterised by the angular frequency $\\omega$ is\nplotted at integer multiples of $2\\pi\/\\omega$. Figure~\\ref{fig:sosa} shows a phenomenon which\nis very characteristic for the non-rigid pendulum on which the BEC in a double well was\nmapped within mean-field: the coexistence between chaotic and  regular regions.\n\\begin{figure}[th]\n\\includegraphics[angle=-90,width=\\linewidth]{sos_a.jpg}\n\\caption{\\label{fig:sosa}Poincar{\\'e} surface of section for the mean-field system. Closed\n  loops characterise stable orbits whereas chaos is represented by irregular dots. The\n  parameters are chosen such that they correspond to a one-photon resonance:  a tilt of $2\\mu_0\/\\Omega=3.0$, a driving frequency of~$\\omega=3\\Omega$, an interaction of $N\\kappa\/\\Omega=0.8$ and a driving amplitude of $2\\mu_1\/\\Omega=0.9$ (cf.\\ Ref.~\\cite{EckardtEtAl05}).}\n\\end{figure}\n\n While each\ninitial condition will, in general, lead to many dots in plots like Fig.~\\ref{fig:sosa}, we\nproceed in the spirit of Ref.~\\cite{WeissTeichmann08} to characterise the $N$-particle\ndynamics: In Fig.~\\ref{fig:entfig1a} each initial condition leads to only one point: the\nentanglement flag~(\\ref{eq:flag}) for this initial condition after a fixed time~$t\\Omega$. Highly entangled states\n(Eq.~(\\ref{eq:highly})) can be found on the boundary of stable regions; already for short\ntimes (Fig.~\\ref{fig:entfig1a}~a) such highly entangled states can occur. For larger times\n(Fig.~\\ref{fig:entfig1a}~b) many features of the Poincar\\'e surface of section in\nFig.~\\ref{fig:sosa} are visible in the entanglement generation which essentially spreads over\nthe entire chaotic part of the initial conditions.\n\\begin{figure}\n\\vspace*{-1cm}\n\\includegraphics[width=\\linewidth]{plotfig1a.jpg}\n\\caption{\\label{fig:entfig1a}(Colour online) The entanglement flag~(\\ref{eq:flag}) in a two-dimensional\n  projection as a function of the initial condition for $N=100$ particles. All other\n  parameter as in Fig.~\\ref{fig:sosa}. The $N$-particle wave-function which corresponds to\n  the mean-field initial conditions ($z_0$,$\\phi_0$) can be found in\n  Eq.~(\\ref{eq:atomic}). In the upper panel, the (experimentally measurable)  entanglement\n  flag~(\\ref{eq:flag}) is shown after a time of $t\\Omega=10$, in the lower panel the time is\n  $t\\Omega=100$. Highly entangled states (cf.\\ Eq.~(\\ref{eq:highly})) occur in the entire\n  chaotic regime.\n  }\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=-90,width=\\linewidth]{sos_b.jpg}\n\\caption{\\label{fig:sosb}Poincar{\\'e} surface of section for the mean-field system (cf.\\\n  Fig.~\\ref{fig:sosa}).  The\n  parameters are chosen such that they correspond to a $3\/2$-photon resonance~\\cite{EckardtEtAl05}  with\n  $N\\kappa\/\\Omega=0.1$,  $2\\mu_0\/\\Omega=3.0$, $\\omega\/\\Omega=2.08$ and~$2\\mu_1\/\\Omega=1.8$.}\n\\end{figure}\n\nFor parameters which display no chaotic parts in the Poincar\\'e surface of section\n(Fig.~\\ref{fig:sosb}), on short time-scales hardly any entanglement emerges\n(Fig.~\\ref{fig:entfig1b} a). For larger time-scales, entanglement generation occurs on the\nboundaries of stable  regions (Fig.~\\ref{fig:entfig1b} b).\n\\begin{figure}\n\\vspace*{-2cm}\n\\includegraphics[width=\\linewidth]{plotfig1b.jpg}\n\\caption{\\label{fig:entfig1b}(Colour online) The entanglement flag~(\\ref{eq:flag}) in a two-dimensional\n  projection as a function of the initial condition (cf.\\ Fig.~\\ref{fig:entfig1a}) for $N=100$ particles; all other\n  parameter as in Fig.~\\ref{fig:sosb}. For the regular regime given by the parameters of\n  Fig.~\\ref{fig:sosb}, after short time-scales of $t\\Omega=10$ (upper panel)\n  entanglement-production is very weak (note that the brightness-entanglement-coding differs by\n  a factor of 10 from Fig.~\\ref{fig:entfig1b} and the longer time-scales $t\\Omega=100$\n  depicted in the lower panel). For larger time-scales, entanglement production mainly occurs\non the boundary of stable regions (lower panel).}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn Ref.~\\cite{WeissTeichmann08} we discovered the relation between mesoscopic ``Schr\\\"odinger-cat''\nlike superpositions in phase space and mean-field entanglement. In the present paper, we\ndemonstrated that also for more general entangled states, multi-particle entanglement can be\na quantum signature of chaos. For regular systems, the general entangled states also\noccur. However, it is restricted to the boundaries of stable regions and only occurs on\nlonger time-scales. While the focus in\nRef.~\\cite{WeissTeichmann08} was on finding particularly highly entangled states for each\ninitial condition, in the present manuscript the entanglement production was shown for each\ninitial condition at the same point in time, both for short and longer times between the\nonset of the computer experiment and the read-out. \n\nAs an entanglement-flag, we apply the quantum Fisher information for pure states.\nIn this paper we use it in a way which is\nparticularly easy to measure experimentally. However, using Eq.~(\\ref{eq:flag}) assumes a pure state and\nwould thus only be valid in an ideal system without decoherence. For\nmore realistic situations, \nexperimental signatures as in Refs.~\\cite{PiazzaEtAl2008,WeissCastin08} should be\ninvestigated in the future.\n\n\n\n\n\\acknowledgments\n\nWe would like to thank A.~Smerzi for useful discussions and M.~Holthaus for his\ncontinuous support. N.T.\\ acknowledges funding by the Studienstiftung des deutschen Volkes.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhile within-category demand effects of the marketing mix have been studied extensively, cross-category effects are less well understood \\citep{leeflang:12}.  Nevertheless, cross-category effects might be substantial.  Some categories are complements, e.g.\\ bacon and eggs studied by \\cite{Niraj08} or cake mix and cake frosting studied by \\cite{Manchanda99}, while others are substitutes, e.g.\\ frozen, refrigerated and shelf-stable juices \\citep{Wedel2004}. But cross-effects also exist among categories that are not complements or substitutes for several reasons. First, as a result of brand extensions, brands are no longer limited to one category \\citep{Erdem98,Kamakura2007,Ma2012}. So advertising and promotion of a brand within one category might spill over to own brand sales in other categories.\nSecond, advertising and promotions generate more store traffic and therefore more sales in other categories \\citep{Bell1998}. \nAnd third, lower expenditures in one category alleviate the budget constraint such that consumers are able to spend more on other, seemingly unrelated, categories \\citep{song:07, Lee2013}.\n\nWhile cross-category effects might be substantial for these reasons, we do not expect that each category's marketing mix variables influence each and every other category.  Instead, we expect some cross-category effects to be zero -- or very close to zero -- but we can not a priori exclude them. Therefore, we use an exploratory modeling approach for parsimonious estimation of a product category network. The network allows us to easily identify categories that are influential for or responsive to changes in other categories. Building on a widely used category typology of destination, routine, occasional and convenience categories \\citep{Blattberg95,Briesch2013}, we find that destination categories are most influential, convenience and occasional categories most responsive, and routine categories moderately influential and moderately responsive. \n\nIn order to estimate the cross-category network, this paper presents sparse estimation of the Vector AutoRegressive (VAR) model.  The estimation is {\\it sparse} in the sense that some of the within-and cross-category effects in the model can be estimated as exactly zero. Initiated by the work of \\cite{Baghestani91} and \\cite{Dekimpe95}, the VAR Market Response Model has become a standard, flexible tool to measure own- and cross-effects of marketing actions in a competitive environment. The main drawback of the VAR model is the risk of overparametrization because the number of parameters increases quadratically with the number of included categories. \nEarlier studies using the VAR model, like e.g. \\citeauthor{Nijs01} (\\citeyear{Nijs01}; \\citeyear{Nijs2007}); \\cite{Pauwels2002}; \\citeauthor{Srinivasan00} (\\citeyear{Srinivasan00}; \\citeyear{Srinivasan04});  \\cite{Steenkamp05}, were often limited by this overparametrization problem.\nTo overcome this problem, previous research on cross-category effects has limited its attention to a small number of categories by studying substitutes or complements \\citep{Kamakura2007,song:07,Leeflang:08,Bandyopadhyay2009,Ma2012}.  We present an estimation technique for cross-category effects in much larger product category networks. The technique allows many parameters to be estimated even with short observation periods. Short observation periods are commonplace in marketing practice since many firms discard data that are older than one year \\citep{Lodish07}. \n\nThis paper contributes to the extant retail literature in a number of important ways.  \n(1) Previous cross-category literature largely limits attention to categories that are directly related through substitution, complementarity or brand extensions. We provide  evidence that cross-category effects go beyond such directly related categories. \n(2) We introduce the concepts of influence and responsiveness of a product category and position different category types (destination, routine, occasional and convenience)  according to these dimensions.  \n(3) To identify the cross-category effects, we estimate a large VAR model using an extension of the lasso approach of \\cite{Tibshirani96}. \n\n\nThe remainder of this article is organized as follows. Section 2 positions this paper in the cross-category management literature and describes the conceptual framework that positions category types according to their influence and responsiveness. Section 3 discusses the methodology.  We describe the sparse estimator of the VAR model, discuss how to construct impulse response functions ans compare the sparse estimation technique with two Bayesian estimators. In Section 4, a simulation study shows the excellent performance of the proposed methodology in terms of estimation reliability and prediction accuracy.  Section 5 presents our data and model, Section 6 our findings on cross-category demand effects. \nWe first identify which categories are most influential and which are most responsive to changes in other categories. Then, we identify the main cross-category effects based on estimated cross-price, promotion and sales elasticities.\n\n\\section{Cross-Category Management}\nThe importance of category management for retailers is widely acknowledged, both as a marketing tool for category performance \\citep{Fader1990, Basuroy2001, Dhar2002} and as an operational tool for planning and logistics \\citep{Rajagopalan2012}. \nSuccessful  category management requires retailers to understand cross-category effects of prices, promotions and sales.\nAmong these, the cross-category effects of prices on sales  -- which define substitutes and complements -- are the most extensively studied \\citep{Song:06, Bandyopadhyay2009, leeflang:12, Sinistyn2012}.  \nCross-category effects of promotions, e.g.\\ feature and display promotions, on sales result from many brands being active in multiple categories \\citep{Erdem02}. Brand associations carry over to products of the same brand in other categories, e.g.\\ through umbrella branding \\citep{Erdem98} or horizontal product line extensions \\citep{Aaker90}. \nLess well understood than the effects of prices and promotions, are the effects of sales in one category on sales in other categories.  Such effects might exist because categories are related based on affinity in consumption \\citep{Shankar2014}, because products from various categories are placed close to each other in the shelves \\citep{Bezawada09, Shankar2014}, or because of the budget constraint \\citep{Du2008}.  If consumers spend more in a certain category they might, all else equal, spend less in other categories simply because they hit their budget constraint.  As a result, cross-category effects might exist between seemingly unrelated categories. \n\nWhen studying these cross-category effects of price, promotion and sales on sales, several asymmetries might arise. \nA first asymmetry concerns within- versus cross-category effects. We expect within-category effects to be more prevalent and larger in size than cross-category effects (e.g. \\citealp{Song:06}; \\citealp{Bezawada09}).\nA second asymmetry concerns category influence versus category responsiveness.\nInfluential categories are important drivers of other category's sales, while sales of responsive categories react to changes in other categories.  To identify which categories are more influential or more responsive, we build on a widely used typology of categories described in \\cite{Blattberg95}. \n\n\\cite{Blattberg95} define 4 category types from the consumer perspective: destination, routine, occasional and convenience. \nDestination categories contain goods that consumers plan to buy before they go on a shopping trip, such as soft drinks. \\cite{Briesch2013} show that destination categories are generally categories in which consumers spend a lot of their budget. Retailers typically use a price aggressive promotion strategy and high promotion intensity for these destination categories with the goal of increasing store traffic.\nBecause consumers shop to buy products in the destination categories, destination categories are likely to influence sales in other categories. However, since consumers already plan to buy in the destination categories before entering the store, destination category sales will not be highly responsive \\citep{Shankar2014}.  \n\nAbout 55\\% to 60\\% of categories are routine categories \\citep{Pradhan09}. Routine categories are regularly and routinely purchased, such as juices and biscuits.  Retailers typically use a consistent pricing strategy and average level of promotion intensity. Because purchases in routine categories can more easily be delayed than purchases in destination categories, we expect routine categories to be more responsive.  But, since purchases in routine categories altogether still account for a large portion of the budget, they are also likely to influence sales in other categories. \n\nOccasional categories follow a seasonal pattern or are purchased infrequently. These categories comprise a small proportion of retail expenditures while they contain typically more expensive items, like oatmeal.  We therefore expect occasional categories to be less influential and more responsive than destination or routine categories. \n\nFinally, convenience categories are categories that consumers find convenient to pick up during their one-stop shopping trip, like ready-to-eat-meals. These purchase decisions are typically made in the store. Since convenience categories are geared towards consumer convenience and filling impulse needs, we expect them to be highly responsive.\n\n\n\\section{Sparse Vector Auto-Regressive Modeling}\n\n\\subsection{Motivation}\n\nThe aim of this paper is to identify cross-category demand effects in a large product category network.\nTo this end, we use the Vector AutoRegressive (VAR) model. The VAR is ideal for measuring within- and cross-category effects of marketing actions since it accounts for both inertia in marketing spending and performance feedback effects by treating marketing variables as endogenous \\citep{Dekimpe95}. \nOther studies on cross-category effects, like e.g.\\ \\cite{Wedel2004} use a demand model with exogenous prices, or a simultaneous equations model without lagged effects like \\cite{Shankar2014}. However, managers may set marketing instruments strategically in response to market performance and market response expectations. Not accounting for time inertia or feedback effects limits our understanding of how the market functions and misleads managerial insights and prediction.\n\nIdentifying cross-category demand effects using VAR analysis remains challenging because the sheer number of such effects makes them hard to estimate. The number of parameters to be estimated in the  VAR rapidly explodes, making standard estimation inaccurate. This undermines the ability to identify important relationships in the data.\nTo overcome an explosion of the number of parameters in the VAR, marketing researchers have used pre-estimation dimension reduction techniques, i.e.\\ they first impose restrictions on the model and then estimate the reduced model. Four such common techniques are (i) treating marketing variables as exogenous (e.g. \\citealp{Nijs01}; \\citealp{Pauwels2002} and \\citealp{Nijs2007}), (ii) estimating submodels rather than a full model (e.g. \\citealp{Srinivasan00}; \\citealp{Srinivasan04}), (iii) aggregating or pooling over, for instance, stores or competitors (e.g. \\citealp{Horvath08}; \\citealp{Slotegraaf2008}), and (iv) applying Least Squares to a restricted model (e.g. \\citealp{Dekimpe95, Dekimpe99_b}; \\citealp{Nijs2007}). Most researchers applying  pre-estimation dimension reduction techniques recognize that they do so because of the practical limitations of standard estimation techniques rather than for theoretical reasons (e.g. \\citealp{Srinivasan04} and \\citealp{Bandyopadhyay2009}).  \n\nTo address the overparametrization of the VAR, we use sparse estimation.  Sparsity means that some of the within- and cross-category effects in the VAR are estimated as exactly zero. \nAs argued in the previous section, from a substantive perspective, we cannot exclude cross-category effects before estimation because cross-category effects might occur between seemingly unrelated categories. \nFrom a methodological perspective, sparse estimation is a powerful solution to handle the overparametrization of the VAR.\nIn our cross-category model, we endogenously model sales, promotion and prices of 17 product categories. Hence, already in a VAR model with one lag, as much as $(3 \\times 17) \\times (3 \\times 17) = 2601$ within- and cross-category effects need to be estimated. Since the sparse estimation procedure puts some of these effects to zero, a more parsimonious model is obtained. Results are easier to interpret and, therefore, the sparse estimation procedure provides actionable insights to managers.\n\n\n\n\\subsection{Extending the Lasso to the VAR model}\nIn situations where the number of parameters to estimate is large relative to the sample size, the Lasso proposed by \\cite{Tibshirani96} provides a solution within the multiple regression model.  The Lasso minimizes the least squares criterion penalized for the sum of the absolute values of the regression parameters.  This penalization forces some of the estimated regression coefficients to be exactly zero, which results in selection of the pertinent variables in the model. The Lasso method is well established \\citep{Buhlmann2010, Chatterjee2011} and shows good performance in various applied fields \\citep{Wu2009, Fan2011}.\n\nThe Lasso technique can not be directly applied to the VAR model because the VAR model differs from a multiple regression model in two important aspects. First, a VAR model contains several equations, corresponding to a multivariate regression model. Correlations between the error terms of the different equations need to be taken into account.\nSecond, a VAR model is dynamic, containing lagged versions of the same time series as right-hand side variables of the regression equation. Both aspects of VAR models make it necessary to extend the lasso to the VAR context, what the sparse estimator in this paper does.\n\nIt builds further on a sparse estimator of the multivariate regression model \\citep{Rothman10}, and the groupwise lasso for categorical variables \\citep{Yuan06, Meier07}. The estimator is consistent for the unknown model parameters, see \\citet{Meier07} and \\citet{Friedman07}. \n\n\n\\subsection{Model Specification}\nSales, price and promotion are measured for several categories over a certain time period. We collect all these time series in a multivariate time series ${\\bf y}_t$ with $q$ components. In our cross-category demand effects study, ${\\bf y}_t$ contains sales, price and promotion for 17 product categories, hence $q= 3 \\times 17 = 51$. The VAR Market Response Model is given by\n\\begin{equation}\\label{varp}\n{\\bf y}_t = B_1 {\\bf y}_{t-1}  + B_2 {\\bf y}_{t-2}  + \\ldots + B_p {\\bf y}_{t-p}  + {\\bf e}_t \\, ,\n\\end{equation}\nwhere $p$ is the lag length. The autoregressive parameters $B_1$ to $B_p$ are $(q \\times q)$ matrices, which capture both within- and cross-category effects. The elements of these matrices measure the effect of sales, price and promotion in one category on the sales, price and promotion in other categories (including its own). The error term ${\\bf e}_t$ is assumed to follow a $N_q(0,\\Sigma)$ distribution. We assume, without loss of generality, that all time series are mean centered such that no intercept is included.\n\nIf the number of components $q$ in the multivariate time series is large,  the number of unknown elements  in the sequence of matrices $B_1,\\ldots,B_p$ explodes to $p q^2$, and accurate estimation by standard methods is no longer possible. Sparse estimation, with many elements of the matrices  $B_1,\\ldots,B_p$ estimated as zero,  brings an outcome: it will not only provide  estimates with smaller mean squared error, but also substantially improve model interpretability.\nThe method we propose does not require the researcher to prespecify which entries in the $B_j$ matrices are zero and which are not. Instead, the estimation and variable selection are  simultaneously performed. This is particularly of interest in situations where there is no a priori information on which time series is driving which.\n\nThe instantaneous correlations in model \\eqref{varp} are captured in the error covariance matrix $\\Sigma$. If the dimension $q$ is large relative to the number of observations, estimation of $\\Sigma$ becomes problematic. The estimated covariance matrix risks getting singular, i.e.\\ its inverse does not exist.  Hence, we also induce sparsity in the estimation of the inverse error covariance matrix  $\\Omega=\\Sigma^{-1}$. The  elements of $\\Omega$ have a natural interpretation as  partial correlations between the error components of the $q$ equations in model \\eqref{varp}. If the $ij$-th element of the inverse covariance matrix is zero this means that, conditional on the other error terms, there is no correlation between the error terms of equations $i$ and $j$.\n\n\n\\subsection{Penalized Likelihood Estimation}\nThis section defines the sparse estimation procedure for the VAR model. The Sparse VAR estimator is defined by minimizing a measure of goodness-of-fit to the data  combined with a {\\it penalty} for the magnitude of the model parameters.  It is convenient to first recast model \\eqref{varp} in stacked form as\n\\begin{equation}\\label{stacked}\ny = X \\beta + e \\, ,\n\\end{equation}\nwhere $y$ is a vector of length $n q$ containing the stacked  values of the time series. If the multivariate time series has length $T$, then $n=T-p$ is the number of time points for which all current and lagged observations are available. The vector $\\beta$ contains the stacked\nvectorized matrices $B_1,\\ldots,B_p$, and $e$ the vector of stacked error terms.\nThe matrix $X=I_q \\otimes X_0$, with $ X_0 = (\\underline{{\\bf Y}}_1, \\ldots, \\underline{{\\bf Y}}_p)$, is of dimension $(n q \\times p q^2)$.\nHere $\\underline{{\\bf Y}}_j$ is an $(n \\times  q)$ matrix, containing the values of the $q$  series at lag $j$ in its columns,  for $1 \\leq j \\leq p$, with $p$ the maximum lag. The symbol  $\\otimes$ stands for the Kronecker product.\n\nThe sparse estimator of the autoregressive parameters $\\beta$ and the inverse covariance matrix $\\Omega=\\Sigma^{-1}$ are obtained by minimizing the negative log likelihood with a groupwise penalization on the $\\beta$ and a penalization on the off-diagonal elements of $\\Omega$:\n\\begin{equation}\\label{mincrit}\n(\\hat{\\beta},\\hat{\\Omega}) = \\underset{(\\beta,\\Omega)}{\\operatorname{argmin}} \\, \\frac{1}{n} (y-X \\beta)^{\\prime} \\tilde{\\Omega} (y-X \\beta) - \\log|\\Omega|  + \\lambda_1 \\sum_{g=1}^{G} ||\\beta_g|| + \\lambda_2 \\sum_{k \\neq k'} |\\Omega_{kk'}| \\, ,\n\\end{equation}\nwhere $||u||= (\\sum_{i=1}^{n} u_i^2)^{1\/2}$ is the Euclidean norm and $\\tilde{\\Omega}= \\Omega \\otimes I_n$. \nBy simultaneously estimating $\\beta$ and $\\Omega$, we take the correlation structure between the error terms into account.\nThe vector $\\beta_g$ in \\eqref{mincrit} is a subvector of $\\beta$,  containing the  regression coefficients for the  lagged values of  the same time series in one of the $q$ equations in model \\eqref{varp}. The coefficients of the lagged values of the same time series form a group. The total number of groups is $G=q^2$ because there are $q$ groups within each of the $q$ equations.\nThe penalty on the regression coefficients enforces that either \\textit{all} elements of the group $\\hat\\beta_g$ are zero or \\textit{none}. As a result, we take the dynamic nature of the VAR model into account since the estimated $B_j$ matrices, for $j=1,\\ldots,p$, have their zero elements in exactly the same cells.\nThe penalization on the off-diagonal elements of $\\Omega$ induces sparsity in the estimate $\\hat\\Omega$.  Finally, the scalars $\\lambda_1$ and $\\lambda_2$ control the degree of sparsity of the regression estimator and the inverse covariance matrix estimator, respectively. The larger these values, the more sparsity is imposed.\nDetails on the algorithm  to perform penalized likelihood estimation and the selection of the sparsity parameters $\\lambda_1$ and $\\lambda_2$ can be found in Appendix A.\n\nOur approach is similar to \\cite{Hsu08} who use the Lasso within a VAR context. However, they do not account for the group-structure in the VAR model, nor do they impose sparsity on the error covariance matrix. \n\\cite{Davis12} propose another sparse estimation procedure for the VAR. They infer the sparsity structure of the autoregressive parameters from an estimate of the partial spectral coherence using a two-step procedure. Since variable selection is performed prior to model estimation, the resulting estimator suffers from pre-testing bias. Moreover, the number of parameters might still approach the sample size, leading to unstable estimation or even making estimation infeasible if the number of parameters still exceeds the sample size.\nSparse estimation in economics is a growing field, see \\cite{Fan2011} and references therein for an overview. \n\n\\subsection{Alternative: Bayesian Estimators}\nAn alternative to the sparse estimation technique is to impose prior information in a Bayesian setting. Bayesian regularization techniques have been proposed for the VAR model in \\cite{Litterman80} and are used in various applied fields such as macroeconomics \\citep{Gefang14, Banbura10}, finance \\citep{Carriero12} and marketing \\citep{Lenk09,Fok:12,Bandyopadhyay2009}. They are also applicable to a situation like ours where there are many parameters to be estimated with a limited observation period, and are thus a good benchmark.  However, these methods are not sparse, they do not perform variable selection simultaneously with model estimation. The following two paragraphs elaborate on two Bayesian estimators which serve as non-sparse alternatives.\n\\bigskip\n\n{\\it Minnesota Prior.} The original Minnesota prior only specifies a prior distribution for the regression parameters of the VAR model. The  error covariance matrix $\\Sigma$ is assumed to be diagonal, and estimated by $\\hat{\\Sigma}_{ii} = \\hat{\\sigma}_{i}^{2}$ with $\\hat{\\sigma}_{i}^{2}$ the standard OLS estimate of the error variance in an AR$(p)$ model for the $i^{th}$ time series \\citep{koop:09}. The prior distribution of the regression parameters is taken to be multivariate normal:\n\\begin{equation}\n\\beta \\sim N(\\underline{\\beta}_{M},\\underline{V}_{M}) \\label{Minnesotaprior}.\n\\end{equation}\nFor the prior mean, the common choice is $\\underline{\\beta}_{M}=0_{Kq}$ for stationary series. The prior covariance matrix $\\underline{V}_{M}$ is diagonal. The posterior distribution is again multivariate normal. Full technical details can be found in \\cite{koop:09}.\n\nThe main advantage of the Minnesota prior is its ease of implementation, since posterior inference only involves  the multivariate normal distribution. However, imposing the Minnesota prior only ensures that the parameter estimates are \\textit{shrunken} towards zero, while the Sparse VAR ensures that some parameters will be estimated as \\textit{exactly} zero.\n\\bigskip\n\n{\\it Normal Inverted Wishart Prior.}\nThe Minnesota prior takes the error covariance matrix $\\Sigma$ as fixed and diagonal and, hence, not as an unknown parameter. To overcome this problem, \\cite{Banbura10} impose an inverse Wishart prior on the $\\Sigma$ matrix. More precisely,\n\n\\begin{equation}\n\\beta \\mid \\Sigma \\sim N(\\underline{\\beta}_{NIW},\\Sigma \\otimes \\Omega_{0}) \\text{\\ \\ and  \\ \\ } \\Sigma \\sim iW(S_{0},\\nu_{0}), \\label{LBVArRprior}\n\\end{equation}\nwhere $\\underline{\\beta}_{NIW},\\Omega_{0},S_{0}$ and $\\nu_{0}$ are hyperparameters. Under this normal inverted Wishart prior (labeled in the remainder of this paper as ``NIW\"), the posterior for $\\beta$, conditional on $\\Sigma$ is normal, and the posterior for $\\Sigma$ is again inverted Wishart. Full technical details can be found in \\cite{Banbura10}.\n\n\\subsection{Impulse Response Functions}\nImpulse response functions (IRFs) are extensively used to assess the dynamic effect of external shocks to the system such as changes in the marketing mix. An IRF pictures how a change to a certain variable at moment $t$ impacts the value of any other time series at time $t+k$, accounting for interrelations with all other variables. The magnitude of the effect  is plotted as a function of $k$. An extensive discussion on the interpretation of the IRF in marketing modeling can be found in \\cite{Dekimpe95}. We use IRFs to gain insight in the dynamics of within and cross-category sales, promotion and price effects on each of the 17 product category sales. The IRFs are easily computed as a function of  the Sparse VAR estimator (see \\citealp{Hamilton91}). Since we want to account for correlated error terms, we use generalized IRFs \\citep{Pesaran1998, Dekimpe99a}.\n\nTo obtain confidence bounds for the generalized IRFs estimated by Sparse VAR, we use a residual parametric bootstrap procedure \\citep{Chatterjee2011}. We generate $N_{b} = 1000$ time series of length $ T $ from the VAR model (\\ref{stacked}). The invertible estimate of $ \\Sigma $ delivered by the Sparse VAR estimation procedure is needed to draw random numbers for the $ N_{q}(0,\\Sigma) $ error distribution. For each of these $ N_{b} $ multiple time series, the estimates of the regression parameters are computed. We compute the covariance matrix of the $N_{b}$ bootstrap replicates. For each of the $N_b$ generated series impulse response functions are computed; the 90\\% confidence bounds are then obtained by taking the 5\\% and 95\\% percentiles.\n\n\\section{Estimation and prediction performance}\nWe conduct a simulation study to  compare the proposed Sparse VAR with Bayesian methods using the Minnesota and NIW prior.  As benchmarks, we include the classical Least Squares (LS) estimator and two restricted versions of LS which are often used in practice. In the 1-step Restricted LS \\citep{Dekimpe95,Dekimpe99_b}, we estimate the model with classical LS, delete all variables with $|$$t$-statistic$|$  $ \\leq 1 $, and re-estimate the model with the remaining variables. We also consider an iterative Restricted LS method described in \\cite{Lutkepohl04} where we fit the full model using LS and sequentially eliminate the variables leading to the largest reduction of BIC until no further improvement is possible, of which a close variant was used by \\cite{Nijs2007}.\n\nWe simulate from a VAR model with $q=10$ dimensions and $p=2$ lags.  Each time series has an own auto-regressive structure and we include system dynamics among the different series.  The first series leads series two to five, while the sixth series leads time series 7 to 10. Specifically, the data generating processes are given by\n$${\\bf y}_t =\n\\begin{bmatrix}\nB_{1} & 0\\\\\n0 & B_{1}\\\\\n\\end{bmatrix} {\\bf y}_{t-1}\n+\n\\begin{bmatrix}\nB_2 & 0\\\\\n0 & B_2\\\\\n\\end{bmatrix} {\\bf y}_{t-2}\n+ {\\bf e}_t \\, ,\n$$\nwith\n\\footnotesize\n$$\nB_{1} =\n\\begin{bmatrix}\n0.4 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n0.4 & 0.4 & 0.0 & 0.0 & 0.0 \\\\\n0.4 & 0.0 & 0.4 & 0.0 & 0.0 \\\\\n0.4 & 0.0 & 0.0 & 0.4 & 0.0 \\\\\n0.4 & 0.0 & 0.0 & 0.0 & 0.4 \\\\\n\\end{bmatrix} \\text{and}\n\\hspace{0.2cm} \\hspace{0.2cm}\nB_{2} =\n\\begin{bmatrix}\n0.2 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n0.2 & 0.2 & 0.0 & 0.0 & 0.0 \\\\\n0.2 & 0.0 & 0.2 & 0.0 & 0.0 \\\\\n0.2 & 0.0 & 0.0 & 0.2 & 0.0 \\\\\n0.2 & 0.0 & 0.0 & 0.0 & 0.2 \\\\\n\\end{bmatrix}.\n$$\n\\smallskip\n\\normalsize\n\nIn total, there are $p q^2=200$ regression parameters to be estimated with 36 true parameter values different from zero. The 10-dimensional error term ${\\bf e}_t $ is drawn from a multivariate normal with mean zero and covariance matrix $\\Sigma=0.1 I_{10}$. We generate $N_s=1000$ multivariate time series of length 50 according to the above simulation scheme.\n\n\\subsection{Performance measures}\n\nWe evaluate the different estimators in terms of (i) estimation accuracy, (ii) sparsity recognition performance, and (iii) forecast performance.\n\nTo evaluate estimation accuracy, we compute the mean absolute estimation error (MAEE), averaged over the simulation runs and over the 200 parameters\n$$\n\\mbox{MAEE} = \\frac{1}{N_s} \\frac{1}{pq^2} \\sum_{s=1}^{N_s} \\sum_{j=1}^p \\sum_{k,l=1}^q  | \\hat{b}^s_{klj} - b_{klj} |,\n$$\nwhere $\\hat{b}^s_{klj}$ is the estimate of  $b_{klj}$,  the $kl^{th}$\\ element of the  matrix $B_j$ corresponding to lag $j$, for the $s^{th}$ simulation run.\n\nConcerning sparsity recognition, we compute the true positive rate and true negative rate\n\\begin{gather}\n\\text{TPR}(\\hat{b},b) = \\dfrac{  \\# \\{ (k,l,j) :  \\hat{b}_{klj} \\neq 0 \\  and \\ b_{klj} \\neq 0 \\}}{\\# \\{ (k,l,j) :  \\ b_{klj} \\neq 0 \\}} \\nonumber \\\\\n\\text{TNR}(\\hat{b},b) = \\dfrac{  \\# \\{ (k,l,j) :  \\hat{b}_{klj} = 0 \\  and \\ b_{klj} = 0 \\}}{\\# \\{ (k,l,j) :  \\ b_{klj} = 0 \\}}. \\nonumber\n\\label{sparsityperformance} \n\\end{gather}\nThe true positive rate (TPR) gives an indication on the number of true relevant regression parameters detected by the estimation procedure. The true negative rate (TNR) measures the hit rate of detecting a true zero regression parameter. Both should be as large as possible.\n\nFinally, we conduct an out-of-sample rolling window forecasting exercise. Using the same simulation design as before, we generate multivariate time series of length $T=60$, and use a rolling window of length $S=50$. For all estimation methods, 1-step-ahead forecasts are computed for $t=S,\\ldots,T-1$. Next, we compute the Mean Absolute Forecast Error (MAFE), averaged over all time series and across time\n\\begin{equation}\n\\text{MAFE} = \\frac{1}{T-S} \\frac{1}{q} \\sum_{t=S}^{T-1}\\sum_{i=1}^{q} | \\ \\hat{y}^{(i)}_{t+1} -  y^{(i)}_{t+1}  \\ | ,\n\\end{equation}\nwhere $y^{(i)}_{t+1}$ is the value of the  $i^{th}$ time series at time ${t+1}$.\n\n\\subsection{Results}\nTable \\ref{simulationresults} presents the performance measures of the Sparse VAR, the Bayesian and benchmark methods. The Sparse VAR estimator performs best in terms of estimation accuracy. It attains the lowest value of the MAEE (0.041).  A paired $t$-test confirms that the Sparse VAR significantly outperforms the other methods (all $p$-values $<0.001$).\n\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Mean Absolute Estimation Error (MAEE), True Positive Rate (TPR), True Negative Rate (TNR) and Mean Absolute Forecast Error (MAFE), averaged over 1000 simulation runs, are reported for every method. \\label{simulationresults}}\n\\begin{tabular}{lcccccccccccc}\n  \\hline\nMethod  \t\t\t\t\t&&& MAEE &&& TPR &&& TNR &&& MAFE  \\\\ \\hline\nSparse VAR \t\t\t\t\t&&& 0.041 &&& 0.860 &&& 0.848 &&& 0.359 \\\\\nLS \t\t\t\t\t\t\t&&& 0.157 &&& 1 &&& 0 &&& 0.540 \\\\\nRestricted LS: 1-step  \t\t&&& 0.121 &&& 0.709 &&& 0.541 &&& 0.520 \\\\\nRestricted LS: Iterative  \t&&& 0.116 &&& 0.261 &&& 0.775 &&& 0.516 \\\\\nBayesian: Minnesota \t\t&&& 0.044 &&& 1 &&& 0 &&& 0.355 \\\\\nBayesian: NIW \t\t\t\t&&& 0.077 &&& 1 &&& 0 &&& 0.476 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\nSparsity recognition performance is evaluated using the true positive rate and the true negative rate, reported in Table \\ref{simulationresults}. For the LS and Bayesian estimators, all parameters are estimated as non-zero, resulting in a perfect true positive rate and zero true negative rate. Among the variable selection methods, the Sparse VAR performs best. Sparse VAR achieves a value of the true positive rate of 0.86; 0.85 for the true negative rate.\n\nFinally, we evaluate the forecast performance of the different estimators by the Mean Absolute Forecast Error in Table \\ref{simulationresults}. The Sparse VAR and the Bayesian estimator with Minnesota prior achieve the best forecast performance. A Diebold-Mariano test confirms that these two methods perform significantly better than the others ($p$-values $<0.001$). There is no significant difference in forecast performance between Sparse VAR and the Bayesian estimator with Minnesota prior.\n\n\\subsection{Robustness checks}\n\\textit{Alternative penalty function.} We investigate the robustness of Sparse VAR to the choice of the penalty function. We replace the grouplasso penalty on the regression coefficients with the elastic net penalty \\citep{Zou05}. Elastic net is a regularized regression method that linearly combines the $L_1$ and $L_2$ penalties of respectively lasso and ridge regression. Like the grouplasso, elastic net produces a sparse estimate of the regression coefficients. All other steps of the methodology remain unchanged.  We find that the grouplasso penalty performs slightly better than the elastic net  penalty in terms of estimation accuracy, sparsity recognition and prediction performance. \n\n\\medskip\n\n\\textit{Sensitivity to the order of the VAR.} We estimate the model with Sparse VAR for different values of $p$ and evaluate the performance. As expected, Sparse VAR attains the best estimation accuracy for the true value $p=2$. The results are, however, very robust to the choice of the order of the VAR. Selecting $p$ too low is slightly worse than selecting $p$ too high. \n\n\\medskip\n\n\\textit{Sensitivity to the sparsity parameters.} The sparsity parameters are selected according to the BIC and this selection is an integral part of the estimation procedure.\nThe results are not sensitive to the value of $\\lambda_2$, which controls the sparsity of $\\widehat{\\Omega}$. The results are more sensitive to the choice of $\\lambda_1$, since it directly influences the sparsity of the autoregressive parameters. It turms out that Sparse VAR still outperforms the other estimators for a large range of $\\lambda_1$ values.\n\n\n\\section{Data and Model}\nWe use the sparse estimation technique for large VARs described in Section 3 to identify cross-category demand effects across 17 categories in the Dominick's Finer Foods database.  This database is a well-established source of weekly scanner data from a large Midwestern supermarket chain, Dominick's Finer Foods (e.g. \\citealp{Kamakura2007, Pauwels07}).  We first describe the data and model in more detail, and then report on the insights the Sparse VAR generates in the next section.\n\n\nWe use all 17 product categories in the Dominick's Finer Foods database containing food and drink items, a much broader selection of categories than previous studies on cross-category demand effects have considered. A description of each product category can be found in Table \\ref{Categories}. For 15 stores, we obtain weekly sales, pricing  and promotional feature and display data for the 17 product categories.\n\n\\linespread{1.2}\n\\begin{table}\n\\small\n\\begin{center}\n\\caption{Description of the 17 categories from Dominick's Finer Foods database that are analyzed in this paper. For each category, we report the proportion of food and drink expenditures.  \\label{Categories}}\n\\small\n\\begin{tabular}{lclllc} \\hline\nCategory & Expenditures &&&  Category & Expenditures \\\\ \\hline\nSoft Drinks & 22.24\\% \t\t&&& Snack Crackers & 3.04\\%\\\\\nCereals & 13.92\\% \t\t\t&&& Frozen Juices & 2.88\\% \\\\\nCheeses & 10.46\\% \t\t&&& Canned Tuna & 2.80\\% \\\\\nRefrigerated Juices & 7.36\\% \t\t\t&&&  Frozen Dinners & 2.00\\% \\\\\nFrozen Entrees & 6.98\\% \t&&&  Front-end-candies & 2.00\\% \\\\\nBeer & 6.35\\% \t\t\t\t&&&  Cigarettes & 1.49\\% \\\\\nCookies & 6.21\\%\t\t\t\t&&&  Oatmeal & 1.43\\%\\\\\nCanned Soup & 4.82\\% \t\t\t&&&  Crackers & 1.37\\%\\\\\nBottled Juices& 4.66\\% \t\t&&& & \\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\\normalsize\n\n\n\\noindent\n{\\bf Sales.} Category sales volumes for the 17 categories, measured in dollars per week.\n\n\\noindent\n{\\bf Promotion.} The promotional data include the percentage of SKUs of each category that are promoted (feature and display) in a given week, following \\citet{Srinivasan04}.\n\n\\noindent\n{\\bf Prices.} To aggregate pricing data from the SKU level to the product category level, we follow \\citet{Srinivasan04} and \\citet{Pauwels2002} in using SKU market shares as weights.  Prices are not deflated because there is strong evidence that people are sensitive to nominal rather than real price changes  \\citep{Shafir1997} over short time periods.\n\n\\medskip\n\nWe use data from January 1993 to July 1994, 77 weeks in total. We neither use \ndata before 1993 since they contain missing observations, nor\nobservations after 1994 since \\citet{Srinivasan04}  pointed out that manufacturers made extensive use of `pay-for-performance' price promotions as of 1994, which are not fully reflected in the Dominick's database. This data range is short relative to the dimension of the VAR, which calls for a regularization approach such as the Sparse VAR. For all stores, we collect data on sales, promotion and pricing  for all 17 categories. Only for cigarettes, no promotion variable is included in the VAR since none of the SKUs in that category were promoted during the observation period.\n\nWe estimate a separate VAR model for each store, which allows to evaluate the robustness of the findings. The multivariate time series entering the VAR model are the log-differenced sales  ($\\mathbf{Y_t}$), differenced promotion ($\\mathbf{M_t}$), and log-differenced prices  ($\\mathbf{P_t}$).\\footnote{Following standard practice, we first test for stationarity. A stationarity test of all individual time series using the Augmented Dickey-Fuller test indicates that most time series in levels are integrated of order 1.}  The dimensions of the time series are represented in Table \\ref{data}.  We use the Vector Autoregressive model, with endogenous promotion and prices,\n\\begin{equation}\\label{eq: application model}\n\\left[ \\begin{matrix} \\mathbf{Y_t} \\\\ \\mathbf{P_t} \\\\ \\mathbf{M_t}  \\end{matrix}  \\right] =\nB_0 + B_1 \\left[ \\begin{matrix} \\mathbf{Y_{t-1}} \\\\ \\mathbf{P_{t-1}} \\\\ \\mathbf{M_{t-1}}  \\end{matrix}  \\right]\n+ \\ldots + B_p \\left[ \\begin{matrix} \\mathbf{Y_{t-p}} \\\\ \\mathbf{P_{t-p}} \\\\ \\mathbf{M_{t-p}}  \\end{matrix}  \\right] + \\mathbf{e_t}.\n\\end{equation}\nAveraged across stores, the selected value of $p$ is two for the Sparse VAR.\nAlso for the Bayesian estimators, the lag order of the VAR is selected using the BIC criterion, which is one for the majority of the stores. \n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Description of the 15 data sets. Each data set contains multivariate time series for sales ($\\textbf{Y}_{t}$), promotion ($\\textbf{M}_{t}$) and prices ($\\textbf{P}_{t}$). \\label{data}}\n\\small\n\\begin{tabular}{cccccc} \\hline\n\\rule{0pt}{3ex} Store & Number of & \\multicolumn{3}{c}{Dimension} & \\\\\n & Time Points  &  $\\textbf{Y}_{t}$ & $\\textbf{M}_{t}$ & $\\textbf{P}_{t}$ & Total \\\\ \\hline\nStore 1-15 \\rule{0pt}{3ex} & 77 & 17 & 16 & 17 & 50\\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\\section{Empirical Results}\n\nWe focus on the effects of prices, promotions and sales in category A on the sales (or demand) in category B, where A and B belong to the product category network.  We first study the direct effects.   \nFor instance, there is no direct effect of price of A on  sales of B if the corresponding estimated regression coefficients  are equal to zero at all lags.   \nThen we turn to the complete chain of direct and indirect effects using Impulse Response Functions.  \nFor instance, price in category A indirectly influences sales in category B when the price of category A influences the price, promotion or sales in a certain other category C which, in turn, influences the sales of category B. Since we work in a time series setting, both direct and indirect effects are dynamic in the sense that the effect occurs with a certain delay.\n\n\\subsection{A network of product categories}\nWe analyze cross-category demand effects as a network of interlinked product categories of which prices, promotions and sales in one category have an effect on sales in other categories. Recently, network perspectives have been increasingly used by marketing researchers to model, for example, the network value of a product in a product network \\citep{Singer2013} or to investigate the flow of influence in a social network \\citep{Zubcsek11}. In our case, the 17 product categories are the nodes of the network. We estimate the Sparse VAR for 15 stores separately. If the Sparse VAR estimation results indicate, by giving a non-zero estimate, that prices in one category have a direct influence on sales in another category in the majority of the 15 stores, a directed edge is drawn between them. The resulting directed network is plotted in Figure \\ref{crosscatPrice}. Similarly, Figures \\ref{crosscatPromo} and \\ref{crosscatSLS} present cross-category effects of respectively  promotion and sales on sales. If promotion or sales in one category directly influence sales in another category, respectively, this is indicated by a directed edge. \n\n\n\\linespread{1.2}\n\\begin{table}\n\\centering\n\\caption{Proportion of nonzero within and cross-category effects of price, promotion and sales on sales, averaged across 15 stores and 17 product categories.} \\label{Within-Cross}\n\\begin{tabular}{lccccccccc}\n  \\hline\n &&& Price &&& Promotion &&& Sales   \\\\\n  \\hline\nWithin-category &&& 34\\% &&& 30\\%  &&& 96\\%    \\\\\nCross-category &&& 19\\%  &&& 21\\%  &&&  21\\%   \\\\\n   \\hline \n\\end{tabular}\n\\end{table}\n\\linespread{1.5}\n\n\nA first important finding is that the cross-category networks are sparse -- not each category influences each and every other category.  While the sparse VAR estimation favors zero-effects, it does not enforce them. Here, as many as 78\\% of all estimated effects are zero-effects. Table \\ref{Within-Cross} summarizes the prevalence of within-and cross-category effects.  As expected, within-category effects are more common than cross-category effects. For all categories, past values of the own category's sales are selected for almost all stores. Cross-category effects of price on sales (19\\%), promotion on sales (21\\%) and sales on sales (21\\%)  are about equally prevalent.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9cm, angle=-90]{PRICE}\n\t\\end{center}\n\t\\caption{Cross-category effect network of prices on sales: a directed edge is drawn from one category to another if its price influences sales in the other category for the majority of stores. \\label{crosscatPrice}} \n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm, angle=-90]{PROMO}\n\t\\end{center}\n\t\\caption{Cross-category effect network of promotions on sales: a directed edge is drawn from one category to another if its promotion influences sales in the other category for the majority of stores. \\label{crosscatPromo}} \n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm, angle=-90]{SLS}\n\t\\end{center}\n\t\\caption{Cross-category effect network of sales on sales: a directed edge is drawn from one category to another if its sales influences sales in the other category for the majority of stores.\\label{crosscatSLS}} \n\\end{figure}\n\n\n\nNext, we focus on category influence and responsiveness in the cross-category network, measured by the number of edges originating from and pointing to a category respectively.\nAs discussed in Section 2, destination categories are expected to be more influential, while   convenience categories are expected to be more responsive. \n  We discuss which types of categories we find to be most influential and\/or responsive in the cross-category networks of prices on sales, promotion on sales, and sales on sales. \n\nThe most influential categories in the cross-category network of prices on sales are destination categories such as Soft Drinks and Cheeses (cfr.\\ each four outgoing edges in Figure \\ref{crosscatPrice}). This is consistent with our expectations, as Soft Drinks is known to be a destination category \\citep{Briesch2013,Shankar2014,Blattberg95}. Soft Drinks is ranked first and Cheeses third in terms of food and drink expenditures (see Table \\ref{Categories}) and are both heavily promoted by retailers.  A price change in either of these categories thus strongly influences the budget constraint, which in turn influences purchase decisions in other categories.  In the cross-category network of promotions on sales, Cereals is the most influential category (cfr. five outgoing edges in Figure \\ref{crosscatPromo}). \\cite{Briesch2013} identified Cereals as highly ranked among the destination categories.  This is not surprising as cereals are part of daily consumption patterns and are ranked second in terms of food and drink expenditures.  In the cross-category effects network of sales on sales in Figure \\ref{crosscatSLS}, we identify again Cheeses as the most influential category.  \n\nWe find convenience categories to be highly responsive to changes in other categories.\nThe most prominent price effects are observed for Canned Soup (cfr.\\ five incoming edges in Figure \\ref{crosscatPrice});\nthe most prominent promotion effects for Frozen Dinners, Crackers and Canned Soup (cfr. each three incoming edges in Figure \\ref{crosscatPromo}); \nand the most prominent sales effects for Oatmeal and Crackers (cfr.\\ each four incoming edges in Figure \\ref{crosscatSLS}). \nThese categories are typically bought out of convenience, such as Frozen Dinners and Canned Soup; or bought on occasion, such as Oatmeal and Crackers, counting for a very small percentage of food and drink expenditures (see Table \\ref{Categories}). \n\nRoutine categories such as Bottled Juices, Refrigerated Juices, Frozen Juices and Cookies score moderate-to-high on category influence but are also responsive. \nThis is in line with our expectation of many grocery categories being routine categories that are moderately influential and moderately responsive. Finally, the cigarettes category is least responsive and least influential.  This finding is not surprising as cigarettes are addictive, hence, smokers probably have a stable consumption unrelated to food and drinks.\n\nTo confirm the robustness of the results obtained by Sparse VAR, we check whether category responsiveness and influence are consistent across stores. We compute Kendall's coefficient of concordance $W$  for category influence and responsiveness calculated from the graphs in Figures 2-4 at the store level.   As $W$ increases from 0 to 1, there is stronger consistency across stores.  Table \\ref{KendallW} indicates that all values of Kendall's $W$ are significant.\n\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Kendall's coefficient of concordance across stores of cross-category effects of price, promotion and sales on sales for both category influence and responsiveness. $P$-values are indicated between parentheses. \\label{KendallW}}\n\\begin{tabular}{lccccccccc}\n\\hline\n\n  &&& Price &&& Promotion &&& Sales  \\\\ \\hline\n  Influence \t\t&&& $\\underset{(< 0.001)}{0.40}$ &&& $\\underset{(< 0.001)}{0.56}$ &&& $\\underset{(< 0.001)}{0.30}$    \\\\\n Responsiveness \\rule{0pt}{3ex}\t&&& $\\underset{(<0.001)}{0.30}$ &&& $\\underset{(0.001)}{0.16}$ &&&  $\\underset{(< 0.001)}{0.17}$  \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\n\\subsection{Impulse Response Functions}\nFor each store, we estimate the Sparse VAR and compute the corresponding Impulse Response Functions (IRFs). The effect size of an impulse is obtained by summing the absolute values of the responses across the first 10 lags of the IRF, where we take absolute values in order not to average out positive and negative response. We compute effect sizes of impulses in price, promotion or sales in one product category on the sales in the same (within) category or another (cross) category. In Table 6, we report the within and cross-category price, promotion and sales effect sizes, averaged across the 15 stores and the  product categories. \n\n\nTable \\ref{Effectsizes} indicates that, for example, a one standard deviation price shock leads to an accumulated absolute change of .004 in own sales growth over a time period of 10 lags. As for the direct effects, we systematically find that within-category effects are larger in magnitude than cross-category effects, especially for sales and prices. For the marketing mix, promotions exert stronger within- as well as cross-category effects than price changes.\n\n\\linespread{1.2}\n\\begin{table}\n\\centering\n\\caption{Size of within and cross-category effects of price, promotion and sales on sales, summed across 10 lags of the IRF, averaged across stores and product categories, and in absolute value.} \\label{Effectsizes}\n\\begin{tabular}{lccccccccc}\n  \\hline\n &&& Price &&& Promotion &&& Sales    \\\\\n  \\hline\nWithin-category &&& 0.004 &&& 0.006 &&& 0.057   \\\\\nCross-category &&& 0.002 &&& 0.005 &&& 0.002    \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\linespread{1.5}\n\n\\begin{table}\n\\caption{Cross-category price, promotion and sales effects on sales summed across 10 lags of IRFs and averaged across stores. We present only the five largest positive and negative effects. \\label{Crosscateffects}}\n\\footnotesize\n\\resizebox{0.95\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\\centering\n\\begin{tabular}{lllcc|llllcc}\n\\hline\n\\multicolumn{6}{l}{\\textbf{Cross-category price effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\ \\hline\nPrice && Sales   && Effect && Price && Sales   && Effect  \\\\  \nimpulse &&  response  &  &   && impulse &&  response  &  &   \\\\ \\hline\n \\multicolumn{4}{c}{\\underline{Perceived complements}} &&&  \\multicolumn{4}{c}{\\underline{Perceived substitutes}} &\\\\\nSoft Drinks && Canned Tuna   && -0.0209 && Front-end-candies && Bottled Juices  && 0.0120\\\\\nSoft Drinks  && Frozen Entrees  &&-0.0182&& Soft Drinks  && Frozen Juices   && 0.0060\\\\\nCanned Tuna  &&  Canned Soup   &&  -0.0173 && Snack Crackers && Beer  && 0.0058\\\\\nCereals && Frozen Dinners  &&  -0.0104 && Cookies && Oatmeal && 0.0056\\\\\nBottled Juices && Crackers && -0.0074 && Frozen Juices && Bottled Juices && 0.0023\\\\ \\hline\n\\multicolumn{6}{l}{\\textbf{Cross-category promotion effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\  \\hline\nPromotion && Sales  &&  Effect && Promotion && Sales  &&  Effect  \\\\ \nimpulse &&  response &&   && impulse &&  response  &  &   \\\\ \\hline\nBottled Juices && Frozen Entrees &&  0.0586  && Oatmeal && Canned Tuna && -0.0214\\\\\nCheeses && Frozen Entrees &&  0.0421 && Cheeses && Cookies &&  -0.0160 \\\\\nCrackers && Frozen Entrees &&  0.0246 && Bottles Juices && Canned Tuna && -0.0158  \\\\\nFrozen Dinners && Frozen Entrees &&  0.0170&& Refrigerated Juices && Canned Tuna && -0.0128 \\\\\nSnack Crackers && Frozen Entrees &&  0.0127 && Cereals && Cheeses &&  -0.0127 \\\\ \\hline\n\n\\multicolumn{6}{l}{\\textbf{Cross-category sales effects}} & \\multicolumn{5}{l}{\\textbf{ }}\\\\   \\hline\nSales && Sales  &&  Effect && Sales && Sales  && Effect  \\\\\nimpulse &&  response  &  &  && impulse &&  response &&   \\\\  \\hline\nFront-end-candies && Soft Drinks && 0.0191 && Snack Crackers  && Oatmeal  && -0.0154 \\\\\nOatmeal && Frozen Entrees &&  0.0123 && Frozen Juices && Frozen Entrees && -0.0120 \\\\\nCanned Tuna && Crackers && 0.0094  && Cereals  && Frozen Dinners  && -0.0099 \\\\\nFront-end-candies  && Beer  && 0.0086 && Snack Crackers &&  Cookies &&  -0.0087\\\\\nSnack Crackers && Frozen Dinners && 0.0064 && Refrigerated Juices  && Canned Tuna  &&-0.0084  \\\\\\hline\n\\end{tabular}\n\\end{minipage} }\n\\end{table}\n\nTo get more insight in the sign of the cross-category effects, we summarize each IRF by the sum of the first 10 responses, and average this number over all stores. Table \\ref{Crosscateffects} reports the five  largest positive and negative cross-category effects of price, promotion and sales on sales.\n\n\\medskip\n\n\\textit{Cross-category price effects.}  We investigate whether consumers perceive categories as complements or as substitutes. Complementary and substitution effects occur between categories because they are consumed together or separately.  Following the standard economic definition \\citep{Pashigian98}, complements are defined as goods having a negative cross-price elasticity, whereas substitutes are defined as goods having a positive cross-price elasticity. \n We find evidence of two important drivers of cross-category price effects: consumption relatedness and the budget constraint. \n \nAs an example of consumption relatedness, consider Soft Drink prices and Frozen Juices.  An increase in Soft Drink prices makes consumers spend more on other drinks as a compensation, in particular Frozen Juices (see Table \\ref{Crosscateffects}). The joint dynamic effect of a one standard deviation price impulse of Soft Drinks on the sales response growth of Frozen Juices is depicted in Figure \\ref{IRF_PRICE} for the first three stores in the data set. Note that the instantaneous effect is estimated as exactly zero since the Sparse VAR puts the corresponding effect in the $\\widehat\\Sigma$ matrix to zero. We see a sharp increase in Frozen Juices sales growth one week after the soft drink price increase, indicating substitution.  However, the next two weeks, sales growth of Frozen Juices slows down, which could indicate stockpiling behavior \\citep{Gangwar13}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14.1cm]{IRFS_SLStoPRICE}\n\\end{center}\n\\caption{Impulse response function: response of frozen juices sales growth to a one standard deviation impulse in the price of soft drinks. \\label{IRF_PRICE}}\n\\end{figure}\n\nAnother example of consumption relatedness is Soft Drinks and Frozen Entrees.  As can be seen from Table \\ref{Crosscateffects}, we find a strong negative effect of Soft Drink prices on Frozen Entrees.  This might be due to the fact that Soft Drinks and Frozen Entrees are consumed together. We do not find the opposite effect of price changes in Frozen Entrees on the sales of Soft Drinks. This asymmetry arises because Soft Drinks is a destination category (high influence), while Frozen Entrees is a convenience category (highly responsiveness).  \n\n\n\n\nConcerning the budget constraint, prominent cross-category price effects are observed for Soft Drinks and Cereals, both destination categories.  Soft Drinks and Cereals account for a relatively large proportion of the expenditures of US families (respectively 22\\% and 14\\% of spending on food and drinks, see Table \\ref{Categories}), which indicates that the budget constraint is an important source of cross-category effects. \n\n\\medskip\n\n\\textit{Cross-category promotion effects.}\nThe results in Table \\ref{Crosscateffects} indicate that branding and promotion intensity are important drivers of cross-category promotion effects.\nConcerning branding, cross-category promotion effects are observed for categories that share brands such as Frozen Dinners and Frozen Entrees (e.g. the frozen prepared foods brand ``Stouffer's\").\nConcerning promotion intensity, prominent cross-category promotion effects are observed for categories in which a high percentage of the SKUs is promoted, such as Cheeses and Bottled Juices (respectively 28\\% and 26\\% of SKUs, on average, are promoted in our data.) A promotion impulse in such categories might either trigger join consumption (e.g. Bottled Juices and Frozen Entrees),  or deter consumption (e.g. Cheeses and Cookies).\n\n\n\\medskip\n\n\\textit{Cross-category sales effects.} \nIn Table \\ref{Crosscateffects}, we find evidence of two important drivers of cross-category effects of sales on sales: affinity in consumption and the budget constraint. \nProminent cross-category sales effects occur because of affinity in consumption. Some categories are jointly consumed  towards a common goal, such as Front-end-candies and Soft Drinks\/Beer  (for a light meal); while others such as Snack Crackers and Cookies are purchased as replacements since consumers might perceive them to have a similar functionality.\nConcerning the budget constraint, we find some cross-category sales effects between seemingly unrelated categories such as Refrigerated Juices and Canned Tuna.\n\n\n\\medskip\n\n\nImportantly, the results from Table \\ref{Crosscateffects} are in line with our findings on category influence and responsiveness. Destination categories such as Soft Drinks, Cereals and Cheeses mainly influence sales in other categories through their price, promotion or sales impulses. Convenience categories such as Frozen Entrees and Frozen Dinners are more responsive to changes in other categories.  Routine categories, such as Cookies, are moderately influential and moderately responsive, while occasional categories, such as Oatmeal, are highly responsive. \n\n\n\\subsection{Robustness checks}\n\\textit{Alternative penalty function.} We investigate the robustness of the results to the choice of the penalty function. We re-estimate the models using the Sparse VAR with elastic net instead of the grouplasso penalty (a short explanation of the elastic net is given in Section 4). The managerial insights obtained by Sparse VAR with either grouplasso or elastic net are very similar. \nSimilarities are that \n(i) within-category effects are more common and larger in magnitude than cross-category effects, \n(ii) destination categories such as Cheeses and Cereals are very influential,\n(iii) convenience categories such as Frozen Entrees, and occasional categories such as Crackers are very responsive\n(iv) routine categories such as Bottled Juices, Refrigerated Juices and Cookies are both influential and responsive\n(v) the most prominent cross-category  effects of price, promotion and sales on sales are highly overlapping.\n\n\\medskip\n\n\\textit{Alternative data period.} We also check the performance of the Sparse VAR on the post-1994 data. Retailers made extensive use of ``pay-for-performance\" price promotions that are not fully reflected in the Dominick's database. The data generating process might have changed in this period. Therefore, we should not assume constant parameter values. We re-estimate the model on the post-1994 data (data from October 1995 until May 1997) and verify its performance. In the post-1994 period, similar conclusions can be drawn with respect to within versus cross-category effects and category influence and responsiveness.\nSome differences are observed in the post-1994 period concerning the impulse response functions. These differences occur due to an altered strategy concerning average pricing and promotion intensity in the 17 product categories in the post-1994 period compared to the 1993-1994 period. Detailed results are available from the authors upon request.\n\n\\medskip\n\n\\textit{Alternative sparsity parameter selection.}  Our results are based on the BIC to select the penalty parameters.  We also ran the analysis using AIC as a selection criterion for the penalty function.  While the model selected by AIC are slightly less sparse, the substantive insights do not change. \n\n\\subsection{Forecast Performance}\nAlthough prediction is not the main goal of the proposed methodology, we deem it important to show that the Sparse VAR can compete with other methods in terms of prediction accuracy. We estimate model \\eqref{eq: application model} for each store and perform a forecast exercise (cfr. Section 4), using a rolling window of length $S=67$. One-step-ahead forecasts of sales for each product category  are computed for $t=S,\\ldots,T-1$, with $T=77$. The same estimation methods as in Section 4 are used.\n\nResults on the sales predictions are summarized in Table \\ref{Forecasts} by the Mean Absolute Forecast Error (MAFE), averaged across time and over the 17 product categories and 15 stores.  The MAFE should be seen as a measure of forecast accuracy, not as a measure of managerial relevance of the obtained results.\nThe variable selection methods Sparse VAR, 1-step and Iterative Restricted LS perform, on average, better than the methods that don't perform variable selection. This indicates that sparsity improves prediction accuracy. Sparse VAR and Iterative Restricted LS achieve the best forecasting performance. A Diebold-Mariano test \\citep{Diebold95} confirms that latter two methods significantly outperform the other methods. We conclude that the improvement in interpretability of the model obtained by Sparse VAR, as discussed in the previous section, does not come at the cost of lower forecast performance.\n\n\\linespread{1.2}\n\\begin{table}\n\\begin{center}\n\\caption{Mean Absolute Forecast Error (MAFE) for category-specific sales, averaged over the 15 stores and the 17 product categories. $P$-values of a Diebold-Mariano test comparing the Sparse VAR to its alternatives are indicated between parentheses. \\label{Forecasts}}\n\\small\n\\begin{tabular}{lcccccc}\n  \\hline\n   \\rule{0pt}{3ex} & Sparse VAR & & \\multicolumn{2}{c}{Restricted LS} &  \\multicolumn{2}{c}{Bayesian Methods}     \\\\\n  \\rule{0pt}{3ex} &   & LS  & 1-step  & Iterative & Minnesota & NIW     \\\\\n  \\hline\nMAFE & 736.80 &  $\\underset{(<0.01)}{1298.54}$   & $\\underset{(<0.01)}{784.96}$   & $\\underset{(0.38)}{734.82}$ & $\\underset{(<0.01)}{875.47}$ & $\\underset{(<0.01)}{1078.03}$     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\linespread{1.5}\n\n\\section{Discussion}\n\nThis paper presents a Sparse VAR methodology to detect the inter-relationships in a large product category network.  In the cross-category demand effects application, we detect an important number of cross-category demand effects for a large number of categories. We find that categories have asymmetric roles: While destination categories are more influential, convenience categories are more responsive.\nWe identify main perceived cross-category effects but also detect cross-category effects between categories that are not directly related at first sight. Hence, the need to study -- potentially a large number of  -- product categories simultaneously. While cross-category effects are prevalent, many of them are still absent, calling for a sparse estimation procedure that succeeds in highlighting the main inter-relationships in the product category network.\n\nWe identify category influence and responsiveness in our cross-category demand effects application using aggregate store level data. Other cross-category studies, such as \\cite{Russell97,Ainslie98,Russell99,Russell00,Elrod02} use market basket data. Since the availability and use of such market basket data pose difficulties to managers, they rarely use market basket data for category analysis \\citep{Shankar2014}. As managerial decisions are often made at the category level, managers prefer to work with more readily available aggregate store level data. Hence, using aggregate category store level is managerially relevant \\citep{ailawadi:09, leeflang:12}.\n\nA first limitation of our approach is that we use aggregate category data, which might lead to biased estimates when there is heterogeneity on the SKU level \\citep{Dekimpe00}. Second, our model does not allow to estimate cross-category effects on the individual consumer level. Insights into the behavior of consumers are revealed using market basket data, which requires a very different modeling approach. Despite these limitations, aggregate category data are highly relevant from the perspective of category management within the store.\n\nAn important advantage of the Sparse VAR is that it overcomes the dimensionality problem -- it results in a parsimonious model with minimal structural constraints.  We show that this leads to more accurate estimation and prediction results as compared to standard Least Squares methods.\nIf the researcher wishes to restrict some of the parameters to zero a priori, using marketing theory, this is of course still possible to implement with the Sparse VAR. The same holds for the reverse, i.e.\\ forcing some variables to be included in the model, which can be done by adjusting the penalty on the regression coefficients in \\eqref{mincrit}.\n\nThe methodology presented in this paper is relevant in a variety of other settings. First, Sparse VAR  can be used to study competitive demand effects across many competitors. The VAR is ideal for measuring competitive effects since it is able to capture own- and cross-elasticity of sales to both pricing and marketing spending \\citep{Srinivasan04, Horvath08}.\nTypically only three competitors are included  in such studies, while using the Sparse VAR allows for a much larger number to be included. Second, in the field of international marketing research there is an increased interest in studying cross-country spill-over effects, as for example in \\cite{Albuquerque07}, \\cite{VanEverdingen09} and \\cite{Kumar02}. Every country that is added to the data set leads to an increase in the number of cross-country parameters to be estimated. Using the proposed methodology, a large VAR model could be built which allows spill-over effects between many countries.  Finally, the Market Response Model could be extended with data on online word of mouth or online search, which are now readily available. Especially in the Big Data era, most companies collect an abundance of variables \\citep{BigData2013}, such that large VAR models will become even larger as more granular data become available. \n\n\\bigskip \\noindent\n{\\bf Acknowledgments.} The authors thank the Editors, Shankar Ganesan and Murali K. Mantrala, and two anonymous referees for their valuable comments that have improved the paper significantly. Financial support from the FWO (Research Foundation Flanders) is also gratefully acknowledged (FWO, contract number 11N9913N).\n\n\n\\begin{appendices}\n\\numberwithin{equation}{section}\n\\section{Penalized Likelihood Estimation}\n\\noindent\nWe iteratively solve the minimization problem \\eqref{mincrit} for $\\beta$ conditional on $\\Omega$ and then for $\\Omega$ conditional on $\\beta$.\n\n\\medskip \\noindent\n{\\it Solving for $\\beta|\\Omega$:}  When $\\Omega$ is fixed, the minimization problem in \\eqref{mincrit} is equivalent to minimizing\n\\begin{equation}\\label{mincritbeta}\n\\hat{\\beta}|\\Omega = \\underset{\\beta}{\\operatorname{argmin}} \\frac{1}{n} (\\tilde{y}-\\tilde{X} \\beta)^{\\prime} (\\tilde{y}-\\tilde{X} \\beta) + \\lambda_1 \\sum_{g=1}^{G} ||\\beta_g||_2 \\, ,\n\\end{equation}\nwhere $\\tilde{y}= Py$, $\\tilde{X}=PX$, and $P$ is a matrix such that $P^{\\prime}P=\\tilde{\\Omega}$. \nThe transformation of the data to $\\tilde{y}$ and $\\tilde{X}$ ensures that the resulting model has uncorrelated and homoscedastic error terms. The above minimization problem is convex if $\\Omega$ is nonnegative definite. The minimization problem is equivalent to the groupwise lasso of \\cite{Yuan06}, implemented in the R package \\verb+grplasso+ \\citep{Rgrplasso}.\n\n\n\\noindent\n{\\it Solving for $\\Omega|\\beta$:} When  $\\beta$ is fixed, the minimization problem in  \\eqref{mincrit} reduces to\n\\begin{equation}\\label{mincritOmega}\n\\hat{\\Omega}|\\beta = \\underset{\\Omega}{\\operatorname{argmin}} \\, \\frac{1}{n} (y-X \\beta)^{\\prime} \\tilde{\\Omega} (y-X \\beta)- \\log|\\Omega| + \\lambda_2 \\sum_{k \\neq k'} |\\Omega_{kk'}| \\, ,\n\\end{equation}\nwhich corresponds to penalized covariance estimation. Using  the glasso algorithm\nof \\cite{Friedman07}, available in the R package \\verb+glasso+ \\citep{Rglasso},  the optimization problem in \\eqref{mincritOmega}\nis solved.\n\nWe start the algorithm by taking $\\widehat{\\Omega}=I_q$ and iterate until convergence. We iterate until $max_s |\\hat{\\beta}_{s,i}-\\hat{\\beta}_{s,i-1}|<\\epsilon$, with $\\hat{\\beta}_{s,i}$ the $s^{th}$ parameter estimate in iteration $i$ (same for $\\hat{\\Omega}$) and the tolerance $\\epsilon$ set to $10^{-3}$. \n\n\\paragraph{Selecting the Sparsity Parameters and the order of the VAR}\nWe first determine the optimal values of $\\lambda_1$ and $\\lambda_2$ for a fixed value of $p$, the order of the VAR. The sparsity parameters $\\lambda_1$ and $\\lambda_2$ are selected according to a minimal Bayes Information Criterion (BIC).\nIn the iteration step where $\\beta$ is estimated conditional on $\\Omega$,  we solve  \\eqref{mincritbeta} over a range of values for $\\lambda_1$ and select the one with lowest value of\n\\begin{equation}\\label{eq: BICbeta}\nBIC_{\\lambda_1} = -2 \\log L_{\\lambda_1} + k_{\\lambda_1} \\log(n),\n\\end{equation}\nwhere $L_{\\lambda_1}$ is the estimated likelihood, corresponding to the first term in \\eqref{mincritbeta},  using sparsity parameter $\\lambda_1$. Furthermore, $k_{\\lambda_1}$ is the number of non-zero estimated regression coefficients  and $n$ the number of observations.\nSimilarly, for selecting $\\lambda_2$, we use the BIC given by\n\\begin{equation}\\label{eq: BIComega}\nBIC_{\\lambda_2} = -2 \\log L_{\\lambda_2} + k_{\\lambda_2} \\log(n) \\, .\n\\end{equation}\nFinally, we select the order $p$ of the VAR. We estimate the VAR for different values of $p$. The optimal values of  $\\lambda_1$ and $\\lambda_2$ are determined for a each of those values of $p$. We select the order $p$ of the VAR  using BIC:\n\\begin{equation}\\label{eq: BICp}\nBIC_{(p, \\lambda_1(p), \\lambda_2(p))} = -2 \\log L_{(p, \\lambda_1(p), \\lambda_2(p))} + k_{(p, \\lambda_1(p), \\lambda_2(p))} \\log(n) \\, , \n\\end{equation}\nwhere $L_{(p, \\lambda_1(p), \\lambda_2(p))}$ and $k_{(p, \\lambda_1(p), \\lambda_2(p))}$ depend on the value $p$ and the optimally chosen values of $\\lambda_1(p)$ and $\\lambda_2(p)$ for that specific value of $p$.\n\\end{appendices}\n\n\n\\linespread{1}\n\\bibliographystyle{asa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Primitive trace map, local injectivity}\n\nLet $(M,g)$ be a smooth closed Riemannian Anosov manifold such as a manifold of negative sectional curvature \\cite{Anosov-67}. Recall that this means that there exists a continuous flow-invariant splitting of the tangent bundle to the unit tangent bundle $\\mathcal{M} := SM$:\n\\[\nT\\mathcal{M} = \\mathbb{R} X \\oplus E_s \\oplus E_u,\n\\]\nsuch that:\n\\begin{equation}\n\\label{equation:anosov}\n\\begin{array}{l}\n\\forall t \\geq 0, \\forall v \\in E_s, ~~ |\\dd\\varphi_t(v)| \\leq Ce^{-t\\theta}|v|, \\\\\n\\forall t \\leq 0, \\forall v \\in E_u, ~~ |\\dd\\varphi_t(v)| \\leq Ce^{-|t|\\theta}|v|,\n\\end{array}\n\\end{equation}\nwhere $(\\varphi_t)_{t \\in \\mathbb{R}}$ is the geodesic flow on $\\mathcal{M}$, and the constants $C,\\theta > 0$ are uniform and the metric $|\\cdot|$ is arbitrary. \n\nLet $\\E \\rightarrow M$ be a smooth Hermitian vector bundle. We denote by $\\mathcal{A}_{\\E}$ the affine space of smooth unitary connections on $\\E$ and $\\mathbb{A}_{\\E}$ the moduli space of connections up to gauge-equivalence, namely a point $\\mathfrak{a} \\in \\mathbb{A}_{\\E}$ is an orbit $\\mathfrak{a} := \\left\\{p^*\\nabla^{\\E} ~|~p \\in C^\\infty(M,\\mathrm{U}(\\E))\\right\\}$ of gauge-equivalent connections, where $\\nabla^{\\E} \\in \\mathfrak{a}$ is arbitrary. We let $\\mathcal{C} = \\left\\{c_1,c_2,...\\right\\}$ be the set of free homotopy classes of loops on $M$ which is known to be in one-to-one correspondence with closed geodesics \\cite{Klingenberg-74}. More precisely, given $c \\in \\mathcal{C}$, there exists a unique closed geodesic $\\gamma_g(c) \\subset M$ in the class $c$. It will be important to make a difference between \\emph{primitive} and \\emph{non-primitive} homotopy classes (resp. closed geodesics): a free loop is said to be primitive if it cannot be homotoped to a certain power (greater or equal than $2$) of another free loop. The set of primitive classes defines a subset $\\mathcal{C}^\\sharp = \\left\\{c_1^\\sharp,c_2^\\sharp,...\\right\\} \\subset \\mathcal{C}$.\n\nGiven a class $\\mathfrak{a} \\in \\mathbb{A}_{\\E}$, a unitary connection $\\nabla^{\\E} \\in \\mathfrak{a}$ and an arbitrary point $x_{c^\\sharp} \\in \\gamma_g(c^\\sharp)$ (for some $c^\\sharp \\in \\mathcal{C}^\\sharp$), the parallel transport $\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp) \\in \\mathrm{U}(\\E_{x_{c^\\sharp}})$, starting at $x_{c^\\sharp}$, with respect to $\\nabla^{\\E}$ and along $\\gamma_g({c^\\sharp})$ depends on the choice of representative $\\nabla^{\\E} \\in \\mathfrak{a}$ since two gauge-equivalent connections have conjugate holonomies. However, the trace does not depend on a choice of $\\nabla^{\\E} \\in \\mathfrak{a}$ and the \\emph{primitive trace map}:\n\\begin{equation}\n\\label{equation:trace}\n\\mathcal{T}^\\sharp : \\mathbb{A}_{\\E} \\ni \\mathfrak{a} \\mapsto \\left(\\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp_1)\\right), \\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c^\\sharp_2)\\right), ...\\right) \\in \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nis therefore well-defined. Observe that the data of the primitive trace map is a rather weak information: in particular, it is \\emph{not} (\\emph{a priori}) equivalent to the data of the conjugacy class of the holonomy along each closed geodesic (and the latter is the same as the non-primitive trace map, where one considers \\emph{all} closed geodesics). One of the main results of this paper is the following:\n\n\\begin{theorem}\n\\label{theorem:injectivity}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$ and let $\\E \\rightarrow M$ be a smooth Hermitian vector bundle. Let $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$ be a \\emph{generic} point. Then, the primitive trace map is \\emph{locally injective} near $\\mathfrak{a}_0$.\n\\end{theorem}\n\nBy \\emph{local injectivity}, we mean the following: there exists $N \\in \\mathbb{N}$ (independent of $\\mathfrak{a}_0$) such that $\\mathcal{T}^\\sharp$ is locally injective in the $C^N$-quotient topology on $\\mathbb{A}_{\\E}$. In other words, for any element $\\nabla^{\\E}_0 \\in \\mathfrak{a}_0$, there exists $\\varepsilon > 0$ such that the following holds; if $\\nabla_{1,2}^{\\E}$ are two smooth unitary connections such that $\\|p_i^*\\nabla_{i}^{\\E} - \\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$ for some $p_i \\in C^\\infty(M,\\mathrm{U}(\\E))$, and $\\mathcal{T}^\\sharp(\\nabla_1^{\\E}) = \\mathcal{T}^\\sharp(\\nabla_2^{\\E})$, then $\\nabla_1^{\\E}$ and $\\nabla_2^{\\E}$ are gauge-equivalent.\n\nWe say that a point $\\mathfrak{a}$ is \\emph{generic} if it enjoys the following two features: \n\n\\begin{itemize}\\label{def:generic}\n\\item[\\textbf{(A)}]  $\\mathfrak{a}$ is \\textbf{opaque}. By definition (see \\cite[Section 5]{Cekic-Lefeuvre-20}), this means that for all $\\nabla^{\\E} \\in \\mathfrak{a}$, the parallel transport map along geodesics does not preserve any non-trivial subbundle $\\mathcal{F} \\subset \\E$ (i.e. $\\mathcal{F}$ is preserved by parallel transport along geodesics if and only if $\\mathcal{F} = \\left\\{0\\right\\}$ or $\\mathcal{F} = \\E$). This was proved to be equivalent to the fact that the Pollicott-Ruelle resonance at $z=0$ of the operator $\\mathbf{X} := \\pi^* \\nabla^{\\mathrm{End}}_X$ has multiplicity equal to $1$, with resonant space $\\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$ (here $\\pi : SM \\rightarrow M$ is the projection; $\\nabla^{\\mathrm{End}}$ is the induced connection on the endomorphism bundle, see \\S\\ref{ssection:connections} for further details);\n\n\n\\item[\\textbf{(B)}] $\\mathfrak{a}$ has \\textbf{solenoidally injective generalized X-ray transform} $\\Pi^{\\mathrm{End}(\\E)}_1$ on twisted $1$-forms with values in $\\mathrm{End}(\\E)$. This last assumption is less easy to describe in simple geometric terms: roughly speaking, the X-ray transform is an operator of integration of symmetric $m$-tensors along closed geodesics. For vector-valued symmetric $m$-tensors, this might not be well-defined, and one needs a more general (hence, more abstract) definition involving the residue at $z=0$ of the meromorphic extension of the family $\\mathbb{C} \\ni z \\mapsto (-\\mathbf{X}-z)^{-1}$, see \\S\\ref{section:twisted}. \n\\end{itemize}\nIt was shown in previous articles \\cite{Cekic-Lefeuvre-20,Cekic-Lefeuvre-21-2} that in dimension $n \\geq 3$, properties \\textbf{(A)} and \\textbf{(B)} are satisfied on an open dense subset $\\omega \\subset \\mathbb{A}_{\\E}$ with respect to the $C^N$-quotient topology.\\footnote{More precisely, there exists $N \\in \\mathbb{N}$ and a subset $\\Omega \\subset \\mathcal{A}_{\\E}$ of the (affine) Fr\\'echet space of smooth affine connections on $\\E$ such that $\\omega = \\pi_{\\E}(\\Omega)$ (where $\\pi_{\\E} : \\mathcal{A}_{\\E} \\rightarrow \\mathbb{A}_{\\E}$ is the projection) and\n\\begin{itemize}\n\\item $\\Omega$ is invariant by the action of the gauge-group, namely $p^*\\Omega = \\Omega$ for all $p \\in C^\\infty(M,\\mathrm{U}(\\E))$;\n\\item $\\Omega$ is open, namely for all $\\nabla^{\\E}_0 \\in \\Omega$, there exists $\\varepsilon > 0$ such that if $\\nabla^{\\E} \\in \\mathcal{A}_{\\E}$ and $\\|\\nabla^{\\E}-\\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$, then $\\nabla^{\\E} \\in \\Omega$;\n\\item $\\Omega$ is dense, namely for all $\\nabla^{\\E}_0 \\in \\mathcal{A}_{\\E}$, for all $\\varepsilon > 0$, there exists $\\nabla^{\\E} \\in \\Omega$ such that $\\|\\nabla^{\\E}-\\nabla^{\\E}_0\\|_{C^N} < \\varepsilon$;\n\\item Connections in $\\Omega$ satisfy properties \\textbf{(A)} and \\textbf{(B)}.\n\\end{itemize}} When the reference connection $\\mathfrak{a}$ satisfies only the property \\textbf{(A)} (this is the case for the product connection on the trivial bundle for instance), we are able to show a \\emph{weak local injectivity} result, see Theorem \\ref{thm:weaklocal}. \n\nWe note that the gauge class of a connection is uniquely determined from the holonomies along \\emph{all} closed loops \\cite{Barrett-91, Kobayashi-54} and that within the mathematical physics community our primitive trace map $\\mathcal{T}^\\sharp$ is known as the \\emph{Wilson loop} operator \\cite{Beasley-13, Giles-81, Loll-93, Wilson-74}. In stark contrast, our Theorem \\ref{theorem:injectivity} says that the \\emph{restriction to closed geodesics} of this operator already determines (locally) the gauge class of the connection.\n\n\\subsection{Global injectivity}\n\n\\label{ssection:global-inj}\n\nWe now mention some global injectivity results. We let $\\mathbb{A}_r := \\bigsqcup_{\\E_r \\in \\mathrm{Vect}_r(M)} \\mathbb{A}_{\\E_r}$, where the disjoint union runs over all Hermitian vector bundles $\\E_r \\in \\mathrm{Vect}_r(M)$ of rank $r$ over $M$ up to isomorphisms, and we set:\n\\[\n\\mathbb{A} := \\bigsqcup_{r \\geq 0} \\mathbb{A}_r,\n\\]\nand $\\mathrm{Vect}(M) = \\bigsqcup_{r \\geq 0} \\mathrm{Vect}_r(M)$ be the space of all topological vector bundles up to isomorphisms. A point $\\mathrm{x} \\in \\mathbb{A}$ corresponds to a pair $([\\mathcal{E}],\\mathfrak{a})$, where $[\\mathcal{E}] \\in \\mathrm{Vect}(M)$ is an equivalence class of Hermitian vector bundles and $\\mathfrak{a}$ a class of gauge-equivalent unitary connections.\\footnote{Note that if two smooth Hermitian vector bundles $\\E_1$ and $\\E_2$ are isomorphic as topological vector bundles (i.e. there exists an invertible $p \\in C^\\infty(M,\\mathrm{Hom}(\\E_1,\\E_2))$), then they are also isomorphic as Hermitian vector bundles, that is $p$ can be taken unitary; the choice of Hermitian structure is therefore irrelevant.} \n\nThe space $\\mathbb{A}$ has a natural monoid structure given by the $\\oplus$-operator of taking direct sums (both for the vector bundle part and the connection part). The primitive trace map can then be seen as a \\emph{global} (monoid) homomorphism: \n\\begin{equation}\n\\label{equation:trace-total}\n\\mathcal{T}^\\sharp : \\mathbb{A} \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nwhere $\\ell^\\infty(\\mathcal{C}^\\sharp)$ is endowed with the obvious additive structure. We actually conjecture that the generic assumption of Theorem \\ref{theorem:injectivity} is unnecessary and that the primitive trace map \\eqref{equation:trace-total} should be globally injective if $\\dim(M) \\geq 3$ and $\\dim (M)$ is odd. Let us discuss a few partial results supporting the validity of this conjecture:\n\n\\begin{enumerate}\n\t\\item In \\S\\ref{sssection:line}, we show that the primitive trace map is injective when restricted to \\emph{direct sums of line bundles} when $\\dim(M) \\geq 3$, see Theorem \\ref{theorem:sum}. Note that it was proved by Paternain \\cite{Paternain-09} that the primitive trace map restricted to line bundles $\\mathcal{T}_1^\\sharp : \\mathbb{A}_1 \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp)$ is injective when $\\dim(M) \\geq 3$. \n\n\t\\item In \\S\\ref{sssection:flat}, under the restriction of $\\mathcal{T}^\\sharp$ to \\emph{flat} connections, we show that the primitive trace map $\\mathcal{T}^\\sharp$ is globally injective, see Proposition \\ref{proposition:flat}.\n\t\n\t\\item In \\S\\ref{sssection:negative}, we also obtain a global result in negative curvature under an extra \\emph{spectral condition}, see Proposition \\ref{proposition:negative}. This condition is generic (see Appendix \\ref{appendix:ckts}) and is also satisfied by connections with \\emph{small curvature}, i.e. whose curvature is controlled by a constant depending only on the dimension and an upper bound on the sectional curvature of $(M,g)$ (see Lemma \\ref{lemma:small-curvature}).\n\t\n\t\n\t\\item In \\S\\ref{sssection:topology}, as a consequence of Corollary \\ref{corollary:iso} below, we have that the primitive trace map $\\mathcal{T}^{\\sharp}([\\mathcal{E}],\\mathfrak{a})$ allows to recover the isomorphism class $\\pi^*[\\E]$. In particular if $\\dim M$ is odd, this suffices to recover $[\\E]$, see Proposition \\ref{proposition:topology}.\n\n\\end{enumerate}\n\n\n\n\n\n\n\n\nTheorem \\ref{theorem:injectivity} is inspired by earlier work on the subject, see \\cite{Paternain-09,Paternain-12,Paternain-13,Paternain-lecture-notes,Guillarmou-Paternain-Salo-Uhlmann-16} for instance. Nevertheless, it goes beyond the aforementioned literature thanks to an \\emph{exact Liv\\v{s}ic cocycle Theorem} (see Theorem \\ref{theorem:weak-intro}), explained in the next paragraph \\S\\ref{ssection:exact}. It also belongs to a more general family of \\emph{geometric inverse results} which has become a very active field of research in the past twenty years, both on closed manifolds and on manifolds with boundary, see \\cite{Pestov-Uhlmann-05, Stefanov-Uhlmann-04,Paternain-Salo-Uhlmann-13, Uhlmann-Vasy-16,Stefanov-Uhlmann-Vasy-17,Guillarmou-17-2} among other references.\n\n\n\nTheorem \\ref{theorem:injectivity} can also be compared to a similar problem called the \\emph{marked length spectrum} (MLS) rigidity conjecture, also known as the Burns-Katok \\cite{Burns-Katok-85} conjecture. The latter asserts that if $(M,g)$ is Anosov, then the marked length spectrum\n\\begin{equation}\n\\label{equation:mls}\nL_g : \\mathcal{C} \\rightarrow \\mathbb{R}_+, ~~~L_g(c) := \\ell_g(\\gamma_g(c)),\n\\end{equation}\n(where $\\ell_g(\\gamma)$ denotes the Riemannian length of the curve $\\gamma \\subset M$ computed with respect to the metric $g$), namely the length of all closed geodesics marked by the free homotopy classes of $M$, should determine the metric up to isometry. Despite some partial answers \\cite{Katok-88, Croke-90,Otal-90,Besson-Courtois-Gallot-95,Hamenstadt-99,Guillarmou-Lefeuvre-18}, this conjecture is still widely open. Recently, Guillarmou and the second author proved a local version of the Burns-Katok conjecture \\cite{Guillarmou-Lefeuvre-18} using techniques from microlocal analysis and the theory of Pollicott-Ruelle resonances. \n\n\\subsection{Inverse Spectral problem}\n\n\nThe \\emph{length spectrum} of the Riemannian manifold $(M,g)$ is the collection of lengths of closed geodesics \\emph{counted with multiplicities}. It is said to be \\emph{simple} if all closed geodesics have distinct lengths and this is known to be a generic condition (with respect to the metric), see \\cite{Abraham-70,Anosov-82} (even in the non-Anosov case). Given $\\nabla^{\\E} \\in \\mathfrak{a}$, one can form the \\emph{connection Laplacian} $\\Delta_{\\nabla^{\\E}} := (\\nabla^{\\E})^*\\nabla^{\\E}$ (also known as the Bochner Laplacian) which is a differential operator of order $2$, non-negative, formally self-adjoint and elliptic, acting on $C^\\infty(M,\\E)$. While $\\Delta_{\\nabla^{\\E}}$ depends on a choice of representative $\\nabla^{\\E}$ in the class $\\mathfrak{a}$, its spectrum does not and there is a well-defined \\emph{spectrum map}:\n\\begin{equation}\\label{eq:spectrummap}\n\\mathcal{S} : \\mathbb{A}_{\\E} \\ni \\mathfrak{a} \\mapsto \\mathrm{spec}(\\Delta_{\\mathfrak{a}}),\n\\end{equation}\nwhere $\\mathrm{spec}(\\Delta_{\\mathfrak{a}}) = \\left\\{0 \\leq \\lambda_0(\\mathfrak{a}) \\leq \\lambda_1(\\mathfrak{a}) \\leq ...\\right\\}$ is the spectrum counted with multiplicities. Note that more generally, the spectrum map \\eqref{eq:spectrummap} can be defined on the whole moduli space $\\mathbb{A}$ (just as the primitive trace map \\eqref{equation:trace}). The trace formula of Duistermaat-Guillemin \\cite{Duistermaat-Guillemin-75, Guillemin-73} applied to $\\Delta_{\\mathfrak{a}}$ reads (when the length spectrum is simple):\n\\begin{equation}\n\\label{equation:trace-formula}\n\\lim_{t \\to \\ell(\\gamma_g(c))} \\left(t-\\ell(\\gamma_g(c))\\right) \\sum_{j \\geq 0} e^{-i \\sqrt{\\lambda_j}(\\mathfrak{a})t} = \\dfrac{\\ell(\\gamma_g(c^\\sharp)) \\Tr\\left(\\mathrm{Hol}_{\\nabla^{\\E}}(c)\\right)}{2\\pi |\\det(\\mathbbm{1}-P_{\\gamma_g(c)})|^{1\/2}} ,\n\\end{equation}\nwhere $\\sharp : \\mathcal{C} \\rightarrow \\mathcal{C}^\\sharp$ is the operator giving the primitive orbit associated to an orbit; $P_\\gamma$ is the Poincar\\'e map associated to the orbit $\\gamma$ and $\\ell(\\gamma)$ its length. Theorem \\ref{theorem:injectivity} therefore has the following straightforward consequence:\n\n\\begin{corollary}\n\\label{corollary:spectral}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$ \\emph{with simple length spectrum}. Then:\n\\begin{itemize}\n\\item Let $\\E \\rightarrow M$ be a smooth Hermitian vector bundle and $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$ be a generic point. Then, the spectrum map $\\mathcal{S}$ is locally injective near $\\mathfrak{a}_0$.\n\\item The spectrum map $\\mathcal{S}$ is also globally injective when restricted to the cases \\emph{(1)-(4)} of the previous paragraph \\S\\ref{ssection:global-inj}.\n\\end{itemize}\n\\end{corollary}\n\nThis corollary simply follows from Theorem \\ref{theorem:injectivity} by observing that under the simple length spectrum assumption, the primitive trace map can be recovered from the equality \\eqref{equation:trace-formula}. Corollary \\ref{corollary:spectral} is analogous to the Guillemin-Kazhdan \\cite{Guillemin-Kazhdan-80,Guillemin-Kazhdan-80-2} rigidity result in which a potential $q \\in C^\\infty(M)$ is recovered from the knowledge of the spectrum of $-\\Delta_g + q$ (see also \\cite{Croke-Sharafutdinov-98,Paternain-Salo-Uhlmann-14-1}). As far as the connection Laplacian is concerned, it seems that Corollary \\ref{corollary:spectral} is the first positive result in this direction. Counter-examples were constructed by Kuwabara \\cite{Kuwabara-90} using the Sunada method \\cite{Sunada-85} but on coverings of a given Riemannian manifolds; hence the simple length spectrum condition is clearly violated. Up to our knowledge, it is also the first positive general result in an inverse spectral problem on a closed manifold of dimension $> 1$ with an \\emph{infinite} gauge-group. \n\nThis gives hope that similar methods could be used in the classical problem of recovering the isometry class of a metric from the spectrum of its Laplace-Beltrami operator \\emph{locally} (similarly to a conjecture of Sarnak for planar domains \\cite{Sarnak-90}). Such a result was already obtained in a neighbourhood of negatively-curved locally symmetric spaces by Sharafutdinov \\cite{Sharafutdinov-09}. See also \\cite{Croke-Sharafutdinov-98} for the weaker deformational spectral rigidity results or \\cite{DeSimoi-Kaloshin-Wei-17, Hezari-Zelditch-19} for recent results in the plane.\n\n\\subsection{Exact Liv\\v{s}ic cocycle theorem}\n\n\\label{ssection:exact}\n\nThe main ingredient in the proof of Theorem \\ref{theorem:injectivity} is the following Liv\\v{s}ic-type result in hyperbolic dynamical systems, which may be of independent interest. It shows that the cohomology class of a unitary cocycle over a transitive Anosov flow is determined by its trace along primitive periodic orbits. We phrase it in a somewhat more general context where we allow non-trivial vector bundles\n\n\\begin{theorem}\n\\label{theorem:weak-intro}\nLet $\\mathcal{M}$ be a smooth manifold endowed with a smooth transitive Anosov flow $(\\varphi_t)_{t \\in \\mathbb{R}}$. For $i\\in \\left\\{1,2\\right\\}$, let $\\E_i \\rightarrow \\mathcal{M}$ be a Hermitian vector bundle over $\\mathcal{M}$ equipped with a unitary connection $\\nabla^{\\mathcal{E}_i}$, and denote by $C_i(x,t) : (\\E_i)_x \\rightarrow (\\E_i)_{\\varphi_t(x)}$ the parallel transport along the flowlines with respect to $\\nabla^{\\E_i}$. If the connections have \\emph{trace-equivalent holonomies} in the sense that for all \\emph{primitive} periodic orbits $\\gamma$, one has\n\\begin{equation}\n\\label{equation:trace-intro}\n\\Tr\\left(C_1(x_\\gamma,\\ell(\\gamma))\\right) = \\Tr\\left(C_2(x_\\gamma,\\ell(\\gamma))\\right),\n\\end{equation}\nwhere $x_\\gamma \\in \\gamma$ is arbitrary and $\\ell(\\gamma)$ is the period of $\\gamma$, then the following holds: there exists $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2,\\E_1))$ such that for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$,\n\\begin{equation}\n\\label{equation:cohomologous}\nC_1(x,t) = p(\\varphi_t x) C_2(x,t) p(x)^{-1}.\n\\end{equation}\n\\end{theorem}\n\nIn the vocabulary of dynamical systems, note that each unitary cocycle is given by parallel transport along some unitary connection and \\eqref{equation:cohomologous} says that the cocycles induced by parallel transport are \\emph{cohomologous}. In particular, in the case of the trivial principal bundle $\\mathrm{U}(r) \\times \\mathcal{M} \\rightarrow \\mathcal{M}$ our theorem can be restated just in terms of $\\mathrm{U}(r)$-cocycles. Note that the bundles $\\E_1$ and $\\E_2$ could be \\emph{a priori} distinct (and have different ranks) but Theorem \\ref{theorem:weak-intro} shows that they are actually isomorphic:\n\n\\begin{corollary}\n\\label{corollary:iso}\nLet $\\E_1,\\E_2 \\rightarrow \\mathcal{M}$ be two Hermitian vector bundles equipped with respective unitary connection $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$. If the traces of the holonomy maps agree as in \\eqref{equation:trace-intro}, then $\\E_1 \\simeq \\E_2$ are isomorphic.\n\n\\end{corollary}\n\nTheorem \\ref{theorem:weak-intro} has other geometric consequences which are further detailed in \\S\\ref{ssection:intro}. Liv\\v{s}ic-type theorems have a long history in hyperbolic dynamical systems going back to the seminal paper of Liv\\v{s}ic \\cite{Livsic-72} and appear in various settings. They were both developed in the Abelian case i.e. for functions (see \\cite{Livsic-72,DeLaLlave-Marco-Moryon-86, Lopes-Thieullen-05,Guillarmou-17-1,Gouezel-Lefeuvre-19} for instance) and in the cocycle case.\n\nSurprisingly, we could not locate any result such as Theorem \\ref{theorem:weak-intro} in the literature. The closest works (in the discrete-time case) are that of Parry \\cite{Parry-99} and Schmidt \\cite{Schmidt-99} which mainly inspired the proof of Theorem \\ref{theorem:weak-intro}. Nevertheless, when considering compact Lie groups, Parry's and Schmidt's results seem to be weaker as they need to assume that the conjugacy classes of the cocycles agree (and not only the traces) and that a certain additional cocycle is transitive in order to derive the same conclusion. The literature is mostly concerned with the discrete-time case, namely hyperbolic diffeomorphisms: in that case, a lot of articles are devoted to studying cocycles with values in a non-compact Lie group (and sometimes satisfying a ``slow-growth\" assumption), see \\cite{DeLaLlave-Windsor-10,Kalinin-11, Sadovskaya-13, Sadvoskaya-17, Avila-Kocsard-Liu-18}. One can also wonder if Theorem \\ref{theorem:weak-intro} could be proved in the non-unitary setting. Other articles such as \\cite{Nitica-Torok-95,Nitica-Torok-98,Nicol-Pollicott-99,Walkden-00,Pollicott-Walkden-01} seem to have been concerned with regularity issues on the map $p$, namely bootstrapping its regularity under some weak \\emph{a priori} assumption (such as measurability only). Let us also point out at this stage that some regularity issues will appear while proving Theorem \\ref{theorem:weak-intro} but this will be bypassed by the use of a recent regularity statement \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20} in hyperbolic dynamics.\n\n\n\\subsection{Organization of the paper}\n\nThe paper is divided in three parts:\n\n\\begin{itemize}\n\n\\item First of all, we prove in Section \\S\\ref{section:livsic} the \\emph{exact Liv\\v{s}ic cocycle} Theorem \\ref{theorem:weak-intro} for general Anosov flows showing that the cohomology class of a unitary cocycle is determined by its trace along closed orbits. The proof is based on the introduction of a new tool which we call \\emph{Parry's free monoid}, denoted by $\\mathbf{G}$, and formally generated by orbits homoclinic to a given closed orbit. We show that any unitary connection induces a unitary representation of the monoid $\\mathbf{G}$ and that trace-equivalent connections have the same character; we can then apply tools from representation theory to conclude. We believe that this notion could be used in other contexts. It is reinvested in a companion paper \\cite{Cekic-Lefeuvre-21-3} to study \\emph{transparent manifolds}, namely Riemannian manifolds with trivial holonomy along closed geodesics.\n\n\\item In subsequent sections, we develop a microlocal framework, based on the theory of Pollicott-Ruelle resonances. We define a notion of \\emph{generalized X-ray transform with values in a vector bundle} which is mainly inspired by \\cite{Guillarmou-17-1,Guillarmou-Lefeuvre-18,Gouezel-Lefeuvre-19}. In \\S\\ref{section:geometry}, we relate the geometry of the moduli space of gauge-equivalent connections with the leading Pollicott-Ruelle resonance of a certain natural operator (called the mixed connection).\n\n\\item Eventually, the main results such as Theorem \\ref{theorem:injectivity} are proved in \\S\\ref{section:proofs}, where we also deduce the global properties of $\\mathcal{T}^\\sharp$ involving line bundles, flat bundles, negatively curved base manifolds, and the topology of bundles.\n\n\n\\end{itemize}\nSome technical preliminaries are provided in Section \\S\\ref{section:tools}. \n\n\\subsection{Perspectives}\n\nWe intend to pursue this work in different directions:\n\n\\begin{itemize}\n\n\\item We expect to be able to show a stability estimate version of Theorem \\ref{theorem:injectivity}, namely that the distance between connections up to gauge-equivalence is controlled (at least locally) by the distance between their images under the primitive trace map $\\mathcal{T}^\\sharp$. We believe that such an estimate should prove that the moduli space of connections up to gauge-equivalence $\\mathbb{A}_{\\E}$ (locally) embeds naturally via the primitive trace map as a closed topological (H\\\"older-continuous) Banach submanifold of $\\ell^\\infty(\\mathcal{C}^\\sharp)$.\n\n\\item We believe that the representation-theoretic approach developed in \\S\\ref{section:livsic} using Parry's free monoid $\\mathbf{G}$ could be pushed further. In a subsequent article \\cite{Cekic-Lefeuvre-21-3}, we will study the particular case of the Levi-Civita connection on the tangent bundle of Anosov manifolds and try to classify the possible images $\\rho(\\mathbf{G}) \\subset \\mathrm{O}(n)$. This problem is intimately connected to the dynamics of the frame flow.\n\n\\item Eventually, the arguments developed in \\S\\ref{section:livsic} mainly rely on the use of homoclinic orbits; to these orbits, we will associate a notion of \\emph{length} which is well-defined as an element of $\\mathbb{R}\/T_\\star\\mathbb{Z}$, for some real number $T_\\star > 0$. We believe that, similarly to the set of periodic orbits where one defines the \\emph{Ruelle zeta function}\n\\[\n\\zeta(s) := \\prod_{\\gamma^\\sharp \\in \\mathcal{G}^\\sharp} \\left(1 - e^{-s \\ell(\\gamma^\\sharp)}\\right), \n\\]\nwhere the product runs over all primitive periodic orbits $\\gamma^\\sharp \\in \\mathcal{G}^\\sharp$ (and $\\ell(\\gamma^\\sharp)$ denotes the orbit period) and shows that this extends meromorphically from $\\left\\{\\Re(s) \\gg 0 \\right\\}$ to $\\mathbb{C}$ (see \\cite{Giulietti-Liverani-Pollicott-13,Dyatlov-Zworski-16}), one could also define a complex function for homoclinic orbits by means of a Poincar\\'e series (rather than a product). It should be a consequence of \\cite[Theorem 4.15]{Dang-Riviere-20} that this function extends meromorphically to $\\mathbb{C}$. It might then be interesting to compute its value at $0$; the latter might be independent of the choice of representatives for the length of homoclinic orbits (two representatives differ by $mT_\\star$, for some $m \\in \\mathbb{Z}$) and could be (at least in some particular cases) an interesting topological invariant as for the Ruelle zeta function on surfaces, see \\cite{Dyatlov-Zworski-17}.  \\\\\n\n\\end{itemize} \n\n\\noindent \\textbf{Acknowledgement:} M.C. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 725967). We warmly thank Yannick Guedes Bonthonneau, Yann Chaubet, Colin Guillarmou, Julien March\\'e, Gabriel Paternain, Steve Zelditch for fruitful discussions. We also thank S\\'ebastien Gou\\\"ezel, Boris Kalinin, Mark Pollicott, Klaus Schmidt for answering our questions on Liv\\v{s}ic theory, and Nikhil Savale for providing us with the reference \\cite{Guillemin-73}. Special thanks to Danica Kosanovi\\'c for helping out with the topology part.\n\n\n\\section{Setting up the tools}\n\n\\label{section:tools}\n\n\\subsection{Microlocal calculus and functional analysis} \n\nLet $\\mathcal{M}$ be a smooth closed manifold. Given a smooth vector bundle $\\E \\rightarrow \\mathcal{M}$, we denote by $\\Psi^m(\\mathcal{M}, \\E)$ the space of pseudodifferential operators of order $m$ acting on $\\E$. When $\\E$ is the trivial line bundle, such an operator $P \\in \\Psi(\\mathcal{M})$ can be written (up to a smoothing remainder) in local coordinates as\n\\begin{equation}\n\\label{equation:quantization}\nPf(x) = \\int_{\\mathbb{R}^{n}} \\int_{\\mathbb{R}^{n}} e^{i\\xi \\cdot(x-y)}p(x,\\xi)f(y)dyd\\xi,\n\\end{equation}\nwhere $f$ is compactly supported in the local patch and $p \\in S^m(T^*\\mathbb{R}^{n})$ is a \\emph{symbol} i.e. it satisfies the following estimates in coordinates:\n\\begin{equation}\n\\label{equation:bounds}\n\\sup_{|\\alpha'| \\leq \\alpha, |\\beta'|\\leq \\beta} \\sup_{(x,\\xi) \\in T^*\\mathbb{R}^{n+1}}  \\langle \\xi \\rangle^{m - |\\alpha'|} |\\partial_\\xi^{\\alpha'} \\partial_x^{\\beta'} p(x,\\xi)| < \\infty,\n\\end{equation}\nfor all $\\alpha, \\beta \\in \\mathbb{N}^n$, with $\\langle \\xi \\rangle = \\sqrt{1+|\\xi|^2}$. When $\\E$ is not the trivial line bundle, the symbol $p$ is a matrix-valued symbol. Given $p \\in S^m(T^*\\mathcal{M})$, one can define a (non-canonical) \\emph{quantization procedure} $\\Op : S^m(T^*\\mathcal{M}) \\rightarrow \\Psi^m(\\mathcal{M})$ thanks to \\eqref{equation:quantization} in coordinates patches. This also works more generally with a vector bundle $\\E \\rightarrow \\mathcal{M}$ and one has a quantization map $\\Op : S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E)) \\rightarrow \\Psi^m(\\mathcal{M},\\E)$ (note that the symbol is then a section of the pullback bundle $\\mathrm{End}(\\E) \\rightarrow T^*\\mathcal{M}$ satisfying the bounds \\eqref{equation:bounds} in coordinates). There is a well-defined (partial) inverse map $\\sigma_{\\mathrm{princ}} : \\Psi(\\mathcal{M},\\E) \\rightarrow S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E))\/S^{m-1}(T^*\\mathcal{M}, \\mathrm{End}(\\E))$ called the \\emph{principal symbol} and satisfying $\\sigma_{\\mathrm{princ}}(\\Op(p)) = [p]$ (the equivalence class as an element of $S^m(T^*\\mathcal{M}, \\mathrm{End}(\\E))\/S^{m-1}(T^*\\mathcal{M}, \\mathrm{End}(\\E))$).\n\nWe denote by $H^s(\\mathcal{M}, \\E)$ the space of Sobolev sections of order $s \\in \\mathbb{R}$ and by $C^s_*(\\mathcal{M}, \\E)$ the space of H\\\"older-Zygmund sections of order $s \\in \\mathbb{R}$. It is well-known that for $s \\in \\mathbb{R}_+ \\setminus \\mathbb{N}$, $C^s_*$ coincide with the space of H\\\"older-continuous sections $C^s$ of order $s$. Recall that $C^s_*$ is an algebra as long as $s > 0$ and $H^s$ is an algebra for $s > n\/2$.\nIf $P \\in \\Psi^m(\\mathcal{M}, \\E)$ is a pseudodifferential operator of order $m \\in \\mathbb{R}$, then $P : X^{s+m}(\\mathcal{M}, \\E) \\rightarrow X^s(\\mathcal{M}, \\E)$, where $X = H$, $C_*$, is bounded. We refer to \\cite{Shubin-01,Taylor-91} for further details.\n\n\\subsection{Connections on vector bundles}\n\n\\label{ssection:connections}\n\nWe refer the reader to \\cite[Chapter 2]{Donaldson-Kronheimer-90} for the background on connections on vector bundles.\n\n\n\n\n\n\n\\subsubsection{Mixed connection on the homomorphism bundle}\n\n\\label{sssection:connection-induced}\n\nIn this paragraph, we consider two Hermitian vector bundles $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$ equipped with respective unitary connections $\\nabla_1 = \\nabla^{\\E_1}$ and $\\nabla_2 = \\nabla^{\\E_2}$ which can be written in some local patch $U \\subset \\mathbb{R}^n$ of coordinates $\\nabla^{\\E_i} = d +\\Gamma_i$, for some $\\Gamma_i \\in C^\\infty(U,T^*U \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E_i))$. Let $\\Hom(\\E_1,\\E_2)$ be the vector bundle of homomorphisms from $\\E_1$ to $\\E_2$, endowed with the natural Hermitian structure.\n\n\\begin{definition}\\label{definition:mixed}\nWe define the (unitary) \\emph{homomorphism} or \\emph{mixed} connection $\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$ on $\\Hom(\\E_1, \\E_2)$, induced by $\\nabla^{\\E_1}$ and $\\nabla^{\\E_2}$, by the Leibnitz property:\n\\[\n\\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1, \\E_2)), \\forall s \\in C^\\infty(\\mathcal{M}, \\E_1), \\quad \\nabla^{\\E_2}(us) = (\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})}u) \\cdot s + u\\cdot  (\\nabla^{\\E_1}s).\n\\]\n\\end{definition}\nEquivalently, it is straightforward to check that this is the canonical tensor product connection induced on $\\Hom(\\E_1, \\E_2) \\cong \\E_2 \\otimes \\E_1^*$ and that in local coordinates we have\n\\begin{equation}\\label{eq:localhom}\n\t\\nabla^{\\Hom(\\nabla^{\\E_1}, \\nabla^{\\E_2})} u := du + \\Gamma_2 (\\bullet) u - u \\Gamma_1(\\bullet).\n\\end{equation}\nNote that this definition does not require the bundles to have same rank;\nwe insist on the fact that the mixed connection $\\nabla^{\\Hom(\\nabla_1, \\nabla_2)}$ \\emph{depends on a choice} of connections $\\nabla_1$ and $\\nabla_2$. In the particular case when $\\E_1 = \\E_2 = \\E$ and $\\nabla_1^{\\E} = \\nabla_2^{\\E} = \\nabla^{\\E}$ we will write $\\nabla^{\\mathrm{End}(\\nabla^{\\E})} = \\nabla^{\\Hom(\\nabla^{\\E}, \\nabla^{\\E})}$ for the induced \\emph{endomorphism} connection on $\\mathrm{End}(\\E)$. When clear from the context, we will also write $\\nabla^{\\mathrm{End}(\\E)}$ for the endomorphism connection induced by $\\nabla^{\\E}$.\n\n\n\nGiven a flow $(\\varphi_t)_{t \\in \\mathbb{R}}$, we will denote by $P(x,t) : \\Hom({\\E_1}_x, {\\E_2}_x) \\rightarrow \\Hom({\\E_1}_{\\varphi_t(x)}, {\\E_2}_{\\varphi_t(x)})$ the parallel transport with respect to the mixed connection along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$. Observe that for $u \\in \\Hom({\\E_1}_x, {\\E_2}_x)$, we have:\n\\begin{equation}\n\\label{equation:tp-mixed}\nP(x,t) u = C_2(x,t) u C_1(x,t)^{-1},\n\\end{equation}\nwhere $C_i(x,t) : {\\E_i}_x \\rightarrow {\\E_i}_{\\varphi_t(x)}$ is the parallel transport with respect to $\\nabla^{\\E_i}$ along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$.\n\nRecall that the curvature tensor $F_\\nabla \\in C^\\infty(\\mathcal{M}, \\Lambda^2(T^*M) \\otimes \\mathrm{End}(\\E))$ of $\\nabla = \\nabla^{\\E}$ is defined as, for any vector fields $X, Y$ on $\\mathcal{M}$ and sections $S$ of $\\E$\n\\begin{equation*\n\tF_\\nabla(X, Y)S = \\nabla_X \\nabla_Y S - \\nabla_Y \\nabla_X S - \\nabla_{[X, Y]}S.\n\\end{equation*} \nThen a quick computation using \\eqref{eq:localhom} reveals that:\n\\begin{equation}\n\\label{equation:induced-curvature}\nF_{\\nabla^{\\mathrm{Hom}(\\nabla_1,\\nabla_2)}} u = F_{\\nabla_2} \\cdot u - u \\cdot F_{\\nabla_1}.\n\\end{equation}\nEventually, using the Leibnitz property, we observe that if $p_i \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_i',\\E_i))$ is an isomorphism, then:\n\\begin{align}\n\\label{equation:lien}\n\\begin{split}\n\\nabla^{\\Hom(\\nabla_1, p_2^*\\nabla_2)} u &= (p_2)^{-1} \\nabla^{\\Hom(\\nabla_1, \\nabla_2)} (p_2 u), \\quad \\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1, \\E_2')),\\\\\n\\nabla^{\\Hom(p_1^*\\nabla_1, \\nabla_2)} u &= \\nabla^{\\Hom(\\nabla_1, \\nabla_2)} (u p_1^{-1}) \\cdot p_1,\\quad\\,\\, \\forall u \\in C^\\infty(\\mathcal{M}, \\Hom(\\E_1', \\E_2)).\n\\end{split}\n\\end{align}\n\n\\subsubsection{Ambrose-Singer formula}\n\nRecall that the celebrated Ambrose-Singer formula (see eg. \\cite[Theorem 8.1]{Kobayashi-Nomizu-69}) determines the tangent space at the identity of the holonomy group with respect to an arbitrary connection, in terms of its curvature tensor. Here we give an integral version of this fact. We start with a Hermitian vector bundle $\\E$ over the Riemannian manifold $(\\mathcal{M}, g)$. Equip $\\E$ with unitary connection $\\nabla = \\nabla^{\\E}$. \n\n\n\nConsider\na smooth homotopy $\\Gamma: [0, 1]^2 \\to \\mathcal{M}$ such that $\\Gamma(0, 0) = p$.\nThe ``vertical'' map $C_{\\uparrow}(s, t): \\E_p \\to \\E_{\\Gamma(s, t)}$ is obtained by parallel transporting with respect to $\\nabla$ from $\\E_p$ to $\\E_{\\Gamma(0, 1)}$, then $\\E_{\\Gamma(0, 1)}$ to $\\E_{\\Gamma(s, 1)}$ and $\\E_{\\Gamma(s, 1)}$ to $\\E_{\\Gamma(s, t)}$, along $\\Gamma(0, \\bullet)$, $\\Gamma(\\bullet, 1)$ and $\\Gamma(s, \\bullet)$, respectively. Next, define the ``horizontal'' map $C_{\\rightarrow}(s, t): \\E_p \\to \\E_{\\Gamma(s, t)}$ by parallel transport with respect to $\\nabla$ from $\\E_p$ to $\\E_{\\Gamma(s, 0)}$ and $\\E_{\\Gamma(s, 0)}$ to $\\E_{\\Gamma(s, t)}$, along $\\Gamma(\\bullet, 0)$ and $\\Gamma(s, \\bullet)$, respectively. For a better understanding, see Figure \\ref{fig:AS1}.\n\t\t\n\t\t\n\t\t\n\nWe are ready to prove the formula:\n\n\n\\begin{lemma}\\label{lemma:ambrosesinger}\n\tThe following formula holds\n\t\\begin{equation}\n\t\tC_{\\uparrow}^{-1}(1, 1) C_{\\rightarrow}(1, 1) - \\mathbbm{1}_{\\E_{p}} = \\int_0^1 \\int_0^1 C_{\\uparrow}(s, t)^{-1} F_\\nabla(\\partial_t, \\partial_s) C_{\\rightarrow}(s, t) \\, dt \\, ds.\n\t\\end{equation}\t\n\\end{lemma}\n\n\n\n\n\n\\begin{proof}\n\tLet $w_1, w_2 \\in \\E_p$; formally, we will identify the connection $\\nabla$ with its pullback $\\Gamma^*\\nabla$ on the pullback bundle $\\Gamma^*\\E$ over $[0, 1]^2$, as well as the curvature $F_\\nabla$ with $\\Gamma^*F_\\nabla$. Then we have the following chain of equalities:\n\t\\begin{align*}\n\t\t&\\langle{w_1, (C_{\\uparrow}(1, 1)^{-1} C_{\\rightarrow}(1, 1) - \\mathbbm{1}_{\\E_p}) w_2}\\rangle = \\langle{C_{\\uparrow}(1, 1)w_1, C_{\\rightarrow}(1, 1)w_2}\\rangle - \\langle{C_{\\uparrow}(0, 1)w_1, C_{\\rightarrow}(0, 1)w_2}\\rangle\\\\\n\t\t &= \\int_0^1 \\partial_s \\langle{C_{\\uparrow}(s, 1) w_1, C_{\\rightarrow}(s, 1) w_2}\\rangle ds\\\\\n\t\t&= \\int_0^1 \\langle{C_{\\uparrow}(s, 1) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, 1) w_2}\\rangle ds\\\\\n\t\t&= \\int_0^1 \\Big[\\int_0^1 \\Big(\\partial_t \\langle{C_{\\uparrow}(s, t) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, t)w_2}\\rangle + \\langle{C_{\\uparrow}(s, 0) w_1, \\nabla_{\\partial_s} C_{\\rightarrow}(s, 0) w_2}\\rangle\\Big)dt\\Big] ds\\\\\n\t\t&= \\int_0^1 \\int_0^1 \\langle{C_{\\uparrow}(s, t) w_1, \\nabla_{\\partial_t} \\nabla_{\\partial_s} C_{\\rightarrow}(s, t) w_2}\\rangle \\,ds\\, dt\\\\\n\t\t&= \\int_0^1 \\int_0^1 \\langle{w_1, C_{\\uparrow}(s, t)^{-1}\\underbrace{(\\nabla_{\\partial_t} \\nabla_{\\partial_s} - \\nabla_{\\partial_s} \\nabla_{\\partial_t})}_{=F_\\nabla(\\partial_t, \\partial_s)} C_{\\rightarrow}(s, t) w_2}\\rangle \\,ds\\, dt,\n\t\\end{align*}\n\tas the Lie bracket $[\\partial_s, \\partial_t] = 0$ and we used the unitarity of $\\nabla$ throughout. This completes the proof, since $w_1$ and $w_2$ were arbitrary.\n\\end{proof}\n\n\\begin{figure}\n             \\centering\n\\begin{tikzpicture}[scale = 0.8, everynode\/.style={scale=0.5}]\n\\tikzset{cross\/.style={cross out, draw=black, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt},\ncross\/.default={1pt}}\n\n      \t\\draw[thick, ->] (0, 0) -- (6,0) node[right] {\\small $s$};\n\t\t\\draw[thick, ->] (0, 0) -- (0,6) node[left] {\\small $t$};\n\t\t\n\t\n\t\t\\draw[thick] (3, 5) -- (5, 5) -- (5, 0);\n\t\t\n\t\n\t\t\\draw[thick, blue] (0, 0) -- (0, 5) -- (3, 5) -- (3, 3);\n\t\t\n\t\n\t\t\\draw[thick, red] (0, 0) -- (3, 0) -- (3, 3);\n\t\t\n\t\n\t\t\\draw[thick, ->] (3, 1.5)--(3, 1.501);\n\t\t\\draw[thick, ->] (1.5, 0)--(1.501, 0);\n\t\t\n\t\n\t\t\\draw[thick, ->] (0, 2.5)--(0, 2.501);\n\t\t\\draw[thick, ->] (1.5, 5)--(1.501, 5);\n\t\t\\draw[thick, ->] (3, 4)--(3, 3.999);\n\t\t\n\t\n\t\t\\fill (0, 0) node[below left] {\\tiny $p$} circle (1.5pt);\n\t\t\\fill (5, 0) node[below] {\\tiny $\\Gamma(1, 0)$}  circle (1.5pt);\n\t\t\\fill (0, 5) node[left] {\\tiny $\\Gamma(0, 1)$}  circle (1.5pt);\n\t\t\\fill (5, 5) node[right] {\\tiny $\\Gamma(1, 1)$}  circle (1.5pt);\n\t\t\n\t\n\t\t\\fill (3, 3) node[right] {\\small $\\Gamma(s, t)$}  circle (1.5pt);\n\t\t\\fill (3, 0) node[below] {\\small $\\Gamma(s, 0)$}  circle (1.5pt);\n\t\t\\fill (3, 5) node[above] {\\small $\\Gamma(s, 1)$}  circle (1.5pt);\n\\end{tikzpicture}\n             \\caption{\\small The homotopy $\\Gamma$ in Lemma \\ref{lemma:ambrosesinger} with the corresponding points in $\\mathcal{M}$: in blue and red are the trajectories along which the parallel transport maps $C_{\\uparrow}$ (vertical) and $C_{\\rightarrow}$ (horizontal) are taken, respectively.}\n             \\label{fig:AS1}\n\\end{figure}\n\n\nWe have two applications in mind for this lemma: one if $\\gamma$ is in a neighbourhood of $p$ and we use the radial homotopy via geodesics emanating from $p$, and the second one for the ``thin rectangle'' obtained by shadowing a piece of the flow orbit, see Lemma \\ref{lemma:ASgeometry}.\n\n\n\n\\subsection{Fourier analysis in the fibers} In this paragraph, we recall some elements of Fourier analysis in the fibers and refer to \\cite{Guillemin-Kazhdan-80, Guillemin-Kazhdan-80-2, Paternain-Salo-Uhlmann-14-2, Paternain-Salo-Uhlmann-15, Guillarmou-Paternain-Salo-Uhlmann-16} for further details.\n\n\n\n\n\\subsubsection{Analysis on the trivial line bundle}\n\n\\label{sssection:line-bundle}\n\n\nLet $(M,g)$ be a smooth Riemannian manifold of arbitrary dimension $n \\geq 2$. The unit tangent bundle is endowed with the natural Sasaki metric and we let $\\pi : SM \\rightarrow M$ be the projection on the base. There is a canonical splitting of the tangent bundle to $SM$ as:\n\\[\nT(SM) = \\mathbb{H} \\oplus \\mathbb{V} \\oplus \\mathbb{R} X,\n\\] \nwhere $X$ is the geodesic vector field, $\\mathbb{V} := \\ker \\dd \\pi$ is the vertical space and $\\mathbb{H}$ is the horizontal space: it can be defined as the orthogonal to $\\mathbb{V} \\oplus \\mathbb{R} X$ with respect to the Sasaki metric (see \\cite[Chapter 1]{Paternain-99}). Any vector $Z \\in T(SM)$ can be decomposed according to the splitting\n\\[\nZ = \\alpha(Z) X + Z_{\\mathbb{H}} + Z_{\\mathbb{V}},\n\\]\nwhere $\\alpha$ is the Liouville $1$-form, $Z_{\\mathbb{H}} \\in \\mathbb{H}, Z_{\\mathbb{V}} \\in \\mathbb{V}$. If $f \\in C^\\infty(SM)$, its gradient computed with respect to the Sasaki metric can be written as:\n\\[\n\\nabla_{\\mathrm{Sas}}f = (Xf) X + \\nabla_{\\mathbb{H}} f + \\nabla_{\\mathbb{V}} f,\n\\]\nwhere $\\nabla_{\\mathbb{H}} f \\in \\mathbb{H}$ is the horizontal gradient, $\\nabla_{\\mathbb{V}} f \\in \\mathbb{V}$ is the vertical gradient. We also let $\\mathcal{N} \\to SM$ be the \\emph{normal bundle} whose fiber over each $(x,v) \\in SM$ is given by $(\\mathbb{R} \\cdot v)^\\bot$. The bundles $\\mathbb{H}$ and $\\mathbb{V}$ may be naturally identified with the bundle $\\mathcal{N}$ (see \\cite[Section 1]{Paternain-99}).\n\nFor every $x \\in M$, the sphere $S_xM = \\left\\{ v \\in T_xM ~|~ |v|^2_x = 1\\right\\} \\subset SM$ endowed with the Sasaki metric is isometric to the canonical sphere $(\\mathbb{S}^{n-1},g_{\\mathrm{can}})$. We denote by $\\Delta_{\\mathbb{V}}$ the vertical Laplacian obtained for $f \\in C^\\infty(SM)$ as $\\Delta_{\\mathbb{V}} f(x,v) = \\Delta_{g_{\\mathrm{can}}}(f|_{S_xM})(v)$, where $\\Delta_{g_{\\mathrm{can}}}$ is the spherical Laplacian. For $m \\geq 0$, we denote by $\\Omega_m$ the (finite-dimensional) vector space of spherical harmonics of degree $m$ for the spherical Laplacian $\\Delta_{g_{\\mathrm{can}}}$: they are defined as the elements of $\\ker(\\Delta_{g_{\\mathrm{can}}} + m(m+n-2))$. We will use the convention that $\\Omega_m = \\left\\{0\\right\\}$ if $m< 0$. If $f \\in C^\\infty(SM)$, it can then be decomposed as $f = \\sum_{m \\geq 0} f_m$, where $f_m \\in C^\\infty(M,\\Omega_m)$ is the $L^2$-orthogonal projection of $f$ onto the spherical harmonics of degree $m$.\n\nThere is a one-to-one correspondence between trace-free symmetric tensors of degree $m$ and spherical harmonics of degree $m$. More precisely, the map\n\\[\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr}) \\rightarrow C^\\infty(M,\\Omega_m),\n\\]\ngiven by $\\pi_m^*f(x,v) = f_x(v,...,v)$ is an isomorphism. Here, the index $0-\\Tr$ denotes the space of trace-free symmetric tensors, namely tensors such that, if $(\\e_1,...,\\e_n)$ denotes a local orthonormal frame of $TM$:\n\\[\n\\Tr(f) := \\sum_{i=1}^n f(\\e_i,\\e_i,\\cdot, ...,\\cdot) = 0.\n\\]\nWe will denote by ${\\pi_m}_* : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr})$ the adjoint of this map. More generally, the mapping\n\\begin{equation}\\label{eq:pi_muntwisted}\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M) \\rightarrow \\oplus_{k \\geq 0}  C^\\infty(M,\\Omega_{m-2k})\n\\end{equation}\nis an isomorphism. We refer to \\cite[Section 2]{Cekic-Lefeuvre-20} for further details.\n\nThe geodesic vector field acts as $X : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m-1}) \\oplus C^\\infty(M,\\Omega_{m+1})$ (see \\cite{Guillemin-Kazhdan-80-2, Paternain-Salo-Uhlmann-15}). We define $X_+$ as the $L^2$-orthogonal projection of $X$ on the higher modes $\\Omega_{m+1}$, namely if $u \\in C^\\infty(M,\\Omega_m)$, then $X_+u := (Xu)_{m+1}$ and $X_-$ as the $L^2$-orthogonal projection of $X$ on the lower modes $\\Omega_{m-1}$. For $m \\geq 0$, the operator $X_+ : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m+1})$ is elliptic and thus has a finite dimensional kernel (see \\cite{Dairbekov-Sharafutdinov-10}). The operator $X_- : C^\\infty(M,\\Omega_m) \\rightarrow C^\\infty(M,\\Omega_{m-1})$ is of divergence type. The elements in the kernel of $X_+$ are called \\emph{Conformal Killing Tensors (CKTs)}, associated to the trivial line bundle. For $m=0$, the kernel of $X_+$ on $C^\\infty(M,\\Omega_0)$ always contains the constant functions. We call \\emph{non trivial CKTs} elements in $\\ker X_+$ which are not constant functions on $SM$. The kernel of $X_+$ is invariant by changing the metric by a conformal factor (see \\cite[Section 3.6]{Guillarmou-Paternain-Salo-Uhlmann-16}). It is known (see \\cite{Paternain-Salo-Uhlmann-15}) that there are no non trivial CKTs in negative curvature and for Anosov surfaces but the question remains open for general Anosov manifolds. We provide a positive answer to this question \\emph{generically} as a byproduct of our work \\cite{Cekic-Lefeuvre-21-2}.\n\n\n\\subsubsection{Twisted Fourier analysis}\n\n\\label{sssection:twisted-fourier}\n\nWe now consider a Hermitian vector bundle with a unitary connection $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ over $(M,g)$ and define the operator $\\mathbf{X} := (\\pi^*\\nabla^{\\mathcal{E}})_X$ acting on $C^\\infty(SM,\\pi^*\\mathcal{E})$, where $\\pi : SM \\rightarrow M$ is the projection. For the sake of simplicity, we will drop the $\\pi^*$ in the following. If $f \\in C^\\infty(SM,\\mathcal{E})$, then $\\nabla^{\\mathcal{E}}f \\in C^\\infty(SM,T^*(SM) \\otimes \\mathcal{E})$ and we can write\n\\[\n\\nabla^{\\mathcal{E}}f = (\\mathbf{X} f, \\nabla_{\\mathbb{H}} f, \\nabla_{\\mathbb{V}} f),\n\\]\nwhere $\\nabla_{\\mathbb{H}} f \\in C^\\infty(SM, \\mathbb{H}^* \\otimes \\mathcal{E}), \\nabla_{\\mathbb{V}} f \\in C^\\infty(SM, \\mathbb{V}^* \\otimes \\mathcal{E})$. For future reference, we introduce a bundle endomorphism map $R$ on $\\mathcal{N} \\otimes \\E$, derived from the Riemann curvature tensor via the formula $R(x, v)(w \\otimes e) = (R_x(w, v)v) \\otimes e$.\n\nIf $(e_1,...,e_r)$ is a local orthonormal frame of $\\mathcal{E}$, then we define the vertical Laplacian as\n\\[\n\\Delta_{\\mathbb{V}}^{\\mathcal{E}}(\\sum_{k=1}^r u_k e_k) := \\sum_{k=1}^r (\\Delta_{\\mathbb{V}} u_k) e_k.\n\\]\nAny section $f \\in C^\\infty(SM,\\mathcal{E})$ can be decomposed according to $f = \\sum_{m \\geq 0} f_m$, where $f_m \\in \\ker(\\Delta_{\\mathbb{V}}^{\\mathcal{E}} + m(m+n-2))$ and we define $C^\\infty(M,\\Omega_m \\otimes \\mathcal{E}) := \\ker(\\Delta_{\\mathbb{V}}^{\\mathcal{E}} + m(m+n-2)) \\cap C^\\infty(SM,\\mathcal{E})$.\n\nHere again, the operator $\\mathbf{X} : C^\\infty(M,\\Omega_m \\otimes \\E) \\rightarrow C^\\infty(M,\\Omega_{m-1} \\otimes \\E) \\oplus C^\\infty(M,\\Omega_{m+1} \\otimes \\E)$ can be split into the corresponding sum $\\mathbf{X} = \\mathbf{X}_- + \\mathbf{X}_-$. For every $m \\geq 0$, the operator $\\mathbf{X}_+$ is elliptic and has finite dimensional kernel, whereas $\\mathbf{X}_-$ is of divergence type. The kernel of $\\mathbf{X}_+$ is invariant by a conformal change of the metric (see \\cite[Section 3.6]{Guillarmou-Paternain-Salo-Uhlmann-16}) and elements in its kernel are called \\emph{twisted Conformal Killing Tensors} (CKTs). There are examples of vector bundles with CKTs on manifolds of arbitrary dimension. We proved in a companion paper \\cite{Cekic-Lefeuvre-20} that the non existence of CKTs is a generic condition, no matter the curvature of the manifold (generic with respect to the connection, i.e. there is a residual set of the space of all unitary connections with regularity $C^k$, $k \\geq 2$, which has no CKTs)\n\nIt is also known by \\cite{Guillarmou-Paternain-Salo-Uhlmann-16}, that in negative curvature, there is always a \\emph{finite number of degrees} with CKTs (and this number can be estimated thanks to a lower bound on the curvature of the manifold and the curvature of the vector bundle). In other words, $\\ker \\mathbf{X}_+|_{C^\\infty(SM,\\mathcal{E})}$ is finite-dimensional. The proof relies on an energy identity called the Pestov identity. This is also known for Anosov surfaces since any Anosov surface is conformally equivalent to a negatively-curved surface and CKTs are conformally invariant. Nevertheless, and to the best of our knowledge, it is still an open question to show that for Anosov manifolds of dimension $n \\geq 3$, there is at most a finite number of CKTs.\n\n\n\n\n\n\\subsection{Twisted symmetric tensors}\\label{section:twisted}\n\nGiven a section $u \\in C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$, the connection $\\nabla^{\\mathcal{E}}$ produces an element $\\nabla^{\\mathcal{E}}u \\in C^\\infty(M, T^*M \\otimes (\\otimes^m_S T^*M) \\otimes \\mathcal{E})$. In coordinates, if $(e_1, ..., e_r)$ is a local orthonormal frame for $\\mathcal{E}$ and $\\nabla^{\\mathcal{E}} = d + \\Gamma$, for some one-form with values in skew-Hermitian matrices $\\Gamma$, such that  $\\nabla^{\\E}e_k = \\sum_{i = 1}^n\\sum_{l = 1}^r \\Gamma_{ik}^{l} dx_i \\otimes e_l$, we have:\n\\begin{equation}\n\\label{equation:nabla-e}\n\\begin{split}\n\\nabla^{\\mathcal{E}}(\\sum_{k=1}^r u_k \\otimes e_k) & = \\sum_{k=1}^r \\big(\\nabla u_k \\otimes e_k + u_k \\otimes \\nabla^{\\mathcal{E}} e_k\\big)\\\\\n& = \\sum_{k=1}^r \\left(\\nabla u_k +   \\sum_{l=1}^r \\sum_{i=1}^n \\Gamma_{il}^k u_l \\otimes dx_i \\right) \\otimes e_k,\n\\end{split}\n\\end{equation}\nwhere $u_k \\in C^\\infty(M,\\otimes^m_S T^*M)$ and $\\nabla$ is the Levi-Civita connection. The symmetrization operator $\\mathcal{S}^{\\mathcal{E}} : C^\\infty(M,\\otimes^m T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$ is defined by:\n\\[\n\\mathcal{S}^{\\mathcal{E}}\\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) = \\sum_{k=1}^r \\mathcal{S}(u_k) \\otimes e_k,\n\\]\nwhere $u_k \\in C^\\infty(M,\\otimes^m_S T^*M)$ and in coordinates, writing $u_k = \\sum_{i_1, ..., i_m=1}^n u_{i_1...i_m}^{(k)} dx_{i_1} \\otimes ... \\otimes dx_{i_m}$, we have\n\\[\n\\mathcal{S}(dx_{i_1} \\otimes ... \\otimes dx_{i_m}) = \\dfrac{1}{m!} \\sum_{\\pi \\in \\mathfrak{S}_m} dx_{\\pi(i_1)} \\otimes ... \\otimes dx_{\\pi(i_m)},\n\\]\nwhere $\\mathfrak{S}_m$ denotes the group of permutations of order $m$. For the sake of simplicity, we will write $\\mathcal{S}$ instead of $\\mathcal{S}^{\\mathcal{E}}$. We can symmetrize \\eqref{equation:nabla-e} to produce an element $D^{\\E} := \\mathcal{S} \\nabla^{\\mathcal{E}}u \\in C^\\infty(M, \\otimes^{m+1}_S T^*M \\otimes \\mathcal{E})$ given in coordinates by:\n\\begin{equation}\n\\label{equation:formula-de}\nD^{\\E} \\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) = \\sum_{k=1}^r \\left( Du_k + \\sum_{l=1}^r  \\sum_{i=1}^n \\Gamma_{il}^k \\sigma(u_l \\otimes dx_i) \\right) \\otimes e_k,\n\\end{equation}\nwhere $D := \\mathcal{S} \\nabla$ is the usual symmetric derivative of symmetric tensors\\footnote{Beware of the notation: $\\nabla^{\\mathcal{E}}$ is for the connection, $D^{\\E}$ for the symmetric derivative of tensors and $\\nabla^{\\mathrm{End}(\\E)}$ is the connection induced by $\\nabla^{\\mathcal{E}}$ on the endomorphism bundle.}. Elements of the form $Du \\in C^\\infty(M,\\otimes^{m+1}_S T^*M)$ are called \\emph{potential tensors}. By comparison, we will call elements of the form $D^{\\E}f \\in C^\\infty(M,\\otimes^{m+1}_S T^*M \\otimes \\mathcal{E})$ \\emph{twisted potential tensors}. The operator $D^{\\E}$ is a first order differential operator and its expression can be read off from \\eqref{equation:formula-de}, namely:\n\\[\n\\begin{split}\n\\sigma_{\\mathrm{princ}}(D^{\\E})(x,\\xi) \\cdot \\left(\\sum_{k=1}^r u_k(x) \\otimes e_k(x) \\right) & = \\sum_{k=1}^r \\left(\\sigma_{\\mathrm{princ}}(D)(x,\\xi) \\cdot u_k(x)\\right) \\otimes e_k(x) \\\\\n& = i \\sum_{k=1}^r \\sigma(\\xi \\otimes u_k(x)) \\otimes e_k(x),\n\\end{split}\n\\]\nwhere $e_k(x) \\in \\mathcal{E}_x, u_k(x) \\in \\otimes^m_S T^*_xM$ and the basis $(e_1(x),...,e_r(x))$ is assumed to be orthonormal for the metric $h$ on $\\mathcal{E}$. One can check that this is an injective map, which means that $D^{\\E}$ is a left-elliptic operator and can be inverted on the left modulo a smoothing remainder. Its kernel is finite-dimensional and consists of smooth elements\n\nBefore that, we introduce for $m \\in \\mathbb{N}$, the operator\n\\[\n\\pi_m^* : C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(SM,\\pi^*\\mathcal{E}), \n\\]\ndefined by\n\\[\n\\pi_m^*\\left(\\sum_{k=1}^r u_k \\otimes e_k\\right) (x,v) := \\sum_{k=1}^r ({u_k})_x(v,...,v) e_k(x).\n\\]\nSimilarly to \\eqref{eq:pi_muntwisted}, the following mappings are isomorphisms (see \\cite[Section 2]{Cekic-Lefeuvre-20}):\n\\begin{align*}\n\\pi_m^* &: C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow \\oplus_{k \\geq 0} C^\\infty(M,\\Omega_{m-2k} \\otimes \\E),\\\\\n\\pi_m^* &: C^\\infty(M,\\otimes^m_S T^*M|_{0-\\Tr} \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\Omega_m \\otimes \\E).\n\\end{align*}\nWe recall the notation $(\\pi^* \\nabla^{\\mathcal{E}})_X := \\mathbf{X}$. The following remarkable commutation property holds (see \\cite[Section 2]{Cekic-Lefeuvre-20}):\n\\begin{equation}\\label{eq:pullback}\n\t\\forall m \\in \\mathbb{Z}_{\\geq 0}, \\quad \\pi_{m+1}^* D^{\\E} = \\mathbf{X} \\pi_m^*.\n\\end{equation}\n\n\nThe vector bundle $\\otimes^m_S T^*M \\otimes \\mathcal{E}$ is naturally endowed with a canonical fiberwise metric induced by the metrics $g$ and $h$ which allows to define a natural $L^2$ scalar product. The $L^2$ formal adjoint $(D^{\\E})^*$ of $D^{\\E}$ is of divergence type (in the sense that its principal symbol is surjective for every $(x,\\xi) \\in T^*M \\setminus \\left\\{ 0 \\right\\}$, see \\cite[Definition 3.1]{Cekic-Lefeuvre-20} for further details). We call \\emph{twisted solenoidal tensors} the elements in its kernel.\n\nBy ellipticity of $D^{\\E}$, for any twisted $m$-tensor $f$ there exists a unique $p \\in (\\ker D^{\\E})^\\perp \\cap C^\\infty(M,\\otimes^{m-1}_S T^*M \\otimes \\mathcal{E}), h \\in C^\\infty(M,\\otimes^{m}_S T^*M \\otimes \\mathcal{E})$ such that:\n\\begin{equation}\\label{eq:decomposition-tt}\nf = D^{\\E}p + h, \\quad (D^{\\E})^*h = 0.\n\\end{equation}\nThe previous decomposition could be extended to other regularities. We define $\\pi_{\\ker (D^{\\E})^*}f := h$ as the $L^2$-orthogonal projection on twisted solenoidal tensors. This can be expressed as:\n\\begin{equation}\n\\label{equation:projection}\n\\pi_{\\ker (D^{\\E})^*} = \\mathbbm{1} - D^{\\E} [(D^{\\E})^*D^{\\E}]^{-1} (D^{\\E})^*,\n\\end{equation}\nwhere $[(D^{\\E})^*D^{\\E}]^{-1}$ is the resolvent of the operator.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Pollicott-Ruelle resonances}\n\nWe explain the link between the widely studied notion of Pollicott-Ruelle resonances (see for instance \\cite{Liverani-04, Gouezel-Liverani-06,Butterley-Liverani-07,Faure-Roy-Sjostrand-08,Faure-Sjostrand-11,Faure-Tsuji-13,Dyatlov-Zworski-16}) and the notion of (twisted) Conformal Killing Tensors introduced in the last paragraph. We also refer to \\cite{Cekic-Lefeuvre-20} for an extensive discussion about this.\n\n\n\n\\subsubsection{Definition of the resolvents}\n\n\n\\label{ssection:resonances}\n\nLet $\\mathcal{M}$ be a smooth manifold endowed with a vector field $X \\in C^\\infty(\\mathcal{M},T\\mathcal{M})$ generating an Anosov flow in the sense of \\eqref{equation:anosov}. Throughout this paragraph, we will always assume that the flow is volume-preserving. It will be important to consider the dual decomposition to \\eqref{equation:anosov}, namely\n\\[\nT^*(\\mathcal{M}) = \\mathbb{R} E_0^* \\oplus E_s^* \\oplus E_u^*,\n\\]\nwhere $E_0^*(E_s \\oplus E_u) = 0, E_s^*(E_s \\oplus \\mathbb{R} X) = 0, E_u^*(E_u \\oplus \\mathbb{R} X) = 0$. As before, we consider a vector bundle $\\mathcal{E} \\rightarrow \\mathcal{M}$ equipped with a unitary connection $\\nabla^{\\E}$ and set $\\mathbf{X} := \\nabla^{\\mathcal{E}}_X$. Since $X$ preserves a smooth measure $\\dd \\mu$ and $\\nabla^{\\mathcal{E}}$ is unitary, the operator $\\mathbf{X}$ is skew-adjoint on $L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu)$, with dense domain\n\\begin{equation}\n\\label{equation:domaine-p}\n\\mathcal{D}_{L^2} := \\left\\{ u \\in L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu) ~|~ \\mathbf{X} u \\in L^2(\\mathcal{M},\\mathcal{E};\\dd\\mu)\\right\\}.\n\\end{equation}\nIts $L^2$-spectrum consists of absolutely continuous spectrum on $i\\mathbb{R}$ and on embedded eigenvalues. We introduce the resolvents\n\\begin{equation}\n\\label{equation:resolvent}\n\\begin{split}\n&\\RR_+(z) := (-\\mathbf{X}-z)^{-1} = - \\int_0^{+\\infty} e^{-t z} e^{-t\\mathbf{X}} \\dd t, \\\\\n&\\RR_-(z) := (\\mathbf{X}-z)^{-1} = - \\int_{-\\infty}^0 e^{z t} e^{-t\\mathbf{X}} \\dd t,\n\\end{split}\n\\end{equation}\ninitially defined for $\\Re(z) > 0$. (Let us stress on the conventions here: $-\\mathbf{X}$ is associated to the positive resolvent $\\RR_+(z)$ whereas $\\mathbf{X}$ is associated to the negative one $\\RR_-(z)$.) Here $e^{-t\\mathbf{X}}$ denotes the propagator of $\\mathbf{X}$, namely the parallel transport by $\\nabla^{\\mathcal{E}}$ along the flowlines of $X$. If $\\mathbf{X} = X$ is simply the vector field acting on functions (i.e. $\\mathcal{E}$ is the trivial line bundle), then $e^{-tX}f(x) = f(\\varphi_{-t}(x))$ is nothing but the composition with the flow.\n\nThere exists a family $\\mathcal{H}^s_\\pm$ of Hilbert spaces called \\emph{anisotropic Sobolev spaces}, indexed by $s > 0$, such that the resolvents can be meromorphically extended to the whole complex plane by making $\\mathbf{X}$ act on $\\mathcal{H}^s_\\pm$. The poles of the resolvents are called the \\emph{Pollicott-Ruelle resonances} and have been widely studied in the aforementioned literature \\cite{Liverani-04, Gouezel-Liverani-06,Butterley-Liverani-07,Faure-Roy-Sjostrand-08,Faure-Sjostrand-11,Faure-Tsuji-13,Dyatlov-Zworski-16}. Note that the resonances and the resonant states associated to them are intrinsic to the flow and do not depend on any choice of construction of the anisotropic Sobolev spaces. More precisely, there exists a constant $c > 0$ such that $\\RR_\\pm(z) \\in \\mathcal{L}(\\mathcal{H}_\\pm^s)$ are meromorphic in $\\left\\{\\Re(z) > -cs\\right\\}$. For $\\RR_+(z)$ (resp. $\\RR_-(z)$), the space $\\mathcal{H}^s_+$ (resp. $\\mathcal{H}^s_-$) consists of distributions which are microlocally $H^s$ in a neighborhood of $E_s^*$ (resp. microlocally $H^{s}$ in a neighborhood of $E_u^*$) and microlocally $H^{-s}$ in a neighborhood of $E_u^*$ (resp. microlocally $H^{-s}$ in a neighborhood of $E_s^*$), see \\cite{Faure-Sjostrand-11,Dyatlov-Zworski-16}. These spaces also satisfy $(\\mathcal{H}^s_+)' = \\mathcal{H}^s_-$ (where one identifies the spaces using the $L^2$-pairing).\n\nFrom now on, we will assume that $s$ is fixed and small enough, and set $\\mathcal{H}_\\pm := \\mathcal{H}_\\pm^s$. We have\n\\begin{equation}\n\\label{equation:betaan}\nH^{s} \\subset \\mathcal{H}_{\\pm} \\subset H^{-s}.\n\\end{equation}\nand there is a certain strip $\\left\\{\\Re(z) > -\\varepsilon_{\\mathrm{strip}}\\right\\}$ (for some $\\varepsilon_{\\mathrm{strip}} > 0$) on which $z \\mapsto \\RR_{\\pm}(z) \\in \\mathcal{L}(\\mathcal{H}_{\\pm})$ is meromorphic (and the same holds for small perturbations of $\\mathbf{X}$).\n\nThese resolvents satisfy the following equalities on $\\mathcal{H}_\\pm$, for $z$ not a resonance:\n\\begin{equation}\n\\label{equation:resolvent-identity}\n\\RR_\\pm(z)(\\mp\\mathbf{X}- z) = (\\mp\\mathbf{X}- z) \\RR_\\pm(z) = \\mathbbm{1}_{\\mathcal{E}}.\n\\end{equation}\nGiven $z \\in \\mathbb{C}$ which not a resonance, we have:\n\\begin{equation}\n\\label{equation:adjoint}\n\\RR_+(z)^* = \\RR_-(\\overline{z}),\n\\end{equation}\nwhere this is understood in the following way: given $f_1, f_2 \\in C^\\infty(\\mathcal{M},\\mathcal{E})$, we have\n\\[\n\\langle \\RR_+(z) f_1, f_2 \\rangle_{L^2} = \\langle f_1,  \\RR_-(\\overline{z}) f_2 \\rangle_{L^2}.\n\\]\n(We will always use this convention for the definition of the adjoint.) Since the operators are skew-adjoint on $L^2$, all the resonances (for both the positive and the negative resolvents $\\RR_\\pm$) are contained in $\\left\\{\\Re(z) \\leq 0 \\right\\}$. A point $z_0 \\in \\mathbb{C}$ is a resonance for $-\\mathbf{X}$ (resp. $\\mathbf{X}$) i.e. is a pole of $z \\mapsto \\RR_+(z)$ (resp. $\\RR_-(z)$) if and only if there exists a non-zero $u \\in \\mathcal{H}^s_+$ (resp. $\\mathcal{H}^s_-$) for some $s > 0$ such that $-\\mathbf{X} u = z_0 u$ (resp. $\\mathbf{X} u = z_0 u$). If $\\gamma$ is a small counter clock-wise oriented circle around $z_0$, then the spectral projector onto the resonant states is\n\\[\n\\Pi_{z_0}^{\\pm} = - \\dfrac{1}{2\\pi i} \\int_{\\gamma} \\RR_{\\pm}(z) \\dd z =  \\dfrac{1}{2\\pi i} \\int_{\\gamma} (z \\pm \\mathbf{X})^{-1} \\dd z,\n\\]\nwhere we use the abuse of notation that $-(\\mathbf{X}+z)^{-1}$ (resp. $(\\mathbf{X}-z)^{-1}$) to denote the meromorphic extension of $\\RR_+(z)$ (resp. $\\RR_-(z)$). \n\n\n\\subsubsection{Resonances at $z=0$}\n\n\\label{sssection:resonances-zero}\n\nBy the previous paragraph, we can write in a neighborhood of $z=0$ the following Laurent expansion (beware the conventions):\n\\[\n\\RR_+(z) = - \\RR_0^+ - \\dfrac{\\Pi_0^+}{z} + \\mathcal{O}(z).\n\\]\n(Or in other words, using our abuse of notations, $(\\mathbf{X}+z)^{-1} = \\RR_0^+ + \\Pi_0^+\/z + \\mathcal{O}(z)$.) And:\n\\[\n\\RR_-(z) = -\\RR_0^- - \\dfrac{\\Pi_0^-}{z} + \\mathcal{O}(z).\n\\]\n(Or in other words, $(z - \\mathbf{X})^{-1} = \\RR_0^- + \\Pi_0^-\/z + \\mathcal{O}(z)$.) As a consequence, these equalities define the two operators $\\RR_0^{\\pm}$ as the holomorphic part (at $z=0$) of the resolvents $-\\RR_\\pm(z)$. We introduce:\n\\begin{equation}\n\\label{equation:pi}\n\\Pi := \\RR_0^+ + \\RR_0^-.\n\\end{equation}\nWe have:\n\n\\begin{lemma}\n\\label{lemma:relations-resolvent}\nWe have $(\\RR_0^+)^* = \\RR_0^{-}, (\\Pi_0^+)^* = \\Pi_0^- = \\Pi_0^+$. Thus $\\Pi$ is formally self-adjoint. Moreover, it is nonnegative in the sense that for all $f \\in C^\\infty(\\mathcal{M},\\mathcal{E})$, $\\langle \\Pi f, f \\rangle_{L^2} = \\langle f, \\Pi f \\rangle_{L^2} \\geq 0$. Also, $\\langle \\Pi f, f \\rangle = 0$ if and only if $\\Pi f = 0$ if and only if $f = \\mathbf{X} u + v $, for some $u \\in C^\\infty(\\mathcal{M},\\E), v \\in \\ker(\\mathbf{X}|_{\\mathcal{H}_\\pm})$.\n\\end{lemma}\n\n\\begin{proof} See \\cite[Lemma 5.1]{Cekic-Lefeuvre-20}. \\end{proof}\n\n\nWe also record here for the sake of clarity the following identities:\n\\begin{equation}\\label{eq:resolventrelations}\n\\begin{split}\n& \\Pi_0^{\\pm} \\RR_0^+ = \\RR_0^+ \\Pi_0^{\\pm} = 0, \\Pi_0^{\\pm} \\RR_0^- = \\RR_0^- \\Pi_0^{\\pm} = 0,\\\\\n& \\mathbf{X} \\Pi_0^\\pm = \\Pi_0^\\pm \\mathbf{X} = 0, \\mathbf{X} \\RR_0^+ = \\RR_0^+ \\mathbf{X} = \\mathbbm{1} - \\Pi_0^+, -\\mathbf{X} \\RR_0^- = -\\RR_0^- \\mathbf{X} = \\mathbbm{1} - \\Pi_0^-.\n\\end{split}\n\\end{equation}\n\n\n\n\n\\subsection{Generalized X-ray transform}\n\\label{ssection:generalized-x-ray}\nThe discussion is carried out here in the closed case, but could also be generalized to the case of a manifold with boundary.\nWe introduce the operator\n\\[\n\\Pi := \\RR_0^+  + \\RR_0^-,\n\\]\nwhere $\\RR_0^+$ (resp. $\\RR_0^-$) denotes the holomorphic part at $0$ of $-\\RR_+(z)$ (resp. $-\\RR_-(z)$) and $\\Pi^+_0$ is the $L^2$-orthogonal projection on the (smooth) resonant states at $0$. Such an operator was first introduced in the non-twisted case by Guillarmou \\cite{Guillarmou-17-1}. The operator $\\Pi + \\Pi^+_0$ is the derivative of the (total) $L^2$-spectral measure at $0$ of the skew-adjoint operator $\\mathbf{X}$.\n\n\\begin{definition}[Generalized X-ray transform of twisted symmetric tensors]\n\\label{definition:generalized-xray}\nWe define the generalized X-ray transform of twisted symmetric tensors as the operator:\n\\[\n\\Pi_m := {\\pi_m}_* \\left(\\Pi + \\Pi^+_0\\right) \\pi_m^*.\n\\]\n\\end{definition}\n\n\\begin{remark}\\rm\n\tThis allows to define a generalized (twisted) X-ray transform $\\Pi_m$ for an arbitrary unitary connection $\\nabla^{\\mathcal{E}}$ on $\\mathcal{E}$. Indeed, it is not clear a priori if one sticks to the usual definition of the X-ray transform that one can find a ``natural'' candidate for the X-ray transform on twisted tensors. For instance, one could consider the map\n\\[\n\\mathcal{C} \\ni \\gamma \\mapsto I_m^{\\nabla^{\\E}}f (\\gamma) := \\dfrac{1}{\\ell(\\gamma)} \\int_0^{\\ell(\\gamma)} (e^{-t \\mathbf{X}} f) (x_\\gamma, v_\\gamma) \\dd t \\in \\mathcal{E}_{x_\\gamma},\n\\]\nwhere $\\gamma \\in \\mathcal{C}$ is a closed geodesic and $(x_\\gamma,v_\\gamma) \\in \\gamma$. However, this definition does depend on the choice of base point $(x_\\gamma,v_\\gamma) \\in \\gamma$ and it would no longer be true that $I^{\\nabla^{\\mathcal{E}}}_m (D^{\\E}p) (\\gamma) = 0$ unless the connection is transparent.\n\\end{remark}\n\n\nBy \\eqref{eq:pullback} and \\eqref{eq:resolventrelations}, we have the equalities:\n\\begin{equation}\\label{eq:Pi_mproperties}\n(D^{\\E})^* \\Pi_m = 0 = \\Pi_m D^{\\E},\n\\end{equation}\nshowing that $\\Pi_m$ maps the set of twisted solenoidal tensors to itself. We say that the generalised $X$-ray transform is \\emph{solenoidally injective} ($s$-injective) on $m$-tensors, if for all $u \\in C^\\infty(SM, \\E)$ and $f \\in C^\\infty(M, \\otimes_S^mT^*M \\otimes \\E)$\n\\begin{equation}\\label{eq:cohomeqn}\n\t\\mathbf{X} u = \\pi_m^* f \\implies \\exists p \\in C^\\infty(M, \\otimes_S^{m-1}T^*M \\otimes \\E)\\,\\, \\mathrm{such\\,\\, that}\\,\\,f = D^{\\E} p.\n\\end{equation}\nWe have the following:\n\n\\begin{lemma}\\label{lemma:x-ray}\n\tThe generalised $X$-ray transform is $s$-injective of $m$-tensors if and only if $\\Pi_m$ is injective on solenoidal tensors (if this holds, we say $\\Pi_m$ is \\emph{$s$-injective}).\n\\end{lemma}\n\\begin{proof}\n\tAssume that $\\Pi_m f = 0$ and $f$ is a twisted solenoidal $m$-tensor. Then\n\\[\n\\langle \\Pi_m f, f \\rangle_{L^2} = \\langle \\Pi \\pi_m^*f , \\pi_m^*f \\rangle_{L^2} +  \\langle \\Pi^+_0 \\pi_m^*f, \\pi_m^* f \\rangle_{L^2} = 0.\n\\]\nBoth terms on the right hand side are non-negative by Lemma \\ref{lemma:relations-resolvent}, hence both of them vanish, and the same Lemma implies that $\\Pi \\pi_m^* f = 0$ and $\\Pi^+_0 \\pi_m^*f = 0$. Thus $\\mathbf{X} u = \\pi_m^*f$ for some smooth $u$, so by the $s$-injectivity of generalised $X$-ray transform we obtain $f$ is potential, which implies $f = 0$.\n\nThe other direction is obvious by \\eqref{eq:resolventrelations}.\n\\end{proof}\n\nNext, we show $\\Pi_m$ enjoys good analytical properties:\n\n\\begin{lemma}\n\\label{lemma:generalized-xray}\nThe operator $\\Pi_m : C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\rightarrow C^\\infty(M,\\otimes^m_S T^*M \\otimes \\mathcal{E})$ is:\n\\begin{enumerate}\n\\item A pseudodifferential operator of order $-1$,\n\\item Formally self-adjoint and elliptic on twisted solenoidal tensors (its Fredholm index is thus equal to $0$ and its kernel\/cokernel are finite-dimensional),\n\\item Under the assumption that $\\Pi_m$ is $s$-injective, the following stability estimates hold:\n\\[\n\\forall s \\in \\mathbb{R},\\,\\, \\forall f \\in H^s(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\cap \\ker (D^{\\E})^*, ~~~ \\|f\\|_{H^s} \\leq C_s \\|\\Pi_m f \\|_{H^{s+1}},\n\\]\nfor some $C_s > 0$ and for some $C > 0$:\n\\[\n\\forall f \\in H^{-1\/2}(M,\\otimes^m_S T^*M \\otimes \\mathcal{E}) \\cap \\ker (D^{\\E})^*, ~~~ \\langle \\Pi_m f, f \\rangle_{L^2} \\geq C \\|\\pi_{\\ker (D^{\\E})^*} f\\|^2_{H^{-1\/2}}.\n\\]\nIn particular, these estimate hold if $(M,g)$ has negative curvature and $\\nabla^{\\E}$ has no twisted CKTs.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nThe proof of the first two points follows from a rather straightforward adaptation of the proof of \\cite[Theorem 2.5.1]{Lefeuvre-thesis} (see also \\cite{Guillarmou-17-1} for the original arguments); we omit it. It remains to prove the third point. \n\nThe first estimate follows from $(2)$, the elliptic estimate and the fact that $\\Pi_m$ is $s$-injective. The last estimate in the non-twisted case follows from \\cite[Lemma 2.1]{Guillarmou-Knieper-Lefeuvre-19} (or \\cite[Theorem 2.5.6]{Lefeuvre-thesis}) and subsequent remarks; the twisted case follows by minor adaptations.\n\n\nIf $(M,g)$ has negative curvature and $\\nabla^{\\E}$ has no twisted CKTs, using Lemma \\ref{lemma:x-ray} and by \\cite[Sections 4, 5]{Guillarmou-Paternain-Salo-Uhlmann-16} we get that $\\Pi_m$ is $s$-injective, proving the claim.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Exact Liv\\v{s}ic cocycle theory}\n\nWe phrase this section in a very general context which is that of a transitive Anosov flow on a smooth manifold. It is of independent interest to the rest of the article. \n\n\\label{section:livsic}\n\n\\subsection{Statement of the results}\n\n\\label{ssection:intro}\n\n\\subsubsection{A weak exact Liv\\v{s}ic cocycle theorem}\n\nLet $\\mathcal{M}$ be a smooth closed manifold endowed with a flow $(\\varphi_t)_{t \\in \\mathbb{R}}$ with infinitesimal generator $X \\in C^\\infty(\\mathcal{M},T\\mathcal{M})$. We assume that the flow is Anosov in the sense of \\eqref{equation:anosov} and that it is \\emph{transitive}, namely it admits a dense orbit\\footnote{Note that there are examples of non-transitive Anosov flows, see \\cite{Franks-Williamas-80}.}. We denote by $\\mathcal{G}$ the set of all periodic orbits for the flow and by $\\mathcal{G}^\\sharp$ the set of all \\emph{primitive} orbits, namely orbits which cannot be written as a shorter orbit to some positive power greater or equal than $2$. \n\n\n\n\nLet $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ be a smooth Hermitian vector bundle of rank $r$ equipped with a unitary connection $\\nabla^{\\mathcal{E}}$. We will denote by \n\\[\nC(x,t) : \\mathcal{E}_x \\rightarrow \\mathcal{E}_{\\varphi_t(x)},\n\\]\n the parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to the connection $\\nabla^{\\mathcal{E}}$. In the more general setting, we may consider $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$, two Hermitian vector bundles, equipped with two respective unitary connections $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$.\nRecall that if $\\nabla^{\\E_2} = p^*\\nabla^{\\E_1}$, for some unitary map $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2, \\E_1))$\\footnote{Here, we denote by $\\mathrm{U}(\\E_2, \\E_1) \\rightarrow \\mathcal{M}$ the bundle of unitary maps from $\\E_2 \\rightarrow \\E_1$. Of course, it may be empty if the bundles are not isomorphic.}, i.e. the connections are gauge-equivalent, then\nparallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ satisfies the commutation relation:\n\\[\nC_1(x,t)= p(\\varphi_t x) C_2(x,t) p(x)^{-1}.\n\\]\nWe say that such cocycles are \\emph{cohomologous}. In particular, given a closed orbit $\\gamma = (\\varphi_t x_0)_{t \\in [0,T]}$ of the flow, one has\n\\[\nC_1(x_0,T) = p(x_0) C_2(x_0,T) p(x_0)^{-1},\n\\]\ni.e. the parallel transport map are conjugate.\n\n\\begin{definition}\n\\label{definition:equivalence}\nWe say that the connections $\\nabla^{\\E_{1,2}}$ have \n\\emph{trace-equivalent holonomies} if for all primitive closed orbits $\\gamma \\in \\mathcal{G}^\\sharp$, we have:\n\\begin{equation}\n\\label{equation:trace-equivalence}\n\\Tr(C_1(x_\\gamma,\\ell(\\gamma))) = \\Tr(C_2(x_\\gamma,\\ell(\\gamma))),\n\\end{equation}\nwhere $x_\\gamma \\in \\gamma$ is arbitrary and $\\ell(\\gamma)$ is the period of $\\gamma$.\n\\end{definition}\n\nThis condition could be \\emph{a priori} obtained with $\\rk(\\E_1) \\neq \\rk(\\E_2)$. We shall see that this cannot be the case.\nThe following result is one of the main theorems of this paper. It seems to improve known results on Liv\\v{s}ic cocycle theory (in particular \\cite{Parry-99,Schmidt-99}), see \\S\\ref{ssection:exact} for a more extensive discussion on the literature.\n\n\\begin{theorem}\n\\label{theorem:weak}\nAssume $\\mathcal{M}$ is endowed with a smooth transitive Anosov flow. Let $\\E_1, \\E_2 \\rightarrow \\mathcal{M}$ be two Hermitian vector bundles over $\\mathcal{M}$ equipped with respective unitary connections $\\nabla^{\\mathcal{E}_1}$ and $\\nabla^{\\mathcal{E}_2}$. If the connections have \\emph{trace-equivalent holonomies} in the sense of Definition \\ref{definition:equivalence}, then there exists $p \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\E_2,\\E_1))$ such that: for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$,\n\\begin{equation*}\nC_1(x,t) = p(\\varphi_t x) C_2(x,t) p(x)^{-1},\n\\end{equation*}\ni.e. the cocycles induced by parallel transport are cohomologous. Moreover, $\\E_2 \\simeq \\E_1$ are isomorphic.\n\\end{theorem}\n\nAs we shall see in the proof, for any given $L > 0$, it suffices to assume that the trace-equivalent holonomy condition \\eqref{equation:trace} holds for all primitive periodic orbits of length $\\geq L$ in order to get the conclusion of the theorem. Surprisingly, the rather weak condition \\eqref{equation:trace} implies in particular that the bundles are isomorphic as stated in Corollary \\ref{corollary:iso} and the trace of the holonomy of unitary connections along closed orbits should allow one in practice to classify vector bundles over manifolds carrying Anosov flows. Even more suprisingly, the rank of $\\E_1$ and $\\E_2$ might be \\emph{a priori} different and Theorem \\ref{theorem:weak} actually shows that the ranks have to coincide.\n\nThe idea relies on a key notion which we call \\emph{Parry's free monoid}, whose introduction goes back to Parry \\cite{Parry-99}. This free monoid $\\mathbf{G}$ corresponds (at least formally) to the free monoid generated by the set of homoclinic orbits to a given periodic orbit of a point $x_{\\star}$ (see \\S\\ref{sssection:homoclinic} for a definition) and we shall see that a connection induces a unitary representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ (it is not canonical but we shall see that its important properties are). Geometric properties of the connection can be read off this representation, see Theorem \\ref{theorem:iso} below. Moreover, tools from representation theory can be applied and this is eventually how we will prove Theorem \\ref{theorem:weak}. \n\n\n\\subsubsection{Opaque and transparent connections}\n\nTheorem \\ref{theorem:weak} has an interesting straightforward corollary. Recall that a unitary connection is said to be \\emph{transparent} if the holonomy along all closed orbits is trivial.\n\n\\begin{corollary}\nAssume $\\mathcal{M}$ is endowed with a smooth transitive Anosov flow. Let $\\E \\rightarrow \\mathcal{M}$ be a Hermitian vector bundle over $\\mathcal{M}$ of rank $r$ equipped with a unitary connection $\\nabla^{\\mathcal{E}}$. If the connection is transparent, then $\\E$ is trivial and trivialized by a smooth orthonormal family $e_1,...,e_r \\in C^\\infty(\\mathcal{M},\\E)$ such that $\\nabla^{\\E}_X e_i = 0$.\n\\end{corollary}\n\nThis result will be used in \\cite{Cekic-Lefeuvre-21-3} to study the Levi-Civita connection on Anosov manifolds and the notion of \\emph{transparent manifolds}, namely manifolds with trivial holonomy along closed geodesics. In order to prove the previous corollary, it suffices to apply Theorem \\ref{theorem:weak} with $\\E_1 =  \\E$ equipped with $\\nabla^{\\E}$ and $\\E_2 = \\mathbb{C}^r \\times M$ equipped with the trivial flat connection. Then $C_2(x,t) = \\mathbf{1}$ and $(e_1,...,e_n)$ is obtained as the image by $p$ of the canonical basis of $\\mathbb{C}^n$. This corollary seems to be known in the folklore but nowhere written. It is stated in \\cite[Proposition 9.2]{Paternain-12} under the extra-assumption that $\\E \\oplus \\E^*$ is trivial.\n\nThe ``opposite\" notion of transparent connections is that of \\emph{opaque} connections which are connections that do not preserve any non-trivial subbundle $\\mathcal{F} \\subset \\E$ by parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$. It was shown in \\cite[Section 5]{Cekic-Lefeuvre-20} that the opacity of a connection is equivalent to the fact that\n\\[\n\\ker (\\nabla^{\\mathrm{End}}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}) = \\mathbb{C} \\cdot \\mathbf{1}_{\\E}.\n\\]\nAlso note that when $X$ is volume-preserving, this corresponds to the Pollicott-Ruelle (co)resonant states at $0$ associated to the operator $\\nabla^{\\mathrm{End}}_X$. We shall also connect this notion with the representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ of the free monoid:\n\n\\begin{prop}\n\\label{proposition:opaque}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item The connection $\\nabla^{\\E}$ is opaque;\n\\item $\\ker (\\nabla^{\\mathrm{End}}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}) = \\mathbb{C} \\cdot \\mathbf{1}_{\\E}$;\n\\item The representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ is irreducible.\n\\end{enumerate}\n\\end{prop}\n\n\n\n\\subsubsection{Kernel of the endomorphism connection}\n\nThe previous proposition actually follows from a more general statement which we now describe. The representation $\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}})$ gives rise to an orthogonal splitting \n\\[\n\\E_{x_{\\star}} = \\oplus_{i=1}^K \\E_i^{\\oplus n_i},\n\\]\nwhere $\\E_i \\subset \\E_{x_\\star}$ and $n_i \\geq 1$; each factor $\\E_i$ is $\\mathbf{G}$-invariant and the induced representation on each factor is irreducible; furthermore, for $i \\neq j$, the induced representations on $\\E_i$ and $\\E_j$ are not isomorphic. Let $\\mathbb{C}[\\mathbf{G}]$ be the formal algebra generated by $\\mathbf{G}$ over $\\mathbb{C}$ and let $\\mathbf{R} := \\rho\\left( \\mathbb{C}[\\mathbf{G}] \\right)$. By Burnside's Theorem (see \\cite[Corollary 3.3]{Lang-02} for instance), one has that:\n\\[\n\\mathbf{R} = \\oplus_{i=1}^K \\Delta_{n_i} \\mathrm{End}(\\E_i),\n\\]\nwhere $\\Delta_{n_i} u = u \\oplus ... \\oplus u$ for $u \\in \\mathrm{End}(\\E_i)$, the sum being repeated $n_i$-times. We introduce the \\emph{commutant} $\\mathbf{R}'$ of $\\mathbf{R}$, defined as:\n\\[\n\\mathbf{R}' := \\left\\{ u \\in \\mathrm{End}(\\E_{x_\\star}) ~|~ \\forall v \\in \\mathbf{R}, uv = vu\\right\\}.\n\\]\nWe then have:\n\n\\begin{theorem}\n\\label{theorem:iso}\nThere exists a natural isomorphism:\n\\[\n\\Phi : \\mathbf{R}' \\rightarrow \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}.\n\\]\nIn particular these spaces have same dimension, that is\n\\[\n\\dim\\left( \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))} \\right) = \\dim(\\mathbf{R}') = \\sum_{i=1}^K n_i^2.\n\\]\n\\end{theorem}\n\n\n\n\\subsubsection{Invariant sections} To conclude this paragraph, we now investigate the existence of smooth \\emph{invariant} sections of the bundle $\\E \\to \\mathcal{M}$, namely elements of $\\ker \\nabla^{\\E}_X|_{C^\\infty(\\mathcal{M},\\E)}$. First of all, observe that if $u \\in C^\\infty(\\mathcal{M},\\E)$ is an invariant section, then $u_\\star:= u(x_\\star)$ is invariant by the $\\mathbf{G}$-action. The converse is also true:\n\n\n\n\\begin{lemma}\n\\label{lemma:invariant-section}\nAssume that there exists $u_\\star \\in \\E_{x_\\star}$ such that $\\rho(g)u_\\star = u_\\star$ for all $g \\in \\mathbf{G}$. Then, there exists (a unique) $u \\in C^\\infty(\\mathcal{M},\\E)$ such that $u(x_\\star)=u_\\star$ and $\\nabla^{\\E}_X u = 0$.\n\\end{lemma}\n\nSuch an approach turns out to be useful when trying to understand a sort of \\emph{weak version} of Liv\\v{s}ic theory, such as the following: if $\\E \\rightarrow \\mathcal{M}$ is a vector bundle equipped with the unitary connection $\\nabla^{\\E}$ and for each periodic orbit $\\gamma \\in \\mathcal{G}$, there exists a section $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$, then one can wonder if this implies the existence of a global invariant smooth section $u \\in C^\\infty(\\mathcal{M},\\E)$? It turns out that the answer depends on the rank of $\\E$:\n\n\\begin{lemma}\n\\label{lemma:rank2}\nAssume that $\\rk(\\E) \\leq 2$ and that for all periodic orbits $\\gamma \\in \\mathcal{G}$, there exists $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$. Then, there exists $u \\in C^\\infty(\\mathcal{M},\\E)$ such that $\\nabla^{\\E}_X u = 0$.\n\\end{lemma}\n\nWe shall see that the proof of the previous Lemma is purely representation-theoretic and completely avoids the need to understand dynamics and the distribution of periodic orbits. We leave as an exercise for the reader the fact that Lemma \\ref{lemma:rank2} does not hold when $\\rk(\\E) \\geq 3$. A simple counter-example can be built using the following argument: any matrix in $\\mathrm{SO}(3)$ preserves an axis; hence, taking any $\\mathrm{SO}(3)$-connection on a \\emph{real} vector bundle of rank $3$ and then complexifying the bundle, one gets a vector bundle and a connection satisfying the assumptions of Lemma \\ref{lemma:rank2}; it then suffices to produce an $\\mathrm{SO}(3)$-connection without any invariant sections.\n\nWe believe that other links between properties of the representation $\\rho$ and the geometry and\/or dynamics of the parallel transport along the flowlines could be discovered. To conclude, let us also mention that all the results are presented here for complex vector bundles; most of them could be naturally restated for real vector bundles modulo the obvious modifications in the statements.\n\n\n\\subsection{Dynamical preliminaries on Anosov flows}\n\n\\label{section:anosov}\n\n\\subsubsection{Shadowing lemma and homoclinic orbits}\n\n\n\nFix an arbitrary Riemannian metric $g$ on $\\mathcal{M}$. As usual, we define the \\emph{local strong (un)stable manifolds} as:\n\\[\n\\begin{array}{l}\nW^{s}_{\\delta}(x) :=\\left\\{y \\in \\mathcal{M}, ~ \\forall t \\geq 0, d(\\varphi_t y, \\varphi_t x) < \\delta, d(\\varphi_t x, \\varphi_t y) \\rightarrow_{t \\rightarrow +\\infty} 0 \\right\\}, \\\\\nW^{u}_{\\delta}(x) :=\\left\\{y \\in \\mathcal{M}, ~ \\forall t \\leq 0, d(\\varphi_t y, \\varphi_t x) < \\delta, d(\\varphi_t x, \\varphi_t y) \\rightarrow_{t \\rightarrow -\\infty} 0 \\right\\},\n\\end{array}\n\\]\nwhere $\\delta > 0$ is chosen small enough. For $\\delta = \\infty$, we obtain the sets $W^{s,u}(x)$ which are the strong stable\/unstable manifolds of $x$. We also set $W^{s,u}_{\\mathrm{loc}}(x) =:  W^{s,u}_{\\delta_0}(x)$ for some fixed $\\delta_0 > 0$ small enough. The \\emph{local weak (un)stable manifolds} $W^{ws,wu}_{\\delta}(x)$ are the set of points $y \\in B(x,\\delta)$ such that there exists $t \\in \\mathbb{R}$ with $|t| < \\delta$ and $\\varphi_t y \\in W^{s,u}_{\\mathrm{loc}}(x)$.\nThe following lemma is known as the \\emph{local product structure} (see \\cite[Theorem 5.1.1]{Fisher-Hasselblatt-19} for more details):\n\n\n\\begin{lemma}\n\\label{lemma:product-structure}\nThere exists $\\varepsilon_0, \\delta_0 > 0$ small enough such that for all $x,y \\in \\mathcal{M}$ such that $d(x,y) < \\varepsilon_0$, the intersection $W^{wu}_{\\delta_0}(x) \\cap W^{s}_{\\delta_0}(y)$ is precisely equal to a unique point $\\left\\{z\\right\\}$. We write $z := [x,y]$.\n\\end{lemma}\n\n\nThe main tool we will use to construct suitable \\emph{homoclinic orbits} is the following classical shadowing property of Anosov flows for which we refer to \\cite[Corollary 18.1.8]{Hasselblatt-Katok-95}, \\cite[Theorem\n5.3.2]{Fisher-Hasselblatt-19} and \\cite[Proposition 6.2.4]{Fisher-Hasselblatt-19}. For the sake of simplicity, we now write $\\gamma=[xy]$ if $\\gamma$ is an orbit segment of the flow with endpoints $x$ and $y$.\n\n\\begin{theorem}\n\\label{theorem:shadowing} There exist $\\varepsilon_0>0$, $\\theta>0$ and $C>0$ with the\nfollowing property. Consider $\\varepsilon<\\varepsilon_0$, and a finite or infinite sequence\nof orbit segments $\\gamma_i = [x_iy_i]$ of length $T_i$ greater than $1$ such\nthat for any $n$, $d(y_n,x_{n+1}) \\leq \\varepsilon$. Then there exists a genuine\norbit $\\gamma$ and times $\\tau_i$ such that $\\gamma$ restricted to $[\\tau_i,\n\\tau_i+T_i]$ \\emph{shadows} $\\gamma_i$ up to $C\\varepsilon$. More precisely, for all $t\\in\n[0, T_i]$, one has\n\\begin{equation}\n\\label{eq:d_hyperbolic}\n  d(\\gamma(\\tau_i+t), \\gamma_i(t)) \\leq C \\varepsilon e^{-\\theta \\min(t, T_i-t)}.\n\\end{equation}\nMoreover, $|\\tau_{i+1} - (\\tau_i + T_i)| \\leq C \\varepsilon$. Finally, if the\nsequence of orbit segments $\\gamma_i$ is periodic, then the orbit $\\gamma$ is\nperiodic.\n\\end{theorem}\n\nLet us also make the following important comment. In the previous theorem, one can also allow the first orbit segment $\\gamma_i$ to be\ninfinite on the left, and the last orbit segment $\\gamma_j$ to be infinite on\nthe right. In this case,~\\eqref{eq:d_hyperbolic} should be replaced by: assuming that $\\gamma_i$ is defined on $(-\\infty, 0]$\nand $\\gamma_j$ on $[0,+\\infty)$, we would get for some $\\tilde\\tau_{i+1}$\nwithin $C\\varepsilon$ of $\\tau_{i+1}$, and all $t\\geq 0$\n\\begin{equation*}\n  d(\\gamma(\\tilde\\tau_{i+1}-t), \\gamma_i(-t)) \\leq C \\varepsilon e^{-\\theta t}, \\quad d(\\gamma(\\tau_{j}+t), \\gamma_j(t)) \\leq C \\varepsilon e^{-\\theta t}.\n\\end{equation*}\n\n\n\n\n\n\n\\label{sssection:homoclinic}\n\nFix an arbitrary periodic point $x_{\\star} \\in \\mathcal{M}$ of period $T_{\\star}$ and denote by $\\gamma_{\\star}$ its primitive orbit.\n\n\\begin{definition}[Homoclinic orbits]\n\\label{definition:homoclinic}\nA point $p \\in \\mathcal{M}$ is said to be \\emph{homoclinic} to $x_{\\star}$ if $p \\in W^{ws}(x_{\\star}) \\cap W^{wu}(x_{\\star})$ (in other words, $d(\\varphi_{t+t^\\pm_0} p, \\varphi_t x_{\\star}) \\rightarrow_{t \\rightarrow \\pm \\infty} 0$ for some $t^\\pm_0 \\in \\mathbb{R}$). We say that an orbit $\\gamma$ is homoclinic to $x_{\\star}$ if it contains a point $p \\in \\gamma$ that is homoclinic to $x_{\\star}$ and we denote by $\\mathcal{H} \\subset \\mathcal{M}$ the set of homoclinic orbits to $x_{\\star}$.\n\\end{definition}\n\n\nNote that due to the hyperbolicity, the convergence of the point $p$ to $x_{\\star}$ is exponentially fast. More precisely, let $\\gamma$ be the orbit of $p$ and let $\\mathbb{R} \\ni t \\mapsto \\gamma(t)$ be the flow parametrization of $\\gamma$. Then, there exists uniform constants $C,\\theta > 0$ (independent of $\\gamma$) and $A_\\pm \\in \\mathbb{R}$ (depending on $\\gamma$) such that the following holds:\n\\begin{equation}\n\\label{equation:distance}\nd(\\gamma(A_\\pm \\pm n T_{\\star}), x_{\\star}) \\leq C e^{-\\theta n}.\n\\end{equation}\nThe points $\\gamma(A_\\pm)$ correspond to an arbitrary choice of points in $W^{s,u}_{\\delta_0}(x_{\\star})\\cap \\gamma$ (for some arbitrary $\\delta_0 > 0$ small enough). Homoclinic orbits have infinite length (except the orbit of $x_{\\star}$ itself) but it will be convenient to introduce a notion of \\emph{length} $T_\\gamma$ which we define to be equal to $T_\\gamma:=A_+-A_-$ (note that this is a highly non-canonical definition). We define the trunk to be equal to the central segment $\\gamma([A_-,A_+])$. In other words, the length of $\\gamma$ is equal to the length of its trunk. We also define the points: $x_n^\\pm := \\gamma(A_\\pm \\pm nT_{\\star})$. Note that another choice of values $A'_\\pm$ has to differ from $A_\\pm$ by $mT_{\\star}$ for some $m \\in \\mathbb{Z}$. Homoclinic orbits will play a key role as we shall see in due course. \n\n\\begin{lemma}\n\\label{lemma:dense}\nAssume that the flow is transitive. Then the set $\\mathcal{W}$ of points belonging to a homoclinic orbit in $\\mathcal{H}$ is dense in $\\mathcal{M}$.\n\\end{lemma}\n\n\n\\begin{proof}\nThis is a straightforward consequence of the shadowing Theorem \\ref{theorem:shadowing}: one concatenates a long segment $S$ of a transitive orbit with $\\gamma_{\\star}$ i.e. one applies Theorem \\ref{theorem:shadowing} with $... \\gamma_{\\star} \\gamma_{\\star} S \\gamma_{\\star} \\gamma_{\\star} ...$.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nIn the particular case of an Anosov geodesic flow on the unit tangent bundle, one can check that $\\mathcal{H}$ is in one-to-one correspondence with $\\pi_1(M)\/\\langle \\widetilde{\\gamma_{\\star}} \\rangle$, where $\\widetilde{\\gamma_{\\star}} \\in \\pi_1(M)$ is any element such that the conjugacy class of $\\widetilde{\\gamma_{\\star}}$ in $\\pi_1(M)$ corresponds\\footnote{Recall that the set of free homotopy classes $\\mathcal{C}$ is in one-to-one correspondence with conjugacy classes of $\\pi_1(M)$, see \\cite[Chapter 1]{Hatcher-02}.} to the free homotopy class $c \\in \\mathcal{C}$ whose unique geodesic representative is $\\gamma_{\\star}$. \n\\end{remark}\n\n\n\n\n\\subsubsection{Applications of the Ambrose-Singer formula}\n\n\nConsider a Hermitian vector bundle $\\E$ over $(\\mathcal{M}, g)$ equipped with a unitary connection $\\nabla = \\nabla^{\\E}$. If $x, y \\in \\mathcal{M}$ are at a distance less than the injectivity radius of $\\mathcal{M}$, denote by $C_{x \\to y}: \\E_x \\to \\E_y$ the parallel transport with respect to $\\nabla^{\\E}$ along the shortest geodesic from $x$ to $y$, by $C(x, t): \\E_x \\to \\E_{\\varphi_tx}$ the parallel transport along the flow and by $C_\\gamma$ the parallel transport along a curve $\\gamma$. For $U \\in C^\\infty(\\mathcal{M},\\mathrm{End}(\\mathcal{E}))$, we define, for all $x \\in \\mathcal{M}$, \n\\[\n\\|U\\|_x := \\Tr(U^*(x)U(x))^{1\/2},\n\\]\nand $\\|U\\|_{L^\\infty} := \\sup_{x \\in \\mathcal{M}} \\|U\\|_x$. In particular, if $U \\in C^\\infty(\\mathcal{M},\\mathrm{U}(\\mathcal{E}))$ is unitary, then $\\|U\\|_{L^\\infty} = \\sqrt{\\rk(\\E)}$. We record the following consequences of Lemma \\ref{lemma:ambrosesinger}:\n\n\n\n\\begin{lemma}\\label{lemma:ASgeometry}\nThe following consequences of the Ambrose-Singer formula hold:\n\\begin{enumerate}\n\t\\item Assume we are in the setting of Theorem \\ref{theorem:shadowing}: for some $C, \\varepsilon, T > 0$, let $x, p \\in \\mathcal{M}$ satisfy $d(\\varphi_t x, \\varphi_t p) \\leq C \\varepsilon e^{-\\theta \\min(t,T-t)}$ for all $t \\in [0, T]$. Then for any $0 \\leq T_1 \\leq T$:\n\t\\[\\|C(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\|_x \\leq \\frac{c_0C \\varepsilon}{\\theta} \\times \\|F_\\nabla\\|_{C^0},\\]\n\twhere $c_0 = c_0(X, g) > 0$ depends only on the flow $X$ and the metric.\n\t\n\t\\item Assume $\\gamma \\subset B(p, \\imath\/2)$ is a closed piecewise smooth curve at $p$ of length $L$. Then for some $C = C(g) > 0$ depending on the metric:\n\t\\[\\|C_\\gamma - \\id_p\\|_p \\leq C L \\times \\sup_{y \\in \\gamma} d(p, y) \\times \\|F_\\nabla\\|_{C^0}.\\]\n\t\n\t\\item Let $\\gamma: [0, L] \\to M$ be a unit speed curve based at $p$, and $\\nabla'$ be a second unitary connection on $\\E$, whose parallel transport along $\\gamma$ we denote by $C'_\\gamma$. Then:\n\t\\[\\|C_{\\gamma}^{-1}C'_{\\gamma} - \\id_p\\|_p \\leq L \\times \\|\\nabla - \\nabla'\\|_{C^0}.\\]\n\t\\end{enumerate}\n\\end{lemma}\nThe geometries appearing in (1), (2) and (3) are depicted in Figure \\ref{fig:ASgeometry} (A), (B) and (C), respectively.\n \n \\begin{figure}\n \\centering\n\\begin{subfigure}{0.39\\linewidth}\n             \\centering\n\\begin{tikzpicture}[scale = 0.55, everynode\/.style={scale=0.5}]\n\\tikzset{cross\/.style={cross out, draw=black, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt},\ncross\/.default={1pt}}\n\n      \n      \t\n      \n\t\t\\fill[pattern=vertical lines, pattern color=blue, draw = black] (0, 0) -- (0, 1.75) to[out = -35, in= -145, distance=75] (10, 1.75) -- (10, 0) --  (0,0);\n\t\t\n\t\t\n\t\t\\fill (0, 0) node[below left] {\\small $x$} circle (1.5pt);\n\t\t\\fill (10, 0) node[below right] {\\small $\\varphi_Tx$}  circle (1.5pt);\n\t\t\\fill (0, 1.75) node[left] {\\small $p$}  circle (1.5pt);\n\t\t\\fill (10, 1.75) node[right] {\\small $\\varphi_T p$}  circle (1.5pt);\n\n\t\t\\draw[thick, ->] (5, 1.1) -- (5, 0.6) node[above left] {\\tiny $\\varphi_t p$};\n\t\t\\draw (5, 1.25) node[right] {$\\mathcal{O}(e^{-\\theta t})$};\n\t\t\\fill (5, 0.6) circle (1pt);\n\t\t\\draw[thick, ->] (5, -0.5) -- (5, 0) node[below left] {\\tiny $\\varphi_t x$};\t\t\n\t\t\\fill (5, 0) circle (1pt);\n\\end{tikzpicture}\n             \\caption{\\small Shadowing homotopy.}\n            \n             \\end{subfigure}\n             \\begin{subfigure}{0.29\\linewidth}\n              \\centering\n             \t\t\\begin{tikzpicture}[scale=0.8]\n             \t\t\\draw (0, 0) circle (1.5);\n             \t\t\t\\draw (0, 0) node[below]{$p$}--(-1, 1)--(1,1)--(0,0);\n             \t\t\t\\fill (0, 0) circle (1pt) (1, 1) circle (1pt) (-1, 1) circle (1pt);\n             \t\t\t\\draw[blue] (0, 0)--(-0.8, 1) (0, 0)--(-0.6,1) (0, 0)--(-0.4,1) (0, 0)--(-0.2,1) (0, 0)--(-0,1) (0, 0)--(0.2,1) (0, 0)--(0.4,1) (0, 0)--(0.6,1) (0, 0)--(0.8,1);\n             \t\t\\draw (0.5, 0.4) node[right]{$\\gamma$};\n             \t\t\\end{tikzpicture}\n             \t\t \\caption{\\small Radial homotopy.}\n            \n             \\end{subfigure}\n             \\begin{subfigure}{0.29\\linewidth}\n             \\centering\n             \t\t\\begin{tikzpicture}\n             \t\t\t\\draw[thick, ->] (0, 0) node[below]{\\small $p = \\gamma(0)$}--(0.5, 0.5);\n             \t\t\t\\draw[thick] (0.5, 0.5)--(1, 1) node[right]{\\small $\\gamma(L)$};\n             \t\t\t\\fill (0, 0) circle(1pt) (1, 1) circle(1pt);\n             \t\t\n             \t\t\t\\draw[->] (0.5, 0.2)--(0.8, 0.5) node[below right]{\\small $C'_\\gamma$};\n             \t\t\t\\draw[->] (0.5, 0.8)--(0.2, 0.5) node[above left]{\\small $C_\\gamma^{-1}$};\n             \t\t\\end{tikzpicture}\n             \t\t             \t\t \\caption{\\small Straight-line homotopy.}\n            \n             \\end{subfigure}\n              \\caption{\\small Presentation of the geometries considered in Lemma \\ref{lemma:ASgeometry}.}\n             \\label{fig:ASgeometry}\n\\end{figure}\n\\begin{proof\n\tWe first prove (1). For $C, \\varepsilon$ small enough, for all $t \\in [0, T]$ we denote by $\\tau_t$ the unit speed shortest geodesic, of length $\\ell(t)$, from $\\varphi_tx$ to $\\varphi_t p$. Define a smooth homotopy $\\Gamma: [0, 1]^2 \\to \\mathcal{M}$ by setting:\n\t\\[\\Gamma(s, t) := \\tau_{tT_1}(s \\ell(t)),\\]\n\tand note that by assumption $\\ell(t) \\leq C\\varepsilon e^{-\\theta \\min(t, T - t)}$. We apply Lemma \\ref{lemma:ambrosesinger} to the homotopy $\\Gamma$ to obtain, after a rescaling of parameters $s$ and $t$:\n\t\\begin{multline*}\n\t\tC(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\\\\n\t\t= \\int_0^{T_1} \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, ds \\, dt.\n\t\\end{multline*}\n\tHere we recall $C_{\\uparrow}$ and $C_{\\rightarrow}$ are parallel transport maps obtained by parallel transport along curves as in Figure \\ref{fig:AS1}. Since $C_{\\uparrow}$ and $C_{\\rightarrow}$ are isometries, and since by compactness $|\\partial_t \\tau_t(s)| \\leq D$ for some $0 < D = D(X, g)$, we have:\n\t\\begin{multline*}\n\t\t\\|C(\\varphi_{T_1}x, -T_1)C_{\\varphi_{T_1}p \\to \\varphi_{T_1}x}C(p, T_1) C_{x \\to p} - \\mathbbm{1}_{\\E_x}\\|_x\\\\\n\t\t \\leq CD\\varepsilon \\|F_{\\nabla}\\|_{C^0} \\int_0^T e^{-\\theta \\min(t, T - t)} dt \\leq \\frac{2D C}{\\theta} \\times \\varepsilon \\|F_\\nabla\\|_{C^0}.\n\t\\end{multline*}\n\t\n\tFor (2), we may assume by approximation that $\\gamma$ is smooth. Then taking the homotopy \n\t\\[\\Gamma(s, t) = \\exp_x( t \\exp_x^{-1} (\\gamma(sL))),\\]\n\tand applying Lemma \\ref{lemma:ambrosesinger}, we obtain by a rescaling of $s$ and writing $\\widetilde{\\Gamma}(s, t) = \\Gamma(s\/L, t)$:\n\t\\[C_\\gamma - \\id_x = \\int_0^L \\int_0^1 C_1(s, t)^{-1} F_\\nabla(\\partial_s \\widetilde{\\Gamma}, \\partial_t \\widetilde{\\Gamma}) C_2(s, t) \\,dt \\,ds.\\]\n\tThe estimate now follows by using $\\|\\partial_t \\widetilde{\\Gamma}\\| \\leq C d(x, \\gamma(s))$, where we introduce the positive constant $C = \\sup_{x \\in M} \\sup_{|y|_{g_x} < \\imath\/2}\\|d\\exp_x(y)\\|_{T_xM \\to T_{\\exp_x(y)}}$.\n\t\n\tFor the final item, denote by $C_t, C_t'$ the parallel transports along $\\gamma|_{[0, t]}$ with the connection $\\nabla, \\nabla'$, respectively. Then it is straightforward that $\\partial_t(C_t^{-1}C'_t) = C_t^{-1}(\\nabla- \\nabla')(\\dot{\\gamma}(t)) C_t'$, so\n\t\\[C_\\gamma^{-1} C_\\gamma' - \\id_p = \\int_0^L C_t^{-1}(\\nabla- \\nabla')(\\dot{\\gamma}(t)) C_t' \\,dt.\\]\n\tThe required estimate follows\n\\end{proof}\n\nWe also have the following result to which we will refer to as the \\emph{spiral Lemma}:\n\n\\begin{lemma}\n\\label{lemma:spiral}\nLet $x_{\\star} \\in \\mathcal{M}$ be a periodic point of period $T_{\\star}$ and let $x_0 \\in W^s_{\\mathrm{loc}}(x_{\\star})$. Define $x_n := \\varphi_{nT_{\\star}}x_0$ and write $q_n :=  C(x_{\\star},nT_{\\star})^{-1}C_{x_n \\rightarrow x_{\\star}} C(x_0, n T_{\\star})C_{x_{\\star} \\rightarrow x_0}$. Then:\n\\[\n\\rho(x_0) := \\lim_{n \\rightarrow +\\infty} q_n \\in \\mathrm{U}(\\E_{x_{\\star}})\n\\]\nexists. Moreover, there exist some uniform constants $C,\\theta > 0$ such that\n\\[\n|q_n - \\rho(x_0)| \\leq C e^{-\\theta n}.\n\\]\n\\end{lemma}\n\n\\begin{center}\n\\begin{figure}[htbp!]\n\\includegraphics{spiral.eps}\n\\caption{The spiral Lemma: the set $\\Omega_1$ corresponds to the area over which the integral in the Ambrose-Singer formula is computed for $n=1$.}\n\\label{fig:spiral}\n\\end{figure}\n\\end{center} \n\n\\begin{proof}\nApply the Ambrose-Singer formula as in the first item of the previous Lemma (same notations as in the previous proof):\n\\[\nq_n - \\mathbbm{1}_{\\E_{x_{\\star}}} = \\int_0^{nT_{\\star}} \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, ds \\, dt,\n\\]\nwhere $\\tau_t$ is the unit speed shortest geodesic of length $\\ell(t)$ from $\\varphi_t x_0$ and $\\varphi_t x_{\\star}$. Observe that this integral converges absolutely as (see \\eqref{equation:distance}):\n\\[\n\\int_0^{nT_{\\star}} \\norm{ \\int_0^{\\ell(t)} C_{\\uparrow}^{-1}(s, t) F_\\nabla(\\partial_t \\tau_t(s), \\partial_s \\tau_t(s)) C_{\\rightarrow}(s, t) \\, \\dd s \\, } \\dd t \\leq \\int_0^{n T_{\\star}} C \\|F_{\\nabla}\\|_{C^0} e^{-\\theta t} \\dd t < \\infty,\n\\]\nand thus the limit exists. Moreover, it is clear that the convergence is exponential.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of the exact Liv\\v{s}ic cocycle Theorem}\n\n\\label{section:weak-strong}\n\n\n\n\\subsubsection{Parry's free monoid}\n\nAs we shall see, Parry's free monoid is the key notion to understand the holonomy of unitary connections.\nWhereas flat connections up to gauge equivalence correspond to representations of the fundamental group up to conjugacy, in the setting of hyperbolic dynamics, we will show that \\emph{arbitrary} connections up to cocycle equivalence correspond to representations of Parry's free monoid. \nRecall from \\S\\ref{sssection:homoclinic} that $x_{\\star} \\in \\mathcal{M}$ is a periodic point of period $T_{\\star}$. Let $\\mathbf{G}$ be the free monoid generated by $\\mathcal{H}$ (homoclinic orbits to $x_{\\star}$), namely the formal set of words\n\\[\n\\mathbf{G} := \\left\\{\\gamma_1^{m_1} ... \\gamma_k^{m_k} ~|~k \\in \\mathbb{N}, m_1,...,m_k \\in \\mathbb{N}_0, \\gamma_1,...,\\gamma_k \\in \\mathcal{H}\\right\\},\n\\]\nendowed with the obvious monoid structure. The empty word corresponds to the identity element denoted by $\\mathbf{1}_{\\mathbf{G}}$. Note the periodic orbit corresponding to $x_{\\star}$ also belongs to the set of homoclinic orbits. We call $\\mathbf{G}$ \\emph{Parry's free monoid} as the idea (although not written like this) was first introduced in his work \\cite{Parry-99} (see also \\cite{Schmidt-99} for a related approach).\nThe main result of this paragraph is the following:\n\n\\begin{prop}\n\\label{proposition:representation0}\nLet $\\nabla^{\\E}$ be a unitary connection on the Hermitian vector bundle $\\E \\rightarrow \\mathcal{M}$. Then $\\nabla^{\\E}$ induces a representation\n\\[\n\\rho : \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_{\\star}}).\n\\]\n\\end{prop}\n\nFormally, this proposition could have also been stated as a definition.\n\n\\begin{proof}\nSince $\\mathbf{G}$ is a free monoid, it suffices to define $\\rho$ on the set of generators of $\\mathbf{G}$, namely for all homoclinic orbits $\\gamma \\in \\mathcal{H}$. For the neutral element we  set $\\rho(\\mathbf{1}_{\\mathbf{G}}) = \\mathbbm{1}_{\\E_{x_{\\star}}}$. For the periodic orbit $\\gamma_{\\star}$ of $x_{\\star}$, we set $\\rho(\\gamma_{\\star}) := C(x_{\\star},T_{\\star})$.\n\nLet $\\gamma \\in \\mathcal{H}$ (and $\\gamma \\neq \\gamma_{\\star}$) and consider a parametrization $\\mathbb{R} \\ni t \\mapsto \\gamma(t)$. Following the notations of \\S\\ref{sssection:homoclinic}, we let $x_n^\\pm := \\gamma(A_\\pm \\pm nT_{\\star}), x_n^+ = \\varphi_{T_n}(x_n^-)$ for some $T_n = A_+-A_-+2n T_{\\star}$, where $T_\\gamma := A_+ - A_-$ (length of the trunk), and the points $(x_n^\\pm)_{n \\in \\mathbb{N}}$ converge exponentially fast to $x_{\\star}$ as $n \\rightarrow \\infty$. As we shall see, there is a small technical issue coming from the fact that $C(x_{\\star},T_{\\star})$ is not trivial and this can be overcome by considering a subsequence $k_n \\rightarrow \\infty$ such that\\footnote{For any compact metric group $G$, if $g \\in G$, there exists a sequence $k_n \\in \\mathbb{N}$ such that $g^{k_n} \\rightarrow_{n \\rightarrow \\infty} \\mathbf{1}_G$.}\n\\begin{equation}\\label{eq:k_n}\n\tC(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\E_{x_{\\star}}}, \\quad n \\to \\infty.\\end{equation}\n\n\n\nFor $n,m \\in \\mathbb{N}$, we define $\\rho_{m, n}(\\gamma) \\in \\mathrm{U}(\\E_{x_{\\star}})$ as follows: \n\\begin{equation}\\label{eq:rhomn}\n\\rho_{m,n}(\\gamma) :=  C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x_0^+, k_m T_{\\star}) C(x_0^-, T_\\gamma)C(x_{k_n}^-,k_nT_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-},\n\\end{equation}\nand we will write $\\rho_{n}(\\gamma) := \\rho_{n,n}(\\gamma)$.\n\n\\begin{lemma}\n\\label{lemma:convergence}\nThere exists $\\rho(\\gamma) \\in \\mathrm{U}(\\E_{x_{\\star}})$ such that:\n\\[\n\\rho_{m,n}(\\gamma) \\rightarrow_{n,m \\rightarrow \\infty} \\rho(\\gamma),\n\\]\nand $\\rho(\\gamma)$ does not depend in which sense the limit in $n,m$ is taken. \n\\end{lemma}\n\n\n\\begin{proof}\nWe have by construction:\n\\begin{equation}\n\\label{equation:cv0}\n\\begin{split}\n\\rho_{m,n}(\\gamma) &   = C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x^+_0, k_m T_{\\star})C(x^-_0, T_\\gamma)C(x_{k_n}^-,k_n T_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-}\\\\\n& = \\left[C_{x_{k_m}^+ \\rightarrow x_{\\star}} C(x^+_0, k_m T_{\\star}) C_{x_{\\star} \\rightarrow x^+_0} C(x_{\\star},k_mT_{\\star})^{-1}\\right] \\\\\n& \\hspace{2cm}  \\times C(x_{\\star},k_mT_{\\star}) C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} C(x_{\\star},k_nT_{\\star}) \\\\\n& \\hspace{2cm} \\times \\left[C(x_{\\star},k_nT_{\\star})^{-1}C_{x^-_0 \\rightarrow x_{\\star}}C(x_{k_n}^-,k_n T_{\\star}) C_{x_{\\star} \\rightarrow x_{k_n}^-}\\right],\n\\end{split}\n\\end{equation}\nwhere $T_\\gamma$ is independent of $n$ (trunk of $\\gamma$). For the middle term we have by \\eqref{eq:k_n}\n\\[\nC(x_{\\star},k_mT_{\\star}) C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} C(x_{\\star},k_nT_{\\star}) = C_{x^+_0 \\rightarrow x_{\\star}} C(x^-_0, T_\\gamma)C_{x_{\\star} \\rightarrow x^-_0} + o(1),\n\\]\nas $n,m$ go to $+\\infty$. Moreover the convergence of the terms between brackets follow from the spiral Lemma \\ref{lemma:spiral} (the convergence is exponentially fast).\n\\end{proof}\n\nThis concludes the proof.\n\n\\end{proof}\n\n\\begin{remark}\n\\rm\nFor $\\gamma \\in \\mathcal{H}$, \\eqref{equation:cv0} shows that $\\rho(\\gamma)$ does not depend on the choice of subsequence $(k_n)_{n \\in \\mathbb{N}}$ as long as it satisfies $C(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}$. However, $\\rho(\\gamma)$ does depend on the choice of trunk $[x_0^-x_0^+]$ for $\\gamma$ and another choice of trunk produces a $\\rho'(\\gamma)$ which differs from $\\rho(\\gamma)$ by:\n\\begin{equation}\n\\label{equation:differs}\n\\rho'(\\gamma) = C(x_{\\star},T_{\\star})^{m_L(\\gamma)} \\rho(\\gamma) C(x_{\\star},T_{\\star})^{m_R(\\gamma)},\n\\end{equation}\nwhere $m_L(\\gamma), m_R(\\gamma) \\in \\mathbb{Z}$. \n\\end{remark}\n\n\n\\subsubsection{Conjugate representations}\n\nWe introduce the submonoid $\\mathbf{G}^* := \\mathbf{G} \\setminus \\left\\{ \\gamma_{\\star}^k, k \\geq 1\\right\\}$ that is $\\mathbf{G}$ minus powers of $\\gamma_{\\star}$. Recall that the character of a representation $\\rho$ is defined by $\\chi_\\rho(\\bullet) := \\Tr(\\rho(\\bullet))$. This paragraph is devoted to proving the following:\n\n\\begin{prop}\n\\label{proposition:representation}\nLet $\\nabla^{\\E_{1,2}}$ be two unitary connections on the Hermitian vector bundles $\\E_1,\\E_2 \\rightarrow \\mathcal{M}$. Assume that the connections have trace-equivalent holonomies in the sense of Definition \\ref{definition:equivalence}. Then, the induced representations $\\rho_{1,2} : \\mathbf{G}^* \\rightarrow \\mathrm{U}({\\E_{1,2}}_{x_{\\star}})$ have the same character. In particular, this implies that they are isomorphic, i.e. there exists $p_{\\star} \\in \\mathrm{U}({\\E_2}_{x_{\\star}}, {\\E_1}_{x_{\\star}})$ such that: \n\\begin{equation}\\label{eq:repconj}\n\\forall \\gamma \\in \\mathbf{G}, ~~ \\rho_1(\\gamma) = p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}.\n\\end{equation}\n\\end{prop}\n\nFollowing Lemma \\ref{lemma:convergence}, we consider a subsequence $(k_n)_{n \\in \\mathbb{N}}$ such that $C_{1,2}(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}$. \n\n\\begin{proof}\nOnce we know that the representations have the same character, the conclusion is a straightforward consequence of a general fact of representation theory, see \\cite[Corollary 3.8]{Lang-02}. \nFor the sake of simplicity, we take $\\gamma = \\gamma_1 \\cdot \\gamma_2$, where $\\gamma_{1,2} \\in \\mathcal{H}$ (and both $\\gamma_{1,2}$ cannot be equal to $\\gamma_{\\star}$ at the same time since the word $\\gamma$ is in $\\mathbf{G}^*$) but the generalization to longer words is straightforward as we shall see and words of length $1$ are also handled similarly (one does not even need to concatenate orbits in this case). The empty word (corresponding to the identity element in $\\mathbf{G}^*)$ will also be dealt with separately. This proposition is based on the shadowing Theorem \\ref{theorem:shadowing} and the fact that one can concatenate orbits. But we will have to be careful to produce periodic orbits which are primitive.\n\nWe have by Lemma \\ref{lemma:convergence}:\n\\[\n\\rho_1(\\gamma) = \\rho_1(\\gamma_1) \\rho_1(\\gamma_2) = \\rho_{1 ; n,N}(\\gamma_1)\\rho_{1 ; n,n}(\\gamma_2) + o(1), \\quad n \\to \\infty,\n\\]\nwhere we use the convention $\\rho_{i;a,b}$ to denote the expression in \\eqref{eq:rhomn} with respect to $\\nabla^{\\E_i}$, for $i=1,2$. The term $N = N(n) \\geq n$ will ensure that a certain orbit is primitive as we shall see below. Let $x_{n}^\\pm(i)$ be the points on the orbit $\\gamma_i$ that are exponentially close to $x_{\\star}$, given by \\S\\ref{sssection:homoclinic}. Consider the concatenation of the orbits $S :=  [x_{k_{N}}^-(1) x_{k_{n}}^+(1)] \\cup  [x_{k_n}^-(2) x_{k_n}^+(2)]$. Note that the starting points and endpoints of these segments are at distance at most $\\mathcal{O}(e^{-\\theta k_n})$. Thus by the shadowing Theorem \\ref{theorem:shadowing}, there exists a genuine periodic orbit $\\widetilde{\\gamma_n}$ and a point $y_n \\in \\widetilde{\\gamma_n}$ (of period $T'_n$) which $\\mathcal{O}(e^{-\\theta k_n})$-shadows the concatenation $S$ (here, if we have a longer word of length $k$, it suffices to apply the shadowing Theorem \\ref{theorem:shadowing} with $k$ segments). \n\nWe claim that $\\widetilde{\\gamma_n}$ is primitive for all $N$ large enough. Indeed, observe that $\\widetilde{\\gamma_n}$ can be decomposed into the following six subsegments:\n\n\\begin{center}\n\\begin{figure}[htbp!]\n\\includegraphics{primitive.eps}\n\\caption{The orbit $\\widetilde{\\gamma_n}$ is made of six segments: in the first segment (of length $k_{n}T_{\\star}$), it shadows the first portion $[x_{k_{n}}^-(2) x_{0}^-(2)]$ which wraps around $\\gamma_{\\star}$; in the second (of length $T_{\\gamma_2}$), it shadows the trunk $[x_{0}^-(2) x_{0}^+(2)]$, in the third (of length $k_{n}T_{\\star}$), it shadows the last portion of $[x_{0}^+(2) x_{k_{n}}^+(2)]$ which also wraps around $\\gamma_{\\star}$; then this process is repeated but for the second orbit $\\gamma_1$.}\n\\label{fig:primitive}\n\\end{figure}\n\\end{center} \n\nMoreover, the total length of $\\widetilde{\\gamma_n}$ is\n\\[\nT'_n = T_{\\gamma_2} + 2 k_n T_{\\star} + T_{\\gamma_1} + (k_{N}+k_n) T_{\\star} + \\mathcal{O}(e^{-\\theta k_n}).\n\\]\nTake $x \\in \\gamma_1 \\cup \\gamma_2$ with $x \\not \\in \\gamma_{\\star}$, and consider a small $\\varepsilon > 0$ such that $d(x, \\gamma_{\\star}) > 3\\varepsilon$. Let $n$ large enough so that for all $m \\geq n$ we have the tail $[x_m^-(1) x_n^-(1)] \\subset B(\\gamma_{\\star}, \\varepsilon)$, $\\widetilde{\\gamma_n}$ satisfies $d(\\widetilde{\\gamma_n}, x) < \\varepsilon$ and finally, so that the shadowing factor of Theorem \\ref{theorem:shadowing} satisfies $\\mathcal{O}(e^{-\\theta k_n}) < \\varepsilon$. Pick $N \\geq n$ such that $(k_{N} - k_n) T_{\\star} > T'_n\/2$. We argue by contradiction and assume that $\\widetilde{\\gamma_n} = \\gamma_0^k$ for some $k \\geq 2$ and $\\gamma_0 \\in \\mathcal{G}^\\sharp$, a primitive orbit.\n\nThis implies that there is a copy of $\\gamma_0$ in the central red segment of Figure \\ref{fig:primitive} which $\\mathcal{O}(e^{-\\theta k_n})$-shadows the orbit of $x_N^-(1)$ and this forces $\\widetilde{\\gamma_n} \\subset B(\\gamma_{\\star}, 2\\varepsilon)$. Thus $d(\\widetilde{\\gamma_n}, x) > \\varepsilon$, which is a contradiction.\n\n\nBy the first and second items of Lemma \\ref{lemma:ASgeometry}, we have:\n\\[\n\\rho_{1;n,N}(\\gamma_1)\\rho_{1;n,n}(\\gamma_2) = C_{1,y_n \\rightarrow x_{\\star}} C_1(y_n, T'_n)C_{1,y_n \\rightarrow x_{\\star}}^{-1} + \\mathcal{O}(e^{-\\theta k_n}).\n\\]\nBy assumption, we have $\\Tr(C_1(y_n, T'_n)) = \\Tr(C_2(y_n,T'_n))$. This yields:\n\\[\n\\begin{split}\n\\Tr(\\rho_1(\\gamma)) & = \\Tr(C_{1,y_n \\rightarrow x_{\\star}} C_1(y_n, T'_n)C_{1,y_n \\rightarrow x_{\\star}}^{-1} ) + o(1) \\\\\n& = \\Tr(C_1(y_n, T'_n)) + o(1) \\\\\n& = \\Tr(C_2(y_n, T'_n)) + o(1)  = \\Tr(\\rho_2(\\gamma)) + o(1).\n\\end{split}\n\\]\nTaking the limit as $n \\rightarrow \\infty$, we obtain the claimed result about characters for all non-empty words $\\gamma \\in \\mathbf{G}^*$. \n\nIt remains to deal with the empty word. For that, take any $\\gamma \\in \\mathbf{G}^*$, and consider $n_i \\in \\mathbb{N}$, a subsequence such that $\\rho_1(\\gamma)^{n_i} \\rightarrow \\mathbbm{1}_{{\\E_1}_{x_{\\star}}}$ and $\\rho_2(\\gamma)^{n_i} \\rightarrow \\mathbbm{1}_{{\\E_2}_{x_{\\star}}}$. Then:\n\\[\n\\Tr\\left(\\rho_1(\\gamma)^{n_i}\\right) =\\Tr\\left(\\rho_2(\\gamma)^{n_i}\\right),\n\\]\nand taking the limit as $i \\rightarrow \\infty$ gives\n\\[\n\\Tr(\\rho_1(\\mathbf{1}_{\\mathbf{G}^*}))=\\rk(\\E_1)=\\rk(\\E_2) =\\Tr(\\rho_2(\\mathbf{1}_{\\mathbf{G}^*})).\n\\]\nBy the mentioned \\cite[Corollary 3.8]{Lang-02}, there is a $p_{\\star}$ satisfying \\eqref{eq:repconj} for $\\gamma \\in \\mathbf{G}^*$.\n\nIt is now straightforward to show \\eqref{eq:repconj} for all $\\gamma \\in \\mathbf{G}$. Applying \\eqref{eq:repconj} with $\\gamma_{\\star} \\gamma \\in \\mathbf{G}^*$, where $\\gamma \\in \\mathcal{H}\\setminus \\{\\gamma_{\\star}\\}$ is arbitrary, we get that:\n\\[\n\\begin{split}\n\\rho_1(\\gamma_{\\star} \\gamma)&  = \\rho_1(\\gamma_{\\star}) \\rho_1(\\gamma) \\\\\n& = p_{\\star} \\rho_2(\\gamma_{\\star} \\gamma) p_{\\star}^{-1} = p_{\\star} \\rho_2(\\gamma_{\\star}) p_{\\star}^{-1} p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}.\n\\end{split}\n\\]\nSince $\\rho_1(\\gamma) = p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1}$ (because $\\gamma \\in \\mathbf{G}^*$), we get that $\\rho_1(\\gamma_{\\star}) = p_{\\star} \\rho_2(\\gamma_{\\star}) p_{\\star}^{-1}$, that is $C_1(x_{\\star},T_{\\star}) = p_{\\star} C_2(x_{\\star},T_{\\star}) p_{\\star}^{-1}$ or equivalently $P(x_{\\star},T_{\\star})p_{\\star} = p_{\\star}$ (where $P$ denotes the parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to the mixed connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X$, as in \\eqref{equation:tp-mixed}), concluding the proof.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nAlthough the representations $\\rho_{1,2}$ depend on choices (namely on a choice of trunk for each homoclinic orbit $\\gamma \\in \\mathcal{H}$), the map $p_{\\star} \\in \\mathrm{U}(\\E_{x_{\\star}})$ does not. Indeed, taking $\\rho_{1,2}'$ two other representations (for some other choices of trunks), one gets by \\eqref{equation:differs}:\n\\[\n\\begin{split}\n\\rho'_1(\\gamma) & = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} \\rho_1(\\gamma) C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n&  = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} p_{\\star} \\rho_2(\\gamma) p_{\\star}^{-1} C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n& = C_1(x_{\\star},T_{\\star})^{m_L(\\gamma)} p_{\\star} C_2(x_{\\star},T_{\\star})^{-m_L(\\gamma)} \\rho'_2(\\gamma) C_2(x_{\\star},T_{\\star})^{-m_R(\\gamma)} p_{\\star}^{-1} C_1(x_{\\star},T_{\\star})^{m_R(\\gamma)} \\\\\n& = \\left(P(x_{\\star},m_L(\\gamma)T_{\\star})p_{\\star}\\right) \\rho'_2(\\gamma) \\left(P(x_{\\star},m_R(\\gamma)T_{\\star})p_{\\star}\\right)^{-1}  = p_{\\star} \\rho'_2(\\gamma) p_{\\star}^{-1},\n\\end{split}\n\\]\nsince $P(x_{\\star},T_{\\star})p_{\\star}=p_{\\star}$, that is $p_{\\star}$ also conjugates the representations $\\rho'_{1,2}$. Note that the map $p_{\\star}$ given by \\cite[Corollary 3.8]{Lang-02} is generally not unique. Nevertheless, if the representation is irreducible, it is unique modulo the trivial $\\mathbb{S}^{1}$-action.\n\\end{remark}\n\n\n\\subsubsection{Proof of Theorem \\ref{theorem:weak}}\n\nWe can now complete the proof of Theorem \\ref{theorem:weak}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:weak}]\nLet $\\mathcal{W}$ be the set of all points belonging to homoclinic orbits in $\\mathcal{H}$. By Lemma \\ref{lemma:dense}, $\\mathcal{W}$ is dense in $\\mathcal{M}$ and we are going to define the map $p$ (which will conjugate the cocycles) on $\\mathcal{W}$ and then show that $p$ is Lipschitz-continuous on $\\mathcal{W}$ so that it extends naturally to $\\mathcal{M}$. The map $p$ is defined as the parallel transport of $p_{\\star}$ with respect to the mixed connection.\n\nBy assumptions, we have $C_i(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\E_*}$, and thus $P(x_{\\star},T_{\\star})^{k_n} \\rightarrow \\mathbbm{1}_{\\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}} )}$ (where we use the notation $\\mathbbm{1}_{\\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}} )}(q) = q$ for $q \\in \\mathrm{Hom}({\\E_2}_{x_{\\star}},{\\E_1}_{x_{\\star}})$). Consider a point $x \\in \\gamma$, where $\\gamma \\in \\mathcal{H}$ is a homoclinic orbit and also consider a parametrization of $\\gamma$ as in \\S\\ref{sssection:homoclinic}. For $n \\in \\mathbb{N}$ large enough, consider the point $x_n^- \\in \\gamma$ (which is exponentially close to $x_{\\star}$) and write $x = \\varphi_{T^-_n}(x_n^-)$ for some $T^-_n > 0$. Define:\n\\[\np^-_n(x) := P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x}).\n\\]\n\n\\begin{lemma}\n\\label{lemma:p-minus}\nFix $\\gamma \\in \\mathcal{H}$. Then for all $x \\in \\gamma$, there exists $p_-(x) \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x} )$ such that $p^-_n(x) \\rightarrow_{n \\rightarrow \\infty} p_-(x)$. There exists $C > 0$ such that: $|p^-_n(x)-p_-(x)| \\leq C\/n$. Moreover, $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X p_- = 0$ on $\\gamma$. \n\\end{lemma}\n\nIn particular, this shows that $p_-$ is smooth in restriction to $\\gamma$ as $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X$ is elliptic on $\\gamma$.\n\n\\begin{proof}\nBy construction, the differential equation is clearly satisfied if the limit exists. Moreover, we have for some time $T_0$ (independent of $n$, $T^-_{k_n} = T_0 + k_n T_{\\star}$):\n\\[\n\\begin{split}\np^-_n(x) & = P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star}\\\\\n&  = P(x_0^-, T_0)P(x_{k_n}^-,k_n T_{\\star}) P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\\\\n& = P(x_0^-, T_0) P_{x_{\\star} \\rightarrow x_0^-} P(x_{\\star},T_{\\star})^{k_n} \\left[P(x_{\\star},T_{\\star})^{-k_n}P_{x_0^- \\rightarrow x_{\\star}} P(x_{k_n}^-,k_n T_{\\star}) P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star}\\right].\n\\end{split}\n\\]\nBy assumption, the term outside the bracket converges as $n \\rightarrow \\infty$ and the term between brackets converges by the spiral Lemma \\ref{lemma:spiral}.\n\\end{proof}\n\nWe now claim the following:\n\n\\begin{lemma}\n\\label{lemma:stable}\nThere exists a uniform constant $C > 0$ such that the following holds. Assume that $x$ and $z$ belong to two homoclinic orbits in $\\mathcal{H}$ and $z \\in W^u_{\\mathrm{loc}}(x)$. Then:\n\\[\n\\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\leq C d(x,z).\n\\]\n\\end{lemma}\n\nBy the previous proofs, the point $x$ is associated to points $x^-_n$ on the homoclinic orbit and we will use the same notations for the point $z$ associated to the points $z^-_n$.\n\n\\begin{proof}\nThere is here a slight subtlety coming from the fact that the parametrizations of the homoclinic orbits $\\gamma$ were chosen in a non-canonical way (via a choice of $A_\\pm$). In particular, it is not true that the flowlines of $z_{k_n}^-$ and $x_{k_n}^-$ shadow each other; in other words, we might not have $T^-_{k_n}(z) = T^-_{k_n}(x)$ but we rather have $T^-_{k_n}(z) = T^-_{k_n}(x) + mT_{\\star}$ for some integer $m \\in \\mathbb{Z}$ depending on both $x$ and $z$.\n\nWe have:\n\\[\n\\begin{split}\n& \\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\\\\n&  = \\|P_{x \\rightarrow z} p^-_n(x) - p^-_n(z)\\| + o(1) \\\\\n& = \\|P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} - P(z_{k_n}^-,T^-_{k_n}(z))P_{x_{\\star} \\rightarrow z_{k_n}^-} p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n}^-,T^-_{k_n}(z))^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n},mT_{\\star})^{-1}P_{x_{\\star} \\rightarrow z_{k_n-m}^- } \\\\\n& \\hspace{1.2cm} \\times \\left[P_{z_{k_n-m}^- \\rightarrow x_{\\star}}P(z_{k_n-m}^-,T^-_{k_n}(z)-mT_{\\star})^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-}\\right]p_{\\star} - p_{\\star}\\| + o(1).\n\\end{split}\n\\]\nApplying the first item of Lemma \\ref{lemma:ASgeometry}, we have that:\n\\[\n\\|P_{z_{k_n-m}^- \\rightarrow x_{\\star}}P(z_{k_n-m}^-,T^-_{k_n}(z)-mT_{\\star})^{-1} P_{x \\rightarrow z} P(x_{k_n}^-,T^-_{k_n}(x))P_{x_{\\star} \\rightarrow x_{k_n}^-} - \\mathbbm{1}_{\\mathrm{End}(\\E_{x_{\\star}})}\\| \\leq Cd(x,z),\n\\]\nwhere the constant $C > 0$ is uniform in $n$. Moreover, observe that\n\\[\n\\lim_{n \\rightarrow \\infty} P_{z_{k_n}^- \\rightarrow x_{\\star}}P(z_{k_n},mT_{\\star})^{-1}P_{x_{\\star} \\rightarrow z_{k_n-m}^- } = P(x_{\\star},mT_{\\star})^{-1}.\n\\]\n\n\nHence:\n\\[\n \\|P_{x \\rightarrow z}p_-(x) - p_-(z)\\| \\leq C\\left(\\| P(x_{\\star},mT_{\\star})^{-1}p_{\\star} - p_{\\star}\\|  + d(x,z) + o(1)\\right).\n \\]\nUsing that $P(x_{\\star},T_{\\star})p_{\\star}=p_{\\star}$, we get that the first term on the right-hand side vanishes. Taking the limit as $n \\rightarrow +\\infty$, we obtain the announced result.\n\\end{proof}\n\nNote that we could have done the same construction ``in the future\" by considering instead:\n\\[\np_+(x) = \\lim_{n \\rightarrow \\infty} P(x,T^+_{k_n})^{-1} P_{x_{\\star} \\rightarrow x_{k_n}^+} p_{\\star} \\in \\mathrm{U}({\\E_2}_{x},{\\E_1}_{x}),\n\\]\nwhere $x_n^+ := \\varphi_{T^+_n}(x)$ is exponentially closed to $x_{\\star}$ as in \\S\\ref{sssection:homoclinic}. A similar statement as Lemma \\ref{lemma:stable} holds with the unstable manifold being replaced by the stable one. We have:\n\n\\begin{lemma}\n\\label{lemma:equality}\nFor all $x \\in \\mathcal{W}$, $p_-(x) = p_+(x)$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from Proposition \\ref{proposition:representation}. Indeed, we have:\n\\[\n\\begin{split}\n\\|p_-(x)-p_+(x)\\| & = \\|P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} - P(x,T^+_{k_n})^{-1} P_{x_{\\star} \\rightarrow x_{k_n}^+} p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x,T^+_{k_n})P(x_{k_n}^-,T^-_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1) \\\\\n& \\leq C \\| P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x_{k_n}^-,T_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} - p_{\\star}\\| + o(1),\\\\\n\\end{split}\n\\]\nwhere $T_n := T_n^- + T_n^+$. Observe that:\n\\[\n\\begin{split}\n& P_{x_{k_n}^+ \\rightarrow x_{\\star} }P(x_{k_n}^-,T_{k_n})P_{x_{\\star} \\rightarrow x_{k_n}^-}p_{\\star} \\\\\n&  \\hspace{2cm}= C_{1,x_{k_n}^+ \\rightarrow x_{\\star}} C_1(x_{k_n}^-,T_{k_n}) C_{1,x_{\\star} \\rightarrow x_{k_n}^-} p_{\\star} \\left(C_{2,x_{k_n}^+ \\rightarrow x_{\\star}} C_2(x_{k_n}^-,T_{k_n}) C_{2,x_{\\star} \\rightarrow x_{k_n}^-} \\right)^{-1} \\\\\n&  \\hspace{2cm} = \\rho_{1,n}(\\gamma) p_{\\star} \\rho_{2,n}(\\gamma)^{-1}  = \\rho_1(\\gamma)p_{\\star} \\rho_2(\\gamma)^{-1} + o(1) = p_{\\star} + o(1),\n\\end{split}\n\\]\nby Proposition \\ref{proposition:representation}. Hence $\\|p_-(x)-p_+(x)\\| = o(1)$, that is $p_-(x)=p_+(x)$.\n\\end{proof}\n\nWe can now prove the following lemma:\n\n\\begin{lemma}\n\\label{lemma:lipschitz}\nThe map $p_-$ is Lipschitz-continuous.\n\\end{lemma}\n\n\\begin{proof}\nConsider $x, y \\in \\mathcal{W}$ which are close enough. Let $z := [x,y] = W^{wu}_{\\mathrm{loc}}(x) \\cap W^{s}_{\\mathrm{loc}}(y)$ and define $\\tau$ such that $\\varphi_\\tau(z) \\in W^u_{\\mathrm{loc}}(x)$. Note that $|\\tau| \\leq Cd(x,y)$ for some uniform constant $C > 0$; also observe that the point $z$ is homoclinic to the periodic orbit $x_{\\star}$. We have:\n\\[\n\\begin{split}\n& \\|p_-(x)-P_{y \\rightarrow x}p_-(y)\\| \\\\\n& \\leq \\|p_-(x)-P_{z \\rightarrow x}p_-(z)\\| + \\|p_-(z)-p_+(z)\\| + \\|P_{z \\rightarrow x}p_+(z)-P_{y \\rightarrow x}p_+(y)\\| + \\|p_+(y)-p_-(y)\\| \\\\\n& \\leq \\|p_-(x)-P_{z \\rightarrow x}p_-(z)\\| + \\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\| \\\\\n& \\leq  \\|p_-(x)-P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))\\| + \\|P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))- P_{z \\rightarrow x}p_-(z)\\| + \\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\| \n\\end{split}\n\\]\nwhere the terms disappear between the second and third line by Lemma \\ref{lemma:equality}. By Lemma \\ref{lemma:stable}, the first term is controlled by:\n\\[\n\\|p_-(x)-P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))\\| \\leq Cd(x,\\varphi_{\\tau}(z)) \\leq Cd(x,y).\n\\]\nAs to the second term, using the second item of Lemma \\ref{lemma:ASgeometry}, we have:\n\\[\n\\|P_{\\varphi_{\\tau}(z) \\rightarrow x} p_-(\\varphi_{\\tau}(z))- P_{z \\rightarrow x}p_-(z)\\| = \\|P_{x \\rightarrow z} P_{\\varphi_{\\tau}(z) \\rightarrow x} P(z,\\tau)p_-(z)- p_-(z)\\| \\leq Cd(x,y).\n\\]\nEventually, the last term $\\|P_{x \\rightarrow y}P_{z \\rightarrow x}p_+(z)-p_+(y)\\|$ is controlled similarly to the first term by applying Lemma \\ref{lemma:stable} (but with the stable manifold instead of unstable).\n\\end{proof}\n\nAs $\\mathcal{W}$ is dense, $p_-$ extends to a Lipschitz-continuous map on $\\mathcal{M}$ which satisfies the equation $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_2},\\nabla^{\\E_1})}_X p_- = 0$ and by \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, this implies that $p_-$ is smooth. This concludes the proof of the Theorem.\n\n\\end{proof}\n\n\n\n\\subsubsection{Proof of the geometric properties}\n\nWe now prove Proposition \\ref{proposition:opaque}.\n\n\\begin{proof}[Proof of Proposition \\ref{proposition:opaque}]\nThe equivalence between (1) and (2) can be found in \\cite[Section 5]{Cekic-Lefeuvre-20}. If $\\mathcal{F} \\subset \\mathcal{E}$ is a non-trivial subbundle that is invariant by parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$, it is clear that $\\rho$ will leave the space $\\mathcal{F}_{x_{\\star}}$ invariant and thus is not irreducible. Conversely, if $\\rho$ is not irreducible, then there exists a non-trivial $\\mathcal{F}_{x_{\\star}} \\subset \\E_{x_{\\star}}$ preserved by $\\rho$. Let $\\pi_{\\star} : \\mathcal{E}_{x_{\\star}} \\rightarrow \\mathcal{F}_{x_{\\star}}$ be the orthogonal projection. For $x$ on a homoclinic orbit, define $\\pi(x) : \\E_x \\rightarrow \\E_x$ similarly to $p_-$ in Lemma \\ref{lemma:p-minus} by parallel transport of the section $\\pi_{\\star}$ with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$. Following the previous proofs (we only use $\\rho \\pi_\\star = \\pi_\\star \\rho$), one shows that $\\pi$ extends to a Lipschitz-continuous section on homoclinic orbits which satisfies $\\pi^2 = \\pi$ and $\\nabla^{\\mathrm{End}}_X \\pi = 0$. By \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, $\\pi$ extends to a smooth section i.e. $\\pi \\in C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))$. Moreover, $\\pi(x_{\\star})=\\pi_{\\star}$, hence $\\pi$ is the projection onto a non-trivial subbundle $\\mathcal{F} \\subset \\mathcal{E}$.\n\\end{proof}\n\n\n\nWe now prove Theorem \\ref{theorem:iso}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:iso}]\nThe linear map $\\Phi : \\mathbf{R}' \\rightarrow \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$ is defined in the following way. Consider $u_{\\star} \\in \\mathbf{R}'$ and define, as in Lemma \\ref{lemma:p-minus}, for $x$ on a homoclinic orbit, $u_-(x)$ as the parallel transport of $u_{\\star}$ from $x_{\\star}$ to $x$ along the orbit (with respect to the endomorphism connection $\\nabla^{\\mathrm{End}(\\E)}$). Similarly, one can define $u_+(x)$ by parallel transport from the future. The fact that $u_{\\star} \\in \\mathbf{R}'$ is then used in the following observation (see Lemma \\ref{lemma:equality}):\n\\[\n\\|u_-(x)-u_+(x)\\| = \\|\\rho(\\gamma)u_{\\star} \\rho(\\gamma)^{-1} - u_\\star\\| = 0.\n\\]\n(Note that $\\rho(\\gamma)u_{\\star} \\rho(\\gamma)^{-1}$ corresponds formally to the parallel transport of $u_{\\star}$ with respect to $\\nabla^{\\mathrm{End}(\\E)}$ from $x_{\\star}$ to $x_{\\star}$ along the homoclinic orbit $\\gamma$.) Hence, following Lemma \\ref{lemma:lipschitz}, we get that $u_-$ is Lipschitz-continuous and satisfies $\\nabla^{\\mathrm{End}(\\E)}_X u_- = 0$. By \\cite[Theorem 4.1]{Bonthonneau-Lefeuvre-20}, it is smooth and we set $u_- := \\Phi(u_{\\star}) \\in \\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$.\n\nAlso observe that this construction is done by using parallel transport with respect to the unitary connection $\\nabla^{\\mathrm{End}(\\E)}$. As a consequence, if $u_\\star, u_\\star' \\in \\mathbf{R}$ are orthogonal (i.e. $\\Tr(u_\\star^* u_\\star') = 0$), then $\\Phi(u_{\\star})$ and $\\Phi(u'_\\star)$ are also pointwise orthogonal. This proves that $\\Phi$ is injective.\n\nIt now remains to show the surjectivity of $\\Phi$. Let $u \\in\\ker \\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty(\\mathcal{M},\\mathrm{End}(\\E))}$. Following \\cite[Section 5]{Cekic-Lefeuvre-20}, we can write $u = u_R + i u_I$, where $u_R^*=u_R,u_I^*=u_I$ and $\\nabla^{\\mathrm{End}(\\E)}_X u_R = \\nabla^{\\mathrm{End}(\\E)}_X u_I = 0$. By \\cite[Lemma 5.6]{Cekic-Lefeuvre-20}, we can then further decompose $u_R = \\sum_{i=1}^p \\lambda_i \\pi_{\\mathcal{F}_i}$ (and same for $u_I$), where $\\lambda_i \\in \\mathbb{R}$, $p \\in \\mathbb{N}$ and $\\mathcal{F}_i \\subset \\E$ is a maximally invariant subbundle of $\\E$ (i.e. it does not contain any non-trivial subbundle that is invariant under parallel transport along the flowlines of $(\\varphi_t)_{t \\in \\mathbb{R}}$ with respect to $\\nabla^{\\E}$), and $\\pi_{\\mathcal{F}_i}$ is the orthogonal projection onto $\\mathcal{F}_i$. Setting $(\\pi_{\\mathcal{F}_i})_\\star := \\pi_{\\mathcal{F}_i}(x_\\star)$, invariance of $\\mathcal{F}_i$ by parallel transport implies that $\\rho(\\gamma)(\\pi_{\\mathcal{F}_i})_\\star = (\\pi_{\\mathcal{F}_i})_\\star\\rho(\\gamma)$, for all $\\gamma \\in \\mathbf{G}$, that is $(\\pi_{\\mathcal{F}_i})_\\star \\in \\mathbf{R}'$. Moreover, we have $\\Phi((\\pi_{\\mathcal{F}_i})_\\star) = \\pi_{\\mathcal{F}_i}$. This proves that both $u_R$ and $u_I$ are in $\\mathrm{ran}(\\Phi)$. This concludes the proof.\n\\end{proof}\n\nIt remains to prove the results concerning invariant sections:\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:invariant-section}]\nUniqueness is immediate since $\\nabla^{\\E}_X u = 0$ implies that\n\\[\nX |u|^2 = \\langle \\nabla^{\\E}_X u, u \\rangle = \\langle u , \\nabla^{\\E}_X u \\rangle = 0,\n\\]\nthat is $|u|$ is constant. Now, given $u_\\star$ which is $\\mathbf{G}$-invariant, we can define $u_-(x)$ for $x$ on a homoclinic orbit $\\gamma$ by parallel transport of $u_\\star$ from $x_\\star$ to $x$ along $\\gamma$ with respect to $\\nabla^{\\E}$, similarly to Lemma \\ref{lemma:p-minus} and to the proof of Theorem \\ref{theorem:iso}. We can also define $u_+(x)$ in the same fashion (by parallel transport in the other direction). Then one gets that $\\|u_-(x)-u_+(x)\\| = \\|u_\\star - \\rho(\\gamma)u_\\star\\|=0$ and the same arguments as before show that $u_-$ extends to a smooth function in the kernel of $\\nabla^{\\E}_X$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:rank2}]\nThis is based on the following:\n\n\\begin{lemma}\nAssume that for all periodic orbits $\\gamma \\in \\mathcal{G}$, there exists $u_\\gamma \\in C^\\infty(\\gamma,\\E|_{\\gamma})$ such that $\\nabla^{\\E}_X u_{\\gamma} = 0$. Then for all $g \\in \\mathbf{G}$, there exists $u_g \\in \\E_{x_\\star}$ such that $\\rho(g)u_g = u_g$. \n\\end{lemma}\n\n\\begin{proof}\nRecall that by the construction of Proposition \\ref{proposition:representation}, each element $\\rho(g) \\in \\mathrm{U}(\\E_{x_\\star})$ can be approximated by the holonomy $C_{y_n \\to x_\\star}C(y_n,T'_n)C_{x_\\star \\to y_n}$ along a sequence of periodic orbits of points $y_n$ converging to $x_\\star$. Now, each $C(y_n,T'_n)$ has $1$ as eigenvalue by assumption and taking the limit as $n \\rightarrow \\infty$, we deduce that $1$ is an eigenvalue of $\\rho(g)$.\n\\end{proof}\n\nAs a consequence, we can write for all $g \\in \\mathbf{G}$, in a fixed orthonormal basis of $\\E_{x_\\star}$\t:\n\\[\n\\rho(g) = \\alpha_g \\begin{pmatrix} 1 & 0 \\\\ 0 & s(g) \\end{pmatrix} \\alpha_g^{-1},\n\\]\nfor some $\\alpha_g \\in \\mathrm{U}(\\E_{x_\\star})$ and $s(g)$ is an $(r-1) \\times (r-1)$ matrix. For $\\rk(\\E)=1$, the Lemma is then a straightforward consequence of Lemma \\ref{lemma:invariant-section} since the conjugacy $\\alpha_g$ does not appear. For $\\rk(\\E)=2$, one has the remarkable property that $s(g)$ is \\emph{still} a representation of $ \\mathbf{G}$ since $\\det \\rho(g) = s(g) \\in \\mathrm{U}(1)$. As a consequence, $\\rho :  \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_\\star})$ has the same character as $\\rho' :  \\mathbf{G} \\rightarrow \\mathrm{U}(\\E_{x_\\star})$ defined by:\n\\[\n\\rho'(g) :=  \\begin{pmatrix} 1 & 0 \\\\ 0 & s(g) \\end{pmatrix}.\n\\]\nBy \\cite[Corollary 3.8]{Lang-02}, we then conclude that these representations are isomorphic, that is there exists $p_\\star \\in \\mathrm{U}(\\E_{x_\\star})$ such that $\\rho(g) = p_\\star \\rho'(g) p_\\star^{-1}$. If $u_\\star' \\in \\E_{x_\\star}$ denotes the vector fixed by $\\rho'(\\mathbf{G})$, then $u_\\star := p_\\star u'_\\star$ is fixed by $\\rho(\\mathbf{G})$. We then conclude by Lemma \\ref{lemma:invariant-section}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Pollicott-Ruelle resonances and local geometry on the moduli space of connections}\n\n\\label{section:geometry}\n\nThis section is devoted to the study of the moduli space of connections, with the point of view of Pollicott-Ruelle resonances. We will first deal with the opaque case and then outline the main distinctions with the non-opaque case. We consider a Hermitian vector bundle $(\\mathcal{E},\\nabla^{\\mathcal{E}})$ endowed with a unitary connection over the Anosov Riemannian manifold $(M,g)$. Recall the notation of \\S \\ref{section:twisted}: we write $\\mathbf{X} = (\\pi^*\\nabla^{\\E})_X$, $\\RR_\\pm(z) = (\\pm \\mathbf{X} + z)^{-1}$ for its resolvent and $\\RR^\\pm_0$, $\\Pi_0^\\pm$ for the holomorphic parts and the spectral projector at zero, respectively.\n\n\\subsection{The Coulomb gauge}\n\nWe study the geometry of the space of connections (and of the moduli space of gauge-equivalent connections) in a neighborhood of a given unitary connection $\\nabla^{\\mathcal{E}}$ of regularity $C^s_*$ (for ${\\color{red}1 < }s < \\infty$\\footnote{It is very likely that the case $s=\\infty$ still works. This would require to use the Nash-Moser Theorem.}) such that $\\ker(\\nabla^{\\mathrm{End}(\\E)}) = \\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$. For the standard differential topology of Banach manifolds, we refer the reader to \\cite{Lang-99}. We denote by\n\\[\n\\mathcal{O}_s(\\nabla^{\\mathcal{E}}) := \\left\\{ \\nabla^{\\mathcal{E}}  + p^{-1} \\nabla^{\\mathrm{End}(\\E)}p ~|~ p \\in C_*^{s+1}(M,\\mathrm{U}(\\E)),\\,\\, \\|p - \\mathbbm{1}\\|_{C^{s+1}_*} < \n\\delta\\right\\}\n\\]\nthe orbit of gauge-equivalent connections of $C^{s}_*$ regularity, where $\\delta > 0$ is small enough so that $\\mathcal{O}_s(\\nabla^{\\E})$ is a smooth Banach submanifold. We also define the slice at $\\nabla^{\\E}$ by\n\\[\n\\mathcal{S}_s(\\nabla^{\\E}) := \\nabla^{\\E} + \\ker (\\nabla^{\\mathrm{End}(\\E)})^* \\cap \\big\\{A \\in C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E)): \\|A\\|_{C_*^s} < \\delta\\big\\}.\n\\]\nNote that $\\mathbb{S}^1$ acts by multiplication freely and properly on $C_*^s(M, \\mathrm{U}(\\E))$ and hence we may form the quotient Banach manifold, denoted by $C_*^s(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$, which in particular satisfies\n\\begin{equation}\n\tT_{\\mathbbm{1}_{\\E}} \\Big(C_*^s(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1\\Big) = C_*^s(M, \\mathrm{End}_{\\mathrm{sk}}(\\E))\/\\big(\\mathbb{R}\\cdot (i\\mathbbm{1}_{\\E})\\big), \n\\end{equation}\nwhere we used the identification of tangent spaces given by the exponential map.\n\tNext, observe that the map $O: p \\mapsto p^*\\nabla^{\\E}$ is injective modulo the multiplication action of $\\mathbb{S}^1$ on $C^{s + 1}_*(M, \\mathrm{U}(\\E))$ and that it is an immersion at $p = \\mathbbm{1}$ with $\\dd_{\\mathbbm{1}} O (\\Gamma) = \\nabla^{\\mathrm{End}(\\E)} \\Gamma$. Therefore by equation \\eqref{eq:decomposition-tt}, $\\mathcal{O}_s(\\nabla^{\\E})$ and $\\mathcal{S}_s$ are smooth transverse Banach manifolds with:\n\t\\[T_{\\nabla^{\\E}} \\mathcal{O}(\\nabla^{\\mathrm{End}(\\E)}) = \\ran(\\nabla^{\\mathrm{End}(\\E)}), \\quad T_{\\nabla^{\\mathrm{End}(\\E)}} \\mathcal{S}_s = \\ker(\\nabla^{\\mathrm{End}(\\E)})^*.\\]\n\t\n\n\nWe will say that a connection $\\nabla_2^{\\E}$ is in the \\emph{Coulomb gauge} with respect to $\\nabla_1^{\\E}$ if $(\\nabla_1^{\\mathrm{End}(\\E)})^*(\\nabla_2^{\\E} - \\nabla_1^{\\E}) =0$. The following lemma shows that, near $\\nabla^{\\E}$, we may always put a pair of connections in the Coulomb gauge with respect to each other. It is a slight generalisation of the usual claim (see \\cite[Proposition 2.3.4]{Donaldson-Kronheimer-90}).\n\n\\begin{lemma}[Coulomb gauge]\n\\label{lemma:coulomb}\nLet $s > 1$. There exists $\\varepsilon = \\varepsilon(s, \\nabla^{\\E}) > 0$ and a neighbourhood $\\mathbbm{1}_{\\E} \\in \\mathcal{U} \\subset C_*^{s + 1}(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$ such that for any $A_i \\in C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A_i\\|_{C^s_*} < \\varepsilon$, after setting $\\nabla_i^{\\E} = \\nabla^{\\E} + A_i$  for $i = 1, 2$, there exists a unique $p_{A_1, A_2} \\in \\mathcal{U}$ such that $p_{A_1, A_2}^*\\nabla_2^{\\E} - \\nabla_1^{\\E} \\in \\ker (\\nabla_1^{\\mathrm{End}(\\E)})^*$. Furthermore, if $A_i$ are smooth, then $p_{A_1, A_2}$ is smooth.\nMoreover, the map \n\\[\n\\big(C^{s}(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))\\big)^2 \\ni (A_1, A_2) \\mapsto \\phi(A_1, A_2) := p_{A_1, A_2}^*\\nabla_2^{\\E} \\in \\mathcal{S}_s(\\nabla_1^{\\E}),\n\\]\nis smooth. Setting $\\phi(A) := \\phi(0, A)$, we have:\n \\[\n \\dd \\phi|_{A = 0} = \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\E)})^*}.\n \\]\n\\end{lemma}\n\\begin{proof}\nNote that the exponential map $\\exp: C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i \\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}} \\to C^{s + 1}_*(M, \\mathrm{U}(\\E))\/\\mathbb{S}^1$ is well-defined and a local diffeomorphism at zero, so we reduce the claim to finding a neighbourhood $0 \\in \\mathcal{V} \\subset C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}$ and setting $p = p_{A_1, A_2} = \\exp(\\chi_{A_1, A_2})$ for $\\chi = \\chi_{A_1, A_2} \\in \\mathcal{V}$, that is $\\mathcal{U} = \\exp(\\mathcal{V})$. Define the functional\n\t\\begin{align*}\n\t\t&F: \\big(C^s_*(M, T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))\\big)^2 \\times C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}\\to C^{s - 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E))\\\\\n\t\t&\\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}}, F(A_1, A_2, \\chi) :=\n\t\t(\\nabla_1^{\\mathrm{End}(\\E)})^*\\Big(\\exp(-\\chi) \\nabla^{\\mathrm{End}(\\E)} \\exp(\\chi) + \\exp(-\\chi) A_2 \\exp(\\chi) - A_1\\Big).\n\t\\end{align*}\n\tWe have $F$ well-defined, i.e. with values in skew-Hermitian endomorphisms, since $\\nabla^{\\E}$ is unitary, and integrating by parts $\\langle{F(A_1, A_2, \\chi), \\mathbbm{1}_{\\E}}\\rangle_{L^2} = 0$; note that $F$ is smooth in its entries. Next, we compute the partial derivative with respect to the $\\chi$ variable at $A_1 = A_2 = 0$ and $\\chi = 0$:\n\t\\[\\dd_{\\chi}F(0, 0, 0) (\\Gamma) = \\partial_t|_{t = 0} F(0, 0, t\\Gamma) = (\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)} \\Gamma.\\]\n\tThis derivative is an isomorphism on \n\t\\[(\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)}: C^{s + 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i \\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}} \\to C^{s - 1}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{i\\mathbbm{1}_{\\E}\\}^{\\perp_{L^2}},\\]\n\tby the Fredholm property of $(\\nabla^{\\mathrm{End}(\\E)})^* \\nabla^{\\mathrm{End}(\\E)}$ and since $\\ker (\\nabla^{\\mathrm{End}(\\E)}) = \\mathbb{C} \\cdot \\mathbbm{1}_{\\E}$ by assumption. The first claim then follows by an application of the implicit function theorem for Banach spaces. \n\t\n\tThe fact that $p$ is smooth if $(A_1, A_2)$ is, is a consequence of elliptic regularity and the fact that $C_*^s$ is an algebra, along with the Coulomb property:\n\t\\[(\\nabla_1^{\\mathrm{End}(\\E)})^* \\nabla_1^{\\mathrm{End}(\\E)} p = (\\nabla_1^{\\mathrm{End}(\\E)}p) p^{-1} \\bullet \\nabla_1^{\\mathrm{End}(\\E)} p + p(\\nabla_1^{\\mathrm{End}(\\E)})^* \\big(p^{-1} (A_1 - A_2) p\\big) \\in C_*^{s}\\]\n\timplies $p \\in C_*^{s+2}$. Bootstrapping we obtain $p_{A_1, A_2} \\in C^\\infty$. Here $\\bullet$ denotes the operation of taking the inner product on the differential form side and multiplication on the endomorphism side.\n\t\n\tEventually, we compute the derivative of $\\phi(A)$. Write $p_A := p_{0, A}$ and $\\chi_A := \\chi_{0, A}$, where $\\chi_A$ is orthogonal to $i \\mathbbm{1}_{\\E}$ with respect to the $L^2$-scalar product, so that by definition\n\t\\begin{equation}\n\t\\label{equation:phi}\n\t\\phi(A) = \\nabla^{\\E} + p_A^{-1} \\nabla^{\\mathrm{End}(\\E)} p_A + p_A^{-1} A p_A.\n\t\\end{equation}\n\tBy differentiating the relation $F(A, \\chi_A) := F(0, A,\\chi_A)=0$ at $A=0$, we obtain for every $\\Gamma \\in C^s_*(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$:\n\t\\begin{align*}\n\t0 &= \\dd_A F|_{A=0,\\chi=0}(\\Gamma) + \\dd_\\chi F|_{A=0,\\chi=0}(\\dd \\chi_A|_{A=0}(\\Gamma))\\\\\n\t&= (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma + (\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)} \\dd \\chi_A|_{A=0}(\\Gamma),\n\t\\end{align*}\n\tthat is $\\dd \\chi_A|_{A=0}(\\Gamma) = - [(\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)}]^{-1} (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma$. \t\n\t Observe that $\\dd p_A|_{A=0} = \\dd \\chi_A|_{A=0}$ via the exponential map and by \\eqref{equation:phi}, we obtain:\n\t\\[\n\t\\dd \\phi|_{A=0}(\\Gamma) = \\nabla^{\\mathrm{End}(\\E)} \\dd \\chi_A|_{A=0}(\\Gamma) + \\Gamma = \\Gamma -  \\nabla^{\\mathrm{End}(\\E)}[(\\nabla^{\\mathrm{End}(\\E)})^*\\nabla^{\\mathrm{End}(\\E)}]^{-1} (\\nabla^{\\mathrm{End}(\\E)})^*\\Gamma.\n\t\\]\n\tWe then conclude by \\eqref{equation:projection}\n\\end{proof}\n\nIn particular, the proof also gives that the map\n\\[\nC^{s}_*(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\ni A \\mapsto \\phi(A) \\in \\mathcal{S}_s := \\mathcal{S}_{s}(\\nabla^{\\E}),\n\\]\nis constant along orbits of gauge-equivalent connections (by construction)\n\n\n\n\n\\subsection{Resonances at $z=0$: finer remarks}\n\\label{ssection:finer}\n\n\n\\begin{lemma}\\label{lemma:symmetricspectrum}\nThe Pollicott-Ruelle resonance spectrum of $\\mathbf{X}$ is symmetric with respect to the real axis.\n\\end{lemma}\n\n\\begin{proof}\nIf $z_0$ is a resonance associated to $-\\mathbf{X}$ i.e. a pole of $z \\mapsto \\RR_+(z)$ then $\\overline{z}_0$ is a resonance associated to $+\\mathbf{X}$ i.e. a pole of $z \\mapsto \\RR_-(z)$ by \\eqref{equation:adjoint}. Let $u \\in \\mathcal{H}_+^s$ be a non-zero resonant state such that $-\\mathbf{X} u = z_0 u$, for some $s > 0$. Let $R : (x,v) \\mapsto (x,-v)$ be the antipodal map. By inspecting the construction of the anisotropic Sobolev space in \\cite{Faure-Sjostrand-11}, we see that we may assume $R^* : \\mathcal{H}_\\pm^s \\rightarrow \\mathcal{H}_\\mp^s$ is an isomorphism.\\footnote{Simply replace the degree function $m$ in the construction by $\\frac{m - R^*m}{2}$, which then implies that $R^*m = -m$.}\nThen $-R^* \\mathbf{X} u = \\mathbf{X} R^* u = z_0 R^* u$ and $R^*u \\in \\mathcal{H}_-^s$. Thus $R^*u$ is a resonant state associated to the resonance $z_0$. So both $z_0$ and $\\overline{z}_0$ are resonances for $+\\mathbf{X}$ and the same holds for $-\\mathbf{X}$.\n\\end{proof}\n\nConsider a contour $\\gamma \\subset \\mathbb{C}$ such that $-\\mathbf{X}$ has no resonances other than zero inside or on $\\gamma$. By continuity of resonances (see \\cite{Bonthonneau-19}), there is an $\\varepsilon > 0$ such that for all skew-Hermitian $1$-forms $A$ with $\\|A\\|_{C_*^s} < \\varepsilon$ the operator $-\\mathbf{X}_A := (\\pi^*(\\nabla^{\\E} + A))_X$ has no resonances on $\\gamma$. Here we need to take $s$ large enough (depending on the dimension), so that the framework of microlocal analysis applies.\n\nIn the specific case where $\\dim \\ker(\\mathbf{X}|_{\\mathcal{H}_+}) = 1$, we denote by $\\lambda_A$ the unique resonance of $-\\mathbf{X}_A$ within $\\gamma$. Note that the map $A \\mapsto \\lambda_A$ is $C^3$-regular for $\\varepsilon > 0$ small enough.\n\n\n\\begin{lemma}\n\\label{lemma:diff-2}\nAssume that $\\dim \\ker(\\mathbf{X}|_{\\mathcal{H}_+}) = 1$. Then $\\lambda_A \\in \\mathbb{R}$ and for $\\Gamma \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$:\n\\[\n\\dd \\lambda_A|_{A=0} = 0, ~~ \\dd^2 \\lambda_A|_{A=0}(\\Gamma,\\Gamma) = - \\langle \\Pi \\pi_1^*\\Gamma u_0, \\pi_1^*\\Gamma u_0 \\rangle_{L^2},\n\\]\nwhere $u_0$ is a resonant state associated to $A=0$ and $\\|u_0\\|_{L^2}=1$.\n\\end{lemma}\n\n\\begin{proof}\nBy the symmetry property of Lemma \\ref{lemma:symmetricspectrum} and continuity of resonances we know $\\lambda_A \\in \\mathbb{R}$. Also, observe that $u_0$ is either pure odd or pure even with respect to $v$ (i.e. $R^*u_0 = u_0$ or $R^*u_0 = -u_0$) because $R^*$ keeps $\\ker (\\mathbf{X})$ fixed and $\\ker (\\mathbf{X})$ is assumed to be one dimensional.\n\nFor the second claim, it is sufficient to start with the equality $-\\mathbf{X}_{\\tau \\Gamma} u_{\\tau \\Gamma} = \\lambda_{\\tau \\Gamma} u_{\\tau \\Gamma}$, where $\\Gamma \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$, $\\tau \\in (-\\delta, \\delta)$ is small enough so that $\\tau \\mapsto \\lambda_{\\tau \\Gamma}$ and $\\tau \\mapsto u_{\\tau \\Gamma} \\in \\mathcal{H}_+$ are $C^3$, and to compute the derivatives at $\\tau=0$. Observe that $\\dot{\\mathbf{X}}_0 = \\pi_1^*\\Gamma,\\, \\ddot{\\mathbf{X}}_0 = 0$. We obtain: $-\\dot{\\mathbf{X}}_0 u_0 -\\mathbf{X}_0 \\dot{u}_0 = \\dot{\\lambda}_0 u_0$ and taking the $L^2$ scalar product with $u_0$, we find $\\dot{\\lambda}_0 = 0$, using that $u_0$ is either pure odd or pure even. Thus $\\dot{u}_0- \\Pi_0^+ \\dot{u}_0 = -\\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0$. Then, taking the second derivative at $\\tau = 0$, we get: $-2 \\pi_1^*\\Gamma \\dot{u}_0 - \\mathbf{X}_0 \\ddot{u}_0 = \\ddot{\\lambda}_0 u_0$ and taking once again the scalar product with $u_0$, we find $\\ddot{\\lambda}_0 = - 2 \\langle \\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0,  \\pi_1^*\\Gamma u_0 \\rangle_{L^2}$. It is then sufficient to observe that by symmetry (using $(\\mathbf{R}_0^+)^* = \\mathbf{R}_0^-$ and $\\ddot{\\lambda}_0 \\in \\mathbb{R}$)\n\\[\n\\langle \\mathbf{R}_0^+ \\pi_1^*\\Gamma u_0,  \\pi_1^*\\Gamma u_0 \\rangle_{L^2} = \\langle \\mathbf{R}_0^- \\pi_1^*\\Gamma u_0,  \\pi_1^*\\Gamma u_0 \\rangle_{L^2}.\n\\]\nThis proves the result.\n\\end{proof}\n\n\n\\subsection{P-R resonance at $0$ of the mixed connection: opaque case}\n\n\\label{ssection:pollicott-ruelle-dea}\n\nWe now further assume that  $\\mathbf{X} := (\\pi^* \\nabla^{\\mathrm{End}(\\E)})_X$ has the resonant space at $0$ spanned by $\\mathbbm{1}_{\\E}$. This condition is known as the \\emph{opacity} of the connection $\\pi^*\\nabla^{\\E}$. When $(M,g)$ is Anosov, this is known to be a generic condition, see \\cite[Theorem 1.6]{Cekic-Lefeuvre-20}.\n\n\nAs in \\S\\ref{sssection:connection-induced}, we assume that $s \\gg 1$ (so that standard microlocal analysis applies) and we introduce the mixed connection induced by $\\nabla_1^{\\E} = \\nabla^{\\E} + A_1$ and $\\nabla_2^{\\E} = \\nabla^{\\E} + A_2$, namely\n\\[\n\\nabla^{\\Hom(\\nabla_1^{\\E}, \\nabla_2^{\\E})} u = \\nabla^{\\mathrm{End}(\\E)} u + A_2u - u A_1,\n\\]\nand we set $\\mathbf{X}_{A_1, A_2} :=  (\\pi^* \\nabla^{\\Hom(\\nabla_1^{\\E}, \\nabla_2^{\\E})})_X$ and $z \\mapsto \\RR_{\\pm}(z, A_i)$ for the resolvents. The operator $\\mathbf{X} := \\mathbf{X}_{0, 0}$ has the resonant space at $z=0$ spanned by $\\mathbbm{1}_{\\E}$ \nand we denote by $\\lambda_{A_1, A_2}$ the unique resonance in $\\left\\{\\Re(z) \\leq 0\\right\\}$ close to $0$, namely the unique pole of $\\RR_{\\pm}(z,A_i)$ close to $0$. For $\\|A_1\\|_{C_*^s}, \\|A_2\\|_{C_*^s}$ small enough we know that the map $(A_1, A_2) \\mapsto \\lambda_{A_1, A_2}$ is $C^3$ (see \\S \\ref{ssection:finer}) and by Lemma \\ref{lemma:diff-2} that $\\lambda_{A_1, A_2} \\in \\mathbb{R}$. In fact $\\lambda_{A_1, A_2}$ descends to the moduli space: if $p_i^*(\\nabla^{\\E} + A_i') = \\nabla^{\\E} + A_i$ for $\\|A_i'\\|_{C_*^s}$ small enough, then using \\eqref{equation:lien} we get\n\\begin{equation}\\label{eq:mixedunitaryequiv}\n\t\\mathbf{X}_{A_1', A_2'}u = (p_2)^{-1} \\cdot \\mathbf{X}_{A_1, A_2}(p_2 u (p_1)^{-1}) \\cdot p_1, \\quad u \\in \\mathcal{H}_+.\n\\end{equation}\nHere we used that $\\mathcal{H}_+$ is stable under multiplication by $C_*^s$ for $s$ large enough; hence $\\mathbf{X}_{A_1, A_2}$ and $\\mathbf{X}_{A_1', A_2'}$ have equal P-R spectra and so $\\lambda_{A_1, A_2} = \\lambda_{A_1', A_2'}$.\n\n\n\nNext, we need a uniform estimate for the $\\Pi_1^{\\mathrm{End}(\\E)}$ operator in a neighbourhood of $\\nabla^{\\E}$:\n\n\\begin{lemma}\\label{lemma:Pi_1uniform}\n\tAssume $\\Pi_1^{\\mathrm{End}(\\E)}$ is $s$-injective. There are constants $\\varepsilon, C > 0$ depending only on $\\nabla^{\\E}$ such that for all skew-Hermitian $1$-forms with $\\|A\\|_{C_*^s} < \\varepsilon$:\n\t\\[\n\\forall f \\in H^{-1\/2}(M, T^*M \\otimes \\mathrm{End}(\\mathcal{E})), ~~~ \\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} f, f \\rangle_{L^2} \\geq C \\|\\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A})^*}f\\|^2_{H^{-1\/2}}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\n\tObserve firstly that the left hand side of the inequality vanishes on potential tensors by \\eqref{eq:Pi_mproperties} and hence it suffices to consider $f \\in \\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*$. Then we obtain:\n\t\\begin{align*}\n\t\t\t\\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} &f, f \\rangle_{L^2} = \\langle \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})} f, f \\rangle_{L^2} + \\langle (\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}) f, f \\rangle_{L^2}\\\\\n\t\t\t&\\geq C_0 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*}f\\|_{H^{-1\/2}}^2 - \\|\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}\\|_{H^{-1\/2} \\to H^{1\/2}} \\|f\\|_{H^{-1\/2}}^2\\\\\n\t\t\t&\\geq \\frac{1}{4} C_0 \\|f\\|_{H^{-1\/2}}^2 - \\frac{1}{2} C_0 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E})})^*} - \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A})^*}\\|_{H^{-1\/2} \\to H^{-1\/2}} \\|f\\|_{H^{-1\/2}}^2\\\\\n\t\t\t&\\geq \\frac{1}{8}C_0 \\|f\\|_{H^{-1\/2}}^2.\n\t\\end{align*}\n\tIn the second line, we used Lemma \\ref{lemma:generalized-xray} (item 3) with a constant $C_0$. In the next line we used that the map $A \\mapsto \\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} \\in \\Psi^{-1}$ is continuous; the proof of this fact is analogous to the proof of \\cite[Proposition 4.1]{Guillarmou-Knieper-Lefeuvre-19} and we omit it. Thus for $\\varepsilon > 0$ small enough $\\|\\Pi_1^{\\mathrm{End}(\\nabla^{\\E} + A)} - \\Pi_1^{\\mathrm{End}(\\nabla^{\\E})}\\|_{H^{-1\/2} \\to H^{1\/2}} \\leq C_0\/4$. Similarly in the last line we used that $A \\mapsto \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E) + A})^*} \\in \\Psi^0$ is continuous which follows from standard microlocal analysis from \\eqref{equation:projection}, so again we choose $\\varepsilon > 0$ small enough so that $\\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E})})^*} - \\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*}\\|_{H^{-1\/2} \\to H^{-1\/2}} \\leq C_0\/4$. The claim follows by setting $C = C_0\/8$.\n\\end{proof}\n\nRecall that $\\lambda_{A_1, A_2} \\leq 0$ in the following:\n\n\\begin{lemma}\n\\label{lemma:borne-lambda-a-1}\nAssume that the generalized X-ray transform $\\Pi^{\\mathrm{End}(\\E)}_1$ defined with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$ is $s$-injective. For $s \\gg 1$ large enough, there exist constants $\\varepsilon, C > 0$ such that for all $A_i \\in C^s(M,T^*M \\otimes  \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A_i\\|_{C^s_*} < \\varepsilon$ for $i = 1, 2$, we have:\n\\[\n0 \\leq  \\|\\phi(A_1, A_2) - \\nabla_1^{\\E}\\|^2_{H^{-1\/2}(M,T^*M\\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq C |\\lambda_{A_1, A_2}|.\n\\]\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWe introduce two functionals in the vicinity of $\\nabla^{\\E}$, for small enough $\\varepsilon > 0$: \n\\begin{align*}\n\tF_1, F_2 &: \\big(C^{s_0}_*(M, \\mathrm{End}_{\\mathrm{sk}}(\\E)) \\cap \\{\\|A\\|_{C_*^{s_0}} < \\varepsilon\\}\\big)^2 \\to \\mathbb{R},\\\\\n\t F_1(A_1, A_2) &:=\\lambda_{A_1, A_2}, \\quad F_2(A_1, A_2) := -\\|\\phi(A_1, A_2)-\\nabla_1^{\\E}\\|^2_{H^{-1\/2}}.\n\\end{align*}\nThey are well-defined and restrict as $C^{3}$-regular maps on $\\mathcal{S}_{s_0}$ for some $s_0 \\gg 1$ large enough by Lemma \\ref{lemma:coulomb} and the discussion above. Moreover, using Lemma \\ref{lemma:diff-2}, we have for all $A$:\n\\[F_1(A, A) = F_2(A, A) = 0 \\quad \\mathrm{and}\\quad \\dd F_1|_{(A, A)} = \\dd F_2|_{(A, A)} = 0.\\] \nWe will compare the second partial derivatives in the variable $A_2$ at a point $(A, A)$. Given $\\Gamma \\in T_{\\nabla^{\\E}} \\mathcal{S}_{s_0} \\simeq \\ker (\\nabla^{\\mathrm{End}(\\E)})^*$, we have by Lemma \\ref{lemma:coulomb}:\n\\begin{equation}\\label{eq:F_2}\n\\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma) = -2\\|\\pi_{\\ker (\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\Gamma\\|^2_{H^{-1\/2}}\n\\end{equation}\nBy Lemma \\ref{lemma:diff-2}, we have\n\\[\n\\dd^2_{A_2} F_1|_{(A, A)}(\\Gamma,\\Gamma) = -c^2 \\langle \\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)} \\pi_1^* \\Gamma \\mathbbm{1}_{\\E}, \\pi_1^* \\Gamma \\mathbbm{1}_{\\E} \\rangle_{L^2} = - c^2 \\langle \\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)}_1 \\Gamma, \\Gamma \\rangle_{L^2},\n\\]\nfor some constant $c > 0$, where $\\Pi^{\\mathrm{End}(\\nabla^{\\E} +A)}$ denotes the $\\Pi$ operator with respect to the endomorphism connection induced by $\\nabla^{\\E} + A$. We used here that the orthogonal projection to the resonant space $\\mathbb{C} \\mathbbm{1}_{\\E}$ of $\\mathbf{X}_{A, A}$ at zero vanishes, because $\\langle{\\pi_1^*\\Gamma, \\mathbbm{1}_{\\E}}\\rangle_{L^2} = 0$ as $\\pi_1^*\\Gamma$ is odd. For $\\varepsilon > 0$ small enough, by Lemma \\ref{lemma:Pi_1uniform} we know $\\Pi^{\\mathrm{End}(\\nabla^{\\E} + A)}_1$ is $s$-injective and there is a constant $C' = C'(\\nabla^{\\E}) > 0$ such that:\n\\begin{equation}\\label{eq:F_1}\n\\dd^2_{A_2} F_1|_{(A, A)}(\\Gamma,\\Gamma) \\leq - C' \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*}\\Gamma\\|^2_{H^{-1\/2}} = C'\/2  \\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma).\n\\end{equation}\nAs a consequence, writing $G(A_2) := F_1(A, A_2) - C'\/4 \\times F_2(A, A_2)$, we have $G(A)=0, \\dd G|_{A_2 = A} = 0$ and by \\eqref{eq:F_1}, \\eqref{eq:F_2}\n\\[\\dd^2 G|_{A_2 = A}(\\Gamma,\\Gamma) \\leq C'\/4  \\dd^2_{A_2} F_2|_{(A, A)}(\\Gamma,\\Gamma) = -C'\/2 \\|\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\Gamma\\|^2_{H^{-1\/2}}.\\]\nIf we now Taylor expand the $C^3$-map $\\mathcal{S}_{s_0} \\ni A_2 \\mapsto G(A_2)$ at $A_2 = A$, we obtain:\n\\begin{align*}\nG(A + \\Gamma) &= \\dfrac{1}{2} \\dd^2 G|_{A_2=A}(\\Gamma, \\Gamma) + \\mathcal{O}(\\|\\Gamma\\|_{C_*^{s_0}}^3)\\\\\n &\\leq -C'\/4 \\|\\Gamma\\|_{H^{-1\/2}}^2 + C'\/4 \\|(\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*} - \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*})\\Gamma\\|_{H^{-1\/2}}^2 + C''\\|\\Gamma\\|^3_{C_*^{s_0}}\\\\\n &\\leq -C'\/8\\|\\Gamma\\|_{H^{-1\/2}}^2 + C''\\|\\Gamma\\|_{C_*^{s_0}}^3.\n\\end{align*}\nIn the second line we introduced a uniform constant $C'' = C''(\\nabla^{\\E}) > 0$ using the $C^3$-regular property and $\\pi_{\\ker(\\nabla^{\\mathrm{End}(\\E)})^*} \\Gamma = \\Gamma$. For the last line, we observed that $A \\mapsto \\pi_{\\ker(\\nabla^{\\mathrm{End}(\\nabla^{\\E} + A)})^*} \\in \\Psi^0$ is a continuous map by equation \\eqref{equation:projection} and hence by standard microlocal analysis the $H^{-1\/2} \\to H^{-1\/2}$ estimate is arbitrarily small for $\\varepsilon$ small enough. This estimate holds for all $\\|A\\|_{C_*^{s_0}}, \\|\\Gamma\\|_{C_*^{s_0}} < \\varepsilon\/2$.\n\nChoosing $s \\gg s_0$ and assuming that $A \\in C_*^s(M,T^*M \\otimes  \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C^s_*} < \\varepsilon$, there is a $C'''(\\nabla^{\\E}) > 0$, such that for $\\varepsilon > 0$ with $C''' \\varepsilon \\leq C'\/16$, we then obtain by interpolation:\n\\[\nG(A + \\Gamma) \\leq -C'\/8 \\|\\Gamma\\|^2_{H^{-1\/2}} + \\underbrace{C''' \\|\\Gamma\\|_{C^s_*}}_{\\leq C'\/16} \\|\\Gamma\\|_{H^{-1\/2}}^2 \\leq - C'\/16 \\|\\Gamma\\|_{H^{-1\/2}}^2 \\leq 0.\n\\]\nAfter taking $\\varepsilon > 0$ small enough, the statement holds with $C=C'\/2$.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{remark}\n\\rm\nIt was proved in \\cite{Guillarmou-Knieper-Lefeuvre-19} that there exists a metric $G$ on the moduli space of isometry classes (of metrics with negative sectional curvature) which generalizes the usual Weil-Petersson metric on Teichm\\\"uller space in the sense that, in the case of a surface, the restriction of $G$ to Teichm\\\"uller space is equal to the Weil-Petersson metric. We point out that the operator $\\Pi_1$ also allows to define a metric $G$ at a generic point $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$, similarly to \\cite{Guillarmou-Knieper-Lefeuvre-19}. Indeed, if $\\mathfrak{a}_0 \\in \\mathbb{A}_{\\E}$, taking a representative $\\nabla^{\\E} \\in \\mathfrak{a}_0$, one has $T_{\\mathfrak{a}_0}\\mathbb{A}_{\\E} \\simeq \\ker (\\nabla^{\\mathrm{End}})^*$ and thus, given $\\Gamma \\in \\ker (\\nabla^{\\mathrm{End}})^*$, one can consider:\n\\[\nG_{\\mathfrak{a}_0}(\\Gamma,\\Gamma) := \\langle \\Pi_1 \\Gamma, \\Gamma \\rangle_{L^2(M,T^*M\\otimes\\mathrm{End}(\\E))} \\geq c \\|\\Gamma\\|^2_{H^{-1\/2}},\n\\]\nfor some constant $c>0$. Lemma \\ref{lemma:Pi_1uniform} shows that the constant $c$ is locally uniform with respect to $\\mathfrak{a}_0$.\n\\end{remark}\n\n\n\\subsection{P-R resonance at $0$ of the mixed connection: non-opaque case}\n\\label{ssection:non-opaque}\n\n\nThe aim of this paragraph is to deal with neighbourhoods of connections that are not necessarily opaque, and only assume $\\Pi_1^{\\mathrm{End}(\\E)}$ is injective. In other words, we do not want to assume the resonant space of $-(\\pi^*\\nabla^{\\mathrm{End}(\\E)})_X$ at zero is spanned by $\\mathbbm{1}_{\\E}$ necessarily.\n\nNext, as in \\S \\ref{ssection:pollicott-ruelle-dea}, we introduce the mixed connection with respect to $\\nabla^{\\E} + A$ and $\\nabla^{\\E}$, denoted by $\\nabla^{\\Hom(\\nabla^{\\E} + A, \\nabla^{\\E})}$, and set $\\mathbf{X}_A :=(\\pi^*\\nabla^{\\Hom(\\nabla^{\\E} + A, \\nabla^{\\E})})_X$. We assume $s \\gg 1$. As before, consider a contour $\\gamma \\subset \\mathbb{C}$ around zero such that $\\mathbf{X} := \\mathbf{X}_0$ has only the resonance zero within $\\gamma$ and $\\varepsilon > 0$ such that $-\\mathbf{X}_A$ has no resonances on $\\gamma$ for all $\\|A\\|_{C_*^s} < \\varepsilon$. We introduce\n\\[\\Pi_A^+ := \\frac{1}{2\\pi i} \\int_\\gamma (z + \\mathbf{X}_A)^{-1} dz, \\quad \\lambda_A := \\Tr(-\\mathbf{X}_A\\Pi_A^+).\\]\nThis generalises the quantity studied in \\S \\ref{ssection:finer}, where it was assumed that the multiplicity of $\\mathbf{X}$ at zero is equal to one. By \\cite{Bonthonneau-19} it follows that $A \\mapsto \\Pi_A^+$ is $C^3$-regular and hence $A \\mapsto \\lambda_A$ is also $C^3$-regular. As in \\eqref{eq:mixedunitaryequiv}, we observe that the operators $-\\mathbf{X}_A$ and $-\\mathbf{X}_{A'}$ are unitarily equivalent on $\\mathcal{H}_+$ whenever $\\nabla^{\\E} + A$ and $\\nabla^{\\E} + A'$ are gauge equivalent; hence $\\lambda_A = \\lambda_{A'}$ and so $\\lambda_A$ descends to the local moduli space\n\n\n\nNote also that $\\re(\\lambda_A) \\leq 0$, since all resonances of $-\\mathbf{X}_A$ lie in the half-plane $\\{\\re z \\leq 0\\}$ and this gives us hope that $\\re(\\lambda_A)$ controls the distance between the connections. Assume that the resonant space of $-\\mathbf{X}$ at zero is spanned by smooth $L^2$-orthonormal resonant states $\\{u_i\\}_{i = 1}^p$. We have the following generalisation of Lemma \\ref{lemma:diff-2}:\n\n\\begin{lemma}\n\\label{lemma:variations}\nFor $A \\in C^\\infty(M,T^*M \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C_*^s} < \\varepsilon$, we have $\\lambda_A \\in \\mathbb{R}$ and the following perturbation formulas hold:\n\\[\n\\dd \\lambda_A|_{A=0} = 0, ~~ \\dd^2 \\lambda_A|_{A=0}(\\Gamma,\\Gamma) = - \\sum_{i = 1}^p \\langle \\Pi u_i \\pi_1^*\\Gamma, u_i \\pi_1^*\\Gamma\\rangle_{L^2}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\tThe fact that $\\lambda_A$ is real follows from the symmetry of the spectrum of $-\\mathbf{X}_A$ shown in Lemma \\ref{lemma:symmetricspectrum}. The first derivative formula is obvious as $\\lambda_A \\leq 0$; the second one follows from minor adaptations of \\cite[Lemma 5.9]{Cekic-Lefeuvre-20}, where the analogous case of endomorphisms was considered.\n\\end{proof}\n\nNext, by a straightforward adaptation of the Lemma \\ref{lemma:coulomb}, we obtain the existence of $\\varepsilon > 0$, such that for all $A$ with $\\|A\\|_{C_*^s} < \\varepsilon$, there is a smooth map $A \\mapsto \\phi(A) \\in \\mathcal{S}_s$ that sends $\\nabla^{\\E} + A$ to Coulomb gauge with respect to $\\nabla^{\\E}$, that is it satisfies $\\phi(A) - \\nabla^{\\E} \\in \\ker (\\nabla^{\\mathrm{End}(\\E)})^*$.\n\\begin{remark}\\rm\n\tWe cannot get the analogous statement to Lemma \\ref{lemma:coulomb} for parameters $(A_1, A_2)$, because the range of $F(A_1, A_2, \\chi)$ equals $\\ker(\\nabla_1^{\\mathrm{End}(\\E)})^\\perp$ and this is not uniform in $A_1$ (i.e. $\\ker \\nabla_1^{\\mathrm{End}(\\E)}$ changes as we move $A_1$); the space $\\mathbb{A}_{\\E}$ is not a smooth manifold at reducible connections.\n\\end{remark}\n\n\n\n\n\n\nIn the following lemma, we will need to assume that $\\pi_{1*} \\Pi_0^+ = 0$. Equivalently, this means that the resonant states $u_i \\in \\ker(\\mathbf{X}|_{\\mathcal{H}_+})$ satisfy $\\pi_{1*} u_i = 0$, for $i = 1, \\dotso, p$, i.e. the degree $1$ Fourier mode of all the $u_i$ vanishes.\n\n\\begin{lemma}\n\\label{lemma:maininequalitygeneral}\nAssume that the generalized X-ray transform $\\Pi^{\\mathrm{End}(\\E)}_1$ defined with respect to the connection $\\nabla^{\\mathrm{End}(\\E)}$ is $s$-injective and additionally that $\\pi_{1*} \\Pi_0^+ = 0$. For $s \\gg 1$ large enough, there exist constants $\\varepsilon, C > 0$ such that for all $A \\in C^s(M,T^*M \\otimes  \\mathrm{End}_{\\mathrm{sk}}(\\E))$ with $\\|A\\|_{C^s_*} < \\varepsilon$:\n\\[\n0 \\leq  \\|\\phi(A)-\\nabla^{\\E}\\|^2_{H^{-1\/2}(M,T^*M\\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq C |\\lambda_A|.\n\\]\n\\end{lemma}\n\\begin{proof}\nThis is straightforward from the proof of Lemma \\ref{lemma:borne-lambda-a-1}. With the same functionals $F_1(A) = \\lambda_A$ and $F_2(A) = -\\|\\phi(A) - \\nabla^{\\E}\\|^2_{H^{-1\/2}}$, the only slight difference is the computation of $\\dd^2 F_1$. Pick an $L^2$-orthonormal basis $u_1, \\dotso, u_p$ of the resonant space of $-\\mathbf{X}$ at zero such that $u_1 = c\\mathbbm{1}_{\\E}$, where $c$ is a fixed constant. By Lemmas \\ref{lemma:variations} and \\ref{lemma:relations-resolvent}, we have\n\\[\n\\dd^2 F_1|_{A=0}(\\Gamma,\\Gamma) = -\\sum_{i = 1}^p \\langle \\Pi u_i \\pi_1^* \\Gamma, u_i \\pi_1^* \\Gamma \\rangle_{L^2} \\leq -c^2 \\langle \\Pi_1^{\\mathrm{End}(\\E)} \\Gamma, \\Gamma \\rangle_{L^2}.\n\\]\nNote importantly that we have used $\\Pi_0^+ \\pi_1^*\\Gamma = 0$. This follows from the expression for the projector $\\Pi_0^+ = \\sum_{i = 1}^p \\langle{\\bullet, u_i}\\rangle_{L^2} u_i$ and $\\pi_{1*}u_i = 0$ for all $i$. This suffices to run the proof in the same manner. \n\\end{proof}\n\n\n\n\n\\section{Injectivity of the primitive trace map}\n\nWe can now prove the main results stated in the introduction.\n\n\n\n\\label{section:proofs}\n\n\n\n\n\\subsection{The local injectivity result}\n\n\\label{ssection:proof-injectivity}\n\nWe now prove the injectivity result of Theorem \\ref{theorem:injectivity}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:injectivity}]\nWe fix a regularity exponent $N \\gg 1$ large enough so that the results of \\S\\ref{section:geometry} apply. We fix $\\nabla^{\\mathcal{E}}$, a smooth unitary connection on $\\E$ and assume that it is \\emph{generic} (see page \\pageref{def:generic}). \nBy mere continuity, the same properties hold for every connection $\\nabla^{\\E} + A$ such that $\\|A\\|_{C^N_*} < \\varepsilon$, where $\\varepsilon > 0$ is small enough depending on $\\nabla^{\\E}$.\n\nConsider two smooth unitary connections $\\nabla_i^{\\E} = \\nabla^{\\E} + A_i$ such that $\\|A_i\\|_{C^N_*} <  \\varepsilon$ for $i = 1, 2$. Assume that $\\mathcal{T}^\\sharp(\\nabla_1^{\\E})=\\mathcal{T}^\\sharp(\\nabla_2^{\\E})$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} yields the existence of a smooth map $p \\in C^\\infty(SM,\\mathrm{U}(\\E))$ such that:\n\\[\n\\pi^* \\nabla^{\\Hom(\\nabla_1, \\nabla_2)}_X u = 0\n\\]\nthat is $u$ is a resonant state for the operator $\\mathbf{X}_{A_1, A_2}$ associated to the eigenvalue $0$. Assumptions \\textbf{(A)} and  \\textbf{(B)} allow us to apply Lemma \\ref{lemma:borne-lambda-a-1}. We therefore obtain:\n\\[\n\\lambda_{A_1, A_2} = 0 \\leq - C \\|\\phi(A_1, A_2) - \\nabla_1^{\\E}\\|^2_{H^{-1\/2}(M,T^*M \\otimes \\mathrm{End}(\\E))} \\leq 0,\n\\]\nwhere $C = C(\\nabla^{\\E}) > 0$ only depends on $\\nabla^{\\E}$. Hence $\\phi(A_1, A_2) = p_{A_1, A_2}^*\\nabla_2^{\\E} = \\nabla_1^{\\E}$. In other words, the connections are gauge-equivalent.\n\\end{proof}\t\n\nNext, we discuss a version of local injectivity in a neighbourhood of a connection which is non-opaque. We will say a map $f: X \\to Y$ of topological spaces is \\emph{weakly locally injective at $x_0 \\in X$} if there exists a neighbourhood $U \\ni x$ such that $f(x) = f(x_0)$ for $x \\in U$ implies $x = x_0$. This notion appears in non-linear inverse problems where the linearisation is not continuous, see \\cite[Section 2]{Stefanov-Uhlmann-08}. We have:\n\n\\begin{theorem}\\label{thm:weaklocal}\n\tIf $N \\gg 1$ and $[\\nabla^{\\E}] \\in \\mathbb{A}_{\\E}$ is such that the generalised $X$-ray transform $\\Pi_1^{\\mathrm{End}(\\E)}$ is $s$-injective, as well as $\\pi_{1*} \\ker(\\pi^*\\nabla^{\\mathrm{End}(\\E)}_X|_{C^\\infty}) = 0$, then the primitive trace map $\\mathcal{T}^\\sharp$ is weakly locally injective at $[\\nabla^{\\E}]$ in the $C^N$-quotient topology.\n\\end{theorem}\n\\begin{proof}\n\tThe proof is analogous to the proof of Theorem \\ref{theorem:injectivity}, by using the results of \\S \\ref{ssection:non-opaque}. We omit the details.\n\\end{proof}\n\nWe shall see below (see Lemma \\ref{lemma:Pi_1inj}) that flat connections have an injective generalised $X$-ray transform $\\Pi_1^{\\mathrm{End}(\\E)}$ and satisfy the additional condition that $\\ker (\\pi^*\\nabla^{\\E})_X|_{C^\\infty}$ consists of elements of degree zero (but might not be opaque). The previous Theorem therefore shows that the primitive trace map is weakly locally injective near such connections. Let us state this as a corollary for the trivial connection, as it partially answers an open question of Paternain \\cite[p33, Question (3)]{Paternain-13}.\n\n\n\n\\begin{corollary}\n\tLet $\\E = M \\times \\mathbb{C}^r$ be the trivial Hermitian vector bundle equipped with the trivial flat connection $d$. Then there exists a neighbourhood $\\mathcal{U} \\ni [d]$ in $\\mathbb{A}_{\\E}$ with $C^N$-quotient topology such that $[d]$ is the unique gauge class of transparent connections in $\\mathcal{U}$.\n\\end{corollary}\t\n\n\\subsection{Global injectivity results}\n\nWe now detail some cases in which Theorem \\ref{theorem:injectivity} can be upgraded.\n\n\n\\subsubsection{Line bundles}\n\n\\label{sssection:line}\n\nWe let $\\mathcal{T}^\\sharp_1$ be the restriction of the total primitive trace map \\eqref{equation:trace-total} to line bundles. The moduli space of all connections on line bundles $\\mathbb{A}_1$ carries a natural Abelian group structure using the tensor product.\nWhen restricted to line bundles, the primitive trace map $\\mathcal{T}^\\sharp_1$ takes value in $\\ell(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$, namely the set of sequences indexed by primitive free homotopy classes. We have:\n\n\\begin{lemma}\nThe map $\\mathcal{T}^\\sharp_1 : \\mathbb{A}_1 \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$ is a multiplicative group homomorphism.\n\\end{lemma}\n\\begin{proof}\n\tLeft as an exercise to the reader.\n\\end{proof}\n\n\n\\begin{remark}\n\\rm\nThere also exists a group homomorphism for higher rank vector bundles by taking the determinant instead of the trace. More precisely, writing $\\mathbb{A}_r$ for the set of all unitary connections on all possible Hermitian vector bundles of rank $r$ (up to isomorphisms), one can set\n\\begin{equation}\n\\label{equation:det}\n\\det{}^\\sharp : \\mathbb{A} \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1)),\n\\end{equation}\nby taking the determinant of the holonomy along each closed primitive geodesic. This map is also a group homomorphism (where the group structure on $\\bigsqcup_{r \\geq 0} \\mathbb{A}_r$) is also obtained by tensor product. Nevertheless, the determinant map \\eqref{equation:det} cannot be injective as all trivial bundles (of different ranks) have same image.\n\\end{remark}\n\n\n\nWe have the following result, mainly due to Paternain \\cite{Paternain-09}:\n\n\n\\begin{prop}[Paternain]\n\\label{proposition:line}\nLet $(M,g)$ be a smooth Anosov $n$-manifold. If $n \\geq 3$, then the restriction of the primitive trace map to line bundles\n\\begin{equation}\n\\label{equation:trace-line}\n\\mathcal{T}_1^\\sharp : \\mathbb{A}_1 \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp),\n\\end{equation}\nis globally injective. Moreover, if $n = 2$ then:\n\\[\n\\ker \\mathcal{T}^\\sharp_1 = \\left\\{ ([\\kappa^{\\otimes n}], [{\\nabla^{\\mathrm{LC}}}^{\\otimes n}]), n \\in \\mathbb{Z}\\right\\},\n\\]\nwhere $\\kappa \\rightarrow M$ denotes the canonical line bundle and $\\nabla^{\\mathrm{LC}}$ is connection induced on $\\kappa$ by the Levi-Civita connection.\n\\end{prop}\n\n\n\nObserve that on surfaces, the trivial line bundle $\\mathbb{C} \\times M \\rightarrow M$ (with trivial connection) and the canonical line bundle $\\kappa \\rightarrow M$ (with the Levi-Civita connection) both have trivial holonomy but are not isomorphic. This explains the existence of a non-trivial kernel for $n=2$. We will need this preliminary lemma:\n\n\\begin{lemma}\n\\label{lemma:iso}\nLet $(M,g)$ be a smooth closed Riemannian manifold of dimension $\\geq 3$ and let $\\pi : SM \\rightarrow M$ be the projection. Let $\\mathcal{L}_1 \\rightarrow M$ and $\\mathcal{L}_2 \\rightarrow M$ be two Hermitian line bundles. If $\\pi^*\\mathcal{L}_1 \\simeq \\pi^*\\mathcal{L}_2$ are isomorphic, then $\\mathcal{L}_1 \\simeq \\mathcal{L}_2$ are isomorphic.\n\\end{lemma}\n\n\\begin{proof}\nThe topology of line bundles is determined by their first Chern class. As a consequence, it suffices to show that $c_1(\\mathcal{L}_1) = c_1(\\mathcal{L}_2)$. By assumption, we have $c_1(\\pi^*\\mathcal{L}_1) = \\pi^* c_1(\\mathcal{L}_1) =  c_1(\\pi^*\\mathcal{L}_2) = \\pi^* c_1(\\mathcal{L}_2)$ and thus it suffices to show that $\\pi^* : H^2(M,\\mathbb{Z}) \\rightarrow H^2(SM,\\mathbb{Z})$ is injective when $\\dim(M) \\geq 3$. But this is then a mere consequence of the Gysin exact sequence \\cite[Proposition 14.33]{Bott-Tu-82}. \n\\end{proof}\n\n\nWe can now prove Proposition \\ref{proposition:line}:\n\n\\begin{proof}[Proof of Proposition \\ref{proposition:line}]\nAssume that $\\mathcal{T}^\\sharp_1(\\mathfrak{a}_1) = \\mathcal{T}^\\sharp_1(\\mathfrak{a}_2)$, where $\\mathfrak{a}_1 \\in \\mathbb{A}_{\\mathcal{L}_1}$ and $\\mathfrak{a}_2 \\in \\mathbb{A}_{\\mathcal{L}_2}$ are two classes of connections defined on two (classes of) line bundles. By Theorem \\ref{theorem:weak}, we obtain that the pullback bundles $\\pi^*\\mathcal{L}_1$ and $\\pi^*\\mathcal{L}_2$ are isomorphic, hence $\\mathcal{L}_1 \\simeq \\mathcal{L}_2$ are isomorphic by Lemma \\ref{lemma:iso}. Up to composing by a first bundle (unitary) isomorphism, we can therefore assume that $\\mathcal{L}_1 = \\mathcal{L}_2 =: \\mathcal{L}$. Let $\\nabla^\\mathcal{L}_1 \\in \\mathfrak{a}_1$ and $\\nabla^\\mathcal{L}_2 \\in \\mathfrak{a}_2$ be two representatives of these classes. They satisfy $\\mathcal{T}^\\sharp(\\nabla^{\\mathcal{L}}_1) = \\mathcal{T}^\\sharp(\\nabla^{\\mathcal{L}}_2)$. Combing Theorem \\ref{theorem:weak} with \\cite[Theorem 3.2]{Paternain-09}, the primitive trace map $\\mathcal{T}^\\sharp_{\\mathcal{L}}$ is known to be globally injective for connections on the same fixed bundle. Hence $\\nabla^{\\mathcal{L}}_1$ and $\\nabla^{\\mathcal{L}}_2$ are gauge-equivalent.\n\nFor the second claim, $\\mathrm{x} = ([\\mathcal{L}],\\mathfrak{a})$. If $\\mathcal{T}^\\sharp_1(\\mathrm{x}) = (1,1,...)$ (i.e. the connection is transparent), then by Theorem \\ref{theorem:weak}, one has that $\\pi^*\\mathcal{L} \\rightarrow SM$ is trivial. By the Gysin sequence \\cite[Proposition 14.33]{Bott-Tu-82}, this implies that $c_1(\\mathcal{L})$ is divisible by $2g-2$, where $g$ is the genus of $M$ (see \\cite[Theorem 3.1]{Paternain-09}), hence $[\\mathcal{L}] = [\\kappa^{\\otimes n}]$ for some $n \\in \\mathbb{Z}$. Moreover, the Levi-Civita connection on $\\kappa^{\\otimes n}$ is transparent and by uniqueness (see \\cite[Theorem 3.2]{Paternain-09}), this implies that $\\mathfrak{a} = [{\\nabla^{\\mathrm{LC}}}^{\\otimes n}]$.\n\\end{proof}\n\n\\begin{remark}\n\\rm\nThe target space in \\eqref{equation:trace-line} is actually $\\ell^\\infty(\\mathcal{C}^\\sharp,\\mathrm{U}(1))$ (sequences indexed by $\\mathcal{C}^\\sharp$ and taking values in $\\mathrm{U}(1)$) which can be seen as a subset of $\\mathrm{U}(\\ell^\\infty(\\mathcal{C}^\\sharp))$, the group of unitary operators of the Banach space $\\ell^\\infty(\\mathcal{C}^\\sharp)$ (equipped with the sup norm). Then $\\mathcal{T}_1^{\\sharp}$ is a group homomorphism and Proposition \\ref{proposition:line} asserts that\n\\[\n\\mathcal{T}_1^{\\sharp} : \\mathbb{A}_1 \\rightarrow \\mathrm{U}(\\ell^\\infty(\\mathcal{C}^\\sharp))\n\\]\nis a faithful unitary representation of the Abelian group $\\mathbb{A}_1$.\n\\end{remark}\n\n\nWe end this paragraph with a generalization of Proposition \\ref{proposition:line}. There is a natural submonoid $\\mathbb{A}' \\subset \\mathbb{A}$ which is obtained by considering sums of lines bundles equipped with unitary connections, that is:\n\\[\n\\mathbb{A}' := \\left\\{ \\mathrm{x}_1 \\oplus ... \\oplus \\mathrm{x}_k ~\\mid~ k \\in \\mathbb{N}, \\mathrm{x}_i \\in \\mathbb{A}_1\\right\\}.\n\\]\nWe then have the following:\n\n\\begin{theorem}\n\\label{theorem:sum}\nLet $(M,g)$ be a smooth Anosov Riemannian manifold of dimension $\\geq 3$. Then the restriction of the primitive trace map to $\\mathbb{A}'$:\n\\[\n\\mathcal{T}^{\\sharp} : \\mathbb{A}' \\longrightarrow \\ell^\\infty(\\mathcal{C}^\\sharp)\n\\]\nis globally injective.\n\\end{theorem}\n\n\n\\begin{proof}\nWe consider $\\mathcal{L} := \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_k$ and $\\mathcal{J} = \\mathcal{J}_1 \\oplus ... \\oplus \\mathcal{J}_{k'}$, two Hermitian vector bundles over $M$, equipped with the respective connections $\\nabla^{\\mathcal{L}_1} \\oplus ... \\oplus \\nabla^{\\mathcal{L}_k}$ and $\\nabla^{\\mathcal{J}_1} \\oplus ... \\oplus \\nabla^{\\mathcal{J}_{k'}}$ and we assume that they have same image by the primitive trace map. Fixing a periodic point $(x_\\star,v_\\star)$ and applying Proposition \\ref{proposition:representation}, we obtain that $k=k'$ and the existence of two isomorphic representations $\\rho_{\\mathcal{L}} : \\mathbf{G} \\to \\mathrm{U}(\\pi^*\\mathcal{L}_{(x_\\star,v_\\star)})$ and $\\rho_{\\mathcal{J}} : \\mathbf{G} \\to \\mathrm{U}(\\pi^*\\mathcal{J}_{(x_\\star,v_\\star)})$, where $\\mathbf{G}$ denotes Parry's free monoid at $(x_\\star, v_\\star)$. Since these representations are sums of $1$-dimensional representations, there is a unitary isomorphism $p_\\star : \\pi^*\\mathcal{L}_{(x_\\star,v_\\star)} \\to \\pi^*\\mathcal{J}_{(x_\\star,v_\\star)}$ such that for each $i \\in \\left\\{1,...,k\\right\\}$, there exists $\\sigma(i) \\in \\left\\{1,...,k\\right\\}$ with $p^{(i)}_\\star := p_\\star|_{\\pi^*\\mathcal{L}_{i, (x_\\star, v_\\star)}}$ is a representation isomorphism $p^{(i)}_\\star: \\pi^* \\mathcal{L}_{i, (x_\\star,v_\\star)} \\to \\pi^* \\mathcal{J}_{\\sigma(i), (x_\\star,v_\\star)}$.\n\nNow, following the arguments of Lemma \\ref{lemma:p-minus}, we parallel-transport $p^{(i)}_\\star$ along the homoclinic orbits with respect to the pullback of the mixed connection $\\pi^*\\nabla^{\\mathrm{Hom}(\\nabla^{\\mathcal{L}_i}, \\nabla^{\\mathcal{J}_{\\sigma(i)}})}_X$ (induced by the connections $\\nabla^{\\mathcal{L}_i}$ on $\\mathcal{L}_i$ and $\\nabla^{\\mathcal{J}_{\\sigma(i)}}$ on $\\mathcal{J}_{\\sigma(i)}$); the Lipschitz-regularity of the obtained section follows, as in Lemma \\ref{lemma:lipschitz}, from the fact that $p_\\star^{(i)} \\rho_{\\mathcal{L}_i}(g) = \\rho_{\\mathcal{J}_{\\sigma(i)}}(g) p_\\star^{(i)}$ for all $g \\in \\mathbf{G}$.\nUsing the regularity result of \\cite{Bonthonneau-Lefeuvre-20}, we thus obtain a unitary section\n\\[\np^{(i)} \\in C^\\infty(SM,\\pi^*\\mathrm{Hom}(\\mathcal{L}_i, \\mathcal{J}_{\\sigma(i)}))\n\\]\nconjugating the parallel transports along geodesic flowlines with respect to the connections $\\pi^* \\nabla^{\\mathcal{L}_i}$ and $\\pi^* \\nabla^{\\mathcal{J}_{\\sigma(i)}}$. In particular, the existence of such $p^{(i)}$ ensures that $\\mathcal{T}^{\\sharp}_1(\\mathcal{L}_i,\\nabla^{\\mathcal{L}_i}) = \\mathcal{T}^{\\sharp}_1(\\mathcal{J}_{\\sigma(i)},\\nabla^{\\mathcal{J}_{\\sigma(i)}})$. We then conclude by Proposition \\ref{proposition:line}, showing that each pair $(\\mathcal{L}_i, \\nabla^{\\mathcal{L}_i})$ is isomorphic to $(\\mathcal{J}_{\\sigma(i)}, \\nabla^{\\mathcal{J}_{\\sigma(i)}})$, for $i = 1, \\dotso, k$.\n\\end{proof}\n\n\n\n\\subsubsection{Flat bundles}\n\n\\label{sssection:flat}\n\nWe discuss the particular case of flat vector bundles. It is well-known that the data of a vector bundle equipped with a unitary connection (modulo isomorphism) is equivalent to a unitary representation of the fundamental group (modulo inner automorphisms of the unitary group). More precisely, given $\\rho \\in \\Hom(\\pi_1(M),\\mathrm{U}(r))$, one can associate a Hermitian bundle $\\E \\rightarrow M$ equipped with a flat unitary connection $\\nabla^{\\E}$ by the following process: let $\\widetilde{M}$ be the universal cover of $M$; consider the trivial bundle $\\mathbb{C}^r \\times \\widetilde{M}$ equipped with the flat connection $d$ and define the relation $(x,v) \\sim (x',v')$ if and only if $x' = \\gamma(x), v' = \\rho(\\gamma)v$, for some $\\gamma \\in \\pi_1(M)$; then $(\\E,\\nabla^{\\E})$ is obtained by taking the quotient $\\mathbb{C}^r \\times \\widetilde{M}\/\\sim$. Changing $\\rho$ by an isomorphic representation $\\rho' = p \\cdot \\rho \\cdot p^{-1}$ (for $p \\in \\mathrm{U}(r)$) changes the connection by a gauge-equivalent connection and this process gives a one-to-one correspondence between the moduli spaces. \n\n\nFor $r \\geq 0$, we let\n\\[\n\\mathcal{M}_r := \\mathrm{Hom}(\\pi_1(M),\\mathrm{U}(r))\/\\sim,\n\\]\nbe the moduli space of unitary representations of the fundamental group, where two representations are equivalent $\\sim$ whenever they are isomorphic. The space $\\mathcal{M}_r$ is called the \\emph{character variety}, see \\cite{Labourie-13} for instance. For $r=0$, it is reduced to a point; for $r=1$, it is given by $\\mathcal{M}_1 = \\mathrm{U}(1)^{b_1(M)}$, where $b_1(M)$ denotes the first Betti number of $M$. Given $\\mathrm{x} \\in \\mathcal{M}_r$, we let $\\Psi(\\mathrm{x}) = (\\E_{\\mathrm{x}}, \\nabla^{\\E_{\\mathrm{x}}})$ be the data of a Hermitian vector bundle equipped with a unitary connection (up to gauge-equivalence) described by the above process. The primitive trace map $\\mathcal{T}^\\sharp$ can then be seen as a map:\n\\[\n\\mathcal{T}^\\sharp : \\bigsqcup_{r \\geq 0} \\mathcal{M}_r \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp), \\quad \\mathcal{T}^\\sharp(\\mathrm{x}) := \\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}}})\n\\]\nwhere the right-hand side is understood by \\eqref{equation:trace}. We then have the following:\n\n\n\\begin{prop}\n\\label{proposition:flat}\nLet $(M, g)$ be an Anosov manifold of dimension $\\geq 2$. Then the primitive trace map\n\\[\n\\mathcal{T}^\\sharp : \\bigsqcup_{r \\geq 0} \\mathcal{M}_r \\rightarrow \\ell^\\infty(\\mathcal{C}^\\sharp), \n\\]\nis globally injective. Moreover, given $\\mathrm{x}_0 = ([\\E_0],[\\nabla^{\\E}_0]) \\in \\mathcal{M}_r$, the primitive trace map is weakly locally injective (in the sense of Theorem \\ref{thm:weaklocal}) near $\\mathrm{x}_0$ in the space $\\mathbb{A}_{[\\E_0]}$ of all unitary connections on $[\\E_0]$.\n\\end{prop}\n\nThe previous Proposition \\ref{proposition:flat} will be strengthened below when further assuming that $(M,g)$ has negative curvature (see Lemma \\ref{lemma:small-curvature}): we will show that the primitive trace map is globally injective on connections with \\emph{small} curvature.\nThe first part of Proposition \\ref{proposition:flat} could be proved by purely algebraic arguments; nevertheless, we provide a proof with dynamical flavour, which is more in the spirit of the present article. We need a preliminary result:\n\n\\begin{lemma}\\label{lemma:Pi_1inj}\nAssume $(M,g)$ is Anosov and $\\nabla^{\\E}$ is a flat and unitary connection on the Hermitian vector bundle $\\E \\rightarrow M$. If $\\mathbf{X} := (\\pi^*\\nabla^{\\E})_X$, then:\n\\begin{itemize}\n\\item If $\\mathbf{X} u = f$ with $f=f_0 + f_1 \\in C^\\infty(M,(\\Omega_0 \\oplus \\Omega_1) \\otimes \\E)$ and $u \\in C^\\infty(SM,\\pi^*\\E)$, then $f_0 = 0$ and $u$ is of degree $0$. \n\\item In particular, smooth invariant sections $u \\in \\ker \\mathbf{X}|_{C^\\infty(SM,\\pi^*\\E)}$ are of degree $0$.\n\\item The operator $\\Pi_1^{\\E}$ is s-injective.\n\\end{itemize}\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nThe proof is based on the twisted Pestov identity for flat connections. \n\n\\begin{lemma}[Twisted Pestov identity]\nLet $u\\in H^2(SM,\\pi^*\\E)$. Then\n\\[\n\\|\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u \\|_{L^2}^2 = \\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u  \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u, \\nabla^{\\E} _{\\mathbb{V}} u \\rangle_{L^2} + (n -1) \\| \\mathbf{X} u \\|_{L^2}^2.\n\\]\n\\end{lemma}\n\nFor the notation, see \\S \\ref{sssection:twisted-fourier}; for a proof, we refer to \\cite[Proposition 3.3]{Guillarmou-Paternain-Salo-Uhlmann-16}. An important point is that the following inequality holds for Anosov manifolds:\n\\[\n\\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u  \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u, \\nabla^{\\E} _{\\mathbb{V}} u \\rangle_{L^2} \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u\\|^2_{L^2},\n\\]\nwhere $C > 0$ is independent of $u$, see \\cite[Theorem 7.2]{Paternain-Salo-Uhlmann-15} for the case of the trivial line bundle (the generalization to the twisted case is straightforward). We thus obtain:\n\\begin{equation}\n\\label{equation:inegalite}\n\\|\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u \\|_{L^2}^2 \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u\\|^2_{L^2} + (n -1) \\| \\mathbf{X} u \\|_{L^2}^2.\n\\end{equation}\n\nBy assumption, $\\mathbf{X} u = f_0 + f_1  \\in C^\\infty(M, (\\Omega_0 \\oplus \\Omega_1) \\otimes \\E)$. Observe that this equation can be split into odd\/even parts, namely: $\\mathbf{X} u_{\\mathrm{even}} = f_1, \\mathbf{X} u_{\\mathrm{odd}} = f_0$, and $u_{\\mathrm{even},\\mathrm{odd}} \\in C^\\infty(SM,\\pi^*\\E)$ have respective even\/odd Fourier components. Applying \\eqref{equation:inegalite} with $u_{\\mathrm{odd}}$, we obtain $f_0 = 0$, $\\mathbf{X} u_{\\mathrm{odd}} = 0$ and $\\nabla^{\\E}_{\\mathbb{V}} u_{\\mathrm{odd}} = 0$, that is $u_{\\mathrm{odd}}$ is of degree $0$ but $0$ is even so $u_{\\mathrm{odd}} = 0$. As far as $u_{\\mathrm{even}}$ is concerned, observe that $\\nabla^{\\E} _{\\mathbb{V}} \\mathbf{X} u_{\\mathrm{even}} = \\nabla^{\\E} _{\\mathbb{V}} f_1$ and:\n\\[\n\\|\\nabla^{\\E} _{\\mathbb{V}} f_1\\|_{L^2}^2 = \\langle -\\Delta_{\\mathbb{V}}^{\\E} f_1, f_1 \\rangle_{L^2} = (n-1) \\|f_1\\|^2_{L^2}.\n\\]\nHence, applying the twisted Pestov identity with $u_{\\mathrm{even}}$, we obtain:\n\\[\n0 = \\|\\mathbf{X} \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}}  \\|_{L^2}^2 - \\langle R \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}}, \\nabla^{\\E} _{\\mathbb{V}} u_{\\mathrm{even}} \\rangle_{L^2} \\geq C\\|\\nabla^{\\E}_{\\mathbb{V}} u_{\\mathrm{even}}\\|^2_{L^2},\n\\]\nthat is $u_{\\mathrm{even}}$ is of degree $0$. This proves the first point and the second point is a direct consequence of the first point.\n\nFor the last point, consider the equation $\\mathbf{X} u = \\pi_1^*f$. By the first point, $u$ is of degree zero, so $u = \\pi_0^*u'$ for some $u' \\in C^\\infty(M, \\E)$. Hence by \\eqref{eq:pullback} we get $f = \\nabla^{\\E}u'$ and the conclusion follows by Lemma \\ref{lemma:x-ray}.\n\\end{proof}\n\n\nWe can now prove Proposition \\ref{proposition:flat}.\n\n\\begin{proof}\nWe prove the first part of Proposition \\ref{proposition:flat}. We assume that $\\mathcal{T}^\\sharp(\\mathrm{x}_1) =\\mathcal{T}^\\sharp(\\mathrm{x}_2)$ or, equivalently, that $\\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}_1}}) = \\mathcal{T}^\\sharp(\\nabla^{\\E_{\\mathrm{x}_2}})$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} implies that the bundles $\\pi^*\\E_{\\mathrm{x}_1}$ and $\\pi^*\\E_{\\mathrm{x}_2}$ are isomorphic and yields the existence of a section $p \\in C^\\infty(SM,\\mathrm{U}(\\pi^*\\E_{\\mathrm{x}_2}, \\pi^*\\E_{\\mathrm{x}_1}))$ such that: $C_{\\mathrm{x}_1}(x,t) = p(\\varphi_t x)C_{\\mathrm{x}_2}(x,t)p(x)^{-1}$ for all $x \\in \\mathcal{M}, t \\in \\mathbb{R}$, which is equivalent to\n\\[\n\\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}_X p = 0,\n\\]\nwhere $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}$ is the mixed connection induced by $\\nabla^{\\E_{\\mathrm{x}_2}}$ and $\\nabla^{\\E_{\\mathrm{x}_1}}$ on $\\mathrm{Hom}(\\E_{\\mathrm{x}_2}, \\E_{\\mathrm{x}_1})$. Observe that by \\eqref{equation:induced-curvature}, the curvature of $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}$ vanishes as both curvatures $F_{\\nabla^{\\E_{\\mathrm{x}_{1,2}}}}$ vanish. Applying Lemma \\ref{lemma:Pi_1inj} with $\\mathbf{X} := \\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{\\mathrm{x}_2}}, \\nabla^{\\E_{\\mathrm{x}_1}})}_X$ acting on the pullback bundle $\\pi^*\\mathrm{Hom}(\\E_{\\mathrm{x}_2}, \\E_{\\mathrm{x}_1})$, we get that $p$ is of degree $0$, which is equivalent to the fact that the connections are gauge-equivalent.\n\nAs to the second part of Proposition \\ref{proposition:flat}, by Theorem \\ref{thm:weaklocal} it is a straightforward consequence of the s-injectivity of $\\Pi_1^{\\mathrm{End}(\\E_0)}$ and the fact that elements of $\\ker(\\pi^*\\nabla^{\\mathrm{End}(\\E_0)})_X|_{C^\\infty})$ are of degree zero, which follows from Lemma \\ref{lemma:Pi_1inj} (items 2 and 3).\n\\end{proof}\n\n\n\\subsubsection{Negative sectional curvature}\n\n\\label{sssection:negative}\n\nWe now assume further that the Riemannian manifold $(M,g)$ has negative sectional curvature. We introduce the following condition:\n\n\\begin{definition}\nWe say that the pair of connections $(\\nabla^{\\E_1},\\nabla^{\\E_2})$ satisfies the \\emph{spectral condition} if the mixed connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1},\\nabla^{\\E_2})}$ has no non-trivial twisted CKTs. \n\\end{definition}\n\nThis condition is symmetric in the pair $(\\nabla^{\\E_1},\\nabla^{\\E_2})$. Observe that by \\eqref{equation:lien}, the previous condition is invariant by changing one of the two connections by $p^* \\nabla^{\\E_i}$, for some vector bundle isomorphism $p$, and thus this condition descends to the moduli space. We then define\n\\begin{equation}\\label{eq:setS}\n\\mathbf{S} \\subset \\mathbb{A} \\times \\mathbb{A},\n\\end{equation}\nthe subspace of all pairs of equivalence classes of connections satisfying the spectral condition. The set $\\mathbf{S}$ is open and dense (for the $C^N_*$-topology, $N \\gg 1$) as shown in Appendix \\ref{appendix:ckts}. Moreover, it also contains all pairs of connections with \\emph{small curvature}, that is if\n\\[\n\\Omega_\\varepsilon := \\left\\{ \\mathrm{x} = ([\\E],[\\nabla^{\\E}]) \\in \\mathbb{A}, ~~~ \\|F_{\\nabla^{\\E}}\\|_{L^\\infty(M,\\Lambda^2 T^*M \\otimes \\mathrm{End}(\\E))} < \\varepsilon \\right\\} \\subset \\mathbb{A},\n\\]\nthen we have the following:\n\n\n\\begin{lemma}\n\\label{lemma:small-curvature}\nLet $(M,g)$ be a negatively-curved Riemannian manifold of dimension $\\geq 2$ and let $-\\kappa < 0$ be an upper bound for the sectional curvature. There exists $\\varepsilon(n,\\kappa) > 0$ such that:\n\\[\n\\Omega_{\\varepsilon(n,\\kappa)} \\times \\Omega_{\\varepsilon(n,\\kappa)} \\subset \\mathbf{S}.\n\\]\nOne can take $\\varepsilon(n,\\kappa) =\\dfrac{\\kappa \\sqrt{n-1}}{4}$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWe start by a preliminary discussion. Given a Hermitian vector bundle $\\E \\to M$ with metric $\\langle \\bullet, \\bullet \\rangle$, a unitary connection $\\nabla^{\\E}$, we introduce, following \\cite[Section 3]{Guillarmou-Paternain-Salo-Uhlmann-16}, an operator $\\mathcal{F}^{\\E} \\in C^\\infty(SM,\\mathcal{N} \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))$ (recall that $\\mathcal{N}$ is the normal bundle, see \\S\\ref{sssection:line-bundle}) defined by the equality:\n\\begin{equation}\n\\label{equation:twisted-curvature}\n \\langle \\mathcal{F}^{\\E}(x,v) e, w \\otimes e' \\rangle := \\langle F_{\\nabla^{\\E}}(v,w)e,e' \\rangle,\n\\end{equation}\nwhere $F_{\\nabla^{\\E}}$ is the connection of $\\nabla^{\\E}$, and $(x,v) \\in SM, e, e' \\in \\E_x, w \\in \\mathcal{N}(x,v)$, and the metric on the left-hand side is the natural extension of the metric $\\langle \\bullet, \\bullet \\rangle$ on $\\E$ to $\\mathcal{N} \\otimes \\E$ by tensoring with the metric $g$. A straightforward computation shows that:\n\\begin{equation}\n\\label{equation:bad-bound}\n\\|\\mathcal{F}^{\\E}\\|_{L^\\infty(SM, \\mathcal{N} \\otimes \\mathrm{End}_{\\mathrm{sk}}(\\E))} \\leq \\|F_{\\nabla^{\\E}}\\|_{L^\\infty(M,\\Lambda^2 T^*M \\otimes \\mathrm{End}(\\E))}. \n\\end{equation}\nNow, let $\\nabla^{\\E_1}$ and $\\nabla^{\\E_2}$ be two unitary connections, $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$ be the mixed connection and $\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}$ be the operator induced by the mixed connection as in \\eqref{equation:twisted-curvature}. Observe that by \\eqref{equation:induced-curvature} and \\eqref{equation:bad-bound}, we get:\n\\begin{equation}\n\\label{equation:bad-bound-2}\n\\begin{split}\n\\|\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}\\|_{L^\\infty}  \\leq \\|F_{\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}}\\|_{L^\\infty} \\leq \\|F_{\\nabla^{\\E_1}}\\|_{L^\\infty} + \\|F_{\\nabla^{\\E_2}}\\|_{L^\\infty} < 2\\varepsilon(n,\\kappa).\n\\end{split}\n\\end{equation}\nBy \\cite[Theorem 4.5]{Guillarmou-Paternain-Salo-Uhlmann-16}, if $m \\geq 1$ satisfies:\n\\begin{equation}\n\\label{equation:ckts}\nm(m+n-2) \\geq 4 \\dfrac{\\|\\mathcal{F}^{\\mathrm{Hom(\\E_1,\\E_2)}}\\|^2_{L^\\infty}}{\\kappa^2},\n\\end{equation}\nthen there are no twisted CKTs of degree $m$ (for the connection $\\nabla^{\\mathrm{Hom}(\\nabla^{\\E_1}, \\nabla^{\\E_2})}$). Now, the choice of $\\varepsilon(n,\\kappa) > 0$ combined with \\eqref{equation:bad-bound-2} guarantees that \\eqref{equation:ckts} is satisfied for any $m \\geq 1$.\n\\end{proof}\n\n\n\n\nWe then have the following statement:\n\n\\begin{prop}\n\\label{proposition:negative}\nLet $(M,g)$ be a negatively-curved Riemannian manifold of dimension $\\geq 2$. Let $(\\mathfrak{a}, \\mathfrak{a}') \\in \\mathbf{S}$ such that $\\mathcal{T}^\\sharp(\\mathfrak{a}) = \\mathcal{T}^\\sharp(\\mathfrak{a}')$. Then $\\mathfrak{a} = \\mathfrak{a}'$.\n\\end{prop}\n\nIn other words, two connections satisfying the spectral condition and whose images by the primitive trace map are equal, are actually gauge-equivalent.\n\n\\begin{proof}\nConsider two representatives $\\nabla^{\\E_1} \\in \\mathfrak{a}$ and $\\nabla^{\\E_2} \\in \\mathfrak{a}'$. The exact Liv\\v{s}ic cocycle Theorem \\ref{theorem:weak} provides a section $p \\in C^\\infty(SM,\\mathrm{U}(\\pi^*\\E_2,\\pi^*\\E_1))$ such that:\n\\[\n\\pi^* \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{2}}, \\nabla^{\\E_{1}})}_X p = 0.\n\\]\nBy assumption, $(M,g)$ has negative curvature and thus $p$ has finite Fourier degree by \\cite[Theorem 4.1]{Guillarmou-Paternain-Salo-Uhlmann-16}. Moreover, since $ \\nabla^{\\mathrm{Hom}(\\nabla^{\\E_{2}}, \\nabla^{\\E_{1}})}$ has no non-trivial twisted CKTs, $p$ is of degree $0$ (see \\cite[Theorem 5.1]{Guillarmou-Paternain-Salo-Uhlmann-16}). This shows that the connections are gauge-equivalent.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Topological results}\n\n\\label{sssection:topology}\n\nIn this section we prove a global \\emph{topological} uniqueness result for the primitive trace map. \n\n\\begin{prop}\\label{proposition:topology}\n\tLet $(M, g)$ be an orientable Anosov manifold. If $\\mathrm{x}_i = ([\\E_i], [\\nabla^{\\E_i}]) \\in \\mathbb{A}$ for $i = 1, 2$, then $\\mathcal{T}^\\sharp(\\mathrm{x}_1) = \\mathcal{T}^\\sharp(\\mathrm{x}_2)$ implies:\n\t\n\t\\begin{itemize}\n\t\t\t\\item If $\\dim M$ is odd or more generally the Euler characteristic $\\chi(M) = 0$ vanishes, then $\\E_1 \\simeq \\E_2$ are isomorphic as vector bundles.\n\t\t\t\\item If $\\dim M = 2d$ for some $d \\in \\mathbb{N}$ and $\\chi(M) \\neq 0$, then\n\t\t\t\t\\begin{itemize}\n\t\t\t\t\t\\item The Chern classes satisfy $c_i(\\E_1) = c_i(\\E_2)$ for $i = 1, \\dotso, d - 1$; also $c_d(\\E_1) - c_d(\\E_2) \\in H^{2d}(M; \\mathbb{Z}) \\cong \\mathbb{Z}$ is a multiple of $\\chi(M)$.\n\t\t\t\t\t\\item If the even cohomology ring $H^{\\mathrm{even}}(M; \\mathbb{Z})$ is torsion-free, and the rank of the bundles is less than $d$ or more generally $c_d(\\E_1) = c_d(\\E_2)$, then $\\E_1$ and $\\E_2$ are stably isomorphic, i.e. there is an $m \\geq 0$ such that $\\E_1 \\oplus \\mathbb{C}^m \\simeq \\E_2 \\oplus \\mathbb{C}^m$\n\t\t\t\t\\end{itemize}\n\t\\end{itemize}\t\n\\end{prop}\n\\begin{proof}\n\tAs a direct consequence of Theorem \\ref{theorem:weak}, from $\\mathcal{T}^\\sharp(\\mathrm{x}_1) = \\mathcal{T}^\\sharp(\\mathrm{x}_2)$ we obtain that $\\pi^*\\E_1 \\simeq \\pi^*\\E_2$ are isomorphic.\n\t\n\tIf $(M, g)$ has a vanishing Euler characteristic, there is a non-vanishing vector field $V \\in C^\\infty(M, TM)$ (see \\cite[Chapter 11]{Bott-Tu-82}), that we normalise to unit norm using the metric $g$ and hence see as a section of $SM$. Then since $\\pi \\circ V = \\id_M$, we get\n\t\\[\\E_1 \\simeq V^*\\pi^*\\E_1 \\simeq V^*\\pi^* \\E_2 \\simeq \\E_2,\\]\n\tcompleting the proof of the first item. \n\t\n\tThe first point of the second item is immediate after an application of the Gysin exact sequence \\cite[Proposition 14.33]{Bott-Tu-82} for the sphere bundle $SM$. The second point follows from the first one and the fact that the Chern character gives an isomorphism between the rational $K$-theory and even rational cohomology, see \\cite[Proposition 4.5]{Hatcher-17}.\n\\end{proof}\n\n\n\nIt is not known to the authors if further results hold as to the injectivity of $\\pi^* : \\mathrm{Vect}(M) \\rightarrow \\mathrm{Vect}(SM)$ in even dimensions ($\\dim M \\geq 4$). \n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOne of the fundamental problems in the study of complex networks~\\cite{Dorogovtsev_book1,NBW,BBB,Dorogovtsev_book2,NetsCrowdsMarkets} is to identify\nevolution mechanisms that shape the structure and dynamics of large\nreal networks such as the Internet, the world wide web, and various biological and\nsocial networks. In particular, how do complex networks grow so that\nmany of them are scale-free and have strong clustering and non-trivial community structure?\nThe preferential attachment (PA) mechanism~\\cite{PA,Krapivsky,Dorogovtsev2}, where new \\textit{connections} are made\npreferentially to more popular nodes, is widely accepted as the\nplausible explanation for the emergence of the scale-free structures\n(i.e.\\ the power-law degree distributions) in large networks.\nPA has been empirically validated for many real growing\nnetworks~\\cite{Newman,BarabasiJeongNeda,Vazquez,Jeong} using statistical analysis of a sequence of network\nsnapshots, demonstrating that it is indeed a key element of network evolution. Moreover, there is some evidence that the evolution of the community graph --- a graph where nodes represent communities and links refer to members shared by two communities --- is also driven by PA~\\cite{Pollner}.\n\nNevertheless, PA alone cannot explain two other empirically observed universal properties of complex networks: strong clustering~\\cite{WS} and significant community structure~\\cite{GirvanNewman}. Namely, in synthetic networks generated by standard PA, clustering\nis asymptotically zero~\\cite{Bollobas} and there are no communities~\\cite{Fortunato}. To resolve the zero-clustering problem, several modifications of the original PA mechanism  have been proposed~\\cite{Dorogovtsev3,Klemm,Vazquez2,Jackson}. To the best of our knowledge, however, none of these models capture all three fundamental properties of complex networks: heavy-tail degree distribution, high clustering, and community structure.\n\n\nIn social networks, the presence of communities, that might represent node clusters based on certain social factors such as economic status or political beliefs, is intuitively expected. A remarkable observation~\\cite{GirvanNewman,www,metabolic,biochemical,financial,metabolic2} is that many other networks, including food webs, the world wide web, metabolic, biochemical, and financial networks, also admit a reasonable division into informative communities. Since that discovery, community detection has become one of the main tools for the analysis and understanding of network data~\\cite{Fortunato,Danon}. \n\nDespite an enormous amount of attention to community detection algorithms and their efficiency, there were very few attempts to answer a more fundamental question: what is the actual mechanism that induces community structure in real networks? For social networks, where  there is a strong relationship between a high concentration of triangles and the existence of community structure~\\cite{NP}, triadic closure~\\cite{Rapoport} has been proposed as a plausible mechanism for generating communities~\\cite{Bianconi}. It was also shown by means of a simple agent-based acquaintance model that a large-scale community structure can emerge from the underlying social dynamics~\\cite{Bhat}. There also exist other contributions in this direction, where proposed mechanisms and generative models are specifically tailored for social networks~\\cite{Marian,Toivonen,Lambiotte,Lambiotte2}.\n\n\nHere we show how latent network geometry coupled with preferential attachment of \\textit{nodes} to this geometry induces community structure as well as power-law degree distributions and strong clustering. We prove that these universal properties of complex networks naturally emerge from the new mechanism that we call \\textit{geometric preferential attachment} (GPA), without appealing to the specific nature (e.g.\\ social) of networks. Using the Internet as an example, we demonstrate that GPA generates networks that are in many ways similar to real networks.\n\n\n\\section{Results}\n\\subsection{Geometric Preferential Attachment}\n\n\nIn growing networks the concept of popularity that PA exploits is just one aspect of node attractiveness; another important aspect is similarity~\\cite{PSO}.  Namely, if nodes are similar (``birds of feather''), then they have a higher chance of being\nconnected (``flock together''), even if they are not popular. This effect, known as homophily in social sciences~\\cite{McPherson}, has been observed in many  real networks of various nature~\\cite{Redner,Watts}.\n\nThe GPA mechanism utilizes the idea that both popularity and similarity are important. We take the node birth time $t=1,2,\\ldots$ as a proxy for node's popularity: all other things being equal, the older the node (i.e.\\ the smaller $t$), the more popular it is. The similarity attribute of node $t$ is modeled by a random variable $\\theta_t$ distributed over a circle $\\mathbb{S}^1$ that abstracts the ``similarity'' space. One can think of the similarity space as an image of a certain projection $p: \\mathcal{A}\\rightarrow\\mathbb{S}^1$ from a space of unknown or not easily measurable  attributes $(a^1,\\ldots,a^k)\\in\\mathcal{A}$ of nodes. For social networks, these attributes could be political beliefs, education, and social status, whereas for biological networks, $\\{a^i\\}$ may represent chemical properties of metabolites or geometric properties of protein shapes. While the absolute value of the similarity coordinate $\\theta_t=p(a_t^1,\\ldots,a_t^k)$ does not have any specific meaning, the angular distance $\\theta_{st}=\\pi-|\\pi-|\\theta_s-\\theta_t||$ quantifies the similarity between two nodes. Upon its birth, a new node $t$ connects to an existing node $s<t$ if $s$ is both popular enough and similar to $t$, that is if $s^\\beta\\theta_{st}$ is small, where $\\beta\\in[0,1]$ is a parameter that controls the relative contributions of popularity and similarity.\n\nThe described rule for establishing new connections admits a simple geometric interpretation which is very useful for analytical treatment of the model. Let us define the radial coordinate of node $s$ at time $s$ as $r_s=2\\ln s$, and let it grow\nwith time, so that at time $t>s$ it is $r_s(t)=\\beta r_s + (1-\\beta)r_t$. The distance $x_{st}$ between two points in the hyperbolic\nplane of curvature $K=-1$ with polar coordinates\n$(r_s(t),\\theta_s)$ and $(r_t,\\theta_t)$ is approximately~\\cite{Bonahon} $x_{st}=r_s(t)+r_t+2\\ln\\frac{\\theta_{st}}{2}=2\\ln\\left(\\frac{s^\\beta\tt^{2-\\beta}\\theta_{st}}{2}\\right)$. Since for any given $t$, the sets of nodes $s<t$ that minimize $s^\\beta\\theta_{st}$ and $x_{st}$ are the same, new nodes simply connect to the hyperbolically closest existing nodes. Note that the increase of the radial coordinate $r_s(t)$\ndecreases the effective age of the node, and thus models the effect of\npopularity fading observed in many real networks~\\cite{Raan}.\n\n\nBut how do new nodes find their positions in this similarity space? The main assumption of our model is that the hidden attribute space $\\mathcal{A}$ of a growing network is likely to contain ``hot'' regions (e.g.\\ of\nhuman activity), and that the hotter the region, the more attractive it is for new nodes.\nHot regions can for instance represent some hot areas in science.  When these regions are projected onto the similarity space $\\mathbb{S}^1$, the hotness\nmanifests itself by a higher node density, more scientists working in a hot area. The higher attractiveness of a hot region is then modeled by placing a new node in this region with the higher probability, the hotter this region is, i.e.\\ the higher the node density in it. That is, new scientists are expected to begin their careers working in hot areas where many existing scientists are already active, versus jumping onto some obscure developments that nobody understands. Therefore the higher the node density in a particular section of our similarity space $\\mathbb{S}^1$, the higher the probability that a new node is placed in this section. Intuitively we would expect that this process should lead to heterogeneous distributions of node coordinates in the similarity space. This intuition is confirmed by empirical results: if we map real networks to their hyperbolic spaces~\\cite{SustainingInt,HyperMap}, we observe that the resulting empirical angular node density is not uniform (e.g.\\ see Fig.~5(a)), and nodes tend to cluster into tight communities.  In the Internet, for example, these communities are groups of Autonomous Systems belonging to the same country.\n\nThere are many ways to implement this general idea. For a variety of reasons we found that the most natural and consistent one is as follows. First we define the attractiveness of any location $\\varphi\\in\\mathbb{S}^1$ for a new node $t$ with radial coordinate $r_t$ as the number of existing nodes $s<t$ lying in the hyperbolic disk\n$D_\\varphi(r_t)$ of radius $r_t$ centered at\n($r_t$,$\\varphi$). The higher the attractiveness of a location $\\varphi$, the higher the probability that a new node $t$ will chose this location as its place $\\theta_t=\\varphi$ in the similarity space. We refer to this mechanism as the geometric preferential attachment (GPA) of nodes to the similarity space. This mechanism is illustrated in Fig.~1.\n\n\\begin{figure}[t]\n\t\\centerline{\\includegraphics[width=90mm]{Fig01.pdf}}\n\t\\caption{\\textbf{Geometric preferential attachment}. At time $t$, a new node appears at distance $r_{t}$ from the center of the hyperbolic disk denoted by cross. Points $\\varphi_1$ and $\\varphi_2$ represent two potential locations of the new node, and the drop-shaped curves are the boundaries of the hyperbolic disks $D_{\\varphi_1}(r_t)$ and $D_{\\varphi_2}(r_t)$ of radius $r_t$ centered at $\\varphi_1$ and $\\varphi_2$. Since similarity is attractive and $D_{\\varphi_1}(r_t)$ contains more nodes (five) than $D_{\\varphi_2}(r_t)$ (none), the new node is more likely to appear at $\\theta_t=\\varphi_1$.}\n\\end{figure}\n\n\nThe exact definition of the GPA model is:\n\\begin{enumerate}\n\t\\item[0.] Initially the network is empty. New nodes $t$ appear one\n\tat a time, $t=1,\\ldots$, and for each $t$:\n\t \\item[$1.$] The angular (similarity) coordinate $\\theta_t$ of a new node $t$\n\t is determined as follows:\n    \\begin{enumerate}\\item[(a)] Sample $\\varphi_i\\sim U[0,2\\pi]$, $i=1,\\ldots,t$, uniformly at random. The set of points $\\hat{t}_1=(r_t,\\varphi_1),\\ldots,\\hat{t}_t=(r_t,\\varphi_t)$ in the hyperbolic plane are the ``candidate'' positions for the newborn node;  \n\t \\item[(b)] Define the attractiveness $A_t(\\varphi_i)$ of the $i^{\\textrm{th}}$ candidate as the number of existing nodes that  lie within hyperbolic distance $r_t$ from it; \n\t \\item[(c)] Set $\\theta_t=\\varphi_i$ with probability\n\t \\begin{equation}\\label{probability}\n\t \\Pi_t(i)=\\frac{A_t(\\varphi_i)+\\Lambda}{\\sum_{j=1}^t(A_t(\\varphi_j)+\\Lambda)},\n\t \\end{equation}\n\t where $\\Lambda\\geq0$ is a parameter, called the initial attractiveness.\n\t\\end{enumerate}\n    \\item[2.] The radial (popularity) coordinate of node $t$ is set to $r_t=2\\ln t$. The radial coordinates of existing nodes $s<t$ are updated to $r_s(t)=\\beta r_s+(1-\\beta)r_t$.\n\t\\item[3.] Node $t$ connects to $m$ hyperbolically closest\n\texisting nodes (if $t\\leq m$, then node $t$ connects to all existing nodes).\n\\end{enumerate}\n\nThe GPA model has thus three parameters: the number of links $m$ established by every new node, the speed of popularity fading $\\beta$, and the initial attractiveness $\\Lambda$. A moment's thought shows that $m$ controls the average degree of the network, $\\bar{k}=2m$. We prove in Methods that the model generates scale-free networks and $\\beta$ controls the power-law exponent $\\gamma$. The initial attractiveness $\\Lambda$ controls the heterogeneity of the angular node density, namely, the heterogeneity is a decreasing function of $\\Lambda$. When $\\Lambda\\rightarrow\\infty$, the GPA model becomes manifestly identical to the homogeneous popularity$\\times$similarity (PS) model~\\cite{PSO}, where the angular coordinate $\\theta_t$ of a new node $t$ is sampled uniformly at random on $[0,2\\pi]$. Note, however, that in GPA, choosing a position in the similarity space is an active decision made by a node based on the attractiveness of different locations, as opposed to ``passive'' uniform randomness in PS. In standard PA, the initial attractiveness term is used  to control the exponent of the power-law degree distribution~\\cite{Krapivsky,Dorogovtsev2}. In what follows we show that in GPA, $\\Lambda$ controls certain properties of the community size distribution.\n\nFigure~2 shows the simulation results for networks of size $n=10^3$ generated by the GPA model with $m=3$ (i.e.\\ each new node connects to the three hyperbolically closest nodes), $\\beta=2\/3$, and different values of $\\Lambda$. As expected, the smaller the value of $\\Lambda$, the more heterogeneous the distribution of angular coordinates. To quantify the difference between the empirical distribution of the angular coordinates and the uniform distribution on $[0,2\\pi]$, we use the Kolmogorov-Smirnov (KS) statistic, one of the standard distances that measures the difference between two probability distributions. Recall that the KS statistic $\\rho$ is defined as the maximum difference between the values of the empirical distribution $\\hat{F}_n(\\theta)$ of the sample $\\theta_1,\\ldots,\\theta_n$ and the uniform distribution $F_{U[0,2\\pi]}(\\theta)=\\theta\/2\\pi$,\n\\begin{equation}\\label{KS_def}\n\\rho=\\max_{\\theta\\in[0,2\\pi]}\\left|\\hat{F}_n(\\theta)-\\frac{\\theta}{2\\pi}\\right|\n\\end{equation}\nThe KS statistic as a function of $\\Lambda$ is shown in the bottom panel of Fig.~2. As expected, $\\rho(\\Lambda)$ is a decreasing function of $\\Lambda$.\n\n\\subsection{Degree Distribution}\n\nFor each of the three networks depicted in Fig.~2, the statistical procedure for quantifying power-law behavior in empirical data proposed in~\\cite{powerlaws} accepts the hypothesis that the network is scale-free. It estimates the lower cutoff for the scaling region as $k_{{\\rm min}}=3$, which is consistent with the minimum degree in the networks $m=3$. Figure~3(a) shows a doubly logarithmic plot of the empirical degree distributions $P(k)\\sim k^{-\\gamma}$ along with the fitted power-law with exponent $\\gamma=2.5$.\n\nThese empirical results show that the degree distribution of a network generated by GPA appears to be a power-law. Moreover, quite unexpectedly, the power-law exponent $\\gamma$ remains similar for different values of $\\Lambda$. These results can be proved analytically (see Methods for details). Remarkably, for any value of $\\Lambda$, the GPA model produces scale-free networks with the power-law degree distribution identical to the degree distribution in networks growing according to PA, and having power-law exponent $\\gamma=1+1\/\\beta$.\n\n\\begin{figure}\n\t\\centerline{\\includegraphics[width=85mm]{Fig02.pdf}}\n\t\\caption{\\textbf{GPA networks.} Synthetic networks of size $n=10^3$ generated according to the GPA model with $m=3$, $\\beta=2\/3$, and $\\Lambda=0.1$ (first row), $\\Lambda=1$ (second row), and $\\Lambda=10$ (third row). The right column shows the corresponding histograms of the angular nodes densities. The bottom panel plots the expected KS statistic $\\rho$ (\\ref{KS_def}), as a function of $\\Lambda$. For each value of $\\Lambda$, $\\rho(\\Lambda)$ is computed by averaging the KS statistics for $100$ independently generated networks.}\n\\end{figure}\n\\begin{figure\n\t\\centerline{\\includegraphics[width=75mm]{Fig03.pdf}}\n\t\\caption{\\textbf{Degree distribution and clustering.} Panel (a) shows the empirical complementary cumulative degree distribution functions (CCDF) $P_c(k)=\\sum_{k'=k} P(k')$ for the networks shown in Fig.~2 versus the corresponding power-law fit. The average clustering coefficient $\\bar{c}(k)$ as a function of node degree $k$ for these networks is shown in panel (b). The mean clustering $\\bar{c}=0.88$ for all networks.\n\t}\n\\end{figure}\n\n\\subsection{Clustering Coefficient}\n\nThe concept of clustering~\\cite{smallworlds} quantifies the tendency to form\ncliques (complete subgraphs) in the neighborhood of\na given node. Specifically, the local clustering coefficient of node $s$ is defined as the probability that two nodes $s'$ and $s''$, adjacent to $s$, are also connected to each other.\nFigure~3(b) shows the average value of the clustering coefficient $\\bar{c}(k)$ for nodes of degree $k$  as a function of $k$ for the three networks in Fig.~2. Interestingly, clustering does not depend on $\\Lambda$ either (a proof is in the Methods), and scales approximately as $k^{-1}$. This means that, on average, the nodes with higher degree have lower clustering, which is consistent with empirical observations of clustering in real complex networks~\\cite{Vazquez,RavaszBara}.\nFor all the three PGA networks, the mean clustering (the average of the local clustering coefficients) is high, $\\bar{c}=0.88$.\n\n\\subsection{Soft Communities}\n\nThe hyperbolic space underlying a network and the GPA mechanism of node appearance in that space naturally induce community\nstructure and allow to detect communities in a very intuitive and simple way. A higher density of links within a community indicates that its nodes are more similar to each other than to the other nodes, because links connect only nodes located within a certain similarity distance threshold. All such densely linked nodes are thus close to each other in some area of the similarity space, meaning that the spatial node density is high in this area. Therefore a community becomes a cluster of spatially close nodes, and the community structure is encoded in a non-uniform distribution of angular (similarity) coordinates of nodes.\n\n\\begin{figure}\n\t\\centerline{\\includegraphics[width=85mm]{Fig04.pdf}}\n\t\\caption{\\textbf{Statistics of the rescaled gaps.} Top panel shows the values of rescaled gaps $\\Delta\\theta\/\\Delta\\theta_c$ for three networks of size $n=10^4$ generated by the GPA model with $\\Lambda=0.1$ (left), $\\Lambda=1$ (middle), and $\\Lambda=10$ (right). The bottom panel shows the sample autocorrelation function of the series in the top panel.}\n\\end{figure}\n\n\nFollowing the approach in~\\cite{Serrano}, let us consider the angular gaps $\\Delta\\theta$ between consecutive nodes, and define a \\textit{soft community} as a group of nodes separated from the rest of the network  by two gaps that exceed a  certain critical value $\\Delta\\theta_c$. If a network has a total number of $n$ nodes, then the critical gap $\\Delta\\theta_c$ is defined as the expected value of the largest gap $\\Delta\\theta_{(n)}=\\max\\{\\Delta\\theta_1,\\ldots,\\Delta\\theta_n\\}$, where $\\theta_1,\\ldots,\\theta_n$ are distributed uniformly at random on $[0,2\\pi]$. The rationale behind this definition is that if nodes are distributed uniformly in the similarity space, and there are no communities, then we do not expect to find any pair of nodes separated by a gap larger than this $\\Delta\\theta_c$. The calculations in the Methods show that the critical gap is approximately\n\\begin{equation}\\label{criticalgap}\n\\Delta\\theta_c=\\frac{2\\pi\\ln n}{n}.\n\\end{equation}\n\nFigure~4 shows the statistics of the rescaled gaps $\\Delta\\theta\/\\Delta\\theta_c$ for three GPA-generated networks of size $n=10^4$ with $\\Lambda=0.1, 1,$ and $10$. In the top panel, we can see the organization of nodes on the circle with many consecutive small gaps ($\\Delta\\theta_i<\\Delta\\theta_c$) indicating groups of similar nodes (communities) separated by large gaps ($\\Delta\\theta_i>\\Delta\\theta_c$) which constitute boundaries between communities, so-called ``fault lines''~\\cite{Newman}. As expected, smaller values of $\\Lambda$ result into more heterogeneous distribution of gaps with strong long range correlations. This effect is clearly visible in the bottom panel, where the sample autocorrelation function is shown: the smaller the $\\Lambda$, the slower  the autocorrelation decays.\n\nHaving a geometric interpretation of the community structure, it is now easy to quantity how well communities are separated from each other. For each community $\\mathcal{C}$, we define its \\textit{separation} from the rest of the network $\\mathcal{S(C)}$ as the rescaled average of two gaps $\\Delta\\theta_1, \\Delta\\theta_2>\\Delta\\theta_c$ that separate $\\mathcal{C}$ from its neighboring communities,\n\\begin{equation}\\label{sep}\n\\mathcal{S(C)}=\\frac{\\Delta\\theta_1+\\Delta\\theta_2}{2\\Delta\\theta_c}\n\\end{equation}\nThe mean community separation, i.e.\\ the expected separation of a community that a randomly chosen node belongs to, can then be computed as follows:\n\\begin{equation}\\label{mean_sep}\n\\bar{\\mathcal{S}}=\\sum_{i=1}^{n_c} \\frac{n_i}{n} \\mathcal{S}(\\mathcal{C}_i),\n\\end{equation}\nwhere $n_i$ is the size of community $\\mathcal{C}_i$ and $n_c$ is total number of communities. The network metric $\\bar{\\mathcal{S}}$ can also be viewed as a measure of narrowness (or specialization) of communities. For example, in scientific collaboration network, where nodes represent scientists and communities correspond to groups with similar research interests, $\\bar{\\mathcal{S}}$ quantifies the degree of interdisciplinarity in the network. When $\\bar{\\mathcal{S}}$ is large, the boundaries between communities are sharp and each community focuses on its narrow, specific topic. On the other hand, if $\\bar{\\mathcal{S}}$ is close to one, then the boundaries are blur, communities are wide spread, and the network is highly interdisciplinary.\n\nThe difference in the stochastic behavior of the rescaled gaps in Fig.~4 suggests that the initial attractiveness $\\Lambda$ controls the mean community separation $\\bar{\\mathcal{S}}$ in the GPA-generated networks. This is confirmed by simulation results shown in Fig.~5(c), where $\\bar{\\mathcal{S}}$ is shown as a function of $\\Lambda$. As expected, $\\bar{\\mathcal{S}}(\\Lambda)$ is a monotonically decreasing function, approaching one when $\\Lambda$ is large.\n\n\n\\subsection{The Internet}\n\nTo demonstrate the ability of the GPA mechanism to generate graphs that are similar to real networks, and, in particular, to reproduce real non-uniform distributions of similarity node coordinates, we consider the Autonomous Systems (AS) Internet topology~\\cite{AS_Internet} of December 2009. The network consists of $N=25910$ nodes, ASs, and $M=63435$ links that represent  logical relationships between ASs. We embed the AS Internet into its hyperbolic space, i.e compute the popularity and similarity  coordinates $\\{r_i,\\theta_i\\}$, using HyperMap~\\cite{HyperMap}, an efficient network mapping algorithm that estimates the latent hyperbolic coordinates of nodes. The network topology has a power-law degree distribution with $\\gamma=2.1$ and average node degree $\\bar{k}\\approx5$. This automatically determines two out of three parameters of the GPA model: $m=\\bar{k}\/2$ and $\\beta=1\/(\\gamma-1)$. In Methods, we explain how to infer the value of $\\Lambda$ from network data using the maximum likelihood method. \nHere we consider the snapshot of the AS Internet based on the first $n=10^3$ nodes. The corresponding estimated value of the initial attractiveness is $\\Lambda_{\\mathrm{Int}}=0.7$.\n\n\\begin{figure}\n\t\\centerline{\\includegraphics[width=85mm]{Fig05.pdf}}\n\t\\caption{\\textbf{AS Internet vs GPA networks.} Panel (a) shows the histogram of the angular (similarity) coordinates $\\{\\theta_i\\}$ for the snapshot of the AS Internet consisting of the first $n=10^3$ nodes. All $\\{\\theta_i\\}$ are inferred by HyperMap~\\cite{HyperMap}. Panel (b) compares the KS statistics for the Internet and synthetic networks generated by the GPA model (box plot) with $\\gamma=2.1$ and $\\Lambda=0.7$. The central red mark is the median, the blue horizontal edges of the box are the 25$^{\\mathrm{th}}$ and 75$^{\\mathrm{th}}$ percentiles, the black whiskers extend to the most extreme data points not considered outliers. The box plot is obtained from 100 independent generated networks. Panel (c) shows the perfect match between real and synthetic values of the mean community separation (\\ref{mean_sep}). Error bars represent plus and minus one standard deviation. Panel (d) juxtaposes the empirical CCDF of the soft community sizes in the Internet against CCDFs obtained for the three GPA-generated networks. Panel (e) shows the temporal evolution of the maximum likelihood estimate $\\widehat{\\Lambda}_{\\mathrm{MLE}}(t)$ for the AS Internet, where the node birth times are their ranks in the decreasing degree order. The yellow star corresponds to the considered snapshot with $n=10^3$ nodes and $\\widehat{\\Lambda}_{\\mathrm{MLE}}=0.7$.}\n\\end{figure}\n\n\nFigure~5(a) shows the histogram of the angular node density for the AS Internet snapshot. We note that it is far from uniform, which is a direct indication of the presence of soft communities. We quantify the degree of heterogeneity of the angular density by the KS distance from the uniform distribution (\\ref{KS_def}) and juxtapose it against the KS distances computed for networks generated by the GPA model with $\\Lambda=0.7$  (Fig.~5(b)). The Internet value lies within the 25th and 75th percentiles of the synthetic values, which shows that the degrees of non-uniformity in the Internet and GPA networks are comparable. Fig.~5(c) compares the real network with its synthetic counterpart in terms of the expected mean community separation (\\ref{mean_sep}). The GPA mechanism generates networks with $\\bar{\\mathcal{S}}$ that match the Internet value very well. In Fig.~5(d), we compare the community size distributions in the Internet snapshot and  prediction given by the GPA model. Whereas $\\bar{\\mathcal{S}}$ for the Internet and GPA networks are essentially identical, the KS statistics and community size distributions are similar, but the match is not perfect. This effect is explained by the systematic bias present in the inferred values of the angular coordinates $\\{\\theta_i\\}$. Indeed, the HyperMap method first assumes that all angular coordinates are uniformly distributed over the similarity space $\\mathbb{S}^1$, i.e.\\ $\\Lambda=\\infty$, and then perturbs them to maximize a certain likelihood function. This \t``smoothes'' the inferred angular node density and makes it more homogeneous than the true distribution. Nevertheless, although the inferred value of $\\Lambda$ is only an approximation for the true value, the GPA model still captures well the degree of heterogeneity in the real network.\n\nFinally we note that GPA defined in Eq.~\\eqref{probability} admits an interesting interpretation that suggests a model extension that may be useful for real network analysis. The probability of a new node born at time $t$ to chose the angular position $\\varphi_i$ can be written as\n\\begin{equation}\n\\label{probability_2}\n\\Pi_t(\\varphi_i)=p_f \\frac{A_t(\\varphi_i)}{\\sum_{j=1}^t A_t(\\varphi_j)}+(1-p_f) \\frac{1}{t},\n\\end{equation}\nwhere\n\\begin{equation}\np_f=\\frac{\\langle A_t \\rangle}{\\langle A_t \\rangle+\\Lambda} \\; \\; \\mbox{and} \\; \\; \\langle A_t \\rangle=\\frac{1}{t}\\sum_{j=1}^t A_t(\\varphi_j).\n\\end{equation}\nTherefore the event of choosing a position on the circle can be understood as follows. With probability $p_f$ the new node is a follower and chooses its position according to pure GPA ($\\Lambda=0$). With the remaining probability $1-p_f$ the new node chooses its position uniformly at random among the $t$ available positions. We note that $\\Lambda$ controls $p_f$, since $\\langle A_t \\rangle \\approx 1$. When $\\Lambda$ is constant, $p_f$ is also constant, and consequently there is always a fraction of nodes that are placed at random locations. At long times, these random nodes diminish the effect of pure GPA, and eventually the angular distribution of nodes become indistinguishable from a Poisson point process on the circle.\nWe can then wonder whether a constant value of $\\Lambda$ is a realistic assumption for dealing with real networks.\nIn scientific citation networks, for example, when a new field of science is being formed, and not much work has yet been done in it, scientists may decide either to explore a new line of research within the field, or to follow one of the mainstream existing lines. The former case can be modeled by a random choice of the angular position, assuming that subfields are homogeneously distributed. The latter is modeled by the pure GPA term in Eq.~\\eqref{probability_2}. However, there is a payoff that does not remain constant during the evolution of the field. At early times, the chances to find an interesting result that would be highly cited and followed by others are very high. At late times, the topic space is crowded and the chances to find something fundamentally new are very slim. Therefore, there is a higher incentive for scientists to take higher risks at early times. This can be modeled by $p_f$ increasing with time, converging to a value close to $1$ as time grows to infinity. In turn, this means that $\\Lambda$ is a decreasing function of time, having a large value at the beginning of network evolution, and decreasing to small values afterwards.\n\nUnfortunately, measuring the temporal evolution of $\\Lambda$ in a real network is not yet possible because there currently exists no parametric theory describing such evolution that could be used for statistical inference of $\\Lambda$. However, it is fairly easy to find an approximate value of $\\Lambda$ as a function of time as follows. If timestamps of a real complex network are available, we can pretend that $\\Lambda$ is constant, and infer its value using the MLE techniques described in the Methods for subgraphs made of nodes that were born before a given time $t$, $\\widehat{\\Lambda}_{\\mathrm{MLE}}(t)$. This value can be thought of as a (possibly weighted) average of $\\Lambda(t)$ in time window $(0,t)$. By increasing the value of $t$, we can detect whether $\\Lambda$ is constant (if $\\widehat{\\Lambda}_{\\mathrm{MLE}}(t)$ does not change with time, beyond statistical fluctuations), or a decreasing function of time. Figure~5(e) shows $\\widehat{\\Lambda}_{\\mathrm{MLE}}(t)$ for the AS Internet where the strong temporal dependence of $\\Lambda$ is evident.\n\n\n\\section{Discussion}\nIn summary, hyperbolic network geometry, combining popularity and similarity forces driving network evolution, and coupled with preferential attachment of nodes to this geometry (GPA), naturally yields scale-free, strongly clustered growing networks with emergent soft community structure. The GPA model has three parameters that can be readily inferred from network data. Using the AS Internet topology as example, we have seen that the GPA mechanism generates heterogeneous networks that are similar to real networks with respect to key properties, including key aspects of the community size distribution and separation. The mean community separation, a new metric that quantifies the narrowness of communities in a network, is controlled in GPA by initial attractiveness $\\Lambda$, which controls the power-law exponent in standard PA.\n\nIn the context of the asymptotic equivalence between de Sitter causal sets and popularity$\\times$similarity (PS) hyperbolic networks established in~\\cite{NetCosm}, we note that $\\Lambda$ is conceptually similar to the cosmological constant $\\Lambda$ in Einstein's equations in general relativity (GR), where it is also an additive term in the proportionality between the energy-momentum tensor and spacetime curvature. Causal sets~\\cite{Bombelli,Dowker} are random graphs obtained by Poisson sprinkling a collection of nodes onto (a patch) of a Lorentzian manifold; edges in these graphs connect all timelike-separated pairs of nodes. If there is no matter (empty spacetime) but there is only dark energy (positive $\\Lambda$), then the solution of Einstein's equations is the de Sitter spacetime, and the main theorem in~\\cite{NetCosm} states that the ensemble of PS graphs is asymptotically ($n\\to\\infty$) identical to an ensemble of causal sets sprinkled onto de Sitter spacetime, which is one of the three maximally symmetric, homogeneous and isotropic Lorentzian manifolds (the other two are Minkowski and anti-de Sitter spacetimes). In this context, the GPA model considered here is a model with cosmological constant $\\Lambda$ and matter. Modeled by high node density, this matter, as in GR, ``attracts more matter,'' thus increasing the spacetime curvature of which the node density is a proxy. Indeed the main feature of the model is that the higher the node density in a particular region of space, the more nodes will appear in this region later. The main difference with GR is that here we essentially have an analogy with only the $00$-component of Einstein's equations. One can envision that other components should describe the coupled dynamics of the similarity space and nodes in it. In case of scientific collaboration network, for example, that would be the co-evolution of science (space) itself, and interests of scientists (node dynamics in this space). In the model considered here nodes do not move. Finding the laws of their spatial dynamics that may further strengthen the analogy with general relativity is a promising but challenging research direction.\n\n\nIn that context, the decay of initial attractiveness $\\Lambda$ that we found in the Internet must be analogous to the decay of cosmological constant $\\Lambda$ in modern cosmological theories. Cosmic inflation~\\cite{Lyth,Mukhanov} is widely accepted as the most plausible resolution of many problems with the classical big bang theory, including the flatness problem, the horizon problem, and the magnetic-monopole problem. Inflation is an initial period of accelerated expansion of the universe during which gravity was repulsive. Inflation does not last long, and can be modeled as a time dependent cosmological ``constant'' $\\Lambda$ that initially has a high value and then decays to zero. The analogies between GPA with decaying $\\Lambda$ and inflation go even further, producing similar outcomes as far as the spatial distribution of events is concerned. Indeed, cosmic inflation has the effect of smoothing out inhomogeneities so that once inflation is over, the universe is nearly flat, isotropic, and homogeneous, except for quantum fluctuations of the inflaton field. These fluctuations are the seeds of future inhomogeneities that we observe in the universe at scales smaller than 100 Mpc. In the GPA context, a high value of $\\Lambda$ has also a homogenizing effect. Indeed, if $\\Lambda$ is large, then $p_f$ is small, and new nodes chose their angular positions at random, producing a Poisson point process on the circle. Once $\\Lambda$ is small enough, we are left with a random distribution of points with Poisson fluctuations that, as in the universe, are the seeds of future communities in the network (galaxies in the universe), because once $\\Lambda$ is nearly zero, these initial fluctuations are reinforced by pure preferential attachment.\n\n\n\\section{Methods}\n\n\\subsection{Invariance of the degree distribution and clustering}\n\nHere we prove that the degree distribution and clustering coefficient in the networks generated by the GPA model do not depend of the initial attractiveness $\\Lambda$. Moreover, the degree distribution is power-law with exponent $\\gamma=1+1\/\\beta$. The proof can be reduced to the proof for the homogeneous PS model~\\cite{PSO} (Supplementary Information, Section IV). Consider a new node $t$, and let\n$R_t$ be the radius of a hyperbolic disk centered at this node such\nthat $t$ is connected to all nodes $s<t$ that lie in this disc. Then the probability $\\mathbb{P}_{\\rm{GPA}}(s,t)$ that nodes $t$ and $s<t$ in the GPA model are connected can be computed as follows:\n\\begin{equation}\\label{link prob1}\n\\begin{split}\n\\mathbb{P}_{\\rm{GPA}}(s,t)&=\\mathbb{P}(x_{st}\\leq R_t)\\\\\n&= \\mathbb{P}\\left(\\theta_{st}\\leq 2e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\right),\n\\end{split}\n\\end{equation}\nwhere $x_{st}=r_s(t)+r_t+2\\ln\\frac{\\theta_{st}}{2}$ is the hyperbolic distance between nodes $s=(r_s(t),\\theta_s)$ and $t=(r_t,\\theta_t)$ at time $t$. Using the total probability theorem,\n\\begin{equation}\\label{link prob2}\n\\begin{split}\n\\mathbb{P}_{\\rm{GPA}}(s,t)=&\n\\sum_{i=1}^t\\mathbb{P}\\left(\\left.\\theta_{st}\\leq\n2e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\right| t=\\hat{t}_i\\right)\\mathbb{P}(t=\\hat{t}_i) \\\\\n=&\\sum_{i=1}^t\\mathbb{P}\\left(\\theta_{s\\hat{t}_i}\\leq\n2e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\right)\\Pi_t(i),\n\\end{split}\n\\end{equation}\nwhere $\\hat{t}_i$ are the candidate positions generated at Step $1(a)$, and $\\Pi_t(i)$ are the corresponding acceptance probabilities (\\ref{probability}). Applying the total probability theorem with respect to node $s$, we have:\n\\begin{equation}\\label{link prob3}\n\\begin{split}\n&\\mathbb{P}_{\\rm{GPA}}(s,t)=\\\\\n&=\\sum_{i=1}^t\\sum_{j=1}^s\\mathbb{P}\\left(\\left.\\theta_{s\\hat{t}_i}\\leq\n2e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\right|s=\\hat{s}_j\\right)\\mathbb{P}(s=\\hat{s}_j)\\Pi_t(i)\\\\\n&=\\sum_{i=1}^t\\sum_{j=1}^s\\mathbb{P}\\left(\\theta_{\\hat{s}_j\\hat{t}_i}\\leq\n2e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\right)\\Pi_s(j)\\Pi_t(i)\n\\end{split}\n\\end{equation}\nSince the angular coordinates of the candidate positions $\\hat{s}_j$ and $\\hat{t}_i$ are uniformly distributed on $[0,2\\pi]$, the probability $\\mathbb{P}(\\theta_{\\hat{s}_j\\hat{t}_i}\\leq\\alpha)$ is simply $\\alpha\/\\pi$. Therefore,\n\\begin{equation}\\label{link prob4}\n\\begin{split}\n\\mathbb{P}_{\\rm{GPA}}(s,t)=&\\frac{2}{\\pi}e^{-\\frac{r_s(t)+r_t-R_t}{2}}\\sum_{i=1}^t\\Pi_t(i)\\sum_{j=1}^s\\Pi_s(j)\\\\\n=&\\frac{2}{\\pi}e^{-\\frac{r_s(t)+r_t-R_t}{2}},\n\\end{split}\n\\end{equation}\nwhere the last equality holds because $\\sum_{i=1}^t\\Pi_t(i)=\\sum_{j=1}^s\\Pi_s(j)=1$. We note that $\\mathbb{P}_{\\rm{GPA}}(s,t)$ does not depend on $\\Lambda$, and that it is exactly the same as the probability $\\mathbb{P}_{\\rm{PS}}(s,t)$ of having a link between nodes $t$ and $s<t$ in the homogeneous PS model. The rest of the proof repeats\nthe proof in~\\cite{PSO} without a change. This  leads to\n\\begin{equation}\\label{PSO_L=PSO=PA}\n\\mathbb{P}_{\\rm{GPA}}(s,t)=\\mathbb{P}_{\\rm{PS}}(s,t)=\\mathbb{P}_{\\rm{PA}}(s,t)=m\\frac{\\left(\\frac{s}{t}\\right)^{-\\beta}}{\\int_1^t\\left(\\frac{s}{t}\\right)^{-\\beta}ds},\n\\end{equation}\nwhich means that the resulting degree distribution in\nGPA is identical to PA: it is the power-law with exponent $\\gamma=1+1\/\\beta$. Since the connection probability  $\\mathbb{P}_{\\rm{GPA}}(s,t)$  does not depend on $\\Lambda$, neither does clustering.\n\n\\subsection{Critical gap}\n\nTo obtain a closed-form expression for the critical gap, we note that for large $n$, the sequence $\\theta_1,\\ldots,\\theta_n\\sim U[0,2\\pi]$ can be  approximately viewed as a realization of the Poisson point process on the circle of unit radius with density $\\lambda=n\/2\\pi$. In this case, the distribution of the angular gaps is approximately exponential with rate $\\lambda$. The maximum gap $\\Delta\\theta_{(n)}$ has then the following PDF\n$f_{\\Delta\\theta_{(n)}}(x)=\n\\frac{n^2}{2\\pi}e^{-\\frac{n}{2\\pi}x}\\left(1-e^{-\\frac{n}{2\\pi}x}\\right)^{n-1}$,\nand its expected value can be calculated as follows:\n\\begin{equation}\n\\begin{split}\n\\Delta\\theta_c=&~\\frac{n^2}{2\\pi}\\int_0^{\\infty}xe^{-\\frac{n}{2\\pi}x}\\left(1-e^{-\\frac{n}{2\\pi}x}\\right)^{n-1}dx\\\\\n=&-2\\pi\\int_0^1y^{n-1}\\ln(1-y)dy\\\\\n=&2\\pi\\int_0^1y^{n-1}\\sum_{k=1}^{\\infty}\\frac{y^k}{k}dy=2\\pi\\sum_{k=1}^{\\infty}\\frac{1}{k(n+k)}\\\\\n=&\\frac{2\\pi H_n}{n}\\approx\\frac{2\\pi(\\ln n +\\gamma)}{n}\\approx \\frac{2\\pi\\ln n}{n},\n\\end{split}\n\\end{equation}\nwhere $H_n$ is the $n^{\\mathrm{th}}$ harmonic number, and $\\gamma$ is Euler's constant.\n\n\\subsection{Inference of $\\Lambda$}\n\nThe initial attractiveness $\\Lambda$ controls the distribution of angular coordinates $\\theta_1,\\ldots,\\theta_n$ of the nodes. We therefore first infer $\\theta_i$ using the HyperMap method~\\cite{HyperMap}.  Given the network embedding $\\{(r_i,\\theta_i)\\}_{i=1}^n$ into its hyperbolic space, the likelihood function $\\mathcal{L}(\\Lambda|\\theta_1,\\ldots,\\theta_n)$ can be written as follows:\n\\begin{equation}\\label{Liklihood}\n\\begin{split}\n&\\mathcal{L}(\\Lambda|\\theta_1,\\ldots,\\theta_n)=\\mathbb{P}(\\theta_1,\\ldots,\\theta_n|\\Lambda)\\\\\n&=\\mathbb{P}(\\theta_1|\\Lambda)\\mathbb{P}(\\theta_2|\\Lambda,\\theta_1)\\ldots\\mathbb{P}(\\theta_n|\\Lambda,\\theta_1,\\ldots,\\theta_{n-1})\\\\\n&\\propto\\int\\limits_0^{2\\pi}\\frac{(A_2(\\theta_2)+\\Lambda)d\\varphi_1}{A_2(\\theta_2)+A_2(\\varphi_1)+2\\Lambda}\\times\\ldots \\\\\n&\\times\\int\\limits_0^{2\\pi}\\ldots\\int\\limits_0^{2\\pi}\\frac{(A_n(\\theta_n)+\\Lambda)d\\varphi_1\\ldots d\\varphi_{n-1}}{A_n(\\theta_n)+\\sum_{i=1}^{n-1}A_n(\\varphi_i)+n\\Lambda},\\\\\n\\end{split}\n\\end{equation}\nwhere $A_t(\\varphi)$ is the attractiveness of location $\\varphi\\in\\mathbb{S}^1$, \nthat is the number of existing nodes at time $(t-1)$ that lie within distance $r_t$ from $(r_t,\\varphi)$. The log-likelihood is then (up to an additive constant):\n\\begin{equation}\\label{logLiklihood}\n\\begin{split}\n&l(\\Lambda|\\theta_1,\\ldots,\\theta_n)=\\\\\n&=\\sum\\limits_{t=2}^n\\log\\int\\limits_0^{2 \\pi}\\ldots\\int\\limits_0^{2\\pi}\\frac{(A_t(\\theta_t)+\\Lambda)d\\varphi_1\\ldots d\\varphi_{t-1}}{A_t(\\theta_t)+\\sum_{i=1}^{t-1}A_t(\\varphi_i)+t\\Lambda}\n\\end{split}\n\\end{equation}\nThe multiple integrals in (\\ref{logLiklihood}) cannot be calculated analytically, since the attractiveness function cannot be written in closed-form. Nevertheless, the log-likelihood can be efficiently estimated be the Monte Carlo method. First, generate $N$ Monte Carlo samples, $\\varphi_1^{(j)},\\ldots,\\varphi_{n-1}^{(j)}\\sim U[0,2\\pi]$, $j=1,\\ldots,N$. The ``truncated'' samples  $\\varphi_1^{(j)},\\ldots,\\varphi_{t-1}^{(j)}$  will be used for estimating the $(t-1)$-dimensional integral in (\\ref{logLiklihood}). Next, precompute all needed attractivenesses, $A_t(\\varphi_i^{(j)})$, where $t=2,\\ldots,n$ and $i=1,\\ldots,t-1$.  Then for each value of $\\Lambda$, the log-likelihood can be\nestimated as follows (up to a constant):\n\\begin{equation}\\label{LogLikMC}\n\\begin{split}\n&l(\\Lambda|\\theta_1,\\ldots,\\theta_n)\\approx\\\\\n&\\approx\\sum\\limits_{t=2}^n\\log\\left(\\frac{1}{N}\\sum\\limits_{j=1}^N\\frac{A_t(\\theta_t)+\\Lambda}{A_t(\\theta_t)+\\sum_{i=1}^{t-1}A_t(\\varphi_i^{(j)})+t\\Lambda}\\right)\n\\end{split}\n\\end{equation}\n\n\\begin{table}\n\t\\centerline{\n\t\\begin{tabular}{l c c c c c c}\n\t\t\\hline\\hline\n\t\tTrue $\\Lambda$\t& $0$ & $0.2$ & $0.5$ & $0.7$ & $1$ & $2$ \\\\ [0.5ex]\n\t\n\t\t\\hline\n\t\t$\\widehat{\\Lambda}_{\\mathrm{MLE}}(100)$& 0& 0.3 & 0.4 & 0.7 & 0.7 & 1.4  \\\\\n\t\t$\\widehat{\\Lambda}_{\\mathrm{MLE}}(200)$& 0& 0.2 & 0.5 & 0.8 & 1.3 & 1.8  \\\\\n\t\t$\\widehat{\\Lambda}_{\\mathrm{MLE}}(500)$& 0& 0.2 & 0.6 & 0.7 & 1.1 & 1.9 \\\\\n\t\t$\\widehat{\\Lambda}_{\\mathrm{MLE}}(1000)$& 0& 0.2 & 0.5 & 0.7 & 1.1 & 1.8  \\\\\n\t\t\\hline\n\t\\end{tabular}}\n\t\t\\caption{\\textbf{Maximum likelihood estimates.} True values of the initial attractiveness parameter $\\Lambda$ and its MLEs $\\widehat{\\Lambda}_{\\mathrm{MLE}}(n_0)$ based on the first $n_0=100, 200, 500,$ and $1000$ nodes. In all simulations, $N=100$ Monte Carlo samples were used in (\\ref{LogLikMC}).}\n\\end{table}\n\\begin{figure\n\t\\centerline{\\includegraphics[width=80mm]{Fig06.pdf}}\n\t\\caption{\\textbf{Log-likelihood functions.} The estimated log-likelihood functions $l(\\Lambda|\\theta_1,\\ldots,\\theta_n)$ for synthetic networks of size $n=10^3$ generated by the GPA model with $\\Lambda=0, 0.2, 0.5, 0.7, 1,$ and $2$. Each log-likelihood is estimated by (\\ref{LogLikMC}) using $N=100$ Monte Carlo samples and $n_0=500$ first nodes.}\n\\end{figure}\n\n\n\nComputing attractivenesses of the Monte Carlo samples $A_t(\\varphi_i^{(j)})$ involves computing $O(n^3N)$ hyperbolic distances, which is the most computationally intensive part of the algorithm. Having all attractivenesses computed, we can then estimate $l(\\Lambda)$ for any $\\Lambda\\in[0:\\Delta\\Lambda:\\Lambda_{\\mathrm{max}}]$, and find the maximum likelihood estimate (MLE) $\\widehat{\\Lambda}_{\\mathrm{MLE}}$. An important observation that drastically improves the efficiency of the algorithm is that we do not have to use the entire network to accurately estimate  $\\widehat{\\Lambda}_{\\mathrm{MLE}}$, the first $n_0\\ll n$ nodes are often enough. Table~1 shows the MLEs $\\widehat{\\Lambda}_{\\mathrm{MLE}}(n_0)$ obtained from the first $n_0=100, 200, 500$, and $1000$ nodes of the networks generated by the GPA model with  $\\Lambda=0, 0.2, 0.5, 0.7, 1,$ and $2$. The corresponding log-likelihood functions are shown in Fig.~6. These simulation results show that  the smaller the true value of $\\Lambda$ ---  and we expect it to be small in real networks since most of them have community structure --- the less network data we need to pin $\\widehat{\\Lambda}_{\\mathrm{MLE}}$ down. If, for example, $\\Lambda=0$, then the MLE of $\\Lambda$  based on the first $n_0=100$ nodes is already zero. The larger the true value of $\\Lambda$, however, the flatter the log-likelihood is around its maximum, which makes inference more challenging.\n\n\\begin{acknowledgments}\nThis work was supported by DARPA grant No.\\ HR0011-12-1-0012; NSF grants No.\\ CNS-1344289,\nCNS-1442999, CNS-0964236, CNS-1441828, CNS-1039646, and CNS-1345286; by Cisco Systems; by the James S. McDonnell Foundation Scholar Award in Complex Systems; by the Icrea Foundation, funded by the {\\it Generalitat de Catalunya}; by the MINECO project No.\\ FIS2013-47282-C2-1-P; and by the {\\it Generalitat de Catalunya} grant No.\\ 2014SGR608.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCataclysmic variables (CVs) are interacting binary stars \ncontaining a white dwarf (WD) primary and a low mass \nsecondary \\citep{warner95}.  The evolution of these \nbinaries is believed to proceed as follows (see \n\\citealt{howell01}):\\ after a common envelope phase \nfollowing the post-main sequence evolution of the WD \nprogenitor, the low mass secondary star eventually \n(over)fills its Roche lobe and the binary commences \nmass transfer.  Due to angular momentum losses, primarily \nvia magnetic braking and gravitational radiation, the \ntwo component stars move closer together over time; that \nis, their orbital period decreases.  For the oldest CVs, \nthe orbital periods are very short (near 80 minutes) and \nthe secondary stars are very low mass, being \n$\\sim$0.06 M$_{\\odot}$ stars or lower mass degenerate \nbrown dwarf-like objects.\n\nEF Eridani contains a strongly magnetic WD ($B\\approx13$--14 MG; \n\\citealt{WR98,howell06b,beuermann07}), making it a member of the \npolar class of CV (named for the highly polarized nature of \ntheir emitted light).  Unlike CVs containing non-magnetic WDs, \npolars have no accretion disks (instead, accretion proceeds \ndirectly from the inner Lagrangian point onto the magnetic \nfield lines of the WD) and generally do not undergo \ndwarf-nova-type (i.e., disk instability) outbursts. They do, \nhowever, experience periods of normal mass transfer from the \nsecondary star (high states) interspersed with times when \nthis accretion flow stops or is significantly decreased \n(low states), possibly related to stellar activity on the \nsecondary star.  EF Eri has been in a low state for the past \n10 years \\citep{howell06b}.  Interestingly, recent high energy \nobservations in the UV \\citep{szkody06} and X-rays \n\\citep{schwope07} have shown that EF Eri still has a \n$\\sim20,000$~K hot spot remaining on the surface of its \n$\\sim10,000$~K WD even after a decade of very low mass \naccretion ($\\dot{M}\\sim10^{-14} M_{\\odot}$ yr$^{-1}$). \nThis hot region is presumably at or near the active accretion \npole, which is best modelled as a non-uniform spot covering \n10--20\\% of one side of the WD \\citep{beuermann07}.\n\nOur initial {\\em Spitzer Space Telescope} \\citep{werner04} \nphotometric observations of a small sample of polars using \nthe Infrared Array Camera (IRAC; \\citealt{fazio04}) included \nEF Eri as the brightest sample member, and revealed a nearly \nflat mid-IR (3.6--8 $\\mu$m) flux density level near 700 $\\mu$Jy \n\\citep{howell06a,brinkworth07}.  In \\citet{howell06a}, we showed that \nthe IRAC spectral energy distributions (SEDs) of EF Eri and \nthree other polars with similar orbital periods \nhad flux density in excess of that produced by the two \ncomponent stars alone.  It was concluded that the best \ncandidate for the excess emission was a circumbinary dust disk \nwith an inner edge temperature ($T_{\\rm in}$) near 800~K.  \nIn \\citet{brinkworth07} (hereafter, B07), we \nused these same data, as well as IRAC observations of two \nadditional polars (making a total of seven systems), \nto examine a larger suite of more \nsophisticated SED models.  For EF Eri, we \nfound that the most plausible way to produce the observed \nbright 8-$\\mu$m flux density, without exceeding the observed \nflux densities at shorter wavelengths, was via a geometrically \nthin, optically thick circumbinary disk (CBD) with \nT$_{in}\\approx650$~K; however, we could not completely rule \nout the possibility that at least some of the long wavelength \nflux density in the SED is due to cyclotron emission.\nThe B07 model results were generally consistent with the need\nfor a prominent cyclotron emission component to explain the \nnear-IR (e.g., 2MASS) portion of the EF Eri SED.\n\nThe {\\em Spitzer} photometric observations alone could not \nprovide any further clarification as to the exact origin of \nthe 3.6--8-$\\mu$m SED, as they had limited wavelength \nresolution and coverage.  Consequently, those data were not \nable to strongly constrain models for the origin of the \nobserved mid-IR flux density of EF Eri (as described in more \ndetail in B07).  Therefore, in an effort \nto better understand the true nature and extent of the mid-IR \nemitting source(s) in EF Eri, we obtained a spectrum spanning \n5.5--14.5 $\\mu$m using the Infrared Spectrograph \n(IRS; \\citealt{houck04}) on {\\em Spitzer}.\n\n\n\\section{Observations and Data Processing}\n\\subsection{Mid-IR Spectrum}\n\nOur spectroscopic observations of EF Eri were obtained using \nthe ``Short Low'' module of the IRS, which covers 5.2--8.7 $\\mu$m \nin second order (SL2) and 7.4--14.5 $\\mu$m in first order (SL1) \nat a resolution of $R\\sim60$--120.  \nA third order (the SL3 ``bonus'' spectrum) is obtained with each \nSL2 observation; it spans 7.4--8.6 $\\mu$m and is primarily used \nto ensure that the flux calibration from SL2 to SL1 is consistent \n(in our case, all three orders were in excellent agreement in the \noverlapping wavelength region, so we did not apply any offsets \nbetween orders, and used the data from all three orders in \nconstructing a final spectrum -- see below).\nWe obtained four cycles of 240~s \neach for both SL1 and SL2 (i.e., a total of sixteen individual \nexposures after counting the two nod positions obtained in each cycle).  \nThe corresponding AOR reqkey number is 17052928.\n\nWe used the Spitzer Science Center post-BCD software SPICE \n(v1.4.1)\\footnote{See \n\\url{http:\/\/ssc.spitzer.caltech.edu\/postbcd\/spice.html}.} to extract \nthe IRS spectra from the two-dimensonal images.  \nThe input images consisted of the four S15.3.0 pipeline-combined \npost-BCD images (i.e., the $*$bksub.fits files), each of which is \nconstructed from four appropriately background-subtracted, masked, \nand co-added sub-exposures to give two images each for the SL1 and \nSL2+SL3 orders (i.e., one image per order at each nod position).  \nWe extracted the spectra from these two-dimensional images using \nthe optimal extraction algorithm with the standard aperture width. \nThis results in two extracted spectra (one for each nod position) \nspanning the three SL orders.\n\nNext, we performed a weighted average of the two (or three) points \nat each wavelength from the two nod spectra to obtain a preliminary \naverage spectrum.  We calculated the weighted mean, \n$\\langle f_{\\rm w} \\rangle$, and standard \ndeviation of the weighted mean, $\\sigma_{\\rm wavg}$, \nof the flux density over the full \nwavelength range for the preliminary average spectrum.  Then, we \nrejected any point in an individual nod spectrum that was more \nthan $3\\sigma_{\\rm wavg}$ away from $\\langle f_{\\rm w} \\rangle$ \nwhen the corresponding point(s) at the same wavelength in the\nother nod was not (i.e., an outlier rejection).  \nFinally, we recalculated the average spectrum using the \noutlier-rejected nod spectra, via a weighted average when two or \nmore data points were available for a given wavelength \nor using the remaining data point when \nonly one survived rejection.\n\nThe IRS spectrum of EF Eri is shown in Figure \\ref{f:spectrum}.  \nIt is characterized by a generally flat continuum with a slight \ndip at the short wavelength end.  There are no obvious emission \nfeatures, and only a few potential absorption features.  However, \nwe note that the IRS was designed to achieve high sensitivity \nat the cost of reduced dynamic range; hence, great care must be \nused in the interpretation of weak spectral features\\footnote{See \nthe IRS chapter of the Spitzer Observer's Manual, available at \n\\url{http:\/\/ssc.spitzer.caltech.edu\/documents\/som\/}.}, especially \nin spectra of faint targets like EF Eri. \n The spectrum of EF Eri used in this work has not been scaled \nfrom its original flux calibration -- the excellent agreement \nin flux density between the IRAC photometric points and the IRS \nspectrum (see \\S\\ref{s:otherdata} and Figure \\ref{f:bigspec}), \nas well as the smooth transition from the SL2 to SL1 data sections \n(at $\\lambda\\approx7.5$ $\\mu$m), attests that the overall continuum \nshape of the spectrum is reliable.  Consequently, our analysis \npresented here will rely on looking at the gross properties of the \nspectrum as a whole (i.e., the continuum shape and flux density \nlevel), rather than focusing on specific features with low \nstatistical probability of being real.\n\n\n\\subsection{Other Data}\n\\label{s:otherdata}\n\nWe also utilized the 2MASS and IRAC photometry of EF Eri reported \nin B07, and the near-IR spectrum shown in Figure 1 of \\citet{harrison07}.  \nFigure \\ref{f:bigspec} shows all of the spectroscopic and photometric \ndata plotted together.  Other than the points from IRAC channels 1 \nand 2 (3.6 and 4.5 $\\mu$m, for which there are no overlapping \nspectroscopic data), all of the photometric data are in \nexcellent agreement with the spectroscopic data.  A slight exception \nto this is the 2MASS $K_{\\rm s}$-band point, which is significantly \nlower in flux density (by more than $1\\sigma$) compared to the \nnear-IR spectrum.  This is likely due to the $K$-band variability \nnoted by \\citet{harrison04}, which has been linked to cyclotron \nemission that varies with orbital phase.\n\n\n\\section{Spectral Energy Distribution Models}\n\\subsection{The Code}\n\\label{s:code}\n\nIn B07, we introduced our IR SED modeling code for CVs, which we \napplied to photometric data for several polars, including EF Eri.  \nWe have now made a number of improvements to the modeling code \nfor use here.  First, we have modified the code to allow SEDs to \nbe calculated at higher wavelength resolution, which is more \nappropriate for comparing to spectral data.  This modification \nhas little effect on the WD and CBD components, but provides \nthe necessary resolution to resolve cyclotron humps in the \ncyclotron component.  For the models presented here, we have \nused a wavelength increment of 0.05 $\\mu$m in the range 1--14.5 $\\mu$m.\n\nSecond, we have modified the handling of the secondary star to \naccommodate data at longer wavelengths.  In B07, the secondary \nstar was represented by average 2MASS and IRAC photometry for \nspectal type templates from \\citet{patten06}.  We have now added \nthe capability to combine the 2MASS and IRAC photometry of a single, \nrepresentative spectral type star from \\citet{patten06} with \nthe IRS SL spectrum of the same star from \\citet{cushing06}.  \nFor the EF Eri models presented here, \nthe secondary star model component (see Table \\ref{t:shared-params}) \nis represented by \nthe L5 star 2MASS~J15074769$-$1627386 \\citep{reid00} scaled to \na distance of $d=132$ pc (see \\S\\ref{s:models}).  \n\nThird, we have made a number of minor improvements to the \ncalculation of the optically thin CBD component SED, which is \nprimarily used in this work instead of the optically thick CBD used in B07.  \nIn general, the procedure remains as described in B07.  \nThe SED of the CBD is obtained by summing the contribution of \n1000 annular rings, each of which has the same width and the \ncorrect temperature for its average radial distance from the \ncenter-of-mass of the CV  \n(based on the $T \\propto [1\/r]^{3\/4}$ profile explained in \nmore detail in B07).\nThe volume of a ring increases as its radial \ndistance increases, and we assume equal mass in each ring, which \nresults in rings that are successively less dense at larger \nradial distances.  \n\nWe have imposed an \nupper limit of 1000 K for the temperature of the inner edge of \nthe CBD.  Not only is it difficult to devise a mechanism that \nwould heat the inner edge of the CBD in EF Eri above 1000 K, but \ndust sublimation (hence, destruction of the CBD) likely occurs \nat temperatures above 1000--2000 K.\nBased purely on the WD effective temperature, we would expect the \ntemperature at the radius of the inner edge of the CBD to be \n$T_{\\rm in}\\approx250$ K; including the contribution from the \nsecondary star, this temperature increases to \n$T_{\\rm in}\\approx400$--500 K.\nHowever, as noted in B07, the upper limit \nto this temperature is difficult to assess because of the \nuncertainties in how to properly account for the \ncontributions from the secondary star and accretion luminosity; \nin addition, the conversion from the ambient temperature of \nirradiation at a particular radius to temperature of the CBD \nmaterial at that radius is uncertain and highly dependent on the \nphysical parameters of the CBD and the WD accretion spot(s).\nEspecially considering the evidence from recent UV and X-ray \nobservations \\citep{szkody06,schwope07} that there is a \nsignificant hot spot on the WD in EF Eri, the temperature at \nthe inner edge of the CBD could be up to \nseveral hundreds of K higher.  Effectively, then, we define an \n``allowed'' range of $T_{\\rm in}=400$--1000 K. \n\nFinally, and most significantly, we have replaced the purely \nmorphological, template-based cyclotron component from B07 with \na physical model based on the formulation discussed and used in, \nfor example, \\citet{chanmugam80,TC87,schwope90} and references therein.\nThe calculation of the cyclotron SED is influenced by several \nparameters:\\ the WD magnetic field strength ($B$), the viewing \nangle ($\\theta$), the electron temperature ($kT$), a dimensionless \n``size'' parameter ($\\Lambda$), and a normalization factor ($A$).  \nNone of these parameters is strictly independent from the \nothers -- the change in the cyclotron SED produced by adjusting \none parameter can usually be offset by adjusting one or more of the \nother parameters.  This makes it difficult to produce a unique \nsolution without having reliable, very narrow constraints for as \nmany of the parameters as possible.  We will describe the constraints \nused to narrow down the possible cyclotron SEDs for EF Eri \nin \\S\\ref{s:models}.  In the remainder of this section, we will \ndiscuss, in general terms, the manner in which each parameter \ninfluences the cyclotron SED.\n\nTo first order, the WD magnetic field strength ($B$) determines \nthe wavelengths at which cyclotron humps appear in the SED for \nsuccessively higher harmonic numbers.  Smaller values of $B$ \nproduce more redshifted humps.  However, the wavelengths of the \nharmonics are also influenced by the electron temperature ($kT$) \nand the viewing angle ($\\theta$).  Increasing $kT$ or decreasing \n$\\theta$ redshifts the humps.\nThe humps become broader with increasing $kT$, increasing size \nparameter $\\Lambda$ (however, see below for additional effects \nof changing $\\Lambda$), or decreasing $\\theta$.  At very high \nvalues of these parameters (especially $kT$), the humps are so \nbroad as to be completely overlapping and blended together, \neffectively forming a humpless pseudo-continuum.  At very low \nvalues, the cyclotron humps are very narrow, resulting in the \ncyclotron spectrum consisting of multiple discrete hump profiles \nwith regions of zero intensity between them.\nThe normalization factor ($A$) is a simple scaling factor given \nby the ratio of the effective emitting area of the cyclotron \ncomponent to the square of the distance to the CV.\n\nThe size parameter $\\Lambda$ deserves additional explanation.  \nAlthough its value is set in the cyclotron SED calculation as \na single number, this parameter actually incorporates several \nother parameters.  It is defined (see \\citealt{schwope90}) as\n\\begin{equation}\n\\Lambda = 2.01\\times10^{5} \\left(\\frac{s}{10^5\\,{\\rm [cm]}}\\right) \\left(\\frac{N_{\\rm e}}{10^{16}\\,{\\rm [cm^{-3}]}}\\right) \\left(\\frac{3\\times10^7\\,{\\rm [G]}}{B}\\right),\n\\end{equation}\n\\noindent where $s$ is the geometric path length through the \ncyclotron emitting region and $N_{\\rm e}$ is the electron number \ndensity.  For a fixed value of $B$, the value of $\\Lambda$ is \ndependent on only the product of $s$ and $N_{\\rm e}$.  \nFunctionally, larger values of $\\Lambda$ correspond to higher \noptical depths, which influences the harmonic number at which \nthe cyclotron humps transition from being optically thick \n(at lower harmonics) to optically thin (at higher harmonics).  \nOptically thick cyclotron humps have a truncated, flat-topped \nappearance compared to the rounded optically thin humps.  \nThe value of $\\Lambda$ also has a large effect on the relative \namplitudes of the cyclotron humps.  The lower harmonic, optically \nthick humps at longer wavelengths rapidly decline in amplitude as \nharmonic number decreases compared to the optically thin humps at \nhigher harmonics (shorter wavelengths).\n\n\n\\subsection{The Models}\n\\label{s:models}\n\n\\subsubsection{Distance to EF Eri}\n\nThe distance used to scale the model components in this work is \nthe $1\\sigma$ upper limit of a trigonometric parallax-derived \ndistance from \\citet{thor03}, and is approximately halfway between \nthe two possible nominal parallax distance values of 113 and 162 pc \nreported in that work.  We increased the distance from the 105 pc \nused in B07 (which was based on non-parallax estimates) because \nit is now clear that cyclotron emission makes a non-negligible \ncontribution in the $J$ band (see discussion of models below), \nwhereas at the smaller distance, the observed $J$-band flux \ndensity was accounted for completely by the WD and secondary star \ncomponents.  The value of 132 pc used here is the minimum distance \nfor which the summed WD, secondary star, and cyclotron emission \ndo not exceed the observed photometric and spectroscopic data \nat $J$ band.  \n\n\n\\subsubsection{Cyclotron Component Constraints}\n\\label{s:cyccon}\n\nAs noted in \\S\\ref{s:code}, our revised cyclotron model component \nis calculated using a number of non-independent parameters, so it \nis helpful to constrain, as much as possible, the range of valid \nparameter space.  To that end, we describe in this section the \nvarious constraints that we used to limit the cyclotron model \nparameters. \n\n\\begin{description}\n\\item[$B$:] The WD magnetic field in EF Eri is frequently derived \nas $B=13$--$14$ MG (e.g., \\citealt{WR98,howell06b,beuermann07}).  \nA value of $B$ in this range is consistent with the location of \nthe cyclotron humps in the near-IR spectrum of EF Eri \n(\\citealt{harrison07}; also see Figure \\ref{f:bigspec}).  \nWe do not consider the much more complex scenario in which the \ncyclotron spectrum of EF Eri results from two or more magnetic \naccretion regions with significantly different field strengths.\n\n\\item[$kT$:] The near-IR spectrum of EF Eri also provides useful \nconstraints for this parameter.  Even after subtracting the WD \nand secondary star components, there is residual flux between the \ncyclotron humps.  This requires $kT\\gtrsim5$ keV (at lower values \nof $kT$ and for reasonable values of $\\Lambda$ -- see below -- the \ncyclotron spectrum is composed of discrete humps with zero flux \nlevel between them).  \nIn addition, for $kT\\lesssim5$ keV, and for values of $\\Lambda$ \nthat reproduce the observed optically thick to thin transition \nand relative hump peak amplitudes, the cyclotron humps are too \nnarrow compared to the observed near-IR spectrum.\nThe fact that discrete cyclotron humps are observed requires \n$kT\\lesssim20$ keV.  \n\n\\item[$\\theta$:] In the absence of consistent information \nconstraining this parameter, we chose to set $\\theta=75^{\\circ}$ \nfor all models.  In any case, $\\theta$ cannot be much lower or \nhigher than this, or the widths and relative amplitudes of the \noptically thin humps will not match the observed near-IR spectrum.\n\n\\item[$\\Lambda$:] For any given values of the other parameters, \n$\\Lambda$ is the most constrained parameter.  This is because \nonly a narrow range of its values will reproduce the observed \nnear-IR spectrum, which shows the $n=4$ harmonic as optically \nthick, and higher harmonics as optically thin.  The value of \n$\\Lambda$ is further fine-tuned by matching the relative peak \namplitudes of both the optically thick and thin cyclotron humps.\n\n\\item[$A$:] The main constraint on the scaling parameter is that \nfor a given distance, $A$ should correspond to an effective \nemitting area of the cyclotron radiation ($a_{\\rm cyc}$) that \nis (much) smaller than the projected surface area of the WD.\n\\end{description}\n\nMost of these constraints are derived from the near-IR spectrum \nof EF Eri, which shows the cyclotron hump at the $n=4$ harmonic \nto be flat-topped and much lower in peak amplitude than the humps \nat the $n\\geq5$ harmonics (see Figure \\ref{f:bigspec}).  \nThis indicates that the transition \nfrom optically thick to thin cyclotron emission occurs in the \nvicinity of the $n=4$ harmonic.  In particular, satisfying the \nobserved transition from optically thick to thin humps results \nin the mid-IR contribution of the cyclotron emission being \nincreasingly negligible at longer wavelengths.  That is, the \ncyclotron component is only important at relatively short IR \nwavelengths, and does not contribute significantly to the mid-IR \nregion spanned by the Spitzer data.\n\nAs expected from the description in \\S\\ref{s:code}, we found it \ndifficult to determine a unique, ``best'' solution for the \ncyclotron model component.  We determined the relative goodness \nof different model cyclotron components by calculating the \n$\\chi^2$ and standard deviation of the residuals ($\\sigma_{\\rm res}$) \nin the 1--2.5 $\\mu$m wavelength region for the summed cyclotron, \nWD, and secondary star model components compared to the observed \nnear-IR spectrum.  \nTable \\ref{t:cycmodels} lists several ``best'' cyclotron models \ndetermined in this way for a range of $kT$ values.  \nAs $kT$ decreases, we had to decrease $B$ and increase $\\Lambda$ \nin order for the cyclotron component to continue to match the \nwavelength spacing, widths, and relative peak amplitudes of the \nobserved cyclotron humps.\nThe goodness of the model also tends to decrease with decreasing \n$kT$, although none of the models has particularly poor agreement \nwith the observed SED.\nIn the end, for the purposes of this work, the specific cyclotron \ncomponent that we use is not particularly important since all of \nthem contribute negligibly at mid-IR wavelengths.  We will use \nthe $kT=10$ keV cyclotron model in the remainder of this work.\n\nIn B07, we utilized a ``sum-over-fields'' approach to calculating \nthe cyclotron component, which considered the summed contributions \nfrom cyclotron emission of electrons encountering a successively \nstronger magnetic field as they approached the WD.  In part, this \nwas an attempt to set a ``worst case'' limit for the contribution \nof cyclotron emission at long wavelengths (since the strength of \ncyclotron emission at long wavelengths is increased relative to \nshort wavelengths through this approach).  However, another effect \nof the sum-over-fields approach is to smear out the individual \ncyclotron humps, which is clearly inconsistent with the observed \nnear-IR spectrum of EF Eri.  Consequently, we have not used that \napproach here.  In any case, even if we considered a two-part \ncyclotron component consisting of the single-field {\\em and} \nsummed-fields cases, the contribution of the latter would have \nto be extremely small in order to not dilute the strong observed \nsingle-field spectrum (and, regardless of strength, would not \ncontribute significantly at $\\lambda\\gtrsim4$ $\\mu$m).\n\n\n\\subsubsection{Comparison with Brinkworth et al.\\ (2007)}\n\\label{s:comparison}\n\nFigure \\ref{f:models}a shows the observational data from \nFigure \\ref{f:bigspec} with a model containing an optically thick \nCBD similar to the best optically thick CBD component from B07 \n(see Model 1 in Table 4 of that work).  It has been adusted \nslightly to account for the larger distance used here by changing \nthe inner edge temperature from 655 K to 755 K.  \nThe criterion used in B07, of best reproducing the 8-$\\mu$m IRAC \npoint without exceeding any shorter wavelength point, has been \npreserved.  This model has the wrong spectral shape and \nsignificantly exceeds the {\\em longer} wavelength SED of EF Eri \nthat is revealed by our IRS spectrum.\n\n\n\\subsubsection{New Model Results}\n\\label{s:newmodels}\n\nFigure \\ref{f:models}b shows a new model that utilizes the \nadditional constraints on flux density at long wavelengths \nprovided by our IRS spectrum of EF Eri.  \nThe parameters common to this model and the one discussed \nbelow are listed in Table \\ref{t:shared-params}, while parameters \nspecific to this model are listed in Table \\ref{t:specific-params} \n(Model 1).  For this model, we have utilized an optically thin CBD \ncomposed of spherical dust grains that radiate as blackbodies \naccording to the radial temperature profile calculated as for \nthe optically thick CBD case (see \\S\\ref{s:code} and B07).  \n\nThis model shows significant improvement over that shown in \nFigure \\ref{f:models}a, especially at the short and long wavelength \nends.  However, the model flux density is too low at the middle \nwavelengths (i.e., IRAC channels 1 and 2 at 3.6 and 4.5 $\\mu$m).\nThe total mass of the CBD is $\\approx 10^{21}$ g, consistent with \nthe finding from B07 that the masses of CBDs in magnetic CVs are \nmany orders of magnitude smaller than predicted to be required to \ninfluence the angular momentum loss history of these systems \\citep{taam03}.  \nFor lower and higher inner edge CBD temperatures, the overall \nflux density level of the observed SED can be matched by increasing \nor decreasing, respectively, the total disk mass (i.e., the number \nof radiating dust grains).  \nIf the temperature of the inner edge of the CBD is decreased, then \nthe match between the model and observed SEDs becomes worse -- the \nCBD profile does not reach peak flux density until an even longer \nwavelength, which exacerbates the problem of missing flux density \nat the IRAC wavelengths.\nA higher temperature for the inner edge of the CBD produces a \nbetter match at the middle wavelengths.  However, the model is \nthen too faint at the long wavelength end, since the CBD SED peaks \nand begins to decline at a shorter wavelength.  \n(See \\S\\ref{s:cbdcon} for more discussion of the constraints on CBD \nmodel parameters.)\n\nWe have explored a possible means of reconciling \nthis CBD model with the 3.6 and 4.5 $\\mu$m data.\nThe model in Figure \\ref{f:models}c is similar to that shown in \nFigure \\ref{f:models}b, but uses a blackbody component to account \nfor the ``missing'' flux at 3.6 and 4.5 $\\mu$m.  The model \nparameters are listed in Tables \\ref{t:shared-params} and \n\\ref{t:specific-params} (Model 2).  This model has the advantage \nthat it requires a CBD with a low inner edge temperature that \ncould easily be produced by irradiation from the stellar \ncomponents in EF Eri.\nOn the other hand, it has the disadvantage that the physical \norigin of the additional component is unclear.  The required \nequivalent emitting area (corresponding to a radius of $46R_{\\rm wd}$) \nis too large to be contained in the stellar Roche lobes, which \npoints to the CBD.  The required temperature is 1000 K, which is \nsomewhat uncomfortably warm from considerations of both the \norigin of the heating and potential destruction of the dust grains.  \nHowever, it might suggest that a more complex radial temperature \nprofile in the CBD could produce an SED shape that is more \nconsistent with the observations.  We have not explored this \npossibility in detail because it would introduce yet more free \nparameters into our already barely constrained model.\n\n\n\\subsubsection{Uniqueness, Plausibility, and Constraints of the Circumbinary Disk Models}\n\\label{s:cbdcon}\n\nMuch like the cyclotron emission component (see \\S\\ref{s:code} \nand \\S\\ref{s:cyccon}), the model CBD SEDs shown in this work are not, \nstrictly speaking, unique solutions, in the sense that very similar \nresults can be achieved from somewhat different combinations of input \nparameters.  We have tried to minimize this as much as possible by \nconstraining plausible parameter ranges based on whatever other \ninformation, observational data, and reasonable assumptions are available.  \nIn this section, we describe in detail the justification for the \nconstraints that have been assumed in fixing the exponent in the \nradial temperature profile calculation (see \\S\\ref{s:code}) and \nthe inner radius, $R_{\\rm in}$, of the model CBD.  We also expand \nupon the discussion of the failure of the optically thick CBD \nmodel first mentioned in \\S\\ref{s:comparison}.\n\nIn general terms, the influence of the exponent in the radial \ntemperature profile can be described as follows:\\   \na larger (smaller) exponent in the CBD radial temperature profile \nleads to a steeper (shallower) temperature gradient near the \ninner edge of the disk and overall lower (higher) temperature \nthroughout the disk, which corresponds to large (small) dust grains \nthat do (do not) cool efficiently.  In practice, we have found that \nan exponent of 3\/4, as used here and in B07, produces the most \nviable results in comparison with our observations of CVs.  \nSmaller exponents produce CBD SEDs that rise too steeply and are, \noverall, too bright to reproduce the observed SEDs without \narbitrarily increasing the distances to the CVs to many \nhundreds or thousands of pc.\nLarger exponents produce CBD SEDs that rise too shallowly at longer \nwavelengths and are, overall, too faint to match the observed mid-IR \nflux densities without arbitrarily increasing the temperature of \nthe disk's inner edge to unrealistic levels.  \nFor example, we can obtain a model optically thin CBD SED that \nis essentially indistinguishable from that in Model 1 \n(Figure \\ref{f:models}b) by using a larger radial \ntemperature profile exponent (1 instead of 3\/4; see \\S\\ref{s:code}) \nand smaller total disk mass ($4.41\\times10^{20}$ g instead \nof $9.55\\times10^{20}$ g); however, one objection to this \napproach is that departure from an exponent of 3\/4 implies \ndust grains that are not blackbodies \\citep{FKR}, whereas this is \nan implicit assumption of the CBD model calculations.\n\nThe inner edge radius of the CBD models calculated in this work \n(for both optically thick and thin cases) is fixed \nat $R_{\\rm in}=73R_{\\rm wd}$, \nwhich is a lower limit set by the tidal truncation radius of EF Eri.  \nIncreasing the inner radius of the CBD from this \nvalue worsens the agreement between the CBD models and the \nobserved SED, because the irradiation-induced temperature of \nthe inner edge will then be lower.  This effectively removes flux \nfrom the short IR wavelengths of the model SED, whereas the main \nproblem we have in reproducing the observations is that the models \nalready have too little flux at short IR wavelengths.  (The outer \nradius of the CBD is calculated to correspond to a temperature of \n20 K according to the radial temperature profile in use, but at \nwavelengths shortward of 15 $\\mu$m the resultant profile is \ninsensitive to increasing the temperature of the outer radius \ncut-off by as much as an order of magnitude.)\nEven if we arbitrarily (and unphysically) decrease the inner \nedge radius of the CBD to $50R_{\\rm wd}$, which is barely larger \nthan the distance from the CV's center-of-mass to the back \nof the secondary star's Roche lobe ($49R_{\\rm wd}$), \nthe resultant model CBD SED (for both \noptically thick and thin cases) has only a few percent improvement \n(increase) in flux density at IRAC channels 1 and 2.  Similarly, \nthe agreement with the IRAC channels 3 and 4 and IRS data is not \nsignificantly better than that achieved by the CBD component used \nin Model 1 (Figure \\ref{f:models}b).\n\nThe optically thick CBD models are parameterized solely by the \nradial temperature profile (including the value of $T_{\\rm in}$) \nand size (i.e., inner and outer radii).  As such, the variety \nof possible model optically thick CBD SEDs is rather more limited \nthan for the optically thin case.  Changing either the radial \ntemperature profile exponent or $R_{\\rm in}$ does not yield an \noptically thick CBD model that reproduces the observed SED any \nbetter than the model shown in Figure \\ref{f:models}a \nand discussed in \\S\\ref{s:comparison}.\nFor example, using an exponent of 1, we can produce an optically \nthick CBD SED whose shape (i.e., relative intensity at each \nwavelength) is almost indistinguishable from that of the optically \nthin model CBD SED in Model 1.  However, it is overall 15--30\\% \nfainter than the optically thin model, so produces much worse \nagreement with the observations.  Smaller exponents produce model \nCBD SEDs that are far too bright, especially at longer wavelengths.\nIncreasing $T_{\\rm in}$ for the optically thick CBD produces more \nflux at shorter wavelengths, but also increases the steepness of \nthe SED and the excess flux at longer wavelengths.  Decreasing \n$T_{\\rm in}$ begins to flatten the SED, but also makes it overall \ntoo faint at all wavelengths, especially the short IR wavelengths.\nIncreasing $R_{\\rm in}$ also fails, since (as described above) \nthis results in even less flux at the short IR wavelengths.  \nIn light of the flat shape of the EF Eri SED at wavelengths \nlonger than the IRAC regime, we conclude from the behavior of\nthe optically thick CBD models described in this section that they \nare much less likely than the more flexible optically thin CBD models\nas viable explanations of the mid-IR SED of EF Eri.  \n\n\n\\section{Conclusions}\n\nOur newly obtained mid-IR spectrum of EF Eri has allowed us to \nfurther constrain and refine the SED model for this system first \npresented in B07.  Based on the B07 model, which was constrained \nby only 2MASS and IRAC photometric data, we would have expected \nthe 8--14 $\\mu$m SED of EF Eri to either rise \n(if dominated by an optically thick CBD) or fall (if dominated \nby short wavelength cyclotron emission).  However, the observed \nspectrum defied both of our expectations, by remaining almost \nflat compared to the IRAC data.  This does allow us to eliminate \ncyclotron emission as a dominant component in the mid-IR SED of \nEF Eri beyond $\\lambda\\approx3$ $\\mu$m.  At the same time, we \nalso show that a geometrically thin, but optically thick, \nCBD is unlikely as a viable explanation for the spatial \ndistribution of dust in EF Eri.  Instead, the \ndust is more likely present as a geometrically and optically thin CBD.  \nIn all cases, however, there are inconsistencies between our CBD \nmodels and the observed SED of EF Eri at the IRAC channel 1 and 2 \nwavelengths (3.6 and 4.5 $\\mu$m) that imply a more complex situation \nis present than represented by our simple CBD models.\nAs also found in B07, the total mass of dust in the CBD is still \nseveral orders of magnitude too small to strongly affect the \nsecular evolution of CVs in the context of current models of \nangular momentum loss mechanisms in CVs. \n\nBased on our study of EF Eri, we can make several generalizations \nregarding the mid-IR observational properties of CVs that contain dust.  \nFirst, EF Eri is a low-field polar.  In polars with moderate to \nstrong WD magnetic fields of several tens of MG or more, cyclotron \nemission will be shifted to even shorter wavelengths and be even \nless important for understanding the mid-IR SED.\nSecond, EF Eri contains a very low mass, faint secondary star.  \nLonger orbital period systems will have correspondingly more \nmassive, brighter secondary stars.  However, in the IRAC bands, \nan M5 dwarf is only $\\approx10$ times brighter than an L5 brown \ndwarf \\citep{patten06}.  Although the observed 3.6 $\\mu$m flux \ndensity in EF Eri is comparable to that of an M5 star, at 8 $\\mu$m \nthe observed flux density is $\\approx4$ times that of an M5 star.  \nSo, even in systems containing a more massive secondary star, dust \nemission at a comparable level to that in EF Eri produces a mid-IR \nSED far in excess of the stellar components.\nFinally, EF Eri lacks an accretion disk.  In non-magnetic CVs, the \nhot accretion disk will appear in the IR as a Rayleigh-Jeans-like \ntail similar in shape to the WD SED, but likely significantly \nbrighter than both stellar components.  However, the maximum \npossible emitting area for the accretion disk is limited by the \nsize of the WD Roche lobe.  The CBD, on the other hand, can have \nan emitting area many orders of magnitude larger.  Hence, even in \nthe presence of an accretion disk, the system SED at the longest \nmid-IR wavelengths could still be dominated by dust emission.\n\nAn extrapolation of our current CBD model for EF Eri to even \nlonger wavelengths predicts a continued gradual decline in the \noverall flux density, with (for example) the total flux density \nof EF Eri at 24 $\\mu$m being about 90\\% of the 8-$\\mu$m value.  \nLonger wavelength observations (e.g., at the {\\em Spitzer} \nPeak-up Imaging 22-$\\mu$m or MIPS 24-$\\mu$m bands) could test \nthis prediction.\n\n\n\n\\acknowledgments\n\nThis work is based in part on observations made with the \n{\\em Spitzer Space Telescope}, which is operated by the Jet \nPropulsion Laboratory, California Institute of Technology, \nunder a contract with the National Aeronautics and Space \nAdministration (NASA).\nSupport for this work was provided by NASA.\nWe thank the Spitzer Science Center (SSC) Director for his \ngenerous allocation of observing time for the NASA\/NOAO\/{\\em Spitzer \nSpace Telescope} Observing Program for Students and Teachers. \nThe National Optical Astronomy Observatory (NOAO), which is \noperated by the Association of Universities for Research \nin Astronomy (AURA), Inc., under cooperative agreement with the \nNational Science Foundation (NSF), has provided many in kind \ncontributions for which SBH is grateful. \nThis work makes use of data products from the \nTwo Micron All Sky Survey, which is a joint project of the \nUniversity of Massachusetts and the Infrared Processing and \nAnalysis Center\/Caltech, funded by NASA and the NSF. \nCSB acknowledges support from the SSC Enhanced Science Fund \nand NASA's Michelson Science Center.\nDWH thanks Axel Schwope for helpful advice on calculating \ncyclotron spectra.\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe 21cm transition of neutral hydrogen (HI) is predicted to trace cold diffuse gas during Cosmic Dawn, the epoch during which the first generation of stars formed, $\\sim 100$ million years after the Big Bang.  Prior to this, the spin temperature of the transition was likely in equilibrium with the Cosmic Microwave Background, well above the gas kinetic temperature.  The rise of Ly-$\\alpha$ background radiation from pockets of star formation coupled the spin temperature to the gas kinetic temperature via the Wouthuysen-Field effect \\citep[WF,][]{Wouthuysen1952, Field1958}.  The growing population of stellar remnants created an X-ray background that drove the gas kinetic and the spin temperatures higher \\citep[e.g.][]{Furlanetto2006,Pritchard2010}. Averaged over the sky, the relative proportion of Ly-$\\alpha$ coupling and X-ray heating varied with redshift and could create a broad absorption trough in the spectrum of the Cosmic Microwave Background.\n\nDetection of the predicted trough would provide unique information about the formation of the first luminous structures in the Universe \\citep[e.g.,][]{Barkana2005,Furlanetto2006,Fialkov2013,Mirocha2014,Mesinger2016,Mirocha2019, Reis2020,Magg2021, Gessey2022, Reis2022}, and of the thermal history of the intergalactic medium \\citep{Pritchard2007,Mesinger2013,Fialkov2014,Reis2021}. \n\nThe signal redshifted to radio frequencies $\\lesssim 100$~MHz and of order $-100$~mK in amplitude would in principle be detectable using meter-scale antennas and an integration time on the order of 100 hours, for sufficiently accurate radiometric calibration, and well understood celestial foregrounds (1000-10.000~K) and antenna gain patterns as a function of frequency.\n\n\nThe scientific importance of a measurement of the 21cm global signal absorption feature has motivated the effort to build several experiments in different locations and with different technical approaches.\n\n\nThe Experiment to Detect the Global Epoch-of-Reionization Signature (EDGES) relies upon a broad-band horizontal 'blade' dipole design and is deployed at the Murchison Radio-astronomy Observatory (MRO) in Western Australia. It reported the detection of a broad $-520$~mK absorption profile centered at $\\sim78$~MHz \\citep{Bowman2018} supported by validation tests also described by \\citet{Mahesh2021}. \nThe profile, more than a factor two deeper than predicted from theory based on standard physics \\citep[e.g. ][]{Cohen2017,Reis2021}, has triggered several studies aimed at an understanding of its origin.\nFrom a theoretical point of view, such a result could imply that either the temperature of the radio background is higher than the  CMB \\citep[e.g.][]{Bowman2018, Feng2018, Reis2020} or the neutral gas at redshift $\\sim 17$ was colder than expected, possibly due to the interaction with dark matter  \\citep[e.g.][]{Barkana2018, Munoz2018, Fialkov2019, Liu2019}. Other studies suggested the presence of un-modelled systematics \\citep{Hills2018,Singh2019,Spinelli2019,Bevins2020}, weaknesses in the analysis pipeline due to inaccurate modelling of the foregrounds \\citep[e.g.][]{Singh2019}, along with flaws in the statistical interpretation of the data \\citep[e.g.][]{Sims2020}. \n\nThe third generation of the Shaped Antenna measurement of the background RAdio Spectrum (SARAS) features a spectral radiometer based on a monocone antenna and receiver floated on a lake in Southern India \\citep{Nambissan2021}, has recently presented an analysis of their data \\citep{Singh2021}, rejecting the EDGES absorption  profile at 95.3$\\%$ confidence level.\n\n\nBesides EDGES and SARAS, there are several 21cm global signal experiments underway, such as the Probing Radio Intensity at high-Z from Marion (PRIZM) experiment \\citep{Philip2019}, the Broadband Instrument for Global HydrOgen ReioNisation Signal \\citep[BIGHORNS;][]{Sokolowski2015}, the Radio Experiment for the Analysis of Cosmic Hydrogen (REACH)\n\\citep{Cumner2021} and the  Mapper of the IGM Spin Temperature (MIST)\\footnote{\\hyperlink{http:\/\/www.physics.mcgill.ca\/mist\/}{http:\/\/www.physics.mcgill.ca\/mist\/}}.\n\n\n\\medskip\nThis work focuses on the Large-aperture Experiment to detect the Dark Age \\citep[LEDA;][]{Price2018} equipped and operated between two and five dual polarisation antennas within the Owens Valley Radio Observatory Long Wavelength Array station (OVRO-LWA) for radiometry, using custom RF and digital signal processing to enable the requisite timing, calibration, and stability (2013-2020).\n\nUsing early radiometric data, \\citet{Bernardi2016} set a coarse upper limit for the amplitude and the width for the anticipated absorption trough.  Using later data (Dec. 2018 - May 2019),\n\\citet{Spinelli2021} constrained the spectral index, $\\beta$, of Galactic radio emission in the northern sky (60-87~MHz), obtaining values compatible with expectation from simulations and other measurements.\nWe note that study by \\citet{Garsden2021} was distinct, using contemporaneous interferometric data, generated by the LEDA correlator that was part of OVRO-LWA, to characterize the effect of systematic errors in calibration on dynamic range for 2D cylindrical spatial power spectra.  (See also \\citet{Eastwood2018} regarding an alternate approach, also using LEDA interferometric data.)\n\n\n\nThe critical challenge in analysing radiometric data is \naccurate\nsubtraction of the bright foregrounds, normally attempted by modelling the spectrally smooth\nemission with an N-term log-polynomial and performing Bayesian\ninference for both foreground and background signals \\citep[e.g.,][]{Bernardi2015,Bernardi2016,Bowman2018,Singh2021}.\n\n\n\nChromaticity is a key factor  in antenna design \\citep[e.g.,][]{Mozdzen2016,Nambissan2021,Cumner2021}\nand chromatic beam effects  can limit the effectiveness of foreground subtraction \\citep[e.g.][]{Vedantham2014}. \nAccurate antenna beam simulations,  including realistic ground plane and soil descriptions, are thus fundamental to understand\nradiometric data. \n\\citet{Mahesh2021} explored antenna beam modelling as a source of uncertainty in global signal measurement, checking the \nstability of the signal reported by EDGES\nwith respect to choice of numerical electromagnetic solver code. \\citet{Bradley2019} instead investigated how a possible systematic artefact within\nthe antenna ground plane may produce broad absorption\nfeatures in the \nspectra. \n\nIn preparation for future data analysis, the REACH project \n has developed a software pipeline that can incorporate more efficiently the effect of beam chromaticity coupled with a non-trivial scaling in frequency of the foreground \\citep[][]{Anstey2020}. This \nstrategy, although computationally costly, leads to robust 21cm global signal extraction in\nsimulations and can thus \nenable better\ninformed optimisation of antenna design \\citep{Cumner2021,Anstey2022}. Other \ntechniques that incorporate beam effects in the foreground model using machine learning methods have been\nproposed by \\citet[e.g.,][]{Tauscher2020,Tauscher2021}.\n\n\\medskip In this paper, we make use of the\ncommon log-polynomial model to parametrise LEDA mock simulated spectra and we analyse in detail its limitations when  \na refined soil modelling with realistic values of its dielectric properties and conductivity is considered.  We explore different ground planes, focusing on the ones used in the field\nduring data acquisition and discuss the\nimpact on the Bayesian reconstruction of signal and foreground parameters. \n\n\nThis paper is organised as follows. We describe {\\it in situ} \nmeasurements of soil complex-permittivity and improved electromagnetic simulations in \\autoref{sec:beam}, including \nthe details of the multi-layer modelling of the ground.\nWe describe the construction of simulated spectra in \\secref{sec:Tobs} and the computation of the\nchromaticity correction factor in \\secref{sec:chrom}. In \\secref{sec:model}, we discuss the accuracy of the smooth foreground approximation in the presence of realistic beam modelling.  We analyse in \\secref{sec:recon} the impact of our realistic beam on the reconstruction of the parameters describing a neutral hydrogen absorption feature.\nA discussion of the results and our conclusions are presented in \\secref{sec:conclusions}.\n\n\n\\section{Modelling LEDA antennas}\\label{sec:beam}\n\nIn this section we present in detail improved electromagnetic models for antennas used for radiometry by LEDA. \nDriven by science requirements, the frequency range of interest is between 50 and 87 MHz (although some of the simulation results are shown in a larger frequency range). \nIn \\secref{sec:ground}, we present the antenna geometry and the ground planes we have considered. Note that the models reflect as-built designs. In \\secref{sec:soilmes}, we describe the {\\it in situ} soil permittivty data gathering. In \\secref{sec:oldbeam}, we review the analytic beam model used in previous analyses of LEDA data, while the focus of \\secref{sec:FEKO} is incorporation of the soil permittivty data and alternate ground-plane geometries.\n\n\\subsection{Antenna and ground plane description}\\label{sec:ground}\n\n\\begin{figure}\n        \\includegraphics[width=\\columnwidth]{figures\/LEDA_geometry2.pdf}\n        \\caption{LEDA antenna 3D view of geometry with the ground planes to be used in the analysis. Dimensions are shown in meters.}\n\\label{fig:example_ground}\n\\end{figure}\n\n\nWe outline here the basic geometrical properties of the antennas used by LEDA for radiometry, which excluding the ground plane are similar to those used in general for construction of Long Wavelength Array stations \\citep[][and references therein]{Taylor2012} and OVRO-LWA specifically.\n\nEach antenna comprises two pairs of triangular dipole arms 1.50~m long, angled downward by $45^\\circ$ (\\autoref{fig:example_ground}). \nWe focus our analysis on the east-west orientation.\n\nThe default OVRO-LWA \nantenna ground plane \\citep{Paravastu2007, Schmitt2009} is a 3\\,m $\\times$ 3\\,m galvanized welded steel mesh with 2.51~mm wire diameter (12.5 gauge) and $10.2$~cm spacing. \nIn this analysis, we modelled the beam for one antenna, numbered 252 \\citep{Price2018}, with three ground plane configurations that corresponded to a series of test modifications made in the field \\citep{Spinelli2021}. The 3\\,m $\\times$ 3\\,m ground plane was replaced first by a 10\\,m $\\times$ 10\\,m patch of mesh comprising the same material but with a 3.06~mm wire diameter (11 gauge), and later by the same but with serrations as represented by four 5~m long, 1.25~m wide isosceles triangles positioned on each side (referred to as the serrated ground plane). \nThe arrangement is shown schematically \nin \\autoref{fig:example_ground}.\nAs in \\citet{Bowman2018}, peripheral serrations are sometimes added to ground planes to reduce coherence among currents proximate to the edge discontinuity.\n\n\n\\subsection{Characterization of soil complex permittivity}\\label{sec:soilmes}\n\nMeasurements of complex permittivity, using coaxial impedance dielectric reflectometry \\citep{Seyfried2004}, were made at three depths (4, 14, and 21$\\pm0.5$ inches, corresponding to 10.16 cm, 35.56 cm and 53.34 cm) in a test pit dug at a midpoint $\\sim 170$~m from antennas 252 and 254 and backfilled.  Three 50~MHz sensors were inserted into the undisturbed strata revealed along one wall of the pit.  Sensor firmware provided temperature-corrected estimates of permittivity using the US Department of Agriculture calibration for soil comprising sand, silt, and clay with conductivity $<1.5$ S\\,m$^{-1}$  \\citep{Seyfried2005}.  \n\nData were collected episodically from May 2019 to Jan. 2020, at epochs spaced in time so as to track complex permittivity during the precipitation-free summer and fall seasons, as it approached baseline values. Scatter in conductivity measurements was 0.001 S\\,m$^{-1}$ and $\\leq 0.02$ in the real part of permittivity.\n\nThe soil composition observed at the test pit above 53~cm depth was similar to that removed from a 1.5~m deep vertical auger hole drilled $\\sim 10$\\,m SE of antenna 252: sand and a mix of sand and pebbles ($<5$\\,mm), depending on depth.  These observations are consistent with the broader geological context of this part of the Owens Valley. \\citet{Tallyn2002} classify the soil deposits on level terrain near the steep front of the Inyo range to the east as deep, well-drained Mazourka-Cajon-Hessica formations of sandy soil.  With finer positional resolution, \\citet{Danskin1998} describe well-sorted and unconsolidated lacustrine deposits of sand, gravelly sand, and silty sand to depths on the order of 100~m in the vicinity of Black Mountain, which overlooks the site from the ENE. This layering and favourable drainage across a broad area, and the absence of bedrock, buried boulders, and volcanic debris suggest that even without the benefit of ground penetrating radar analyses, it is reasonable to anticipate that the antenna sites are likely to be good ones from a geological perspective.\n\n\n\\subsection{Analytic beam model}\\label{sec:oldbeam}\nWe summarise in this section the analytical beam model of the LWA dipole used in previous works. For a more extensive description see \\citet{Dowell2011,Taylor2012,Ellingson2013}. Previous LEDA studies \\citep{Bernardi2015, Spinelli2019} used this beam modelling.\nThis model allows the reconstruction of the antenna beam pattern in every azimuth direction, using two principle antenna planes ($E$ and $H$, effectively taking into account the antenna symmetries) and reads:\n\\begin{equation}\nA(\\theta,\\phi,\\nu)=\\sqrt{[p_E(\\theta,\\nu) \\cos{\\phi}]^2 + [p_H(\\theta,\\nu) \\sin{\\phi}]^2}.\n\\notag\n\\end{equation}\nThe pattern in the $E$- and $H$-plane is given by:\n\\begin{equation}\np_i(\\nu,\\theta)=\\left[ 1 - \\left( \\frac{\\theta}{\\pi\/2} \\right)^{\\alpha_i(\\nu)} \\right](\\cos{\\theta})^{\\beta_i(\\nu)} + \\gamma_i(\\nu) \\left(\\frac{\\theta}{\\pi\/2} \\right)(\\cos\\theta)^{\\delta_i(\\nu)}\n\\label{eq:analytic_beam}\n\\end{equation}\nwhere $i=E, H$ and $\\theta$ is the elevation angle. The behavior of the coefficients $[\\alpha_i,\\beta_i,\\gamma_i, \\delta_i]$ with respect to frequency is fitted with a polynomial of $n^{\\rm th}$-order\\footnote{Different works throughout the years have use different value for the polynomial order. \\citet{Bernardi2016,Spinelli2019} used a 3$^{\\rm rd}$ order polynomial while \\citet{Spinelli2021} used a 13$^{\\rm th}$ order polynomial.} to NEC4 simulations \\citep{Hicks2012}. \nNote that for this simulation only the $3\\times3$ ground plane case is available.\n\n\n\n\n\n\\begin{table}\n\\centering\n\\caption{Soil parameters for the one-layer and the multi-layer model, extracted from measurements of soil at the LWA site during both dry and wet conditions at depths ${\\rm z}_i,\\ i=1,2,3$.}\\label{tab:soil_params}\n\\begin{tabular}{ |p{2cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}| }\n \\multicolumn{5}{|c|}{Soil layer parameters (\\( \\sigma \\) in S\/m, \\( \\epsilon_r \\) dimensionless)} \\\\\n \\hline\n  & \\( \\sigma_{\\rm dry} \\) & \\( \\sigma_{\\rm wet} \\) & \\( \\epsilon_{ r, {\\rm dry}} \\) & \\( \\epsilon_{ r,{\\rm wet}} \\) \\\\\n \\hline \n one layer & 0.004 & 0.01 & 4.4 & 6.5 \\\\\n \\hline \\hline\n$ {\\rm z}_1=10.16\\ cm $ & 0.0013 & 0.005 & 3.73 & 8.09 \\\\\n \\hline\n$  {\\rm z}_2=35.56\\ cm $ & 0.004 & 0.0068 & 4.25 & 6.45 \\\\\n \\hline\n$ {\\rm z}_3=53.34\\ cm $ & 0.0187 & 0.0388 & 7.58 & 20.56 \\\\\n\n\\end{tabular}\n\\end{table}\n\n\\subsection{Improving the beam model}\\label{sec:FEKO}\n\\begin{figure}\n        \\includegraphics[width=\\columnwidth,height=0.2\\textheight]{figures\/multi-layer}\n        \\caption{A 2D schematic view of\n        the multi-layer splitting of soil half-infinite space $ \\rm z<0 $,\n        described in \\autoref{sec:FEKO}.\n        A number of different measurements were made at 3 specific depths $ \\rm z_i$ (10.16 cm, 35.56 cm and 53.34 cm) and then ensemble averaged. A numerical EM driven multi-layer modelling is then constructed with an iterative sub-layering scheme. See text and \\autoref{alg:sublayers} for details.}\n\\label{fig:example_layers}\n\\end{figure}\n\nIn this section, we incorporate the soil properties measured in-situ, describing more realistically the environmental conditions in which the data have been collected. \nAlthough not previously explored for the LEDA antennas, there are examples in the literature of similar studies where the soil is analysed as an homogeneous dielectric semi-infinite volume, discriminating only between dry and wet soil conditions \\citep[e.g.,][]{Sutinjo2015}, or a combination of both, as in \\citet{Bradley2019}.\n\n\\paragraph*{One layer baseline model.} \nFor our new simulation we use FEKO\\footnote{\\url{https:\/\/altairhyperworks.com\/feko\/}}, a commercial software widely employed in numerical EM simulations and based on the Method of Moments (MoM, e.g. \\citet{Davidson2010}).\n\nWe model the soil as a single half-infinite layer of constant complex permittivity and conductivity, namely: \n\\begin{equation}\n    \\epsilon=\\epsilon_0\\left(\\epsilon_r + i\\frac{\\sigma}{2\\pi\\nu\\epsilon_0}\\right)\n    \\label{eqn:complex_epsilon}\n\\end{equation}\nwhere $\\epsilon_0$ is the electrical permittivity of vacuum, $\\epsilon_r$ is the dimensionless relative permittivity, $\\sigma$ is the conductivity in S\/m, and $\\nu$ is the frequency in Hz. This quantity had been measured on site for different soil moisture condition corresponding to both dry and wet soil (see \\autoref{tab:soil_params}). We use this one-layer model as our baseline one since it is simple and relatively immune to numerical artefacts. We simulate the three different ground planes - $3\\times3$, $10\\times10$ and serrated - described in  \\secref{sec:ground} and \\autoref{fig:example_ground}.\n\n\n\\paragraph*{Measurement driven three-layer modelling.} \nWe update the modelling just described adding the new available measurements for the soil complex permittivity. In constructing our first layered geometrical model, we adopted  values at the nominal depths of \\autoref{tab:soil_params}, according to \\secref{sec:soilmes}. A thickness was also chosen for the simulated layers according to the measurement depths, and we ensemble averaged the permittivities of the available measurements.\n\nNote that the thickness of these three consecutive layers does not match the standard numerical accuracy criteria required by EM solvers. Despite this, we use these measurements to construct a three-layer soil model for the FEKO simulation (``measurements'' column of \\autoref{tab:beam_type}).\n\n\\paragraph*{Numerical EM driven multi-layer modelling.}\nThe thicknesses of the three-layer model are imposed by the depth of the available measurements and might not be sufficient for accurately representing a varying soil permittivity gradient. In order to solve the possible numerical issues connected to the thicknesses of the three-layer model we refine our electromagnetic simulations using more layers. Although the separation between two consecutive layers is subject to different discretisation rules according to each method, requesting a separation smaller than $\\lambda\/10$ is a widely used criterion.\nThe standard FEKO method for any multi-layer substrate is a planar Green's function analytical solution embedded in the MoM formalism. A continuously varying permittivity is treated by means of properly discretising in layers, and a Transmission Line Green's Function (TLGF) approach is used by the solver. Details for this can be found in \\citet{Michalski1997}.\n\n\\medskip\nWhen considering the $\\lambda\/10$ rule to discretise the half-infinite space, we have to take into account the different phase velocity which implies a different wavelength in each layer. For a low-loss medium, such as soil ($\\sigma\\ll 2\\pi\\nu\\epsilon_0$ when considering our available measurements), the phase velocity is $c_p=c_0\/\\epsilon_r$, where $c_0$ is the speed of light in vacuum. We shall split each of the previous three layers into sub-layers such that their thickness is less than or equal to $\\lambda_p\/10$, and then double these layers iteratively until a certain convergence criterion is satisfied. The algorithm which implements this iteration is presented in \\autoref{alg:sublayers}. In each iteration, we interpolate linearly to compute the new values of complex permittivity (\\texttt{linInterp}), we merge them with the values of the previous iteration (\\texttt{merge}), and we run a new FEKO simulation to calculate the gain pattern (\\texttt{Gain}).\nMore details on the choice of the sub-layering scheme are explained in \\autoref{app:multi}. \n\nThis iteration is applied both for relative permittivity and for conductivity in a square lattice fashion, and a maximum frequency $\\nu_{\\rm max}=100$~MHz is used which provides a strong bound. By performing the first step of discretization, we conclude that each of the previous three layers of the dry soil should be divided into one or two sub-layers (see the grey dashed lines in \\autoref{fig:example_layers}). We then follow an iterative splitting by doubling these sub-layers (red dotted-dash lines in \\autoref{fig:example_layers}), in order to make an ever finer discretization. Wet soil conditions require a slightly finer splitting and are not reported in the schematic figure. The number of sub-layers used in the first iteration is reported in \\autoref{tab:beam_type}. \n\n\\begin{algorithm}\n\\caption{Sub-layer splitting algorithm (see also \\autoref{fig:example_layers}) to achieve gain at zenith convergence of 0.1 dBi. [arg] indicates a vector-valued argument, while a subscript is used to index these values ($i_1:i_2$ also indicates parsing from initial index $i_1$ to final index $i_2$). See text and appendix for details.}\\label{alg:sublayers}\n$[\\epsilon]\\gets [\\epsilon_1, \\epsilon_2, \\epsilon_3]$\\;\n$[\\lambda_p]\\gets\\frac{c_0\\epsilon_0}{\\nu_{\\rm max}\\Re\\{[\\epsilon]\\}}$\\;\n$[\\Delta\\epsilon]\\gets [\\epsilon_1, \\epsilon_2-\\epsilon_1, \\epsilon_3- \\epsilon_2]$\\;\n$[N] \\gets [1, 1, 1]$\\;\n$G_{0}\\gets$ Gain($[\\epsilon]$)\\;\n\\For{$i=0,1,2$}{\n$N_{init}\\gets\\left\\lceil\\frac{|z_{i+1}-z_i|}{[\\lambda_p]_{i+1}\/10}\\right\\rceil$\\;\n$[\\Delta\\epsilon]_{i+1}\\gets\\frac{[\\Delta\\epsilon]_{i+1}}{N_{init}}$\\;\n$[N]_{i+1} \\gets N_{init}$\\;\n$[\\epsilon]\\gets$ merge($[\\epsilon]_{1:i}$,linInterp($[\\epsilon]_{i+1},[N]_{i+1},[\\Delta\\epsilon]_{i+1}$)\\;\n}\n$G_1 \\gets$ Gain($[\\epsilon]$)\\;\n\\While{$ |G_1-G_0| \\begingroup>\\endgroup 0.1 $ dBi}{\n  $G_0\\gets G_1$\\;\n  \\For{$i=0,1,2$}{\n    \\begin{align}\n    [\\epsilon]\\gets \\text{merge}&([\\epsilon]_{1:[N]_{i}-1},\\hdots \\nonumber \\\\\n    \\text{linInterp}&([\\epsilon]_{[N]_{i}:[N]_{i+1}-1},2[N]_{i+1},[\\Delta\\epsilon]_{i+1}\/2))\\text{\\;} \\nonumber\n    \\end{align}\n  }\n  $[\\Delta\\epsilon]\\gets\\frac{[\\Delta\\epsilon]}{2}$\\;\n  $[N]\\gets 2[N]$\\;\n  $G_1\\gets$ Gain($[\\epsilon]$)\n}\n\\end{algorithm}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, keepaspectratio]{figures\/fig3b.pdf}\n\\caption{Gain at zenith dB difference of multi-layer implementations, calculated between the converged solution with 40 and 56 layers for dry and wet soil conditions, respectively, and solutions with fewer layers. Six cases are examined: different types of soil conditions (dry, wet) and different types of ground plane $3 \\times 3$,$10 \\times 10$, serrated). The vertical dashed lines highlight the region where LEDA data are available.}\n\\label{fig:dgaindB_convergence}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/fig4b.pdf}\n\\caption{Gain at zenith dB difference of ground plane implementations, calculated between a reference model of an infinite PEC plane and the converged model of each ground plane as reported in the legend, for dry and wet soil conditions, respectively. An intermediate case of a $5 \\times 5$ ground plane and a large $20 \\times 20$ one, not examined elsewhere, are also shown. The vertical dashed lines in each panel highlight the region where LEDA data are available.}\n\\label{fig:groundplanedB_convergence}\n\\end{figure}\n\n\\medskip \n\nThe iterative algorithm stops when a convergence threshold for the value of the gain at zenith is satisfied. We choose 0.1 dB which should be roughly similar to the numerical accuracy of FEKO. The number of layers needed for convergence are 40 and 56 layers for dry and wet conditions, respectively. \nThe high number of layers to reach convergence is needed only for the serrated case but we assume it for all ground planes for convenience. We refer to this implementation as the converged model.\n\\autoref{tab:beam_type} summarises the different multi-layer approaches listing their number of layers. \\autoref{fig:dgaindB_convergence} shows the gain differences (in dB) between the converged case and the solution obtained with a smaller number of layers at all previous iterations. The analysis is repeated for all three different ground planes. We note the existence of periodic oscillations for the case of the $10 \\times 10$ ground plane, with or without serrations. This fact highlights that successive implementations of sub-layers differ roughly by sinusoidal factors, which are still present when the ground plane gets bigger. This is a counter-intuitive conclusion, which might originate from sharp permittivity value transitions between consecutive layers (expressed in boundary conditions of the TLGF). The $3 \\times 3$ ground plane performs better with respect to the oscillation, apart from a low-frequency drift off. \n\n\n\n\\begin{table}\n\\caption{Number of soil layers that were used in each of our 3 multi-layer models: $\\lambda_{ p}\/10$ is for the first iteration of the algorithm, and converged is the final step of doubling that allows gain convergence to 0.1dB.}\\label{tab:beam_type}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \n & \\multicolumn{3}{c}{number of soil layers for multi-layer schemes}\\\\\n\\hline\n& measurements & $\\lambda_{ p}\/10$ & converged\\\\\n\\hline\ndry & 3 & 5 & 40\\\\\n \\hline\nwet & 3 & 7 & 56\\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\paragraph*{Finite size of the ground plane.}\nOscillation effects are also observed when comparing all finite sized ground planes with respect to an infinite Perfect Electric Conductor (PEC) plane solution. To demonstrate this, in \\autoref{fig:groundplanedB_convergence} we subtract from the converged case the solution for an infinite PEC plane, for each of the considered ground planes. We also include a $5 \\times 5$ and a $20 \\times 20$ ground plane, as an intermediate and extreme case. The sinusoidal variations that can be seen imply that there are finite-size truncation\/diffraction effects which need long electrical distances (i.e., distances as multiples of $\\lambda$) to diminish. The amplitude and periodicity of these oscillations are different for each ground plane and depend on their size. \n\n\n\n\n\\medskip\nHaving examined the effect of different ground planes, soil conditions as well as soil layering options, a comparative plot of zenith gain can best illustrate their effects before any chromaticity correction is calculated. For each ground plane and soil conditions, we present in \\autoref{fig:gaindb_zenith} the baseline one-layer, the three-layer, and the converged multi-layer FEKO model. As expected, the 3x3 ground plane provides in most cases a lower gain, since there are more losses related to the part of the soil not covered by the ground plane. It is, however, the best one in terms of ground plane induced ripple, which has a lower frequency due to its smaller dimension. Note that concerns over the spectral structure of an underlying ground plane have already been addressed in other experiments such as SARAS3 \\citep{Nambissan2021,Singh2021} that moved the antenna on a lake to minimise the chromatic response. The MIST experiment has opted instead for not deploying a ground plane and carefully characterise the ground properties. For an antenna configuration less sensitive to the ground properties than LEDA (which is facing downwards and is more coupled to any underlying structure), a larger ground plane is also expected to alleviate the problem. The EDGES team, for example, has used a $30\\times 30$ ground plane for their results \\citep{Bowman2018}.\n\n\nWe include in \\autoref{fig:gaindb_zenith} for comparison the $3 \\times 3$ ground plane solution obtained with the NEC4 pattern used in previous analysis. The NEC4 patterns have been scaled down using the radiation efficiency as calculated with FEKO data (see the next paragraph) since the NEC4 model was overestimating the gain by omitting the inclusion of soil losses in its gain calculation \\citep{Weiner1991}. Despite this correction, the FEKO and NEC4 models differ by a factor of up to 1 dB across the frequency range of interest, a result that showcases the differences between numerical solvers. \n\n\\paragraph*{Radiation efficiency.} \nWe present here the radiation efficiency $\\eta_{\\rm rad}$ as a function of frequency, which takes into account losses over all of the 3D antenna pattern. The radiation efficiency is calculated using the integral of the upper hemisphere far field patterns and the input power at the antenna port, given by FEKO.\nIn \\autoref{fig:rad_eff}, $\\eta_{\\rm rad}$ is shown for the three examined ground planes. As expected, the $3\\times 3$ ground plane presents more losses, with a clearer frequency dependence as well, since the efficiency is poorer at lower frequencies. A $10\\times 10$ ground plane with or without serrations is more appropriate to keep losses smaller than $10\\%$ in most of the frequency band, and quite more stable at the extremes, as they have a larger extent and allow for more power radiated below the horizon ($>90^\\circ$) to be reflected back, contributing to the gain. The radiation efficiency is not constant and this affects the gain integral which is not $4\\pi$ and varies with frequency. The beam integral is expected to vary significantly in the case of $3 \\times 3$ ground planes, such that any normalisation of the beam pattern should be made separately for each frequency (see also \\secref{sec:chrom}).\n\n\\paragraph*{Beam gain variation.} As a final assessment of the radiation pattern spectral robustness, we present a number of 2D colour-maps of the beam dB change in gain with respect to frequency in \\autoref{fig:beamdir}, for 4 azimuth angles $\\phi=0^{\\circ}$,$30^{\\circ}$, $60^{\\circ}$,$90^{\\circ}$, a criterion similar to that evaluated by \\citet{Mahesh2021}. This kind of plot is useful not only to confirm the sinusoidal-factor spectral periodicity due to the ground plane structure, but any other spatial effects which predominantly appear across zenith angle $\\theta$. It can be seen from the figure that these spatial variations are significant even for $\\theta < 40^\\circ$ , which compares with the Half-Power Beam Width of the antenna. Another interesting phenomenon is that the ``phase'' of this sinusoidal-factor variation is different for each elevation, which means that any analytic approach such as that of \\autoref{eq:analytic_beam} with $\\theta\\times\\cos(\\theta)$ polynomial terms is not adequate to describe the complexity of the beam pattern.\n\n\nThe amplitude in azimuth angle diminishes when we cross from the $E$-plane ($ \\phi=0^{\\circ}$) to the $H$-plane ($ \\phi=90^{\\circ}$). The greatest variation both in terms of amplitude and number of complete cycles in the frequency range of interest is found for the serrated ground plane, which reaches as high as 0.2 dB\/MHz in many $\\nu -\\theta$ sample points.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, keepaspectratio]{figures\/fig5b.pdf} \n\\caption{Gain at zenith of different multi-layer implementations, including the baseline model, a three-layer model, and the converged multi-layer model (40 and 56 layers for dry and wet soil conditions, respectively). Six cases are examined, as in \\autoref{fig:groundplanedB_convergence}. For the $3\\times3$ ground plane, the NEC4 simulated model is also shown. The vertical dashed lines in each panel highlight the region where LEDA data are available.}\n\\label{fig:gaindb_zenith}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, keepaspectratio]{figures\/fig6b.pdf}\n\\caption{Radiation efficiency over frequency calculated for the three different cases of ground plane, using the FEKO solution outputs. Dry (solid lines) and wet (dotted lines) conditions for the baseline one-layer model are presented. The vertical dashed lines highlight the region where LEDA data are available.}\n\\label{fig:rad_eff}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/beam_drectivity_dry.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/beam_drectivity_dry10.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/beam_drectivity_dry_ser.pdf}\n\\caption{Beam gain change per unit frequency as a function of the angle $\\theta$ and for four selected $\\phi$ values (0, 30,60 and 90 degrees). We show here the one-layer model for dry soil condition.}\n\\label{fig:beamdir}\n\\end{figure}\n\n\n\n\n\\section{Simulated observations}\\label{sec:sim}\n\nIn this section we describe the construction of LEDA mock measured spectra, the correction factor for the beam-induced chromaticity of the simulated spectra and discuss qualitatively its impact on the accuracy of the smooth foreground model. As we already mentioned, we make use of a baseline, single-layer model, since the previous analysis has shown that the three-layer and multilayer models suffer from some uncertainties: lack of more measurements for the former, and interpolations assumptions for the latter.\n\n\n\\subsection{Modelling the sky observed temperature}\\label{sec:Tobs}\n\nIn order to simulate the spectra measured by LEDA we compute the beam-averaged sky brightness temperature as seen by a single antenna $T(\\hat{\\boldsymbol{n}}_0,\\nu,t)$ at the time $t$ and direction $\\hat{\\boldsymbol{n}}_0$ \\citep[e.g.,][]{Bernardi2015}:\n\\begin{equation}\\label{eq:meas}\nT_{\\rm obs}(\\hat{\\boldsymbol{r}}_0,\\nu,t) = \\frac{\\int_{\\Omega} B(\\hat{\\boldsymbol{n}}',\\nu) \\, T_{\\rm sky}( \\hat{\\boldsymbol{n}}',\\nu,t) \\, d\\hat{\\boldsymbol{n}}'}{\\int_{\\Omega} B(\\hat{\\boldsymbol{n}}',\\nu) \\, d\\hat{\\boldsymbol{n}}'} + T_{\\rm N}(\\nu) + T_{21}(\\nu)\n\\end{equation}\nwhere \n$B$ is the antenna gain pattern and $T_{\\rm N}$ the instrumental noise. \n$T_{\\rm sky}$ is the sky brightness temperature, which changes with time as the sky drifts over the dipole.\nTo model this latter we simply consider the Haslam $408$~MHz full-sky map $T^{{\\rm H}}(\\mathbf{\\hat{n}})$ \\citep{Haslam1982} and scale it to different frequencies assuming a constant spectral index $\\beta$\n\\begin{equation}\\label{eq:T_H}\n    T^{{\\rm H}}_{{\\rm sky}}(\\nu,\\mathbf{\\hat{n}})=[T_{{\\rm H}}(\\mathbf{\\hat{n}})-T_{{\\rm cmb}}]\\left( \\frac{\\nu}{408}\\right)^{\\beta}+T_{{\\rm cmb}}.\n\\end{equation}\nwhere $T_{{\\rm cmb}}=2.725$~K and $\\beta=-2.5$.\nNote that other different sky models such as the GSM \\citep{Zheng2017} or the GMOSS \\citep{GMOSS2017} could be considered, including the effect of a spatially varying spectral index. Nevertheless, in this work we are mainly interested in varying the antenna simulations and we postpone the investigation of sky modelling uncertainties to a future work.\nMoreover, due to the large beam, a spatially constant spectral index is a reasonable approximation \\citep{Cumner2021}. \n\n\n\n\n\nWe show the resulting antenna temperature averaged over 4~h LST bins in \\autoref{fig:tobs}, that can be compared to Figure 13 in \\citet{Mahesh2021} and shows the low foreground contamination in the LST range 8-12~h, corresponding roughly to the chosen one for our data analysis in \\citet{Spinelli2021}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/tobs_vsfreq.pdf}\n\\caption{Simulated antenna temperature as seen through the LEDA beam as a function of frequency in the 50-87 MHz range. The different colours correspond to different 4~h LST bins.}\n\\label{fig:tobs}\n\\end{figure}\n\n\nWe model the 21cm global signal $T_{21}(\\nu)$ with a Gaussian absorption profile \\citep{Bernardi2015,Presley2015,Bernardi2016,Monsalve2017}:\n\\begin{equation}\\label{eq:gauss}\nT_{21}(\\nu)=A_{21} \\, e^{-\\frac{(\\nu-\\nu_{21})^2}{2\\sigma_{21}^2}},\n\\end{equation}\nwhere $A_{21}$, $\\nu_{21}$ and $\\sigma_{21}$ are the amplitude, peak position and standard deviation of the 21cm trough, respectively.\nThis is an approximation and more realistic shapes for the absorption feature could be computed from numerical or semi-analytical simulations  \\citep[e.g.,][]{Mirocha2014,Mirocha2015,Cohen2016,Cohen2017,Mirocha2017, Reis2021}, however, analytic expressions are useful for fast evaluation of likelihood functions.\nWe adopt in this work a simplistic assumption for the signal strength using $A_{21}=-180.0$~mK, $\\nu_{21}=70$~MHz and $\\sigma_{21}=2$~MHz. \nWe do expect our results to be dependent on the parameters chosen for the signal. With these values, the absorption profile is narrow (i.e. easier to disentangle from the smooth foregrounds) and well inside our observed band. In this study we are, however, mostly interested in relative behaviours for the beam modelling allowing us to fix the input cosmological model. We discuss this approximation further in \\appref{app:diffHI}.\n\n\n\nIn order to limit the parameter space to explore, we construct mock data as faithful as possible to the actual LEDA data \\citep[see][]{Spinelli2021} and we construct the mock spectra $\\bar{T}_{\\rm ant}(\\nu)$ by averaging \\autoref{eq:meas} in the LST range 8.5-12~h.\n\nTo compute the noise $T_N$, we assume that it is given by the radiometer equation and it is uncorrelated in frequency and time. We assume that for each frequency channel it follows a Gaussian distribution with standard deviation: \n\\begin{equation}\\label{eq:noise}\n\\sigma^N(\\nu)=\\frac{\\bar{T}_{\\rm ant}(\\nu)}{\\sqrt{\\Delta t \\, \\Delta \\nu}},\n\\end{equation}\nwhere we consider a $\\Delta \\nu = 1$~MHz channel width and a $\\Delta t = 100$~hours of total integration time, in agreement to real LEDA data specifics. \n\n\n\n\\subsection{Chromaticity correction}\\label{sec:chrom}\n\nAssuming a well-behaved beam with a small degree of chromaticity, the resulting spectra of \\autoref{eq:meas} are expected to be smooth in frequency. Although not visible by eye in \\autoref{fig:tobs}, realistic beam shapes induce a non-smooth frequency structure in the measured sky temperature that, as we will discuss more extensively later on, complicates the signal reconstruction. As discussed in other works \\citep[e.g.,][]{Mozdzen2019,Monsalve2020,Anstey2020}\na possible approach to alleviate the effect of the chromatic beam on the measured spectra is to correct the original spectra using the following factor:\n\\begin{equation}\\label{eq:beam_chromaticity}\nB_{{\\rm c}}(\\nu,{\\rm LST})=\\frac{\\int_{\\Omega} T_{{\\rm sky}}(\\nu_0,{\\rm LST},\\mathbf{\\hat{n}'}) B(\\nu,\\mathbf{\\hat{n}}') d\\mathbf{\\hat{n}}'}{\\int_{\\Omega} T_{{\\rm sky}}(\\nu_0,{\\rm LST},\\mathbf{\\hat{n}}') B(\\nu_0,\\mathbf{\\hat{n}}') d\\mathbf{\\hat{n}}'} \\frac{\\int_{\\Omega}  B(\\nu_0,\\mathbf{\\hat{n}}') d\\mathbf{\\hat{n}}'}{\\int_{\\Omega} B(\\nu,\\mathbf{\\hat{n}}') d\\mathbf{\\hat{n}}'}  .\n\\end{equation}\nNote that $T_{{\\rm sky}}$ is a function of LST since the sky drifts with time over the antenna.\nWe choose $\\nu_0=75$~MHz as a reference frequency  \\citep{Mozdzen2019} since it is approximately central in our range. In this formulation it should also be pointed out that the beam pattern $B(\\nu,\\mathbf{\\hat{n}}')$ used is gain, so the second term on the right hand side of the equation\nis ${\\eta_{\\rm rad}(\\nu)}\/{\\eta_{\\rm rad}(\\nu_0)}$.\n\nWe compute the beam chromaticity correction using the various beam models presented in \\autoref{sec:beam}. Note that we consider LST bins of 10 minutes and the full 24~h range for completeness when computing the correction. Since our goal is to investigate the effect of the beam, we use the same sky model of equation~\\ref{eq:T_H} to disentangle the two effects and to avoid introducing complications due to a different sky model than the one assumed to compute the simulated spectra. An example of the correction is reported in \\autoref{fig:beamchromdry_example} where we show our baseline case of one-layer, dry soil conditions and $3\\times3$~m ground plane (see \\autoref{sec:beam}).\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/ref75_50_85_LEDA_beam_corr_dry_lst0_24_new.png}\n\\caption{The chromaticity correction (\\autoref{eq:beam_chromaticity}) computed using the sky model of  \\autoref{eq:T_H} and the baseline beam model from FEKO (one-layer, dry soil) considering the $3\\times3$~m ground plane. The dashed vertical lines highlight the LST range preferred for LEDA data analysis.}\n\\label{fig:beamchromdry_example}\n\\end{figure}\n\nIt is interesting to evaluate, as shown in \\autoref{fig:gaindb_zenith}, the difference  between the analytic beam model previously used and the new FEKO baseline simulation, computing the chromaticity correction. This is reported in \\autoref{fig:chrom_diff_old_new}.\nDifferences reach a few percent especially around LST$\\sim 18$\nwhen the Galactic center is transiting.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/ref75_50_85_LEDA_beam_corr_old_new_diff_lst0_24_etacorr.png}\n\\caption{Percentage difference between the chromaticity correction factor $B_c$  (\\autoref{eq:beam_chromaticity}) computed for the new baseline beam model and the analytic beam used in previous analysis. The dashed vertical lines highlight the LST range preferred for LEDA data analysis.}\n\\label{fig:chrom_diff_old_new}\n\\end{figure} \n\nWe can repeat the same exercise for the various FEKO beam models computed for this analysis. We show in \\autoref{fig:chromdiffwet} the difference between the dry and wet soil conditions for the three different ground planes. Differences are a fractional of percent but their structure present different patterns for different ground plane shapes  (see \\autoref{fig:dgaindB_convergence} and \\autoref{fig:gaindb_zenith}). The same structure of periodic peaks\/troughs across frequency is seen in each constant LST line, as was observed for the dry\/wet layer simulation, as well as the sub-layer convergence study. There is additionally a variation along LSTs in constant frequency lines that has to do with the beam coupling to a fainter\/brighter sky. \n\n\n\n\nIn \\appref{app:chrom_diff}, we report for completeness the resulting chromaticity difference arising from small variation in soil properties and for the multi-layer parametrisation discussed in \\secref{sec:FEKO}.\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/ref75_50_85_LEDA_beam_corr_dry_wet_diff_lst0_24_new.png}\\\\\n\\includegraphics[width=\\columnwidth]{figures\/ref75_50_85_LEDA_beam_corr_dry10_wet10_diff_lst0_24_new.png}\\\\\n\\includegraphics[width=\\columnwidth]{figures\/ref75_50_85_LEDA_beam_corr_dryser_wetser_diff_lst0_24_new.png}\n\\caption{The difference in percentage between the chromaticity correction factor $B_c$ of \\autoref{eq:beam_chromaticity} computed for dry and wet soil conditions. We consider here the one-layer FEKO model and we compute the difference for the various ground planes: $3\\times3$~m (top panel), $10\\times10$~m (middle panel) and serrated (bottom panel). The dashed vertical lines highlight the LST range preferred for LEDA data analysis.}\n\\label{fig:chromdiffwet}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/NOHI_infground.pdf}\\\\\n\\includegraphics[width=\\columnwidth]{figures\/NOHI_33ground.pdf}\n\\caption{Residual difference (in K) as a function of frequency between the simulated spectra of equation \\secref{eq:meas} and the best fit model of \\autoref{eq:fg}, presented for different orders of the log-polynomial ($N=6$ in solid lines and $N=7$ with dashed lines) and for different LST bins. Different ground planes are considered: an ideal infinite ground plane (top panel) and the $3\\times 3$ ground plane (bottom panel). Note that the residual are computed without any chromaticity correction.}\n\\label{fig:inf_lst}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_wet.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_wet.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_wet.pdf}\\\\\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_wet10.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_wet10.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_wet10.pdf}\\\\\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_wet_ser.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_wet_ser.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_wet_ser.pdf}\n\\caption{Residual difference (in K) as a function of frequency between the best fit model of \\autoref{eq:fg} and the simulated spectra obtained as described in \\secref{sec:Tobs}, presented for different polynomial orders ($N=5,6,7$, left, central and right panels, respectively) and for different ground planes ($3\\times 3$ top panels, $10\\times 10$ central panels and serrated bottom panels). The simulated spectra are obtained considering dry condition and a one-layer description for the soil. The residuals are then computed without any chromaticity correction (grey solid lines), with an exact correction using the same beam model as for the spectra in input (dark red solid line) and for a chromaticity correction computed with wet soil condition (blue dashed lines). Note that the vertical scale changes for the different ground plane examined. Rms value for these residuals can be found in \\autoref{tab:rms_soil}.}\n\\label{fig:res_soil}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_layers.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_layers.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_layers.pdf}\\\\\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_layers10.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_layers10.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_layers10.pdf}\\\\\n\\includegraphics[width=0.65\\columnwidth]{figures\/N5_layers_ser.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N6_layers_ser.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/N7_layers_ser.pdf}\n\\caption{Residual difference (in K) as a function of frequency between the best fit model of \\autoref{eq:fg} and the simulated spectra obtained as described in \\secref{sec:Tobs}, presented for different polynomial orders ($N=5,6,7$, left, central and right panels, respectively) and for different ground planes ($3\\times 3$ top panels, $10\\times 10$ central panels and serrated bottom panels). The simulated spectra are obtained considering dry condition and a one-layer description for the soil. The residuals are then computed without any chromaticity correction (grey solid lines), with an exact correction using the same beam model as for the input spectra (dark red solid line), for a chromaticity correction computed instead with the three-layer model (blue dashed lines) or the converged layer model (pink dashed line). Note that the vertical scale changes for the different ground planes examined. Rms values for these residuals can be found in \\autoref{tab:rms_layers}.}\n\\label{fig:res_layers}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{Mock data modelling}\\label{sec:model}\n\nTo construct a model for our mock data we assume that the Galactic foreground spectrum can be described as a $N$-term log-polynomial \\citep[e.g.,][]{Bowman2010,Pritchard2010,Harker2012,Bernardi2015,Presley2015,Bernardi2016}:\n\\begin{equation}\\label{eq:fg}\n\\log_{10}{T_{\\rm fg}(\\nu)} = \\sum_{n=1}^{N} p_{n-1} \\left[ \\log_{10}{ \\left(\\frac{\\nu}{\\nu_0}\\right)} \\right]^{(n-1)}\n\\end{equation}\nwith $\\nu_0=60$~MHz. Note that $p_1$ corresponds to the spectral index $\\beta$ of \\autoref{eq:T_H} since for $N=2$ we can rewrite \\autoref{eq:fg} as $T_{\\rm fg}(\\nu)=p_0 (\\nu\/\\nu_0)^{p_1}$.\n\n\\medskip\nWe are interested in assessing the impact of the beam chromaticity on this assumption, examining its effect on the frequency smoothness of our simulated spectra computed as in \\autoref{eq:meas} and assuming no absorption feature, i.e. $T_{\\rm HI}=0$. \nWe compute the deviation of the mock measured sky temperature with respect to the smooth foreground model of \\autoref{eq:fg}. Note that the best fit values for the foreground parameters are obtained with a non-linear least squares solver. \n\n\\medskip\nWe first analyse the ideal case of an infinite ground plane for completeness. This antenna pattern is an ideal case which has only been used as reference for subtraction in \\autoref{fig:groundplanedB_convergence}, and produces the smoothest spectral response for gain. We show in the top panel of \\autoref{fig:inf_lst} that the log-polynomial model with $N=6$ is already capable of describing the spectra very well, in particular for LST$<8$~h. A higher log-polynomial order is required instead for the LST range of the 2018\/2019 LEDA observing campaign (i.e. $9-12$~h). When the beam modelling includes the finite ground plane (the $3 \\times 3$ ground plane is shown as an example in the bottom panel of \\autoref{fig:inf_lst}) there are residual structures that are not captured by the model of \\autoref{eq:fg}. These residuals again depend on the LST range considered, as the beam couples with the sky structures. \n\n\\medskip\nWe investigate this further as a function of the ground plane type in \\autoref{fig:res_soil}, where we use our baseline case, i.e. the dry soil one-layer model.  If we do not correct for the effect of beam chromaticity, the induced structures in the spectra prevent the smooth model from accurately describing the foregrounds, and the residuals are highly oscillating in frequency. \nThe rms values for the results can be found in \\autoref{tab:rms_soil} for different choices for the order of the log-polynomial. \nThe effect is more prominent for the serrated ground plane (where we found residual rms values around 1K) and still important for the $10 \\times 10$ ground plane. An exact chromaticity correction solves the problem for all types of ground planes describing the resulting simulated spectra with residuals of only a few mK.\n\nIf we attempt a correction assuming wet properties for the soil parameters instead of dry conditions, we obtain better residuals but still with strong features as a function of frequency.\n\nWe repeat  this same analysis considering now the impact of the multi-layer description of the soil. We show in \\autoref{fig:res_layers} the residual structures for the baseline model (one-layer, dry soil condition) when correcting the effect of chromaticity assuming either the three-layer or the converged multi-layer model, for different N-term log-polynomial models (5,6 and 7) and different ground planes ($3\\times 3$, $10\\times10$ and serrated). The rms of the residuals are reported in  \\autoref{tab:rms_layers} in \\appref{app:FoM_multi}. Note that the non-corrected and the exact correction cases are the same of \\autoref{fig:res_soil}.\n\n\n\nAttempting a chromaticity correction with the three-layer model (that is similar to the baseline one) always improves the rms and the smoothness of the resulting spectra. The converged layer model instead worsens the structures in the residuals and, for the serrated case, there is very small improvement increasing the order of the log-polynomial model. These results can be compared with \\autoref{fig:chromdiffgrounds}. Despite the differences in the chromaticity patterns between the three-layer and the converged layer cases not being strong, the higher contrast of the structures in the right column of  \\autoref{fig:chromdiffgrounds}, foreshadows the possible struggles of the correction procedure for the converged case.\n\n\\medskip\nWe can in each case confirm that the presence or not of a ground plane is more important than the parametrisation of soil both for dry\/wet conditions and different layer models, since the residuals are high for the more oscillating larger ground planes, and these oscillations are inherent to the antenna pattern as a function of frequency.\n\n\\medskip\nThese type of structures in the real data could prevent the detection of the cosmological signal or produce an erroneous detection and thus need to be investigated further. The rest of this work is dedicated to this problem.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/HI_diffgroundplane_a.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/HI_diffgroundplane_b.pdf}\n\\caption{The reconstructed $21$cm absorption profile for different ground planes considered. In the upper panel the infinite case and $3\\times3$ are shown in green and dark red respectively.  The $10\\times10$ and the serrated cases are instead shown in the lower panel in pink and blue, respectively. Note the different scale for the y-axis. The simulated spectra are generated for the one-layer dry soil conditions and are not corrected for the effect of chromaticity. The input model that we would like to reconstruct is shown as a black dotted line. The serrated result is shown for completeness but the amplitude of the absorption feature converged to the edge of the prior.}\n\\label{fig:diffground}\n\\end{figure}\n\n\n\\section{Foreground and cosmological parameter reconstruction}\\label{sec:recon}\n\nIn the last session we have discussed qualitatively the impact of a realistic beam model for the LEDA antennas, which can compromise the smoothness of the measured spectra, and is thus an important assumption for discerning the 21cm signal from the foregrounds.\n\nIn this section we investigate how much these spurious structures in the simulated spectra impact the Bayesian extraction of the 21cm absorption feature \\citep[e.g.][]{Bowman2018,Bernardi2016,Singh2021}.\n\n\\medskip\nOur results were obtained running the \\textsc{hibayes} code \\citep{Bernardi2016,Zwart2016}, a fully Bayesian framework where the posterior probability distribution is explored through the \\textsc{multinest} sampler \\citep{Feroz2008,Feroz2009}. We assume a Gaussian Likelihood for the data and make use of the model described in \\secref{sec:model}. The covariance matrix is assumed diagonal in frequency and the diagonal terms are computed with \\autoref{eq:noise}. For each analysed case we run the pipeline for different order of the log-polynomial and present the results with the highest evidence. \n\n\n\\subsection{Finite ground plane effect}\n\nWe analyse the structure induced on the simulated spectra by the different ground planes used in LEDA observations and modelled in this work.\nWe show in \\autoref{fig:diffground} the reconstructed $T_{\\rm HI}$ obtained from the mean values of the posterior distributions, in comparison with the input HI model used for the simulated spectra. While for an infinite ground plane it is possible to reconstruct the correct input, in presence of a finite ground plane the algorithm converges towards biased solutions, preventing a correct detection of the cosmological signal.\nIn \\autoref{fig:diffground} no correction for the effect of chromaticity is applied. \nIf we divide the simulated spectra with the beam factor of \\autoref{eq:beam_chromaticity}, computing it with the same beam model used for the simulated spectra, we are exactly correcting for the effect of chromaticity. The corrected spectra are much smoother as a function of frequency and they are well modelled by a low order of the log-polynomial. The input HI parameters are successfully reconstructed with residuals of the order of a few mK using only a 5-term polynomial model.\n\n\n\\subsection{Soil properties}\n\nThe results of the previous section suggest that, when attempting a reconstruction of the absorption feature in the LEDA data we need to correct for the effect of chromaticity. It is, however, not realistic to assume that our beam model agrees perfectly with the true beam. We investigate here the effect of a non-perfect reconstruction by varying the properties of the soil.\n\n\\medskip We analyse in \\autoref{fig:diffsmall}, for the $3\\times3$ ground plane, the effect on the reconstructed absorption feature of small variations in the assumed value for the conductivity ($\\sigma$) or the permittivity ($\\epsilon_r$) of the soil. We consider a 10\\% shift with respect to the dry soil condition to be compared with a factor $\\sim2$ difference between the dry and wet condition (presented in \\autoref{tab:soil_params}).  We report for completeness in \\autoref{fig:chromdiff3x3} the variation of the beam factor in these analysed cases with respect to the baseline beam model. As can be seen also from \\autoref{fig:diffsmall}, the change in conductivity biases the reconstructed absorption feature for both higher and lower values of $\\sigma$. A lower value of the conductivity leads to a 20\\% lower amplitude for the absorption signal while a higher conductivity bias the reconstruction of its central frequency.  \nIncreasing the permittivity has a similar effect while the bias gets stronger for a lower value of $\\epsilon_r$, resulting in a factor $\\sim2$ enhanced amplitude. These results are consistent with \\autoref{fig:chromdiff3x3}. Note that a different assumption for the input model would have produced slightly different results. It is however still instructive to estimate the expected magnitude of the bias.\n\n\\medskip\nWe investigate also a more drastic situation, where the soil conditions for the correcting chromaticity factor are the ``wet'' case presented  in \\autoref{tab:soil_params} and \\autoref{fig:chromdiffwet}. We report the reconstructed $T_{\\rm HI}$ in \\autoref{fig:diffgroundwet}. For the $3\\times3$ ground plane, already analysed, the result is similar to the small variations in permittivity and conductivity just discussed, as could have been anticipated comparing the top panel of \\autoref{fig:chromdiffwet} with \\autoref{fig:chromdiff3x3}. \nThe reconstruction is completely biased for the case of the larger ground planes, where the ``dry'' and ``wet'' chromaticity corrections present more structured differences (see the  middle and lower panel of \\autoref{fig:chromdiffwet}). \n\n\\medskip \nWe analyse the impact of soil modelling further in \\autoref{fig:difflayers} where we compare the three-layer and converged layer models against the baseline. As was hinted in \\autoref{fig:res_layers}, for the $3\\times 3$ case, the three-layer correction is better than no correction, while the converged layers description of the soil is already too much different from the one layer to offer better correction. The other ground plane cases show various biased results, proving that the correction is not solving the problem of the residual structures in the simulated spectra.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/HI_smallvar_3times3.pdf}\n\\caption{The reconstructed $21$cm absorption profile\nobtained when the simulated spectra (generated for the one-layer dry soil conditions) are corrected for the effect of chromaticity considering a $\\pm 10\\%$ variation for the permittivity and conductivity. The input model that we would like to reconstruct is shown as a black dotted line.}\n\\label{fig:diffsmall}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/HI_wet_grounds.pdf}\n\\caption{The reconstructed $21$cm absorption profile\nobtained when the simulated spectra (generated for the one-layer dry soil conditions) are corrected for the effect of chromaticity considering a wet soil moisture. We present the results for the three different ground planes considered ($3\\times3$ in dark red, $10\\times10$ in pink, and the serrated case in blue).}\n\\label{fig:diffgroundwet}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.65\\columnwidth]{figures\/HI_difflayers1.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/HI_difflayers2.pdf}\n\\includegraphics[width=0.65\\columnwidth]{figures\/HI_difflayers3.pdf}\n\\caption{The reconstructed absorption profile for the $3\\times3$ (left panel), $10\\times10$ (central panel) and serrated ground plane (right panel) for the one-layer dry soil baseline case spectra when we are correcting with a chromaticity factor computed using the three- or the converged layer model. Note the different vertical axis scale for the three panels. The dotted line is the input profile. For completeness, also the non-corrected case presented in \\autoref{fig:diffground} is shown in grey in each panel.}\n\\label{fig:difflayers}\n\\end{figure*}\n\n\\subsection{Spectral index reconstruction}\n\nWhile studying the correct reconstruction of the HI absorption feature, we also discuss the reconstructed foreground parameters. The most informative one is the spectral index $\\beta$ that can be compared with its input value $-2.5$ (see \\secref{sec:Tobs}). We report the peak of the posterior distribution of the spectral index for each case in \\autoref{fig:betavar}. We find that, when we do not correct for the effect of chromaticity, we recover up to a 10\\% flatter $\\beta$. The effect is stronger for larger ground planes. We instead always recover the right spectral index for the exact correction, or when the beam factor is computed varying the soil moisture properties. Finally, there is a tendency for a slightly flatter $\\beta$ (a few \\%) when we use different multi-layer modelling.\nThese flatter values of the spectral index are associated with strongly biased values of the amplitude of the absorption feature as can be seen in \\autoref{fig:diffground} and \\autoref{fig:difflayers}.\n\n\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{figures\/beta_variation.pdf}\n\\caption{Mean value for the posterior of the spectral index $\\beta$ (corresponding to the second term of the log-polynomial model) for the different cases analysed in this work. The true input value of $\\beta=-2.5$ is indicated with a dashed line to guide the eye. The results are colour-coded to distinguished the three different ground planes considered.}\n\\label{fig:betavar}\n\\end{figure*}\n\n\n\n\\section{Discussion and conclusions}\n\\label{sec:conclusions}\n\nIn this work, we presented the characterisation of the LEDA antenna beam,\nwith emphasis on the role of the ground plane and of the soil modelling both in terms of discretizing the semi-infinite volume it occupies, and varying its electromagnetic parameters. We used FEKO for our simulations and constructed a multi-layer model of the terrain relying on in-situ measurements of the soil complex permittivity. We used a more standard one-layer model under dry\/wet soil conditions as our baseline and discussed the variations in the absolute gain pattern. when exploring more sophisticated models. The characterisation of the antenna beam is of primary importance in assuring an accurate enough control of the systematic effects in 21cm global signal analysis and has been studied in detail for other experiments \\citep[e.g. ][]{Mahesh2021,Raghunathan2021}.\n\n\nWe explored the impact of beam modelling uncertainties due to soil moisture on the antenna chromaticity, focusing on three different ground planes used in the actual LEDA observations \\citep{Spinelli2021}: a 3~m $\\times$ 3~m, a 10~m $\\times$ 10~m and a 10~m $\\times$ 10~m with 10~m long triangular serrations. The addition of the ground plane induces a frequency ripple in the beam pattern that compromises its smoothness. The ripple amplitude is smaller for larger ground planes while its oscillation frequency is larger. The amplitude of the ripple depends also, to a lesser extent, on the number of layers that describe the permittivity as a function of depth. However, this effect saturates for a sufficiently large number of layers.\n\nWe then compared the gain at zenith for values of the complex permittivity corresponding to dry or wet terrain at the LEDA site. Surprisingly, the impact of different soil conditions is not suppressed by the presence of a larger ground plane and appears as a slight shift in the oscillatory pattern. The shift is more evident for the serrated ground plane case and affects the full shape of the beam. This makes the correction of the beam chromaticity more difficult and sensitive to electromagnetic property assumptions or measurement errors.\n\nIn future works, antenna beam simulations could be improved by\ncollecting more data to model the moisture conditions, leading to a better description of the complex permittivity as a function of depth \\citep{Campbell1990,Bobrov2015}. Moreover, a more complex frequency dependence with respect to \\autoref{eqn:complex_epsilon} could be adopted \\citep[for example a Cole-Cole model as in][]{Sternberg2001}.\nNote also that the finite accuracy of the numerical solver limits the refinement of the multi-layer model and different EM software solutions should be compared for benchmarking our model.\n\n\nWe followed with computing the beam chromaticity\nfactor and simulated observed beam-averaged sky spectra as\na function of frequency and LST, assuming the sky can be described as diffuse synchrotron emission scaled with a constant power law $\\beta=-2.5$ across our frequency range. The differences in complex permittivity typical of dry or wet soil conditions translate to a few \\textperthousand\\ variations in the beam chromaticty factor with similar amplitude across ground planes, but increasingly more structured in frequency and LST for larger ground planes, as expected.\n\nWe modelled the simulated spectra with an $N$-term log-polynomial exploring N values from 5 to 7, and studied the behaviour of the residuals as a function of $N$, LST and beam model, finding negligible frequency structure only for the ideal case of an infinite ground plane.\n\nWe added a Gaussian absorption feature to the various simulated spectra to mimic the high redshift 21cm signal. The final model for the mock measured spectra consisted of the $N^{\\rm th}$ term log-polynomial model plus three parameters to describe the Gaussian absorption feature (the central frequency $\\nu_{21}$, the amplitude $A_{21}$ and the width $\\sigma_{21}$).\nWe studied how beam model uncertainties propagate in the analysis, and bias the Bayesian model parameter reconstruction.\n\nWhen the exact beam chromaticity correction is applied to the simulated spectra, the model parameters are reconstructed without any bias and residuals are around the mK level. \nHowever, we found much larger residuals when the chromaticity is not accounted for, or the beam model for the correction is obtained for different soil conditions or for the different multi-layer implementations. These residuals can reach up to a few hundreds (thousands) of mK for the $10{\\rm m}\\times 10{\\rm m}$ (serrated) ground plane and bias the absorption signal reconstruction. Our results disfavour the use of a large ground plane coupled with the LEDA antenna \nas it seems that the interaction of the soil and the antenna itself lead to significant oscillating factors on the gain spectral response for all realistic ground plane sizes. \n\nThe $3{\\rm m}\\times 3{\\rm m}$ case behaves better and beam model uncertainties results in smaller parameter bias. Note however that a 10\\% changes in permittivity or conductivity can enhance\/reduce the parameters $A_{21}$ and $\\sigma_{21}$ up to a factor of two or shift by few \\% the recovered central frequency $\\nu_{21}$.\n\nFinally, we checked the effect of the various beam models on the reconstructed spectral index $\\beta$ of the simulated spectra and find results in agreement with the input value. We observed \na maximum 6\\% flattening only when the correction for beam chromaticity is not applied.\n\nApart from some ideal cases, the smooth foreground log-polynomial model is found not to be an accurate description of the frequency structures induced in the observed spectra by realistic LEDA gain patterns, preventing the Bayesian exploration of the parameter space to converge to the expected result. In the future we will investigate how this effect can be mitigated by increasing the number of model parameters used to describe the foreground spatial distribution \\citep[e.g.,][]{Anstey2020}.\n\n\n\n\n\n\n\\section*{acknowledgements}\nLEDA research herein was supported in part by NSF grant AST\/1616709. MS acknowledges support from the AstroSignals Synergia grant CRSII5\\_193826 from the Swiss National Science Foundation. \nThis research made use of \\texttt{Numpy} \\citep{Numpy2020}, \\texttt{Astropy} \\citep{Astropy} and \\texttt{Scipy} \\citep{Scipy2020}.\nSome of the results in this paper have been derived using the \\texttt{healpy} \\citep{Zonca2019} and \\textit{HEALPix} \\citep{2005ApJ...622..759G} package.\n\n\n\n\\section*{Data availability}\n\nThe simulated antenna gain patterns used in this work can be found tabulated in \\texttt{.mat} format in \\href{https:\/\/drive.google.com\/file\/d\/1xkxdh7N6B9w8f12ZG_UNuTzIBAKjrP5X\/view?usp=sharing}{this drive folder}.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCold dark matter (CDM) simulations (both with and without the\ninclusion of baryonic physics) are a crucial tool and proving ground\nfor understanding the physics of the universe on nonlinear scales.\nOne of the most active aspects of research in this area concerns the\nform of the dark matter density profile. Key questions raised in\nrecent years include: Is there a universal dark matter density profile that \nspans a wide range of halo masses? What is the form of this profile\nand how uniform is it from one halo to another? To what extent\ndo baryons modify the dark matter distribution?\n\nDrawing on a suite of N-body simulations, \\citet{NFW97} originally proposed\nthat the dark matter density profiles in halos ranging in size from those\nhosting dwarf galaxies to those with galaxy clusters have a\nuniversal form. This 3-D density distribution, termed the NFW profile, \nfollows $\\rho_{DM}\\propto r^{-1}$ within some scale radius, $r_{sc}$, and \nfalls off as $\\rho_{DM}\\propto r^{-3}$ beyond.  Subsequent simulations \nindicated that the inner density profile could be yet steeper - \n$\\rho_{DM}\\propto r^{-1.5}$ \\citep{M98,Ghigna00}.  As computing power \nincreases and numerical techniques improve, it is now unclear whether \nthe inner dark matter distribution converges to a power law form rather \nthan becoming progressively shallower in slope at smaller radii\n\\citep{P03,Navarro04,Diemand04,Diemand05}.  \n\nFor comparisons with data, such simulations need to account for the \npresence of baryons. This is particularly the case in the cores of rich \nclusters. Although baryons represent only a small fraction of the overall \ncluster mass, they may be crucially important on scales comparable to \nthe extent of typical brightest cluster galaxies. Much work is being done \nto understand the likely interactions between baryons and DM\n\\citep{Gnedin04,Nagai05,Faltenbacher05}.  These simulations will\nprovide refined predictions of the relative distributions of baryons\nand DM. \n\nThis paper is a further step in a series which aims to present an \nobservational analog to progress described above in the numerical \nsimulations. At each stage it is desirable to confront numerical \npredictions with observations. Whereas some workers have\nmade good progress in constraining the {\\em total} density profile \n(e.g.~\\citet{Broadhurst05b}), in order to address the relevance of the\nnumerical simulations we consider it important to develop and test \ntechniques capable of  separating the distributions of dark and \nbaryonic components (e.g. ~\\citet{Sand02,Zappacosta06,Biviano06,Mahdavi07}).  \n\nThis paper presents a refined version of the method first proposed by\n\\citet{Sand02}, exploited more fully in \\citet{Sand04} (hereafter\nS04). S04 sought to combine constraints from the velocity dispersion\nprofile of a central brightest cluster galaxy (BCG) with a strong\ngravitational lensing analysis in six carefully selected galaxy\nclusters in order to separate the baryonic and dark matter\ndistributions.  S04 carefully selected clusters to have simple,\napparently 'relaxed' gravitational potentials in order to match\nbroadly the 'equilibrium' status of the simulated dark matter halos\noriginally analyzed by \\citet{NFW97} and subsequent simulators.  For\nexample, Abell 383 and MS2137-23 have almost circular BCGs ($b\/a$=0.90\nand 0.83 respectively), require a single cluster dark matter halo to\nfit the strong lensing constraints (in contrast to the more typical\nclusters that require a multi-modal dark matter morphology -- Smith et\nal. 2005), have previously published lens models with a relatively\nround dark matter halo ($b\/a$=0.88 and 0.78 respectively - Smith et\nal. 2001; Gavazzi 2005), and display no evidence for dynamical\ndisturbance in the X-ray morphology of the clusters (Smith et\nal. 2005; Schmidt \\& Allen 2006). \n\nThe merit of the approach resides in combining two techniques that\ncollectively probe scales from the inner $\\sim$10 kpc (using the BCG\nkinematics) to the $\\sim$100 kpc scales typical of strong\nlensing. Whereas three of the clusters contained tangential arcs,\nconstraining the total enclosed mass within the Einstein radius, three\ncontained both radial and tangential gravitational arcs, the former\nproviding additional constraints on the derivative of the total mass\nprofile. In their analysis, S04 found the gradient of the inner dark\nmatter density distribution varied considerably from cluster to\ncluster, with a mean value substantially flatter than that predicted\nin the numerical simulations.\n\nS04 adopted a number of assumptions in their analysis whose effect on\nthe derived mass profiles were discussed at the time. The most\nimportant of these included ignoring cluster substructure and adopting\nspherically-symmetric mass distributions centered on the BCG. The\nsimplifying assumptions were considered sources of systematic\nuncertainties, of order 0.2 on the inner slope. Although the six\nclusters studied by S04 were carefully chosen to be smooth and round,\nseveral workers attributed the discrepancy between the final results\nand those of the simulations as likely to arise from these simplifying\nassumptions \\citep{Bartelmann04,Dalal04b,Meneghetti05}.\n\nThe goal of this paper is to refine the data analysis for two of the\nclusters (MS2137-23 and Abell 383) originally introduced by S04 using\nfully 2-D strong gravitational lensing models, thus avoiding any\nassumptions about substructure or spherical symmetry. The lensing\nmodels are based on an improved version of the LENSTOOL program\n(\\citealt{Kneibphd,Kneib96}; see Appendix;\nhttp:\/\/www.oamp.fr\/cosmology\/lenstool\/). A major development is the\nimplementation, in the code, of a pseudo-elliptical parameterization\nfor the NFW mass profile, i.e. a generalization of those seen in CDM\nsimulations, viz:\n\\begin{equation}\\label{eqn:gnfw}\n\\label{eq:gnfw}\n\\rho_d(r)=\\frac{\\rho_{c} \\delta_{c}}{(r\/r_{sc})^{\\beta}(1+(r\/r_{sc}))^{3-\\beta}}\n\\end{equation}\nwhere the asymptotic DM inner slope is $\\beta$. This formalism allows \nus to overcome an important limitation of previous work and takes into \naccount the ellipticity of the DM halo and the presence of galaxy-scale \nsubhalos. Furthermore the 2-D lensing model fully exploits the \nnumerous multiply-imaged lensing constraints available for MS2137-23 \nand Abell 383.\n\nThe combination of gravitational lensing and stellar dynamics is the\nmost powerful way to separate baryons and dark matter in the inner\nregions of clusters.  However, it is important to keep a few caveats\nin mind.  Galaxy clusters are structurally heterogeneous objects that\nare possibly not well-represented by simple parameterized mass\nmodels. To gain a full picture of their mass distribution and the\nrelative contribution of their major mass components will ultimately\nrequire a variety of measurements applied simultaneously across a\nrange of radii.  Steps in this direction are already being made with\nthe combined use of strong and weak gravitational lensing\n(e.g.~\\citet{Limousin06,Bradac06}), which may be able to benefit\nfurther from information provided from X-ray analyses\n(e.g.~\\citet{Schmidt06}) and kinematic studies (e.g.~\\citet{Lokas03}).\nA recent analysis has synthesized weak-lensing, X-ray and\nSunyaev-Zeldovich observations in the cluster Abell 478 -- similar\ncross-disciplinary work will lend further insights into the mass\ndistribution of clusters \\citep{Mahdavi07}.\n\nOf equal importance are mass models with an appropriate amount of \nflexibility and sophistication.  For instance, incorporating models that \ntake into account the interaction of baryons and dark matter may shed \nlight into the halo formation process and provide more accurate \nrepresentations of dark matter structure.  Halo triaxiality, multiple \nstructures along the line of sight and other geometric effects will \nalso be important to characterize. At the moment, incorporating these \ncomplexities and securing good parameter estimates is computationally \nexpensive even with sophisticated techniques such as the Markov-Chain \nMonte Carlo method. \n\nNumerical simulation results are often presented as the average\nprofile found in the suite of calculations performed. Instead, the\ndistribution of inner slopes would be a more useful quantity for\ncomparison with individual cluster observations.  Also, comparisons\nbetween simulations and observations would be simplified if {\\it\nprojected} density profiles of simulated halos along multiple lines of\nsight were to be made available.  These issues should be resolvable as\nlarge samples of observed mass profiles are obtained.\n\nFor the reasons above, comparing observational results with\nnumerical simulations is nontrivial.  The observational task should\nbe regarded as one of developing mass modeling techniques of\nincreasing sophistication that separate dark and baryonic matter, \nso as to provide the most stringent constraints to high resolution\nsimulations which include baryons as they also increase in\nsophistication.  The combination of stellar dynamics and strong\nlensing is the first crucial step in this process.  Its diagnostic\npower will be further enhanced by including other major mass\ncomponents (i.e.~the hot gas of the intracluster medium or the stellar\ncontribution from galaxies) out to larger radii.\n\nA plan of the paper follows.  In \\S~\\ref{sec:methods} we explain the\nmethodology used to model the cluster density profile by combining\nstrong lensing with the BCG velocity dispersion profile.  In\n\\S~\\ref{sec:obsresults} we focus on translating observational\nmeasurements into strong lensing input parameters.  This section\nincludes the final strong lensing interpretation of MS2137-23 and\nAbell 383.  In \\S~\\ref{sec:stronglens} we present the results of our\ncombined lensing and dynamical analysis.  In \\S~\\ref{sec:systematics}\nwe discuss further systematic effects, limitations and degeneracies\nthat our technique is susceptible to -- with an eye towards future\nrefinements.  Finally, in \\S~\\ref{sec:finale} we summarize and discuss\nour conclusions. Throughout this paper, we adopt $r$ as the radial\ncoordinate in 3-D space and $R$ as the radial coordinate in 2-D\nprojected space.  When necessary, we assume $H_{0}$=65 km~$\\tx{s}^{-1}$\nMpc$^{-1}$, $\\Omega_{m}$=0.3, and $\\Omega_{\\Lambda}$=0.7.\n\n\\section{Methods}\\label{sec:methods}\n\nThe intent of this work is to use the full 2D information provided by\nthe deep Hubble Space Telescope (HST) imaging in two strong lensing\nclusters (MS2137-23 and Abell 383) in conjunction with the BCG stellar\nvelocity dispersion profile in order to constrain the distribution of\nbaryonic and dark matter.  These two clusters were selected for\nfurther study from the larger sample presented by S04 because, of the\nthree systems with both radial and tangential gravitational arcs,\nthese two presented the shallowest DM inner slopes.\n\n\\subsection{Lens Modeling}\\label{sec:massmodel}\n\nWe use the updated LENSTOOL ray-tracing code to construct models of\nthe cluster mass distribution.  Our implementation of the mass\nprofiles is identical to that of \\citet{Golse02}, with the exception\nthat we have generalized their pseudo-elliptical parameterization to\ninclude ones with arbitrary inner logarithmic slopes.  For the\ndetails, the reader is referred to both \\citet{Golse02}, and the\nAppendix. Here we briefly explain the lens modeling process and\nparameterization of the cluster mass model.\n\nIdentifying mass model components and multiple-image candidates is an\niterative process. Initially, multiple images are spectroscopically\nconfirmed systems with counter images identified by visual inspection\nand with the aid of preliminary lens models, taking into account that\ngravitational lensing conserves surface brightness.  Multiple images\nwithout spectroscopic confirmation were used in the case of Abell 383,\nsince these additional constraints helped clarify the role that galaxy\nperturber \\#1 (Table~\\ref{tab:lensmodel}) played in the central regions\nof the cluster (see \\S~\\ref{sec:lensinterpa383}).  If the location of\na counter image is tentative, especially if there are several\npossibilities or an intervening cluster galaxy confuses the situation,\nthe system is not included in deriving the mass model.\n\\S~\\ref{sec:lensinterpms2137} and \\S~\\ref{sec:lensinterpa383} present\na detailed description of the final multiple image list adopted.\n\nOnce the multiple images are determined, the cluster mass model is\nrefined and perturber galaxy properties are fixed.  In general, a lens\nmass model will have both cluster and galaxy scale mass components.\nThe cluster scale mass component represents the DM associated with the\ncluster as a whole plus the hot gas in the intracluster medium.  In\nthe limit that the cluster DM halo is spherical (see\nEqn~\\ref{rho_ell}), its density profile has the form of\nEqn~\\ref{eqn:gnfw}.  In the adopted parameterization, the DM halo also\nhas a position angle ($\\theta$) and associated pseudo-ellipticity\n($\\epsilon$) (see Eqn~\\ref{sigma_ell} \\& \\ref{rho_ell}).\n\nGalaxy scale mass components are necessary to account for\nperturbations to the cluster potential that seem plausible based on\nthe HST imaging and are demanded by the observed multiple image\npositions.  These components are described by pseudo-isothermal\nelliptical mass distributions (PIEMD; \\citet{Kassiola93}).  Each PIEMD\nmass component is parametrized by its position ($\\mathrel{x_{\\rm c}}$, $\\mathrel{y_{\\rm c}}$),\nellipticity ($e$), position angle ($\\theta$), core radius ($\\mathrel{r_{\\rm core}}$),\ncut-off radius ($\\mathrel{r_{\\rm cut}}$) and central velocity dispersion ($\\mathrel{\\sigma_{\\rm o}}$).  The\nprojected mass density, $\\Sigma$, is given by:\n\n\\begin{equation}\\label{eq:piemd}\n\\Sigma(x,y){=}\\frac{\\mathrel{\\sigma_{\\rm o}}^2}{2G}\\,\\frac{\\mathrel{r_{\\rm cut}}}{\\mathrel{r_{\\rm cut}}{-}\\mathrel{r_{\\rm core}}}\\left(\\frac{1}{(\\mathrel{r_{\\rm core}}^2{+}\\rho^2)^{1\/2}}{-}\\frac{1}{(\\mathrel{r_{\\rm cut}}^2{+}\\rho^2)^{1\/2}}\\right)\n\\end{equation}\n\n\\noindent where\n$\\rho^2{=}[(x{-}\\mathrel{x_{\\rm c}})\/(1{+}e)]^2{+}[(y{-}\\mathrel{y_{\\rm c}})\/(1{-}e)]^2$\nand the ellipticity of the lens is defined as $e{=}(a{-}b)\/(a{+}b)$\n\\footnote{This quantity should not be confused with the quite\ndifferent definition used for the pseudo-elliptical generalized NFW\nprofile, see the Appendix.}.  The total mass of the PIEMD is thus\n$3\/2\\pi \\sigma_0^2 r_{\\rm cut}\/G$. In order to relate\nEquation~\\ref{eq:piemd} to the observed surface brightness of the BCG\nin particular, we take $\\Sigma=(M_*\/L)I$, where $M_*\/L$ is the stellar\nmass to light ratio and $I$ is the surface brightness, and find the\nfollowing relation\n\n\\begin{equation} \\label{eq:mlpiemd}\nM_*\/L = 1.50 \\pi\\mathrel{\\sigma_{\\rm o}}^{2}\\mathrel{r_{\\rm cut}} \/ (GL)\n\\end{equation}\n\n\\noindent where $L$ is the total luminosity of the BCG.  The $M_*\/L$\nof the central BCG will be used as a free parameter in our mass\nmodeling analysis.  Further details and properties of the truncated\nPIEMD model can be found in \\citet{Natarajan97} and\n\\citet{Limousin05}.\n\nThe relevant parameters of the perturber galaxies (position,\nellipticity, core radius, cutoff radius and position angle) are\nassumed to be those provided via examination of the HST imaging (see\n\\S~\\ref{sec:galgeom} for details). Only the central stellar velocity\ndispersion, $\\mathrel{\\sigma_{\\rm o}}$, is determined by optimization. At a particular\nstage in the process, the predicted multiple image positions are\ncompared with those observed, and a $\\chi^{2}$ value calculated (see\n\\S~\\ref{sec:stats}).  The iteration stops when a $\\chi^{2}$ minimum is\nreached.  The sole criteria for adding a perturber galaxy was\nwhether or not it was necessary for the lens model to match the\nmultiple image positions.  If adding an additional perturber did not\nalter our interim minimum $\\chi^{2}$, we did not include it in our\nsubsequent analysis.\n\nTwo special cases were encountered during the above procedure.\nFirst, one of the galaxy perturbers in Abell~383 required a larger\ncutoff radius ($r_{cut}$) than implied from the light distribution.\nAs is described in \\S~\\ref{sec:lensinterpa383}, this concentration is\nnecessary to account for several of the multiple image positions.  For\nthis mass concentration, not only do we determine the optimum\n$\\sigma_{0}$ parameter, but also the cutoff radius ($r_{cut}$).\n\nThe other special case concerns the BCG in both galaxy clusters.\nThese are assumed to be coincident with the center of the cluster DM\nhalo -- one justified by the co-location of the BCG and central X-ray\nisophotes \\citep{gps05,gavazzi03}.  The BCG mass distribution,\nrepresented by a PIEMD model, comprises only the stellar mass.  In\nthis case the HST imaging is used to fix the BCG ellipticity and\nposition angle, but since the measured stellar velocity dispersion is\nto be used as a constraint on the cluster mass profile, we leave the\ncentral velocity dispersion parameter (and hence the stellar $M_*\/L$,\nEqn~\\ref{eq:mlpiemd}) free {\\it in the lensing analysis}.  As the\nJaffe density profile is used for the BCG dynamical analysis (S04),\nthe PIEMD core and cutoff radius which best match the Jaffe surface\ndensity are adopted.  \\S~\\ref{sec:galgeom} discusses further the\nresults of the surface brightness matching between the Jaffe and PIEMD\nmodels in the clusters.\n\n\\subsection{Incorporating the Dynamical Constraints}\\label{sec:dynconsts}\n\nApart from the use of the pseudo-elliptical gNFW profile for the dark\nmatter component, the observational data and analysis methods adopted\nhere are identical to those used by S04. In that work, the observed\nvelocity dispersion profile of the BCG was interpreted via the\nspherical Jeans equation (see Appendix of S04) to assist in the\ndecomposition of the dark and baryonic mass components.  This portion\nof the $\\chi^{2}$ was calculated by comparing the expected velocity\ndispersion profile of the BCG (which depends on the mass enclosed at a\ngiven radius and the relative contribution of dark and luminous\nmatter) given a mass model with the observed velocity dispersion\nprofile, taking into account the effects of seeing and the long slit\nshape used for the observations. Ellipticity in the BCG and its dark\nmatter halo can be ignored as its effect on the velocity dispersion\nprofile will be small (e.g.~\\citet{Kronawitter00}).\n\nThe reader is referred to S04 for the observational details pertaining\nto the velocity dispersion profile, the surface brightness profile of\nthe BCG and the subsequent dynamical modeling to constrain the cluster\nDM inner slope.\n\n\\subsection{Statistical Methods}\\label{sec:stats}\n\nA $\\chi^{2}$-estimator is used to constrain the acceptable range of\nparameters compatible with the observational data. First we use the\nstrong lens model to calculate the likelihood of the lensing\nconstraints and then we combine it with a dynamical model to include\nthe kinematic information into the likelihood.\n\nThe first step is the strong lensing likelihood.  Once the lensing\ninterpretation is finalized and the perturbing galaxy parameters are\nfixed, the remaining free parameters are constrained by calculating a\nlensing $\\chi^2$ over a hypercube which encompasses the full range of\nacceptable models, modulo a prior placed on the dark matter scale\nradius (see \\S~\\ref{sec:rscprior}).  In Bayesian terms this\ncorresponds to adopting a uniform prior.  The lensing $\\chi^2$ value\nis calculated in the source plane identically to that of\n\\citep{gps05}.  For each multiply-imaged system, the source\nlocation for each noted image ($x_{model,i}$,$y_{model,i}$) is\ncalculated using the lens equation.  Since there should be only one\nsource for each multiple image set (with N images), the difference\nbetween the source positions should be minimized, hence:\n\n\\begin{equation}\n\\chi^2_{pos}{=}\\sum_{i{=}1}^{N}\\frac{(x_{model,i}{-}x_{model,i+1})^2{+}(y_{model,i}{-}y_{model,i+1})^2}{\\sigma_{S}^2}\n\\label{theory:chipos}\n\\end{equation}\n\n\\noindent $\\sigma_{S}$ is calculated by scaling the positional error\nassociated with a muliple image knot, $\\sigma_{I}$, by the\namplification of the source $A$ so that $\\sigma_{I}^{2}=A\n\\sigma_{S}^{2}$.  The following analysis will assume two different\npositional errors for the multiply imaged knots, using uncertainties\nof $\\sigma_{I}=$0\\farcs2 and 0\\farcs5, referred to hereafter as the\n`fine' and `coarse' analyses, respectively. The case for each is\njustified below.\n\n\n\n\nThe finer 0\\farcs2 error bar corresponds to the uncertainty in the\nmultiple image knot positions as defined by the resolution and pixel\nsize of the HST\/WFPC2 images.  Excellent strong lensing fits are\nachieved with the finer 0\\farcs2 error bar ($\\chi^{2}\/dof\\sim 1$), so\nthat technically there is no need for increased uncertainties.  The\nuncertainty is dominated by the spatial extension of the multiple\nimage knots employed and the ability to identify surface brightness\npeaks.\n\nIn contrast to our ability to match the image positions down to the\nresolution of the HST\/WFPC2 images, recent combined strong and weak\nlensing analyses of Abell 1689 have been unable to do so\n(e.g.~\\citet{Broadhurst05,Halkola06,Limousin06}).  Although Abell 1689\nis a more complex cluster than those studied here, it does have the\nmost identified multiple images of any other cluster to date, and so\ncan probe the overall mass profile on smaller scales and with many\nmore constraints.  As espoused in \\S~1, real galaxy clusters are\ncomplex systems that are likely not easily parameterized by simple\nmass models, and as the number and density of mass probes increases\nthe more refined and complete the mass model necessary to match the\ndata.  In our case, where we have relatively few mass profile\nconstraints (at least in comparison to Abell 1689), we adopt a coarser\n0\\farcs5 positional error which allows us to account for complexities\nin the actual mass distribution of our clusters that our small number\nof mass probes are insensitive to.  This, plus the fact that we\ncarefully chose our perturbing galaxies such that a lensing\n$\\chi^{2}\/dof\\sim1$ was found should account for reasonable situations\nwhere we have missed an interesting perturber galaxy.  By adopting\ntoo small a multiple image position uncertainty, the region in\nparameter space explored may be overly confined (such that, for\nexample, the observed BCG velocity dispersion profile cannot be\nreproduced).\n\n\nThe strong lensing analysis is performed with 5 free parameters,\nanalogous to those adopted in S04.  These are the dark matter inner\nlogarithmic slope ($\\beta$), the pseudo-ellipticity of the potential\n($\\epsilon$), the amplitude of the DM halo ($\\delta_{c}$), the dark\nmatter scale radius ($r_{sc}$), and the mass-to-light ratio of the BCG\n($M_*\/L$).  We choose to place a uniform prior on the dark matter\nscale radius, ($r_{sc}$) based on past mass profile analyses of these\nclusters and results from CDM simulations in order to reduced\ncomputation time (see \\S~\\ref{sec:rscprior}).  In practice, to\nevaluate the $\\chi^2_{pos}$ at each point in the hypercube, the\npseudo-ellipticity of the cluster dark matter halo is optimized while\nsimply looping over the remaining free parameters.\n\nOnce the strong lensing $\\chi^{2}$ values are computed, attention is\nturned to the dynamical data.  In contrast to the strong lensing\nmodel, the dynamical model is spherically symmetric and follows that\npresented by S04, with the $\\chi^{2}$ value being:\n\n\\begin{equation}\n\\chi^{2}_{\\sigma}{=}\\sum_{i{=}1}^{N}\\frac{(\\sigma_{i,obs}{-}\\sigma_{i,model})^{2}}{\\Delta_{i}^{2}}\n\\label{eqn:chisigma}\n\\end{equation}\n\n\\noindent where $\\Delta_{i}$ is the uncertainty in the observed\nvelocity dispersion measurements.  \n\nThe lensing and velocity dispersion $\\chi^{2}$ values are summed, \nallowing for standard marginalization of nuisance parameters and \nthe calculation of confidence regions.\n\n\\subsection{Dark matter scale radius prior}\\label{sec:rscprior}\n\nAs mentioned in the previous sections, in order to limit computation\ntime, a prior was placed on the dark matter scale radius\n$r_{sc}$. This is justified both on previous mass profile analyses of\nthese clusters and the results of CDM simulations.\n\nAn array of CDM simulations has provided information not only on the\ninner dark matter density profile, but on the expected value of the\nscale radius, $r_{sc}$, and its intrinsic scatter at the galaxy\ncluster scale (e.g.~\\citealt{Bullocketal01,Tasitsiomi04,Dolag04}).\nFor example, \\citet{Bullocketal01} found that dark matter halos the\nsize of small galaxy clusters have scale radii between 240 and 550 kpc\n(68\\% CL).  \\citet{Tasitsiomi04}, using higher resolution simulations\nwith fewer dark matter halos found $r_{sc}$ of 450$\\pm$300 kpc.\nFinally, \\citet{Dolag04} studied the DM concentrations of galaxy\nclusters in a $\\Lambda CDM$ cosmology and found a typical range of\nscale radii between $r_{sc}$ of 150 and 400 kpc.  These results\nrepresent a selection of the extensive numerical work being done on\nthe concentration of dark matter halos.\n\nPrevious combined strong and weak lensing analyses of MS2137-23 have\nprovided approximate values for the scale radius\n\\citep{gavazzi03,Gavazzi05}.  \\citet{gavazzi03} found a best fitting\nscale radius of $\\sim$130 kpc (and hints that the scale radius may be\nas low as $\\sim$70 kpc from their weak lensing data) for their\nanalysis of MS2137-23.  A more recent analysis \\citep{Gavazzi05} found a\nbest fitting radius of $\\sim$170 kpc. Similarly for Abell 383, a\nrecent X-ray analysis found a best-fitting dark matter scale radius of\n$\\sim130$ kpc (Zhang et al. 2007).\n\nTaking these observational studies into account, we chose a uniform\nscale radius prior between 100 and 200 kpc for MS2137-23 and Abell\n383, both for simplicity and to bracket the extant observational\nresults which often have constraints at larger radii (and thus\nconstrain the scale radius better) than the current work.  It is worth\nnoting that the extant observations of these two clusters indicate a\nscale radius which is on the low end of that predicted from CDM\nsimulations.  For a fixed virial mass, a smaller scale radius\nindicates a higher concentration, $c=r_{vir}\/r_{sc}$.  This could be\ndue to the effect of baryonic cooling, which could increase halo\nconcentration (as well as inner slope perhaps). It has been suggested\nthat those halos with the highest concentration (again for a fixed\nmass) are those that are the oldest and with the least substructure,\nproviding more indirect evidence that we have chosen 'relaxed' galaxy\nclusters to study \\citep{Zentner05}. We briefly explore the\nconsequences of changing our assumed scale radius prior range in\n\\S~\\ref{sec:highrad}.\n\n\\section{Application to Data}\\label{sec:obsresults}\n\nWe now turn to the observational data for MS2137-23 and Abell 383\nand describe our methods for analyzing these in the context\nof lensing input parameters.\n\n\\subsection{BCG and Perturber Galaxy Parameters}\\label{sec:galgeom}\n\nIn order to fix the position angle and ellipticity of the perturber\ngalaxies and BCG components, the IRAF task {\\sc ellipse} is used to\nestimate the surface brightness profile at typically the effective\nradius.  The measured parameters are fixed in the lensing\nanalysis. The galaxy position, core radius ($\\mathrel{r_{\\rm core}}$) and cutoff radius\n($\\mathrel{r_{\\rm cut}}$) are each chosen to match those fitting the photometry\n(Table~\\ref{tab:lensmodel}).  For perturbing galaxies, this leaves\nonly the PIEMD parameter velocity dispersion ($\\mathrel{\\sigma_{\\rm o}}$) which must be\nadjusted to match the multiple imaging constraints, as explained in\n\\S~\\ref{sec:massmodel}.\n\nFor the BCG, following S04, it is preferable to use the Jaffe stellar\ndensity profile for the dynamical analysis since this function\nprovides an analytic solution to the spherical Jeans equation.\nHowever, the PIEMD model implemented in {\\sc lenstool} offers numerous\nadvantages for the lensing analysis. To use the most advantageous\nmodel in each application, a correspondence is established between the\ntwo by fitting with a PIEMD model the Jaffe surface brightness fit\npresented by S04.  An appropriate combination of the PIEMD $\\mathrel{r_{\\rm core}}$ and\n$\\mathrel{r_{\\rm cut}}$ model parameters matches the Jaffe profile found by S04 with no\nsignificant residuals (PSF smearing was also taken into account).\nTable~\\ref{tab:lensfixed} lists the PIEMD parameters used for our\nlensing analysis, as well as the Jaffe profile parameters used by S04.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Fixed Parameters in Abell 383 and MS2137-23 Lens Model\\label{tab:lensfixed}}\n\\begin{tabular}{lcccccccc}\n\\tableline\\tableline\nCluster&Parameters& $x_{c}$&$y_{c}$& b\/a & $\\theta$&$r_{core}$&$\\sigma_{0}$& $r_{cut}\/R_{e}$\\\\\n&&(arcsec)&(arcsec)&&(deg)&(kpc)&($km s^{-1}$)&(kpc)\\\\\n\\tableline\nMS2137&Cluster-scale DM halo&0.0&0.0&Free&5.0 & -&-&-\\\\\n&${\\rm BCG_{PIEMD}}$&0.0&0.0&0.83&17.75&5$\\times10^{-6}$&Free&22.23\\\\\n&${\\rm BCG_{Jaffe}}$&0.0&0.0&-&-&-&-&24.80\\\\\n&Galaxy Perturber&16.2&-5.46&0.66&159.9&0.05&173.0&4.8\\\\\nAbell 383&Cluster-scale DM halo&0.0&0.0&Free&104.3&-&-&-\\\\\n&${\\rm BCG_{PIEMD}}$&0.0&0.0&0.90&107.2&3$\\times10^{-6}$&Free&25.96\\\\\n&${\\rm BCG_{Jaffe}}$&0.0&0.0&-&-&-&-&46.75\\\\\n&Perturber 1&14.92&-16.78&0.804&-20.9&0.67&230.0&26.98\\\\\n&Perturber 2&9.15&-1.92&0.708&10.3&0.51&140.0&10.79\\\\\n&Perturber 3&0.17&-24.26&0.67&65.2&0.24&124.8&9.10\\\\\n&Perturber 4&-4.10&-13.46&0.645&27.7&0.17&125.7&2.19\\\\\n\\tableline\n\\end{tabular}\n\\tablecomments{The position angle, $\\theta$ is measured from North\ntowards East.  The DM halo is parameterized with the pseudo-gNFW\nprofile.  All other mass components are parameterized by a PIEMD\nmodel. Note that $R_{e}=0.76r_{jaffe}$ \\citep{Jaffe83}.}\n\\end{center}\n\\end{table*}\n\n\n\n\n\\subsection{MS2137-23 Lens Model}\\label{sec:lensinterpms2137}\n\nThe strong lensing properties of MS2137-23 have been studied\nextensively by many workers\n\\citep{Mellier93,Miralda95,Hammer97,gavazzi03,Gavazzi05}.  The most\ndetailed model \\citep{gavazzi03} used 26 multiply-imaged knots from\ntwo different background sources.  The model adopted here is more\nconservative and based only on those multiple images confirmed via\nspectroscopy or suggested on the grounds of surface brightness or\ninterim lens models.  Despite having some multiple images in common\nwith \\citet{gavazzi03}, we have retained our own nomenclature.\n\nFollowing \\citet{Sand02}, the tangential and radial arcs arise from\nseparate sources, at $z=1.501$ and $z=1.502$, respectively.  The\nmultiple image interpretation is detailed in Figure~\\ref{fig:mulplot}\nand Table~\\ref{tab:lensinterp}.  There are two separate features (1\nand 3) on the source giving rise to the tangential arc which is\nmultiply-imaged four and three times respectively.  It has not been\npossible to confidently locate the fourth image of feature 3, since it\nis adjacent to the perturbing galaxy.  Also, it is expected that a\nfifth, central image would be associated with the giant tangential\narc.  Although the position of this fifth central image has been\ntentatively reported \\citep{gavazzi03}, we do not include it in our\nmodel because we were unable to clearly identify it due to BCG\nsubtraction residuals.  Two images of the source giving rise to the\nradial arc were also identified.  The mirror image of feature 2a\nnearer the center of the BCG could not be recovered, most likely\nbecause of residuals arising from the subtraction of the BCG.\n\n\\begin{figure*}  \n\\begin{center}\n\n\\mbox{\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f1a.eps}}\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f1b.eps}}\n}\n\\mbox{\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f1c.eps}}\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f1d.eps}}\n}\n\n\n\\caption[Multiple image interpretation of the cluster\nMS2137.]{\\label{fig:mulplot} Multiple image interpretation of the\ncluster MS2137-23.  The exact positions used are shown in\nTable~\\ref{tab:lensinterp}.  Three sets of multiple images are\nidentified, one with the radial arc system (2a \\& 2b) and two with the\ntangential arc system (1abcd \\& 3abc).  The perturbing galaxy is the\nelliptical S0 next to the lensed feature 1b. }\n\\end{center}\n\\end{figure*}\n\nAs in \\citet{gavazzi03}, only one perturbing galaxy is included in the\nlens model (see Table~\\ref{tab:lensfixed}). For this system, the\nbest-fitting $\\chi^{2}$ value occurred near $\\mathrel{\\sigma_{\\rm o}}$=173 km~$\\tx{s}^{-1}$.\n\n\nIn the initial modeling of MS2137-23, some experimentation was\nundertaken with different cluster DM ellipticities and position\nangles.  While some variation in ellipticity is permitted by the\nlensing interpretation, a robust position angle offset was detected\nbetween the BCG and that of the DM halo of $\\Delta \\theta \\sim$13\ndegrees, in agreement with \\citet{gavazzi03}.  In the following,\nresults are presented with the DM position angle fixed at $\\theta$=5\ndegrees.  This optimal position angle was determined during the\ninitial lens modeling process by fixing all cluster mass parameters to\nvalues corresponding to a model with $\\chi^{2}\/dof\\sim 1$ and letting\nthe DM position angle vary until a $\\chi^{2}$ minimum was reached.\nAs a consistency check, we repeated our calculations with a\nfixed DM position angle of 4.0 and 6.0 degrees.  Varying the DM\nposition angle had very little effect on our other parameter\nconstraints, but results in a slightly larger overall $\\chi^{2}$\n(lensing + velocity dispersion profile; $\\Delta \\chi^{2} <$1) value.\nFor this reason, we only present our DM position angle of 5 degree\nresults.\n\n\n\n\\subsection{Abell 383 Lens Model}\\label{sec:lensinterpa383}\n\nDetailed lens models for Abell 383 have been published in\n\\citet{gps01,gps05}, which we will largely adopt in this work.\nMultiple image sets 1 and 2 are based on the in-depth lensing\ninterpretation of \\citet{gps05}.  The reader is referred to that work\nfor a detailed description of this radial and tangential gravitational\narc.  Multiple image sets 3, 4, 5 and 6 (for which there are no\nspectroscopic redshifts, but for which their distinctive morphology is\nreassuring) are included largely to constrain the properties of\nperturbing galaxies 1, 3, and 4 (see Fig~\\ref{fig:mulplota383} and\nTable~\\ref{tab:lensinterp}).  Since these images have no spectroscopic\nconfirmation, a redshift $z\\sim$3 was assumed; the mass model is very\ninsensitive to the exact choice.\n\n\\begin{figure*}  \n\\begin{center}\n\\mbox{\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f2a.eps}}\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f2b.eps}}\n}\n\\mbox{\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f2c.eps}}\n\\mbox{\\epsfysize=6.2cm \\epsfbox{f2d.eps}}\n}\n\\caption[Multiple image interpretation of the cluster Abell\n383.]{\\label{fig:mulplota383} Multiple image interpretation of the\ncluster Abell 383.  The exact positions used are shown in\nTable~\\ref{tab:lensinterp}.  In all figures except for the top left we\nhave subtracted cluster galaxies in order to more clearly see the\nmultiple image features. }\n\\end{center}\n\\end{figure*}\n\n\nThe Abell 383 cluster mass model is more complex than that for\nMS2137-23, but only in the sense that there are more perturbing\ngalaxies that must be put into the mass model to match the image\npositions.  The bright cluster elliptical southwest of the BCG (\\#2 in\n\\citet{gps05}; Perturber \\#1 in this work) requires a DM halo more\nextended than the light, as mentioned in \\S~\\ref{sec:massmodel}.\nAfter some iteration, it was found that the parameters of this\nimportant perturber could be fixed to those listed in\nTable~\\ref{tab:lensfixed}.  Multiple image sets 3, 4 and 5 play a\ncrucial role (see Table~\\ref{tab:lensinterp}) in constraining the\nperturber. Although other perturbing galaxies were added, none has a\ncomparable effect on the lensing $\\chi^2$.  Table~\\ref{tab:lensfixed}\nprovides the full model parameter list.\n\nA slight ($\\sim$3 degree) offset between the position angle of the BCG\nand the cluster DM halo was noted.  We found the best-fitting DM\nposition angle in the same way as in MS2137-23\n(\\S~\\ref{sec:lensinterpms2137}).  The position angle of the DM halo\nwas kept fixed but the ellipticity was left as a free parameter. As in\nMS2137-23, we also reran our analysis with a DM position angle of\n103.3 and 105.3, but only present the results with a DM PA of 104.3.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Multiple Image Interpretation of MS2137 and Abell 383\\label{tab:lensinterp}}\n\\begin{tabular}{lcccc}\n\\tableline\\tableline\nCluster&Multiple Image& $x_{c}$&$y_{c}$&Redshift\\\\\n&ID&(arcsec)&(arcsec)\\\\\n\\tableline\nMS2137&1a&6.92&-13.40&1.501\\\\\n&1b&12.40&-7.94&1.501\\\\\n&1c&0.07&19.31&1.501\\\\\n&1d&-11.57&-7.49&1.501\\\\\n&2a&3.96&-5.51&1.502\\\\\n&2b&-8.01&22.10&1.502\\\\\n&3a&5.16&-14.68&1.501\\\\\n&3b&0.11&18.91&1.501\\\\\n&3c&-12.30&-6.74&1.501\\\\\nAbell 383&1A&-1.74&2.56&1.0\\\\\n&1B&-1.03&1.20&1.0\\\\\n&1C&16.37&-4.03&1.0\\\\\n&2A&7.00&-14.01&1.0\\\\\n&2B&8.23&-13.20&1.0\\\\\n&2C&14.11&-8.19&1.0\\\\\n&3A&5.88&-22.02&3.0\\*\\\\\n&3B&14.69&-14.68&3.0\\*\\\\\n&3C&16.49&-14.39&3.0\\*\\\\\n&4A&8.35&-23.96&3.0\\*\\\\\n&4B&17.45&-17.28&3.0\\*\\\\\n&4C&17.92&-15.43&3.0\\*\\\\\n&5A&6.64&-21.75&3.0\\*\\\\\n&5B&16.98&-14.09&3.0\\*\\\\\n&6A&7.05&-21.75&3.0\\*\\\\\n&6B&17.27&-14.17&3.0\\*\\\\\n\\tableline\n\\end{tabular}\n\\tablecomments{Multiple Image Interpretation.  All image positions are with\nrespect to the BCG center.}\n\\end{center}\n\\end{table*}\n\n\n\\section{Results}\\label{sec:stronglens}\n\nWe now analyze the refined 2D lens models of MS2137-23 and Abell 383\ntogether with the velocity dispersion profiles presented in S04.  We\npresent our analysis with both a multiple image position uncertainty\nof 0\\farcs2 and 0\\farcs5, described as the fine and coarse fits in\n\\S~\\ref{sec:stats}.  The results are summarized in\nTable~\\ref{tab:lensmodel} and Figures~\\ref{fig:ms2137rscprior} and\n\\ref{fig:a383rscprior}.\n\n\\subsection{MS2137-23}\n\nFigure~\\ref{fig:ms2137rscprior} and the discussion below summarizes\nthe results for the fine and coarse positional cases. One thing to\nnote is the number of degrees of freedom involved, i.e. the difference\nbetween the number of constraints and the number of free parameters,\nin order to quantify the goodness of fit.  The mass model has 5 free\nparameters, as detailed in \\S~\\ref{sec:lensinterpms2137}: the DM inner\nslope $\\beta$, the DM pseudo-ellipticity $\\epsilon$, the DM amplitude\n$\\delta_{c}$, the BCG stellar $M_{*}\/L$, and the dark matter scale\nradius $r_{sc}$ which is allowed to vary in the 100-200 kpc range.\nConsidering Table~\\ref{tab:lensinterp}, the multiple images provide 12\nconstraints, while the velocity dispersion data provides 8, giving a\ntotal of 20 data constraints.  The resulting number of degrees of\nfreedom is thus 15.\n\n\n\\begin{figure*}\n\\begin{center}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f3a.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f3b.eps}}\n}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f3c.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f3d.eps}}\n}\n\\caption{The combined lensing+dynamics results for the cluster\n  MS2137-23.  The top row summarizes the results for the 0\\farcs2\n  lensing position uncertainty scenario while the bottom row\n  encapsulates the 0\\farcs5 scenario.  Top Left--Lensing+dynamics\n  likelihood contours (68\\%,95\\%, and 99\\%) in the $M\/L-\\beta$ plane\n  after marginalizing over the other free parameters with the 0\\farcs2\n  lensing multiple image uncertainty.  Top Right-- Best fitting\n  velocity dispersion profile from the combined lensing+dynamics\n  analysis with the 0\\farcs2 lensing multiple image uncertainty.  No\n  models could be found that fit both the lensing and observed\n  velocity dispersion constraints.  Bottom Left -- Lensing+dynamics\n  likelihood contours (68\\%,95\\%, and 99\\%) with the 0\\farcs5 lensing\n  multiple image uncertainty in the $M\/L-\\beta$ plane after\n  marginalizing or optimizing over the other free parameters.  Bottom\n  Right -- Best fitting velocity dispersion profile from the combined\n  lensing+dynamics analysis with the 0\\farcs5 lensing multiple image\n  uncertainty.  While the best-fitting model velocity dispersion is a\n  better fit to the data than in the 0\\farcs2 lensing scenario, it\n  still cannot reproduce the observed decline in the velocity\n  dispersion profile in our highest radial bins -- suggesting a\n  problem with our current mass model parameterization.\n\\label{fig:ms2137rscprior}}\n\\end{center}\n\\end{figure*}\n\n\\subsubsection{The fine positional accuracy lensing case}\n\nThe two top panels of Figure~\\ref{fig:ms2137rscprior} and the\nappropriate line in Table~\\ref{tab:lensmodel} encapsulate the results\nof the fine positional analysis.  DM inner slopes between $\\beta =\n0.65 - 1.0$ lie within the 68\\% confidence limit (after marginalizing\nover all other free parameters), although the best fitting DM scale\nradius sits at the edge of the allowed prior ($r_{sc,best}$=200 kpc).\nA scale radius of 200 kpc is higher than that seen in previous lensing\nanalyses of MS2137-23 that have used a similar mass parameterization\nto our own (e.g.\\citet{gavazzi03,Gavazzi05}), so there is no case to\nalter the prior.\n\nThe parameter constraints are not particularly tight because the total\n$\\chi^2\\sim$54, larger than expected given the number of degrees of\nfreedom.  Such a value may indicate that the form of the mass profile\nused in the fit is inappropriate. Indeed, the model velocity\ndispersion profile is a poor match to that observed\n(Figure~\\ref{fig:ms2137rscprior}).  In fact, if the BCG velocity\ndispersion results are ignored, acceptable lens models ($\\chi^{2}\/dof\n\\lesssim 1$) can be recovered with a variety of inner DM slopes, scale\nradii and BCG stellar $M\/L$, although these parameters have correlated\nvalues.  We postpone discussion of the possible reasons for this\nmismatch until later.\n\n\\subsubsection{The coarse positional accuracy lensing case}\n\nThe two bottom panels of Figure~\\ref{fig:ms2137rscprior}, along with\nTable~\\ref{tab:lensmodel} summarize our results for the coarse\npositional analysis.  DM inner slopes between $\\beta =$0.4-0.75 lie\nwithin our 68\\% CL, and we again find DM scale radii at the high end\nof our prior range, as expected.  The shift in the BCG M\/L vs. DM\ninner slope contour fine positional case indicates that the increased\nparameter space in the lensing models has led to a slightly improved\nvelocity dispersion profile (bottom right panel of\nFigure~\\ref{fig:ms2137rscprior}). The overall $\\chi^{2}\\simeq$31 is\nimproved, although the probability for 15 degrees of freedom is less\nthan 1\\%, assuming measurements are governed by Gaussian\nstatistics. Thus the model remains a marginal fit to the data.\n\n\\subsubsection{Comparison with Gavazzi 2005}\n\nWe now briefly compare our results with those of \\citet{Gavazzi05}.\nGavazzi's analysis used a similar strong lensing model to our own, \nincluding what we consider to be somewhat less secure multiple\nimages. However, he extended the analysis to larger scales including\nweak lensing data and incorporated the BCG velocity dispersion profile \npresented by S04.  Gavazzi adopted a strict NFW profile for the cluster \nDM halo and a Hernquist profile for the stellar component of the BCG.  \n\nDespite these differences, Gavazzi's conclusions are very similar to\nthose of the present paper. Models with NFW-like DM haloes (regardless\nof whether the inner slopes were varied) were uniformly poor fits to\nthe observational data.  In particular, the falling velocity\ndispersion profile observed at $R \\gtrsim 5$kpc cannot be reproduced,\ndespite experimenting with the effect of anisotropic orbits in the\nstellar distribution.  A major conclusion of Gavazzi's study is that\nhalo triaxiality, an effect not typically included, may play an\nimportant role in the central regions of galaxy clusters.  We will\nreturn to this topic in \\S~\\ref{sec:ac}.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Acceptable Parameter Range\\label{tab:lensmodel}}\n\\begin{tabular}{lcccccccc}\n\\tableline\\tableline\nPrior Setup&Cluster& Inner DM Slope& $\\epsilon$& $\\delta_{c}$ & $r_{sc}$ & $M_*\/L_{V}$ \\\\\n&& $\\beta$ & & & (kpc) &\\\\\n\\tableline\n100 - 200 kpc $r_{sc}$ Prior&\\\\\n$\\sigma_{lens}=$0\\farcs2& MS2137 & $0.95^{+0.05}_{-0.30}$&$0.08^{+0.01}_{-0.01}$&$29420^{+98310}_{-1760}$&$200^{-42}$&$1.58^{+0.52}_{-0.635}$\\\\\n& Abell 383 & $0.55^{+0.2}_{-0.05}$&$0.08^{+0.01}_{-0.02}$&$140000^{+8500}_{-60600}$&$100^{+28}$& $2.4^{+0.42}_{-0.42}$\\\\\n$\\sigma_{lens}=$0\\farcs5& MS2137 & $0.6^{+0.15}_{-0.2}$&$0.06^{+0.01}_{-0.01}$&$44600^{+35500}_{-7175}$&$200^{-31}$&$2.45^{+0.75}_{-0.65}$\\\\\n& Abell 383 & $0.45^{+0.2}_{-0.25}$&$0.06^{+0.03}_{-0.01}$&$156000^{+38500}_{-67150}$&$100^{+21}$& $2.34^{+1.02}_{-0.54}$\\\\\n\\tableline\n\\end{tabular}\n\\tablecomments{Best-fitting parameters and\/or confidence limits for\nthe different prior scenarios present in this paper.   }\n\\end{center}\n\\end{table*}\n\n\\subsection{Abell 383}\n\nOur results for Abell 383 are shown in Figure~\\ref{fig:a383rscprior}\nand Table~\\ref{tab:lensmodel} for both the fine and coarse positional\nuncertainty cases.  We discuss each separately below.\nCalculating the number of degrees of freedom in a similar way to that\ndone for MS2137-23, we again have 5 free parameters in our mass model.\nConsidering Table~\\ref{tab:lensinterp}, multiple images provide 16\nconstraints, taking into account that those related to multiple image\nsets 3,4,5 \\& 6 do not have known redshift information.  The velocity\ndispersion data provide 3 additional constraints.  Thus, the\nresulting number of degrees of freedom is 14.\n\n\\subsubsection{The fine positional accuracy lensing case}\n\nThe top two panels of Figure~\\ref{fig:a383rscprior} and the\nappropriate line in Table~\\ref{tab:lensmodel} summarize the results in\nthis case.  DM inner slopes between $\\beta = 0.5 - 0.7$ lie within the\n68\\% confidence limit of our analysis (after marginalizing over all\nother free parameters).  The best fitting scale radius sits again at\nthe edge of the allowed $r_{sc}$ prior range ($r_{sc}$=100 kpc).  An\nX-ray analysis of Abell 383, which was able to probe to higher radius\nthan the current analysis, indicates that the DM scale radius is well\nabove 100 kpc (Zhang et al. 2007). For these reasons, and those\ndiscussed earlier, there is no case for adjusting the DM scale radius\nprior.\n\nThe total $\\chi^{2}$=40.4, high given the 14 degrees of freedom in\nthe analysis.\n\n\\subsubsection{The coarse positional accuracy lensing case}\n\nThe two bottom panels of Figure~\\ref{fig:a383rscprior}, along with\nTable~\\ref{tab:lensmodel} summarize the results for the coarse\npositional accuracy case.  DM inner slopes between $\\beta =$0.2 - 0.65 \nlie within our 68\\% CL, along with a best fitting DM scale radius of \n100 kpc.  Our parameter constraints encompass the values found in \nthe fine accuracy case with no shift in parameter space (unlike the\ncase for MS2137-23).  This suggests that although we should expect \na lower $\\chi^{2}$ due to the increased uncertainties allowed, no \nsignificant improvement to the best-fitting velocity dispersion profile \nshould be expected.  As we can see in the bottom right panel of\nFigure~\\ref{fig:a383rscprior}, the best fitting velocity dispersion\nprofile is very similar to that obtained in the fine case.\nThe total $\\chi^{2}$=22, acceptable given 14 degrees of freedom.\n\n\\begin{figure*}\n\\begin{center}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f4a.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f4b.eps}}\n}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f4c.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f4d.eps}}\n}\n\\caption{The combined lensing+dynamics results for the cluster Abell 383.\n  The top row summarizes the results for the 0\\farcs2 lensing position\n  uncertainty scenario while the bottom row encapsulates the 0\\farcs5\n  scenario.  Top Left--Lensing+dynamics likelihood contours\n  (68\\%,95\\%, and 99\\%) in the $M\/L-\\beta$ plane with the 0\\farcs2\n  lensing multiple image uncertainty after marginalizing over the\n  other free parameters.  Top Right-- Best fitting velocity dispersion\n  profile from the combined lensing+dynamics analysis with the\n  0\\farcs2 lensing multiple image uncertainty.  Bottom Left --\n  Lensing+dynamics contours (68\\%,95\\%, and 99\\%) in the $M\/L-\\beta$\n  plane with the 0\\farcs5 lensing multiple image uncertainty after\n  marginalization over the other free parameters.  Bottom Right --\n  Best fitting velocity dispersion profile from the combined\n  lensing+dynamics analysis with the 0\\farcs5 lensing multiple image\n  uncertainty.  The 0\\farcs5 lensing multiple image scenario provides\n  a better overall fit to the observations, although we are limited by\n  the relatively poor quality of the observed Abell 383 velocity dispersion\n  profile.\n\\label{fig:a383rscprior}}\n\\end{center}\n\\end{figure*}\n\n\\section{Discussion}\\label{sec:systematics}\n\nIn the previous section we have presented the results of our analysis,\nwhich showed that a mass model comprising a stellar component for the\nBCG following a Jaffe profile together with a generalized NFW DM\ncluster halo is able to adequately reproduce the observations for\nAbell 383 (albeit {\\it only} for the coarse lensing positional\naccuracy scenario) but is unable to simultaneously reproduce the\nobserved multiple image configuration and BCG velocity dispersion\nprofile for MS2137-23.  In the case of Abell 383, the inner DM profile\nis flatter than $\\beta$=1, supporting the earlier work of S04.  This\nindicates that at least some galaxy clusters have inner DM slopes\nwhich are shallower than those seen in numerical simulations --\nbut only if the mass parameterization used in the current work is\nreflective of reality.  Further work in this interesting cluster using\nother observational probes will further refine the mass model, \nand determine if the generalized NFW DM form is a good fit to the\ncluster profile.\n\nIn this section we discuss systematic uncertainties in\nour method and possible refinements that could be made to\nreconcile the mass model with the observations for MS2137-23.  \nWe hope that many of these suggestions will become important \nas cluster mass models improve and thus will present \nfruitful avenues of research.\n\n\\subsection{Systematic Errors}\n\nWe focus first on systematic errors associated particularly with the\ntroublesome stellar velocity dispersion profile for MS2137-23.  Errors\ncould conceivably arise from (i) significant non-Gaussianity in the\nabsorption lines (which are fit by Gaussians), (ii) uncertain\nmeasurement of the instrumental resolution used to calibrate the\nvelocity dispersion scale, and (iii) template mismatch.\n\nNon-gaussianity introduces an error that we consider too small to\nsignificantly alter the goodness of fit \\citep{Gavazzi05}. The\ninstrumental resolution of ESI (the Keck II instrument used to measure\nthe velocity dispersion profile; \\citet{Sheinis02}) is $\\sim$30 km~$\\tx{s}^{-1}$;\nthis is much smaller than the measured dispersion. Even if the\ninstrumental resolution was in error by a factor of two, the\nsystematic shift in $\\sigma$ would only be 3 km~$\\tx{s}^{-1}$ (using Eq~3 in Treu\net al.\\ 1999). This would affect all measurements and not reverse the\ntrend with radius.\n\nConcerning template mismatch, S04 estimated a possible systematic\nshift of up to 15-20 km~$\\tx{s}^{-1}$ . This could play a role especially as the\nsignal to noise diminishes at large radii, where the discrepancy with\nthe model profile is greatest. To test this hypothesis, we added 20\nkm~$\\tx{s}^{-1}$ in quadrature to only those velocity dispersion data points in\nMS2137-23 at $R > 4$ kpc and recalculated the best-fitting $\\chi^{2}$\nvalues. $\\chi^{2}$ is reduced from 31 to 28.8, a modest reduction\nwhich fails to explain the poor fit.\n\nAlthough selectively increasing the error bars on those data points\nmost discrepant with the model is somewhat contrived, our result does\nhighlight the need for high S\/N velocity dispersion measures out to\nlarge radii.  A high quality velocity dispersion profile has been\nmeasured locally for Abell 2199 to $\\sim$20 kpc \\citep{Kelsonetal02}.\nInterestingly, these high S\/N measures display similar trends to those\nfor MS2137-23 in the overlap regime, i.e. a slightly decreasing\nprofile at $R\\lesssim 10$kpc.  The dip witnessed in MS2137-23 is thus\nnot a unique feature, although with deeper measurements we might\nexpect to see a rise at larger radii as a result of the shallow DM\nprofile.\n                                                                 \nA final potential limitation in the dynamical analysis is the assumption\nof orbital isotropy. Both S04 and \\citet{Gavazzi05} explored the \nconsequences of mild orbital anisotropy, concluding a possible offset \nof $\\Delta \\beta \\sim0.15$ might result.  Even including orbital\nanisotropy into his analysis, \\citet{Gavazzi05} was unable to fit the \nobserved velocity dispersion profile.  \n\nSince we determine our lensing $\\chi^{2}$ values in the source\nplane, we checked to make sure that no extra images were seen after\nremapping our best-fit lensing + velocity dispersion models back to the\nimage plane. No unexpected images were found, although several images\nthat were explicitly not used as constraints were found, such as the\nmirror image of radial arc image 2a in MS2137 and the complex of\nmultiple images associated with 3abc, 5ab, and 6ab in Abell 383 (see\nFigures~\\ref{fig:mulplot} and \\ref{fig:mulplota383}).  As discussed in\n\\S~\\ref{sec:lensinterpms2137} and \\ref{sec:lensinterpa383}, some of\nthese multiple images were not used as constraints because we could\nnot confidently identify their position either due to galaxy\nsubtraction residuals or blending with other possible multiple image\nsystems.  \n\n\\begin{figure*}\n\\begin{center}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f5a.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f5b.eps}}\n}\n\\mbox{\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f5c.eps}}\n\\mbox{\\epsfysize=4.5cm \\epsfbox{f5d.eps}}\n}\n\\caption{Confidence contours (68\\%,95\\%, and 99\\%) when we allow\n  the dark matter scale radius to be fixed at values a factor of two\n  beyond our observationally motivated prior.  Top Row -- Contours\n  when we fix the dark matter scale radius to $r_{sc}$=50 and 400 kpc\n  in MS2137.  Although the $r_{sc}$=400 kpc scenario provides a\n  relatively good fit to the data ($\\chi^{2}\\sim$26), this value for\n  the scale radius is much larger than that observed from weak lensing\n  data.  The $r_{sc}$=50 kpc scenario is a significantly worse fit to\n  the data, with $\\chi^{2}\\sim$39.  Note that the DM inner slope is\n  $\\beta < 1$ in both scenarios.  Bottom Row -- Contours when we fix\n  the dark matter scale radius to $r_{sc}$=50 and 400 kpc in A383.\n  The large discrepancy in inner slope values obtained emphasize the\n  need for a mass probe at larger radii.  The best-fitting model for\n  either fixed scale radius is significantly worse than the\n  best-fitting $r_{sc}$=100 kpc result ($\\chi^{2}\\sim$26.5 and 31.3\n  for $r_{sc}$=50 and 400 kpc respectively).\n\\label{fig:diffrsc}}\n\\end{center}\n\\end{figure*}\n\nWe finally comment on the uncertainties assigned to the multiple image\nsystems for our lens models.  We have presented two sets of results in\nthis work; with assigned image positional accuracies of\n$\\sigma_{I}$=0\\farcs2 and 0\\farcs5.  We find a variety of lens models\nare compatible with the $\\sigma_{I}$=0\\farcs2 case and only when the\nvelocity dispersion data is included into the analysis does the data\nfail to be reproduced by the model.  Certainly if we were to further\nincrease the positional errors, at some point a good velocity\ndispersion fit could conceivably be obtained, but we will refrain from\ndoing so in the present work.\n\nIncreasing the positional uncertainties is only justified if there is\nevidence that there are significant missing components in the mass\nmodels.  Further observations that can probe the mass distribution on\nfine scales to larger radii and higher quality models which can\naccount for phenomena such as adiabatic contraction in the inner\nregions of galaxy clusters and triaxiality represent the best way to\nobtain a more precise picture of the cluster mass distribution.\n\n\n\\subsection{Improving the Mass Model}\n\nWe now turn our attention to possible inadequacies in the mass\nmodel. It is important to stress that the two diagnostics (lensing and\ndynamics) adopted in this study probe different scales.  The lensing\ndata tightly constrains the mass profile at and outside the radial arc\n($\\sim$20 kpc), while the velocity dispersion constrains the mass\nprofile inside $R \\lesssim 10$ kpc.  Since multiple images are\nnumerous and their positions can be more precisely measured than\nvelocity dispersion \\footnote{The error on the astrometry with\nrespect to the relevant scale, the Einstein Radius $\\theta_{\\rm E}$ is\nmuch smaller than the relative error on velocity dispersion,\ni.e. $\\delta \\theta \/ \\theta_{\\rm E} << \\delta \\sigma\/\\sigma$}, they\ncarry more weight in the $\\chi^2$ statistic than the kinematic points,\nproducing a best overall fitting model (which is lensing dominated)\nthat is a relatively poor fit to the kinematic data.  To improve the\nmodel, one must admit that either one of the two components of the\nmodeling is incorrect, or that the functional form of the mass profile\nchosen to extrapolate the lensing information at the scales relevant\nfor dynamics is insufficient.  In this section we discuss several\nareas where the mass model presented in this paper could be improved.\n\n\\subsubsection{The Contribution of the Brightest Cluster Galaxy}\n\nWe might query the assumption of a Jaffe density profile for the BCG.  \nThis seems an unlikely avenue for improvement given the Jaffe profile \nfits the observed BCG surface brightness profile remarkably well \n(see Figure 2 of S04).  Moreover, \\citet{Gavazzi05} utilized a Hernquist \nmass profile in his analysis of MS2137-23, which also matches the \nobservations, and \\citet{Gavazzi05} was likewise unable to reproduce \nthe observed S04 velocity dispersion profile.\n\nWe have additionally checked our assumptions by altering the\nPIEMD fit to the BCG surface brightness data so that it is matched not\nto the derived Jaffe profile fit to the BCG but directly to the HST\nsurface brightness profile.  With this setup, we found a $r_{cut}$ value\nof 23.70 kpc for MS2137 and 28.65 kpc for Abell 383 (compare this with\nthe numbers in Table~\\ref{tab:lensfixed}).  Redoing our analysis for\nthe best-fitting $r_{sc}$ scenario only, our constraints on $\\beta$\nfor both Abell 383 and MS2137 did not change by more than 0.05, and so\nit is not likely that our method for constraining the BCG mass\ncontribution is the root cause of our inability to fit the data to a\nBCG + gNFW cluster DM halo mass model.\n\nConceivably the BCG may not be coincident with the center of the\ncluster DM halo, as has been assumed throughout this work.  It is\noften the case that small subarcsecond off sets between BCGs and\ncluster DM halos are necessary to fit lensing constraints\n(e.g.~\\citet{gps05}).  There is strong evidence that the BCG is\nnearly coincident with the general cluster DM halo {\\it in projection}\nfrom the strong lensing work presented here and by others\n\\citep{gavazzi03,Gavazzi05}.  However, an offset could be responsible\nfor the flat to falling observed velocity dispersion profile if the\nBCG were actually in a less dark matter dominated portion of the\ncluster.  Another possibility is that there are multiple massive\nstructures along the line of sight, which would be probed by the\nstrong lensing analysis, but not with the velocity dispersion profile\nof the BCG.  A comprehensive redshift survey of MS2137-23 could\nprovide further information on structures along the line of sight.\n\n\n\\subsubsection{The Advantage of a Mass Probe at Larger Radii}\\label{sec:highrad}\n\nWith our presented data set, we have seen that it is difficult to\nconstrain the DM scale radius, $r_{sc}$ because both of our mass\nprobes are only effective within the central $\\sim$100 kpc of the\nclusters -- within the typical DM scale radius observed and seen in\nCDM simulations.  For this reason, the inferred DM scale radius for\nboth Abell 383 and MS2137-23 lay at the boundary of our assumed prior\nrange.  Future work will benefit from weak lensing data, along with\ngalaxy kinematics and X-ray data of the hot ICM which can each probe\nout to large clustercentric radii.\n\nAlthough not the focus of the current work, pinning down the correct\nDM scale radius will be crucial for constraining other DM mass\nparameters.  For instance, there is a well-known degeneracy between\n$r_{sc}$ and the inner slope $\\beta$\n(e.g.~\\citet{gavazzi03,Gavazzi05}). To briefly explore this, we\nhave reran our analysis (for the coarse positioning lensing case) for\nboth clusters with a $r_{sc}$ of 50 and 400 kpc -- factors of two\nbeyond our chosen $r_{sc}$ prior.  We show our confidence contours in\nFigure~\\ref{fig:diffrsc}, which are noteworthy.  For example in the\ncase of MS2137-23, if we fix $r_{sc}$=50 kpc, then the best-fitting\n$\\beta = 0.05$.  However, if $r_{sc}$=400 kpc then $\\beta=0.7$, more\nin accordance with simulations.  Interestingly, the $r_{sc}$=400kpc\nscenario returns a better overall $\\chi^{2}\\sim26$ than any model with\n$r_{sc}$=100-200 kpc -- even though a $r_{sc}$ of 400 kpc is clearly\nruled our by extant weak lensing observations.  None of the other\n$r_{sc}$=50,400 kpc scenarios produced $\\chi^{2}$ values that were\ncomparable to those seen with $r_{sc}$=100-200 kpc.  Any further\nknowledge of the DM scale radius would aid greatly in constraining\n$\\beta$ and determining the overall goodness of fit of the generalized\nNFW DM profile to the cluster data.\n\nX-ray studies assuming hydrostatic equilibrium\n\\citep{Allen01vir,Schmidt06} and a combined strong and weak lensing\nanalysis \\citep{Gavazzi05} have presented data on MS2137-23 to radii\nmuch larger than that probed in this study. To check that the mass\nmodel derived from data within $\\sim$ 100 kpc do not lead to results\nat variance with published data at larger radii, we have taken the\n\\citet{Gavazzi05} results and compared their derived mass at large\nradii with an extrapolation of our mass models.\n\n\nExamining Figure~3 from \\citet{Gavazzi05} we estimate that from his\nweak lensing analysis a 2D projected mass enclosed between $1.6 \\times\n10^{14}$ and $1.1 \\times 10^{15} M_{\\odot}$ at $\\sim 1.08$ Mpc using\nthe cosmology adopted in this paper.  Correspondingly, if we take all\nof the $\\Delta \\chi^{2}<1.0$ models using our analysis method (the\ncoarse positional accuracy case was use) and calculate the expected 2D\nprojected mass enclosed at 1.08 Mpc we find values between $6.9 \\times\n10^{14}$ and $8.4 \\times 10^{14} M_{\\odot}$, well within the expected\nrange.\n\n\nNote that no attempt was made to extrapolate the mass {\\it profiles}\nderived in our analysis to larger radii than the data in this paper\nallow, although we are acquring weak lensing data for a large\nsample of galaxy clusters to perform a more extensive analysis. The\npurpose of this consistency check is to only ensure that the masses we\nderive for such large radii are not too discrepant with existing\nanalyses.  The consistency check is satisfied and lends some credence\nto the models.\n\n\\subsubsection{Dark matter baryons interactions and Triaxiality}\\label{sec:ac}\n\nThe central regions of DM halos can be strongly affected by the\ngravitational interaction with baryons during halo formation.  If\nstars form and condense much earlier than the DM, it is expected that\nthe baryons will adiabatically compress the DM resulting in a halo\nthat is {\\it steeper} than that of the original\n\\citep{Blumenthal86,Gnedin04}.  Alternatively, dark matter heating\nthrough dynamical friction with cluster galaxies can counteract\nadiabatic contraction, leading to a shallower DM profile\n\\citep{Elzant04,Nipoti03}.  The present work takes into account\nneither of the above scenarios, and if any baryon-DM interaction\ngreatly changes the cluster density profile, our assumed parameterized\ngNFW profile may be inappropriate.  Recently, \\citet{Zappacosta06}\nhave used X-ray mass measurements in the cluster Abell 2589 to\nconclude that processes in galaxy cluster formation serve to\ncounteract adiabatic contraction in the cluster environment.\nCertainly, more observational work is needed to understand the\ninterplay between baryons and DM in clusters, and extended velocity\ndispersion profiles of BCGs in conjunction with other mass tracers at\nlarger radii could serve as the best testing ground for the interplay\nof dark and luminous matter.\n\nNot only is there likely significant interplay between baryons and DM\nin the central regions of clusters, but real galaxy clusters are\ncertainly triaxial and, if ignored, this may lead to biased parameter\nestimations and discrepancies when combining mass measurement\ntechniques that are a combination of two- and three-dimensional.\nSeveral recent studies have considered the effects of halo triaxiality\non observations.  Using an N-body hydrodynamical simulation of a disk\ngalaxy and performing a 'long slit' rotation curve observation,\n\\citet{Hayashi04} found that orientation and triaxial effects can\nmistake a cuspy DM profile for one that has a constant density core.\nAt the galaxy cluster scale, \\citet{Clowe04} performed mock weak\nlensing observations of simulated galaxy clusters and found that the\nNFW concentration parameter recovered was correlated with the 3D\ngalaxy cluster orientation.  In order to investigate the recent rash\nof galaxy clusters with observed high concentration parameters in\nseeming contradiction to the CDM paradigm\n\\citep{Kneib03,Gavazzi05,Broadhurst05b}, \\citet{Oguri05} used strong\nand weak lensing data in Abell 1689 along with a set of models that\nincluded halo triaxiality and projection effects.  Again, it was seen\nthat halo shape causes a bias in mass (and mass profile)\ndetermination, although it should be kept in mind that measurements of\nconcentration are extremely difficult (e.g. Halkola et al.\\ 2006), and\nthe recent study of \\citet{Limousin06} has seemed to clear up the\nconcentration parameter controversy for at least Abell 1689.\n\nIn terms of the current work, \\citet{Gavazzi05} has pointed out that\nthe inability of his lensing model to fit the MS2137-23 BCG velocity\ndispersion profile may be due to halo triaxiality or projected mass\nalong the line of sight (which would increase the mass measured in the\nlensing analysis but would not be seen in the stellar velocity\ndispersion).  \\citet{Gavazzi05} showed that an idealized prolate halo\nwith an axis ratio of $\\sim$ 0.4 could explain the velocity dispersion\nprofile in MS2137-23.  Halo triaxiality could also explain the high\nconcentration previously seen in this cluster.  Again, the gap between\nsimulations and observations may be bridged with respect to\ntriaxiality if further steps were taken to compare the two directly.\nOne step in this direction would be the publication of detailed\ndensity profiles for the simulations (in 3-D or along numerous\nprojected sight-lines).\n\nThe most recent DM only simulations have indicated that the\nstandard NFW profile representation of a DM profile (and its\n\\citet{M99} counterpart with an inner slope $\\beta \\sim 1.5$) can be\nsignificantly improved by slightly altering the model to a profile\nwith a slope that becomes systematically shallower at small radii\n(e.g.~\\citet{Navarro04}, but see \\citet{Diemand05}).  While we have\nadopted the traditional generalized NFW profile in this study, future\nwork with parameterized models should move towards the latest fitting\nfunctions along with an implementation of adiabatic contraction as has\nalready been attempted by \\citet{Zappacosta06}.  Note, however, that\nboth \\citet{Navarro04} and \\citet{Diemand04} have stated that all\nfitted functions have their weaknesses when describing complicated\nN-body simulations and that when possible simulations and observations\nshould be compared directly.  \n\n\n\\section{Summary \\& Future Work}\\label{sec:finale}\n\nWe have performed a joint gravitational lensing and dynamical analysis\nin the inner regions of the galaxy clusters Abell 383 and MS2137-23 in\norder to separate luminous baryonic from dark matter in the cluster\ncore.  To achieve this, we implemented a new 2D pseudo-elliptical\ngeneralized NFW mass model in an updated version of the {\\sc LENSTOOL}\nsoftware package. This refinement is a natural progression from\nour earlier attempts to measure the dark matter density profile\n\\citep{Sand02, Sand04}.\n\nFor the study, we adopted an observationally motivated scale radius\nprior of $r_{sc}=100-200$ kpc.  With strong lensing alone, we find\nthat a range of mass parameters and DM inner slopes are compatible\nwith the multiple image data, including those with $\\beta > 1$ as seen\nin CDM simulations.  However, including the BCG kinematic constraints\nfor both systems, the acceptable parameter ranges shrink\nsignificantly.\n\nWe can summarize the results for the two clusters as follows:\n\n\\begin{enumerate}\n\n\\item For the cluster Abell 383 we have found satisfactory BCG +\ngeneralized NFW cluster DM models only for our coarse lensing\npositional accuracy scenario.  Assuming that this is reflective of the\nunderlying cluster DM distribution, the dark matter inner slope is\nfound to be $\\beta=0.45^{0.2}_{-0.25}$, supporting our earlier\ncontention that some clusters have inner DM profiles flatter than\nthose predicted in numerical simulations.\n\n\\item For MS2137-23 our model is unable to reproduce the observed BCG\nvelocity dispersion profile and the range of accepted inner slopes\ntherefore depends sensitively on the adopted uncertainties in the mass\nmodel.  This may suggest an unknown systematic uncertainty in our\nanalysis or that we have adopted an inappropriate mass model.  We\nexplore the former in considerable detail, extending the quite\nextensive discussion of \\citet{Sand04}.  However, no obvious cause can\nbe found. If, as we suspect, the cause lies with our adopted mass\nmodel, it points to the need for further work concerning the\ndistribution of dark matter in the central regions of galaxy\nclusters.\n\n\\end{enumerate}\n\nFuture modeling efforts should include the effects of triaxiality and\nthe influence of baryons on dark matter.  It is also critical to\nobtain high S\/N extended velocity dispersion measurements of more BCGs\nout to larger radii so that, in conjunction with other mass\nmeasurement techniques, the interplay of baryons and dark matter in\ncluster cores can be studied with a real sample.  Some other future\ndirections are straightforward.  For example, the deep multiband ACS\nimaging now being done with galaxy clusters \\citep{Broadhurst05} allow\nfor literally hundreds of multiple images to be found, significantly\nincreasing the number of constraints and allowing for nonparametric\nmass modeling \\citep{Diego04} -- a crucial addition in case the\ncurrently used parameterized models do not correspond to reality.  We\nare eager to find ways to more directly compare simulations with\nobservations so that clearer conclusions can be drawn over whether or\nnot simulations and observations are compatible.  This may\ninvolve measuring other properties of the dark matter halo rather than\na sole emphasis on the inner slope, such as the concentration\nparameter, $c$.  Simulated observations of numerical simulations,\nsuch as that presented recently by \\citet{Meneghetti05}, offer a clear\nway forward in understanding the systematics involved in observational\ntechniques and the kinds of observations required to test the current\nparadigm for structure formation.\n\n\n\\acknowledgements\n\nWe thank Raphael Gavazzi for numerous stimulating conversations. DJS\nacknowledges support provided by NASA through Chandra Postdoctoral\nFellowship grant number PF5-60041.  TT acknowledges support from the\nSloan Foundation through a Sloan Research Fellowship. GPS acknowledges\nfinancial support from a Royal Society University Research Fellowship.\nFinally, the authors wish to recognize and acknowledge the cultural\nrole and reverence that the summit of Mauna Kea has always had within\nthe indigenous Hawaiian community.  We are most fortunate to have the\nopportunity to conduct observations from this mountain.  This research\nhas made use of the NASA\/IPAC Extragalactic Database (NED) which is\noperated by the Jet Propulsion Laboratory, California Institute of\nTechnology, under contract with the National Aeronautics and Space\nAdministration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEnergy density functional (EDF) approaches like the Skyrme-Hartree-Fock \nmodel~\\cite{sto07,BenderHeenenReinhard} have proved to be successful to \ndescribe bulk properties of nuclei over the nuclear chart~\n\\cite{BrinkVautherin}. Recent computer improvements allow large scale \nEDF calculations of nuclear structure~\\cite{ben05a,ben05b} and \nreactions~\\cite{UmarOberacker, CSBA}. \nIn these approaches, one restricts the many-body \nwave-functions to a subset of the Hilbert space \n on the one hand and guess a nuclear EDF on the other hand.\nA commonly used technique is to break symmetries to enrich \nthe variational sub-space and improve the description of nuclear structure.\nAs an example, breaking gauge invariance associated \nto particle number conservation yields the Hartree-Fock-Bogoliubov \n(HFB) formalism~\\cite{Hartree, Fock, Bogoliubov}.\nThis technique allows for the description of superfluidity in \nground states of open-shell nuclei. \n\nExtensions to treat collective excitations in the presence\nof pairing correlations are possible in the framework\nof the Quasiparticle Random Phase Approximation (QRPA)\nwhich has been widely used in nuclear structure \nstudies~\\cite{eng99,ben02,cQRPA,fra05,ter05,ter06,per08}. \nThis approach and its zero pairing counterpart (RPA) \ngive reasonable estimates  of giant resonances\nthough improvements are necessary to reproduce fine structures~\\cite{lac04}. \nIn fact, the QRPA can be obtained from the linearization of the \ntime-dependent Hartree-Fock-Bogoliubov (TDHFB) equation\nwhich provides a self-consistent evolution of an independent quasiparticles state.\nThere is a consistency requirement that QRPA and the static limit of TDHFB\nshould use the same effective interaction.\nThis is a natural feature of TDHFB that  the same EDF\ncan be easily used in the static and dynamical calculations\nthanks to the structure of the TDHFB equation. \nThis is not always the case in (Q)RPA calculations where \nspin-orbit and Coulomb parts of the\nresidual interaction are often omitted \n(see discussion in~\\cite{nak05} and references therein),\nwhich may affect collective modes~\\cite{ter05,colo04,per05,sil06}.\n\nUnlike in condensed matter where, for instance, TDHFB has been applied\n to study dynamics of Bose-Einstein condensates~\\cite{Holland, bul05},\nexplicit time evolutions of nuclei including pairing are sparse\nand usually limited to the BCS ansatz of superfluidity with simple functionals\n~\\cite{neg78, cus79, cus80} or to simple systems~\\cite{BlockiFlocard}.\nOnly recently, a numerical method of solving TDHFB with the Gogny interaction\n~\\cite{dec80} has been proposed to study quadrupole oscillations using a \nharmonic oscillator basis~\\cite{HashimotoNodeki}.\n\nAt the limit where pairing is neglected, however,\nextensive calculations of nuclear dynamics have been performed\n using the time-dependent Hartree-Fock (TDHF) formalism introduced \n by Dirac in 1930~\\cite{dirac}. In this approach, one considers \n the dynamics of independent particles in a self-consistent mean-field generated \n by all the others.\n The use of Skyrme EDF~\\cite{sky56} allowed recent realistic \n calculations of both collision \nmechanisms~\\cite{bon76, Negele, sim01, sim04, UmarOberacker, mar06, sim07, CSBA} \nand giant resonances~\\cite{sim03,nak05,uma05,mar05}.\nFor instance, TDHF has been used to study anharmonicities in collective motion \nwhich are beyond the RPA range of applications~\\cite{sim03, rei07}. \nIt is then an appealing challenge to repeat these works using TDHFB \nto investigate the role of pairing correlations in nuclear dynamics.\nIn particular, ''How do they affect low-energy reaction \nmechanisms~?'' is still an open question.\nHowever, full three-dimensional TDHFB codes for collisions are still prohibitive at moment. \nFor nuclear structure purposes, pairing vibrations \nand rotations~\\cite{BohrMottelson, RnS, BesBroglia}\nalso demand theoretical investigations to reach realistic predictions~\\cite{das06}.\nOf particular interest are high-lying pairing collective modes \nlike giant pairing vibrations (GPV)~\\cite{BrogliaHighLying}. \nThese modes can be viewed as coherent sums \nof two-quasiparticle excitations across a major shell.\nThey are probed by two-nucleon transfer \nreactions~\\cite{RipkaPadjen,BrogliaHansenRiedel}. \nThe GPVs are still unobserved experimentally but they are \nexperiencing a renewed interest\nsince recent theoretical developments predicted that the use of \nradioactive ion beams could provide better conditions \nfor their studies~\\cite{Fortunato, VonOertzenVitturi, Khan}.\n\nIn this article, we solve for the first time the TDHFB equation with a \nfull Skyrme functional in the normal part of the EDF\nand a local density dependent one in its pairing part. \nAs a first application, we investigate pairing vibrations \n using the linear response theory. \nOur model is then equivalent to a fully consistent QRPA including\nspin-orbit and Coulomb interactions also in the particle-hole channel. \nIn section~\\ref{sec:formalism}, \nwe recall the TDHFB formalism and the choice of the EDF. \nWe present in section~\\ref{sec:numerical}  \nnumerical implementations in spherical symmetry.\nThen, we discuss in section~\\ref{sec:linear} our choice of \nobservables associated to pairing vibrations in the framework\nof the linear response theory.\nFinally, we present the results for  $\\Ox{18,20,22}$ and $\\Ca{42,44,46}$\n isotopes in section~\\ref{sec:results}\nbefore to conclude in section~\\ref{sec:conclusion}.\n\n\\section{Formalism \\label{sec:formalism}}\n\n\\subsection{TDHFB equation}\n\nThe TDHFB equation can be derived starting from the action\nbetween an initial and final time $t_i$ and $t_f$\n\\begin{equation}\nS = \\int_{t_i}^{t_f} \\mbox{d} t \\, \\, \\bra{\\Psi(t)} i \\hbar \\frac{\\der}{\\dt} -\\hat{H} \\ket{\\Psi(t)}\n\\label{PVfunc}\n\\end{equation}\nand writing the variational principle $\\delta S=0$ \nin the sub-space of quasiparticle vacua.\nFor each state $\\ket{\\Psi}$ \nof this sub-space, one can find a basis of quasiparticle annihilators\n$\\{\\hat{\\beta}\\}$ such that\n $ \\hat{\\beta}_\\mu \\ket{\\Psi} = 0$ for all $\\mu$~\\cite{RnS}.\nThe latter can be related to the particle creation and annihilation operators \n$\\{\\hat{a}^\\dagger,\\hat{a}\\}$ through the Bogoliubov transformation~\\cite{Bogoliubov}\n\\begin{equation}\n\\hat{\\beta}_\\mu = \\sum_\\nu \\left( U^*_{\\nu \\mu} \\hat{a}_\\nu \n+ V^*_{\\nu \\mu} \\hat{a}^\\dagger_\\nu\\right)\n\\label{eq:betadag}\n\\end{equation}\nwhere the matrices $U$ and $V$ are such that the quasiparticle operators \n$\\{\\hat{\\beta}^\\dagger,\\hat{\\beta}\\}$\nfulfill the canonical anti-commutation rules for fermions.\n\nThe variational principle leads to the TDHFB equation~\\cite{BlaizotRipka}\n\\begin{eqnarray}\ni \\hbar \\frac{\\der}{\\dt} \\mathcal{R}= \\com{\\mathcal{H}, \\mathcal{R}}.\n\\label{eq:TDHFB}\n\\end{eqnarray}\nThe generalized one-body density matrix reads\n\\begin{eqnarray}\n\\mathcal{R}(t)= \\Smat{ \\rho(t) }{ \\kappa(t) }{ -\\kappa^*(t) }{ 1-\\rho^*(t) },\n\\end{eqnarray}\nwhere $\\rho_{\\mu\\nu}=\\left<\\psi| \\hat{a}^\\dagger_\\nu \\hat{a}_\\mu|\\psi\\right>$ \nare the matrix elements of the normal density\nand  $\\kappa_{\\mu\\nu}= \\left< \\psi|\\hat{a}_\\nu \\hat{a}_\\mu |\\psi\\right>$ \nare the elements of the pairing tensor.\nThe generalized one-body Hamiltonian $\\mathcal{H}$ has a block structure\nand can be writen in terms of the Hartree-Fock (HF) Hamiltonian $h$\n and the pairing field $\\Delta$\n \\begin{eqnarray}\n\\mathcal{H}=\\Smat{ h }{ \\Delta }{ -\\Delta^* }{ -h^* },\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nh_{\\mu\\nu}=\\frac{\\delta \\mathcal{E}[\\rho,\\kappa,\\kappa^*]}{\\delta \\rho_{\\nu\\mu}} \\mbox{~~and~~} \n\\Delta_{\\mu\\nu}=\\frac{\\delta \\mathcal{E}[\\rho,\\kappa,\\kappa^*]}{\\delta\n \\kappa^*_{\\mu\\nu}} .\n \\label{eq:field}\n\\end{eqnarray}\nIn the above HFB formalism, the functional $\\mathcal{E}[\\rho,\\kappa,\\kappa^*]$\nis  the expectation value of the exact Hamiltonian $\\hat{H}$\non the quasiparticle vacuum $\\ket{\\Psi}$. \n\nA more practical form of the TDHFB equation is found by\nrecasting these equations in terms of the quasiparticle components $U$ and $V$ \nintroduced in Eq.~(\\ref{eq:betadag})\n\\begin{eqnarray}\ni \\hbar \\frac{\\der}{\\dt} \\Svec{U_{\\nu \\mu}}{V_{\\nu \\mu}} = \n\\sum_{\\eta} \\Smat{h_{\\nu \\eta}}{\\Delta_{\\nu \\eta}}{-\\Delta_{\\nu \\eta}^*}{-h_{\\nu \\eta}^*}\n\\Svec{ U_{\\eta \\mu} }{ V_{\\eta \\mu} }.\n\\label{eq:TDHFB_QPcomp}\n\\end{eqnarray}\n\n\\subsection{EDF approach}\n\\label{sec:EDF}\n\nIn nuclear physics, the above formalism should be modified, in particular\nto take into account the short range repulsive part of the nuclear interaction\nwhich makes the mean-field HFB approach irrelevant with the bare interaction.\nThis is a reason why one generally replaces $\\mathcal{E}[\\rho,\\kappa,\\kappa^*]$\nby an effective EDF fitted on nuclear properties\nwithout invoking directly the underlying exact Hamiltonian.\nMoreover, in the spirit of the density functional theory~\\cite{DFT_HK, \nDFT_KS, TDDFT_RG}, this procedure allows to include many-body correlations.\n\nLet us decompose the total energy into kinetic, Skyrme, Coulomb and pairing parts\n(see appendix~\\ref{TDHFB_sphe} for an explicit expression of each component)\n\\begin{eqnarray}\n\\mathcal{E}=\\mathcal{E}_{kin}+\\mathcal{E}_{Sk}\n+ \\mathcal{E}_{Coul} + \\mathcal{E}_{pair}.\n\\label{TDEDF}\n\\end{eqnarray}\nWe choose the SLy$4$ parameterization~\\cite{Chabanat2} \nof the Skyrme functional~\\cite{sky56} \n including time-odd densities~\\cite{Cranking}. \nThe Coulomb energy includes direct and exchange terms. \nThe latter is estimated using the Slater approximation~\\cite{neg73}. \nThe three first terms of Eq.~(\\ref{TDEDF}) depend only on the normal densities.\n It is convenient to express the pairing energy $\\mathcal{E}_{pair}$\nusing the anomalous density \n\\begin{equation}\n\\tilde{\\rho}_q(\\mathbf{r}s,\\mathbf{r}'s') = -2s'\\kappa_q(\\mathbf{r}s,\\mathbf{r}'-s')\n\\label{eq:rhotilde}\n\\end{equation}\nwhere $s$ the projection of the spin and $q$ the isospin.\nWe use a local pairing functional  \n (see, e.g.,~\\cite{BenderHeenenReinhard} and references therein)\n\\begin{eqnarray}\n\\mathcal{E}_{pair} &=& \\int \\!\\!\\! \\mbox{d}\\mathbf{r}\n\\,\\,\\,  \\frac{g}{4} \\,\n\\left[ 1 - \\left(\\frac{\\rho_0(\\mathbf{r})}{\\rho_c}\\right)^\\gamma \\right\n \\sum_{q} \\tilde{\\rho}_q^*(\\mathbf{r})\\,  \\tilde{\\rho}_q(\\mathbf{r}) \n\\label{pairfunc}\n\\end{eqnarray}\nwhere $\\rho_q(\\mathbf{r})=\\sum_{s} \\rho_q(\\mathbf{r}s,\\mathbf{r}s)$\nand $\\tilde{\\rho}_q(\\mathbf{r})=\\sum_{s} \\tilde{\\rho}_q(\\mathbf{r}s,\\mathbf{r}s)$ \nare the local parts of the normal and anomalous densities with isospin $q$\nrespectively, $\\rho_0(\\mathbf{r})=\\sum_{q} \\rho_q(\\mathbf{r})$ is the scalar-\nisoscalar density and $g$ is the pairing coupling constant.\nThe parameters $\\rho_c$ and $\\gamma$ are adjusted to generate pairing\ncorrelations preferably at the surface and\/or in the bulk of the nucleus.\nSuch a pairing scheme yields pairs of nucleons of the same isospin\ncoupled to angular momentum zero. The simplicity of such an EDF made \nsystematic three-dimensional HFB calculations over the whole nuclear \nchart possible~\\cite{ben05a,ben05b}.\n\nHowever, one has to face the divergence of local pairing densities~\\cite{UVdiv}. \nIt is then necessary either to \nregularize the equations by introducing a cutoff in the quasiparticle spectrum\nor eventually to perform a more complex renormalization scheme \n(for an example based on the Thomas-Fermi approximation, see~\\cite{bru99,Bulgac1}). \n\nIn our calculations, we use a cutoff to regularize the TDHFB equation in a quasiparticle\nenergy window of $80$~MeV. \nThis value allows two-quasiparticle excitations up to 160 MeV.\nThe pairing parameters are fitted to reproduce a neutron \nspectral gap of $1.25$~MeV in $^{120}$Sn, the pairing acting both in the bulk and at the \nsurface of the nucleus~\\cite{dob01}. The obtained pairing coupling constant is \n$g=-275.25$~MeV with the parameters $\\rho_c = 0.32$~fm$^{-3}$ and~$\\gamma=1$. \n\n\\subsection{Particle number conservation \\label{sec:particle}}\n\nDue to the broken $U(1)$ gauge invariance associated to the \nparticle number conservation, the HFB states are not eigenstate \nof the particle number operator $\\hat{N}$. In static calculations, \none adds a Lagrange multiplier~$\\lambda$, \ninterpreted as a chemical potential,\nin order to  fix the number of particles in average.\nIn TDHFB dynamical simulations, however, the particle number obeys to\n\\begin{eqnarray}\n  i \\hbar \\frac{\\der}{\\dt} \\langle \\hat{N} \\rangle = \n    \\mbox{Tr}\\left( \\kappa \\Delta^* - \\Delta \\kappa^* \\right).\n  \\label{eq:pairing_functional}\n\\end{eqnarray}\nThe definitions of $\\Delta$  and  $\\tilde{\\rho}$ in Eqs.~(\\ref{eq:field}) \nand~(\\ref{eq:rhotilde}) respectively,\ntogether with the choice of the pairing functional in Eq.~(\\ref{pairfunc}) \nensure that the right hand side of Eq.~(\\ref{eq:pairing_functional}) vanishes.\nAs a consequence, we do not need to enforce the conservation of the average \nnumber of particles with a chemical potential in the TDHFB equation. \n\nNevertheless, dropping this constraint in the dynamics \ninduces a rotation of the Bogoliubov vacuum in gauge \nspace~\\cite{TDHFBBulgac}. \nIn particular, the anomalous density of a stationary state will \ncarry a phase $\\exp{\\left(-{2i\\lambda t}\/{\\hbar}\\right)}$.\nThen, the ground state expectation values of observables which are linear \nin the anomalous density will evolve in time.\nFor instance, this is the case of the observable we use \nto study pairing vibrations (see  Eq.~(\\ref{eq:Fdet})).\nThe time-Fourier analysis of such an observable in the linear response theory \n(see section \\ref{sec:linear}) will then contain a spurious peak \nat an energy $\\hbar \\omega = 2 \\lambda$.\nThis is a manifestation of the Goldstone mode due to \nthe broken symmetry associated to particle number conservation.\nIn QRPA calculations, it induces a spurious mode at zero energy.\nIn order to avoid such a spurious mode in the linear response of TDHFB, we have to\n keep the static ground state chemical potential in the particle-hole field during \nthe evolution.\nFinally, we note that the chemical potential $\\lambda$ is easy to compute  \nonly for nuclei with pairing. Thus we do not consider doubly magic nuclei\nas initial states in the present work.\n\n\\section{Numerical implementation of TDHFB \\label{sec:numerical}}\n\n\\subsection{Spherical symmetry}\n\nThe pairing functional defined in Eq.~(\\ref{pairfunc}) couples only nucleons of the same isospin.\nWe can then focus on semi-magical nuclei for which \nthe spherical assumption is a good approximation.\nAs a consequence, we solve the TDHFB equation using spherical symmetry.\n\nLet us recast the problem with purely local fields in space, spin and isospin\nusing this symmetry. \nThe total many-body wave-function being rotational invariant,\nit is convenient to write the Bogoliubov transformation \nin the spherical basis using the standard notation for quantum numbers\n$n$, $l$, $j$ and $m$ (we omit the isospin $q$ in the notation for simplicity)\n\\begin{eqnarray}\n  \\hat{\\beta}^\\dagger_{nljm} = \\sum_k(\\mathcal{V}^{(ljm)}_{kn}(-1)^{j-m}\\hat{a}_{klj-m} + \\mathcal{U}^{(ljm)}_{kn}\\hat{a}^\\dagger_{kljm}).\n  \\label{eq:beta_spher}\n\\end{eqnarray}\nThis definition \nensures that the component $m$ of these quasiparticle operators \ntransforms under rotation as  a tensor of rank $j$ (see Eq.~(8.79) \nof Ref.~\\cite{BlaizotRipka}).\n\nWe choose to solve the TDHFB equation in coordinate space \nusing quasiparticle wave functions \n$U_\\nu( \\mathbf{r} s)\\equiv U_{ \\mathbf{r}s, \\nu}$ and\n$V_\\nu( \\mathbf{r} s)\\equiv V_{ \\mathbf{r} s,\\nu}$ \n defined as components of quasiparticle spinors\n\\begin{eqnarray}\n  \\Svec{{U}_{nljm} \\left( \\mathbf{r} s\\right)}{{V}_{nljm} \\left( \\mathbf{r} s \\right)} = \n\\Svec{ \\bra{\\mathbf{r} s}\\hat{\\beta}^\\dagger_{nljm}\\ket{-} }{ \\bra{-}\\hat{\\beta}^\\dagger_{nljm}\\ket{\\mathbf{r} s} },\n  \\label{eq:qp_spher}\n\\end{eqnarray}\nwhere $\\ket{-}$ is the particle vacuum.\nThe standard decomposition of single particle orbitals in spherical coordinates writes \n$\\langle {\\mathbf{r} s} \\ket{nljm}=R_{nlj}(r) \\,\\Omega_{ljms}(\\theta \\phi)$.\nThe angular part is expressed in terms of Clebsch-Gordan coefficients \nand spherical harmonics \n$\\Omega_{ljms} = \\bran{l (m-s)\\frac{1}{2}s}\\ket{jm} Y_l^{m-s}$.\nDefining the radial quasiparticle wave functions\n$u_{njl}(r) = r \\sum_k \\mathcal{U}_{kn}^{(ljm)} R_{klj}(r)$ and\n$v_{njl}(r) = (-1)^{l+1} r \\sum_k \\mathcal{V}_{kn}^{(ljm)} R^*_{klj}(r)$,\nand using the property \n$\\Omega^*_{lj-ms}= -2s (-1)^{m+l-j} \\Omega_{ljm-s}$,\nEq.~(\\ref{eq:qp_spher}) becomes\n\\begin{eqnarray}\n \\Svec{{U}_{nljm} \\left( \\mathbf{r} s \\right) }\n      {{V}_{nljm} \\left( \\mathbf{r} s \\right) } = \n \\frac{1}{r}\\Svec{ u_{nlj}\\left(r\\right)\\, \\, \\Omega_{ljms}\\left(\\theta \\phi\\right) }\n                 { 2\\sigma \\, \\, v_{nlj}\\left(r\\right) \\, \\, \\Omega_{ljm-s}\\left(\\theta \\phi\\right) }. \\nonumber\n\\end{eqnarray}\n\nFollowing the same way as Dobaczewski {\\it et al.} \nfor the static HFB problem~\\cite{NPA422}, \nwe introduce  the anomalous field $\\tilde{h}_q(\\mathbf{r}s,\\mathbf{r}'s')\n= \\frac{\\delta \\mathcal{E}[\\rho,\\tilde{\\rho},\\tilde{\\rho}^*]}{\\delta\n \\tilde{\\rho}^*_q(\\mathbf{r}s,\\mathbf{r'}s')}$\n where only the pairing energy in Eq.~(\\ref{pairfunc})\ncontributes. \nThe EDF considered here contains only local densities \n(see appendix~\\ref{TDHFB_sphe}).\nTherefore, the HF and anomalous fields are also local in space. \nFinally, it is  possible to recast the TDHFB equation~(\\ref{eq:TDHFB_QPcomp}) as a set \nof Schroedinger like equations for the quasiparticle radial wave functions $u$ and~$v$\n\\begin{eqnarray}\n  i \\hbar \\frac{\\der}{\\dt}\n \\Svec{u_{nlj}}{v_{nlj}}  =\n \\Smat{h_{lj}-\\lambda}{\\tilde{h}}{\\tilde{h}^*}{-h_{lj}^*+\\lambda}\n \\Svec{u_{nlj}}{v_{nlj}}\n\\label{eq:TDHFB_qp}\n \\end{eqnarray}\nwhere $\\lambda$ is the chemical potential (see section~\\ref{sec:particle}).\nThe expressions of the fields $h(r)$ \nand~$\\tilde{h}(r)$ and those of the various densities\nentering the EDF solved in spherical symmetry\ncan be found in appendix~\\ref{TDHFB_sphe}.\n\n\\subsection{Computational details}\n\nThe initial condition is obtained \nwith the \\textsc{hfbrad} code~\\cite{hfbrad} which solves the static HFB \nequation in spherical symmetry. \nWe have constructed a time-dependent version of this code \nto solve the TDHFB equation with the functional described \nin section \\ref{sec:EDF}.\nThe set of equations~(\\ref{eq:TDHFB_qp}) is solved iteratively\nusing a one-step predictor-corrector method~\\cite{FKW}\nand a truncation of the time propagator\n\\begin{eqnarray}\n  U\\left(t,t+\\delta t\\right)= \\exp{\\left( \\frac{-i \\delta t}{\\hbar}\\mathcal{H}\n  (t+\\delta t\/{2})\\right)}\n\\end{eqnarray}\n at $4$th order in $\\delta t$.\n\nSpatial derivatives are calculated in a discretized $r$-space using seven-points formula. \nThe numerical accuracy of this method decreases increasing\nquasi-particle energy. As a consequence, this approximate derivation\nformula induces a small periodic variation of the HFB ground state\n(of the order of few tens of keV) due to high energy quasiparticles.\nWe checked \nthat this numerical artifact disappears if we\ndevelop the wave-functions on a \nconstant step Lagrange mesh~\\cite{BayeLM}.\nHowever, the latter method increases the numerical effort. \nThe amplitude of these variations being linked \nto the mesh discretization, a mesh step of $0.15$~fm \nhas been found to ensure a good numerical precision \nwith a reasonable computational effort.\nIn particular, the energy is conserved up to $15$~keV\nand deviations of the total number of particles are of the order of $10^{-7}$ \nin the present calculations. \nThough this latter value is one order of magnitude higher than in the TDHF case \nwith the same numerical conditions, it is small enough to \nleave the observables of interest unaffected. \nMoreover, to avoid any unphysical contribution\ndue to the approximate spatial derivative \nin the evolution of observables, \nwe subtract from their expectation values \nthe one obtained without external field.\nWe checked that this procedure does not affect the physical content\nof the spectra presented in Sec.~\\ref{sec:results}.\n\nLet us precise that we use hard box boundary conditions. \nThe latter are not optimized for a proper treatment of the continuum\nbecause they lead to a discretized quasiparticle spectrum.\nIn addition, particles which are reflected on the boundaries of the box \nmay interact with the nucleus and induce unphysical effects\non the evolution of observables.\nThis problem has been tackled in TDHF with the help of \nabsorbing boundary conditions~\\cite{nak05, ABC_PRE}.\nHowever, we did not found this technique to be appropriate \nto treat the TDHFB continuum because of the non-vanishing \nasymptotic nature of the upper component of the quasiparticle \nwave-functions in Eq.~(\\ref{eq:qp_spher})~\\cite{NPA422}. Though needed \nto refine description of unbound \nstates~\\cite{cQRPA}, further improvements of the boundary conditions \nare beyond the scope of this exploratory work.\n\nWe consider nuclei with magic proton numbers and \nexcitations acting on neutrons only. \nThen the calculations include pairing for neutrons only.\nThe local pairing functional couples to high angular momenta and high energy quasiparticle states up to the energy cutoff of $80$~MeV.\nIn the calculations presented here, convergence \nof the static solutions have been obtained \nwith a maximum total angular momentum for neutrons, \nfor which the Bogoliubov transformation is achieved,\n of $j_c=19\/2$ ($23\/2$) for Oxygen (Calcium) isotopes respectively. \n For Oxygen isotopes, a box radius $22.5$~fm was used, in \norder to be in conditions as close as possible to the discrete QRPA results of Ref.~\\cite{Khan}. \nCalculations for Calcium isotopes have been performed in a box of radius $30$~fm.\\\\\t\nBoth the mesh step, through the derivation formulae, and the maximum angular momentum, through \nthe centrifugal part of the kinetic operator, constrain very much the time step for which the \ncalculations are stable. The adopted time step used in these calculations are $0.003$ and \n$0.002$~fm\/c for Oxygen and Calcium isotopes respectively. \n\n\\section{Linear response framework for pairing excitations\\label{sec:linear}}\n\n\\subsection{Linear response theory}\n\nThe linear response theory has been widely used with TDHF\nto study collective vibrations in nuclei~\\cite{blo79,str79,uma86,cho87,chi96,sim03,nak05,uma05,mar05,alm05,rei07,ste07}.\nIn this theory, one computes the time evolution of an observable \n\\begin{equation}\n\\Delta Q(t)=\\bra{\\Psi (t)}\\hat{Q}\\ket{\\Psi (t)} - \\bra{0}\\hat{Q}\\ket{0}\n\\end{equation}\nafter an excitation induced by a small external potential\n$\\hat{V}_{ext}(t) = \\epsilon \\hat{F} \\xi(t)$ \non the ground state $\\ket{0}$ of the system.\nThe parameter $\\epsilon$ quantifies the intensity of the excitation.\nIt has to be small enough to ensure the linear regime, \n{ i.e.}, the amplitude $\\Delta Q_{max}$ of the response \nmust be proportional to $\\epsilon$. In this study, the time dependence \nof the external potential is chosen to be a Dirac function $\\xi(t)=\\delta(t)$.\nThe excitation is then equivalent to a boost applied on the \nground state at the initial time\n\\begin{equation}\n\\ket{\\Psi(0)} = e^{-i\\epsilon \\hat{F} \/\\hbar} \\ket{0}.\n\\label{eq:initial_condition}\n\\end{equation}\n\nThe response $\\Delta Q(t)$ to this excitation can be decomposed\ninto various frequencies $\\omega$ using\n\\begin{eqnarray}\nR_{Q}(\\omega) &=& \\frac{-\\hbar }{\\pi \\epsilon}\\,\n\\int_{0}^{\\infty} \\!\\!\\! \\mbox{d} t \\,\\,\\, \\Delta{Q}(t) \\, \\sin (\\omega t).\n\\label{eq:spectral_response}\n\\end{eqnarray}\nIn the particular case where the operators used for the excitation \nand the observations are the same, i.e., \n $\\hat{F} = \\hat{Q}$, Eq.~(\\ref{eq:spectral_response}) gives the strength distribution\n\\begin{eqnarray}\nR_{F}(\\omega) &=& \\sum_\\alpha \\, | \\bra{\\alpha} \\hat{F} \\ket{0} |^2 \n\\,\\, \\delta (\\omega - \\omega_\\alpha)\n\\label{eq:strength}\n\\end{eqnarray}\nwhere $ \\bra{\\alpha} \\hat{F} \\ket{0}$ is the transition amplitude between \nthe ground state and the eigenstate $\\ket{\\alpha}$ of the Hamiltonian\n and $\\hbar \\omega_\\alpha = E_\\alpha-E_0$ is their energy difference.\n\nWhen the excitation generates  a transition to neighboring nuclei, \n$E_0$ and $E_ \\alpha$ are the energies of the \nground and excited states of the final nucleus\nif one keeps the static chemical potential in the Hamiltonian \n(see section 10.1 of Ref.~\\cite{BlaizotRipka}).\nFor instance, in the case of an addition of two nucleons\non a nucleus of $A$ nucleons,\nthe energy of the mode reads \n$\\hbar \\omega_\\alpha = E^{(A+2)}_\\alpha-E_0^{(A+2)}$\nwhere the ground state energy of the final\nnucleus is approximated by $E_0^{(A+2)}=E_0^{(A)}+2\\lambda$.\nNote that this energy may differ from the one obtained\nin a HFB calculation in the $A+2$ nucleus because of a possible \nrearrangement of the HFB field.\n  \n\\subsection{Application to pairing vibrations}\n\nPairing vibrations of quantum numbers $0^+$ \ncan be excited by two-nucleon transfer reactions \nand have been studied in the small amplitude limit \nwithin the QRPA framework~\\cite{Khan}. \nA Hermitean pair-transfer operator is given by~\\cite{BesBroglia}\n\\begin{eqnarray}\n\\hat{F}= \\sum_{\\nu}  \\left(  f_\\nu\\,\\hat{a}^\\dagger_\\nu \\hat{a}^\\dagger_{\\bar{\\nu}} \n+ f^*_\\nu\\, \\hat{a}_{\\bar{\\nu}} \\hat{a}_\\nu \\right)\n\\label{pairvibOBS}\n\\end{eqnarray}\nwhere $\\bar{\\nu}$ denotes the time-reversed state of~$\\nu$.\n\nIn this paper, we consider local excitations acting on neutrons only and, \nfor the sake of simple notations, we do not write explicitly \nthe isospin quantum number in the following.\nThe pair-transfer operator then writes in coordinate space\n\\begin{eqnarray}\n\\hat{F} = \n\\int \\!\\!\\! \\mbox{d}\\mathbf{r} \\,\\,\\, f(\\mathbf{r}) \\left( \\hat{a}^\\dagger_{\\mathbf{r},\\downarrow} \n\\hat{a}^\\dagger_{\\mathbf{r},\\uparrow} \n+ \\hat{a}_{\\mathbf{r},\\uparrow} \\hat{a}_{\\mathbf{r},\\downarrow} \\right).\n\\label{pairvibOBS2}\n\\end{eqnarray}\nIn this particular choice, the spatial distribution $f(\\mathbf{r})$ is real. \nUsing Eq.~(\\ref{eq:rhotilde}), the expectation value of $\\hat{F}$ simply writes\n\\begin{equation}\n\\bra{\\Psi (t)}\\hat{F}\\ket{\\Psi (t)} = \\frac{1}{2}\\int\\!\\!\\! \\mbox{d}\\mathbf{r} \\,\\,\\,f(\\mathbf{r})\\,\\, \\left(\\tilde{\\rho}_0(\\mathbf{r}; t)+\\tilde{\\rho}^*_0(\\mathbf{r}; t)\\right)\n\\label{eq:Fdet}\n\\end{equation}\nwhere $\\tilde{\\rho}_0 = \\sum_q\\tilde{\\rho}_q$.\n\nTo preserve spherical symmetry, we focus on monopole pairing modes,\nrequiring a radial dependence only, { i.e.},  $f(\\mathbf{r})\\equiv f(r)$.\nWe choose a Fermi-Dirac spatial \ndistribution $f(r)=\\left(1+\\exp\\left(\\frac{r-R_c}{d}\\right)\\right)^{-1}$\nwhere the parameters $R_c=(1.27~A^{1\/3}+4)$~fm \nand $d=0.5$~fm are chosen to allow for pair transfer on the whole nucleus\non the one hand, and to remove unphysical high energy modes associated \nto pair creation outside of the nucleus on the other hand.\nFinally, this excitation may change the number of neutrons\nat the initial time. \nHowever, deviations are small\nin the present calculations ($\\sim 10^{-3}$ neutrons).\n\nWe choose to follow the excitation operator $\\hat{F}$ itself \nto get its strength distribution defined in Eq.~(\\ref{eq:strength}). \nWe also decompose the excitation operator $\\hat{F}= \\sum_l \\hat{F}_l$ \ninto components of single particle angular momentum $l$\n\\begin{eqnarray}\n\\hat{F}_l &=& \\sum_{nn'jm} \\int \\!\\!\\! \\mbox{d} \\mathbf{r} \\,f(r)\\,\n \\langle nljm |\\mathbf{r} \\downarrow\\rangle\\,\n\\langle n'lj(-m) |\\mathbf{r} \\uparrow\\rangle \\nonumber \\\\\n && \\times \\hat{a}^\\dagger_{nljm} \\hat{a}^\\dagger_{n'lj(-m)} + h.c.\n \\label{eq:Fl}\n\\end{eqnarray}\nwhere $h.c.$ denotes the {\\it Hermitean conjugate} \nof the entire expression.\nComputing the response of the $\\hat{F}_l$\n helps us interpret the spectra in terms of specific \nquasiparticle excitations. \n\nWe perform the calculations over a total time interval of $T=1200$~fm\/c. \nIn order to minimize the effects of the time gate on the Fourier transforms, \nwe follow the protocol given in Ref.~\\cite{ABC_PRE}, \nmultiplying the observables by a time filter \n$\\cos^2\\left(\\frac{\\pi t}{2T}\\right)$.\nThis procedure induces an additional width of~$\\sim 1$~MeV.\n \n\\section{Results \\label{sec:results}}\n\n\nIn this section, two-neutron transfer in several nuclei is studied.\nTo illustrate the method described in the previous section,\nwe first detail the analysis in $^{18}$O. \nThen, we present the results on neutron-rich Oxygen isotopes\nand on $f$-shell Calcium isotopes.\n\n\\subsection{Detailed analysis on $^{18}$O}\n\n\\begin{figure}\n\\includegraphics[height=8cm,angle=-90]{fig1.ps}\n\\caption{Evolution of $\\Delta F(t)$\nafter a pair transfer type\nexcitation on $\\Ox{18}$. The inset shows the same quantity at early times.\n\\label{evolA_O18}}\n\\end{figure}\n\nWe apply the boost of Eq.~(\\ref{eq:initial_condition}) \nin the linear regime on the HFB ground state of $\\Ox{18}$.\nThe pair-transfer operator $\\hat{F}$ is defined in Eq.~(\\ref{pairvibOBS2}). \nThe variation of the expectation value of $\\hat{F}$, obtained from \n Eq.~(\\ref{eq:Fdet}), is plotted in \n{Fig.}~\\ref{evolA_O18} as a function of time.\nWe observe a complex evolution due the excitation of several modes\nat different energies. We see in the inset that a strong variation\nof $\\Delta F(t)$ occurs at early times \nbecause all modes are initially in phase~\\cite{sim03}.\n\n\\begin{figure}\n\\includegraphics[height=8cm,angle=-90\n]{fig2.ps}\n\\caption{Decomposition of the responses (in arbitrary units)\ninto frequencies $\\omega$ for a two-neutron pair transfer \nexcitation in $^{18}$O. (a) Strength distribution of $\\hat{F}$ \nobtained from TDHFB (solid line) and the unperturbed approximation \n(dotted line).\nThe arrows indicate the $1p_{3\/2}$ (solid) and $1p_{1\/2}$ (dotted)\ndeep hole states. \n(b) TDHFB responses of $\\hat{F}_l$ with $l=0, 1, 2$ and 3.\n\\label{TF_evolA_O18}}\n\\end{figure}\n\nThe evolution in {Fig.}~\\ref{evolA_O18} is used to compute\n the strength distribution of $\\hat{F}$ according to Eq.~(\\ref{eq:spectral_response}) \n and using the time filter procedure described in the previous section.\nWe have controlled that the extracted strength is independent of the \nexcitation amplitude $\\epsilon$.\n The resulting spectrum is shown in {Fig.}~\\ref{TF_evolA_O18}(a) \n in solid line. We see two separated peaks at 6.5 and 14.3 MeV \n and several peaks between 20 and 26 MeV.\n\nTo understand the effect of the residual interaction \nwhich is included in TDHFB,\nwe have computed the so-called unperturbed response to the pair transfer excitation.\nThe latter is equivalent to Eq.~(\\ref{eq:strength}) \nif one assumes that the states $\\ket{\\alpha}$ are two-quasiparticle \nexcitations of the type $\\ket{\\mu \\nu} =  \\hat{\\beta}^\\dagger_\\nu \\hat{\\beta}^\\dagger_\\mu\\ket{0}$\nwhere $\\hat{\\beta}^\\dagger_{\\nu}$ creates a quasiparticle eigenstate \nof the static  HFB Hamiltonian on its ground state $|0\\rangle$.\nIn this approximation, the energy of the transition\nto the state $|{\\mu \\nu}\\rangle$ \nis the sum of the quasiparticle energies \n$\\hbar \\omega_{\\mu \\nu} = e_\\mu + e_\\nu$.\nTo allow for a quantitative comparison between the strength distributions obtained\nfrom TDHFB and unperturbed approximations, we compute the latter \nusing the same time Fourier technique as for the TDHFB case. \nFirst, we determine the transition amplitudes $\\langle{\\mu \\nu}|\\hat{F}|0\\rangle$\nand then the time evolution of the observable $\\hat{F}-\\langle 0|\\hat{F}|0\\rangle$ \nwithin the unperturbed approximation. The latter reads\n\\begin{equation}\n\\Delta F^{0}(t) = \\sum_{\\mu \\nu} |\\langle{ \\mu \\nu}|\\hat{F}|0\\rangle|^2 \n\\sin (\\omega_{{\\mu\\nu}} t).\n\\end{equation}\nFinally, we apply exactly the same procedure \n as for the TDHFB evolution to extract its strength distribution.\nThe resulting unperturbed spectrum is represented by a dotted line \nin {Fig.}~\\ref{TF_evolA_O18}(a). We see clearly that the effect of \nthe residual interaction is to increase the strength on the one hand\nand, on the other hand, to shift down the positions of the peaks.\n\nTo get a deeper insight into the nature of the peaks, \nwe decompose the response into components of the \nsingle particle orbital momentum $l$\nusing the observables $\\hat{F}_l$ defined in Eq.~(\\ref{eq:Fl}).\nThe response for $l=0,1,2$ and $3$ are plotted in \n{Fig.}~\\ref{TF_evolA_O18}(b).\nThese spectra, together with the quasiparticle HFB spectrum,\nallow us to characterize the peaks in terms of dominating \ntwo-quasiparticle excitations.\nAs one can see in {Fig.}~\\ref{TF_evolA_O18}(b), the first peak located \nat $6.5$~MeV is associated to the $l=0$ component of $\\hat{F}$.\nIt corresponds mainly to a pair transfer toward the almost empty $2s_{1\/2}$ orbitals.\n\nThe next peak, located at $14.3$~MeV, is mainly a mixture of two contributions:\n the transfer of a pair towards $1d_{3\/2}$ orbitals and\nthe removal of the $1p_{1\/2}$ occupied neutrons\nindicated by a dotted arrow in {Fig.}~\\ref{TF_evolA_O18}(a). \nThe fact that these two modes have the same energy is fortuitous.\n(This is also the case at the unperturbed level.)\nAs we will see later, they are well separated in the other Oxygen isotopes.\nWe also see in Fig.~\\ref{TF_evolA_O18}(b)\nthat there is a $l=3$ contribution to this peak\ndue to a coupling to $f_{7\/2}$ orbitals in the continuum.\n\nLet us now focus on the group of peaks at higher energies. \nAs one can see in {Fig.}~\\ref{TF_evolA_O18}(b), \nthey are mostly populated by $l=1$ and 3 components.\nIn fact, the peaks between 20 and 24 MeV are mainly \nassociated to the excitation of $f_{7\/2}$ quasiparticle resonant states\nwhile the peak at 24.7 MeV, indicated by a solid arrow in \n{Fig.}~\\ref{TF_evolA_O18}(a), corresponds to the deep hole\n$1p_{3\/2}$ state.\nExcept for the latter contribution, which is due to the removal of \ntwo occupied neutrons, these peaks belong to the GPV~\\cite{Khan}.\n{Indeed, they correspond to excitations of resonant states\nbelonging to the next major shell and the enhancement of the \nstrength as compared to the unperturbed spectrum is a sign of their \ncollectivity}~\\cite{BesBroglia}.\n\n\\subsection{neutron-rich Oxygen isotopes}\n\n\\begin{figure}\n\\includegraphics[height=8cm,angle=-90\n]{fig3.ps}\n\\caption{Strength distributions of the two-neutron transfer operator $\\hat{F}$ for \n$^{18,20,22}$O (in arbitrary units with the same scale on each plot).\nTDHFB results (solid lines) and the unperturbed approximation (dotted lines) are shown.\nThe arrows indicate the $1p_{3\/2}$ (solid) and $1p_{1\/2}$ (dotted)\ndeep hole states and the $1d_{5\/2}$ (dashed) pair removal. \nThe filled regions correspond to GPV candidates (see text).\n\\label{GPV_Ox}}\n\\end{figure}\n\n\n \\begin{table}\n \\caption{\\label{Ox_energy}\nEnergies and main quasiparticle contributions of the \nmost important peaks appearing in the strength distribution of \nthe two-neutron pair transfer operator $\\hat{F}$ extracted from\nTDHFB calculations for various Oxygen isotopes.\nNumbers in parentheses are the centroids of the continuum-QRPA \nenergies of Ref.~\\cite{Khan}.\nLabels in brackets indicate two-neutron removal contributions.}\n \\begin{ruledtabular}\n \\begin{tabular}{ccc}\nnucleus & E (MeV) & main orbital contribution\\\\\n\\hline\n$^{18}$O        & 6.5{\\footnotesize\\it (6.5)}     & $2s_{1\/2}$ \\\\\n                & 14.3{\\footnotesize\\it (14)}     & $1d_{3\/2}$, $[1p_{1\/2}]$ \\\\\n                & 20-24{\\footnotesize\\it (21.5)}  & $f_{7\/2}$ \\\\\n                & 24.7                            & $[1p_{3\/2}]$ \\\\\n\\hline\n$^{20}$O        & 4.3{\\footnotesize\\it (4)}       & $2s_{1\/2}$ \\\\\n                & 12.1{\\footnotesize\\it (11)}     & $1d_{3\/2}$ \\\\\n                & 17.5                            & $[1p_{1\/2}]$ \\\\\n                & 18-22{\\footnotesize\\it (19)}    & $f_{7\/2}$ \\\\\n                & 28.0                            & $[1p_{3\/2}]$ \\\\\n\\hline\n$^{22}$O        & 2.0                             & $2s_{1\/2}$, $[1d_{5\/2}]$ \\\\\n                & 9.2{\\footnotesize\\it (8)}       & $1d_{3\/2}$ \\\\\n                & 16-20{\\footnotesize\\it (16)}    & $f_{7\/2}$ \\\\\n                & 21.0                            & $[1p_{1\/2}]$ \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n\nIn addition to $^{18}$O, we also studied two-neutron \npair transfer in $^{20,22}$O nuclei.\nThe spectra are shown in {Fig.}~\\ref{GPV_Ox} \nwhile the energies and most important quasiparticle contributions \nto the main peaks are summarized in table~\\ref{Ox_energy}. \nComparing strength distributions obtained from TDHFB (solid lines) \nwith the unperturbed approximation (dotted lines) \nin {Fig.}~\\ref{GPV_Ox} leads to the same conclusions for all isotopes, \n{ i.e.}, an increase of the strength and a lowering of the peak energies\ndue to the TDHFB residual interaction. \nWe also see in {Fig.}~\\ref{GPV_Ox} that the energies \nof the $1p_{3\/2}$ and $1p_{1\/2}$ deep-hole states increase \nwith the number of neutrons. The $1p_{3\/2}$ peak in $^{22}$O\nis located outside of the figure at 31.3 MeV.\nThe occupied single particle orbitals are indeed deeper\nas compared to the Fermi level\nwhen increasing the neutron number, corresponding to higher \nquasiparticle energies.\nWe also note that transitions associated to the $1d_{5\/2}$ orbital appear only  \nin the $^{22}$O spectrum. \nThis is due to the fact that the operator $\\hat{F}$ leaves \nthe strongly paired levels almost unchanged~\\cite{BesBroglia}, \nand then no significant strength is associated to \naddition and removal of nucleons into\na partially occupied level at the Fermi surface.\nThis is not the case in $^{22}$O where the \n $1d_{5\/2}$ single particle orbital is almost fully occupied.\n \nLast but not least, we see that the GPV, indicated by a filled region, \nis present in the three isotopes with similar amplitudes.\nIn all cases, the most important contributions to the GPV \nare the excitation of $f_{7\/2}$ quasiparticle resonant states.\nIn the present calculations, continuum states are discretized \ndue to the finite size of the box, inducing a fragmentation of the GPV.\nWe also note that the GPV and the other two-neutron additional modes,\ncontrary to the removal ones, have a decreasing energy \nwith the neutron number. This is due to the fact that the \nFermi level is less deep for neutron-rich nuclei, \nwhich decreases the quasiparticle energy of states \nwith small or zero occupation number.\n\nLet us now compare the energies predicted by the present TDHFB calculations \nwith the continuum-QRPA results of Khan {\\it et al.}~\\cite{Khan}, \nalso reported in table~\\ref{Ox_energy}. \nThe latter have been computed for two-neutron additional modes only.\nCompared to QRPA results, TDHFB calculations globally predict slightly higher energies \nwhen going to more neutron-rich nuclei. In the GPV region, the centroid energies \nare 0.5, 1 and 2 MeV higher with TDHFB for $^{18,20,22}$O respectively.\nHowever, this overall agreement can be considered as good in regard to the \ndifferences between the two approaches.\nBoth calculations use the SLy4 Skyrme functional, but with different pairing schemes.\nIn the present work, we use a mixed volume-surface effective coupling.\nIn addition, our calculations are performed in wide quasiparticle energy \nand angular momentum windows, with cutoff values $E_c=80$ MeV \nand $j_c=19\/2$ respectively, while the QRPA calculations \nhave been performed in smaller windows\n($E_{c}=50$~MeV and $j_c=9\/2$) with a surface type pairing functional.\nThe parameters of the latter have been determined using a \ndifferent prescription than ours~\\cite{cQRPA}, in particular to reproduce \nthe trend of the experimental gap in neutron-rich Oxygen isotopes.\nAnother possible source of discrepancies is the fact that \nthe QRPA calculations of Ref.~\\cite{Khan} do not take into account \n the Coulomb and spin-orbit parts of the residual interaction whereas\nthe TDHFB approach uses the same EDF as the underlying HFB field.\nThis assumption may induce a slight shift in the energy of collective \nmodes~\\cite{colo04,per05,sil06}.\n\n\n\n\\subsection{Calcium isotopes}\n\nIn this section we discuss two-neutron pair transfer on\n$^{42,44,46}$Ca.\nWe have plotted in~{Fig}.~\\ref{Ca0}\nthe corresponding TDHFB strength distributions (solid lines).\nThe spectra are roughly similar for the three isotopes.\nThey exhibit several discrete transitions to bound states\ntogether with excitations of resonant two-quasiparticle states. \nThe energy threshold for the latter can be estimated \nby twice the Fermi energy, {i.e.}, $2 E_{F}= 20.6$, 19.6 and 17.8 MeV\nfor $^{42}$Ca, $^{44}$Ca and $^{46}$Ca respectively.\n\nWe performed the same analysis as for the Oxygen isotopes, \n{ i.e.}, we decomposed the strength distribution in $l$-components \nwhich, together with the HFB quasiparticle spectra,\nhelped us assign the main quasiparticle contributions to each peak.\nA summary of the results is given in table~\\ref{Ca_energy} where\nthe transitions associated to the removal of two neutrons are indicated in brackets.\n\n\\begin{figure}\n\\includegraphics[height=8cm,angle=-90\n]{fig4.ps}\n\\caption{Strength distributions of the two-neutron pair transfer operator $\\hat{F}$ for \n $^{42,44,46}$Ca (in arbitrary units with the same scale on each plot).\nTDHFB results (solid lines) and the unperturbed approximation (dotted lines) are shown.\nThe arrows indicate the $1d_{5\/2}$ (solid), $2s_{1\/2}$ (dashed)\nand $1d_{3\/2}$ (dotted) two deep-hole states. \n\\label{Ca0}}\n\\end{figure}\n\n \\begin{table}\n \\caption{\\label{Ca_energy}\nEnergies and main quasiparticle contributions of the \nmost important peaks appearing in the strength distribution of \nthe two-neutron pair transfer operator $\\hat{F}$ extracted from\nTDHFB calculations for Calcium isotopes.\nLabels in brackets indicate two-neutron removal contributions.\\\\\n}\n \\begin{ruledtabular}\n \\begin{tabular}{ccc}\nnucleus & E (MeV) & main orbital contribution\\\\\n\\hline\n$^{42}$Ca \n& 9.1     & $2p_{3\/2}$, $[1d_{3\/2}]$ \\\\\n& 13.7   & $2p_{1\/2}$, $[2s_{1\/2}]$ \\\\\n& 16.6   & $1f_{5\/2}$\\\\\n& 22.0   & $[1d_{5\/2}]$ \\\\\n& 23.4   & $g_{9\/2}$ \\\\\n\\hline\n$^{44}$Ca \n& 7.5     & $2p_{3\/2}$ \\\\\n& 11.6   & $2p_{1\/2}$, $[1d_{3\/2}]$ \\\\\n& 15.2   & $1f_{5\/2}$, $[2s_{1\/2}]$ \\\\\n& 22.1   & $g_{9\/2}$ \\\\\n& 23.9   & $[1d_{5\/2}]$ \\\\\n\\hline\n$^{46}$Ca\n& 6.0     & $2p_{3\/2}$ \\\\\n& 10.7   & $2p_{1\/2}$ \\\\\n& 13.6   & $1f_{5\/2}$, $[1d_{3\/2}]$ \\\\\n& 16.4   & $[2s_{1\/2}]$ \\\\\n& 20.8   & $g_{9\/2}$ \\\\\n& 25.4   & $[1d_{5\/2}]$ \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n\nFor the three isotopes, the lowest mode is interpreted in terms of the addition of a \nneutron pair in the $2p_{3\/2}$ orbitals.\nIn $^{42}$Ca, an additional $l=2$ quasiparticle component contributes to this mode\nand corresponds to the removal of a $1d_{3\/2}$ neutron pair. \nAs for Oxygen isotopes, the appearance of removal modes (in brackets in \ntable~\\ref{Ca_energy}) together with additional modes at the same energy \nis fortuitous. In the Calcium isotopes, the removal modes are built of neutrons \nfrom the $s-d$ shell, the major shell below the Fermi energy.\nAs expected, one finally notes that energies of the removal \n(resp. additional) modes increase (decrease) with the neutron number. \nAlthough 2 $g_{9\/2}$ quasiparticles excitations are forbidden below\nthe $2 E_F$ threshold in the unperturbed spectrum, it gets mixed\nto the last bound 2 ($f_{5\/2}$) quasiparticles excitation because of \nthe residual interaction.\n\nWe have also plotted the unperturbed spectra \nin~{Fig}.~\\ref{Ca0} (dotted lines). \nAs in the case of Oxygen isotopes, \nwe observe that the TDHFB residual interaction \nlowers the energies on the one hand and increases the \nstrength on the other hand, though this second effect is \nless pronounced in the high energy part of the spectra. \nHere, the pairing residual interaction\nis not strong enough to gather high energy states together \nand to generate a well identified collective pairing vibration\nin the continuum.\nIn a pure independent particles shell model picture,\nthe last occupied neutron level of these Calcium isotopes is \nthe $1f_{7\/2}$ orbital.\nThen, one expects the GPV to be built mainly on low-lying $g_{9\/2}$\nresonant quasiparticle states. However, as we clearly\nsee in~{Fig}.~\\ref{Ca0}, the strength associated\nto the two-quasiparticle excitation of $g_{9\/2}$ levels,\nlocated at $23.4$, $22.1$ and $20.8$~MeV for \n$\\Ca{42}$, $\\Ca{44}$ and $\\Ca{46}$ respectively, \nis only slightly enhanced by the TDHFB residual interaction.\nWe checked that employing other parameterizations \nof the spatial distribution $f(r)$ of the excitation operator \nin Eq.~(\\ref{pairvibOBS2}) does not alter these conclusions.\nWe note that this lack of collectivity of the $g_{9\/2}$ has already\nbeen observed in hole pairing giant resonances studies \nwithin a more schematic formalism~\\cite{her85}.\n\n\\section{Conclusion \\label{sec:conclusion}}\n\nWe solved the TDHFB equation\nin coordinate space with spherical symmetry\nfor the evolution of a single nucleus in an external field.\nFor the normal part of the energy density functional, \nwe used the SLy4 Skyrme functional.\nFor its pairing part, we chose a local density dependent functional.\nSpecial care has been taken regarding the convergence of the static HFB solutions\nand the energy and particle number conservations in the TDHFB calculations.\n\nAs a first application, we studied $0^+$ pairing modes excited \nby a two-neutron pair transfer type operator.\nThe linear response theory has been used to compute the strength \ndistributions of this operator in $^{18,20,22}$O and $^{42,44,46}$Ca nuclei.\nBoth transitions to bound states and to the continuum are observed in all nuclei.\nIn particular, the GPV is observed in all Oxygen isotopes\nwhereas no significant enhancement of the strength \ndue to dynamical pairing correlations appears in the continuum \nof the studied Calcium isotopes. In the latter, the $g_{9\/2}$ \nquasiparticle excitations are not collective enough to generate a GPV.\n\nA detailed comparison with previous QRPA calculations have been performed \nin the Oxygen isotopes. Though there is a good agreement \nfor the most stable isotope, we find slightly higher energies for the\npairing vibrations when going to more neutron-rich nuclei.\nDifferent pairing schemes and implementations of the residual interaction \nin both calculations are invoked to explain these differences.\n\nIn addition, there is room for a better treatment of the continuum, \nfor instance in the spirit of continuum-QRPA calculations,\nbut such an improvement is not straightforward. Indeed, one cannot \nextrapolate the absorbing boundary conditions used in TDHF calculations\nto the TDHFB case because of the delocalized upper components \nof the Bogoliubov spinors.\n\nFinally, TDHFB calculations are much more demanding \nin terms of computational time than standard TDHF calculations, \nby about two orders of magnitude in the one dimensional case. \nHowever, thanks to recent increase of computational power, \nsome of the standard TDHF applications to nuclear structure and reactions\nshould be repeated with the inclusion of dynamical pairing correlations\nin the framework of TDHFB.\n\n\\begin{acknowledgments} \nWe thank K. Bennaceur for his help on the \\textsc{hfbrad} code \nand the associated formalism. \nDiscussions with M. Bender, T. Duguet, H. Flocard, E. Khan, \nD. Lacroix and V. Rotival are gratefully acknowledged. \nOne of us (B.A.) also thanks R. Broglia and G. Ripka for useful discussions \nat the early stage of this work. \nCalculations have been performed at the Centre de Calcul CC-IN2P3.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{intro}\nLattice regularization of chiral gauge theories has remained a long standing\nproblem of nonperturbative investigation of quantum field theory. Lack of\nchiral gauge invariance in $L\\chi GT$ proposals is responsible for\nthe longitudinal gauge degrees of freedom ({\\em dof}) coupling to\nfermionic {\\em dof} and eventually spoiling the chiral nature of the theory.\nThe well-known example is the Smit-Swift proposal of $L\\chi GT$ \\cite{smit}.\nAlthough in a recent development using a Dirac operator that\nsatisfies the Ginsparg-Wilson relation, it was possible to formulate a\n$L\\chi GT$ without violating gauge-invariance or locality \\cite{luscher}, an\nexplicit model for nonperturbative numerical studies is still not \navailable.\n \nIn this paper, we follow the gauge fixing approach to $L\\chi GT$\n\\cite{golter1}. The obvious remedy to control the longitudinal gauge {\\em dof}\nis to gauge fix with a target theory in mind. The Roma proposal \\cite{roma}\ninvolving gauge fixing passed perturbative tests but does not address the\nproblem of gauge fixing of compact gauge fields and the associated problem of\nlattice artifact Gribov copies. The formal problem is that\nfor compact\ngauge fixing a BRST-invariant partition function as well as (unnormalized)\nexpectation values of BRST invariant operators vanish as a consequence of\nlattice Gribov copies \\cite{neuberger}. Shamir and Golterman \\cite{golter1}\nhave proposed to\nkeep the gauge fixing part of the action BRST noninvariant and tune\ncounterterms to recover BRST in the continuum. In their formalism, the\ncontinuum limit is to be taken from within the broken ferromagnetic (FM) phase\napproaching another broken phase which is called ferromagnetic directional\n(FMD) phase, with the mass of the gauge field vanishing at the FM-FMD\ntransition. This was tried out in a $U(1)$ Smit-Swift model and so\nfar the results show that in the pure gauge sector, QED is recovered in\nthe continuum limit \\cite{recent} and in the {\\em reduced} model limit \n(to be defined below) free chiral fermions in the appropriate chiral \nrepresentation are obtained \\cite{bock1}. Tuning with counterterms has \nalso not posed any practical problem, actually very little tuning is \nnecessary. Efforts are currently underway to extend this gauge fixing \nproposal to include nonabelian gauge groups \\cite{nab}. \n      \nWithout gauge fixing the longitudinal gauge {\\em dof}, which are \nradially frozen scalar fields, are rough and nonperturbative even if the \ntransverse gauge coupling may be weak (this is because with the standard \nlattice measure, each point on the gauge orbit has equal weight). \nThe theory in the continuum limit, taken at the transition \nbetween the broken symmetry ferromagnetic (FM) phase and the \nsymmetric paramagnetic (PM) phase, displays undesired nonperturbative \neffects of the scalar-fermion coupling that usually spells disaster for \nthe chiral theory. The job of the gauge fixing is to introduce a new \ncontinuous phase transition, from the FM phase to a new broken symmetry \nphase (FMD), at which the gauge symmetry is recovered and at the same \ntime the gauge fields become smooth.\n  \nThe problem can be cleanly studied in the reduced model as explained in the\nfollowing. When one gauge transforms a gauge non-invariant theory, one picks\nup the longitudinal gauge degrees of freedom (radially frozen scalars)\nexplicitly in the action. The reduced model is then obtained   \nby making the lattice gauge field unity for \nall links, {\\em i.e.}, by switching off the transverse gauge coupling. \nThe action becomes that of a chiral Yukawa theory with interaction \nbetween the fermions and the longitudinal gauge {\\em dof}. \nThe reduced model would have a phase structure similar to the full \ntheory, {\\em e.g.}, the gauge fixed theory in the reduced limit \nwill have a FM-FMD transition in addition to the FM-PM transition. \nNow for the gauge fixing proposal to work, the scalars need \nto decouple from the fermions at the FM-FMD transition leaving the \nfermions free in the appropriate chiral representation. Passing the \nreduced model test is an important first step for any $L\\chi GT$ \nproposal that breaks gauge invariance.  \n\nIn the reduced model derived from the gauge fixed theory the \nscalar fields become smooth and expandable in a perturbative \nseries as $1+{\\cal O}(\\mbox{coupling constant})$ at the FM-FMD \ntransition. If continuum limit can be taken near the point in the \ncoupling parameter space around which this perturbative expansion is \ndefined, the scalar fields will decouple from the theory. \nThe parameterization of the gauge fixing action turns out to be a good \none, because this continuum limit can be taken ($i$) very easily by \napproaching the FM-FMD transition almost perpendicularly by tuning \nessentially one counterterm, and ($ii$) at a point on this transition \nline which is reasonably far away from the expansion point. This has \nbeen possible in \\cite{bock1} and again in the present work.  \n\nA central claim of the gauge fixing proposal \nis that it is universal, {\\em i.e.}, it should work with any lattice \nfermion action that has the correct classical continuum limit. This is \nbecause the central idea as discussed above is independent of the \nparticular lattice fermion regularization. In the \npresent paper we want to confirm the universality claim by applying the \nproposal to domain wall fermions \\cite{kaplan} with $U(1)$ gauge group. \nFor this purpose we have chosen the waveguide formulation \\cite{kaplan2} \nof the domain wall fermion and investigate in the reduced model. This \nmodel was investigated before without gauge fixing and the free domain \nwall spectrum was not obtained in the reduced limit \\cite{golter2}.  \nMirror chiral modes were found at the waveguide boundaries in addition \nto the chiral modes at the domain wall or anti-domain wall.\n \nIn section II we present the gauge-fixed domain wall fermion action for a\n$U(1)$ chiral gauge theory and then go to the so-called reduced\nmodel by switching off the transverse gauge coupling. In section III we\nperform a weak coupling perturbation theory in the reduced model for the\nfermion propagators and mass matrix to 1-loop. However, in sections\n\\ref{ferm_m} and \\ref{ferm_m1} we have used special boundary conditions\n(instead of the actual Kaplan boundary conditions) to arrive at explicit\nexpressions for the overlap of the opposite chiral modes. Our numerical \nresults for \nthe quenched phase diagram and chiral fermion propagators at the domain \nwall and anti-domain wall and at the waveguide boundaries are presented \nand compared with the perturbative results in section IV. We summarize \nin the concluding section V. In Appendix A, we describe the special \nboundary conditions used in sections \\ref{ferm_m} and \\ref{ferm_m1}.\nIn Appendix B we schematically discuss how using Kaplan boundary conditions\none can arrive at the same qualitative conclusion about the 1-loop\noverlap of the opposite chiral modes.\n \n\\section{Gauge-fixed Domain Wall Action} \\label{gfdwa}\n\nKaplan's free domain wall fermion action \\cite{kaplan} on a\n$4+1$-dimensional lattice is given by (lattice constant is taken to be\nunity throughout this paper),\n\\begin{equation}\nS_F = \\sum_{XY} \\overline{\\psi}_X \\left[ \\partial\\!\\!\\!\/_5 - w_5 +\n{\\bf M}\\right]_{XY} \\psi_Y  \\label{dwact}\n\\end{equation}\nwhere $\\overline{\\psi}$ and $\\psi$ are the fermion fields, and  \n$\\partial\\!\\!\\!\/_5$ and $w_5$ are respectively the 5-dimensional\nDirac operator and the Wilson term,\n\\begin{eqnarray}\n(\\partial\\!\\!\\!\/_5)_{XY} &=&\n\\frac{1}{2}\\sum_{\\alpha=1}^5 \\gamma_\\alpha \\left( \n\\delta_{X+\\hat{\\alpha},Y} - \\delta_{X-\\hat{\\alpha},Y} \\right),            \n\\nonumber \\\\ (w_5)_{XY} &=& \\frac{r}{2} \\sum_{\\alpha=1}^5\n\\left( \\delta_{X+\\hat{\\alpha},Y} + \\delta_{X-\\hat{\\alpha},Y} - 2\\delta_{XY}\n\\right),  \\label{par5}\n\\end{eqnarray}\nThe $\\gamma_\\alpha$'s are the five hermitian euclidean gamma matrices,\n$r$ is the Wilson parameter, $X=(x,s),~Y=(y,t)$ label the sites of\nthe $L^4 L_s$ lattice and $L_s$ is the extent of the 5th dimension:\n$0 \\leq s,t \\leq L_s-1$. We are interested in\ntaking the continuum limit in the 4 space-time dimensions only. It is\nconvenient to look at the 5-th dimension as a flavor space.\n\n\n\nWith periodic boundary conditions in the 5th or\n$s$-direction  ($s,t = L_s \\Rightarrow s,t=0$) and the domain wall mass\n${\\bf M}$ taken as\n\\begin{equation}\n{\\bf M}_{XY}=m(s)\\delta_{XY},\\;\\; {\\rm where},\n\\end{equation}\n\\begin{equation}\nm(s) =  \\begin{array}{rl}\n-m_0, &  0 <s< L_s\/2 \\\\\n 0,   &  s = 0, L_s\/2 \\\\\n m_0, &  L_s\/2 <s< L_s\n\\end{array} \\label{dwmass}\n\\end{equation}\nthe model possesses a lefthanded (LH) chiral mode bound to the domain wall\nat $s=0$ and a righthanded (RH) chiral mode bound to the anti-domain wall\nat $s=L_s\/2$. For $m_0 L_s\\gg 1$, these modes have exponentially\nsmall overlap. The chiral modes exist for momenta $p$\nbelow a critical momentum $p_c$, i.e. $\\vert \\hat{p} \\vert < p_c$, where\n$\\hat{p}^2 = 2\\sum_\\mu[1-\\cos(p_\\mu)]$ and $p_c^2 = 4 - 2m_0\/r$. Taking\nthe Wilson parameter $r=1$ the choice of $m_0$ is then restricted to\n$0<m_0<2$.\n\nA 4-dimensional gauge field which is same for all\n$s$-slices can be coupled to fermions only for a restricted number of\n$s$-slices around the anti-domain wall \\cite{golter2} with a view to\ncoupling only to the RH mode at the anti-domain wall. The gauge field\nis thus confined within a {\\em waveguide},\n\\begin{eqnarray}\nWG &=& (s: s_0 < s \\leq s_1) \\nonumber \\\\\n{\\rm with} \\hspace{0.5cm} s_0 &=& \\frac{L_s+2}{4}-1, \\hspace{0.7cm} s_1 =\n\\frac{3L_s+2}{4}-1.\n\\end{eqnarray}\nWith this choice, $(L_s-2)$ has to be a multiple of 4. For convenience,\nthe boundaries at ($s_0,s_0+1$) and ($s_1,s_1+1$) are denoted waveguide\nboundary $I$ and $II$ respectively.\n\nThe gauge transformations on the fermion fields are defined as follows:\n\\begin{eqnarray}\n\\psi_x^s \\rightarrow g_x\\psi_x^s, \\hspace{0.8cm} &\\overline{\\psi}_x^s &\n\\rightarrow \\overline{\\psi}_x^s g_x^\\dagger \\hspace{0.8cm} s \\in WG\n\\nonumber \\\\\n\\psi_x^s \\rightarrow \\psi_x^s, \\hspace{0.8cm} &\\overline{\\psi}_x^s&\n\\rightarrow \\overline{\\psi}_x^s \\hspace{1.15cm} s \\in\\!\\!\\!\\!\\!\/ \\;WG\n\\label{gsym}\n\\end{eqnarray}   \nwhere $g_x\\in G$, the gauge group. Other symmetries of the model remain \nthe same as in \\cite{golter2}.\n\nObviously, the hopping terms from $s_0$ to $s_0+1$ and that\nfrom $s_1$ to $s_1+1$ would break the local gauge invariance of the\naction. This is taken care of by gauge transforming the action and\nthereby picking up the pure gauge {\\em dof} or a radially frozen scalar\nfield $\\varphi$ (St\\\"{u}ckelberg field) at the waveguide boundary,\nleading to the gauge-invariant action (with $\\varphi_x\\rightarrow g_x\n\\varphi_x$ and (\\ref{gsym})) :\n\\begin{eqnarray}\nS_{F} & = & \\sum_{s \\in WG}\n\\overline{\\psi}^s \\left( D\\!\\!\\!\\!\/\\,(U) - W(U) + m(s) \\right) \\psi^s\n+ \\sum_{s\\not\\in WG} \\overline{\\psi}^s \\left( \\partial\\!\\!\\!\/\n- w + m(s) \\right) \\psi^s \\nonumber \\\\\n&+& \\;\\sum_s \\overline{\\psi}^s \\psi^s - \\sum_{s\\neq s_0,s_1}\n\\left( \\overline{\\psi}^s P_L \\psi^{s+1} + \\overline{\\psi}^{s+1} P_R\n\\psi^s \\right) \\nonumber \\\\\n&-& \\;y \\left( \\overline{\\psi}^{s_0} \\varphi^\\dagger P_L \\psi^{s_0+1} +\n\\overline{\\psi}^{s_0+1} \\varphi P_R \\psi^{s_0} \\right)\n- y \\left( \\overline{\\psi}^{s_1} \\varphi P_L\n\\psi^{s_1+1} + \\overline{\\psi}^{s_1+1} \\varphi^\\dagger P_R \\psi^{s_1}\n\\right) \\label{wgact}\n\\end{eqnarray}\nwhere we have taken the Wilson parameter $r=1$\nand have suppressed all indices other than $s$. The projector $P_{L(R)}$\nis $(1\\mp \\gamma_5)\/2$ and $y$ is the Yukawa coupling introduced by hand at\nthe waveguide boundaries. $D\\!\\!\\!\\!\/\\;(U)$ and $W(U)$ are respectively the\ngauge covariant Dirac operator and the Wilson term in 4 space-time dimensions.\n$\\partial\\!\\!\\!\/$ and $w$ are the 4-dimensional versions of (\\ref{par5}).\n\nThe gauge-fixed pure gauge action for $U(1)$, where the ghosts are free\nand decoupled, is:\n\\begin{equation}\nS_B(U) = S_g(U) + S_{gf}(U) + S_{ct}(U) \\label{ggact}\n\\end{equation}\nwhere, $S_g$ is the usual Wilson plaquette action;\nthe gauge fixing term $S_{gf}$ (as proposed by Shamir and Golterman) and \nthe gauge field mass counter term $S_{ct}$ are given by (for a \ndiscussion of relevant counterterms see \\cite{golter1,bock2}),\n\\begin{eqnarray}\nS_{gf}(U) & = & \\tilde{\\kappa} \\left( \\sum_{xyz} \\Box(U)_{xy}\n\\Box (U)_{yz} - \\sum_x B_x^2 \\right), \\label{gfact} \\\\\nS_{ct}(U)  & = &  - \\kappa \\sum_{x\\mu} \\left( U_{\\mu x} +\nU_{\\mu x}^\\dagger \\right),\n\\end{eqnarray}\nwhere $\\Box(U)$ is the covariant lattice laplacian and\n\\begin{equation}\nB_x  = \\sum_\\mu \\left( \\frac{V_{\\mu x-\\hat{\\mu}} +\nV_{\\mu x}}{2} \\right)^2  \\label{bxsq}\n\\end{equation}\nwith\n$V_{\\mu x} = \\frac{1}{2i} \\left( U_{\\mu x} - U_{\\mu x}^\\dagger \\right)$\nand $\\tilde{\\kappa} = 1\/(2\\xi g^2)$.\n\n$S_{gf}$ is not just a naive lattice transcription of the continuum \ncovariant gauge fixing term, it has in addition appropriate irrelevant \nterms. As a result, $S_{gf}$ has a unique absolute minimum at $U_{\\mu \nx}=1$, validating weak coupling perturbation theory (WCPT) around $g=0$ \nor $\\tilde{\\kappa}=\\infty$ and in the naive continuum limit it reduces \nto $\\frac{1}{2\\xi} \\int d^4x (\\partial_\\mu A_\\mu)^2$. \n\nObviously, the action $S_B(U)$ is not gauge invariant. By giving it a gauge\ntransformation the resulting action $S_B(\\varphi^\\dagger_x U_{\\mu x}\n\\varphi_{x+\\hat{\\mu}})$ is gauge-invariant with $U_{\\mu x}\\rightarrow g_x U_{\\mu\nx} g^\\dagger_{x+\\hat{\\mu}}$ and $\\varphi_x \\rightarrow g_x \\varphi_x$, $g_x \\in\nU(1)$. By restricting to the trivial orbit, we arrive at the so-called\n{\\bf reduced model} action\n\\begin{equation}\nS_{reduced} = S_F(U=1) + S_B(\\varphi^\\dagger_x \\;1 \\;\\;\\varphi_{x+\\hat{\\mu}})\n\\label{reduced}\n\\end{equation}\nwhere $S_F(U=1)$ is obtained quite easily from eq.(\\ref{wgact}) and\n\\begin{equation}\nS_B(\\varphi^\\dagger_x \\;1 \\;\\;\\varphi_{x+\\hat{\\mu}})\n= -\\kappa \\sum_x \\varphi^\\dagger_x(\\Box \\varphi)_x + \\tilde{\\kappa}\n\\sum_x \\left[\\varphi^\\dagger_x(\\Box^2 \\varphi)_x - B^2_x \\right] \\label{redB}\n\\end{equation}\nnow is a higher-derivative scalar field theory action. $B_x$ in (\\ref{redB})\nis same as in (\\ref{bxsq}) with\n\\begin{equation}\nV_{\\mu x} = \\frac{1}{2i}\\left(\\varphi^\\dagger_x\\varphi_{x+\\hat{\\mu}} -\n\\varphi^\\dagger_{x+\\hat{\\mu}} \\varphi_x\\right).\n\\end{equation}\n\nIn the following, we investigate the action (\\ref{reduced}) at $y=1$ by\nanalytical and numerical methods. Some numerical results with other values of\n$y$ have been presented in \\cite{npbps}. The waveguide model strictly\nat $y=0$ would give rise to opposite\nchiral modes at the waveguide boundaries as can be seen from fermion current\nconsiderations \\cite{golter2} (and also from numerical simulation) \nand would thereby spoil the chiral nature of the theory. It is an \ninteresting question to investigate the model for $0<y<1$. Analysis of \nthe results for small values of $y (<1)$ is tricky and will be discussed \nin a separate article \\cite{big}.\n\n\\section{Weak Coupling Perturbation Theory in the Reduced Model} \\label{wcpt}\nAt $y=1$, we\ncarry out a WCPT in the coupling $1\/\\tilde{\\kappa}$ for the fermion\npropagators to 1-loop. In order to develop perturbation theory, in reduced\nmodel, we expand,\n\\begin{equation}\n\\varphi_x = \\exp(ib\\theta_x) = 1 +ib \\theta_x  -\\frac{1}{2} b^2 \\theta_x^2\n+ {\\cal O}(b^3)\n\\end{equation}\nwhere,\n$b=1\/\\sqrt{2\\tilde{\\kappa}} $  and $\\theta_x$ is dimensionless, \nleading to\n\\begin{equation}\nS = S_F^{(0)}(\\psi,\\overline{\\psi};y) + S_B^{(0)}(\\theta) + S^{({\\rm\nint})}(\\psi,\\overline{\\psi},\\theta;y)  \\label{Sint}\n\\end{equation}\nwhere $S^{(0)}$'s are free actions and $S^{({\\rm int})}$ is the interaction\npart.\n\n\\subsection{Scalar propagator at tree level} \\label{scal_p}\nFrom $S_B^{(0)}(\\theta)$ one gets the free propagator for the compact scalar\n$\\theta$ \\cite{bock2},\n\\begin{equation}\n{\\cal G}(k) = \\frac{1}{\\hat{k^2}(\\hat{k^2} + \\omega^2)},\n\\hspace{1.0cm} \\omega^2=\\frac{\\kappa}{\\tilde{\\kappa}}\n\\end{equation}\nwhere, $\\hat{k}_\\mu = 2\\sin(k_\\mu\/2)$.\n\n\\subsection{$LL$ and $RR$ fermion propagators at tree level and at 1-loop}\n\\label{ferm_p}\n\\subsubsection{Tree level}\nWith $y=1$, $S_F^{(0)}(\\psi,\\overline{\\psi};y=1)$ is the free domain wall\naction (\\ref{dwact}).\nFree fermion propagators at $y=1$\nare obtained in momentum space for 4-spacetime dimensions while staying\nin the coordinate space for the 5th dimension following \\cite{aoki1}\n(results in \\cite{aoki1,aoki2,blum} cannot be directly used because of\ndifference in implementation of the domain wall (\\ref{dwmass})). The\nfree action is written as,\n\\begin{equation}\nS_F^{(0)}(y=1) = \\sum_{p,s,t} \\overline{\\widetilde{\\psi_p^s}} \\left[\ni\\overline{p\\!\\!\\!\/} \\delta_{s,t} + M_{st}P_L + M^\\dagger_{st} P_R\n\\right] \\widetilde{\\psi_p^t}\n\\end{equation}\nwhere, $M_{st} = F(p)\\delta_{s,t} + (M_0)_{st}$, $(M_0)_{st} =\n[1+m(s)]\\delta_{s,t} - \\delta_{s+1,t}$,\n$F(p) = \\sum_\\mu (1 - \\cos(p_\\mu))$,\n$\\overline{p}_\\mu = \\sin(p_\\mu)$ and $\\overline{p\\!\\!\\!\/} =\n\\gamma_\\mu\\overline{p}_\\mu$.\nThe free fermion propagator can formally be written as,\n\\begin{eqnarray}\n\\Delta(p) &=& \\left[ i\\overline{p\\!\\!\\!\/} + M P_L +\n                                  M^\\dagger P_R \\right]^{-1} \\nonumber \\\\\n &=& ( -i\\overline{p\\!\\!\\!\/} + M^\\dagger ) P_L G_L(p) +\n   ( -i\\overline{p\\!\\!\\!\/} + M ) P_R G_R(p)  \\label{fprop}\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray}\nG_L(p) & = & \\frac{1}{\\sum_\\mu \\overline{p}_\\mu^2 + M M^\\dagger}\n\\label{gl} \\\\\nG_R(p) & = & \\frac{1}{\\sum_\\mu \\overline{p}_\\mu^2 +\nM^\\dagger M}. \\label{gr}\n\\end{eqnarray}\nSolution of $G_L$ is obtained by writing (\\ref{gl}) explicitly:\n\\begin{equation}\n\\left[ \\overline{p}^2+1+B(s)^2 \\right] (G_L)_{s,t} - B(s+1)\n(G_L)_{s+1,t} - B(s) (G_L)_{s-1,t} = \\delta_{s,t} \\label{green}\n\\end{equation}\nand similarly for $G_R$. In (\\ref{green}), $B(s)= F(p)+1+m(s)$.\nWe show only the calculations for obtaining $G_L$ and henceforth drop\nthe subscript $L$.\n\nSetting the notation as follows:\n\\begin{eqnarray}\nG &=& G^-, \\;\\;\\;\\;B(s)=F(p)+1-m_0=a_- \\hspace{0.8cm} {\\rm for}\\;\\;\\;\\;\n0 <s \\leq L_s\/2-1 \\label{nomen1}\\\\\n{\\rm and} \\hspace{0.4cm} G &=& G^+, \\;\\;\\;\\;B(s)=F(p)+1+m_0=a_+\n\\hspace{0.8cm} {\\rm for}\\;\\;\\;\\;L_s\/2 <s\\leq L_s-1, \\label{nomen2}\n\\end{eqnarray}\nthe equations for $G^\\pm$ are given by,\n\\begin{eqnarray}\n\\left( \\overline{p}^2+1+a^2_{-} \\right) G^{-}_{s,t} - a_{-} G^{-}_{s+1,t}\n- a_{-} G^{-}_{s-1,t} &=& \\delta_{s,t}, \\label{lmslice} \\\\\n\\left( \\overline{p}^2+1+a^2_{+} \\right) G^{+}_{s,t} - a_{+} G^{+}_{s+1,t}\n- a_{+} G^{+}_{s-1,t} &=& \\delta_{s,t}. \\label{lpslice}\n\\end{eqnarray}\n\nThe ranges of $s$ in eqs.(\\ref{nomen1}, \\ref{nomen2}) for which $G^-$\nand $G^+$ are defined, are applicable\nonly to the translationally invariant eqs.(\\ref{lmslice},\n\\ref{lpslice}). In general for the translationally noninvariant\neq.(\\ref{green}) we also define $G^-$ and $G^+$ at $s=0$ and $L_s\/2$,\nthe ones excluded by (\\ref{lmslice}, \\ref{lpslice}). The $\\pm$\nsuperscript to $G$ at $s=0,\\;L_s\/2$ is decided by the translationally\ninvariant $s$-sector from which $s=0$ or $L_s\/2$ is approached in\neq.(\\ref{green}). We have used this notation for the boundary conditions\neq.(\\ref{dwbc}) below.\n\nThe solutions of the eqs.(\\ref{lmslice}, \\ref{lpslice}) are expressed\nas sum of homogeneous and inhomogeneous solutions:\n\\begin{equation}\nG^{\\pm}_{s,t}(p) = g_{\\pm}^{(1)}(t)e^{-\\alpha_{\\pm}(p)s} +\n                   g_{\\pm}^{(2)}(t)e^{\\alpha_{\\pm}(p)s}\n+ \\frac{\\cosh[\\alpha_{\\pm}(p)(\\vert s-t \\vert - l\/2)]}\n{2a_{\\pm}\\sinh(\\alpha_{\\pm}(p))\\sinh(\\alpha_{\\pm}(p)l\/2)}, \\label{lmsol}\n\\end{equation}\nwhere, $l=L_s\/2$ and\n\\begin{equation}\n\\cosh(\\alpha_{\\pm}(p))=\\frac{1}{2}\\left(\na_{\\pm}+\\frac{1+\\overline{p}^2} {a_{\\pm}} \\right).\n\\end{equation}\nThe third term in (\\ref{lmsol})\nis the inhomogeneous solution. To avoid singularities in\n$\\alpha_{\\pm}(p)$ when $a_{\\pm}$ is zero further restricts the allowed range\nof $m_0$ to $0<m_0<1$. In this paper we have taken $m_0=0.5$.\n\nIn order to get the complete\nsolution we need to determine the unknown functions $g_{\\pm}^{(1)}(t)$\nand $g_{\\pm}^{(2)}(t)$ in (\\ref{lmsol}), which are obtained by\nconsidering boundary conditions from eqs.(\\ref{lmslice}, \\ref{lpslice})\nat $s=0,\\;1,\\;L_s\/2-1,\\;L_s\/2,\\;L_s\/2+1,\\;L_s-1$,\n\\begin{eqnarray}\na_0 G^{-}_{0,t}(p) &=& a_{+} G^{+}_{L_s,t}(p), \\nonumber \\\\\na_0 G^{+}_{L_s\/2,t}(p) &=& a_{-} G^{-}_{L_s\/2,t}(p), \\nonumber \\\\\n(\\overline{p}^2+1+a_0^2) G^{-}_{0,t}(p) - a_{-}G^{-}_{1,t}(p)\n&=& \\delta_{0,t} + a_0 G^{+}_{L_s-1,t}(p), \\nonumber \\\\\n(\\overline{p}^2+1+a_0^2) G^{+}_{L_s\/2,t}(p) - a_{+}G^{+}_{L_s\/2+1,t}\n&=& \\delta_{L_s\/2,t} + a_0 G^{-}_{L_s\/2-1,t}. \\label{dwbc}\n\\end{eqnarray}\nwith $B(s)=F(p)+1=a_0$ at $s=0,L_s\/2$. It is to be\nnoted that these boundary conditions are significantly different from\nthe ones given in \\cite{aoki1} because of the difference in implementation\nof the domain wall. $G^-_{0,t}$ and $G^+_{L_s\/2,t}$ (and the corresponding\nones from $G_R$) are used to determine\nthe free chiral propagators at the domain wall and anti-domain wall for\ncomparison with numerical data in Fig.\\ref{waw}. We will see later that\nthese chiral propagators do not receive any 1-loop self-energy corrections.\n\nSubstituting\n$\\overline{p}^2+1+a_0^2=F_0$ and $2a_{\\pm}\\sinh(\\alpha_{\\pm}(p))\n\\sinh(\\alpha_{\\pm}(p)l\/2)=X_{\\pm}$ and using the boundary conditions,\neqs.(\\ref{dwbc}), we arrive at an equation of the form,\n\\begin{equation}\n{\\bf A}\\cdot{\\bf g}(t) = {\\bf X}(t), \\label{mateq}\n\\end{equation}\nwhere, ${\\bf g}(t) = (g_{-}^{(1)}\\;\\;\\;g_{-}^{(2)}\\;\\;\\;g_{+}^{(1)}\\;\\;\\;\ng_{+}^{(2)})$ is a 4-component vector, ${\\bf X}(t)$ is another 4-component\nvector and ${\\bf A}$ is a $4\\times 4$ matrix as given below,\n\n\\[ {\\bf A} = \\left( \\begin{array}{cccc}\na_0 & a_0 & -a_{+}e^{-\\alpha_{+}L_s} & -a_{+}e^{\\alpha_{+}L_s} \\\\\na_{-}e^{-\\alpha_{-}L_s\/2} & a_{-}e^{\\alpha_{-}L_s\/2} & -a_0e^{-\\alpha_{+}\nL_s\/2} & -a_0e^{\\alpha_{+}L_s\/2} \\\\\nF_0-a_{-}e^{-\\alpha_{-}} & F_0-a_{-}e^{\\alpha_{-}} & -a_0e^{-\\alpha_{+}\n(L_s-1)} & -a_0e^{\\alpha_{+}(L_s-1)} \\\\\n-a_0e^{-\\alpha_{-}(L_s\/2-1)} & -a_0e^{\\alpha_{-}(L_s\/2-1)} &\n(F_0-a_{+}e^{-\\alpha_{+}})e^{-\\alpha_{+}L_s\/2} &\n(F_0-a_{+}e^{\\alpha_{+}})e^{\\alpha_{+}L_s\/2}\n\\end{array} \\right) \\]\nand\n\\[ {\\bf X}(t) = \\left( \\begin{array}{ll}\na_{+}X_{+}\\cosh[\\alpha_{+}(\\vert L_s-t \\vert -l\/2)] & \\hspace{-0.9cm} -\\,\na_0X_{-}\\cosh[\\alpha_{-}(\\vert -t \\vert -l\/2)] \\\\\n& \\\\\na_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2-t \\vert -l\/2)] & \\hspace{-0.7cm} -\\,\na_{-}X_{-}\\cosh[\\alpha_{-}(\\vert L_s\/2-t \\vert -l\/2)] \\\\\n& \\\\\n\\delta_{0,t} + a_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s-1-t \\vert\\, - &\n\\hspace{-0.3cm} l\/2)] - F_0X_{-}\\cosh[\\alpha_{-}(\\vert -t \\vert -l\/2)] + \\\\\n& \\hspace{3.5cm} a_{-}X_{-}\\cosh[\\alpha_{-}(\\vert 1-t \\vert -l\/2)] \\\\\n& \\\\\n\\delta_{L_s\/2,t} + a_{+}X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2+1-&\\!\\!t \\vert\n-l\/2)] - F_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2-t \\vert -l\/2)] + \\\\\n& \\hspace{2.4cm} a_0X_{-}\\cosh[\\alpha_{-}(\\vert L_s\/2-1-t \\vert -l\/2)]\n\\end{array} \\right). \\]\n\nThe explicit expressions for ${\\bf A}$ and ${\\bf X}(t)$ are obviously\ndifferent from similar expressions given in \\cite{aoki1,aoki2,blum} because\nof the differences in domain wall implementation as already discussed\nearlier.\n\nThe solution to the eqs.(\\ref{mateq}) is very complicated in general,\nparticularly for finite $L_s$. However, $g_{\\pm}(t)$ can be obtained for\nfinite $L_s$ by solving the above equations numerically for different $t$\nvalues. This way we can easily construct the free fermion propagators at any\ngiven $s$-slice, including the zero mode propagators at $s=t=0,L_s\/2$.\n\nThe solutions for $(G_R)_{s,t}$ and the resulting propagators are \nobtained in exactly the same way. However, in this case the explicit \nforms for (\\ref{green}), (\\ref{lmsol}), (\\ref{dwbc}) and matrices ${\\bf \nA}$ and ${\\bf X}(t)$ in (\\ref{mateq}) are obviously different.\n\n\\subsubsection{1-loop} \\label{ferm_p1}\nNext we calculate the chiral fermion propagators to 1-loop. {\\em Half-circle}\ndiagrams which are diagonal in flavor space contributes to $LL$ and $RR$\npropagator self-energies. However, the\nself-energies are nonzero only at the waveguide boundaries $I$ and $II$. \n\n\nRetaining up to ${\\cal O}(b^2)$ in the interaction term\n$S^{({\\rm int})}(\\psi,\\overline{\\psi},\\theta;y=1)$ in (\\ref{Sint}),\nwe find the vertices necessary to calculate the self-energies to\n1-loop.\n\n\\vspace{-0.8cm}\n\\begin{center}\n\\begin{figure}\n\\begin{picture}(300,85)(0,65)\n\\ArrowLine(75,85)(125,85)\n\\ArrowLine(125,85)(175,85)\n\\ArrowLine(175,85)(225,85)\n\\DashArrowArcn(150,85)(35,180,0){3}\n\\Text(80,92)[]{\\footnotesize{$L$}}\n\\Text(92,76)[]{$p$, $s_0+1$}\n\\Text(125,92)[]{\\footnotesize{$R$}}\n\\Text(175,92)[]{\\footnotesize{$R$}}\n\\Text(150,76)[]{$p-k$, $s_0$}\n\\Text(220,92)[]{\\footnotesize{$L$}}\n\\Text(208,76)[]{$p$, $s_0+1$}\n\\Text(150,110)[]{$k$}\n\\end{picture}\n\\caption{1-loop self-energy contribution to $LL$ propagator at $WG$ boundary\n$I$.} \\label{hcirc}\n\\end{figure}\n\\end{center}\n\nThe $LL$ propagator on the $(s_0+1)$-th slice at the waveguide boundary\n$I$ receives a nonzero  self-energy contribution from the\nhalf-circle diagram,\n\\begin{eqnarray}\n-\\left(\\Sigma_{LL}^I(p)\\right)_{st} &= &\\int_{BZ} \\frac{d^4k}{(2\\pi)^4}\nb^2 \\left[-i\\gamma_\\mu (\\overline{p-k})_\\mu P_L\nG_L(p-k)\\right]_{s_0,s_0} {\\cal G}(k) \\;\\delta_{s,s_0+1}\\delta_{t,s_0+1}\n\\label{fselfl} \\\\\n& \\rightarrow & \\frac{b^2}{L^4} \\sum_k \\left[{\\cal\nS}^{(0)}_{RR}(p-k)\\right]_{s_0,s_0}\\;\\; \\frac{1}{\\hat{k}^2 \n(\\hat{k}^2+\\omega^2)} \\;\\delta_{s,s_0+1}\\delta_{t,s_0+1} \\label{fselfd}\n\\end{eqnarray}\nwhere  the expression in the square bracket in\n(\\ref{fselfl}) is the free $RR$ propagator \n$[{\\cal S}^{(0)}_{RR}]_{s_0,s_0}$ on the $s_0$-slice. \nEq.(\\ref{fselfl}) assumes \ninfinite 4 space-time volume while in eq.(\\ref{fselfd}) a finite \nspace-time volume $L^4$ is considered. \n\nUsing (\\ref{fselfd}) we numerically evaluate the analytic \n1-loop propagator on a given finite lattice, using\n\\begin{equation}\n{\\cal S}_{LL} = {\\cal S}^{(0)}_{LL} + {\\cal S}^{(0)}_{LL}  \n\\left[-\\Sigma^{I,II}_{LL}\\right] {\\cal S}^{(0)}_{LL},\n\\end{equation}\nand in the section $IV$ \ncompare with nonperturbative numerical results. To avoid the infra-red \nproblem in the scalar propagator, we use anti-periodic boundary \ncondition in one of the space-time directions in evaluating \n(\\ref{fselfd}). \n\nIn a similar\nway, 1-loop corrected $RR$ or $LL$ propagators are obtained at all the\n$s$-slices of the waveguide boundaries $I$ and $II$, {\\em i.e.}, at the\nslices $s_0$, $s_0+1$, $s_1$ and $s_1+1$.\n\n\\subsection{Fermion mass matrix at tree level and 1-loop} \\label{ferm_m}\nAnother issue of interest is the spread of the\nwavefunctions of the two chiral zero mode solutions along the discrete\n$s$-direction and their possible overlap. A finite overlap would\nmean an induced Dirac mass. The extra dimension, as already pointed out\nin the discussion following (\\ref{par5}), can be interpreted as a flavor\nspace with one LH chiral fermion, one RH chiral fermion and $(L_s\/2-1)$\nheavy fermions on each sector of $1\\leq s\\leq L_s\/2-1$ and $ L_s\/2+1\\leq\ns\\leq L_s-1$. \n\\subsubsection{Tree level}\nFor the spread of the zero modes at the tree level, \none needs only to solve\n$M_0 u_L =0$ and $M_0^\\dagger u_R=0$ \\cite{kaplan,aoki0} where $M_0=\nM(p=0)$ (keeping the momentum $p$ non-zero is unnecessary in this\ndiscussion). However, for radiative correction on the domain wall mass\n$m(s)$, the heavy mode spreads are also needed. Accordingly we consider\nflavor diagonalization of $M^\\dagger_0 M_0$ ($M_0$ is not hermitian) as\nin \\cite{aoki2,blum}:\n\\begin{eqnarray}\n\\left( M^\\dagger_0 M_0 \\right)_{st} \\left( \\phi^{(0)}_j \\right)_t &=& \\left[\n\\lambda_j^{(0)} \\right]^2 \\left( \\phi^{(0)}_j \\right)_s \\label{eveq1}\\\\\n\\left( M_0 M^\\dagger_0 \\right)_{st} \\left( \\Phi^{(0)}_j \\right)_t &=& \\left[\n\\lambda_j^{(0)} \\right]^2 \\left( \\Phi^{(0)}_j \\right)_s. \\label{eveq2}\n\\end{eqnarray}\n\nThe index $j$ for the eigenvalues and the eigenvectors is basically a\nflavor index, but unlike $s$ and $t$ which vary from $0$ to $L_s-1$, it is\ntaken symmetric around the domain wall $j=0$. Index $j$ varies from\n$-L_s\/2$ to $L_s\/2$. From periodic boundary condition, $L_s\/2$ and \n$-L_s\/2$  are the same point in flavor space corresponding to the \nanti-domain wall. It is to be noted that $j$ appears \nexplicitly in the heavy mode solutions below. However, it may be pointed \nout that the index $j$ need not be chosen this way. One could also \ndefine $j$ in the same way as the flavor indices $s$ and $t$, only in \nthat case the explicit solutions below would have a different \nappearance and to our taste less tractable. \n\nTo solve the above tree level eigenequations (later also at 1-loop), we \nshould ideally use the Kaplan boundary conditions because we used in our\naction the Kaplan way of implementing the domain wall. However, that would\nmake an explicit calculation of the heavy eigenmodes quite complicated and\nalmost intractable. We have hence devised a special set of boundary\nconditions (look at Appendix A for details on the boundary conditions) \nwith the property that\nno information can be passed through the wall and the antiwall. This \nmakes the calculation less cumbersome and explicit expressions can be \nobtained in manageable forms. We stress that these are not \nthe actual boundary conditions in the particular domain wall \nimplementation we have taken. However, we show in Appendix B that using \nthe correct boundary conditions ({\\em i.e.}, the Kaplan boundary \nconditions) also would lead to the same qualitative \nconclusions about the nature of 1-loop corrections to the eigenvalues, \nstability of the zero modes, and particularly the 1-loop overlap of the \nopposite chiral zero modes. \n  \nWith our special boundary conditions (used only in sections \\ref{ferm_m}\nand \\ref{ferm_m1}), the explicit form of the \neigenfunctions are obviously different from \\cite{aoki2,blum} and are \nobtained as, \n\n\\vspace{0.3cm}\n\\leftline{Zero ($L$-handed) mode:}\n\\begin{eqnarray}\n(j=0) \\hspace{2.0cm}\n\\left( \\phi^{(0)}_j \\right)_s &=& {\\cal A}\\;\n\\exp(-\\tilde{\\alpha}s) \\hspace{1.9cm} 0\\leq s\\leq L_s\/2 \\\\\n(j=L_s\/2) \\hspace{2.7cm}\n&=& {\\cal A}\\; \\exp\\left[-\\tilde{\\alpha}(L_s-s)\\right] \n\\hspace{0.8cm} L_s\/2\\leq s\\leq L_s \\\\\n{\\cal A} &=& \\left(\n\\frac{1-\\exp(-2\\tilde{\\alpha})}{1-\\exp(-\\tilde{\\alpha}L_s)}\n\\right)^{1\/2} \\nonumber \\\\\n{\\rm with}, \\hspace{2.0cm} \\exp(-\\tilde{\\alpha}) &=& 1-m_0\n\\end{eqnarray}\n\n\\leftline{Heavy mode $(\\vert j \\vert \\neq 0,\\,L_s\/2)$:}\n\\begin{eqnarray}\n(0<j<L_s\/2) \\hspace{1.0cm} \\left( \\phi^{(0)}_j \\right)_s\n&=& \\sqrt{\\frac{4}{L_s}} \\,\\sin\\beta_j(s-1)\n\\hspace{2.1cm} 0\\leq s\\leq L_s\/2 \\\\\n(-L_s\/2<j<0) \\hspace{2.0cm} &=& \\sqrt{\\frac{4}{L_s}} \n\\,\\sin\\beta_{-j}(L_s\/2-s+1) \\hspace{0.7cm} L_s\/2\\leq s\\leq L_s .\n\\end{eqnarray}\nThe parameters involved are as follows:\n\\begin{eqnarray}\n\\left(\\lambda^{(0)}_j \\right)^2 &= & 1 + \\tilde{a}(s)^2 \n-2\\tilde{a}(s)\\cos \\beta_j, \\label{eigv} \\\\  \n\\tilde{a}(s) & = & 1 + m(s) \\label{apm} \n\\end{eqnarray}\nand for nontrivial heavy mode solutions, \n$\\beta_j = 2\\pi j\/L_s$ with $\\vert j \\vert \\ne 0,\\,L_s\/2$.\n\nSolutions to (\\ref{eveq2}) for all $j$ and $s$ easily follow:\n\\begin{equation}         \n\\left( \\Phi^{(0)}_j \\right)_s = \\left(\n\\phi^{(0)}_j \\right)_{L_s\/2-s}, \\label{modes}\n\\end{equation}\n\nIt is obvious that the\neigenvectors $\\phi^{(0)}_{j=0}$ and $\\Phi^{(0)}_{j=0}$ correspond \nrespectively to the\nLH and RH chiral zero modes at the domain wall and the anti-domain wall\nin the region $0\\leq s\\leq L_s\/2$. Similarly the\neigenvectors $\\phi^{(0)}_{j=L_s\/2}$ and $\\Phi^{(0)}_{j=L_s\/2}$ correspond to\nthe same chiral zero modes at the domain wall and the anti-domain wall \nin the region\n$L_s\/2\\leq s \\leq L_s$. The $\\vert j \\vert \\neq 0,\\,L_s\/2$ eigenvectors \nare for the heavy flavor modes. All solutions are real.\n\nThe overlap of the opposite chiral modes in the region $0\\leq s\\leq L_s\/2$\ndoes not depend explicitly on $s$ and is given at the tree level by,\n\\begin{equation}\n\\Phi^{(0)}_{j=0}\\; \\phi^{(0)}_{j=0} = {\\cal A}^2\\;\n\\exp(-\\tilde{\\alpha}L_s\/2) \n\\end{equation}\nand similarly in the region $L_s\/2\\leq s\\leq L_s$. The exponentially \ndamped mixing of the LH and RH chiral modes does not induce any Dirac \nmass at the tree level for large $L_s$.\n \nIt is also noted that $\\phi^{(0)}$ and \n$\\Phi^{(0)}$ diagonalize $M_0$ and $M_0^\\dagger$ in the following \nmanner: \n\\begin{equation}\n\\left( \\Phi^{(0)}_j\\right)_s \\left( M_0 \\right)_{st}\n\\left(\\phi^{(0)}_{j^\\prime} \\right)_t = \\lambda_j^{(0)} \\delta_{j,j^\\prime}\n= \\left( \\phi^{(0)}_j\\right)_s \\left(M_0^\\dagger\\right)_{st}\n\\left(\\Phi^{(0)}_{j^\\prime} \\right)_t   \\label{diag}\n\\end{equation}\nThe way the formalism is set up, the absolute sign of $\\lambda_j^{(0)}$ \nin eq.(\\ref{diag}) is arbitrary.\n\n\\subsubsection{1-loop}\nFlavor off-diagonal {\\em tadpole} diagrams\nproduce the self-energies for the $LR$ and $RL$ parts of the fermion\npropagator. Again, the\nself-energies are nonzero only at the waveguide boundaries $I$ and $II$.\n\n\\vspace{-0.7cm}\n\\begin{center}\n\\begin{figure}\n\\begin{picture}(300,85)(0,65)\n\\ArrowLine(90,85)(150,85)\n\\ArrowLine(150,85)(210,85)\n\\DashArrowArcn(150,105)(20,180,0){3}\n\\DashCArc(150,105)(20,180,360){3}\n\\Text(95,92)[]{\\footnotesize{$L$}}\n\\Text(112,76)[]{$p$, $s_0+1$}\n\\Text(205,92)[]{\\footnotesize{$R$}}\n\\Text(198,76)[]{$p$, $s_0$}\n\\Text(150,115)[]{$k$}\n\\end{picture}\n\\caption{Tadpole contribution to 1-loop $LR$ propagator at $WG$ \nboundary $I$.} \\label{tadp}\n\\end{figure}\n\\end{center}\n\nFor the $LR$ propagator connecting $s_0$ and $s_0+1$ at the waveguide\nboundary $I$, the self-energy contribution from the tadpole diagram is\ngiven by,\n\\begin{eqnarray}\n-\\left(\\Sigma_{LR}^I(p)\\right)_{st} &=& \\frac{1}{2}b^2P_L \\int_{BZ}\n\\frac{d^4k}{(2\\pi)^4} \\frac{1}{\\hat{k}^2 (\\hat{k}^2+\\omega^2)}\\;\n\\delta_{s,s_0}\\delta_{t,s_0+1} \\\\\n&=& \\frac{1}{2}b^2P_L \\,{\\cal T}\n\\;\\delta_{s,s_0}\\delta_{t,s_0+1}  \\label{tadpld}\n\\end{eqnarray}\nwhere ${\\cal T}\\sim 0.04$ is the tadpole loop integral.\n\nSimilarly the self-energy contribution to the $LR$ propagator at the\nwaveguide boundary $II$ connecting $s_1$ and $s_1+1$ comes from a\ntadpole diagram and is given by,\n\\begin{equation}\n-\\left(\\Sigma_{LR}^{II}(p)\\right)_{st}\n= \\frac{1}{2}b^2P_L \\,{\\cal T}\n\\;\\delta_{s,s_1}\\delta_{t,s_1+1}.  \\label{tadpld2}\n\\end{equation}\n\nThe mass parameter $M_0$ gets modified at 1-loop as:\n\\begin{eqnarray}\n(M_0)_{st}P_L \\rightarrow (\\widetilde{M}_0)_{st}P_L &=&\n(M_0)_{st}P_L+\\left[-(\\Sigma^I_{LR}(0))_{st} \\right] +\n\\left[-(\\Sigma^{II}_{LR}(0))_{st}\\right]   \\nonumber \\\\\n&\\equiv & (M_0)_{st}P_L + b^2 ({\\bf \\Sigma}_{LR})_{st} P_L\\,.\n\\end{eqnarray}\n$\\Sigma^{I,\\,II}_{LL,RR}(0) =0$ identically. $(M_0^\\dagger)_{st}P_R$\ngets modified accordingly. \n\nBecause of our use of the special boundary conditions (Appendix A) which do\nnot allow any communication through the walls, 1-loop correction to \nthe eigenvectors and eigenvalues involves either \n$s_0$ (waveguide boundary $I$) or $s_1$ (waveguide boundary $II$) \ndepending on which sector of $j$ is considered. In Appendix B we show that \nif the actual (Kaplan) boundary conditions are used, one gets contribution \nat 1-loop level also from another diagram, called there the {\\em global\nloop} diagram. However, the conclusions are qualitatively the same.  \n\n\\subsection{Mass matrix diagonalization at 1-loop} \\label{ferm_m1}\nAt 1-loop level we organize the corrections to the eigenvectors and the\neigenvalues of the fermion mass matrix squared as follows:\n\\begin{eqnarray}\n\\phi^{(0)} \\rightarrow \\phi &=& (1 + b^2 \\phi^{(1)})\\phi^{(0)}, \n\\nonumber \\\\\n\\Phi^{(0)} \\rightarrow \\Phi &=& (1 + b^2 \\Phi^{(1)})\\Phi^{(0)}, \n\\nonumber \\\\\n\\left(\\lambda^{(0)}\\right)^2 \\rightarrow \\lambda^2 & = &\n\\left(\\lambda^{(0)} + b^2 \\lambda^{(1)}\\right)^2.\n\\end{eqnarray}\n$\\lambda^{(1)}$  is found to be\n\\begin{equation}\n\\lambda^{(1)}_j = \\left(\\Phi^{(0)}_j\\right)_s ({\\bf\n\\Sigma}_{LR})_{st} \\left(\\phi^{(0)}_j\\right)_t\n\\end{equation}\nThe eq.(\\ref{diag}) gets modified in 1-loop as\n\\begin{equation}\n(\\Phi_j)_s (\\widetilde{M}_0)_{st} (\\phi_{j^\\prime})_t =\n\\lambda_j \\,\\delta_{j,j^\\prime} + {\\cal O}(b^4) =\n(\\phi_j)_s (\\widetilde{M}_0^\\dagger)_{st}\n(\\Phi_{j^\\prime})_t.\n\\end{equation}\n\nThe 1-loop correction to the eigenvalues, \n$(\\delta \\lambda)_j=b^2 \\lambda^{(1)}_j$,  is given by:\n\\begin{eqnarray}\n(\\delta \\lambda)_j &=& \\frac{2b^2}{L_s}\\,{\\cal T} \\sin\\beta_j(L_s\/2-s_0-1)\n\\sin\\beta_js_0 + {\\cal O}(b^4) \\hspace{2.0cm} {\\rm for}\\;\\;1\\leq j \\leq\n(L_s\/2-1),   \\label{delm1}  \\\\\n &=& \\frac{2b^2}{L_s}\\,{\\cal T} \\sin\\beta_j(s_1+1) \n\\sin\\beta_j(L_s\/2-s_1) + {\\cal O}(b^4) \\hspace{1.75cm} {\\rm \nfor}\\;\\;-(L_s\/2-1) \\leq j \\leq -1, \\label{delm2} \\\\  \n&=& \\frac{b^2}{2}\\;\n{\\cal A}^2\\,{\\cal T} \\exp[-\\tilde{\\alpha} (L_s\/2+1)]+ {\\cal O}(b^4) \n\\;\\;\\;\\;\\stackrel{L_s\\rightarrow \\infty} {\\longrightarrow} \\;0 \n\\hspace{1.2cm} {\\rm for}\\;\\;j=0,L_s\/2. \\label{delm3}\n\\end{eqnarray}\nWe notice in \neq.(\\ref{delm3}) that the 1-loop correction to zero mode eigenvalue is \nexponentially damped and the zero modes are hence perturbatively stable.\n \nThe 1-loop expression for the chiral zero mode at the domain wall, in \nthe region $0\\leq s\\leq L_s\/2$, is,\n\\begin{equation}\n(\\phi_{j=0})_s = {\\cal A} \\left[ \\exp(-\\tilde{\\alpha} s) -\n\\frac{2b^2}{L_s}{\\cal T} \\exp(-\\tilde{\\alpha} (s_0+1))\n\\sum_{j^\\prime=1}^{L_s\/2-1} \\frac{1}{\\lambda^{(0)}_{j^\\prime}}\n\\sin\\beta_{j^\\prime}(L_s\/2-s_0-1)\\sin\\beta_{j^\\prime}(s-1) \\right].\n\\end{equation}\nWe obtain a similar 1-loop expression for the chiral mode at the \nanti-domain wall. \n\nThe overlap of the opposite chiral modes in the region $0\\leq \ns\\leq L_s\/2$ at 1-loop is given by,\n\\begin{eqnarray}\n\\Phi_{j=0}\\;\\phi_{j=0} = {\\cal A}^2 \\exp(-\\tilde{\\alpha}L_s\/2)\n&-& \\left(\\frac{2b^2}{L_s}\\right){\\cal A}^2{\\cal T}\\,{\\cal F}_1 \\,\n\\exp[-\\tilde{\\alpha}(s_0+1)]\n\\nonumber\\\\\n&-& \\left(\\frac{2b^2}{L_s}\\right){\\cal A}^2{\\cal T}\\,{\\cal F}_2 \\,\n\\exp[-\\tilde{\\alpha}(L_s\/2-s_0)], \\label{mixing}\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray*}\n{\\cal F}_1 &=& \\sum_{j^\\prime,s} \\frac{1}{\\lambda^{(0)}_{j^\\prime}}\n\\exp[-\\tilde{\\alpha}(L_s\/2-s)]\n\\sin\\beta_{j^\\prime}(s-1) \\sin\\beta_{j^\\prime}(L_s\/2-s_0-1) \\\\\n{\\cal F}_2 &=& \\sum_{j^\\prime,s} \\frac{1}{\\lambda^{(0)}_{j^\\prime}} \n\\exp(-\\tilde{\\alpha}s)\n\\sin\\beta_{j^\\prime}s_0 \\sin\\beta_{j^\\prime}(L_s\/2-s-1).\n\\end{eqnarray*}\nEq.(\\ref{mixing}) clearly shows that 1-loop corrections to the mixing of \nthe LH and RH modes are also exponentially damped. This guards against \nany induced Dirac mass in the domain wall model for large enough $L_s$ \nand the waveguide boundaries $I$ and $II$ chosen approximately \nequidistant from the domain wall and the anti-domain wall.   \nIn this context we point out that in the Smit-Swift model a shift \nsymmetry for the singlet fermion ensures that no fermion mass counter \nterm is needed.\n\nUsing the actual Kaplan boundary conditions would make the \nrelatively nice explicit expressions in eqs.(\\ref{delm1} -- \\ref{mixing})\nmuch less manageable, to say the least. Diagonalization of the fermion mass\nmatrix upto 1-loop  using the Kaplan boundary condition is discussed in\nAppendix B. \n   \n\\section{Numerical Results} \\label{numr}\nIn the quenched approximation, we have first numerically confirmed the\nphase diagram in \\cite{bock3} of the reduced model in\n($\\kappa,\\tilde{\\kappa}$) plane. The phase diagram shown schematically in\nFig.\\ref{phases} has the interesting feature that for large enough\n$\\tilde{\\kappa}$, there is a continuous phase transition between the\nbroken phases FM and FMD. FMD phase is\ncharacterized by loss of rotational invariance and the continuum limit is to\nbe taken from the FM side of the transition. In the full theory with \ngauge fields, the gauge symmetry reappears at this transition and the gauge\nboson mass vanishes, but the longitudinal gauge {\\em dof} remain decoupled.\nIn Fig.\\ref{phases} PM is the symmetric phase and AM is the broken\nanti-ferromagnetic\nphase. The numerical details involved in reconstruction of the phase \ndiagram and the fermionic measurements that follow will be \navailable in \\cite{big}. \n\n\\begin{center}\n\\begin{figure}\n\\hspace{0.1cm}\\psfig{figure=ph1.ps,width=6.5cm,height=5.0cm}\\\\\n\n\\caption{Schematic quenched phase diagram.} \\label{phases}\n\\end{figure}\n\\end{center}\n\\vspace{0.1cm}\n\nFor calculating the fermion propagators, as in \\cite{bock1} we have \nchosen the point $\\kappa =0.05$, $\\tilde{\\kappa}=0.2$ (gray blob in \nFig.\\ref{phases}). Although this point is far away from $\\tilde{\\kappa}=\\infty$, \naround which we did our perturbation theory in the previous section, the \nimportant issue here is to choose a point near the FM-FMD transition and \naway from the FM-PM transition. The results below show that for the \nfermion propagators there is \nexcellent agreement between numerical results obtained at $\\kappa =0.05$, \n$\\tilde{\\kappa}=0.2$ and perturbation theory. \n   \nNumerically  on $4^3 16$ and $6^3 16$ \nlattices with $L_s=22$ and $m_0=0.5$ we look for chiral modes at the \ndomain wall ($s=0$), the anti-domain wall ($s=11$), and at the waveguide \nboundaries ($s=5,6$ and $s=16,17$). Error bars in all the figures are \nsmaller than the symbols. \n\nFig.\\ref{waw} shows the $RR$ propagator $|S_{RR}|$ and the $LL$ propagator\n$|S_{LL}|$ at the domain and anti-domain wall as a function of a component of\nmomentum $p_4$ for both $\\vec{p}=(0,0,0)$ (physical mode) and $(0,0,\\pi)$\n(first doubler mode) at $y=1$. From the figures, it is clear that\nthe doubler does not exist, only the physical $RR$ ($LL$) propagator seems to\nhave a pole at $p=(0,0,0,0)$ at the anti-domain (domain) wall. In all the\nfigures, NS, PT and FF respectively indicate data from numerical simulation,\nfrom perturbation theory and from free fermion propagator by direct inversion\nof the free fermion matrix.\n\n\\begin{figure}\n\\vspace{-1.0cm}\n\\parbox{8cm}{\\psfig{figure=y1s0br1.ps,width=8.6cm,height=10.6cm}} \\ \\\n\\parbox{8cm}{\\psfig{figure=y1s11br1.ps,width=8.6cm,height=10.6cm}}\n\\vspace{-2.7cm}\n\\caption{Chiral propagators at domain wall $s=0$ and at anti-domain\nwall $s=11$ ($L_s=22$; a.p.b.c. in $L_4$, $y=1.0$).} \\label{waw}\n\\end{figure}\n\\vspace{0.5cm}\n\n\\begin{figure}\n\\vspace{-1.5cm}\n\\parbox{8cm}{\\psfig{figure=y1s5br1.ps,width=8.6cm,height=10.6cm}} \\ \\\n\\parbox{8cm}{\\psfig{figure=y1s6br1.ps,width=8.6cm,height=10.6cm}}\n\\vspace{-2.9cm}\n\\caption{$RR$ propagator at waveguide boundary $s=5$ and\n$LL$ propagator at waveguide boundary $s=6$ ($L_s=22$; a.p.b.c. in\n$L_4$, $y=1.0$).} \\label{wgb}\n\\end{figure}\n\nFor Fig.\\ref{waw}, PT also means zeroth order perturbation theory, {\\em i.e.},\nnumerical solution of propagator following eq.(\\ref{mateq}) (as noted\nbefore in subsubsection \\ref{ferm_p1} the self-energy contributions to\nthe $LL$ and $RR$ propagators are nonzero only at the waveguide boundaries,\nthe propagators in Fig.\\ref{waw} do not get any 1-loop correction).\nWe have PT\nresults also for $6^3\\times 16$ lattice but have chosen not to show them\nbecause they fall right on top of the numerical data. Instead PT results are\nshown for $8^3\\times 20$ lattice for which the $p_4$ points are distinct. The\ndotted line in all figures refer to the propagator from PT using a\n$256^3\\times 1024$ lattice. {\\em The curves stay the same irrespective of\nmethods or lattice size}. Based on the above, we can conclude that there are\n{\\em only free RH fermions} at the anti-domain wall, and at the domain wall\nthere are {\\em only free LH fermions}.\n\nFig.\\ref{wgb} show no evidence of a chiral mode at the waveguide boundaries\n$s=5$ and 6 and excellent agreement with 1-loop perturbation\ntheory. Here too doublers do not exist. Actually the agreement with the FF\nmethod (direct inversion of the free domain wall fermion matrix on a given\nfinite lattice) is also excellent, because the 1-loop corrections are almost\ninsignificant. For clarity, in Fig.\\ref{wgb} we have not shown the\n$LL$ propagator on $s=5$ and the $RR$ propagator at $s=6$, but conclusions are\nthe same.\n\nSimilar investigation at the other waveguide boundary $s=16, 17$ also does\nnot show any  chiral modes. Previous investigations of the domain wall \nwaveguide model without gauge fixing \\cite{golter2} have shown that the \nwaveguide boundaries are the most likely places to have the unwanted \nmirror modes. This is why we have mostly concentrated in showing that \nthere are no mirror chiral modes at these boundaries, although we have \nlooked for chiral modes everywhere along the flavor dimension. In fact, \nwe do not see any evidence of a chiral mode anywhere other than at the \ndomain wall and the anti-domain wall. \n\n\\section{Discussion} \\label{disc}\n\nWe have followed the gauge fixing proposal of Shamir and Golterman and \napplied it to domain wall fermions for a $U(1)$ $L\\chi GT$. By switching \noff the transverse gauge {\\em dof}, we arrive at the so-called  \nreduced model. We have determined the quenched phase diagram of the \nmodel and confirmed that there is a continuous phase transition from a \nbroken symmetry ferromagnetic (FM) phase to a broken symmetry \nrotationally noninvariant ferromagnetic directional (FMD) phase with the \nproperties that at this transition the longitudinal gauge {\\em dof}, \n{\\em i.e.}, the $\\varphi$ fields get decoupled. We have come to this \nconclusion by performing a WCPT for the fermion propagators and \ncomparing them to nonperturbative numerical simulations.\n\nLet us now contrast this with the previous attempt \\cite{golter2} of \nlattice regularizing a chiral gauge theory with domain wall fermions. No \ngauge fixing was done and in the reduced model, a combination of \nanalytic and numerical methods showed that mirror chiral modes were \ndynamically generated in the theory.\n\nAs in the Smit-Swift model \\cite{smit}, the mirror chiral modes are a \nreflection of the undesired presence of the longitudinal gauge {\\em \ndof} in the continuum limit. Without gauge fixing, these radially \nfrozen group-valued scalar $\\varphi$ fields are generally \nnonperturbative or rough for any value of the transverse gauge coupling, \neven in the reduced model limit. If the continuum limit is taken at the \nFM to a symmetric paramagnetic (PM) phase transition, the $\\varphi$ \nfields survive with radial modes and physical effects of them coupling \nwith the rest of the theory become manifest in the form of mirror chiral \nmodes etc..\n\nIn the gauge fixing approach of Shamir and Golterman, care has been \nexercised so that the gauge fixing term is not just a naive lattice \ntranscription of the continuum gauge fixing condition. There are \nappropriate additions of irrelevant terms in the covariant gauge fixing \nterm so that a unique perturbative vacuum exists. With this gauge \nfixing in the reduced model limit the $\\varphi$ fields \nbecome smooth and can be perturbatively expanded as $1 + {\\cal \nO}(1\/\\sqrt{\\tilde{\\kappa}}) + \\cdots$ around $\\tilde{\\kappa}=\\infty$. \nWe have found in our investigation that as long as $\\tilde{\\kappa}$ is \ntaken sufficiently large so that a continuum limit exists from the FM \nphase to the FMD phase (not the PM phase), the model is as good as at \nthe perturbative limit $\\tilde{\\kappa}=\\infty$. This seems to hold true \nin our particular implementation, {\\em i.e., the domain wall fermion \ncase}, in a {\\em strong sense}, because the perturbative 1-loop \ncorrections ({\\em i.e.}, the leading nontrivial corrections) to the \nfermion propagators are found to be negligible (Figs. 4 and 5). In gauge \nfixing the Smit-Swift model, however, 1-loop corrections to the fermion \npropagators were small but not negligible \\cite{bock1}. In the present \ninvestigation with domain wall fermions, to the accuracy of our \ncalculations, the reduced model spectrum is that of a free domain wall \nmodel with absolutely no trace of the $\\varphi$ fields.\n\nIn our investigation of the reduced model, only a counterterm with \ncoefficient $\\kappa$ was sufficient to reach the FM-FMD phase transition \nby decreasing $\\kappa$ from the FM-side. A dimension-three fermion mass \ncounterterm was not needed, because domain wall fermions have the robust \nproperty that mixing between the opposite chiral modes at the domain \nwall and the anti-domain wall is exponentially damped and is \nnegligible for large $L_s$. \n\nWe have carried out and presented the \nperturbative results for fermion propagators and mass matrix in \nreasonable detail, because, although the technique employed is not new, \nthe explicit results are available mostly for only the QCD (wall) \nimplementation rather than the wall-antiwall implementation suitable for \na chiral gauge theory. With transverse gauge fields back on (and \nfermions in an anomaly-free representation), the full gauge-fixed domain \nwall model of $L\\chi GT$ is ready for a perturbative treatment with the \nsame techniques as used in this paper if the transverse gauge coupling \nis perturbative. The model is also ready for a nonperturbative treatment \nby numerical simulation for strong transverse gauge coupling. \n\nOur numerical computations have been done in the quenched approximation \nand they agree for the fermion propagators very well with zeroth order \nperturbation theory with the 1-loop corrections almost negligible in \nour case. Effects of fermion loops would enter these calculations at \nleast at the 2-loop level.  Inclusion of dynamical fermions in our \nnonperturbative numerical investigation should not affect our results \nabout the spectrum of the model at all if the relevant part of the phase \ndiagram remains qualitatively the same.\n \nSo far the gauge fixing method is applicable only to the abelian theory. \nExtension to a nonabelian gauge group is nontrivial and  is \nbeing pursued at the current time \\cite{nab}.\n\nAs commented at the end of section II, the reduced gauge fixed \ndomain wall model  with $0<y<1$ is interesting because at $y=0$ the \nmodel is known to have mirror modes at the waveguide boundaries. This \nwill be taken up in a separate publication \\cite{big}. \n\n                                                                         \n\\acknowledgements \n \nThe authors thank Yigal Shamir, Maarten Golterman, Palash Pal and \nKrishnendu Mukherjee for useful discussions. One of the authors (SB) \nwould like to thank the Theory Group of Saha Institute of Nuclear \nPhysics for providing the facilities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDark energy has been modelled by a large variety of theories since decades. Among these, many rely on the introduction of additional scalar fields whose dynamics, at the origin of the late-time acceleration of the expansion of the Universe, is determined by arbitrary parametric functions, potentials and\/or coupling (see \\textit{e.g.} \\cite{Brax_2018}). These are the so-called scalar-tensor theories of modified gravity. In particular, the class of Horndeski theories is of great interest as it contains all models of modified gravity with a single additional scalar field leading to second-order equations of motion  \\cite{Horndeski:1974wa,Deffayet:2011gz}. Extensions of Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of higher orders have also been explored \\cite{Langlois:2015skt,Langlois:2018dxi}. Particular Horndeski theories are described by the specification of four arbitrary functions of the scalar field and its kinetic energy, leading to a huge variety of models and phenomenological behaviours.\n\nAmong these wide classes of models, some can be built from first physical principles or arguments of symmetry. For instance, the Galileon model \\cite{Nicolis:2008in} and its covariant extension \\cite{Deffayet:2009wt} was built by imposing a galilean symmetry for the scalar field, leaving only five free numerical parameters. We can also cite, among many others, the pure kinetic gravity theory \\cite{GUBITOSI2011113}, massive gravity in the non-relativistic limit \\cite{deRham:2010ik,deRham:2010kj} and the DBI-Galileon \\cite{deRham:2010eu} which is the main object of this paper. \n\nThe DBI-Galileon is a model that falls into the class of Brane-world scenarios of extra-dimension theories, where the matter fields are confined on a 4D brane while gravity can propagate into the additional spatial dimensions. Of most interest for the DBI-Galileon is the case of a single extra-dimension as it has been shown that theories with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter solution \\cite{Hinterbichler:2010xn}. The action include a volume term for the 4D brane in the 5D bulk which leads to the well-known Dirac-Born-Infeld (DBI) action. This action, and DBI-like extensions, can lead to a self-accelerating solution and has been thoroughly studied as a candidate model in the early Universe cosmic inflation paradigm \\cite{Silverstein:2003hf,Langlois:2008qf}. In addition, the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit \\cite{deRham:2010eu} but giving a physical meaning to their free parameters: the Planck  mass in the brane, the Planck mass in the bulk, etc. In particular, the brane tension here plays the role of the cosmological constant which brings a possible interpretation of its nature.\n\nThe original probe brane construction has been revisited in \\cite{Zumalacarregui:2012us} where the matter metric is disformally related to a standard gravitational metric, or in \\cite{Goon:2012dy} in the framework of spontaneous symmetry breaking for the 5D space-time symmetries broken by the presence of the brane, bridging the gap with Brane-world scenarios developed in the context of quantum field theory and an interpretation of the scalar field as a Nambu-Goldstone boson \\cite{Dobado:2000gr,Cembranos:2001my}. The DBI-Galileon model has been studied extensively in special cases of the maximally symmetric bulk geometry \\cite{Burrage:2011bt,Goon:2011qf}. However, to our knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential candidate for Dark Energy has been performed so far.\n\nIn this paper we develop the DBI-Galileon theory in the non relativistic limit (Section~\\ref{sec:dbigalileon}) and study its dynamics in the Friedmann-Lema\u00eetre-Robertson-Walker flat metric (Section~\\ref{sec:background}). The perturbation stability is explored in Section~\\ref{sec:perturbations} and then discussed in Section~\\ref{sec:discussion}.\n\n\\section{DBI-Galileon in the non-relativistic limit}\\label{sec:dbigalileon}\n\n\\subsubsection*{DBI-Galileon \\\\}\n\nWe are interested in the description of a four dimensional brane universe embedded in a five dimensional bulk from the cosmological perspective. In this context, it has been shown in \\cite{deRham:2010eu} that the most general action on the brane is given by the 4D Lovelock terms \\cite{Lovelock:1971yv} inside the brane and the boundary terms associated to the 5D Lovelock terms in the bulk:\n\\begin{equation}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-g} \\left( -\\Lambda - M_{5}^{3}K + \\frac{M_{P}^{2}}{2}R - \\beta\\frac{M_{5}^{3}}{m^{2}}\\mathcal{K}_{GB} + \\mathcal{L}_{m} \\left( \\tilde{q}_{\\mu\\nu}, \\psi_{m} \\right) \\right), \\label{eq:dRT action}\n\\end{equation}\nwhere $g$ is the induced metric on the brane, $\\Lambda$ the brane cosmological constant, $K$ is the extrinsic curvature of the brane, $R$ is the Ricci scalar on the brane, $\\mathcal{K}_{GB}$ is the boundary term on the brane of the Gauss-Bonnet scalar in the bulk and $\\mathcal{L}_{m}$ is the Lagrangian density of matter that lives confined in the brane. In the above action has been defined the 4D Planck mass $M_{P}$, its 5D counterpart $M_{5}$ and their ratio $m=M_{5}^{3}\/M_{P}^{2}$, and $\\beta$ is an arbitrary constant parameter.\n\nBecause the action is defined on the 4-dimensional brane, the above quantities are expressed in terms of the induced metric $g$, as opposed to the 5-dimensional bulk metric $G$. Assuming we have a coordinate system $\\left(x^{\\mu}, y \\right)$ in the bulk, Greek letters being defined to span from 0 to 3, such that the length element in this frame is $ds^{2} = q_{\\mu\\nu} dx^{\\mu}dx^{\\nu} + \\left( dy \\right)^{2}$. The brane position in the bulk is defined by $y = \\pi \\left( x^{\\mu} \\right)$ such that we can express the induced metric from the bulk metric:\n\\begin{equation}\n    g_{\\mu\\nu} = q_{\\mu\\nu} + \\partial_{\\mu} \\pi \\partial_{\\nu} \\pi \\qquad \\mathrm{and} \\qquad g^{\\mu\\nu} = q^{\\mu\\nu} - \\gamma^{2} \\partial^{\\mu} \\pi \\partial^{\\nu} \\pi\n\\end{equation}\nwith the Lorentz factor $\\gamma = \\left( 1+ \\left( \\partial \\pi \\right)^{2} \\right)^{-1\/2}$. From this expression of the induced metric, we can explicitly write the action \\eqref{eq:dRT action} using the bulk metric $q$ and the scalar field $\\pi$. As an illustration, see how the cosmological constant part of the action on the brane can be expressed:\n\\begin{equation}\n    \\mathcal{S}_{\\Lambda} = - \\Lambda \\int d^{4}x \\sqrt{-g} = - \\Lambda \\int d^{4}x \\sqrt{-q} \\sqrt{1 + \\left( \\partial\\pi \\right)^{2}}\n\\end{equation}\nAs was pointed out in the original paper by de Rham and Tolley \\cite{deRham:2010eu}, we recover a DBI term in the action. This leads us to associate the cosmological constant $\\Lambda$ with a brane tension $f$, following $\\Lambda = f^{4}$ for unit convenience.\n\n\\subsubsection*{Non-relativistic limit \\\\}\n\nIf the derivatives of the field vanish, $\\partial_{\\mu} \\pi = 0$, then we recover standard GR. Because we are interested in the cosmological setting, where predictions from the $\\Lambda$CDM model based on GR match precisely a large range of observations, we consider only small corrections to GR. Therefore, we consider the DBI-Galileon in the so-called non-relativistic limit where $\\left( \\partial \\pi \\right)^{2} \\ll 1$. In particular, the DBI part of the action in the non-relativistic limit becomes:\n\\begin{equation}\n    \\mathcal{S}_{f} = -f^{4} \\int d^{4}x \\sqrt{-q} \\left[ 1 + \\frac{\\nabla_{\\mu} \\pi \\nabla^{\\mu} \\pi}{2} - \\frac{\\left( \\nabla_{\\mu} \\pi \\nabla^{\\mu} \\pi \\right)^{2}}{8} + \\ldots \\right]\n\\end{equation}\nWe see that, to have a canonically normalized field, we can proceed to the field redefinition $\\pi \\rightarrow \\varphi= f^{2} \\pi $. Up to degree 5 in $\\partial \\varphi\/f^{2}$, we find the following Lagrangian operators for the DBI-Galileon in the non-relativistic limit:\n\\begin{eqnarray}\n    && \\mathcal{L}_{f} = 1 + \\frac{\\mathcal{L}_{2}}{2f^{4}} - \\frac{X^{2}}{2} + \\ldots \\\\\n    && \\mathcal{L}_{K} = - \\frac{\\mathcal{L}_{3}}{2f^{6}} - \\frac{2X}{f^{6}}\\left[ \\psi \\right] + \\ldots \\\\\n    && \\mathcal{L}_{R} = \\bar{R} + \\frac{1}{f^{4}} \\left( \\left[ \\Phi \\right]^{2} - \\left[ \\Phi^{2} \\right] - \\frac{f^{4}X}{2}\\bar{R} - 2\\bar{R}_{\\mu\\nu}\\nabla^{\\mu}\\varphi \\nabla^{\\nu}\\varphi \\right) + \\frac{\\mathcal{L}_{4}}{4f^{8}} + \\ldots \\\\\n    && \\mathcal{L}_{\\mathcal{K}_{GB}} = \\frac{2}{f^{6}} \\left( \\left( - \\bar{R}_{\\mu\\nu}\\left[ \\Phi \\right] + 2 \\bar{R}_{\\mu\\rho} \\Phi_{\\nu}^{\\rho} + \\bar{R}_{\\mu\\rho\\nu\\lambda}\\Phi^{\\rho\\lambda} \\right) \\nabla^{\\mu}\\varphi \\nabla^{\\nu}\\varphi + \\frac{1}{3}\\left[ \\Phi \\right]^{3} - \\left[ \\Phi \\right]\\left[ \\Phi^{2} \\right] + \\frac{2}{3}\\left[ \\Phi^{3} \\right] \\right. \\nonumber \\\\\n    && \\left.  - \\frac{1}{2}\\bar{R}\\left[ \\psi \\right] \\right) + \\frac{\\mathcal{L}_{5}}{3f^{10}} + \\ldots\n\\end{eqnarray}\nWe defined on the above expressions the tensor $\\Phi_{\\mu\\nu} = \\nabla_{\\mu}\\nabla_{\\nu} \\varphi$, and the three scalars $\\left[ \\Phi \\right] = \\Phi_{\\mu}^{\\mu}$, $\\left[ \\psi \\right] = \\partial^{\\mu} \\varphi \\cdot \\Phi_{\\mu\\nu} \\cdot \\partial^{\\nu} \\varphi$ and $X = -\\left( \\partial \\varphi \\right)^{2}\/2f^{4}$. Furthermore, the $\\mathcal{L}_{2 \\ldots 5}$ Lagrangian operators are the covariant Galileon model Lagrangian operators as defined in \\cite{Nicolis:2008in,Deffayet:2009wt}. Therefore, the theory described here is a generalization of the Galileon model. We can note that the additional terms in $\\mathcal{L}_{R}$ and $\\mathcal{L}_{\\mathcal{K}_{GB}}$ compared to the Galileon Lagrangian operators vanish in the particular case of a flat geometry. We conclude that the DBI-Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon than the covariant Galileon \\cite{Deffayet:2009wt} or dRGT massive gravity \\cite{deRham:2010ik,deRham:2010kj}, with an additional terms in $\\mathcal{L}_{f}$ and another one in $\\mathcal{L}_{K}$. That point noted, we will stay at leading order in $X$ in the following. Being a theory of a scalar field interacting with a metric, with equations of motion of at most second order, it can be described as a Horndeski theory \\cite{Horndeski:1974wa,Deffayet:2011gz} with the following Horndeski functions:\n\\begin{align}\nG_{2} & = \\mathcal{A} \\left( \\varphi \\right) - f^{4} \\left( 1 - X + \\ldots \\right) \\label{eq:Horndeski G2} \\\\\nG_{3} & = \\frac{M_{5}^{3}}{f^{2}} \\left( X + \\ldots \\right) \\\\\nG_{4} & = \\frac{M_{P}^{2}}{2} \\left( 1 - X -  X^2\/2 \\ldots \\right) \\\\\nG_{5} & = -2\\beta\\frac{M_{5}^{3}}{m^{2}f^{2}} \\left( X + X^2 + \\ldots \\right) \\label{eq:Horndeski G5}\n\\end{align}\n\n\n\\section{Cosmological background evolution}\\label{sec:background}\n\nWe expand the dynamics of the fields around a flat FLRW background on the 4D slice of the bulk along $x^{\\mu}$ where the properties of the brane, $g_{\\mu\\nu}$ and $\\varphi$, are defined. On this slice, the length element is:\n\\begin{equation}\n    ds^{2} = q_{\\mu\\nu}dx^{\\mu}dx^{\\nu} = -dt^{2} + a^{2} \\left( t \\right)\\delta_{ij}dx^{i}dx^{j} \\label{eq:FLRW bulk}\n\\end{equation}\n\nIt might appear more natural to expand around a flat FLRW background on the brane with its induced metric $g_{\\mu\\nu}$. However, the two metrics $q_{\\mu\\nu}$ and $g_{\\mu\\nu}$ are related by a disformal transformation involving the scalar field which, at the background level, depends only on the physical time:\n\\begin{equation}\n    \\bar{g}_{\\mu\\nu} = \\bar{q}_{\\mu\\nu} + \\frac{\\left( \\dot{\\bar{\\varphi}} \\left( t \\right) \\right)^{2}}{f^{4}} \\delta_{\\mu 0}\\delta_{\\nu 0}\n\\end{equation}\nwhere barred quantities are taken at the background level. Therefore, the geometry on the brane is also of the FLRW type, but with a different definition for the physical time, leading to a different scale factor and expansion history. Because in our case $\\dot{\\bar{\\varphi}} \\ll f^{2}$, the expansion history on the 4D slice and on the brane will, then, be approximately the same. In addition, equations~\\eqref{eq:Horndeski G2}-\\eqref{eq:Horndeski G5} determine a self-consistent Horndeski theory of gravity with the metric $q_{\\mu\\nu}$ and the scalar field $\\varphi$ without referring to the induced metric $g_{\\mu\\nu}$.\nThus, in the following we apply the well-known techniques used in the context of Horndeski theories.\n\nIn the non-relativistic limit, action~\\eqref{eq:dRT action} reads:\n\\begin{equation}\\label{eq:action_galdbi}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-q} \\left(\\mathcal{L}_f + \\mathcal{L}_K + \\mathcal{L}_R + \\mathcal{L}_{\\mathcal{K}_{GB}} + \\mathcal{L}_{m} \\left( q_{\\mu\\nu}, \\psi_{m} \\right) \\right).\n\\end{equation}\nWe define $\\Omega_m^0$ and $\\Omega_r^0$ the standard present energy density parameters for pressureless matter and radiation respectively, and $\\bar{H}$ the normalized Hubble rate $H\/H_0$ with $H_0$ the present Hubble constant. Prime symbol denotes the derivative with respect to $\\ln a$. We set: \n\\begin{equation}\n\\tilde{x} = \\frac{\\varphi' H_0}{f^2}, \\quad \\Omega_\\Lambda^0 = \\frac{\\Lambda}{3 H_0^2 M_P^2}= \\frac{f^4}{3 H_0^2 M_P^2}, \\quad \\eta = \\frac{M_5^3}{M_P^2 H_0}, \\quad \\xi = \\frac{\\beta}{\\eta}, \\quad \\kappa = \\frac{M_P H_0}{f^2}.\n\\end{equation}\nThen, the two Friedmann equations derived from action~\\eqref{eq:action_galdbi} are:\n\\begin{eqnarray}\n    && \\bar{H}^2 =  - \\frac{3}{2} \\bar{H}^4 \\tilde{x}^2- \\frac{15}{8} \\bar{H}^6 \\tilde{x}^4  + \\eta \\bar{H}^4 \\tilde{x}^3 - \\frac{10}{3} \\xi \\bar{H}^6 \\tilde{x}^3 - \\frac{14}{3} \\xi \\bar{H}^8 \\tilde{x}^5 \\nonumber \\\\\n    && + \\frac{\\Omega_m^0}{a^3} + \\frac{\\Omega_r^0}{a^4} + \\Omega_\\Lambda^0 \\left( 1 + \\frac{1}{2}\\bar{H}^2 \\tilde{x}^2 \\right) \\\\[5pt]\n    && \\bar{H}^2 +\\frac{2}{3}\\bar{H} \\bar{H}' = - \\frac{2}{3} \\xi\\left( 2\\bar{H}^6 \\tilde{x}^3 + 5\\bar{H}^5 \\tilde{x}^3 \\bar{H}'  + 3\\bar{H}^6 \\tilde{x}^2 \\tilde{x}' + 2\\bar{H}^8 \\tilde{x}^5 + 5\\bar{H}^8 \\tilde{x}^4 \\tilde{x}'  + 7\\bar{H}^7 \\tilde{x}^5 \\bar{H}'\\right) \\nonumber \\\\    \n    && - \\frac{1}{2}\\bar{H}^4 \\tilde{x}^2 - \\bar{H}^3 \\tilde{x}^2 \\bar{H}'  - \\frac{2}{3} \\bar{H}^4 \\tilde{x} \\tilde{x}' + \\frac{1}{3} \\eta \\bar{H}^3 \\tilde{x}^2 (\\bar{H}\\tilde{x})' - \\frac{3}{8} \\bar{H}^6 \\tilde{x}^4 - \\frac{5}{4} \\bar{H}^5 \\tilde{x}^4 \\bar{H}'- \\bar{H}^6 \\tilde{x}^3 \\tilde{x}' \\nonumber \\\\\n    && - \\frac{\\Omega_r^0}{3a^{4}} + \\Omega_\\Lambda^0 \\left( 1 - \\frac{1}{2}\\bar{H}^2 \\tilde{x}^2 \\right)\n\\end{eqnarray}\nWe see that we recover the $\\Lambda$CDM equations when setting $\\tilde{x}$ to zero, but with a physical interpretation of the cosmological constant as the brane tension. The DBI model proposed here is then an extension of the standard model of cosmology, and as such follows a late accelerated expansion but with a physical interpretation of the origin of $\\Lambda$ as a brane tension.\nUsing the same methodology and similar notations as in \\cite{Stephen-Appleby_2012}, we derive the field $\\varphi$ equation of motion from action~\\eqref{eq:action_galdbi}. This leads to a system of two coupled equations\n\\begin{equation}\n    \\left.\\begin{array}{ll}\n\\tilde x' & = - \\tilde x + \\dfrac{\\alpha \\lambda - \\sigma \\gamma}{\\sigma \\beta - \\alpha \\omega}  \\\\\n\\bar H' & = \\dfrac{\\omega \\gamma - \\beta\\lambda}{\\sigma \\beta - \\alpha\\omega}\n\\end{array}\\right.\n\\end{equation}\nwith\n\\begin{align*}\n\t\\beta &= 2\\bar{H}^4+ \\frac{9}{2} \\bar{H}^6 \\tilde{x}^2 - \\Omega_\\Lambda^0 \\bar{H}^2  - 2 \\eta \\bar{H}^4 \\tilde{x} + 4 \\xi \\bar{H}^6 \\tilde{x} + \\frac{40}{3} \\xi \\bar{H}^8 \\tilde{x}^3\n \\\\\n\t\\alpha & = 6 \\bar{H}^3 \\tilde{x} -  \\Omega_\\Lambda^0 \\bar{H} \\tilde{x} + \\frac{15}{2} \\bar{H}^5 \\tilde{x}^3 - 3 \\eta \\bar{H}^3 \\tilde{x}^2 + 10 \\xi \\bar{H}^5 \\tilde{x}^2  + \\frac{70}{3} \\xi \\bar{H}^7 \\tilde{x}^4 \n \\\\\n\t\\gamma & = 4 \\bar{H}^4 \\tilde{x} -2  \\Omega_\\Lambda^0 \\bar{H}^2\\tilde{x}  -  \\eta \\bar{H}^4 \\tilde{x}^2 + 2 \\xi \\bar{H}^6 \\tilde{x}^2 - \\frac{10}{3} \\xi \\bar{H}^8 \\tilde{x}^4 \n \\\\\n\t\\omega & = \\frac{4}{3} \\bar{H}^4 \\tilde{x} + \\bar{H}^6 \\tilde{x}^3 - \\frac{1}{3} \\eta \\bar{H}^4 x^2 + 2 \\xi \\bar{H}^6 \\tilde{x}^2 + \\frac{10}{3} \\xi \\bar{H}^8 \\tilde{x}^4\n \\\\\n\t\\sigma &=  \\frac{2}{3} \\bar{H} + 2 \\bar{H}^3 \\tilde{x}^2 + \\frac{15}{12} \\bar{H}^5 \\tilde{x}^4 - \\frac{1}{3} \\eta \\bar{H}^3 \\tilde{x}^3 + \\frac{10}{3} \\xi \\bar{H}^5 \\tilde{x}^3 + \\frac{14}{3} \\xi \\bar{H}^7 \\tilde{x}^5 \n \\\\\n\t\\lambda &= \\bar{H}^2  + \\frac{\\Omega_r^0 }{3 a^4} - \\Omega_\\Lambda^0 + \\frac{1}{2} \\Omega_\\Lambda^0 \\bar{H}^2 \\tilde{x}^2 - \\frac{1}{3} \\bar{H}^4 \\tilde{x}^2 - \\frac{5}{8} \\bar{H}^6 \\tilde{x}^4 + \\frac{1}{3} \\eta \\bar{H}^4 \\tilde{x}^3  -\\frac{2}{3} \\xi \\bar{H}^6 \\tilde{x}^3  - 2 \\xi \\bar{H}^8 \\tilde{x}^5\n\\end{align*}\nGiven values for the parameters $\\Omega_m^0$, $ \\eta$, $\\xi$ and $\\kappa$, and initial conditions $\\tilde x_0$ and $H_0$, this system can be integrated to compute background cosmology observables like the distance moduli of type Ia supernovae. In Figure~\\ref{fig:hubble_diagram} we illustrate this with a Hubble diagram prediction compared with recent type Ia supernova data \\cite{Scolnic2018}. As an initial condition for $\\tilde{x}$, we chose to set $\\tilde{x}_0 = 6\\times 10^{-8}$ today. This is the maximum value allowed by the constraint on gravitational wave speed (see Section~\\ref{sec:tensorial}) coming from the quasi simultaneous observation of photons and gravitational waves after neutron star merger event GW170817A~\\cite{LIGOScientific:2017zic}.  Nevertheless, before discussing more the cosmological scenarios proposed by the DBI-Galileon model, stability conditions much be computed first to assess the viability of the models for any set of parameters at the perturbation level.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hubble_diagramme.pdf}\n\\caption{Hubble diagram prediction for the non-relativistic DBI-Galileon model for $\\Omega_\\Lambda^0 = 0.7$, $\\tilde x_0 = 6 \\times 10^{-8}$, $\\eta = \\xi=0$ (blue) compared with binned Pantheon data (black points) \\cite{Scolnic2018}. We used $\\mathcal{M}=23.81$ for the offset magnitude of the diagram. Residuals to the fit are presented in the bottom panel.\n}\\label{fig:hubble_diagram}\n\\end{figure}\n\n\n\\section{Stability conditions}\\label{sec:perturbations}\n\nTo be viable as a description of our Universe, the model has to fulfill stability conditions. These requirements apply to degrees of freedom propagating around the fixed background, \\textit{i.e.} to the cosmological perturbations, that are capable of undermining the stability of the Universe. In the determination of the stability conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism described in \\cite{DeFelice:2011bh} for Horndeski theories. In our case, these stability conditions are defined from the following quantities derived from the particular Horndeski functions \\eqref{eq:Horndeski G2} to \\eqref{eq:Horndeski G5}:\n\\begin{align}\n    \\omega_{1} & = 1 +  \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  + \\frac{3}{8} \\bar{H}^4 \\tilde{x}^4  + 2 \\xi \\bar{H}^4 \\tilde{x}^3  + 2\\xi \\bar{H}^6 \\tilde{x}^5 \\\\\n    \\omega_{2} &  = 2\\bar{H} + 3 \\bar{H}^3 \\tilde{x}^2   + \\frac{15}{4} \\bar{H}^5 \\tilde{x}^4  -\\eta \\bar{H}^3 \\tilde{x}^3  + 10 \\xi \\bar{H}^5\\tilde{x}^3  + 14 \\xi \\bar{H}^7 \\tilde{x}^5\\\\\n    \\omega_{3} & = 9\\left( -\\bar{H}^2 + \\frac{1}{2}\\Omega_\\Lambda^0 \\bar{H}^2\\tilde{x}^2 - 3 \\bar{H}^4 \\tilde{x}^2  + 2 \\eta \\bar{H}^4 \\tilde{x}^3 -10 \\xi \\bar{H}^6 \\tilde{x}^3- \\frac{45}{8} \\bar{H}^6 \\tilde{x}^4   -\\frac{56}{3}\\xi \\bar{H}^8 \\tilde{x}^5 \\right)  \\\\\n    \\omega_{4} & = 1 - \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  - \\frac{1}{8}\\bar{H}^4 \\tilde{x}^4 + 2 \\xi \\bar{H}^3 \\tilde{x}^2 (\\bar{H} \\tilde{x})'  + 2 \\xi \\bar{H}^5 \\tilde{x}^4 (\\bar{H}\\tilde{x})'\n\\end{align}\n\n\n\\subsection*{Tensorial stability conditions}\\label{sec:tensorial}\n\nIn order to avoid ghosts and Laplacian instabilities, we impose the following constraints on the sign of the kinetic term and on the sign of the gravitational waves speed squared:\n\\begin{align}\n    Q_{t} & \\equiv \\frac{\\omega_{1}}{4} =  \\frac{1}{4} + \\frac{1}{8}\\bar{H}^2\\tilde{x}^2   + \\frac{3}{32}\\bar{H}^4\\tilde{x}^4 + \\frac{1}{2} \\xi \\bar{H}^4 \\tilde{x}^3  + \\frac{1}{2}\\xi \\bar{H}^6 \\tilde{x}^5   > 0 \\\\\n    c_{t}^{2} & \\equiv \\frac{\\omega_{4}}{\\omega_{1}}  = \\frac{1 - \\frac{1}{2}\\bar{H}^2\\tilde{x}^2  - \\frac{1}{8}\\bar{H}^4 \\tilde{x}^4 + 2 \\xi \\bar{H}^3 \\tilde{x}^2 (\\bar{H} \\tilde{x})'  + 2 \\xi \\bar{H}^5 \\tilde{x}^4 (\\bar{H}\\tilde{x})'}{1 +  \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  + \\frac{3}{8} \\bar{H}^4 \\tilde{x}^4  + 2 \\xi \\bar{H}^4 \\tilde{x}^3  + 2\\xi \\bar{H}^6 \\tilde{x}^5} \\geq 0  \n\\end{align}\n\nIn particular, we see that the gravitational wave speed depends on $\\tilde{x}$, and tends to 1 when  $\\tilde{x} \\rightarrow 0$ :\n\\begin{align}\n4 Q_t & \\simeq 1 + 2(\\bar{H} \\tilde{x})^2 \\\\\n    c_{t} & \\simeq 1 - \\frac{1}{2} (\\bar{H} \\tilde{x})^2\n\\end{align}\nGiven the very tight constraint on the speed of gravitational waves, equal to the speed of light up to a $\\sim 10^{-15}$ difference \\cite{LIGOScientific:2017zic,Creminelli:2017sry,Ezquiaga:2017ekz}, this justifies \\textit{a posteriori} the relevance of the non-relativistic limit where $\\tilde{x} \\ll 1$. Moreover, we see that tensorial perturbations are stable in this limit since $Q_t > 0$.\n\n\\subsection*{Scalar stability conditions}\n\nSimilar stability conditions apply to the scalar degrees of freedom, here including the scalar perturbations of matter components:\n\\begin{align}\n    Q_{s} & \\equiv \\frac{\\omega_{1} \\left( 4\\omega_{1}\\omega_{3} + 9\\omega_{2}^{2} \\right)}{3\\omega_{2}^{2}} > 0 \\label{eq:Qs} \\\\\n    c_{s}^{2} & \\equiv \\frac{3 \\left( 2\\omega_{1}^{2}\\omega_{2}H - \\omega_{2}^{2}\\omega_{4} + 4\\omega_{1}\\omega_{2}\\dot{\\omega}_{1} - 2\\omega_{1}^{2}\\dot{\\omega}_{2} \\right) - 6\\omega_{1}^{2} \\sum \\left( 1+w_{i} \\right) \\rho_{i}}{\\omega_{1} \\left( 4\\omega_{1}\\omega_{3} + 9\\omega_{2}^{2} \\right)} \\geq 0\n\\end{align}\nwhere $w_{i}$ and $\\rho_{i}$ are respectively the equation of state parameter and the energy density of the fluid $i$, and the sum runs over all the components of the Universe (here only pressureless matter and radiation). At the lowest order in $\\tilde{x}$, we get:\n\\begin{align}\nQ_s & \\simeq \\frac{3}{2} (\\Omega_\\Lambda^0 - \\bar{H}^2) \\tilde{x}^2 \\\\\nc_{s}^{2} & \\simeq 1 + \\frac{2 \\left( \\eta \\bar{H}^{2}\\tilde{x}' - \\bar{H}\\h' - 2\\xi\\bar{H}^{4}\\tilde{x}' \\right)}{3 \\left( \\Omega_{\\Lambda}^{0} - \\bar{H}^{2} \\right)}\n\\end{align}\n\nWith $\\tilde{x} \\ll 1$, a fit of the DBI-Galileon model to data leads to cosmological parameters close to the standard model ones: $\\Omega_m^0 \\approx 0.3$ and $\\Omega_\\Lambda^0 \\approx 0.7$ \\cite{Planck2018}. Therefore, from the first Friedmann equation, we get $\\Omega_\\Lambda^0 < \\bar{H}^2$ for all relevant models in agreement with cosmological observations. As $Q_s \\leq 0$, the DBI-Galileon model contains scalar instabilities unless it reduces to GR. One way to avoid this would be to add a spatial curvature to the metric, but with a strong energy density (at least $\\sim 0.3$) which is also excluded by observations \\cite{Planck2018}. \n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection*{Physical interpretation}\n\nFrom the definition \\eqref{eq:Qs}, we see that the dominant terms come $\\G2$ (giving the $\\Omega_\\Lambda^0$ term) and $\\G4$ (giving the $\\bar{H}^2$ term). The competition between the two terms leads to the ghost-like behaviour in a cosmological setting: $Q_s \\leq 0$. In other words, it is the result of the competition between the DBI and the Einstein-Hilbert terms. The DBI action will have the effect of stretching the brane towards an extremal surface, whereas the Einstein-Hilbert term on the brane will tend to make the brane contract on itself from the effect of curvature. However, in the non-relativistic limit of the DBI-Galileon, the Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect from the cosmological constant is not strong enough to counterbalance, leading to an instantaneous decay of the vacuum state.\n\nBecause the DBI-Galileon action is the most general one can find of a 4D probe brane in a 5D bulk, we expect this statement to be quite general for all such theories studied in the current context.  Indeed, this is true in a standard cosmological setting which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW background on the brane). Therefore, the only way to evade this ghostly behaviour in cosmology is to include the full relativistic dynamics of the theory ($\\tilde{x} \\sim 1$). We have seen that, in this case, we expect significant deviations of the speed of gravitational waves $c_t$ from $c$. This is not a definitive impossibility though if the full DBI-Galileon is viewed as an effective theory valid only at cosmological scales for which the speed of gravitational waves has not been probed \\cite{deRham:2018red,Ezquiaga:2018btd}. Indeed, the constraint on the gravitational speed from the observation of GW170817 in coincidence with GRB170817A \\cite{LIGOScientific:2017zic} is only valid on small scales probed by LIGO and Virgo. A modification of the dispersion relation of gravitational waves at small scales from operators present in the UV complete theory could allow $c_{t} \\neq c$ on cosmological scales while being compatible with current astrophysical observations. Waiting for the next generation of gravitational wave interferometers, in particular LISA, which will be able to probe this relation at larger scales \\cite{Barausse:2020rsu}, this possibility remains open.\n\n\\subsection*{Direct coupling to matter}\n\nIn the context of cosmology, where standard model matter is present, there might be direct coupling to the scalar field. In that case, the metric $\\tilde{q}$ to which matter is sensitive is different than the space-time metric $q$:\n\\begin{equation}\\label{eq:action_galdbi-qtilde}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-q} \\left(\\mathcal{L}_f + \\mathcal{L}_K + \\mathcal{L}_R + \\mathcal{L}_{\\mathcal{K}_{GB}}\\right) + \\int dx^{4} \\sqrt{-\\tilde q} \\mathcal{L}_{m} \\left(\\tilde q_{\\mu\\nu}, \\psi_{m} \\right).\n\\end{equation}\nIt has been shown in \\cite{Bekenstein:1992pj} that the two metrics are related by a disformal transformation of the following form:\n\\begin{equation}\n    q_{\\mu\\nu} = A \\left( \\varphi, \\tilde{X} \\right) \\tilde{q}_{\\mu\\nu} + B \\left( \\varphi, \\tilde{X} \\right) \\frac{\\partial_{\\mu}\\varphi \\partial_{\\nu}\\varphi}{f^{4}} \\label{eq:disformal transformation}\n\\end{equation}\nwhere $A$ and $B$ are arbitrary functions of the scalar field and $\\tilde{X} = -\\tilde{q}_{\\mu\\nu}\\partial_{\\mu}\\varphi\\partial_{\\nu}\\varphi\/2f^{4}$. For simplicity and following the treatment of the covariant Galileon \\cite{Stephen-Appleby_2012}, we assume that $A$ and $B$ are constant parameters. This can be further justified by the fact that, a dependency on $X$ would introduce, in general, higher order terms which would go beyond the framework of Horndeski theories \\cite{Bettoni:2013diz}, and a dependency on $\\varphi$ would, in general, break the shift symmetry followed by the scalar field $\\varphi$ in the probe brane context. Note that, when $A = -B$, matter is coupled to the induced metric on the brane.\n\nContrary to the covariant Galileon, the DBI-Galileon action is not invariant by such a change of reference frame. However, new terms that can not be absorbed into a redefinition of the parameters arise only at higher order in $\\tilde{X}$. Therefore, the non-relativistic dynamics is not change by the introduction of a direct coupling between the scalar field and matter of the form \\eqref{eq:disformal transformation} with constant parameters. In particular, this does not prevent the perturbations around the FLRW background from showing ghost instabilities.\n\\subsection*{Generalization}\n\nThe DBI-Galileon is a particular example of the more general class of shift-symmetric Horndeski theories.  These are subclass of Horndeski theories which are invariant under a shift symmetry of the scalar field $\\varphi \\rightarrow \\varphi + c$ \\cite{Sotiriou:2013qea,Sotiriou:2014pfa}. In these theories, the arbitrary Horndeski functions are restricted to be functions of $X$ alone. In order to make the non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR:\n\\begin{align}\n    G_{2} & \\equiv \\Lambda + \\sum_{n=1}^{+\\infty} g_{2}^{\\left( n \\right)} X^{n} \\\\\n    G_{3} & \\equiv \\sum_{n=1}^{+\\infty} g_{3}^{\\left( n \\right)} X^{n} \\\\\n    G_{4} & \\equiv \\frac{M_{P}^{2}}{2} + \\sum_{n=1}^{+\\infty} g_{4}^{\\left( n \\right)} X^{n} \\\\\n    G_{5} & \\equiv \\sum_{n=1}^{+\\infty} g_{5}^{\\left( n \\right)} X^{n}\n\\end{align}\n\nThe constant terms in $G_{3}$ and $G_{5}$ do not appear in the expansion as they lead to total derivative terms. Because the Horndeski functions depend only on $X$, the $\\omega$ functions that determine the stability conditions reduce to:\n\\begin{eqnarray}\n    \\omega_{1} & \\equiv & 2 \\G4 - 2X\\left( 2\\G{4,X} + \\dot{\\phi}H\\G{5,X} \\right) \\\\\n    \\omega_{2} & \\equiv & 4 H\\G4 - 2X \\left( \\dot{\\phi}\\G{3,X} + 8 H\\G{4,X} + 5 \\dot{\\phi}H^{2} \\G{5,X} \\right) - 4X^{2}H \\left( 4 \\G{4,XX} + \\dot{\\phi}H\\G{5,XX} \\right) \\\\\n    \\omega_{3} & \\equiv & -18H^{2}\\G4 + 3X \\left( \\G{2,X} + 12 \\dot{\\phi}H\\G{3,X} + 42H^{2} \\G{4,X} + 30 \\dot{\\phi}H^{3} \\G{5,X} \\right) \\nonumber \\\\\n    && + 6X^{2} \\left( \\G{2,XX} + 3\\dot{\\phi}H\\G{3,XX} + 48 H^{2}\\G{4, XX} + 13H^{3}\\dot{\\phi}\\G{5,XX} \\right) \\nonumber \\\\\n    && + 12 X^{3}H^{2} \\left( 6\\G{4,XXX} + H\\dot{\\phi}\\G{5,XXX} \\right) \\\\\n    \\omega_{4} & \\equiv & 2\\G4 - 2X\\ddot{\\phi} \\G{5,X}\n\\end{eqnarray}\n\nFrom these, we can compute the quantity $Q_{s}$ up to first order in $X$:\n\\begin{equation}\n    Q_{s} = \\frac{X}{H^{2}} \\left( \\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)} \\right) + O \\left( X^{\\frac{3}{2}} \\right)\n\\end{equation}\n\nThis leads to a very simple formulation of the no-ghost condition, independent of $X$, in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit:\n\\begin{equation}\n     \\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)} > 0\n\\end{equation}\n\nIn the context of the brane galileon, where $\\g4^{\\left( 1 \\right)} = -M_{P}^{2}\/2$ and $\\g2^{\\left( 1 \\right)} = \\Lambda$, this is equivalent to the inequality which is never fulfilled in flat space:\n\\begin{equation}\n    \\Lambda - 3M_{P}^{2}H^{2} > 0 \\quad \\Leftrightarrow \\quad \\Omega_{\\Lambda}^{0} > \\bar{H}^{2}\n\\end{equation}\n\nOther stability conditions are given by:\n\\begin{align}\n    c_{s}^{2} & \\simeq 1 + \\frac{2\\Ddot{\\phi} \\g3^{\\left( 1 \\right)} + 4\\Dot{H}\\g4^{\\left( 1 \\right)} + 2\\Ddot{\\phi}H^{2}\\g5^{\\left( 1 \\right)}}{\\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)}} > 0\\\\\n    Q_t & \\simeq \\frac{M_{P}^{2}}{4} \\\\\n    c_t^{2} & \\simeq 1\n\\end{align}\nwhere we expressed these quantities at the lowest order. The two tensorial conditions are, thus, automatically satisfied in this context. On the other hand, the stability conditions for scalar perturbations at the lowest order give a simple inequality involving the parameters of the Taylor expansion, that can be easily checked at the background level.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nWe described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk space-time in the non-relativistic limit where its local kinetic energy is small compared to its tension. This model belongs to the class of shift-symmetric Horndeski theories, themselves being a subclass of the more general family of Horndeski theories. From the construction of the DBI-Galileon model, the free parameters of the model acquire a physical meaning. In particular, the interpretation of the cosmological constant is linked to the brane tension energy density. We derived the equations driving the evolution of the late-time Universe around a spatially flat FLRW cosmological background and studied the stability of scalar and tensorial perturbations. This model reduces to an expansion around standard GR, and therefore around standard $\\Lambda$CDM in the cosmological context. As such, it is naturally compatible with data in the non-relativistic limit provided the effect of the scalar field is small enough, even considering the speed of the gravitational waves. However, it revealed fatal ghostly behaviour for scalar perturbations  around the FLRW background. From there, we derived the corresponding stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit in the cosmological context and found very simple formulations for these conditions.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Marc Besan\u00e7on, Arnaud de Mattia and Vanina Ruhlmann-Kleider for their comments on the present paper. We also want to thank David Langlois for useful and interesting comments and suggestions.\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe supersymmetry technique is a powerful method in random matrix theory and disordered systems. For a long time it was thought to be applicable for Gaussian probability densities only \\cite{Efe83,VerZir85,VWZ85,Efe97}. Due to universality on the local scale of the mean level spacing \\cite{BreZee93a,BreZee93b,HacWei95,GMW98}, this restriction was not a limitation for calculating in quantum chaos and disordered systems. Indeed, results of Gaussian ensembles are identical for large matrix dimension with other invariant matrix ensembles on this scale. In the Wigner--Dyson theory \\cite{Bee97} and its corrections for systems with diffusive dynamics \\cite{Mir00}, Gaussian ensembles are sufficient. Furthermore, universality was found on large scale, too \\cite{AJM90}. This is of paramount importance when investigating matrix models in high--energy physics.\n\nThere are, however, situations in which one can not simply resort to Gaussian random matrix ensembles. The level densities in high--energy physics \\cite{BIPZ78} and finance \\cite{LCBP99} are needed for non-Gaussian ensembles. But these one--point functions strongly depend on the matrix ensemble. Other examples are bound--trace and fixed--trace ensembles \\cite{Meh04}, which are both norm--dependent ensembles \\cite{Guh06}, as well as ensembles derived from a non-extensive entropy principle \\cite{TVT04,BBP04,Abu04}. In all these cases one is interested in the non-universal behavior on special scales.\n\nRecently, the supersymmetry method was extended to general rotation invariant probability densities \\cite{Guh06,Som07,LSZ07,KGG08}. There are two approaches. The first one is the generalized Hubbard--Stratonovich transformation \\cite{Guh06,KGG08}. With help of a proper Dirac--distribution in superspace an integral over rectangular supermatrices was mapped to a supermatrix integral with non-compact domain in the Fermion--Fermion block. The second approach is the superbosonization formula \\cite{Som07,LSZ07} mapping the same integral over rectangular matrices as before to a supermatrix integral with compact domain in the Fermion--Fermion block.\n\nIn this work, we prove the equivalence of the generalized Hubbard--Stratonovich transformation with the superbosonization formula. The proof is based on integral identities between supersymmetric Wishart--matrices and quadratic supermatrices. The orthogonal, unitary and unitary-symplectic classes are dealt with in a unifying way.\n\nThe article is organized as follows. In Sec. \\ref{sec2}, we give a motivation and introduce our notation. In Sec. \\ref{sec3}, we define rectangular supermatrices and the supersymmetric version of Wishart-matrices built up by supervectors. We also give a helpful corollary for the case of arbitrary matrix dimension discussed in Sec. \\ref{sec7}. In Secs. \\ref{sec4} and \\ref{sec5}, we present and further generalize the superbosonization formula and the generalized Hubbard--Stratonovich transformation, respectively. The theorem stating the equivalence of both approaches is given in Sec. \\ref{sec6} including a clarification of their mutual connection. In Sec. \\ref{sec7}, we extend both theorems given in Secs. \\ref{sec4} and \\ref{sec5} to arbitrary matrix dimension. Details of the proofs are given in the appendices.\n\n\\section{Ratios of characteristic polynomials}\\label{sec2}\n\nWe employ the notation defined in Refs.  \\cite{KKG08,KGG08}. ${\\rm Herm\\,}(\\beta,N)$ is either the set of $N\\times N$ real symmetric ($\\beta=1$), $N\\times N$ hermitian ($\\beta=2$) or $2N\\times 2N$ self-dual ($\\beta=4$) matrices, according to the Dyson--index $\\beta$. We use the complex representation of the quaternionic numbers $\\mathbb{H}$. Also, we define\n\\begin{equation}\\label{2.0}\n \\gamma_1=\\left\\{\\begin{array}{ll} 1 & ,\\ \\beta\\in\\{2,4\\} \\\\ 2 & ,\\ \\beta=1 \\end{array}\\right. \\quad ,\\quad \\gamma_2=\\left\\{\\begin{array}{ll} 1 & ,\\ \\beta\\in\\{1,2\\} \\\\ 2 & ,\\ \\beta=4 \\end{array}\\right.\n\\end{equation}\nand $\\tilde{\\gamma}=\\gamma_1\\gamma_2$.\n\nThe central objects in many applications of supersymmetry are averages over ratios of characteristic polynomials \\cite{AkeFyo03,AkePot04,BorStr05}\n\\begin{eqnarray}\n  \\fl Z_{k_1k_2}(E^-) & = & \\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\frac{\\prod\\limits_{n=1}^{k_2}\\det\\left(H-(E_{n2}-\\imath\\varepsilon)\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\right)}{\\prod\\limits_{n=1}^{k_1}\\det\\left(H-(E_{n1}-\\imath\\varepsilon)\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\right)}d[H]\\nonumber\\\\\n  & = & \\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H){\\rm Sdet\\,}^{-1\/\\tilde{\\gamma}}\\left(H\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}(k_1+k_2)}- \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\otimes E^-\\right)d[H]\\label{2.1}\n\\end{eqnarray}\nwhere $P$ is a sufficiently integrable probability density on the matrix set ${\\rm Herm\\,}(\\beta,N)$ invariant under the group\n\\begin{equation}\\label{2.2}\n {\\rm U\\,}^{(\\beta)}(N)=\\left\\{\\begin{array}{ll} \n              \\Or(N)\t& ,\\ \\beta=1\\\\\n\t      {\\rm U\\,}(N)\t& ,\\ \\beta=2\\\\\n\t      {\\rm USp\\,}(2N)\t& ,\\ \\beta=4\n             \\end{array}\\right. .\n\\end{equation}\nHere, we assume that $P$ is analytic in its real independent variables. We use the same measure for $d[H]$ as in Ref.  \\cite{KKG08} which is the product over all real independent differentials, see also Eq.~\\eref{t1.4}. Also, we define $E={\\rm diag\\,}(E_{11},\\ldots,E_{k_11},E_{12},\\ldots,E_{k_22})\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}}$ and $E^-=E-\\imath\\varepsilon \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}(k_1+k_2)}$.\n\nThe generating function of the $k$--point correlation function \\cite{BreHik00,Zir06,Guh06,KGG08}\n\\begin{equation}\\label{2.3}\n R_{k}(x)=\\gamma_2^{-k}\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\prod\\limits_{p=1}^k\\tr\\delta(x_p-H)d[H]\n\\end{equation}\nis one application and can be computed starting from the matrix Green function and Eq.~\\eref{2.1} with $k_1=k_2=k$. Another example is the $n$--th moment of the characteristic polynomial \\cite{MehNor01,Fyo02,Zir06}\n\\begin{equation}\\label{2.4}\n \\widehat{Z}_{n}(x,\\mu)=\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\Theta(H){\\det}^n\\left(H-E \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2k}\\right)d[H],\n\\end{equation}\nwhere the Heavyside--function for matrices $\\Theta(H)$ is unity if $H$ is positive definite and zero otherwise. \\cite{KGG08} \n\nWith help of Gaussian integrals, we get an integral expression for the determinants in Eq.~\\eref{2.1}. Let $\\Lambda_j$ be the Grassmann space of $j$--forms. We consider a complex Grassmann algebra \\cite{Ber87} $\\Lambda=\\bigoplus\\limits_{j=0}^{2\\gamma_2Nk_2}\\Lambda_j$ with $\\gamma_2Nk_2$ pairs $\\{\\zeta_{jn},\\zeta_{jn}^*\\}$, $1\\leq n\\leq k_2,\\ 1\\leq j\\leq \\gamma_2N$, of Grassmann variables and use the conventions of Ref.  \\cite{KKG08} for integrations over Grassmann variables. Due to the $\\mathbb{Z}_2$--grading, $\\Lambda$ is a direct sum of the set of commuting variables $\\Lambda^0$ and of anticommuting variables $\\Lambda^1$. The body of an element in $\\Lambda$ lies in $\\Lambda_0$ while the Grassmann generators are elements in $\\Lambda_1$.\n\nLet $\\imath$ be the imaginary unit. We take $\\gamma_2Nk_1$ pairs $\\{z_{jn},z_{jn}^*\\}$, $1\\leq n\\leq k_1,\\ 1\\leq j\\leq \\gamma_2N$, of complex numbers and find for Eq.~\\eref{2.1}\n\\begin{equation}\\label{2.5}\n \\fl Z_{k_1k_2}(E^-)= (2\\pi)^{\\gamma_2N(k_2-k_1)}\\imath^{\\gamma_2Nk_2}\\int\\limits_{\\mathfrak{C}} \\mathcal{F}P(K)\\exp\\left(-\\imath{\\rm Str\\,} BE^-\\right)d[\\zeta]d[z]\n\\end{equation}\nwhere $d[z]=\\prod\\limits_{p=1}^{k_1}\\prod\\limits_{j=1}^{\\gamma_2N}dz_{jp}dz_{jp}^*$ , $d[\\zeta]=\\prod\\limits_{p=1}^{k_2}\\prod\\limits_{j=1}^{\\gamma_2N}(d\\zeta_{jp}d\\zeta_{jp}^*)$ and $\\mathfrak{C}=\\mathbb{C}^{\\gamma_2k_1N}\\times\\Lambda_{2\\gamma_2Nk_2}$. The characteristic function appearing in \\eref{2.5} is defined as\n\\begin{equation}\\label{2.6}\n \\mathcal{F}P(K)=\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\exp\\left(\\imath\\tr HK\\right)d[H] .\n\\end{equation}\nThe two matrices\n\\begin{equation}\\label{2.7}\n K = \\frac{1}{\\tilde{\\gamma}}V^\\dagger V\\qquad{\\rm and}\\qquad\n B = \\frac{1}{\\tilde{\\gamma}}VV^\\dagger\n\\end{equation}\nare crucial for the duality between ordinary and superspace. While $K$ is a $\\gamma_2N\\times\\gamma_2N$ ordinary matrix whose entries have nilpotent parts, $B$ is a $\\tilde{\\gamma}(k_1+k_2)\\times\\tilde{\\gamma}(k_1+k_2)$ supermatrix. They are composed of the rectangular $\\gamma_2N\\times\\tilde{\\gamma}(k_1+k_2)$ supermatrix\n\\begin{eqnarray}\n V^\\dagger|_{\\beta\\neq2} & = & (z_1,\\ldots,z_{k_1},Yz_1^*,\\ldots,Yz_{k_1}^*,\\zeta_1,\\ldots,\\zeta_{k_2},Y\\zeta_1^*,\\ldots,Y\\zeta_{k_2}^*) ,\\nonumber\\\\\n V|_{\\beta\\neq2} & = & (z_1^*,\\ldots,z_{k_1}^*,Yz_1,\\ldots,Yz_{k_1},-\\zeta_1^*,\\ldots,-\\zeta_{k_2}^*,Y\\zeta_1,\\ldots,Y\\zeta_{k_2})^T ,\\nonumber\\\\\n V^\\dagger|_{\\beta=2} & = & (z_1,\\ldots,z_{k_1},\\zeta_1,\\ldots,\\zeta_{k_2}) ,\\nonumber\\\\\n V|_{\\beta=2} & = & (z_1^*,\\ldots,z_{k_1}^*,-\\zeta_1^*,\\ldots,-\\zeta_{k_2}^*)^T .\\label{2.9}\n\\end{eqnarray}\nThe transposition ``$T$'' is the ordinary transposition and is not the supersymmetric one. However, the adjoint ``$\\dagger$'' is the complex conjugation with the supersymmetric transposition ``$T_{\\rm S}$''\n\\begin{equation}\\label{2.9b}\n \\sigma^{T_{\\rm S}}=\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]^{T_{\\rm S}}=\\left[\\begin{array}{cc} \\sigma_{11}^T & \\sigma_{21}^T \\\\ -\\sigma_{12}^T & \\sigma_{22}^T \\end{array}\\right],\n\\end{equation}\nwhere $\\sigma$ is an arbitrary rectangular supermatrix. We introduce the constant $\\gamma_2N\\times\\gamma_2N$ matrix\n\\begin{equation}\\label{2.10}\n Y=\\left\\{\\begin{array}{ll}\n             \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_N & ,\\ \\beta=1\\\\\n\t     Y_s^T\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_N & ,\\ \\beta=4\n   \\end{array}\\right.\\ ,\\ \\ Y_s=\\left[\\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array}\\right] .\n\\end{equation}\nThe crucial duality relation \\cite{Guh06,KGG08}\n\\begin{equation}\\label{2.11}\n \\tr K^m={\\rm Str\\,} B^m\\ ,\\ \\ m\\in\\mathbb{N} ,\n\\end{equation}\nholds, connecting invariants in ordinary and superspace. As $\\mathcal{F}P$ inherits the rotation invariance of $P$, the duality relation \\eref{2.11} yields\n\\begin{equation}\\label{2.12}\n \\fl Z_{k_1k_2}(E^-)= (2\\pi)^{\\gamma_2N(k_2-k_1)}\\imath^{\\gamma_2Nk_2}\\int\\limits_{\\mathfrak{C}} \\Phi(B)\\exp\\left(-\\imath{\\rm Str\\,} BE^-\\right)d[\\zeta]d[z] .\n\\end{equation}\nHere, $\\Phi$ is a supersymmetric extension of a representation $\\mathcal{F}P_0$ of the characteristic function,\n\\begin{equation}\\label{2.13}\n \\Phi(B)=\\mathcal{F}P_0({\\rm Str\\,} B^m|m\\in\\mathbb{N})=\\mathcal{F}P_0(\\tr K^m|m\\in\\mathbb{N})=\\mathcal{F}P(K) .\n\\end{equation}\nThe representation $\\mathcal{F}P_0$ is not unique \\cite{BEKYZ07}. However, the integral \\eref{2.12} is independent of a particular choice \\cite{KGG08}.\n\nThe supermatrix $B$ fulfills the symmetry\n\\begin{equation}\\label{2.14}\n B^*=\\left\\{\\begin{array}{ll}\n                \\widetilde{Y}B\\widetilde{Y}^T &,\\ \\beta\\in\\{1,4\\}, \\\\\n\t\t\\widetilde{Y}B^*\\widetilde{Y}^T &,\\ \\beta=2\n               \\end{array}\\right.\n\\end{equation}\nwith the supermatrices\n\\begin{equation}\\label{2.15}\n \\fl\\widetilde{Y}|_{\\beta=1}=\\left[\\begin{array}{ccc} 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 & 0 \\\\ 0 & 0 & Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\end{array}\\right]\\quad,\\quad\\widetilde{Y}|_{\\beta=4}=\\left[\\begin{array}{ccc} Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 & 0 \\\\ 0 & 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} & 0 \\end{array}\\right]\n\\end{equation}\nand $\\widetilde{Y}|_{\\beta=2}=  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1+k_2}$ and is self-adjoint for every $\\beta$. Using the $\\pi\/4$--rotations\n\\begin{equation}\\label{2.16}\n \\fl U|_{\\beta=1}=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{ccc}  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\ -\\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & \\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\ 0 & 0 & \\sqrt{2}\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{2k_2} \\end{array}\\right],\\ U|_{\\beta=4}=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{ccc} \\sqrt{2}\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{2k_1} & 0 & 0 \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2}\\\\ 0 & -\\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} & \\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\end{array}\\right]\n\\end{equation}\nand $U|_{\\beta=2}= \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1+k_2}$, $\\widehat{B}=UBU^\\dagger$ lies in the well-known symmetric superspaces \\cite{Zir96},\n\\begin{eqnarray}\n \\fl\\widetilde{\\Sigma}_{\\beta,\\gamma_1k_1,\\gamma_2k_2}^{(\\dagger)} & = &\\Biggl\\{\\sigma\\in{\\rm Mat}(\\tilde{\\gamma}k_1\/\\tilde{\\gamma}k_2)\\Biggl|\\sigma^\\dagger=\\sigma,\\nonumber\\\\\n & & \\left.\\sigma^*=\\left\\{\\begin{array}[c]{ll}\n                \\widehat{Y}_{\\gamma_1k_1,\\gamma_2k_2}\\sigma\\widehat{Y}_{\\gamma_1k_1,\\gamma_2k_2}^{T} & ,\\ \\beta\\in\\{1,4\\} \\\\\n\t\t\\widehat{Y}_{k_1k_2}\\sigma^*\\widehat{Y}_{k_1k_2}^{T} & ,\\ \\beta=2\n          \\end{array}\\right\\}\\right\\}\\label{2.17}\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{2.18}\n \\fl\\left.\\widehat{Y}_{pq}\\right|_{\\beta=1}=\\left[\\begin{array}{cc}   \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p} & 0 \\\\ 0 & Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{q} \\end{array}\\right]\\ ,\\ \\ \\left.\\widehat{Y}_{pq}\\right|_{\\beta=2}=  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p+q}\\ \\ \\ {\\rm and}\\ \\ \\ \\left.\\widehat{Y}_{pq}\\right|_{\\beta=4}=\\left[\\begin{array}{cc} Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p} & 0 \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{q} \\end{array}\\right].\n\\end{equation}\nThe set ${\\rm Mat}(p\/q)$ is the set of $(p+q)\\times(p+q)$ supermatrices on the complex Grassmann algebra $\\bigoplus\\limits_{j=0}^{2pq}\\Lambda_j$. The entries of the diagonal blocks of an element in ${\\rm Mat}(p\/q)$ lie in $\\Lambda^0$ whereas the entries of the off-diagonal block are elements in $\\Lambda^1$.\n\nThe rectangular supermatrix $\\widehat{V}^\\dagger=V^\\dagger U^\\dagger$ is composed of real, complex or quaternionic supervectors whose adjoints form the rows. They are given by\n\\begin{equation}\\label{2.19}\n \\fl\\Psi_j^\\dagger=\\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\sqrt{2}{\\rm Re\\,} z_{jn},\\sqrt{2}{\\rm Im\\,} z_{jn}\\right\\}_{1\\leq n\\leq k_1},\\left\\{\\zeta_{jn},\\zeta^*_{jn}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{jn}\\right\\}_{1\\leq n\\leq k_1},\\left\\{\\zeta_{jn}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{jn} & -z_{j+N,n}^* \\\\ z_{j+N,n} & z_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq k_1},\\displaystyle\\left\\{\\begin{array}{cc} \\zeta_{jn}^{(-)} & \\zeta_{jn}^{(+)} \\\\ \\zeta_{jn}^{(-)*} & \\zeta_{jn}^{(+)*} \\end{array}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\n\\end{equation}\nrespectively, where $\\zeta_{jn}^{(\\pm)}=\\imath^{(1\\pm1)\/2}(\\zeta_{jn}\\pm\\zeta_{j+N,n}^*)\/\\sqrt{2}$. Then, the supermatrix $\\widehat{B}$ acquires the form\n\\begin{equation}\\label{2.20}\n \\widehat{B}=\\frac{1}{\\tilde{\\gamma}}\\sum\\limits_{j=1}^N\\Psi_j\\Psi_j^\\dagger .\n\\end{equation}\nThe integrand in Eq.~\\eref{2.12}\n\\begin{equation}\\label{2.21}\n F\\left(\\widehat{B}\\right)=\\Phi\\left(\\widehat{B}\\right)\\exp\\left(-\\imath{\\rm Str\\,} E\\widehat{B}\\right)\n\\end{equation}\ncomprises a symmetry breaking term,\n\\begin{equation}\\label{2.22}\n \\exists\\ U\\in{\\rm U\\,}^{(\\beta)}(\\gamma_1k_1\/\\gamma_2k_2)\\ {\\rm\\ that\\ \\ }F\\left(\\widehat{B}\\right)\\neq F\\left(U\\widehat{B}U^\\dagger\\right) ,\n\\end{equation}\naccording to the supergroup\n\\begin{equation}\\label{2.23}\n {\\rm U\\,}^{(\\beta)}(\\gamma_1k_1\/\\gamma_2k_2)=\\left\\{\\begin{array}{ll}\n                 {\\rm UOSp\\,}^{(+)}(2k_1\/2k_2) & ,\\ \\beta=1\\\\\n                 {\\rm U\\,}(k_1\/k_2) & ,\\ \\beta=2\\\\\n                 {\\rm UOSp\\,}^{(-)}(2k_1\/2k_2) & ,\\ \\beta=4\n                \\end{array}\\right. .\n\\end{equation}\nWe use the notation of Refs.~\\cite{KohGuh05,KKG08} for the representations ${\\rm UOSp\\,}^{(\\pm)}$ of the supergroup ${\\rm UOSp\\,}$. These representations are related to the classification of Riemannian symmetric superspaces by Zirnbauer \\cite{Zir96}. The index ``$+$'' in Eq.~\\eref{2.23} refers to real entries in the Boson--Boson block and to quaternionic entries in the Fermion--Fermion block and ``$-$'' indicates the other way around.\n\n\\section{Supersymmetric Wishart--matrices and some of their properties}\\label{sec3}\n\nWe generalize the integrand \\eref{2.21} to arbitrary sufficiently integrable superfunctions on rectangular $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2a+\\gamma_1b)$ supermatrices $\\widehat{V}$ on the complex Grassmann--algebra $\\Lambda=\\bigoplus\\limits_{j=0}^{2(ad+bc)}\\Lambda_j$. Such a supermatrix\n\\begin{equation}\\label{3.1}\n \\fl\\widehat{V}=\\left(\\Psi_{11}^{({\\rm C})},\\ldots,\\Psi_{a1}^{({\\rm C})},\\Psi_{12}^{({\\rm C})},\\ldots\\Psi_{b2}^{({\\rm C})}\\right)=\\left(\\Psi_{11}^{({\\rm R})*}\\ldots,\\Psi_{c1}^{({\\rm R})*},\\Psi_{12}^{({\\rm R})*},\\ldots\\Psi_{d2}^{({\\rm R})*}\\right)^{T_{\\rm S}}\n\\end{equation}\nis defined by its columns\n\\begin{eqnarray}\n \\Psi_{j1}^{({\\rm C})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{x_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\chi_{jn},\\chi_{jn}^*\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\chi_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{jn1} & -z_{jn2}^* \\\\ z_{jn2} & z_{jn1}^* \\end{array}\\right\\}_{1\\leq n\\leq c},\\left\\{\\begin{array}{c} \\chi_{jn} \\\\ \\chi_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.2a}\\\\\n \\Psi_{j2}^{({\\rm C})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\begin{array}{c} \\zeta_{jn} \\\\ \\zeta_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq c},\\left\\{\\begin{array}{cc} \\tilde{z}_{jn1} & -\\tilde{z}_{jn2}^* \\\\ \\tilde{z}_{jn2} & \\tilde{z}_{jn1}^* \\end{array}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{\\zeta_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\tilde{z}_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\zeta_{jn},\\zeta^*_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{y_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.2b}\n\\end{eqnarray}\nor by its rows\n\\begin{eqnarray}\n \\Psi_{j1}^{({\\rm R})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{x_{nj}\\right\\}_{1\\leq n\\leq a},\\left\\{ \\zeta_{nj}^*, -\\zeta_{nj} \\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{nj}^*\\right\\}_{1\\leq n\\leq a},\\left\\{\\zeta_{nj}^*\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{nj1}^* & z_{nj2}^* \\\\ -z_{nj2} & z_{nj1} \\end{array}\\right\\}_{1\\leq n\\leq a},\\left\\{\\begin{array}{c} \\zeta_{nj}^* \\\\ -\\zeta_{nj} \\end{array}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.3a}\\\\\n \\Psi_{j2}^{({\\rm R})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\begin{array}{c} -\\chi_{nj}^* \\\\ \\chi_{nj} \\end{array}\\right\\}_{1\\leq n\\leq a},\\left\\{\\begin{array}{cc} \\tilde{z}_{nj1}^* & \\tilde{z}_{nj2}^* \\\\ -\\tilde{z}_{nj2} & \\tilde{z}_{nj1} \\end{array}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{-\\chi_{nj}^*\\right\\}_{1\\leq n\\leq a},\\left\\{\\tilde{z}_{nj}^*\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{-\\chi_{nj}^*,\\chi_{nj}\\right\\}_{1\\leq n\\leq a},\\left\\{y_{nj}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=4\n                       \\end{array}\\right.\\label{3.3b}\n\\end{eqnarray}\nwhich are real, complex and quaternionic supervectors. We use the complex Grassmann variables $\\chi_{mn}$ and $\\zeta_{mn}$ and the real numbers $x_{mn}$ and $y_{mn}$. Also, we introduce the complex numbers $z_{mn}$, $\\tilde{z}_{mn}$, $z_{mnl}$ and $\\tilde{z}_{mnl}$. The $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2c+\\gamma_1d)$ supermatrix $\\widehat{B}=\\tilde{\\gamma}^{-1}\\widehat{V}\\widehat{V}^\\dagger$ can be written in the columns of $\\widehat{V}$ as in Eq.~\\eref{2.20}. As this supermatrix has a form similar to the ordinary Wishart--matrices, we refer to it as supersymmetric Wishart--matrix. The rectangular supermatrix above fulfills the property\n\\begin{equation}\\label{3.4}\n \\widehat{V}^*=\\widehat{Y}_{cd}\\widehat{V}\\widehat{Y}_{ab}^T .\n\\end{equation}\nThe corresponding generating function \\eref{2.1} is an integral over a rotation invariant superfunction $P$ on a superspace, which is sufficiently convergent and analytic in its real independent variables,\n\\begin{equation}\\label{3.5}\n  Z_{cd}^{ab}(E^-) =\\int\\limits_{\\widetilde{\\Sigma}_{\\beta,ab}^{(-\\psi)}} P(\\sigma) {\\rm Sdet\\,}^{-1\/\\tilde{\\gamma}}\\left(\\sigma\\otimes\\widehat{\\Pi}_{2\\psi}^{({\\rm C})}- \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2a+\\gamma_1b}\\otimes E^-\\right)d[\\sigma],\n\\end{equation}\nwhere\n\\begin{equation}\\label{3.5b}\n \\fl E^-={\\rm diag\\,}\\left(E_{11}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2},\\ldots,E_{c1}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2},E_{12}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1},\\ldots,E_{d2}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1}\\right)-\\imath\\varepsilon \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2c+\\gamma_1d} .\n\\end{equation}\nLet $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ be a subset of $\\widetilde{\\Sigma}_{\\beta,ab}^{(\\dagger)}$. The entries of elements in $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ lie in $\\Lambda_0$ and $\\Lambda_1$. The set $\\widetilde{\\Sigma}_{\\beta,ab}^{(-\\psi)}=\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}$ is the Wick--rotated set of $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ by the generalized Wick--rotation $\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}={\\rm diag\\,}( \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2a},e^{-\\imath\\psi\/2} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1b})$. As in Ref.~\\cite{KGG08}, we introduce such a rotation for the convergence of the integral \\eref{3.5}. The matrix $\\widehat{\\Pi}_{2\\psi}^{({\\rm C})}={\\rm diag\\,}( \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2c},e^{\\imath\\psi} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1d})$ is also a Wick--rotation.\n\nIn the rest of our work, we restrict the calculations to a class of superfunctions. These superfunctions has a Wick--rotation such that the integrals are convergent. We have not explicitly analysed the class of such functions. However, this class is very large and sufficient for physical interests. We consider the probability distribution\n\\begin{equation}\\label{3.5c}\n P(\\sigma)=f(\\sigma)\\exp(-{\\rm Str\\,}\\sigma^{2m}),\n\\end{equation}\nwhere $m\\in\\mathbb{N}$ and $f$ is a superfunction which does not increase so fast as $\\exp({\\rm Str\\,}\\sigma^{2m})$ in the infinty, in particular\n\\begin{equation}\\label{3.5d}\n \\underset{\\epsilon\\to\\infty}{\\lim}P\\left(\\epsilon e^{\\imath\\alpha}\\sigma\\right)=0 \\Leftrightarrow \\underset{\\epsilon\\to\\infty}{\\lim}\\exp\\left(-\\epsilon e^{\\imath\\alpha}{\\rm Str\\,}\\sigma^{2m}\\right)=0\n\\end{equation}\nfor every angle $\\alpha\\in[0,2\\pi]$. Then, a Wick--rotation exists for $P$.\n\nTo guarantee the convergence of the integrals below, let $\\widehat{V}_{\\psi}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{V}$, $\\widehat{V}_{-\\psi}^\\dagger=\\widehat{V}^\\dagger\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$ and $\\widehat{B}_{\\psi}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{B}\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$. Considering a function $f$ on the set of supersymmetric Wishart--matrices, we give a lemma and a corollary which are of equal importance for the superbosonization formula and the generalized Hubbard--Startonovich transformation. For $b=0$, the lemma presents the duality relation between the ordinary and superspace \\eref{2.11} which is crucial for the calculation of \\eref{2.1}. This lemma was proven in Ref.~\\cite{LSZ07} by representation theory. Here, we only state it.\n\\begin{lemma}\\label{c1}\\ \\\\\n Let $f$ be a superfunction on rectangular supermatrices of the form \\eref{3.1} and invariant under\n \\begin{equation}\\label{c1.1}\n  f(\\widehat{V}_{\\psi},\\widehat{V}_{-\\psi}^\\dagger)=f\\left(\\widehat{V}_{\\psi}U^\\dagger,U\\widehat{V}_{-\\psi}^\\dagger\\right)\\ ,\n \\end{equation}\n for all $\\widehat{V}$ and $U\\in{\\rm U\\,}^{(\\beta)}(a\/b)$. Then, there is a superfunction $F$ on the ${\\rm U\\,}^{(\\beta)}(c\/d)$--symmetric supermatrices with\n \\begin{equation}\\label{c1.2}\n  F(\\widehat{B}_{\\psi})=f(\\widehat{V}_{\\psi},\\widehat{V}_{-\\psi}^\\dagger)\\ .\n \\end{equation}\n\\end{lemma}\n\nThe ${\\rm U\\,}^{(\\beta)}(c\/d)$--symmetric supermatrices are elements of $\\widetilde{\\Sigma}_{\\beta,ab}^{(\\dagger)}$. The invariance condition \\eref{c1.1} implies that $f$ only depends on the rows of $\\widehat{V}_{\\psi}$ by $\\Psi_{nr}^{({\\rm R})\\dagger}\\Psi_{ms}^{({\\rm R})}$ for arbitrary $n,m,r$ and $s$. These scalar products are the entries of the supermatrix $\\widehat{V}_{\\psi}\\widehat{V}_{-\\psi}^\\dagger$ which leads to the statement.\n\nThe corollary below is an application of integral theorems by Wegner \\cite{Weg83} worked out in Refs.~\\cite{Con88,ConGro89} and of the Theorems III.1, III.2 and III.3 in Ref.~\\cite{KKG08}. It states that an integration over supersymmetric Wishart--matrices can be reduced to integrations over supersymmetric Wishart--matrices comprising a lower dimensional rectangular supermatrix. In particular for the generating function, it reflects the equivalence of the integral \\eref{3.5} with an integration over smaller supermatrices \\cite{KKG08}. We assume that $\\tilde{a}=a-2(b-\\tilde{b})\/\\beta\\geq0$ with\n\\begin{equation}\\label{3.6}\n \\tilde{b}=\\left\\{\\begin{array}{ll}\n\t\t1 & ,\\ \\beta=4\\ {\\rm and}\\ b\\in2\\mathbb{N}_0+1\\\\\n\t\t0 & ,\\ {\\rm else}\n\t   \\end{array}\\right. .\n\\end{equation}\n\\begin{corollary}\\label{c2}\\ \\\\\n Let $F$ be the superfunction of Lemma \\ref{c1}, real analytic in its real independent entries and a Schwartz--function. Then, we find\n \\begin{equation}\\label{c2.1}\n  \\displaystyle\\int\\limits_{\\mathfrak{R}}F(\\widehat{B}_{\\psi})d[\\widehat{V}]=C\\int\\limits_{\\widetilde{\\mathfrak{R}}}F(\\widetilde{B}_{\\psi})d[\\widetilde{V}]\n \\end{equation}\n where $\\widetilde{B}=\\tilde{\\gamma}^{-1}\\widetilde{V}\\widetilde{V}$. The sets are $\\mathfrak{R}=\\mathbb{R}^{\\beta ac+4bd\/\\beta}\\times\\Lambda_{2(ad+bc)}$ and $\\widetilde{\\mathfrak{R}}=\\mathbb{R}^{\\beta\\tilde{a}c+4\\tilde{b}d\/\\beta}\\times\\Lambda_{2(\\tilde{a}d+\\tilde{b}c)}$, the constant is\n \\begin{equation}\\label{c2.2}\n  C=\\left[-\\frac{\\gamma_1}{2}\\right]^{(b-\\tilde{b})c}\\left[\\frac{\\gamma_2}{2}\\right]^{(a-\\tilde{a})d}\n \\end{equation}\n and the measure\n\\begin{equation}\\label{c2.3}\n d[\\widehat{V}]=\\underset{1\\leq l\\leq \\beta}{\\underset{1\\leq n\\leq c}{\\underset{1\\leq m\\leq a}{\\prod}}}dx_{mnl}\\underset{1\\leq l\\leq 4\/\\beta}{\\underset{1\\leq n\\leq d}{\\underset{1\\leq m\\leq b}{\\prod}}}dy_{mnl}\\underset{1\\leq n\\leq c}{\\underset{1\\leq m\\leq b}{\\prod }}d\\zeta_{mn}d\\zeta_{mn}^*\\underset{1\\leq n\\leq d}{\\underset{1\\leq m\\leq a}{\\prod }}d\\chi_{mn}d\\chi_{mn}^*\\ .\n\\end{equation}\nThe $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2\\tilde{a}+\\gamma_1\\tilde{b})$ supermatrix $\\widetilde{V}$ and its measure $d[\\widetilde{V}]$ is defined analogous to $\\widehat{V}$ and $d[\\widehat{V}]$, respectively. Here, $x_{mna}$ and $y_{mna}$ are the independent real components of the real, complex and quaternionic numbers of the supervectors $\\Psi_{j1}^{({\\rm R})}$ and $\\Psi_{j2}^{({\\rm R})}$, respectively.\n\\end{corollary}\n\\textbf{Proof:}\\\\\nWe integrate $F$ over all supervectors $\\Psi_{j1}^{({\\rm R})}$ and $\\Psi_{j2}^{({\\rm R})}$ except $\\Psi_{11}^{({\\rm R})}$. Then, \n \\begin{equation}\\label{c2.4}\n  \\displaystyle\\int\\limits_{\\mathfrak{R}^\\prime}F(V_{\\psi}V_{-\\psi}^\\dagger)d[\\widehat{V}_{\\neq11}]\n \\end{equation}\nonly depends on $\\Psi_{11}^{({\\rm R})\\dagger}\\Psi_{11}^{({\\rm R})}$. The integration set is $\\mathfrak{R}^\\prime=\\mathbb{R}^{\\beta a(c-1)+4bd\/\\beta}\\times\\Lambda_{2(ad+b(c-1))}$ and the measure $d[\\widehat{V}_{\\neq11}]$ is $d[\\widehat{V}]$ without the measure for the supervector $\\Psi_{11}^{({\\rm R})}$. With help of the Theorems in Ref.~\\cite{Weg83,Con88,ConGro89,KKG08}, the integration over $\\Psi_{11}^{({\\rm R})}$ is up to a constant equivalent to an integration over a supervector $\\widetilde{\\Psi}_{11}^{({\\rm R})}$. This supervector is equal to $\\Psi_{11}^{({\\rm R})}$ in the first $\\tilde{a}$--th entries and else zero. We repeat this procedure for all other supervectors reminding that we only need the invariance under the supergroup action ${\\rm U\\,}^{(\\beta)}\\left(b-\\tilde{b}\/b-\\tilde{b}\\right)$ on $f$ as in Eq.~\\eref{c1.1} embedded in ${\\rm U\\,}^{(\\beta)}(a\/b)$. This invariance is preserved in each step due to the zero entries in the new supervectors. \\hfill$\\square$\n\nThis corollary allows us to restrict our calculation on supermatrices with $b=1$ only to $\\beta=4$ and $b=0$ for all $\\beta$. Only the latter case is of physical interest. Thus, we give the computation for $b=0$ in the following sections and consider the case $b=1$ in Sec. \\ref{sec7}. For $b=0$ we omit the Wick--rotation for $\\widehat{B}$ as it is done in Refs.~\\cite{Guh06,KGG08} due to the convergence of the integral \\eref{3.5}.\n\n\\section{The superbosonization formula}\\label{sec4}\n\nWe need for the following theorem the definition of the sets\n\\begin{eqnarray}\n \\fl\\Sigma_{1,pq} = \\left\\{\\left.\\sigma=\\left[\\begin{array}{ccc} \\sigma_1 & \\eta & \\eta^* \\\\ -\\eta^\\dagger & \\sigma_{21} & \\sigma_{22}^{(1)} \\\\ \\eta^T  & \\sigma_{22}^{(2)} & \\sigma_{21}^T \\end{array}\\right]\\in{\\rm Mat}(p\/2q)\\right|\\sigma_1^\\dagger=\\sigma_1^*=\\sigma_1\\ {\\rm with\\ positive}\\right.\\nonumber\\\\\n \\left.{\\rm definite\\ body},\\ \\sigma_{22}^{(1)T}=-\\sigma_{22}^{(1)},\\ \\sigma_{22}^{(2)T}=-\\sigma_{22}^{(2)}\\right\\},\\label{4.1}\\\\\n \\fl\\Sigma_{2,pq} = \\left\\{\\left.\\sigma=\\left[\\begin{array}{cc} \\sigma_1 & \\eta \\\\ -\\eta^\\dagger & \\sigma_2 \\end{array}\\right]\\in{\\rm Mat}(p\/q)\\right|\\sigma_1^\\dagger=\\sigma_1\\ {\\rm with\\ positive\\ definite\\ body}\\right\\},\\label{4.2}\\\\\n \\fl\\Sigma_{4,pq} = \\left\\{\\sigma=\\left[\\begin{array}{ccc} \\sigma_{11} & \\sigma_{12} & \\eta \\\\ -\\sigma_{12}^* & \\sigma_{11}^* & \\eta^* \\\\ -\\eta^\\dagger  & \\eta^T & \\sigma_2 \\end{array}\\right]\\in{\\rm Mat}(2p\/q)\\right|\\sigma_1^\\dagger=\\sigma_1=\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ -\\sigma_{12}^* & \\sigma_{11}^* \\end{array}\\right]\\nonumber\\\\\n {\\rm with\\ positive\\ definite\\ body},\\ \\sigma_2=\\sigma_2^T\\Biggl\\}.\\label{4.3}\n\\end{eqnarray}\nAlso, we will use the sets\n\\begin{equation}\\label{4.4}\n \\Sigma_{\\beta,pq}^{(\\dagger)} = \\left\\{\\left.\\sigma\\in\\Sigma_{\\beta,pq}\\right|\\sigma_2^\\dagger=\\sigma_2\\right\\}=\\widetilde{\\Sigma}_{\\beta,pq}^{(\\dagger)}\\cap\\Sigma_{\\beta,pq}\n\\end{equation}\nand\n\\begin{equation}\\label{4.5}\n \\Sigma_{\\beta,pq}^{({\\rm c})} = \\left\\{\\left.\\sigma\\in\\Sigma_{\\beta,pq}\\right|\\sigma_2\\in{\\rm CU\\,}^{(4\/\\beta)}\\left(q\\right)\\right\\}\n\\end{equation}\nwhere ${\\rm CU\\,}^{(\\beta)}\\left(q\\right)$ is the set of the circular orthogonal (COE, $\\beta=1$), unitary (CUE, $\\beta=2$) or unitary-symplectic (CSE, $\\beta=4$) ensembles,\n\\begin{equation}\\label{4.6}\n \\fl{\\rm CU\\,}^{(\\beta)}\\left(q\\right)=\\left\\{A\\in{\\rm Gl}(\\gamma_2q,\\mathbb{C})\\left| \\begin{array}{ll} A=A^T\\in{\\rm U\\,}^{(2)}(q) & ,\\ \\beta=1 \\\\ A\\in{\\rm U\\,}^{(2)}(q) & ,\\ \\beta=2 \\\\ A=(Y_s\\otimes\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_q)A^T(Y_s^T\\otimes\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_q)\\in{\\rm U\\,}^{(2)}(2q) & ,\\ \\beta=4 \\end{array}\\right.\\right\\}\n\\end{equation}\nThe index ``$\\dagger$'' in Eq.~\\eref{4.4} refers to the self-adjointness of the supermatrices and the index ``${\\rm c}$'' indicates the relation to the circular ensembles. We notice that the set classes presented above differ in the Fermion--Fermion block. In Sec. \\ref{sec6}, we show that this is the crucial difference between both methods. Due to the nilpotence of $B$'s Fermion--Fermion block, we can change the set in this block for the Fourier--transformation. The sets of matrices in the sets above with entries in $\\Lambda_0$ and $\\Lambda_1$ are denoted by $\\Sigma_{\\beta,pq}^{0}$, $\\Sigma_{\\beta,pq}^{0(\\dagger)}$ and $\\Sigma_{\\beta,pq}^{0({\\rm c})}$, respectively.\n\nThe proof of the superbosonization formula \\cite{Som07,LSZ07} given below is based on the proofs of the superbosonization formula for arbitrary superfunctions on real supersymmetric Wishart--matrices in Ref.~\\cite{Som07} and for Gaussian functions on real, complex and quaternionic Wishart--matrices in Ref.~\\cite{Som08}. This theorem extends the superbosonization formula of Ref.~\\cite{LSZ07} to averages of square roots of determinants over unitary-symplectically invariant ensembles, i.e. $\\beta=4$, $b=c=0$ and $d$ odd in Eq.~\\eref{3.5}. The proof of this theorem is given in  \\ref{app1}.\n\\begin{theorem}[Superbosonization formula]\\label{t1}\\ \\\\\n Let $F$ be a conveniently integrable and analytic superfunction on the set of $\\left(\\gamma_2c+\\gamma_1d\\right)\\times\\left(\\gamma_2c+\\gamma_1d\\right)$ supermatrices and\n\\begin{equation}\\label{t1.3}\n \\kappa=\\frac{a-c+1}{\\gamma_1}+\\frac{d-1}{\\gamma_2} .\n\\end{equation}\nWith\n\\begin{equation}\\label{t1.0}\n a\\geq c\\ ,\n\\end{equation}\nwe find\n \\begin{equation}\\label{t1.1}\n  \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}\\right)d[\\widehat{V}]=C_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{0({\\rm c})}}F(\\rho)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\rho\\right){\\rm Sdet\\,}\\rho^{\\kappa}d[\\rho] ,\n \\end{equation}\n where the constant is\n\\begin{eqnarray}\n \\fl C_{acd}^{(\\beta)}\n =  \\left(-2\\pi\\gamma_1\\right)^{-ad}\\left(-\\frac{2\\pi}{\\gamma_2}\\right)^{cd}2^{-c}\\tilde{\\gamma}^{\\beta ac\/2}\\frac{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a)\\right)}{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a-c)\\right)}\\times\\nonumber\\\\\n \\times\\prod\\limits_{n=1}^d\\frac{\\Gamma\\left(\\gamma_1\\kappa+2(n-d)\/\\beta\\right)}{\\imath^{4(n-1)\/\\beta}\\pi^{2(n-1)\/\\beta}} .\\label{t1.2}\n\\end{eqnarray}\nWe define the measure $d[\\widehat{V}]$ as in Corollary \\ref{c2} and the measure on the right hand side is $d[\\rho]=d[\\rho_1]d[\\rho_2]d[\\eta]$ where\n\\begin{eqnarray}\n \\fl d[\\rho_1] & = & \\prod\\limits_{n=1}^{c}d\\rho_{nn1}\\times\\left\\{\\begin{array}{ll}\n                  \\prod\\limits_{1\\leq n< m\\leq c}d\\rho_{nm1} & ,\\ \\beta=1,\\\\\n                  \\prod\\limits_{1\\leq n< m\\leq c}d{\\rm Re\\,}\\rho_{nm1}d{\\rm Im\\,}\\rho_{nm1} & ,\\ \\beta=2,\\\\\n                  \\prod\\limits_{1\\leq n< m\\leq c}d{\\rm Re\\,}\\rho_{nm11}d{\\rm Im\\,}\\rho_{nm11}d{\\rm Re\\,}\\rho_{nm12}d{\\rm Im\\,}\\rho_{nm12} & ,\\ \\beta=4,\n                 \\end{array}\\right.\\label{t1.4}\\\\\n \\fl d[\\rho_2] & = & {\\rm FU}_{d}^{(4\/\\beta)}|\\Delta_{d}(e^{\\imath\\varphi_j})|^{4\/\\beta}\\prod\\limits_{n=1}^{d}\\frac{de^{\\imath\\varphi_n}}{2\\pi\\imath}d\\mu(U)\\label{t1.5} ,\\\\\n \\fl d[\\eta]    & = & \\prod\\limits_{n=1}^{c}\\prod\\limits_{m=1}^{d}(d\\eta_{nm}d\\eta_{nm}^*)\\label{t1.6}.\n\\end{eqnarray}\nHere, $\\rho_2=U{\\rm diag\\,}\\left(e^{\\imath\\varphi_1},\\ldots,e^{\\imath\\varphi_{d}}\\right)U^\\dagger$, $U\\in{\\rm U\\,}^{(4\/\\beta)}\\left(d\\right)$ and $d\\mu(U)$ is the normalized Haar-measure of ${\\rm U\\,}^{(4\/\\beta)}\\left(d\\right)$. We introduce the volumes of the rotation groups\n\\begin{equation}\\label{t1.7}\n {\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(n)\\right)=\\prod\\limits_{j=1}^n\\frac{2\\pi^{\\beta j\/2}}{\\Gamma\\left(\\beta j\/2\\right)}\n\\end{equation}\nand the ratio of volumes of the group flag manifold and the permutation group\n\\begin{equation}\\label{t1.8}\n {\\rm FU}_{d}^{(4\/\\beta)}=\\frac{1}{d!}\\prod\\limits_{j=1}^d\\frac{\\pi^{2(j-1)\/\\beta}\\Gamma(2\/\\beta)}{\\Gamma(2j\/\\beta)}\\ .\n\\end{equation}\nThe absolute value of the Vandermonde determinant $\\Delta_{d}(e^{\\imath\\varphi_j})=\\prod\\limits_{1\\leq n<m\\leq d}\\left(e^{\\imath\\varphi_n}-e^{\\imath\\varphi_m}\\right)$ refers to a change of sign in every single difference $\\left(e^{\\imath\\varphi_n}-e^{\\imath\\varphi_m}\\right)$ with ``$+$'' if $\\varphi_m<\\varphi_n$ and with ``$-$'' otherwise. Thus, it is not an absolute value in the complex plane.\n\\end{theorem}\n\nThe exponential term can also be shifted in the superfunction $F$. We need this additional term to regularize an intermediate step in the proof.\n\nThe inequality \\eref{t1.0} is crucial. For example, let $\\beta=2$ and $F(\\rho)=1$. Then, the left hand side of Eq.~\\eref{t1.1} is not equal to zero. On the right hand side of Eq.~\\eref{t1.1}, the dependence on the Grassmann variables  only stems from the superdeterminant and we find\n\\begin{equation}\\label{4.7}\n  \\int\\limits_{\\Lambda_{2cd}}{\\rm Sdet\\,}\\rho^\\kappa d[\\eta]=\\int\\limits_{\\Lambda_{2cd}}\\frac{\\det\\left(\\rho_1+\\eta\\rho_2^{-1}\\eta^\\dagger\\right)^\\kappa}{\\det\\rho_2^\\kappa} d[\\eta]=0\n\\end{equation}\nfor $\\kappa<d$. The superdeterminant ${\\rm Sdet\\,}\\rho$ is a polynomial of order $2c$ in the Grassmann variables $\\{\\eta_{nm},\\eta_{nm}^*\\}$ and the integral over the remaining variables is finite for $\\kappa\\geq0$. Hence, it is easy to see that the right hand side of Eq.~\\eref{t1.1} is zero for $\\kappa<d$. This inequality is equivalent to $a<c$.\n\nThis problem was also discussed in Ref.~\\cite{BEKYZ07}. These authors gave a solution for the case that \\eref{t1.0} is violated. This solution differs from our approach in Sec. \\ref{sec7}.\n\n\\section{The generalized Hubbard--Stratonovich transformation}\\label{sec5}\n\nThe following theorem is proven in a way similar to Refs.~\\cite{Guh06,KGG08}. The proof is given in \\ref{app2}. We need the Wick--rotated set $\\Sigma_{\\beta,cd}^{(\\psi)}=\\widehat{\\Pi}_\\psi^{({\\rm C})}\\Sigma_{\\beta,cd}^{0(\\dagger)}\\widehat{\\Pi}_\\psi^{({\\rm C})}$, particularly $\\Sigma_{\\beta,cd}^{(0)}=\\Sigma_{\\beta,cd}^{0(\\dagger)}$. The original extension of the Hubbard--Stratonovich transformation \\cite{Guh06,KGG08} was only given for $\\gamma_2c=\\gamma_1d=\\tilde{\\gamma}k$. Here, we generalize it to arbitrary $c$ and $d$.\n\\begin{theorem}[Generalized Hubbard--Stratonovich transformation]\\ \\label{t2}\\\\\n Let $F$ and $\\kappa$ be the same as in Theorem \\ref{t1}. If the inequality \\eref{t1.0} holds, we have\n \\begin{eqnarray}\n   \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}\\right)d[\\Psi]=\\nonumber\\\\\n   \\fl= \\widetilde{C}_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{(\\psi)}}F\\left(\\hat{\\rho}\\right)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\hat{\\rho}\\right)\\det\\rho_1^\\kappa \\left(e^{-\\imath\\psi d}D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\frac{\\delta(r_2)}{|\\Delta_d(e^{\\imath\\psi}r_2)|^{4\/\\beta}}e^{-\\imath\\psi cd}d[\\rho]=\\nonumber\\\\\n    \\fl= \\widetilde{C}_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{(0)}}\\det\\rho_1^\\kappa\\frac{\\delta(r_2)}{|\\Delta_d(r_2)|^{4\/\\beta}} \\left((-1)^dD_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\left.F\\left(\\hat{\\rho}\\right)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\hat{\\rho}\\right)\\right|_{\\psi=0}d[\\rho]\\label{t2.1}\n \\end{eqnarray}\n with\n \\begin{equation}\\label{t2.0}\n  \\hat{\\rho}=\\left[\\begin{array}{c|c} \\rho_1 & e^{\\imath\\psi\/2}\\rho_\\eta \\\\ \\hline -e^{\\imath\\psi\/2}\\rho_\\eta^\\dagger & e^{\\imath\\psi}\\left(\\rho_2-\\rho_\\eta^\\dagger\\rho_1^{-1}\\rho_\\eta\\right) \\end{array}\\right].\n \\end{equation}\n The variables $r_2$ are the eigenvalues of the supermatrix $\\rho_2$. The measure $d[\\rho]=d[\\rho_1]d[\\rho_2]d[\\eta]$ is defined by Eqs. \\eref{t1.4} and \\eref{t1.6}. For the measure $d[\\rho_2]$ we take the definition \\eref{t1.4} for $4\/\\beta$. The differential operator in Eq.~\\eref{t2.1} is an analog of the Sekiguchi--differential operator \\cite{OkoOls97} and has the form \\cite{KGG08}\n \\begin{equation}\\label{t2.2}\n  D_{dr_2}^{(4\/\\beta)}=\\frac{1}{\\Delta_d(r_2)}\\det\\left[r_{n2}^{d-m}\\left(\\frac{\\partial}{\\partial r_{n2}}+(d-m)\\frac{2}{\\beta}\\frac{1}{r_{n2}}\\right)\\right]_{1\\leq n,m\\leq d} .\n \\end{equation}\n The constant is\n \\begin{equation}\\label{t2.3}\n  \\widetilde{C}_{acd}^{(\\beta)}=2^{-c}\\left(2\\pi\\gamma_1\\right)^{-ad}\\left(\\frac{2\\pi}{\\gamma_2}\\right)^{cd}\\tilde{\\gamma}^{\\beta ac\/2}\\frac{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a)\\right)}{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a-c)\\right){\\rm FU}_d^{(4\/\\beta)}} .\n \\end{equation}\n\\end{theorem}\n\nSince the diagonalization of $\\rho_2$ yields an $|\\Delta_d(r_2)|^{4\/\\beta}$ in the measure, the ratio of the Dirac--distribution with the Vandermonde--determinant is for Schwartz--functions on ${\\rm Herm\\,}(4\/\\beta,d)$ well--defined. Also, the action of $D_{dr_2}^{(4\/\\beta)}$ on such a Schwartz--function integrated over the corresponding rotation group is finite at zero.\n\nThe distribution in the Fermion--Fermion block in Eq.~\\eref{t2.1} takes for $\\beta\\in\\{1,2\\}$ the simpler form \\cite{Guh06,KGG08}\n\\begin{eqnarray}\n \\fl\\left(D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\frac{\\delta(r_2)}{|\\Delta_d(r_2)|^{4\/\\beta}} =\\nonumber\\\\\n ={\\rm FU}_d^{(4\/\\beta)}\\prod\\limits_{n=1}^{d}\\frac{\\Gamma\\left(a-c+1+2(n-1)\/\\beta \\right)}{(-\\pi)^{2(n-1)\/\\beta}\\Gamma\\left(\\gamma_1\\kappa\\right)}\\prod\\limits_{n=1}^{d}\\frac{\\partial^{\\gamma_1\\kappa-1}}{\\partial r_{n2}^{\\gamma_1\\kappa-1}}\\delta(r_{2n})\\label{5.1}\\ .\n\\end{eqnarray}\nThis expression written as a contour integral is the superbosonization formula \\cite{BasAke07}. For $\\beta=4$, we do not find such a simplification due to the term $|\\Delta(r_2)|$ as the Jacobian in the eigenvalue--angle coordinates.\n\n\\section{Equivalence of and connections between the two approaches}\\label{sec6}\n\nAbove, we have argued that both expressions in Theorems \\ref{t1} and \\ref{t2} are equivalent for $\\beta\\in\\{1,2\\}$. Now we address all $\\beta\\in\\{1,2,4\\}$. The Theorem below is proven in  \\ref{app3}. The proof treats all three cases in a unifying way. Properties of the ordinary matrix Bessel--functions are used.\n\\begin{theorem}[Equivalence of Theorems \\ref{t1} and \\ref{t2}]\\ \\label{t3}\\\\\n The superbosonization formula, \\ref{t1}, and the generalized Hubbard--Stratonovich transformation, \\ref{t2}, are equivalent for superfunctions which are Schwartz--functions and analytic in the fermionic eigenvalues.\n\\end{theorem}\n\nThe compact integral in the Fermion--Fermion block of the superbosonization formula can be considered as a contour integral. In the proof of Theorem \\ref{t3}, we find the integral identity\n\\begin{eqnarray}\n  & & \\int\\limits_{[0,2\\pi]^{d}}\\widetilde{F}\\left(e^{\\imath\\varphi_j}\\right)\\left|\\Delta_d\\left(e^{\\imath\\varphi_j}\\right)\\right|^{4\/\\beta}\\prod\\limits_{n=1}^d\\frac{e^{\\imath(1-\\gamma_1\\kappa)\\varphi_n}d\\varphi_n}{2\\pi}=\\nonumber\\\\\n  & = & \\prod\\limits_{n=1}^d\\frac{\\imath^{4(n-1)\/\\beta}\\Gamma(1+2n\/\\beta)}{\\Gamma(2\/\\beta+1)\\Gamma(\\gamma_1\\kappa-2(n-1)\/\\beta)}\\left.\\left(D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\widetilde{F}(r_2)\\right|_{r_2=0}\\label{6.1}\n\\end{eqnarray}\nfor an analytic function $\\widetilde{F}$ on $\\mathbb{C}^d$ with permutation invariance. Hence, we can relate both constants \\eref{t1.2} and \\eref{t2.3},\n\\begin{equation}\\label{6.2}\n \\frac{\\widetilde{C}_{acd}^{(\\beta)}}{C_{acd}^{(\\beta)}}=(-1)^{d(a-c)}\\prod\\limits_{n=1}^d\\frac{\\imath^{4(n-1)\/\\beta}\\Gamma(1+2n\/\\beta)}{\\Gamma(2\/\\beta+1)\\Gamma(\\gamma_1\\kappa-2(n-1)\/\\beta)}.\n\\end{equation}\nThe integral identity \\eref{6.1} is a reminiscent of the residue theorem. It is the analog of the connection between the contour integral and the differential operator in the cases $\\beta\\in\\{1,2\\}$, see Fig. \\ref{f1}. Thus, the differential Operator with the Dirac--distribution in the generalized Hubbard--Stratonovich transformation restricts the non-compact integral in the Fermion--Fermion block to the point zero and its neighborhood. Therefore it is equivalent to a compact Fermion--Fermion block integral as appearing in the superbosonization formula.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{Superbosonization.eps}\n \\caption{In the superbosonization formula, the integration of the fermionic eigenvalues is along the unit circle in the complex plane (dotted circle). The eigenvalue integrals in the generalized Hubbard--Stratonovich transformation are integrations over the real axis (bold line) or on the Wick--rotated real axis (thin line at angle $\\psi$) if the differential operator acts on the superfunction or on the Dirac--distribution at zero (bold dot, $0$), respectively.}\\label{f1}\n\\end{figure}\n\n\\section{The general case for arbitrary positive integers $a$, $b$, $c$, $d$ and arbitrary Dyson--index $\\beta\\in\\{1,2,4\\}$}\\label{sec7}\n\nWe consider an application of our results. The inequality \\eref{t1.0} reads\n\\begin{equation}\\label{7.1}\n  N\\geq\\gamma_1k\n\\end{equation}\nfor the calculation of the $k$--point correlation function \\eref{2.3} with help of the matrix Green function. For $\\beta=1$ , a $N\\times N$ real symmetric matrix has in the absence of degeneracies $N$ different eigenvalues. However, we can only calculate $k$--point correlation functions with $k<N\/2$. For $N\\to\\infty$, this restriction does not matter. But for exact finite $N$ calculations, we have to modify the line of reasoning.\n\nWe construct the symmetry operator\n\\begin{equation}\\label{7.2}\n \\mathfrak{S}\\left(\\sigma\\right)=\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\right)=\\left[\\begin{array}{cc} -\\sigma_{22} & -\\sigma_{21} \\\\ \\sigma_{12} & \\sigma_{11} \\end{array}\\right]\n\\end{equation}\nfrom $(m_1+m_2)\\times(n_1+n_2)$ supermatrix to $(m_2+m_1)\\times(n_2+n_1)$ supermatrix. This operator has the properties\n\\begin{eqnarray}\n \\mathfrak{S}(\\sigma^\\dagger) & = & \\mathfrak{S}(\\sigma)^\\dagger\\quad, \\label{7.3}\\\\\n \\mathfrak{S}(\\sigma^*) & = & \\mathfrak{S}(\\sigma)^*, \\label{7.4}\\\\\n \\mathfrak{S}^2(\\sigma) & = & -\\sigma, \\label{7.5}\\\\\n \\fl\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\left[\\begin{array}{cc} \\rho_{11} & \\rho_{12} \\\\ 0 & 0 \\end{array}\\right]\\right) & = & \\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\right)\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\rho_{11} & \\rho_{12} \\\\ 0 & 0 \\end{array}\\right]\\right). \\label{7.6}\n\\end{eqnarray}\n\nLet $a,\\ b,\\ c$, $d$ be arbitrary positive integers and $\\beta\\in\\{1,2,4\\}$. Then, the equation \\eref{7.6} reads for a matrix product of a $(\\gamma_2c+\\gamma_1d)\\times(0+\\gamma_1b)$ supermatrix with a $(0+\\gamma_1b)\\times(\\gamma_2c+\\gamma_1d)$ supermatrix\n\\begin{equation}\\label{7.7}\n \\fl\\left[\\begin{array}{c} \\zeta^\\dagger \\\\ \\tilde{z}^\\dagger \\end{array}\\right]\\left[\\begin{array}{cc} \\zeta & \\tilde{z} \\end{array}\\right]=\\mathfrak{S}\\left(\\left[\\begin{array}{c} \\tilde{z}^\\dagger \\\\ -\\zeta^\\dagger \\end{array}\\right]\\right)\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\tilde{z} & \\zeta \\end{array}\\right]\\right)=\\mathfrak{S}\\left(\\left[\\begin{array}{c} \\tilde{z}^\\dagger \\\\ -\\zeta^\\dagger \\end{array}\\right]\\left[\\begin{array}{cc} \\tilde{z} & \\zeta \\end{array}\\right]\\right).\n\\end{equation}\nWith help of the operator $\\mathfrak{S}$, we split the supersymmetric Wishart--matrix $\\widehat{B}$ into two parts,\n\\begin{equation}\\label{7.8}\n \\widehat{B}=\\widehat{B}_1+\\mathfrak{S}(\\widehat{B}_2)\n\\end{equation}\nsuch that\n\\begin{equation}\\label{7.9}\n \\fl\\widehat{B}_1=\\tilde{\\gamma}^{-1}\\sum\\limits_{j=1}^{a}\\Psi_{j1}^{({\\rm C})}\\Psi_{j1}^{({\\rm C})\\dagger}\\qquad{\\rm and}\\qquad\\widehat{B}_2=\\tilde{\\gamma}^{-1}\\sum\\limits_{j=1}^{b}\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)^\\dagger .\n\\end{equation}\nThe supervectors $\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)$ are of the same form as $\\Psi_{j1}^{({\\rm C})}$. Let $\\sigma$ be a quadratic supermatrix, i.e. $m_1=n_1$ and $m_2=n_2$. Then, we find the additional property\n\\begin{equation}\\label{7.10}\n {\\rm Sdet\\,}\\mathfrak{S}(\\sigma)=(-1)^{m_2}{\\rm Sdet\\,}^{-1}\\sigma .\n\\end{equation}\n\nLet $\\widehat{\\Sigma}_{\\beta,pq}^{(0)}=\\mathfrak{S}\\left(\\Sigma_{\\beta,pq}^{(0)}\\right)$, $\\widehat{\\Sigma}_{\\beta,pq}^{0({\\rm c})}=\\mathfrak{S}\\left(\\Sigma_{\\beta,pq}^{0({\\rm c})}\\right)$ and the Wick--rotated set $\\widehat{\\Sigma}_{\\beta,pq}^{(\\psi)}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{\\Sigma}_{\\beta,pq}^{(0)}\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$. Then, we construct the analog of the superbosonization formula and the generalized Hubbard--Stratonovich transformation.\n\\begin{theorem}\\label{t4}\\ \\\\\n Let $F$ be the superfunction as in Theorem \\ref{t1} and \n \\begin{equation}\\label{t4.1}\n  \\kappa=\\frac{a-c+1}{\\gamma_1}-\\frac{b-d+1}{\\gamma_2} .\n \\end{equation}\n Also, let $e\\in\\mathbb{N}_0$ and\n \\begin{equation}\\label{t4.2}\n \\tilde{a}=a+\\gamma_1e\\qquad{\\rm and}\\qquad\\tilde{b}=b+\\gamma_2e\n \\end{equation}\n with\n \\begin{equation}\\label{t4.3}\n \\tilde{a}\\geq c\\qquad\\tilde{b}\\geq d .\n \\end{equation}\n We choose the Wick--rotation  $e^{\\imath\\psi}$ such that all integrals are convergent. Then, we have\n \\begin{eqnarray}\n   \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B}_{\\psi})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}_{\\psi}\\right)d[\\widehat{V}]=\\nonumber\\\\\n   \\fl= \\left(-\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed} \\int\\limits_{\\widetilde{\\mathfrak{R}}}F(\\widetilde{B}_{\\psi})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widetilde{B}_{\\psi}\\right)d[\\widetilde{V}]=\\nonumber\\\\\n  \\fl= C_{{\\rm SF}}\\int\\limits_{\\Sigma_{\\beta,cd}^{0({\\rm c})}}\\int\\limits_{\\widehat{\\Sigma}_{4\/\\beta,dc}^{0({\\rm c})}}d[\\rho^{(2)}]d[\\rho^{(1)}]F(\\rho^{(1)}+e^{\\imath\\psi}\\rho^{(2)})\\exp\\left[-\\varepsilon{\\rm Str\\,}(\\rho^{(1)}+e^{\\imath\\psi}\\rho^{(2)})\\right]\\times\\nonumber\\\\\n  \\fl\\times {\\rm Sdet\\,}^{\\kappa+\\tilde{b}\/\\gamma_2}\\rho^{(1)}{\\rm Sdet\\,}^{\\kappa-\\tilde{a}\/\\gamma_1}\\rho^{(2)}=\\label{t4.5a}\\\\\n  \\fl =C_{{\\rm HS}}\\int\\limits_{\\Sigma_{\\beta,cd}^{(0)}}\\int\\limits_{\\widehat{\\Sigma}_{4\/\\beta,cd}^{(0)}}d[\\rho^{(2)}]d[\\rho^{(1)}]\\frac{\\delta\\left(r_2^{(1)}\\right)}{\\left|\\Delta_d\\left(r_2^{(1)}\\right)\\right|^{4\/\\beta}}\\frac{\\delta\\left(r_1^{(2)}\\right)}{\\left|\\Delta_c\\left(r_1^{(2)}\\right)\\right|^{\\beta}}{\\det}^{\\kappa+b\/\\gamma_2}\\rho_1^{(1)}{\\det}^{a\/\\gamma_1-\\kappa}\\rho_2^{(2)}\\times\\nonumber\\\\\n  \\fl\\times \\left(D_{dr_2^{(1)}}^{(4\/\\beta)}\\right)^{\\tilde{a}-c}\\left(D_{cr_1^{(2)}}^{(\\beta)}\\right)^{\\tilde{b}-d}F(\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)})\\exp\\left[-\\varepsilon{\\rm Str\\,}(\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)})\\right], \\label{t4.5b}\n \\end{eqnarray}\n where the constants are\n \\begin{eqnarray}\n  C_{{\\rm SF}} & = & (-1)^{c(b-d)}e^{\\imath\\psi(\\tilde{a}d-\\tilde{b}c)}\\left(\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed}C_{\\tilde{a}cd}^{(\\beta)}C_{\\tilde{b}dc}^{(4\/\\beta)} ,\\\\\n  C_{{\\rm HS}} & = & (-1)^{d(a-c)}e^{\\imath\\psi(\\tilde{a}d-\\tilde{b}c)}\\left(-\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(-\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed}\\widetilde{C}_{\\tilde{a}cd}^{(\\beta)}\\widetilde{C}_{\\tilde{b}dc}^{(4\/\\beta)} .\n \\end{eqnarray}\n Here, we define the supermatrix\n \\begin{equation}\\label{t4.6}\n  \\fl\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)}=\\left[\\begin{array}{c|c} \\rho_1^{(1)}+e^{\\imath\\psi}\\left(\\rho_1^{(2)}-\\rho_{\\tilde{\\eta}}^{(2)}\\rho_2^{(2)-1}\\rho_{\\tilde{\\eta}}^{(2)\\dagger}\\right) & \\rho_\\eta^{(1)}+e^{\\imath\\psi}\\rho_{\\tilde{\\eta}}^{(2)} \\\\ \\hline -\\rho_\\eta^{(1)\\dagger}-e^{\\imath\\psi}\\rho_{\\tilde{\\eta}}^{(2)\\dagger} & \\rho_2^{(1)}-\\rho_\\eta^{(1)\\dagger}\\rho_1^{(1)-1}\\rho_\\eta^{(1)}+e^{\\imath\\psi}\\rho_2^{(2)} \\end{array}\\right]\n \\end{equation}\n The set $\\widetilde{\\mathfrak{R}}$ is given as in corollary \\ref{c2}. The measures $d[\\rho^{(1)}]=d[\\rho_1^{(1)}]d[\\rho_2^{(1)}]d[\\eta]$ and $d[\\rho^{(2)}]=d[\\rho_1^{(2)}]d[\\rho_2^{(2)}]d[\\tilde{\\eta}]$ are given by Theorem \\ref{t1}. The measures \\eref{t1.4} for $\\beta$ and $4\/\\beta$ assign $d[\\rho_1^{(1)}]$ and $d[\\rho_2^{(2)}]$ in Eqs. \\eref{t4.5a} and \\eref{t4.5b}, respectively. In Eq.~\\eref{t4.5a}, $d[\\rho_2^{(1)}]$ and $d[\\rho_1^{(2)}]$ are defined by the measure \\eref{t1.5} for the cases $\\beta$ and $4\/\\beta$, respectively, and, in Eq.~\\eref{t4.5b}, they are defined by the measure \\eref{t1.4} for the cases $4\/\\beta$ and $\\beta$, respectively. The measures $d[\\eta]$ and $d[\\tilde{\\eta}]$ are the product of all complex Grassmann pairs as in Eq.~\\eref{t1.6}.\n\\end{theorem}\nSince this Theorem is a consequence of corollary \\ref{c2} and Theorems \\ref{t1} and \\ref{t2}, the proof is quite simple.\\\\\n\\textbf{Proof:}\\\\\nLet $e\\in\\mathbb{N}_0$ as in Eq.~\\eref{t4.2}. Then, we use corollary \\ref{c2} to extend the integral over $\\widehat{V}$ to an integral over $\\widetilde{V}$. We split the supersymmetric Wishart--matrix $\\widehat{B}$ as in Eq.~\\eref{7.8}. Both Wishart--matrices $\\widehat{B}_{1}$ and $\\widehat{B}_{2}$ fulfill the requirement \\eref{t1.0} according to their dimension. Thus, we singly apply both Theorems \\ref{t1} and \\ref{t2} to $\\widehat{B}_{1}$ and $\\widehat{B}_{2}$. \\hfill$\\square$\n\nOur approach of a violation of inequality \\eref{t1.0} is quite different from the solution given in Ref.~\\cite{BEKYZ07}. These authors introduce a matrix which projects the Boson--Boson block and the bosonic side of the off-diagonal blocks onto a space of the smaller dimension $a$. Then, they integrate over all of such orthogonal projectors. This integral becomes more difficult due to an additional measure on a curved, compact space. We use a second symmetric supermatrix. Hence, we have up to the dimensions of the supermatrices a symmetry between both supermatrices produced by $\\mathfrak{S}$. There is no additional complication for the integration, since the measures of both supermatrices are of the same kind. Moreover, our approach extends the results to the case of $\\beta=4$ and odd $b$ which is not considered in Ref.~\\cite{BEKYZ07}.\n\n\\section{Remarks and conclusions}\\label{sec8}\n\nWe proved the equivalence of the generalized Hubbard--Stratonovich transformation \\cite{Guh06,KGG08} and the superbosonization formula \\cite{Som07,LSZ07}. Thereby, we generalized both approaches. The superbosonization formula was proven in a new way and is now extended to odd dimensional supersymmetric Wishart--matrices in the Fermion--Fermion block for the quaternionic case. The generalized Hubbard--Stratonovich transformation was here extended to arbitrary dimensional supersymmetric Wishart--matrices which not only stem of averages over the matrix Green functions. \\cite{GMW98,Zir06,Guh06,KKG08} Furthermore, we got an integral identity beyond the restriction of the matrix dimension, see Eq.~\\eref{t1.0}. This approach distinguishes from the method presented in Ref.~\\cite{BEKYZ07} by the integration of an additional matrix. It is, also, applicable on the artificial example $\\beta=4$ and odd $b$ which has not been considered in Ref.~\\cite{BEKYZ07}.\n\nThe generalized Hubbard--Stratonovich transformation and the superbosonization formula reduce in the absence of Grassmann variables to the ordinary integral identity for ordinary Wishart--matrices. \\cite{Fyo02,LSZ07} In the general case with the restriction \\eref{t1.0}, both approaches differ in the Fermion--Fermion block integration. Due to the Dirac--distribution and the differential operator, the integration over the non-compact domain in the generalized Hubbard--Stratonovich transformation is equal with help of the residue theorem to a contour integral. This contour integral is equivalent to the integration over the compact domain in the superbosonization formula. Hence, we found an integral identity between a compact integral and a differentiated Dirac--distribution.\n\n\\section*{Acknowledgements}\n\nWe thank Heiner Kohler for fruitful discussions. This work was supported by Deutsche Forschungsgemeinschaft within Sonderforschungsbereich Transregio 12 ``Symmetries and Universality in Mesoscopic Systems''.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe most successful method for detecting faint companions around nearby stars is undoubtedly the radial velocity (RV) technique \\citep{Perryman00}, with more than 500 extrasolar planets detected up to date. Nevertheless, RV has three major drawbacks. First, it cannot be applied to all kind of stars: pulsating stars, as well as young, rapidly rotating and\/or active stars have an intrinsic radial velocity jitter that generally precludes planet-search programmes \\citep{Udry07}. The discovery of extrasolar planets orbiting high-mass main-sequence stars (O, B, and A spectral types) and young solar-type stars (T Tauri stars) therefore remains rare in the literature. Second, RV can only access companions with periods shorter than a few tens of years, owing to the limited time span of the observations obtained with high-precision spectrographs. Third, RV is inherently insensitive to the orbital inclination $i$. This implies that additional astrometric observations are mandatory to unveil the real companion mass from the measured quantity $M_p\\sin{i}$. For these reasons, the development of direct imaging techniques for low-mass companions is currently one of the highest priorities in instrumental astronomy \\citep{Oppenheimer09,Absil10b}.\n\nDirect imaging techniques should be tailored to the typical angular separations for which low-mass companions are to be found. For instance, the search for young planets in nearby star-forming regions ($\\geq 100$\\,pc) should be optimised for typical linear separations from 0.1 to a few 10\\,AU for Sun-like stars, where planets are supposed to be forming and migrating in their early history \\citep{Bodenheimer02}. This translates into typical angular separations of 1--100\\,milli-arcsecs (mas). In a more general context, more than 90\\% of the sub-stellar companions detected to date by all techniques have expected apparent separations smaller than 100\\,mas \\citep{Schneider11}. This domain is hardly accessible to single-pupil imaging techniques, even with adaptive-optics (AO) assisted facilities \\citep{Burrows05}. Nowadays, with 10-m class telescopes, only interferometric aperture-masking observations routinely break the 100\\,mas limit. However, their search space is still restricted to angular separations larger than about 40\\,mas for dynamic ranges of the order of 500:1 \\citep{Kraus08,Lacour:2011}.\n\nThanks to its higher angular resolution, optical long baseline interferometry is the ideal tool for exploring separations in the range 1--50\\,mas. The simplest strategy for faint companion detection with optical\/infrared interferometry is to obtain closure phase measurements that are directly sensitive to the asymmetry in the brightness distribution of the target source and hence to possible off-axis companions. However, owing to the sparse structure of the point spread function associated with the diluted aperture of an interferometer, the depth to which a companion can be detected strongly depends on the relative orientation of the companion and the interferometric baselines. Consequently, the sensitivity limit should be defined as a two-dimensional map and\/or for various completeness levels.\n\nThe goal of this paper is to provide quantitative estimations of the detection efficiency versus the companion contrast and separation considering realistic observations. In Sect.~\\ref{sec:theory}, we derive a linear expression for the closure phase signature of a faint companion. We also define a simple, general expression for the sensitivity limit in terms of companion contrast, which we use in Sect.~\\ref{sec:array} to compute the efficiency of various interferometric configurations. We compare the efficiency of the four configurations currently offered at the Very Large Telescope Interferometer. To add generality, we also discuss the efficiency of three ``standard'' interferometric arrays (two linear and one Y-shaped). In Sect.~\\ref{sec:accuracy}, we discuss the typical achievable dynamic range for the closure phase accuracy provided by existing instruments.\n\n\n\n\\section{Detecting faint companions with closure phase}\n\\label{sec:theory}\n\nOptical long baseline interferometers provide several observables, which are all sensitive to the presence of a faint companion: visibilities, differential phases, and closure phases. The latter has the main advantage of being, to first order, uncorrupted by telescope-specific phase errors, including pointing errors, atmospheric piston, and longitudinal dispersion due to air and water vapour. The noise floor is only limited by the accuracy to which the instrumental systematics can be mastered. As a consequence, the high-precision closure phase is the favoured observational strategy to directly detect faint companions with modern interferometers \\citep{Vannier06,Zhao08,Renard08}.\n\n\n\\subsection{Closure-phase signal of a faint companion} \\label{sec:signal}\n\nMeasuring a closure phase requires the use of an interferometric array composed of (at least) three telescopes. Each pair of telescopes defines a geometrical vector referred to as the interferometric baseline. The closure phase is then defined as the sum of the phases measured on the three baselines, or equivalently, as the argument of the bispectrum, formed through the triple product of the measured complex visibilities around the triangle \\citep{Monnier03}. Assuming that the stellar diameters are not resolved at the considered interferometric baseline lengths, the closure phase signature of a binary object can be expressed as a function of the three baseline vectors projected onto the sky \\{\\ensuremath{\\vec{B}_{12}},\\ensuremath{\\vec{B}_{23}},\\ensuremath{\\vec{B}_{31}}\\}, the wavelength of observation \\ensuremath{\\lambda{}}{}, the binary flux ratio \\ensuremath{\\rho}{} (referred to as the contrast in the following), and the apparent binary separation vector $\\ensuremath{\\vec\\Delta}$\n\\begin{equation}\n\\ensuremath{\\phi} = \\arg \\left( \\frac{(1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{12}}}) (1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{23}}}) (1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{31}}})} {(1+\\ensuremath{\\rho})^3} \\right) \\, , \\label{eq:1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\ensuremath{\\alpha_{12}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{12}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; ; \\;\\;\n\\ensuremath{\\alpha_{23}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{23}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; ; \\;\\;\n\\ensuremath{\\alpha_{31}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{31}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; . \\label{eq:1b}\n\\end{equation}\nThe geometrical terms \\ensuremath{\\alpha_{12}}{}, \\ensuremath{\\alpha_{23}}{}, and \\ensuremath{\\alpha_{31}}{} describe the relative orientation of the companion compared to the spatial frequencies explored by the interferometric array. We note that $\\ensuremath{\\alpha_{31}}=-(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})$ since the vectorial sum of the three interferometric baselines is zero by definition.\n\nTo model the signal of a high contrast binary, we can either use Eq.~\\ref{eq:1} with small ($\\ll$$1$) or large ($\\gg$$1$) values of $\\ensuremath{\\rho}$. In the first case, the interferometer points toward the primary, while in the second case it points toward the companion. Since the closure-phase is independent of the pointing position, these two approaches are formally identical. Assuming $\\ensuremath{\\rho}\\ll1$, we can develop and simplify Eq.~\\ref{eq:1} to be\n\\begin{equation}\n\\ensuremath{\\phi} \\approx \\ensuremath{\\rho} \\, (\\sin\\ensuremath{\\alpha_{12}} + \\sin\\ensuremath{\\alpha_{23}} - \\sin(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})) \\, . \\label{eq:2}\n\\end{equation}\nThe quantity $\\ensuremath{m}=\\sin\\ensuremath{\\alpha_{12}} + \\sin\\ensuremath{\\alpha_{23}} - \\sin(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})$ is referred to as the magnification factor because a high value of $\\ensuremath{m}$ corresponds to a stronger companion signature in the closure phase signal.\n\nThe validity range of our approximate formula can be explored numerically by computing the difference between Eq.~\\ref{eq:1} and~\\ref{eq:2} for various values of \\{\\ensuremath{\\rho},\\ensuremath{\\alpha_{12}},\\ensuremath{\\alpha_{23}}\\}. We found that Eq.~\\ref{eq:2} stays within a factor $\\lesssim 1.5$ of the value given by Eq.~\\ref{eq:1} as long as $\\ensuremath{\\rho} \\leq 10^{-1}$, which we consider to be the validity range of our work. Therefore, the following results apply only to relatively faint companions, and cannot be immediately transposed to binaries with flux ratios close to unity.\n\nApart from the computational gain, the main advantage of using Eq.~\\ref{eq:2} is to define a proportional relation between the closure phase and the companion contrast. This reduces the number of parameters to be explored and allows the results to be presented in a synthetic way. Additionally, we note that in Eq.~\\ref{eq:2}, the magnification factor $m$ takes a maximum value of about 2.6\\,rad ($\\approx149\\deg$) for purely geometrical reasons. This magnification is only achieved in the optimal geometrical case, where all interferometric baselines add together to amplify the companion signal. Therefore, as a quantitative example, we can already conclude that reaching a dynamic range of $10^{-3}$ with one single closure phase measurement requires a closure phase accuracy \\emph{better} than $0.15\\deg$. In practice, this accuracy is generally reached after appropriate uv-plane or temporal averaging of a series of individual closure phase measurements.\n\n\\subsection{The close-companion limit}\nIn the specific case where the binary is not resolved by any individual interferometric baselines (all $\\alpha<1$), it is possible to simplify Eq.~\\ref{eq:2} by expanding the sines for small values of $\\alpha$\n\\begin{equation}\n  \\frac{\\phi}{\\ensuremath{\\rho}} \\approx \\frac{\\ensuremath{\\alpha_{12}}^2\\,\\ensuremath{\\alpha_{23}} + \\ensuremath{\\alpha_{12}}\\,\\ensuremath{\\alpha_{23}}^2}{2} \\;+\\; o(\\alpha^5) \\; .\n\\label{eq:close1}\n\\end{equation}\nWe can arbitrarily choose $\\ensuremath{\\alpha_{12}}$ to be associated with the most resolving baseline and $\\ensuremath{\\alpha_{23}}{}$ with the least resolving baseline in the direction of the considered binary. We introduce the $r$ ratio of the two geometrical terms: $\\ensuremath{\\alpha_{23}}=-r\\,\\ensuremath{\\alpha_{12}}$ (the negative sign is introduced for the triangle to close with positive values of $r$). Rewriting Eq.~\\ref{eq:close1}, we obtain\n\\begin{equation}\n  \\frac{\\phi}{\\ensuremath{\\rho}} \\approx \\frac{1}{2}\\,\\ensuremath{\\alpha_{12}}^3\\,r\\,(1-r)\n  \\label{eq:close2}\n\\end{equation}\nThis equation already shows two interesting aspects: (i) the closure phase signature of an unresolved companion is proportional to the baseline length at the third power, and (ii) the arrays that are redundant when projected onto the binary direction $(r=0.5)$ seem optimal for detecting a close companion using a given long baseline.\n\n\n\\subsection{Deriving sensitivity limits}\nA typical interferometric observation is not composed of a single closure phase measurement. We now derive sensitivity limits in terms of companion contrast for a set of $n$ linearly independent\\footnote{Using $m$ apertures, one can form $(m-1)(m-2)\/2$ linearly independent closure phases. This is equivalent to holding one aperture fixed and forming all possible triangles with that aperture \\citep{Monnier03}.} closure phase measurements $\\ensuremath{\\phi}_i$ of accuracy $\\sigma_i$. We base our analysis on the $\\chi^2$ of the data with respect to a model of an unresolved source with no companion, for which all closure phases are zero\n\\begin{equation}\n\\chi^2 = \\sum_{i=1}^n \\frac{\\ensuremath{\\phi}^2_i}{\\sigma_i^2} \\; .\n\\end{equation}\nWe introduce the simplification presented in Eq.~\\ref{eq:2} and assume that all the measurements have similar accuracies (a valid assumption when observing an unresolved target) to obtain\n\\begin{equation}\n\\chi^2 = \\frac{\\rho^2}{\\sigma^2} \\; \\sum_{i=1}^n m_i^2 \\; .\n\\end{equation}\nThe probability that the data set is compatible with the single-star model is given by the complement of the cumulative probability distribution function (CDF) with $n$ degrees of freedom\n\\begin{equation}\nP = 1-\\mathrm{CDF}_n(\\chi^2) \\; .\n\\end{equation}\nIf $P$ is below a predefined threshold, the data set allows the model with no companion to be rejected. The threshold can generally be fixed at a 3-$\\sigma{}$ level, i.e., at a probability of 0.27\\%. The choice of the threshold actually depends on the context of the observations. If a large region of parameter space (or indeed number of targets) is searched for a companion, then a 5-$\\sigma{}$ threshold may be more appropriate. A higher threshold may even be needed in the case of sparse data sets dominated by non-Gaussian systematic errors. Taking the 3-$\\sigma{}$ level as an example, this threshold can be converted into a sensitivity limit in terms of companion contrast\n\\begin{equation}\n  \\rho = \\sigma \\sqrt{\\frac{\\mathrm{CDF}^{-1}_n(1-0.27\\%)}{\\sum_{i=1}^n m_i^2}} \\; ,\n\\end{equation}\nwhere $\\mathrm{CDF}^{-1}_n(1-0.27\\%)$ is simply the $\\chi^2$ value for a 3-$\\sigma{}$ detection with $n$ degrees of freedom. As a consequence of the approximation presented in Eq.~\\ref{eq:2}, the sensitivity limit is directly proportional to the accuracy of the closure phase measurements.\n\nBecause of the sparse structure of the point spread function associated with the diluted aperture of an interferometer, the depth to which a companion can be detected strongly depends on the relative orientation between the companion and the interferometric baselines (information embedded in the magnification factors $m_i$). For some lucky separations, the greatest dynamic range is achieved, while in the worst cases even obvious binaries with equal brightnesses can be missed. Consequently, the sensitivity limit should be defined as a two-dimensional map or for various completeness levels on a given search region.\n\n\\subsection{Validity limit of this study}\nIn the case of wide companions, care should be taken regarding the chromaticity limit of our study. The results presented here are indeed formally valid only for monochromatic light. The main effect of wavelength smearing inside Eq.~\\ref{eq:1} is to degrade the dynamic range. To avoid significant smearing, the spectral resolution should be higher than the $\\alpha$ quantities defined in Eq.~\\ref{eq:1b}. This translates into a spectral resolution $R>10$ for a $40\\,$mas binary observed with a $100\\,$m baseline at $1.7\\,\\mu$m (H band). Such a low spectral resolution is available in most modern interferometric beam combiners. In the case of spatially filtered beam combiners, a similar effect may occur because of baseline smearing, in the case where the telescope size cannot be neglected in Eq.~\\ref{eq:1}. For the $1.8$-m Auxiliary Telescopes, this corresponds to angular separations of about $175\\,$mas for H-band observations. For $10$-m class telescopes, this corresponds to angular separations of about $45\\,$mas. In practice, these limitations are not very relevant to our study because AO-assisted spare aperture-masking imaging on 10-m class telescopes becomes more efficient than long-baseline interferometry for separations larger than about $40\\,$mas \\citep[see e.g.][]{Kraus08,Lacour:2011}.\n\nFinally, Eq.~\\ref{eq:1} assumes that the angular diameters of the binary components are unresolved by the interferometric baselines. Resolving the diameter of the faint component indeed appears unrealistic, although maybe not for the central star, especially in the case of bright late-type giants or very nearby stars. In this latter situation, Eq.~\\ref{eq:1} underestimates the closure phase signal, which peaks for a fully resolved primary star. This effect, referred to as \\emph{closure phase nulling}, can lead to larger magnification factors than the limit $m<149\\deg$ presented in Sect.~\\ref{sec:signal}. This specific observing technique is discussed in detail in \\citet{Chelli:2009}, while on-sky applications can be found in \\citet{Monnier:2006}, \\citet{Lacour08}, \\citet{Zhao08}, and \\citet{Duvert:2010}.\n\n\n\\section{Optical interferometric array}\n\\label{sec:array}\n\nWe use the formalism introduced previously to compute and compare the capabilities of various four-telescope interferometric configurations.\n\n\\subsection{Performances of VLTI configurations}\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[scale=0.58]{visa}\n    \\caption{Interferometric configuration offered at VLTI. The configurations using the four relocatable Auxiliary Telescopes are represented by colours. The configuration using the four fixed Unit Telescopes is represented in black.}\n    \\label{fig:visa}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering \n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle1}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance1}\n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle3}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance3}\n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle5}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance5}\n  \\caption{{\\it Left:} Map of the 3-$\\sigma{}$ sensitivity with the A1-K0-G1-I1 configuration from VLTI.  Radial zones are the bins in separation used for the plots in the right panel. {\\it Right:}  Sensitivity as a function of angular distance, for various completeness levels. {\\it From top to botom:}  Simulations for a single snapshot pointing (top), and for three pointing (middle), and for five pointing (bottom). The contrast axes can be scaled for any accuracy on the closure phase (here $\\sigma=0.25\\deg$).}\n  \\label{fig:det_map}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering \n    \\includegraphics[scale=0.55]{all_contdist}\\hspace{2cm}\n    \\includegraphics[scale=0.55]{all_compcont}\n    \\caption{{\\it Left:} 3-$\\sigma{}$ sensitivity versus separation for a completeness level of 80\\%. {\\it Right:} Completeness in the separation range $6-40\\,$mas versus contrast. Simulations are for a single snapshot pointing (dashed lines) and for five pointings separated by one hour (solid lines). Colours are for the four configurations of VLTI displayed in Fig.~\\ref{fig:visa}. The contrast axes can be scaled for any given accuracy of the closure phase measurements (here $\\sigma=0.25\\deg$).}\n    \\label{fig:cont_sep}\n    \\label{fig:comp_cont}\n\\end{figure*}\n\nWe compute the map of the 3-$\\sigma{}$ sensitivity limit using all the VLTI configurations displayed in Fig.~\\ref{fig:visa}, assuming a target at declination $-35\\,$deg. The limits are computed considering data sets consisting of respectively one pointing at an hour angle $\\mathrm{HA}=-2\\,$h, three pointings at hour angles $\\mathrm{HA}=-2\\,$h, $-1\\,$h and $0\\,$h, and five pointings at hour angles $\\mathrm{HA}=-2\\,$h, $-1\\,$h, $0\\,$h, $+1\\,$h, and $+2\\,$h. A closure phase accuracy of $0.25\\deg$ is assumed (see Sect.~\\ref{sec:accuracy}). We consider a maximum binary separation of $40\\,$mas, which is the separation where AO-assisted spare aperture-masking imaging on 10-m class telescopes becomes more efficient than long-baseline interferometry.\n\nDetailed results of the widest AT configuration (A1-K0-G1-I1) are displayed in Fig.~\\ref{fig:det_map} for illustration. The figure shows that the detection completeness for a given sensitivity level increases drastically with the number of pointings. This is clearly illustrated by the decreasing number of ``blind spots'' (white zones) in the left-hand side plots. The sensitivity level is mostly flat for angular separations larger than 2\\,mas, while the detection performance drops considerably within this inner working angle (IWA). The median sensitivity levels in the region 2--40 mas are respectively about $6\\times 10^{-3}$, $4.5\\times 10^{-3}$ and $4\\times 10^{-3}$ for the three considered data sets. For more than five pointings, the median sensitivity level would continue to improve slightly, but the shape of the sensitivity curve would no longer significantly change.\n\nThe relative performances of the different configurations are illustrated in Fig.~\\ref{fig:cont_sep} (left: sensitivity versus separation, right: completeness level versus contrast). All configurations provide flat performances for large separations, down to their respective IWA where the performances dramatically drop. The IWA are respectively $2\\,$mas for the A1-K0-G1-I1 and U1-U2-U3-U4 configurations, $3\\,$mas for the D0-H0-I1-G1 configuration, and $6\\,$mas for the A1-B2-C1-D0 configuration. They correspond to the spatial resolution of the smallest baseline of the array. Given the similarity of the results for all configurations, we discuss them together in two different regimes: (i) the close-companion and (ii) the wide-companion regimes.\n\n\\subsection{Close-companion regime}\n\nClose companions are defined here as companions with angular separations that are not fully resolved by \\emph{at least} one baseline (that is $\\vec{B}\\cdot\\vec{\\Delta}\/\\lambda<1$). In this regime, the achievable contrast for a given completeness follows a power law of the angular separation $\\ensuremath{\\rho} \\propto \\Delta^{-3}$, as predicted by Eq.~\\ref{eq:close2}. The exact factor entering into this law depends on the array geometry but, as expected, the longest arrays provide the highest spatial resolution and are thus able to detect both the deepest and the closest binaries.\n\nThis well-known result was discussed by \\citet{Lachaume03} in the context of partially resolved interferometric observations. We emphasize that our study additionally provides a quantitative estimation. As a typical example, we now detail the case of a faint companion with a contrast of $5\\times10^{-3}$. We first consider the companion to be located at 2\\,mas from the central star.  A closure-phase accuracy of $0.25\\deg$ results in a detection efficiency of about 50\\% using three pointings with the configuration A0-K0-G1-I1, according to Fig.~\\ref{fig:det_map} (middle-right plot). However, if we now consider the companion to be located at $1\\,$mas from the central star, the closure-phase accuracy should be $0.025\\deg$ to reach the same efficiency. Since the angular separation is only marginally resolved by the interferometer, the lack of spatial resolution has to be compensated for by an increase in the accuracy on the signal (super-resolution effect).\n\n\\subsection{Wide-companion regime}\n\nWe note that those companions are \\emph{wide} only in the interferometric sense, corresponding to separations larger than about 4\\,mas for the typical $\\sim100\\,$m baselines available in modern interferometric facilities. \n\nIn this regime, the detection efficiency becomes independent of the companion separation. Interestingly, all arrays have the same efficiency. In other words, as long as the companion is expected to be resolved by the interferometric baselines, the choice of array configuration does not matter. We conclude that there is no reason to favour a given VLTI configuration when looking for faint unknown companion with separations in the range $6-40\\,$mas. More quantitatively, Fig.~\\ref{fig:cont_sep} (right) displays the detection efficiency in this annular region for the four VLTI configurations versus the companion contrast, and for two observing scenarios (snapshot and long integration). The combination of five observations separated by one hour provides a detection efficiency higher than 95\\% for companion contrasts of $10^{-2}$, assuming a realistic closure phase accuracy of $0.25\\deg$. \n\nWe note that the curve of completeness versus contrast become significantly sharper when accumulating observations. As shown by the solid lines in the right panel of Fig.~\\ref{fig:cont_sep}, when accumulating five pointings, the efficiency drops from 80\\% for a contrast of $5\\times10^{-3}$ to less than 10\\% for a contrast of $3\\times10^{-3}$. The constraints provided by this dataset can thus be presented as a sensitivity limit and an inner working-angle, as for a classical imaging observation.\n\nQuantitatively, when accumulating several pointings, these detection limits computed from the derivation of Sect.~\\ref{sec:theory} are compatible with the blind-test analyses presented by \\citet[Fig.~5]{Absil:2011} and the Monte-Carlo simulations of \\citet[Fig.~4 and Eq.~2]{Lacour:2011}.\n\n\\subsection{Performances of standard configurations}\n\\label{sec:fake}\n\nTo add generality to the results presented in the previous section, we now study configurations that are not specifically linked to any existing interferometric array. We select the configurations presented in Fig.~\\ref{fig:fake}, which is a non-redundant linear configuration, a fully redundant linear configuration and a Y-shaped configuration. All configurations have their longest baseline of the same size. The results are the following:\n\n\\begin{enumerate}\n\\item The linear non-redundant and Y-shaped configurations have similar detection limits as the currently offered (irregular) VLTI configurations for snapshot observations. Surprisingly, they also have similar performances when accumulating several pointings, while we may have expected that the Y-shape would unveil faster the remaining blind-spots.\n\\item For snapshot observations, the linear redundant configuration favours the \\emph{dynamic range} with respect to the \\emph{completeness}: it has a fainter detection limit for completeness levels below 50\\%, but becomes significantly worse for higher completeness levels. When accumulating several pointings, both the highest completeness and largest dynamic range are reached, although the gain is never higher than 20\\%.\n\\item Y-shaped arrays have smaller inner working angles than linear configurations of identical maximum baseline length, even when considering the accumulation of several pointings. The gain is almost a factor of two in terms of angular resolution.\n\\end{enumerate}\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[scale=0.58]{fake}\n    \\caption{Fake VLTI interferometric configurations used in this paper: non-redundant linear D0-E0-H0-K0 (red), redundant linear D9-G2-H9-K9 (green, fake stations), and Y-shaped E0-J3-J2-H0 (blue).}\n    \\label{fig:fake}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering \n    \\includegraphics[scale=0.55]{all_contdist_fake}\\hspace{2cm}\n    \\includegraphics[scale=0.55]{all_compcont_fake}\n    \\caption{{\\it Left:} 3-$\\sigma{}$ sensitivity versus separation for a completeness level of 80\\%. {\\it Right:} Completeness in the separation range $6-40\\,$mas versus contrast. Simulations are for a single snapshot pointing (dashed lines) and for five pointing separated by one hour (solid lines). Colours are for the four configurations displayed in Fig.~\\ref{fig:fake}. The black curves are for the U1-U2-U3-U4 configuration displayed in Fig.~\\ref{fig:visa}. The contrast axes can be scaled for any accuracy on the closure phase (here $\\sigma=0.25\\deg$).   }\n    \\label{fig:cont_sep_fake}\n    \\label{fig:comp_cont_fake}\n\\end{figure*}\n\n\n\n\n\\section{Closure phase accuracy and achievable dynamic range}\n\\label{sec:accuracy}\n\n\\subsection{Photon and piston noises}\nWe consider the theoretical photon noise limit for an observation of 1\\,h on a star of sixth magnitude using one-metre class telescopes (e.g., the $1.8\\,$m auxiliary telescopes of the VLTI). The choice of a sixth-magnitude star is driven by the current sensitivity limit of most interferometric instruments world-wide. A crude estimation of the photon noise is given by\n\\begin{equation}\n\\sigma_{phot} \\approx \\frac{360\\deg}{\\sqrt{N}} \\; ,\n\\end{equation}\nwhere $N$ is the total number of detected photons. If we consider that the 1\\,h observing time should include the overheads and the observation of a calibration star, the effective integration time on target will be of the order of 20\\,min. Assuming a realistic total transmission of 2\\%, including both the instrumental and atmospheric contributions, the number of detected photons is $N\\approx 10^8$ and the resulting photon noise is $\\sigma_{phot} \\approx 0.1\\deg$. For a $1^\\mathrm{mag}$ star, the resulting photon noise would be $\\sigma_{phot} \\approx 0.01\\deg$.\n\nIn theory, closure phase is a robust observable against the telescope phase errors \\citep{Monnier03}. However, in the context of non-zero exposure times and the presence of atmospheric turbulence, closure phase measurements are also affected by piston noise. A proper estimation of its amplitude is beyond the scope of this paper but it is still possible to provide a rough upper limit. Since piston noise is independent of the number of incident photons, it is expected to dominate the final statistical uncertainty for very bright stars. As an example, in the case of the PIONIER instrument at VLTI, the statistical uncertainty for bright stars is typically of the order of $0.25\\deg$ to $2.5\\deg$ for an integration time of 1\\,min. This uncertainty depends on the atmospheric conditions as expected for piston noise. In decent atmospheric conditions, integrating over 20\\,min allows the piston noise contribution to be reduced below $0.2\\deg$, as it decreases with the square root of the integration time.\n\n\n\t\\subsection{Calibration accuracy}\n\nIt is interesting to compare these fundamental limits to published accuracies that include the calibration of the instrumental closure phase (also called the transfer function):\n\\begin{description}\n\\item[VLTI\/AMBER:] \\citet{Absil:2010} reported calibration errors of between $0.20\\deg$ and $0.37\\deg$ depending on the night, using this three-telescope combiner in its medium spectral resolution mode ($R=1\\,500$). With the low spectral resolution mode ($R=35$), typical calibration errors range from one to a few degrees \\citep[see for instance][]{kraus:2009apr,le-bouquin:2009mar}.\n\\item[VLTI\/PIONIER:] Typical calibration errors range from $0.25\\deg$ to $1\\deg$ \\citep{LeBouquin:2011,Absil:2011} for this four-telescope combiner. Sequences with closure phases stable down to $0.1\\deg$ have been recorded. Systematic discrepancies have been noted when calibration stars were separated by more than $10\\deg$ on the sky.\n\\item[CHARA\/MIRC:] The typical accuracy obtained with this four-telescope combiner is between $0.1$ and $0.2\\deg$, which makes this instrument the most accurate of the currently available suite. Calibration uncertainties dominate the final accuracy at this level \\citep{Zhao08,Zhao:2010,Zhao11}.\n\\end{description}\n\nAltogether, a typical noise floor of $\\sim0.25\\deg$ seems to appear for the calibration of the closure phases in long baseline interferometry. Two results indicate that the major cause is probably longitudinal dispersion: (i) that the accuracy depends on the spectral resolution and (ii) the dependence on position of the calibration star on the sky. This is also the finding of \\citet{Zhao11}, who proposed an elaborate calibration scheme for MIRC. Although this is clearly a very promising way of characterizing already known substellar companions, this method is probably not suited to surveying a large number of stars with a standard calibration procedure. Interestingly, $0.25\\deg$ is also the noise floor reported by \\citet{Lacour:2011} for the calibration of the closure phase of the spare aperture masking mode of NACO at VLT. This calibration noise floor of $0.25\\deg$ theoretically does not prevent us from reaching very high dynamic ranges, by accumulating a large number of observations and\/or baselines, as for instance in sparse aperture masking or spectrally dispersed observations. Care should however be taken to ensure that individual closure phase measurements are statistically independent. In particular, one should avoid repeating the same systematic errors in individual data sets, e.g., by choosing different calibrator stars, instrumental setups, etc., in order not to reach a true noise floor in the observations.\n\nConcerning future instruments, the announced accuracy on the closure phases is $1\\deg$ for the K-band four-telescope combiner GRAVITY \\citep[document \\mbox{VLT-SPE-ESO-15880-4853} and][]{Gillessen:2010} and from $1\\deg$ to $5\\deg$ for the L-band four-telescope combiner MATISSE \\citep[Florentin Millour, private communication and][]{Lopez:2008}. Although these performances may be conservative, we conclude that the next generation of VLTI instruments is unlikely to break the $0.25\\deg$ limit.\n\n\n\t\\subsection{Discussion}\n\nIt appears realistic to reach a closure phase accuracy of $0.25\\deg$ within less than one hour on one-metre class telescopes for stars of magnitude six and brighter. According to the results of Sect.~\\ref{sec:array}, such performances allow a dynamic range of $5\\times 10^{-3}$ ($\\Delta \\mathrm{mag}=5.75$) to be reached with 80\\% completeness when five pointings are obtained with a four-telescope interferometer. The same performance would probably be reached within a snapshot using an interferometric instrument combining six telescopes or more at a time.\n\nThis result can be compared with the survey for stellar and sub-stellar companions on the ten-metre Keck and five-metre Palomar telescopes using aperture masking techniques in the K band \\citep{Kraus08,Kraus:2011}. The achieved dynamic range is $\\Delta K\\approx5.5$ for separation as small as $25\\,$mas (slightly worse for Palomar). A similar dynamic range and inner working angle have  been achieved within an ongoing survey of massive stars using aperture masking at VLT\/NACO in the H band, e.g. $\\Delta H\\approx5$ down to $25\\,$mas with this eight-metre diameter telescope (Hugues Sana, private communication). All close companions presented in these near-infrared surveys would have been detected by interferometry with an efficiency higher than 90\\%, provided that they reside within the interferometric field-of-view. In addition, this efficiency would have been achieved down to about 2\\,mas. At longer wavelengths, the aperture masking technique has a dynamic range of about $\\Delta L\\approx 7.5$ \\citep{Hinkley:2011}, although the inner working angle in that case is only $70\\,$mas. There is currently no L-band interferometric beam-combiner with closure phase capabilities to which these performances could be compared.\n\nSeveral observing programs related to faint companion detection would benefit significantly from the capabilities of closure phase measurements on long-baseline interferometric instruments. An example is the search for low-mass (sub)-stellar companions around main-sequence stars residing in nearby young associations. Considering associations with ages between 10\\,Myr and 200\\,Myr, and a limiting magnitude $K=6$ for the instrument, one could survey stars up to about 15-20\\,pc for stellar type M0V, 40\\,pc for type G0V, and 120\\,pc for type A0V. With an estimated median dynamic range of $\\Delta K \\simeq 6$, we computed the masses of the faintest companions that could be detected within a survey of nearby moving groups, using the (sub-)stellar cooling models of \\citet{Baraffe98,Baraffe03}. The results are given in Table~\\ref{tab:associations}, showing that the $13 M_{\\rm Jup}$ ($=0.012 M_{\\odot}$) limit between the brown dwarf and planetary regimes can be reached for young late-type dwarfs. In the case of A-type stars, one would be sensitive to companions in the range M3V-M7V depending on the age. For even younger stars, located in nearby star forming regions, closure phase measurements have the potential to reveal the formation of planetary-mass objects, as suggested by \\citet{Kraus12}.\n\n\\begin{table}[t]\n\\caption{Sensitivity limits in terms of companion masses around young main-sequence stars.}\n\\centering\n\\begin{tabular}{c c c c}\n\\hline\\hline\nAge & A0V & G0V & M0V \\\\\n\\hline\n10 Myr & $0.09 M_{\\odot}$ & $0.017 M_{\\odot}$ & $0.012 M_{\\odot}$ \\\\\n50 Myr & $0.22 M_{\\odot}$ & $0.043 M_{\\odot}$ & $0.013 M_{\\odot}$ \\\\\n200 Myr & $0.35 M_{\\odot}$ & $0.08 M_{\\odot}$ & $0.030 M_{\\odot}$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:associations}\n\\end{table}\n\nAnother program is the determination of the binary fraction for massive stars. The interest is that despite the preponderance of multiple stars, the mechanism that produces multiple stars rather than single stars is still uncertain. The measurement of the mass distribution, and how it evolves with the mass of the primary, is an appropriate tool for disentangling between \\emph{capture} and \\emph{fragmentation} models. Stellar companions to B-type stars have been investigated using AO \\citep[e.g. ][]{Roberts:2007}, although the stars observed typically have a large range of distances, limiting the statistical significance of the results. Radial velocity measurement of massive stars is challenging owing to the lack of suitable spectral lines and their intrinsic broadening. With the limiting magnitude $K=6$ presented in this paper, it is possible to observe interferometrically all the B stars within a distance of 75\\,pc ($\\sim$$100$ objects for the southern hemisphere), providing the first comprehensive study of massive binaries in the $0.25-5\\,$AU separation range. \n\nLast but not least, one of the main selling argument for high-precision closure phases in optical interferometry has been the direct detection of hot extrasolar giant planets (EGP). Several hot EGP host stars are indeed bright enough to be observed with state-of-the-art interferometric instruments. For mature planetary systems ($>10$\\,Myr), the expected contrast between the planet and the star is however generally too low ($<10^{-3}$) to be currently accessible with closure phase measurements \\citep{Zhao08,Zhao11}. To routinely reach the hot EGP regime ($\\Delta K \\simeq 8 - 10$), a gain of two to four magnitudes is required in the dynamic range, which would translate into a noise floor of between $0.04\\deg$ and $0.006\\deg$ on the closure phase. Achieving such an accuracy would probably require a significant breakthrough in the instrumental domain.\n\n\n\\section{Conclusions}\n\nIn summary, optical interferometric surveys designed to detect faint companions have the following properties:\n\\begin{enumerate}\n\\item  The observable (closure phase) is robust against unstable atmospheric seeing conditions \\citep{Monnier03}. Integrating over $20\\,$min is sufficient to consistently reduce the photon and atmospheric noises below $0.25\\deg$, which appears as a hard limit for the calibration of current instruments.\n\\item A single snapshot with four telescopes provides a 80\\% detection efficiency at \\mbox{$\\Delta\\mathrm{mag}=4.5$} as soon as the binary separation is fully resolved. The only requirement of the interferometric array is to use baselines as long as possible to improve the inner working angle, which is typically of the order of a few milli-arcseconds.\n\\item Accumulating more observations (several pointing and\/or recombining more telescopes) allows a dynamic range \\mbox{$\\Delta\\mathrm{mag}=6$} to be reached, which appears to be a realistic limit in respect to published performances. Going deeper would require us to break the current limit of $0.25\\deg$ on the closure phase accuracy, or to massively increase the number of observations.\n\\item The achievable dynamic range scales linearly with the closure phase accuracy.\n\\end{enumerate}\n\nIn conclusion, interferometric closure phase surveys would be well-suited as filler programs for service-mode interferometric facilities, such as the VLTI. They can be considered as a useful complement to the AO-assisted imaging surveys currently carried out on ten-metre class telescopes. In particular, the search space of long-baseline interferometry bridges the gap between the wide companions found in direct imaging and the close companions detected by RV measurements. Moreover, interferometry could nicely complement RV studies in the particular cases where RV measurements are quite inappropriate. Young stars, for instance, are especially promising targets since their (sub)stellar companions are supposed to be relatively bright compared to their host stars.\n\n\\begin{acknowledgements} \nThe authors thank the referee for his helpful comments. This work has made use of the Smithsonian\/NASA Astrophysics Data System (ADS) and of the Centre de Donnees astronomiques de Strasbourg (CDS). All calculations and graphics were performed with the freeware \\texttt{Yorick}\\footnote{\\texttt{http:\/\/yorick.sourceforge.net}}.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDiscovering new particles would entail the Standard Model (SM) being falsified in Popper's sense~\\cite{Popper:1934} and force us to extend it. Absent such discovery, \nthe SM is still falsifiable upon finding new forces among the known particles. Because the SM has a characteristic energy scale of 100 GeV (the mass of the Higgs boson at $m_h=125$ GeV, as well as the vacuum constant $v=246$ GeV exemplify it), but no new particles below 1000 GeV have been found, there is a scale separation that begs the use of Effective Field Theory.\n\nThe popular SMEFT extension of the SM Electroweak Symmetry Breaking Sector (EWSBS) adds to it operators classified by their mass dimension,\n\\begin{equation}\n    \\mathcal{L}_{\\rm SMEFT} =\n    \\mathcal{L}_{\\rm SM} + \n    \\sum_{n=5}^{\\infty}\n    \\sum_i\n    \\frac{c_i^{(n)}}{\\Lambda^{n-4}} \\mathcal{O}_i^{(n)}(H) \\ .\n \\label{SMEFTL}\n \\end{equation}\nThese operators $\\mathcal{O}_i^{(n)}(H)$,  whose intensity is controlled by the  Wilson coefficients $c_i^{(n)}$ for a given reference scale $\\Lambda$, are the potentials of those new forces being sought. \nA nonzero $c_i^{(n)}$ would signal departure from the SM, that then would need to be extended,\n{perhaps by new resonances~\\cite{Dobado:2017lwg}}.\n\nBut it is easy to ask oneself how the whole framework of SMEFT can be tested. \nEffective theories include all the possible interactions that are compatible with the known particle content and symmetries believed to hold. Would it not be that any separation from the SM could be recast in SMEFT form? In that case, absent some new light particle, any phenomena could be described by adding an operator with a parameter to the SM. This is not so, as we will detail. \n\nThe particle content of the electroweak sector is packaged in a Higgs doublet field in the SM as well as in SMEFT\n\\begin{equation}\nH = \n\\frac{1}{\\sqrt{2}} \\begin{pmatrix} \\varphi_1+i\\varphi_2 \\\\   \\varphi_0 + i\\varphi_3\n\\end{pmatrix} \n= \nU(\\boldsymbol{\\omega}) \\begin{pmatrix} 0 \\\\   (v+h_{\\rm SMEFT})\/\\sqrt{2}\n\\end{pmatrix} \\, ,\n\\end{equation}     \nwhere the Cartesian coordinates $\\varphi_a$ can be rearranged to the polar decomposition in terms the $\\omega_i$ Goldstone bosons (which set the orientation of $H$ through the unitary matrix $U(\\boldsymbol{\\omega})$) and the radial coordinate $h_{\\rm SMEFT}$ (with $|H|=(v+h_{\\rm SMEFT})\/\\sqrt{2}$). \n\n\n\n\\begin{figure}[!b] \n\\centering\n     \\begin{tikzpicture}[scale=1]\n     \\draw[decoration={aspect=0, segment length=1.8mm, amplitude=0.7mm,coil},decorate] (-1,1) -- (0,0)-- (-1,-1);\n     \\draw[] (0,0)-- (1,1);\n     \\draw[] (1.25,1) node {$h_1$};\n          \\draw[] (1.25,-1) node {$h_n$};\n     \\draw[] (0,0)-- (.98,0.6);\n          \\draw[] (1.25,0.6) node {$h_2$};\n     \\draw[] (.75,-.15) node {.};\n     \\draw[] (.75,0.) node {.};\n     \\draw[] (.75,-0.3) node {.};\n     \\draw[] (0,0)-- (1,-1);\n       \\draw[] (3.25,0) node {$\\displaystyle{  \\, =\\,  -\\frac{n!a_n}{2v^n} \\,  s }$};  \n\\end{tikzpicture}\n\\caption{\\label{fig:Feynman}\\small\nThe $\\omega\\omega\\to nh$ processes can be the key to disentangling the nature of the EWSBS. They give direct access to the $a_i$ coefficients of the flare function $\\mathcal{F}$, and hence to their correlations, as listed in Table~\\ref{tab:correlations}.\n}\n\\end{figure}\n\n\n\nAn additional non-linear redefinition of $h_{\\rm SMEFT}$ allows us to rearrange the SMEFT Lagrangian in the form of a  more general theory, HEFT: \n\\begin{align}  \\label{HEFTL}\n{\\cal L}_{\\rm HEFT} = \\frac{1}{2}\\partial_\\mu h_{\\rm HEFT}\\partial^\\mu h_{\\rm HEFT}-V(h_{\\rm HEFT}) + \\nonumber \\\\ \n\\frac{1}{2}\\!\\mathcal{F}(h_{\\rm HEFT})\n\\partial_\\mu \\omega^i \\partial^\\mu \\omega^j\\!\\left(\\!\\delta_{ij}\\!+\\!\\frac{\\omega^i\\!\\omega^j}{v^2\\!-\\!\\boldsymbol{\\omega}^2}\\!\\right)\\ .\n\\end{align}\nOf current focus therein is the flare function~\\cite{Alonso:2016oah, Grinstein:2007iv} \n\\begin{equation} \\label{Fexpansion}\n    {\\mathcal F}(h_{\\rm HEFT})=1+\\sum_{n=1}^{\\infty}{a_n}\\Big(\\frac{h_{\\rm HEFT}}{v}\\Big)^n \\,,\n\\end{equation}\nwhich amounts to a radial ``scale'' (think of $a(t)$ in a Friedmann-Robertson-Walker cosmology) in the field space of the $(h,\\omega_i)$ electroweak bosons (with $\\omega_i$ analogous to the spatial coordinates).  \nWhat we call attention to in this letter is that the Taylor-series coefficients of $\\mathcal{F}$ as defined in Eq.~(\\ref{Fexpansion}) must satisfy experimental correlations or constraints as given in Table~\\ref{tab:correlations} and in the companion extended manuscript~\\cite{Gomez-Ambrosio:2022giw} \n{\\it if SMEFT is a valid description}. It is clear that an experimental program aimed at these correlations\nvia the key process to access $\\mathcal{F}$, $\\omega\\omega\\to nh$ as sketched in Figure~\\ref{fig:Feynman}, or $mh\\to nh$ to access $V(h_{\\rm HEFT})$, can test the validity of SMEFT itself, and not only its parameters.\n\n\n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{Bandasa1a2.png}\n    \\caption{\\small The correlation $a_2=2a_1-3$ that SMEFT predicts at order $1\/\\Lambda^2$ is plotted against the current 95\\% confidence intervals for these two HEFT parameters~\\cite{ATLAS:2020qdt,CMS:2022cpr}.}\n    \\label{fig:a1a2}\n\\end{figure}\n\n\n\n\\begin{table}[!t] \n\\setlength{\\arrayrulewidth}{0.3mm}\n\\setlength{\\tabcolsep}{0.2cm} \n\\renewcommand{\\arraystretch}{1.4}\n    \\caption{\\small Correlations between the $a_i$ HEFT coefficients necessary for SMEFT to exist, at order $\\Lambda^{-2}$ (first and second columns, with the numbers in the second consistent with 95\\% confidence-level experimental bounds $a_1\/2 \\in [0.97,1.09]$~\\cite{ATLAS:2020qdt}).\n    The right column provides the corresponding numerical values at the next order~\\cite{Gomez-Ambrosio:2022giw}. \n   \n    They are quoted in terms  of $\\Delta a_1:=a_1-2$ and $\\Delta a_2:=a_2-1$, so that all  objects in the table vanish in the Standard Model, with all the equalities becoming $0=0$.    \n    \\label{tab:correlations}}\n    \\centering\n    \\begin{tabular}{|c|c|c|}\\hline\n        $\\mathcal{O}(1\/\\Lambda^2)$ & $\\mathcal{O}(1\/\\Lambda^2)$ & $\\mathcal{O}(1\/\\Lambda^4)$ \\\\ \\hline \n        $\\Delta a_2=2\\Delta a_1$ & $\\Delta a_2\\in [-0.12,0.36] $  &  \\\\\n        $a_3=\\frac{4}{3} \\Delta a_1$ &  $a_3\\in[-0.08,0.24]$  & $a_3\\in [-3.1,1.7]$  \\\\  \n        $a_4=\\frac{1}{3} \\Delta a_1$ &  $a_4\\in [-0.02,0.06]    $ &  $a_4\\in [-3.3,1.5]$ \\\\\n        $a_5=0$ &  & $a_5 \\in[-1.5,0.6]$ \\\\\n        $a_6=0$ &  & $ a_6=a_5$ \\\\ \n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\n\nAs an example, the correlation predicted by SMEFT among $a_1$ and $a_2$ is shown in Figure~\\ref{fig:a1a2}.\n\n\n\n\n\n\nThere are several reasons why the experimental tests of those coefficients need large energies and statistics, at the limit of what is possible today at the LHC and beyond.\nFirst, the $\\mathcal{F}$ function multiplies terms with derivatives of the Goldstone bosons $\\partial_\\mu \\omega_i\\to q_\\mu \\omega_i$ that yield couplings proportional to their four-momenta, and become more relevant at higher energies. \nSecond, \n the equivalence theorem~\\cite{Veltman:1989ud,Dobado:1993dg} tells us that the scattering of longitudinal gauge bosons is related to scattering of Goldstone boson ($\\omega_i \\sim W^\\pm_L,\\ Z_L$): the EW gauging of the HEFT Lagrangian~(\\ref{HEFTL}) leads to the $WW\\to n h$ interaction $\\Delta\\mathcal{L}_{\\rm HEFT}^{WW\\to n h} = \\left(\\frac{1}{2}m_Z^2 Z_\\mu Z^\\mu + m_W^2 W_\\mu^+ W_\\mu^-\\right) \\mathcal{F}(h_{\\rm HEFT})$, which for longitudinal gauge bosons clearly dominates over the non-derivative interactions from $V$ only  at high energies~\\cite{Dicus:1987ez,Kallianpur:1988cs}.  \nAnd third, an increasing number of Higgs bosons  (necessary to access each $h^n$ order of $\\mathcal{F}$, the Higgs-flare function) requires an ample phase space and, thus, high energy.  \n\nHowever, as we shortly show after\nEq.~(\\ref{changeofvariable}) below, the correlations arise from the need for\nconsistency of the SMEFT formulation when a change of variable $h_{\\rm HEFT}\\to h_{\\rm SMEFT}$ is performed. This change affects any other piece of the Lagrangian that involves the Higgs bosons, such as the Yukawa couplings to fermions (that we leave for future works) but also the interactions among Higgs bosons themselves. Such interactions do not require derivative couplings and first appear even in the renormalizable SM, \n\\begin{equation}\n{\\mathcal{L}}_{\\rm SM} = |\\partial H|^2 -   \\underbrace{\\left( \\mu^2 |H|^2 +\\lambda|H|^4\\right)}_{V(H)} \\, ,\n\\end{equation}\nin the much discussed $V(H)$  \nHiggs-potential, accessible already at low $\\sqrt{s}$. In HEFT, this potential has additional non-renormalizable couplings and can be organized in a power-series expansion\n\\begin{align}\\label{expandV}\n    V_{\\rm HEFT}= \\frac{m_h^2 v^2}{2}  \\Bigg[&   \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^2 +  v_3   \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^3 \\nonumber+ \\\\  &+ v_4    \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^4 + \\dots \\Bigg]\\,,\n\\end{align}\nwith $v_3=1$, $v_4=1\/4$ and $v_{n\\geq 5}=0$ in the SM. \nIts coefficients also need to satisfy constraints that are exposed in Table~\\ref{tab:corV} and Figure~\\ref{fig:a1a22} if and when SMEFT applies.\n\n\nLet us then see, very briefly, how these correlations come about. Instead of relying on the powerful geometric methods of~\\cite{Alonso:2016oah,Alonso:2015fsp,Alonso:2016btr,Alonso:2021rac,Alonso:2022ffe} we use the more pedestrian coordinate-dependent approach, more familiar to phenomenologists working on LHC physics.\nThe goal is to see when is it possible to cast  \nEq.~(\\ref{HEFTL}) into the specific SMEFT one, Eq.~(\\ref{SMEFTL}).\nThis we write as \n\\begin{align}\n \\mathcal{L}_{\\rm SMEFT}= |\\partial H|^2-V(|H|^2)\n+\\frac{1}{2} B(|H|^2)(\\partial (|H|^2))^2 +\\dots  \n\\label{eq:SMEFTL-kin+V+B}\n\\end{align}\nwhere the  non-derivative and derivative terms, respectively given by $V$ and $B$, \ncollect typical SMEFT operators of Eq.~(\\ref{SMEFTL}) such as, at lowest $1\/\\Lambda^2$ order,\n\\begin{equation}\\label{operatorsSMEFT}\n  \\mathcal{O}_H := (H^\\dagger H)^3  \\, , \\ \\ \\ \n  \\mathcal{O}_{H \\Box} := (H^\\dagger H) \\Box (H^\\dagger H)\\ .\n\\end{equation} \n There are also other operators, such as, e.g., \n$  \\mathcal{O}_{HD} = (H^\\dagger D_{\\mu} H)^*  (H^\\dagger D^{\\mu} H) $, but they break custodial symmetry, and LEP studies suggest that the $SU(2)\\times SU(2)\\to SU(2)$ electroweak symmetry breaking mechanism is the appropriate pattern, leaving the residual custodial $SU(2)$ as a good approximate global symmetry of the scalar sector.  \nThe additional $A(H)$ structure pointed out in~\\cite{Cohen:2020xca} for Lagrangian~(\\ref{eq:SMEFTL-kin+V+B}) can be eliminated through partial integration and the use of the equations of motion~\\cite{Gomez-Ambrosio:2022giw}. \n\n \n \n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{Bandasv3v4.png}\n    \\caption{\\small The correlation $v_4=\\frac{3}{2} v_3 -\\frac{5}{4} -\\frac{1}{6}\\Delta a_1$ that SMEFT predicts at $\\mathcal{O}(1\/\\Lambda^2)$ is plotted making use of current 95\\% confidence interval for $v_3\\in[-2.5,5.7]$~\\cite{ATLAS:2021jki}. The experimental $a_1$ uncertainty~\\cite{ATLAS:2020qdt,CMS:2022cpr},  $a_1\/2\\in[0.97,1.09]$, is numerically negligible and allows to  predict a SMEFT band given by the solid black line. An experimental measurement for $v_4$ is still missing. }\n    \\label{fig:a1a22}\n\\end{figure}\n\\begin{table}[!t]\n\\setlength{\\arrayrulewidth}{0.3mm} \n\\setlength{\\tabcolsep}{0.2cm}  \n\\renewcommand{\\arraystretch}{1.4} \n\\caption{\\small Correlations among the coefficients $\\Delta v_3:=v_3-1$, $\\Delta v_4:=v_4-1\/4$, $v_5$ and $v_6$ of the HEFT Higgs potential expansion in Eq.~(\\ref{expandV}) that need to hold, at $\\mathcal{O}(1\/\\Lambda^2)$, if SMEFT is a valid description of the electroweak sector.\nBased  on the current bound $\\Delta v_3\\in[-2.5,5.7]$ in Ref.~\\cite{ATLAS:2021jki}, $\\mathcal{O}(1\/\\Lambda^2)$ SMEFT predicts the coefficient intervals in the last column, testable in few-Higgs final states. A coupling \n$c_{H\\Box}\\neq 0$ induces the correction $\\Delta a_1\\propto c_{H\\Box}$, nevertheless numerically negligible since $v_3$ experimental uncertainties much exceed those of $a_1$.}\n\\label{tab:corV}\n\\begin{center}\n \\begin{tabular}{|c|c|} \\hline \n        $\\Delta v_4=\\frac{3}{2}\\Delta v_3  -\\frac{1}{6}\\Delta a_1$ & $\\Delta v_4\\in[-3.8,8.6]$\\\\[2ex]\n        $ v_5=6v_6=\\frac{3}{4}\\Delta v_3 -\\frac{1}{8}\\Delta a_1 $ \n        & $v_5=6 v_6 \\in[-1.9,4.3]$\n           \\\\\n         \\hline    \n    \\end{tabular}\n\\end{center}  \n\\end{table}\n \nTo proceed, we need to perform the following conversion to pass from SMEFT to HEFT and viceversa:  \n \\begin{align}\n& |\\partial H|^2 +  \n\\frac{1}{2} B(|H|^2)(\\partial (|H|^2))^2 \\longleftrightarrow \\nonumber \\\\\n&\\frac{v^2}{4} \\mathcal{F}(h_{\\rm HEFT}) \\,   {\\rm Tr}\\{ \\partial_\\mu U^\\dagger \\partial^\\mu U\\} \n+\n\\frac{1}{2} (\\partial h_{\\rm HEFT})^2 \\,,\n\\end{align}  \nThe change from SMEFT to HEFT is straightforward and always possible, with the canonical, nonlinear change of variables given in differential form as\n\\begin{equation}  \\label{changeofvariable}  \ndh_{\\rm HEFT}\\, =\\, \\sqrt{1+(v+h_{\\rm SMEFT})^2 B(h_{\\rm SMEFT})}\\,\\, dh_{\\rm SMEFT} \\,,\n\\end{equation}\nwhere the flare-function is provided by the relation \n\\begin{equation}\n\\mathcal{F}(h_{\\rm HEFT}) \\,=\\, \\left(1+h_{\\rm SMEFT}\/v\\right)^2\\, .\n\\end{equation}  \nHowever,  \nthe reverse conversion from HEFT to SMEFT, \n\\begin{equation}\nh_{\\rm HEFT}   \\,=\\, \\mathcal{F}^{-1}\\left((1+h_{\\rm SMEFT}\/v)^2\\right) \\,,\n\\end{equation}\nruns into difficulty. \nThis is because of the need to reconstruct squared operators of the Higgs doublet field $H$ that is the basis of SMEFT, such as\n\\begin{eqnarray}\n|H|^2 &=& \\frac{(v+h_{\\rm SMEFT})^2}{2}\\, ,\n\\nonumber \\\\\n(\\partial|H|^2)^2 &=& (v+h_{\\rm SMEFT})^2 \\,   (\\partial h_{\\rm SMEFT})^2 \\nonumber \\\\ &=&\\, 2 |H|^2 \\,   \n(\\partial h_{\\rm SMEFT})^2 \\, .\n\\end{eqnarray} \nThe extra $|H|^2$ on the right hand side of the second equation \nends in a denominator\n\\begin{align}\n&\\mathcal{L}_{\\rm SMEFT} =  \\underbrace{ |\\partial H|^2}_{= \\mathcal{L}_{\\rm SM}} \\quad +\\quad\n\\label{eq:HEFT2SMEFT}\n\\\\ \\nonumber \n&\\underbrace{ \\frac{1}{2} \\bigg[  \\frac{8|H|^2}{v^2}\\bigg(   (\\mathcal{F}^{-1})'\\left(2| H |^2\/v^2\\right)  \\bigg)^2 \\,\\,\\,-\\,\\,\\, 1\\bigg] \\, \\frac{(\\partial| H |^2)^2}{2| H |^2}  }_{=\\Delta \\mathcal{L}_{\\rm BSM}}\\, .\n\\end{align}\nAs SMEFT is assumed to have the analytical power expansion\nin Eq.~(\\ref{SMEFTL}), such singularity precludes its existence and needs to be cancelled by the preceding bracket in the second line of Eq.~(\\ref{eq:HEFT2SMEFT}). \n\nThe result is the same as that obtained by geometric methods~\\cite{Cohen:2020xca}, there must be a double zero of $\\mathcal{F}$, a symmetric point with respect to the global $SU(2)\\times SU(2)$ group so that the SMEFT expansion can be performed.  Furthermore,\nanalyticity requires that all its odd derivatives vanish at the symmetric point.\n\nThe particular case of the SM is given by $\\mathcal{F}=(1+h_{\\rm SMEFT}\/v)^2 $. As already pointed out,\nat higher orders in $h\/v$, the existence of SMEFT requires that the odd derivatives of $\\mathcal{F}$ at the symmetric point $h_\\ast$ vanish. \n\nThe correlations from Table~\\ref{tab:correlations} can then be obtained by matching the Taylor expansion of $\\mathcal{F}$ around such symmetric point \n$ h_{\\rm HEFT}=  h_\\ast$ with the expansion around our physical vacuum $h_{\\rm HEFT}=0$.\nInstead of that matching, one can also obtain the correlations by eliminating the SMEFT Wilson coefficients order by order. For example, at $\\mathcal{O}(1\/\\Lambda^2)$ there is only one operator, $\\mathcal{O}_{H\\Box}$,  in Eq.~(\\ref{operatorsSMEFT}), that controls all the HEFT coefficients of $\\mathcal{F}$: \n\\begin{align}\n&&\na_1 = 2a=2\\left(1 + v^2\\frac{c_{H\\Box}}{\\Lambda^2}\\right)\\,, \\quad \na_2 = b=1+{4v^2}\\frac{c_{H\\Box}}{\\Lambda^2} \\,, \n\\nonumber \\\\ \n&&\na_3 = \\frac{8v^2}{3}\\frac{c_{H\\Box}}{\\Lambda^2}\\,, \\qquad \na_4 = \\frac{2v^2}{3}\\frac{c_{H\\Box}}{\\Lambda^2}\\,,\\qquad \na_{n\\geq 5} = 0 \\, .\n\\end{align}\nThe potential $V(h_{\\rm HEFT})$ is in turn also affected by $\\mathcal{O}_H$,\n\\begin{eqnarray}\n&&\nv_3 =1 +  \\frac{3v^2 c_{H\\Box}}{\\Lambda^2} +\\epsilon_{c_H}\n\\,, \\,\\,\\, \nv_4 = \\frac{1}{4} +  \\frac{25v^2c_{H\\Box} }{6\\Lambda^2} +\\frac{3}{2}\\epsilon_{c_H}\n\\,,\n\\nonumber\\\\\n&& \nv_5 =    \\frac{2v^2c_{H\\Box} }{\\Lambda^2} +\\frac{3}{4}\\epsilon_{c_H}\n\\,,\\,\\,\\, \nv_6 =  \\frac{v^2c_{H\\Box} }{3\\Lambda^2} +\\frac{1}{8}\\epsilon_{c_H}\n\\,,\n\\nonumber\\\\\n&& \nv_{n\\geq 7} = 0 \\,, \\qquad\\qquad\\qquad \n\\end{eqnarray}\nwith  $m_h^2=\\, -2\\mu^2 \\left(1+\\frac{2 c_{H\\Box}v^2}{\\Lambda^2}+ \\frac{3}{4}\\epsilon_{c_H}\\right)$, $2\\langle |H|^2\\rangle =v^2= -\\frac{\\mu^2}{\\lambda}\\left(1-\\frac{3}{4}\\epsilon_{c_H}\\right) $ and $\\epsilon_{c_H}= - \\frac{2 v^4 c_H}{m_h^2\\Lambda^2}= \\frac{\\mu^2 c_H}{\\lambda^2  \\Lambda^2}$. \n\nVarious authors, see {\\it e.g.}~\\cite{Brivio:2016fzo} have pointed out to differences between the  SMEFT and HEFT formulations~\\cite{Dobado:2019fxe}.\nFor example, in SMEFT the Goldstone\n $\\omega_i$  and Higgs  $h_{\\rm SMEFT}$ bosons are arranged in a left-$SU(2)$ doublet while in HEFT\n$h_{\\rm HEFT}$ is an $SU(2)\\times SU(2)$ singlet, independent of the Goldstone triplet $\\omega_i$.  \nAlso, in SMEFT the Higgs field always appears  in the combination $ (h_{\\rm SMEFT} + v)$ \nand thus, HEFT deploys more independent higher-dimension effective operators  (in exchange, it is less model dependent).\nThis means that SMEFT is natural when $h_{\\rm SMEFT}$ is a fundamental field while HEFT is typical for composite models of the EWSBS (such as those with $h_{\\rm HEFT}$ as a Goldstone boson).\nAnd finally, the counting of SMEFT is based in a cutoff $\\Lambda$ expansion taking the canonical operator dimensions, $\\mathcal{O}(d)\/\\Lambda^{d-4}$ (independently of $N_{\\rm loops}$) whereas HEFT is a derivative expansion (independently of $N_{\\rm particles})$ like the older Electroweak Chiral Lagrangian, with $\\mathcal{F}(h)$ inserted in the derivative Goldstone term.\n\n\n\nAmong the two types of correlations that we have presented in tables~\\ref{tab:correlations} and~\\ref{tab:corV}, the first ones for the coefficients of $\\mathcal{F}$ are more interesting for large values of the energy $\\sqrt{s}\\gg m_h\\sim m_W \\sim m_Z$ whereas the second, that do not involve Goldstone bosons, are therefore of greater interest at low energies, when  and additionally \nthe potential competes with the derivative operators on equal ground, as $\\sqrt{s} \\sim m_i$.  \n\\\\\nNevertheless, a lot of this is cosmetic and can be reorganized by changing variables $h_{\\rm SMEFT}\\leftrightarrow h_{\\rm HEFT}$. What is key is the San Diego criterion~\\cite{Alonso:2015fsp,Alonso:2016oah}:  \n$\\mathcal{F}(h_{\\rm HEFT})$ must have a point $h_\\ast$ symmetric under the global $SU(2)\\times SU(2)$  group  and due to its existence and convergence in the $h$ field space, SMEFT is deployable if and only if  (which is a statement about the HEFT Lagrangian)\n\\begin{itemize}\n\\item   $\\exists h_\\ast \\in\\mathbb{R}$ where $\\mathcal{F} (h_\\ast)=0$, and \n\\item  Because of the need for $\\mathcal{L}_{\\rm SMEFT}$ analyticity, $\\mathcal{F}$ is analytic between our vacuum $h=0$ and $h_\\ast$, particularly around $h_\\ast$. Moreover its odd derivatives vanish.\n\\end{itemize}\nWe have presented new relations that implement this criterion \nup to $\\mathcal{O}(1\/\\Lambda^2)$ and $\\mathcal{O}(1\/\\Lambda^4)$ in the $1\/\\Lambda$ counting;\nmore precision is unnecessary until (if) separations from the Standard Model are found. Then only, with the scale $\\Lambda$ at hand, out of separations of EFT coefficients from the SM, can we decide how relevant the corrections due to the higher orders are expected to be, and whether further work is warranted.\n\nIn conclusion, we have newly translated these conditions into correlations among HEFT coefficients \nwhose violation falsifies SMEFT. \nMoreover, since many extensions of the Standard Model incorporating supersymmetry, supergravity, or other possibilities, can be cast as a SMEFT, they can be likewise simultaneously falsified.\n\n\nFor the time being, \n{no separations from the SM have been found~\\cite{Eboli:2021unw} and} one can only infer direct experimental bounds on  the first  terms, $a_1$ and, perhaps, $a_2$, so we have to wait for data with a larger number of Higgs boson before assessing them. But when this will be done, the correlations will allow to falsify SMEFT in experiment\neven without new particles. We believe that this possibility improves the standing of SMEFT as a scientific theory.\n\n\n\\vspace{.2cm}\n\\begin{acknowledgments}\nSupported by spanish MICINN PID2019-108655GB-I00 grant, and Universidad Complutense de Madrid under research group 910309 and the IPARCOS institute; \nERC Starting Grant REINVENT-714788; UCM CT42\/18-CT43\/18;\nthe Fondazione Cariplo and Regione Lombardia, grant 2017-2070.\n\\end{acknowledgments}\n\\vspace*{-0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\nAntimonene is a recent addition~\\cite{Zhang2015,Lei2016} to a growing family of elemental two-dimensional (2D) materials. This monolayer phase of antimony adopts a buckled honeycomb lattice ($D_{3d}$ point group), similar to those of \nsilicene~\\cite{Vogt2012} and germanene~\\cite{Zhang2016}, or more recently discovered blue phosphorene~\\cite{Gu2017}.\nStructurally, antimonene is identical to a single layer of bulk antimony, which possesses layered rhombohedral structure with a $D_{3d}$ point group.\nAntimonene was successfully obtained by various experimental techniques~\\cite{Ares2018}, including but not limited to, epitaxial growth~\\cite{Wu2017}, liquid- and solid-phase exfoliation~\\cite{Gibaja2016,Ares2016}.\nBased on both experimental observation and \\textit{ab initio} modeling~\\cite{Ares2016,Zhang2015}, antimonene is believed to be a material with remarkable stability in air and in water~\\cite{Ares2016,Wu2017}. \nThis alone makes antimonene an appealing candidate for various applications because instability at realistic conditions is one of the main factors limiting the application of 2D materials~\\cite{Morishita2015,Kuriakose2018}.\nFrom a fundamental point of view, characteristic feature of antimonene is a strong spin-orbit coupling (SOC)~\\cite{Rudenko2017}, with the intraatomic SOC constant $\\lambda_{\\text{SOC}}=0.34$ eV. A proper account of SOC is important for a correct description of the electronic structure and effective masses~\\cite{Rudenko2017}. Antimonene is expected to have high charge carrier mobility~\\cite{Pizzi2016} comparable to or exceeding that of the other 2D semiconductors. Besides, monolayer Sb is an indirect band semiconductor with a theoretically estimated band gap of 1.2 eV \\cite{Singh2016}. \nThese properties make antimonene suitable for application in electronic~\\cite{Pizzi2016} and optical devices~\\cite{Singh2016}, where thickness- and strain-tunable band gap could provide an additional control over the material's properties~\\cite{Zhao2015,Zhang2015}.\n\nSuperconductivity is not uncommon for two-dimensional materials~\\cite{Uchihashi2017}. It is observed in thin films~\\cite{Ozer2006}, atomic sheets~\\cite{Ludbrook2015}, surface atomic layers~\\cite{Ge2015} and other 2D structures of various chemical compositions.\nRecent experimental works~\\cite{Chapman2016,Ludbrook2015} revealed superconductivity in graphene laminates with the critical temperatures ${T_{\\text{c}}}$~in the range of 4--6~K.\nMagnetization measurements of intercalated black phosphorus reveal ${T_{\\text{c}}}$~$=3.8\\pm$0.1~K, which was shown to be the same for different intercalants~\\cite{Zhang2017}.\nAlso, a large number of experimental measurements of ${T_{\\text{c}}}$~were recently performed for doped 2D transition metal dichalcogenides: WS$_{2}$ and NbSe$_{2}$ were demonstrated to be superconducting below 3~K \\cite{Lu2018,Xi2015}, while MoS$_2$ was reported to have ${T_{\\text{c}}}$~in the range of 7--11~K \\cite{Lu2015,Ye1193}.\nSuperconductivity in two dimensions is especially interesting in view of the possibility of controlling the electronic structure of 2D materials by\nstrain, electric field, thickness, or substrate. Superconducting 2D materials also appear as an appealing testbed for studying interface phenomena and proximity effects. In particular, 2D materials can be implemented as a part of the Josephson junction~\\cite{Heersche2007,Yabuki2016}. \n\nComputational studies predicting the superconductivity in monolayer graphene~\\cite{Profeta2012,Margine2014} and phosphorene~\\cite{Shao2014} were preceding the experimental observations~\\cite{Ludbrook2015,Chapman2016,Zhang2017}.\nAlso, superconductivity at nonzero charge doping has already been predicted for recently proposed arsenene~\\cite{Kong2018} (monolayer As), as well as for silicene \\cite{Durajski2014}. Experimental data on superconductivity of these materials are not available yet.\nWhile theoretical studies lack realistic accounts of the experimental setup, they allow study of the underlying mechanisms of superconductivity, for example, the key contributions to electron-phonon coupling or important changes of electronic structure due to charge carrier doping~\\cite{Ge2013}.\nAdditionally, there is an opportunity to study modifications of the electronic structure and electron-phonon coupling properties including ${T_{\\text{c}}}$~in the presence of strain~\\cite{Shao2014} or other external factors.  Neither calculations nor measurements of ${T_{\\text{c}}}$~are available in the literature for doped antimonene.\n\nIn this paper, we present a systematic study of the electron-phonon coupling and conventional superconductivity in $n$- and $p$-doped antimonene. To this end, we use a combination of Density Functional Theory (DFT)~\\cite{Hohenberg1964,Kohn1965}  and Density Functional Perturbation Theory (DFPT)~\\cite{Gonze1997} in conjunction with the formalism of Maximally Localized Wannier Functions (MLWF)~\\cite{Giustino2007,Marzari2012}.\nBesides charge doping, we also consider the role of an electric field applied in the direction perpendicular to the atomic layer, which allows us to modify the band structure.\nConsidering the interplay of these effects is interesting for a number of reasons.\nFor example, the perpendicular electric field allows one to control the band gap and effective masses in silicene and germanene~\\cite{Ni2012,Acun2015}, as well is in few-layer phosphorene~\\cite{Rudenko2015,Kim723,Liu2015}, where it is possible to transform material from normal insulator to topological insulator or metal.\nBesides this, perpendicular electric field breaks inversion symmetry, which, in combination with strong SOC, induces spin-splitting of both valence and conduction bands~\\cite{Kormanyos2013}.  \nThis case is observed, for example, in experimental work~\\cite{Ye1193}, where setup combining liquid and solid gating is used to study superconductivity in MoS$_{2}$ flakes, and ionic liquid naturally creates the environment for emergence of perpendicular electric field.\nSince SOC in antimonene is strong, this situation is particularly interesting to study.\nFor these reasons, we study the dependence of the electron-phonon coupling on bias voltage.\n\nThe rest of the paper is organized as follows. We first present the theory and computational methods used to calculate the electron-phonon coupling and superconducting critical temperatures, and also discuss the relevant approximations (Sec.~II). We then give a description of the calculated electronic structure and phonon dispersion of antimonene (Sec.~III~A). Main results are presented in Sec.~III~B, where we discuss the obtained dependencies of ${T_{\\text{c}}}$~and electron-phonon coupling strength, as well as superconducting transition temperatures on charge carrier concentrations, with a detailed consideration of the most important cases. The effect of a perpendicular electric field (bias voltage) is analyzed and discussed in Sec.~III~C. \nIn Sec.~IV, we summarize our results and conclude the paper.\n\n\n\\section{Theoretical background and computational details}\n\n\\subsection{Electron-phonon coupling and Allen-Dynes-McMillan equation}\nIn this section, we give a short theoretical description of the electron-phonon coupling and its relation to superconductivity.\nMore detailed description of the theory from the viewpoint of \\textit{ab initio} calculations can be found in Ref.~\\onlinecite{Giustino2017}.\n\nThe interaction of electrons with phonons is described by the matrix elements\n\\begin{equation}\ng_{mn,\\nu}({\\bf k,q}) = \\bigg( \\frac{\\hbar}{2m_0\\omega_{{\\bf q}\\nu} }\n\\bigg)^{1\/2} M_{mn}^{\\nu}({\\bf k},{\\bf q}),\n\\label{eq:gmn}\n\\end{equation}\nwhere \n\\begin{equation}\t\t\nM_{mn}^{\\nu}({\\bf k},{\\bf q}) =  \n\\langle \\psi_{m{\\bf k+q}} | \\partial_{{\\bf q}\\nu}V | \\psi_{n{\\bf k}}\\rangle,\n\\label{eq:mmn}\n\\end{equation}\nand $\\psi_{n{\\bf k}}$ is the electronic Bloch function for band $n$ and wavevector ${\\bf k}$; $\\partial_{{\\bf q}\\nu}V$ is the derivative of the self-consistent potential associated with  the phonon of wavevector ${\\bf q}$, branch index $\\nu$; and frequency $\\omega_{{\\bf q}\\nu}$; and $m_{0}$ is the atomic mass. \n\nIt is useful to define a dimensionless representation of the electron-phonon coupling associated with the single phonon mode $\\nu$ and wavevector ${\\bf q}$ as an average over the Fermi surface:\n\n\\begin{equation}\n\\label{eq:lambda}\n\\begin{split}\n\\lambda_{{\\bf q}\\nu} = \n\\frac{1}{N_{\\rm F}\\omega_{{\\bf q}\\nu}}\\sum_{mn,{\\bf k}} \n\\mathrm{w}_{{\\bf k}} |g_{mn,\\nu}({\\bf k,q})|^2 \\\\ \n\\times\\delta(\\varepsilon_{n{\\bf k}}-\\varepsilon_F)\\delta(\\varepsilon_{m{\\bf k}+{\\bf q}}-\\varepsilon_F),\n\\end{split}\n\\end{equation}\t\t\t\t\t\t\t\nwhich is closely related to the Eliashberg electron-phonon spectral function $\\alpha^{2}F(\\omega)$\n\\begin{equation}\n\\alpha^{2}F( \\omega) = \\frac{1}{2} \\sum_{{\\bf q}\\nu} \n\\mathrm{w}_{\\bf q} \\omega_{{\\bf q}\\nu} \\lambda_{{\\bf q}\\nu}\n\\delta (\\omega - \\omega_{{\\bf q}\\nu}) \n, \n\\label{eq:a2f}\n\\end{equation}\t\t\t\t\t\t\t\t\t\t\t\nwhere $\\mathrm{w}_{\\bf k}$ ($\\mathrm{w}_{\\bf q}$) is the symmetry-dependent weight of ${\\bf k}$ (${\\bf q}$) points, $\\varepsilon_{n{\\bf k}} (\\varepsilon_{m{\\bf k+q}})$ is the electronic energy, and $N_{\\text{F}}$~ is the density of states (DOS) at the Fermi energy, $\\varepsilon_F$.\n\n\nBesides $\\lambda_{{\\bf q},\\nu}$, we also consider the nesting function $\\xi_{\\bf q}$ as defined in~\\cite{Bazhirov2010}:\n\\begin{equation}\n\\xi_{\\bf q}= \n\\sum_{mn,{\\bf k}} \n\\mathrm{w}_{{\\bf k}}\n\\delta(\\varepsilon_{n{\\bf k}}-\\varepsilon_F)\\delta(\\varepsilon_{m{\\bf k}+{\\bf q}}-\\varepsilon_F).\n\\label{eq:nesting}\n\\end{equation}\t\t\nWhile the nesting function is independent of electron-phonon coupling matrix elements, it provides the understanding of effects of Fermi surface topology on effective electron-phonon coupling. It is important to mention, that the singularities of nesting function, associated with such topological phenomena as Fermi surface nesting may result in a high electron susceptibility at specific \\textbf{q}, thus increasing the value of $\\lambda$. Additional effects can be related to softening of phonon frequencies near this \\textbf{q}-points, due to the giant Kohn anomaly~\\cite{Katsnelson1994}.\n\nThe critical temperature of the superconducting transition according to the McMillan equation~\\cite{McMillan1968}, modified by Allen and Dynes~\\cite{Allen1975} can be estimated as:\n\\begin{equation}\nT_{\\text{c}} = \n\\frac{\\omega_{\\text{log}}}{1.2}\n\\exp\\left(\n-\\frac{1.04(1+\\lambda)}\n{\\lambda-\\mu_{\\text{c}}^{*}(1+0.62\\lambda)}\n\\right), \n\\label{eq:ADMC}\n\\end{equation}\t\nwhich is based on the superconductivity theory of Migdal and Eliashberg~\\cite{Migdal1958,Eliashberg1960}. Although derivation of Eq.~(\\ref{eq:ADMC}) contains a number of approximations, it is a reasonable starting point for the estimation of~${T_{\\text{c}}}$.\nIn Eq.~(\\ref{eq:ADMC}), ${\\mu^{*}_{\\text{c}}}$~is the Morel-Anderson effective Coulomb potential~\\cite{Morel1962},\n\n\\begin{equation}\n\\omega_{\\text{log}}=\n\\exp\\left[\n\\frac{2}{\\lambda}\\int_{0}^{\\omega_{\\text{max}}}\nd\\omega \\frac{\\alpha^{2}F( \\omega)}{\\omega} \\log \\omega\n\\right]\n\\label{eq:wlog}\n\\end{equation}\nis the logarithmically averaged phonon frequency,\n\\begin{equation}\n\\lambda = \\sum_{{\\bf q}\\nu} \\mathrm{w}_{{\\bf q}} \\lambda_{{\\bf q}\\nu}\n\\label{eq:lambtot}\n\\end{equation}\nis the total Fermi energy-dependent electron-phonon coupling strength, which is also known as the mass-enhancement factor~\\cite{Peters2018}. In the present paper, ${\\mu^{*}_{\\text{c}}}$~is considered as a phenomenological parameter with the typical values in the range of 0.1--0.2 \\cite{McMillan1968,Allen1975,Giustino2017}.\nAmong 2D materials the Coulomb pseudopotential was estimated more rigorously for monolayer graphene in Ref.~\\onlinecite{Margine2014} (${\\mu^{*}_{\\text{c}}}$~$=0.16$) and Ref.~\\onlinecite{Profeta2012} (${\\mu^{*}_{\\text{c}}}$~$=0.1$),  {\\color{black} bilayer graphene (${\\mu^{*}_{\\text{c}}}$~$=0.155$) in Ref.~\\onlinecite{Margine2016}, as well as NbS$_2$ (${\\mu^{*}_{\\text{c}}}$~$=0.2$) in Ref.~\\onlinecite{Heil2017}.} We note that an accurate estimate of ${\\mu^{*}_{\\text{c}}}$~requires an account of dielectric screening, and, therefore, would be strongly influenced by the underlying substrate.\n\n\\subsection{Calculation details}\n\\label{subsec:methods}\n\nThe initial electronic structure and crystal structure optimization calculations were performed at the DFT level as implemented in the plane-wave {\\sc Quantum ESPRESSO} ({\\sc QE}) code~\\cite{Giannozzi2009,Giannozzi2017},\nusing fully relativistic norm conserving pseudopotentials.\nExchange and correlation were treated with the local density approximation (LDA).\nThe kinetic energy cutoff for plane waves was set to 90 Ry, the Brillouin zone was sampled with a (16$\\times$16) Monkhorst-Pack \\textbf{k}-point mesh~\\cite{Monkhorst1976}, and the electronic state occupancies were treated as fixed.\nThe crystal structure was fully relaxed with a threshold of \\pten{1}{-12}~eV for total energies and \\pten{1}{-12}~eV\/\\AA~ for forces.\nThe vacuum thickness of 30 \\AA~was used to avoid spurious interactions between the supercell periodic images in the direction perpendicular to the 2D plane.\nThe Brillouin zone for phonons within DFPT was sampled by a (16$\\times$16) \\textbf{q}-point mesh, and the self-consistency threshold of \\pten{1}{-16}~eV was used.\n\nElectronic structure, dynamical, matrices and electron-phonon matrix elements obtained from DFT and DFPT calculations were used as the initial data for Wannier interpolation within the MLWF formalism, as implemented in {\\sc EPW}~\\cite{Giustino2007,Ponce2016}.\nThe calculation of the electron-phonon-related properties was performed on dense grids of (432$\\times$432) \\textbf{k}- and (208$\\times$208) \\textbf{q}-points, which ensures the numerical convergence of the results presented in this work.\n\n\\subsection{Role of charge doping, out-of-plane acoustic phonons, and bias voltage}\n\\label{subsec:approx}\n\nIn this work, we simulate the charge carrier doping of antimonene using the rigid band shift approximation, that leaves the electronic structure and phonon dispersion unchanged.\nTypically, charge doping in 2D materials mainly affects the out-of-plane mode and optical phonons~\\cite{Margine2014,Kong2018,Shao2014}.\nAt the same time, out-of-plane acoustic mode (ZA) is not taken into account in our calculations. \nAlthough consideration of ZA mode may be important for fundamental understanding of the associated effects, it is of little practical interest for the superconductivity studies: Interaction of 2D materials with a substrate would suppress out-of-plane vibrations making the already small coupling of electrons with these modes negligible. Furthermore, a correct description of such effects is not trivial within the slab geometry.\nAn accurate description of electrons coupling with out-of-plane phonon modes is presented in Ref.~\\onlinecite{Sohier2017} for doped graphene, where it is also shown, that the contribution from ZA mode is negligible. \nOptical phonons, as we show below, have a minor effect on the electron-phonon coupling and~${T_{\\text{c}}}$. Therefore, the rigid band shift approximation is justified for the purpose of our study.\nThe charge carrier concentration $\\delta \\rho$ is thus chosen in accordance with the ground state DOS and the Fermi energy, $\\delta \\rho(\\varepsilon_F)=\\int_0^{\\varepsilon_F}d\\varepsilon \\, \\rho(\\varepsilon)$.\n\nWe consider both $n$- and $p$-doping cases. We limit ourselves to the concentrations less than~\\con{1}{15}.\nAlthough such charge carrier concentrations correspond to a heavy doping regime, they are not unrealistic. Electron concentrations of the order of~\\con{1}{15} are achievable in thin metallic films by means of electrochemical techniques~\\cite{Daghero2012}. \nSmaller electron and hole concentrations of the order of \\con{1}{14} can be achieved by liquid gating~\\cite{Efetov2010} and  solid-electrolyte gating~\\cite{Xu2017}.\n{\\color{black}At the same time, the presence of the strong Fermi-surface nesting~\\cite{Antropov1988,Vaks1989,Katsnelson1994}, as well as high charge carrier concentrations may lead to the loss of structural stability of the system.\nSince such instabilities cannot be detected within the rigid band approximation, we also consider lattice dynamics of antimonene using the jellium doping method, as implemented in the DFT code used.\nFor every doping case, relaxation of atomic positions is performed, while the lattice parameter of undoped antimonene was used to model the behavior of the system on a substrate.}\n\n\nTo simulate the effect of a perpendicular electric field, we consider a tight-binding Hamiltonian obtained in the MLWF basis, and add position-dependent bias voltage, yielding the following Hamiltonian\n\\begin{equation}\nH=\\sum_{ij}t_{ij}c_i^{\\dag}c_j+V_{\\text{b}}\/d\\sum_{i}z_ic_i^{\\dag}c_i,\n\\end{equation}\nwhere the sum runs over the real-space MLWF orbitals $i$ and $j$, $c_i^{\\dag}$,$c_i$ ($c_{j}$) are the creation and annihilation operators of electrons on the corresponding orbitals, $z_i$ is the $z$-component of the position operator of the orbital $i$, $t_{ij}$ is the corresponding hopping integral, $d$ is the buckling constant, and $V_{\\text{b}}$ is bias voltage applied to the upper and lower planes of antimonene. \nIt is worth mentioning that since the consideration of electric field is not performed in the self-consistent manner, as is done, for example, in~Ref.~\\onlinecite{Li2018}, additional screening arising due to the charge carrier doping is not taken into account in our calculations; thus, the bias effect is quantitatively overestimated.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[]{struct.pdf} \n\t\\caption{Crystal structure of hexagonal antimonene; red lines denote the hexagonal unit cell, used in calculations. Lattice parameter, buckling constant, and bond length, denoted as $a$, $d$, and $b$ respectively, are also given on the figure.}\n\t\\label{fig:structure}\n\n\\end{figure}\n\t\n\\section{Results and discussion}\n\\subsection{Electronic structure and phonon dispersion}\n\\label{subsec:dft}\n\n\nWe find the calculated relaxed lattice parameter of free-standing antimonene to be $a = 4.0$~\\AA~with the bond length $b = 2.82$~\\AA~and buckling constant $d = 1.6$~\\AA.\nThe structure and calculated parameters are illustrated in Fig.~\\ref{fig:structure}.\nThe found values are consistent with previously reported data~\\cite{Wang2015}.\nThe corresponding electronic band structure is given in Fig.~\\ref{fig:electrons}(a), from which one can see that antimonene is an indirect gap semiconductor with the gap width of 0.7~eV. The obtained value is less than 1.0 and 1.2~eV obtained in Refs.~\\onlinecite{Rudenko2017,Singh2016}, due to the differences in exchange-correlation functional, but agrees well with the value reported in~Ref.~\\onlinecite{Wang2015}. \nThe band structure and DOS [Fig.~\\ref{fig:electrons}(b)] display high degrees of electron-hole asymmetry even at small Fermi energies, unlike those of typical 2D materials including graphene and phosphorene.\n\n\n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[]{{fig1_a_b.pdf}}\\vspace*{0.3cm}\n\t\\includegraphics[]{{fig1_c_f.pdf}} \n    \\caption{Electronic structure of antimonene. (a) Band structure plotted along high-symmetry directions of the Brillouin zone. (b) DOS in the relevant charge carrier concentration range. Horizontal lines on panels (a) and (b) mark the following charge carrier concentrations (from bottom to top): $N_{\\text{h}}=$~\\con{1}{15}, $N_{\\text{h}}=$~\\con{3.7}{14}, $N_{\\text{e}}=$~\\con{4}{14}, and $N_{\\text{e}}=$~\\con{1}{15}. The corresponding Fermi surface contours are given on subplots (c)--(f).}\n\t\\label{fig:electrons}\n   \n\\end{figure}\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[]{{fig2.pdf}}\n\t\\caption{Phonon specturm of antimonene. (a) Phonon dispersion plotted along high-symmetry directions of the Brillouin zone. (b) Phonon DOS.}\n\t\\label{fig:phonons}\n\n\\end{figure}\n\nAt negative Fermi energies (hole-doping) the Fermi surface is initially formed by one valence band. As the Fermi energy reduces, second and third occupied bands get involved, forming three concentric pockets [see Fig.~\\ref{fig:electrons}(c)].\nIn the \\textit{K}--$\\Gamma$ direction of the electronic structure one can see a flat region in the valence band, which gives rise to a van Hove singularity (VHS) in DOS ($\\varepsilon=-1.3$~eV in Fig.~\\ref{fig:electrons}). The corresponding\nconstant-energy contour is shown in Fig.~\\ref{fig:electrons}(d), which exhibits a hexagonal warping.\nThis topology opens up a possibility for the scattering between the parallel regions of the surface, known as the Fermi surface nesting~\\cite{Katsnelson1994,Landa2018}. In this case one can expect an increase of the electron-phonon coupling strength at the corresponding charge carrier concentrations ($N_{\\mathrm h}=~$\\con{3.7}{14}).\nFurther reduction of the Fermi energy leads to a widening of the contours \naccompanied by the bending of the hexagons, suppressing the nesting effect. \nThe topology of the Fermi surface at positive Fermi energies (electron-doping) is determined entirely by a single conduction band, yet involving multiple valleys.\nAt small energies the surface is formed by six closed droplet-shaped pockets, originating from the valleys centered along the $\\Gamma$--\\textit{M} direction.\nAdditional circle-shaped pockets appear as the Fermi energy reaches the valley at the \\textit{K} point [Fig.~\\ref{fig:electrons}(e)].\nFurther increase of the Fermi energy results in a VHS originating from the band bending around the \\textit{M} point.\n\n\n\nAt even higher conduction band fillings there is a distinctive change in the Fermi surface topology. Namely, previously closed pockets become connected, and an additional valley emerges around $\\Gamma$ [circle region in Fig.~\\ref{fig:electrons}(f)]. Similar to the valence band, one can see a hexagonal warping around the \\textit{K} point, which gives rise to a VHS ($\\varepsilon=1.2$ eV in Fig.~\\ref{fig:electrons}), corresponding to a heavy electron-doping with $N_{\\mathrm e}=~$\\con{1.0}{15}.\n\nCalculated phonon spectra is given on Fig.~\\ref{fig:phonons}.\nWhile the impact of SOC on the electronic structure is notable, it is significantly less prominent in the context of lattice dynamics.\nIn the long-wavelength limit, the frequency changes are nearly unnoticeable, introducing the difference below~1\\%. At higher $\\mathbf{q}$, particularly in vicinity of \\textit{K} high-symmetry point, the difference in acoustic phonon frequency reaches~5\\%.\nFrequencies of optical phonon modes, particularly ZO and LO, demonstrate comparable changes.\nOverall phonon spectra with SOC taken into account only slightly differs from that of without SOC (see the SM, Fig.~S1).\nIn comparison to other elemental 2D materials, antimonene has considerably lower phonon frequencies, which results, for example, in low thermal conductivity, as seen in~\\cite{SWang2016}.\nElastic constants can be estimated from the frequencies of long-wavelength phonons, using the expression $\\omega_\\nu=q\\sqrt{C_{ij}\/\\rho_{2D}}$, where $C_{ij}$ is the elastic constant, related to acoustic phonon mode $\\nu$, $\\rho_{2D}$ is the mass density of 2D material.\nOut-of-plane phonon mode ZA with quadratic dispersion at low ${\\mathbf q}$ is related to bending rigidity $\\kappa$ in the following way $\\omega_{ZA}=q^{2}\\sqrt{\\kappa\/\\rho_{2D}}$.\nThus, independent 2D elastic constants of antimonene would have the following values: $C_{11}=2.1$~eV\/\\AA$^{2}$, associated with LA phonon mode, and $C_{66}=0.8$~eV\/\\AA$^{2}$, associated with TA phonon mode, while $C_{12}=C_{11}-2C_{66}=0.5$~eV\/\\AA$^{2}$ and Young modulus $E = (C_{11}^{2}-C_{12}^2)\/C_{11} = 2.17~\\text{eV\/\\AA}^2$. The corresponding sound velocities are $v_{\\text{s},LA}=3.4$~km\/s and $v_{\\text{s},TA}=2.1$~km\/s. The bending rigidity associated with ZA phonons is $\\kappa=0.3$~eV.\nThese values are significantly smaller than the elastic moduli and speed of sound of graphene, as well as black and blue phosphorene~\\cite{DLiu2016,Zhu2014}.\nIndeed, in contrast to light elements, heavy antimony atoms suppress vibrations, leading to smaller frequencies.\nFinally, it is interesting to make a note of the role of anharmonic effects in antimonene. The characteristic cutoff wavevector below which anharmonic corrections become dominant is given by $q^{*}= \\sqrt{\n\\frac{3T_{\\text{R}}E}{16\\pi\\cdot\\kappa^2}\n}$.\nAt room temperature $T_{\\text{R}}=300$~K it can be estimated as $q^{*}=0.2~\\text{\\AA}^{-1}$, which is an order of magnitude larger than for black phosphorus~\\cite{Rudenko2016} and comparable with graphene~\\cite{Katsnelson2010}.\nIn the context of superconductivity, however, these effects are not relevant.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{{fig3.pdf}} \n\t\\caption{DOS [(a), (b)] and critical temperature [(c), (d)]  dependencies on charge carrier concentration for antimonene. Vertical dotted lines mark considered charge carrier concentrations. Multiple points for single $N_{\\text{h\/e}}$ on (c) and (d) represent ${T_{\\text{c}}}$~at different values of ${\\mu^{*}_{\\text{c}}}$~(0.1, 0.12,...0.2) from top to bottom. Red symbols and lines on panels (c) and (d) mark ${T_{\\text{c}}}$~values at ${\\mu^{*}_{\\text{c}}}$~$=0.1$. Dashed lines on bottom panels serve as a guide to the eye.}\n\t\\label{fig:dos_tc_bias0}\n    \\vspace*{-0.3cm}\n\\end{figure*}\t\n\n\\subsection{Superconductivity and electron-phonon coupling}\\label{subsec:main_res}\n\nCalculated values of ${T_{\\text{c}}}$~at various charge carrier concentrations are presented in Fig.~\\ref{fig:dos_tc_bias0}. \nAs can be seen, the concentrations resulting in the highest critical temperatures for both holes and electrons correlate with VHS of the corresponding charge carrier DOS. The highest value ${T_{\\text{c}}}$~$=17$ K is achieved for the hole-doping with $N_\\mathrm{h}=$\\con{3.7}{14}.\nIn the case of electron-doped antimonene, the highest value is comparable (${T_{\\text{c}}}$~$\\approx16$~K) yet it is observed at significantly higher charge carrier concentration $N_{\\text{e}}=$~\\con{1}{15}. In all considered cases, ${T_{\\text{c}}}$~decreases by 2--3 K at the maximum chosen ${\\mu^{*}_{\\text{c}}}$~$=0.2$.\n\nFor the sake of comparison with experimental results, let us consider reference experimental concentrations reported recently.\nIn the context of superconductivity of intercalated graphene laminates, chemical doping was utilized leading to ${T_{\\text{c}}}$~in the range 6--6.4~K at $N_{e}\\approx$~\\con{1}{14}~\\cite{Ludbrook2015,Chapman2016}.\nAt this concentration antimonene demonstrates slightly lower ${T_{\\text{c}}}$~right above the liquid helium temperatures (4.2~K). \nHowever, charge carrier concentrations achievable by the electrostatic doping~\\cite{Zhao2017,Efetov2010,Xu2017}   yield higher critical temperatures: 6.6--10.4~K at 2--\\pten{5}{14}~cm$^{-2}$ with a local maximum at~\\con{4.1}{14}.\nIn case of the hole-doping, ${T_{\\text{c}}}$~vanishes rapidly with increasing ${\\mu^{*}_{\\text{c}}}$~at charge carrier concentrations below \\con{2}{14}, mainly due to small DOS.\nIn the vicinity of VHS, ${T_{\\text{c}}}$~increases rapidly: At \\con{3}{14} we find ${T_{\\text{c}}}$~$=15.4$ K.\nOur estimation for ${T_{\\text{c}}}$~in antimonene is comparable with that of phosphorene, according to the calculations reported in Refs.~\\onlinecite{Shao2014,Ge2015}.\nAt $N_{\\text e}=$~\\con{7}{14} both materials yield ${T_{\\text{c}}}$~$\\approx10~$ K with phosphorene showing slightly higher values.\nAt smaller concentrations antimonene shows better results: At 1--\\con{4}{14} electron-doped phosphorene is predicted to have ${T_{\\text{c}}}$~in the range of 0.5--5~K, while antimonene exhibits ${T_{\\text{c}}}$~$=10$~K.\nApplication of strain to phosphorene can change the situation: ${T_{\\text{c}}}$~of phosphorene in this case can be tuned to exceed those of monolayer Sb in the aforementioned concentration range \\cite{Shao2014,Ge2015}.\nCalculated values of ${T_{\\text{c}}}$~for the other group V 2D material, namely arsenene, closely resemble those of antimonene, with a slight shift to lower concentrations~\\cite{Kong2018}.\nParticularly, at concentrations ranging from \\pten{0.9}{14} to \\con{3.7}{14} $n$-doped arsenene shows ${T_{\\text{c}}}$~in the range of 4.7--10.1~K, with the maximum value at \\con{2.8}{14}, which is just 1~K higher, than for antimonene at the same concentration.\nAs for phosphorene, a strain tuning of ${T_{\\text{c}}}$~is proposed for arsenene, which can increase these values.\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{{fig4.pdf}}\n\t\\caption{Contributions to total electron-phonon coupling ($\\lambda_\\mathrm{p}$) and critical temperature ($T_{\\mathrm{c}}$) as functions of hole (a) and electron (b) concentrations.\n\t\tGray pentagons denote critical temperatures at $\\mu^{*}$ = 0.1. Gray shaded region represents variability of ${T_{\\text{c}}}$~with respect to $\\mu^{*}$. Red triangles denote total electron phonon-coupling. Contributions to $\\lambda_{\\mathrm{p}}$ from acoustic and optical phonon modes are shown by green squares and blue circles, respectively.\n\t\tDashed horizontal line corresponds to $\\lambda=1.3$.\n        }\n\t\\label{fig:lambda_contrib_nza}\n\\end{figure*}\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]\n\t{{fig5.pdf}}\n\t\\caption{Electron-phonon coupling and nesting function of doped antimonene. Panels (a) and (c) show nesting function resolved in $\\bf q$-space for $N_{\\mathrm{h}}=$~\\con{3.7}{14} and $N_{\\mathrm{e}}=$~\\con{1}{15}, respectively. Heatmaps are given on the same scale for clarity, whereas actual maximum of $\\xi$ is marked by a yellow arrow in panel (c). Panel (b) shows the Fermi surface contour for $N_{\\mathrm{h}}=$~\\con{3.7}{14}, where arrows mark two distinct electron scattering channels discussed in the text. In dashed frame, panels (d)--(g) show electron-phonon coupling contributions from different phonon modes resolved in $\\bf q$-space. Green labels in panels (c) and (d) mark the Brillouin zone and high-symmetry directions. \n    }\n\t\\label{fig:lambda_single}\n    \\vspace*{-0.3cm}\n\\end{figure*}\n\nLet us analyze electron-phonon coupling in antimonene and its role in the superconducting properties in more details.\nData on the averaged and doping-dependent electron-phonon coupling $\\lambda$ [Eq.~(\\ref{eq:lambtot})] of antimonene is given in Fig.~\\ref{fig:lambda_contrib_nza} (red triangles).\nIt is worth mentioning that electron-phonon coupling strength of doped-antimonene is in general higher than that for other elemental monolayer materials discussed above. At comparable charge carrier concentrations, $\\lambda$ in the range of 0.5--1.4 was obtained for phosphorene~\\cite{Shao2014}, in the range of 0.76--1.27 for arsenene~\\cite{Kong2018}, while $\\lambda=0.6$ was measured experimentally for Li-doped graphene at $N_{\\text{e}}=$~\\con{1}{14}~\\cite{Ludbrook2015}.\nElectron-phonon coupling strength of antimonene exceeds these values already at $N_{\\text{e}}=$~\\con{1}{14}.\nDespite significantly higher values of $\\lambda$, this does not lead to a significant increase of ${T_{\\text{c}}}$~in comparison with the other materials discussed.\nIt can be explained by the difference in characteristic phonon frequencies for these systems (see Sec.~\\ref{subsec:dft}), and particularly smaller values of $\\omega_{\\text{log}}$ [Eq.~(\\ref{eq:wlog})].\nAs a result, in the context of superconductivity this effect compensates for higher $\\lambda$.\n{\\color{black}\nThe maximum value $\\lambda=5$ is observed at $N_{\\text{h}}=$~\\con{3.7}{14}.\nThus, the case of strong electronic nesting is observed for the hole-doped antimonene, which leads to a strong electron-phonon coupling for certain wave vectors, and to a higher $T_{\\text{c}}$.\nAt the same time, it has to be taken into account that the Fermi surface nesting, as well as Van Hove singularities in the electron energy spectrum leads to a general destabilization of the system~\\cite{Antropov1988,Vaks1989,Katsnelson1994}.\nThe most obvious manifestation of this effect is the loss of structural stability. \nWe discuss this aspect in Sec.~\\ref{subsec:stability} in details.\n}\n\nElectron-doped antimonene demonstrates considerably smaller $\\lambda$, yet larger than in the other elemental 2D materials: At $N_{\\text{e}}>$~\\con{3}{14} one has $\\lambda>1.3$, particularly $N_{\\text{e}}$=\\con{1}{15} and ~\\con{4}{14} give $\\lambda=1.8$ and $\\lambda=2.3$ respectively.\nWe note that in the original work where Eq.~(\\ref{eq:ADMC}) was derived~\\cite{Allen1975}, as well as in later works~\\cite{Kresin1987}, it is pointed out that Eq.~(\\ref{eq:ADMC}) underestimates ${T_{\\text{c}}}$~for materials with strong electron-phonon coupling, characterized by $\\lambda>1.3$.\nTherefore, the critical temperatures reported here should be considered as a lower limit.\n\nAs can be seen from Fig.~\\ref{fig:lambda_contrib_nza}, coupling with acoustic in-plane phonons (green squares) is the dominant contribution to $\\lambda$. \nThe coupling with optical phonons (blue circles) is small, but not negligible, at least in case of $p$-doping.\nWe now consider $\\lambda_{\\text{p}}$, contributions from individual phonon modes to the electron-phonon coupling at $N_{\\text{h}}=$~\\con{3.7}{14}, which are presented in Fig.~\\ref{fig:lambda_single}.\nThe main contribution is provided by longitudinal acoustic mode, while coupling with TA mode gives the second highest value.\nAs it was mentioned before, at $N_{\\text{h}}=$~\\con{3.7}{14} there is an indication of a strong Fermi surface nesting. \nTo further investigate these phenomena, we calculate the nesting function $\\xi$ [Eq.~(\\ref{eq:nesting})] shown in Fig.~\\ref{fig:lambda_single}(a).\nIndeed, $\\xi$ exhibits maxima in the $\\Gamma$--\\textit{K} direction of ${\\bf q}$, corresponding to the momentum transfer between parallel sections of the Fermi surface [dotted line in Fig.~\\ref{fig:lambda_single}(b)]. As can be seen from Fig.~\\ref{fig:lambda_single}(d), this mechanism provides a dominant contribution to the coupling with TA phonons. At the same time, one can also see another set of peaks in $\\xi_{\\bf q}$ along the $\\Gamma$--\\textit{M} direction. These peaks correspond to the momentum transfer between nonparallel parts of the Fermi surface [dashed line in Fig.~\\ref{fig:lambda_single}(b)]. This mechanism turns out to be more important for the coupling with LA phonons,\nwhich is shown in Fig.~\\ref{fig:lambda_single}(e), resulting in a higher value of $\\lambda_{\\text{LA}}$.\nThe electron-phonon coupling strengths for optical modes are two orders of magnitude smaller than those for the acoustic ones, with the exception of optical out-of-plane ZO mode [Figs.~\\ref{fig:lambda_single}(f)-\\ref{fig:lambda_single}(h)].\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\begin{minipage}[c]{8.6cm}\n\t\t\\centering\n\t\t\\includegraphics{{fig6_a.pdf}}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{6.7cm}\n\t\t\\centering\n\t\t\\includegraphics{{fig6_b.pdf}}\n\t\\end{minipage}\n\t\\caption{Effect of applied bias voltage on the electronic structure of antimonene. \n\t\t(a) DOS at various values of bias voltage. \n\t\t(b) Comparison of the electronic band structure for ${V_{\\text{b}}}$=1.0~V (red line) and ${V_{\\text{b}}}$=0.0~V (gray line).\n\t}\n\t\\label{fig:dos_bias}\n\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\t\\includegraphics[width=0.9\\textwidth]{fig7}\\vspace*{-0.2cm}\n\t\\caption{Critical temperature dependence on charge carrier concentration at bias voltage of 1~V. Vertical lines mark relevant hole (a) and electron (b) concentrations. Six points for a single $N_{\\text{h\/e}}$ represent different values of ${\\mu^{*}_{\\text{c}}}$~(0.1, 0.12,...0.2). Blue solid line represents ${T_{\\text{c}}}$~at ${\\mu^{*}_{\\text{c}}}$~$=0.1$ and ${V_{\\text{b}}}$~$=0.0$~V, and is given for comparison. Dashed lines serve as guide to the eye.} \n\t\\label{fig:dos_tc_bias1}\n\n\\end{figure*}\n\n\nTabulated values of $\\lambda_{\\text{p}}$ for different charge carrier concentrations is given in Supplemental Material (SM), Tables~S1 and~S2.\nIt is seen, that main contribution in case of electron-doping arises from acoustic phonons, for both $n$ and $p$-doping cases.\nHowever, for $N_{e}<$~\\con{2}{14} individual contributions of acoustic and optical phonons are of the same order.\nDistribution of the nesting function in ${\\bf q}$-space for $N_{\\text{e}}=$~\\con{1}{15} is more complex [Fig.~\\ref{fig:lambda_single}(b)] than for the hole-doping [Fig.~\\ref{fig:lambda_single}(a)]. This is because electron scattering now involves considerable intraband transitions.\nTherefore, it is not possible to determine specific scattering directions in this case. Here, the dominant contribution to $\\lambda$ arises from small ${\\bf q}$ (see SM, Fig.~S2), contrary to a short-wavelength character of the electron-phonon coupling at $N_{\\text{h}}=$~\\con{3.7}{14}. \n\n\n\n\n\\subsection{Effects of bias voltage}\n\n\\begin{figure*}[ht]\n\t\\begin{minipage}[c]{8.0cm}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{{fig8_a_c.pdf}} \n\t\\end{minipage}\\hspace{0.1cm}\n\t\\begin{minipage}[c]{8.0cm}\t\n\t\t\\includegraphics[width=\\textwidth]{{fig8_d_e_r.pdf}}\n\t\\end{minipage}\n\t\\caption{Dependence of critical temperature and electron-phonon coupling in doped antimonene on the bias voltage. Filled and open symbols correspond to $n$- and $p$-doping cases: (a) maximal critical temperature $T_{\\text{c}}^{\\text{max}}$; (b) charge carrier concentration $N_{\\text{e\/h}}$ corresponding to $T_{\\text{c}}^{\\text{max}}$; (c) total electron-phonon coupling strength $\\lambda$. (d) and (e) show variation of critical temperature and electron-phonon coupling strength with bias voltage at fixed concentrations  $N_{\\text{e\/h}}=$~\\con{2}{14}. }\n\t\\label{fig:tcmax}\n\\end{figure*}\n\nThe electronic bands of pristine antimonene are doubly degenerate with respect to spin~\\cite{Rudenko2017}, which is governed by the inversion symmetry.\nWhen an electric field is applied, the inversion symmetry is broken and the spin degeneracy is lifted, which gives rise to the band splitting~\\cite{Prishchenko2018}.\nThis can be clearly seen from Fig.~\\ref{fig:dos_bias}, where we show the effect of bias voltage on the electronic structure of antimonene.\nApart from the SOC related splitting, the band gap is enhanced by the field, unlike, for example, few-layer phosphorene~\\cite{Kim723,Liu2015,Rudenko2015}, where an opposite trend is observed.\nA small gap change of 0.1 eV is observed already at ${V_{\\text{b}}}$=1.0~V, while the highest considered bias voltage of 1.6 V results in a gap of around 30\\% larger than the original one. DOS changes accordingly, as one can see from Fig.~\\ref{fig:dos_bias}(a).\nTaking into account strong correlation of $\\lambda$ with DOS at the Fermi level, bias voltage is expected to have an effect on the superconducting critical temperatures.\nDOS at $n$- and $p$-doping behaves differently with the application of electric field. The most significant change of ${T_{\\text{c}}}$~is expected for $p$-doping at the concentrations corresponding to VHS at ${V_{\\text{b}}}$~$=$~0~V.\nBecause of the strong band splitting in the $\\Gamma$--\\textit{K} direction, the flat band at $\\varepsilon_F\\approx -1.3$~eV splits, resulting in two separate peaks in DOS. In contrast to the original VHS, the two resulting peaks have significantly smaller DOS, which becomes more clear at larger voltages.\nAt the same time, DOS at lower charge carrier concentrations slowly increases with ${V_{\\text{b}}}$, which is mostly observed for $p$-doping. The splitting of the VHS in the conduction band ($n$-doping) is less prominent.\n\n\n\nLet us now consider how ${T_{\\text{c}}}$~depends on the charge carrier concentration in the presence of ${V_{\\text{b}}}$~$=1.0$~V.\nAt this voltage one can clearly see qualitative changes of the carrier DOS [Fig.~\\ref{fig:dos_bias}(a)], i.e. the splitting of a peak at $\\varepsilon_F\\approx -1.3$~eV. \nAs can be seen from Fig.~\\ref{fig:dos_tc_bias1}(b), the change of ${T_{\\text{c}}}$~for the electron-doping is nearly negligible at concentrations $N_{\\text{e}}<$~\\con{3}{14}. At higher concentrations, ${T_{\\text{c}}}$~increase of the order of 1.5 K is observed, with the exception of the highest considered concentration, which results in $T_\\mathrm{c}$ being 1.5~K smaller than in the ${V_{\\text{b}}}$~$=0$~V case.\nMaximum achievable ${T_{\\text{c}}}$~is now 14.6 K for high electron concentration \\con{9.4}{14}, while hole-doping gives comparable values of 13.4 and 12.1~K at $N_{\\text{h}}$ of \\pten{2.7}{14}~and \\con{4.5}{14}, respectively. This corresponds to two new local maxima in DOS. Interestingly, ${T_{\\text{c}}}$~at these points are in good agreement with those observed at the same concentrations at ${V_{\\text{b}}}$$~=~$0~V.\n\nHigh holes concentration of~\\con{8.4}{14}~yields the critical temperature of 13~K, approximately 3~K higher than the highest considered concentration of~\\con{1}{15}~in the absence of the bias voltage. \nLower holes concentrations also yield higher ${T_{\\text{c}}}$~than in the absence of electric field: \\con{2}{14} now gives ${T_{\\text{c}}}$~of 3.2~K, which is nearly twice as large as it was at ${V_{\\text{b}}}$~$=0$~V.\nWhile the absolute value of ${T_{\\text{c}}}$~enhancement at this concentration is small, it represents a practically important trend: Increase of the bias voltage allows us to achieve higher values of ${T_{\\text{c}}}$~at lower concentrations.\n\n\nWe now turn to the dependencies of maximal critical temperature $T_{\\text{c}}^{\\text{max}}$ on the bias voltage [Fig.~\\ref{fig:tcmax}(a)] in the chosen charge carrier concentration range.\nAs expected, the observed trends are different for different doping types. In the case of $n$-doping, $T_{\\text{c}}^{\\text{max}}$ does not change significantly with bias, ranging from 14.6 to 16.4 K with the lowest value corresponding to ${V_{\\text{b}}}$~$=0.4$~V.   \nConcentrations required to achieve $T_{\\text{c}}^{\\text{max}}$ slowly decrease with bias voltage yet remain high: $N_{\\text{e}}=$~\\con{9}{14} at ${V_{\\text{b}}}$~$=1.6$~V. \nAs shown in Fig.~\\ref{fig:tcmax}(c), strong electron phonon coupling is observed for all considered concentrations with $\\lambda$ in the range of 2--3. At low bias voltages, $\\lambda$ decreases first until ${V_{\\text{b}}}$~$=0.4$ V and then increases in the rest of the range. The opposite trend is observed for $p$-doping.\nIn this case, $T_{\\text{c}}^{\\text{max}}$ decreases monotonously with ${V_{\\text{b}}}$~from 17 to 10~K.\nThe required hole concentration, on the contrary, increases, but does not exceed \\con{4.7}{14}, which is still significantly smaller than for electrons. In the presence of bias voltage, $\\lambda$ reduces significantly from 5 to 1.8, indicating a sufficiently strong electron-phonon coupling.\n\n\nCharge carrier concentrations presented in Fig.~\\ref{fig:tcmax}(b) are high, \nand thus may be difficult to achieve in practice. At the same time, \nbias voltage could increase DOS, as well as $\\lambda$ and ${T_{\\text{c}}}$~at moderate concentrations, i.e., below \\con{4}{14}.\nIn Figs.~\\ref{fig:tcmax}(d)-\\ref{fig:tcmax}(e), we consider $\\lambda$ and ${T_{\\text{c}}}$~at more realistic concentrations for both doping cases $N=$~\\con{2}{14}, which are typical values in the medium concentration range. In case of electron-doping, both $\\lambda$ and ${T_{\\text{c}}}$~increase monotonously with ${V_{\\text{b}}}$, yet for the highest bias voltage considered, ${T_{\\text{c}}}$~increases by only $\\approx$1~K.\nHole-doping yields $\\approx$2 K enhancement of ${T_{\\text{c}}}$~already at ${V_{\\text{b}}}$~$=1.0$~V, while $\\lambda$ reaches the value of 2.0.\nIn contrast to the case of electrons, $\\lambda$ and ${T_{\\text{c}}}$~exhibit a maximum at ${V_{\\text{b}}}$~$\\approx1.2$ V, after which one can see a gradual decrease of their values. This behavior can be attributed to a decreasing DOS, shown in Fig.~\\ref{fig:dos_bias}.\nAlthough in absolute values the effect of bias voltage is not large, it provides a possibility to\nincrease ${T_{\\text{c}}}$~and $\\lambda$ in antimonene by up to 20\\%.\n\n\n\n\n\t\\subsection{Dynamical stability of doped antimonene}\\label{subsec:stability}\n\t\n\n\tIn this section, we discuss the dynamical stability of doped antimonene.\n    To model the doping effect beyond the rigid band approximation, we use the jellium doping technique.\n    We are interested in studying the stability at doping concentrations corresponding to the strong Fermi-nesting, considered in the previous section, as well as to the highest charge carrier concentrations in case of $n$-doping. \n    In both cases the loss of structural stability would naturally provide the limiting factor for the applicability of used approximations.\n\nThe results of phonon spectra calculations are presented in Fig.~\\ref{fig:fig9}.\nFor brevity, we consider the $\\Gamma$--\\textit{M} direction only. Qualitatively similar effects are observed in the $\\Gamma$--\\textit{K} direction.\nLet us first consider the hole-doping case. At low enough doping, the system demonstrates the enhancement of acoustic phonon frequencies in the long-wavelength limit, which is especially evident for TA mode. At the same time, out-of-plane phonon mode ZA becomes nearly linear at low ${\\bf q}$.\n\tThese effects persist in the concentration range from~\\con{0.7}{14} to~\\con{1.8}{14}.\n\tAt~\\con{1.8}{14}, a pronounced softening of ZA mode in the $\\Gamma$--\\textit{M} direction is observed. At the concentration of~\\con{2.0}{14}, this mode becomes imaginary in a wide range of ${\\bf q}$, including the vicinity of high-symmetry \\textit{M} point. The TA mode also demonstrates pronounced softening.\n\tFurther imaginary frequencies appear for the TA mode at a slightly higher hole concentration of~\\con{2.1}{14} (+0.3 e\/cell).\n\tIt is important to distinguish out-of-plane mode related instability and in-plane instability.\n\tThe appearance of the former is reported for other doped 2D materials~\\cite{Margine2014,Kong2018} and is not considered as a sign of instability in the case of realistic applications, mainly due to the presence of the substrate in experimental setup. \n\tAt the same time, the presence of in-plane instability is an explicit indication of instability. \n\tTherefore, we conclude that antimonene at concentrations $N_{\\text{h}}>$~\\con{2}{14} is structurally unstable.\n\tIt is worth mentioning that to a certain extent strain can be used to stabilize the system~\\cite{Zeng2016}, even if in-plane instability appears in the phonon spectrum.\n\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[]{{fig9.pdf}}\n\t\t\\caption{Phonon spectra of hole- and electron-doped antimonene (left and right columns respectively) in $\\Gamma$--M high-symmetry direction at various charge carrier concentrations, simulated using jellium doping technique. Dashed line corresponds to phonon spectra in undoped case.}\\label{fig:fig9}\n\t\\end{figure}\n\t\nOn the contrary, electron-doping would only lead to out-of-plane instability, even at the highest concentration considered. \nUnlike the $p$-doping case, electron-doped antimonene is stable at~\\con{2}{14} and higher. At lower concentration considered, i.e.,~\\con{0.7}{14}, only the ZA mode demonstrates imaginary frequencies at low ${\\bf q}$, while in-plane modes demonstrate stiffening. \nAt~\\con{2.1}{14}, however, softening of TA mode is observed, which further increases at higher concentrations\nThe doping effect on longitudinal modes is not as pronounced, with significant softening taking place only at highest charge carrier concentrations considered. This behavior is different from the hole-doping case, where the LA frequencies decrease significantly, especially away from the zone center.  \nThe reason for the higher stability of electron-doped antimonene can partially be associated with a more distributed electron nesting function [Fig.~\\ref{fig:lambda_contrib_nza}(a)], as well as $\\lambda$~(Fig.~S1 in Supplemental Materials).\n\n\t\n\tLet us now discuss possible effect of bias voltage on the stability and ${T_{\\text{c}}}$~in light of the results presented above. \n\tFirst of all, in the case of hole-doping, the effect of bias voltage on the electronic structure remains an important factor, since the increase of ${T_{\\text{c}}}$~is already observed at $N_{\\text{h}}$=~\\con{2}{14}, where the system remains stable.\n\tFurthermore, the splitting of the DOS peak in the presence of $V_{\\text{b}}$ allows us to assume that bias voltage may actually increase the stability of doped antimonene at $N_{\\text{h}}>$~\\con{2}{14} due to the modification of DOS and suppression of VHS (see Fig.~\\ref{fig:dos_bias}).\n    While the effect of bias voltage on the electronic structure of $n$-doped antimonene is less prominent, we expect the tendencies described for this case in the previous section to remain. \n\n\n\n\\FloatBarrier\n\\section{Summary and Conclusion}\n\nWe performed \\textit{ab initio} calculations of electron-phonon coupling for $n$- and $p$-doped antimonene using state-of-the-art computational techniques.\nWe estimated critical temperature of the superconducting transition at various charge carrier concentrations in a wide range from \\pten{5}{13} to \\con{1}{15}~using the Allen-Dynes-McMillan equation.\n{\\color{black}Doing so, we considered the dynamical stability of studied configurations.}\n\nDependence of the electron-phonon coupling strength from charge carrier concentration is essentially conditioned by the doping type and is mainly determined by the Fermi surface topologies.\nWe find that at hole concentrations below \\con{2}{14} and electron concentration below \\con{5}{13} superconductivity is not expected above 0.5~K. \nHigher charge carrier concentrations yield ${T_{\\text{c}}}$~in a wide range.\n{\\color{black}However, the increase of charge carrier concentration in the case of hole-doping is limited by the dynamical instability, which occurs for $N_{\\text{h}}>$\\con{2.0}{14}. This behavior can be explained by the presence of van Hove singularity close to dynamical stability limit in the density of hole states. Among the stable configurations, the maximum value of~${T_{\\text{c}}}$~is estimated to be 15~K, and corresponds to the electron-doping case. The hole-doping yields $T_{\\text{c}}^{\\text{max}}\\approx1.5$~K at the concentrations in the range of 1.8--\\con{2.0}{14}.}\nThese value are comparable or exceed those reported for other doped elemental monolayer materials.\nThe value of strong electron-phonon coupling strength as high as $\\lambda=2.3$ is predicted for the highest considered electron concentration of \\con{1}{15}.\nFor all studied charge carrier concentrations the main contribution to electron-phonon coupling arises from the interaction of carriers with in-plane acoustic phonons.\n\nWe also studied the effects of electric field applied in the direction perpendicular to the atomic sheet, on electron-phonon coupling and critical temperature of antimonene.\nIn the bias voltage range of 0.0--1.6~V, the electronic structure of antimonene undergoes spin-orbit-assisted bands splitting, as well as enhancement of the band gap width. \nThis effect allows us to increase the ${T_{\\text{c}}}$~for the stable hole-doped configurations; thus, in the vicinity of destabilization point at~\\con{2}{14} the maximum increase of ${T_{\\text{c}}}$~reaches 3.5 K at the highest considered bias voltage.\nIn contrast, the electron-doping case is less affected by the application of bias voltage, leading to a slight variation of the critical temperature and electron-phonon coupling over the whole concentration range, which potentially simplifies the application of gating-based doping methods for the system and making results more predictable.\nOverall, bias voltage allows us to control electron-phonon coupling as well as related properties in heavy-element 2D semiconductors, making them interesting objects for further studies.\n\n\n\n\n\n\\section{Acknowledgments}\n\nThis work is part of the research programme ``Two-dimensional semiconductor crystals'' with project number 14TWOD01, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).\nThe calculations were preformed at computational cluster ``TCM'', Radboud University, Nijmegen, Nederlands and at computational cluster NUST ``MISIS'', Moscow, Russia. A.N.R. acknowledges  support from the Russian Science Foundation, Grant No. 17-72-20041.\n\n\\nocite{apsrev41Control}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper, we are concerned with disturbed nonlinear reaction-diffusion\nequations of the form\n\\begin{align} \\label{eq:react-diffus-eq}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) + h(\\zeta) u(t) \\qquad (\\zeta \\in \\Omega) \\\\\ny(t,\\zeta) &= 0 \\qquad (\\zeta \\in \\partial \\Omega)\n\\end{split} \n\\end{align}\non a bounded domain $\\Omega \\subset \\mathbb{R}^d$\nwith smooth boundary $\\partial \\Omega$, \nwhere $g \\in C^1(\\mathbb{R},\\mathbb{R})$ and $h \\in L^2(\\Omega,\\mathbb{R})$ and the disturbance $u$ belongs to $\\mathcal{U} := L^{\\infty}([0,\\infty),\\mathbb{R})$. It is well-known~\\cite{Robinson} that the corresponding undisturbed equation\n\\begin{align} \\label{eq:react-diffus-eq, undisturbed}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) \\qquad (\\zeta \\in \\Omega) \\\\\ny(t,\\zeta) &= 0 \\qquad (\\zeta \\in \\partial \\Omega)\n\\end{split} \n\\end{align}\nhas a unique global attractor $\\Theta \\subset X := L^2(\\Omega,\\mathbb{R})$ under suitable growth and upper-boundedness conditions on the nonlinearity $g$ and its derivative $g'$ respectively. As usual, a global attractor for~\\eqref{eq:react-diffus-eq, undisturbed} is defined to be a compact subset of $X$ that is invariant and uniformly attractive for~\\eqref{eq:react-diffus-eq, undisturbed}. Also, it can be shown~\\cite{KaVa09} that the global attractor $\\Theta$ of~\\eqref{eq:react-diffus-eq, undisturbed} is a stable set for~\\eqref{eq:react-diffus-eq, undisturbed}. \n\\smallskip\n\nWhat we show in this paper is that the disturbed reaction-diffusion equations~\\eqref{eq:react-diffus-eq} are locally input-to-state stable w.r.t.~the global attractor $\\Theta$ of the undisturbed equation~\\eqref{eq:react-diffus-eq, undisturbed}.\nSo,\nwe show that there exist comparison functions $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and radii $r_{0x}, r_{0u} > 0$ such that for every initial value $y_0 \\in X$ with $\\norm{y_0}_{\\Theta} \\le r_{0x}$ and every disturbance $u \\in \\mathcal{U}$ with $\\norm{u}_{\\infty} \\le r_{0u}$ the\nglobal weak solution $$[0,\\infty) \\ni t \\mapsto y(t,\\cdot) = y(t,y_0,u) \\in X$$\nof the boundary value problem~\\eqref{eq:react-diffus-eq}  \nwith initial condition $y(0,\\cdot) = y_0 \\in X$ satisfies the following estimate:\n\\begin{align} \\label{eq:lISS-estimate,intro}\n\\norm{y(t,y_0,u)}_{\\Theta} \\le \\beta(\\norm{y_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in [0,\\infty)). \n\\end{align}\nSee~\\cite{Mi16} for the analogous definition in the special case~$\\Theta = \\{0\\}$. \nIn the above relations, we use the standard notation\n\\begin{align} \\label{eq:norm-Theta,def}\n\\norm{x}_{\\Theta} := \\operatorname{dist}(x,\\Theta) := \\inf_{\\theta \\in \\Theta} \\norm{x-\\theta} \\qquad (x \\in X)\n\\end{align}\nand the standard definitions for the comparison function classes $\\mathcal{KL}$ and $\\mathcal{K}$, which are recalled in~\\eqref{eq:comparison-fct-classes} below. \nIn words,\nthe local input-to-state stability estimate~\\eqref{eq:lISS-estimate,intro} means\nthat  \n\\begin{itemize}\n\\item[(i)] the\ninvariant set $\\Theta$ for~\\eqref{eq:react-diffus-eq, undisturbed} is locally stable and\nattractive for the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} and \n\\item[(ii)] these local stability and attractivity properties are affected only slightly in the presence of disturbances of small magnitude $\\norm{u}_{\\infty}$.\n\\end{itemize}\nIn order to achieve the estimate~\\eqref{eq:lISS-estimate,intro}, we will construct a suitable local input-to-state Lyapunov function $V$.\n\\smallskip\n\nAs far as we know, our result is the first (local) input-to-state stability result w.r.t.~attractors $\\Theta$ of concrete partial differential equation systems. All previous concrete pde results\nwe are aware of -- like those from~\\cite{DaMi13}, \\cite{JaNaPaSc16}, \\cite{JaSc18}, \\cite{KaKr16}, \\cite{KaKr17}, \\cite{MaPr11}, \\cite{MiKaKr17}, \\cite{ScZw18}, \\cite{TaPrTa17}, \\cite{ZhZh17a}, \\cite{ZhZh17b}, for instance --\nestablish input-to-state stability only w.r.t.~an equilibrium point $\\theta$, which without loss of generality is assumed to be $\\theta = 0$. In particular, all these previous results require their   nonlinearity $g$ to be such that $g(\\theta) = g(0) = 0$ and such that the undisturbed system has the singleton $\\Theta := \\{\\theta\\} = \\{0\\}$ as an attractor. \nWith our result, by contrast, we can treat much more general nonlinearities:\nwe can treat nonlinearities $g$ with $g(0) \\ne 0$ and, more importantly, nonlinearities $g$ for which the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} has only a non-singleton attractor $\\Theta \\supsetneq \\{0\\}$. A simple example of such a nonlinearity is given by $g(r) := -r^3 + r$, which leads to the Chaffee--Infante equation.\nWe refer to~\\cite{KaKaVa15}, \\cite{GoKaKaPa14}, \\cite{GoKakaKh15} \\cite{DaKaRo17} for other interesting results about non-trivial global attractors of nonlinear, impulsive, or even multi-valued semigroups. \n\\smallskip\n\nIn the entire paper, we will use the following conventions and notations. As above, $X := L^2(\\Omega,\\mathbb{R})$ and $\\mathcal{U} := L^{\\infty}(\\mathbb{R}^+_0,\\mathbb{R})$ with $\\mathbb{R}^+_0 := [0,\\infty)$ and with the standard norm of $X$ being denoted simply by $\\norm{\\cdot} := \\norm{\\cdot}_{L^2(\\Omega)}$. As usual,\n\\begin{align*}\nB_r(x_0) = B_r^{X}(x_0), \\quad \\overline{B}_r(x_0) = \\overline{B}_r^{X}(x_0) \\quad \\text{and} \\quad B_r(u_0) = B_r^{\\mathcal{U}}(u_0), \\quad  \\overline{B}_r(u_0) = \\overline{B}_r^{\\mathcal{U}}(u_0)\n\\end{align*}\ndenote the open and closed balls in $X$ or $\\mathcal{U}$ of radius $r$ around $x_0 \\in X$ or $u_0 \\in \\mathcal{U}$ respectively. We will often use the notation~\\eqref{eq:norm-Theta,def} and\n\\begin{align*}\nB_r(\\Theta) := \\{x \\in X: \\norm{x}_{\\Theta} < r \\} \n\\qquad \\text{and} \\qquad\n\\overline{B}_r(\\Theta) := \\{x \\in X: \\norm{x}_{\\Theta} \\le r \\},\n\\end{align*}\nas well as the notation \n$\\operatorname{dist}(M,\\Theta) := \\sup_{x\\in M} \\norm{x}_{\\Theta}$\nfor subsets $M, \\Theta \\subset X$. Also, $\\mathcal{K}$, $\\mathcal{K}_{\\infty}$ and $\\mathcal{KL}$ will denote the following standard classes of comparison functions:\n\\begin{gather}\n\\mathcal{K} := \\{ \\gamma \\in C(\\mathbb{R}^+_0,\\mathbb{R}^+_0): \\gamma \\text{ strictly increasing with } \\gamma(0) = 0 \\} \\notag \\\\\n\\mathcal{K}_{\\infty} := \\{ \\gamma \\in \\mathcal{K}: \\gamma \\text{ unbounded} \\}  \\label{eq:comparison-fct-classes}\\\\\n\\mathcal{KL} := \\{ \\beta \\in C(\\mathbb{R}^+_0 \\times \\mathbb{R}^+_0,\\mathbb{R}^+_0): \\beta(\\cdot,t) \\in \\mathcal{K} \\text{ for } t \\ge 0 \\text{ and } \\beta(s,\\cdot) \\in \\mathcal{L} \\text{ for } s > 0 \\}, \\notag\n\\end{gather}\nwhere $\\mathcal{L} := \\{ \\gamma \\in C(\\mathbb{R}^+_0,\\mathbb{R}^+_0): \\gamma \\text{ strictly decreasing with } \\lim_{t\\to\\infty} \\gamma(t) = 0 \\}$. And finally, upper right-hand Dini derivatives will be denoted by\n\\begin{align*}\n\\overline{\\partial}_t^+ v(t) := \\varlimsup_{\\tau \\to 0+} \\frac{v(t+\\tau)-v(t)}{\\tau}.\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Some preliminaries}\n\nIn this section, we provide\nthe necessary preliminaries for our local input-to-state stability result. \nWe begin by recalling the definition of weak solutions of initial boundary value problems of the form \n\\begin{align} \\label{eq:ibvp}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) + h(\\zeta) u(t) \\qquad ((t,\\zeta) \\in [s,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(s,\\cdot) = y_s \\qquad (t \\in [s,\\infty)).\n\\end{split} \n\\end{align}\nIn fact, we will have to consider initial boundary value problems with more general inhomogeneities of the form\n\\begin{align} \\label{eq:ibvp-general}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + \\overline{g}(y(t,\\zeta)) + \\overline{h}(t,\\zeta)  \\qquad ((t,\\zeta) \\in [s,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(s,\\cdot) = y_s \\qquad (t \\in [s,\\infty)),\n\\end{split} \n\\end{align}\nwhere $\\overline{g}$, $\\overline{h}$ satisfy the following conditions. \n\n\n\n\n\\begin{cond} \\label{cond:ol-g,ol-h}\n\\begin{itemize}\n\\item[(i)] $\\Omega$ is a bounded domain in $\\mathbb{R}^d$ for some $d \\in \\N$ with smooth boundary $\\partial \\Omega$ \nand, moreover, $p \\in [2,\\infty)$, $q \\in (1,2]$ are dual exponents: $1\/p + 1\/q = 1$\n\\item[(ii)] $\\overline{g} \\in C^1(\\mathbb{R},\\mathbb{R})$ and there exist constants $\\alpha_1, \\alpha_2, \\kappa, \\lambda \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:growth-cond-ol-g}\n-\\kappa - \\alpha_1 |r|^p \\le \\overline{g}(r)r \\le \\kappa - \\alpha_2 |r|^p\n\\qquad \\text{and} \\qquad\n\\overline{g}'(r) \\le \\lambda\n\\qquad (r \\in \\mathbb{R})\n\\end{align}\nand, moreover, $\\overline{h} \\in L^q_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^q(\\Omega))$. \n\\end{itemize}\n\\end{cond}\n\n\nA bit more explicitly, the first two inequalities in~\\eqref{eq:growth-cond-ol-g} mean that $\\overline{g}|_{(0,\\infty)}$\nlies between $r \\mapsto -\\kappa\/|r| - \\alpha_1 |r|^{p-1}$ and $r \\mapsto \\kappa\/|r| - \\alpha_2 |r|^{p-1}$ \nand that $\\overline{g}|_{(-\\infty,0)}$ lies between $r \\mapsto -\\kappa\/|r| + \\alpha_2 |r|^{p-1}$ and $r \\mapsto \\kappa\/|r| + \\alpha_1 |r|^{p-1}$. \nA simple class of functions $\\overline{g}$ satisfying the three inequalities from~\\eqref{eq:growth-cond-ol-g}\nis given by the polynomials of odd degree with negative leading coefficient:\n\\begin{align*}\n\\overline{g}(r) = \\sum_{i=0}^{2m-1} c_i r^{i} \\qquad (r \\in \\mathbb{R})\n\\end{align*} \nwith $c_{2m-1} < 0$, where $m \\in \\N$. (Choose $p := 2m$.)\nIn particular, the nonlinearity of the Chaffee--Infante equation given by $\\overline{g}(r) := -r^3 + r$ falls into that class (Section~11.5 of~\\cite{Robinson}). \n\\smallskip \n\nSuppose that Condition~\\ref{cond:ol-g,ol-h} is satisfied and let $s \\in \\mathbb{R}^+_0$ and $y_s \\in X$. A function $y \\in C([s,\\infty),X)$ is\ncalled a \\emph{global weak solution of\n\\eqref{eq:ibvp-general}} iff $y(s) = y_s$ and for every $T \\in (s,\\infty)$ one has\n\\begin{align}\ny|_{[s,T]} \\in L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))\n\\end{align}\nand there exists a (then unique) $z \\in L^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega))$ such that\n\\begin{align} \\label{eq:def-weak-sol}\n\\int_s^T \\big( z(t), \\phi(t)\\big) \\d t &= -\\int_s^T \\int_{\\Omega} \\nabla y(t)(\\zeta) \\cdot \\nabla \\phi(t)(\\zeta) \\d \\zeta \\d t + \\int_s^T \\int_{\\Omega} \\overline{g}\\big( y(t)(\\zeta) \\big) \\, \\phi(t)(\\zeta) \\d \\zeta \\d t  \\notag \\\\\n&\\quad + \\int_s^T \\int_{\\Omega} \\overline{h}(t)(\\zeta) \\, \\phi(t)(\\zeta)  \\d \\zeta \\d t \n\\end{align}\nfor every $\\phi \\in L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))$. See~\\cite{VaKa06} or~\\cite{KaVa09} and, for more background information, \\cite{ChVi96} or~\\cite{ChVi}. In this equation, $(\\cdot,\\cdot \\cdot)$ stands for the dual pairing of $H^1_0(\\Omega)^* + L^q(\\Omega)$ and $H^1_0(\\Omega) \\cap L^p(\\Omega)$, that is, \n\\begin{align} \\label{eq:dual-pairing}\n(z,\\phi) = (z_1,\\phi)_{H^1_0(\\Omega)^*,H^1_0(\\Omega)} + (z_2,\\phi)_{L^q(\\Omega),L^p(\\Omega)}\n\\end{align}\nfor every $z = z_1+z_2 \\in H^1_0(\\Omega)^* + L^q(\\Omega)$ and $\\phi \\in H^1_0(\\Omega) \\cap L^p(\\Omega)$, where $(\\cdot,\\cdot\\cdot)_{H^1_0(\\Omega)^*,H^1_0(\\Omega)}$ and $(\\cdot,\\cdot\\cdot)_{L^q(\\Omega),L^p(\\Omega)}$ denote the respective dual pairings. See~\\cite{BeLoe} (Theorem~2.7.1) and \\cite{DiUh} (Theorem~IV.1.1 and Corollary~III.2.13), for instance, to get that $H^1_0(\\Omega)^* + L^q(\\Omega)$, $H^1_0(\\Omega) \\cap L^p(\\Omega)$ and\n\\begin{align*}\nL^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega)), \\quad L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))\n\\end{align*} \nare dual to each other. \nWe point out that if $y$ is a global weak solution to~\\eqref{eq:ibvp-general}, then for every $T \\in (s,\\infty)$ there is only one $z \\in L^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega))$ satisfying~\\eqref{eq:def-weak-sol}. And this $z$ is called the \\emph{weak or generalized derivative} of $y|_{[s,T]}$. It is denoted by $\\partial_t y|_{[s,T]}$ or simply by $\\partial_t y$ in the following. \n\n\n\\begin{lm} \\label{lm:weak-sol-general}\nSuppose that Condition~\\ref{cond:ol-g,ol-h} is satisfied and let $s \\in \\mathbb{R}^+_0$ and $y_s \\in X$. Then the initial boundary value problem~\\eqref{eq:ibvp-general} has a unique global weak solution $y$ and, moreover, $t \\mapsto \\norm{y(t)}^2$ is absolutely continuous (hence differentiable almost everywhere) with\n\\begin{align}\n\\ddt \\norm{y(t)}^2 = 2 \\big( \\partial_t y(t),y(t) \\big) \n\\end{align} \nfor almost every $t \\in [s,\\infty)$, where $(\\cdot,\\cdot \\cdot)$ is the dual pairing from~\\eqref{eq:dual-pairing}. \n\\end{lm}\n\n\n\\begin{proof}\nIt is clear from the first two inequalities in~\\eqref{eq:growth-cond-ol-g} that\n\\begin{align} \\label{eq:estimate-g(r)r}\n|\\overline{g}(r)r| \\le \\kappa + \\alpha_1|r|^p \\qquad (r \\in \\mathbb{R}).\n\\end{align} \nSince $\\sup_{|r|\\le 1} |\\overline{g}(r)| < \\infty$ by the continuity of $\\overline{g}$, it follows from~\\eqref{eq:estimate-g(r)r} that for some constant $C_1 \\in (0,\\infty)$\n\\begin{align} \\label{eq:g(y)-in-Lq}\n|\\overline{g}(r)| \\le C_1 (1+|r|^{p-1}) \\qquad (r \\in \\mathbb{R})\n\\end{align}\nand therefore condition~(2) from~\\cite{VaKa06} is satisfied. Also, in view of the second and third inequalities in~\\eqref{eq:growth-cond-ol-g}, condition~(3) and condition~(4) from~\\cite{VaKa06} with $M=0$ are satisfied. \nConsequently, the assertions of the lemma follow from the remarks made in Section~2 (up to Remark~1) of~\\cite{VaKa06}. \n\\end{proof}\n\n\n\nWith this lemma at hand, it is easy to see that the initial boundary value problem~\\eqref{eq:ibvp} generates a semiprocess family $(S_u)_{u\\in\\mathcal{U}}$ on $X$ (Lemma~\\ref{lm:semiproc}).  A \\emph{semiprocess family on $X$} is a family of maps $S_u: \\Delta \\times X \\to X$ for every $u \\in \\mathcal{U}$ such that\n\\begin{gather}\nS_u(s,s,x) = x \\qquad \\text{and} \\qquad S_u\\big(t,s, S_u(s,r,x) \\big) = S(t,r,x) \\label{eq:def-semiproc-1}\\\\\nS_u(t+\\tau,s+\\tau,x) = S_{u(\\cdot+\\tau)}(t,s,x) \\label{eq:def-semiproc-2}\n\\end{gather} \nfor all $(t,s), (s,r) \\in \\Delta$, $\\tau \\in \\mathbb{R}^+_0$, $x \\in X$ and $u \\in \\mathcal{U}$, where we used the abbreviation $\\Delta := \\{(s,t) \\in \\mathbb{R}^+_0 \\times \\mathbb{R}^+_0: t \\ge s\\}$. See~\\cite{ChVi}, for instance, for more information on semiprocess families. \n\n\n\n\n\\begin{cond} \\label{cond:g,h}\n\\begin{itemize}\n\\item[(i)] $\\Omega$ is a bounded domain in $\\mathbb{R}^d$ for some $d \\in \\N$ with smooth boundary $\\partial \\Omega$ \nand, moreover, $p \\in [2,\\infty)$\n\\item[(ii)] $g \\in C^1(\\mathbb{R},\\mathbb{R})$ and there exist constants $\\alpha_1, \\alpha_2, \\kappa, \\lambda \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:cond-g}\n-\\kappa - \\alpha_1 |r|^p \\le g(r)r \\le \\kappa - \\alpha_2 |r|^p\n\\qquad \\text{and} \\qquad\ng'(r) \\le \\lambda\n\\qquad (r \\in \\mathbb{R})\n\\end{align}\nand, moreover, $h \\in X \\setminus \\{0\\}$.\n\\end{itemize}\n\\end{cond}\n\n\n\n\\begin{lm} \\label{lm:semiproc}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then for every $s \\in \\mathbb{R}^+_0$ and every $(y_s,u) \\in X \\times \\mathcal{U}$ the initial boundary value problem~\\eqref{eq:ibvp} has a unique global weak solution $y(\\cdot,s,y_s,u)$. Additionally, $(S_u)_{u\\in\\mathcal{U}}$ defined by\n\\begin{align} \\label{eq:S_u-def}\nS_u(t,s,y_s) := y(t,s,y_s,u)\n\\end{align}\nis a semiprocess family on $X$. \n\\end{lm}\n\n\n\\begin{proof}\nIn order to see the unique global weak solvability, simply apply Lemma~\\ref{lm:weak-sol-general} with $\\overline{g}:=g$ and with $\\overline{h} \\in L^2_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^2(\\Omega)) \\subset L^q_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^q(\\Omega))$ defined by $\\overline{h}(t)(\\zeta) := h(\\zeta) u(t)$. \nIn order to see the semiprocess property, use the definition of weak solutions and the uniqueness statement from Lemma~\\ref{lm:weak-sol-general}. \n\\end{proof}\n\n\n\n\n\n\nIn the following, $(S_u)_{u\\in\\mathcal{U}}$ will always denote the semiprocess family from the previous lemma. Also, we will often refer to $(S_u)_{u\\in \\mathcal{U}}$ and $S_0$ as the disturbed and the undisturbed system, respectively. \nIn proving our local input-to-state stability result, the following estimates will play an important role. \n\n\\begin{lm} \\label{lm:semiproc-estimates}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then\n\\begin{align}\n\\norm{S_0(t,0,y_{01})-S_0(t,0,y_{02})} &\\le \\e^{\\lambda t} \\norm{y_{01}-y_{02}} \\qquad (t \\in \\mathbb{R}^+_0) \\label{eq:semiproc-estimate-1}\\\\\n\\norm{S_u(t,0,y_0)-S_0(t,0,y_0)} &\\le 2 \\e^{2 \\lambda} \\norm{h} \\norm{u}_{\\infty} t \\qquad (t \\in [0,1]) \\label{eq:semiproc-estimate-2}\n\\end{align} \nfor all $y_0, y_{01}, y_{02} \\in X$ and all $u \\in \\mathcal{U}$.\n\\end{lm}\n\n\n\\begin{proof}\nAs a first step, we show that for every $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ the function \n\\begin{align} \\label{eq:y-12-u}\ny_{12}^{u} := y_1^{u}-y_2^0 \\qquad \\text{with} \\qquad y_1^{u} := S_u(\\cdot,0,y_{01}) \\qquad \\text{and} \\qquad y_2^0 := S_0(\\cdot,0,y_{02})\n\\end{align}\nis a global weak solution of the initial boundary value problem\n\\begin{align} \\label{eq:semiproc-estimates-ibvp-general}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + \\overline{g}(y(t,\\zeta)) + \\overline{h}(t,\\zeta)  \\qquad ((t,\\zeta) \\in [0,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(0,\\cdot) = y_{01}-y_{02} \\qquad (t \\in [0,\\infty)),\n\\end{split} \n\\end{align}\nwhere $\\overline{g} := g$ and $\\overline{h}(t)(\\zeta) := g(y_1^{u}(t)(\\zeta)) - g(y_2^{0}(t)(\\zeta)) - g(y_{12}^{u}(t)(\\zeta)) + h(\\zeta)u(t)$. \nSo, let $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ and adopt the abbreviations from~\\eqref{eq:y-12-u}. It is not difficult -- using~\\eqref{eq:g(y)-in-Lq} and $q(p-1)=p$ -- to see from Condition~\\ref{cond:g,h} that with $\\overline{g}$, $\\overline{h}$ as defined above, Condition~\\ref{cond:ol-g,ol-h} is satisfied. \nSince $y_1^{u}$, $y_2^0$ are global weak solutions, we have $y_{12}^{u} \\in C(\\mathbb{R}^+_0,X)$ and for every $T \\in (0,\\infty)$ we have\n\\begin{align*}\ny_{12}^{u}|_{[0,T]} \\in L^2([0,T],H^1_0(\\Omega)) \\cap L^p([0,T], L^p(\\Omega))\n\\end{align*}\nand $\\partial_t y_{1}^{u}|_{[0,T]} - \\partial_t y_{2}^{0}|_{[0,T]} \\in L^2([0,T],H^1_0(\\Omega)^*) + L^q([0,T], L^q(\\Omega))$ as well as\n\\begin{align} \\label{eq:semiproc-estimates-step-1}\n\\int_0^T \\big( \\partial_t y_{1}^{u}(t) &- \\partial_t y_{2}^{0}(t), \\phi(t) \\big) \\d t \n= - \\int_0^T \\int_{\\Omega} \\nabla y_{12}^{u}(t)(\\zeta) \\cdot \\nabla \\phi(t)(\\zeta) \\d\\zeta \\d t \\notag \\\\\n&+  \\int_0^T \\int_{\\Omega} \\overline{g}\\big( y_{12}^{u}(t)(\\zeta) \\big) \\, \\phi(t)(\\zeta) \\d \\zeta \\d t  + \\int_0^T \\int_{\\Omega} \\overline{h}(t)(\\zeta) \\, \\phi(t)(\\zeta)  \\d \\zeta \\d t\n\\end{align}\nfor every $\\phi \\in L^2([0,T],H^1_0(\\Omega)) \\cap L^p([0,T], L^p(\\Omega))$. And therefore, $y_{12}^{u}$ is a weak solution of~\\eqref{eq:semiproc-estimates-ibvp-general}, as desired.\n\\smallskip\n\nAs a second step, we show that for every $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ the function $y_{12}^{u}$ from~\\eqref{eq:y-12-u} satisfies the estimate\n\\begin{align} \\label{eq:semiproc-estimates-step-2}\n\\sup_{T \\in [0,t]} \\norm{y_{12}^{u}(T)}^2 \\le \\e^{2\\lambda t} \\Big( \\norm{y_{01}-y_{02}}^2 + 2\\norm{h}\\norm{u}_{\\infty} \\cdot t \\cdot \\sup_{T \\in [0,t]}  \\norm{y_{12}^{u}(T)} \\Big) \n\\end{align}\nfor every $t \\in \\mathbb{R}^+_0$. \nIndeed, by the first step and Lemma~\\ref{lm:weak-sol-general}, the function $t \\mapsto \\norm{y_{12}^{u}(t)}^2$ is absolutely continuous with \n\\begin{align*}\n\\ddt \\frac{\\norm{y_{12}^{u}(t)}^2}{2} = \\big( \\partial_t y_{12}^{u}(t), y_{12}^{u}(t) \\big) = \\big( \\partial_t y_{1}^{u}(t) - \\partial_t y_{2}^{0}(t), y_{12}^{u}(t) \\big) \n\\end{align*}\nfor almost every $t \\in \\mathbb{R}^+_0$. And therefore, by virtue of~\\eqref{eq:semiproc-estimates-step-1} with $\\phi := y_{12}^{u}$, we get\n\\begin{align} \\label{eq:semiproc-estimates-step-2,1}\n&\\frac{\\norm{y_{12}^{u}(T)}^2}{2} - \\frac{\\norm{y_{12}^{u}(0)}^2}{2}\n=\n\\int_0^T \\big( \\partial_t y_{1}^{u}(t) - \\partial_t y_{2}^{0}(t) , y_{12}^{u}(t) \\big)  \\d t \\notag \\\\\n&\\qquad \\le \n\\int_0^T \\int_{\\Omega} \\big( g(y_1^{u}(t)(\\zeta)) - g(y_2^{0}(t)(\\zeta)) \\big) y_{12}^{u}(t)(\\zeta) \\d\\zeta \\d t + \\int_0^T \\int_{\\Omega} h(\\zeta) u(t) y_{12}^{u}(t)(\\zeta) \\d\\zeta \\d t \\notag \\\\\n&\\qquad \\le \n\\lambda \\int_0^T \\norm{y_{12}^{u}(t)}^2 \\d t + \\norm{h}\\norm{u}_{\\infty} \\int_0^T \\norm{y_{12}^{u}(t)} \\d t\n\\end{align}\nfor every $T \\in (0,\\infty)$. In the last inequality, we used that $(g(r)-g(s)) (r-s) \\le \\lambda |r-s|^2$ for all $r,s \\in \\mathbb{R}$ due to~\\eqref{eq:cond-g}.\nSo, for every $t_0 \\in (0,\\infty)$, we obtain\n\\begin{align*}\n\\norm{y_{12}^{u}(T)}^2 \\le \\norm{y_{01}-y_{02}}^2 + 2\\norm{h}\\norm{u}_{\\infty} \\cdot t_0 \\cdot \\sup_{t\\in[0,t_0]} \\norm{y_{12}^{u}(t)} + 2 \\lambda \\int_0^T \\norm{y_{12}^{u}(t)}^2 \\d t\n\\end{align*}\nfor every $T \\in [0,t_0]$. And from this, in turn,\nthe claimed estimate~\\eqref{eq:semiproc-estimates-step-2} immediately follows by Gr\\\"onwall's lemma.  \n\\smallskip\n\nAs a third step, it is now easy to conclude the desired estimates~\\eqref{eq:semiproc-estimate-1} and~\\eqref{eq:semiproc-estimate-2} from the second step.\nIndeed, \\eqref{eq:semiproc-estimate-1} immediately follows from~\\eqref{eq:semiproc-estimates-step-2} with the special choice $u := 0 \\in \\mathcal{U}$ and~\\eqref{eq:semiproc-estimate-2} follows from~\\eqref{eq:semiproc-estimates-step-2} with the special choice $y_{01} = y_{02} := y_0 \\in X$.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe remark for later reference that our semiprocess family $(S_u)_{u\\in\\mathcal{U}}$, like any other semiprocess family~\\cite{ScKaDa19-wAG},\nsatisfies the following so-called cocycle property:\n\\begin{align}\nS_u(t+\\tau,0,x) = S_{u(\\cdot+\\tau)}\\big(t,0,S_u(\\tau,0,x)\\big)\n\\end{align}\nfor all $t, \\tau \\in \\mathbb{R}^+_0$, $x \\in X$ and $u \\in \\mathcal{U}$. (Just combine~\\eqref{eq:def-semiproc-1} and~\\eqref{eq:def-semiproc-2} to see this.) In particular, $S_0$ satisfies the following (nonlinear) semigroup property~\\cite{Miyadera}:\n\\begin{align} \\label{eq:S0-sgr}\nS_0(t+\\tau,0,x) = S_0\\big(t,0,S_0(\\tau,0,x)\\big) \\qquad (t,\\tau \\in \\mathbb{R}^+_0 \\text{ and } x \\in X).\n\\end{align} \n\nWe conclude this section with some remarks on the asymptotic behavior\nof this semigroup $S_0$ in terms of attractors~\\cite{Robinson}, \\cite{Temam}.\nA \\emph{global attractor} of $S_0$ is a compact subset $\\Theta$ of $X$ such that\n\\begin{itemize}\n\\item[(i)] $\\Theta$ is invariant under $S_0$, that is, $S_0(t,0,\\Theta) = \\Theta$ for every $t \\in \\mathbb{R}^+_0$\n\\item[(ii)] $\\Theta$ is uniformly attractive for $S_0$, that is, for every bounded subset $B \\subset X$ one has\n\\begin{align} \\label{eq:uniform-attractivity-def}\n\\operatorname{dist}\\big( S_0(t,0,B), \\Theta \\big) = \\sup_{x\\in B} \\norm{S_0(t,0,x)}_{\\Theta} \\longrightarrow 0 \\qquad (t \\to \\infty).\n\\end{align}\n\\end{itemize} \nIt directly follows from this definition that a global attractor of $S_0$ is minimal among all closed uniformly attractive sets of $S_0$ and maximal among all bounded invariant sets of $S_0$. And from this, in turn, it immediately follows that if $S_0$ has any global attractor then it is already unique. \n\n\n\n\\begin{lm} \\label{lm:S0-UGAS}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then the undisturbed system $S_0$ has a unique global attractor $\\Theta$ and, moreover, $\\Theta$ is uniformly globally asymptotically stable for $S_0$, that is, there exists a comparison function $\\beta_0 \\in \\mathcal{KL}$ such that\n\\begin{align} \\label{eq:S0-UGAS}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\beta_0(\\norm{x}_{\\Theta},t) \\qquad (t\\in \\mathbb{R}^+_0 \\text{ and } x \\in X). \n\\end{align}\n\\end{lm}\n\n\n\n\\begin{proof}\nIt is well-known that $S_0$ has a global attractor $\\Theta$ (by Theorem~11.4 of~\\cite{Robinson}, for instance) and that global attractors when existent are already unique (by the remarks preceding the lemma). \nSo, we have only\nto show that $\\Theta$ is uniformly globally asymptotically stable for $S_0$. And in order to do so, we will proceed in three steps, applying results from~\\cite{Mi17} to the system $S_0 = (S_u)_{u\\in\\mathcal{U}_0}$ with trivial disturbance space $\\mathcal{U}_0 := \\{0\\}$. (In this context, it should be noticed that by~\\eqref{eq:S0-sgr} and the continuity of weak solutions, $(S_u)_{u\\in\\mathcal{U}_0}$ is a forward-complete system in the sense of~\\cite{Mi17}, \\cite{MiWi18}, \\cite{Sc19-wISS}.)\n\\smallskip\n\nAs a first step, we show that $\\Theta$ is uniformly globally stable for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$,\nthat is, there exists a comparison function $\\sigma_0 \\in \\mathcal{K}$ such that\n\\begin{align}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\sigma_0(\\norm{x}_{\\Theta}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor every $x \\in X$ (Definition~2.8 of~\\cite{Mi17}).\nIndeed, it immediately follows from the invariance of $\\Theta$ under $S_0$ and from the estimate~\\eqref{eq:semiproc-estimate-1} that for every $\\varepsilon > 0$ and every $T \\in (0,\\infty)$ there exists a $\\delta \\in (0,1]$ such that\n\\begin{align*}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\inf_{\\theta \\in\\Theta} \\norm{S_0(t,0,x)-S_0(t,0,\\theta)} < \\varepsilon \\qquad (t \\in [0,T] \\text{ and } x \\in B_{\\delta}(\\Theta)).  \n\\end{align*}\nAnd from this and the uniform attractivity~\\eqref{eq:uniform-attractivity-def} of $\\Theta$ for $S_0$ (with $B := B_1(\\Theta)$), in turn, it follows that for every $\\varepsilon > 0$ there exists a $\\delta > 0$ such that\n\\begin{align} \\label{eq:Theta-ULS}\n\\norm{S_0(t,0,x)}_{\\Theta} < \\varepsilon \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor every $x \\in B_{\\delta}(\\Theta)$. \nAlso, it is well-known that \n\\begin{align} \\label{eq:dissip-estimate-S0}\n\\norm{S_0(t,0,x)}^2 \\le \\e^{-2\\omega t} \\norm{x}^2 + \\frac{\\lambda |\\Omega|}{\\omega} \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $x \\in X$, where $\\omega \\in (0,\\infty)$ is the smallest eigenvalue of $-\\Delta$, the negative Dirichlet Laplacian on $\\Omega$. (See the very last equation on page 286 of~\\cite{Robinson}, for instance.) Since $\\norm{S_0(t,0,x)}_{\\Theta} \\le \\norm{S_0(t,0,x)} + \\norm{\\Theta}$ and $\\norm{x} \\le \\norm{x}_{\\Theta} + \\norm{\\Theta}$ with $\\norm{\\Theta} := \\sup_{\\theta \\in \\Theta} \\norm{\\theta}$, it follows from~\\eqref{eq:dissip-estimate-S0}  that there exists a comparison function $\\sigma \\in \\mathcal{K}$ and a constant $c \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:Theta-Lagrange-stable}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\sigma(\\norm{x}_{\\Theta}) + c \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align} \nfor every $x \\in X$. \nIn the terminology of~\\cite{Mi17}, the relations~\\eqref{eq:Theta-ULS} and~\\eqref{eq:Theta-Lagrange-stable} mean that $\\Theta$ is uniformly locally stable and Lagrange-stable for $(S_u)_{u\\in\\mathcal{U}_0}$, respectively. And therefore, $\\Theta$ is uniformly globally stable for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$ by virtue of Remark~2.9 of~\\cite{Mi17}, as desired. \n\\smallskip\n\nAs a second step, we show\nthat $\\Theta$ is uniformly globally attractive for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$, that is, for every $\\varepsilon >0$ and $r>0$ there exists a time $\\tau(\\varepsilon,r) \\in \\mathbb{R}^+_0$ such that \n\\begin{align} \\label{eq:Theta-UGATT}\n\\norm{S_0(t,0,x)}_{\\Theta} < \\varepsilon \\qquad (t \\ge \\tau(\\varepsilon,r))\n\\end{align} \nfor every $x \\in \\overline{B}_r(\\Theta)$ (Definition~2.8 of~\\cite{Mi17}). \nIndeed, this immediately follows from the uniform attractivity~\\eqref{eq:uniform-attractivity-def} of $\\Theta$ for $S_0$ with $B := \\overline{B}_r(\\Theta)$. \n\\smallskip\n\nAs a third step, we can now conclude the desired uniform global asymptotic stability of $\\Theta$ for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$\nfrom Theorem~4.2 of~\\cite{Mi17} and the first two steps.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A local input-to-state stability result}\n\n\n\nIn this section, we establish our local input-to-state stability result for the disturbed reaction-diffusion system~\\eqref{eq:react-diffus-eq}. We begin by showing that the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} has a local Lyapunov function and, for that purpose, we will\nargue in a similar way as~\\cite{Henry} (Theorem~4.2.1). \n\n\n\\begin{lm} \\label{lm:L-fct}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$.\nThen for every $r_0 > 0$ there exists a Lipschitz continuous function $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ with Lipschitz constant $1$ and comparison functions $\\underline{\\psi}, \\overline{\\psi}, \\alpha \\in \\mathcal{K}_{\\infty}$ such that\n\\begin{gather}\n\\underline{\\psi}(\\norm{x}_{\\Theta}) \\le V(x) \\le \\overline{\\psi}(\\norm{x}_{\\Theta}) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)) \\label{eq:V-coercive-Lfct}\\\\\n\\dot{V}_0(x) := \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_0(t,0,x))-V(x) \\big)  \\le -\\alpha(\\norm{x}_{\\Theta}) \\qquad (x \\in B_{r_0}(\\Theta)). \\label{eq:V-Dini-derivative}\n\\end{gather}\n\\end{lm}\n\n\n\n\\begin{proof}\nChoose an arbitrary $r_0 \\in (0,\\infty)$ and fix it for the rest of the proof. Also, choose $\\beta_0 \\in \\mathcal{KL}$ as in Lemma~\\ref{lm:S0-UGAS} and, for every $\\varepsilon > 0$,\nlet $T(\\varepsilon) = T_{r_0}(\\varepsilon)$ be a time such that\n\\begin{align} \\label{eq:Lfct-def-T(eps)}\n\\beta_0(r_0,t) \\le \\varepsilon \\qquad (t \\in [T(\\varepsilon),\\infty)).\n\\end{align} \nSet now, for every given $\\varepsilon > 0$,\n\\begin{align} \\label{eq:Lfct-def-V-eps}\nV^{\\varepsilon}(x) := \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,\\infty)} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta)),\n\\end{align}\nwhere $c_0 \\in (0,\\infty)$ is an arbitrary constant (which is fixed throughout the proof) and $\\eta_{\\varepsilon}(r) := \\max\\{ 0,r-\\varepsilon\\}$ for every $r \\in \\mathbb{R}^+_0$. \nIn view of~\\eqref{eq:S0-UGAS} and~\\eqref{eq:Lfct-def-T(eps)}, the supremum in~\\eqref{eq:Lfct-def-V-eps} for $x \\in \\overline{B}_{r_0}(\\Theta)$ actually extends only over a compact interval, namely\n\\begin{align} \\label{eq:Lfct-V-eps-supremum-only-over-compact-interval}\nV^{\\varepsilon}(x) = \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta)).\n\\end{align}\nIn particular, $V^{\\varepsilon}: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ is a well-defined map (with finite values) and\n\\begin{align} \\label{eq:Lfct-V-eps-le-beta0}\nV^{\\varepsilon}(x) \\le \\e^{-\\lambda T(\\varepsilon)}  \\sup_{t \\in [0,T(\\varepsilon)]} \\Big(  \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\le \\beta_0(\\norm{x}_{\\Theta},0) \n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta))\n\\end{align}\nbecause $\\eta_{\\varepsilon}(r) \\le r$ for all $r \\in \\mathbb{R}^+_0$. \nSince, moreover,\n$|\\eta_{\\varepsilon}(r)-\\eta_{\\varepsilon}(s)| \\le |r-s|$ for all $r,s \\in \\mathbb{R}^+_0$, \nwe see from~\\eqref{eq:Lfct-V-eps-supremum-only-over-compact-interval} and~\\eqref{eq:semiproc-estimate-1} that \n\\begin{align} \\label{eq:Lfct-V-eps-Lipschitz-with-const-1}\n|V^{\\varepsilon}(x) &- V^{\\varepsilon}(y)| \n\\le \n\\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big| \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta}\\big) - \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,y)}_{\\Theta} \\big) \\Big| \\notag \\\\\n&\\le \n\\e^{-\\lambda T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big|  \\norm{S_0(t,0,x)}_{\\Theta} - \\norm{S_0(t,0,y)}_{\\Theta}  \\Big| \\\\\n&\\le \n\\e^{-\\lambda T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\norm{ S_0(t,0,x) - S_0(t,0,y) }\n\\le \\norm{x-y}  \n\\qquad (x,y \\in \\overline{B}_{r_0}(\\Theta)).  \\notag\n\\end{align}\n(In the first inequality above, we used the elementary fact that $|\\sup_{t \\in I} a_t - \\sup_{t\\in I} b_t| \\le \\sup_{t\\in I} |a_t-b_t|$ for arbitrary bounded functions $t \\mapsto a_t, b_t$ on an arbitrary set $I$, and in the third inequality above, we used the elementary fact that $|\\norm{\\xi}_{\\Theta}-\\norm{\\eta}_{\\Theta}| \\le \\norm{\\xi-\\eta}$ for arbitrary $\\xi,\\eta \\in X$.) \nAdditionally, for every $x \\in B_{r_0}(\\Theta)$, we have $S_0(\\tau,0,x) \\in B_{r_0}(\\Theta)$ for $\\tau$ small enough and thus, by~\\eqref{eq:Lfct-def-V-eps} and the semigroup property~\\eqref{eq:S0-sgr},\n\\begin{align*}\nV^{\\varepsilon}(S_0(\\tau,0,x)) = \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,\\infty)} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t+\\tau,0,x)}_{\\Theta} \\big) \\Big) \n\\le \\e^{-c_0 \\tau} V^{\\varepsilon}(x)\n\\end{align*}\nfor every $x \\in B_{r_0}(\\Theta)$ and all sufficiently small times $\\tau$. Consequently,\n\\begin{align} \\label{eq:Lfct-Dini-derivative-V-eps}\n\\dot{V}_0^{\\varepsilon}(x) = \\varlimsup_{\\tau \\to 0+} \\frac{1}{\\tau} \\big( V^{\\varepsilon}(S_0(\\tau,0,x)) - V^{\\varepsilon}(x) \\big) \\le -c_0 V^{\\varepsilon}(x)\n\\qquad (x \\in B_{r_0}(\\Theta)). \n\\end{align}\nWith the help of the auxiliary functions $V^{\\varepsilon}$, we can now construct a function $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ with the desired properties. Indeed, let \n\\begin{align} \\label{eq:Lfct-def-V}\nV(x) := \\sum_{k=1}^{\\infty} 2^{-k} V^{1\/k}(x) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)).\n\\end{align}\nWe then conclude from~\\eqref{eq:Lfct-V-eps-le-beta0}, \\eqref{eq:Lfct-V-eps-Lipschitz-with-const-1}, \\eqref{eq:Lfct-Dini-derivative-V-eps} that\n\\begin{gather}\nV(x) \\le \\beta_0(\\norm{x}_{\\Theta},0) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)), \\label{eq:Lfct-bounded-above-by-beta0}\\\\\n|V(x)-V(y)| \\le \\sum_{k=1}^{\\infty} 2^{-k} |V^{1\/k}(x)-V^{1\/k}(y)| \\le \\norm{x-y}\n\\qquad (x,y \\in \\overline{B}_{r_0}(\\Theta)),  \\label{eq:Lfct-Lipschitz-with-constant-1}\\\\\n\\dot{V}_0(x) \\le \\sum_{k=1}^{\\infty} 2^{-k} \\dot{V}_0^{1\/k}(x) \\le -c_0 V(x)  \\qquad (x \\in B_{r_0}(\\Theta)). \\label{eq:Lfct-Dini-derivative}\n\\end{gather}\nSince $\\sup_{t\\in[0,\\infty)} (\\e^{c_0 t} \\eta_{1\/k}(\\norm{S_0(t,0,x)}_{\\Theta})) \\ge \\eta_{1\/k}(\\norm{x}_{\\Theta})$ for all $x \\in X$, we also conclude from~\\eqref{eq:Lfct-def-V-eps} and~\\eqref{eq:Lfct-def-V} that\n\\begin{align} \\label{eq:Lfct-bounded-below}\nV(x) \\ge \\sum_{k=1}^{\\infty} 2^{-k} \\e^{-(\\lambda + c_0) T(1\/k)} \\eta_{1\/k}(\\norm{x}_{\\Theta}) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta).\n\\end{align}\nIn view of these estimates, we now define the comparison functions $\\overline{\\psi}$, $\\underline{\\psi}$ and $\\alpha$  in the following way:\n\\begin{align*}\n\\overline{\\psi}(r) := \\beta_0(r,0) + r \\qquad \\text{and} \\qquad \\underline{\\psi}(r) := \\sum_{k=1}^{\\infty} 2^{-k} \\e^{-(\\lambda + c_0) T(1\/k)} \\eta_{1\/k}(r)\n\\end{align*}\nand $\\alpha(r) := c_0 \\underline{\\psi}(r)$ for $r \\in \\mathbb{R}^+_0$. It is easy to verify that $\\overline{\\psi}$, $\\underline{\\psi}$ and hence $\\alpha$ belong to $\\mathcal{K}_{\\infty}$. And, moreover, by virtue of~\\eqref{eq:Lfct-bounded-above-by-beta0}, \\eqref{eq:Lfct-Lipschitz-with-constant-1}, \\eqref{eq:Lfct-Dini-derivative}, \\eqref{eq:Lfct-bounded-below}, the desired estimates~\\eqref{eq:V-coercive-Lfct} and~\\eqref{eq:V-Dini-derivative} follow. \n\\end{proof}\n\n\n\nIt should be noticed that the functions $V, \\underline{\\psi}, \\alpha$ constructed in the proof above all depend on the chosen radius $r_0 \\in (0,\\infty)$ because these functions are defined in terms of the times $T(\\varepsilon) = T_{r_0}(\\varepsilon)$ from~\\eqref{eq:Lfct-def-T(eps)}. \nWith the next lemma, we show that the local Lyapunov function $V$ for the undisturbed system is also a local input-to-state Lyapunov function\nfor the disturbed system w.r.t.~$\\Theta$. (See~\\cite{DaMi13} for the definition of local input-to-state Lyapunov functions w.r.t.~an equilibrium point.) \n\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{lm} \\label{lm:lISS-L-fct}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$. Also, let $r_0 > 0$ and let $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}_0^+$ be chosen as in the previous lemma. Then there exist comparison functions $\\alpha, \\sigma \\in \\mathcal{K}$ such that\nfor every $u \\in \\mathcal{U}$\n\\begin{align*}\n\\dot{V}_u(x) := \\varlimsup_{t\\to0+} \\frac{1}{t}\\big( V(S_u(t,0,x))-V(x) \\big) \\le -\\alpha(\\norm{x}_{\\Theta}) + \\sigma(\\norm{u}_{\\infty})\n\\qquad (x \\in B_{r_0}(\\Theta)).\n\\end{align*}\n\\end{lm}\n\n\n\\begin{proof}\nChoose $\\alpha = \\alpha_{r_0} \\in \\mathcal{K}_{\\infty}$ as in Lemma~\\ref{lm:L-fct} and define $\\sigma \\in \\mathcal{K}_{\\infty}$\nby $\\sigma(r) := 2\\e^{2\\lambda} \\norm{h} r$ for all $r \\in \\mathbb{R}^+_0$. \nWe then see from Lemma~\\ref{lm:L-fct} and from \\eqref{eq:semiproc-estimate-2}\nthat for every $x \\in B_{r_0}(\\Theta)$ and every $u \\in \\mathcal{U}$\n\\begin{align}\n\\dot{V}_u(x) \\le \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_0(t,0,x))-V(x) \\big) + \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_u(t,0,x))-V(S_0(t,0,x)) \\big) \\notag \\\\\n\\le -\\alpha(\\norm{x}_{\\Theta}) + \\varlimsup_{t\\to 0+} \\frac{1}{t} \\norm{ S_u(t,0,x) - S_0(t,0,x) } \n\\le -\\alpha(\\norm{x}_{\\Theta}) + \\sigma(\\norm{u}_{\\infty}),\n\\end{align}\nas desired. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWith these lemmas at hand, we can now establish the local input-to-state stability of the disturbed reaction-diffusion system~\\eqref{eq:react-diffus-eq} w.r.t.~the global attractor of the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed}. \nIt is an open question -- left to future research -- whether this result can actually be extended to a semi-global\ninput-to-state stability\nresult. See the remarks after the proof for a discussion of the obstacles to such an extension. \n\n\n\\begin{thm} \\label{thm:lISS}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$. Then the disturbed system $(S_u)_{u\\in\\mathcal{U}}$ is locally input-to-state stable w.r.t.~$\\Theta$, that is, there exist comparison functions $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and radii $r_{0x}, r_{0u} > 0$ such that\n\\begin{align}\n\\norm{S_u(t,0,x_0)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $(x_0,u) \\in X \\times \\mathcal{U}$ with $\\norm{x_0}_{\\Theta} \\le r_{0x}$ and  $\\norm{u}_{\\infty} \\le r_{0u}$.\n\\end{thm}\n\n\n\\begin{proof}\nChoose an arbitrary $r_0 \\in (0,\\infty)$ and fix it for the entire proof. Also,  take $V = V_{r_0}$ and $\\underline{\\psi} = \\underline{\\psi}_{r_0}$, $\\overline{\\psi}$ as in Lemma~\\ref{lm:L-fct}.\nIt then immediately follows from Lemma~\\ref{lm:lISS-L-fct} that there exist comparison functions $\\alpha = \\alpha_{r_0} \\in \\mathcal{K}$ and $\\chi = \\chi_{r_0} \\in \\mathcal{K}$ such that for all $(x_0,u) \\in X \\times \\mathcal{U}$ with $r_0 \\ge \\norm{x_0}_{\\Theta} \\ge \\chi(\\norm{u}_{\\infty})$ one has\n\\begin{align} \\label{eq:lISS-L-fct,implicative}\n\\dot{V}_u(x_0) \\le -\\alpha(\\norm{x_0}_{\\Theta}). \n\\end{align}\n(Simply choose $\\chi(r) := \\alpha_0^{-1}(2\\sigma_0(r))$ and $\\alpha(r) := \\alpha_0(r)\/2$, where $\\alpha_0, \\sigma_0 \\in \\mathcal{K}_{\\infty}$ are as in Lemma~\\ref{lm:lISS-L-fct}.)\nAccording to the comparison lemma from~\\cite{MiIt16} (Corollary~1), we can then choose a comparison function $\\overline{\\beta} = \\overline{\\beta}_{\\alpha\\circ \\overline{\\psi}^{-1}}$ in such a way that for every $T \\in (0,\\infty]$ and every function $v \\in C([0,T),\\mathbb{R}^+_0)$ with\n\\begin{align*}\n\\overline{\\partial}_t^+ v(t) \\le -(\\alpha \\circ \\overline{\\psi}^{-1})(v(t)) \\qquad (t \\in [0,T))\n\\end{align*} \none has $v(t) \\le \\overline{\\beta}(v(0),t)$ for all $t\\in[0,T)$. \nWe now define\n\\begin{align} \\label{eq:def-beta-gamma}\n\\beta(r,t) := \\underline{\\psi}^{-1}\\big( \\overline{\\beta}(\\overline{\\psi}(r),t) \\big) \n\\qquad \\text{and} \\qquad\n\\gamma(r) := \\underline{\\psi}^{-1}\\big( \\overline{\\psi}(\\chi(r)) \\big)\n\\end{align} \nfor $r,t \\in \\mathbb{R}^+_0$ and choose $r_{0x}, r_{0u} \\in (0,\\infty)$ so small that\n\\begin{align} \\label{eq:def-r0x-and-r0u}\nr_{0x} < r_0 \\qquad \\text{and} \\qquad \\beta(r_{0x},0) < r_0 \n\\qquad \\text{and} \\qquad\n\\gamma(r_{0u}) < r_0. \n\\end{align}\nAlso, we will write\n\\begin{align} \\label{eq:M_u-def}\nM_u := \\big\\{ x \\in \\overline{B}_{r_0}(\\Theta): V(x) \\le \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\big\\}\n\\end{align}\nfor $u \\in \\mathcal{U}$.\nClearly, $\\beta \\in \\mathcal{KL}$, $\\gamma \\in \\mathcal{K}$ and $M_u$ is closed for every $u \\in \\mathcal{U}$. Additionally, for every $u \\in \\overline{B}_{r_{0u}}(0)$ we have by~\\eqref{eq:def-r0x-and-r0u} that\n\\begin{align} \\label{eq:M_u-contained-in-better-sets}\nM_u \\subset \\big\\{ x \\in \\overline{B}_{r_0}(\\Theta): \\norm{x}_{\\Theta} \\le \\gamma(\\norm{u}_{\\infty}) \\big\\} \\subset B_{r_0}(\\Theta).\n\\end{align}\n\nAfter these preliminary considerations, we now prove that\n\\begin{align} \\label{eq:lISS-estimate}\n\\norm{S_u(t,0,x_0)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $(x_0,u) \\in \\overline{B}_{r_{0x}}(\\Theta) \\times \\overline{B}_{r_{0u}}(0)$ and thus obtain the desired local input-to-state stability. So, let $(x_0,u) \\in \\overline{B}_{r_{0x}}(\\Theta) \\times \\overline{B}_{r_{0u}}(0)$ be fixed for the rest of the proof.\nWe will distinguish two cases in the following, namely the case where $x_0 \\in M_u$ (part (i) of the proof) and the case where $x_0 \\notin M_u$ (part (ii) of the proof). \n\\smallskip\n\n(i) Suppose we are in the case $x_0 \\in M_u$. In order to establish~\\eqref{eq:lISS-estimate} in that case,\nwe will show -- in two steps -- that for every $t_0 \\in [0,\\infty)$ one has\n\\begin{align} \\label{eq:M_u-invariant}\nS_u(t,t_0,M_u) \\in M_u \\qquad (t \\in [t_0,\\infty)). \n\\end{align}\nSo, let $t_0 \\in [0,\\infty)$ and $x_{t_0} \\in M_u$\nand \n\\begin{align} \\label{eq:def-T-case(i)}\nT := \\sup \\big\\{ T' \\in (t_0,\\infty): \\norm{x(t)}_{\\Theta} < r_0 \\text{ for all } t \\in [t_0,T') \\big\\}, \n\\end{align}\nwhere we use the abbreviation $x(t) := S_u(t,t_0,x_{t_0})$. \nSince $x(t_0) = x_{t_0} \\in M_u$ and thus $\\norm{x(t_0)}_{\\Theta} < r_0$ by~\\eqref{eq:M_u-contained-in-better-sets}, we observe\nthat $T \\in (t_0,\\infty]$ and that\n\\begin{align} \\label{eq:norm-initially-less-than-r0,case(i)}\n\\norm{x(t)}_{\\Theta} < r_0 \\qquad (t \\in [t_0,T)).\n\\end{align}\n\nAs a first step, we show that $x(t) \\in M_u$ at least for all $[t_0,T)$. \nAssuming the contrary, we find a $t \\in [t_0,T)$ and an $\\varepsilon > 0$ such that $V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$. Since $x(t_0) = x_{t_0} \\in M_u$ and thus $V(x(t_0)) \\le \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$,\nwe observe\nthat\n\\begin{align}\nt_1 := \\inf \\big\\{ t \\in [t_0,T): V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon \\big\\}\n\\end{align}\nbelongs to the interval $(t_0,T)$ and, moreover, $V(x(t_1)) = \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$. So,\n\\begin{align*}\n\\overline{\\psi}(\\norm{x(t_1)}_{\\Theta}) \\ge V(x(t_1)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\ge \\overline{\\psi}\\big(\\chi(\\norm{u(\\cdot+t_1)}_{\\infty})\\big)\n\\end{align*} \nand therefore we get by virtue of~\\eqref{eq:lISS-L-fct,implicative} that\n\\begin{align}\n\\varlimsup_{t\\to 0+} \\frac{1}{t}\\Big( V(x(t_1+t))-V(x(t_1)) \\Big) \n&= \\varlimsup_{t\\to 0+} \\frac{1}{t}\\Big( V\\big(S_{u(\\cdot + t_1)}(t,0,x(t_1))\\big)-V(x(t_1)) \\Big)  \\notag \\\\\n&= \\dot{V}_{u(\\cdot + t_1)}(x(t_1)) \\le -\\alpha(\\norm{x(t_1)}_{\\Theta}) < 0.\n\\end{align}\nConsequently, there exists a $\\delta >0$ such that $V(x(t_1+t)) \\le V(x(t_1)) = \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$ for all $t \\in [0,\\delta)$. Contradiction to the definition of $t_1$!\n\\smallskip\n\nAs a second step, we show that $T = \\infty$. Indeed, assuming $T < \\infty$, we would get by the first step and continuity\nthat even $x(T) \\in M_u$ and thus $\\norm{x(T)}_{\\Theta} < r_0$ by~\\eqref{eq:M_u-contained-in-better-sets}. And from this, in turn, it would follow again by continuity that $\\norm{x(t)}_{\\Theta} < r_0$ for all $t \\in [T,T+\\delta)$ with some $\\delta > 0$. In conjunction with~\\eqref{eq:norm-initially-less-than-r0,case(i)}, this would yield a contradiction to the definition~\\eqref{eq:def-T-case(i)} of $T$!\n\\smallskip\n\nCombining now the first and the second step, we finally obtain the desired invariance~\\eqref{eq:M_u-invariant}, which clearly implies~\\eqref{eq:lISS-estimate} in the case $x_0 \\in M_u$.\n\\smallskip\n\n(ii) Suppose we are in the case $x_0 \\notin M_u$. In order to establish~\\eqref{eq:lISS-estimate} in that case,\nwe will show -- in three steps -- that for some $t_0 \\in (0,\\infty]$ one has\n\\begin{align} \n\\norm{S_u(t,0,x_0)}_{\\Theta} &\\le \\beta(\\norm{x_0}_{\\Theta},t) \\qquad (t\\in [0,t_0)) \\label{eq:part-(ii),1} \\\\\n\\norm{S_u(t,0,x_0)}_{\\Theta} &\\le \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in (t_0,\\infty)). \\label{eq:part-(ii),2}\n\\end{align}\nIndeed, let $t_0 := \\inf \\{t \\in \\mathbb{R}^+_0: x(t) \\in M_u\\}$\nand \n\\begin{align} \\label{eq:def-T-case(ii)}\nT := \\sup \\big\\{ T' \\in (0,t_0): \\norm{x(t)}_{\\Theta} < r_0 \\text{ for all } t \\in [0,T') \\big\\},\n\\end{align}\nwhere we use the abbreviation $x(t) := S_u(t,0,x_0)$. (In view of the standard convention $\\inf \\emptyset := \\infty$, we have $t_0 = \\infty$ in case $x(t) \\notin M_u$ for all $t \\in \\mathbb{R}^+_0$.) Since $x(0) = x_0 \\in (X\\setminus M_u) \\cap \\overline{B}_{r_{0x}}(\\Theta)$ and thus $\\norm{x(0)}_{\\Theta} < r_0$ by~\\eqref{eq:def-r0x-and-r0u}, we observe that $t_0 \\in (0,\\infty]$ and $T \\in (0,t_0]$ and that\n\\begin{align} \\label{eq:norm-initially-less-than-r0,case(ii)}\nx(t) \\notin M_u \\qquad (t \\in [0,t_0)) \\qquad \\text{and} \\qquad \\norm{x(t)}_{\\Theta} < r_0 \\qquad (t \\in [0,T)). \n\\end{align} \n\nAs a first step, we show that $\\norm{x(t)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t)$ at least for all $t \\in [0,T)$. \nIndeed, in view of~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.a) and (\\ref{eq:norm-initially-less-than-r0,case(ii)}.b) we have\n\\begin{align*}\n\\overline{\\psi}(\\norm{x(t)}_{\\Theta}) \\ge V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\ge \\overline{\\psi}\\big(\\chi(\\norm{u(\\cdot+t)}_{\\infty})\\big)\n\\qquad (t \\in [0,T))\n\\end{align*}\nand therefore we get by virtue of~\\eqref{eq:lISS-L-fct,implicative} that\n\\begin{align}\n\\overline{\\partial}_t^+ V(x(t)) &= \\varlimsup_{\\tau\\to 0+} \\frac{1}{\\tau}\\Big( V(x(t+\\tau))-V(x(t)) \\Big) \\notag \\\\\n&= \\varlimsup_{\\tau\\to 0+} \\frac{1}{\\tau}\\Big( V\\big(S_{u(\\cdot + t)}(\\tau,0,x(t))\\big)-V(x(t)) \\Big) = \\dot{V}_{u(\\cdot + t)}(x(t)) \\notag \\\\\n&\\le -\\alpha(\\norm{x(t)}_{\\Theta}) \\le -\\big(\\alpha \\circ \\overline{\\psi}^{-1}\\big)\\big(V(x(t))\\big)\n\\qquad (t \\in [0,T)).\n\\end{align}\nConsequently, by our choice of $\\overline{\\beta}$ we see that\n\\begin{align*}\nV(x(t)) \\le \\overline{\\beta}(V(x(0)),t) \\qquad (t \\in [0,T)).\n\\end{align*}\nIn view of~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.b) and our definition~\\eqref{eq:def-beta-gamma} of $\\beta$, the assertion of the first step is then clear. \n\\smallskip\n\nAs a second step, we show that $T = t_0$. \nIndeed, assuming $T < t_0$, we would get by the first step and continuity that even $\\norm{x(T)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},T) \\le \\beta(r_{0x},0)$ and thus $\\norm{x(T)}_{\\Theta} < r_0$ by~\\eqref{eq:def-r0x-and-r0u}. And from this, in turn, it would follow that $\\norm{x(t)}_{\\Theta} < r_0$ for all $t \\in [T,T+\\delta)$ with some $\\delta > 0$. In conjunction with~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.b), this would yield a contradiction to the definition~\\eqref{eq:def-T-case(ii)} of $T$!\n\\smallskip\n\nAs a third step, we show that $\\norm{x(t)}_{\\Theta} \\le \\gamma(\\norm{u}_{\\infty})$ for all $t \\in [t_0,\\infty)$. \nWe can assume $t_0 < \\infty$\nbecause in the case $t_0 = \\infty$ the assertion is empty. So, by\nthe definition of $t_0$ it then follows that $x(t_0) \\in M_u$ and therefore by virtue of~\\eqref{eq:M_u-invariant} \n\\begin{align*}\nx(t) = S_u(t,0,x_0) = S_u(t,t_0,x(t_0)) \\in M_u \\qquad (t\\in[t_0,\\infty)).\n\\end{align*} \nIn view of~\\eqref{eq:M_u-contained-in-better-sets}, the assertion of the third step is then clear.\n\\smallskip\n\nCombining now the first, second and third step, we finally obtain the desired estimates~\\eqref{eq:part-(ii),1} and~\\eqref{eq:part-(ii),2}, which clearly imply~\\eqref{eq:lISS-estimate} in the case $x_0 \\notin M_u$.\n\\end{proof}\n\n\n\n\n\n\n\nAn inspection of the above proof shows that we actually proved a bit more than local input-to-state stability, namely we have: for every $r_0 > 0$ there exist $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and $r_{0x}, r_{0u} > 0$ such that the estimate~\\eqref{eq:lISS-estimate} holds true for all $\\norm{x}_{\\Theta} \\le r_{0x}$ and $\\norm{u}_{\\infty} \\le r_{0u}$. So, if by choosing $r_0$ large enough, we could also ensure that $r_{0x}$ and $r_{0u}$ with~\\eqref{eq:def-r0x-and-r0u} can be chosen arbitrarily large, we would even have semi-global input-to-state stability.\nYet, this is not so clear because the functions $\\beta = \\beta_{r_0}$ and $\\gamma = \\gamma_{r_0}$ from~\\eqref{eq:def-r0x-and-r0u} which determine our choice of $r_{0x}$ and $r_{0u}$ depend on $r_0$ themselves (basically because $V = V_{r_0}$ and $\\underline{\\psi} = \\underline{\\psi}_{r_0}$ depend on $r_0$ as was pointed out after Lemma~\\ref{lm:L-fct}).\nWe therefore leave the question of semi-global input-to-state stability to future research. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nS. Dashkovskiy and O. Kapustyan are partially supported by the German Research Foundation (DFG) and the State Fund for Fundamental Research of Ukraine (SFFRU) through the joint German-Ukrainian grant ``Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances'' (DA 767\/12-1).\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{small}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe consider the existence and properties of radially symmetric weak solutions to the following system of differential equations:\n\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial F} {\\partial t}(t, p)=n(t) I_3(F(t))(p)\\qquad t>0,\\; p\\in \\mathbb{R}^3, \\label{PA}\\\\\n&n'(t)=-n(t)\\int _{ \\mathbb{R}^3 } I_3(F(t))(p)dp \\qquad t>0,\\label{PB}\n\\end{empheq}\nwhere\n\\begin{align}\n&I_3(F(t))(p)=\\!\\!\\iint _{(\\mathbb{R}^3)^2}\\!\\!\\big[R(p, p_1, p_2)\\!-\\!R(p_1, p, p_2)\\!-\\!R(p_2, p_1, p) \\big]dp_1dp_2, \\label{E1BCD} \\\\\n&R(p, p_1, p_2)\\,=\\left[\\delta (|p|^2-|p_1|^2-|p_2|^2)  \\delta (p-p_1-p_2)\\right]\\,\\times \\nonumber \\\\\n&\\hskip 2.5cm \\times\\left[ F_1F_2(1+F)-(1+F_1)(1+F_2)F \\right],  \\label{S1EA4BC}\n\\end{align}\nand we denote $F=F(t,p)$ and $F_{\\ell}=F(t,p_{\\ell})$ for $\\ell=1, 2$.\n\nThe system (\\ref{PA}), (\\ref{PB}) is motivated by the mathematical description of a weakly interacting dilute gas of bosons. Given such a gas at equilibrium, if its temperature is below the so-called critical temperature $T_c$, a macroscopic density of bosons, called a condensate, appears at the lowest quantum state (cf.\\cite{lieb}). A description of the system of particles out of equilibrium at zero temperature has also been rigorously obtained  (\\cite{ESY}). The system (\\ref{PA}), (\\ref{PB})  is more directly related to a gas out of equilibrium and at non zero temperature. The equations that, in the physic's literature, describe a gas in such a situation have not been the object of a mathematical proof; they have rather been deduced on the basis of physical arguments  (cf. \\cite{Zoller}, \\cite{GNZB, ZNG}, \\cite{Stoof2} for example). \nWe are particularly interested in the kinetic description of the interaction between the condensate and the particles in the dilute gas, when most of the particles are still in the gas, and so when the system is at a temperature close to  $T_c$.  \n\n\\subsection{The Nordheim equation}\n\nThe kinetic equation consistently used to describe the evolution of the  distribution function for a spatially homogeneous, weakly interacting dilute gas of bosons of momentum $p_1$ is\n\\begin{align}\n&\\frac{\\partial F}{\\partial t}(t, p_1)=I_4(F(t))(p_1)\\qquad  t>0,\\; p_1\\in \\mathbb{R}^3, \\label{S1E0a}\n\\end{align}\nwhere\n\\begin{align}\n&I_4(F(t))(p_1)=\\iiint_{\\left(\\mathbb{R}^{3}\\right)  ^{3}}q(F)  d\\nu (p_2, p_3, p_4),  \\label{S1Eb}\\\\\n&q(F)= F_{3}F_{4}( 1+ F_{1})(1+F_{2}) -F_{1}F_{2}(1+F_{3})(1+F_{4}), \\label{S1Eq}\\\\\n&d\\nu (p_1, p_2, p_3)=2a^2 \\pi ^{-3} \\delta\\left(  p_{1}+p_{2}-p_{3}-p_{4}\\right)\\times \\nonumber \\\\\n&\\hskip 2.5cm \\delta \\left(  E(p_1)+ E(p_2)- E(p_3)- E(p_4)\\right) dp_{2}dp_{3}dp_{4}.\n\\end{align}\nsometimes called Nordheim equation (\\cite{Nordheim1}), (cf. for example  \\cite{Zoller}, \\cite{GNZB}, \\cite{Stoof2}). We are assuming that the particles have mass $m=1\/2$ and $E(p)$ denotes  the energy of a particle of momentum $p$.  The constant $a$ is the scattering length that parametrizes the Fermi pseudopotential  of scattering. In the absence of condensate, the energy of the particles is taken to be $E(p)=|p|^2$.  \n\nFor a condensed Bose gas, it is necessary to include  the collisions involving the condensate. A kinetic equation is derived in \\cite{Eckern} and \\cite{Kirkpatrick} describing such processes.  More recently, \\cite{ZNG}  extended the treatment to a trapped Bose gas by including Hartree-Fock\ncorrections to the energy of the excitations, and  have derived coupled kinetic equations for the\ndistribution functions of the normal and superfluid components. Later on the results where generalized  to low temperatures in \\cite{IG}  using  the Bogoliubov-Popov approximation to describe the energy particle. The system is as follows\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial F} {\\partial t}(t, p)=I_4(F(t))(p)+ 32 a^2 n(t) \\widetilde I_3(F(t))(p)\\quad t>0,\\; p\\in \\mathbb{R}^3,  \\label{S1EA1}\\\\\n& n'(t)=-n(t)\\int  _{ \\mathbb{R}^3 } \\widetilde I_3(F(t))(p)dp\\qquad t>0. \\label{S1EA1B}\n\\end{empheq}\n(cf. \\cite{Eckern}, \\cite{GNZB}, \\cite{Kirkpatrick} for a deduction based on physic's arguments). The term $I_4(F)$ is exactly as in (\\ref{S1Eb}) and   the constant $32 a^2$ comes from the approximation of  the transition probability:\n$|\\mathcal M(p, p_1, p_2)|^2\\approx 32 a^2 n(t)$. The integral collision $\\widetilde I_3$ is given by an expression similar to (\\ref{E1BCD}), (\\ref{S1EA4BC})\n but where the corresponding terms $\\widetilde R(p, p_1, p_2)$ are as follows,\n \\begin{align}\n&\\widetilde R(p, p_1, p_2)\\,=\\left[\\delta (E(p)-E(p_1)-E(p_2))  \\delta (p-p_1-p_2)\\right]\\,\\times \\nonumber \\\\\n&\\hskip 1cm \\times\\left[ F(p_1)F(p_2)(1+F(p))-(1+F(p_1))(1+F(p_2))F(p) \\right].   \\label{S1EBog}\n\\end{align}\nIn presence of a condensate, the energy $E(t, p)$ of the particles at time $t$ is now taken as $E(t, p)=\\sqrt{|p|^4+16a\\,n(t)|p|^2}$,  where $n(t)$ is the condensate density (\\cite{Chiara}, \\cite{GNZB}). \nOnce equation (\\ref{S1EA1}) has been obtained, the equation (\\ref{S1EA1B}) is just what is needed in order to ensure that the total number of particles $n(t)+\\int  _{ \\mathbb{R}^3 }F(t, p)dp$ in the system is constant in time.  \n\nWe are particularly interested in a situation where most of the particles are in the gas, and the condensate density $n$ is very small. The energy of the particles is then usually approximated as $E(t, p)\\approx |p|^2+4 a\\pi n(t)$ (cf.\\cite{GNZB}).\nIn all what follows we need  the strongest simplification  $E(t, p)\\approx |p|^2$ to have the collision integral $I_3$ in (\\ref{E1BCD}).\n \nMoreover, in the problem (\\ref{PA}), (\\ref{PB})  only the term that in the equation  (\\ref{S1EA1})  describes the interactions involving one particle of the condensate has been kept. The term $I_4$, the same as in equation (\\ref{S1E0a}), that only considers interactions between particles in the gas, has been dropped. The term $I_4$ has been studied with detail to prove the existence of solutions to the Nordheim equation (\\ref{S1E0a}) and describe some of their properties. The problem (\\ref{PA}), (\\ref{PB}) only takes into account the collision processes involving a particle of the condensate. \n\nSince we are only concerned with radial solutions $(F,n)$ of (\\ref{PA}), (\\ref{PB}), a very natural independent variable is  $x=|p|^2$. But  this introduces a jacobian and then, the most suitable quantity is not always $f(x)=F(p)$ but may be sometimes  $\\sqrt x f(x)$.\n\n\n\\subsection{The term $I_4$ and the Nordheim equation}\n\nThe local existence of bounded solutions for Nordheim equation (\\ref{S1E0a}) was proved in \\cite{BE}. Global existence of bounded solutions has been proved in \\cite{Lu5} for bounded and suitably small initial data. The existence of radially symmetric weak solutions was first proved in \\cite{Lu1} for all initial data $f_0$ in the space of nonnegative radially symmetric measures on $[0, \\infty)$. \n\nFor radially symmetric solutions $F(p)=f(x)$, $x=|p|^2$, the expression of the Nordheim equation simplifies because it is possible to perform the angular variables in the collision integral. After rescaling the time variable $t$ (in order to absorb some constants), the Nordheim equation reads:\n\n\\begin{align}\n&\\frac{\\partial f}{\\partial t}(t, x_1)= J_4(f(t))(x_1),\\qquad t>0,\\;x_1\\geq0, \\label{S1E1}\n\\end{align}\nwhere\n\\begin{align}\n&J_4(f)(x_1)=\\iint _{[0,\\infty)^2}\\hskip -0.4cm  \\frac{w(x_1, x_2, x_3)}{\\sqrt{x_1}} q(f)(x_1,x_2,x_3)dx_{2}dx_{3}, \\label{S1E2}\\\\ \n&q(f)=(1+  f_{1})(1+f_{2})  f_{3}f_{4}-( 1+ f_{3})(1+f_{4})  f_{1}f_{2},  \\label{S1E3}\\\\\n&w(x_1,x_2,x_3)=\\min\\left\\{\\sqrt{x_{1}},\\sqrt{x_{2}},\\sqrt{x_{3}},\\sqrt{x_{4}}\\right\\},\\,\\,x_{4}=(x_{1}+x_{2}-x_{3})_+. \\label{E2'}\n\\end{align}\n\nThe factor $\\frac{w}{\\sqrt{x_1}}$ in the collision integral comes from the angular integration of the Dirac's delta of the energies $|p _{ \\ell }|^2$.\n\nIf we denote $\\mathscr M _+([0, \\infty))$ the space of positive and finite Radon measures on $[0, \\infty)$, and define for all $\\alpha \\in \\mathbb{R}$\n\\begin{align}\n\\label{S1E17}\n&\\mathscr{M}^{\\alpha} _+([0, \\infty))=\\left\\{G\\in  \\mathscr{M}_+([0, \\infty)):\\; M_{\\alpha}(G)<\\infty\\right\\},\\\\\n&M_{\\alpha}(G)=\\int_{[0,\\infty)}x^{\\alpha}G(x)dx\\qquad\\text{(moment of order $\\alpha$)},\\label{S1E17'}\n\\end{align}\nthe definition of weak solution introduced in \\cite{Lu1} is the following.\n\n\\begin{definition}[Weak radial solutions of (\\ref{S1E0a})]\n\\label{S1D0}\nLet $G$ be a  map from $[0, \\infty)$ into $\\mathscr M_+^1([0, \\infty))$ and consider $f$ defined as $\\sqrt xf(t)=G(t)$. We say that $f$ is a weak radial solution of (\\ref{S1E0a}) if $G$ satisfies:\n\\begin{align}\n&\\forall t>0:\\,\\,\\,G(t)\\in \\mathscr M_+^1([0, \\infty)), \\label{S1ED1}\\\\\n&\\forall  T>0:\\,\\,\\,\\sup _{ 0\\le t<T } \\int  _{ [0, \\infty) }(1+x)G(t, x)dx<\\infty, \\label{S1ED2} \\\\\n& \\forall \\,\\varphi \\in C^{1, 1}_b([0, \\infty)): \\,\\, \\int  _{ [0, \\infty) } \\varphi (x)G(t , x) dx\\in C^1([0, \\infty)),  \\label{S1ED3} \n\\\\\n&\\frac {d} {dt} \\int_{\\left[  0,\\infty\\right)}\\varphi(t,x)G(t,x)dx =\\mathcal{Q}_4(\\varphi,G(t)), \\label{S1E4}\\\\\n&\\mathcal Q_4(\\varphi,G)=\\iiint_{\\left[  0,\\infty\\right)^3} \\frac{G_{1}G_{2}G_{3}\n}{\\sqrt{x_{1}x_{2}x_{3}}}w\\Delta\\varphi \\;dx_1dx_2dx_3+ \\nonumber \\\\\n&\\hskip 2cm +\\frac{1}{2}\\iiint_{\\left[  0,\\infty\\right)^3} \\frac{G_{1}G_{2}}{\\sqrt{x_{1}x_{2}}}w \\Delta\\varphi \\;dx_1dx_2dx_3  \\label{S1E5}\\\\\n&\\Delta\\varphi (x_1, x_2, x_3)=\\varphi (x_4)+\\varphi (x_3)-\\varphi (x_2)-\\varphi (x_1),\\label{S2E1}\\\\\n&w(x_1,x_2,x_3)=\\min\\{\\sqrt{x_1}, \\sqrt{x_2}, \\sqrt{x_3}, \\sqrt{x_4}\\},\\,\\,\\,x_4=(x_1+x_2-x_3)_+. \\label{S1E6'}\n\\end{align}\n\\end{definition}\n\nFor all initial data $f_0$ such that $G_0=\\sqrt x f_0\\in \\mathscr M_+^1([0, \\infty))$, the existence of a weak solution was proved in \\cite{Lu1}.\nThe moments of order zero and one of $G$ where shown to be constant in time. \nIt was shown in \\cite{Lu3} that  a definition equivalent to Definition \\ref{S1D0} would be to impose $\\varphi (0)=0$ to the test functions in Definition \\ref{S1D0} and impose the conservation of mass on $G(t)$ for all $t>0$. Further properties of the solutions, such as the gain of moments, asymptotic behavior,   where obtained in a series of articles  \\cite{Lu1,Lu2, Lu3, Lu4}\n\nIt is proved in Proposition \\ref{S1P0} below that if the measure $G$ is written as $G(t)=n(t)\\delta _0+g(t)$, where\n$n(t)=G(t,\\{0\\})$, then for all $\\varphi \\in C^{1,1}_b([0, \\infty))$ the term $\\mathcal Q_4(\\varphi,G)$ may be decomposed as follows:\n\\begin{align}\n\\mathcal Q_4(\\varphi,G(t))=\\mathscr Q _4(\\varphi,g(t))+n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\label{S1E13B}\n\\end{align}\nwhere\n\\begin{align}\n&\\mathscr Q_4(\\varphi,g)=\\iiint_{(0,\\infty)^3}\\frac{g_{1}g_{2}g_{3}\n}{\\sqrt{x_{1}x_{2}x_{3}}}w \\Delta\\varphi \\;dx_1dx_2dx_3 \\nonumber \\\\\n&\\hskip 2cm +\\frac{1}{2}\\iiint_{(0,\\infty)^3}\\frac{g_{1}g_{2}}{\\sqrt{x_{1}x_{2}}}w \\Delta\\varphi \\;dx_1dx_2dx_3, \\label{S1E15}\\\\\n&\\mathscr Q _3(\\varphi,g)=\\mathscr{Q}_3^{(2)}(\\varphi,g)-\\mathscr{Q}_3^{(1)}(\\varphi,g),\\label{S1E1Q3}\\\\\n&\\mathscr{Q}_3^{(2)}(\\varphi,g)=\\iint_{(0, \\infty)^2} \\frac {\\Lambda(\\varphi)(x, y)} {\\sqrt{x y}}g(x)g(y)dxdy,\\label{S1E1Q32}\\\\\n&\\mathscr{Q}_3^{(1)}(\\varphi,g)=\\int_{ (0, \\infty)}\\frac {\\mathcal{L}_0(\\varphi)(x)} {\\sqrt x}g(x)dx, \\label{S1E1Q31}\\\\\n&\\Lambda(\\varphi)(x, y)=\\varphi (x+y)+\\varphi (|x-y|)-2\\varphi (\\max\\{x, y\\}), \\label{S1E154}\\\\\n&\\mathcal {L}_0(\\varphi )(x)=x\\big(\\varphi (0)+\\varphi (x)\\big)-2\\int _0^x \\varphi (y)dy. \\label{S1E155}\n\\end{align}\n\n\nIt was also proved in \\cite{Lu1} that as $t\\to \\infty$, the measure $G$ converges in the weak sense of measures  to one of the measures:\n\\begin{eqnarray}\n\\label{BED}\nG _{ \\beta, \\mu, C} =\\frac {\\sqrt x} {e^{\\beta x-\\mu  }-1}+C\\delta _0,\\,\\,\\,\\beta >0,\\,\\,\\mu  \\le 0, \\,\\,\\,C\\ge 0\n\\end{eqnarray}\nwhere the constants $C$ and $\\mu  $ are such that $C\\mu  =0$.  \n\nWhen $C=0$ and $\\mu \\le 0$, the function  $F_{ \\beta , \\mu , 0  }(p)=|p|^{-1} G _{ \\beta , \\mu , 0  }(|p|^2)$ is an equilibrium of the Nordheim equation (\\ref{S1E0a}) because\n$q(F _{ \\beta , \\mu , 0 })d\\nu \\equiv 0$. When $C>0$ and $\\mu =0$, then $F _{ \\beta , 0, C  }(p)=|p|^{-1} G _{ \\beta , 0, C  }(|p|^2)$ is an equilibria of (\\ref{S1EA1}) because\n$q(f _{ \\beta , 0 , 0 }) \\equiv 0$ and  $R(p, p', p'')\\equiv 0$ for all $(p, p', p'')\\in (\\mathbb{R}^3)^3$ for  $f _{ \\beta , 0 , 0} $, where  $R(p, p', p'')$ is defined in  (\\ref{S1EA4BC}).  It was proved in \\cite{Lu1} that $F_{ \\beta , \\mu, C  }$ is a weak solution of the Nordheim equation  (\\ref{S1E1}) if and only if $\\mu C=0$. \n\nOn the other hand, it was proved in \\cite{EV1} that, given any  $N>0$, $E>0$ there exists initial data $f_{0}\\in L^{\\infty}\\left( \n\\mathbb{R}_+;\\left( 1+x\\right) ^{\\gamma}\\right) $ with $\\gamma >3$, satisfying\n$$\n\\int_{\\mathbb{R}^{+}}f_{0}(x)\\sqrt{x }dx=N,\\qquad\\int_{\\mathbb{R}^{+}}f_{0}(x)\\sqrt{x^{3}}dx=E,\n$$\nand such that there exists a global weak solution $f$  and  positive times  $0<T_{\\ast}<T^*$ such that:\n\\begin{eqnarray}\n\\label{S1EC1}\n\\sup_{0< t\\leq T_{\\ast}}\\left\\Vert f\\left( t,\\cdot\\right) \\right\\Vert\n_{L^{\\infty}\\left( \\mathbb{R}^{+}\\right) }<\\infty,\\,\\,\\,\\,\\,\\sup _{T_*<t\\le T^* }\\int_{\\left\\{ 0\\right\\} }\\sqrt{x}f\\left( t,x\\right)\ndx>0.\n\\end{eqnarray}\nProperty (\\ref{S1EC1}) shows that the  solution $G=\\sqrt x f$ of  (\\ref{S1ED1})--(\\ref{S1E6'}) is a bounded function on the time interval $[0, T^*)$ and a Dirac mass is formed at the origin at some time $T_0$ between $T_*$ and $T^*$. After that time $T_0$, the solution $G$ is such that $G(t, \\{0\\})>0$. \n\nIn the simplified description of the physical system of particles that we are using, where only the radial density  $G$ of particles of momentum $p$  is considered, the description of the physical Bose-Einstein condensate can just be given by a  Dirac measure at the origin. \n\nNotwithstanding the similarity of these two phenomena, the extent to which the first one is a truthful mathematical description of the second is not clear.  Nevertheless, we refer to the term $n(t)\\delta _0$  that appears in finite time in some of the weak solutions of the Nordheim equation as ``condensate'', with some abuse of language.\n\n\\subsection{The term $I_3$ in radial variables.}\n\nThe results briefly presented in the previous sub Section describe some of the properties of the weak solutions to the Nordheim equation in terms of the measure $G$.  In particular, the weak convergence of $G$ to the measures defined in (\\ref{BED}) shows what is the limit of  $G(t, \\{0\\})$ as $t\\to \\infty$.   \nTo understand better  the dynamics of $G(t, \\{0\\})\\delta _0$ and its interaction with $G(t)-G(t, \\{0\\})\\delta _0$ it seems suitable to  write  $G(t)=G(t, \\{0\\})\\delta _0+g(t)$ and consider the system (\\ref{S1EA1}), (\\ref{S1EA1B}).  \n\nFor radially symmetric functions $F(p)=f(x)$, $x=|p^2|$, the  system (\\ref{PA}), (\\ref{PB}) reads, after a suitable time rescaling to absorb some constants:\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial f} {\\partial t}(t, x)=\\frac {n(t)} {\\sqrt x} J_3(f(t))(x)\\qquad t>0,\\;x>0, \\label{PR}\\\\\n&n'(t)=-n(t)\\int _0^\\infty J_3(f(t))(x)dx\\qquad t>0, \\label{PR19}\n\\end{empheq}  \nwhere\n\\begin{align}\n&J_3(f)(x)=\\int_0^x \\Big(f(x-y)f(y)-f(x)\\big[1+f(x-y)+f(y)\\big]\\Big)dy +\\nonumber\\\\\n&\\hskip 1cm +2\\int_x^\\infty \\Big(f(y)\\big[1+f(y-x)+f(x)\\big]-f(y-x) f(x)\\Big) dy.\\label{PR2}\n\\end{align}\n(cf. \\cite{ST1} and \\cite{Svis} for the isotropic system that also contains the term $J_4(f)$, that comes from $I_4$ in  (\\ref{S1EA1})).\nNotice that\n\\begin{align}\n&\\int _0^\\infty J_3(f(t))(x)dx\\nonumber\\\\\n&=\\int _0^\\infty\\int _0^\\infty \\Big(f(t, x)f(t, y)- f(t, x+y)\\big[1+f(t, x)+f(t, y)\\big]\\Big)dxdy \\label{S1E9}\n\\end{align}\nwhenever the integral in the right hand side is finite, for example, if\n$f\\in L^1\\big(\\mathbb{R}_+, (1+x)dx\\big)$. In that case we also have,\n\\begin{equation}\n\\int _0^\\infty J_3(f(t))(x)dx=M _1(f(t)).\n\\end{equation}\n\nThe factor $x^{-1\/2}$ in the right hand side  of (\\ref{PR}) comes from the angular integration of the Dirac's measure of energies of $I_3$, just as the \n$\\frac{w}{\\sqrt{x_1}}$ term of (\\ref{S1E2}) in $I_4$.  But since $\\frac{w}{\\sqrt{x_1}}$ is a bounded function,\nit appears that the operator $I_3$ is more singular than $I_4$ for small values of $x$. \n\nIf we denote  $F(t, p)=f(t, |p|^2)=|p|^{-1}g(t, |p|^2)$ and $x=|p^2|$, from the original motivation of the Nordheim equation  it is very natural to expect\n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)dp= 2\\pi\\int _0^\\infty f(t, x)\\sqrt x dx= 2\\pi\\int_0^\\infty g(t, x)dx<\\infty, \n$$\n(that corresponds to the  number of particles in the normal fluid), and\n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)|p|^2dp= 2\\pi\\int _0^\\infty f(t, x)x^{3\/2}dx=2\\pi\\int_0^\\infty g(t, x)xdx<\\infty,\n$$\n(corresponding to the total energy in the system).  But there is no particular reason to expect \n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)\\frac {dp} {|p|}= 2\\pi\\int _0^\\infty f(t, x)dx=2\\pi\\int_0^\\infty g(t, x)\\frac {dx} {\\sqrt x}<\\infty.\n$$\nWithout that last condition, the convergence of the integrals in the term $I_3(F(t))$ (cf. (\\ref{E1BCD}), (\\ref{S1EA4BC})), or in (\\ref{PR}), (\\ref{PR2}), is delicate.\nThat difficulty is usually avoided using a suitable weak formulation. \n\nIf we suppose that  $f=x^{-1\/2}g\\in L^1\\big(\\mathbb{R}_+, (1+x)dx\\big)$, and multiply the equation (\\ref{PR}) by $\\sqrt x\\, \\varphi $, we obtain by Fubini's Theorem, \n\\begin{equation}\n\\label{g11}\n\\frac {d} {dt}\\int_{ [0, \\infty) } \\varphi (x) g(t, x)dx=n(t)\\widetilde{\\mathscr Q}_3(\\varphi,g(t))\\quad\\forall  \\varphi \\in C^1_b([0, \\infty)),\n\\end{equation}\nwhere\n\\begin{align}\n&\\widetilde{\\mathscr Q}_3(\\varphi,g)=\\mathscr{Q}_3^{(2)}(\\varphi,g)-\\widetilde{\\mathscr Q}_3^{(1)}(\\varphi,g),\\label{S1EB2}\\\\\n&\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,g)=\\int_{ (0, \\infty)} \\frac {\\mathcal{L}(\\varphi )(x)} {\\sqrt x}g(x)dx, \\label{S1E20R}\\\\\n&\\mathcal{L}(\\varphi)(x)=x\\varphi (x)-2\\int _0^x \\varphi (y)dy. \\label{S1E21R}\n\\end{align}\nNotice that, by (\\ref{S1E1Q3}),\n\\begin{align}\n\\mathscr{Q}_3(\\varphi,g)=\\widetilde{\\mathscr{Q}}_3(\\varphi,g)-\\varphi(0)M_{1\/2}(g).\\label{S1EB1}\n\\end{align}\n\nA natural weak formulation for $G=n(t)\\delta _0+g$ is then obtained by adding (\\ref{PR19}) to (\\ref{g11}). \nWe  then define a weak radially symmetric solution of the Problem (\\ref{PA}), (\\ref{PB}) as follows.\n\\begin{definition}[Weak radial solution of (\\ref{PA}), (\\ref{PB})]\n\\label{S1D1}\nConsider a map $G:[0, T) \\to \\mathscr M_+^1([0, \\infty))$  for some $T\\in (0, \\infty]$, that we decompose as follows:\n$$\n\\forall t\\in [0, T):\\,\\,\\,\\,\\,G(t)=n(t)\\delta _0+g(t), \\,\\,\\,\\, \\hbox{where}\\,\\,\\,\\, n(t)=G(t, \\{0\\});\n$$\nand  define  $ F(t, p)=|p|^{-1}g(t, |p|^2)$ for all $t>0$ and $p\\in \\mathbb{R}^3$. We say that $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}) on $(0, T)$ if:\n\n\\begin{align}\n&\\forall T'\\in (0, T]:\\qquad\\sup _{ 0\\le t<T' } \\int  _{ [0, \\infty) }(1+x)G(t, x)dx<\\infty, \\label{S1ED2S} \\\\\n& \\forall \\varphi \\in C^{1}_b([0, \\infty)): \\quad t\\mapsto \\int  _{ [0, \\infty) } \\varphi (x)G(t, x) dx \\in W^{1,\\infty } _{ loc }([0, T)),  \\label{S1ED3S}\n\\end{align}\nand for a.e. $t\\in (0, T)$\n\\begin{equation}\n\\frac {d} {dt}\\int_{ [0, \\infty) } \\varphi (x) G(t, x)dx=n(t) \\mathscr{Q}_3(\\varphi,g(t))\\quad\\forall  \\varphi \\in C^1_b([0, \\infty)), \\label{S1E16}\\\\\n\\end{equation}\nwhere $\\mathscr Q _3(\\varphi,g)$ is defined by (\\ref{S1E1Q3})-(\\ref{S1E1Q31}).\n\\end{definition}\n\nWe  show in Proposition \\ref{S1P0} that the Definition \\ref{S1D1} substantially  coincides with the Definition \\ref{S1D0} of radial weak solution of (\\ref{S1E0a}) when the term $\\mathscr Q_4(\\varphi,g)$ in (\\ref{S1E13}) is dropped (cf. Remark \\ref{SXR1}). As a  consequence, the measures $f _{ \\beta , 0, C  }(p)$\ndefined above are weak radial solutions of (\\ref{PA}), (\\ref{PB}) (cf. Proposition \\ref{equilibria}).\n\n\n\n\\subsection{Main results}\nThe existence of weak radial solutions for the Cauchy problem associated with the system (\\ref{PA}), (\\ref{PB}) is given in the following Theorem.\n\n\\begin{theorem}[Existence result]\n\\label{S1T1}\nSuppose that $G_0\\in \\mathscr{M}^1_+([0,\\infty))$ satisfies $G_0(\\{0\\})>0$, and define $F_0(p)=|p|^{-1}g_0(|p|^2)$, where $g_0=G_0-G_0(\\{0\\})\\delta _0$. Then, there exists  a weak radial solution $(F, n)$ of  (\\ref{PA}), (\\ref{PB}) on $(0, \\infty)$ such that $F(t, p)=|p|^{-1}g(t, |p|^2)$, where $G=n\\delta _0+g$  satisfies:\n\\begin{eqnarray}\n\\label{S1T1E0}\nG\\in C\\big([0,\\infty),\\mathscr{M}_+^1([0,\\infty))\\big),\\quad G(0)=G_0\n\\end{eqnarray} \nand:\n\\begin{enumerate}[(i)]\n\\item$G$ conserves the total number of particles $N$ and energy $E$:\n\\begin{align}\n&M_0(G(t))=M_0(G_0)=N\\qquad\\forall t\\geq 0,\\label{S1E210}\\\\\n&M_1(G(t))=M_1(G_0)=E\\qquad\\forall t\\geq 0.\\label{S1E220}\n\\end{align}\n\n\\item For all $\\alpha \\geq 3$,  if $M_\\alpha (G_0)<\\infty$, then $G\\in C \\big((0, \\infty), \\mathscr{M}_+^{\\alpha}([0,\\infty))\\big)$ and \n\\begin{flalign}\n\\label{S1E23}\n&M_{\\alpha}(G(t))\\le \\left(M_{\\alpha}(G_0)^{\\frac{2}{\\alpha-1}}\n+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau(t)\\right)^{\\frac{\\alpha-1}{2}}\\quad\\forall t>0,&\\\\\n&\\text{where}\\quad\\tau (t)=\\int _0^t G(s, \\{0\\})ds.& \\label{E657tau}\n\\end{flalign}\n\n\\item For all $\\alpha \\geq3$,\n\\begin{equation}\n\\label{S5Ealphahh}\n\\quad M_{\\alpha}(G(t))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau(t)}}\\right)^{2(\\alpha-1)}\\quad\\forall t >0,\n\\end{equation}\nwhere $\\tau(t)$ is given by (\\ref{E657tau}), and the constants $C(\\alpha,E)$ and $\\gamma (\\alpha,E)$ are defined in Theorem \\ref{S5T5R}.\n\n\\item If $\\alpha \\in (1, 3]$ and\n\\begin{align}\nE> C(\\alpha)N^{5\/3},\\label{PRO112}\n\\end{align}\n\\begin{flalign}\n&\\text{where}\\qquad\n\\label{PRO115}\nC(\\alpha)=\n\\begin{cases}\n\\Big(\\frac{(2^{\\alpha}-2)(\\alpha+1)}{(\\alpha-1)}\\Big)^{\\frac{2}{3}}&\\text{if}\\quad\\alpha\\in(1,2],\\\\\n\\big(\\alpha(\\alpha+1)\\big)^{\\frac{2}{3}}&\\text{if}\\quad\\alpha\\in(2,3],\n\\end{cases}\n&\n\\end{flalign}\nthen $M_{\\alpha}(G(t))$ is a decreasing function on $(0,\\infty)$.\n\\end{enumerate}\n\\end{theorem}\n\nThe next result is a property satisfied by all  the weak radial solutions of  (\\ref{PA}), (\\ref{PB}).\n\\begin{theorem}\n\\label{S1Treg}\nLet $G_0$ be  as in Theorem \\ref{S1T1}, and $G$ a weak radial solution of  (\\ref{PA}), (\\ref{PB}).\nThen for all $T>0$, $R>0$ and $\\alpha\\in\\left(-\\frac{1}{2},\\infty\\right)$,\n\\begin{align}\n&\\int_0^TG(t,\\{0\\})\\int_{(0,R]}x^{\\alpha}G(t,x) dxdt\\leq \\nonumber \\\\\n&\\,\\,\\, \\le \\frac{2R^{\\frac{1}{2}+\\alpha}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}}\\left(\\int_0^T\\!\\! G(t,\\{0\\})dt\\right)^{\\frac{1}{2}}\\left(\\frac{\\sqrt{E}}{2}\\int_0^T\\!\\!G(t,\\{0\\})dt+\\sqrt{N}\\right).\\label{MNEG2}\n\\end{align}\n\\end{theorem}\nThe only\npossible algebraic behavior for such a measure $G$ near the origin is then $x^{-1\/2}$.\n\\begin{remark}\n\\label{S1R1}\nThe functions $F _{ \\beta , 0, C }$ defined above are weak radial solutions of (\\ref{PA}), (\\ref{PB}) for all $\\beta >0$ and $C\\ge 0$ (cf. Proposition \\ref{equilibria}). Since\n\\begin{eqnarray*}\n\\int  _{ (0, \\infty) } x^{\\alpha}  G _{ \\beta , 0, C}\\;dx<\\infty\\quad\\Longleftrightarrow\\quad\\alpha >-1\/2,\n\\end{eqnarray*}\nthe estimate (\\ref{MNEG2}) can not hold for all radial weak solutions if $\\alpha\\leq -1\/2$.\n\\end{remark}\n\nIn the next two results we describe the evolution of the measure at the origin $n(t)=G(t,\\{0\\})$ by taking the limit $\\varepsilon\\to 0$ in the weak formulation \n(\\ref{S1E16}) for test functions $\\varphi_{\\varepsilon}$ as follows:\n\\begin{remark}\n\\label{TEST}\nGiven $\\varphi\\in C^1_b([0,\\infty))$ nonnegative, convex, with $\\varphi(0)=1$ and \n$\\lim_{x\\to\\infty}\\sqrt{x}\\varphi(x)=0$, denote $\\varphi_{\\varepsilon}(x)=\\varphi(x\/\\varepsilon)$ for $\\varepsilon>0$. Notice that for any \n$G\\in\\mathscr{M}_+([0,\\infty))$,\n\\begin{align}\n\\label{TEST2}\nG(\\{0\\})=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)dG(x).\n\\end{align}\nThe standard example is $\\varphi_{\\varepsilon}(x)=(1-x\/\\varepsilon)^2_+$.\n\\end{remark}\n\n\\begin{theorem}\n\\label{THn01}\nLet $G$ be the solution of (\\ref{S1E16}) obtained in Theorem \\ref{S1T1}, with initial data $G_0\\in \\mathscr{M}^1_+([0,\\infty))$ such that $N=M_0(G_0)>0$, $E=M_1(G_0)>0$ and $G_0(\\{0\\})>0$. Denote $G(t)=n(t)\\delta _0+g(t)$, with\n$n(t)=G(t, \\{0\\})$. Then $n$ is right continuous and a.e. differentiable on $[0,\\infty)$. Moreover, there exists a positive measure $\\mu$ on $[0,\\infty)$ whose cumulative distribution function is given by\n\\begin{align}\n\\label{ZE01}\n\\mu((0,t])=\\lim_{\\varepsilon\\to 0}\\int_{0}^{t}n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds\n\\end{align}\nfor any $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST},\nand  such that:\n\\begin{align}\n\\label{ZE02}\nn(t)-n(0)+\\int_{0}^{t}n(s)M_{1\/2}(g(s))ds=\\mu((0,t])\\qquad\\forall t>0.\n\\end{align}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{MU1}\nLet $G$ and $\\mu$ be as in Theorem \\ref{THn01}. Then\n\\begin{align}\n\\label{ZE00}\n0<\\mu((0,t]))<\\infty\\qquad\\forall t>0.\n\\end{align}\n\\end{theorem}\nThe measure $\\mu$ in (\\ref{ZE01}) depends on the atomic part of  $g$, and on the behaviour of $g$ at the origin (it seems to be actually related with its moment of order $-1\/2$ c.f. Proposition \\ref{LM-1\/2} and Remark \\ref{HH}). This measure $\\mu $ appears as a source term in the equation (\\ref{ZE02}) for $n$.\nGiven  the function $n$, the equation (\\ref{PR}) satisfied by $g$ on $(0, \\infty)$ has also a natural weak formulation by itself. In terms of $g(t)$, where $g(t)=G(t)-G(t, \\{0\\})\\delta _0$ and $\\sqrt x f(t, x)=g(t, x)$ it reads \n\n\\begin{eqnarray}\n\\label{2E765}\n\\frac{d}{dt}\\int_{[0,\\infty)} \\!\\!\\!\\!\\!\\varphi(x)g(t,x)dx=n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\;\\forall \\varphi\\in C_b^1([0, \\infty)),\\,\\varphi (0)=0.\n\\end{eqnarray}\n\nIn the next result we describe the relation between a weak solution $(F,n)$ of (\\ref{PA}), (\\ref{PB}), where $F(t, p)=|p|^{-1}g(t, |p|^2) $, $G(t)=n(t)\\delta _0+g(t)$, $n(t)=G(t, \\{0\\})$, and a pair $(g, n)$ where $g$ is a weak radial solution of the equation (\\ref{PA}) and $n$ satisfies (\\ref{PB}). \n\\begin{theorem}\n\\label{EQUIV}\nSuppose that $G\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ is such that $G(0)=G_0\\in \\mathscr{M}^1_+([0,\\infty))$ with $G_0(\\{0\\})>0$, and denote  $G(t)=n(t)\\delta _0+g(t)$ with $n(t)=G(t, \\{0\\})$.\\\\\n(i) If $(F, n)$  is a weak radial solution of (\\ref{PA}), (\\ref{PB}) and $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $n$ is given by (\\ref{ZE02}), (\\ref{ZE01}),\nand $g$ satisfies (\\ref{2E765}) for a.e. $t>0$.\\\\\n(ii) On the other hand, if  $g$ satisfies (\\ref{S1ED2S}), (\\ref{S1ED3S}) and (\\ref{2E765}) for some nonnegative bounded function $n$,  \nthen the limit in (\\ref{ZE01}) exists. If $n$ also satisfies\n\\begin{equation}\n\\label{ene1}\nn(t)=n(0)+\\lim_{\\varepsilon\\to 0}\\int_0^t n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds-\\int_0^t n(s)M_{1\/2}(g(s))ds\n\\end{equation}\nand $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}).\n\\end{theorem}\n\nIf in the Definition \\ref{S1D1} only test functions satisfying $\\varphi (0)=0$ are taken, it becomes necessary to introduce some other condition to the system. Otherwise the system would be reduced to  find $g$ satisfying (\\ref{S1ED3S})--(\\ref{S1E16})  for a given function $n(t)$ and  for test functions such that  $\\varphi (0)=0$. If we impose just the conservation of mass, we prove below (Corollary \\ref{S1C31}) that we recover a solution that satisfies the Definition \\ref{S1D1}.\n\n\\begin{corollary}\n\\label{S1C31}\nIf $g$ satisfies (\\ref{S1ED2S}), (\\ref{S1ED3S}) and (\\ref{2E765}) for some nonnegative bounded function $n=n(t)$ such that \n\\begin{equation}\n\\label{MBC}\nn(t)+\\int  _{ (0, \\infty) }g(t, x)dx= constant\n\\end{equation}\nand $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}).\n\\end{corollary}\n\nIn our last  result we show that, under some sufficient conditions, the condensate density $n(t)$ tends to zero as $t\\to\\infty$, fast enough to be  integrable.\n\\begin{theorem} \n\\label{S1T5}\nSuppose that  $G_0\\in \\mathscr M_+^1([0, \\infty))$  satisfies $G_0(\\{0\\})>0$ and let $(F,n)$ be the weak radial solution of (\\ref{PA}), (\\ref{PB}) obtained in Theorem \\ref{S1T1}. Let us call $N=M_0(G_0)$ and $E=M_1(G_0)$. If condition (\\ref{PRO112}), (\\ref{PRO115}) hold for some $\\alpha \\in (1, 3]$,\nthen, for all $t_0>0$,\n\\begin{equation}\n\\label{S1ET5B}\n\\int _{ t_0 }^{\\infty} n(t)dt\\leq M_{\\alpha}(G(t_0)) C(N,E,\\alpha)\n\\end{equation}\nfor some explicit constant $C(N,E,\\alpha)$ given in (\\ref{PRO116}), and\n\\begin{equation}\n\\label{S1ET5C}\n\\lim_{t\\to \\infty }n(t)=0.\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n\\label{S1R2}\nThe quantity $E\/N^{5\/3}$ has a very precise  interpretation in physical terms.  Suppose that $T$ is the temperature of  a system of  particles at equilibrium with total number of particles $N$ and total energy $E$. And denote $T_c$ the critical temperature, that is the temperature at which the ground state of the system becomes macroscopically occupied. Then:\n\\begin{eqnarray*}\n\\frac {E} {N^{5\/3}} = b\\, \\frac {T} {T_c},\\qquad\\hbox{where}\\qquad\nb=\\frac {3}{(2\\pi )^{\\frac {1} {3}}}\\frac  {\\zeta (5\/2)}{\\zeta (3\/2)^{5\/3}}.\n\\end{eqnarray*}\nand  condition  (\\ref{PRO112}) implies\n$$\n\\frac {T} {T_c}=\\frac {1} {b}\\,\\frac {E} {N^{5\/3}}> \\frac {C(\\alpha )} {b}.\n$$\nThe function $C(\\alpha )\/b$ is continuous and strictly increasing on $[1, 3]$ and its limit as $\\alpha \\to 1^+$ is $\\log(16)^{2\/3}\/b\\approx 4.48403$. Condition (\\ref{PRO112}) means that, when at equilibrium, the  system of particles would be at a temperature  clearly above the critical temperature. Anyway,  the solution $F$ of the problem (\\ref{PA}), (\\ref{PB}) may be far from any real distribution of particles of the original system of particles.\n\\end{remark}\n\n\\subsection{Some arguments of the proofs.}\nIt is very natural to make the following change of variables in  problem (\\ref{S1E16}). Given $G(t)=n(t)\\delta _0+g(t)$, where $n(t)=G(t, \\{0\\})$, we define\n\\begin{eqnarray}\n\\label{S1E45}\n&&H(\\tau)=G(t),\\qquad\\text{where}\\qquad\\tau =\\int _0^t n(s) ds. \\label{S1E45b}\n\\end{eqnarray}\nIn terms of $H$, (\\ref{S1E16}) reads\n\\begin{equation}\n\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)H(\\tau , x)dx= \\mathscr Q _3(\\varphi,H(\\tau))\\quad\\forall \\varphi \\in C^1_b([0, \\infty)). \\label{S1E16Ha}\n\\end{equation}\nTo obtain a measure $H$ that satisfies (\\ref{S1E16Ha}), we first find $h$ satisfying\n\\begin{align}\n\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)h(\\tau , x)dx= \\widetilde{\\mathscr Q}_3(\\varphi,h(\\tau))\n\\quad\\forall \\varphi \\in C^1_b([0, \\infty)),\\label{S1E16ha}\n\\end{align}\nwhere $\\widetilde{\\mathscr Q}_3$ is given in (\\ref{S1EB2})--(\\ref{S1E21R}).\nThen we define $H$ as\n\\begin{eqnarray}\n\\label{S1EdecompH}\nH(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0.\n\\end{eqnarray}\nBy (\\ref{S1EB1}), the measure  $H$ will satisfy (\\ref{S1E16Ha}).\n\nAs it will be seen in Section \\ref{existenceH}, all the arguments are much simpler and clear in the equation for $H$ than in the equation for $G$. In particular, the measure $\\lambda$, that corresponds to the measure $\\mu$ of Theorem \\ref{THn01}, appears as the Lebesgue-Stieltjes measure associated to $m(\\tau )=h(\\tau , \\{0\\})$.\n\nThe proofs of Theorem \\ref{S1T1} and Theorem \\ref{S1Treg} make great use of the change of variables (\\ref{S1E45b}).\nSeveral of our arguments will need the measure $h(\\tau )$ to satisfy only one inequality in (\\ref{S1E16ha}). This requires the following:\n\\begin{definition}\n\\label{SUPER}\nA  function $h:[0,\\infty)\\to\\mathscr{M}_+([0,\\infty))$ is said to be a super solution of (\\ref{S1E16ha}) if\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\forall  \\varphi \\in C^1_b([0, \\infty))\\;\\hbox{nonnegative, convex and decreasing} :\\nonumber\\\\\n&\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)h(\\tau,x)dx\\ge \\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))\\qquad a.e.\\;\\tau>0.\\label{S1E16haB}\n\\end{empheq}\n\\end{definition}\nThe operator $\\mathscr Q_3^{(2)}$ is considered in \\cite{KIER} and \\cite{AV1}, where a problem similar to (\\ref{S1E16ha}) is studied, with\n$\\widetilde{\\mathscr{Q}}_3$  replaced by $\\mathscr Q_3^{(2)}$ and for which, the property of instantaneous condensation is proved. We extend this result to the solutions $h$ of the problem (\\ref{S1E16ha}) with the whole $\\widetilde{\\mathscr{Q}}_3$, using similar arguments (monotonicity,  convexity of  test functions) and taking care of the linear term.\n \nTheorem \\ref{S1T1} is deduced from the corresponding existence result of $h$, that is proved using very classical arguments: regularization of the problem,  fixed point, a priori estimates and passage to the limit. Then, the delicate point is to invert the change of variables (\\ref{S1E45b}) in order to obtain a global in time nonnegative solution $G$.\n\nThe Plan of the article is the following. In Section \\ref{model} we prove Proposition \\ref{S1P0}. Section \\ref{existenceH} is devoted to the proof of the existence of the measure $H$. In Section \\ref{SectionC} we obtain several properties of $h(\\tau , \\{0\\})$. In Section \\ref{sectionG} we prove Theorem \\ref{S1T1} (existence for the measure $G$) and Theorem \\ref{S1Treg}. The contents of Section \\ref{SectionK} are the\nproofs of Theorem \\ref{THn01}, Theorem \\ref{MU1}, Theorem \\ref{EQUIV} and Corollary \\ref{S1C31}. Finally in Section \\ref{SectionD} we prove Theorem \\ref{S1T5}. Several technical results are presented in an Appendix.\n\n\\section{On weak formulations.}\n\\label{model}\n\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe  deduce first a  detailed  expression of  the weak formulation of (\\ref{S1E1}) for a radial measure $G$.\n\\begin{proposition}\n\\label{S1P0}\nLet $G$ satisfy (\\ref{S1ED1})--(\\ref{S1E6'}) for some $T>0$, and write \n$G(t)=n(t)\\delta_0+g(t)$, where $n(t)=G(t,\\{0\\})$. Then, for all $\\varphi \\in C^{1,1}_b([0, \\infty))$ and for all $t\\in (0, T)$:\n\\begin{align}\n&\\frac {d} {dt}\\int _{ [0, \\infty) } \\varphi (x)G(t, x)dx=\\mathscr{Q}_4(\\varphi,g(t))+n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\label{S1E13}\n\\end{align}\nwhere $\\mathscr Q_4(\\varphi,g)$ and $\\mathscr Q _3(\\varphi,g)$ are defined in (\\ref{S1E15})--(\\ref{S1E155}).\n\\end{proposition}\n\n\n\\begin{remark}\n\\label{SXR1}\nIf the term $\\mathscr Q_4(\\varphi,g)$ in (\\ref{S1E13}) is dropped, we recover the equation (\\ref{S1E16})  that defines a radial weak solution of (\\ref{PA}), (\\ref{PB}).\n\\end{remark}\n\n\\begin{proof}\n[\\upshape\\bfseries{Proof of Proposition \\ref{S1P0}}]\nWe may rewrite $\\mathcal{Q}_4(\\varphi,G)$ in (\\ref{S1E5}) as \n\\begin{align*}\n\\mathcal{Q}_4(\\varphi,G)&=\\iiint_{[0,\\infty)^3}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\n+\\frac{1}{2}\\iiint_{[0,\\infty)^3}\\sqrt{x_3}\\Phi_{\\varphi}\\;dG_1dG_2dx_3,\n\\end{align*}\nwhere $\\Phi_{\\varphi}$ is as in Lemma \\ref{S2L1}, and we have used notation $dG$ instead of $Gdx$. Then we decompose \n$[0,\\infty)^3=(0,\\infty)^3\\cup A\\cup P$, where, for $\\{i,j,k\\}=\\{1,2,3\\}$,\n\\begin{align*}\n&A=\\{(x_1,x_2,x_3)\\in\\partial[0,\\infty)^3\\;:\\; x_i=x_j=0,\\; x_k> 0\\}\\cup\\{(0,0,0)\\},\\\\\n&P=\\{(x_1,x_2,x_3)\\in\\partial[0,\\infty)^3\\;:\\; x_i=0,\\; (x_j,x_k)\\in (0,\\infty)^2\\}.\n\\end{align*}\nLet $\\varphi\\in C^{1.1}_b([0,\\infty)$. By (\\ref{S2E2}) in Lemma \\ref{representation of Deltavarphi} and the definition (\\ref{S2E3}) of $W$, it follows that $\\Phi_{\\varphi}\\equiv 0$ on $A$. Hence, recalling the definition (\\ref{S1E15}) of $\\mathscr{Q}_4(\\varphi,g)$ and the definition of $\\Phi_{\\varphi}$ in Lemma \\ref{representation of Deltavarphi}, we have\n\\begin{align}\n\\label{999C}\n\\mathcal{Q}_4(\\varphi,G)=\\mathscr{Q}_4(\\varphi,g)+\\iiint_{P}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\n+\\frac{1}{2}\\iiint_{P}\\sqrt{x_3}\\Phi_{\\varphi}\\;dG_1dG_2dx_3.\n\\end{align}\nWe now study the integral over $P$ for the cubic and the quadratic terms in (\\ref{999C}).\\\\\n(a) The cubic term.\nSince $\\Phi_{\\varphi}$ is symmetric in the $x_1$, $x_2$ variables, and $\\Phi_{\\varphi}$ is uniformly continuous on $[0,\\infty)^3$ by Lemma\n\\ref{S2L1}, then\n\\begin{align}\n\\iiint_{P}\\Phi_{\\varphi}\\;dG_1dG_2dG_3=&2 \\iiint_{\\{x_2=0,\\;x_1>0,\\,x_3>0\\}}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\\nonumber\\\\\n&+\\iiint_{\\{x_3=0,\\;x_1>0,\\,x_2>0\\}}\\Phi_{\\varphi}\\;dG_1 dG_2 dG_3\\nonumber\\\\\n=&2G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,0,x_3)\\;dG_1dG_3\\nonumber\\\\\n&+G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,x_2,0)\\;dG_1 dG_2.\\label{mid step}\n\\end{align}\nUsing now the definition of $\\Phi_{\\varphi}$, we have\n\\begin{align}\n\\label{line 1}\n&2\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,0,x_3)\\;dG_1dG_3\\\\\n&=2\\iint_{\\{x_1>x_3>0\\}}\\big[\\varphi(x_1-x_3)+\\varphi(x_3)-\\varphi(0)-\\varphi(x_1)\\big] \\frac{dG_1dG_3}{\\sqrt{x_1x_3}}\\nonumber\\\\\n&=\\iint\\limits_{(0,\\infty)^2}\\big[\\varphi(|x_1-x_3|)+\\varphi(\\min\\{x_1,x_3\\})-\\varphi(0)-\\varphi(\\max\\{x_1,x_3\\})\\big] \\frac{dG_1dG_3}{\\sqrt{x_1x_3}}.\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\label{line 2}\n&\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,x_2,0)\\,dG_1 dG_2\\\\\n&=\\iint\\limits_{(0,\\infty)^2}\\big[\\varphi(x_1+x_2)+\\varphi(0)-\\varphi(\\min\\{x_1,x_2\\})-\\varphi(\\max\\{x_1,x_2\\})\\big]\\frac{dG_1 dG_2}{\\sqrt{x_1x_2}}.\\nonumber\n\\end{align}\nNotice in (\\ref{line 1}) that $\\varphi(|x_1-x_3|)+\\varphi(\\min\\{x_1,x_3\\})-\\varphi(0)-\\varphi(\\max\\{x_1,x_3\\})=0$ on the diagonal $\\{x_1=x_3>0\\}$.\nThen, using (\\ref{line 1}) (changing the labels $x_3$ by $x_2$) and (\\ref{line 2}) in (\\ref{mid step}), and recalling the definition (\\ref{S1E154}) of $\\Lambda(\\varphi)$, we obtain\n\\begin{align}\n\\label{999A}\n\\iiint_{P}&\\Phi_\\varphi\\;dG_1dG_2dG_3=G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi)(x_1,x_2)}{\\sqrt{x_1x_2}}dG_1 dG_2.\n\\end{align}\n(b) The quadratic term.\nAgain, by the symmetry of $\\Phi_{\\varphi}$ in $x_1$, $x_2$, and the continuity of $\\Phi_{\\varphi}$ on $[0,\\infty)^3$, we obtain\n\\begin{align}\n\\label{999B}\n\\frac{1}{2}\\iiint_{P}\\sqrt{x_3}\\,\\Phi_{\\varphi}\\;dG_1dG_2d x_3\n&=\\iiint_{\\{x_2=0,\\;x_1>0,\\,x_3>0\\}}\\sqrt{x_3}\\,\\Phi_{\\varphi} \\,dG_1dG_2d x_3\\nonumber\\\\\n&=G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\sqrt{x_3}\\,\\Phi_{\\varphi}(x_1,0,x_3)\\,dG_1d x_3\\nonumber\\\\\n&=G(t,\\{0\\})\\iint_{\\{x_1>x_3>0\\}}\\frac{\\Delta\\varphi(x_1,0,x_3)}{\\sqrt{x_1}}dG_1d x_3\\nonumber\\\\\n&=-G(t,\\{0\\})\\int_{(0,\\infty)}\\frac{\\mathcal{L}_0(\\varphi)(x_1)}{\\sqrt{x_1}}dG_1,\n\\end{align}\nwhere $\\mathcal{L}_0(\\varphi)$ is given in (\\ref{S1E155}).\nUsing (\\ref{999A}) and (\\ref{999B}) in (\\ref{999C}), the result follows.\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{equilibria}\nFor all $C> 0$ and all $\\beta >0$, the measure $f _{ \\beta , 0, C }$ is a radial weak solutions of (\\ref{PA}),(\\ref{PB}).\n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{S1P0},\n\\begin{eqnarray*}\n\\mathcal Q_4(\\varphi,G _{ \\beta , 0, C })=\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })+C\\mathscr{Q}_3(\\varphi,G _{ \\beta , 0, 0 }).\n\\end{eqnarray*}\nWe already know by Theorem  5 of \\cite{Lu1} that\n$\\mathcal{Q}_4(\\varphi,G _{ \\beta , 0, C })=0$ for all  $\\varphi \\in  C^{1,1}([0, \\infty))$. \nSince $\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })\\equiv \\mathcal{Q}_4(\\varphi,G _{ \\beta , 0, 0 })$, we deduce\n$\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })=0$ for all $\\varphi \\in  C^{1,1}([0, \\infty))$. Then, since $C>0$,\n$$\\mathscr{Q}_3(\\varphi,G _{ \\beta , 0, 0 })=0\\quad\\forall \\varphi \\in  C^{1,1}([0, \\infty)).$$\n\\end{proof}\n\n\\section{Existence of solutions $H$ to (\\ref{S1E16Ha})}\n\\label{existenceH}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nThe main result of this Section is the following,\n\\begin{theorem}\n\\label{S5T5R}\nLet $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$. Then, there exists $h\\in C\\big((0,\\infty), \\mathscr{M}_+^\\alpha([0,\\infty))\\big)$ for any $\\alpha\\geq1$,  \nthat satisfies the following properties: for all $\\varphi\\in C^1_b([0,\\infty)$\n\\begin{align}\n\\label{lip loc h}\n(i)\\quad&  \\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx\\in W^{1,\\infty} _{loc}([0,\\infty)),\\\\\n\\label{AUXW}  \n(ii)\\quad&\\frac{d}{d \\tau}\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)\\,dx=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))\\quad a.e.\\,\\tau>0,\\\\\n(iii)\\quad&  h(0)=h_0,\\\\\n\\label{MMI}\n(iv)\\quad& M_0(h(\\tau))\\le\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2\\quad\\forall\\tau \\ge 0,\\\\\n \\label{EE}\n(v)\\quad&M_1(h(\\tau))=E\\quad\\forall\\tau \\ge 0,\\\\\n(vi)\\quad& \\text{For all }\\alpha \\geq 3,\\,\\,\\hbox{if}\\,\\,M_{\\alpha}(h_0)<\\infty,\\,\\,\\hbox{then}  \\nonumber\\\\\n\\label{MAh}\n&\\quad M_{\\alpha}(h(\\tau))\n\\leq \\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\\right)^{\\frac{\\alpha-1}{2}}\\quad\\forall\\tau\\geq 0,\\\\\n\\label{S5Ealpha }\n(vii)\\quad& M_{\\alpha}(h(\\tau))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau}}\\right)^{2(\\alpha-1)}\\,\\,\\forall \\alpha\\geq 3,\\;\\forall\\tau>0,\n\\end{align}\nwhere $C=C(\\alpha,E)$ is the unique positive root of the algebraic equation\n\\begin{equation}\n\\label{algebraic equation}\n2^{\\alpha-2}(\\alpha+1)E^{\\frac{2\\alpha+3}{2(\\alpha-1)}}(1+C)=C^{\\frac{2\\alpha-1}{2(\\alpha-1)}},\n\\end{equation}\nand $\\gamma=\\gamma(\\alpha,E)$:\n\\begin{equation}\n\\label{gamma}\n\\gamma=\\frac{1}{2(\\alpha+1)}\\left(\\frac{C}{E}\\right)^{\\frac{1}{2(\\alpha-1)}}.\n\\end{equation}\n\n\\end{theorem}\nThe proof of Theorem \\ref{S5T5R} is in three steps. In the first, a regularized problem is solved (Theorem \\ref{Ex1T2}). Then, using an approximation argument, a solution is obtained that satisfies (\\ref{lip loc h})--(\\ref{MAh}) but not yet (\\ref{S5Ealpha }) (Theorem \\ref{Ex1T1}). The Theorem  \\ref{S5T5R} is proved with  a second approximation argument on the initial data.\n\nAs a Corollary, we obtain the measure $H$ (not necessarily positive). \n\n\\begin{corollary}\n\\label{S5C52R}\nSuppose that $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$, consider $h$ given by Theorem \\ref{S5T5R},\nand define, for $\\tau\\geq 0$\n\\begin{align}\n\\label{DEFH}\nH(\\tau)=h(\\tau)-\\left(\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0.\n\\end{align}\nThen $H\\in C\\big([0,\\infty),\\mathscr{M}^1([0,\\infty))\\big)$ and for all $\\tau\\in[0,\\infty)$ and $\\varphi\\in C^1_b([0,\\infty))$:\n\\begin{align}\n\\label{lip loc H}\n(i)\\quad&\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)dx\\in W^{1,\\infty} _{loc}([0,\\infty)),\\\\\n(ii)\\quad&\n\\label{AUXWH}\n\\frac{d}{d \\tau}\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)\\,dx=\\mathscr{Q}_3(\\varphi,H(\\tau))\\quad a.e.\\,\\tau>0,\\\\\n\\label{S5IDh}\n(iii)\\quad& H(0)=h_0,\\\\\n(iv)\\quad& \\label{MMH}M_0(H(\\tau))=N\\quad\\forall \\tau \\ge 0,\\\\\n(v)\\quad& \\label{EEH} M_1(H(\\tau))=E\\quad\\forall \\tau \\ge 0,\\\\\n(vi)\\quad& \\forall \\alpha \\geq3, \\,\\,\\hbox{if}\\,\\,  M_{\\alpha}(h_0)<\\infty\\,\\hbox{ then, for all}\\,\\tau >0, \\nonumber\\\\\n&\\label{MAH} \\hskip 1.5cm  M_{\\alpha}(H(\\tau))\\leq \\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\\right)^{\\frac{\\alpha-1}{2}},\\\\\n(vii)\\quad&\\label{S5EalphaR }\nM_{\\alpha}(H(\\tau))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau}}\\right)^{2(\\alpha-1)},\\,\\,\\forall \\alpha\\geq 3,\n\\end{align}\nwhere the constants $C(\\alpha,E)$ and $\\gamma (\\alpha,E)$ are defined in Theorem \\ref{S5T5R}.\n\\end{corollary}\n\n\\begin{remark}\nUnder the hypothesis that all the moments of the initial data $h_0$  are bounded it is easy to obtain the estimate (\\ref{S5Ealpha }) using the weak formulation  (\\ref{AUXW}). However, it is not so easy using the regularized weak formulation (\\ref{REGWEAK}) below. For that reason, we first want to obtain a solution $h$ satisfying (\\ref{AUXW}) with an initial data with bounded moments of all order.\n\\end{remark}\n\n\\subsection{A first result.}\n\\begin{theorem}\n\\label{Ex1T1}\nFor any $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)$ and $E=M_1(h_0)$, there exists\n$h\\in C\\big([0,\\infty), \\mathscr{M}_+^1 ([0,\\infty))\\big)$ that satisfies (\\ref{lip loc h})--(\\ref{MAh}).\n\\end{theorem}\n\nThe proof of  Theorem \\ref{Ex1T1} is made in two steps. We first solve a regularised version of (\\ref{AUXW}). Then, in a second step, we use an approximation argument. More precisely, we consider the following cutoff:\n\\begin{cutoff}\n\\label{cut-off}\nFor every $n\\in \\mathbb{N}$ let $\\phi _n\\in  C_c([0,\\infty))$ be such that $\\operatorname{supp}\\phi_n=[0,n+1]$,\n$\\phi _n(x)\\le x^{-1\/2}$ for all $x>0$ and $\\phi_n(x)=x^{-1\/2}$ for all $x\\in\\left(\\frac {1} {n}, n \\right)$,\nin such a way that:\n\\begin{align}\n\\forall x>0\\,\\,\\,\\,\\,\\lim _{ n\\to \\infty }\\phi _n(x)=\\frac {1} {\\sqrt x}.\n\\end{align}\n\\end{cutoff}\n\n\\subsection{Regularised problem}\nWe now solve in Theorem \\ref{Ex1T2} a regularised version of (\\ref{AUXW}) with the operator $\\widetilde{\\mathscr{Q}}_{3,n}$ defined in \n(\\ref{Aq3tilden})--(\\ref{Q3n1w}). The solution $h_n$ is obtained as a mild solution to the equation\n\\begin{align}\n\\label{Ex1EApn}\n\\frac {\\partial h_n} {\\partial\\tau}(\\tau,x)=J_{3,n}(h_n(\\tau))(x),\n\\end{align} \nwhere $J_{3,n}$ is defined in (\\ref{A1E32})-(\\ref{A1E35}), and corresponds to a regularised version of the term $J_3$ defined in (\\ref{PR2}). Namely,\n$J_{3,n}(h)=J_3(h\\phi_n)$, where $\\phi_n$ is as in Cutoff \\ref{cut-off}.\n\\begin{theorem}\n\\label{Ex1T2}\nFor any $n\\in\\mathbb{N}$ and any nonnegative function  $h_0\\in C_c([0,\\infty))$,\nthere exists a unique nonnegative function $h_n\\in C\\big([0,\\infty), L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{x}(\\mathbb{R}_+)\\big)$ such that for all $\\tau\\in[0,\\infty)$ \nand all $\\varphi\\in  L^1_{loc}(\\mathbb{R}_+)$:\n\\begin{align}\n&\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx\\in W^{1,\\infty}_{ loc  }([0,\\infty)) \\label{S5E765}\\\\\n&\\frac{d}{d\\tau}\\int_0^{\\infty}\\varphi(x)h_n(\\tau,x)dx=\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau)).\\label{REGWEAK}\\\\\n&h_n(0,x)=h_0(x)\\label{datan}\n\\end{align}\nMoreover, if we denote by $N=M_0(h_0)$ and $E=M_1(h_0)$, then for every $\\tau\\in[0,\\infty)$ and $\\alpha\\geq 3$:\n\\begin{align}\n&M_0(h_n(\\tau))\\leq\\bigg(\\frac{E}{2}\\tau+\\sqrt{N}\\bigg)^2, \\label{mass inequality} \\\\\n&M_1(h_n(\\tau))=E, \\label{conservation of energy}\\\\\n&M_{\\alpha}(h_n(\\tau))\\leq\\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}. \\label{MAhn}\n\\end{align}\nFurthermore, there exist two positive constants $C _{ 1, n }$ and $C _{ 2, n }$ depending on $n$ and $\\|h_0\\| _{ L^\\infty\\cap L^1_x }$ such that for all $\\tau >0$:\n\\begin{equation}\n\\label{a priori sup norm}\\|h_n(\\tau)\\|_{\\infty}\\leq C _{ 1, n }e^{C _{ 2, n }(\\tau^2+\\tau)}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nUsing\n(\\ref{A1E32}) we write equation (\\ref{Ex1EApn}) as\n\\begin{eqnarray}\n\\label{Ex1EApn2}\n\\frac {\\partial h_n} {\\partial \\tau }+h_nA_n(h_n)=K_n(h_n)+L_n(h_n),\n\\end{eqnarray}\nand the solution $h_n$ is obtained as a fixed point of the operator:\n\\begin{align}\nR_n(h_n)(\\tau,x)=&h_0(x)S_n(0, \\tau; x)\\nonumber \\\\\n&+\\int_0^\\tau S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)d\\sigma, \\label{Ex1E1}\\\\\nS_n(\\sigma,\\tau; x)=&e^{-\\int_\\sigma ^\\tau A_n(h_n)(\\sigma,x)d\\sigma}\\label{Ex1E4}\n\\end{align} \non\n\\begin{align}\nB(T):=\\Big\\{h\\in C\\big([0,T],L^{\\infty}&(\\mathbb{R}_+)\\cap L^1_x(\\mathbb{R}_+)\\big): h\\geq 0\\quad\\text{and} \\nonumber\\\\\n&\\sup_{\\tau\\in[0,T]}\\|h(\\tau)\\|_{L^{\\infty}\\cap L^1_x}\\leq 2\\|h_0\\|_{L^{\\infty}\\cap L^1_x}\\Big\\}. \\label{Ex1E2}\n\\end{align}\nLet us show first that $R_n$ sends $B(T)$ into itself. Let $r_0:=\\|h_0\\|_{L^\\infty\\cap L^1_x}$ and for an arbitrary $T>0$, let $h\\in  B(T)$. \nBy Proposition \\ref{well defined operators}  with $\\rho (x)=x$, \n\\begin{align*}\n&R_n(h)(\\tau,x)\\geq 0\\qquad\\forall\\tau\\in[0,T],\\;\\forall x\\in\\mathbb{R}_+,\\\\\n&R_n(h)\\in C\\big([0,T],L^{\\infty}(\\mathbb{R}_+)\\cap L^1_x(\\mathbb{R}_+)\\big).\n\\end{align*}\nMoreover, using (\\ref{SAE100}) and  (\\ref{bound L}):\n\\begin{align*}\n\\sup _{ \\tau \\in [0,T] }\\|R_n(h)(\\tau)\\|_{L^{\\infty}\\cap L^1_x}\n\\leq r_0+T\\,C(n)(4r_0^2+2r_0).\n\\end{align*}\nIf $T$ satisfies:\n\\begin{eqnarray}\nT\\le \\frac {1} {C(n)(4r_0+2)}\n\\label{Ex1E257}\n\\end{eqnarray}\nthen \n$R_n(h)\\in B(T)$. \n\n\nTo prove that $R_n$ is a contraction, let $h_1\\in B(T)$, $h_2\\in B(T) $ and write:\n\n\\begin{align*}\n&\\big|R_n(h_1)(\\tau,x)-R_n(h_2)(\\tau,x)\\big|\\le h_0(x)\\left|  S_1(0, \\tau ; x)-S_2(0, \\tau ; x)\\right| +\\\\\n&+\\int_0^{\\tau}\\left|  S_1(\\sigma , \\tau ; x)-S_2(\\sigma , \\tau ; x) \\right|\n\\big(K_n(h_1)(\\sigma,x)+L_n(h_1)(\\sigma,x)\\big)d\\sigma\\\\\n&+\\int_0^{\\tau}\\big|K_n(h_1)(\\sigma,x)-K_n(h_2)(\\sigma,x)\\big|d\\sigma\\\\\n&+\\int_0^{\\tau} \\big|L_n(h_1)(\\sigma,x)-L_n(h_2)(\\sigma,x)\\big|d\\sigma.\n\\end{align*}\nBy (\\ref{SaE121}), for all $\\sigma \\ge 0$ and $\\tau \\ge0$\n\\begin{align}\n\\left|  S_1(\\sigma , \\tau ; x)-S_2(\\sigma , \\tau ; x) \\right| & \\le   \\int_0^\\tau |A_n(h_1)(\\sigma,x)-A_n(h_2)(\\sigma,x)|d\\sigma \\nonumber \\\\\n& \\le C(n)\\,\\tau  \\sup_{\\tau\\in[0,T]}\\|h_1(\\tau)-h_2(\\tau)\\|_{\\infty}.   \\label{Ex1E258}\n\\end{align}\nUsing now (\\ref{Ex1E258}) and (\\ref{SAE100})--(\\ref{SaE121}), we deduce:\n\\begin{align*}\n&\\|R_n(h_1)(\\tau)-R_n(h_2)(\\tau)\\| _{ L^\\infty \\cap L^1_x }\\le  C_1\n\\sup_{\\tau\\in[0,T]}\\|h_1(\\tau)-h_2(\\tau)\\|_{\\infty},\\\\\n&C_1\\equiv C_1(n, T, r_0)=C(n)T\\left(1+ 3r_0+2Tr_0(1+2r_0)\\right).\n\\end{align*}\nIf (\\ref{Ex1E257}) holds and\n\\begin{align*}\nC(n)T\\left(1+ 3r_0+2Tr_0(1+2r_0)\\right)<1,\n\\end{align*}\n$R_n$ will be a contraction from $B(T)$ into itself. This is achieved, for example, as soon as:\n\\begin{align*}\nT<\\min \\left\\{ \\frac {1} {2r_0(1+2r_0)}, \\frac {1} {2C(n) (1+2r_0)}\\right\\}=\\kappa _{ r_0 }.\n\\end{align*}\nThe fixed point $h_n$  of $R_n$ in $B(T)$  is then a  mild solution of  (\\ref{Ex1EApn}), that can be extended to a maximal interval of existence $[0, T _{ n, \\max })$. \n\nWe claim now that $h_n$ satisfies (\\ref{S5E765}), (\\ref{REGWEAK}).\nSince $h_n$ is a mild solution of (\\ref{Ex1EApn}):\n\\begin{equation}\n\\label{Ex1mildE}\nh_n(\\tau , x)=h_0(x)S_n(0, \\tau; x)+\\int_0^\\tau S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)d\\sigma \n\\end{equation}\nWe multiply this equation by $\\varphi\\in L^1_{loc}(\\mathbb{R}_+)$ and integrate on $(0, \\infty)$:\n\\begin{align*}\n\\int _0^\\infty & h_n(\\tau , x)\\varphi ( x)dx=\\int _0^\\infty h_0(x)S_n(0, \\tau ; x) \\varphi (x)dx+\\\\\n&+\\int _0^\\tau \\int _0^\\infty S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)\\varphi (x)dx d\\sigma.\n\\end{align*}\nUsing Lemma \\ref{well defined operators} and $h_0\\in C_c([0,\\infty))$, it follows that the integrals above are well define.\nIt also follows from Lemma \\ref{well defined operators} and (\\ref{Ex1E4}) that $\\tau \\mapsto \\int _0^\\infty h_n(\\tau , x)\\varphi (x)dx$ is  locally Lipschitz on $(0, T _{ n, \\max })$, and:\n\\begin{align*}\n\\frac {d} {dt}\\int _0^\\infty &h_n(\\tau , x)\\varphi (x)dx= \\int _0^\\infty h_0(x)(S_n(0, \\tau ; x)) _{ \\tau  } \\varphi  (x)dx+\\\\\n&+\\int _0^\\infty\\big(K_n(h_n)(\\tau ,x)+L_n(h_n)(\\tau ,x)\\big)\\varphi (x) dx+\\\\\n&+\\int _0^\\tau \\int _0^\\infty (S_n(\\sigma , \\tau; x))_\\tau \\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)\\varphi (x)dx d\\sigma.\n\\end{align*}\nWe use now that $(S_n(\\sigma , \\tau; x))_\\tau=-A_n(h_n)(\\tau , x)S_n(\\sigma , \\tau; x)$\n and the identity (\\ref{Ex1mildE}) to deduce:\n\\begin{align*}\n\\frac {d} {dt}\\int _0^\\infty h_n(\\tau , x)\\varphi (x)dx=&\\int _0^\\infty\\!\\!\\big(K_n(h_n)(\\tau ,x)+L_n(h_n)(\\tau, x)\\big)\\varphi (x) dx- \\nonumber\\\\\n&-\\int _0^\\infty A_n(h_n)h_n(\\tau , x)\\varphi (\\tau , x)dx,\n\\end{align*}\nthat is (\\ref{REGWEAK}).\n\nSuppose now that $T _{ n, \\max }<\\infty$ and \n$$\n\\sup_{\\tau \\in [0, T _{ n, \\max })}\\|h_n(\\tau)\\|_{L^{\\infty}\\cap L^1_x}<\\infty.\n$$\nThen there is an increasing sequence \n$\\tau_j\\rightarrow T_{n,\\max}$  as $j\\rightarrow\\infty$  and $L>0$ such that\n$$\n\\sup_{j }\\|h_n(\\tau_j)\\|_{L^{\\infty}\\cap L^1_x}\\le L<\\infty.\n$$\n\nFix $\\delta >0$ such that $\n\\delta <\\kappa _{ r_0+1 }$.\nStarting with the initial value $h(\\tau _j)$ we have a mild solution $h_j$ defined on $[0, \\delta]$. Gluing together $h$ with $h_j$ we obtain a mild solution on\n$[0, t_j+\\delta]$. For $j$ large enough, $t_j+\\delta >T _{ n, \\max }$, and this is a contradiction. Therefore, either $T _{ n, \\max }=\\infty$ or, if \n$T _{ n, \\max }=\\infty$, then $\\limsup \\|h_n(\\tau)\\|_{L^{\\infty}\\cap L^1_x}=\\infty$ as $\\tau\\to T_{n,\\max}$.\n \nLet us prove now the estimates (\\ref{mass inequality}), (\\ref{conservation of energy}) and (\\ref{a priori sup norm}), first for all $\\tau\\in (0, T _{ n, \\max })$. Then,  the property $T _{ n, \\max }=\\infty$ will follow. We start proving (\\ref{conservation of energy}). To this end we use (\\ref{REGWEAK}) with $\\varphi=x$. Since in that case \n$\\Lambda(\\varphi)(x,y)=0$ and $\\mathcal{L}(\\varphi)(x)=0$, \n(\\ref{conservation of energy}) is immediate. To prove (\\ref {mass inequality}), we use  (\\ref{REGWEAK})  with $\\varphi =1$. \nThen, $\\Lambda(\\varphi)(x,y)=0$ and $\\mathcal{L}(\\varphi)(x)=-x$\n and then, using $\\phi _n\\le x^{-1\/2}$, H\\\"older inequality and  (\\ref{conservation of energy}): \n\\begin{align*}\n\\frac{d}{d\\tau}\\left(\\int_0^\\infty h_n(\\tau,x)d x\\right)^{1\/2}\\leq\\frac{\\sqrt{E}}{2},\n\\end{align*}\nfrom where (\\ref {mass inequality}) follows. \n\nIn order to prove \\eqref{a priori sup norm} we use \\eqref{mass inequality}:\n\\begin{align*}\n\\|K_n(h_n)(\\sigma)\\|_{\\infty}&\\leq\\|\\phi_n\\|_{\\infty}^2\\|h_n(\\sigma)\\|_1\\|h_n(\\sigma)\\|_{\\infty}\\\\\n&\\leq\\|\\phi_n\\|_{\\infty}^2\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)^2\\|h_n(\\sigma)\\|_{\\infty},\n\\end{align*}\nwhich combined with the estimate $\\|L_n(h_n)(\\sigma)\\|_{\\infty}\\leq 2\\|\\phi_n\\|_1\\|h_n(\\sigma)\\|_{\\infty}$, gives \n\\begin{align*}\n\\|h_n(\\tau)\\|_{\\infty}&\\leq\\|h_0\\|_{\\infty}\n+\\int_0^{\\tau}\\big(\\|K_n(h_n)(\\sigma)\\|_{\\infty}+\\|L_n(h_n)(\\sigma)\\|_{\\infty}\\big)d\\sigma\\\\\n&\\leq \\|h_0\\|_{\\infty}+C(n,h_0)\\int_0^{\\tau}(\\sigma^2+1)\\|h_n(\\sigma)\\|_{\\infty}d\\sigma.\n\\end{align*}\nwhere\n\\begin{eqnarray*}\nC(n,h_0)=\\max\\left\\{\\|\\phi _n\\|_1\\|\\phi _n\\| ^2_{ \\infty } \\|h_0\\|_1,\\, \\frac {\\|\\phi _n\\| ^2_{ \\infty }} {4}\\|h_0\\| _{ L^1_x }\\right\\}.\n\\end{eqnarray*}\nThen \\eqref{a priori sup norm} follows from Gronwall's inequality.\n\nFor the proof of (\\ref{MAhn}) we use (\\ref{REGWEAK}) with $\\varphi(x)=x^{\\alpha}$ for $\\alpha\\geq 3$:\n\\begin{eqnarray}\n\\label{S5E902}\n\\frac {d} {d\\tau }M_{\\alpha}(h_n(\\tau))=\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau)).\n\\end{eqnarray}\nSince:\n\\begin{align}\n\\label{MAL}\n\\mathcal{L}(\\varphi)(x)=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)x^{\\alpha+1}\\geq 0,\n\\end{align}\nwe have,\n\\begin{eqnarray*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau))\\leq \n 2\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^x \\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)h_n(\\tau,x)h_n(\\tau,x) dydx.\n\\end{eqnarray*}\nThen, we write\n$\\Lambda(\\varphi)(x,y)=x^{\\alpha}\\big((1+z)^{\\alpha}+(1-z)^{\\alpha}-2\\big),$\nwhere $z=y\/x$, and by Taylor's expansion around $z=0$:\n\\begin{align*}\nu(z)\\leq\\frac{\\|u''\\|_{\\infty}}{2}z^2\\leq \\alpha(\\alpha-1)2^{\\alpha-3}z^2.\n\\end{align*}\nHence for all $0\\leq y\\leq x$:\n\\begin{align}\n\\label{MAQ}\n\\Lambda(\\varphi)(x,y)&\\leq C_{\\alpha}x^{\\alpha-2}y^2,\\qquad\\text{where}\\qquad C_{\\alpha}=\\alpha(\\alpha-1)2^{\\alpha-3},\n\\end{align}\nand then, using $\\phi_n(x)\\phi_n(y)\\leq y^{-1}$ and (\\ref{conservation of energy}),\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau) \\leq 2C_{\\alpha}M_{\\alpha-2}(h_n(\\tau))E.\n\\end{align*}\nSince  by Holder's inequality and (\\ref{conservation of energy})\n\\begin{align*}\nM_{\\alpha-2}(h_n(\\tau))\\leq E^{\\frac{2}{\\alpha-1}}M_{\\alpha}(h_n(\\tau))^{\\frac{\\alpha-3}{\\alpha-1}},\n\\end{align*} \nwe deduce\n\\begin{align*}\n\\frac{d}{d\\tau}\\left(M_{\\alpha}(h_n(\\tau))^{\\frac{2}{\\alpha-1}}\\right)\\leq\\frac{4C_{\\alpha}}{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}},\n\\end{align*}\nand (\\ref{MAhn}) follows.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{Ex1T1}.}\n\nThe solution $h$ whose existence is claimed in Theorem \\ref{Ex1T1}  is obtained as the limit of a subsequence of solutions  $(h_n) _{ n\\in \\mathbb{N} }$  to the regularized problems obtained in Theorem \\ref{Ex1T2}. We first prove the following Lemma.\n\n\\begin{lemma}\n\\label{precomp}\nLet $h_0\\in C_c([0,\\infty))$ be nonnegative with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$, and consider $(h_n)_{ n\\in \\mathbb{N} }$ the sequence of functions  given by Theorem \\ref{Ex1T2}. Then for every $\\tau\\in[0,\\infty)$ there exists a subsequence, still denoted $(h_n(\\tau))_{n\\in\\mathbb{N}}$, and a measure $h(\\tau)\\in\\mathscr{M}^1_+([0,\\infty))$ such that, as $n\\to \\infty$,  \n$h_n(\\tau)$ converges to $h(\\tau)$ in the following sense:\n\\begin{align}\n\\label{growth condition}\n&\\forall\\varphi \\in C([0, \\infty));\\;\\exists\\theta\\in [0, 1): \\quad \\sup _{ x\\ge 0} \\frac {\\varphi (x)} {1+x^\\theta}<\\infty,\\\\\n\\label{sense of convergence}\n&\\lim _{ n\\to \\infty }\\int_{[0,\\infty)}\\varphi(x)h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{align}\nMoreover, for every $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{pre conservation laws for widetildeG}\n&M_0(h(\\tau))\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2,\\\\\n\\label{pre energy}\n&M_1(h(\\tau))\\leq E.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nLet us prove first the convergence for a subsequence of $(h_n(\\tau))_{n\\in\\mathbb{N}}$. For every $\\tau\\geq 0$ we have by (\\ref{mass inequality}) that \n\\begin{align*}\n\\sup_{n\\in\\mathbb{N}}\\int_0^\\infty h_n(\\tau,x)d x\\leq \\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align*}\nTherefore, there exists a subsequence, still denoted $(h_n(\\tau))_{n\\in\\mathbb{N}}$, and a measure $h(\\tau)$ such that \n $(h_n(\\tau))_{n\\in\\mathbb{N}}$  converges to $h(\\tau)$ in the weak* topology of $\\mathscr{M}([0,\\infty))$, as $n\\to \\infty$: \n\\begin{equation}\n\\label{weak* 1}\n\\lim _{ n\\to \\infty  }\\int_{[0,\\infty)}\\!\\!\\!\\varphi(x)h_{n}(\\tau,x)d x=\n\\int_{[0,\\infty)}\\!\\!\\varphi(x)h(\\tau,x)d x,\\,\\,\n\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{equation}\nSince for all $n\\in\\mathbb{N}$, $h_n(\\tau)$ is nonnegative, then $h(\\tau)$ is a positive measure. Also by weak* convergence and \\eqref{mass inequality} we deduce that $h(\\tau)$ is a finite measure:\n\\begin{align}\n\\label{weak mass}\n\\int_{[0,\\infty)}h(\\tau,x)d x\\leq\\liminf_{n\\rightarrow\\infty}\\int_0^\\infty h_{n}(\\tau,x)d x\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align}\nMoreover, by (\\ref{conservation of energy}) we also have that the sequence $(h_n(\\tau))_{n\\in\\mathbb{N}}$ is bounded in $L^1_x(\\mathbb{R}_+)$. \nHence there exists a subsequence (not relabelled) that converges to a measure \n$\\nu(\\tau)$ in the weak* topology of $\\mathscr{M}([0,\\infty))$, i.e., such that\n\\begin{equation}\n\\label{weak* 2}\n\\lim _{ n\\to \\infty  }\\int_0^\\infty\\varphi(x)\\,x\\,h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\!\\!\\!\\varphi(x)\\nu(\\tau,x)dx,\\,\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{equation}\nAgain, since $h_n(\\tau)$ is nonnegative for all $n\\in\\mathbb{N}$ then $\\nu(\\tau)$ is a positive measure. Also by weak* convergence and \\eqref{conservation of energy} we have\n\\begin{align}\n\\label{weak energy}\n\\int_{[0,\\infty)}\\nu(\\tau,x)d x\\leq\\liminf_{n\\rightarrow\\infty}\\int_0^\\infty\\,x\\,h_{n}(\\tau,x)d x=E.\n\\end{align}\nLet us show now that $\\nu(\\tau)=x\\,h(\\tau)$. This will follow from \n\\begin{align}\n\\label{equality measures}\n\\forall \\varphi\\in C_0([0,\\infty)):  \\int_{[0,\\infty)}\\varphi(x)\\nu(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\n\\end{align}\n In a first step we show that (\\ref{equality measures}) holds for $\\varphi\\in C_c([0,\\infty))$ and then we use a density argument. \nLet $\\varepsilon>0$ and \n$\\varphi\\in C_c([0,\\infty))$. Using \\eqref{weak* 2} with test function $\\varphi$, and \\eqref{weak* 1} with test function $x\\varphi(x)$, we deduce that \n\\begin{align*}\n\\bigg|\\int_{[0,\\infty)}&\\varphi(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\\bigg |\\\\\n&\\leq \\bigg|\\int_0^\\infty\\varphi(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h_n(\\tau,x)d x\\bigg|\\\\\n&+\\bigg|\\int_0^\\infty\\varphi(x)\\,x\\,h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\\bigg|<\\varepsilon\n\\end{align*}\nfor $n$ large enough. Hence \\eqref{equality measures} holds for all $\\varphi\\in C_c([0,\\infty))$. Now let $\\varphi\\in C_0([0,\\infty))$ and consider a sequence \n$(\\varphi_k)_{k\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ such that \\\\$\\|\\varphi_k-\\varphi\\|_{\\infty}\\rightarrow0$ as $k\\rightarrow\\infty$. \nUsing \\eqref{equality measures} with $\\varphi_k$ and the bounds \\eqref{weak mass} and \\eqref{weak energy}, we deduce that\n\n\\begin{align*}\n\\bigg|\\int_{[0,\\infty)}&\\varphi(x)\t\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x \\bigg|\\\\\n&\\leq\\int_{[0,\\infty)}\\big|\\varphi(x)-\\varphi_k(x)\\big|\\nu(\\tau,x)d x\\\\\n&+\\bigg|\\int_{[0,\\infty)}\\varphi_k(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi_k(x)\\,x\\,h(\\tau,x)d x \\bigg|\\\\\n&+\\int_{[0,\\infty)}\\big|\\varphi_k(x)-\\varphi(x)\\big|\\,x\\,h(\\tau,x)d x<\\varepsilon\n\\end{align*}\nfor $k$ large enough. Therefore \\eqref{equality measures} holds for all $\\varphi \\in C_0([0,\\infty))$, i.e., $\\nu(\\tau)=x\\,h(\\tau)$. Hence we rewrite \\eqref{weak* 2} as \n\\begin{align}\n\\lim _{ n\\to \\infty }\\int_0^{\\infty}\\varphi(x)x\\,h_n(\\tau,x)d x =\\int_{[0,\\infty)} \\!\\!\\!\\!\\!\\!\\varphi(x)x\\,h(\\tau,x)dx, \\,\\,\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{align}\nLet us show now (\\ref{growth condition}), (\\ref{sense of convergence}).\nLet  then  $\\varphi\\in C([0, \\infty))$  be any nonnegative test function that satisfies (\\ref{growth condition}). We denote $(\\zeta _j) _{ j\\in \\mathbb{N} }$ a sequence of nonnegative and nonincreasing functions  of  $C_c^\\infty([0, \\infty))$ such that:\n\\begin{equation*}\n\\zeta_j(x)= 1\\,\\,\\hbox{if}\\,\\,\\,x\\in [0, j),\\qquad\n\\zeta_j(x)= 0\\,\\,\\,\\hbox{if}\\,\\,x>j+1,\n\\end{equation*}\nand  define $\\varphi_j=\\varphi\\,\\zeta_j$. Then for every $n,\\,j\\in\\mathbb{N}$:\n\\begin{align}\n\\label{est 1. lemma convergence for tau fixed}\n\\bigg|\\int_0^{\\infty}&\\varphi(x)h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x \\bigg|\\\\\n&\\leq\\int_0^{\\infty}\\big|\\varphi(x)-\\varphi_j(x)\\big|h_n(\\tau,x)d x\\nonumber\\\\\n&+\\bigg|\\int_0^{\\infty}\\varphi_j(x)h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi_j(x)h(\\tau,x)d x \\bigg|\\nonumber\\\\\n&+\\int_{[0,\\infty)}\\big|\\varphi_j(x)-\\varphi(x)\\big|h(\\tau,x)d x\\nonumber\n\\end{align}\nSince $\\varphi_j\\in C_0([0,\\infty))$, using \\eqref{weak* 2}, the second term in the right hand side of \n\\eqref{est 1. lemma convergence for tau fixed} converges to zero as $n\\rightarrow\\infty$ for every $j\\in\\mathbb{N}$. The first and the third term in the right hand side of \\eqref{est 1. lemma convergence for tau fixed} are treated in the same way, using that $\\varphi _j(x)=\\varphi (x)$ for all $x\\in [0, j)$. For instance, in the first term:\n\\begin{align*}\n\\int_0^{\\infty}\\big|\\varphi(x)&-\\varphi_j(x)\\big|h_n(\\tau,x)d x\n=\\int_j^{\\infty}\\big|\\varphi(x)-\\varphi_j(x)\\big|h_n(\\tau,x)d x\\\\\n&\\leq 2\\int_j^{\\infty}|\\varphi(x)| h_n(\\tau,x)d x \\leq 2C\\int_j^{\\infty}(1+x^{\\theta})h_n(\\tau,x)d x\\\\\n&\\leq 2C\\left(\\frac{1+j^{\\theta}}{j}\\right)\\int_j^{\\infty}x\\,h_n(\\tau,x)d x\\leq 2C\\left(\\frac{1+j^{\\theta}}{j}\\right)E.\n\\end{align*}\nTherefore this term is small provided $j$ is large enough. In conclusion, the difference in \\eqref{est 1. lemma convergence for tau fixed} is less than $\\varepsilon$ for $n$ sufficiently large, i.e., \\eqref{sense of convergence} holds.\n\\end{proof}\n\n\n\\begin{remark}\n\\label{remark. weak* topology generated by a distance}\nThe so-called narrow topology $\\sigma(\\mathscr{M}([0,\\infty)),C_b([0,\\infty)))$ on $\\mathscr{M}_+([0,\\infty))$ is generated by the metric\n$d(\\mu,\\nu)=\\|\\mu-\\nu\\|_0$, where\n\\begin{align*}\n\\|\\mu\\|_0=\\sup\\left\\{\\int_{[0,\\infty)}\\varphi d\\mu:\\varphi\\in\\text{Lip}_1([0,\\infty)),\\;\\|\\varphi\\|_{\\infty}\\leq 1\\right\\},\n\\end{align*}\n(cf. \\cite{BOG} Theorem 8.3.2).\n\\end{remark}\n\nUsing this Remark, Lemma \\ref{precomp} and the Arzel\\`{a}-Ascoli's Theorem we prove now the following:\n\\begin{proposition}\n\\label{equicont}\nLet $h_0$ and $(h_n) _{ n\\in \\mathbb{N} }$ be as in Lemma \\ref{precomp}.\nThen there exist a subsequence (not relabelled) and $h\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ such that \n\\begin{align}\n\\label{S5E90}\nh_n\\xrightarrow[n\\rightarrow\\infty]{}h\\quad\\text{in}\\quad C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big).\n\\end{align}\nMoreover, if we denote by $N=M_0(h_0)$ and $E=M_1(h_0)$, then for all $\\tau\\geq 0$ \n\\begin{align}\n\\label{MASS IN}\n&M_0(h(\\tau))\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2,\\\\\n\\label{EN IN}\n&M_1(h(\\tau))\\leq E,\n\\end{align}\nand for all $\\varphi\\in C([0,\\infty))$ satisfying the growth condition (\\ref{growth condition}):\n\\begin{equation}\n\\label{sense of convergence 2}\n\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Proposition \\ref{equicont}}]\nBy Lemma \\ref{precomp} the sequence \n$(h_n(\\tau))_{n\\in\\mathbb{N}}$ is relatively compact in $\\mathscr{M}([0,\\infty))$ for every $\\tau\\in[0,\\infty)$. Let us show now that $(h_n)_{n\\in\\mathbb{N}}$ is also equicontinuous. To this end let $\\tau_2\\geq\\tau_1\\geq 0$, and consider $\\varphi$ as in Remark \\ref{remark. weak* topology generated by a distance}, i.e., \n$\\varphi\\in\\text{Lip}([0,\\infty))$ with Lipschitz constant $\\text{Lip}(\\varphi)\\leq 1$, and $\\|\\varphi\\|_{\\infty}\\leq 1$.\nThen, using $\\phi_n(x)\\leq x^{-1\/2}$, (\\ref{lemma regularity 1}) and (\\ref{lemma regularity 4}) in Lemma \\ref{lemma regularity}, we have\n\\begin{align}\n&\\bigg|\\int_0^{\\infty}\\varphi(x)h_n(\\tau_1,x)d x-\\int_0^{\\infty}\\varphi(x)h_n(\\tau_2,x)d x\\bigg|\\nonumber \\\\\n&\\leq\\int_{\\tau_1}^{\\tau_2}\\big|\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))\\big|d\\sigma\\leq 2\\int_{\\tau_1}^{\\tau_2}\\bigg(\\int_0^{\\infty}h_n(\\sigma,x)d x\\bigg)^2d\\sigma\\nonumber\\\\\n&+4\\int_{\\tau_1}^{\\tau_2}\\int_0^{\\infty}\\sqrt{x}\\,h_n(\\sigma,x)d xd\\sigma.\\label{equicontinuity 1}\n\\end{align}\nUsing H\\\"{o}lder's inequality and the estimates (\\ref{mass inequality}) and (\\ref{conservation of energy}) in (\\ref{equicontinuity 1}), it follows that \n\\begin{align*}\n&\\bigg|\\int_0^{\\infty}\\varphi(x)h_n(\\tau_1,x)d x-\\int_0^{\\infty}\\varphi(x)h_n(\\tau_2,x)d x\\bigg|\\\\\n&\\leq 2\\int_{\\tau_1}^{\\tau_2}\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)^4d\\sigma\n+4\\sqrt{E}\\int_{\\tau_1}^{\\tau_2}\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)d\\sigma\\quad\\forall n\\in\\mathbb{N}.\\nonumber\n\\end{align*}\nWe then deduce using Remark \\ref{remark. weak* topology generated by a distance} that $(h_n)_{n\\in\\mathbb{N}}$ is equicontinuous.\nIt then follows from Arzel\\`{a}-Ascoli's Theorem (cf. for example \\cite{Roy})  that there exists $h\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ such that\n$h_n\\rightarrow h$ in $C\\big([0,T], \\mathscr{M}_+([0,\\infty))\\big)$, for every $T>0$, as $n\\rightarrow\\infty$. \n\nThe estimates \n(\\ref{MASS IN}), (\\ref{EN IN}) and the convergence (\\ref{sense of convergence 2}) are deduced in the same way as in the Proof of  Lemma \\ref{precomp}.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{Ex1T1}}]\nBy Corollary \\ref{APD1}, there exists a sequence of nonnegative function $(h_{0,n})_{n\\in\\mathbb{N}}\\in C_c([0,\\infty))$ that approximate $h_0$ in the weak* topology of the space $C_b([0,\\infty))^*$.\nLet then  $(h_n) _{ n\\in \\mathbb{N} }\\subset  C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ be the sequence of solutions to (\\ref{S5E765}), (\\ref{REGWEAK})\nobtained by Theorem \\ref{Ex1T2} with the initial data $h_{0,n}$.  By Proposition \\ref{equicont}  there exists a subsequence, still denoted $(h_n) _{ n\\in \\mathbb{N} }$, and $h\\in  C\\big([0,\\infty), \\mathscr{M}_+([0,\\infty))\\big)$ such that\n$h_n$ converges to $h$ in the topology of $C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$. \n\nBy (\\ref{REGWEAK}) and (\\ref{datan}), for all  $\\varphi \\in C^1_b([0, \\infty))$ and $\\tau >0$:\n\\begin{equation}\n\\label{S5E456}\n\\int_0^{\\infty}\\varphi (x)h_n(\\tau , x)dx-\\int_0^{\\infty}\\varphi (x)h _{ 0, n }( x)dx=\\int _0^\\tau \\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))d\\sigma. \n\\end{equation}\nBy construction, for every $\\varphi\\in C_b^1([0,\\infty))$ and every $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{LIM55}\n\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)h_n(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{align}\nWe prove now the convergence of the linear term: for all $\\varphi\\in C_b^1([0,\\infty))$ and $\\tau\\in[0,\\infty)$\n\\begin{align}\n\\label{linear limit 3}\n\\lim_{n\\to\\infty}\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi, h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi, h_n(\\tau)).\n\\end{align}\nBy definition:\n\\begin{align}\n&\\Big|\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h(\\tau))- \\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h_n(\\tau))\\Big|\\nonumber \\\\\n&\\leq  \\bigg|\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h(\\tau,x)d x-\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h_n(\\tau,x)d x \\bigg|\\nonumber\\\\\n&+\\int_0^{\\infty}\\bigg|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg|h_n(\\tau,x)d x. \\label{linear limit 1}\n\\end{align}\nFrom Lemma \\ref{lemma regularity} (iii) and (\\ref{sense of convergence 2}):\n\\begin{equation}\n\\lim _{ n\\to \\infty } \\bigg|\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h(\\tau,x)d x-\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h_n(\\tau,x)d x \\bigg|=0\n\\end{equation}\nFor the second term  in the right hand side of (\\ref {linear limit 1}) we split the integral $\\int _0^\\infty$ in two:  $\\int_0^R$ and  $\\int_R^{\\infty}$ for   $R>0$,  \nand apply (\\ref{lemma regularity 4}). We obtain:\n\\begin{align}\n\\label{linear limit 2}\n\\int_0^{\\infty}&\\bigg|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg|h_n(\\tau,x)d x  \\\\\n&\\leq \\bigg\\|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg\\|_{C([0,R])}\\int_0^R h_n(\\tau,x)d x+\\nonumber\\\\\n&\\hskip 4.5cm +4\\|\\varphi\\|_{\\infty}\\int_R^{\\infty}\\sqrt{x}\\;h_n(\\tau,x)d x\\nonumber.\n\\end{align}\nBy (\\ref{conservation of energy}), for any $\\varepsilon >0$ and $R> (E\/\\varepsilon )^2$:\n\\begin{align*}\n\\int_R^{\\infty}\\sqrt{x}\\;h_n(\\tau,x)d x \\leq\\frac{E}{\\sqrt{R}}<\\varepsilon \\qquad\\forall n\\in\\mathbb{N}.\n\\end{align*}\nThen by Lemma \\ref{regularised operators converge uniformly} and (\\ref{mass inequality}), the part on $[0,R]$ converges to zero as $n\\to\\infty$. \nSince $R>0$ is arbitrary we finally deduce that (\\ref{linear limit 2}) converges to zero as $n\\to\\infty$.\nTherefore (\\ref{linear limit 3}) holds.\n\nLet us prove now the convergence of the quadratic term: for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{quadratic limit 2}\n\\lim_{n\\to\\infty}\\mathscr{Q}_{3,n}^{(2)}(\\varphi, h_n(\\tau))=\\mathscr{Q}_3^{(2)}(\\varphi, h_n(\\tau)).\n\\end{align}\nAs before\n\\begin{align}\n\\label{quadratic limit 1}\n&\\bigg|\\mathscr{Q}_{3}^{(2)}(\\varphi,h(\\tau))-\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h_n(\\tau))\\bigg|\\\\\n&\\leq\\bigg|\\mathscr{Q}_{3}^{(2)}(\\varphi,h(\\tau))-\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}h_n(\\tau,x)h_n(\\tau,y)d xd y\\bigg|\\nonumber\\\\\n&+\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\nonumber.\n\\end{align}\nIt follows from Lemma \\ref{lemma regularity} (ii) and (\\ref{sense of convergence 2})\nthat the first term in the right hand side above converges to zero as $n\\rightarrow\\infty$.\nFor the second term we proceed as before. For any $R>0$ we split the double integral:\n\\begin{align*}\n&\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq \\bigg\\|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg\\|_{C([0,R]^2)}\\left(\\int_0^R h_n(\\tau,x)dx\\right)^2\\nonumber\\\\\n&+\\iint_{(0,\\infty)^2\\setminus (0,R)^2}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&=I_1+I_2.\n\\end{align*}\nBy Lemma \\ref{regularised operators converge uniformly} and (\\ref{mass inequality}), $I_1$ converges to zero as \n$n\\to\\infty$.\nFor the term $I_2$ we use (\\ref{lemma regularity 1}) in Lemma \\ref{lemma regularity} and the estimates (\\ref{conservation of energy}) and (\\ref{mass inequality}):\n\\begin{align*}\n&\\int_R^{\\infty}\\!\\!\\!\\!\\int_R^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|\nh_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq 4\\|\\varphi'\\|_{\\infty}\\bigg(\\int_R^{\\infty}h_n(\\tau,x)d x\\bigg)^2\n\\leq \\frac{4\\|\\varphi'\\|_{\\infty}E^2}{R^2}\\qquad\\forall n\\in\\mathbb{N}\n\\end{align*}\nand\n\\begin{align*}\n&2\\int_R^{\\infty}\\!\\!\\!\\int_0^R\\bigg|\\Lambda(\\varphi)(x)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x)}{\\sqrt{xy}}\\bigg|\nh_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq4\\|\\varphi'\\|_{\\infty}\\int_R^\\infty\\int_0^Rh_n(\\tau,x)h_n(\\tau,y)d xd y\n\\leq \\frac{4 \\|\\varphi'\\|_{\\infty}E}{R}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align*}\nSince $R>0$ is arbitrary we deduce that $I_2$ also converges to zero as $n\\to\\infty$.\nWe then conclude that (\\ref{quadratic limit 2}) holds.\n\nCombining (\\ref{linear limit 3}) and (\\ref{quadratic limit 2}) it follows that for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$: \n\\begin{align}\n\\label{LIMQ3N}\n\\lim _{ n\\to \\infty  }\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau)).\n\\end{align}\nMoreover, using $\\phi_n(x)\\leq x^{-1\/2}$, (\\ref{lemma regularity 1}) and (\\ref{lemma regularity 4}) in Lemma \\ref{lemma regularity}, and the estimates\n(\\ref{mass inequality}) and (\\ref{conservation of energy}), we have for all $\\varphi\\in C_b^1([0,\\infty))$, all $\\tau\\in[0,\\infty)$ and all $n\\in\\mathbb{N}$:\n\n\\begin{align*}\n&\\Big|\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau))\\Big|\\leq\\\\\n&\\leq 2\\|\\varphi'\\|_{\\infty}\\bigg(\\int_0^{\\infty}h_n(\\tau,x)dx\\bigg)^2+4\\|\\varphi\\|_{\\infty}\\int_0^{\\infty}\\sqrt{x}\\,h_n(\\tau,x)dx\\\\\n&\\leq2\\|\\varphi'\\|_{\\infty}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^4+4\\|\\varphi\\|_{\\infty}\\sqrt{E}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg).\n\\end{align*}\nBy  (\\ref{LIMQ3N}) and the dominated convergence Theorem:\n\\begin{align}\n\\label{CQ3I}\n\\lim _{ n\\to \\infty  }\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))d\\sigma=\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{align}\nUsing now (\\ref{LIM55}) and (\\ref{CQ3I}), we may pass to the limit as $n\\to \\infty$ in (\\ref{S5E456}) for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$ to obtain:\n\\begin{equation}\n\\label{S5E963}\n\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{equation}\nThe map $\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx$ is then locally Lipschitz on $[0,\\infty)$, and  $h$ satisfies  (\\ref{lip loc h}), (\\ref{AUXW}) for all\n$\\varphi\\in C_b^1([0,\\infty))$ and for a.e. $\\tau\\in[0,\\infty)$. It also follows from (\\ref{S5E963}) that $h(0)=h_0$ in $\\mathscr M_+$.\n\nThe property (\\ref{MMI}) follows from (\\ref{MASS IN}). The  conservation of energy (\\ref{EE}) is obtained as follows. We already know by (\\ref{EN IN}) that $M_1(h(\\tau ))\\le E$. On the other hand, let  $\\varphi_k\\in C^1_b([0,\\infty))$ be a concave test function such that $\\varphi_k(x)=x$ for $x\\in[0,k)$ and $\\varphi_k(x)=k+1$ for $x\\geq k+2$. Notice that there exists a positive constant $C$ such that\n\\begin{align}\n\\label{SUP9}\n\\sup_{k\\in\\mathbb{N}}\\|\\varphi_k'\\|_{\\infty}\\leq C.\n\\end{align}\nBy Remark \\ref{concave-negativity}, $\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_k,h)\\leq 0$ and $\\mathscr{Q}_3^{(2)}(\\varphi_k,h)\\leq 0$ for all $k\\in\\mathbb{N}$,\nand then, from (\\ref{S5E963}):\n\\begin{align}\n\\label{INW1}\n\\int_{[0,\\infty)}\\!\\!\\!\\varphi_k(x)h(\\tau,x)dx\\geq\\int_{[0,\\infty)}\\!\\!\\!\\varphi_k(x)h_0(x)dx+\\int_0^{\\tau}\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))d\\sigma.\n\\end{align}\nWe now prove that for all $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{DOMQ32}\n\\lim _{ k\\to \\infty }\\int_0^{\\tau}\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))d\\sigma =0.\n\\end{align}\nNotice that $\\Lambda(\\varphi_k)(x,y)\\to 0$ as $k\\to\\infty$, since $\\varphi_k(x)\\to x$. Then, using (\\ref{lemma regularity 1}) in Lemma \\ref{lemma regularity}, (\\ref{SUP9}) and (\\ref{mass inequality}), we deduce for all $\\tau\\in[0,\\infty)$ and $\\sigma \\in (0, \\tau )$:\n\\begin{align}\n&\\lim _{ k\\to \\infty }\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))=0\\\\\n&\\Big| \\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\tau))\\Big|\\leq 2C\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^4\\qquad\\forall k\\in\\mathbb{N}.\n\\end{align}\nand  (\\ref{DOMQ32}) follows from the dominated convergence Theorem. We take now limits in (\\ref{INW1}) as $k\\to \\infty$. By (\\ref{DOMQ32}) and the monotone convergence Theorem we obtain that $M_1(h(\\tau ))\\ge E$ and then $M_1(h(\\tau ))=E$ for all $\\tau >0$.\n\nWe assume now that $M_\\alpha (h_0)<\\infty $ for some $\\alpha \\geq 3$ and prove (\\ref{MAh}). By  (\\ref{MAhn}) and \n(\\ref{APD56}) in Corollary \\ref{APD1}:\n\\begin{align}\nM_{\\alpha}(h(\\tau))&\\leq  \\liminf_{ n\\to \\infty } \\left(M_{\\alpha}(h _{ 0, n })^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}M_1(h_{0,n})^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}\\\\\n&\\leq\\left(M_{\\alpha}(h _{ 0})^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}.\n\\end{align}\n\\end{proof}\n\n\\subsection{ Proof of Theorem \\ref{S5T5R}}\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S5T5R}}] \nConsider again the sequence of initial data $h _{ 0, n }$  used in the proof of Theorem \\ref{Ex1T1} and the sequence of solutions  $h _{ n }$  obtained  by Theorem \\ref{Ex1T1}. Using  (\\ref{MAhn})  we know that $M_\\alpha (h_n(\\tau ))<\\infty$ for $\\tau >0$ and $n\\in \\mathbb{N}$. \n\nOur first step is to prove that (\\ref{AUXW}) holds also true for $\\varphi (x)=x^\\alpha $. Notice that $h_n$ solves now the equation (\\ref{AUXW}), with the operator $\\widetilde{\\mathscr Q}_3$ in the right-hand side, whose kernel is not compactly supported and the argument in the proof of (\\ref{MAhn}) must be slightly modified.\n\nIn order to use (\\ref{AUXW}) we consider a sequence $(\\varphi_k)\\subset C^1_b([0,\\infty))$ such that:\\begin{align}\n\\label{M7a5}\n&\\varphi_k\\to\\varphi\\quad\\text{as}\\quad k\\to\\infty\\\\\n\\label{M7a6}\n&\\varphi_k\\leq\\varphi_{k+1}\\leq \\varphi\\\\\n\\label{M7a7}\n&\\varphi'\\geq\\varphi_k'\\geq 0.\n\\end{align}\nLet us prove by the dominated convergence Theorem  that for all $\\tau\\geq 0$:\n\\begin{align}\n\\label{Q332}\n(i)\\quad&\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n)\\in L^1(0,\\tau),\\\\\n\\label{Q333}\n(ii)\\quad&\\lim _{ k\\to \\infty }\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi_k,h_n(\\sigma))d\\sigma=\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\sigma))d\\sigma.\n\\end{align}\nTo this end we first observe that, for $x\\ge y>0$:\n\\begin{align}\n\\label{MAJ1}\n\\lim _{ k\\to \\infty }\\Lambda(\\varphi_k)(x, y)=\\Lambda(\\varphi)\\qquad\\text{and}\\qquad \\lim _{ n\\to \\infty  }\\mathcal{L}(\\varphi_k)=\\mathcal{L}(\\varphi)(x)\n\\end{align}\nand, by the mean value Theorem:\n\\begin{align*}\n\\frac{\\Lambda(\\varphi_k)(x,y)}{\\sqrt{xy}}\\leq \\varphi_k'(\\xi_1)-\\varphi_k'(\\xi_2)\n\\end{align*}\nfor some $\\xi_1\\in (x,x+y)$ and $\\xi_2\\in(x-y,x)$. Using then (\\ref{M7a7}):\n\\begin{align}\n\\label{MAJ2}\n\\frac{|\\Lambda(\\varphi_k)(x,y)|}{\\sqrt{xy}}\\leq \\alpha\\big(2^{\\alpha-1}+1\\big)x^{\\alpha-1}\\qquad\\forall k\\in\\mathbb{N},\n\\end{align}\nand by (\\ref{M7a6}):\n\\begin{align*}\n\\frac{|\\mathcal{L}(\\varphi_k)(x)|}{\\sqrt{x}}\\leq \\left(\\frac{\\alpha+3}{\\alpha+1}\\right)x^{\\alpha+\\frac{1}{2}}\\qquad\\forall k\\in\\mathbb{N}.\n\\end{align*}\nSince by Theorem \\ref{Ex1T1}: $M_{\\alpha-1}(h_n(\\tau))<\\infty$ and $M_{\\alpha+1\/2}(h_n(\\tau))<\\infty$, for every fixed $n$  we may apply the Lebesgue's convergence Theorem \nto  the sequences $\\left\\{\\frac{\\Lambda(\\varphi_k)(x,y)}{\\sqrt{xy}}h_n(\\sigma , x)h_n(\\sigma , y)\\right\\} _{ k\\in \\mathbb{N} }$ and\n$\\left\\{\\frac{\\mathcal{L}(\\varphi_k)(x)}{\\sqrt{x}}h_n(\\sigma , x)\\right\\} _{ k\\in \\mathbb{N} }$ and obtain (\\ref{Q332}), (\\ref{Q333}).\n\nWe use now $\\varphi_k$ in (\\ref{AUXW}) and take the limit $k\\to\\infty$. We obtain from (\\ref{M7a5}), (\\ref{M7a6}), (\\ref{Q333}) and monotone convergence:\n\\begin{align}\nM_{\\alpha}(h_n(\\tau))=M_{\\alpha}(h _{ 0, n })+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\sigma))d\\sigma\\qquad\\forall\\tau\\geq 0,\n\\end{align}\nand then, using (\\ref{Q332}):\n\\begin{align}\n\\label{EQMA}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\tau))\\qquad a.e.\\;\\tau>0.\n\\end{align}\nIf we use  (\\ref{MAL}) and (\\ref{MAQ}) in the right hand side of (\\ref{EQMA}), we obtain\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n M_{\\alpha-2}(h_n)-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)M_{\\alpha+\\frac{1}{2}}(h_n),\n\\end{align*}\nwhere $E_n=M_1(h_{0,n})$.\nNow by H\\\"{o}lder's inequality:\n\\begin{align*}\n&M_{\\alpha-2}(h_n)\\leq E_n^{2\/(\\alpha-1)}M_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\\\\n&M_{\\alpha}(h_n)\\leq E_n^{1\/(2\\alpha-1)}M_{\\alpha+\\frac{1}{2}}(h_n)^{2(\\alpha-1)\/(2\\alpha-1)}.\n\\end{align*}\nThen we obtain\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)&\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n^{1+2\/(\\alpha-1)}M_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\\\\n&-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E_n^{-1\/(2(\\alpha-1))}M_{\\alpha}(h_n)^{(2\\alpha-1)\/(2(\\alpha-1))}.\\nonumber\n\\end{align*}\nSince $(\\alpha-3)\/(\\alpha-1)\\in [0,1)$ then\n\\begin{align*}\nM_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\leq 1+M_{\\alpha}(h_n),\n\\end{align*}\nand :\n\\begin{align}\n\\label{MOM19}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)&\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n^{1+2\/(\\alpha-1)}\\big(1+M_{\\alpha}(h_n)\\big)\\\\\n&-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E_n^{-1\/(2(\\alpha-1))}M_{\\alpha}(h_n)^{(2\\alpha-1)\/(2(\\alpha-1))},\\nonumber\n\\end{align}\nwhere $(2\\alpha-1)\/(2(\\alpha-1))>1$. If we define:\n\\begin{align*}\n&u(\\sigma)=M_{\\alpha}(h_n(\\tau)),\\quad\\sigma=C_1\\tau,\\quad q=2(\\alpha-1),\\\\\n&C_1=2^{\\alpha-2}\\alpha(\\alpha-1)E^{1+2\/(\\alpha-1)}\\\\\n&C_2=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E^{-1\/(2(\\alpha-1))},\\quad C=\\frac {C_2} {C_1}.\n\\end{align*}\nWe deduce from (\\ref{MOM19}) that\n\\begin{align}\nu'\\leq1+u-Cu^{1+1\/q},\n\\end{align}\nand then by  Lemma 6.3 in \\cite{Bob}, for every $n\\in \\mathbb{N}$:\n\\begin{equation}\nM_{\\alpha}(h_n(\\tau))\\leq C(\\alpha,E_n)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E_n)\\tau}}\\right)^{2(\\alpha-1)},\n\\end{equation}\nwhere the constants $C(\\alpha,E_n)$ and $\\gamma (\\alpha,E_n)$ are defined as in Theorem \\ref{S5T5R}. We may argue  now as in the proof of Theorem \\ref{Ex1T1}\nand  pass to the limit along a subsequence to obtain a limit $h\\in C\\big([0,\\infty), \\mathscr{M}_+ ([0,\\infty))\\big)$ satisfying (\\ref{lip loc h})--(\\ref{EE}) and (\\ref{S5Ealpha }). Using (\\ref{S5Ealpha }) and $h\\in C\\big([0,\\infty), \\mathscr{M}_+ ([0,\\infty))\\big)$ we deduce  as in the proof of Theorem \\ref{Ex1T1} that \n$h\\in C\\big((0,\\infty), \\mathscr{M}_+^\\alpha  ([0,\\infty))\\big)$ for all $\\alpha\\geq 1$.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Corollary \\ref{S5C52R}}] \nWe first observe that  the map $\\tau\\mapsto M_{1\/2}(h(\\tau))$ is locally bounded. Indeed by H\\\"{o}lder's inequality, (\\ref{MMI}) and (\\ref{EE}):\n$$\nM_{1\/2}(h(\\tau))\\leq\\sqrt{M_1(h(\\tau))M_0(h(\\tau))}\\leq\\sqrt{E}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg).\n$$\nThen by (\\ref{lip loc h}) it follows (\\ref{lip loc H}). Now for all $\\varphi\\in C^1_b([0,\\infty))$ and for a.e. $\\tau> 0$, we deduce from (\\ref{AUXW}): \n\\begin{align*}\n\\quad\\frac{d}{d\\tau}\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)dx&=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))-\\varphi(0)M_{1\/2}(h(\\tau))\\\\\n&=\\mathscr{Q}_3(\\varphi,h(\\tau)).\n\\end{align*}\nSince $H=h$ on $(0,\\infty)$ then $\\mathscr{Q}_3(\\varphi,H)\\equiv\\mathscr{Q}_3(\\varphi,h)$, and therefore (\\ref{AUXWH}) holds.\n\nNow for the initial data: $H(0)=h(0)=h_0$.\nThe conservation of mass (\\ref{MMH}) follows from (\\ref{AUXWH}) for $\\varphi=1$, since $\\Lambda(\\varphi)=0=\\mathcal{L}_0(\\varphi)$. The conservation of energy (\\ref{EEH}) follows directly from (\\ref{EE}) since $H=h$ on $(0,\\infty)$. \n\\end{proof}\n\n\\section{Properties of $h(\\tau , \\{0\\})$.}\n\\label{SectionC}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nIn all this Section we denote\n\\begin{equation}\n\\label{mZ}\nm(\\tau )=h(\\tau , \\{0\\}).\n\\end{equation}\nThe main result of this Section  is the following.\n\\begin{theorem}\n\\label{S1T4h}\nSuppose that $h\\in C([0, \\infty); \\mathscr{M}_+^1([0,\\infty))$ is a solution of (\\ref{S1E16ha}) with  $h(0)=h_0\\in\\mathscr{M}_+^1([0,\\infty))$, $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$. \nThen $m$ is right continuous, a.e. differentiable and strictly increasing on $[0,\\infty)$.\n\\end{theorem}\nWe begin with the following properties of the function $m$ defined in (\\ref{mZ}).\n\\begin{lemma}\n\\label{S6L1'}\nThe function $m$ is nondecreasing, a.e. differentiable and right continuous on $[0,\\infty)$.\n\\end{lemma}\n\n\\begin{proof}\nGiven any $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, then for all $\\tau \\ge 0$\n\\begin{align}\n&m(\\tau)=\\lim _{ \\varepsilon \\to 0 }\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx,\\label{S6L1E1}\n\\end{align}\nand by (\\ref{S1E16ha})-(\\ref{S1EB2})\n\\begin{align}\n\\frac{d}{d\\tau}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx=\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))\n-\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h(\\tau)). \\label{S6E47}\n\\end{align}\nSince $\\varphi_{\\varepsilon}$ is convex, nonnegative and decreasing, it follows from Lemma \\ref{convex-positivity} \nthat $\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\geq 0$ and \n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h)\\leq 0$ for all $\\varepsilon>0$. \nThen by (\\ref{S6E47})\n\\begin{equation*}\n\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau_2,x)d x\\geq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau_1,x)dx\\qquad\\forall\\tau_2\\geq\\tau_1\\geq 0.\n\\end{equation*}\nLetting $\\varepsilon\\to 0$ it follows from (\\ref{S6L1E1}) that $m$ is nondecreasing on $[0,\\infty)$ and, for any $\\tau\\geq 0$ and $\\delta>0$,\n\\begin{align}\n\\label{right continuity 1}\n\\liminf_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)\\geq m(\\tau).\n\\end{align}\nUsing Lebesgue's Theorem (cf. \\cite{Roy}), $m$ is a.e. differentiable on $[0,\\infty)$. \nOn the other hand, if in (\\ref{S6E47}) the term $\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h)$ is dropped,\n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau+\\delta,x)&d x\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\n+\\int_{\\tau}^{\\tau+\\delta}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma.\n\\end{align*}\nUsing $\\mathds{1}_{\\{0\\}}\\leq\\varphi_{\\varepsilon}$ for all $\\varepsilon>0$, and  the bound \\eqref{lemma regularity 1}, we deduce\n\\begin{equation*}\nm(\\tau+\\delta)\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\n+\\frac {2\\delta} {\\varepsilon }(M_0(h(\\tau)))^2.\n\\end{equation*}\nIf we take now superior limits as $\\delta\\to 0^+$\nat $\\varepsilon >0$ fixed,\n\\begin{align*}\n\\limsup_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)&\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\\qquad\\forall\\varepsilon>0.\n\\end{align*}\nWe may pass now to the limit  as $\\varepsilon \\to 0$ in the right hand side above\nand use (\\ref{S6L1E1}) to get,\n\\begin{align}\n\\label{right continuity 2}\n\\limsup_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)\\leq m(\\tau).\n\\end{align}\nThe right continuity then follows from \\eqref{right continuity 1} and \\eqref{right continuity 2}.\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{S6L1}\nThe map $\\tau\\mapsto H(\\tau,\\{0\\})$, defined for all $\\tau \\ge 0$, is right continuous on $[0,\\infty)$ and \n\\begin{align}\n\\label{JIM}\n\\limsup_{\\delta\\rightarrow 0^+}H(\\tau-\\delta,\\{0\\})\\leq H(\\tau,\\{0\\})\\qquad\\forall\\tau> 0.\n\\end{align}\n\\end{corollary}\n\n\\begin{proof}\nBy construction (cf.(\\ref{S1EdecompH})) it follows\n\\begin{align*}\nH(\\tau,\\{0\\})=m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{align*}\nSince $M_{1\/2}(h)\\in L^1_{loc}(\\mathbb{R}_+)$ then $\\tau\\mapsto\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma$ is absolutely continuous, \nand since $m$ is right continuous by Lemma \\ref{S6L1'}, it follows that $\\tau\\mapsto H(\\tau,\\{0\\})$ is also right continuous. \nTo prove (\\ref{JIM}) we use the continuity of $\\tau\\mapsto\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma$ and the monotonicity of $h(\\tau,\\{0\\})$:\nfor all $\\tau>0$ and $\\delta\\in(0,\\tau)$,\n\\begin{align*}\n\\limsup_{\\delta\\to 0^+} H(\\tau-\\delta,\\{0\\})&=\\limsup_{\\delta\\to 0^+} m(\\tau-\\delta)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\\\\n&\\leq m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma=H(\\tau,\\{0\\}).\n\\end{align*}\n\\end{proof} \n\n\\begin{remark}\n\\label{S6Ej}\nWe do not know if the map $\\tau\\mapsto H(\\tau,\\{0\\})$ is continuous. By property \\eqref{JIM} however, $H(\\tau,\\{0\\})$ does not decrease through the possible discontinuities.\n\\end{remark}\n\nThe proof of Theorem \\ref{S1T4h} closely follows the proof of Proposition 1.21 in \\cite{AV1} (see also \\cite{KIER}, Ch. 3), where the authors proved the same result for the equation without the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$. The main arguments in the proof are, on the one hand, the invariance of the problem (\\ref{S1E16ha}) with respect to time translation and under a suitable  scaling transformation. On the other hand, the fact that \n$\\Lambda(\\varphi)\\ge 0$ on $\\mathbb{R}_+^2$ for convex test functions $\\varphi $, and that the map $\\tau\\mapsto\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))$ is locally integrable on $[0, T)$. When the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$ is added, a slight modification of these argument still leads to the proof. \nSince by Lemma \\ref{convex-positivity}, for all nonnegative, convex decreasing test function $\\varphi \\in C_b^1([0, \\infty))$, we have\n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h)\\leq 0$, then solutions $h$ to (\\ref{S1E16ha}) are also super solutions (cf. Definition \\ref{SUPER}).\n\n\\begin{proposition}\n\\label{S5P1}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Then for any $R>0$ and $\\theta\\in(0,1)$\n\\begin{align}\n\\int_{[0,R]}h(\\tau ,x)d x\\geq (1-\\theta)\\int_{[0,\\theta R]}h(\\tau_0 ,x)d x\\qquad\\forall \\tau\\geq\\tau_0\\geq 0.\\label{S5EP12}\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nChose $\\varphi _R(x)=(1-x\/R)_+$ for $R>0$, and consider a sequence $(\\varphi_{R,n})_{n\\in\\mathbb{N}}\\subset C_b^1([0, \\infty))$ such that $\\varphi_{R,n}\\to\\varphi_R$,\n$\\varphi _{R,n}\\le\\varphi_{R}$ and $\\varphi_{R,n}(0)=1$ for all $n\\in\\mathbb{N}$. Since by convexity $\\mathscr{Q}_3^{(2)}(\\varphi_{R,n},h)\\geq 0$, then for all\n$\\tau$ and $\\tau_0$ with $\\tau\\geq\\tau_0\\geq 0$,\n\\begin{align*}\n\\int_{ [0, \\infty) } &\\varphi_{ R, n } (x)h(\\tau  , x)dx\\ge \\int_{ [0, \\infty) } \\varphi _{ R, n }(x)h(\\tau_0  , x)dx\\\\\n&\\ge  \\int_{ [0,\\theta R] } \\varphi _{ R, n }(x)h(\\tau_0  , x)dx\n\\ge \\varphi_{ R, n }(\\theta R) \\int_{ [0,\\theta R]}h(\\tau_0  , x)dx,\n\\end{align*}\nand (\\ref{S5EP12}) follows since, if we let $n\\to\\infty$,\n$$\n\\int_{[0,R]}h(\\tau ,x)d x\\geq \\int_{ [0,\\infty) }\\varphi_R(x)h(\\tau  , x)dx\n\\ge \\varphi _R(\\theta R) \\int_{ [0,\\theta R]}h(\\tau_0, x)dx.\n$$\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{S5L1}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Let $R>0$ and consider a sequence $R:=a_0< a_1< a_2<...< a_n<...$ such that \n$|a_i-a_{i-1}|\\leq\\frac{R}{2}$ for all $i\\in\\{1, 2, 3,... \\}$. Then for all $\\tau \\geq\\tau_0 \\geq 0$ there holds\n\\begin{align}\n\\int_{[0,R]}h(\\tau,x)d x\\geq\\sum_{i=1}^\\infty \\frac{1}{2a_i}\\int_{\\tau _0}^{\\tau}\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)d x\\bigg)^2d\\sigma. \\label{S5E97}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nWe chose $\\varphi _R$ and $\\varphi _{R, n }$ as in the proof of Proposition \\ref{S5P1} above. Since $h$ is a super solution of (\\ref{S1E16ha}), then\nfor all $n\\in\\mathbb{N}$,\n$$\n\\frac {d} {d\\tau }\\int  _{ [0, \\infty) }h(\\tau , x)\\varphi  _{ R, n }(x)dx\\ge \n{\\mathscr Q}_3^{(2)}(\\varphi  _{ R, n },h(\\tau)).\n$$\nWe have now:\n\\begin{eqnarray*}\n&&{\\mathscr Q}_3^{(2)}(\\varphi  _{ R, n },h(\\tau)) \\ge \\iint _{ (R, \\infty)^2 }h(\\tau, x)h(\\tau , y)\n\\frac {\\varphi  _{ R, n }(|x-y|)} {\\sqrt {x y}}dxdy\\\\\n&&\\ge \\sum_{ i=1 } ^\\infty\\frac { \\varphi  _{ R, n }(R\/2)} {a_i} \\iint _{(a _{ i-1 }, a_i]^2}h(\\tau, x)h(\\tau , y)dxdy\\\\\n&&=\\sum_{ i=1 } ^\\infty\\frac { \\varphi  _{ R, n }(R\/2)} {a_i} \\left(\\int _{(a _{ i-1 }, a_i]}h(\\tau, x)dx\\right)^2.\n\\end{eqnarray*}\nEstimate (\\ref{S5E97}) follows in the limit $n\\to\\infty$, since $\\varphi_{ R, n }(R\/2)\\to1\/2.$\n\\end{proof}\n\n\\begin{proposition}\n\\label{S5P2}\nLet $h$ be a super solution of (\\ref{S1E16ha}) with initial data $h_0\\in\\mathscr{M}_+^1([0,\\infty))$, and denote $N=M_0(h_0)$ and $E=M_1(h_0)$. Then for all $R>0$, \n$\\alpha\\in\\left(-\\frac{1}{2},\\infty\\right)$, and $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1\\leq\\tau_2$:\n\\begin{align}\n\\label{MNEG6}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq\\frac{2R^{\\frac{1}{2}+\\alpha}\\sqrt{\\tau_2-\\tau_1}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}}\\left(\\frac{\\sqrt{E}}{2}\\tau_2+\\sqrt{N}\\right).\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nSince h is a super solution of (\\ref{S1E16ha}), if we chose $\\varphi(x)=(1-x\/r)^2_+$ for $r>0$, then \n\\begin{align}\n\\label{MNEG4}\n\\int_{[0,\\infty)}\\varphi(x)h(\\tau_2,x)dx\\geq\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))d\\tau.\n\\end{align}\nSince \n$\\operatorname{supp}\\Lambda(\\varphi)=\\{(x,y)\\in[0,\\infty)^2:|x-y|\\leq r\\}$\nand\n$\\Lambda(\\varphi)(x,y)=\\varphi(|x-y|)$ for all $(x,y)\\in[r,\\infty)^2$,\nthen for all $\\tau\\geq 0$:\n\\begin{align*}\n\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))&\\geq \\iint_{\\left(r,\\frac{3r}{2}\\right]^2}\\frac{\\varphi(|x-y|)}{\\sqrt{xy}}h(\\tau,x)h(\\tau,y)dxdy\\\\\n&\\geq\\frac{1}{4}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\\right)^2.\n\\end{align*}\nIf we use that $\\varphi\\leq1$ \nin the left hand side of (\\ref{MNEG4}), and  the estimate above in the right hand side, then\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\\right)^2d\\tau\\leq 4 M_0(h(\\tau_2)).\n\\end{align*} \nSince for any $\\alpha\\in(-1\/2,\\infty)$\n\\begin{align*}\n\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\n\\geq \\left(\\frac{3r}{2}\\right)^{-\\alpha-\\frac{1}{2}}\\int_{\\left(r,\\frac{3r}{2}\\right]}x^{\\alpha}h(\\tau,x)dx,\n\\end{align*}\nwe then obtain\n\\begin{align}\n\\label{MNEG5}\n\\int_{\\tau_1}^{\\tau_2}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}x^{\\alpha}h(\\tau,x)dx\\right)^2d\\tau\\leq 4M_0(h(\\tau_2))\\left(\\frac{3r}{2}\\right)^{1+2\\alpha}.\n\\end{align}\nFor any given $R>0$, using the decomposition \n$$(0,R]=\\bigcup_{k=0}^\\infty(a_{k+1},a_k],\\qquad a_k=\\left(\\frac{2}{3}\\right)^{k}\\!\\!\\!\\!R,$$\nand Cauchy-Schwarz inequality we obtain\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq\\sqrt{\\tau_2-\\tau_1}\\sum_{k=0}^{\\infty}\\left(\\int_{\\tau_1}^{\\tau_2}\\bigg(\\int_{(a_{k+1},a_k]}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!x^{\\alpha}h(\\tau,x)dx\\bigg)^2d\\tau\\right)^{\\frac{1}{2}}.\n\\end{align*}\nIf we chose $r=a_{k+1}$ so that $(a_{k+1},a_k]=(r,(3\/2)r]$ for every $k\\in\\mathbb{N}$, then by (\\ref{MNEG5}) we deduce\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq2\\sqrt{(\\tau_2-\\tau_1)M_0(h(\\tau_2))}\\sum_{k=0}^{\\infty}a_k^{\\frac{1}{2}+\\alpha}.\n\\end{align*}\nUsing the estimate (\\ref{MMI}) for $M_0(h(\\tau_2))$ and \n\\begin{align*}\n\\sum_{k=0}^{\\infty}a_k^{\\frac{1}{2}+\\alpha}\n=\\frac{R^{\\frac{1}{2}+\\alpha}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}},\n\\end{align*}\nwe finally obtain (\\ref{MNEG6}).\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lemma 2^n}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Then for all $r>0$, $\\tau\\geq \\tau_0\\geq 0$ and $n\\in\\mathbb{N}$:\n\\begin{equation}\n\\label{e1}\n\\int_{[0,r]}h(\\tau,x)dx \\geq\\frac{1}{4^{n+1}r}\\int_{\\tau_0}^{\\tau}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nConsider the decomposition\n$$\n\\left(r,2^nr\\right]=\\bigcup_{i=3}^{2^{n+1}}\\left(\\frac{r}{2}(i-1),\\frac{r}{2}i\\right].\n$$\nThen by Lemma \\ref{S5L1}, and Lemma 3.12 in \\cite{AV1}, we have\n\\begin{align*}\n\\int_{[0,r]}h(\\tau,x)dx&\\geq\\int_{\\tau_0}^{\\tau}\\sum_{i=3}^{2^{n+1}}\\frac{1}{ri}\\bigg(\\int_{\\left(\\frac{r}{2}(i-1),\\frac{r}{2}i\\right]}\nh(\\sigma,x)dx\\bigg)^2d\\sigma\\\\\n&\\geq\\int_{\\tau_0}^{\\tau}\\frac{1}{r}\\Bigg(\\sum_{i=3}^{2^{n+1}}i\\Bigg)^{-1}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma\\\\\n&\\geq\\frac{1}{(2^n-1)(2^{n+1}+3)r}\\int_{\\tau_0}^{\\tau}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{align*}\nNotice that $(2^n-1)(2^{n+1}+3)\\leq 4^{n+1}$.\n\\end{proof}\n\nThe next Lemma takes into account the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$.\n\\begin{lemma}\n\\label{S5L47} \nLet $h$ be a solution of (\\ref{S1E16ha}) with initial data $h_0\\in \\mathscr{M}_+^1([0,\\infty))$ satisfying\n\\begin{equation}\nm_0=\\int_{(0,\\infty)}h_0(x)dx>0. \\label{S5EL467}\n\\end{equation}\nThen, for any $\\tau _0\\ge 0$  there exist $R_1>0$, $C_1>0$  such that\n\\begin{equation}\n\\int_{[0,r]}h(\\tau ,x)d x\\geq C_1\\,r\\qquad\\forall r\\in[0,R_1],\\quad\\forall\\tau\\geq \\tau _0.\n\\end{equation}\n\\end{lemma} \n\n\\begin{proof}\nBy (\\ref{S5EL467}), there exist $0<a\\leq b<\\infty$ such that \n\\begin{align}\n\\label{mass of g0}\n\\int_{(a,b]}h_0(x)dx>\\frac{m_0}{2}.\n\\end{align}\nWe prove now\n\\begin{eqnarray}\n\\exists T'>0;\\,\\forall \\tau\\in [0, T'):\\,\\,\\,\\int_{\\left(\\frac{a}{2},2b\\right]}h(\\tau,x)dx\\geq\\frac{m_0}{4}. \\label{S5EL468}\n\\end{eqnarray}\nTo this end we use  (\\ref{S1E16ha}) with a test function\n $\\varphi\\in C^1_c([0,\\infty))$ such that $0\\leq\\varphi\\leq 1$, $\\varphi=1$ on $(a,b]$ and $\\varphi=0$ on $[0,\\infty)\\setminus\\left(\\frac{a}{2},2b\\right]$ and  (\\ref{mass of g0}) to obtain:\n\\begin{align}\n\\label{es1}\n\\int_{\\left(\\frac{a}{2},2b\\right]} h(\\tau,x)dx\n&\\geq\\frac{m_0}{2}+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{align}\nNow using (\\ref{lemma regularity 1}) and (\\ref{MMI}) we deduce\n$$\n\\left|\\mathscr{Q}_3^{(2)}(\\varphi,h(\\sigma))\\right|\\leq 2\\|\\varphi'\\|_{\\infty}\\bigg(\\frac{\\sqrt{M_1(h_0)}}{2}\\sigma+\\sqrt{M_0(h_0)}\\bigg)^4.\n$$\nUsing now $\\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq 3\\|\\varphi\\|_{\\infty}\\sqrt{x}$\nand $M_{1\/2}(h)\\leq \\sqrt{M_0(h)M_1(h)}$,\nwe have by the conservation of energy and the mass inequality\n$$\n\\left|\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h(\\sigma))\\right|\\leq 2\\|\\varphi\\|_{\\infty}\\sqrt{M_1(h_0)}\\left(\\frac{M_1(h_0)}{2}\\sigma+\\sqrt{M_0(h_0)}\\right).\n$$\nIt follows that  $ \\widetilde{\\mathscr{Q}}_3(\\varphi,h)\\in L^1_{loc}(\\mathbb{R}_+)$\nand  we deduce  (\\ref{S5EL468}) from (\\ref{es1}).\\\\\n\n\nBy Lemma \\ref{lemma 2^n} and (\\ref{S5EL468}), for any $r\\in\\left(0,\\frac{a}{2}\\right]$ and $n\\in\\mathbb{N}$ such that $r2^n\\in(2b,3b]$ we have\n\\begin{align}\n\\int_{[0,r]}h(\\tau,x)dx\n&\\geq\\int_0^\\tau\\frac{1}{4^{n+1}r}\\left(\\int_{\\left(\\frac{a}{2},2b\\right]}h(\\sigma,x)dx\\right)^2d\\sigma \\nonumber\\\\\n&\\geq\\frac{\\tau}{4^{n+1}r}\\left(\\frac{m_0}{4}\\right)^2  \\ge  \\frac{m_0^2}{4^3(3b)^2} \\tau\\,r \\qquad\\forall\\tau\\in[0,T']. \\label{1}\n\\end{align}\nwhere $\\left(\\frac{a}{2},2b\\right]\\subset (r,r2^n]$ has been used. \n\nFor any given   $\\tau _0\\ge 0$  define $\\tau'=\\min\\{\\tau_0,T'\\}$. Then by (\\ref{S5EP12}) in Proposition \\ref{S5P1} \nwith $\\theta=\\frac{1}{2}$ and $R=2r$, we deduce from (\\ref{1}):\n\\begin{equation}\n\\label{2}\n\\int_{[0,2r]}h(\\tau,x)dx \\geq\\frac{C\\tau'}{2}r \\qquad\\forall\\tau\\geq \\tau'.\n\\end{equation}\nand this proves the Lemma, where $R_1=a\/2$ and $C_1=C\\tau '\/4$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{S4P47}\nLet $h$ and $h_0$ be as in Lemma \\ref{S5L47}. For all $L>0$ and  every $\\tau _1>0$  there exists $R_0=R_0(h,L,\\tau _1)>0$ such that\n\\begin{align}\n\\label{S4EP47}\n\\int_{[0,R_0]}h(\\tau ,x)dx\\geq LR_0\\qquad\\forall\\tau\\geq\\tau_1.\n\\end{align}\n\\end{proposition}\n\n\n\\begin{proof}\nBy Lemma \\ref{S5L47} for $\\tau _0=\\frac{\\tau_1}{2}$\n\\begin{align}\n\\label{by the lemma}\n\\exists C_1>0, \\, \\exists R_1>0;\\,\\,\\int_{[0,r]}h(\\tau,x)dx &\\geq C_1r,\\quad\\forall r\\in[0,R_1],\\,\\,\\,\\forall\\tau\\geq\\frac{\\tau_1}{2}.\n\\end{align}\nNow fix an integer $p\\geq 2$ such that $C_1p \\geq 8 L$. We divide the proof in two parts.  Assume first :\n\\begin{equation}\n\\label{assumption 1}\n\\exists r'\\in (0, R_1],\\; \\exists \\tau'\\in\\left[\\frac{\\tau_1}{2},\\tau_1\\right]:\\,\\,\\,\\int_{\\left[0,\\frac{r'}{p}\\right]}h(\\tau',x)dx\\geq \\frac{C_1r'}{2}.\n\\end{equation}\nIt follows from lemma \\ref{S5P1} with $\\theta=\\frac{1}{2}$ and $R=\\frac{2r'}{p}$ that\n$$\n\\int_{\\left[0,\\frac{2r'}{p}\\right]}h(\\tau,x)dx\\geq\\frac{C_1r'}{4}\\qquad\\forall \\tau\\geq\\tau',\n$$\nIf we take $R_0:=\\frac{2r'}{p}$, we have, by our choice of $p$,\n$$\n\\int_{[0,R_0]}h(\\tau,x)dx\\geq \\frac{C_1p}{8}R_0\\geq LR_0\\qquad\\forall \\tau\\geq\\tau',\n$$\nso  (\\ref{S4EP47})  holds.\n\nAssume now that \\eqref{assumption 1} does not hold,\nthen, by (\\ref{by the lemma}):\n\\begin{equation}\n\\label{assumption 2}\n\\int_{\\left(\\frac{r}{p},r\\right]}h(\\tau,x)dx\\geq\\frac{C_1r}{2}\\qquad\\forall r\\in(0,R_1],\\quad\\forall\\tau\\in\\left[\\frac{\\tau_1}{2},\\tau_1\\right].\n\\end{equation}\nTake now any $r\\in\\left(0,\\frac{R_1}{p}\\right]$, let $n\\in\\mathbb{N}$ be the largest integer such that \n$r p^n\\in\\left(\\frac{R_1}{p},R_1\\right]$, and consider now the following decomposition\n$$(r,rp^n]=\\bigcup_{i=p+1}^{p^{n+1}}\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]\n=\\bigcup_{k=1}^n\\bigcup_{i=p^k+1}^{p^{k+1}}\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right].$$\nBy lemma \\ref{S5L1} on $(\\tau _1\/2, \\tau _1)$ with $a_i=ri\/p$, $i=p+1, \\cdots, p^{n+1}$:\n\\begin{align}\n\\label{ue1}\n\\int_{[0,r]}h(&\\tau_1,x)dx\n\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{p}{2r}\\sum_{i=p+1}^{p^{n+1}}\\frac{1}{i}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma\\nonumber\\\\\n&=\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{p}{2r}\\sum_{k=1}^n\\sum_{i=p^k+1}^{p^{k+1}}\\frac{1}{i}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma\\nonumber\\\\\n&\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^k}\\sum_{i=p^k+1}^{p^{k+1}}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma.\n\\end{align}\nWe use now  Lemma  3.12 in \\cite{AV1}\n\\begin{align*}\n&\\sum_{i=p^k+1}^{p^{k+1}}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\n\\geq\\frac{1}{p^k(p-1)}\\times \\\\ \n&\\times \\left(\\sum_{i=p^k+1}^{p^{k+1}}\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\n\\geq\\frac{1}{p^{k+1}}\\left(\\int_{(rp^{k-1},rp^k]}h(\\sigma,x)dx\\right)^2\n\\end{align*}\nand deduce\n\\begin{align*}\n\\int_{[0,r]}h(\\tau_1,x)dx\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^{2k+1}}\\left(\\int_{(rp^{k-1},rp^k]}h(\\sigma,x)dx\\right)^2\\right]d\\sigma.\n\\end{align*}\nDue to the choice of the integer $n$, $r p^k\\in (0, R_1]$ for all $k=1, \\cdots, n$, and we can use (\\ref{assumption 2}) on each interval\n$(rp^{k-1},rp^k]$ to obtain:\n\\begin{align*}\n\\int_{[0,r]}h(\\tau_1,x)dx\n&\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[ \\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^{2k+1}}\\left(\\frac{C_1rp^k}{2}\\right)^2\\right]d\\sigma\n=\\frac{\\tau_1C_1^2n}{16p}\\;r.\n\\end{align*}\nIt then follows from lemma \\ref{S5P1} with $\\theta=\\frac{1}{2}$ and $R=2r$ that\n\\begin{align}\n\\label{estimate unbounded}\n\\int_{[0,2r]}h(\\tau,x)dx\\geq\\frac{\\tau_1C_1^2n}{32p}\\;r\\qquad\\forall\\tau\\geq\\tau_1.\n\\end{align}\nSince $r p^n\\in\\left(\\frac{R_1}{p},R_1\\right]$, then $n\\geq\\frac{\\log\\left(\\frac{R_1}{rp}\\right)}{\\log(p)}$, and\nwe chose $r>0$ small enough in order to have $r\\in (0, R_1\/p)$ and \n$$\\frac{\\tau_1C_1^2}{64p}\\frac{\\log\\left(\\frac{R_1}{rp}\\right)}{\\log p}\\geq L;$$\nand set $R_0:=2r$. The result then follows from (\\ref{estimate unbounded}).\n\\end{proof}\n\n\\begin{lemma}\n\\label{rescaled}\nLet $h$ be a solution of (\\ref{S1E16ha}) and, for any $\\kappa>0$ and $\\lambda>0$, consider the rescaled measure\n$h_{\\kappa,\\lambda}$ defined as:\n\\begin{equation}\n\\label{S5Escaled}\n\\int_{[0,\\infty)} \\!\\!\\!\\! h_{\\kappa,\\lambda}(\\tau,x)\\varphi(x)dx\n=\\kappa\\!\\int_{[0,\\infty)}\\!\\!\\!\\!h(\\kappa\\lambda\\tau, x)\\varphi\\left(\\frac{x}{\\lambda}\\right)dx,\\;\\forall \\varphi \\in C_b([0, \\infty)).\n\\end{equation}\nThen $h_{\\kappa,\\lambda}$ is a super solution of (\\ref{S1E16ha}).\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi\\in C^1_b([0,\\infty))$ be nonnegative, convex and decreasing, $\\psi(x)=\\varphi(x\/\\lambda)$, and $\\eta=\\kappa\\lambda\\tau$.\nBy Lemma \\ref{convex-positivity}, \n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\psi,h)\\leq 0$, and by (\\ref{S1E16ha})\n\\begin{align*}\n\\frac{d}{d\\eta}&\\int_{[0,\\infty)}\\psi(x)h(\\eta, x)dx\\geq \\mathscr{Q}_3^{(2)}(\\psi,h(\\eta)).\n\\end{align*}\nSince $\\mathscr{Q}_3^{(2)}(\\psi,h(\\eta))=\\kappa^{-2}\\lambda^{-1}\\mathscr{Q}_3^{(2)}(\\varphi,h_{\\kappa,\\lambda}(\\tau))$, then\n\\begin{align*}\n\\frac{d}{d\\tau}\\int_{[0,\\infty)} \\varphi(x)h_{\\kappa,\\lambda}(\\tau,x)dx&=\\kappa^2\\lambda\\frac{d}{d\\eta}\\int_{[0,\\infty)}\\psi(x)h(\\eta, x)dx\n\\geq\\mathscr{Q}_3^{(2)}(\\varphi,h_{\\kappa,\\lambda}(\\tau)).\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\n\\label{concentration lemma}\nLet $h$ be a super solution of (\\ref{S1E16ha}).\nSuppose that there exists $\\tau'>0$ such that\n\\begin{equation}\n\\label{S5HC}\n\\int_{[0,1]}h(\\tau,x)dx\\geq 1\\qquad\\forall \\tau\\geq\\tau'.\n\\end{equation}\nThen for any given $\\delta>0$ there exist  $\\tau _0$ such that\n\\begin{align}\n&\\tau '\\leq\\tau _0\\leq\\tau '+T_0(\\delta),\\qquad T_0(\\delta )=\\frac{64}{\\delta^3}\\left(1-\\frac{\\delta}{2}\\right) \\label{S5EX2}\\\\\n&\\hbox{and}\\,\\,\\,\\,\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau_0,x)dx\\geq 1-\\frac{\\delta}{2}. \\label{S5EX3}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n The statement of the Lemma is  equivalent to show that the following set\n$$\nA:=\\left\\{\\tau\\in[\\tau', \\tau '+T_0( \\delta )]:\\int_ {\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\\geq 1-\\frac{\\delta}{2}\\right\\}.\n$$\nis non empty, where $T_0(\\delta )$ is defined in (\\ref{S5EX2}). \nTo this end we first apply  Lemma \\ref{S5L1}  with  $a_0=\\frac{\\delta}{4}$, $a_i=\\frac{\\delta}{4}\\left(1+\\frac{i}{2}\\right)$ for $i\\in\\{1,...,n-1\\}$ and $a_n=1$. The number $n$ is chosen to be the largest integer such that $a_{n-1}<1$, which implies\n\\begin{equation}\n\\label{S5e0}\n\\frac{1}{n+1}>\\frac{\\delta}{8}.\n\\end{equation}\nThen, using ${a_i}^{-1}\\geq 1$ for all $i\\in\\{1,...,n\\}$:\n\\begin{align*}\n\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\n\\geq\\frac{1}{2}\\int_{\\tau'}^{\\tau} \\sum_{i=1}^n\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)dx\\bigg)^2d\\sigma,\\,\\,\\,\\forall \\tau >\\tau '.\n\\end{align*}\nSince by Lemma 3.12 in \\cite{AV1} and (\\ref{S5e0}):\n\\begin{align*}\n\\sum_{i=1}^n\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)dx\\bigg)^2\n\\geq\\frac{\\delta}{8}\\bigg(\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\sigma,x)dx\\bigg)^2,\n\\end{align*}\nwe obtain, for all $\\tau >\\tau '$\n\\begin{align}\n\\label{estimate for contradiction}\n\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\n\\geq\\frac{\\delta}{16}\\int_{\\tau'}^{\\tau}\\bigg(\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{align}\nArguing by contradiction suppose that $A=\\emptyset$:\n\\begin{align*}\n\\int_{\\left(0,\\frac{\\delta}{4}\\right]}h(\\tau ,x)dx< 1-\\frac{\\delta}{2}\\qquad\\forall\\tau \\in[\\tau',\\tau '+T_0(\\delta )]\n\\end{align*}\nand by (\\ref{S5HC}):\n$$\n\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\tau ,x)dx\\geq \\frac{\\delta}{2}\\qquad\\forall\\tau \\in[\\tau',\\tau '+T_0(\\delta )].\n$$\nIt  follows from (\\ref{estimate for contradiction}) that\n$\n1-\\frac{\\delta}{2}>\\frac{\\delta^3}{64}(\\tau-\\tau')$ for all $\\tau\\in[\\tau',\\tau '+T_0(\\delta )]$\nwhich is a contradiction for $\\tau=\\tau '+T_0(\\delta )$. \n\\end{proof}\n\n\n\\begin{proposition}\n\\label{S5LB}\nLet $h$ be a solution of  (\\ref{S1E16ha}). Suppose that there exist $m$, $R>0$ such that\n\\begin{equation}\n\\int_{[0,R]}h(\\tau,x)dx\\geq m\\qquad\\forall\\tau\\in[0,\\infty).\n\\end{equation}\nThen given any $\\alpha\\in(0,1)$ there exists $T_*=T_*(\\alpha)>0$ such that\n\\begin{align}\n\\label{S5ELB}\n\\int_{[0,r]}h(\\tau,x)dx\\geq \\frac{m}{(2R)^{\\alpha}}\\,r^{\\alpha}\\qquad\\forall r\\in[0,R],\\quad\\forall\\tau\\in\\bigg[\\frac{R T_*}{m},\\infty\\bigg).\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nWe argue by induction and define first the scaled measure  $h_1=h _{ \\kappa_1, \\lambda _1 } $, defined as in (\\ref{S5Escaled}), that satisfies condition (\\ref{S5HC}) for $\\kappa_1=\\frac {1} {m},\\,\\,\\,\\lambda _1=R.$\nFrom Lemma \\ref{rescaled}, and Lemma \\ref{concentration lemma} with $\\tau '=0$, we deduce that for all $\\delta \\in (0, 1)$ there exists  $\\tau _1>0$ such that:\n\\begin{align*}\n&0\\leq\\tau _1\\leq T_0(\\delta),\\quad \\int_{\\left[0,\\frac{\\delta}{4}\\right]}h_1(\\tau_1,x)dx\\geq 1-\\frac{\\delta}{2}.\n\\end{align*}\nThen from Lemma \\ref{rescaled}, and Proposition \\ref{S5P1} with $\\theta=\\delta \/2$ and $R=1\/2$,\n\\begin{align}\n\\int_{\\left[0,\\frac{1}{2}\\right]}h_1(\\tau,x)dx\\geq \\left(1-\\frac{\\delta}{2}\\right)^2,\\,\\,\\,\\forall \\tau \\geq T_0(\\delta ),\\nonumber\\\\\n\\label{S5EIt1}\n\\int_{\\left[0,\\frac{R}{2}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right),\\,\\,\\,\\forall \\tau \\geq \\frac {R} {m}T_0(\\delta ).\n\\end{align}\nExactly as before we now define $h_2=h _{ \\kappa_2, \\lambda _2 } $ as in (\\ref{S5Escaled}), that satisfies condition (\\ref{S5HC}) for\n$\\kappa_2=\\frac {1} {m(1-\\delta )^2},\\,\\,\\,\\lambda _2=\\frac {R} {2},\\,\\,\\,\\tau '= 2(1-\\delta )T_0(\\delta ).$\nThe same argument gives then:\n\\begin{equation}\n\\label{S5EIt2}\n\\int_{\\left[0,\\frac{R}{4}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right)^2,\\quad\\forall \\tau \\geq \\frac {R T_0(\\delta )} {m}\\left(1+\\frac {1} {2(1-\\delta )} \\right).\n\\end{equation}\nWe deduce after $n$ iterations\n\\begin{align}\n\\label{S5EItn}\n\\int_{\\left[0,\\frac{R}{2^n}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right)^{n},\\quad\\forall \\tau \\geq \\frac {R T_0(\\delta )} {m}\n\\sum_{ k=0 }^{n-1}\\frac {1} {2^{k}(1-\\delta )^k}\n\\end{align}\nIf we chose $\\delta=1-2^{-\\alpha }$, for any $0<\\alpha <1$, we may  define\n\\begin{eqnarray}\n\\label{S5ET*}\nT_*=T_0(\\delta )\\sum_{ k=0 }^{\\infty}2^{-(1-\\alpha)  k}= \\frac {T_0(\\delta )} {1-2^{-(1-\\alpha) }}.\n\\end{eqnarray}\nSince for any $r\\in (0, R)$ there exists $n\\in \\mathbb{N}$ such that $r\\in \\left(\\frac{R}{2^n} ,\\frac{R}{2^{n-1}}\\right]$,\n\\begin{align*}\n\\int_{\\left[0, r \\right]}h\\left(\\tau , x\\right)dx\\geq m2^{-\\alpha n} \n,\\,\\,\\,\\forall \\tau > \\frac {R T_*} {m}\n\\end{align*}\nand using $2^{-n}>r\/2R$, (\\ref{S5ELB}) follows.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{S5PLB}\nLet $h$ be a solution of  (\\ref{S1E16ha}).  Then, for all $\\tau_0>0$ and for any  $\\alpha\\in(0,1)$ there exists \n$R_*=R_*(h,\\tau_0,\\alpha)>0$ such that\n\\begin{align}\n\\int_{[0,r]}h(\\tau,x)\\,\\mathrm{d} x\\geq C\\,r^{\\alpha}\\qquad\\forall r\\in[0,R_*]\\quad\\forall \\tau\\in[\\tau_0,\\infty),\n\\end{align} \nwhere $C=\\frac{T_*(\\alpha)}{\\tau_0}(2R_*)^{1-\\alpha}$, and $T_*(\\alpha)$ is given by Proposition \\ref{S5LB}. \n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{S4P47} with $L>0$ and for $\\tau _1=\\tau _0\/2 $, there exists $R_0(h,L,\\tau _1)>0$ such that\n\\begin{align*}\n\\int_{[0,R_0]}h(\\tau ,x)dx\\geq LR_0\\qquad\\forall\\tau\\geq\\frac{\\tau_0}{2}.\n\\end{align*}\nThen by Proposition \\ref{S5LB}, with $m=LR_0$ and $R=R_0$, we obtain that for any given $\\alpha\\in(0,1)$ there exists $T_*=T_*(\\alpha)>0$ such that\n\\begin{align*}\n\\int_{[0,r]} h(\\tau,x)\\,\\mathrm{d} x\\geq \\frac{LR_0}{(2R_0)^{\\alpha}}\\,r^{\\alpha}\\qquad\\forall r\\in[0,R_0],\\quad\\forall \\tau\\in\\bigg[\\frac{\\tau_0}{2}+\\frac{ T_*}{L},\\infty\\bigg).\n\\end{align*}\nIf we chose $L=2T_*\/\\tau_0$, then the Proposition follows with $R_*=R_0$.\n\\end{proof}\n\n\\begin{proof}[\\upshape{\\bfseries{Proof of Theorem \\ref{S1T4h}}}]\nBy Lemma \\ref{S6L1'} the map $\\tau\\mapsto h(\\tau,\\{0\\})$ is right continuous, nondecreasing and a.e. differentiable on $[0,\\infty)$. It remains to prove that it is actually strictly increasing.\nWe  first suppose that  $h_0$ is such that \n\\begin{eqnarray}\n\\label{S5E54}\n\\int_{\\{0\\}}h_0(x)dx=0,\\qquad \\int_{(0,\\infty)}h_0(x)dx>0,\n\\end{eqnarray}\nand prove \n\\begin{equation}\n\\label{S1ET4h}\nh(\\tau,\\{0\\})>0\\qquad\\forall \\tau>0.\n\\end{equation}\nArguing by contradiction, if we suppose that there exists $\\tau_0>0$ such that $h(\\tau_0,\\{0\\})=0$, by monotonicity \n$h(\\tau,\\{0\\})=0$ for all $\\tau\\in[0,\\tau_0]$. In particular\n\\begin{align}\n\\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{[0,r]}h(\\sigma,x)dxd\\sigma=\\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{(0,r]}h(\\sigma,x) dxd\\sigma\n\\end{align}\nfor all $r>0$. Now using  Proposition \\ref{S5P2} with $\\alpha=0$, and Proposition \\ref{S5PLB},  we deduce that, for any $\\alpha\\in(0,1\/2)$, there exists $R_*=R_*(h, \\tau _0\/2, \\alpha )$ such that\n\\begin{align*}\n&C_2\\,r^{\\alpha}\\leq \\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{(0,r]}h(\\sigma,x)dxd\\sigma \\leq C_1\\sqrt{r},\n\\qquad\\forall r\\in[0,R_*];\\\\\n&C_1=8\\, \\sqrt{\\frac{\\tau_0}{2}} \\left( \\frac {\\sqrt{M_1(h_0)}} {2}\\tau_0 +\\sqrt{M_0(h_0)}\\right),\\quad C_2=\\frac{T_*(\\alpha)}{2}(2R_*)^{1-\\alpha},\n\\end{align*}\nand that leads to a contradiction for $r$ small enough.\n\nConsider now a general initial data $h_0$ such that $\\int_{\\{0\\}}h_0(x)dx>0$. Let $h$ be a solution of (\\ref{S1E16ha}) with initial data $h_0$ and define\n$$\n\\tilde h(\\tau )=h(\\tau )-h_0(\\{0\\})\\delta _0.\n$$\nThen, on the one hand, the initial data of $\\tilde h$  satisfies $\\tilde h(0, \\{0\\})=0$. On the other hand we claim that $\\tilde h$ is still a solution of   (\\ref{S1E16ha}). Notice indeed that $\\tilde h_\\tau \\equiv h_\\tau $ and, moreover,\n$\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,\\tilde h(\\tau))$. Using the previous case\n$$\n\\int  _{ \\{0\\}}\\tilde h(\\tau, x)dx>0,\\quad\\forall \\tau >0,\n$$\nand then\n$$\n\\int  _{ \\{0\\}} h(\\tau, x)dx>\\int  _{ \\{0\\}} h_0(x)dx,\\quad\\forall \\tau >0.\n$$\nThe Theorem follows using now the time translation invariance of the equation.\n\\end{proof}\n\n\nThe last result of this section describes the relation between the Lebesgue-Stieltjes measure associated to the (right continuous and strictly increasing) function\n$m(\\tau)=h(\\tau,\\{0\\})$, and the equation for $h$ (\\ref{S1E16ha}).\n\n\\begin{proposition}\n\\label{Stieltjes1}\nLet $h$ be a solution of (\\ref{S1E16ha}) for a initial data $h_0\\in\\mathscr{M}_+^1([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$.\nIf we denote $m(\\tau)=h(\\tau,\\{0\\})$ and $\\lambda$ is the Lebesgue-Stieltjes measure associated to $m$, then for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST} and for all $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1<\\tau_2$:\n\\begin{align}\n\\label{mm1}\n&m(\\tau_2)-m(\\tau_1)=\\lambda((\\tau_1,\\tau_2]),\\\\\n\\label{Stieltjes3}\n&\\lambda((\\tau_1,\\tau_2])=\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))d\\tau,\n\\end{align}\n\\begin{flalign}\n\\label{Stieltjes0}\n&\\text{and}&&0<\\lambda((\\tau_1,\\tau_2]))<\\infty.&&\n\\end{flalign} \nFurthermore, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}\n\\begin{align}\n\\label{Stieltjes7}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\in\\mathscr{D}'(0,\\infty),\n\\end{align}\nand if we denote $m'$ the  derivative in the sense of Distributions of $m$, then \n\\begin{align}\n\\label{Stieltjes2}\nm'=\\lambda=\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty).\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{S6L1'}, $m$ is right continuous and nondecreasing on $[0,\\infty)$. Then it has a Lebesgue-Stieltjes measure associated to it, $\\lambda$, that satisfies (\\ref{mm1}) (c.f.  \\cite{Fo} Ch.1).\n\nOn the other hand, since $h$ is a solution of (\\ref{S1E16ha}), using $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST} and taking the limit $\\varepsilon\\to 0$, it follows from (\\ref{limQ31b}) in Lemma \\ref{convergence lemma} that for all $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1<\\tau_2$:\n\\begin{align}\n\\label{ZE3}\nm(\\tau_2)-m(\\tau_1)=\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma,\n\\end{align}\nand then (\\ref{Stieltjes3}) follows from (\\ref{mm1}). Moreover, since by Theorem \\ref{S1T4h} $m$ is strictly increasing, then (\\ref{Stieltjes0}) holds.\n\nNotice that the limit in (\\ref{ZE3}) is independent of the choice of the test function $\\varphi_{\\varepsilon}$. Indeed, if $\\psi_{\\varepsilon}$ is another test function as in Remark \\ref{TEST}, since for all $\\tau\\geq 0$\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\psi_{\\varepsilon}(x)h(\\tau,x)dx=m(\\tau)=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx,\n\\end{align*}\nit follows from (\\ref{ZE3}) that for all $0\\leq \\tau_1\\leq\\tau_2$\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\psi_{\\varepsilon},h(\\sigma))d\\sigma=\n\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma.\n\\end{align*}\n\nNow, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, consider the absolutely continuous function \n\\begin{align*}\n\\theta_{\\varepsilon}(\\tau)=\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx.\n\\end{align*}\nThen the equation in (\\ref{AUXW}) reads $\\theta_{\\varepsilon}'(\\tau)=\\widetilde{\\mathscr{Q}}_3(\\varphi_{\\varepsilon},h(\\tau))\n$.\nUsing integration by parts we deduce that for all $\\varepsilon>0$:\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(\\tau)\\theta_{\\varepsilon}(\\tau)d\\tau=\\int_0^{\\infty}\\phi(\\tau)\\widetilde{\\mathscr{Q}}_3(\\varphi_{\\varepsilon},h(\\tau))d\\tau\n\\quad\\forall \\phi\\in C^{\\infty}_c(0,\\infty).\n\\end{align*}\nTaking the limit $\\varepsilon\\to 0$ it then follows from Lemma \\ref{convergence lemma} that\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(\\tau)m(\\tau)d\\tau=\\lim_{\\varepsilon\\to 0}\\int_0^{\\infty}\\phi(\\tau)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))d\\tau,\n\\end{align*}\nhence, $m'=\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)$. On the other hand, by Fubini's theorem\n\\begin{align*}\n\\int_0^{\\infty}\\phi(\\tau)d\\lambda(\\tau)=\\int_0^{\\infty}\\int_0^{\\tau}\\phi'(\\sigma)d\\sigma d\\lambda(\\tau)\n=-\\int_0^{\\infty}\\phi'(\\sigma)m(\\sigma)d\\sigma\n\\end{align*}\nfor all $\\phi\\in C^{\\infty}_c(0,\\infty)$ (cf. \\cite{WR3}, Example 6.14), thus $m'=\\lambda$.\n\\end{proof}\n\n\\section{Existence of solutions $G$, proof of  Theorem \\ref{S1T1}.}\n\\label{sectionG}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nGiven a initial data $G_0\\in \\mathscr M_+^1$ as in Theorem \\ref{S1T1}, let $h\\in C\\big([0,\\infty), \\mathscr{M}_+([0,\\infty))\\big)$ satisfy (\\ref{lip loc h})--(\\ref{EE}),  (\\ref{S5Ealpha }), given by (\\ref{S5C52R}) and $H$ defined by (\\ref{DEFH}) and satisfying  (\\ref{lip loc H})--(\\ref{EEH}), (\\ref{S5EalphaR }) by Corollary \\ref{S5C52R}.  It is natural, in view of the change of variables  (\\ref{S1E45b})  to define now, \n\\begin{equation}\n\\label{S6EG1}\nG(t) =H(\\tau ),\\quad\\tau =\\int _0^tG(s, \\{0\\})ds.\n\\end{equation}\nNotice nevertheless that since $G(s, \\{0\\})$ is still unknown,  (\\ref{S6EG1}) does not define $G(t)$ actually.  What we know is rather, given $\\tau >0$, what would be the value of $t$ such that \n\\begin{equation}\n\\label{S6EtCh}\nt =\\int _0^\\tau \\frac {d\\sigma  } {H(\\sigma  , \\{0\\})},\n\\end{equation}\nsince we expect to have $G(s,\\{0\\})=H(\\sigma , \\{0\\})$ for $s$ and $\\sigma $ such that\n\\begin{eqnarray*}\n\\sigma =\\int _0^sG(r, \\{0\\})dr,\\quad\\hbox{or}\\quad s=\\int _0^\\sigma \\frac {d\\rho } {H(\\rho , \\{0\\})}.\n\\end{eqnarray*}\nIf $G$ is going to be defined in that way it is then necessary first to check that the range of values taken by the variable $t$ in (\\ref{S6EtCh}) is all of $[0, \\infty)$. By definition (\\ref{DEFH}),\n\\begin{equation}\n\\label{{DEFH}2}\nH(\\tau,\\{0\\})=h(\\tau, \\{0\\})-\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{equation}\nSince both terms in the right hand side are nonnegative, $H(\\tau, \\{0\\})$ has no a priori  definite sign. We must then consider that question in some detail.\nOur first step is to prove the following\n\n\\begin{lemma}\n\\label{S6C1}\nIf $G_0(\\{0\\})>0$, then\n\\begin{align}\n\\label{TAU1}\n&\\tau_*=\\inf\\{\\tau>0:H(\\tau,\\{0\\})=0\\}>0,\\\\\n\\label{TAU2}\n&H(\\tau_*,\\{0\\})=0,\\\\\n\\label{TAU3}\n&H(\\tau,\\{0\\})>0\\qquad\\forall\\tau\\in[0,\\tau_*).\n\\end{align}\n\\end{lemma}\n\\begin{proof}\n$H(0)=G_0$ by (\\ref{S5IDh}), and then, using $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, we deduce $H(0,\\{0\\})=G_0(\\{0\\})$, which is strictly positive by hypothesis. Then (\\ref{TAU1}) follows from the right continuity of $H(\\tau,\\{0\\})$ (cf. Corollary \\ref{S6L1}).\n\nIn order to prove (\\ref{TAU2}) we use a minimizing sequence $(\\tau _n)_{ n\\in \\mathbb{N} }$, i.e., $\\tau _n\\ge \\tau _*$, $H(\\tau _n,\\{0\\})=0$ for every $n\\in \\mathbb{N}$,  and $\\tau _n\\to \\tau _*$ as $n\\to \\infty$. Then from the right continuity (\\ref{TAU2}) holds.\n\nLet us prove now (\\ref{TAU3}). If $H(\\tau_0 , \\{0\\})<0$ for some $\\tau_0 \\in (0, \\tau_*)$, then $\\tau _0$ must be a left discontinuity point of $H(\\tau, \\{0\\})$ and\n\\begin{equation*}\n\\limsup_{\\delta\\rightarrow 0^+}H(\\tau_0-\\delta,\\{0\\})> H(\\tau_0,\\{0\\}),\n\\end{equation*}\nand this would contradict (\\ref{JIM}). That proves (\\ref{TAU3}).\n\\end{proof}\n\nIt follows from Lemma \\ref{S6C1} that the function:\n\\begin{equation}\nt =\\xi (\\tau )=\\int _0^\\tau \\frac {d\\sigma  } {H(\\sigma  , \\{0\\})}\n\\end{equation}\nintroduced  in (\\ref{S6EtCh}) is well defined, monotone nondecreasing  and continuous on  the interval $ [0, \\tau _*)$. We  then define,\n\\begin{align}\n\\label{S6DG2}\n\\forall t\\in [0, \\xi (\\tau_*)):\\quad G(t)=H(\\xi ^{-1}(t)).\n\\end{align}\nBy (\\ref{S6DG2}) and (\\ref{{DEFH}2}), if $G(t)=G(t,\\{0\\})\\delta _0+g(t)$ and $H(\\tau)=H(\\tau , \\{0\\})\\delta _0+\\tilde h(\\tau )$, then\n\\begin{align}\n&G(t,\\{0\\})=H(\\tau , \\{0\\}), \\label{S6DG2bis}\\\\\n&\\tilde h(\\tau )=h(\\tau )-h(\\tau , \\{0\\})\\delta_0, \\label{S6DG20}\\\\\n&g(t)=\\tilde h(\\tau ).\\label{ght}\n\\end{align}\n\n\\begin{remark}\nFormula (\\ref{S6DG2}) defines the function $G$ at time $t\\in (0, \\xi (\\tau _*))$ from the knowledge of the function $H(\\tau )$ for $\\tau >0$ such that $\\tau =\\xi ^{-1}(t)$. Moreover,\n\\begin{equation}\n\\forall t\\in (0, \\xi (\\tau _*)): \\quad \\xi ^{-1}(t)=\\int _0^t G(s, \\{0\\})ds.\n\\end{equation}\n\\end{remark}\nWe have now,\n\n\\begin{proposition}\n\\label{S6P3G1}\nThe function  $G$ defined by (\\ref{S6DG2}) is  such that\n\\begin{eqnarray}\nG\\in C\\big([0, \\xi (\\tau _* )),\\mathscr{M}_+^1([0,\\infty))\\big),\\,\\,\\,G(0)=G_0\n\\end{eqnarray} \nand satisfies (\\ref{S1ED3S}), (\\ref{S1E16}), (\\ref{S1E210}) and (\\ref{S1E220}) on the time interval $[0, \\xi (\\tau_* ))$.\n\\end{proposition}\n\n\\begin{proof}\nWe first prove that $G(t)$ is a positive measure for all $t\\in[0,\\xi(\\tau_*))$. By (\\ref{TAU3}) and (\\ref{S6DG2bis}) we have $G(t,\\{0\\})>0$ for all $t\\in[0,\\xi(\\tau_*))$. Then, since $h(\\tau)$ is a positive measure for all $\\tau\\in[0,\\infty)$, we deduce from (\\ref{ght}) and (\\ref{S6DG20}) that $g(t)$ is a positive measure for all $t\\in[0,\\xi(\\tau_*)).$ Hence $G(t)=G(t,\\{0\\})\\delta_0+g(t)$ is also positive. \n\n\nAll the properties of $G(t)$ at $t\\in [0, \\xi (\\tau_*))$ fixed follow from the corresponding property of $H(\\tau )$ with $t=\\xi(\\tau )$. The only property where $t$ is not fixed are (\\ref{S1ED2S}) and (\\ref{S1ED3S}). Since\n\\begin{equation*}\n\\left|\\frac {\\partial G(t)} {\\partial t}\\right|= \\left|\\frac {\\partial \\tau } {\\partial t}\\frac {\\partial H(\\tau )} {\\partial \\tau }\\right|\n\\le |H(\\tau , \\{0\\})|\\left|\\frac {\\partial H(\\tau )} {\\partial \\tau }\\right|\n\\end{equation*}\nBy definition,\n\\begin{eqnarray*}\n|H(\\tau , \\{0\\})|\\le |h(\\tau, \\{0\\})|+\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{eqnarray*}\nSince $h\\in C([0, \\infty), \\mathscr M_+^1)$ it follows using also (\\ref{MMI}),  (\\ref{EE}) and H\\\"older inequality that \n$H(\\tau , \\{0\\})\\in L^\\infty _{ loc }([0, \\infty))$.Then, by  (\\ref{lip loc h}), $G(t)$ is locally Lipschitz on $[0, \\xi (\\tau _*))$ and satisfies (\\ref{S1ED3S}).\nSince $H$ satisfies (\\ref{AUXW}) the change of variables ensures that $G$ satisfies (\\ref{S1E16}).\n\\end{proof}\nWe prove now the following property of the function $G$ defined in (\\ref{S6DG2}).\n\\begin{proposition}\n\\label{origin G}\nLet  $G$ be the function defined in  (\\ref{S6DG2}) for $t\\in (0, \\xi (\\tau _*))$. Then the map $t\\mapsto G(t,\\{0\\})$ is right continuous and differentiable for almost every   $ t\\in [0,\\xi (\\tau _*))$ and, for all $t_0\\in (0, \\xi (\\tau _*))$ \n\\begin{align}\n\\label{CMI}\nG(t,\\{0\\})\\geq G(t_0,\\{0\\})e^{-\\int_{t_0}^tM_{1\/2}(g(s))d s}\\quad\\forall t\\in (t_0, \\xi (\\tau _*)).\n\\end{align} \nIn particular, if $G(0, \\{0\\})>0$, then $G(t,\\{0\\})> 0$ for all $t\\in (0, \\xi (\\tau _*))$.  \n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{S1E16}) and (\\ref{S1EB1}) with $\\varphi _{\\varepsilon}$ as in Remark \\ref{TEST}, we have\n\\begin{align}\n\\frac{d}{dt}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x+G(t,\\{0\\})M_{1\/2}&(G(t))\n=G(t,\\{0\\}) \\widetilde{\\mathscr Q}_3(\\varphi _\\varepsilon,G(t))\n.\\label{S6E78}\n\\end{align}\nWe use now that for all $\\varepsilon >0$:\n\\begin{align}\n\\label{ZAS}\nG(t,\\{0\\})\\leq \\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x,\n\\end{align}\nand we deduce from (\\ref{S6E78}), using  $J(t)=\\exp\\left({\\int_0^t M_{1\/2}(G(s))d s}\\right)$,\n\\begin{align}\n\\label{S6E79}\n&\\frac{d}{d t}\\bigg(J(t)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\\bigg) \\geq G(t,\\{0\\})J(t)  \\widetilde{\\mathscr Q}_3(\\varphi _\\varepsilon,G(t)).\n\\end{align}\nBy Lemma \\ref{convex-positivity} the right hand side of (\\ref{S6E79}) is nonnegative, and we deduce\n\\begin{equation*}\n\\label{S6E80}\nJ(t)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x \\geq J(t_0)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t_0,x)d x\n\\end{equation*}\nfor all $t\\in (t_0, \\xi (\\tau _*))$ and all $\\varepsilon >0$.  If we pass now to the limit as $\\varepsilon \\to 0$:\n\\begin{equation}\n\\label{S6E80}\nJ(t)G(t,\\{0\\}) \\geq J(t_0)G(t_0,\\{0\\}),\n\\end{equation}\nand this proves the estimate (\\ref{CMI}). It  also follows from Lebesgue's Theorem that $J(t)G(t,\\{0\\})$ is differentiable for almost every $t\\in (0, \\xi (\\tau _*))$ (cf. \\cite{Roy},  Theorem 2). On the other hand, since $J(t)$ is a.e differentiable and $J(t)>0$ for all $t\\in [0, \\xi (\\tau _*))$, we deduce that $G(t,\\{0\\})$ is also differentiable for almost every \n$t\\in [0, \\xi (\\tau _*))$.\n\nWe prove now the  right continuity  of $G(t, \\{0\\})$. It follows from  (\\ref{S6E80}),\n\\begin{equation*}\nJ(t+\\delta)G(t+\\delta,\\{0\\}) \\geq J(t)G(t,\\{0\\}),\\quad\\forall \\delta>0\\,\\,\\forall t>0.\n\\end{equation*}\nIf we take inferior limits and use that $J$ is continuous and strictly positive we obtain,\n\\begin{equation}\n\\label{S6E578}\n\\liminf _{ \\delta\\to 0 }G(t+\\delta,\\{0\\}) \\geq G(t,\\{0\\}),\\quad\\forall t>0.\n\\end{equation}\n\nSince $\\mathcal L_0(\\varphi_{\\varepsilon})\\ge 0$ by convexity (cf. Lemma \\ref{convex-positivity}), we deduce \n\\begin{align*}\n\\frac{d}{d t}&\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\n\\leq G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi_{\\varepsilon})(x,y)}{\\sqrt{x y}}G(t,x)G(t,y)d xd y,\n\\end{align*}\nand the argument  follows now as in the proof of the right continuity of $H$. \nFrom the inequality (\\ref{ZAS}), the bound \\eqref{lemma regularity 1} and the conservation of mass, we deduce for all  $t\\in [0, \\xi (\\tau _*))$ fixed  and  $\\delta\\in[0, \\xi (\\tau _*)-t)$,\n$$\nG(t+\\delta,\\{0\\})\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\n+\\frac {2N^2\\delta} {\\varepsilon }\\int_0^t G(s,\\{0\\})ds.\n$$\nIf we take superior limits as $\\delta\\to 0$, and then let $\\varepsilon \\to 0$ we obtain, using  (\\ref{S6L1E1}) with $G$ instead of $H$:\n$$\n\\limsup _{ \\delta\\to 0 }G(t+\\delta,\\{0\\})\\leq G(t, \\{0\\}).\n$$\nand this combined with (\\ref{S6E578}) proves that  $G(t, \\{0\\})$ is right continuous on $[0, \\xi (\\tau _*))$.\n\\end{proof}\n\n\nIn the next Lemma we prove that the function $G$ defined by (\\ref{S6DG2}) is actually well defined for all $t>0$.\n\\begin{lemma}\n\\label{S6LG}\n\\begin{equation}\n\\lim _{ \\tau \\to \\tau _*^- }\\xi (\\tau )=\\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince the function $\\xi (\\tau )$ is monotone  nondecreasing and continuous on $[0, \\tau _*)$, its limit as $\\tau \\to \\tau _*^-$ exists in $\\overline{\\mathbb{R}}_+$. Let us call it $\\ell$ and  suppose  $\\ell\\in \\mathbb{R}_+$. Now, from (\\ref{CMI}) and the fact that $G$ satisfies :\n$0\\le M _{ 1\/2 }(G(s))\\le \\sqrt {N E}$,\nwe deduce \n\\begin{equation}\n\\label{S6C56E1}\n\\limsup _{t\\to \\ell^-  }G(t, \\{0\\})\\ge e^{-\\sqrt{NE} \\ell}G(0, \\{0\\})>0,\n\\end{equation}\nand by (\\ref{JIM})\n$$\nH(\\tau_*, \\{0\\})\\ge \\limsup _{\\tau \\to \\tau _*^-  }H(\\tau, \\{0\\})=\\limsup _{ t\\to \\ell^- }G(t, \\{0\\})>0,\n$$\nand this contradicts (\\ref{TAU2}). This proves that $\\ell=\\infty$.\n\\end{proof}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S1T1}}] \n By Lemma \\ref{S6LG} the function $G$ is defined for all $t>0$. As we have seen in the proof of Lemma \\ref{S6LG}, $G(t)\\in \\mathscr M_+([0, \\infty))$ for all $t>0$. It then follows from Proposition \\ref{S6P3G1} that $G$ satisfies now all the conditions (\\ref{S1ED2S})--(\\ref{S1E16}) and (\\ref{S1T1E0})--(\\ref{S1E220}).\nProperty (\\ref{S1E23}) follows from the corresponding estimate (\\ref{MAh}) for $h$. Similarly, property (\\ref{S5Ealphahh}) follows from the property (\\ref{S5Ealpha }) of $h$.\nWe prove now the point (iv). Suppose then $\\alpha \\in (1, 3]$ and condition (\\ref{PRO112}). For $\\varphi(x)=x^{\\alpha}$ we have,\n$$\n\\mathscr{Q}_3^{(1)}(\\varphi,G(t))=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right) M_{\\alpha+\\frac{1}{2}}(G(t)).\n$$\nOn the other hand, for $0\\le y\\le x$,\n$$\n\\Lambda(\\varphi)(x,y)=x^{\\alpha}\\left(\\big(1+z\\big)^{\\alpha}+\\big(1-z\\big)^{\\alpha}-2\\right),\\quad z=\\frac{y}{x}\\in[0,1],\n$$\nIf  $\\alpha\\in(1,2]$, for all $x\\ge y >0$,\n$$\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\leq (2^{\\alpha}-2) x^{\\alpha-\\frac{3}{2}}y^{\\frac{1}{2}}\\le  (2^{\\alpha}-2) (xy)^{\\frac{\\alpha-1}{2}}.\n$$\nWe deduce\n$$\n\\mathscr{Q}_3^{(2)}(\\varphi,G(t))\\leq (2^{\\alpha}-2) \\Big( M_{\\frac{\\alpha-1}{2}}(G(t))\\Big)^2.\n$$\nand obtain\n\\begin{align*}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\left[C_{1,1}\\Big(M_{\\frac{\\alpha-1}{2}}(G(t))\\Big)^2-C_2M_{\\alpha+\\frac{1}{2}}(G(t))\\right],\n\\end{align*}\nwhere $C_{1,1}=2^{\\alpha}-2$ and $C_2=(\\alpha-1)\/(\\alpha+1).$\nUsing H\\\"{o}lder's inequality\n\\begin{align}\n\\label{CT98}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\Big[C_{1,1}N^{3-\\alpha}E^{\\alpha-1}\n-C_2E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}\\Big].\n\\end{align}\nBy (\\ref{PRO112}), the right hand side of (\\ref{CT98}) is negative, and then $M_{\\alpha}(G(t))$ is decreasing on $(0,\\infty)$.\n\nFor $\\alpha\\in[2,3]$ we use the estimate (\\ref{MAQ}) with $C_{1,2}=\\alpha(\\alpha-1)$ instead of $C_{\\alpha}$. Then we proceed as in the previous case to obtain\n\\begin{equation}\n\\label{CT99}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\Big[C _{ 1,2 }N^{3-\\alpha}E^{\\alpha-1}\n-C_2E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}\\Big].\n\\end{equation}\nAs before, (\\ref{PRO112}) implies that the right hand side of (\\ref{CT99}) is negative, and then $M_{\\alpha}(G(t))$ is decreasing.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S1Treg}}] \nBy construction\n\\begin{align*}\nG(t)=H(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0,\n\\end{align*}\nwhere $\\tau$ and $t$ are related by\n\\begin{align}\n\\label{CHANGE}\nt=\\xi(\\tau)=\\int_0^{\\tau}\\frac{d\\sigma}{H(\\sigma,\\{0\\})};\\qquad \\tau=\\xi^{-1}(t)=\\int_0^t G(s,\\{0\\})ds.\n\\end{align}\nTherefore $G(t,x)=h(\\tau,x)$ for $x\\in(0,\\infty)$, and \n\\begin{align*}\n\\int_0^{T}G(t,\\{0\\})\\int_{(0,\\infty)}x^{\\alpha}G(t,x)dxdt\n=\\int_0^{\\xi^{-1}(T)}\\int_{(0,\\infty)}x^{\\alpha}h(\\tau,x)dxd\\tau.\n\\end{align*} \nThe result  then follows from Proposition \\ref{S5P2}.\n\\end{proof}\n\n\\begin{remark}\nOne could try  to directly solve the  system (\\ref{PR}), (\\ref{PR19}), written in $(g, n)$ variables. First, to obtain a sequence of solutions $(g_k, n_k)$ of an approximated system where the factor $x^{-1\/2}$ is modified by truncation and regularization, and then pass to the limit. However, the limit obtained in that way, say $(g, n)$ is not a solution of (\\ref{PR}), (\\ref{PR19}). The reason is that all the solutions $g_k$ of the approximated system will be functions with a bounded moment of order $-1\/2$. Then, the right hand side of the equation (\\ref{S1E9}) is equal to $M _{ 1\/2 }(g_k)$ and by passage to the limit the equation for $n$ will be $n'(t)=-n(t)M _{ 1\/2 }(g(t))$, and  the total mass will not be conserved. \n\\end{remark}\n\n\n\\section{Proofs of Theorems \\ref{THn01}, \\ref{MU1} and \\ref{EQUIV}.}\n\\label{SectionK}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe first prove Theorem \\ref{THn01}. \n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{THn01} }]\nWe already know by Proposition \\ref{origin G} and Lemma \\ref{S6LG} that $n$ is right continuous and a.e. differentiable on $[0,\\infty)$.\nThen, by construction\n\\begin{align*}\nG(t)=H(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0,\n\\end{align*}\nwhere $\\tau\\in[0,\\tau^*)$ and $t\\in[0,\\infty)$ are related by (\\ref{CHANGE}).\nHence\n\\begin{align}\n\\label{ZE2}\nn(t)=m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma=m(\\tau)-\\int_0^t n(s)M_{1\/2}(g(s))ds.\n\\end{align}\nSince $n(0)=m(0)$, it then follows from Proposition \\ref{Stieltjes1} that for all $t>0$:\n\\begin{align}\n\\label{ZEaa}\nn(t)-n(0)+\\int_0^t n(s)M_{1\/2}(g(s))ds=\\lambda((0,\\tau]),\n\\end{align}\nand using (\\ref{CHANGE})\n\\begin{align}\n\\label{ZEbb}\n\\lambda((0,\\tau])=\\lim_{\\varepsilon\\to 0}\\int_0^t n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds.\n\\end{align}\nIf we denote $\\mu=\\xi_{\\#}\\lambda$ (c.f. \\cite{AMB}, Ch. 5), i.e., the push-forward  of $\\lambda$ through the function $\\xi:[0,\\tau^*)\\to[0,\\infty)$ in (\\ref{CHANGE}), then from the definition of $\\mu$ we obtain\n\\begin{equation}\n\\label{Mul}\n\\mu((0,t])=\\lambda((0,\\tau])\\qquad\\forall t>0.\n\\end{equation}\nThen (\\ref{ZE02}) and  (\\ref{ZE01}) follows from (\\ref{ZEaa}), (\\ref{ZEbb}) and (\\ref{Mul}). Moreover, (\\ref{ZE00}) follows from (\\ref{Stieltjes0}) in Proposition \\ref{Stieltjes1}. \n\\end{proof}\n\nThe following properties of $n(t)$ follows by   the same arguments used in the proofs of properties (\\ref{Stieltjes7}) and (\\ref{Stieltjes2}) of Proposition \\ref{Stieltjes1} \n\n\\begin{proposition} \nLet $G$, $g$, and $n(t)$ be as in Theorem \\ref{THn01}.  Then, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, the following limit exists \nin $\\mathscr{D}'(0,\\infty)$:\n\\begin{flalign}\n\\label{ZE04}\n&&&\\lim_{\\varepsilon\\to 0} n\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g)=T(G),&&\\\\\n\\text{and}&&& \n\\label{ZE05}\nn'+nM_{1\/2}(g)=T(G)\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty).&&\n\\end{flalign}\n\\end{proposition}\n\n\\begin{proof}\nConsider, for all  $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, the absolutely continuous functions \n\\begin{align}\n\\eta_{\\varepsilon}(t)=\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)dx.\n\\end{align}\nThen equation (\\ref{S1E16}) becomes $\n\\eta_{\\varepsilon}'=n\\mathscr{Q}_3(\\varphi_{\\varepsilon},g).$\nUsing integration by parts,\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(t)\\eta_{\\varepsilon}(t)dt=\\int_0^{\\infty}\\phi(t)n(t)\\mathscr{Q}_3(\\varphi_{\\varepsilon},g(t))dt\n\\quad\\forall \\phi\\in C^{\\infty}_c(0,\\infty).\n\\end{align*}\nTaking the limit $\\varepsilon\\to 0$ we deduce, using Lemma \\ref{convergence lemma}, that\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(t)n(t)dt=&\\lim_{\\varepsilon\\to 0}\\int_0^{\\infty}\\phi(t)n(t)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(t))dt\\\\\n&-\\int_0^{\\infty}\\phi(t)n(t)M_{1\/2}(g(t))dt,\n\\end{align*}\nand then (\\ref{ZE04}), (\\ref{ZE05}) follows. \n\\end{proof}\n\n\\begin{remark}\nIf we take distributional derivatives in both sides of  (\\ref{ZE02}) we obtain:\n$$\nn'+n M _{ 1\/2 }(g)=\\mu\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty),$$\nand by (\\ref{ZE05}), $\\mu =T(G)$.\n\\end{remark}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem  \\ref{MU1}}]\nThe statement of the Theorem follows from (\\ref{Stieltjes0}) in Proposition \\ref{Stieltjes1} and (\\ref{Mul}).\n\\end{proof}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{EQUIV}}]\nProof of part (i).\nBy Theorem \\ref{THn01}, $n$ is given by (\\ref{ZE02}) and (\\ref{ZE01}). On the other hand, since $G$ satisfies (\\ref{S1E16}), and for all\n$\\varphi\\in C^1_b([0,\\infty))$ such that $\\varphi(0)=0$:\n\\begin{align}\n\\label{G=g}\n\\int_{[0,\\infty)}\\varphi(x)G(t,x)dx=\\int_{[0,\\infty)}\\varphi(x)g(t,x)dx,\n\\end{align}\nthen $g$ satisfies (\\ref{2E765}). In order to prove part (ii) we first show the existence of the limit in (\\ref{ZE01}). To this end we write $\\varphi_\\varepsilon  = (1-\\psi _\\varepsilon )$, where  $\\psi _\\varepsilon$ is as in Remark \\ref{TEST}. Then $\\varphi _\\varepsilon (0)=0$, and by (\\ref{2E765}) and (\\ref{S1EB1}),\nusing that $\\mathscr{Q}_3(1-\\psi_{\\varepsilon},g)=\\mathscr{Q}_3(1,g)-\\mathscr{Q}_3(\\psi_{\\varepsilon},g)$, and $\\mathscr{Q}_3(1,g)=0$, we deduce\n\\begin{align}\n\\label{Bht}\n\\int _0^tn(s)\\widetilde{\\mathscr Q}_3(\\psi _\\varepsilon,g(s))ds&=\\int  _{ (0, \\infty) }\\varphi _\\varepsilon (x)\\left(g(0, x)-g(t, x)\\right) dx\\nonumber \\\\\n&+\\int _0^tn(s)M _{ 1\/2 }(g(s))ds.\n\\end{align}\nThe existence of the limit in (\\ref{ZE01}) follows and, if we pass to the limit,\n\\begin{align}\n\\lim _{ \\varepsilon \\to 0 }\\int _0^tn(s) \\mathscr Q_3^{(2)}(\\psi _\\varepsilon,g(s))ds&=\\int  _{ (0, \\infty) }(g(0, x)-g(t, x))dx\\nonumber\\\\\n&+\\int _0^tn(s)M _{ 1\/2 }(g(s))ds. \\label{Bhtw}\n\\end{align}\nWe now check that, if $n$ satisfies the equation (\\ref{ene1}) then  $G$ satisfies equation (\\ref{S1E16}) for a.e. $t>0$ and for every $\\varphi \\in C_b^{1}([0, \\infty))$. \nIf $\\varphi (0)=0$ this follows from (\\ref{2E765}) and (\\ref{G=g}).\n\n\nFor $\\varphi (0)\\not= 0$ we may assume without loss of generality that $\\varphi (0)=1$, and write $\\varphi =(\\varphi -\\psi _\\varepsilon )+\\psi _\\varepsilon $, where \n$\\psi _\\varepsilon$ is as in Remark \\ref{TEST}. \nSince  $(\\varphi -\\psi _\\varepsilon )(0)=0$, using (\\ref{2E765}) and  (\\ref{S1ED3S})\n\\begin{align}\n\\label{S5P10E1'}\n\\int_{[0,\\infty)}(\\varphi-\\psi_{\\varepsilon})(x)g(t,x)dx&=\\int_{[0,\\infty)}(\\varphi-\\psi_{\\varepsilon})(x)g(0,x)dx\\nonumber\\\\\n&+\\int_0^t n(s)\\widetilde{\\mathscr{Q}}_3((\\varphi-\\psi_{\\varepsilon}),g(s))ds.\n\\end{align}\nIn order to pas to the limit as $\\varepsilon\\to 0$, we first use\n$\\widetilde {\\mathscr{Q}}_3((\\varphi -\\psi _\\varepsilon ),g)=\\widetilde {\\mathscr{Q}}_3(\\varphi,g)-\\widetilde{\\mathscr{Q}}_3(\\psi_{\\varepsilon},g).$\nThen, since for all $t\\geq 0$\n\\begin{align}\n\\label{gzero}\n\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\psi_{\\varepsilon}(x)g(t,x)dx=0,\n\\end{align}\nand $n$ satisfies (\\ref{ene1}), we deduce from (\\ref{S5P10E1'}) and Lemma \\ref{convergence lemma}:\n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi(x)g(t,x)dx=&\\int_{[0,\\infty)}\\varphi(x)g(0,x)dx+\\int_0^t n(s)\\widetilde{\\mathscr{Q}}_3(\\varphi,g(s))ds\\\\\n&+n(0)-n(t)-\\int_0^t n(s)M_{1\/2}(g(s))ds.\n\\end{align*}\nSince $\\widetilde {\\mathscr{Q}}_3(\\varphi,G)-M_{1\/2}(g)=\\mathscr Q _3(\\varphi,G)$,\nit follows that $G$ satisfies \n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi(x)G(t,x)dx=\\int_{[0,\\infty)}\\varphi(x)G(0,x)dx+\\int_0^t n(s)\\mathscr{Q}_3(\\varphi,g(s))ds,\n\\end{align*}\nthus (\\ref{S1E16}) holds for a.e. $t>0$.\n\nIn order to check that $G$ satisfies (\\ref{S1ED2S}) we first use (\\ref{S1E16})  with $\\varphi =1\\in C_b^1([0, \\infty))$. For that choice of $\\varphi $ we have $\\Lambda(\\varphi)=\\mathcal L_0(\\varphi )\\equiv 0$\nand then:\n$$\n\\int  _{ [0, \\infty) }G(t, x)dx=\\int  _{ [0, \\infty) }G_0(x)dx.\n$$\nBecause:\n$$\n\\int  _{ [0, \\infty) }x\\,G(t, x)dx=\\int  _{ [0, \\infty) }x\\,g(t, x)dx,\n$$\n$G$ satisfies (\\ref{S1ED2S}) since by hypothesis so does $g$.\n\\end{proof}\n\n\\begin{remark} If $G$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}), we know by  Theorem \\ref{EQUIV} that  $g$ satisfies (\\ref{2E765}). It is straightforward to check that it also satisfies,\n\\begin{equation*}\n\\frac{d}{dt}\\int_{(0,\\infty)}\\varphi(x)g(t,x)dx=n(t)\\widetilde{\\mathscr{Q}} _3(\\varphi,g(t))-\\varphi (0)\\frac{d}{dt}\\mu((0,t]), \\label{LE02'}\n\\end{equation*}\nwhere $\\mu$ is as in Theorem \\ref{THn01}, and $\\widetilde{\\mathscr{Q}}_3$ is defined in (\\ref{S1EB2})--(\\ref{S1E21R}).\n\\end{remark}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Corollary \\ref{S1C31}}]  If we prove that $n$ satisfies  (\\ref{ene1}), the conclusion of the Corollary will follow from part (ii) of Theorem \\ref{EQUIV}. \nBy the hypothesis and part (ii) of Theorem \\ref{EQUIV}, the limit in (\\ref{ZE01}) exists, and (\\ref{Bhtw}) holds, that we write:\n\\begin{align*}\n\\lim _{ \\varepsilon \\to 0 }\\int _0^tn(s) \\mathscr Q_3^{(2)}(\\psi _\\varepsilon,g(s))ds-\\int _0^tn(s)M _{1\/2}(g(s))ds=\\\\\n=\\int  _{ [0, \\infty) }(G(0, x)-G(t, x))dx+n(t)-n(0).\n\\end{align*}\nUsing  the conservation of mass (\\ref{MBC}) it follows that $n$ satisfies equation (\\ref{ene1}).\n\\end{proof}\n\n\\begin{proposition}\n\\label{LM-1\/2}\nLet $G\\in\\mathscr{M}_+([0,\\infty))$. If $G$ has no atoms on $(0,\\infty)$ and $\\int_{(0,\\infty)}\\frac{G(x)}{\\sqrt{x}}dx<\\infty$, \nthen, for all $\\varphi _\\varepsilon $ as in Remarrk \\ref{TEST},\n\\begin{equation*}\n\\mathscr{T}(G)=\\lim_{\\varepsilon\\to 0}  \\mathscr Q_3^{(2)}(\\varphi_{\\varepsilon},G)=0.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\nBy definition \n$$\n\\mathscr{T}(G)=\\lim_{\\varepsilon\\to 0}\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi_{\\varepsilon})(x,y)}{\\sqrt{xy}}G(x)G(y)dxdy,\n$$\nSince $\\Lambda(\\varphi_{\\varepsilon})\\leq 1$ for all $\\varepsilon>0$ and \n\\begin{align*}\n\\lim _{ \\varepsilon \\to 0 }\\Lambda(\\varphi_{\\varepsilon})(x,y)=\\mathds{1}_{\\{x=y>0\\}}(x,y)\\qquad\\forall (x,y)\\in(0,\\infty)^2,\n\\end{align*}\nand $\\int_{(0,\\infty)}\\frac{G(x)}{\\sqrt{x}}dx<\\infty$, then by dominated convergence\n$$\n\\mathscr{T}(G)=\\iint_{\\{x=y>0\\}}\\frac{G(x)G(y)}{\\sqrt{xy}}dxdy.\n$$\nSince $G$ has no atoms on $(0,\\infty)$, i.e., $G(\\{x\\})=0$ for all $x>0$, by Fubini's theorem\n\\begin{align*}\n\\iint_{\\{x=y>0\\}}\\frac{G(x)G(y)}{\\sqrt{xy}}dxdy\n&=\\int_{(0,\\infty)}\\frac{G(x)}{x}G(\\{x\\})dx=0.\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\n\\label{HH}\nFrom Proposition \\ref{LM-1\/2}, if  $M_{-1\/2}(g)<\\infty$ and $g$ has no atoms, then $\\mu((0,t])=0$ for all $t>0$. \nIf  $g\\in L^1(0, \\infty)$ and $x=0$  is a Lebesgue point of $g$ then $\\mathscr{T}(g)=0$\n(cf. \\cite{Nouri1}) and again $\\mu((0,t])=0$ for all $t>0$. \nIf $g(x)=x^{-1\/2}$, then $\\mathscr{T}(g)=\\pi^2\/6$,\n(cf. \\cite{Lu3}), and a similar result holds if \n$\\lim _{ x\\to 0 }\\sqrt x g(x)=C>0$ (cf. \\cite{Spohn}). In that case,  $\\mu((0,t])=\\pi ^2\/6\\int _0^t n(s)ds$.\n\\end{remark}\n\n\\section{Proof of Theorem \\ref{S1T5}}\n\\label{SectionD}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\n\\begin{proof}\nBy (\\ref{CT98})  and (\\ref{CT99}), we deduce that for all $t>t_0>0$:\n\\begin{align}\n&\\int _{ t_0 }^t G(s,\\{0\\})ds\\leq \\big(M_{\\alpha}(G(t_0))-M_{\\alpha}(G(t))\\big)C(N,E,\\alpha) \\nonumber \\\\\n\\label{PRO116}\n&C(N,E,\\alpha)=\\left[\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}-C_{1}N^{3-\\alpha}E^{\\alpha-1}\\right]^{-1},\n\\end{align}\nwhere $C_{1}=2^{\\alpha}-2$ for $\\alpha\\in(1,2]$ and $C_1=\\alpha(\\alpha-1)$ for $\\alpha\\in[2,3]$.\nSince by part (i), $0\\leq M_{\\alpha}(G(t_0))-M{\\alpha}(G(t))\\leq M_{\\alpha}(G(t_0))$ for every $t>t_0$, we immediately deduce (\\ref{S1ET5B}).\n\n\nWe prove now (\\ref{S1ET5C}). \nSince, as we have seen in (\\ref{S6E80}), the function $n(t)J(t)$ is monotone nondecreasing, from where, for all $t>0$ and $s\\in (0, t)$:\n\\begin{equation*}\nn(t)\\ge e^{-\\int _s^tM _{ 1\/2 }(g(r))dr}n(s).\n\\end{equation*}\nAs we have $M _{ 1\/2 }(g(r))\\le \\sqrt{NE}$ for all $r\\ge 0$,\n\\begin{equation}\n\\label{S80E20}\nn(t)\\ge e^{-\\sqrt {NE}(t-s)}n(s).\n\\end{equation}\nBy (\\ref{S1ET5B}) we already have a sequence of times $\\theta_k$ such that $\\theta_k \\to \\infty $ and  $n(\\theta_k)\\to 0$ as $k\\to \\infty$.\nSuppose that there exists, for some $\\rho >0$, an increasing sequence of times $(s_k) _{ k\\in \\mathbb{N} }$ such that $s_k\\to\\infty$ as $k\\to\\infty$ and :\n\\begin{align*}\n\\forall k,\\;n(s_k)\\ge \\rho\\quad\\hbox{and}\\quad s _{ k+1 }-s_k>\\frac {\\log 2} {\\sqrt {NE}}.\n\\end{align*} \nThen, if we denote $t_k=s_k+\\frac {\\log 2} {\\sqrt {NE}}$,\nwe deduce from (\\ref{S80E20}) that for all $t\\in (s_k, t_k)$:\n\\begin{eqnarray*}\nn(t)\\ge e^{-\\sqrt {NE}(t-s_k)}n(s_k)\\ge e^{-\\sqrt {NE}(t_k-s_k)}\\rho =\\frac {\\rho } {2}.\n\\end{eqnarray*}\nThis would imply\n\\begin{equation*}\n\\int _0^\\infty n(t)dt\\ge \\sum_{ k=0 } ^\\infty \\int  _{ s_k }^{t_k}n(t)dt=\\infty,\n\\end{equation*}\nand this contradiction proves (\\ref{S1ET5C}).\n\\end{proof}\n\n\n\n\n\\section{Appendix}\n\\label{Appendix}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\nWe have gathered in this Section several results that are important and useful, but not directly related to the main results. For the sake of clarity, we present them in two different Sub Sections. In the first one, we find results that are used all along the manuscript, perhaps several times. In the second, we present results that are needed in Section \\ref{model}. \n\n\\subsection{A1}\n\n\\begin{lemma}[Convex-positivity]\n\\label{convex-positivity}\nLet $\\varphi\\in C([0,\\infty))$. If $\\varphi$ is convex then $\\Lambda(\\varphi)(x,y)\\geq 0$ for all $(x,y)\\in[0,\\infty)^2$ and $\\mathcal{L}_0(\\varphi)(x)\\geq 0$ for all $x\\in[0,\\infty)$. If $\\varphi$ is nonnegative and nonincreasing, then $\\mathcal{L}(\\varphi)(x)\\leq 0$ for all $x\\in[0,\\infty)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Lambda(\\varphi)(x,y)$ is symmetric we may reduce the proof to the case $0\\leq y\\leq x$. Putting\n$x=\\frac{x+y}{2}+\\frac{x-y}{2},$\nthen by the very definition of convexity\n$$\\varphi(x)\\leq\\frac{\\varphi(x+y)}{2}+\\frac{\\varphi(x-y)}{2},$$\ntherefore $\\Lambda(\\varphi)(x,y)\\geq 0$. \n\n\nThe positivity of $\\mathcal{L}_0(\\varphi)$ is equivalent to prove \n\\begin{align}\n\\label{positivity weak L}\n\\frac{1}{x}\\int_0^x\\varphi(y)d y\\leq\\frac{\\varphi(0)+\\varphi(x)}{2}\\qquad\\forall x\\in[0,\\infty).\n\\end{align}\nSince for any $0\\leq y\\leq x$ we may trivially write $y=\\left(1-\\frac{y}{x}\\right)\\, 0+\\frac{y}{x}\\, x$, then by convexity \n$\\varphi(y)\\leq \\left(1-\\frac{y}{x}\\right)\\varphi(0)+\\frac{y}{x}\\varphi(x)$, which implies\n\\eqref{positivity weak L}.\\\\\nIf $\\varphi$ is nonnegative and nonincreasing, then  $\\mathcal{L}(\\varphi)(x)\\leq -x\\,\\varphi(x)\\leq 0$ for all\\ $x\\in[0,\\infty).$\n\\end{proof}\n\n\\begin{remark}\n\\label{concave-negativity}\nBy linearity and Lemma \\ref{convex-positivity}, it follows that for all $\\varphi\\in C([0,\\infty))$ concave,\n$\\Lambda(\\varphi)(x,y)\\leq 0$ for all $(x,y)\\in[0,\\infty)^2$ and $\\mathcal{L}_0(\\varphi)(x)\\leq 0$ for all $x\\in[0,\\infty)$.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma regularity}\nConsider the operators $\\Lambda(\\cdot)$, $\\mathcal{L}_0(\\cdot)$ and $\\mathcal{L}(\\cdot)$ given in (\\ref{S1E154}), (\\ref{S1E155}) and (\\ref{S1E21R}) respectively. Then\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in\\emph{Lip}([0,\\infty))$ with Lipschitz constant $L$, then  \n\\begin{align}\n\\label{lemma regularity 1}\n\\frac{|\\Lambda(\\varphi)(x,y)|}{\\sqrt{xy}}\\leq 2L\\qquad\\forall (x,y)\\in[0,\\infty)^2.\n\\end{align}\n\\item\nIf $\\varphi\\in C^1([0,\\infty))$, then the map\n$(x,y)\\mapsto\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ belongs to $C([0,\\infty)^2)$ and \n\\begin{align}\n\\label{lemma regularity 2}\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}=0\\qquad\\forall (x,y)\\in\\partial[0,\\infty)^2.\n\\end{align}\n\\item \nIf $\\varphi\\in C([0,\\infty))$ then the maps $x\\mapsto\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}$ and  \n $x\\mapsto\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ belong to $C([0,\\infty))$ and \n$\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}= \\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}=0$ at $x=0$. If in addition \n$\\varphi$ is bounded, then\n\\begin{align}\n\\frac{|\\mathcal{L}_0(\\varphi)(x)|}{\\sqrt{x}}\\leq 4\\|\\varphi\\|_{\\infty}\\sqrt{x}\\qquad\\forall x\\in[0,\\infty), \\label{lemma regularity 4}\\\\\n\\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq 3\\|\\varphi\\|_{\\infty}\\sqrt{x}\\qquad\\forall x\\in[0,\\infty). \\label{lemma regularity 4B}\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) By the symmetry of $\\Lambda(\\varphi)$ we can assume that $0\\leq y\\leq x$, and directly from the Lipschitz continuity\n\\begin{align*}\n|\\Lambda(\\varphi)(x,y)|&\\leq |\\varphi(x+y)-\\varphi(x)|+|\\varphi(x-y)-\\varphi(x)|\\leq 2L\\,y,\n\\end{align*}\nwhich implies \\eqref{lemma regularity 1}.\\\\\n(ii) The only possible problem for the continuity is on the boundary of $[0,\\infty)^2$.\nAgain by the symmetry of $\\Lambda(\\varphi)$ we can assume $0\\leq y\\leq x$. Then by the mean value theorem\n$\\Lambda(\\varphi)(x,y)=y\\,(\\varphi'(\\xi_1)-\\varphi'(\\xi_2))$ for some $\\xi_1\\in(x,x+y)$ and $\\xi_2\\in(x-y,x)$. \nHence\n\\begin{align*}\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\leq \\varphi'(\\xi_1)-\\varphi'(\\xi_2),\n\\end{align*}\nand the continuity of $\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ on $[0,\\infty)^2$ and \\eqref{lemma regularity 2} follow from the continuity of $\\varphi'$.\\\\\n(iii) The continuity of $\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}$  and $\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ are clear for $x>0$.  Using that $\\frac{1}{x}\\int_0^x \\varphi(y)d y\\rightarrow \\varphi(0)$ as $x\\rightarrow 0$ by Lebesgue differentiation Theorem, it follows the continuity at $x=0$ and that $\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}=\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}=0$ for $x=0$. The bounds (\\ref{lemma regularity 4}) and (\\ref{lemma regularity 4B})   are straightforward  for $\\varphi\\in C_b([0,\\infty))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{regularised operators converge uniformly}\nConsider the operators $\\Lambda(\\cdot)$ and $\\mathcal{L}_0(\\cdot)$ given in (\\ref{S1E154}) and (\\ref{S1E155}), and a sequence \n$(\\phi_n)_{n\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ as in Cutoff \\ref{cut-off}.\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in C^1([0,\\infty))$ then $\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)\\xrightarrow[n\\rightarrow\\infty]{}\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ uniformly on the compact sets of $[0,\\infty)^2$.\n\\item\nIf $\\varphi\\in C([0,\\infty))$ then $\\mathcal{L}(\\varphi)(x)\\phi_n(x)\\xrightarrow[n\\rightarrow\\infty]{}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$\nuniformly on the compact sets of $[0,\\infty)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) The pointwise convergence on $[0,\\infty)^2$ is trivial since $\\phi_n(x)\\rightarrow x^{-1\/2}$ as $n\\rightarrow\\infty$. Then, let $\\varepsilon>0$ and $R>0$. For $n\\geq R$ there holds $\\phi_n(x)=x^{-1\/2}$ for all \n$x\\in[1\/n,R]$, so we only need to show the uniform convergence on the regions $(x,y)\\in[0,R]\\times[0,1\/n]$ and $(x,y)\\in[0,1\/n]\\times[0,R]$. \nBy the symmetry of $\\Lambda(\\varphi)$, we may study only one region. \n\nUsing that $\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ is continuous (hence uniformly continuous on compacts) and vanishes when  \n$(x,y)\\in\\partial[0,\\infty)^2$\n(c.f. Lemma \\ref{lemma regularity}), there holds for all $(x,y)\\in[0,R]\\times [0,1\/n]$ that, for $n$ large enough,\n\\begin{align*}\n\\left|\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}-\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)\\right|\\leq\\frac{|\\Lambda(\\varphi)(x,y)|}{\\sqrt{xy}}\\leq \\varepsilon\n\\end{align*}\n\n(ii) Let $\\varepsilon>0$ and $R>0$. Since for $n\\geq R$ there holds $\\phi_n(x)=x^{-1\/2}$ for all $x\\in[1\/n,R]$, we only need to prove the uniform convergence on the region $[0,1\/n]$. Using that $\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ is continuous (hence uniformly continuous on compacts) and vanishes when $x\\rightarrow 0$ (cf. Lemma \\ref{lemma regularity}), we have\n\\begin{align*}\n\\left|\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}-\\mathcal{L}(\\varphi)(x)\\phi_n(x)\\right|\\leq \\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq\\varepsilon\n\\qquad\\forall x\\in[0,1\/n]\n\\end{align*}\nfor $n$ large enough.\n\\end{proof}\n\n\n\n\n\nThe following Lemma is about the approximation of a measure by functions, keeping the mass and the energy constants. It is taken from \\cite{Lu1} with  minor modifications.\n\\begin{lemma}\n\\label{APROXDATA}\nLet $h_0\\in \\mathscr{M}_+^{\\alpha}([0,\\infty))$ for some $\\alpha\\geq 1$.\nThen there exists a sequence of functions $(f_n) _{n\\in\\mathbb{N}}\\subset C([0,\\infty))\\cap L^1\\big(\\mathbb{R}_+,(1+x^{\\alpha})dx\\big)$ with $f_n>0$ such that\n\\begin{align}\n\\label{GROWTHA}\n&\\forall \\varphi \\in C([0,\\infty)):\\quad\\sup_{x\\geq 0}\\frac{|\\varphi(x)|}{1+x^{\\alpha}}<\\infty,\\\\\n\\label{APROXDATA1}\n&\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)f_n(x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx.\n\\end{align}\nMoreover, if $M_1(h_0)>0$, then for all $n\\in\\mathbb{N}$:\n\\begin{align}\n\\label{APROXDATA2}\nM_0(f_n)=M_0(h_0)\\qquad\\text{and}\\qquad M_1(f_n)=M_1(h_0).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nFor $a>0$ and $b>0$, let\n\\begin{align*}\nJ_{a,b}(x)=a e^{-bx^2},\\qquad(x\\geq 0)\n\\end{align*}\nand let, for $n\\in\\mathbb{N}$,\n\\begin{align*}\nf_n(x)=e^n\\int_0^{\\frac{x}{1-e^{-n}}}J_{a,b}\\left(e^n\\big(x-y(1-e^{-n})\\big)\\right)h(y)dy,\\qquad(x\\geq0).\n\\end{align*}\nSince $J_{a,b}$ is bounded and $M_0(h_0)<\\infty$, then $f_n$ is well defined. The continuity and the strict positivity of $J_{a,b}$, together with $M_0(h)>0$, implies that $f_n$ is continuous and $f_n>0$.\nNow for any $\\varphi\\in C([0,\\infty))$ satisfying (\\ref{GROWTHA}), using Fubini and the change of variables $z=e^n\\big(x-y(1-e^{-n})\\big)$:\n\\begin{align}\n\\label{APROXDATA3}\n&\\int_0^{\\infty}\\varphi(x)f_n(x)dx=\\int_{[0,\\infty)}I_n(\\varphi)(y)h_0(y)dy,\\\\\n\\label{APROXDATA33}\n&I_n(\\varphi)(y)=\\int_0^{\\infty}\\varphi\\big(ze^{-n}+y(1-e^{-n})\\big)J_{a,b}(z)dz.\n\\end{align}\nSince by (\\ref{GROWTHA}):\n\\begin{align*}\n\\big|\\varphi\\big(ze^{-n}+y(1-e^{-n})\\big)\\big|&\\leq C\\Big(1+\\big(ze^{-n}+y(1-e^{-n})\\big)^{\\alpha}\\Big)\\\\\n&\\leq C2^{\\alpha}\\big(1+y^{\\alpha}+z^{\\alpha}\\big)\\\\\n&\\leq C2^{\\alpha}\\big(1+y^{\\alpha}\\big) \\big(1+z^{\\alpha}\\big),\n\\end{align*}\nand $h_0\\in\\mathscr{M}_+^{\\alpha}([0,\\infty))$, we deduce from (\\ref{APROXDATA3})-(\\ref{APROXDATA33}) that \n$f_n\\in L^1\\big(\\mathbb{R}_+,(1+x^{\\alpha})\\big)$. \nWe also deduce using dominated convergence that\n\\begin{align}\n&\\lim _{ n\\to \\infty }I_n(\\varphi)(y)=\\varphi(y),\\,\\,\\forall y\\geq 0,\\nonumber \\\\\n\\label{APROXDATA4}\n&\\lim _{ n\\to \\infty  }\\int_{[0,\\infty)}I_n(\\varphi)(y)h_0(y)dy=\\bigg(\\int_0^{\\infty} J_{a,b}(z)dz\\bigg)\\!\\!\\int_{[0,\\infty)}\\varphi(y)h_0(y)dy.\n\\end{align}\nWe now fix $a>0$ so that $\\int_0^{\\infty}J_{a,b}(z)dz=1$. Namely, $a=2\\sqrt{b\/\\pi}$. Then (\\ref{APROXDATA1}) follows from (\\ref{APROXDATA3}) and (\\ref{APROXDATA4}). \n\nIf we chose $\\varphi=1$ in (\\ref{APROXDATA3})-(\\ref{APROXDATA33}), the first part of (\\ref{APROXDATA2}) follows.\nIf we chose now $\\varphi(y)=y$ then\n\\begin{align*}\nM_1(f_n)=M_1(h)+e^{-n}\\left(\\frac{M_0(h_0)}{\\sqrt{b\\pi}}-M_1(h_0)\\right).\n\\end{align*}\nWe now fix $b=\\pi^{-1}(M_0(h_0)\/M_1(h_0))^2$ to obtain the second part of (\\ref{APROXDATA2}).\n\\end{proof}\n\n\\begin{corollary}\n\\label{APD1}\nLet $h_0\\in\\mathscr{M}^{\\alpha} _+([0,\\infty))$ for some $\\alpha \\ge 1$. Then, there exists a sequence of nonnegative functions \n$(h_{0,n})_{n\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ such that\n\\begin{align}\n\\label{APD56}\n\\limsup_{n\\to\\infty}M_{\\alpha}(h_{0,n})\\leq M_{\\alpha}(h_0),\n\\end{align}\nand for all $\\varphi\\in C_b([0,\\infty))$\n\\begin{align}\n\\label{APD3}\n&\\lim _{ n\\to \\infty }\\int_0^{\\infty}\\varphi(x)h_{0,n}(x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx.\n\\end{align}\n\n\\end{corollary}\n\n\\begin{proof}\nWe consider the sequence $(f_n)$ given by Lemma \\ref{APROXDATA} and a smooth cutoff $\\zeta_n\\in C([0,\\infty))$ such that $0\\leq\\zeta_n\\leq 1$, $\\zeta_n(x)=1$ for $x\\in[0,n]$ and $\\zeta_n(x)=0$ for $x\\geq n+1$. Then we define for all $n\\in\\mathbb{N}$:\n\\begin{align}\n\\label{APD33}\nh_{0,n}(x)=f_n(x)\\zeta_n(x)\n\\end{align}\nIt then follows that $h_{0,n}$ is continuous, nonnegative and with compact support. \nThe property (\\ref{APD56}) follows directly from (\\ref{APROXDATA1}) in Lemma \\ref{APROXDATA} since $h_{0,n}\\leq f_n$.\nNow let $\\varphi\\in C_b([0,\\infty))$. Since $f_n$ satisfies (\\ref{APROXDATA1}), in order to prove (\\ref{APD3}) it is sufficient to prove\n\\begin{align}\n\\label{APD4}\n\\lim _{ n\\to \\infty }\\bigg|\\int_0^{\\infty}\\varphi(x)h_{0,n}(x)dx-\\int_0^{\\infty}\\varphi(x)f_n(x)(x)dx\\bigg|=0,\n\\end{align}\nand (\\ref{APD4}) follows from\n\\begin{align*}\n\\lim_{n\\to\\infty}\\int_n^{\\infty}\\varphi(x)f_n(x)dx\\leq\\lim_{n\\to\\infty} \\frac{\\|\\varphi\\|_{\\infty}M_1(f_n)}{n}=0.\n\\end{align*}\n\\end{proof}\n\n\\begin{definition}\n\\label{definition operators strong}\nLet $h$, $\\phi_n$ and $\\varphi$ be real-valued functions with domain $\\mathbb{R}_+$. Then, let\n\\begin{align}\n\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h)=\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h)-\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h), \\label{Aq3tilden}\n\\end{align}\nwhere\n\\begin{align}\n\\label{Q3n2}\n&\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h)=\\int_0^{\\infty}\\!\\!\\!\\int_0^{\\infty} \\Lambda(\\varphi) (x, y)\\phi_n(x)\\phi_n(y)h(x)h(y)dxdy,\\\\\n\\label{Q3n1w}\n&\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h)=\\int_0^{\\infty}\\mathcal {L}(\\varphi )(x)\\phi_n(x)h(x)dx,\n\\end{align}\nand let, for $x\\in\\mathbb{R}_+$:\n\\begin{align}\n\\label{A1E32}\nJ_{3,n}(h)(x)&=K_n(h)(x)+L_n(h)(x)-h(x)A_n(h)(x),\n\\end{align} \nwhere\n\\begin{align}\nK_n(h)(x)&=\\int_0^x h(x-y)h(y)\\phi_n(x-y)\\phi_n(y)d y \\nonumber\\\\\n&+2\\int_x^{\\infty} h(y)h(y-x)\\phi_n(y)\\phi_n(y-x)d y,\\label{A1E33}\\\\\nL_n(h)(x)&=2\\int_x^{\\infty}h(y)\\phi_n(y)d y,\\label{A1E34}\\\\\nA_n(h)(x)&=\\phi_n(x)\\Big(x+4\\int_0^x h(y)\\phi_n(y)d y\\Big).\\label{A1E35}\n\\end{align}\n\\end{definition}\n\n\\begin{lemma}\n\\label{convergence lemma}\nLet $G\\in\\mathscr{M}_+([0,\\infty))$, $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, and $\\phi_n$ as in Cutoff \\ref{cut-off}. Then\n\\begin{align}\n&G(\\{0\\})=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(x)dx,\\label{convergence 1}\\\\\n&\\lim_{\\varepsilon\\to 0}\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi_{\\varepsilon},G)=0\\qquad\\forall n\\in\\mathbb{N}.\\label{convergence 3}\n\\end{align}\nIf in addition $G$ has no singular part in $(0,\\infty)$, then\n\\begin{align}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_{3,n}^{(2)}(\\varphi_{\\varepsilon},G)=0\\qquad\\forall n\\in\\mathbb{N}.\\label{convergence 4}\n\\end{align}\nFurthermore, if $G\\in\\mathscr{M}_+^{1\/2}([0,\\infty))$, then\n\\begin{align}\n\\label{limQ31a}\n&\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(1)}(\\varphi_{\\varepsilon},G)=M_{1\/2}(G),\\\\\n\\label{limQ31b}\n&\\lim_{\\varepsilon\\to 0}\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},G)=0,\n\\end{align}\nwhere $\\mathscr{Q}_3^{(1)}$ and $\\widetilde{\\mathscr{Q}}_3^{(1)}$ are defined in (\\ref{S1E1Q31}) and (\\ref{S1E20R}) respectively.\n\\end{lemma}\n\n\\begin{proof}\nThe proof only uses dominated convergence. Since $\\varphi_{\\varepsilon}\\leq 1$ for all $\\varepsilon>0$, and $M_0(G)<\\infty$, and \n$\\varphi_{\\varepsilon}\\to\\mathds{1}_{\\{0\\}}$ as $\\varepsilon\\to 0$, then (\\ref{convergence 1}) holds.\nThen, since for all $x\\in[0,\\infty)$ it follows from dominated convergence that\n\\begin{align}\n\\label{LIN1}\n\\lim_{\\varepsilon\\to 0}\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)=x\n\\qquad\\text{and}\\qquad\n\\lim_{\\varepsilon\\to 0}\\mathcal{L}(\\varphi_{\\varepsilon})(x)=0,\n\\end{align}\nand $\\phi_n$ is compactly supported, then (\\ref{convergence 3}) follows. \nAlso, since for all $(x,y)\\in[0,\\infty)^2$, $\\Lambda(\\varphi_{\\varepsilon})(x,y)\\leq 1$ for all $\\varepsilon>0$, and \n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\Lambda(\\varphi_{\\varepsilon})(x,y)=\\mathds{1}_{\\{x=y>0\\}}(x,y),\n\\end{align*}\nthen\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_{3,n}^{(2)}(\\varphi_{\\varepsilon},G)=\\iint_{\\{x=y>0\\}}\\phi_n(x)\\phi_n(y)G(x)G(y)d xd y,\n\\end{align*}\nUsing that $G$ has no singular part on $(0,\\infty)$, (\\ref{convergence 4}) follows.\n\nLastly, since \n\\begin{align}\n\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},G)\\leq \\mathscr{Q}_3^{(1)}(\\varphi_{\\varepsilon},G)=\\int_{(0,\\infty)}\\frac{\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)}{\\sqrt{x}}G(x)dx,\n\\end{align}\nand by (\\ref{lemma regularity 4})\n\\begin{align*}\n\\int_{(0,\\infty)}\\frac{|\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)|}{\\sqrt{x}}G(x)dx\\leq 4M_{1\/2}(G)\\qquad\\forall\\varepsilon>0.\n\\end{align*}\nthen (\\ref{limQ31a}) and (\\ref{limQ31b}) follows from (\\ref{LIN1}) and dominated convergence.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{well defined operators}\nConsider $n\\in\\mathbb{N}$, $\\phi_n\\in C_c([0,\\infty))$ nonnegative and  $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$ nonnegative. Then for every nonnegative functions \n$h$, $h_1$ and $h_2$ in $L^{\\infty}(\\mathbb{R}_+)$, the functions $K_n(h)$, $L_n(h)$, $A_n(h)$ and $hA_n(h)$ are also nonnegative, belong to \n$L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$, and there exists a positive constant $C(n,\\rho)$ such that:\n\\begin{align}\n&\\|K_n(h_1)-K_n(h_2)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h_1\\|_{\\infty}\\|h_1-h_2\\|_{\\infty}\\label{SAE100}\\\\\n&\\|L_n(h)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h\\|_{\\infty}\\label{bound L}\\\\\n&\\|A_n(h)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\big(1+\\|h\\|_{\\infty}\\big)\\label{SaE120}\\\\\n&\\|A_n(h_1)-A_n(h_2)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h_1-h_2\\|_{\\infty}.\\label{SaE121}\n\\end{align}\n\\begin{flalign}\n\\text{Moreover}&&J_{3,n}(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+).&&\n\\end{flalign}\n\\end{lemma}\n\n\\begin{proof}\nThe positivity of the operators is clear from their definitions. Notice that since $\\phi_n$ is bounded and compactly supported on $\\mathbb{R}_+$ and $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$, there exist two positive constants $C(n)$ and $C(n,\\rho)$ such that\n\\begin{align*}\n&\\sup_{x\\geq 0}\\int_0^{\\infty} \\phi_n(|x-y|)\\phi_n(y)d y\\leq C(n),\\\\\n&\\int_0^{\\infty}\\!\\!\\!\\int_0^{\\infty}\\rho(x)\\phi_n(|x-y|)\\phi_n(y)d yd x\\leq C(n,\\rho).\n\\end{align*}\n\n\\noindent1. Estimates for $K_n$. For all $x\\geq 0$:\n\\begin{align*}\nK_n(h)(x)&\\leq 3\\|h\\|_{\\infty}^2\\int_0^{\\infty} \\phi_n(|x-y|)\\phi_n(y)d y\\leq 3\\|h\\|_{\\infty}^2C(n),\n\\end{align*}\nand\n\\begin{align*}\n\\|K_n(h)\\|_{L^1_{\\rho}}\n&\\leq 3\\|h\\|_{\\infty}^2\\int_0^{\\infty}\\!\\!\\! \\int_0^{\\infty}  \\rho(x)\\phi_n(|x-y|)\\phi_n(y)d yd x\\leq 3\\|h\\|_{\\infty}^2 C(n,\\rho).\n\\end{align*}\nThen for all $x\\geq 0$:\n\\begin{align}\n\\label{KNs}\n&\\big|K_n(h_1)(x)-K_n(h_2)(x)\\big|\\\\\n&\\leq3\\int_0^{\\infty}\\phi_n(|x-y|)\\phi_n(y)\\big|h_1(|x-y|)h_1(y)-h_2(|x-y|)h_2(y)\\big|d y.\\nonumber\n\\end{align}\nWithout loss of generality we assume that $\\|h_1\\|_{\\infty}\\geq \\|h_2\\|_{\\infty}$. Using \n\\begin{align*}\n&\\big|h_1(|x-y|)h_1(y)-h_2(|x-y|)h_2(y)\\big|\\leq 2\\|h_1\\|_{\\infty}\\|h_1-h_2\\|_{\\infty}\n\\end{align*}\nin (\\ref{KNs}) then (\\ref{SAE100}) follows.\\\\\n2. Estimates for $L_n$. Since $\\phi_n$ is bounded and compactly supported and $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$, there exist two positive constants $C(n)$ and \n$C(n,\\rho)$ such that\n\\begin{align*}\n\\int_0^{\\infty}\\phi_n(x)d x\\leq C(n)\\quad\\text{and}\\quad\n\\int_0^{\\infty}\\rho(x)\\int_x^{\\infty}\\phi_n(y) dy dx\\leq C(n,\\rho)\n\\end{align*}\nand (\\ref{bound L}) follows.\\\\\n3. Estimates for $A_n$.\nThe estimate  (\\ref{SaE120}) follows from\n\\begin{align*}\n\\|A_n(h)\\|_{\\infty}&\\leq \\|x\\,\\phi_n(x)\\|_{\\infty}+4\\|\\phi_n\\|_{\\infty}^2\\|h\\|_{\\infty}|\\operatorname{supp}(\\phi_n)|\\leq C(n)(1+\\|h\\|_{\\infty}),\n\\end{align*}\nand\n\\begin{align*}\n\\|A_n(h)\\|_{L^1_{\\rho}}&\\leq \\int_0^{\\infty}\\rho(x)\\,x\\,\\phi_n(x)d x+4\\,\\|h\\|_{\\infty}\\int_0^{\\infty}\\rho(x)\\,\\phi_n(x)\\int_0^x \\phi_n(y)d yd x\\\\\n&\\leq C(n,\\rho)(1+\\|h\\|_{\\infty}).\n\\end{align*}\nFor all $x\\geq 0$,\n\\begin{align*}\n|A_n(h_1)(x)-A_n(h_2)(x)|&\\leq 4\\|h_1-h_2\\|_{\\infty}\\phi_n(x)\\int_0^x\\phi_n(y)d y\\\\\n&\\leq C(n) \\|h_1-h_2\\|_{\\infty}.\n\\end{align*}\nWe also have, \n\\begin{align*}\n\\|A_n(h_1)-A_n(h_2)\\|_{L^1_{\\rho}}&\\leq 4\\|h_1-h_2\\|_{\\infty}\\int_0^{\\infty}\\rho(x)\\phi_n(x)\\int_0^x\\phi_n(y)d yd x\\\\\n&\\leq C(n,\\rho)\\,\\|h_1-h_2\\|_{\\infty},\n\\end{align*}\nand then, (\\ref{SaE121}) follows.\\\\\n4. Since $h\\in L^{\\infty}(\\mathbb{R}_+)$ and $A_n(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$, then $hA_n(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$.\\\\\n5. It also follows from points 1 to 4 that $J_{3,n}(h)$ has the desired regularity.\n\\end{proof}\n\n\\subsection{A2}\n\n\\begin{lemma}\n\\label{representation of Deltavarphi}\nLet $\\varphi\\in C ^{1.1}([0,\\infty))$. Then, for all $(x_1,x_2,x_3)\\in[0,\\infty)^3$ such that $x_1+x_2\\geq x_3$:\n\\begin{align*}\n&\\Delta\\varphi(x_1,x_2,x_3)\n=(x_1-x_3)(x_2-x_3)\\times  \\nonumber \\\\\n&\\qquad \\times \\int_0^1\\int_0^1\\varphi''\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt.\n\\end{align*}\nMoreover, if $\\varphi\\in C^{1.1}_b([0,\\infty))$, then for all $(x_1,x_2,x_3)\\in [0,\\infty)^3$\n\\begin{align}\n|\\Delta\\varphi(x_1,x_2,x_3)| \\leq\\min\\left\\{A, B, C, D\\right\\}. \\label{S2E2}\n\\end{align}\n\\begin{flalign}\n\\text{where} &&A&=4\\|\\varphi\\|_{\\infty},\\quad B=2\\|\\varphi'\\|_{\\infty}|x_1-x_3|,\\quad C=2\\|\\varphi'\\|_{\\infty}|x_2-x_3|,\\nonumber\\\\\n&&D&=\\|\\varphi''\\|_{\\infty}|x_1-x_3||x_2-x_3|.\\nonumber \n\\end{flalign}\n\\end{lemma}\n\n\\begin{proof}\nLet $(x_1,x_2,x_3)\\in[0,\\infty)^3$ be such that $x_1+x_2\\geq x_3$. By the fundamental Theorem of calculus\n\\begin{align*}\n\\Delta &\\varphi(x_1,x_2,x_3)=\\big[\\varphi(x_4)-\\varphi(x_2)\\big]-\\big[\\varphi(x_1)-\\varphi(x_3)\\big]\\\\\n&  =\\int_0^1\\frac{d}{dt}\\varphi\\big(x_2+t(x_1-x_3)\\big)dt-\\int_0^1\\frac{d}{dt}\\varphi\\big(x_3+t(x_1-x_3)\\big)dt\\\\\n&=(x_1-x_3)\\int_0^1\\big[\\varphi'\\big(x_2+t(x_1-x_3)\\big)-\\varphi'\\big(x_3+t(x_1-x_3)\\big)\\big]dt\\\\\n&=(x_1-x_3)\\int_0^1\\int_0^1\\frac{d}{ds}\\varphi'\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt\\\\\n&=(x_1-x_3)(x_2-x_3)\\int_0^1\\int_0^1\\varphi''\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt.\n\\end{align*}\nAssume now that $\\varphi\\in C^{1.1}_b([0,\\infty))$. Using the first, the third, and the fifth line above, estimate (\\ref{S2E2}) follows. \n\\end{proof}\n\nWe now consider the function $w$ given in (\\ref{S1E6'}) and define\n\\begin{align}\n\\label{S2E3}\nW(x_1,x_2,x_3)=\\left\\{\n\\begin{array}{ll}\n\\frac{w(x_1,x_2,x_3)}{\\sqrt{x_1x_2x_3}}&\\!\\!\\text{if}\\quad(x_1,x_2,x_3)\\in (0,\\infty)^3\\\\\n\\frac{1}{\\sqrt{x_1x_2}}&\\!\\!\\!\\!\\!\\!\\text{if}\\quad x_3=0,\\quad (x_1,x_2)\\in(0,\\infty)^2\\\\\n\\frac{1}{\\sqrt{x_ix_3}}&\\!\\!\\!\\!\\!\\!\\text{if}\\; x_j=0,\\, x_i>x_3>0;\\,\\{i, j\\}=\\{1,2\\} \\\\\n0&\\!\\!\\!\\text{otherwise}.\n\\end{array}\n\\right.\n\\end{align}\n\nWe then have:\n\\begin{lemma}\n\\label{S2L1}\nConsider the function $\\Phi_{\\varphi}=W\\Delta\\varphi$, where $\\Delta\\varphi$  and $W$  are defined in (\\ref{S2E1}) and (\\ref{S2E3}) respectively.\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in C^{1.1}([0,\\infty))$ then $\\Phi_{\\varphi}\\in C([0,\\infty)^3)$.\n\\item\nIf $\\varphi\\in C^{1.1}_b([0,\\infty))$ then $\\Phi_{\\varphi}\\in C_0([0,\\infty)^3)$. In particular $\\Phi_{\\varphi}$ is uniformly continuous on $[0,\\infty)^3$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\textbf{Proof of (i).} By definition $\\Phi _\\varphi\\in C ((0, \\infty)^3)$. Therefore it only remains to study the behaviour of $\\Phi _\\varphi$ in a neighborhood of the boundary $\\partial [0, \\infty)^3$ of $[0, \\infty)^3$. First we show that $\\Phi_{\\varphi}$ is continuous on $\\partial [0, \\infty)^3$. \\\\\nThanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables, we just need to prove:\\\\\n(i)for all $(x_1,x_2)\\in (0,\\infty)^2$, \n\\begin{align}\n\\label{boundary1}\n\\Phi_{\\varphi}(x_1,x_2,0)=\\frac{\\Delta\\varphi(x_1,x_2,0)}{\\sqrt{x_1x_2}}\\longrightarrow 0\n\\end{align}\nwhenever $x_1\\rightarrow 0$ or $x_2\\rightarrow 0$ or $(x_1,x_2)\\rightarrow (0,0)$, and\\\\\n(ii) for all $x_1>x_3>0$,\n\\begin{align}\n\\label{boundary2}\n\\Phi_{\\varphi}(x_1,0,x_3)=\\frac{\\Delta\\varphi(x_1,0,x_3)}{\\sqrt{x_1x_3}}\\longrightarrow 0\n\\end{align}\nwhenever $x_1\\rightarrow x_3$ or $x_3\\rightarrow 0$ or $(x_1,x_3)\\rightarrow (0,0)$.\n\nBy (\\ref{S2E2}) $|\\Delta\\varphi(x_1,x_2,0)|\\leq \\|\\varphi''\\|_{\\infty}x_1x_2$ for all $(x_1,x_2)\\in (0,\\infty)^2$, which implies (\\ref{boundary1}). Also $|\\Delta\\varphi(x_1,0,x_3)|\\leq \\|\\varphi''\\|_{\\infty}x_3(x_1-x_3)$ for all $x_1>x_3>0$. Hence\n\\begin{align*}\n\\frac{|\\Delta\\varphi(x_1,0,x_3)|}{\\sqrt{x_1x_3}}\\leq  \\|\\varphi''\\|_{\\infty}\\sqrt{\\frac{x_3}{x_1}}(x_1-x_3)\\leq \\|\\varphi''\\|_{\\infty}(x_1-x_3),\n\\end{align*}\nwhich implies (\\ref{boundary2}).\n\nThen we prove that for any $x\\in\\partial[0,\\infty)^3$ and for any \n$(x_n)_{n\\in\\mathbb{N}}\\subset (0,\\infty)^3$ such that $x_n\\rightarrow x$, then $\\Phi_{\\varphi}(x_n)\\rightarrow\\Phi_{\\varphi}(x)$ as $n\\to\\infty$. Let us denote\n$$\\Omega=\\{(x_1,x_2,x_3)\\in(0,\\infty)^3:x_1+x_2\\leq x_3\\}.$$ Since $x_4$ is defined as $x_4=(x_1+x_2-x_3)_+$, then for all $(x_1,x_2,x_3)\\in(0,\\infty)^3$,\n$$(x_1,x_2,x_3)\\in\\Omega\\quad\\text{if and only if}\\quad x_4=0.$$\nIt might happen that the sequence $(x_n)_{n\\in\\mathbb{N}}$ ``jumps'' from $\\Omega$ to $\\Omega^c$. If in every neighbourhood of $x$ the sequence has points in both regions, then we may consider two subsequences, each one contained in one region only. For the sequel, the main estimate is the following:\nif we denote $x_n=(x_1^n,x_2^n,x_3^n)$ and $w(x_n)=\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_2^n},\\sqrt{x_3^n},\\sqrt{x_4^n}\\right\\}$, then by (\\ref{S2E2}) \n\\begin{align}\n\\label{main estimate to prove continuity}\n|\\Phi_{\\varphi}(x_n)|\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{w(x_n)}{\\sqrt{x_1^nx_2^nx_3^n}}\n\\big|x_1^n-x_3^n\\big|\\big|x_2^n-x_3^n\\big|.\n\\end{align}\nWe study case by case depending on where $x$ lies.\n\nCase $x=(0,0,0)$. If $(x_n)\\subset\\Omega$ then $x_4^n=0$, \n$w(x_n)=\\sqrt{x_4^n}=0$ and thus $\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\\\\\nIf $\\{x_n\\}\\subset\\Omega^c$ then $x_4^n>0$ and we study case by case depending on the relative order of $x_1^n$, $x_2^n$, and $x_3^n$. \nSince $\\Phi_{\\varphi}$\nis symmetric in the $x_1$, $x_2$ variables, we may assume without loss of generality that $x_1^n\\leq x_2^n$.  \nNote by \\eqref{main estimate to prove continuity} that we also may assume $x_3^n\\neq x_1^n$, $x_3^n\\neq x_2^n$; otherwise the result follows directly.\n\nIf $x_1^n\\leq x_2^n<x_3^n$,\nthen $w(x_n)=\\sqrt{x_4^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{\\sqrt{x_4^n}}{\\sqrt{x_1^nx_2^nx_3^n}}\\big(x_3^n-x_1^n\\big)\\big(x_3^n-x_2^n\\big)\\\\\n&\\leq\\|\\varphi''\\|_{\\infty}\\left(\\frac{\\sqrt{x_4^n}\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_1^nx_2^n}}+\\frac{\\sqrt{x_4^n x_1^n x_2^n}}{\\sqrt{x_3^n}}\\right)\\\\\n&\\leq\\|\\varphi''\\|_{\\infty}\\left( \\frac{\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_2^n}}+\\sqrt{x_1^n x_2^n}\\right).\n\\end{align*}\nSince $x_n\\to x=0$, then $\\sqrt{x_1^n x_2^n}\\to 0$. Moreover, since $x_n\\in\\Omega^c$ and $x_1^n\\leq x_2^n$, then \n$x_3^n<2 x_2^n$, and so \n$$ \\frac{\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_2^n}}\\leq 2^{3\/2} x_2^n\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.$$\n\nIf $x_1^n<x_3^n< x_2^n$, then $w(x_n)=\\sqrt{x_1^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\\frac{(x_2^n-x_3^n)(x_3^n -x_1^n)}{\\sqrt{x_2^n x_3^n}}\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_2^n x_3^n}+\\frac{x_1^n\\sqrt{x_3^n}}{\\sqrt{x_2^n}}\\right)\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_2^n x_3^n}+\\sqrt{x_1^n x_3^n}\\right)\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.\n\\end{align*}\n\nLastly, if $x_3^n<x_1^n\\leq x_2^n$, then $w(x_n)=\\sqrt{x_3^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\\frac{(x_1^n-x_3^n)(x_2^n-x_3^n)}{\\sqrt{x_1^n x_2^n}}\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_1^n x_2^n}+\\frac{\\big(x_3^n\\big)^2}{\\sqrt{x_1^n x_2^n}}\\right)\\\\\n&\\leq 2\\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_1^n x_2^n}+x_1\\right)\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.\n\\end{align*}\nHence, in the three cases above $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x).$\n\nCase $x=(x_1,0,0)$ with $x_1>0$. \nThen $w(x_n)=\n\\min\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}$ for $n$ large enough.\nOn the other hand\n\\begin{align*}\n\\big|x_2^n-x_3^n\\big|&=\\big(\\sqrt{x_2^n}+\\sqrt{x_3^n}\\big)\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|\\\\\n&\\leq 2\\max\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|.\n\\end{align*}\nSince $\\min\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}\\max\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}=\\sqrt{x_2^nx_3^n}$, then by \\eqref{main estimate to prove continuity} \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq2\\|\\varphi''\\|_{\\infty}\\frac{\\big|x_1^n-x_3^n\\big|}{\\sqrt{x_1^n}}\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|\n\\end{align*}\nfor $n$ large enough. It then follows $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x)$ as $n\\to\\infty$.\n\nThe case $x=(0,x_2,0)$ with $x_2>0$ is analogous to the previous one thanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables.\n\nCase $x=(0,0,x_3)$ with $x_3>0$. Then $x_n\\in\\Omega$ for $n$ large enough, $x_4^n=0$ and\n$w(x_n)=\\sqrt{x_4^n}=0$. Thus\n$\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$ for $n$ large enough.\n\nCase $x=(0,x_2,x_3)$ with $x_2>0$ and $x_3>0$. If $x_2>x_3$ then\n$w(x_n)=\\sqrt{x_1^n}$ for $n$ large enough and \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)-\\Phi_{\\varphi}(x)|=\\left|\\frac{1}{\\sqrt{x_2^nx_3^n}}\\Delta\\varphi(x_1^n,x_2^n,x_3^n)\n-\\frac{1}{\\sqrt{x_2x_3}}\\Delta\\varphi(0,x_2,x_3)\\right|,\n\\end{align*}\nwhich clearly goes to zero as $n\\rightarrow\\infty$. If $x_2<x_3$ then $x_4^n=0$ for $n$ large enough and \n$w(x_n)=\\sqrt{x_4^n}=0$, thus $\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\nIf $x_2=x_3$ and $(x_n)\\subset\\Omega$ for $n$ large enough, then $x_4^n=0$, thus\n$\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\\\\ \nIf $x_2=x_3$ and $(x_n)\\subset\\Omega^c$ for $n$ large enough, then\n$w(x_n)\n=\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}$, and by \\eqref{main estimate to prove continuity} \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}}{\\sqrt{x_1^nx_2^nx_3^n}}\n\\big|x_1^n-x_3^n\\big|\\big|x_2^n-x_3^n\\big|.\n\\end{align*}\nOn the one hand \n\\begin{align*}\n\\big|x_1^n-x_3^n\\big|\n\\leq 2\\max\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}\\big|\\sqrt{x_1^n}-\\sqrt{x_3^n}\\big|.\n\\end{align*}\nOn the other hand \n$\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}\\leq\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}$ for $n$ large enough. \nSince $\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}\\max\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}=\\sqrt{x_1^n x_3^n}$, then\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq2\\|\\varphi''\\|_{\\infty}\\frac{\\big|x_2^n-x_3^n\\big|}{\\sqrt{x_2^n}}\\big|\\sqrt{x_1^n}-\\sqrt{x_3^n}\\big|,\n\\end{align*}\nwhich goes to zero as $n\\rightarrow\\infty$ since $x_2=x_3$. Thus $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x)$.\n\nThe case $x=(x_1,0,x_3)$ with $x_1>0$ and $x_3>0$ is analogous to the previous one thanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables.\n\nCase $x=(x_1,x_2,0)$ with $(x_1,x_2)\\in (0,\\infty)^2$. Then $w(x_n)=\\sqrt{x_3^n}$ for $n$ large enough and\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)-\\Phi_{\\varphi}(x)|=\\left|\\frac{1}{\\sqrt{x_1^nx_2^n}}\\Delta\\varphi(x_1^n,x_2^n,x_3^n)\n-\\frac{1}{\\sqrt{x_1x_2}}\\Delta\\varphi(x_1,x_2,0)\\right|,\n\\end{align*}\nwhich clearly goes to zero as $n\\rightarrow\\infty$.\n\n\n\\textbf{Proof of (ii).}\nBy part (i) $\\Phi_{\\varphi}\\in C([0,\\infty)^3)$. Let us show now that for any given $\\varepsilon>0$ there exists $R(\\varepsilon)>0$ such that \n$|\\Phi_{\\varphi}(x)|\\leq\\varepsilon$ for all $x\\in[0,\\infty)^3\\setminus[0,R(\\varepsilon)]^3$. \n\nGiven $R>0$ and $\\alpha>0$, let $(x_1,x_2,x_3)\\in[0,\\infty)^3\\setminus[0,R]^3$ and denote\n$x_i=\\min\\{x_1,x_2,x_3\\}$, $x_k=\\max\\{x_1,x_2,x_3\\}$ and $x_j$ neither $x_i$ nor $x_k$.\nNotice that $x_k>R$ and the function $W$ defined in (\\ref{S2E3}) satisfies $W(x_1,x_2,x_3)\\leq\\frac{1}{\\sqrt{x_jx_k}}$.\nIf $x_i>\\alpha$ or $x_j>\\alpha$ then by (\\ref{S2E2}) \n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|\\leq\\frac{|\\Delta\\varphi(x_1,x_2,x_3)|}{\\sqrt{x_j x_k}}\\leq\\frac{4\\|\\varphi\\|_{\\infty}}{\\sqrt{\\alpha R}}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq\\frac{16\\|\\varphi\\|^2_{\\infty}}{\\alpha\\varepsilon^2}.$\nIf $x_i\\leq\\alpha$ and $x_j\\leq\\alpha$ we study case by case depending on the relative position of $x_1$, $x_2$, $x_3$. Since $\\Phi_{\\varphi}$ is symmetric in variables $x_1$ and $x_2$, we may assume without loss of generality that $x_2\\leq x_1$.\nIf $x_k=x_1$, using (\\ref{S2E2})\n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|&\\leq\\frac{2\\|\\varphi'\\|_{\\infty}(x_j-x_i)}{\\sqrt{x_1 x_j}}\n\\leq\\frac{2\\|\\varphi'\\|_{\\infty}\\sqrt{x_j}}{\\sqrt{x_1}}\n\\leq\\frac{2\\|\\varphi'\\|_{\\infty}\\sqrt{\\alpha}}{\\sqrt{R}}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq\\frac{4\\|\\varphi'\\|^2_{\\infty}\\alpha}{\\varepsilon^2}.$\nIf $x_k=x_3$ and $x\\in\\Omega$ then $x_4=0$ and $\\Phi_{\\varphi}(x)=0$.\nIf $x_k=x_3$ and $x\\in\\Omega^c$, then\n$x_1\\geq R\/2$ and\n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|&\\leq\\frac{4\\|\\varphi\\|_{\\infty}}{\\sqrt{x_1x_3}}\\leq\\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{R}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq \\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{\\varepsilon}$.\n\nFinally, if we chose $R\\geq\\max\\left\\{\\frac{16\\|\\varphi\\|^2_{\\infty}}{\\alpha\\varepsilon^2},\\frac{4\\|\\varphi'\\|^2_{\\infty}\\alpha}{\\varepsilon^2},\n\\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{\\varepsilon}\\right\\}$ \nthen $\\Phi_{\\varphi}\\in C_0([0,\\infty)^3)$ and in particular, $\\Phi_{\\varphi}$ is uniformly continuous in $[0,\\infty)^3$.\n\\end{proof}\n\n\\noindent\n\\textbf{Acknowledgments.}\nThe research of the first author is supported by the Basque Government through the BERC 2014-2017 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323, and by MTM2014-52347-C2-1-R of DGES. The research of the second author is supported by grants MTM2014-52347-C2-1-R of DGES and  IT641-13 of the Basque Government. The authors acknowledge the valuable remarks and helpful comments received from A. H. M. Kierkels and Pr. J. J. L. Vel\\'azquez, as well as the hospitality of  IAM at the University of Bonn.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAll existing experimental data in particle physics are in good agreement\nwith the Standard Model predictions. However, the problems exist which\ncould not be resolved within the SM and it is obviously not a complete\nor final theory. It is unquestionable that the SM should be the low-energy\nlimit of some higher symmetry. The question is what could be this symmetry.\nAnd the main question is what is the mass scale of this symmetry restoration.\nA gloomy prospect is the restoration of this higher symmetry at once on\na very high mass scale, the so-called 'gauge desert'. A concept of a\nconsecutive symmetry restoration is much more attractive. It looks\nnatural in this case to suppose a correspondence of the hierarchy\nof symmetries and the hierarchy of the mass scales of their restoration.\nNow we are on the first step of some stairway of symmetries\nand we try to guess what could be the next one.\nIf we consider some well--known higher symmetries from this point of view,\ntwo questions are pertinent. First, isn't the supersymmetry as the\nsymmetry of bosons and fermions, higher than the symmetry within the\nfermion sector, namely, the quark--lepton symmetry\\cite{3}, or the\nsymmetry within the boson sector, namely, the left--right\nsymmetry\\cite{4}? Second, wouldn't the supersymmetry restoration be\nconnected with a higher mass scale than the others?\n\nWe should like to analyse a possibility when the quark-lepton symmetry\nis the next step beyond the SM. We take a minimal symmetry of the\nPati-Salam type with the lepton number as the fourth color\\cite{3},\n$SU(4)_V \\otimes SU(2)_L \\otimes G_R$. The fermions are combined into the\nfundamental representations of the $SU(4)_V$ group, the neutrinos with\nthe {\\it up} quarks and the charged leptons with the {\\it down} quarks.\nSome attractive features of this symmetry should be pointed out.\n\ni) The renormalizability of the SM demands some quark-lepton symmetry,\nnamely, the fermions have to be combined into generations for the\ncancellation of the triangle anomalies.\n\nii) The proton decay is absent.\n\niii) A natural explanation for the quark fractional hypercharge takes\nplace.\n\n\\noindent Really, the 15-th generator of the $SU(4)$ group can be written\nin the form  $\\qquad \\qquad T_{15} = \\sqrt{3\/8} \\, diag \\, ( 1\/3 \\, , \\,\n1\/3 \\, , \\, 1\/3 \\, , \\, -1 )$.\nIt is traceless and the values of the left hypercharge $Y_L$ appear to\nbe placed on the diagonal. Let us call it the vector hypercharge,\n$Y_L = Y_V$.\n\niv) Let us suppose that $G_R \\, = \\, U(1)_R$. If we take the\nwell--known values\nof the SM hypercharge of the left and right, and $up$ and $down$ quarks and\nleptons, then from the equation $Y_{SM} \\, = \\, Y_V \\, + \\, Y_R$ the values\nof the right\nhypercharge $Y_R$ occur to be equal $\\pm 1$ for the $up$ and $down$ fermions,\nboth quarks and leptons.\nIt is tempting to interpret this fact as the evidence for the\nright hypercharge to be actually the doubled third component of the right\nisospin. Hence the $G_R$ group is possibly $SU(2)_R$.\n\nThe most exotic object of the Pati--Salam type symmetry is the charged and\ncolored gauge $X$ boson named leptoquark.\nIts mass $M_X$ should be the scale of reducing\nof $SU(4)_V$ to $SU(3)_c$. The bounds on the vector leptoquark\nmass\\cite{5} were obtained from the data on the $\\pi \\rightarrow e \\nu$\ndecay and from the upper limit on $K^0_L \\rightarrow \\mu e$ decay.\nIn fact, these estimations\nwere not comprehensive because the phenomenon of a mixing in the lepton-quark\ncurrents was not considered there. It can be shown that such a mixing\ninevitably occurs in the theory.\n\n\\section{Three types of fermion mixing}\n\nThree fermion generations are\ncombined into the \\{4,2\\} representations of the semi-simple group\n$SU(4)_V \\otimes SU(2)_L$ of the type\n\n\\begin{equation}\n\\left ( \\begin{array}{c} u^c \\\\ \\nu \\end{array} \\,\n\\begin{array}{c} d^c \\\\ \\ell \\end{array}\n\\right )_i , \\qquad (i=1,2,3) , \\label{eq:d}\n\\end{equation}\n\n\\medskip\n\n\\noindent where $c$ is the color index.\nThe mixing in the quark interaction with the $W$ bosons being depicted by\nthe Cabibbo-Kobayashi-Maskawa matrix is sure to exist in Nature.\nIf one starts from the diagonal $d, \\nu, \\ell$ states and the $u$\nstates mixed by the CKM matrix than at the one--loop level the $d$\nstates are mixed due to the conversion $d \\to W + u (c,t) \\to d'$ and\nthen the $\\ell$ states are mixed also, $\\ell \\to X + d (s,b) \\to \\ell'$, etc.\nConsequently, it is necessary for the renormalizability of the model\nto include all kinds of mixing at the tree-level.\nIn the general case, none of the $\\, u,d,\\nu,\\ell \\,$ components is the mass\neigenstate.\nDue to the identity of the three representations~(\\ref{eq:d})\nthey always could be regrouped so that one of the components was\ndiagonalized with respect to mass. If we diagonalize the charged\nlepton mass matrix, then the representations~(\\ref{eq:d}) can be rewritten\nto the form where the $\\nu, u, $ and $d$ states are not the mass\neigenstates and are included\ninto the same representations as the charged leptons $\\, \\ell$,\n$\\nu_\\ell  =  {\\cal K}_{\\ell i} \\nu_i , \\; u_{\\ell}  =\n{\\cal U}_{\\ell p} u_p , \\; d_{\\ell}  =  {\\cal D}_{\\ell n} d_n$ .\nHere  $\\nu_i, u_p, $ and $d_n \\, (i, p, n = 1,2,3)$  are the mass\neigenstates, and ${\\cal K}_{\\ell i} , \\, {\\cal U}_{\\ell p}$, and\n${\\cal D}_{\\ell n}$ are the unitary mixing matrices.\nThe standard Cabibbo-Kobayashi-Maskawa\nmatrix is seen to be $V \\, = \\, {\\cal U}^+ \\cal D$.\nThis is as far as we know about $\\cal U$ and $\\cal D$ matrices.\n$\\cal K$ is the mixing matrix in the lepton sector.\n\n\\section{Bounds from the low--energy experiments}\n\nSubsequent to the spontaneous $SU(4)_V$ symmetry breaking up to $SU(3)_c$\non the $M_X$ scale six massive vector bosons are separated from the 15-plet\nof the gauge fields to generate three charged and colored leptoquarks.\nTheir interaction with the fermions has the form\n\n\\begin{equation}\n{\\cal L}_X \\, = \\, \\frac{g_S(M_X)}{\\sqrt 2} \\big [\n{\\cal D}_{\\ell n}\n\\big ( \\bar \\ell \\gamma_{\\alpha} d^c_n \\big ) +\n\\big ( {\\cal K^+ \\; \\cal U} \\big )_{i p}\n\\big ( \\bar{\\nu_i} \\gamma_{\\alpha} u^c_p \\big ) \\big ] X^c_{\\alpha} +\nh.c.\n\\label{eq:Lx}\n\\end{equation}\n\n\\noindent The constant\n\\, $g_S(M_X)$ \\, can be expressed in terms of the strong coupling constant\n\\, $\\alpha_S$ \\, at the leptoquark mass scale\n$M_X, \\quad g_S^2(M_X)\/4 \\pi = \\alpha_S(M_X)$.\n\nIf the momentum transferred is \\, $q \\ll M_X$, \\, then the Lagrangian\n{}~(\\ref{eq:Lx}) in the second order leads to the effective four-fermion\nvector-vector interaction of quarks and leptons. By using the Fiertz\ntransformation, the scalar,\npseudoscalar, vector and axial-vector terms may be separated in the\neffective Lagrangian.\nThe QCD correction amounts to the appearance of the magnifying factor\n$Q(\\mu)$ at the scalar and pseudoscalar terms,\n$Q(\\mu) \\, = \\, ( {\\alpha_S(\\mu)}\/{\\alpha_S(M_X)})^{4\/\\bar b}$.\nHere $\\alpha_S(\\mu)$ is the effective strong coupling constant\nat the hadron mass scale $\\mu \\sim 1~GeV$,\n$\\; \\bar b \\, = \\, 11 \\, - \\, \\frac {2}{3} \\bar n_f, \\; \\bar n_f $\nis the averaged number of the quark\nflavors at the scales $\\mu^2 \\le q^2 \\le M_X^2$. If the condition\n$M_X^2 \\gg m_t^2$ is valid, then we have $\\, \\bar n_f \\, \\simeq \\, 6$,\nand $\\bar b \\, \\simeq \\, 7$.\n\nAs the analysis shows, the tightest restrictions on the leptoquark mass\n$M_X$ and the mixing matrix $\\cal D$ elements\ncan be obtained from the experimental data on rare $\\pi$ and $K$ decays and\n$\\mu^- \\rightarrow e^-$ conversion in nuclei.\nIn the description of the interactions of $\\pi$ and $K$ mesons it is\nsufficient to take the scalar and pseudoscalar terms only. As we\nshall see later, these terms acquire, in addition to the QCD corrections,\nan extra enhancement at the amplitude by the small quark current masses.\n\nOne can easily see that the leptoquark contribution to the $\\pi\n\\rightarrow e \\nu$ decay is not suppressed by the electron mass\nin contrast to the $W-$contribution.\nTaking into account the interference of the leptoquark and $W-$\nexchange amplitudes we get the following expression for the ratio\n\n\\begin{equation}\nR \\, = \\,\n\\frac{\\Gamma (\\pi \\rightarrow e \\nu)}\n{\\Gamma (\\pi \\rightarrow \\mu \\nu)} \\, = \\,\nR_{SM} \\left [ 1 \\, - \\,\n\\frac{2 \\sqrt 2 \\pi \\, \\alpha_S(M_X) \\, m^2_{\\pi} \\, Q(\\mu)}\n{G_F M^2_X m_e (m_u(\\mu) + m_d(\\mu) )} \\; Re \\left ( \\frac{{\\cal D}_{e d}\n{\\cal U}^*_{e u}}{V_{u d}} \\right ) \\right ],\n\\label{eq:R}\n\\end{equation}\n\n\\noindent where $R_{SM} = (1.2345 \\pm 0.0010) \\cdot 10^{-4}$ is\nthe value of the ratio\nin the Standard Model\\cite{6}, $m_{u,d}(\\mu)$ are the running current\nmasses. To the $\\mu \\simeq 1~GeV$ scale there correspond the well-known\nvalues $m_u \\simeq 4~MeV, m_d \\simeq 7~MeV$ and $m_s \\simeq 150~MeV$.\nUsing the experimental data\\cite{7},  $R^{exp} = (1.2310 \\pm 0.0037)\n\\cdot 10^{-4}$, we get\nthe following lower bound on the leptoquark mass\n\n\\begin{equation}\nM_X > (210~TeV) \\cdot |Re  ( {{\\cal D}_{e d}{\\cal U}^*_{e u}} \/\n{V_{u d}} ) |^{1\/2}.\n\\label{eq:X1}\n\\end{equation}\n\n{}From the data on the $K \\to e \\nu$ decay we obtain similarly\n\n\\begin{equation}\nM_X > (55~TeV) \\cdot |Re ({{\\cal D}_{e s}{\\cal U}^*_{e u}}\n\/ {V_{u s}} ) |^{1\/2}.\n\\label{eq:55T}\n\\end{equation}\n\nOne can establish the following limits from the data\\cite{8}\non the rare decays $K^0_L \\rightarrow \\mu e$ and\n$K^0_L \\rightarrow e^+ e^-$\n\n\\begin{equation}\nM_X > (1200~TeV) \\cdot |{\\cal D}_{e d}\n{\\cal D}^*_{\\mu s} \\; + \\;\n{\\cal D}_{e s}\n{\\cal D}^*_{\\mu d} |^{1\/2}, \\qquad\n\\label{eq:X2}\n\\end{equation}\n\n\\begin{equation}\nM_X > (1400~TeV) \\cdot |Re \\big ( {\\cal D}_{e d}\n{\\cal D}^*_{e s} \\big )|^{1\/2}.\n\\label{eq:Kee}\n\\end{equation}\n\nThe situation with another rare $K$ decay,\n$K^0_L \\rightarrow \\mu^+ \\mu^-$, is rather intriguing.\nThe recent measurements of the branching ratio at BNL\\cite{9}\nlowered its value closely to the unitary limit $Br_{abs} = 6.8 \\cdot 10^{-9}$,\nand thus the decay amplitude has no real part. However, it was\nshown\\cite{10} that the real part could not be small in the\nSM with a heavy top quark. Isn't it a signal for a new physics, e.g.\nleptoquark? In this regard the discontinuance of the experiment KEK E137\nwhere the $K^0_L \\rightarrow \\mu^+ \\mu^-$ decay rate was also measured,\nis regrettable.\n\nA low-energy process under an intensive experimental searches, where the\nleptoquark could manifest itself is the $\\mu e$ conversion in nuclei.\nWe estimate the branching ratio of the conversion in titanium and\nestablish the bound on the model parameters\non the base of the experimental data\\cite{11}\n\n\\begin{equation}\nM_X > (680~TeV) \\cdot |{\\cal D}_{e d}\n{\\cal D}^*_{\\mu d} |^{1\/2}.\n\\label{eq:Xmue}\n\\end{equation}\n\nThe above restrictions on the model parameters contain the elements of the\nunknown unitary mixing matrices $\\cal D$ and $\\cal U$, which are connected\nby the condition ${\\cal U}^+ {\\cal D} = V$ only.\nThus the possibility is not excluded, in principle, that the bounds obtained\ndid not restrict $\\, M_X \\,$ at all, e.g. if the elements ${\\cal D}_{e d}$\nand ${\\cal D}_{\\mu d}$ were rather small. It would correspond to the\nconnection of the $\\tau$ lepton largely with the $d$ quark in the ${\\cal D}$\nmatrix, and the electron and the muon with the $s$ and $b$ quarks.\nIn general, it is not contradictory to anything even if it appears to be\nunusual. In this case a leptoquark could give a more noticeable\ncontribution to the flavor-changing decays of the $\\tau$ lepton and\n$B$ mesons. However, an accuracy of these data is relatively poor.\n{}From the experimental limits\\cite{12} on the decays\n$\\tau^- \\to \\mu^- K^0$,\n$\\tau^- \\to e^- K^0$, and\n$B^+ \\to K^+ \\mu^+ e^-$,\n$B^+ \\to K^+ \\mu^- e^+$,\nwhich are possible via the leptoquark exchange without suppression by the\nelements ${\\cal D}_{e d}$ and ${\\cal D}_{\\mu d}$, we obtain\n\n\\begin{equation}\nM_X > (1~TeV) \\cdot |{\\cal D}_{\\mu s}\n{\\cal D}^*_{\\tau d} |^{1\/2},\n\\qquad\nM_X > (1~TeV) \\cdot |{\\cal D}_{e s}\n{\\cal D}^*_{\\tau d} |^{1\/2},\n\\label{eq:tau}\n\\end{equation}\n\n\\begin{equation}\nM_X > (2.4~TeV) \\cdot |{\\cal D}_{e s}\n{\\cal D}^*_{\\mu b} |^{1\/2},\n\\qquad\nM_X > (2.4~TeV) \\cdot |{\\cal D}_{\\mu s}\n{\\cal D}^*_{e b} |^{1\/2}.\n\\label{eq:BK}\n\\end{equation}\n\nIn the recent paper\\cite{13} the limits on the Pati--Salam leptoquark\nwere also considered in the specific cases when every charged\nlepton is connected with only one quark in the currents. For the most part\nthe results of ref.\\cite{13} agree with ours\\cite{1}.\n\n\\section {Mixing--independent bound}\n\nWe could find only one occasion when the mixing-independent lower bound\non the leptoquark mass arises, namely, from the decay\n$\\pi^0 \\rightarrow \\nu \\bar \\nu$. The best laboratory limit\\cite{14}\n on this decay is\n$\\; Br(\\pi^0 \\rightarrow \\nu \\bar \\nu ) < 8.3 \\cdot 10^{-7}$.\nIn the papers\\cite{15} the almost coinciding cosmological and\nastrophysical estimations of the width of this decay were found:\n$Br(\\pi^0 \\rightarrow \\nu \\bar \\nu ) < 3 \\cdot 10^{-13}$ .\nWithin the Standard Model this value is proportional to $m^2_{\\nu}$.\nThe process is also possible through the leptoquark mediation, without the\nsuppression by the smallness of neutrino mass.\nOn summation over all neutrino species the decay probability\nis mixing-independent. As a result the bound on the leptoquark mass occurs\n$M_X > 18~TeV$. However, in the recent paper\\cite{16} a criticism\nhas been expressed on both the cosmological and astrophysical limits.\nTherefore, only the laboratory limit\\cite{14} is reliable to establish\nthe bound $M_X > 440~GeV$.\n\n\\section {Rare muon decays}\n\nThe lepton--number violating decays $\\mu \\rightarrow e \\gamma$,\n$\\mu \\rightarrow e \\gamma \\gamma$, $\\mu \\rightarrow e e \\bar e$\nare under the intensive experimental searches. Let us point out,\nhowever, that these decay modes are strongly suppressed in the SM with\nlepton mixing due to\nthe well--known GIM cancellation\\cite{17} by the factor\n$(m_{\\nu}\/m_W)^4 \\; \\sim \\; 10^{-39} \\cdot\n(m_{\\nu}\/20 \\,eV)^4$.\n\nThese processes arise in the model with vector leptoquark at the loop\nlevel via the virtual $d, s, b$ quarks\\cite{2}.\nAs the analyses of the radiative muon decays show, the two--photon decay\ndominates the one--photon decay\n\n\\begin{equation}\n\\frac{\\Gamma(\\mu\\to e\\gamma\\gamma)}{\\Gamma(\\mu\\to e\\gamma)} \\; \\sim \\;\n\\frac{\\alpha}{\\pi} \\, \\left( \\frac{M_X}{m_b} \\right) ^{4} \\; \\gg \\; 1.\n\\label{eq:GIM1}\n\\end{equation}\n\n\\noindent The magnitude of the $\\mu \\rightarrow e \\gamma \\gamma$ decay width\ncould be estimated using the bound~(\\ref{eq:Xmue}):\n\n\\begin{equation}\nBr(\\mu \\rightarrow e \\gamma \\gamma) \\; < \\; 1.0 \\cdot 10^{-18}.\n\\label{eq:BrSD}\n\\end{equation}\n\n\\noindent A similar analysis of the $\\mu \\to e e \\bar e$ decay\nwithin the above restrictions on the model parameters provides\n\n\\begin{equation}\nBr(\\mu \\rightarrow e e \\bar e)  \\; < \\; 1.0 \\cdot 10^{-15}.\n\\label{eq:Br3e}\n\\end{equation}\n\nAlthough the predicted values of the branches of the $\\mu \\to e \\gamma\n\\gamma$ and $\\mu \\rightarrow e e \\bar e$ decays\nare essentially less then the existing\nexperimental limits\\cite{18}\n$Br(\\mu \\rightarrow e \\gamma \\gamma)_{exp} < 7.2 \\cdot 10^{-11}$,\n$Br(\\mu \\rightarrow e e \\bar e)_{exp} < 1.0 \\cdot 10^{-12}$,\nthey are not as small as the predictions of the SM with lepton mixing,\nand a hope for their observation in the future still remains.\n\n\\section {Conclusions}\n\n$\\bullet$ The bounds on the Pati--Salam leptoquark mass were reexamined\nwith taking account of the mixing in the quark--lepton currents.\n\n\\noindent $\\bullet$\nSome semileptonic meson decays strongly suppressed within the SM\ncould be induced by vector leptoquark. Their further experimental\ninvestigations are very important.\n\n\\noindent $\\bullet$\nThe only mixing independent bound on the vector leptoquark mass arises\nfrom the limits on the invisible $\\pi^0 \\rightarrow \\nu \\bar \\nu$ decay.\nIt is $M_X > 440~GeV$ from the laboratory limit and\n$M_X > 18~TeV$ from the cosmological limit, but the last has to be verified.\n\n\\noindent $\\bullet$\nThe specific hierarchy of the rare muon decay probabilities via vector\nleptoquark takes place and the branching ratios are not as small as the\npredictions of the SM with lepton mixing.\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgements.}\nWe are grateful to L.B.~Okun, J.~Ritchie, V.A.~Rubakov, A.D.~Smirnov,\nK.A.~Ter-Martirosian and S.~Willenbrock for helpful discussions.\n\nThe research described in this publication was made possible in part by\nGrant No. RO3000 from the International Science Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction.}\n\\label{sec01}\n\n\nDynamics of elastic lattices is of exceptional importance in a wide range of applications in problems of structural mechanics. \nThe pioneering work of N.F. Morozov and his co-authors has addressed scale effects and dynamic  trapping for cracks propagating in transient regimes through structured media, as well as formation of nanocracks in crystalline solids \\cite{MPU1, MP, MPU2, Morozov_Petrov,Morozov_Petrov1}. An important concept of the dynamic fracture toughness at a stage of the crack initiation has been fully investigated, and an ``incubation time'' principle has been introduced by N.F. Morozov and his co-authors to describe time-dependent fracture in structured materials for the general types of loading. \n\n\n\n\n\nA significant impact has been made by the investigations of N.F. Morozov {\\em et al.}\\ \\cite{MT1,MT2,MT3,MT4} on the dynamics and stability of elastic rods subjected to transient longitudinal loads.   A new insight has been presented by Morozov and Tovstik, who had addressed the influence of longitudinal loads on the dynamics of elastic rods and elastic waves. This also includes important features such as parametric resonances and the dynamic loss of stability for the loads, whose magnitude is less than the classical Eulerian load. These important studies also extend to the non-linear regimes to analyse the growth of the post-critical deformations. \n\nAlthough most of micro-structured solids with defects are not periodic, they may possess structured interfaces or large subdomains, which include locally periodic patterns. Dynamic response of such elastic systems may be very unusual and sometimes counter-intuitive. The theory of Floquet-Bloch waves has been successfully used to explain the dynamic anisotropy, dispersion in multi-structured solids and localisation.       In particular, Slepyan \\cite{Slepyan_2002} and Slepyan and Ryvkin \\cite{Slepyan_Ryvkin_2010} have studied Floquet-Bloch waves in conjunction with the problems of dynamic fracture.\nStructured interfaces, asymptotic analysis and the transmission problems for solids, containing structured interfaces, have been analysed by  Bigoni and Movchan \\cite{Bigoni_2002}. For dynamic formulations in structured media where the long-wave approximations are not valid, the  high-frequency homogenisation is required, as explained in \\cite{Craster_2014}, and this may also identify the anisotropy and localisation of waveforms in micro-structured solids. The so-called  band gap Green's functions, together with the dynamic defect modes, were  analysed in \\cite{Movchan_Slepyan}.\n\n\n\n\nIn the previous papers by Piccolroaz and Movchan \\cite{PM_2014} and Piccolroaz {\\em et al.} \\cite{PMC2017a}, Floquet-Bloch elastic waves were considered for elastic networks of elastic beams, with an additional rotational inertia. The authors also drew a connection between dynamic models of couple-stress elastic materials and structured Rayleigh beams. In particular, the Rayleigh beam model includes effects of the rotational inertia, that is excluded from the theory of the classical Euler-Bernoulli beams. It was also demonstrated that the rotational inertia is apparently significant for the high-frequency dynamic response, especially in problems involving the Dirac cone steering and analysis of the dispersion degeneracies.   \n\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{honeycomb_lattice_interface_R.pdf}\n\\caption{\\footnotesize A composite honeycomb lattice of flexural beams, which contains a structured interface consisting of the Rayleigh beams with an additional rotational inertia.}\n\\label{honeycomb_lattice_interface}\n\\end{figure}\n\n\n\nFig.~\\ref{honeycomb_lattice_interface} shows a honeycomb flexural lattice containing a structured interface, where the classical Euler-Bernoulli beams are used to construct the ambient network, whereas the structured interface is built of the Rayleigh beams with an appropriately designed rotational inertia. \nWhen a point source is applied in the proximity of such a structured interface, it is demonstrated that the dynamic response of such a multi-scale solid may show localisation, negative refraction as well as dynamic anisotropy and interfacial edge waves.\n\nThe structure of the paper is as follows. The main definitions and the geometry of the Rayleigh beam network are given in Section \n\\ref{sec02}. The elementary cell and the Bloch-Floquet junction conditions are discussed in Section \\ref{BFjc}. \nSection \\ref{sec03} is devoted to the dispersion of waves in the networks of the Euler-Bernoulli and of the Rayleigh beams. The dynamic anisotropy and the slowness contours corresponding to the dispersion diagrams are discussed in Section \\ref{sec04}. The study of the forced vibrations and of the dynamic response of structured interfaces is presented in Section \\ref{forced}. Finally, Section \\ref{sec05} includes concluding remarks and discussion.  \n\n\n\n\\section{The Rayleigh beam lattice.}\n\\label{sec02}\n\n\nWe begin by considering a doubly periodic honeycomb lattice consisting of the Rayleigh beams, which possess a rotational inertia. The geometry of the periodic structure is shown in Fig.~\\ref{fig01}, which includes a periodic honeycomb lattice consisting of Rayleigh beams. The elementary cell, of a parallelogram shape is also shown, together with the basis vectors of the lattice, $\\mbox{\\boldmath $v$}_1=(\\sqrt{3}\/2\\ h, 3\/2\\ h)$ and $\\mbox{\\boldmath $v$}_2=(-\\sqrt{3}\/2\\ h, 3\/2\\ h)$ where $h$ is the length of the beams.\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{Reticolo_R.pdf}\n\\caption{\\footnotesize  A periodic honeycomb lattice consisting of Rayleigh beams. The elementary cell, of a parallelogram shape is shown, together with the basis vectors of the lattice.}\n\\label{fig01}\n\\end{figure}\n\n\n\nWithin a one-dimensional ligament of the lattice, consider a time harmonic flexural wave, with the out-of-plane displacement \n$U(x, t) = u(x) \\exp(i \\omega t),$ with $x$ being a local spatial coordinate and $t$ denoting time. \n\nIn the Rayleigh beam, the amplitude function $u(x)$ satisfies the governing equation\n\\begin{equation}\n\\label{eq:gov}\nEI u''''(x) - (P - \\rho I \\omega^2) u'' + (\\beta - \\rho A \\omega^2) u = 0\n\\end{equation}\nHere the following notations are in use: $E$ is the Young modulus, $\\beta$ is the stiffness of a Winkler type elastic foundation, $P$ the prestress, $\\rho$ the mass density, $A$ the area of the cross-section, and $I$ the area moment of inertia of the cross-section.\n\nLet $M$ and $V$ denote respectively the internal bending moment and the shear force, as follows: \n\\begin{equation}\nM(x) = -EI u''(x), \\quad V(x) = -EI u'''(x) + (P - \\rho I \\omega^2) u'(x)\n\\end{equation}\nIf the Rayleigh beam is of infinite extent and prestress and elastic foundation are absent, then the solution  of (\\ref{eq:gov}) can be written as\n\\begin{equation}\nu(x) = \\sum_{q=1}^{4} C_q e^{i \\kappa_q x} \\label{u_repr}\n\\end{equation}\nwhere \n\\begin{equation}\n\\kappa_{1,2,3,4} = \\pm \\frac{1}{r} \\sqrt{\\alpha\\frac{R\\omega^2}{2} \\pm \\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}}\n\\label{kappa}\n\\end{equation}\nThe following notations are used here:\n\\begin{equation}\nr = \\sqrt{\\frac{I}{A}}, \\quad\nR = \\frac{\\rho r^2}{E} = \\frac{\\rho I}{EA}\n\\end{equation}\nHere $\\alpha=1$ for the Rayleigh beams, and $\\alpha=0$ for the Euler-Bernoulli beams. \n\n\n\n\n\\section{The Floquet-Bloch and junction conditions within the elementary cell.}\n\\label{BFjc}\n\n\nFor the flexural lattice ligaments, shown in Fig.~\\ref{fig01}, it is convenient to introduce local coordinates $x_j$ for each ligament and to refer to the junction points E and F, where three ligaments are connected.  \n\nFor the structure of five beams, shown in Fig.~\\ref{fig01}(c), the local displacement amplitudes $u_q(x_q)$, $q= 1,\\ldots,5,$  (there is no summation with respect to the repeated index $q$) are\n\\begin{equation}\nu_q (x_q) = \\sum_{p=1}^4 C_{pq} \\exp(i \\kappa_{pq} x_q), ~ q=1,\\ldots, 5 \\label{disp_nodal}\n\\end{equation}\nand the quantities $\\kappa_{pq}$ are defined by (\\ref{kappa}) for each flexural ligament of the elementary cell. \nHence, \nthere are 20 constants, $C_{pq}$, $p=1,\\cdots,4$, $q=1,\\cdots,5$, which can be found by solving the system of equations derived from the 8 Floquet-Bloch conditions at the boundary of the unit cell (points A, B, C, D), and from the 12 junction conditions at the points E and F) .\n\nIn particular, for the points $A-B$ of the unit cell, the quasi-periodic conditions at $x_2 = h\/2$ and $x_4 = h\/2$ hold\n\\begin{equation}\nu_2\\left(\\frac{h}{2}\\right) = u_4\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)} \\label{eq6}\n\\end{equation}\n\\begin{equation}\nu_2'\\left(\\frac{h}{2}\\right) = -u_4'\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n- EI u_2''\\left(\\frac{h}{2}\\right) = - EI u_4''\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n-EI u_2'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_2'\\left(\\frac{h}{2}\\right) =\n- \\left[ -EI u_4'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_4'\\left(\\frac{h}{2}\\right) \\right] e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\nwhich prescribe the Floquet-Bloch shift across the unit cell for flexural displacement, rotation, internal moment and internal shear force. \n\nAnalogous quasi-periodic boundary conditions apply to the points $C-D$ of the unit cell,\n\\begin{equation}\nu_3\\left(\\frac{h}{2}\\right) = u_5\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\nu_3'\\left(\\frac{h}{2}\\right) = -u_5'\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n- EI u_3''\\left(\\frac{h}{2}\\right) = - EI u_5''\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n-EI u_3'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_3'\\left(\\frac{h}{2}\\right) =\n- \\left[ -EI u_5'''\\left(\\frac{h}{2}\\right)- \\rho I \\omega^2 u_5'\\left(\\frac{h}{2}\\right) \\right] e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\nThe junction conditions at the node $E$ are as follows.\nContinuity of displacement implies\n\\begin{equation}\nu_2(0) = u_1\\left(\\frac{h}{2}\\right)\n\\end{equation}\n\\begin{equation}\nu_3(0) = u_1\\left(\\frac{h}{2}\\right)\n\\end{equation}\nand the continuity of rotations yields\n\\begin{equation}\nu_1'\\left(\\frac{h}{2}\\right) = u_2'(0)+u_3'(0)\n\\end{equation}\nequations of motion for the node $E$ become\n\\begin{multline}\n\\left[ -EI u_1'''\\left(\\frac{h}{2}\\right)- \\rho I \\omega^2 u_1'\\left(\\frac{h}{2}\\right) \\right]=\n\\left[ -EI u_2'''(0) - \\rho I \\omega^2 u_2'(0) \\right] +\n\\left[ -EI u_3'''(0) - \\rho I \\omega^2 u_3'(0) \\right]\n\\end{multline}\n\\begin{equation}\nEI u_1''\\left(\\frac{h}{2}\\right)=\\frac{EI}{2} u_2''(0)+\\frac{EI}{2} u_3''(0)\n\\end{equation}\n\\begin{equation}\nEI u_2''(0)=EI u_3''(0)\n\\end{equation}\n\n\\noindent\nThe junction conditions at the node $F$ include: continuity of displacement\n\\begin{equation}\nu_4(0) = u_1\\left(-\\frac{h}{2}\\right)\n\\end{equation}\n\\begin{equation}\nu_5(0) = u_1\\left(-\\frac{h}{2}\\right)\n\\end{equation}\ncontinuity of rotations\n\\begin{equation}\nu_1'\\left(-\\frac{h}{2}\\right)+u_4'(0)+u_5'(0)=0\n\\end{equation}\nand the equations of motion for the node $F$\n\\begin{multline}\n\\left[ -EI u_1'''\\left(-\\frac{h}{2}\\right)- \\rho I \\omega^2 u_1'\\left(-\\frac{h}{2}\\right) \\right]=\n\\left[ -EI u_4'''(0) - \\rho I \\omega^2 u_4'(0) \\right] +\n\\left[ -EI u_5'''(0) - \\rho I \\omega^2 u_5'(0) \\right]\n\\end{multline}\n\\begin{equation}\nEI u_1''\\left(-\\frac{h}{2}\\right)=\\frac{EI}{2} u_4''(0)+\\frac{EI}{2} u_5''(0)\n\\end{equation}\n\\begin{equation}\nEI u_4''(0)=EI u_5''(0) \\label{eq25}\n\\end{equation}\n\n\n\n\n\n\n\\noindent\nEquations (\\ref{eq6})--(\\ref{eq25}) comprise a homogeneous linear algebraic system for the 20 unknown constants, and the requirement of the vanishing of the determinant of the matrix of this algebraic system yields the dispersion equation for the Floquet-Bloch waves in the honeycomb periodic lattice of the Rayleigh beams. \n\n\\section{Lower-dimensional model, dispersion properties.}\n\\label{sec03}\n\nIn the earlier paper \\cite{PMC2017a}, which addressed the networks of flexural beams for square and rectangular lattices, it has been noted that the \nanalysis of Floquet-Bloch waves contributes to the studies of the dynamic response of structured solids.\nHere we analyse the dispersion properties of flexural waves in honeycomb systems with rotational inertia, and also make a comparison between periodic networks of different geometries. \n\n\n\\subsection{The algebraic system.}\n\nSubstitution of the representation\n\\eq{disp_nodal} into the 20 relations \n(\\ref{eq6})--(\\ref{eq25}) \nleads to a system of linear algebraic equations with respect to the variables  $C_{pq}$. Introducing a vector $$\\mbox{\\boldmath $C$} = \\Big(C_{11}, C_{12}, C_{13}, C_{14}, \\ldots, C_{51}, C_{52}, C_{53}, C_{54}\\Big)$$ the matrix form of the equations is \n\\begin{equation}\n\\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$}) \\mbox{\\boldmath $C$}^T =0\n\\label{alg_syst}\n\\end{equation}\nwhere $\\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$})$ is a $20 \\times 20$ matrix  function, whose arguments are the radian frequency $\\omega$ and the Bloch vector $\\mbox{\\boldmath $k$}=(k_x,k_y)$. The dispersion equation has the form\n\\begin{equation}\n\\det \\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$}) = 0\n\\label{disp_eq}\n\\end{equation}\nThe above equation defines implicitly the dispersion surfaces, and for any given $\\mbox{\\boldmath $k$}$ from the Brillouin zone of the reciprocal lattice we identify a countable set of values of the spectral parameter $\\omega$.\nThe dispersion equation has been solved and the results concerning the dispersion surfaces and, in particular, the Dirac cones and saddle points are discussed below.\n\n\\subsection{Dirac cones and standing waves.}\n\nHoneycomb lattice of the Rayleigh beams has many attractive features, due to its high-order symmetry and possession of a dynamic response linked to dynamic anisotropy and to standing waves.\n\nIn particular, Figs.~\\ref{fig3dh}, \\ref{fig3d} shows the surfaces, which include a conical point, which corresponds to a finite non-zero frequency. The conical surfaces, adjacent to that point, are referred to as {\\em Dirac cones}. It is also noted that a standing wave occurs, represented by a surface of zero slope traversing through the Dirac cone vertex.\n\n\n\n\n\n\nComparing the cases of the honeycomb networks of the Rayleigh beams, with an additional rotational inertia, and the classical case of the Euler-Bernoulli beams, we observe a clear distinction between the dispersion diagrams.\nThe first three dispersion surfaces for the Floquet-Bloch waves in the lattice, consisting of the Rayleigh beams, are placed at much lower frequencies compared to those for the Euler-Bernoulli beams,  as demonstrated in Fig.~\\ref{fig3dh}. \n\nWe also provide a comparison between the cases of different lattice geometries: the honeycomb lattice and the square lattice of flexural beams are discussed here. The results, concerning the dispersion of the Floquet-Bloch waves in the square flexural lattice were presented in the earlier work \\cite{PMC2017a}. \nIn particular, for the square lattice of elastic beams (both cases of Euler-Bernoulli and Rayleigh beams) the smoothness of dispersion surfaces and their degeneracies were investigated.\nThe effects of rotational inertia, attributed to the Rayleigh beams, deserve special attention, as illustrated in the text below.\n\n\n\n\n\n\nFor the honeycomb lattice, where the normalised physical and geometrical parameters are chosen as $E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$, $P=0$, $\\beta=0$, the dispersion equation, which relates the radian frequency $\\omega$ and the wave vector $(k_x, k_y)$,  takes the form\n\\begin{multline}\n\\sin ({\\Omega_1 h}) \\sinh ({\\Omega_2 h}) \\left[4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}-\\frac{3}{2} {k_y h}\\right) \\right.\n+4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}+\\frac{3}{2} {k_y h}\\right) \\\\ \\left.  +4 \\cos \\left(\\sqrt{3} {k_x h}\\right)-9 \\cos (2 {\\Omega_1 h})-3\\right]\n\\left[4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}-\\frac{3}{2} {k_y h}\\right) \\right. \\\\ \\left. +4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}+\\frac{3}{2} {k_y h}\\right)+4 \\cos \\left(\\sqrt{3} {k_x h}\\right)-9 \\cos (2 {\\Omega_2 h})-3\\right]=0 \\label{disp_honey}\n\\end{multline}\nwhere\n\\begin{equation}\n\\Omega_1 = \\frac{1}{r} \\sqrt{\\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}+\\alpha\\frac{R\\omega^2}{2}},\n\\quad\n\\Omega_2 = \\frac{1}{r} \\sqrt{\\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}-\\alpha\\frac{R\\omega^2}{2}}\n\\end{equation}\nin which $\\alpha=1$ for Rayleigh beams and $\\alpha=0$ for Euler-Bernoulli beams.\n\n\n\n\n\n\n\nThe high degree of symmetry of the honeycomb structure leads to a nicely factorised form of the dispersion equation, as above. In particular, two of the factors in the left-hand side are ${\\mbox{\\boldmath $k$}}$-independent, and there exists a countable set of standing waves characterised by the flat dispersion surfaces at $\\Omega_1(\\omega) h = \\pi k$ for any integer $k$.\nOne of such  flat surfaces is shown in Fig.~\\ref{fig3dh}, together with the two dispersion surfaces that are not smooth and have conical points. This figure includes several parts: (a) the Euler-Bernoulli beam structure, (b) the Rayleigh beam structure. Neither prestress nor elastic foundations are present in this computation. The physical and geometrical parameters are chosen as follows: $E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$. The effect of the rotational inertia, inherent to the Rayleigh beams, is significant,   as the first several dispersion surfaces occur at much lower frequencies compared to the corresponding surfaces for the Euler-Bernoulli's beams as in part (a).\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=48mm]{fig3dVEBh.pdf}\n\\includegraphics[width=48mm]{fig3dVRAh.pdf}\n\\caption\n{\\footnotesize Dispersion surfaces for Floquet-Bloch waves in a periodic honeycomb lattice: (a)  the Euler-Bernoulli beam structure, (b) the Rayleigh beam structure. Neither prestress nor elastic foundations are present in this computation. The physical and geometrical parameters are chosen as follows: \n$E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$.\nThe effect of the rotational inertia, inherent to the Rayleigh beams, is significant,   as the first several dispersion surfaces occur at much lower frequencies compared to the corresponding surfaces for the Euler-Bernoulli's beams as in part (a).\n}\n\\label{fig3dh}\n\\end{figure}\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=48mm]{fig3dVEB.pdf}\n\\includegraphics[width=48mm]{fig3dVRA.pdf}\n\\caption{\\footnotesize The case of a square lattice. Dispersion surfaces are presented for \nthe Euler-Bernoulli beam structure (a), and for the \nnetwork of Rayleigh beams  (b).\nThe parameter values, used in this combination, are $E=1$, $\\rho=1$, $A=1$, $I=1$, ${h}=1$.\n} \n\\label{fig3d}\n\\end{figure}\n\n\nAlthough a square lattice of flexural beams possesses interesting dispersion properties for the Floquet-Bloch waves, as shown in Fig.~\\ref{fig3d}, there are significant differences compared to the case of the honeycomb flexural lattice. Dispersion surfaces are presented for the Euler-Bernoulli beam structure in part (a), and for the network of Rayleigh beams in part (b). The parameter values, used in this combination, are $E=1$, $\\rho=1$, $A=1$, $I=1$, ${h}=1$.\nIn particular, the dispersion equation cannot be factorized to a similar form as in (\\ref{disp_honey}), and the flat dispersion surfaces characterising standing waves are absent. Instead, the emphasis is made on the dynamic anisotropy, and special dispersion features of the Floquet-Bloch waves in the neighbourhoods of the vertices of the cones.  \n Following \\cite{McPhedran_2015} we refer to these conical surfaces as ``Dirac cones'', and the radian frequency $\\omega$ corresponding to the vertex of the cone is identified as a special resonant frequency, corresponding to a multiple root of the dispersion equation.  \n\n\nIn particular, if the rotational inertia, the pre-stress and the stiffness of the elastic foundation are equal to zero, i.e.\\ the Rayleigh beam becomes the classical Euler-Bernoulli beam, the dispersion equation \\eq{disp_eq} for Floquet-Bloch waves in the square lattice takes the form\n\\begin{multline}\n\\sin \\left(\\Omega {h}\\right)\n\\left[\\cos (k_x {h}) + \\cos (k_y {h}) - 2 \\cos \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_x {h}) - \\cosh \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_y {h}) - \\cosh \\left(\\Omega {h}\\right)\\right] \\\\\n-\\sinh \\left(\\Omega {h}\\right)\n\\left[\\cos (k_x {h}) + \\cos (k_y {h}) - 2 \\cosh \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_x {h}) - \\cos \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_y {h}) - \\cos \\left(\\Omega {h}\\right)\\right] = 0\n\\end{multline}\nwhere ${h}$ is the length of the ligaments in the square lattice, $\\Omega = \\sqrt{\\omega} \\sqrt[4]{\\frac{\\rho A}{EI}}$, and the corresponding dispersion diagram is shown in Fig.~\\ref{fig3d}a.\n\n\\subsection{Resonant modes for elementary ligaments.}    \nIt turns out that the frequencies, corresponding to the Dirac cone vertices, and some of the corresponding vibration modes (i.e. standing waves) are identified as the natural frequencies and the eigenmodes of a simply supported beam, respectively. It also explains why when the size of beam ligaments and their physical properties are the same, the Dirac cone vertices for the honeycomb and for the square lattice occur at the same frequencies: \n\n\\begin{equation}\n\\omega_n = \\frac{n^2 \\pi^2}{{h}^2} \\sqrt{\\frac{EI}{\\rho A}} \\qquad \\text{for the Euler-Bernoulli beam} \\label{swEB}\n\\end{equation}\n\n\\begin{equation}\n\\omega_n = \\frac{n^2 \\pi^2}{{h}} \\sqrt{\\frac{EI}{\\rho (A{h}^2+n^2\\pi^2I)}} \\qquad \\text{for the Rayleigh beam} \\label{swR}\n\\end{equation}\n\n\n\nThe honeycomb lattice, even in the low-frequency regime, shows a very different dynamic response compared to the square lattice. It follows from the diagrams of Fig.~\\ref{fig3dh} and Fig.~\\ref{fig3d}, which present the dispersion surfaces for the Floquet-Bloch waves, that the honeycomb lattice is locally isotropic in the neighbourhood of the Dirac cone frequencies as well as at low frequencies. On the contrary, the orthotropy of the square lattice is apparent, and in particular, it leads to an unusual directional preference in the vicinity of the Dirac cones frequencies. \n\n\nAlso the effect of the rotational inertia, which is present in the Rayleigh beam structure, becomes important, and consequently the dispersion surfaces representing structures consisting of the Rayleigh beams and structures consisting of the Euler-Bernoulli beams, become different, as seen in Figs.~\\ref{fig3dh},~\\ref{fig3d}.\nFloquet-Bloch waves in a Rayleigh beam structure would show zero group velocity at much lower frequencies compared to the similar structure of the Euler-Bernoulli beams. At higher frequencies the Rayleigh beams show much richer behaviour, especially when it is concerned with the degeneracies and formation of the Dirac cones. \n\n\n\n\nFigs.~\\ref{figband2h}, \\ref{figband2} complement the  dispersion surfaces diagrams by the cross-sectional plots along the boundaries of the irreducible  Brillouin zones in the reciprocal lattices constructed for the honeycomb and the square networks of flexural beams.\nIn particular, Fig.~\\ref{figband2h} includes the cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the honeycomb lattice comprised of the Euler-Bernoulli beams in part (a) and the Rayleigh beams in part (b). \n{The inset on the right shows the contour $\\Gamma M K \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3dh}.}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{disp_diagrams.pdf}\n\\caption{\n\\footnotesize The cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the honeycomb lattice comprised of the Euler-Bernoulli beams (a) and the Rayleigh beams (b).\n{The inset on the right shows the contour $\\Gamma M K \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3dh}.}\n}\n\\label{figband2h}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{figband2.pdf}\n\\caption{\n\\footnotesize The cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the square networks of the Euler-Bernoulli beams (a) and the Rayleigh beams (b).\n{The inset on the right shows the contour $\\Gamma X M \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3d}.}\n}\n\\label{figband2}\n\\end{figure}\n\n\nSimilar diagrams for square networks are shown in Fig.~\\ref{figband2} where the case of the Euler-Bernoulli beams is shown in part (a) and the Rayleigh beams case is displayed in part (b).\n{The inset on the right shows the contour $\\Gamma X M \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3d}.\n\n\n\n\n\n\nSpecial attention is given to the Dirac cones, represented by the intersecting dispersion curves. In particular, for the first intersection the case of the square lattice shows a triple root of the dispersion equation, with the standing wave being represented by the flat band crossing through the Dirac cone vertex. On the contrary, the Floquet-Bloch waves in the honeycomb network flexural waves also posses the Dirac cone mode, but for the first intersection depicted in Fig.~\\ref{figband2h} the standing wave is absent. We note that the standing wave for honeycomb network of flexural beams occurs at the second intersection, and this mode is represented by the flat band at the frequency defined by formulae \\eq{swEB},  \\eq{swR}.\nAlso, we remark that the additional rotational inertia attributed to the Rayleigh beams leads to the dispersion curves being compressed towards the horizontal axis compared to those on the diagrams presented for the Euler-Bernoulli beams. This feature is clearly visible from the comparison of the parts (a) and (b) on the dispersion diagrams shown in Figs.~\\ref{figband2h} and \\ref{figband2}.\n\n\n\n\n\n\n\n\n\n\\section{Floquet-Bloch waves in a honeycomb network. Saddle points and slowness contours.}\n\\label{sec04}\n\n\nThe slowness contours (often referred to as isofrequency contours) are useful to identify stationary points on the dispersion surface as well as the dynamic anisotropy of the structured medium. %\nA comprehensive  exposition of such an approach for flexural waves in periodic square lattices is discussed in \\cite{PMC2017a}.\n\n\n\n\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{fig1EB.pdf} \\\\[6mm]\n\\includegraphics[width=120mm]{fig1RA.pdf}\n\\caption{\n\\footnotesize\nFirst dispersion surface and the corresponding isofrequencies contours for the Euler-Bernoulli beam honeycomb lattice (parts (a) and (b)) and for the honeycomb lattice of the Rayleigh beams (parts (c) and (d)). The Dirac cone is shown in both configurations. The slowness contours around the origin bound non-convex domains in the diagram (d), for the Rayleigh beams, in contrast with the diagram (b), corresponding to the Euler-Bernoulli beams. \n}\n\\label{slowness_contours}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{fig2EB.pdf} \\\\[6mm]\n\\includegraphics[width=120mm]{fig2RA.pdf}\n\\caption{\\footnotesize\nThe second dispersion surface, including the Dirac cone, presented for the Floquet-Bloch waves in the case of the Euler-Bernoulli beam square lattice (parts (a) and (b)) and of the Rayleigh beam square lattice (parts (c) and (d)). Preferential directions along the coordinate axes are clearly identified.\n}\n\\label{slowness_contours2}\n\\end{figure}\n\n\n\nFor Floquet-Bloch flexural waves in the honeycomb lattice, the dispersion surfaces and the corresponding slowness contours, are presented in Figs.~\\ref{slowness_contours}, \\ref{slowness_contours2} for the cases of the Rayleigh beams and the Euler-Bernoulli beams. \nIn particular, the diagrams in Fig.~\\ref{slowness_contours} represent the first dispersion surface (so-called ``acoustic band'') for the networks of Euler-Bernoulli beams and of the Rayleigh beams, whereas the diagrams in  Fig.~\\ref{slowness_contours2} correspond to the second dispersion surface (the ``optical band'') for the networks of Euler-Bernoulli beams and of the Rayleigh beams. The cases of the Euler-Bernoulli beam honeycomb lattice correspond to parts (a) and (b) and the computations for the honeycomb lattice of the Rayleigh beams are presented in parts (c) and (d). \n\nThese dispersion surfaces show the presence of the conical points (vertices of the Dirac cones) as well as the saddle points. Locally isotropic pattern is observed in the neighbourhood of the Dirac cones vertices, whereas a strong dynamic anisotropy is featured at frequencies corresponding to the saddle points.\n\nThe additional rotational inertia, attributed to the case of the Rayleigh beams, leads to lower frequencies of the Dirac cone vertices and the saddle points compared to similar networks of the Euler-Bernoulli beams.\n\nIn the following section, we give an illustration of the dynamic response of the honeycomb flexural lattice. In particular, we consider the case of a structured interface built of the Rayleigh beams embedded into the ambient geometrically identical lattice comprised of the Euler-Bernoulli beams, as illustrated in Fig.~\\ref{honeycomb_lattice_interface}.\n\n\n\n\n\n\n\n\n\n\\section{Forced vibrations of a honeycomb flexural network.}\n\\label{forced}\n\n\n\n\n\n\n\n\n\nHere we present the results of the finite element  simulations for the Euler-Bernoulli and for the Rayleigh beams programmed in COMSOL Multiphysics for a honeycomb network subjected to a time-harmonic transverse point force or several point forces applied at the lattice junctions\nThe forces are applied in the direction perpendicular to the $(x, y)-$plane.\n\nTo simulate a dynamic response of an infinite lattice with a finite-size computational window and to avoid wave reflection at the boundaries, Rayleigh and Euler-Bernoulli beams with damping were also programmed and introduced at the boundary of the computational domain. This was achieved by replacing the Young modulus $E$ by a complex value, $E(1+i\\eta)$. These beams were used to build a damping layer around the perimeter of the finite-size lattice, and the viscous parameter $\\eta$ was chosen so that to suppress the wave reflection.\n\n\\subsection{A homogeneous flexural network of Rayleigh beams.}\n\nIn Fig.~\\ref{fig_homo_RA} we consider a uniform network where six identical time-harmonic point forces are applied at the junctions of the hexagonal cell.\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=160mm]{fig_homo_RA_R.pdf}\n\\caption{\n\\footnotesize\nThe field patterns in the network of the Rayleigh beams, where  forced vibrations are generated by six time-harmonic identical forces applied at the junction points of the hexagonal cell.\n(a) The radian frequency $\\omega = 2 \\pi f = 0.314$ is sufficiently low, and the dynamic response appears to be isotropic. (b) The radian frequency $\\omega = 0.942$ is in the neighbourhood of the first saddle point frequency, where strong dynamic anisotropy is observed. (c) The radian frequency $\\omega = 1.319$ is in the neighbourhood of the Dirac cone vertex, and the waveform is localised. (d) Another group of saddle points occurs at the radian frequency of  $\\omega = 1.696$, where a strong dynamic anisotropy is observed. Here and in the following figures, inserts represent slowness contour diagrams for Floquet-Bloch waves at the given frequencies.\n}\n\\label{fig_homo_RA}\n\\end{figure}\n\n\nSeveral field patterns are observed for different values of the input frequency.\nNamely, when the frequency is sufficiently low, as in part (a) of Fig.~\\ref{fig_homo_RA}, the radial wave pattern appears to be isotropic, as expected in the low-frequency regime for the honeycomb lattice.\nHowever, the increase of the input frequency leads to radical, but predictable  changes in the frequency response of the network of the Rayleigh beams.  \nIn the cases (b) and (d) of Fig.~\\ref{fig_homo_RA}, the frequency values are close to those of the saddle points on the dispersion surfaces, and hence strong dynamic anisotropy is observed. On the contrary, strong localisation shown in the case (c) corresponds to the vertex of the Dirac cone. \nThe frequency values are chosen as follows. (a) The radian frequency $\\omega = 2 \\pi f = 0.314$ is sufficiently low, and the dynamic response appears to be isotropic. (b) The radian frequency $\\omega = 0.942$ is in the neighbourhood of the first saddle point frequency, where strong dynamic anisotropy is observed. (c) The radian frequency $\\omega = 1.319$ is in the neighbourhood of the Dirac cone vertex, and the waveform is localised. (d) Another group of saddle points occurs at the radian frequency of  $\\omega = 1.696$, where a strong dynamic anisotropy is observed. Here and in the following figures, inserts represent slowness contour diagrams for Floquet-Bloch waves at the given frequencies.\n\n\n\\subsection{Structured interface possessing rotational inertia.}\n\nHere we consider a geometrically homogeneous lattice of flexural beams, but we assume that within a layer of finite thickness the classical Euler-Bernoulli beams are replaced by the Rayleigh beams, which possess a rotational inertia, as described by the governing equation  \\eq{eq:gov}.\nA point force excitation is applied at a junction, outside the structured layer, as depicted in Fig.~\\ref{honeycomb_lattice_interface}. \nIn different frequency regimes, the dynamic response of the structured layer is investigated here.\n\nFirst, in Fig.~\\ref{fig_inter1} we present two cases, which include the low frequency excitation (part (a)) and an excitation at a higher frequency (part (b)), where a negative refraction is observed. \n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{fig_inter1_R.pdf}\n\\caption{\n\\footnotesize\nA dynamic response of the structured layer, built of the Rayleigh beams. The ambient lattice, surrounding the structured interface, consists of the classical Euler-Bernoulli beams.      The field patterns are presented for the two cases, where a point force is applied at the junction separated from the structured interface by the distance equal to the double diameter of the elementary cell of the honeycomb lattice (``distance 2''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) A sufficiently low radian frequency $\\omega = 2 \\pi f = 0.314$ corresponds to the isotropic response, where the action of the interface layer is negligibly small. (b) The normalised radian frequency $\\omega = 4.082$ corresponds to the case of Floquet-Bloch waves, which occur near saddle points on the dispersion diagrams for both the Euler-Bernoulli and the Rayleigh beams.  In this case, the rotational inertia of the interface layer becomes important, as it leads to the negative refraction and consequently focussing of the elastic flexural wave across the interface.\n}\n\\label{fig_inter1}\n\\end{figure}\n\n\n\nThis is understandable, as in the low frequency regime the Euler-Bernoulli and the Rayleigh beams appear to be very similar in terms of their dynamic response to an external load, and hence the structured interface is ``invisible''.\nOn the other hand, as the frequency increases, the corresponding dispersion diagrams in Fig.~\\ref{fig3dh} for the Floquet-Bloch waves  show the presence of the saddle points as well the Dirac cones. Consequently, this leads to an interesting dynamic response of the structured interface:\nat the normalised radian frequency of $\\omega = 4.082$, we observe saddle points on the dispersion diagrams for both the Euler-Bernoulli and for the Rayleigh beam networks. Such a regime implies a strong dynamic anisotropy, and consequently a negative refraction and focussing by the structured interface are observed in this simulation.\nThe field patterns are presented for the two cases, where a point force is applied at the junction separated from the structured interface by the distance equal to the double diameter of the elementary cell of the honeycomb lattice (``distance 2''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) A sufficiently low radian frequency $\\omega = 2 \\pi f = 0.314$ corresponds to the isotropic response, where the action of the interface layer is negligibly small. (b) The normalised radian frequency $\\omega = 4.082$ corresponds to the case of Floquet-Bloch waves, which occur near saddle points on the dispersion diagrams for both the Euler-Bernoulli and the Rayleigh beams.  In this case, the rotational inertia of the interface layer becomes important, as it leads to the negative refraction and consequently focussing of the elastic flexural wave across the interface.\n\n\n\n\n\n\nFurthermore, another Fig.~\\ref{fig_inter2} displays the field plots and shows examples of strong dynamic anisotropy as well localisation within the structured interface consisting of the Rayleigh beams. The time-harmonic point force is placed at the distance 1 from the interface, as illustrated in Fig.~\\ref{honeycomb_lattice_interface}. \nIn particular, the part (a) of  Fig.~\\ref{fig_inter2} shows the localisation within the structured interface, parts (c) and (d) show the edge-wave modes, and the field pattern of the part (b) shows the wave blockage by the structured interface and a strong dynamic anisotropy exhibited by the ambient lattice of the Euler-Bernoulli beams.      \nThe field patterns are presented for four cases, where a point force is applied at the junction separated from the structured interface by the distance equal to a diameter of the elementary cell of the honeycomb lattice (``distance 1''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) The normalised radian frequency $\\omega = 2 \\pi f = 1.005$ corresponds to a frequency below the first saddle point for the ambient lattice, where the dynamic response is almost isotropic, but the same frequency corresponds to the saddle point of the Rayleigh beam network used in the structured interface; hence the strong anisotropy is shown within the structured interface. (b) The normalised radian frequency $\\omega = 1.508$  corresponds to the saddle point on the dispersion diagram for the Floquet-Bloch waves in the ambient lattice, and the same frequency corresponds to the neighbourhood of the Dirac cone vertex for the network of the Rayleigh beams. (c) The normalised radian frequency $\\omega = 2.513$ corresponds to the neighbourhoods of the Dirac cone vertices for both Euler-Bernoulli and the Rayleigh beams on the first dispersion surface and the second dispersion surface, respectively  (d)  The forced excitation at the normalised radian frequency $\\omega = 9.111$ gives a waveguide vibration mode, and it is shown that this regime is close to the Dirac cones of different orders for the Euler-Bernoulli and the Rayleigh beam networks, respectively.\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{fig_inter2_R.pdf}\n\\caption{\n\\footnotesize\nA dynamic response of the structured layer, built of the Rayleigh beams. The ambient lattice, surrounding the structured interface, consists of the classical Euler-Bernoulli beams.      The field patterns are presented for four cases, where a point force is applied at the junction separated from the structured interface by the distance equal to a diameter of the elementary cell of the honeycomb lattice (``distance 1''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) The normalised radian frequency $\\omega = 2 \\pi f = 1.005$ corresponds to a frequency below the first saddle point for the ambient lattice, where the dynamic response is almost isotropic, but the same frequency corresponds to the saddle point of the Rayleigh beam network used in the structured interface; hence the strong anisotropy is shown within the structured interface. (b) The normalised radian frequency $\\omega = 1.508$  corresponds to the saddle point on the dispersion diagram for the Floquet-Bloch waves in the ambient lattice, and the same frequency corresponds to the neighbourhood of the Dirac cone vertex for the network of the Rayleigh beams. (c) The normalised radian frequency $\\omega = 2.513$ corresponds to the neighbourhoods of the Dirac cone vertices for both Euler-Bernoulli and the Rayleigh beams on the first dispersion surface and the second dispersion surface, respectively  (d)  The forced excitation at the normalised radian frequency $\\omega = 9.111$ gives a waveguide vibration mode, and it is shown that this regime is close to the Dirac cones of different orders for the Euler-Bernoulli and the Rayleigh beam networks, respectively.\n}\n\\label{fig_inter2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion.}\n\\label{sec05}\n\nThe modelling of the beam networks which possess rotational inertia has revealed novel and counter-intuitive properties related to the dynamic response of the multi-scale solids.\nSpecifically, we have focused our attention on the regimes, which correspond to the saddle points or neighbourhoods of the Dirac cones on the dispersion diagrams, constructed for the Floquet-Bloch waves in the periodic networks of the Rayleigh or the Euler-Bernoulli beams.\nThe waveforms, and in particular standing waves also depend on the geometry of the periodic network, and we have given the comparative outline for the cases of square and honeycomb lattices, with the emphasis on the dynamic anisotropy and localisation.\nClosed form asymptotic estimates for frequencies corresponding to the vertices of the Dirac cones and of the standing waves provide an additional useful tool, which is essential in problems of optimal design of phononic filters and polarisers of elastic waves.\nFinally, the dynamic response  of the structured interfaces with an additional rotational inertia has been investigated, with the emphasis on edge waves, negative refraction and wave trapping.\nThis work naturally extends to the area of metamaterials design, and to control of flexural waves by multi-scale elastic networks. \n\n\n\\vspace{6mm}\n{\\bf Acknowledgements}. AP would like to acknowledge financial support from the\nEuropean Union's Seventh Framework Programme FP7\/2007-2013\/ under REA grant\nagreement number PCIG13-GA-2013-618375-MeMic. \nMajor part of the work was carried out while AM was visiting the University of Trento in 2016 with the support from the\nEuropean Union's Grant ERC-2013-ADG-340561-INSTABILITIES, which is gratefully acknowledged.\nAM also acknowledges the support from the UK EPSRC Program Grant  EP\/L024926\/1.\nLC acknowledges financial support from the University of Trento, within the research project 2014 entitled ``3D printed metallic foams for biomedical applications: understanding and improving their mechanical behavior''.\n\n\\bibliographystyle{jabbrv_unsrt}\n\\bibliography\n{%\nroaz1}\n\n\n\n\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nPressure-balanced magnetic structures in the form of strong magnetic\nenhancements (humps) and depressions (holes) that are quasi-stationary in\nthe plasma frame, with no or little change in the magnetic field direction,\nare commonly observed in regions of the solar wind and of planetary\nmagnetosheaths with relatively large $\\beta $ and a dominant (generally\nion) temperature in the transverse direction ( see, for instance, \\cite{S11, Genot09b} and\nreferences therein). The origin of these structures is still not fully\nunderstood, but they are usually viewed as nonlinearly saturated states of the\nmirror instability (MI) discovered by Vedenov and Sagdeev \n\\cite{VedenovSagdeev}. It is a kinetic instability whose growth rate\nwas first obtained under the assumption of cold electrons, a regime where\nthe contributions of the parallel electric field $E_{\\Vert }$ can be\nneglected. However, in realistic space plasmas, the electron temperature can\nhardly be ignored \\cite{Stverak08}. The linear theory retaining the electron\ntemperature and its possible anisotropy, in the quasi-hydrodynamic limit\n(which neglects finite Larmor radius corrections), was developed in the case\nof bi-Maxwellian distribution functions by several authors \n\\cite{Stix62}--\\cite{Hell07}. A general estimate of the growth rate \nunder the sole condition that it is small compared with the ion\ngyrofrequency (a condition reflecting  close vicinity to threshold) is\npresented in \\cite{KPS12a}. The instability then develops in\nquasi-perpendicular directions, making the parallel magnetic perturbation\ndominant. This analysis includes in particular regimes with a significant\nelectron temperature anisotropy for which the instability extends beyond the\nion Larmor radius. In the limit where the instability is limited to scales\nlarge compared with the ion Larmor radius, only the leading order\ncontribution in terms of the small parameter $\\gamma \/(|k|_{z}v_{\\|i})$ is\nto be retained in estimating Landau damping, and the growth rate is given by \n\\begin{eqnarray}\n&&\\gamma =\\frac{2}{\\sqrt{\\pi }}\\frac{T_{\\Vert i}}{T_{\\perp i}}\\frac\n|k_{z}|v_{\\Vert i}}{E}\\Big \\{\\Gamma  -\\frac{1}{\\beta _{\\perp }}\\Big (1+\\frac{\\beta _{\\perp }-\\beta\n_{\\Vert }}{2}\\Big )\\frac{k_{z}^{2}}{k_{\\perp }^{2}}  \\nonumber \\\\\n&&\\qquad -\\frac{3}{4(1+\\theta _{\\perp })}\\Big (\\frac{T_{\\perp i}}{T_{\\Vert i\n}-1\\Big )(1+F)k_{\\perp }^{2}r_{L}^{2}\\Big \\},  \\label{growth_rate}\n\\end{eqnarray\nwhere \n\\begin{equation}\n\\Gamma =\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{(\\theta _{\\parallel\n}+\\theta _{\\perp })^{2}+2\\theta _{\\parallel }(\\theta _{\\perp }^{2}+1)}\n2\\theta _{\\parallel }(1+\\theta _{\\perp })(\\theta _{\\parallel }+1)}-1-\\frac{\n}{\\beta _{\\perp }}  \\label{newthreshold}\n\\end{equation\nmeasures the distance to threshold and \n\\begin{eqnarray}\nE &=&\\frac{1+\\theta _{\\perp }}{(1+\\theta _{\\parallel })^{2}}\\left[ 2+\\theta _{\\perp\n}(4+\\theta _{\\perp })+\\theta _{\\parallel }^{2}\\right]  \\nonumber \\\\\nF &=&\\frac{T_{\\Vert e}}{T_{\\Vert e}+T_{\\Vert i}}\\Big \\{-1+\\frac{\\theta\n_{\\perp }}{\\theta _{\\Vert }}  \\nonumber \\\\\n&&-\\frac{2}{3}\\frac{T_{\\Vert i}}{T_{\\perp i}}\\Big [\\Big (\\frac{T_{\\Vert i}}\nT_{\\perp i}}-1\\Big )\\frac{1}{\\beta _{\\perp i}}-\\theta _{\\perp }\\Big (\\frac\nT_{\\perp e}}{T_{\\Vert e}}-1\\Big )\\Big ]\\Big \\}.  \\nonumber\n\\end{eqnarray\nHere, ${T_{\\perp \\alpha }}$ and ${T_{\\Vert \\alpha }}$ are the perpendicular\nand parallel (relative to the ambient magnetic field $\\mathbf{B}_{0}$ \ntaken in the $z$ direction)\ntemperatures of the species $\\alpha $ ($\\alpha =i$ for ions and $\\alpha =e$\nfor electrons ), $\\theta _{\\perp }={T_{\\perp e}}\/{T_{\\perp i}}$, $\\theta\n_{\\Vert }={T_{\\Vert e}}\/{T_{\\Vert i}}$ and $\\beta _{\\perp }=\\beta _{\\perp\ni}+\\beta _{\\perp e}$ with $\\beta _{\\perp \\alpha }=8\\pi p{_{\\perp \\alpha }\/B\n_{0}^{2}$ where $p{_{\\perp \\alpha }}$ is the perpendicular thermal pressure\n(similar definition for $\\beta _{\\Vert }$). Furthermore, the parallel\nthermal velocity is defined as $v_{\\Vert \\alpha}=\\sqrt{{2T_{\\Vert \\alpha}}\/{m_{\\alpha}}}$,\nand $r_{L}=({2{T_{\\perp i}\/m_{p})^{1\/2}\/\\Omega _{i}}}\n$ denotes the ion Larmor radius ($\\Omega _{i}=eB_{0}\/m_{i}c$ is the ion\ngyrofrequency).\n\nThe growth rate given by Eq. (\\ref{growth_rate}) has the same structure \nas in the cold electron regime considered  \nin \\cite{Hall79} in the case of bi-Maxwellian ions \nand generalized in \\cite{Pokho05} and \\cite\n{Hell07} to an arbitrary distribution function. The first term within the\ncurly brackets  provides the threshold condition which \ncoincides with that \ngiven in\n\\cite{Stix62}--\\cite{Hall79}. The second one reflects the magnetic field\nline elasticity and the third one (where $F$ depends on the electron\ntemperatures due to  the coupling between the species induced by the\nparallel electric field which is relevant for hot electrons)\nprovides the arrest of the instability at small scales by finite Larmor\nradius (FLR) effects.\n\nAn aim of this letter is to extend to hot electrons the weakly nonlinear\nanalysis  previously developed for cold electrons \\cite{KPS07a,KPS07b}. \nSince in this asymptotics, FLR contributions appear only at the linear\nlevel, the idea is to use the drift kinetic formalism\nto calculate the nonlinear terms. We show that the equation governing\nthe evolution of  weakly nonlinear mirror modes has\nthe same form as in the case of cold electrons. In particular, the sign of the\nnonlinear coupling coefficient that prescribes the shape of\nmirror structures, is not changed. This equation \nis of gradient type equation with a  free\nenergy (or a Lyapunov functional) which is unbounded  from below.\nThis leads to  finite-time blowing-up solutions \\cite{ZK12}, associated with \nthe existence of a subcritical bifurcation \n\\cite{KPS07a,KPS07b}. To describe subcritical stationary mirror\nstructures in the strongly nonlinear regime, we present an anisotropic MHD model \nwhere the perpendicular and parallel pressures are determined\nfrom the drift kinetic equations in the adiabatic\napproximation, in the form of prescribed functions of the magnetic field amplitude.\n\n\\section{Basic equations}\n\nA main condition characterizing mirror modes,  at least near threshold,  is\nprovided by the force balance equation \n\\begin{eqnarray}\n&&-\\nabla \\Big(p_{\\perp }+\\frac{B^{2}}{8\\pi }\\Big)+\\Big[1+ \\frac{4\\pi }{B^{2\n}(p_{\\perp }-p_{\\Vert })\\Big]\\frac{(\\mathbf{B}\\cdot \\nabla) \\mathbf{B}}{4\\pi \n}  \\nonumber \\\\\n&&+ \\mathbf{B}(\\mathbf{B}\\cdot\\nabla) \\Big (\\frac{p_\\perp -p_\\|}{B^2} \\Big )\n-\\nabla \\cdot \\mathbf{\\Pi}=0,  \\label{balance-nl}\n\\end{eqnarray}\nwhere the pressure tensor, viewed as the  the sum of the contributions of the various species,\nhas been written as the sum of a gyrotropic part\ncharacterized by the parallel ($p_{\\Vert }=\\sum_{\\alpha}p_{\\Vert \\alpha }$) and perpendicular \n($p_{\\perp }=\\sum_{\\alpha }p_{\\perp \\alpha }$) pressures, and \nof a gyroviscous contribution $\\mbox{\\boldmath $\\Pi$}$\noriginating from the sole ion FLR effects when concentrating on scales large compared with the\nelectron Larmor radius.  As mentioned above,\nFLR effects  arising only at the linear level with\nrespect to the amplitude of the perturbations, the other linear and nonlinear \ncontributions can be evaluated from the drift kinetic equation for each particle species\n\\begin{equation}\n\\frac{\\partial f_{\\alpha }}{\\partial t}+v_{\\Vert }\\mathbf{b}\\cdot \\nabla\nf_{\\alpha }+\\Big[-\\mu \\mathbf{b}\\cdot \\nabla B+\\frac{e_{\\alpha }}{m_{\\alpha\n} }E_{\\Vert }\\Big]\\frac{\\partial f_{\\alpha }}{\\partial v_{\\Vert }}=0\n\\label{mainkin}\n\\end{equation\n\n\nWe ignore the transverse electric drift which is subdominant for mirror\nmodes. In this approximation, both ions and electrons move in the direction\nof the magnetic field (defined by the unit vector $\\mathbf{b}=\\mathbf{B}\/B$)\nunder the effect of the magnetic force $\\mu \\mathbf{\\ b}\\cdot\\nabla B$ and\nthe parallel electric field $E_{\\Vert }=-\\mathbf{b}\\cdot \\nabla\\phi$ where\nthe magnetic moment $\\mu=v_{\\perp }^{2}\/(2B)$ is an adiabatic invariant\nwhich plays the role of a parameter in Eq. (\\ref{mainkin}). Here $\\phi$ is\nthe electric potential. The quasi-neutrality condition $n_{e}=n_{i}\\equiv n\n, where $n_{\\alpha }=B\\int f_{\\alpha }d\\mu dv_{\\Vert }d\\varphi \\equiv \\int\nf_{\\alpha}d^{3}v$, is used to close the system and eliminate $E_\\|$.\n\nIn this framework where FLR effects are neglected, the gyrotropic pressures \n are given  by \n$p_{\\alpha \\Vert }\\equiv m_{\\alpha }\\int v_{\\Vert }^{2}f_{\\alpha }d^{3}v\n=m_{\\alpha }B\\int v_{\\Vert }^{2}f_{\\alpha }d\\mu dv_{\\Vert}d\\varphi $,  \nand $p_{\\alpha \\perp } \\equiv \\frac{1}{2}m_{\\alpha }\\int v_{\\perp }^{2}f_{\\alpha }d^{3}v\n=m_{\\alpha }B^{2}\\int \\mu f_{\\alpha }d\\mu dv_{\\Vert}d\\varphi$.\n\nThe asymptotic equation governing the mirror dynamics near threshold is obtained by\nexpanding Eqs. (\\ref{balance-nl}), (\\ref{mainkin}) and the quasi-neutrality\ncondition, with  the pressure tensor elements for each species\ncomputed near a bi-Maxwellian equilibrium state characterized by the\ntemperatures $T_{\\perp\\alpha }$ and $T_{\\Vert\\alpha }$.\n\n\\section{Linear instability}\n\nBefore turning to the nonlinear regime, we briefly review the derivation of\nthe MI linear growth rate in the simplified framework provided by the drift\nkinetic approximation which is only valid at scales large enough for FLR\neffects to be subdominant.\n\nLinearizing  Eq. (\\ref{balance-nl}) about the background field $\\mathbf\nB_{0}}$ and equilibrium pressures $p_\\perp^{(0)}$ and  $p_\\|^{(0)}$, \nand considering  perturbations $\\widetilde{\\mathbf{B}}$ and \n$p_\\perp^{(1)}\\propto  e^{-i\\omega t+i\\mathbf{k\\cdot r}}$, we get \n\\begin{equation}\np_{\\perp }^{(1)}+\\frac{B_{0}\\widetilde{B}_{z}}{4\\pi }=-\\frac{k_{z}^{2}}\nk_{\\perp }^{2}}\\Big(1+\\frac{\\beta _{\\perp }-\\beta _{\\Vert }}{2}\\Big)\\frac\nB_{0}\\widetilde{B}_{z}}{4\\pi }.  \\label{pressures-1}\n\\end{equation\nHere, $p_{\\perp }^{(1)}$ has to be calculated from the linearized drift kinetic\nequation\n\\begin{equation}\n\\frac{\\partial f_{\\alpha }^{(1)}}{\\partial t}+v_{\\Vert }\\frac{\\partial\nf_{\\alpha }^{(1)}}{\\partial z}+\\Big [-\\mu \\frac{\\partial \\widetilde{B}_{z}}\n\\partial z}+\\frac{e_{\\alpha }}{m_{\\alpha }}E_{\\Vert }\\Big]\\frac{\\partial\nf_{\\alpha }^{(0)}}{\\partial v_{\\Vert }}=0,  \\label{kin-1}\n\\end{equation\nwhere we assume each $f_{\\alpha }^{(0)}$ to be a bi-Maxwellian distribution\nfunction \n\\begin{equation}\nf_{\\alpha }^{(0)}=A_{\\alpha }\\exp \\Big [-\\frac{v_{\\parallel }^{2}}\nv_{\\parallel \\alpha }^{2}}-\\frac{\\mu B_{0}m_{\\alpha }}{T_{\\perp \\alpha }\n\\Big ],  \\label{bi-maxwell}\n\\end{equation\nwith $A_{\\alpha }~=~n_{0}m_{\\alpha }\/(2\\pi \\sqrt{\\pi }v_{\\parallel \\alpha\n}T_{\\perp \\alpha })$. \n\n\nEquation (\\ref{kin-1}) is solved in Fourier representation, as \n\\begin{equation}\nf_{\\alpha }^{(1)}=-\\frac{\\mu \\widetilde{B}_{z}+\\frac{e_{\\alpha }} {m_{\\alpha\n}}\\phi }{\\omega -k_{z}v_{\\Vert }}k_{z}\\frac{\\partial f_{\\alpha }^{(0)}}{\n\\partial v_{\\Vert }}.  \\label{kin-gen-1}\n\\end{equation}\nThe neutrality condition \nallows one to express the potential $\\phi$ in terms of $\\widetilde{B}\n_{z}$. Indeed, assuming   $\\displaystyle\n\\zeta ={\\sqrt{\\pi }\\omega }\/(|k_{z}|v_{\\parallel i})\\ll 1}$\n(so that the contribution from the Landau pole is small),\n\\begin{equation}\n\\int f_{i}^{(1)}dv_{z}d\\mu d\\varphi =-\\frac{n_{0}}{B_{0}T_{\\parallel i}}\n\\Big[ T_{\\perp i}\\frac{\\widetilde{B}_{z}}{B_{0}}+e\\phi \\Big ] \\Big [ 1+ \ni\\zeta \\Big ].  \\nonumber\n\\end{equation}\nSimilarly, neglecting the electron\nLandau resonance contribution because of the small mass ratio, \n\\begin{equation}\n\\int f_{e}^{(1)}dv_{\\|}d\\mu d\\varphi =-\\frac{n_{0}}{B_{0}T_{\\Vert e}}\\Big[\nT_{\\perp e}\\frac{\\widetilde{B}_{z}}{B_{0}}-e\\phi \\Big].  \\nonumber\n\\end{equation}\nConsequently,\n\\begin{equation}\ne\\phi \\approx \\frac{T_{\\perp i}}{1+\\theta _{\\parallel }}\\Big[(\\theta _{\\perp\n}-\\theta _{\\parallel })-\\frac{\\theta _{\\parallel }(1+\\theta _{\\perp })}{\n1+\\theta _{\\parallel }}i\\zeta \\Big]\\frac{\\widetilde{B}_{z}}{B_{0}}.\n\\label{phi-1}\n\\end{equation\nWe thus recover that for mirror modes, the parallel electric field\nvanishes when the electrons are cold ($\\theta_\\perp = \\theta_\\| = 0$).\nInterestingly, when $\\theta _{\\perp }=\\theta _{\\parallel }$, only the Landau\npole contributes to $\\phi$.\n\nIt is now necessary to evaluate \n\\begin{equation}\np_{\\perp }^{(1)}=2\\frac{\\widetilde{B}_{z}}{B_{0}}p_{\\perp\n}^{(0)}+B_{0}^{2}\\sum_{\\alpha }m_{\\alpha }\\int \\mu f_{\\alpha }^{(1)}d\\mu\ndv_{\\Vert }d\\varphi .  \\nonumber\n\\end{equation\nUsing \n\\begin{eqnarray}\n\\int \\frac{k_{z}v_{\\Vert }}{\\omega -k_{z}v_{\\Vert }}f_{i}^{(0)}d\\mu\ndv_{\\Vert }d\\varphi  &=&-\\frac{n_{0}}{B_{0}}(1+i\\zeta )  \\nonumber \\\\\n\\int \\frac{k_{z}v_{\\Vert }}{\\omega -k_{z}v_{\\Vert }}f_{e}^{(0)}d\\mu\ndv_{\\Vert }d\\varphi  &=&-\\frac{n_{0}}{B_{0}},  \\nonumber\n\\end{eqnarray\nwe get \n\\begin{equation}\np_{\\perp }^{(1)}=-\\beta _{\\perp }\\frac{B_{0}^{2}}{4\\pi }\\Big[\\frac{1}{\\beta\n_{\\perp }}+\\Gamma +\\frac{T_{\\perp i}}{T_{\\Vert i}}\\frac{i\\zeta D}{2(1+\\theta\n_{\\perp })}\\Big]\\frac{\\widetilde{B}_{z}}{B_{0}}.  \\nonumber\n\\end{equation\nSubstituting this expression into the linearized force balance equation\nyields the linear instability growth rate given by Eq. (\\ref{growth_rate}),\nup to the FLR term which is not captured by the drift kinetic approximation.\nNote that the growth rate given by Eq. (\\ref{growth_rate}) is consistent\nwith the applicability condition $\\gamma \/|k_{z}|\\ll v_{{\\Vert }i}$ near\nthreshold ($\\Gamma \\ll 1$), as \n$k_{z}$ and   $(k_{z}\/k_{\\perp })^2$ scale like  $\\Gamma $,  while \n$\\gamma$ like  $\\Gamma ^{2}$. \n\n\\section{General pressure estimates}\n\nAs demonstrated in \\cite{KPS07a,KPS07b}, the scalings resulting from the\nlinear theory near threshold imply an adiabaticity condition to leading order.\nIt is thus enough to consider the stationary kinetic equation \n\\begin{equation}\nv_{\\Vert }\\mathbf{b}\\cdot \\nabla f_{\\alpha }-(\\mathbf{b}\\cdot \\nabla )\\left[\n\\mu B+\\frac{e_{\\alpha }}{m_{\\alpha }}\\phi \\right] \\frac{\\partial f_{\\alpha } \n}{\\partial v_{\\Vert }}=0.  \\label{Vlasov_stat}\n\\end{equation}\nIt turns out that Eq. (\\ref{Vlasov_stat}) is exactly solvable, the\ngeneral solution being an arbitrary function $f_{\\alpha}=g_{\\alpha }(\\mu\n,W_{\\alpha })$ of the particle energy $\\displaystyle{\\ W_{\\alpha }=\n{v_{\\Vert }^{2}}\/{2}+\\mu B+\\frac{e_{\\alpha }}{m_{\\alpha }} \\phi}$, and of $\\mu$.\nTo find the function $g_{\\alpha }(\\mu ,W_{\\alpha })$, we use the\nadiabaticity argument which means that, to leading order, $g_{\\alpha }$ as a\nfunction of $\\mu$ and $W_{\\alpha }$ retains its form during\nthe evolution. Therefore, the function $g_{\\alpha }(\\mu ,W_{\\alpha })$ is\nfound by matching with the initial distribution function $f_{\\alpha }^{(0)}$\ngiven by Eq. (\\ref{bi-maxwell}) which corresponds to $\\phi =0$ and \nW_{\\alpha }=\\frac{v_{\\Vert }^{2}}{2}+\\mu B_{0}$. We get \n\\begin{eqnarray}\n&&g_{\\alpha }(\\mu ,W_{\\alpha })=A_{\\alpha }\\exp \\Big[ -\\frac{v_{\\parallel\n}^{2}}{v_{\\parallel \\alpha }^{2}}-\\frac{\\mu B_{0}m_{\\alpha }}{T_{\\perp\n\\alpha }}\\Big]  \\nonumber \\\\\n&&\\quad=A_{\\alpha }\\exp \\Big [ -\\frac{2W_{\\alpha }} {v_{\\parallel \\alpha}^{2}}+\n\\mu B_{0}m_{\\alpha }\\Big( \\frac{1}{T_{\\parallel \\alpha }}-\\frac{1}{\nT_{\\perp \\alpha }}\\Big) \\Big] .  \\label{g-alpha}\n\\end{eqnarray}\nThus, $g_{\\alpha }(\\mu ,W_{\\alpha })$ is a Boltzmann distribution function\nwith respect to $W_{\\alpha }$ but, at fixed $W_{\\alpha }$, it displays an\nexponential growth relatively to $\\mu $ if $T_{\\perp \\alpha }>$ \nT_{\\parallel \\alpha }$. This effect can however be compensated by the\ndependence of $W_{\\alpha }$ in $\\mu$. This means that only a fraction of the\nphase space $(\\mu ,W_{\\alpha })$ is accessible, a property possibly related\nwith the existence of trapped and untrapped particles.\n\nNote that expanding Eq. (\\ref{g-alpha}) relatively to $\\widetilde{B}_{z}\/B_0$\nand $e\\phi ^{(1)}\/T_{\\perp i}$ reproduces the first order contribution to the\ndistribution function given by Eq. (\\ref{kin-gen-1}) with $\\omega=0$, and also\nthe second order correction found in \\cite{KPS07a,KPS07b} \nin the case of cold electrons. It should be\nemphasized that Eq. (\\ref{g-alpha}) only assumes adiabaticity and remains\nvalid for finite perturbations.\n\nThe function $g_{\\alpha }$ can also be rewritten in terms of $v_{\\Vert }$, \nv_{\\perp }$ and $\\phi $ as \n\\begin{eqnarray}\n&&g_{\\alpha }=A_{\\alpha }\\exp \\Big[-\\frac{m_{\\alpha }v_{\\Vert }^{2}}{\n2T_{\\parallel \\alpha }}-\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big]\n\\times  \\nonumber \\\\\n&&\\qquad \\exp \\left\\{ -\\frac{m_{\\alpha }v_{\\perp }^{2}}{2T_{\\perp \\alpha }}\n\\Big (\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}-\\frac{B_{0}}{B}\\Big [ \n\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}-1\\Big ]\\Big )\\right\\} , \n\\nonumber\n\\end{eqnarray\nwhich can be viewed as the bi-Maxwellian distribution function with the\nrenormalized transverse temperature \n\\begin{equation}\nT_{\\perp \\alpha }^{(eff)}=T_{\\perp \\alpha }\\left[ \\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}-\\frac{B_{0}}{B}\\Big(\\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}-1\\Big)\\right] ^{-1}.  \\nonumber\n\\end{equation\nNote the Boltzmann factor $\\exp {-[e_{\\alpha }\\phi \/T_{\\parallel \\alpha }]}$\nin the expression of $g_{\\alpha }$. For cold electrons, the ion distribution\nfunction was obtained in \\cite{Const02} by assuming that it \nremains bi-Maxwellian, and owing to the invariance of the kinetic energy and\nof the magnetic moment. This estimate, obtained by neglecting both time\ndependency (and consequently  Landau resonance) and finite Larmor radius\ncorrections, reproduces the closure condition given in \\cite{PRS06}.\n\nAfter rewriting Eq. (\\ref{g-alpha}) in the form \n\\begin{equation}\ng_{\\alpha }=A_{\\alpha }\\exp \\Big[-\\frac{e_{\\alpha }\\phi }{T_{\\parallel\n\\alpha }}-\\frac{v_{\\Vert }^{2}}{v_{\\parallel \\alpha }^{2}}-\\frac{\\mu\nB_{0}m_{\\alpha }}{T_{\\perp \\alpha }}\\Big(1+\\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big)\\Big],  \\nonumber\n\\end{equation}\nthe quasi-neutrality condition gives \n\\begin{eqnarray}\n&&\\left( 1+\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{B-B_{0}}{B_{0}}\\right)\n^{-1}\\exp \\left( -\\frac{e\\phi }{T_{\\parallel i}}\\right) =  \\nonumber \\\\\n&&\\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{B-B_{0}}{B_{0}}\\right)\n^{-1}\\exp \\left( \\frac{e\\phi }{T_{\\parallel e}}\\right)  \\nonumber\n\\end{eqnarray}\nor \n\\begin{eqnarray}\n&&e\\phi =(T_{\\parallel i}^{-1}+T_{\\parallel e}^{-1})^{-1}\\times  \\nonumber \\\\\n&&\\log \\left[ \\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{B-B_{0}}{\nB_{0}}\\right) \\left( 1+\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{B-B_{0}}{\nB_{0}}\\right) ^{-1}\\right] .  \\label{potential}\n\\end{eqnarray\nInterestingly, the electron density (and thus also that of the ions)\n\\begin{equation}\nn_{e}=n_{0}\\frac{B}{B_{0}}\\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{\nB-B_{0}}{B_{0}}\\right) ^{-1}\\exp \\left[ \\frac{e\\phi }{T_{\\parallel e}}\\right]\n\\nonumber\n\\end{equation\nhas the usual Boltzmann factor $\\exp \\left[ e\\phi \/T_{\\parallel e}\\right] $\nand also an algebraic prefactor depending on the magnetic field $B$. In the\ncase of isotropic electron temperature ($T_{\\perp e}=T_{\\parallel e}\\equiv\nT_{e}$), the electron density has the usual Boltzmann form $n_{e}=n_{0}\\exp\n\\left[ e\\phi \/T_{e}\\right] $.\n\nEquation (\\ref{potential}) shows that the potential vanishes in two\ncases: for cold electrons and also when electron and ion temperature anisotropies \n$a_e$ and $a_i$ (with $a_\\alpha = T_{\\perp \\alpha} \/T_{\\|\\alpha}$) \nare equal, a case considered in the linear theory of the\nmirror instability \\cite{Stix62,hasegawa,Hall79}.\n\nIn order  to evaluate explicitly the perpendicular pressure for each species \n\\begin{eqnarray}\n&&p_{\\perp \\alpha }=m_{\\alpha }B^{2}\\int \\mu g_{\\alpha }d\\mu dv_{\\Vert\n}d\\varphi   \\nonumber \\\\\n&&\\ \\ =n_{0}T_{\\perp \\alpha }\\frac{B^{2}}{B_{0}^{2}}\\Big(1+\\frac{T_{\\perp\n\\alpha }}{T_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big)^{-2}\\exp \\Big(\n\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big),  \\nonumber\n\\end{eqnarray\nwhere $e\\phi $ is given by Eq. (\\ref{potential}), it is convenient to\nintroduce the functions \n\\begin{eqnarray}\nS_{\\perp i}(u) &=&\\left( \\frac{1+u}{1+a_{i}u}\\right) ^{2}\\left( \\frac\n1+a_{i}u}{1+a_{e}u}\\right) ^{c_{i}}  \\label{Sperpi} \\\\\nS_{\\perp e}(u) &=&\\left( \\frac{1+u}{1+a_{e}u}\\right) ^{2}\\left( \\frac\n1+a_{e}u}{1+a_{i}u}\\right) ^{c_{e}},  \\label{Sperpe}\n\\end{eqnarray\nwith the notations $u=(B-B_{0})\/B_{0}$ and  $c_{\\alpha }=T_{\\parallel \\alpha\n}^{-1}\/(T_{\\parallel i}^{-1}+T_{\\parallel e}^{-1})$. The two latter\nfunctions transform one into the other by exchanging the subscripts $i$ and \ne$. The ion and electron perpendicular pressures are then written as \np_{\\perp \\alpha }=n_{0}T_{\\perp \\alpha }S_{\\perp \\alpha }(u)$. In the\nspecial case of cold electrons, \n\\begin{equation}\np_{\\perp }=n_{0}T_{\\perp i}\\frac{B^{2}}{B_{0}^{2}}\\Big(1+\\frac{T_{\\perp i}}\nT_{\\parallel i}}\\frac{B-B_{0}}{B_{0}}\\Big)^{-2},  \\label{p_perb-B}\n\\end{equation\nwhich is algebraic relatively to $B$. From  this expression \nas well as from the general formula for  $p_{\\perp }=p_{\\perp i}+p_{\\perp e}$\ngiven by Eqs. (\\ref{Sperpi}) and (\\ref{Sperpe}) it follows that the perpendicular\nand  magnetic pressures are anticorrelated. When $B$  increases (decreases), \nthe ratio of the perpendicular to the magnetic \npressure, i.e. the local $\\beta _{\\perp }$,\ndecreases (increases), which  corresponds to a reduction (an increase)\nof the distance to threshold. This implies that the instability \ncannot  saturate at  small amplitudes. \n\nSimilarly, for the parallel pressure, we have \n\\begin{equation}\np_{_{\\parallel \\alpha }}=n_{0}T_{_{\\parallel \\alpha }}\\frac{B}{B_{0}}\\Big(1\n\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big\n^{-1}\\exp \\Big(-\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big). \n\\nonumber\n\\end{equation\nthat rewrites $p_{\\Vert \\alpha} =n_{0}T_{\\Vert \\alpha}S_{\\Vert \\alpha}(u)$ with \n\\begin{eqnarray}\nS_{\\| i}(u) &=&\\left (\\frac{1+u}{1+a _{i}u}\\right ) \\left (\\frac{1+a _{i}u}\n1+a _{e}u}\\right)^{c_i}  \\label{Sparali} \\\\\nS_{\\| e}(u) &=&\\left(\\frac{1+u}{1+a _{e}u}\\right) \\left (\\frac{1+a _{e}u}{\n1+a _{i}u}\\right)^{c_e}.  \\label{Sparale}\n\\end{eqnarray\n\n\\section{ The weakly nonlinear regime}\n\nAs it follows from Eq. (\\ref{pressures-1}), in the linear regime near \nthreshold, the fluctuations of perpendicular and magnetic\npressures  almost compensate each other. \nIn the weakly nonlinear regime, the second order\ncorrection to the total (perpendicular plus magnetic) pressure is thus relevant\nand leads to a local shift of $\\Gamma$. To find this correction, we consider  the\nexpansions of  the perpendicular pressures of the ions and electrons in the \n$u$ variable. Because of the symmetry between the functions $S_{\\perp\ni}(u)$ and $S_{\\perp e}(u)$, it is enough to consider the expansion \n\\begin{eqnarray}\nS_{\\perp i}(u) &=&1+u\\Big (2-2a _{i}-c_{i}(a _{e}-a _{i})\\Big ) \\nonumber \\\\\n&&+u^{2}\\Big [c_{i}\\Big (a _{e}a _{i}-a _{i}^{2}+\\frac{1}{2}(a_{e}-a _{i})^{2}\\Big)  \n\\nonumber \\\\\n&&-4a _{i}+3a _{i}^{2}+\\frac{1}{2}c_{i}^{2}(a _{e}-a\n_{i})^{2}-2c_{i}(a _{e}-a _{i})  \\nonumber \\\\\n&&+2\\alpha _{i}c_{i}(a _{e}-a _{i})+1\\Big ]+O\\left( u^{3}\\right)  \\nonumber\n\\end{eqnarray\nAs a result, the second order contributions to the  perpendicular ion  pressure\nis  given by \n\\begin{eqnarray}\n&&p_{i\\perp }^{(2)}=n_{0}T_{\\perp i}\\Big \n3a _{i}^{2}-4a _{i}+1+c_{i}(a _{e}-a _{i})  \\nonumber \\\\\n&&\\qquad \\times \\Big(\\frac{1}{2}(c_{i}+1)(a _{e}-a _{i})-2+3a _{i\n\\Big)\\Big ]u^2,  \\nonumber\n\\end{eqnarray\nwith an analogous formula for the perpendicular electron pressure, \nobtained by exchanging the $i$ and $e$ indices.\n   Furthermore,  the threshold condition rewrites\n\\begin{eqnarray}\n&&\\frac{B_{0}^{2}}{4\\pi }+n_{0}\\Big \\{T_{\\perp i}\\left[ 2-2a\n_{i}-c_{i}\\left( a _{e}-a _{i}\\right) \\right]   \\nonumber \\\\\n&&\\qquad +T_{\\perp e}\\left[ 2-2a _{e}+c_{e}\\left( a _{e}-a\n_{i}\\right) \\right] \\Big \\}=0.  \\label{new-threshold}\n\\end{eqnarray\nThe quadratic contributions to the pressure balance (\\ref{balance-nl}), \noriginating from $p_{i\\perp }^{(2)}+p_{e\\perp\n}^{(2)}+\\left( B-B_{0}\\right) ^{2}\/(8\\pi )$, are collected in a  term \n $\\Lambda \\left( \\frac{B-B_{0}}{B_{0}}\\right) ^{2}$ with \n\\begin{eqnarray}\n\\Lambda  &=&n_{0}\\Big \\{T_{\\perp i}\\Big (3a _{i}^{2}-4a _{i}+1 \n\\nonumber \\\\\n&&+c_{i}(a _{e}-a _{i})\\Big [\\frac{1}{2}(1+c_{i})(a _{e}-a\n_{i})-2+3a _{i}\\Big ]\\Big )  \\nonumber \\\\\n&&+T_{\\perp e}\\Big (3a _{e}^{2}-4a _{e}+1+c_{e}(a _{e}-a _{i}) \n\\nonumber \\\\\n&&\\times \\Big [\\frac{1}{2}(1+c_{e})(a _{e}-a _{i})+2-3a _{e}\\Big \n\\Big )\\Big \\}+\\frac{B_{0}^{2}}{8\\pi }.  \\label{eqlambda}\n\\end{eqnarray\nThe value $\\Lambda _{c}$ of  $\\Lambda $ at\nthreshold is obtained  by expressing ${B_{0}^{2}}\/{8\\pi }$ by\nmeans of Eq.  (\\ref{new-threshold}), which gives \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}\\Big \\{T_{\\perp i}\\Big [3a _{i}^{2}-4a _{i}+1 \n\\nonumber \\\\\n&&+c_{i}(a _{e}-a _{i})\\Big (\\frac{1}{2}(1+c_{i})(a _{e}-a\n_{i})-2+3a _{i}\\Big )  \\nonumber \\\\\n&&-\\frac{1}{2}\\Big (2-2a _{i}-c_{i}(a _{e}-a _{i})\\Big )\\Big ] \n\\nonumber \\\\\n&&+T_{\\perp e}\\Big [3a _{e}^{2}-4a _{e}+1+c_{e}(a _{e}-a _{i}) \n\\nonumber \\\\\n&&\\times \\Big (\\frac{1}{2}(1+c_{e})(a _{e}-a _{i})+2-3a _{e}] \n\\nonumber \\\\\n&&-\\frac{1}{2}\\Big (2-2a _{e}+c_{e}(a _{e}-a _{i})\\Big )\\Big ]\\Big \\}. \\nonumber\n\\end{eqnarray\nAfter some algebra, defining $\\lambda _{c}=\\Lambda _{c}\/(n_{0}T_{\\perp i})$, one gets \n\\begin{eqnarray}\n&&\\frac{\\lambda _{c}}{\\alpha _{i}}=\\frac{T_{\\perp i}}{T_{\\parallel i}}\\Big \n3+3\\frac{\\theta _{\\perp }^{3}}{\\theta _{\\parallel }^{2}}-\\frac{1}{2}\\frac\n\\left( \\theta _{\\perp }-\\theta _{\\parallel }\\right) ^{2}}{\\theta _{\\parallel\n}^{2}\\left( 1+\\theta _{\\parallel }\\right) ^{2}}  \\nonumber \\\\\n&&\\qquad \\times \\left( 4\\theta _{\\perp }+4\\theta _{\\parallel\n}^{2}+\\allowbreak 5\\left( \\theta _{\\perp }+1\\right) \\theta _{\\parallel\n}\\right) \\Big]  \\nonumber \\\\\n&&\\qquad -\\frac{3}{2\\theta _{\\parallel }\\left( 1+\\theta _{\\parallel }\\right) \n}\\left[ \\left( \\theta _{\\perp }+\\theta _{\\parallel }\\right) ^{2}+2\\theta\n_{\\parallel }(1+\\theta _{\\perp }^{2})\\right]. \\label{eqlambda_c}\n\\end{eqnarray}\n\nProceeding as in \\cite{KPS07a}, retaining the contribution of the \nabove  quadratic terms to the pressure balance, leads one to supplement \na nonlinear contribution to  Eq. (\\ref{growth_rate}) that becomes \n\\begin{eqnarray}\n&&\\frac{\\partial u}{\\partial t}=\\frac{2}{\\sqrt{\\pi }}\\frac{T_{\\Vert i}}\nT_{\\perp i}}\\frac{v_{\\Vert i}}{D}{\\widehat{{\\cal K}_z }}\n\\Big \\{ \\Gamma u -\\frac{\\chi}{\\beta _{\\perp }}(\\Delta_\\perp )^{-1}\\partial _{zz}u \\nonumber \\\\\n&& +\\frac{3}{4}\\Big (\\frac{T_{\\perp i}}{T_{\\Vert i}}-1\\Big \n\\frac{1+F}{1+\\theta _{\\perp }}r_{L}^{2}\\Delta _{\\perp }u\n-\\frac{\\lambda _{c}}{2(1+\\theta _{\\perp })}u^2 \\Big \\}\\label{NL}\n\\end{eqnarray\nHere the integral operator ${\\widehat{{\\cal K}}_z}$ reduces in Fourier representation\nto $|k_z|$ and  $\\chi = 1+\\frac{\\beta_{\\perp }-\\beta _{\\Vert }}{2}$. Furthermore, within the present\napproximation, $u$ coincides with ${\\widetilde B}_z\/B_0$.\n\n\nEquation (\\ref{NL}) extends the result of \\cite{KPS07a, Calif08} valid for \ncold electrons. As in the latter case,  this equation is a gradient type equation,\n\\[\n\\frac{\\partial u}{\\partial t}=-{\\widehat{{\\cal K}_z }}\\frac{\\delta F}{\\delta u},\n\\]\nfor which the free\nenergy (written in dimensionless variables)\n\\[\nF=\\int \\left \\{  \\frac 12 \\left [ -\\Gamma u^2 +(\\partial_z u)^2 +u\\Delta_\\perp ^{-1}\\partial _{zz}u\\right ] +\\frac 13\\lambda _{c}u^3\\right \\} d{\\bf r}\n\\] \nis  unbounded  from below due to the integral $\\int \\lambda _{c}u^3 d{\\bf r}$.\nThis leads to a blow-up behavior, associated with a subcritical bifurcation \\cit\n{KPS07a}, \\cite{KPS07b}. Saturation at large values of the amplitude \nand formation of stationary structures requires additional nonlinear\neffects such as the influence of resonant particles on the nonlinear coupling \\cite{PSKH09}.\nEquation (\\ref{NL}) that does not include saturation processes is not suitable to address the \nquestion of the reduction of the temperature anisotropy by the development of the mirror\ninstability mentioned in \\cite{VedenovSagdeev}. This effect is reproduced by the \nquasi-linear theory \\cite{SS64}, and was also studied in the context of the \nso-called FLR-Landau fluid model\nthat, like the present asymptotics, retains a linear description of the Landau resonance and \nof FLR effects, but includes all the hydrodynamic nonlinearities and does not a priori\nprescribe a pressure balance condition. It was  observed in this case \nthat during the saturation phase, the mean temperatures rapidly evolve in a way as to   \nreduce the distance to threshold \\cite{Borgogno07}.\n\nAs demonstrated in \\cite{KPS07a,KPS07b}, \nthe sign of the nonlinear coupling $\\lambda _{c}$ \ndefines the type of the mirror structures, namely holes ($\\lambda _{c}>0$) or \nhumps ($\\lambda _{c}<0$), near threshold. This sign is strongly dependent on the\nequilibium distribution function \\cite{HKP09} . It is nevertheless of interest to consider the case\nwhere both  ions and electrons have a bi-Maxwellian distribution function.\nIt turns out that the sign of $\\lambda _{c}$\ncan then be determined analytically in a few special cases.\n\n\\noindent\n{ \\it (i) Limit $\\protect\\theta_\\| \\ll \\protect\\theta_\\perp$:}\n\\begin{equation}\n\\frac{\\Lambda _{c}}{n_{0}T_{\\perp i}a _{i}}=\\frac{\\theta _{\\perp }^{2}}\n\\theta _{\\parallel }}\\left( \\frac{T_{\\perp e}}{T_{\\parallel e}}-\\frac{3}{2\n\\right) >0.\\nonumber\n\\end{equation}\n\n\\noindent\n{\\it (ii) Equal anisotropies ($\\protect\\theta _{\\perp}=\\protect\\theta_{\\parallel }$)}\n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(T_{\\perp i}+T_{\\perp e})\\left( 3a ^{2}-4a\n+1\\right)   \\nonumber \\\\\n&&-n_{0}(T_{\\perp i}+T_{\\perp e})\\left( 1-a \\right) =3a \\frac{B_{0}^{2\n}{8\\pi }>0.\\nonumber\n\\end{eqnarray}\n\n\\noindent\n{\\it (iii) Isotropic electron temperature:}\nThe coefficient\n$\\Lambda _{c}$ can be rewritten in the form \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(a _{i}-1)\\{T_{\\perp i}\\Big((3a _{i}-1)  \\nonumber\n\\\\\n&&+c_{i}\\Big[\\frac{1}{2}(1+c_{i})\\left( \\alpha _{i}-1\\right) +2-3a _{i\n\\Big]\\Big )  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[\\frac{1}{2}\\left( 1+c_{e}\\right) \\left( a _{i}-1\\right)\n+1\\Big]\\}+\\frac{B_{0}^{2}}{8\\pi }.  \\nonumber\n\\end{eqnarray\n   Furthermore, at  threshold, \n\\begin{equation}\n\\frac{1}{2}n_{0}(a _{i}-1)\\left[ T_{\\perp i}\\left( 2-c_{i}\\right)\n+T_{\\perp e}c_{e}\\right] =\\frac{B_{0}^{2}}{8\\pi }>0.\\nonumber\n\\end{equation\nHence, we simultaneously have two inequalities $a _{i}>1$ and $T_{\\perp\ne}c_{e}>T_{\\perp i}(c_{i}-2)$. Therefore, \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(a _{i}-1)\\Big \\{T_{\\perp i}\\Big((3a _{i}-1) \n\\nonumber \\\\\n&&+c_{i}\\Big [\\frac{1}{2}(1+c_{i})\\Big(a _{i}-1\\Big)+2-3a _{i}\\Big\n\\Big)  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[\\frac{1}{2}(1+c_{e})(a _{i}-1)+1\\Big ]\\}  \\nonumber \\\\\n&&+\\frac{1}{2}n_{0}(a _{i}-1)\\left[ T_{\\perp i}\\left( 2-c_{i}\\right)\n+T_{\\perp e}c_{e}\\right]   \\nonumber \\\\\n&=&n_{0}(a _{i}-1)\\Big \\{T_{\\perp i}\\Big(3a _{i}(1-c_{i})  \\nonumber \\\\\n&&+c_{i}\\Big[\\frac{1}{2}(1+c_{i})\\left( a _{i}-1\\right) +\\frac{3}{2}\\Big\n\\Big)  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[2+\\frac{1}{2}\\left( 1+c_{e}\\right) \\left( a\n_{i}-1\\right) \\Big]\\Big \\},\\nonumber\n\\end{eqnarray\nwhich is positive, because $1-c_{i}= c_{e}=\n(1+\\theta _{\\parallel })^{-1}>0$ and $a _{i}>1$.\n\n\n\\begin{figure}[t]\n\\centerline{\n\\includegraphics[width=0.46\\textwidth]{fig1.eps}\n}\n\\caption{Fig. 1. Variation with $\\protect\\theta_\\|$ of the distance to\nthreshold $\\Gamma$ given by Eq. (\\protect\\ref{newthreshold}) (dashed line)\nand of the normalized nonlinear coupling coefficient $\\protect\\lambda$\n(solid line) evaluated from Eq. (\\protect\\ref{eqlambda}) for $\\protect\\theta_{\\perp}=1$\n, $\\protect a_i=1.1$ and $\\protect\\beta_{\\perp i} = 10$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centerline{\n\\includegraphics[width=0.46\\textwidth]{fig2.eps}\n}\n\\caption{Fig. 2. Variation with $\\protect\\beta_{\\perp i}$ of the minimum \n\\mathrm{min}\\,( \\protect\\lambda_c)$ of the normalized nonlinear coupling\ncoefficient taken in an interval of values of $a_p$ between $0$ and $a_{p1}\n\\protect\\beta_{\\perp i})$, defined such that the threshold is obtained for a\nvalue of $\\protect\\theta_\\|$ equal to $100$, for $\\protect\\theta_\\perp= 0.2$\n(solid line), $\\protect\\theta_\\perp=1$ (dotted line) and $\\protect\\thet\n_\\perp=5$ (dashed line).}\n\\label{fig2}\n\\end{figure}\n\n\n\\noindent\n{\\it (iv)  More general conditions:}\nA numerical approach was used in this case. Figure  1 displays, for typical values \nof the parameters (taken\nhere as $\\theta _{\\perp }=1$, $a _{i}=1.1$ and $\\beta _{\\perp i}=10$), the\ndistance to threshold $\\Gamma $ (dashed line) given by Eq. (\\ref\n{newthreshold}) and the non-dimensional nonlinear coupling coefficient \n\\lambda =\\Lambda \/(n_{0}T_{\\perp i})$ (solid line), where $\\Lambda $ is given by\nEq. (\\ref{eqlambda}), as a function of $\\theta _{\\Vert }$. This graph is\ntypical of the general behavior of these functions and shows that they are\nboth decreasing as $\\theta _{\\Vert }$ increases, with $\\lambda $ possibly\nreaching negative values, but only below threshold. In order to show that\nthe value $\\lambda _{c}$, given by Eq. (\\ref{eqlambda_c}), of $\\lambda $ \nat threshold is positive in a wider range of parameters, we display in Fig.\n2, as a function of $\\beta _{\\perp i}$ for $\\theta _{\\perp }=0.2$ (solid\nline), $\\theta _{\\perp }=1$ (dotted line) and $\\theta _{\\perp }=5$ (dashed\nline), the quantity $\\mathrm{{min}\\,(\\lambda _{c})}$ obtained after\nminimizing $\\lambda _{c}$ in an interval of values of $a_{p}$ between $0$\nand $a_{p1}(\\beta _{\\perp i})$. The latter quantity is arbitrarily defined\nsuch that the threshold is obtained for a value of $\\theta _{\\Vert }$ equal\nto $100$. This graph shows that $\\mathrm{min}(\\lambda _{c})$ varies little\nwith $\\theta _{\\perp }$ but is very sensitive to $\\beta _{\\perp i}$. As the\nlatter parameter is increased, $\\mathrm{{min}\\,(\\lambda _{c})}$ decreases\nbut remains always positive. Although \nthis numerical observation is  \nnot a rigorous proof, it convincingly shows that $\\Lambda>0 $\nin the parameter range  of physical interest.\n\n\\section{Stationary nonlinear structures}\n\nSubstituting the explicit expressions of the gyrotropic pressures\nin terms of the magnetic field amplitude\ngiven in the Section IV, within  the equation\nfor the balance of forces \n\\begin{eqnarray}\n&&-\\nabla \\Big( p_{\\perp }+\\frac{B^{2}}{8\\pi }\\Big) +\\Big[ 1+\\frac{4\\pi } \nB^{2}}( p_{\\perp }-p_{\\Vert }) \\Big] \\frac{(\\mathbf{B}\\cdot \\nabla )\\mathbf{B\n}{4\\pi }  \\nonumber \\\\\n&&\\qquad + \\mathbf{B}(\\mathbf{B}\\cdot\\nabla) \\Big (\\frac{p_\\perp -p_\\|}{B^2} \\Big ) \\, \n =0, \\label{equil_noFLR}\n\\end{eqnarray}\nleads to a closed system that seems overdetermined due to the divergenceless condition \n$\\nabla \\cdot {\\bf B}=0$. In fact, it can be checked, after some algebra\nusing the explicit expressions  (\\ref{Sperpi},\\ref{Sperpe}) and (\\ref{Sparali},\\ref{Sparale}),   \nthat the projection of Eq. \n(\\ref{equil_noFLR}) on the magnetic field vanishes identically, thus reducing the \nsystem to three equations for three unknowns.\nThese equations  can be useful  for finding, possibly numerically,  \nstationary profiles of three-dimensional finite-amplitude \nstationary mirror structures. Note that Eq. (\\ref{equil_noFLR}) differs from the \nGrad-Shafranov equation  \\cite{grad,shafra} in  that the parallel and perpendicular\npressures are here prescribed functions of the magnetic field amplitude.\nA main issue concerns the existence of \nstable subcritical solutions, a question that is beyond the scope of this \nletter and will be addressed in forthcoming works. \nSuch structures are reported by satellite observations \\cite{SLD07,Genot09} \nand are also expected from the subcritical character of the\nmirror instability \\cite{KPS07b}. Equilibrium solutions\nwere computed  in one-space dimension in \n\\cite{PRS06}, where they lead to discontinuous profiles. Their regularization would \nrequire that FLR corrections be retained. These  additional contributions are known \nfrom the linear kinetic theory but their extension to the finite-amplitude\ncase remains a challenging problem. \n\n\nThis work was supported by the CNRS PICS programme 6073 and RFBR grant\n12-02-91062-CNRS\\_a. T.P. and P.L.S. benefited from support from INSU-CNRS\n Programme National PNST. The work of\nE.K. was also supported by the RAS Presidium Program \"Fundamental problems\nof nonlinear dynamics in mathematical and physical sciences\",  Grant NSh\n7550.2006.2 and by the French Minist\\`ere de l'Enseignement Sup\\'erieur \net de la Recherche.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPoint cloud registration is a fundamental task in the fields of 3D computer vision, robotics, photogrammetry and remote sensing. It aims to align two point clouds by estimating the optimal rigid transformation (including the rotation and translation) between them, and has been broadly applied in 3D reconstruction~\\cite{henry2012rgb,choi2015robust,zhang2015visual}, object recognition and localization~\\cite{drost2010model,zeng2017multi,wong2017segicp,marion2018label}, SLAM~\\cite{zhang2014loam}, medical imaging~\\cite{audette2000algorithmic}, archaeology~\\cite{chase2012geospatial}, etc.\n\nThough Iterative Closet Point (ICP)~\\cite{besl1992method} is a popular registration method, its performance is still limited since it highly relies on the initial guess provided by users and is prone to converge to the local minima when the initialization is not good enough. To circumvent the need of initial guess, people tend to use 3D keypoint detecting and matching techniques (e.g. FPFH~\\cite{rusu2009fast}, ISS~\\cite{zhong2009intrinsic}, 3DSmoothNet~\\cite{gojcic2019perfect}) to establish correspondences between the point clouds, and then estimate the transformation based on these correspondences. However, 3D keypoint matching is much less robust and accurate than 2D keypoint matching (e.g. SIFT~\\cite{lowe2004distinctive}, SURF~\\cite{bay2006surf}), so it may easily generate a huge number of false matches, usually called \\textit{outliers}, among the putative correspondences. Consequently, robust estimation methods must be adopted to solve the correct transformation from the potentially abundant outliers.\n\nUnfortunately, many existing robust methods have their own nonnegligible limitations. RANSAC~\\cite{fischler1981random} is a well-known hypothesis-and-test consensus maximization paradigm for robust estimation, but it has exponentially growing computational cost w.r.t. the outlier ratio, thus unsuitable for handling high-outlier situations. Note that an outlier ratio of over 95\\% is common after 3D keypoint matching in real scenes~\\cite{bustos2017guaranteed}, so RANSAC is not a generally practical option. Branch-and-Bound (BnB)~\\cite{parra2014fast,horst2013global}, as another famous robust estimator, can yield the globally optimal solution, but it also suffers from the worst-case exponential runtime w.r.t. the problem size. Other robust methods include the non-minimal global solvers such as FGR~\\cite{zhou2016fast}, GNC~\\cite{yang2020graduated} and ADAPT~\\cite{tzoumas2019outlier} which are limited in robustness and generally cannot tolerate outlier ratios higher than 90\\%, the guaranteed outlier removal method GORE~\\cite{bustos2017guaranteed} which may also be too slow for use due to its probable internal use of BnB, and the certifiably optimal solver TEASER~\\cite{yang2019polynomial,yang2020teaser} which is also slow without using parallelism programming. Hence, we can see that almost all of these methods have limited performance in some ways.\n\nTherefore, our goal in this paper is to propose a new robust estimation approach, which can handle registration problems with high or extreme outliers in a time-efficient way.\n\n\\textbf{Our Contributions.} We present a specialized consensus maximization method to realize rapid robust estimation for point cloud registration with even extremely high outliers (e.g. up to 99\\%). \n\nFirst, we abandon the traditional time-consuming single-layered three-point sampling framework used in RANSAC, and present a more efficient strategy by smartly decomposing the three-point layer into: (i) a one-point sampling layer that serves as a raw outlier `filter' on the basis of the 3D-geometric rigidity constraint to diminish the correspondence size, and (ii) a two-point sampling layer that performs random sampling and minimal model estimating. This strategy can significantly reduce the computational cost for obtaining an all-inlier subset from the random samples.\n\nMoreover, for faster consensus maximization, we propose a compatibility-based consensus building strategy by introducing a novel stratified element-wise compatibility checking technique that is merely made up of very simple calculations and boolean conditions, in order to effectively cut down the time cost for the repetitive construction of consensus set as in RANSAC. \n\nThese two contributions lead to our robust solver DANIEL (Double-layered sAmpliNg with consensus maximization based on stratIfied Element-wise compatibiLity). Comprehensive experimental evaluation on various real-world datasets shows that the proposed solver is highly robust against over 99\\% outliers and is also rather fast in practice (running within 3 seconds for solving a 99\\%-outlier registration problem), outperforming existing state-of-the-art robust registration solvers.\n\n\n\n\\section{Related Work}\n\nThis section provides brief reviews on addressing the point cloud registration problem with correspondences.\n\nBefore the application of robust solvers, correspondences have to be first established between the point clouds. 3D keypoint detecting and matching with feature descriptors~\\cite{rusu2009fast,zhong2009intrinsic,zhong2009intrinsic} is a widely-used process for building the correspondences, based on which registration solvers are able to estimate the best transformation.\n\nIn the most ideal case, assuming that there is no outlier among these putative correspondences, closed-form estimators can efficiently solve the optimal transformation based on eigenvector~\\cite{horn1987closed} or Singular Value Decomposition (SVD)~\\cite{arun1987least}. More recent methods include the optimal BnB solver~\\cite{olsson2008branch} and the certifiable Semi-Definite relaxation method~\\cite{briales2017convex}. \n\nHowever, it is known to all that 3D keypoint matching is less robust than its 2D counterpart such as SIFT~\\cite{lowe2004distinctive} or SURF~\\cite{bay2006surf} and could easily generate outliers to corrupt the performance of these outlier-intolerant solvers in practical use. Even worse, the outliers may sometimes occupy a massive majority of the correspondences (as shown in~\\cite{bustos2017guaranteed} and Section~\\ref{Experiments}), making inliers (correct matches) fairly sparse. In this case, we need to apply robust estimation to reject outliers and find the true inliers to make reasonable estimates, where the robustness as well as efficiency of the robust methods could determine the registration results to a great extent.\n\nWe introduce several types of robust solvers as follows.\n\n\\subsection{Consensus Maximization}\n\nConsensus maximization consists in finding a model that can maximize the number of correspondences with residual errors lower than a certain inlier threshold w.r.t. this model. RANSAC~\\cite{fischler1981random} is a very common consensus maximization paradigm via the hypothesize-and-test model-fitting framework. Besides, further techniques, including local optimization~\\cite{chum2003locally,lebeda2012fixing} correspondence sorting~\\cite{chum2005matching}, etc, have been applied to improve the performance of RANSAC. However, the time cost of these RANSAC solvers generally increases exponentially with the outlier ratio. For instance, RANSAC should require more than 4.6$\\times$10$^6$ samples to select one all-inlier subset with 0.99 confidence if the outlier ratio is 99\\% in one registration problem. This would make RANSAC infeasible for use in realistic problems. \n\nAnother typical consensus maximization approach is BnB. It addresses the optimization problem globally optimally by searching in the parameter space (e.g. $SO(3)$ w.r.t. rotation or $SE(3)$ w.r.t. transformation). Unfortunately, BnB runs in exponential time and could not scale to problems with large correspondence numbers, but thousands of correspondences is quite common in reality, which limits the practicality of BnB.\n\nADAPT (Adaptive Trimming)~\\cite{tzoumas2019outlier} provides a non-minimal way to solve the consensus maximization problem. It can be directly in conjunction with standard non-minimal solvers to find the maximum consensus set through iterations, but it has the issue of limited robustness. In the registration problem, ADAPT could hardly tolerate more than 90\\% outliers.\n\nIn essence, our solver DANIEL is also a consensus maximization method based on the RANSAC paradigm. But much differently, DANIEL employs a smarter framework by decomposing the sampling into two layers and applying the stratified compatibility checking before the consensus building, so it exceeds RANSAC in time-efficiency by tens to tens of thousands of times.\n\n\\subsection{M-Estimation}\n\nM-estimation can actively decrease the weights of outliers by incorporating robust loss functions into the optimization process. Local M-estimators (e.g.~\\cite{agarwal2013robust,kummerle2011g,sunderhauf2012towards}) can be used to minimize the object function, but the problem is that they must require the initial guess and hence is liable to converge to local solutions if the initial guess is poor. FGR~\\cite{zhou2016fast} first applied Graduated Non-Convexity (GNC) in the registration problem, and more recently, GNC is incorporated with non-minimal solvers and extended to more robotics and computer vision problems~\\cite{yang2020graduated}. However, these solvers generally have confined robustness against outliers. For example, FGR and GNC-TLS would both become brittle once the outlier ratio exceeds 90\\%. \n\n\n\\subsection{Downside of RANSAC}\n\nWe now provide a more explicit analysis on the limitation of RANSAC. In traditional RANSAC or many of its variants, when we obtain one minimal subset in a certain iteration of random sampling, we must then: (i) estimate the coarse minimal model ($\\boldsymbol{R}^{*},\\boldsymbol{t}^{*}$), and (ii) build the consensus set with this model by computing the residual errors w.r.t. all the correspondences. This consensus maximization strategy has two apparent downsides: (i) when the outlier ratio is high, the probability to sample an all-inlier subset can be extremely low, thus making a great number of iterations necessary (exponentially growing with the outlier ratio), and (ii) it is time-consuming to build the consensus set in every iteration, especially when the problem size is large (e.g. thousands of correspondences). And these two limitations are the main issues that we aim to circumvent by using our solver.\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{ccc}\n\n\\footnotesize{(a) Correspondences with 99\\% outliers}\n&\n\\footnotesize{(b) Registration by DANIEL}\n&\n\\footnotesize{(c) Overview of DANIEL}\n\n\\\\\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{demo-corres-99-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.16\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{demo-res-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\n\n&\n\n\\begin{minipage}[t]{0.5\\linewidth}\n\\centering\n\\includegraphics[width=0.96\\linewidth]{demo-overview.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\\vspace{-1mm}\n\\caption{Illustration of robust point cloud registration using our solver DANIEL. (a) An example of a registration problem with $N=1000$ correspondences and only 10 inliers. (b) DANIEL can robustly estimate the best transformation in only 2.597 seconds. (c) An intuitive overview of DANILE. Please see Section~\\ref{overview} for explicit descriptions.}\n\\label{demo-for-show}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\section{Our Method: DANIEL}\n\nIn this section, we present our robust point cloud registration method DANIEL.\n\n\\subsection{Problem Formulation: Consensus Maximization}\n\nFirst of all, we introduce the consensus maximization formulation for the registration problem.\n\nAssume that we have two sets of 3D points: $\\mathcal{X}=\\{\\boldsymbol{x}_i\\}_{i=1}^{N}$ and $\\mathcal{Y}=\\{\\boldsymbol{y}_i\\}_{i=1}^{N}$ where $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i\\,(\\boldsymbol{x}_i,\\boldsymbol{y}_i\\in\\mathbb{R}^3)$ makes up a putative correspondence. Point cloud registration aims to estimate the best transformation including rotation $\\boldsymbol{R}\\in SO(3)$ and translation $\\boldsymbol{t}\\in \\mathbb{R}^{3}$ that can align point set $\\mathcal{X}$ and $\\mathcal{Y}$. If correspondence $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ is an inlier, then we can have:\n\\begin{equation}\\label{prob-form1}\n\\boldsymbol{y}_i=\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}+\\boldsymbol{\\epsilon}_i,\n\\end{equation}\nwhere $\\boldsymbol{\\epsilon}_i\\in\\mathbb{R}^3$ denotes the noise measurement, so that the following relation \n\\begin{equation}\n\\left\\|\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}-\\boldsymbol{y}_i\\right\\|\\leq \\xi\n\\end{equation}\ncould be satisfied, where $\\xi\\geq\\|\\boldsymbol{\\epsilon}\\|$ denotes the inlier threshold; however, if $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ is an outlier, we can assume:\n\\begin{equation}\n\\left\\|\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}-\\boldsymbol{y}_i\\right\\|>\\xi.\n\\end{equation}\nUsually, we assume the noise to be isotropic Gaussian with standard deviation $\\sigma$, so the inlier threshold can be typically set as: $\\xi=5\\sim 6\\sigma$.\n\nIn this case, letting set $\\mathcal{N}=\\{1,2,\\dots,N\\}$, the registration problem can be written as the following formulation:\n\\begin{equation}\\label{CM}\n\\begin{gathered}\n\\underset{\\mathcal{M}\\subset \\mathcal{N}}{\\max}\\, |\\mathcal{M}|, \\\\\ns.t. \\,\\|\\boldsymbol{R}_{\\mathcal{M}}\\boldsymbol{x}_i+\\boldsymbol{t}_{\\mathcal{M}}-\\boldsymbol{y}_i\\| \\leq \\xi,\\,(\\forall i\\in\\mathcal{M})\n\\end{gathered}\n\\end{equation}\nwhere $(\\boldsymbol{R}_{\\mathcal{M}},\\boldsymbol{t}_{\\mathcal{M}})$ represents a rigid transformation and $\\mathcal{M}$ is its corresponding consensus set. The goal of formulation~\\eqref{CM} is to find the transformation $(\\boldsymbol{R}_{\\mathcal{M}},\\boldsymbol{t}_{\\mathcal{M}})$ that maximizes the size of consensus $\\mathcal{M}$, the process of which is typically called \\textit{consensus maximization}. The goal of our solver is to solve~\\eqref{CM} in a computationally efficient way.\n\n\n\n\n\\subsection{Methodology Overview}\\label{overview}\n\nFig.~\\ref{demo-for-show} illustrates the effectiveness and the overview of our proposed solver DANIEL with descriptions given below.\n\nIn general, DANIEL maximizes the consensus set also by random sampling, but its main operation structure is significantly different from the traditional RANSAC solvers. To be specific, DANIEL consists of two random-sampling layers, where the latter is embedded into the main framework of the former. The first layer adopts one-point sampling in each iteration (in order to maximize the probability to obtain inliers) and then reduces the correspondence set size by using rigidity constraint. (See Section~\\ref{first-layer} for the first layer)\n\nIn one certain iteration of the first layer and after the reduction of the correspondence set, the second layer performs continuous random sampling of two points as well as the computing of the respective minimal models. In the meantime, a new stratified element-wise compatibility checking approach is proposed to conduct rapid compatibility checking between pairs of minimal models obtained, which serves as a prerequisite for consensus building in DANIEL. Only when we can obtain two minimal subsets (models) that are compatible with each other can we average the parameters of these two models and build the consensus set. Furthermore, we can derive probabilistic termination conditions (by computing maximum iteration numbers) for both of the two layers, and the maximized consensus in the second layer will be used to further maximize the consensus in the first layer. Finally, we can return the maximum consensus set in the first layer to further obtain the ultimate full inlier set. (See Section~\\ref{second-layer} for the second layer) \n\n\\subsection{First Layer: One-Point Sampling and Inlier Candidate Searching with Rigidity Constraint}\\label{first-layer}\n\nRigidity~\\cite{michel2017global,zach2015dynamic,quan2020compatibility,yang2019polynomial} (invariance of length) is a common pairwise constraint in 3D geometry. Specifically, for any two correspondences $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ and $\\boldsymbol{x}_j\\leftrightarrow\\boldsymbol{y}_j$ that are both inliers, we can have the rigidity constraint such that\n\\begin{equation}\\label{rigidity}\n\\left|\\left\\|\\boldsymbol{y}_i-\\boldsymbol{y}_j\\right\\|-\\left\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\right\\|\\right| \\leq 2\\xi,\n\\end{equation}\nwhich can be derived, based on triangular inequality and the property that norm is invariant to $\\boldsymbol{R}$, as follows:\n\\begin{equation}\\label{proof-of-rigidity}\n\\begin{gathered}\n\\left|{\\|\\boldsymbol{y}_i-\\boldsymbol{y}_j\\|}-{\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\|}\\right| \\\\\n=\\left|{\\left\\|\\boldsymbol{R} (\\boldsymbol{x}_i-\\boldsymbol{x}_j+\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_i-\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_j)\\right\\|}-{\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\|}\\right| \\\\\n\\leq \\left\\| (\\boldsymbol{x}_i-\\boldsymbol{x}_j)-({\\boldsymbol{x}_i-\\boldsymbol{x}_j})+\\boldsymbol{R}^{\\top}\\boldsymbol{\\varepsilon}_i-\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_j \\right\\| \\\\ \n= {2\\|(\\boldsymbol{\\epsilon}_i-\\boldsymbol{\\epsilon}_j)\\|}\\leq 2\\xi.\n\\end{gathered}\n\\end{equation}\nIntuitively, this constraint indicates that the length between two points (both inliers) remains fixed before and after the rigid transformation, which underlies our strategy to reduce the correspondence set in our first sampling layer.\n\nIn the first layer of DANIEL, we perform one-point sampling, that is, to select only one random correspondence in each iteration, say correspondence $n\\in\\mathcal{N}$ ($\\mathcal{N}=\\{1,2,\\dots,N\\}$ denotes the full correspondence set). Then, we employ rigidity constraint to sift out all the eligible correspondences from $\\mathcal{N}$ that can satisfy~\\eqref{rigidity} with $n$, and then add them to set $\\mathcal{C}$, called the `\\textit{inlier candidate set}'. (See \\textbf{lines 3-9} in Algorithm~\\ref{DANIEL-algo})\n\nThe insight here lies in that if the sampled point $n$ is an inlier, then all the other inliers must fulfill condition~\\eqref{rigidity} with $n$ and must therefore lie within set $\\mathcal{C}$. In this way, we can establish a smaller correspondence set for our second layer. Note that this step could be rather effective especially when the outlier ratio is high, since a large portion of `raw' outliers can be swiftly removed by the rigidity constraint, exponentially increasing the probability of sampling an all-inlier subset later in the second layer.\n\n\\begin{algorithm}[t]\n\\caption{\\textit{CompatibilityStaircase}}\n\\label{Staircase}\n\\SetKwInOut{Input}{\\textbf{Input}}\n\\SetKwInOut{Output}{\\textbf{Output}}\n\\Input{two minimal models: ($\\boldsymbol{R}^*_{a},\\boldsymbol{t}^*_{a}$) and ($\\boldsymbol{R}^*_{b},\\boldsymbol{t}^*_{b}$); element-wise thresholds $\\theta^{r}$ and $\\theta^{t}$\\;}\n\\Output{boolean compatibility status $comp$\\;}\n\\BlankLine\n$\\boldsymbol{T}_{a}\\leftarrow{[{vec(\\boldsymbol{R}^*_{a})}^{\\top},{\\boldsymbol{t}^{*}_{a}}^{\\top}]}^{\\top}$, $\\boldsymbol{T}_{b}\\leftarrow{[{vec(\\boldsymbol{R}^*_{b})}^{\\top},{\\boldsymbol{t}^{*}_{b}}^{\\top}]}^{\\top}$, $\\boldsymbol{\\theta}\\leftarrow{[\\underbrace{\\theta^r,\\dots,\\theta^r}_{9\\times\\theta^r}, \\underbrace{\\theta^t,\\dots,\\theta^t}_{3\\times\\theta^t}]}^{\\top}$, and $comp\\leftarrow 0$\\;\n\\For{$rep=1:12$}{\n\\If{$\\left|\\boldsymbol{T}_{a}(rep)-\\boldsymbol{T}_{b}(rep)\\right|>\\boldsymbol{\\theta}(rep)$}{\n\\textbf{break}}\n\\If{$rep=12$}{\n$comp\\leftarrow 1$\\;\n}\n}\n\\Return boolean compatibility status $comp$\\;\n\\end{algorithm}\n\n\\subsection{Second Layer: Two-Point Sampling and Consensus Maximization based on Stratified Element-wise Compatibility}\\label{second-layer}\n\nIn the second layer of DANIEL, with the correspondence $n$ selected in the first layer, we only need to sample two more points to construct a three-point minimal subset for the estimation of the transformation, which requires much less computational cost in comparison with the traditional three-point RANSAC since: (i) the correspondence set is reduced from the original set $\\mathcal{N}$ to set $\\mathcal{C}$ now, and (ii) the sampling dimension is reduced from 3 to 2 (three-point to two-point). Subsequently, rigidity constraint~\\eqref{rigidity} can be applied once again for `raw' outlier removal. When we select a pair of random correspondences $\\{a,b\\}\\subset\\mathcal{C}$, knowing that both $(a,n)$ and $(b,n)$ have already satisfied rigidity in the first layer, we can further test $(a,b)$ with~\\eqref{rigidity}. If satisfied, the three-point set $\\{n,a,b\\}$ is entirely rigid (each line between any two points has fixed length after the transformation) and can be used to solve the transformation model $(\\boldsymbol{R}^*,\\boldsymbol{t}^*)$ minimally using Horn's triad-based method~\\cite{horn1987closed}. (See \\textbf{lines 12-14} in Algorithm~\\ref{DANIEL-algo})\n\nMore importantly, we depart from the traditional sampling-and-consensus-building technique as applied in RANSAC, and introduce a novel compatibility-based consensus maximization paradigm, which is partially inspired by~\\cite{sun2021ransic}.\n\nDifferent from the standard procedure of first making an estimate with a minimal subset and then establishing the consensus set in every single iteration in traditional RANSAC, when we compute a minimal model ($\\boldsymbol{R}^*,\\boldsymbol{t}^*$) with a random subset, we store its parameters and check the \\textit{mutual compatibility} between this model and every single existing model already in storage. (Intuitively, compatibility checking can be operated by measuring the mathematical error between the two models and then judging whether it is small enough.) Only when we have found two models that fulfill the mutual compatibility could we build the consensus set with these two models by computing the residual errors w.r.t. all the correspondences in set $\\mathcal{C}$. This strategy is inspired by the fact that models estimated with different true inliers should still be sufficiently similar (not equivalent due to the presence of noise), and it can tremendously curtail the time cost of the repeated but `redundant' consensus building in every iteration of sampling. (See \\textbf{lines 15-17} in Algorithm~\\ref{DANIEL-algo})\n\nHowever, when the outlier ratio is high, we may have to sample a large number of random subsets and store many minimal models so as to achieve two all-inlier subsets to satisfy this compatibility condition. Consequently, we further propose a series of \\textit{stratified element-wise compatibility tests} in order to ensure high efficiency in our compatibility checking process.\n\n\\subsubsection{Stratified Element-wise Model Compatibility}\n\nInstead of the holistic checking of compatibility, our idea is that two models must be mutually compatible as long as all their respective parameters are compatible.\nThe parameters of the model involved in the registration problem include 9 scalars from the rotation matrix $\\boldsymbol{R}\\in SO(3)$ plus 3 scalars from the translation vector $\\boldsymbol{t}\\in\\mathbb{R}^3$, which can be represented as $\\boldsymbol{R}=\\left[\\begin{array}{ccc} r_1, &r_4, & r_7 \\\\  r_2, &r_5, & r_8 \\\\  r_3, &r_6, & r_9  \\end{array}\\right]$ and $\\boldsymbol{t}=[t_1,t_2,t_3]^{\\top}$, respectively. \n\nFor a certain translation vector $\\boldsymbol{t}^*$ estimated from a minimal subset, it can be rewritten as:\n\\begin{equation}\n\\boldsymbol{t}^*=\\boldsymbol{t}_{gt}+\\boldsymbol{\\epsilon}^{\\boldsymbol{t}},\n\\end{equation}\nwhere $\\boldsymbol{t}_{gt}$ represents the ground-truth translation and $\\boldsymbol{\\epsilon}^{\\boldsymbol{t}}\\in\\mathbb{R}^3$ denotes the noise measurement on translation. Now assume that we have two such minimal translations, say $\\boldsymbol{t}^*_{a}$ and $\\boldsymbol{t}^*_{b}$. Then, we can derive the following compatibility condition if they are both inliers:\n\\begin{equation}\\label{t-comp}\n\\left\\|\\boldsymbol{t}^*_{a}-\\boldsymbol{t}^*_{b}\\right\\|=\\left\\|\\boldsymbol{\\epsilon}^{\\boldsymbol{t}_a}-\\boldsymbol{\\epsilon}^{\\boldsymbol{t}_b}\\right\\| \\leq 2\\xi^{\\boldsymbol{t}},\n\\end{equation}\nwhere $\\xi^{\\boldsymbol{t}}\\geq\\|\\boldsymbol{\\epsilon}^{\\boldsymbol{t}}\\|$ is the noise upper-bound for translation, and we usually set $\\xi^{\\boldsymbol{t}}=5\\sigma$. Moreover, since the three elements in translation are completely independent, compatibility~\\eqref{t-comp} can be easily decoupled into three errors in L1 norm: $|t^*_{a1}-t^*_{b1}|$, $|t^*_{a2}-t^*_{b2}|$ and $|t^*_{a3}-t^*_{b3}|$. We can then set a scalar threshold for each of them in order to fulfil condition~\\eqref{t-comp} such that\n\\begin{equation}\\label{t-comp-each}\n\\theta^{t}=\\left|{t}^*_{ap}-{t}^*_{bp}\\right| \\leq \\frac{2\\mu}{\\sqrt{3}}\\xi^{\\boldsymbol{t}},\\,(\\forall p\\in\\{1,2,3\\})\n\\end{equation}\n\n\n\\clearpage\n\\begin{algorithm*}[h]\n\\caption{DANIEL}\n\\label{DANIEL-algo}\n\\SetKwInOut{Input}{\\textbf{Input}}\n\\SetKwInOut{Output}{\\textbf{Output}}\n\\Input{correspondences $\\mathcal{P}=\\{(\\mathbf{x}_i,\\mathbf{y}_i)\\}_{i=1}^N$; noise $\\sigma$; minimum inlier number $I_{min}=\\max(5,0.01N)$\\;}\n\\Output{optimal $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$; inlier set $\\mathcal{N}^{\\star}$ \\;}\n\\BlankLine\n$\\mathcal{N}\\leftarrow [1,2,\\dots,N]$, $samp1\\leftarrow 0$, $\\mathcal{N}_{best}\\leftarrow\\emptyset$, $bestSize_1\\leftarrow0$, $maxItr_1\\leftarrow 459$, and get $\\theta^r$ and $\\theta^t$ with~\\eqref{r-comp-each} and~\\eqref{t-comp-each}\\;\n\\While{$samp_1\\leq maxItr_1$}{\nSelect a random $n\\in\\mathcal{N}$, $\\mathcal{C}\\leftarrow \\emptyset$, and $samp_1\\leftarrow samp_1+1$\\;\n\\For{\\textbf{\\textup{all}} $i\\in\\mathcal{N}$ and $i\\neq n$}{\n\\If{correspondence pair $(i,n)$ can satisfy condition~\\eqref{rigidity}}{\n$\\mathcal{C}=\\mathcal{C}\\cup{\\{i\\}}$\\;\n}\n}\n\\If{$|\\mathcal{C}|\\geq I_{min}$}{\n$bestSize_2\\leftarrow0$, $samp_2\\leftarrow 0$ and $\\mathcal{S}_{best}\\leftarrow\\emptyset$\\;\n\\While{$samp_2\\leq maxItr_2$}{\nSelect a random subset $\\{a,b\\}\\subset\\mathcal{C}$, and $samp_2\\leftarrow samp_2+1$\\;\n\\If{correspondence pair $(a,b)$ can satisfy condition~\\eqref{rigidity}}{\nEstimate $(\\boldsymbol{R}^{*},\\boldsymbol{t}^{*})$ minimally using Horn's method~\\cite{horn1987closed}\\;\n$\\mathcal{R}\\leftarrow\\mathcal{R}\\cup\\{\\boldsymbol{R}^{*}\\}$, and $\\mathcal{T}\\leftarrow\\mathcal{T}\\cup\\{\\boldsymbol{t}^{*}\\}$\\;\n\\For{$z=1:\\left(|\\mathcal{R}|-1\\right)$}{\n$comp\\leftarrow$\\textit{CompatibilityStaircase}($\\mathcal{R}_{(z)}$, $\\mathcal{T}_{(z)}$, $\\boldsymbol{R}^{*}$, $\\boldsymbol{t}^{*}$, $\\theta^r$, $\\theta^t$)\\;\n\\If{$comp=1$}{\n$\\boldsymbol{R}^{\\circ}\\leftarrow$\\textit{AverageRotPara}($\\mathcal{R}_{(z)}$, $\\boldsymbol{R}^{*}$), $\\boldsymbol{t}^{\\circ}\\leftarrow$\\textit{AverageTranPara}($\\mathcal{T}_{(z)}$, $\\boldsymbol{t}^{*}$), and $\\mathcal{S}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set of the averaged model: $\\boldsymbol{R}^{\\circ}$ and $\\boldsymbol{t}^{\\circ}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $j\\in\\mathcal{C}$}{\n\\If{$\\|\\boldsymbol{R}^{\\circ}\\boldsymbol{x}_j+\\boldsymbol{t}^{\\circ}-\\boldsymbol{y}_j|\\leq 5\\sigma$}{\n$\\mathcal{S}\\leftarrow\\mathcal{S}\\cup\\{j\\}$\\;\n}\n}\n\\If{$|\\mathcal{S}|\\geq bestSize_2$}{\n$\\mathcal{S}_{best}\\leftarrow\\mathcal{S}$, $bestSize_2\\leftarrow |\\mathcal{S}|$, and update $maxItr_2$ with~\\eqref{max-itr-2}\\;\n}\n}\n}\n}\n}\n} \n\\If{$bestSize_2\\geq bestSize_1$}{\nSolve $(\\boldsymbol{R}^{\\dagger}, \\boldsymbol{t}^{\\dagger})$ non-minimally with $\\mathcal{S}_{best}$ using SVD~\\cite{arun1987least}, and $\\mathcal{N}_{best}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set of current $\\boldsymbol{R}^{\\dagger}$ and $\\boldsymbol{t}^{\\dagger}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $j\\in\\mathcal{N}$}{\n\\If{$\\|\\boldsymbol{R}^{\\dagger}\\boldsymbol{x}_j+\\boldsymbol{t}^{\\dagger}-\\boldsymbol{y}_j\\|\\leq 5\\sigma$}{\n$\\mathcal{N}_{best}\\leftarrow\\mathcal{N}_{best}\\cup\\{j\\}$\\;\n}\n}\n$bestSize_1\\leftarrow |\\mathcal{N}_{best}|$, and update $maxItr_1$ with~\\eqref{max-itr-1}\\;\n}\n}\nSolve $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ non-minimally with $\\mathcal{N}_{best}$ using SVD, and $\\mathcal{N}^{\\star}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set with current $\\boldsymbol{R}^{\\star}$ and $\\boldsymbol{t}^{\\star}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $k\\in\\mathcal{N}$}{\n\\If{$\\|\\boldsymbol{R}^{\\star}\\boldsymbol{x}_k+\\boldsymbol{t}^{\\star}-\\boldsymbol{y}_k\\|\\leq 5\\sigma$}{\n$\\mathcal{N}^{\\star}\\leftarrow\\mathcal{N}^{\\star}\\cup\\{j\\}$\\;\n}\n}\nSolve $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ non-minimally with $\\mathcal{N}^{\\star}$ using SVD\\;\n\\Return $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ and $\\mathcal{N}^{\\star}$\\;\n\\end{algorithm*}\n\\clearpage\n\n\\noindent where $\\mu \\geq1$. Note that if $\\mu=1$,  \\eqref{t-comp-each} $\\Rightarrow$ \\eqref{t-comp}. But we want a slightly more lenient threshold, so $\\mu$ can be set to slightly larger than 1. In practice, we can empirically set $\\mu=1.2$. \n\n\nFor rotation, the widely-used geodesic distance~\\cite{hartley2013rotation} is represented by angle and requires relatively complex computation including $\\arccos$ and $trace$, so it is difficult to be rewritten as element-wise. But according to~\\cite{hartley2013rotation}, the chordal distance, having natural relation to the geodesic distance, enables us to derive a series of element-wise compatibility conditions.\n\nAssume that we have two rotations $\\boldsymbol{R}^*_a$ and $\\boldsymbol{R}^*_b$ estimated from two minimal subsets. $d_{\\text{chord}}$ denotes the chordal distance between them that can be written as:\n\\begin{equation}\\label{chordal}\nd_{\\text{chord}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)=\\left\\|\\boldsymbol{R}^*_a-\\boldsymbol{R}^*_b\\right\\|_{F}=2\\sqrt{2}\\sin (\\frac{d_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)}{2}),\n\\end{equation}\nwhere $\\|\\cdot\\|_F$ denotes the Frobenius norm and $d_\\text{geo}(\\cdot,\\cdot)$ means the geodesic distance~\\cite{hartley2013rotation} such that\n\\begin{equation}\\label{geodesic}\nd_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)=\\left|\\arccos\\left(\\frac{trace({\\boldsymbol{R}_a^*}^{\\top}\\boldsymbol{R}^*_b)-1}{2}\\right)\\right|.\n\\end{equation}\nChordal distance~\\eqref{chordal} intuitively represents the square root of the sum of sqaures of all the 9 parameters in matrix $\\boldsymbol{R}^*_a-\\boldsymbol{R}^*_b$. Now our goal is to set a proper threshold for each of them as in~\\eqref{t-comp-each}. Similarly, a minimal rotation $\\boldsymbol{R}^*$ can be described by:\n\\begin{equation}\n\\boldsymbol{R}^*=\\boldsymbol{R}_{gt}\\cdot Exp\\left([\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}]_{\\times}\\right),\n\\end{equation}\nwhere $\\boldsymbol{R}_{gt}$ is the ground-truth rotation, $\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}\\in\\mathbb{R}^3$ is the noise measurement on rotation, $[\\,\\cdot\\,]_{\\times}$ denotes the skew-symmetric matrix for the size-3 vector, and $Exp(\\,\\cdot\\,)$ is the exponential map of matrix. Then we let $\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}=\\mathit{S}\\cdot\\boldsymbol{e}$ where $\\boldsymbol{e}$ is a random vector of unit length and $\\mathit{S}$ denotes its scale, and according to~\\cite{barfoot2017state}, the geodesic error between $\\boldsymbol{R}^*$ and $\\boldsymbol{I}_3$ can be written as:\n\\begin{equation}\nd_{\\text{geo}}(\\boldsymbol{R}^*,\\boldsymbol{I}_3)=\\mathit{S},\n\\end{equation}\nand based on the triangular inequality of geodesic distances, we can then have that\n\\begin{equation}\nd_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b) \\leq d_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{I}_3)+d_{\\text{geo}}(\\boldsymbol{R}_b^*,\\boldsymbol{I}_3)=2\\mathit{S},\n\\end{equation}\nso that the chordal distance between $\\boldsymbol{R}^*_a$ and $\\boldsymbol{R}^*_b$ satisfies:\n\\begin{equation}\nd_{\\text{chord}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)\\leq 2\\sqrt{2}\\sin \\left( \\mathit{S} \\right).\n\\end{equation}\n\nAfterwards, the threshold for the L1-norm distance of each of the 9 element in rotation should be:\n\\begin{equation}\\label{r-comp-each}\n\\theta^{r}=\\left|r^*_{aq}-r^*_{bq}\\right|\\leq \\mu\\cdot\\frac{2\\sqrt{2}\\sin \\left( \\mathit{S} \\right)}{3},\\,(\\forall q\\in\\{1,2,\\dots,9\\})\n\\end{equation}\nwhere $\\mu\\geq1$ and we can choose $\\mu=1.2$ for practical use, similar to~\\eqref{t-comp-each}. Note that in practice, if the maximum distances (diameters) of the point cloud in the 3 axes (X,Y and Z) are $D_X$, $D_Y$ and $D_Z$ and their mean value is $\\widetilde{D}$, we can provide an empirical choice for $\\mathit{S}$ such that\n\\begin{equation}\n\\mathit{S}=10\\frac{\\sigma}{\\widetilde{D}}.\n\\end{equation}\n\nUp to now, we can render our fast compatibility checking method that we name \\textit{CompatibilityStaircase} because it is operated like a staircase, only when the first condition is satisfied could we proceed to the second condition, as demonstrated in Algorithm~\\ref{Staircase}. (Note that we use $\\boldsymbol{V}(y)$ to denote the $y_{th}$ entry of the vector $\\boldsymbol{V}$.)\n\n\\textbf{Description of Algorithm~\\ref{Staircase}:} With two minimal models ($\\boldsymbol{R}^*_{a},\\boldsymbol{t}^*_{a}$) and ($\\boldsymbol{R}^*_{b},\\boldsymbol{t}^*_{b}$) estimated from two minimal subsets, we vectorize $\\boldsymbol{R}^*_{a}$ (and $\\boldsymbol{R}^*_{b}$) in column-wise and stack it with $\\boldsymbol{t}^*_{a}$ (and $\\boldsymbol{t}^*_{b}$) to form the size-12 vector $\\boldsymbol{T}_{a}$ (and $\\boldsymbol{T}_{b}$) (\\textbf{line 1}). Subsequently, we check the element-wise compatibility for the 12 pairs of elements based on condition~\\eqref{t-comp-each} and~\\eqref{r-comp-each} in sequence. If the $i_{th}$ element pair is not mutually compatible, then the rest $12-i$ element pairs are no longer required to be checked. This operation could be easily realized by \\textbf{lines 2-9}, where `$comp=1$' defines that the two models are compatible with each other whereas `$comp=1$' symbolizes their mutual incompatibility. Note that the operations mainly are subtracting, getting absolute values, and boolean conditions, which are all rather fast to implement.\n\n\n\\subsubsection{Parameter Averaging for Consensus Building}\n\nOnce two mutually compatible models (subsets) are found, implying that two different subsets are approximately pointing to the same model, it is reasonable and necessary to establish and evaluate the consensus set over these two models. Note that in traditional RANSAC, minimal models may be easily affected by noise, hence probably deviating from the ground-truth model to a great extent. But fortunately, since we have two models now, averaging them could reduce the influence of noise. Here, we prefer parameter averaging to applying the non-minimal solvers (e.g. SVD~\\cite{arun1987least}) because: (i) the former is more time-efficient, and (ii) the advantage of non-minimal solvers can be hardly shown here because the correspondence number is relatively small (no greater than 6). \n\nFor translation, we directly adopt the mean vector:\n\\begin{equation}\\label{t-average}\n\\boldsymbol{t}^{\\circ}=\\frac{\\boldsymbol{t}_a+\\boldsymbol{t}_b}{2},\n\\end{equation}\nwhile for rotation, we adopt the geodesic L2-mean~\\cite{hartley2013rotation} such that\n\\begin{equation}\\label{R-average}\n\\boldsymbol{R}^{\\circ}=\\boldsymbol{R}_a\\cdot Exp\\left(\\frac{log(\\boldsymbol{R}_a^{\\top}\\boldsymbol{R}_b)}{2}\\right),\n\\end{equation}\nwhere $log(\\,\\cdot\\,)$ denotes the logarithm of matrix. Here, we introduce functions: \\textit{AverageRotPara}($\\boldsymbol{R}_a,\\boldsymbol{R}_b$), \\textit{AverageTranPara}($\\boldsymbol{t}_a,\\boldsymbol{t}_b$), to represent the parameter averaging operations in~\\eqref{R-average} and~\\eqref{t-average}, respectively.\n\nSubsequently, we can compute the residual errors of the correspondences in set $\\mathcal{C}$ using $\\boldsymbol{R}^{\\circ}$ and $\\boldsymbol{t}^{\\circ}$ and then build the consensus set. (See \\textbf{lines 18-29} in Algorithm~\\ref{DANIEL-algo})\n\n\n\\begin{figure*}[t]\n\\centering\n\n\\begin{tabular}{cc}\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-demo-bunny-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-time-bunny-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-R-error-bunny-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-t-error-bunny-eps-converted-to.pdf}\n\\end{minipage} \n\\end{tabular}\n\n\\caption{Standard benchmarking results in boxplot. We show the rotation and translation estimation accuracy as well as the runtime of different solvers over the `\\textit{bunny}' dataset~\\cite{curless1996volumetric} w.r.t. increasing outlier ratios from 20\\% up to 99\\%. The top-left image exemplifies the correspondences with 95\\% outliers in our experimental environment, where green lines denote the inliers while red lines denote the outliers.}\n\\label{Benchmarking}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Probabilistic Computation of Maximum Iteration Numbers}\\label{probability}\n\nThere exist two maximum iteration numbers involved in DANIEL, one for the one-point sampling in the first layer and the other for the two-point sampling in the second layer. \n\nIn the first layer, according to~\\cite{fischler1981random,chum2003locally}, given the size of the minimal subset $A_1=1$ (since we use one-point sampling), the so-far-the-best consensus set $\\mathcal{N}_{best}$, and the probability of sampling at least one all-inlier subset $P_1=0.99$ (sufficiently close to 1), we can compute the expected maximum number of (random sampling) iteration such that\n\\begin{equation}\\label{max-itr-1}\nmaxItr_1\\geq\\frac{\\log{(1-P_1)}}{\\log{\\left(1-{\\left(\\frac{|\\mathcal{N}_{best}|}{N}\\right)}^{A_1}\\right)}}.\n\\end{equation}\n\\noindent{At} the beginning of the algorithm, we are required to preset a minimum inlier number $I_{min}$ (usually we set $I_{min}=0.01N$ in practice), so replacing $|\\mathcal{N}_{best}|$ with $I_{min}$ in~\\eqref{max-itr-1}, we can have $maxItr_1=459$. During the sampling process, whenever a new consensus set is obtained, we update $maxItr_1$ with~\\eqref{max-itr-1}. (See \\textbf{line 42} in Algorithm~\\ref{DANIEL-algo})\n\nIn the second layer, we set $A_2=2$ (two-point sampling) and require to sample at least two all-inlier minimal subsets, so we can set $P_2=0.995$, and then compute $maxItr_2$ as\n\\begin{equation}\\label{max-itr-2}\nmaxItr_2\\geq2\\cdot\\frac{\\log{(1-P_2)}}{\\log{\\left(1-{\\left(\\frac{I_{min}}{|\\mathcal{C}|}\\right)}^{A_2}\\right)}}.\n\\end{equation}\nIn this case, the probability of sampling two all-inlier subsets to fulfill the mutual compatibility should be $0.995^2\\approx0.99$. Similarly, during the sampling process, when we construct a new consensus set after the compatibility of two subsets, we can update $maxItr_2$ with~\\eqref{max-itr-2}. (See \\textbf{lines 27} in Algorithm~\\ref{DANIEL-algo})\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cccc|ccc}\n\n\\quad & \\,\\, & \\footnotesize{FPFH Correspondences} & \\footnotesize{Registration by DANIEL} & \\quad & \\,\\, & \\footnotesize{FPFH Correspondences} & \\footnotesize{Registration by DANIEL} \\\\\n\\hline \n&&&&&&&\n\\\\\n\\rotatebox{90}{\\scriptsize{bunny, $N=1401$, 92.78\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{bunny-cor-eps-converted-to.pdf}\n\\end{minipage}\n& \n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{bunny-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{armadillo, $N=876$, 89.81\\%}}\\,\n& &\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{armadillo-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.18\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{armadillo-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{dragon, $N=1208$, 89.97\\%\\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{dragon-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{dragon-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{cheff, $N=1500$, 93.09\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{cheff-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.205\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{cheff-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{chicken, $N=1500$, 91.73\\%\\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chicken-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.205\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chicken-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{rhino, $N=1500$, 93.88\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{rhino-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{rhino-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{parasauro, $N=1500$, 92.98\\% \\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{parasauro-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{parasauro-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{T-rex, $N=1500$, 91.95\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{trex-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{trex-res-eps-converted-to.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\n\\caption{Qualitative registration results (boxplot) on multiple real point cloud datasets using our solver DANIEL. The first column shows the correspondences matched by FPFH~\\cite{rusu2009fast} where green lines denote the inliers while red lines denote the outliers. The second column displays the registration (projection) result using the transformation estimated by DANIEL.}\n\\label{qualit-partial}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Main Algorithm}\n\nThe pseudocode of the main algorithm of DANIEL is provided in Algorithm~\\ref{DANIEL-algo}. \n\n\\textbf{Description of Algorithm~\\ref{DANIEL-algo}:} After the initialization setup, we start the first layer of one-point random sampling with rigidity examining, as described in Section~\\ref{first-layer}. If the size of set $\\mathcal{C}$ exceeds $I_{min}$, meaning that $n$ is likely to be an inlier, we then move on to the second layer of two-point random sampling, as described in Section~\\ref{second-layer}. This layer has two sequential steps: (i) continuous sampling and model estimating with mutual compatibility checking, and (ii) parameter averaging to build the consensus set after a pair of compatible models are detected. In both sampling layers, we update the best consensus set (replacing the smaller consensus set with the larger one) as well as the maximum iteration numbers (according to Section~\\ref{probability}) iteratively. Note that the consensus updating and maximizing procedure in the second layer is conducted within set $\\mathcal{C}$, while in the first layer it uses the complete correspondence set $\\mathcal{N}$. Eventually, when the first layer converges\\footnote[1]{Here, convergence means that the actual iteration number reaches the current maximum iteration number computed with the best consensus set so far using formulation~\\eqref{max-itr-1} or~\\eqref{max-itr-2}. It is the stop condition of random sampling.}, the best consensus set can be applied to obtain the final inlier set and to compute the optimal transformation for this registration problem.\n\n\\textbf{`1+2$<$3'? Why is DANIEL Faster?} Our DANIEL uses a one-point sampling layer plus an inner two-point sampling layer, while traditional RANSAC adopts one single three-point sampling layer. Though it seems that 1+2 should be equal to 3, DANIEL can be in fact up to over 10 thousand times faster than RANSAC. The main reasons are given as follows.\n\nFirst, in the RANSAC paradigm, the theoretical maximum iteration number of finding the best consensus set should be\n\\begin{equation}\\label{max-itr}\nmaxItr^{\\star}=\\frac{\\log{(1-0.99)}}{\\log{\\left(1-{\\left(\\frac{N_{inlier}}{N}\\right)}^{A}\\right)}},\n\\end{equation}\nwhere $N_{inlier}$ is the true inlier number and $A$ is the subset size. The function of the first layer is two-fold: (i) to reduce $N$, which exponentially decrease $maxItr^{\\star}$, and (ii) to lower $A$ (from 3 to 2), which also exponentially decrease $maxItr^{\\star}$. And when we reach the second layer, where both $N$ and $A$ are reduced, the maximum iteration number has enormously declined. Besides, the first layer only involves the computation of norms and the judging of boolean conditions, so its own computational cost is fairly low. Using such a `cheap' operation in exchange for the exponential decrease of  iteration number is completely rewarding, and that is why DANIEL is much faster than RANSAC.\n\nSecond, another reason lies in our stratified compatibility checking method. Note that in each iteration, we replace the traditional repeated consensus building operation with the compatibility checking, where the latter also only consists of norm computing and boolean conditions, so the efficiency can be greatly enhanced. Generally, the larger the problem size $N$ is, the more apparent the speed superiority of DANIEL will be. In experiments, we further compare the runtime of DANIEL against that of RANSAC (Fig.~\\ref{RANSAC-vs-DANIEL}).\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{ccc}\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-R-error-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-t-error-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-time-eps-converted-to.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\n\\caption{Quantitative registration results (boxplot) on multiple real point cloud datasets. The performances (estimation errors and runtime) of different solvers over different point clouds tested are shown.}\n\\label{Partial-reg}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\section{Experiments and Applications}\\label{Experiments}\n\nIn this section, we conduct multiple comprehensive experiments on the basis of various realistic datasets in order to fully evaluate the performance of DANIEL, in comparison with other state-of-the-art competitors. All the experiments are implemented in Matlab over a laptop with a 2.8Ghz CPU and a 16GB RAM without using any parallelism programming.\n\n\n\\subsection{Standard Benchmarking on Semi-Synthetic Data}\\label{sec-bench}\n\nWe first evaluate our DANIEL in the standard benchmarking experiment (the point cloud used is realistic while the outliers are artificially generated, thus called \\textit{semi-synthetic}), in comparison with the existing state-of-the-art solvers including: (i) three non-minimal solvers: FGR~\\cite{zhou2016fast}, GNC-TLS~\\cite{yang2020graduated} and ADAPT~\\cite{tzoumas2019outlier}, (ii) three RANSAC solvers: RANSAC~\\cite{fischler1981random}, LO-RANSAC~\\cite{chum2003locally} and FLO-RANSAC~\\cite{lebeda2012fixing} (also called LO$^+$-RANSAC), and (iii) the guaranteed outlier removal solver GORE~\\cite{bustos2017guaranteed} as well as its enhanced version GORE+RANSAC\\footnote[2]{Further using RANSAC to find the best consensus set after the guaranteed outlier removal of GORE.}. Note that the RANSAC solvers are all set with 10000 maximum iterations and 0.99 confidence, and the local optimization is set with 10 iterations. For all solvers, the inlier threshold is set to $\\xi=6\\sigma$. For quantitative evaluation of estimation errors, we use the geodesic error~\\eqref{geodesic} to represent rotation error in degrees such that\n\\begin{equation}\\label{geodesic}\nE_{\\text{rot}}(\\boldsymbol{R}_{gt}, \\boldsymbol{R}^{\\star})=\\left|\\arccos\\left(\\frac{trace({\\boldsymbol{R}_{gt}}^{\\top}\\boldsymbol{R}^{\\star})-1}{2}\\right)\\right|\\cdot\\frac{180}{\\pi}^{\\circ},\n\\end{equation}\nand use L2-norm to represent translation error in meters such that\n\\begin{equation}\\label{geodesic}\nE_{\\text{tran}}(\\boldsymbol{t}_{gt}, \\boldsymbol{t}^{\\star})=\\left\\|\\boldsymbol{t}_{gt}-\\boldsymbol{t}^{\\star}\\right\\|\\, m,\n\\end{equation}\nwhere $\\boldsymbol{R}_{gt}$ and $\\boldsymbol{t}_{gt}$ denotes the ground-truth transformation.\n\n\nWe adopt the `\\textit{bunny}' point cloud dataset from the Stanford 3D Scanning Repository~\\cite{curless1996volumetric}. This point cloud is downsampled to $N=1000$ and resized to be placed a $[-0.5,0.5]^3m$ box as our initial point set $\\mathcal{X}=\\{\\boldsymbol{x}_i\\}_{i=1}^{N}$. Then we transform $\\mathcal{X}$ with a random transformation such that $\\boldsymbol{R}\\in SO(3)$ and $\\boldsymbol{t}\\in\\mathbb{R}^3$ ($||\\boldsymbol{t}||\\leq 3$), and add random noise with $\\sigma=0.01m$ to the transformed point set $\\mathcal{Y}=\\{\\boldsymbol{y}_i\\}_{i=1}^{N}$. To create outliers simulating cluttered scenes, we artificially corrupt (replace) 20\\% to 99\\% of the points in $\\mathcal{Y}$ with completely random points inside a 3D sphere of radius 1. The boxplot benchmarking results are shown in  Fig.~\\ref{Benchmarking}, in which 50 Monte Carlo runs are implemented to obtain the result data.\n\nFrom Fig.~\\ref{Benchmarking}, we can clearly observe that non-minimal solvers (FGR, GNC-TLS and ADAPT) fail at 90-92\\% outliers, and the RANSAC solvers (RANSAC, LO-RANSAC and FLO-RANSAC) with 10000 maximum iterations break at over 95\\% outliers, meanwhile requiring very long computational time with high outlier ratios, whereas GORE, GORE+RANSAC and our DANIEL are robust against as many as 99\\% outliers. Furthermore, DANIEL is the fastest solver at all outlier ratios (from 20\\% to 99\\%) with (at least one of) the highest estimation accuracy, superior to all the other tested competitors in overall performance.\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=0.87\\linewidth]{RANSAC-DANIEL-time-eps-converted-to.pdf}\n\\end{minipage}\n\\caption{Comparison of mean runtime between DANIEL and RANSAC w.r.t. increasing outlier ratios.}\n\\label{RANSAC-vs-DANIEL}\n\\vspace{-1mm}\n\\end{figure}\n\n\\subsection{Runtime Comparison against RANSAC}\n\nSince that RANSAC has been a de-facto standard among all the robust solvers, we exclusively compare the runtime of our DANIEL with that of RANSAC, in order to more clearly observe DANIEL's performance in efficiency. Note that in this experiment, no maximum iteration number is imposed to RANSAC and we simply wait for it until convergence.\n\nWe also adopt the experimental setup in Section~\\ref{sec-bench} and demonstrate the mean runtime (based on 50 data points) of these two solvers w.r.t. increasing outlier ratios from 20\\% up to 95\\%. But when the outlier ratio is 99\\%, RANSAC would take over 7 hours per run, making it difficult to measure its mean runtime. Fortunately, we can reasonably deduce its runtime at 99\\% by computing the theoretical maximum iteration based on~\\eqref{max-itr}. Note that in each iteration of RANSAC, the operations are almost constant and fixed, that is, minimal sampling, model estimating and consensus building, so its runtime should be mainly dependent on its iteration number. For example, in our experiment, RANSAC's mean runtime at 95\\% is found to be 7.97 times greater than that at 90\\%, which fits well with the fact that the maximum iteration number required for 95\\% is 8 times greater than that required for 90\\% according to~\\eqref{max-itr}. As a result, since that the maximum iteration number of RANSAC at 99\\% outliers should be 125 greater than that at 95\\% which is about 206.37 seconds in average, the runtime of RANSAC at 99\\% can be computed as 206.37$\\times$125$\\approx$25,796 seconds.\n\nThus, according to Fig.~\\ref{RANSAC-vs-DANIEL}, we can see that DANIEL is 75 times faster than RANSAC at 90\\%, 357 times faster at 95\\%, and more than 10000 times faster at 99\\%, so that the efficiency superiority of DANIEL is extremely obvious.\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0.1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cc|ccccc}\n\n\\quad &\\,&\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL} \\\\\n\n\\hline\n\n&&\\footnotesize{$N=534$, 97.94\\%}& & \\,\\footnotesize{$133.649^{\\circ},2.206m,0.065s$}\\, &\\,\\footnotesize{$90.351^{\\circ},2.177m,15.391s$}\\, &\\,\\footnotesize{$\\textbf{0.807}^{\\circ},\\textbf{0.024}m,0.998s$}\\,&\\,\\footnotesize{$\\textbf{0.807}^{\\circ},\\textbf{0.024}m,\\textbf{0.259}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-01, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=458$, 95.41\\%}& & \\,\\footnotesize{$101.597^{\\circ},1.632m,0.103s$}\\, &\\,\\footnotesize{$171.168^{\\circ},2.401m,18.079s$}\\, &\\,\\footnotesize{$\\textbf{0.187}^{\\circ},\\textbf{0.003}m,0.827s$}\\,&\\,\\footnotesize{$\\textbf{0.187}^{\\circ},\\textbf{0.003}m,\\textbf{0.291}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-02, box}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=689$, 97.68\\%}& & \\,\\footnotesize{$92.381^{\\circ},0.591m,0.055s$}\\, &\\,\\footnotesize{$110.968^{\\circ},2.199m,15.839s$}\\, &\\,\\footnotesize{${0.198}^{\\circ},\\textbf{0.003}m,1.329s$}\\,&\\,\\footnotesize{$\\textbf{0.139}^{\\circ},\\textbf{0.003}m,\\textbf{0.103}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-03, cap}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=304$, 97.04\\%}& & \\,\\footnotesize{$150.055^{\\circ},1.566m,0.041s$}\\, &\\,\\footnotesize{$33.426^{\\circ},0.749m,12.562s$}\\, &\\,\\footnotesize{$\\textbf{1.969}^{\\circ},\\textbf{0.024}m,0.763s$}\\,&\\,\\footnotesize{$\\textbf{1.969}^{\\circ},\\textbf{0.024}m,\\textbf{0.118}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-04, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=773$, 97.41\\%}& & \\,\\footnotesize{$138.748^{\\circ},2.567m,0.132s$}\\, &\\,\\footnotesize{$152.764^{\\circ},2.566m,29.398s$}\\, &\\,\\footnotesize{$\\textbf{0.427}^{\\circ},\\textbf{0.011}m,1.321s$}\\,&\\,\\footnotesize{$\\textbf{0.427}^{\\circ},\\textbf{0.011}m,\\textbf{0.310}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-05, box}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\n\\end{tabular}\n\n\\caption{Object localization and pose estimation results on the RGB-D scenes dataset~\\cite{lai2011large}. The left-most column shows the raw FPFH correspondences where information regarding the correspondence number and outlier ratio is also provided, and the rest columns show the registration results (reprojecting the object back to the scene with the transformation estimated) using GNC-TLS, FLO-RANSAC, GORE+RANSAC and DANIEL, respectively. On top of the results of each solver, from left to right, we display the rotation error (in degrees), translation error (in meters) and runtime (in seconds), respectively. Best results are shown in \\textbf{bold} font.}\n\\label{obj-local1}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Reslistic Registration on Real Data}\n\nWe conduct realistic point cloud registration experiments for further comparative evaluation over more datasets, including the `\\textit{bunny}', `\\textit{armadillo}' and `\\textit{dragon}' from the Stanford 3D Scanning Repository~\\cite{curless1996volumetric}, as well as the `\\textit{cheff}', `\\textit{chicken}', `\\textit{rhino}', `\\textit{parasaurolophus}' and `\\textit{T-rex}' from Mian's dataset~\\cite{mian2006three,mian2010repeatability} (8 point cloud datasets in total). For each point cloud above, we first resize it to fit in a $[-50,50]^3m$ box, then manually divide its full scan into 10 random partial scans where each scan shares 30\\%-35\\% overlapping with the original full scan, generate random transformations ($\\boldsymbol{R}\\in SO(3)$ and $\\boldsymbol{t}\\in\\mathbb{R}^3$ where $||\\boldsymbol{t}||\\leq 100$) to transform these partial scans, and finally use FPFH~\\cite{rusu2009fast} (Matlab function: \\textit{extractFPFHFeatures}) to establish correspondences between each of the transformed partial scans and the initial full scan. When $N>1500$, we only preserve the first 1500 correspondences. Owing to serious partiality, the FPFH correspondences often contain huge amounts of outliers. Subsequently, these correspondences are fed to our DANIEL as well as the other solvers included in Section~\\ref{sec-bench} to robustly solve the registration problems (10 for each point cloud, hence 80 problems in total), where the noise is set to $\\sigma=0.1m$.\n\nThe qualitative registration examples (one example for each point cloud) are shown in Fig.~\\ref{qualit-partial}, where the average correspondence number and outlier ratio of each point cloud are labeled on the left side of the examples. In addition, the quantitative results in boxplot are displayed in Fig.~\\ref{Partial-reg}. We can find that our DANIEL (i) is the most robust and accurate solver, never yielding any single wrong estimate throughout the tests, and (ii) is fast in practice. Note that though the non-minimal solvers seem to show similar or even better (e.g. GNC-TLS) efficiency compared to DANIEL, they are prone to generate wrong results, unable to keep stably robust in all tests. Besides, the RANSAC solvers require relatively long runtime, and also occasionally return failure cases, while single-handed GORE has poor accuracy despite its promising runtime. GORE+RANSAC is the only solver that can have comparable robustness and accuracy with our DANIEL, but it is apparently slower than DANIEL. So overall, DANIEL manifests the best performance.\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0.1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cc|ccccc}\n\n\\quad &\\,&\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL} \\\\\n\n\\hline\n\n&&\\footnotesize{$N=186$, 95.16\\%}& & \\,\\footnotesize{$131.278^{\\circ},2.784m,0.032s$}\\, &\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,6.024s$}\\, &\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,0.604s$}\\,&\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,\\textbf{0.057}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-06, can}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=214$, 95.33\\%}& & \\,\\footnotesize{$104.921^{\\circ},1.537m,0.031s$}\\, &\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,6.814s$}\\, &\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,0.744s$}\\,&\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,\\textbf{0.059}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-07, cap}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=326$, 96.32\\%}& & \\,\\footnotesize{$102.869^{\\circ},1.769m,0.045s$}\\, &\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,12.510s$}\\, &\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,0.784s$}\\,&\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,\\textbf{0.093}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-11, can}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=339$, 97.05\\%}& & \\,\\footnotesize{$140.652^{\\circ},2.390m,0.038s$}\\, &\\,\\footnotesize{$121.956^{\\circ},4.738m,18.808s$}\\, &\\,\\footnotesize{$\\textbf{0.582}^{\\circ},\\textbf{0.014}m,0.753s$}\\,&\\,\\footnotesize{$\\textbf{0.582}^{\\circ},\\textbf{0.014}m,\\textbf{0.119}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-12, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=382$, 96.07\\%}& & \\,\\footnotesize{$103.388^{\\circ},1.950m,0.047s$}\\, &\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,17.814s$}\\, &\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,0.910s$}\\,&\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,\\textbf{0.103}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-14, bowl}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\\end{tabular}\n\n\\caption{Supplementary object localization and pose estimation results on the RGB-D scenes dataset~\\cite{lai2011large}. The explicit organization of the results is similar to Fig.~\\ref{obj-local1}. Best results are shown in \\textbf{bold} font.}\n\\label{obj-local2}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Real Application: Object Localization}\n\nWe evaluate DANIEL in the real-world 3D object localization and pose estimation application. We adopt the large-scale RGB-D Scenes dataset~\\cite{lai2011large}, where 3D object models (point clouds) in different shapes and different RGB-D scene scans are provided. For each pair of object and scene, we extract the point cloud of the object from the scene according to the ground-truth labels, and then change the pose of this object with a random transformation ($||\\boldsymbol{t}||\\leq 3$). Afterwards, we downsample the object and the scene and apply FPFH to build correspondences between the transformed object and the entire scene, which would easily trigger high outlier ratios. Then, we use one non-minimal solver: GNC-TLS, one RANSAC solver: FLO-RANSAC (with 10000 maximum iterations), one guaranteed outlier removal solver: GORE+RANSAC, and our solver DANIEL to estimate the relative pose (rigid transformation) between the object and scene, where noise is constantly set to $\\sigma=0.001m$.\n\nWe conduct 10 tests over 10 scenes with 5 objects, including the \\textit{coffee mug, cereal box, cap, soda can} and \\textit{bowl}, where all these tests are with extreme outlier ratios (ranging from 95.16\\% to 97.94\\%). We report the qualitative and quantitative object localization results in Fig.~\\ref{obj-local1} and~\\ref{obj-local2}, where each figure shows the results of 5 tests in 5 scenes (with possibly different objects). We can see that in such a high-outlier environment, GNC-TLS fails to render reasonable results in all tests, while FLO-RANSAC, mostly running in tens of seconds, could only succeed in 40\\% tests. GORE+RANSAC and our DANIEL are the two solvers that can successfully localize all the objects (to be specific, estimate the correct transformation and reproject the object back to the scene) in all the 10 tests. More importantly, DANIEL runs only in tens or hundreds of milliseconds and is significantly faster than GORE+RANSAC, showing the most outstanding performance.\n\n\n\\subsection{Scan Matching}\n\nScan matching, or called scene stitching, is another crucial application of point cloud registration, underlying the 3D reconstruction and SLAM technologies. We test DANIEL for scan matching based on the Microsoft 7-scenes dataset~\\cite{shotton2013scene} that provides realistic RGB images with associated depth images. We deliberately select 10 pairs of scans with overlapping scenes from the \\textit{red kitchen, office, chess, stairs} and \\textit{heads}. Since FPFH may generate too many outliers on RGB-D data, \n\n\\clearpage\n\n\\begin{figure*}[h]\n\\centering\n\\setlength\\tabcolsep{0pt}\n\\addtolength{\\tabcolsep}{-0.2pt}\n\\begin{tabular}{c|cc|c|c|c|c}\n\\quad &\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL}\n\\\\\n\\hline\n\n & \\footnotesize{$N=1840$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.245s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.308,8.960s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.283},11.727s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.299,\\textbf{0.182}s$}\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-0-59-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=1273$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.118s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.352,34.662s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.343,8.657s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.335},\\textbf{1.190}s$}\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-55-110-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=900$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.093s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.438,12.606s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.335,7.440s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.314},\\textbf{0.177}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-100-140-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=1984$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.314,0.291s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.516,8.044s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.323,74.158s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.301},\\textbf{0.272}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-290-340-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=1823$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.199s$} &\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,49.098s$}&\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,63.692s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.278},\\textbf{0.781}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-450-520-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=2002$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.288,0.293s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.502,19.645s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.292,70.343s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.287},\\textbf{0.278}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{chess}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chess-0-70-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=1822$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.262s$} &\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,48.811$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.326,12.171s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.324},\\textbf{0.784}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{office}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{office-0-140-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=985$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.112s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.373,26.628$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.360,9.035s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.334},\\textbf{0.249}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{office}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{office-910-980-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=244$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.036s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.430},0.706$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.330,0.418s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.430},\\textbf{0.039}s$}\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{stair}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{stair-0-70-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=2001$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.224},0.294s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.262,5.414$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.225,49.971s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,{0.227},\\textbf{0.269}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{heads}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{heads-0-50-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\end{tabular}\n\n\\caption{Scan matching results on the Microsoft 7-scenes dataset~\\cite{shotton2013scene}. The left-most column demonstrates the putative correspondences matched by SURF in cyan lines (already converted into the 3D space using depth and calibration information), and the rest columns show: (i) the inliers found and (ii) the qualitative scene stitching results, using FLO-RANSAC, GNC-TLS, GORE+RANSAC and DANIEL, respectively. On top of the results of each solver, from left to right, we display the stitching status (either \\textcolor[rgb]{1,0,0}{Fail} or \\textcolor[rgb]{0,0.5,0}{Succeed}), the RMSE and runtime (in seconds), respectively. Note that stitching with visibly distinguishable error should be regarded as \\textcolor[rgb]{1,0,0}{Fail}, and when the stitching is failed, we no longer display the RMSE since it would become meaningless. Best results are shown in \\textbf{bold} font. Best viewed when zoomed-in.}\n\\label{scan-matching}\n\\end{figure*}\n\\clearpage\n\n\\noindent we use SURF~\\cite{bay2006surf} (Matlab function: \\textit{detectSURFFeatures}) to match 2D keypoint correspondences across the two RGB images and then convert them into 3D correspondences by using depth data and camera intrinsic parameters. We also apply the four solvers: GNC-TLS, FLO-RANSAC, GORE+RANSAC and DANIEL, for comparative evaluation.\n\nBoth qualitative and quantitative scan matching results are displayed in Fig.~\\ref{scan-matching}, where we show the raw correspondences matched by SURF, and the inliers found as well as the scan matching results by the respective robust solvers in qualitative results, and show the registration status (\\textcolor[rgb]{1,0,0}{Fail} means that the scenes are wrongly stitched and \\textcolor[rgb]{0,0.5,0}{Succeed} means that the stitching is successful), RMSE (Root Mean Square Error) and runtime in quantitative results. Though 2D keypoint matching is used, we can observe that the inliers are merely a small minority of the putative correspondences. The results show that GNC-TLS is the least robust solver, failing in 70\\% tests, and FLO-RANSAC and GORE-RANSAC both generate a few failure cases but their runtime are too slow for practical use. Our DANIEL remains highly robust in all tests and also most often has the lowest RMSE and the fastest speed, which further validates the promising practicality of DANIEL.\n\n\n\n\\section{Conclusion}\n\nIn this paper, a novel consensus maximization method for correspondence-based robust point cloud registration with high or even extreme outliers is proposed. We design a double-layered random sampling framework with the smart application of 3D rigidity constraint, in order to tremendously reduce the computational cost for sampling pure inlier subsets even in high-outlier registration problems. We further render a stratified element-wise compatibility checking technique during the sampling process, which intends to circumvent the repeated consensus building procedure so as to accelerate the algorithm convergence. These two contributions enable both the high robustness and the high time-efficiency of the resulting solver DANIEL.  Extensive experiments over diverse datasets validate that DANIEL is superior to other existing state-of-the-art methods. To be specific, DANIEL is robust against extremely high outlier ratios as many as 99\\%, is up to four orders of magnitude faster than RANSAC and also faster than other solvers in most cases, and proves to be effective in practical application problems such as object pose estimation and scan matching. The source code of DANIEL is available at: \\url{https:\/\/github.com\/LeiSun-98\/Daniel}.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Acknowledgements}\nNicola Gigante and Luca Geatti acknowledge the support of the Free University of\nBozen-Bolzano, Faculty of Computer Science, by means of the projects TOTA\n(\\emph{Temporal Ontologies and Tableaux Algorithms}) and STAGE (\\emph{Synthesis\nof Timeline-based Planning Games}).\n\n\\bibliographystyle{eptcs}\n\n\\section{Controller synthesis}\n\\label{sec:games}\n\nIn this section we use the deterministic automaton constructed above to\nobtain a deterministic arena where we can solve a simple reachability game\nfor checking the existence of (and, in this case, to synthesize)\na controller for the corresponding timeline-based game.\n\n\\subsection{From the automaton to the arena}\n\nLet $G=(\\SV_C, \\SV_E, \\S, \\D)$ be a timeline-based game. We use the construction\nof the automaton explained in the previous section in order to obtain a game\narena. However, the automaton construction considers a planning problem with a\nsingle set of synchronization rules, while here we have to account for the roles\nof both $\\S$ and $\\D$. \n\nTo do that, let $A_\\S$ and $A_\\D$ be the deterministic automata built over the\ntimeline-based planning problem $P_\\S=(\\SV_C\\cup\\SV_E, \\S)$ and\n$P_\\D=(\\SV_C\\cup\\SV_E, \\D)$, respectively. We define the automaton $A_G$ as\n$\\overline{A_\\D}\\cup A_\\S$, \\ie the union of $A_\\S$ with the complement of\n$A_\\D$. Note that these are all standard automata-theoretic constructions over\nDFAs. Any accepting run of $A_G$ represents either a plan that violates the\ndomain rules or a plan that satisfies both the domain and the system rules, in\nconformance with \\cref{def:games:winning-strategy}. Note that $A_G$ is\ndeterministic and can be built from $A_\\D$ and $A_\\S$ with only a polynomial\nincrease in size.\n\nNow, the $A_G$ automaton is still not suitable as a game arena, because the\nmoves of the timeline-based game are not directly visible in the labels of the\ntransitions. In other words, the $A_G$ automaton reads events, while we need an\nautomaton that reads game \\emph{moves}. In particular, a single transition in\nthe automaton can correspond to different combinations of rounds, since the\npresence of $\\wait(\\delta)$ moves is not explicit in the transition. For\nexample, an event $\\event=(A, 5)$ can be the result of a $\\wait(5)$ move by\n\\charlie followed by a $\\play(5, A)$ move by $\\eve$, or by any $\\wait(\\delta)$\nmove with $\\delta>5$ followed by $\\play(5, A)$. Hence, we need to further adapt\n$A_G$ to obtain a suitable arena.\n\nLet $A_G=(Q,\\Sigma, q_0, F, \\tau)$ be the automaton built as described before.\nLet $\\event=(A,\\delta)$ be an event. If $\\delta>1$, this transition must have\nresulted from \\charlie playing a $\\wait(\\delta')$ move with $\\delta'\\ge\\delta$.\nHowever, if $A$ contains any $\\tokend(x,v)$ action with $x\\in\\SV_C$, this is for\nsure the result of more than one pair of starting\/closing rounds. In order to\nsimplify the construction below, we remove this possibility beforehand. More\nformally, we define a slightly different automaton $A_G'=(Q,\\Sigma,q_0,F,\\tau')$\nwhere $\\tau'$ is now a \\emph{partial} transition function (\\ie the automaton\nbecomes \\emph{incomplete}) that agrees with $\\tau$ on everything excepting that\ntransitions $\\tau(q,(\\actions,\\delta))$ is \\emph{undefined} if $\\delta>1$ and\n$\\actions$ contains any $\\tokend(x,v)$ action with $x\\in\\SV_C$. You can see an\nexample of this operation in \\cref{fig:constructions}, on the left. Note that\nthis removal does not change the plans accepted by the automaton because for\neach transition $\\tau(q,(\\actions,\\delta))=q'$ with $\\delta > 1$ there are two\ntransitions $\\tau(q,(\\emptyset,\\delta-1))=q''$ and $\\tau(q'',(\\actions,1))=q'$.\n\n\\begin{figure}\n  \\begin{tikzpicture}[state\/.style={fill, circle, minimum width=5pt}]\n    \\path (0,0) node[state] (n1) { }\n          (4,4) node[state] (n2) { }\n          (4,0) node[state] (n3) { };\n\n    \\path[draw,dashed,->] (n1) -- (n2) node[midway, above, sloped, font=\\scriptsize] { \n      $\\event=(\\set{\\tokend(x,v)}, 5)$\n    };\n\n    \\path[draw, ->] (n1) -- (n3) node[midway, above, font=\\scriptsize] { \n      $\\event'=(\\emptyset, 4)$\n    };\n    \\path[draw, ->] (n3) -- (n2) node[midway, above, sloped, rotate=180, font=\\scriptsize] { \n      $\\event''=(\\set{\\tokend(x,v)}, 1)$\n    };\n\n    \\begin{scope}[xshift=6cm]\n      \\path (0,0) node[state] (n1) { } node[above, outer sep=5pt] {$q$}\n            (9,4) node[state] (n2) { } node[above, outer sep=5pt] {$w$};\n\n      \\path[draw,dashed,->] (n1) -- (n2) node[midway, above, sloped, font=\\scriptsize] { \n        $\\event=(\\set{\\tokend(x,v_1),\\tokend(y,w_1),\\tokstart(x,v_2),\\tokstart(y,w_2)}, 5)$\n      };\n\n      \\path (3,0) node[state] (q6) { } node[above right] { }\n            (3,0.75) node[state] (q5) { } node[above right] { }\n            (3,-0.75) node[state] (q7) { } node[above right] { }\n            (3,-2.25) node[state] (q10) { } node[above right] { };\n\n      \\path[dashed] (q7) -- (q10);\n\n      \\path[draw,->] (n1) -- (q5) node[above, sloped, near end, font=\\scriptsize] { $\\wait(5)$ };\n      \\path[draw,->] (n1) -- (q6) node[above, sloped, near end, font=\\scriptsize] { $\\wait(6)$ };\n      \\path[draw,->] (n1) -- (q7) node[above, sloped, near end, font=\\scriptsize] { $\\wait(7)$ };\n      \\path[draw,->] (n1) -- (q10) node[above, sloped, near end, font=\\scriptsize] { $\\wait(10)$ };\n\n      \\path (6,0) node[state] (q'') { };\n      \\path[draw,->] (q5) -- (q'') node[above, sloped, midway, font=\\scriptsize] { $\\play\\left(5, \n        \\left\\{\\begin{array}{@{}c@{}}\n          \\tokend(x,v_1)\\\\\n          \\tokend(y,w_1)\n        \\end{array}\\right\\}\\right)$ };\n      \\path[draw,->] (q6) -- (q'');\n      \\path[draw,->] (q7) -- (q'');\n      \\path[draw,->] (q10) -- (q'') node[below, sloped, midway, font=\\scriptsize] { $\\play\\left(5, \n      \\left\\{\\begin{array}{@{}c@{}}\n        \\tokend(x,v_1)\\\\\n        \\tokend(y,w_1)\n      \\end{array}\\right\\}\\right)$ };\n\n      \\path (9,0) node[state] (q''') { };\n      \\path[draw,->] (q'') -- (q''') node[midway, above,font=\\scriptsize] {\n        $\\play(\\set{\\tokstart(x,v_2)})$\n      };\n      \\path[draw,->] (q''') -- (n2) node[midway, sloped,rotate=180, above,font=\\scriptsize] {\n        $\\play(\\set{\\tokstart(y,w_2)})$\n      };\n    \\end{scope}\n  \\end{tikzpicture}\n  \\caption{On the left, the removal of transitions $\\event=(A,\\delta)$ with\n  $\\delta>1$ and ending actions of controllable tokens in $A$. On the right, the\n  transformation of a transition of the $A_G$ into a sequence of transitions in\n  $A^a_G$, with $x\\in\\SV_C$, $y\\in\\SV_E$, and\n  $\\gamma_x(v_1)=\\gamma_y(w_1)=\\mathsf{u}$.}\n  \\label{fig:constructions}\n\\end{figure}\n\nNow we can transform the automaton in order to make the game rounds, and\nespecially $\\wait(\\delta)$ moves, explicit. Intuively, each transition of the\nautomaton is split into four transitions explicitating the four moves of the two\nrounds. Given the automaton $A_G'=(Q,\\Sigma, q_0, F, \\tau')$, we define the\nautomaton $A_G^a=(Q^a,\\Sigma^a, q_0^a, F^a, \\tau^a)$, which will be the arena of\nour game, as follows:\n\\begin{enumerate}\n  \\item $Q^a=Q\\cup\\set{q_\\delta\\mid 1\\le\\delta\\le d}\\cup\\set{q_{\\delta,A}\\mid 1\\le\\delta\\le d, A\\subseteq \\mathsf{A}}$ is the set of states;\n  \\item $\\Sigma^a=\\moves_C\\cup\\moves_E$, \\ie the alphabet is turned into the set\n    of moves of the two players;\n  \\item $q_0^a=q_0$ and $F^a=F$, \\ie initial and final states do not change;\n  \\item the (partial) transition function $\\tau^a$ is defined as follows. Let\n    $w=\\tau(q,\\event)$ with $\\event=(\\actions,\\delta)$. We distinguish the case where $\\delta=1$ or $\\delta>1$.\n    \\begin{enumerate}\n      \\item if $\\delta=1$, let $\\actions_C\\subseteq\\actions$ and\n      $\\actions_E\\subseteq\\actions$ be the set of actions in $\\actions$ playable\n      by \\charlie and by \\eve, respectively. Then:\n      \\begin{enumerate}\n        \\item $\\tau(q,\\play(\\actions_C^e))=q_{1,\\actions_C^e}$, where \n          $\\actions_C^e$ is the set of \\emph{ending} actions in $\\actions_C$;\n        \\item $\\tau(q_{1,\\actions_C^e},\\play(\\actions_E^e))=q_{1,\\actions_C^e\\cup\\actions_E^e}$, where \n          $\\actions_E^e$ is the set of \\emph{ending} actions in $\\actions_E$;\n        \\item $\\tau(q_{1,\\actions_C^e\\cup\\actions_E^e},\\play(\\actions_C^s))=q_{1,\\actions_C^e\\cup\\actions_E^e\\cup\\actions_C^s}$, where \n          $\\actions_C^s$ is the set of \\emph{starting} actions in $\\actions_C$;\n        \\item $\\tau(q_{1,\\actions_C^e\\cup\\actions_E^e\\cup\\actions_C^s},\\play(\\actions_E^s))=w$, where \n          $\\actions_E^s$ is the set of \\emph{starting} actions in $\\actions_E$;\n      \\end{enumerate}\n      where the mentioned states are added to $Q^a$ as needed.\n      \\item if $\\delta>1$, let $\\actions_C\\subseteq\\actions$ and\n      $\\actions_E\\subseteq\\actions$ be the set of actions in $\\actions$ playable\n      by \\charlie and by \\eve, respectively. Note that by construction,\n      $\\actions_C$ only contains \\emph{starting} actions. Then:\n      \\begin{enumerate}\n        \\item $\\tau(q,\\wait(\\delta_C))=q_{\\delta_C}$ for all \n          $\\delta\\le\\delta_C\\le d$;\n        \\item $\\tau(q_{\\delta_C},\\play(\\delta, \\actions_E^e))=q_{\\delta,\\actions_E^e}$\n          where $\\actions_E^e$ is the set of \\emph{ending} actions in\n          $\\actions_E$;\n        \\item $\\tau(q_{\\delta,\\actions_E^e},\\play(\\actions_C))=q_{\\delta,\\actions_E^e\\cup\\actions_C}$;\n        \\item $\\tau(q_{\\delta,\\actions_E^e\\cup\\actions_C},\\play(\\actions_E^s))=w$ where\n          $\\actions_E^s$ is the set of \\emph{starting} actions in $\\actions_E$;\n      \\end{enumerate}\n      where the mentioned states are added to $Q^a$ as needed.\n    \\end{enumerate}\n    All the transitions not explicitly defined above are undefined.\n\\end{enumerate}\n\nA graphical example of the above construction can be seen in\n\\cref{fig:constructions}, on the right. Note that the structure of the original\n$A_G$ automaton is preserved by $A^a_G$. In particular, one can see that for\neach $q\\in Q$ and event $\\event=(A,\\delta)$, any sequence of moves whose outcome\nwould append $\\event$ to the partial plan (see \\cref{def:games:round-outcome})\nreach from $q$ the same state $w$ in $A^a_G$ that is reached in $A_G$ by reading\n$\\event$. Hence, one can consider $A^a_G$ to also being able to \\emph{read}\nevent sequences, even though its alphabet is different. We denote as $[\\evseq]$\nthe state $q\\in Q^a$ reached by reading $\\evseq$ in $A^a_G$.\n\nMoreover, note that, with a minimal abuse of notation, any play $\\bar\\rho$ for\nthe game $G$ can be seen as a word readable by the automaton $A_G^a$. Hence, we\ncan prove the following.\n\\begin{theorem}\n  \\label{thm:arena-soundness}\n  If $G$ is a timeline-based game, for any play $\\bar\\rho$ for $G$, $\\bar\\rho$\n  is successful if and only if it is accepted by $A_G^a$.\n\\end{theorem}\n\n\\subsection{Computing the Winning Strategy}\n\nOnce built the arena, we can focus on computing the winning region $W_C$\nfor \\charlie, that is, the set of states of the arena from which \\charlie\ncan force the play to reach a final state of $A_G^a$, no matter of the\nstrategy of \\eve. These games are called \\emph{reachability\ngames}~\\cite{Thomas2008}.  If the winning region $W_C$ is not empty,\na winning strategy of \\charlie can be simply derived from $W_C$.  As\na consequence of \\cref{thm:soundness-completeness,thm:arena-soundness}, the\ncomputed winning strategy $\\sigma_C$ for $A_G^a$ respects\n\\cref{def:games:winning-strategy}.\n\nAs stated in \\cite[Theorem 4.1]{Thomas2008}, rechability games are\ndetermined, and the winning region $W_C$ along with the corresponding\npositional winning strategy $s$ are computable. Let $A_G^a=(Q^a,\\Sigma^a,\nq_0^a, F^a, \\tau^a)$ be the automaton built from $G$ as described in the\nprevious section. Note that, by construction, in any state $q\\in Q^a$ only\none of the players has available moves. Let $Q^a_C\\subseteq Q^a$ be the set\nof states \\emph{belonging} to \\charlie, \\ie states from which \\charlie can\nmove, and let $Q^a_E=Q^a\\setminus Q^a_C$. Moreover, let $E=\\set{ (q, q')\n\\in Q^a\\times Q^a \\mid \\exists \\event \\mathrel{.} \\tau^a(q, \\event) = q'}$,\n\\ie the set of all the edges of $A_G^a$. \n\nNow, for each $i\\ge0$, we can compute the $i$-th attractor of $F^a$, written $Attr^i_C(F^a)$, that is, the set of states from which \\charlie can win in at most $i$ steps. $Attr^i_C(F^a)$ is defined as follows:\n\\begin{align*} Attr^0_C (F^a) &= F^a \\\\\nAttr^{i+1}_C (F^a) &= Attr^i_C (F^a) \\\\\n&\\cup \\set{ q^a \\in Q^a_C \\, | \\, \\exists r \\big((q^a, r) \\in E \\land r \\in Attr^i_C (F^a)\\big) } \\\\\n&\\cup \\set{ q^a \\in Q^a_E \\, | \\, \\forall r \\big((q^a, r) \\in E \\implies r \\in Attr^i_C (F^a) \\big) }\n\\end{align*}\nAs remarked in \\cite{Thomas2008}, the sequence $Attr^0_C (F^a) \\subseteq\nAttr^1_C (F^a) \\subseteq Attr^2_C (F^a) \\subseteq \\ldots$ becomes\nstationary for some index $k \\leq \\lvert Q^a \\rvert$. Thus, we define\n$Attr_C (F^a) = \\bigcup^{\\lvert Q^a \\rvert}_{i=0} Attr^i_C(F^a)$. \nIn order to prove that $W_C = Attr_C(F^a)$, it suffices to use the proof of\n\\cite[Theorem 4.1]{Thomas2008} for showing that $Attr_C(F^a) \\subseteq W_C$\nand $W_C \\subseteq Attr_C(F^a)$.\n\nTo compute a winning strategy for \\charlie in the case that $q_0^a\\in W_C$,\nit is sufficient to define $s (q) = \\mu$ for any $\\mu$ such that\n$\\tau^a(q,\\mu)=q'$ with $q,q'\\in W_C$ (which is guaranteed to exist by\nconstruction of the attractor). Then, the strategy $\\sigma_C$ for \\charlie in $G$ (see \\cref{def:games:winning-strategy}) is defined as $\\sigma_C(\\evseq)\n= s([\\evseq])$. \n\n\\begin{theorem}\n    \\label{thm:winning-region-soundness-completeness}\n  Given $A_G^a=(Q^a,\\Sigma^a, q_0^a, F^a, \\tau^a)$, $q_0^a\\in W_C$ if and only\n  if \\charlie has a winning strategy $\\sigma_C$ for $G$.\n\\end{theorem}\n\\begin{proof}\n  We first prove \\emph{soundness}, that is, $q_0^a\\in W_C$ implies that\n  \\charlie has a winning strategy $\\sigma_C$ for $G$. If $q_0^a\\in W_C$,\n  then it means that there exists a positional winning strategy $s$ for\n  \\charlie for the reachability game over the arena $A_G^a$. By\n  \\cref{thm:arena-soundness} and by the definition of reachability game, we\n  know that each play generated by $s$ corresponds to a successful play for\n  game $G$. \n  Let $\\sigma_C(\\evseq)=s([\\evseq])$ be the winning strategy for \\charlie\n  in game $G$ as defined above. By construction of $\\sigma_C$ and by\n  \\cref{def:games:winning-strategy}, this means that $\\sigma_C$ is\n  a winning strategy of \\charlie for $G$.\n\n  To prove \\emph{completeness} (\\ie if \\charlie has a winning strategy\n  $\\sigma_C$ for $G$ then $q_0^a\\in W_C$), we proceed as follows. From\n  \\cref{def:games:winning-strategy} we know that a winning strategy\n  $\\sigma_C$ for \\charlie is a strategy such that for every admissible\n  strategy $\\sigma_E$ for \\eve, there exists $n \\ge 0$ such that the play\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is successful.  From\n  \\cref{thm:arena-soundness}, we know that\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is accepted by $A^a_G$.  Therefore,\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ reaches a state in the set\n  $F^a$ starting from $q^a_0$. By definition of reachability game, this\n  means that $q^a_0 \\in W_C$.\n\\end{proof}\n\n\n\\section{A deterministic automaton for timeline-based planning}\n\\label{sec:automaton}\n\nIn this section we encode a timeline-based planning problem into a\n\\emph{deterministic} finite state automaton (DFA) that recognises all and only\nthose event sequences that represent solution plans for such problem. This\nautomaton will form the basis for the game arena solved in the next section. The\nwords accepted by the automaton are \\emph{event sequences} representing solution\nplans.\n\nLet $P=(\\SV, S)$ be a timeline-based planning problem. To get a finite alphabet,\nwe define $d=\\max(L,U)+1$, where $L$ and $U$ are in turn the \\emph{maximum}\nlower and (finite) upper bounds appearing in any rule of $P$, and we account\nonly for event sequences such that the distance between two consecutive events\nis at most $d$. It can be easily seen that this assumption does not loose\ngenerality (for a proof, see Lemma~4.8 in \\cite{Gigante19}). Hence, the symbols\nof the alphabet $\\Sigma$ are \\emph{events} of the form $\\event = \\pair{A,\n\\delta}$, where $A \\subseteq \\actions_\\SV$ and $1\\le\\delta\\le d$. Formally,\n$\\Sigma=2^{\\actions_\\SV}\\times\\ar{d}$, where $\\ar{d}=\\set{1,\\ldots,d}$. Note\nthat the size of $\\Sigma$ is exponential in the size of the problem. Moreover,\nwe define the amount $\\window(P)$ as the product of all the non-zero\ncoefficients appearing as upper bounds in rules of $P$. Intuitively, $\\window(P)$\nis the maximum amount of time a rule of $P$ can \\emph{count} far away from the\noccurrence of the quantified tokens. For example, consider the following rule:\n\\begin{align*}\n  a_0[x_0=v_0]\\to{} &\\exists a_1[x_1=v_1] a_2[x_2=v_2] a_3[x_3=v_3] \\suchdot \\\\\n  &\\tokstart(a_1)\\before_{[4,14]}\\tokend(a_0)\n  \\land \\tokend(a_0)\\before_{[0,+\\infty]}\\tokend(a_2) \\land \\tokstart(a_2)\\before_{[0,3]}\\tokend(a_3) \n\\end{align*}\nIn this case, $\\window(P)$ (assuming this is the only rule of the problem),\nwould be $3\\cdot 14=42$. This means the rule can precisely account for what\nhappens at most $42$ time point from the occurrence of its quantified tokens.\nFor example, if the token $a_1$ appears at a given distance from $a_0$, it has\nto be at less than $42$ time points (less than $14$, in particular), and any\nmodification of the plan that changes such distance has the potential to break\nthe satisfaction of the rule. Instead, what happens further away from $a_0$ only\naffects the satisfaction of the rule \\emph{qualitatively}. Suppose the tokens\n$a_2$ and $a_3$ lie at $100$ time points from $a_0$ (at most $3$ time steps from\neach other). Changing this distance (while maintaining the qualitative order\nbetween tokens) cannot ever break the satisfaction of the rule. See\n\\cite{Gigante19} for a precise account of the properties of $\\window(P)$.\n\nA key observation underlying our construction is that every atomic temporal\nrelation $T \\before_{l,u} T'$ can be rewritten as the conjunction of two\ninequalities $T^\\prime - T \\leq u$ and $T - T^\\prime \\leq -l$, and that the\nclause \\clause of an existential statement \\E can be rewritten as a system of\ndifference constraints $\\nu(\\clause)$ of the form $T - T' \\leq n$, with $n \\in\n\\Z_{+\\infty}$. Then, the system $\\nu(\\clause)$ can be conveniently represented\nby a squared matrix $D$ indexed by terms, where the entry associated with $D[T,\nT']$ gives the upper bound on $T- T'$. Such matrices, which take the name of\n\\emph{Difference Bound Matrices} (DBMs)~\\cite{dill1989timing,peron2007abstract},\ncan be conveniently updated as the plan evolves to keep track of the\nsatisfaction of the atomic temporal relations among terms. In building a DBM for\nthe system of constraints $\\nu(\\clause)$, we augment the system with constraints\nof kind $\\tokstart(a_i)-\\tokend(a_i)\\leq-\\dmin^{x_i=v_i}$ and\n$\\tokend(a_i)-\\tokstart(a_i)\\leq\\dmax^{x_i=v_i}$, for any quantified token\n$a_i[x_i=v_i]$ of $\\E$. Moreover, if two different bounds $T - T'\\le n$ and\n$T-T'\\le n'$ with $n'<n$ belong to $\\nu(\\clause)$, we keep only $T-T'\\le n'$. As\nan example, the DBM for the existential graph of the rule above is the one in\n\\cref{fig:dbm}.\n\\begin{figure}\n  \\begin{equation*}\n    \\begin{array}{rcccccccc}\\toprule\n      &\\tokstart(a_0) & \\tokend(a_0) & \\tokstart(a_1) & \\tokend(a_1) &\n      \\tokstart(a_2) & \\tokend(a_2) & \\tokstart(a_3) & \\tokend(a_3) \\\\\\midrule\n      \\tokstart(a_0) \\\\\n      \\tokend(a_0) & & & -4 & & & \\\\\n      \\tokstart(a_1) & & 14 \\\\\n      \\tokend(a_1) \\\\\n      \\tokstart(a_2)  & & & & & & & & 3 \\\\\n      \\tokend(a_2) & & 0 \\\\\n      \\tokstart(a_3) \\\\\n      \\tokend(a_3)  & & & & & 0 & & & \\\\\\bottomrule\n    \\end{array}\n  \\end{equation*}\n  \\caption{DBM of an example synchronization rule. Missing entries are intended to be $+\\infty$.}\n  \\label{fig:dbm}\n\\end{figure}\nNote that, when the bounds of the temporal relations are translated into a DBM,\nthere is no longer a distinction between \\emph{lower} and \\emph{upper} bounds.\nHowever, for some of the entries we can retrieve their original meaning. Indeed,\nif $D[T,T'] < 0$, then such entry is the lower bound of a temporal relation\n$T\\before_{l,u}T'$, whereas, if $D[T, T'] > 0$, it is the upper bound of a\nrelation $T' \\before_{l,u} T$.\n\nOn top of DBMs, we define the concept of \\emph{matching structure}, a data\nstructure that allows us to manipulate and reason about partially matched\nexistential statements, \\ie existential statements of which only a part of the\nrequests has already been satisfied by the part of the word already read, while\nthe rest can be still potentially matched in the future.\n\n\\begin{definition}[Matching Structure]\n  \\label{def:matching-structure}\n  Let $\\E\\equiv \\exists a_1[x_1 = v_1] \\dots a_m[x_m = v_m] \\,.\\, \\clause$ be\n  the existential statement of a synchronisation rule $\\Rule \\equiv a_0[x_0 =\n  v_0] \\rightarrow \\E_1 \\lor \\dots \\lor \\E_k$ over the set of state variables\n  \\SV.\n\n  The \\emph{matching structure} for $\\E$ is a tuple $\\M_{\\E} = (V, D, M, t)$\n  where:\n  \\begin{itemize}\n  \\item $V$ is the set of terms $\\tokstart(a)$ and $\\tokend(a)$ for\n    $a\\in\\set{a_0, \\dots, a_m}$;\n  \\item $D \\in \\Z_{+\\infty}^{|V|^2}$ is a DBM indexed by terms of $V$ where\n    $D[T,T']=n$ if $(T-T'\\le n) \\in \\nu(\\clause)$, $D[T,T']=0$ if $T=T'$, and $D[T,T']=+\\infty$ otherwise;\n  \\item $M \\subseteq V$ and $0\\le t \\le \\window(P)$.\n  \\end{itemize}\n\\end{definition}\n\nThe set $M$ contains the terms of $V$ that the matching structure has correctly\nmatched over the event sequence read so far. With $\\overline{M} = V \\setminus M$\nwe denote the actions that we have yet to see. Then, we say that a matching\nstructure $\\M$ is \\emph{closed} if $M = V$, it is \\emph{initial} if $M =\n\\emptyset$ and it is \\emph{active} if it is not closed and $\\tokstart(a_0) \\in\nM$. Intuitively, a matching structure is \\emph{active} if its trigger has been\nmatched over the word the automaton is reading. Then, when all the terms have\nbeen matched over the word, the matching structure becomes \\emph{closed}. The\ncomponent $t$ is the time elapsed since $\\tokstart(a_0)$ has been matched.\nWhen time flows, a matching structure can then be updated as follows.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{definition}[Time shifting]\n  \\label{def:time-shift}\n  Let $\\delta > 0$ be a positive amount of time, and $\\M = (V, D, M, t)$ be a\n  matching structure. The result of shifting $\\M$ by $\\delta$ time units,\n  written $\\M + \\delta$, is the matching structure $\\M^\\prime = (V, D^\\prime, M,\n  t')$ where:\n  \\begin{itemize}\n  \\item for all $T, T' \\in V$:\n    \\begin{equation*}\n      D^\\prime[T,T'] =\n      \\begin{cases}\n        D[T,T'] + \\delta &\\text{if } T \\in M \\text{ and } T' \\in\n        \\overline{M}\\\\% D[Tj] \u00e8 lower bound\n        D[T,T'] - \\delta &\\text{if } T \\in \\overline{M} \\text{ and } T' \\in\n        M\\\\% D[ij] \u00e8 upper bound\n        D[T,T'] &\\text{otherwise}\n      \\end{cases}\n    \\end{equation*}\n  \\item and\n    \\[\n      t' =\n      \\begin{cases}\n        t+\\delta & \\text{if } \\M \\text{ is \\emph{active}}\\\\\n        t & \\text{otherwise}\n      \\end{cases}\n    \\]\n  \\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Matching]\n  \\label{def:matching}\n  Let $\\M = (V, D, M, t)$ be a matching structure and $I \\subseteq \\overline{M}$\n  a set of matched terms. A matching structure $\\M^\\prime = (V, D, M^\\prime, t)$\n  is the result of matching the set $I$, written $\\M \\cup I$, if $M^\\prime = M\n  \\cup I$.\n\\end{definition}\n\nBeside updating the reference $t$ to the trigger occurrence of an active\nmatching structure, \\Cref{def:time-shift} dictates how to update the entries of\nthe DBM. In particular, the distance bounds between any pair of terms $T$ and\n$T'$ where one is in $M$ and the other is not are tighten by the elapsing of\ntime: when $T\\in M$ and $T'\\in\\overline{M}$, $D[T,T']$ is a lower bound loosen\nby adding the elapsed time $\\delta$, when $T\\in\\overline{M}$ and $T'\\in M$,\n$D[T,T']$ is an upper bound tighten by subtracting $\\delta$. For example,\nconsider the DBM in \\cref{fig:dbm} and consider the pair of terms\n$\\tokstart(a_1)$ and $\\tokend(a_0)$. $D[\\tokstart(a_1),\\tokend(a_0)]=-4$,\nmeaning that $\\tokend(a_0)-\\tokstart(a_1)\\le 4$ must hold. Suppose\n$\\tokstart(a_1)\\in M$ (\\ie it has been matched), and $\\tokend(a_0)\\not\\in M$ (it\nstill has to). Now, if $1$ time point passes, the entry in the DBM is\nincremented and updated to $-4+1=-3$, which corresponds to the constraint\n$\\tokend(a_0)-\\tokstart(a_1)\\le 3$. This reflects the fact that to be able to\nsatisfy the constraint, $\\tokend(a_0)$ has now only $3$ time steps left before\nit is too late.\n\\Cref{def:matching} tells us how to update the set $M$ of a matching structure.\n\nTo correctly match an existential statement while reading an event sequence, a\nmatching structure is updated only as long as no violations of temporal\nconstraints are witnessed. As such, an event is classified from the standpoint\nof a matching structure as \\emph{admissible} or not.\n\n\\begin{definition}[Admissible Event]\n  An event $\\event = (A, \\delta)$ is \\emph{admissible} for a matching structure\n  $\\M_{\\E} = (V, D, M, t)$ if and only if, for every $T \\in M$\n  and $T' \\in \\overline{M}$, $\\delta \\leq D[T',T]$, \\ie the elapsing of\n  $\\delta$ time units does not exceed the upper bound of some term $T'$ not yet\n  matched by $\\M_{\\E}$.\n\\end{definition}\n\nEach admissible event $\\event$ read from the word can be matched with a subset\nof the terms of the matching structure. There are usually more than one way to\nmatch events and terms. The following definition makes this choice explicit.\n\n\\begin{definition}[$I$-match Event]\\label{def:match-event\n  Let $\\M_{\\E} = (V, D, M, t)$ be a matching structure and  $I \\subseteq\n  \\overline{M}$. An $I$\\emph{-match event} is an admissible event $\\event = (A,\n  \\delta)$ for $\\M_{\\E}$ such that:\n  \\begin{enumerate}\n  \\item for all token names $a \\in \\mathsf{N}$ quantified as $a[x = v]$ in $\\E$\n    we have that:\\label{def:match-event:good-match}\n    \\begin{enumerate}\n    \\item if $\\tokstart(a) \\in I$, then $\\tokstart(x, v) \\in A$;\n      \\label{def:match-event:good-match:start}\n    \\item $\\tokend(a) \\in I$ if and only if $\\tokstart(a) \\in M$ and $\\tokend(x,v) \\in\n      A$;\\label{def:match-event:good-match:end}\n    \\end{enumerate}\n  \\item and for all $T \\in I$ it holds that:\\label{def:match-event:relations}\n    \\begin{enumerate}\n    \\item \\label{def:match-event:preceding-terms} for every other term $T' \\in\n      V$, if $D[T',T] \\leq 0$, then $T' \\in M \\cup I$;\n    \\item \\label{def:match-event:lower-bounds} for all $T' \\in M$, $\\delta \\geq\n      -D[T',T]$, \\ie all the lower bounds on $T$ are satisfied;\n    \\item \\label{def:match-event:zero-no-bounds} for each other term $T' \\in I$,\n      either $D[T',T] = 0$, $D[T,T'] = 0$, or $D[T',T] = D[T, T'] = +\\infty$.\n    \\end{enumerate}\n  \\end{enumerate}\n\\end{definition}\n\nIntuitively, an event is an $I$-match event if the actions in the event\ncorrectly match the terms in $I$. \\Cref{def:match-event:good-match} ensures that\neach term is correctly matched over an action it represents and, most\nimportantly, that the endpoints of a quantified token correctly identify the\nendpoints of a token in the event sequence. \\Cref{def:match-event:relations}\nensures that matching the terms in $I$ does not violate any atomic temporal\nrelation. In particular, \\Cref{def:match-event:preceding-terms} deals with the\nqualitative aspect of an ``happens before'' relation, while \\Cref{%\n  def:match-event:lower-bounds,%\n  def:match-event:zero-no-bounds%\n} deal with the quantitative aspects of the lower bounds of these relations.\nNote that an $\\emptyset$-event is admitted.\n\nLet $\\matchstructs_P$ be the set of all the matching structures for a planning\nproblem $P$. By \\Cref{def:match-event}, a single event can represent several\n$I$-match events for a matching structure, hence a matching structure can evolve\ninto several matching structures, one for each $I$-match event. Such\nevolution is defined as a ternary relation $S \\subseteq\\matchstructs_P \\times\n\\Sigma \\times \\matchstructs_P$ such that $(\\M, (A, \\delta), \\M^\\prime) \\in {S}$\nif and only if $(A, \\delta)$ is an $I$-match event for $\\M$ and $\\M^\\prime = (\\M\n+ \\delta) \\cup I$. To deal with the nondeterministic nature of this relation,\nstates of the automaton will comprise sets of matching structures collecting all\nthe possible outcomes of $S$, so that suitable notation for working with sets of\nmatching structures, denoted by $\\Upsilon$ hereafter, is introduced. We define\n$\\Upsilon^\\Rule_t\\subseteq\\Upsilon$ as the set of all the \\emph{active} matching\nstructures $\\M\\in\\Upsilon$ with timestamp $t$, associated with any existential\nstatement of $\\Rule$. Intuitively, matching structures in $\\Upsilon^\\Rule_t$\ncontribute to the fulfilment of the same triggering event for the rule \\Rule\n(because they have the same timestamp), regardless of the existential statement\nthey represent. We also define $\\Upsilon_\\bot\\subseteq\\Upsilon$ as the set of\n\\emph{non active} matching structures of $\\Upsilon$. A set $\\Upsilon$ is\n\\emph{closed} if there exists $\\M \\in \\Upsilon$ such that $\\M$ is \\emph{closed}.\nLastly, a function $\\step_\\event$ extends the relation $S$ to sets of matching\nstructures: $\\step_\\event(\\Upsilon) = \\set{\\M' | (\\M,\\event,\\M^\\prime)\\in S,\n\\text{ for some } \\M \\in \\Upsilon}$.\n\nWe are now ready to define the automaton. If $\\E$ is an existential statement,\nlet $\\mathbb{E}_\\E$ be the set of all the existential statements of the same\nrule of $\\E$. Let $\\mathbb{F}_P$ be the set of functions mapping each existential\nstatement of $P$ to a set of existential statements, and let $\\mathbb{D}_P$ be the\nset of functions mapping each existential statement to a set of matching\nstructures. A simple automaton $\\TV_P$ that checks the transition function and\nduration functions of the variables is easy to define. Then, given a\ntimeline-based planning problem $P=(\\SV,S)$, the corresponding automaton is\n$A_P=\\TV_P\\cap\\S_P$ where $\\S_P$, the automaton that checks the satisfaction of\nthe synchronization rules, is defined as $\\S_P = (Q, \\Sigma, q_0, F, \\tau)$,\nwhere:\n\\begin{enumerate}\n\\item $Q = 2^\\matchstructs \\times \\mathbb{D} \\times \\F \\cup \\set{\\bot}$ is the\n  finite set of states, \\ie states are tuples of the form $\\langle \\Upsilon,\n  \\Delta, \\Phi \\rangle\\in2^\\matchstructs \\times \\mathbb{D} \\times \\F$, plus a\n  sink state $\\bot$;\n\\item $\\Sigma$ is the input alphabet defined above;\n \n \n\\item the initial state $q_0 = \\langle \\Upsilon_0, \\Delta_0, \\Phi_0 \\rangle$ is\n  such that $\\Upsilon_0$ is the set of initial matching structures of the\n  existential statements of $P$ and, for all existential statements $\\E$ of $P$,\n  we have $\\Delta_0(\\E) = \\emptyset$ and $\\Phi_0(\\E) = \\mathbb{E}_\\E$;\n \n \n \n \n\\item $F \\subseteq Q$ is the set of final states defined as:\n  \\[\n    F = \\Set{ \\langle \\Upsilon, \\Delta, \\Phi \\rangle \\in Q |\n      \\begin{gathered}\n        \\M \\text{ is not \\emph{active} for all } \\M \\in\n        \\Upsilon\\\\\n        \\text{and }\\Delta(\\E)=\\emptyset\\text{ for all }\\E\\text{ of } P\n      \\end{gathered}}\n  \\]\n\\item $\\tau : Q \\times \\Sigma \\rightarrow Q$ is the transition function that\n  given a state $q=\\langle \\Upsilon, \\Delta, \\Phi \\rangle$ and a symbol $\\event\n  = (A, \\delta)$ computes the new state $\\tau(q,\\event)$. Let\n  $\\step^\\E_\\event(\\Upsilon^\\Rule_t)=\\set{\\M_\\E \\mid\n  \\M_\\E\\in\\step_\\event(\\Upsilon^\\Rule_t)}$. Moreover, let $\\Psi^\\Rule_t = \\set{ \\E |\n  \\M_{\\E} \\in \\step_\\event(\\Upsilon^\\Rule_t)}$. Then, the updated components of\n  the state are based on what follows, where $W = \\window(P)$:\n  \\begin{align*}\n    \\Upsilon' &= \\step_\\event(\\Upsilon_\\bot) \\cup \\bigcup \\Set{\n      \\step_\\event(\\Upsilon^\\Rule_t) |\n      \\text{$t<W-\\delta$ and \n      $\\step_\\event(\\Upsilon^\\Rule_t)$ is not \\emph{closed}}} \\\\\n    \\Delta'(\\E) &=\\begin{cases}\n        \\step^\\E_\\event(\\Upsilon^\\Rule_t) & \\text{where $t$ is the minimum such that $t\\ge W-\\delta$ and $\\step^\\E_\\event(\\Upsilon^\\Rule_t)\\ne\\emptyset$} \\\\\n        \\step_\\event(\\Delta(\\E)) & \\text{if such $t$ does not exist}\n      \\end{cases}\\\\\n    \\Phi'(\\E) &= \\begin{cases}\n      \\mathbb{E}_\\E\\quad\\text{if $\\E\\in\\Psi(\\E')$ for some $\\E'$ such that \n      $\\Delta'(\\E')$ is \\emph{closed}}  \\\\\n      \\Phi(\\E) \\setminus \n        \\set{\n          \\E'\\mid \\exists t\\ge W-\\delta \\suchdot \\E'\\in\\Psi^\\Rule_t \n          \\land \\E\\not\\in\\Psi^\\Rule_t\n        } \\quad \\text{otherwise}\n    \\end{cases}\n  \\end{align*}\n\n  Let $\\Delta''(\\E)=\\Delta'(\\E)$ unless there is an $\\E'$ with $\\E\\in\\Phi'(\\E')$\n  such that $\\Delta'(\\E')$ is \\emph{closed}, in which case\n  $\\Delta''(\\E)=\\emptyset$. Then, $\\tau(q,\\event)=\\seq{\\Upsilon', \\Delta'',\n  \\Phi'}$ if the following holds:\n  \\begin{enumerate}\n  \\item for every $\\Upsilon^\\Rule_t$, $\\step_\\event(\\Upsilon^\\Rule_t) \\neq\n    \\emptyset$, and \\label{dfa:delta:no-failed-step}\n  \\item for every synchronisation rule $\\Rule \\equiv a_0[x_0=v_0] \\rightarrow\n    \\E_1 \\lor \\dots \\lor \\E_n$ in $S$, if $\\tokstart(x_0, v_0) \\in A$, then\n    there exists $\\M_{\\E_i} = (V,D,M,0) \\in \\Upsilon'$, \n    with $i \\in \\{1\\dots n\\}$, such that $\\tokstart(a_0) \\in M$;\\label{dfa:delta:trigger-capture}\n  \\end{enumerate}\n  Otherwise, $\\tau(q,\\event)=\\bot$.\n\\end{enumerate}\n\nLet us explain what is going on. The first component $\\Upsilon$ of an automaton\nstate $q=(\\Upsilon, \\Delta,\\Phi)$ is a set of matching structures that keeps\ntrack of what have been tracked so far. Intuitively, the automaton precisely\nkeeps track of what happened to the last $\\window(P)$ time points, and only\nsummarizes what happened before that window, which is what allows us to keep the\nsize under control. Any matching structure in $\\Upsilon$ has $t<\\window(P)$.\nMatching structures in $\\Upsilon$ evolve following the $\\step$ function, until\nthey are closed or the $t$ component reaches $\\window(P)$. Matching structures\nthat reach $\\window(P)$ are promoted to a new role. Their new task is to record\nthe pieces of existential statements that still have to be matched in order to\nsatisfy all the trigger events of $\\Rule$ that no longer fit into the\n\\emph{recent history} of the event sequence (\\ie the last $\\window(P)$ time\npoints). These matching structures are not stored in $\\Upsilon$ though, they are\nsummarized by the function $\\Delta$ that maps each existential statement $\\E$ of\na rule $\\Rule$ to the set of matching structures for $\\E$ with $t=\\window(P)$.\n\nWhen a set $\\Upsilon^\\Rule_t$ exceeds the bound $\\window(P)$, the $\\Delta$\nfunction must be updated by merging the information of $\\Upsilon^\\Rule_t$ to the\ninformation already present in $\\Delta$. Now, it has to be noted that, by\nclosing a set $\\Delta(\\E)$, we can not conclude that every event that triggered\n$\\Rule$ actually satisfies $\\Rule$. Indeed, there can be sets $\\Delta(\\E)$ and\n$\\Delta(\\E')$ that are in charge of the satisfaction of the same rule $\\Rule$,\nbut for different trigger events, and closing $\\Delta(\\E)$ does not imply that\n$\\Rule$ has been satisfied. The opposite case may also arise, in which\n$\\Delta(\\E)$~and~$\\Delta(\\E')$ contribute to the fulfilment of the same trigger\nevents and closing either set suffices to satisfy $\\Rule$. To overcome the\ninformation lost when a set of matching structures gets added to the $\\Delta$\nfunction, the $\\Phi$ function (the third component of the automaton states) maps\neach existential statement $\\E$ to the set of existential statements $\\E'$ such\nthat $\\Delta(\\E')$ tracks the fulfilment of the same trigger events of the set\n$\\Delta(\\E)$. We use $\\Phi$ as follows: when a set $\\Delta(\\E)$ gets closed, we\ncan discard its matching structures and all the matching structures of the sets\n$\\Delta(\\E')$, with $\\E' \\in \\Phi(\\E)$.\n\nOne can prove the soundness and completeness of our construction.\n\n\\begin{theorem}(Soundness and completeness)\n  \\label{thm:soundness-completeness}\n  Let $P=(\\SV, S)$ be a timeline-based planning problem and let $\\A_P$ be the\n  associated automaton. Then, any event sequence $\\evseq$ is a solution plan for\n  $P$ if and only if $\\evseq$ is accepted by $\\A_P$.\n\\end{theorem}\n\nRecall that we assumed the timestamp of each event of event sequences to be\nbounded, but since events can have an empty set of actions,\n\\cref{thm:soundness-completeness} can actually deal with arbitrary event\nsequences, after adding suitable empty events. Now, let us look at the size of\nthe automaton. Let $E$ be the overall number of existential statements in $P$,\nwhich is linear in the size of $P$. It can be seen that $\\abs{\\mathbb{D}_P} \\in\n\\O({(2^{\\abs{\\matchstructs_P}})}^E)= \\O(2^{E\\cdot\\abs{\\matchstructs_P}})$, \\ie\nthe number of $\\Delta$ functions is doubly exponential in the~size~of~$P$. Then,\nobserve that $\\lvert\\mathbb{F}_P\\rvert \\in \\mathcal{O}({(2^E)}^E) =\n\\mathcal{O}(2^{E^2})$. Then, $\\abs{\\S_P} \\in \\O(\\abs{\\Sigma}\\cdot\n2^{\\abs{\\matchstructs_P}})$, that is, the size of $\\S_P$ is at most exponential\nin the number of possible matching structures. To bound this number, let $N$ be\nthe largest finite constant appearing in $P$ as bound in any atom or value\nduration function and let $L$ be the length of the largest existential prefix of\nan existential statement occurring inside a rule of $P$. Notice that $N$ is\nexponential in the size of $P$, since constants are expressed in binary, while\n$L \\in \\O(\\abs{P})$. Then, the entries of a DBM for $P$, of which there is a\nnumber quadratic in $L$, are constrained to take values within the interval\n$\\ar{-N,N}$ (excluding the infinitary value $+\\infty$), whose size is linear in\n$N$. By \\Cref{def:matching-structure}, it follows that, for the planning problem\n$P$, $\\abs{\\matchstructs_P} \\in \\O(N^{L^2} \\cdot 2^L \\cdot \\window(P))$, \\ie the\nnumber of matching structures is at most exponential in the size of $P$. Hence,\nwe proved the following:\n\\begin{theorem}[Size of the automaton]\n  Let $P=(\\SV, S)$ be a timeline-based planning problem and let $\\A_P$ be the\n  associated automaton. Then, the size of $A_P$ is at most doubly-exponential in\n  the size of $P$.\n\\end{theorem}\nNote that this is the same size as the automaton built by\nDella~Monica~\\etal~\\cite{DellaMonicaGMS18}, but their automaton was\n\\emph{nondeterministic}, while ours is by construction deterministic, essential \nfor its use as a game arena.\n\n\n\\section{Timeline-based games}\n\\label{sec:preliminaries}\n\nIn this section, we introduce timeline-based games, as defined in\n\\cite{GiganteMOCR20}. \n\n\\subsection{State variables, event sequences, synchronization rules}\nThe first basic concept is that of \\emph{state variable}.\n\n\\begin{definition}[State variable]\n  \\label{def:timelines:state-variable}\n  A \\emph{state variable} is a tuple $x=(V_x,T_x,D_x,\\gamma)$, where:\n  \\begin{itemize}\n  \\item $V_x$ is the \\emph{finite domain} of $x$;\n  \\item $T_x:V_x\\to2^{V_x}$ is the \\emph{value transition function} of $x$,\n    which maps each value $v\\in V_x$ to the set of values that can immediately\n    follow it;\n  \\item $D_x:V_x\\to\\N\\times\\N$ is the \\emph{duration function} of $x$, mapping\n    each value $v\\in V_x$ to a pair $(\\dmin, \\dmax)$ specifying respectively the\n    minimum and maximum duration of any interval where $x=v$;\n  \\item $\\gamma:V_x\\to\\set{\\mathsf{c},\\mathsf{u}}$ is the \n    \\emph{controllability tag}, that, for each value $v\\in V_x$, specifies whether it\n    \\emph{controllable} ($\\gamma(v)=\\mathsf{c}$) or \\emph{uncontrollable}\n    ($\\gamma(v)=\\mathsf{u}$).\n  \\end{itemize}\n\\end{definition}\n\nIntuitively, a state variable $x$ takes a value from a finite domain and\nrepresents a simple finite-state machine, whose transition function is $T_x$.\nThe behaviour over time of a set of state variables $\\SV$ is defined by a \nset of \\emph{timelines}, one for each variable.\nInstead of reasoning about timelines directly, though, in this paper we follow \nthe approach outlined in \\cite{GiganteMOCR20} and represent the whole execution of a \nsystem, modeled by means of a set of state variables, with a single word, called \\emph{event \nsequence}.\n\n\\begin{definition}[Event sequence \\cite{GiganteMOCR20}]\n  \\label{def:event-sequence}\n  %\n  Let $\\SV$ be a set of state variables. Let $\\actions_\\SV$ be the set of all\n  the terms, called \\emph{actions}, of the form $\\tokstart(x,v)$ or\n  $\\tokend(x,v)$, where $x\\in\\SV$ and $v\\in V_x$.\n\n  An \\emph{event sequence} over $\\SV$ is a sequence\n  $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ of pairs $\\event_i=(A_i,\\delta_i)$,\n  called \\emph{events}, where $A_i\\subseteq\\actions_\\SV$ is a set of actions,\n  and $\\delta_i\\in\\N_+$, such that, for any $x\\in\\SV$:\n  \\begin{enumerate}\n  \\item \\label{def:event-sequence:start}\n        for all $1\\le i\\le n$, if $\\tokstart(x,v)\\in A_i$, for some $v\\in V_x$,\n        then there is no $\\tokstart(x,v')$ in any $\\event_j$ before the\n        closest $\\event_k$, with $k > i$, such that $\\tokend(x,v)\\in A_k$ (if\n        any);\n  \\item \\label{def:event-sequence:end}\n        for all $1\\le i\\le n$, if $\\tokend(x,v)\\in A_i$, for some $v\\in V_x$,\n        then there is no $\\tokend(x,v')$ in any $\\event_j$ after the\n        closest $\\event_k$, with $k < i$, such that $\\tokstart(x,v)\\in A_k$ (if\n        any);\n  \\item \\label{def:event-sequence:gaps-right}\n        for all $1\\le i < n$, if $\\tokend(x,v)\\in A_i$, for some $v\\in V_x$, then\n        $\\tokstart(x,v')\\in A_i$, for some $v'\\in V_x$;\n  \\item \\label{def:event-sequence:gaps-left}\n        for all $1< i \\le n$, if $\\tokstart(x,v)\\in A_i$, for some $v\\in V_x$,\n        then $\\tokend(x,v')\\in A_i$, for some $v'\\in V_x$.\n  \\end{enumerate}\n\\end{definition}\n\nIntuitively, an event sequence represents the evolution over time of the state\nvariables of the system by representing the \\emph{start} and the \\emph{end} of\n\\emph{tokens}, \\ie a sequence of adjacent\nintervals where a given variable takes a given\nvalue. An event $\\event_i=(A_i,\\delta_i)$ consists of a set $A_i$ of actions\ndescribing the start or the end of some tokens, happening $\\delta_i$ time steps\nafter the previous one. In an \\emph{event sequence}, events are collected to\ndescribe a whole plan.\n\n\\Cref{def:event-sequence} intentionally implies that a started token is not\nrequired to end before the end of the sequence, and a token can end without the\ncorresponding starting action to have ever appeared before. In this case we say\nthe event sequence is \\emph{open} (on the right or on the left, respectively).\nOtherwise, it is said to be \\emph{closed}. An event sequence closed on the left\nand open on the right is also called a \\emph{partial plan}. Note that the empty\nevent sequence is closed on both sides for any variable. Moreover, on closed\nevent sequences, the first event only contains $\\tokstart(x,v)$ actions and the\nlast event only contains $\\tokend(x,v)$ actions,\none for each variable $x$.\nGiven an event sequence $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ over a set of\nstate variables $\\SV$, with $\\event_i=(A_i,\\delta_i)$, we define\n$\\delta(\\evseq)=\\sum_{1<i\\le n}\\delta_i$, that is, $\\delta(\\evseq)$ is the time\nelapsed from the start to the end of the sequence (its duration). The amount\nof time spanning a subsequence, written as $\\delta_{i,j}$ when $\\evseq$ is clear\nfrom context, is then $\\delta(\\slice\\evseq_{i,j})=\\sum_{i<k\\le j}\\delta_k$.\nFinally, given an event sequence $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$, for\neach $1<i\\le n$, we define $\\evseq_{<i}$ as $\\seq{\\event_1,\\ldots,\\event_{i-1}}$. \n\nIn timeline-based games, the controller plays to satisfy a set of\n\\emph{synchronization rules}, which describe the desired behavior of the system.\nSynchronization rules relate tokens, possibly belonging to different timelines,\nthrough temporal relations among token endpoints. Let \\SV be a set of state variables\nand $\\toknames = \\set{a,b,\\ldots}$ be an arbitrary set of \\emph{token names}.\nMoreover, let an \\emph{atomic temporal relation}, or simply \\emph{atom}, be an\nexpression of the form $\\production{term} \\before_{l,u} \\production{term}$, \nwhere $l\\in\\N$, $u\\in\\N\\cup\\set{\\infty}$, and a \\emph{term} is either $\\tokstart(a)$ \nor $\\tokend(a)$, for some $a\\in\\toknames$.\nA synchronisation rule \\Rule takes the following form:\n\\begin{equation*}\\label{eq:synchronisation-rules}\n  \\begin{array}{rcl}\n    \\Rule&\\bydef&a_0[x_0=v_0] \\implies \\E_1 \\lor \\E_2 \\lor \\dots \\lor \\E_k\\quad where\\\\[0.4em]\n    \\E_i&\\bydef&\\exists a_1[x_1=v_1] a_2[x_2=v_2]\\ldots a_n[x_n=v_n] \\suchdot \\clause_i\n  \\end{array}\n\\end{equation*}\nwhere $a_0, \\ldots, a_n \\in \\toknames$, $x_0,\\ldots,x_n \\in \\SV$, $v_0,\\ldots,\nv_n$ are such that $v_i \\in V_{x_i}$, for all $0 \\leq i\\leq n$, and\n$\\clause_i$ is a conjunction of atomic temporal relations (a clause). The elements\n$a_i[x_i=v_i]$ are called \\emph{quantifiers} and the quantifier $a_0[v_0=x_0]$\nis called the \\emph{trigger}. The disjuncts in the body are called\n\\emph{existential statements}.\n\nWe say that a token $\\tau = (x, v, d)$ \\emph{satisfies} a quantifier\n$a_i[x_i=v_i]$ if $x = x_i$ and $v = v_i$. The semantics of a synchronisation\nrule \\Rule states that for every token satisfying the trigger, at least one of\nthe existential statements is satisfied. Each existential statement $\\E_i$\nrequires the existence of some tokens, satisfying the quantifiers in its prefix,\nsuch that the clause $\\clause_i$ is satisfied. When a token satisfies the\ntrigger of a rule, it is said to \\emph{trigger} such a rule.\n\nFor space concerns, we do not provide all the details of the semantics of synchronization rules. The reader can find them in \\cite{GiganteMOCR20}. Intuitively, each time there is a token that satisfies the trigger of a rule, one of its existential statements must be satisfied as well. The existential statements in turn assert the existence of other tokens that satisfy a conjunction of atoms.\n\nIf $a$ and $b$ are two token names, then examples of atomic relations are $\\tokstart(a)\\before_{3,7}\\tokend(b)$ and $\\tokstart(a)\\before_{0,+\\infty}\\tokstart(b)$. Intuitively, a token name $a$ refers to a specific token, that is, a pair of $\\tokstart(x,v)$ and $\\tokend(x,v)$ actions in an event sequence, and $\\tokstart(a)$ and $\\tokend(a)$ to its endpoints. Then, an atom such as $\\tokstart(a)\\before_{l,u}\\tokend(b)$ constrains $a$ to start before the end of $b$, with the distance between the two endpoints to be comprised between the lower and upper bounds $l$ and $u$.\n\nExamples of synchronization rules\nare the following,\nwhere the relations $=$ and $\\before$ are respectively syntactic sugar for\n$\\before_{0,0}$ and $\\before_{0,+\\infty}$:\n\\begin{align*}\n  a[x_s=\\mathsf{Comm}] \\implies {}\n  & \\exists b[x_g=\\mathsf{Available}] \\suchdot\n    \\tokstart(b) \\before \\tokstart(a) \\land \\tokend(a) \\before \\tokend(b)\\\\\n  a[x_s=\\mathsf{Science}] \\implies {}\n  & \\exists b[x_s=\\mathsf{Slewing}] c[x_s=\\mathsf{Earth}] d[x_s=\\mathsf{Comm}]\n    \\suchdot {}\\\\\n  & \\tokend(a) = \\tokstart(b) \\land \\tokend(b) = \\tokstart(c) \\land\n    \\tokend(c) = \\tokstart(d)\n\\end{align*}\nwhere the variables $x_s$ and $x_g$ represent respectively the state of a\nspacecraft and the visibility of the communication ground station. The first\nrule requires the satellite and the ground station to synchronise their\ncommunications, so that when the satellite is transmitting the ground station\nis available for reception. The second rule instructs the system to transmit\ndata back to Earth after every measurement session, interleaved by the required\nslewing operation. A rule whose trigger is empty ($\\true$), called \n\\emph{triggerless rule}, can be used to state the \\emph{goal} of the system.\nAs an example, they allow one to force the spacecraft to\nperform some scientific measurement at all:\n\\begin{equation*}\n  \\true \\implies\\exists a[x_s=\\mathsf{Science}]\n\\end{equation*}\n\nTriggerless rules have a trivial universal quantification, which means they only\ndemand the existence of some tokens, as specified by the existential statements.\nAlthough triggerless rules are meant to specify the goals of a planning problem,\nthey can be regarded as syntactic sugar on top of the syntax described above.\nIndeed, triggerless rules can be translated into triggered\nrules~\\cite{GiganteMOCR20}, and thus we do not consider them from here onwards.\n\nFinally, even though our focus is on timeline-based games, we conclude the section \nby formally defining \\emph{timeline-based planning problems}.\n\\begin{definition}[Timeline-based planning problem]\n  A \\emph{timeline-based planning problem} is a pair $P=(\\SV,\\S)$, where $\\SV$ is\n  a set of state variables and $\\S$ is a set of synchronization rules over\n  $\\SV$. An event sequence $\\evseq$ over $\\SV$ is a solution plan for $P$ if all\n  the rules in $\\S$ are satisfied by $\\evseq$.\n\\end{definition}\n\n\\subsection{The game arena}\n\nWe are now ready to introduce \\emph{timeline-based games}. Their definition is quite\ninvolved, as their structure has been designed with the goal of being strictly\nmore general than \\emph{timeline-based planning with\nuncertainty}~\\cite{CialdeaMayerOU16} while being able to capture its semantics\nprecisely. For space concerns, we keep the exposition quite terse, but the\nreader can refer to \\cite{GiganteMOCR20} for details.\n\n\\begin{definition}[Timeline-based game]\n  \\label{def:games:game}\n  A \\emph{timeline-based game} is a tuple $G=(\\SV_C,\\SV_E,\\S,\\D)$,\n  where $\\SV_C$ and $\\SV_E$ are the sets of \\emph{controlled}\n  and \\emph{external} variables, respectively, and $\\S$ and $\\D$ are the sets of\n  \\emph{system}  and \\emph{domain} synchronisation rules, respectively, both \n  involving variables from $\\SV_C$ and $\\SV_E$.\n\\end{definition}\n\nA partial plan for $G$ is a partial plan over the state variables\n$\\SV_C\\cup\\SV_E$. Let $\\partialplans_G$ be the set of all possible partial plans\nfor $G$, simply $\\partialplans$ when there is no ambiguity.\n\nSince $\\epsilon$ is a closed event sequence and $\\delta(\\epsilon)=0$, the\n\\emph{empty} partial plan $\\epsilon$ is a good starting point for the game.\nPlayers incrementally build a richer partial plan, starting from $\\epsilon$, by\nplaying actions that specify which tokens to start and\/or to end, adding an\nevent that extends the event sequence, or complementing the existing\nlast event of the sequence. We partition all the available actions into those\nthat are playable by either of the two players.\\fitpar\n\n\\begin{definition}[Partition of player actions]\n  \\label{def:games:actions-partition}\n  %\n  Let $\\SV=\\SV_C\\cup\\SV_E$. The set $\\actions$ of available actions over $\\SV$\n   is partitioned into the sets $\\actions_C$ of \\charlie's actions and  \n   $\\actions_E$ of \\eve's actions, which are defined as follows:\\fitpar\n  \\begin{align}\n    \\actions_C = {} &\n      \\underbrace{%\n        \\set{\\tokstart(x,v)\\suchthat x\\in\\SV_C,\\; v\\in V_x}%\n      }_{\\text{start tokens on \\charlie's timelines}}\\;\\cup\\;\n      \\underbrace{%\n        \\set{\\tokend(x,v)\\suchthat x\\in\\SV,\\; v\\in V_x,\\; \\gamma_x(v)=\\ctag}%\n      }_{\\text{end controllable tokens}}\\\\\n    \\actions_E = {} &\n      \\underbrace{%\n        \\set{\\tokstart(x,v)\\suchthat x\\in\\SV_E,\\; v\\in V_x}%\n      }_{\\text{start tokens on \\eve's timelines}}\\;\\cup\\;\n      \\underbrace{%\n        \\set{\\tokend(x,v)\\suchthat x\\in\\SV,\\; v\\in V_x,\\; \\gamma_x(v)=\\utag}%\n      }_{\\text{end uncontrollable tokens}}\n  \\end{align}\n  %\n\\end{definition}\n\nHence, players can start tokens for the variables that they own, and end the\ntokens that hold values that they control. Actions are combined into\n\\emph{moves} that can start\/end multiple tokens at once.\n\n\\begin{definition}[Moves]\n  \\label{def:games:moves}\n  %\n  A \\emph{move} $\\move_C$ for \\charlie is a term of the form $\\wait(\\delta_C)$\n  or $\\play(A_C)$, where $\\delta_C\\in\\N^+$ and $\\emptyset\\ne\n  A_C\\subseteq\\actions_C$ is  either a set of \\emph{starting} actions or \n  a set of \\emph{ending} actions.\n\n  A \\emph{move} $\\move_E$ for \\eve is a term of the form $\\play(A_E)$ or\n  $\\play(\\delta_E,A_E)$, where $\\delta_E\\in\\N^+$ and $A_E\\subseteq\\actions_E$ is\n  either a set of \\emph{starting} actions or a set of \\emph{ending} actions.\n  %\n\\end{definition}\n\nWe denote by $\\moves_C$ and $\\moves_E$ the set of moves playable by \\charlie\nand $\\eve$, respectively. Moves such as $\\play(A_C)$ and\n$\\play(\\delta_E,A_E)$ can play either $\\tokstart(x,v)$ actions only or\n$\\tokend(x,v)$ actions only. A move of the former kind is called a\n\\emph{starting} move, while a move of the latter kind is called an \\emph{ending}\nmove. We consider $\\wait$ moves as \\emph{ending} moves. Moreover, Starting and\nending moves have to be alternated during the game.\n\n\\begin{definition}[Round]\n  \\label{def:games:round}\n  %\n  A \\emph{round} $\\round$ is a pair\n  $(\\move_C,\\move_E)\\in\\moves_C\\times\\moves_E$ of moves such that:\n  \\begin{enumerate}\n  \\item \\label{def:games:round:alternation}\n        $\\move_C$ and $\\move_E$ are either both \\emph{starting} or both\n        \\emph{ending} moves;\n  \\item \\label{def:games:round:paring}\n        either $\\round=(\\play(A_C),\\play(A_E))$, or\n        $\\round=(\\wait(\\delta_C),\\play(\\delta_E,A_E))$, with\n        $\\delta_E\\le\\delta_C$;\n  \\end{enumerate}\n  %\n\\end{definition}\n\nA \\emph{starting} (\\emph{ending}) round is one made of starting (ending) moves.\nNote that since \\charlie cannot play empty moves and $\\wait$ moves are\nconsidered ending moves, each round is unambiguously either a starting or an\nending round. Also note that since $\\play(\\delta_E,A_E)$ moves are played only\nin rounds together with $\\wait(\\delta_C)$, and $\\wait(\\delta_C)$ is always an\nending move, then any $\\play(\\delta_E,A_E)$ must be an ending move. We can now\ndefine how a round is applied to the current partial plan to obtain the new one.\nThe game always starts with a single \\emph{starting round}.\n\n\\begin{definition}[Outcome of rounds]\n  \\label{def:games:round-outcome}\n  %\n  Let $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ be an event sequence, with\n  $\\event_n=(A_n,\\delta_n)$ or $\\event_n=(\\emptyset,0)$ if $\\evseq=\\epsilon$.\n  Let $\\round=(\\move_C,\\move_E)$ be a round, let $\\delta_E$ and $\\delta_C$ be\n  the time increments of the moves, with $\\delta_C=\\delta_E=1$ for $\\play(A)$\n  moves, and let $A_E$ and $A_C$ be the set of actions of the two moves ($A_C$\n  is empty if $\\move_C$ is a $\\wait$ move).\n\n  The \\emph{outcome} of $\\round$ on $\\evseq$ is the event sequence\n  $\\round(\\evseq)$ defined as follows:\n  %\n  \\begin{enumerate}\n  \\item \\label{def:games:round-outcome:starting}\n        if $\\rho$ is a starting round, then $\\rho(\\evseq)=\\evseq_{< n}\\event_n'$,\n        where $\\event_n'\\nobreak=\\nobreak(A_n\\cup A_C\\cup A_E,\\delta_n)$;\n  \\item \\label{def:games:round-outcome:ending}\n        if $\\rho$ is an ending round, then $\\rho(\\evseq)=\\evseq\\event'$, where\n        $\\event'=(A_C\\cup A_E,\\delta_E)$;\n  \\end{enumerate}\n  %\n  We say that $\\round$ is \\emph{applicable} to $\\evseq$ if:\n  \\begin{enumerate}[label=\\alph*)]\n  \\item \\label{def:games:round-outcome:integrity}\n        the above construction is well-defined, \\ie $\\round(\\evseq)$ is a\n        valid event sequence by \\cref{def:event-sequence};\n  \\item \\label{def:games:round-outcome:alternation}\n        $\\round$ is an ending round if and only if $\\evseq$ is open for all\n        variables.\n  \\end{enumerate}\n  %\n\\end{definition}\n\nWe say that a single move by either player is applicable to $\\evseq$ if there is\na move for the other player such that the resulting round is applicable to\n$\\evseq$.\n\nThe game starts from the empty partial plan $\\epsilon$, and players play in\nturn, composing a round from the move of each one, which is applied to the\ncurrent partial plan to obtain the new one.\n\nIt is now time to define the notion of \\emph{strategy} for each player, and of\n\\emph{winning strategy} for \\charlie.\n\n\\begin{definition}[Strategies]\n  \\label{def:games:strategies}\n  %\n  A \\emph{strategy for Charlie} is a function\n  $\\strategy_C:\\partialplans\\to\\moves_C$ that maps any given partial plan\n  $\\evseq$ to a move $\\move_C$ applicable to $\\evseq$.\n  A \\emph{strategy for Eve} is a function\n  $\\strategy_E:\\partialplans\\times\\moves_C\\to\\moves_E$ that maps a partial\n  plan $\\evseq$ and a move $\\move_C\\in\\moves_C$ applicable to $\\evseq$, to a\n  $\\move_E$ such that $\\round=(\\move_C,\\move_E)$ is applicable to\n  $\\evseq$.\n  %\n\\end{definition}\n\nA sequence $\\rounds=\\seq{\\round_0,\\ldots,\\round_n}$ of rounds is called a\n\\emph{play} of the game. A play is said to be \\emph{played according to} some\nstrategy $\\strategy_C$ for \\charlie, if, starting from the initial partial plan\n$\\evseq_0=\\epsilon$, it holds that $\\round_i=(\\strategy_C(\\Pi_{i-1}),\n\\move_E^i)$, for some $\\move_E^i$, for all $0<i\\le n$, and to be played\naccording to some strategy $\\strategy_E$ for \\eve if $\\round_i=(\\move_C^i,\n\\strategy_E(\\Pi_{i-1},\\move_C^i))$, for all $0<i\\le n$. It can be seen that for\nany pair of strategies $(\\strategy_C,\\strategy_E)$ and any $n\\ge0$, there is a\nunique run $\\rounds_n(\\strategy_C,\\strategy_E)$  of length $n$ played according\nboth to $\\strategy_C$ and $\\strategy_E$.\n\nThen, we say that a partial plan $\\evseq$, and the play $\\rounds$ such that\n$\\evseq=\\rounds(\\epsilon)$, are \\emph{admissible}, if the partial plan satisfies\nthe domain rules, and are \\emph{successful} if the partial plan satisfies the\nsystem rules.\n\n\\begin{definition}[Admissible strategy for \\eve]\n  \\label{def:games:admissible-strategy}\n  %\n  A strategy $\\strategy_E$ for \\eve is \\emph{admissible} if for each strategy\n  $\\strategy_C$ for \\charlie, there is $k\\ge 0$ such that the play\n  $\\rounds_k(\\strategy_C,\\strategy_E)$ is admissible.\n  %\n\\end{definition}\n\n\\charlie wins if, \\emph{assuming} domain rules are respected, he manages to\nsatisfy the system rules no matter how \\eve plays.\n\n\\begin{definition}[Winning strategy for \\charlie]\n  \\label{def:games:winning-strategy}\n  %\n  Let $\\strategy_C$ be a strategy for \\charlie. We say that $\\strategy_C$ is a\n  \\emph{winning strategy} for \\charlie if for any \\emph{admissible} strategy\n  $\\strategy_E$ for \\eve, there exists $n\\ge0$ such that the play\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is successful.\n  %\n\\end{definition}\n\nWe say that \\charlie \\emph{wins} the game $G$ if he has a winning strategy,\nwhile \\eve \\emph{wins} the game if a winning strategy for \\charlie does not\nexist.\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper, we completed the picture about timeline-based games by providing\nan effective procedure for controller synthesis, whereas before only a proof of\nthe complexity of the strategy existence problem was known. Previous approaches\neither provided a deterministic concurrent game structure which was however not\nbuilt effectively, or an effectively built automata which was, however,\nnondeterministic and thus unsuitable for use as a game arena without a costly\ndeterminization. Our approach surpasses the limits of both previous ones by\nproviding a deterministic construction, of optimal asymptotic size, suitable to\nbe used as a game arena. Then, we solve the reachability game on the arena with\nstandard methods to effectively compute the winning strategy for the game, if it\nexists.\n\nThis work paves the way to interesting future developments. On the one hand, the\neffective procedure shown here can be finally implemented, bringing\ntimeline-based games from theory to practice. On the other hand, developing an\neffective system based on such games requires to answer many interesting\nquestions, from which concrete modeling language to adopt, to which algorithmic\nimprovements are needed to make the approach feasible. For example, it can be\nforeseen that, to solve the fixpoint computation that leads to the strategy with\nreasonable performance, the application of \\emph{symbolic techniques} would be\nneeded.\n\\section{Introduction}\n\\label{sec:introduction}\n\nIn the timeline-based approach to planning, the world is viewed as a system made\nof a set of independent but interacting components whose behaviour over time\n(the timelines) is governed by a set of temporal constraints, called\n\\emph{synchronization rules}. Timeline-based planning has been originally\nintroduced in the space industry~\\cite{Muscettola94}, with timeline-based\nplanners developed and used by space agencies on both sides of the\nAtlantic~\\cite{CestaCFOP06,CestaCDDFOPRS07,FrankJ03,BernardiniS07,ChienRKSEMESFBST00},\nboth for short- to long-term mission planning~\\cite{ChienRTTDNASVGA15} and\non-board autonomy~\\cite{FratiniCORD11}.\n\nWhile successful in practice, only recently timeline-based planning has been\nstudied from a theoretical perspective. The formalism has been at first compared\nwith traditional action-based languages \\ala STRIPS, proving that they can be\nexpressed by means of timeline-based languages~\\cite{GiganteMMO16}. Then, the\ncomplexity of the timeline-based plan existence problem has been studied: the\nproblem is \\EXPSPACE-complete~\\cite{GiganteMMO17} over discrete time in the\ngeneral case, and \\PSPACE-complete with qualitative\nconstraints \\cite{DellaMonicaGTM20}. On dense time, the problem goes\nfrom being \\NP-complete to undecidable, depending on the syntactic restrictions\napplied~\\cite{BozzelliMMPW20}. The expressiveness of timeline-based languages\nhas also been studied from a logical perspective~\\cite{DellaMonicaGMSS17}, and\nan automata-theoretic point of view~\\cite{DellaMonicaGMS18}.\n\nTraditional timeline-based planning systems excel at the integration of planning\nwith \\emph{execution} by treating explicitly the concept of \\emph{temporal\nuncertainty}: the exact timings of the events under control of the environment\nneed not to be precisely known in advance. However, general nondeterminism,\nwhere the environment can also decide \\emph{what} to do (instead of only\n\\emph{when} to do it) is usually not handled by these systems. To overcome this\nlimitation,  the concept of \\emph{timeline-based games} has been recently\nintroduced~\\cite{GiganteMOCR20}. In these games, the state variables are\npartitioned between the controller and the environment, and the latter has the\nfreedom to play arbitrarily as long as a set of \\emph{domain} rules, that define\nthe game arena, are satisfied. The controller plays to satisfy his set of\n\\emph{system} rules. A  strategy for controller is winning if it allows him\/her \nto win independently from the choices of the environment.\n\nEstablishing whether a winning strategy exists for these games has been proved to be\n\\EXPTIME[2]-complete~\\cite{GiganteMOCR20}. However, no concrete way to\nsynthesize a controller implementing such strategies is known. The proof technique\nof the aforementioned complexity result involves the construction of\na huge (doubly exponential) \\emph{concurrent game structure}, which is used to\nmodel check some Alternating-time Temporal Logic (ATL) formulas~\\cite{AlurHK02}.\nWhile this structure is deterministic and can be in principle used as an arena\nto solve a reachability game and synthesize a controller, its construction is\nbased on theoretical nondeterministic procedures which have no hope to\nbe ever concretely implemented. On the other hand, the automata-theoretic\napproach by Della~Monica~\\etal~\\cite{DellaMonicaGMS18} provides a concrete and\neffective construction of an automaton that accepts a word if and only if the\noriginal planning problem has a solution plan. However, the automaton is\n\\emph{nondeterministic} and already doubly exponential, and the determinization\nneeded to use it as an arena would result into a further blow up and a\nnon-optimal procedure.\n\nIn this paper, we provide a\nconcrete and computationally optimal approach\nto controller synthesis for timeline-based games. We overcome the limitations\nof both the above-mentioned approaches by devising a direct construction for a\n\\emph{deterministic} finite-state automaton that recognizes solution plans,\nwhich is doubly exponential in size (thus not requiring the determinization of a\nnondeterministic automaton). This automaton is then used as the arena of a\nreachability game for which plenty of controller synthesis techniques are known\nin the literature.\n\nThe paper is structured as follows. In \\cref{sec:preliminaries} we introduce the\nneeded background on timeline-based planning and timeline-based games. Then,\n\\cref{sec:automaton} provides the core technical contribution of the paper,\nnamely the construction of the deterministic automaton recognizing solution\nplans. \\Cref{sec:games} uses this automaton as the game arena to solve the\ncontroller synthesis problem. Last, \\cref{sec:conclusions} summarizes the\nmain contributions of the work and discuss future developments.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\footnotetext[1]{The data presented herein were obtained at the\n  W.M. Keck Observatory, which is operated as a scientific partnership\n  among the California Institute of Technology, the University of\n  California and the National Aeronautics and Space\n  Administration. The Observatory was made possible by the generous\n  financial support of the W.M. Keck Foundation.}\n  \\footnotetext[2]{\n  Based on observations obtained with MegaCam, a joint\n  project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii\n  Telescope (CFHT) which is operated by the National Research Council\n  (NRC) of Canada, the Institute National des Sciences de l'Univers of\n  the Centre National de la Recherche Scientifique of France, and the\n  University of Hawaii.}\n\n\nRoughly 70\\% of bright galaxies observed at redshift $z=0$ possess\nstellar discs (e.g. \\citealt{hammer05,park07,choi07} and\n\\citet{dserrano10}), including our own Galaxy and its two largest\nneighbours, M31 and M33. From this, we can infer that spiral\nmorphologies are the dominant configuration for galaxies viewed at the\npresent epoch. Under the formalism of hierarchical structure\nformation, galaxies are believed to evolve into their present forms\nvia the accretion of, and mergers with, smaller systems. The effect of\nthis process on the seemingly dynamically stable stellar discs we see\nin Milky Way (MW) type galaxies is still largely uncertain. The\nability of these fragile objects to survive a ``major merger'' event\n(i.e. a merger with a system $>1\/3$ of the host's mass) is something\nthat is still debated. Such major mergers are thought to be\ncosmologically common, with $\\sim70\\%$ of all galaxies with a halo\nmass of $M\\sim10^{12}{\\rm\\,M_\\odot}$ having experienced at least one major\ninteraction within the past 8 Gyrs \\citep{stewart08,purcell09}. Thus\nit has been argued that galaxies that possess thin stellar discs at\n$z=0$ could not have experienced a major merger within the last 10 Gyr\nwithout the disc being destabilized\n\\citep{toth92,walker96,stewart08,purcell09}. This poses a significant\nchallenge to our understanding of the formation of disc galaxies like\nthe MW and M31. Recently, several authors have argued that these thin\ndiscs could survive such an event if the merging system is\nsufficiently gas rich\n\\citep{robertson06,brook07,hopkins09,stewart09,brooks09}, although the\ndisc would still undergo heating, resulting in a thicker disc than\nthat observed presently in the MW.  In addition to the effect of major\nmergers on the structure of discs, galaxies viewed at the present\nepoch have undergone (and are still undergoing) many smaller ``minor''\nmergers which are not sufficiently massive to destroy thin stellar\ndiscs, but are thought to kinematically heat them, causing them to\nflare outwards and create a second, thick disc component\n\\citep{quinn93,robin96,walker96,velazquez99,chen01,sales09,villalobos09,purcell10}.\nOther physical processes are also thought to heat up and thicken the\nthin disc, including the accretion of a satellite on a radial orbit\nabout its host \\citep{abadi03,read08}, internal heating within the\ndisc from massive star clusters, interactions with spiral arms,\netc. (\\citealt{villumsen85,carlberg87,sellwood02,hanninen02,benson04,hayashi06,kazantzidis07,roskar08,schonrich09a,loebman10}). Thick\ndiscs may also have formed thick, with significant star formation\noccurring above the mid-plane of the galaxy or with large initial\nvelocity dispersions \\citep{brook04,kroupa02}. In recent work by\n\\citet{roskar10}, they suggest that in-situ formation\ncould also occur if the stellar disc is misaligned with the hot, gaseous\nhalo. This misalignment results in a significant warping of the\nouter disc, and subsequent star formation within this warp results in a low\nmetallicity thick disc. Finally, it is also possible that a number of\nthese mechanism will act in conjunction. In particular, it has been\nsuggested by a number of authors that secular growth from internal\nheating may be significantly enhanced by minor merger events via swing\namplification (e.g. \\citealt{sellwood98,dubinski08}), as these\nprocesses often occur simultaneously. As such, it makes little sense\nto treat these two scenarios as separate processes. \n\nWith so many potential mechanisms capable of producing thickened\nstellar discs, just how common are thick discs in spiral galaxies at\nthe present epoch?  \\citet{dalcanton02} claim that thick disc\nformation is a universal feature of disc formation, and as such should\nbe observed in all spiral galaxies. Whether such discs are formed\npredominantly via one mechanism, or a mixture of them is still\nuncertain, and disentangling the various formation scenarios from one\nanother in present data sets has proven difficult.\n\nIn the MW, the existence of a thick disc has long been known, and was\nfirst identified by \\citet{gilmore83}. Subsequent spectroscopic\nstudies of this component have shown it to be kinematically distinct\nfrom the thin stellar disc, with the thick disc lagging behind the\nthin disc by $\\sim50{\\rm\\,km\\,s^{-1}}$ \\citep{carollo10} and having a larger\nvelocity dispersion than the thin disc. This thick component also\nseems to be composed of older, more metal deficient stars\n(e.g. \\citealt{chiba00,wyse06}). However the observed properties of\nthe thick disc, such as scale height, length and velocity dispersion,\ntend to vary depending on the survey sample and tracer population used\n\\citep{juric08,ivezic08,carollo10,dejong10}. As such, the origin of\nthe MW thick disc is still a subject of great debate in the\nliterature. Thick discs have also been observed in a number of edge on\nspiral galaxies\n(e.g. \\citealt{burstein79,tsikoudi79,vanderkruit84,shaw89,vandokkum94,dalcanton02,elmegreen06,yoachim06,yoachim08a,yoachim08b}),\nand spectroscopic observations of these objects also show the thick\ndiscs to be composed of older stars than their corresponding thin\ndiscs. However, as these galaxies are all located at distances greater\nthan $\\sim$10 Mpc from the MW, one cannot obtain spectra for\nindividual stars, and must instead rely on the integrated spectral\nproperties of RGB stars. Obtaining spectra with a high enough\nsignal-to-noise (S:N) to discern velocity dispersion profiles and\nreliable metallicities is also challenging, making it impossible to\ndistinguish between the various formation mechanisms for these\nstructures.\n\n\nIf thick stellar discs are universal amongst spiral galaxies, and are\nformed by mergers with, or accretions of, satellites, one might expect\nto see such a structure in M31. This neighbouring galaxy is considered\nto be a ``typical'' spiral galaxy when compared with other local\nexternal disc galaxies \\citep{hammer07}. It is thought to have had an\nactive merger history, and a recent panoramic photometric survey by\nthe Pan-Andromeda Archaeological Survey collaboration (PAndAS,\n\\citealt{mcconnachie09}) has shown the halo of this galaxy to be\nlittered with tidal streams from interactions with in-falling\nsatellites. These include the Giant Southern Stream (GSS,\n\\citealt{ibata01b,gilbert09}) and streams A, B, C, D and E\n\\citep{ibata07,chapman08,mcconnachie09}. The outer disc of M31 is very\nperturbed \\citep{ferguson02,richardson08}, suggestive of some tidal\ninteraction. M31 is also host to 25 known dSph and 4 dE satellites, at\nleast 2 of which (NGC 205 and M32) show evidence for significant tidal\ninteraction \\citep{choi02,mcconnachie04a,geha06,howley08}. Therefore\nthe possibility of numerous interactions between the disc of M31 and\nits satellite population seems highly likely.  \\citet{mcconnachie09}\nalso present evidence for a recent interaction between M31 and its\nneighbouring spiral galaxy, M33, which could have significantly\ndistorted and heated the M31 disc, giving rise to a thick disc\ncomponent or substantial substructure in the outer disc.  Other groups\nhave postulated links between the formation of bulges and thick discs\nin spiral galaxies \\citep{melendez08,hopkins08,bournaud09,bensby10},\nand as M31 is known to have a reasonably massive bulge\n\\citep{saglia10}, it is an interesting candidate for hosting a thick\ndisc. Despite its high inclination to us along the line of sight\n(77$^\\circ$, \\citealt{walterbos88}), M31 is not seen sufficiently\nedge-on to allow us to look for such a population using photometry.\nTherefore to look for evidence of a thick disc in M31, we must search\nfor it via its kinematic signature, using spectroscopy. Given its\nproximity to us (785 kpc, \\citealt{mcconnachie05a}), M31 is an ideal\ntarget for spectroscopic observations as we are able to resolve and\nobtain reliable velocities for individual Red Giant Branch (RGB)\nstars, and it has an advantage over our own galaxy as we are afforded\na panoramic view, whereas in the MW we are hampered along various\nlines of sights by confusion from the disc and the bulge.\n\nSince 2002 our group has been conducting a systematic spectroscopic\nsurvey of M31, including the disc, halo and regions of substructure\nusing the DEIMOS instrument mounted on the Naysmyth focus of Keck II\n(I05,\n\\citealt{chapman05,chapman06,chapman07,chapman08,collins09,collins10}). The\ndata from this survey gives us an ideal opportunity to identify a\nthick disc if present. In fact, in their study of M31's extended disc\nusing this same data set, I05 identified a population lagging behind\nthe thin disc which they excluded from their study that they termed a\n`thick disc-like' population and \\citet{chapman06} briefly examined\nthis component as a function of radius but were unable to comment on\nits global properties. In this work, we discuss the results from an in\ndepth study of this population, analysing its kinematics and\nchemistry, comparing them to M31's thin and extended discs, the thin\nand thick discs in the MW, and those seen in other galaxies, and\ncomment on possible formation scenarios for this component. The paper\nis set out as follows; in \\S2 we discuss the known structure of M31,\n\\S3 focuses on our spectroscopic survey of M31 and discusses field\nselection and analysis techniques. We present our results in \\S4 and\ndiscuss their implications in \\S5. We conclude our findings in \\S6.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{map_thick.eps}\n\\caption{Map showing the location of fields within our M31\n  survey. Fields selected for study in this work are labelled and\n  highlighted in red. The outer ellipse shows a segment of a 55 kpc\n  radius ellipse flattened to c\/a= 0.6, and the major and minor axis\n  are indicated with straight lines out to this ellipse.  The inner\n  ellipse corresponds to a disk of radius 2$^\\circ$, (27 kpc), with the\n  same inclination as the main M31 disk.  }\n\\label{map}\n\\begin{flushleft}\n\n\\end{flushleft}\n\\end{center}\n\\end{figure}\n\n\\section{The bulge, discs and halo of M31}\n\nThe first recorded observation of M31 was made in 964 AD in \nthe `Book of constellations and fixed stars' written by the Persian astronomer, \nAbd al-Rahman al-Sufi, who described it as `a small cloud' in the night sky. \nIn the centuries that have passed since, M31 has been a \npopular target for astronomers, and much has been learned \nabout its structure. M31 is a spiral galaxy of SA(s)b type, \nwith a significant bulge, a classical thin stellar disc, \na vast extended stellar disc and a metal poor halo. In \nthis section, we outline the properties of each of these \ncomponents. \n\nFirst we discuss the bulge component. Numerous studies have show that\nM31 possesses a classical bulge, with a Sersic index of $\\sim2$ and an\neffective radius of 1.93 kpc \\citep{kormendy99,seigar08}. It is\nlargely supported by random motions, although recent work by\n\\citet{saglia10} has found evidence for rotation in the innermost\nregions. Saglia et al.  also find the bulge to be dominated by an old\nstellar population (age $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 12$~Gyrs) of roughly solar metallicity,\nwith a large velocity dispersion of 166-170${\\rm\\,km\\,s^{-1}}$. This component is\ndominant out to about $8'$ ($\\sim2{\\rm\\,kpc}$), at which point the disc\nbegins to dominates the surface brightness profile of the galaxy\n\\citep{saglia10}, however according to \\citet{merrett06} the bulge can\nbe traced out as far as 10 effective radii, equivalent to $\\sim15{\\rm\\,kpc}$\nmeaning that some of our innermost disc fields may be subjected to\nminor contamination from this component.\n\nStudies of the thin stellar disc of M31 have been performed by a\nnumber of authors (e.g. \\citealt{walterbos88,ferguson01}; I05;\n\\citealt{ibata07,mcconnachie09}), and have challenged our previous\nnotions of the structure of classical discs. With a scale length of\n5.9 kpc (\\citealt{walterbos88}, corrected for assumed distance to M31\nof 785 kpc, and \\citealt{merrett06}), it is more extensive than that\nof our Galaxy, and also appears to be forming stars at a lower rate\n\\citep{walterbos94}.  And while it is a characteristic feature of the\nsurface brightness profiles of stellar discs to steeply decline at\n3--4 scale lengths \\citep{vanderkruit81,pohlen00}, which corresponds\nto 18--24 kpc in M31, a spectroscopic study by \\citet{ibata05}\nuncovered a vast, extended disc component that can be traced out to\ndistances of $\\sim40$~kpc ($\\sim8$ scale lengths) from the centre of the\ngalaxy that has an exponential surface density profile that is very\nsimilar to the inner disc. While this structure is rather clumpy, on\naverage it appears to follow on smoothly from the classical inner\ndisc, although perhaps with a slightly larger scale length of\n6.6$\\pm0.6{\\rm\\,kpc}$ and with a slight lag behind circular velocities at\nlarge radii ( $\\langle\\Delta v\\rangle=20{\\rm\\,km\\,s^{-1}}$, I05). It is dynamically\ncold, with a velocity dispersion ranging from 20--40${\\rm\\,km\\,s^{-1}}$, leading\nI05 to conclude that it is likely not a thickened disc. Whether this\nextended component is truly separate from the classical thin disc, or\nmerely an extension of it that shows some evidence of heating and\nwarping at larger radii where the disc is more sensitive to\nperturbations from mergers and interactions, is unclear. Owing to the\nsimilarity of the thin and extended discs, we will refer to them both\nas the `thin disc' throughout this paper. Where we wish to make a\ndistinction between the two, we shall use the terminologies\n`classical' and `extended' disc.\n\nThe presence of a smooth, pressure supported metal-poor halo in M31 eluded\ndetection until very recently. In 2006, two groups\n\\citep{chapman06,kalirai06} independently identified such a component\nusing the DEIMOS instrument on the Keck II telescope.  Centred on the\nsystemic velocity of M31, with a central velocity dispersion of\n152${\\rm\\,km\\,s^{-1}}$, and showing no strong evidence of rotation, both groups\nfound this component to be metal poor with an average metallicity of\n${\\rm[Fe\/H]}=-1.4\\pm0.2$ \\citep{chapman06}. \\citet{kalirai06} were able to\ntrace this component out as far as 165 kpc from the centre of the\ngalaxy although there is an inevitable confusion with the halo of M33\nat these large distances \\citep{ibata07,koch08, mcconnachie09}.\n\nThe halo of M31 is also a known host to a number of kinematic\nsubstructures, such as the GSS, the tangential streams that cross the\nSE minor axis, the western shelf and a wealth of substructure in the\nNE of the galaxy that is thought to be linked to the GSS. In the\nfollowing analysis, we will carefully consider the kinematics of all\nthese components to ensure any thick disc sample that we define is\nfree from contamination by any of these sources. We shall discuss this\nin greater detail in \\S3.\n\n\\section{Observations and field selection}\n\nA detailed description of the observational methodology and target\nselection employed in the survey is given in I05, which we briefly\nsummarize here. Using Colour-Magnitude Diagrams (CMDs) from both the\nCanada France Hawaii (CFHT) and Isaac Newton (INT) telescopes\n\\citep{ferguson02,ibata07,mcconnachie09}, we selected targets for\nobservation by prioritising Red Giant Branch (RGB) stars in M31 with\n$20.5\\leq i\\leq21.2$ and colours $1.0\\leq (V-i)_0\\leq4.0$ (priority\nA), then filling the remainder of the masks with stars with $I\\le22.0$\nthat are unsaturated (priority B), where the V and I colours are\ntransformed from their native $g$ and $i$ colours using the relations\ndescribed in \\citet{mcconnachie04b} and \\citet{ibata07}.  We used a\ncombination of standard DEIMOS multislit mode for low density fields,\nsuch as the halo, and our own minislitlet approach which allowed us to\ntarget $>600$ stars per mask in more crowded regions, such as the\ndisc. Our observational setup covers the range of the Calcium Triplet\n(Ca II) lines at 8498, 8542 and 8662$\\; \\buildrel \\circ \\over {\\mathrm A}$, a prominent absorption\nfeature that can be used both to measure radial velocities, and as a\nmetallicity indicator. To obtain velocities, we cross-correlate all\nobserved stars with a template Ca II spectrum. We estimate the errors\non our velocities by following the procedures of \\citet{simon07} and\n\\citet{kalirai10}. First, we make an estimate of our velocity\nuncertainties for each observed star using a Monte Carlo method,\nwhereby noise is randomly added to each pixel in the spectrum,\nassuming a Poisson distribution for the noise and the velocity is\nrecalculated using the same cross correlation technique described\nabove. This procedure is repeated 1000 times, and then the error is\ncalculated to be the square root of the variance of the resulting mean\nvelocity. We combine this error with a systematic error, $\\epsilon$,\nwhich contains information on any errors we may not have accounted for\n(for example, wavelength calibration error, misalignment of the mask\netc.). For the fields observed with the 600 line\/mm grating, we\nevaluate this error directly by using repeat measurements in fields\n231Dis and 232Dis, a total of 332 stars. We define the normalised\nerror, $\\sigma_N$ as:\n\n\\begin{equation}\n\\sigma_N=\\frac{v_1-v_2}{(\\sigma_1^2+\\sigma_2^2+2\\epsilon^2)^{1\/2}}\n\\end{equation}\n\n\\noindent where $v_1$ and $v_2$, $\\sigma_1$ and $\\sigma_2$ are the\nvelocities and errors of each measurement pair, and $\\epsilon$ is the\nadditional random error required in order to reproduce a unit Gaussian\ndistribution with our data (shown in Fig.~\\ref{errors}) This gives us\na systematic error for this setup of $\\epsilon=5.6{\\rm\\,km\\,s^{-1}}$, slightly\nlower than the value of $\\epsilon=6.2{\\rm\\,km\\,s^{-1}}$ derived by\n\\citet{collins10} for the same setup, though we note that their\nmeasurement was based on repeat observations of 47 stars, compared\nwith our much larger data set of 332 stars. The typical uncertainties\nfor these measurements above a threshold of S:N = 3, are 5-10${\\rm\\,km\\,s^{-1}}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.8\\hsize]{n2nvars.eps}\n\\caption{A histogram showing the normalized error distribution for\n  repeat measurements of the same stars in two of our fields that were\n  observed with the 600 lines\/mm grating. The normalized error,\n  $\\sigma_N$ incorporates the velocity differences between the repeat\n  measurements (v$_1$ and v$_2$), and the Monte-Carlo uncertainties\n  calculated for each observation ($\\sigma_1$ and $\\sigma_2$). In\n  order to reproduce a unit Gaussian distribution for our\n  uncertainties, we also include an additional error term, $\\epsilon$,\n  which accounts for any systematic uncertainties we have not\n  included. We find $\\epsilon$=5.6 kms$^{-1}$ for this setup. For\n  fields using the 1200 lines\/mm setup, we using the \\citet{simon07}\n  value of $\\epsilon=2.2{\\rm\\,km\\,s^{-1}}$.}\n\\label{errors}\n\\end{center}\n\\end{figure}\n\n\nThe Ca II features also provide us with a method for measuring the\nspectroscopic metallicity of our observed sample. Following the\nprocedure of \\citet{rutledge97} and \\citep{battaglia08}, we fit\nGaussian functions to the three Ca II peaks to estimate their\nequivalent widths (EWs), and calculate [Fe\/H] using equation (1)\n\n\\begin{equation}\n[Fe\/H]=-2.66+0.42[\\Sigma Ca+0.64(V_{RGB}-V_{HB})]\n\\end{equation}\n\n\n\\noindent where $\\Sigma$Ca=0.5EW$_{8498}$+EW$_{8542}$+0.6EW$_{8662}$,\n$V_{RGB}$ is the magnitude (or, if using a composite spectrum, the\naverage magnitude) of the RGB star, and $V_{HB}$ is the mean\n$V$-magnitude of the horizontal branch (HB). Using $V_{HB}-V_{RGB}$\nremoves any strong dependence on distance or reddening in our\ncalculated value of [Fe\/H], and gives us the Ca II line strength at\nthe level of the HB. For M31, we set this value to be 25.17\n\\citep{holland96}. We note that this assumed value is sensitive to age\nand metallicity effects, see \\citealt{chen09} for a discussion,\nhowever owing to the large distance of M31, small differences in this\nvalue within the disc of M31 will have a negligable effect on\nmetallicity calculations. For individual stars, these measurements\ncarry large errors ($\\gta0.4$ dex), but the errors are significantly\nreduced when stacking the spectra into a composite in order to measure\nan average metallicity for a given population.\n\n\n\\subsection{Field selection and sample definition}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Properties of fields analysed in this work}\n\\label{fprops}\n\\begin{tabular}{lcccccccc}\n\\hline\nField  & Date observed & $\\alpha_{J2000}(hh:mm:ss)$  & $\\delta_{J2000}(^\\circ:':'')$ & Grating & P.A. & Exp. time (s) & No. targets & R$_{proj}$ (kpc)\\\\\n\\hline\n228Dis & 23\/09\/2006 & 00:40:50.56 & 40:43:54.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 301 & 9.8\\\\\n227Dis & 23\/09\/2006 & 00:39:37.40 & 40:50:42.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 312 & 15.6\\\\\n166Dis & 03\/10\/2005 & 00:39:17.89 & 40:42:18.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 209 & 15.8\\\\\n106Dis & 30\/08\/2005 & 00:39:10.00 & 40:39:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 257 & 16.1\\\\\n105Dis & 30\/08\/2005 & 00:39:00.00 & 40:28:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 271 & 16.2\\\\\n224Dis & 25\/09\/2006 & 00:38:50.00 & 40:20:30.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 265 & 16.6\\\\\n232Dis & 05\/10\/2006 & 00:38:50.00 & 40:20:00.0 & 600 lines\/mm  & 90$^\\circ$ & 16200s& 184 & 16.6\\\\\n104Dis & 30\/08\/2005 & 00:38:50.00 & 40:20:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 271 & 16.7\\\\\n220Dis & 22\/09\/2006 & 00:38:00.00 & 40:06:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 322 & 20.3\\\\\n213Dis & 22\/09\/2006 & 00:38:11.60 & 40:06:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 155 & 20.5\\\\\n102Dis & 30\/08\/2005 & 00:38:00.00 & 40:00:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 268 & 21.5\\\\\n231Dis & 05\/10\/2006 & 00:38:00.00 & 40:00:00.0 & 600 lines\/mm  & 90$^\\circ$ & 16200s & 185 & 21.6\\\\\n223Dis & 25\/09\/2006 & 00:37:12.00 & 39:57:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 304 & 23.2\\\\\n101Dis & 30\/08\/2005 & 00:38:00.00 & 39:54:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 275 & 23.5\\\\\n222Dis & 22\/09\/2006 & 00:37:12.49 & 39:48:06.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 298 & 24.9\\\\\n221Dis & 22\/09\/2006 & 00:37:11.97 & 39:45:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 303 & 25.8\\\\\n50Disk & 16\/09\/2004 & 00:37:35.29 & 39:33:55.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 216 & 30.1\\\\\n107Ext & 30\/08\/2005 & 00:35:28.00 & 39:36:19.1 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 265 &31.0\\\\\nw11old & 30\/09\/2002 & 00:35:27.02 & 39:37:15.3 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 95  &31.0\\\\\n167Hal & 03\/10\/2005 & 00:34:30.24 & 39:23:58.7 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 205 &34.2\\\\\n148Ext & 04\/10\/2005 & 00:37:07.23 & 39:12:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 211 & 39.6\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{rotcurve.eps}\n\\caption{Here we show a number of HI rotation curves for\n  M31. Throughout this work, we use results based on the\n  \\citet{chemin09} work, shown as filled black circles. We also show\n  rotation curves from \\citet{corbelli10} and I05, which differ\n  slightly from the Chemin et al. 2009 curve in the outermost\n  regions. We find that using these curves vs. the \\citet{chemin09}\n  results do not affect our results.}\n\\label{rotcurve}\n\\end{center}\n\\end{figure}\n\nIn order to detect a thick disc component in M31 kinematically, we\nneed to measure the velocities of stars within our sample relative to\nsome model for the velocity of stars within the thin disc of the\ngalaxy. If a thick disc is present, we should observe a population\nthat lags behind the thin disc in terms of its rotational\nvelocity. The component is also expected to have a larger velocity\ndispersion than the thin disc.  This is observed in the MW, where the\nthick disc lags the thin by between 20-50 kms$^{-1}$\n\\citep{chiba00,soubiran03} and has an average rotational velocity\ndispersion of $\\sigma_{V_{\\phi}}$=57 kms$^{-1}$ \\citep{carollo10}. For\nthe purposes of this work, we shall use an updated version to the disc\nmodel of I05. In this model, we assume circular orbits for all stars\nabout the centre of M31 and interpolate their velocities from the HI\nrotation curve of \\citet{chemin09}, which is shown in\nFig.~\\ref{rotcurve} as the solid black points. This rotation curve\ndiffers from that adopted by I05, particularly in the\noutermost regions. They used a compilation of CO data\nfrom \\citet{klypin02} and HI data from \\citet{brinks84}, which we also\nshow in Fig.~\\ref{rotcurve} as red triangles. We also show the HI rotation curve derived by\n\\citet{corbelli10} from a WRST survey \\citep{braun09} as blue\nsquares. Using either of these rotation curves as opposed to that of\nChemin et al. leads to differences in our interpolated velocities of\norder a few--$20{\\rm\\,km\\,s^{-1}}$, however there is a negligible effect on the\ndispersions within particular populations, and so the adoption of any\nof these curves would give us consistent results when analysing the\nglobal properties of the stellar discs in this work. We assume an\ninclination for M31 of 77$^\\circ$ \\citep{walterbos88}, and adopt\nparameters for the thickness of the disc identical to those used in\nI05, with a constant thickness for the disc of 350 pc (which is\nroughly the thickness of the MW disc, \\citet{ivezic08}) out to 16 kpc,\nat which point we assume the stellar disc begins to flare with a scale\nheight that increases linearly with radius. We set a maximum scale\nheight of 1.69 kpc at radius of 30.5 kpc, beyond which we assume that\nthe disc has constant thickness. We then integrate along the line of\nsight through this flaring exponential disc and project the velocities\nof objects on circular orbits about M31 onto the line of sight. This\nproduces an average velocity map for the disc of M31, which we display\nin Fig.~\\ref{velmap}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{rotationmap_fin.eps}\n\\caption{A contour map of the expected velocities of stars in circular\n  orbits in the disc of M31. This was constructed using our simple\n  model as discussed in \\S3.  }\n\\label{velmap}\n\\end{center}\n\\end{figure}\n\nOnce our disc model has been constructed, we need to select\na sample of fields from our DEIMOS survey that will\nprovide the most reliable kinematic comparison with respect to the\nvelocity of the disc. As the disc of M31 is not infinitesimally thin,\nbut possesses some unknown scale height, any line of sight taken\nthrough the galaxy traverses a significant depth. Given the\ninclination of M31, some lines of sight will traverse larger depths\nthan others, which could have the effect of smearing out the\nvelocities of objects with respect to the disc model. This is\nillustrated in Fig.~8 of I05. They find that objects along the major\naxis of M31 are less susceptible to this effect than those that are\nlocated off the major axis, and therefore we limit our initial study to\nfields along the major axis.\n\nA further complication in field selection arises from MW\ncontamination. Our colour selection criteria means that we inevitably\nobserve Galactic K dwarf stars within our sample, as these lie\ncoincident with M31 RGB stars in the CFHT and INT CMDs. Eliminating\nthese stars from our sample is straightforward in the South West (SW)\nregion of M31, as the disc of Andromeda and the halo of the MW occupy\ndistinct positions in heliocentric velocity space. Assuming the\nBesancon model is a good description of the foreground populations in\nthe direction of M31, it can be shown that the Galactic population\npeaks at v$_{hel}$=-61kms$^{-1}$, and the contribution of MW K dwarfs\nto our sample at v$_{hel}\\leq-100$kms$^{-1}$ is very low\n(\\citealt{robin04}, I05). Given that the average rotational velocity\nin the SW of M31 is less than -300 kms$^{-1}$ (I05), we are able to\ncleanly separate M31 stars from MW field stars. However, in the North\nEast (NE) of M31, the average heliocentric velocity of the M31 disc\ntypically ranges between -100 kms$^{-1}$ and -200 kms$^{-1}$,\nresulting in a significant overlap between Galactic and M31\npopulations, making it difficult to distinguish between the two. While\nit is possible to remove some of this contamination by examining the\nstrength of the Sodium doublet (NaI), located at a rest wavelength of\n$\\sim8190\\; \\buildrel \\circ \\over {\\rm A}$, this is not a perfect discriminator. One can also\neliminate some foreground contamination via a comparison of\nphotometric and spectroscopic metallicities \\citep{gilbert06}, but\ngiven the uncertainties on the individual spectroscopic [Fe\/H] of our\nobserved stars (discussed above), we still retain a significant\npopulation of contaminants within our sample. There is also\ncontamination in the NE from M31 substructure\n(I05,\\citealt{chapman06,richardson08}) which can be difficult to\nseparate from the M31 disc in the NE. For these reasons, we limit our\ninitial study to the SW major axis. We hope to analyse the NE\npopulation in a future paper. These criteria leave us with a sample of\n21 fields along the SW major axis, highlighted in red in\nFig.~\\ref{map}. The positions and properties of these fields can be\nfound in Table~\\ref{fprops}. Two of these fields (231Dis and 232Dis)\nwere observed as part of our ultra-deep M31 disc survey (Chapman et\nal. in prep.) and were integrated for 4.5 hours with the 600 line\/mm\ngrating, which allowed us to make more robust measurements of\nindividual stellar metallicities than for our other fields.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{contam.eps}\n\\caption{Histograms for both heliocentric (top) and disc lag (bottom)\n  velocities of all stars within our sample of 21 fields. Regions\n  expected to be inhabited by thin disc (light blue), thick disc\n  (red), halo (green) and MW foreground (grey) are highlighted. }\n\\label{contam}\n\\end{center}\n\\end{figure} \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{velcuts_new.eps}\n\\caption{Our initial sample of fields, selected for their position\n  along the South West major axis as described in the text, are shown\n  in order of increasing (projected) radius. Gaussian fits\n  indicating the thin and (where applicable) thick disc are shown as\n  magenta and blue curves respectively. Our selection criteria are\n  overlaid, with the dashed lines representing our 2$\\sigma$ cuts and\n  the solid lines representing our Gaussian cuts both of which sample\n  roughly the same region of velocity space.  }\n\\label{fields}\n\\end{center}\n\\end{figure*} \n\nTo isolate a potential thick disc population in our sample of 21\nfields, we must first define a statistical measure of what constitutes\na lagging population with respect to the thin and extended discs. We\nmust also ensure that our definition is able to distinguish between\nthis population and that of the metal-poor M31 halo which, as it is a\nnon-rotating component \\citep{chapman06,kalirai06}, also lags behind\nthe disc. In Fig.~\\ref{contam}, we display two histograms, one with\nthe heliocentric velocities ($v_{hel}$) of the stars in our 21 fields,\nand one with their velocities with respect to the disc ($v_{lag}$) and\nwe highlight the regions we expect each of these components to\ninhabit, along with where we expect to see contamination from halo\nK-dwarfs in the MW.\n\n\n We do this using two separate methods. The first is to fit Gaussians\n to both a disc component, located on or around\n v$_{lag}$=0~kms$^{-1}$, and a broad halo component centered on or\n around v$_{lag}$=-300~kms$^{-1}$. We then define a thick disc\n population to encapsulate anything that lags the thin disc by\n $>2\\sigma$ of the thin disc peak value and we implement a lower cut\n on this population by requiring the contribution from the halo would\n be $<$1 star per velocity bin (20 kms$^{-1}$), thus minimizing the\n contamination. In fields where there is no obvious halo component to\n fit to, we use a Gaussian centered on -300 kms$^{-1}$ with a\n dispersion of 90~kms$^{-1}$ \\citep{chapman06}, and normalize it with\n respect to the thin disk by assuming that the halo contributes\n $\\sim$10\\% to the total number of stars within the field (a\n conservative estimate, given that the stellar halo contributes\n $<<10$\\% to the total stellar light in disc galaxies). The second\n method is to fit multiple component Gaussians to each of the\n fields. We apply a Gaussian Mixture Modelling (GMM) technique, which\n allows the number of Gaussians to vary freely between 1 and\n 7 components. To discern which model best fits the data, we apply a\n likelihood ratio test (LRT) to the resulting probabilities of the\n fits.  The use of the LRT in astronomy was popularised by\n \\citet{cash79}, and is often used in the literature to determine\n whether the properties of an observed stellar population can be well\n described by single vs. multiple Gaussian components\n (e.g. \\citealt{ashman94,carollo10}). The LRT compares the likelihoods\n of nested models (in our case, a mixture of Gaussian components) to\n determine whether applying a model with additional parameters\n produces a significantly better fit than a simpler model. This is\n done by calculating the LRT statistic, $-2\\rm ln( \\mathcal{L}_1\/\n \\mathcal{L}_2)$, where $ \\mathcal{L}_1$ and $ \\mathcal{L}_2$\n represent the likelihoods of the simple and complex model\n respectively, and comparing it with a $\\chi^2$ distribution with\n degrees of freedom equal to the difference between the number of\n parameters in the two models (3 in our case). For a model with\n additional parameters to be accepted as a statistically better fit,\n this ratio must be greater than 7.82 which corresponds to a P-value\n of $<0.05$. In general, this technique converges on fits with three\n components (a thin disc, a halo and a thick population) though there\n are a few exceptions. We shall discuss these fits in greater detail\n in the following section. Where this technique converges on fits that\n identify a lagging component that is distinct from both the thin disc\n and halo, we define a sample of highly probable thick disc stars by\n applying a standard Bayesian classification scheme to assign each\n star a probability of being a member of the thin disc $P$(thin),\n thick disc, $P$(thick), or halo, $P$(halo), population based on their\n velocity, and the properties of the Gaussian fits to each population\n on a field by field basis. We define a star as being a highly\n probable member of the thick disc if $P$(thick)$\\geq0.997$. The\n results of both these techniques can be seen in\n Fig.~\\ref{fields}. The velocity cuts for stars selected using our\n 2$\\sigma$ technique are shown as dashed lines, and the range of\n velocities selected using the Bayesian classification technique are\n marked with solid lines. It can be seen that both techniques isolate\n a very similar population. In Fig.~\\ref{cmd}, we plot a CMD showing\n the V-I colours of the thin (blue points) and thick (magenta points)\n populations, and we overlay Dartmouth isochrones \\citep{dart08} of\n ${\\rm[\\alpha\/Fe]}=+0.2$ and an age of 8 Gyrs (in line with the estimated range of\n ages for the thin disc of 4--8 Gyrs, \\citealt{brown06}) with\n metallicities ranging from ${\\rm[Fe\/H]}=-0.4$ to ${\\rm[Fe\/H]}= -1.5$. Both thin and\n thick populations inhabit roughly the same region in this CMD, which\n we shall discuss in more detail in \\S4.3.\n\nFinally, we note that in both selection methods, we expect some cross\ncontamination between the thin and thick disc components, as the two\npopulations significantly overlap. However, we assume this\ncontamination will be lower in our cuts based on the Gaussian fits, as\nthese are more conservative. Therefore, we use these cuts\npredominantly in this paper when referring to clean thin and thick\ndisc samples. We have also assumed both components have symmetric,\nGaussian distributions in velocity, which may not be the case, and\nthis could cause further contamination if the populations are\nskewed. We also expect some contamination from the halo, however,\ngiven that the disc is the dominant population in all our fields, we\nexpect this contamination to be negligible in comparison to the cross\ncontamination between the discs.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{cmd.eps}\n\\caption{CMD for our thin (blue points) and thick (magenta points)\n  populations in standard Landolt V and I colours. Dartmouth\n  isochrones with ${\\rm[\\alpha\/Fe]}=+0.2$ and an age of 8 Gyrs are overlaid with\n  metallicities from left to right of [Fe\/H]=-1.5, -1.0, -0.5,\n  -0.4. The colours of the remaining stars in our DEIMOS survey are\n  also plotted in light blue.}\n\\label{cmd}\n\\end{center}\n\\end{figure}\n\n\\subsection{Testing the statistical significance of our sample}\n\nBefore we analyse our sample, we test the significance of our thick\npopulation to ensure it is not merely consistent with noise above a\nthin disc population plus smooth halo component. To do this, we fit a\nsingle Gaussian to the disc and halo components, as described above,\nthen calculate the deviation of the data from the fit for all\nvelocities greater than the peak disc velocity (i.e. the right hand\nside of the disc fit), normalizing it to the expected contribution\nfrom the Gaussian in this region. We then define the noise to be 1.5\ntimes the median absolute deviation of this sample. We repeat this\nexercise for all velocities less than the thin disc peak and greater\nthan -200~${\\rm\\,km\\,s^{-1}}$ in the lag frame, in this case comparing to the\nexpected contribution from both thin disc and halo fits. This allows\nus to work out the significance of our thick disc population,\n$\\sigma_{conf}$. In all cases where the GMM identified a thick disc\ncomponent, we find that our excess above the thin disc plus halo model\nhas a significance of $>3\\sigma$ (see table~\\ref{kprops}). For the\nfields where the GMM converged on a 2 Gaussian fit (232Dis, 222Dis,\n107Ext and w11old), we find $\\sigma_{conf}<3\\sigma$. We also identify an\nadditional 3 fields (101Dis, 166Dis and 227Dis), where\n$\\sigma_{conf}<3\\sigma$. Two of these fields are located at radii of\n$\\sim15{\\rm\\,kpc}$, where there may be residual contamination from the bulge\ncomponent. This may also explain the large dispersions (of order\n50${\\rm\\,km\\,s^{-1}}$) seen in our innermost fields. Excluding these, we are left\nwith 14 of our 21 ($2\/3$) fields where we confidently detect a\nthick component. We shall focus on these fields in the remainder of\nour analysis, but we shall discuss the significance of the\nnon-detections in \\S5.\n\n\\section{Results}\n\n\n\\subsection{Kinematic and structural properties of the thin and thick discs}\n\nIn this section we present measurements for the kinematic and\nstructural properties of the thin and thick discs. Properties of\nindividual fields can be found in Table~\\ref{kprops}, while the\naverage properties for both components can be found in\nTable~\\ref{avprops}.\n\n\\subsubsection{Velocity lag and dispersion profiles}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{vel_summary.eps}\n\\caption{{\\bf Top panel:} The difference in velocity, $\\Delta v$,\n  between the thin disc and thick component as a function of projected\n  radius. This lag appears to be approximately constant as a function\n  of radius, with an average lag of $46.0\\pm3.9{\\rm\\,km\\,s^{-1}} $. There appears\n  to be a slight in crease in lag in the outermost part, however this\n  is largely driven by fields that lie off the major axis of M31, and\n  therefore the velocities are less reliable. {\\bf Middle panel:\n  }Dispersion, $\\sigma_v$, of both thin (black squares, solid line)\n  and thick (red triangles, dot-dashed line) components are plotted as\n  a function of projected radius. The thin disc appears to maintain a\n  constant dispersion of $\\sigma_{thin}$=35.7$\\pm1.0{\\rm\\,km\\,s^{-1}}$, however the\n  thick component appears to decrease somewhat at larger radii. {\\bf\n    Bottom panel: }Average spectroscopic metallicity of thin and thick\n  components as a function of projected radius. Neither component\n  evolves with radius.}\n\\label{summary}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{slength_thin.eps}\n\\includegraphics[angle=0,width=0.9\\hsize]{slength.eps}\n\\caption{{\\bf Top panel:} Plot of the density of stars\n  (N$_*$\/arcmin$^2$) in the thin disc against R$_{proj}$. The\n  densities are calculated by first separating our sample by their\n  target prioritisation (A or B, see \\S3), then counting all stars\n  with $v_{thin}>v_{lag}>v_{thin}+2\\sigma_{thin}$ and multiplying\n  these values by 2 (i.e. assuming the distributions are symmetric)\n  for both prioritisations. We then calculate\n  $\\rho_*==n_sn_t\/n_o-n_b$, and combine these results from priority A\n  and B. Fitting an exponential profile to these points we deduce\n  $h_r=7.3\\pm1.1$~kpc. Solid line represents the best fit to the data\n  from a weighted-least-squares routine, and the shaded region\n  indicates the 1$\\sigma$ errors from the fit. {\\bf Bottom panel:} As\n  above, for the thick disc. Here we count all stars with\n  $v_{thick}-2\\sigma_{thick}>v_{lag}>v_{thick}$ and multiply by two\n  again. Fitting an exponential profile to these points we deduce\n  $h_r=8.0\\pm1.2$~kpc.}\n\\label{scale}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{scalerel.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{thinscalerel.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{thickscalerel.eps}\n\\caption{Here we compare the scale lengths of the thin and thick discs\n  in M31 to those of other galaxies with observed thick discs in order\n  to infer their scale heights. {\\bf Left panel: }Here we plot thin disc\n  scale lengths against thick disc scale lengths for 34 external\n  galaxies (YD06) as black points, and overplot the same measurements\n  for the MW (blue triangle). We fit a linear relation to these points\n  with a gradient of 1.3. We plot our result for M31 as a red square,\n  and it is in excellent agreement with this relation. {\\bf Centre panel:\n  }Here we plot scale height, $z_0$, against scale height for the thin\n  disc of the YD06 sample plus the MW and derive a best-fit linear\n  function with a gradient of 0.17. From this, we can estimate a scale\n  height for the M31 thin disc of 1.1$\\pm0.2$~kpc, which we overplot\n  on the relation as a red square. {\\bf Right panel: } We now plot the\n  same, but for the thick disc and find a best fit gradient of 0.35,\n  and therefore infer a scale height for the thick disc in M31 of\n  2.8$\\pm0.6$~kpc, which is overplotted as a red square.}\n\\label{height}\n\\end{center}\n\\end{figure*}\n\nIn this first section, we initially address the thin and extended\ndiscs of M31. In I05, the extended disc was identified as a stellar\ndisc that, while appearing in many respects to be similar to the\nclassical thin disc, was a separate entity that was clumpy in terms of\nits structure, and lagged behind the classical disc in terms of its\nkinematics. As we are limiting our study to one slice down the major\naxis of M31, we do not attempt to comment on the global `clumpiness'\nof this extended disc, but we return to the issue of the velocity lag\nand distinction from the thin stellar disc. As we have analysed the\ndisc frame velocities for all our fields using a rotation curve that\ndiffers from the one used in I05, it is useful for us to determine\nwhether the increasing lag with respect to the classical thin disc is\nseen here also. In I05, they split their sample of 21 fields into an\ninner (with R$_{proj}<20{\\rm\\,kpc}$) and outer (with 20$<$R$_{proj}<30{\\rm\\,kpc}$)\nsample to determine the average properties of the disc and extended\ndisc. For their inner (classical) disc sample, they calculated an\naverage velocity for the disc in the disc lag frame of\n$v_{lag}=-17.0{\\rm\\,km\\,s^{-1}}$ and a dispersion of $\\sigma_v=50.0{\\rm\\,km\\,s^{-1}}$. In the\nouter (extended) sample they calculated an average velocity of\n$v_{lag}=-16.0{\\rm\\,km\\,s^{-1}}$ and a dispersion of $\\sigma_v=51.0{\\rm\\,km\\,s^{-1}}$. If we\nperform the same analysis for our study, we find an average lag of\n$v_{lag}=-14.8{\\rm\\,km\\,s^{-1}}$ for our inner fields and $v_{lag}=-25.5{\\rm\\,km\\,s^{-1}}$ for\nour outer fields. However, we note that this value is calculated with\nthe inclusion of fields 107Ext, w11old and 167Hal, which have very large lags of\n$v_{lag}<-55{\\rm\\,km\\,s^{-1}}$ compared with the other fields. We note that these \nfields are located slightly off the semi major axis (see Fig.~\\ref{map}) \nwhere our interpolated disc-frame velocities are subject to larger uncertainties. If\nwe exclude these fields, we find an average lag of $v_{lag}=-14.9{\\rm\\,km\\,s^{-1}}$,\nvery similar to our inner sample. We therefore conclude that there is\na negligible difference in the lags of the classical and extended disc\nbehind circular velocities. For these samples we also calculate\naverage dispersions of $\\sigma_v=42.7{\\rm\\,km\\,s^{-1}}$ and $\\sigma_v=30.0{\\rm\\,km\\,s^{-1}}$,\nimplying that the extended disc has a lower dispersion than the\nclassical disc. However, in our inner sample, we are more likely to see \nresidual contamination from the bulge and we also have a large\nproportion of fields for which we could not cleanly isolate the thick\ndisc ($\\sim40$\\% cf. $\\sim20$\\% in the outer sample). These factors may \ncause us to overestimate the dispersion of the disc in these\nregions. From these results, we therefore find no concrete reason to\nassume that the extended disc is a separate component from the\nclassical disc and we treat these two components as one thin stellar\ndisc in the remainder of our analysis.\n\nBy using the information from our Gaussian fits to the thin and thick\ncomponents, we can comment on their global kinematic properties, and\ndiscuss any variation of these properties with radius. In\nTable~\\ref{kprops} we show the peak velocities and velocity\ndispersions of both thin and thick (where applicable) components in\neach field, with associated errors from the GMM fits. Where both thin\nand thick components are detected, we compute the lag between the two\ncomponents, $\\Delta v=v_{thin}-v_{thick}$, and plot this lag as a\nfunction of radius in Fig.~\\ref{summary}.  The 14 fields for which a\nthick disc component is reliably detected cover a range of radii from\n15.2 to 39.6 kpc. In the top panel of Fig.~\\ref{summary}, we can see\nthat the lag between the two components does not appear to increase\nwith distance from the centre of M31, and shows an average lag of\n$\\langle\\Delta v\\rangle=46.0{\\rm\\,km\\,s^{-1}}$.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new10.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new8.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new5.eps}\n\\caption{Here we display the spectroscopic MDF for all stars in our\n  thin and thick components (shown as blue hatched and red solid\n  histograms respectively) In the left panel, we show the MDF using\n  all stars for which metallicities can be reliably measured (i.e.\n  S:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, and the middle and right panels apply lower\n  quality cuts (S:N$\\ge8$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ and S:N$\\ge5$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$). For our\n  lower S:N cuts, we note that the median [Fe\/H] values for both\n  populations remain similar, and the dispersions (inter-quartile\n  range) begin to increase, losing some of the detail of the shape of\n  the MDF. In all cases the median [Fe\/H] of the thick disc is more\n  metal poor than the thin by $\\sim0.1$ dex.}\n\\label{spechist}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{sumspec_prob.eps}\n\\caption{Composite spectra of both the thin disc and thick component,\n  using both 2$\\sigma$ (top panel) and Gaussian (lower panel) cuts to\n  isolate the thick component. The composite spectra are constructed\n  from stars within the selection regions that possess a S:N $\\geq$\n  3.0$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$.  We find that the average metallicity for the thin\n  component is more metal rich than the thick by 0.2--0.3 dex. These\n  results are consistent with composites formed from spectra with S:N\n  $\\leq$ 3$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ and with our field-by-field metallicity estimates\n  (Fig.~\\ref{summary}). We also show the locations of a number of Fe I\n  lines present in these spectra.}\n\\label{spectra}\n\\end{center}\n\\end{figure}\n   \nWe also plot the dependence of velocity dispersion, $\\sigma_{thin}$ and\n$\\sigma_{thick}$, for both components with radius in the middle panel\nof Fig.~\\ref{summary}. For the thin disc, we fit both a constant\nrelation and a single power law to the data. The linear power law\nsuggests a decrease in dispersion with radius, with a gradient of\n-0.87${\\rm\\,km\\,s^{-1}}{\\rm\\,kpc}^{-1}$, however this fit is not statistically better\nthan a constant fit, with an average dispersion of\n$\\sigma_{thin}=31.6\\pm1.1{\\rm\\,km\\,s^{-1}}$ (reduced $\\chi^2$ of 5.6 vs. 5.2). We\ndo the same for our thick disc results, and we find that a linearly\ndecreasing profile where\n$\\sigma_{thick}=-0.8(\\pm0.2)R_{proj}+66.1(\\pm5.8)$ has a marginally better\nfit to the data than a fit with no evolution, however the difference\nin negligible (reduced $\\chi^2$ of 1.2 vs. 1.4), and deemed\ninsignificant in a $\\chi^2$ significance test. Even if we were to\naccept this fit as preferred, we note that the two outermost fields\nsituated at 34.2 and 39.6~kpc, are perhaps the driving force in the decreasing\ndispersion seen in our thick disc component. As these field are the\nfurthest out in our survey, they also suffers from the greatest chance\nof halo contamination in our sample, and therefore could be\nunreliable. If we exclude these final points from the fit, we find that\n$\\sigma_{thick}$ is best fit with no evolution as a\nfunction of radius, with an average dispersion of 50.8$\\pm1.9{\\rm\\,km\\,s^{-1}}$. We\ntherefore conclude that our data cannot tell us anything reliable\nabout the dependence of these kinematic properties with radius, and\nallow us to merely calculate the average kinematics of both\ncomponents.\n\n\n\\subsubsection{Scale length of the thin and thick disc}\n\nTo determine the scale lengths of our two disc components, we need to\ncalculate the number density of thin and thick disc stars within our\nDEIMOS field of view. There are 2 complications we must consider\nbefore we proceed. Firstly, the two components are not completely\ndistinct from one another, and in all fields, we observe some\noverlap. Secondly, owing to our selection criteria (discussed in\n\\S~3), we prioritise stars of certain colours and magnitudes above\nothers, and this must be considered when calculating densities on a\nfield-by-field basis.\n\n\nWe determine the number of stars associated with the extended thin\ndisc, $n_s$, in each of our fields by integrating the Gaussian we have\nfit to this component. To determine the density of stars contained in\nthe thin disc, we multiply $n_s$ by the total number of available\ntarget stars within our DEIMOS field that fall within our selection\ncriteria, $n_t$, and divide this by the total number of stars that\nwere observed with our DEIMOS mask, $n_o$. We then subtract the\ndensity of background stars $n_b$, which is computed from a number of\nfields on the edge of our survey region;\ni.e. $\\rho_*=n_sn_t\/n_o-n_b$. To account for our prioritised selection\ntechnique, we perform this calculation separately for our priority A\nand priority B stars, then combine these measurements. We repeat this\ncalculation for the thick disc. We plot the results in\nFig.~\\ref{scale}, where we apply a weighted least-squares exponential\nfit to our data points, and determine $h_r=7.3\\pm1.1$~kpc for the thin\ndisc and $h_r=8.0\\pm1.2$~kpc for the thick. Comparing this to previous\ncalculations for the scale length of the thin and extended discs, we\nfind that the extended disc has a larger scale length than the\nexponential thin disc, ($5.1\\pm0.1$~kpc, I05). The value of 7.3~kpc\nthat we derive is slightly higher than that derived in I05 of\n6.6$\\pm0.4$~kpc, and with much larger error bars but the two are\nconsistent within their $1\\sigma$ uncertainties. The difference\nbetween the two values can be attributed to two factors. Firstly, in\nI05, they included fields from the NE of the galaxy, plus fields\nlocated away from the major axis, where we have have sampled fields\nsolely from the SW major axis. Secondly, in I05 they did not fully\naddress any biases that may have been introduced by our two-tiered\nprioritisation system. Finally, we note that the thick disc appears to\nbe more radially extended than either the thin or extended disc,\nalthough it is consistent with the scale length of the extended disc\nwithin its $1\\sigma$-errors.\n\n\nIn previous work \\citet{yoachim06} (hereafter YD06) measured\nthe scale lengths of 34 edge-on disc galaxies using a photometric\nfitting technique, and found that the scale lengths of the thick discs\nwere larger than those of the thin discs by a factor of$\\sim1.3$. We\nplot their results in the left panel of Fig.~\\ref{height}, and overlay\na linear relation with a gradient of 1.3. We add to this our results\nfor M31, using an average value for the thin disc from the range of\nscale lengths derived for the thin and extended discs (5.1--7.3 kpc)\nof 6.3~kpc, and our calculated value of 8.0~kpc. We also overplot the\nresult for the MW (using \\citealt{juric08} values of 2.6 and 3.6 kpc\nfor thin and thick discs respectively). and note that M31 sits in\nexcellent agreement with this relation.\n\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Kinematic properties of thin and thick disc components}\n\\label{kprops}\n\\begin{tabular}{lccccccc}\n\\hline\nField & v$_{thin,lag}$ (${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thin} ({\\rm\\,km\\,s^{-1}})$ & v$_{thick, lag}$ (${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thick} ({\\rm\\,km\\,s^{-1}})$ & $\\sigma_{conf}$ & [Fe\/H]$_{thin}$ & [Fe\/H]$_{thick}$\\\\\n\\hline\n228Dis &  6.8$\\pm5.0$  & 55.2$\\pm3.2$ & -141.9$\\pm7.5$ & 41.2$\\pm11.8$ & 37.0 & -0.8$\\pm$0.1 & -1.0$\\pm$0.1\\\\\\\n227Dis & -3.7$\\pm10.8$ & 68.7$\\pm3.3$ & N\/A            & N\/A           & 2.8  & -0.7$\\pm0.2$ &N\/A\\\\\n166Dis & -7.5$\\pm6.9$  & 48.9$\\pm4.6$ & N\/A            & N\/A           & 0.5 & -0.8$\\pm$0.2 & N\/A\\\\\n106Dis & 11.6$\\pm$8.2  & 47.4$\\pm5.2$ & -85.0$\\pm8.2$  & 52.5$\\pm4.1$  & 3.5 & -0.9$\\pm$0.3 &  -1.0$\\pm0.3$\\\\\n105Dis & -16.4$\\pm7.2$ & 32.0$\\pm4.1$ & -54.8$\\pm8.7$  & 51.0$\\pm5.2$  & 4.0 & -0.7$\\pm$0.2 & -1.0$\\pm$0.2\\\\\n224Dis & -34.4$\\pm6.2$ & 24.3$\\pm2.5$ & -63.8$\\pm10.2$ & 55.0$\\pm12.5$ & 5.1 & -0.9$\\pm$0.1 & -1.1$\\pm$0.2\\\\\n232DiS & -37.2$\\pm3.0$ & 35.0$\\pm3.8$ & N\/A            & N\/A           & 2.0 & -0.8$\\pm$0.1 & N\/A\\\\\n104Dis & -37.9$\\pm6.5$ & 30.0$\\pm4.4$ & -114.3$\\pm9.5$ & 53.3$\\pm7.2$  & 9.0 & -0.7$\\pm$0.2 & -0.9$\\pm$0.1\\\\\n220Dis & -12.5$\\pm5.0$ & 39.6$\\pm2.9$ & -64.8$\\pm12.1$ & 54.7$\\pm10.9$ & 26.7 & -1.1$\\pm$0.3 & -1.6$\\pm$0.3\\\\\n213Dis & -10.9$\\pm3.2$ & 31.9$\\pm6.3$ & -80.5$\\pm10.0$ & 55.0$\\pm8.8$  & 39.2 & -0.8$\\pm$0.3 & -1.5$\\pm$0.3\\\\\n102Dis & -18.8$\\pm7.1$ & 22.2$\\pm5.2$ & -61.5$\\pm11.6$ & 48.1$\\pm5.6$  & 47.1 & -0.9$\\pm$0.2 & -1.0$\\pm$0.2\\\\\n231Dis &   4.7$\\pm15.2$& 22.9$\\pm4.2$ & -35.9$\\pm7.8$  & 53.4$\\pm7.2$  & 43.4 & -0.9$\\pm$0.1 & -1.1$\\pm$0.2\\\\\n223Dis &  -8.1$\\pm5.9$ & 20.1$\\pm4.9$ & -55.7$\\pm9.9$  & 51.1$\\pm6.8$  & 14.7 & -0.7$\\pm$0.2 & -1.0$\\pm$0.3\\\\\n101Dis & -16.2$\\pm3.9$ & 35.8$\\pm5.8$ & -57.3$\\pm8.6$  & 44.9$\\pm6.2$  & 3.1 & -1.1$\\pm$0.2 & -1.3$\\pm$0.3\\\\\n222Dis & -25.9$\\pm$4.8 & 33.6$\\pm2.9$ & N\/A            & N\/A           & 0.8 & -0.7$\\pm$0.1 & N\/A\\\\\n221Dis & -27.5$\\pm5.5$ & 35.0$\\pm3.6$ & -121.2$\\pm7.4$ & 52.2$\\pm6.3$  & 27.1 & -1.1$\\pm$0.2 & -1.4$\\pm$0.2\\\\\n50Disk & -16.9$\\pm8.1$ & 34.2$\\pm5.9$ & -149.2$\\pm13.2$& 42.3$\\pm7.2$  & 12.6 & -1.0$\\pm$0.2 & -1.2$\\pm$0.2\\\\\n107Ext & -55.2$\\pm6.9$ & 24.8$\\pm4.4$ & N\/A            & N\/A           & 2.4 & -1.0$\\pm$0.3 & N\/A\\\\\nw11old & -55.1$\\pm10.2$& 28.8$\\pm5.6$ & N\/A            & N\/A           & 1.4 & -1.0$\\pm$0.3 & N\/A\\\\\n167Hal & -72.6$\\pm8.3$ & 22.9$\\pm5.4$ & -120.3$\\pm12.2$& 35.1$\\pm7.8$  & 23.5 & -0.9$\\pm$0.3 & -1.0$\\pm0.3$\\\\\n148Ext & -17.2$\\pm7.0$ & 41.3$\\pm6.6$ & -154.5$\\pm11.0$& 25.7$\\pm5.1$  & 28.2 & -0.9$\\pm$0.3 & -1.0$\\pm$0.2\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table}\n\\begin{center}\n\\caption{Average properties of thin and thick disc components derived in this work}\n\\label{avprops}\n\\begin{tabular}{lcccc}\n\\hline\nComponent & $\\sigma_v ({\\rm\\,km\\,s^{-1}})$ & $h_r$ (kpc) & z$_0$ (kpc)& [Fe\/H]$_{spec}$\\\\\n\\hline\nThin disc  & 35.7$\\pm1.0$ & 7.3$\\pm1.1$ & 1.1$\\pm0.2$ & -0.7$\\pm0.05$ \\\\\nThick disc & 50.8$\\pm1.9$ & 8.0$\\pm1.2$ & 2.8$\\pm0.6$ & -1.0$\\pm0.1$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Inferring the scale heights of the thin and thick discs}   \n\nOwing to the inclination of M31, we are unable to measure the height\nof either the thin or thick disc components directly. No photometric\nexcess above a typical bulge or extended disc profile is observed when\nperforming minor axis star counts \\citep{irwin05}, suggesting that\nthese components dominates the surface profile out to large radii. In\norder to infer probable scale heights for both components, we make use\nof the properties of the 34 edge-on galaxies measured by YD06. As the\nscale lengths and heights of both thin and thick discs in each of\nthese galaxies were derived, it is possible for us to search for a\nrelation between the scale length, $h_r$, and scale height, $z_0$ of\neach component. In the central panel of Fig.~\\ref{height}, we plot\n$h_r$ vs. $z_0$ for the YD06 sample as well as for the MW\n\\citep{ivezic08}, and fit it with a linear relation, on which we force\nan intercept of (0,0). We find that the data are well fit with a\ngradient for this relation of 0.18$\\pm0.04$, though there is\nsignificant scatter beyond $\\sim9$~kpc. From this, we deduce\n$z_0$=1.1$\\pm0.2$~kpc for the M31 extended disc (using $h_r$=7.3\nkpc). We repeat this for the thick disc (shown in the right panel of\nFig.~\\ref{height}) and find that these values are well fit with a\nlinear relation of gradient 0.35$\\pm0.06$, giving us\n$z_0$=2.8$\\pm0.6$~kpc for the M31 thick disc. If these values are\ncorrect, then not only are the discs of M31 more radially extended\nthan those of the MW by a factor of $\\sim2-3$, they are also\nsignificantly thicker.\n\n\n\\subsubsection{Contrast of the thin and thick discs}\n\nIn the previous sections, we have derived the density in each field of\nboth our components as a means to determine the scale lengths. We now use\nthese densities to work out how much of the total (disc related)\nstellar population is contained within either component. There are\nseveral caveats to such a comparison that should be\nmentioned. Firstly, our sampling of the field is likely to have\nan effect on our field-to-field estimates of the stellar density\n(which we discuss further in \\S~5). Secondly, as the disc is not\nobserved edge-on, we are measuring a 2D projection of the\ndensities which is difficult to interpret. We also note that the\nmeasurement errors associated with the densities of each field (shown\nin table~\\ref{densities}) are significant (of order $\\sim50$\\%). \n\nWith this in mind, we find that, on average, the thick disc component\naccounts for $35$\\% of the total stellar density, with an\ninter-quartile range of $\\pm$10\\%. In the Milky Way, we know that the\nthick disc contributes to $\\sim10$\\% of the stellar density in the\nsolar neighbourhood, and accounts for $\\sim1\/3$ of the {\\it total}\ndisc mass \\citep{juric08,schonrich09a}, comparable to what we derive here. \n\nFrom our calculated contrasts and individual density profiles for the\nthin and thick discs, we can estimate the mass contained within the\nthick disc component using values for the mass of the thin disc from\nthe literature. From our analysis above, we have determined that the\nthick disc contributes 35$\\pm10$\\% of the {\\it total} stellar density,\nmeaning the thick:thin disc density ratio is of order 55$\\pm15$\\%. We\ncan also estimate this fraction by integrating our stellar density\nprofiles (Fig.~\\ref{scale}) over the limits of our data, and from this\nwe calculate a thick:thin disc density ratio of $\\sim$65\\%, which is\nin good agreement with our contrast estimate. If we assume that both\ndiscs are composed of similar stellar populations, we can set the mass\nratio between the disc to be equivalent to the density ratio. In\n\\citet{yin09}, they quote a total mass for the thin stellar disc of\n$M_{*,thin}=5.9\\times10^{10}{\\rm\\,M_\\odot}$, calculated from the mass models of\n\\citet{widrow03} and \\citet{geehan06}. From this we estimate that the\ntotal mass of the M31 thick disc lies in the range\n2.4$\\times10^{10}{\\rm\\,M_\\odot}<M_{*,thick}<4.1\\times10^{10}{\\rm\\,M_\\odot}$. As we are\nunable to determine the full radial and underlying luminosity profile\nfor the thick disc, these values are obviously prone to large errors\nintroduced by our simplifying assumptions, and the mass of the thick\ndisc may be lower than our quoted range. For example, if we just\ncompare the number of stars we detect in the thin disc throughout our\nentire sample with the number we detect in the thick disc by\nintegrating the fitted Gaussians in Fig.~\\ref{fields} we find a\nthick:thin disc ratio of 20\\%. If we assume this value for the ratio\nof the masses between the components, our lower limit on the thick\ndisc is reduced to $M_{*,thin}=1.2\\times10^{10}{\\rm\\,M_\\odot}$. A future study\nof the thick disc which includes fields from the entirety of our\nsurvey will help us to better constrain both the radial profile and\nmass of this component.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Field by field densities of thin and thick disc stars}\n\\label{densities}\n\\begin{tabular}{lcccccc}\n\\hline\nField & $\\rho_{* thin}$ (*\/arcmin) & $\\rho_{* thick}$ (*\/arcmin) \\\\\n\\hline\n228Dis & 57.6$\\pm$22.3 & N\/A\\\\\n227Dis & 56.9$\\pm$22.3 & N\/A\\\\\n166Dis & 20.4$\\pm$9.4  & N\/A\\\\\n106Dis & 36.2$\\pm$15.3 & 23.6$\\pm$10.5\\\\\n105Dis & 45.3$\\pm$18.6 & 41.0$\\pm$16.3\\\\\n224Dis & 20.1$\\pm$8.4  & 17.1 $\\pm$7.1\\\\\n232DiS & 35.9$\\pm$15.7 & N\/A \\\\\n104Dis & 35.7$\\pm$14.6 & 6.6$\\pm$4.5\\\\\n220Dis & 19.9$\\pm$8.9  & 13.5$\\pm$6.4\\\\\n213Dis & 23.0$\\pm$12.2 & 9.8$\\pm$3.0\\\\\n102Dis & 24.0$\\pm$12.8 & 8.9$\\pm$4.5\\\\\n231Dis & 13.0$\\pm$6.6  & 16.6$\\pm$8.1\\\\\n223Dis & 15.1$\\pm$6.7  & 12.2$\\pm$5.6\\\\\n101Dis & 26.5$\\pm$12.9 & 11.1$\\pm$4.8\\\\\n222Dis & 4.9$\\pm$ 2.1  & N\/A \\\\\n221Dis & 14.4$\\pm$6.4  & 4.8$\\pm$3.3\\\\\n50Disk & 8.1$\\pm$3.9   & 2.0$\\pm$0.6\\\\\n107Ext & 9.6$\\pm$7.8   & N\/A\\\\\nw11old & 9.0$\\pm$6.5   & N\/A\\\\\n167Hal & 1.3$\\pm$0.8   & 0.8$\\pm0.5$\\\\\n148Ext & 1.1$\\pm$0.9   & 0.7$\\pm$0.5\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Spectroscopic metallicities}\n\n\nIn this section we present the spectroscopic values of [Fe\/H], both\nfor individual stars, and for the composite spectrum of each\ncomponent. Measuring individual metallicities from the Ca II triplet\nfor the stars in our survey, with S:N typically between\n5--15$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, is quite problematic. In \\citet{battaglia08}, they\nshow that the `best case' errors in measuring the equivalent widths\nof the Ca II lines, $\\Delta EW$, scale with S:N as:\n\n\\begin{equation}\n\\Delta EW=\\frac{\\sqrt{1.5\\times FWHM}}{S:N}\n\\end{equation}\n\n\\noindent assuming no contamination from residual sky lines and no covariance noise, where\nFWHM is the full width at half-maximum of the CaT lines which is\ntypically 2--3$\\; \\buildrel \\circ \\over {\\mathrm A}$. Using this equation, we can determine the average\nerrors in [Fe\/H] for stars in our sample at different S:N, and we find\nthat for spectra with S:N of (5, 8, 10)$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, the errors in their\ncalculated metallicity are of order (0.6, 0.4, 0.3) dex. To\ndemonstrate the effects of these large errors on the metallicity\ndistribution function (MDF) of our sample, we present histograms of\nthe individual spectroscopic metallicities for three quality cuts, one\nat S:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ (left panel), one with S:N$\\ge8$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$\n(middle panel) and one with S:N$\\ge5$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ (right panel) in\nFig.~\\ref{spechist}. In both plots, the blue hatched histogram\nrepresents our thin disc sample and the filled red histogram\nrepresents our thick disc population. For our higher quality spectra,\nwe calculate a median metallicity for the thin disc of [Fe\/H]=-0.9\nwith a dispersion of 0.4 dex (from the inter quartile range, IQR). For\nour thick disc population we calculate a median of [Fe\/H]=-1.0 also\nwith a dispersion of 0.4 dex. We note that the distributions of both\npopulations deviate from a Gaussian distribution, with a kurtosis of\n-0.6 and -0.3 for thin and thick discs respectively, implying a broad\npeaked distribution, with narrow tails. Both distributions are skewed\ntowards lower metallicity with skewness $\\alpha=-0.4$ and\n$\\alpha=-0.3$ for thin and thick disc. For our lowest S:N cut,\nhowever, much of this information is lost. While the median [Fe\/H]\nremains very similar with [Fe\/H]=-1.0 for the thin disc and\n[Fe\/H]=-1.1 for the thick, the distributions begin to broaden, with\ndispersions of 0.5 dex for both populations, and present almost no\nskew ($\\alpha=-0.1$ and $\\alpha=-0.2$ for thin and thick disc). This\nshows that by including data with larger measurement errors, we wash\nout our MDF considerably, and lose any meaningful information. As a\nsanity check, we compare the MDF for all stars with\nS:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ with one for stars with S:N$\\ge15$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, and\nfind both the median values of [Fe\/H] and general distributions to be\ncomparable. We note that by requiring such a high S:N cut on our\nindividual measurements of [Fe\/H], we bias our sample towards more\nmetal-rich stars as these will have intrinsically stronger Ca II\nlines.\n\nOwing to the large errors associated with these measurements, this\nanalysis provides a crude indication of the metallicities of both\ndiscs, and so to get a more accurate estimate of the average\nmetallicities of both populations we construct composite spectra for\nboth components (using both the 2$\\sigma$ and Gaussian velocity cuts)\nby co-adding the individual spectra of all stars with\nS:N$>3$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, weighted by their S:N values. The resulting S:N of\nthe composite is much greater than the individual spectra\n(S:N$\\sim60-100$ cf. S:N$\\sim3-25$), allowing a better fit to the CaII\nlines. We use a cut of S:N$>3$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ as below this the velocity\nuncertainties of our stars begin to significantly increase (as\ndiscussed in \\S~3). As we shift all spectra to the rest frame before\nco-adding, including spectra where the velocity is uncertain could\nsmear out the Ca II lines, resulting in an over-estimate of [Fe\/H] for\nthe composite. We note that the results from our composites are only\nindicative of an average metallicity for each component, and can tell\nus nothing about the metallicity dispersion for the discs. We display\nthe resulting composites in Fig.~\\ref{spectra}. The top 2 panels show\nthe thin and thick spectra for the 2$\\sigma$ velocity cuts, while the\nbottom 2 panels show the same, but for the Gaussian velocity cuts. In\nthe case of the 2$\\sigma$ cuts, our thin composite comprises 511 stars\nthat match our kinematic and quality criteria, while our thick\ncomposite is constructed from 78 stars. For our Gaussian cuts, these\nnumbers fall to 380 and 52 stars respectively. We find an offset of\norder 0.2 dex between the thick and thin components for the 2$\\sigma$\ncuts, with the thick disc being more metal poor at [Fe\/H]$=-0.9\\pm0.1$\ncompared with [Fe\/H]$=-0.7\\pm0.05$ for the thin, inconsistent within\ntheir respective 1$\\sigma$ errors. For our Gaussian cuts, we find the\nthick disc to be more metal poor, giving us a larger difference in\nmetallicity between the two components of 0.3 dex (with\n[Fe\/H]$=-1.0\\pm0.1$ for the thick disc compared with\n[Fe\/H]$=-0.7\\pm0.05$ for the thin), although the two results for the\nthick disc are consistent within their 1$\\sigma$ errors. We also note\nthat we are liable to experience non-negligible thin disc\ncontamination of our thick disc component, which could cause us to\nover estimate the average [Fe\/H], so the true difference could be\nlarger still. We note that these results are consistent with\nperforming the same analysis on composites constructed from spectra\nwith S:N$>10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$.\n\nFinally, the continuum fit to the third line in our composite spectra,\nparticularly for our thick disc selection, gives us some cause for\nconcern. Could this metallicity difference we derive be driven by poor\ncontinuum fitting in this region of the spectrum? To investigate this,\nwe analyse the [Fe\/H] for the thin and thick discs again, using solely\nthe first two lines (CaII$_{8498}$ and CaII$_{8542}$. In the case of\nour simple $2\\sigma$ cut, this narrows our difference in metallicity\nslightly from 0.2 dex to 0.15 dex, with [Fe\/H]$=-0.85\\pm0.1$ compared\nwith [Fe\/H]$=-0.7\\pm0.05$ for the thick and thin discs\nrespectively. However, in the case of our Gaussian cuts, which are\narguably less affected by cross contamination between the components,\nthe metallicity difference of 0.3 dex persists.\n\n\n\n\nWe also perform this composite analysis on a field-by-field basis. The\nresults of this analysis, shown in Table~\\ref{kprops} are again, less\naccurate than our overall composite, but they suggest a similar offset in\nmetallicity exists in the thin and thick components in each field. We\nplot this result as a function of radius in the lower panel of\nFig.~\\ref{summary}. We find no evidence for any evolution of\nmetallicity with radius.\n\nA slight concern in ascertaining the metallicity of a population from\na composite spectrum arises from inaccuracies in the estimate that\ncome from combining spectra with different effective\ntemperatures and $V$-band magnitudes, as the derived metallicities are\nweakly dependent on the apparent $V$-band colours of the stars. The\nrms dispersion in the $V$-band magnitudes within our sample are small\n($<0.5$ mag for both thin and thick discs) as we are sampling only a\nsmall region of the tip of the RGB, so the error introduced by this\neffect will be very small. However, to further assess this, we\nseparate our thin and thick disc spectra into bins of 0.2 mags in the\n$V$-band and create composite spectra for each bin, measuring the\nmetallicity of each. We show a sample of these spectra in\nFig.~\\ref{binned}, labelled with the metallicity and average $V$-band\nmagnitude. The typical errors in metallicities determined for these\ncomposites ranges from 0.1--0.3 dex. What we see is that the composite\nthick disc spectrum in each bin is more metal-poor than the\ncorresponding thin disc composite. We also find that the average\nmetallicities for both thin and thick discs agree with those that we\nderived from the composites for the entire sample.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{bin_sumspec_new.eps}\n\\caption{Composite spectra of our thin (left panel) and thick (right panel) disc samples, binned in $V$-band magnitude. Each bin spans 0.2 mags. We see that in each case, the thick disc is more metal-poor than the thin disc by $\\sim0.2$ dex. The errors in the values of [Fe\/H] for these composites ranges from 0.1--0.3 dex.}\n\\label{binned}\n\\end{center}\n\\end{figure*}\n   \n\n\\subsection{Photometric Metallicities}\n\nWe inspected the photometric metallicities of our sample using the\nDartmouth isochrone models \\citep{dart08}.  We select an age of 8 Gyrs\nas the work of \\citet{brown06} suggests that the age of the disc in\nthese outer regions varies between 4 and 8 Gyrs. We use an\n$\\alpha$-abundance of [$\\alpha$\/Fe]=+0.2 as it has been shown in\nvarious works (e.g. \\citealt{reddy06,abrito10}) that the\n$\\alpha$-enhancement of thin or extended stellar disc populations\ntypically ranges between [$\\alpha$\/Fe]=+0.0 and [$\\alpha$\/Fe]=+0.2. We\nthen interpolate between these isochrone models for every star within\nour sample to determine its metallicity.  We can then compare the\nMDFs for our thin and thick disc sample, selected by both the\n2$\\sigma$ and Gaussian cuts discussed above. The results of this are\nshown in the left panel of Fig.~\\ref{mdf}. This figure shows us that\nwhen using this set of isochrones, the MDFs of both populations trace\neach other remarkably well. We calculate a median metallicity for each\ncomponent and find [Fe\/H]$_{thin}=-0.79$ and [Fe\/H]$_{thick}=-0.80$,\nboth with IQRs of 0.2 dex. Neither population has a Gaussian\ndistribution, with positive kurtosis of +2.2 for both MDFs (i.e. more\npeaked, with broader tails), and both populations are skewed towards\nlower [Fe\/H] with $\\alpha\\sim-1.2$ for both discs. From this analysis,\none might conclude that the two discs are chemically\nindistinguishable. This is in contrast to our findings from the\ncombined spectra in \\S4.2 where we find an offset in the average\nmetallicities of thin and thick components of 0.2 dex. As our\nphotometric data are not deep enough to detect the MSTO of these\nfields, we are exposed to the age-metallicity-[$\\alpha$\/Fe] degeneracy\nproblem. If we analyse our data with isochrones of different ages and\nabundances, we find that the individual metallicities we measure\nchange. Increasing the age by 2 Gyrs has the effect of decreasing\n[Fe\/H] of a star by $\\sim0.05$ dex on average (shown in the centre\npanel of Fig~\\ref{mdf}) and increasing the abundance from\n[$\\alpha$\/Fe]=+0.2 to [$\\alpha$\/Fe]=+0.4 reduces [Fe\/H] by\n$\\sim0.1$~dex, (right hand panel, Fig.~\\ref{mdf}). The dispersions,\nkurtosis and skew remain largely unchanged by these variations. These\nfindings demonstrate that it may be difficult to discern slight\ndifferences in metallicity (such as the 0.2 dex measured in \\S4.2)\nusing photometric isochrones without knowing the ages and\/or\n$\\alpha$-abundances of the thick and thin disc. Studies of the thin\nand thick discs in the MW have shown that the thick disc is both older\nand more $\\alpha$-enriched than the thin disc\n\\citep{reddy06,abrito10}, and many of the formation scenarios of thick\ndiscs suggest this could be true for thick discs in general, including\nM31. Such differences would certainly affect our derived values of\n[Fe\/H] for both discs.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot_age.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot_alpha.eps}\n\\caption{Photometric MDFs derived from Dartmouth isochrones of varying\n  age and [$\\alpha$\/Fe] \\citep{dart08} for the thin and thick\n  components (shown as blue hatched and red filled histograms\n  respectively) as defined by our 2$\\sigma$ cuts. {\\bf Left panel: }\n  Analysis of [Fe\/H] for thin and disc using [$\\alpha$\/Fe]=+0.2 and an\n  age of 8 Gyrs. We detect no significant differences between the two\n  populations, calculating median [Fe\/H] and\n  [Fe\/H]$_{thick}=-0.8\\pm0.2$. {\\bf Centre panel: } Increasing the age\n  of isochrones used to calculate metallicity for the thick disc from\n  8 Gyrs to 10 Gyrs. An offset of $\\sim0.05$ dex in the average [Fe\/H]\n  of the two components is observed, with median\n  [Fe\/H]$_{thick}=-0.85$. The dispersion remains the same as\n  before. {\\bf Right panel: } Increasing [$\\alpha$\/Fe] for the thick\n  disc from +0.2 to +0.4. An offset of $\\sim0.1$ dex between the\n  median metallicities of the populations is now observed, with\n  [Fe\/H]$_{thick}=-0.93$.}\n\\label{mdf}\n\\end{center}\n\\end{figure*}\n\n\\section{Discussion}\n\nIn this section we discuss our findings, and comment on the\nmorphology of this thick component. First we compare our findings with\nan expected thin+thick disc population inclined to us along the line\nof sight by 77$^\\circ$, by creating a model of a galaxy with a\nthin\/extended disc with similar properties to those of M31 that has an\nadditional thick disc component and analysing it in the same way as\nour data. We then compare the M31 thick disc to the MW and the\n\\citet{yoachim06} sample of thick discs. Finally we comment on the\npossible formation mechanisms for this component.\n\n\\subsection{Comparison with thin + thick disc model}\n\nTo lend confidence to our defining the lagging component we isolate in the above\nanalysis as a thick disc, we create a simple kinematic model\nof a galaxy with properties similar to those of a MW-type galaxy,\nwhich has both a thin and thick stellar disc, and analyse this in the\nsame manner as our data. This is done as follows; first, we create a\nthin stellar disc of 9$\\times10^{6}$ stars, randomly generating radii\nfor each assuming the stars are distributed in an exponential disc\nwith a scale length equal to that of M31's (6.6 kpc,\nI05). We assign each particle with a velocity randomly\ndrawn from a Gaussian population centred on 0${\\rm\\,km\\,s^{-1}}$ with a velocity\ndispersion of 25${\\rm\\,km\\,s^{-1}}$ in the disc frame. We repeat this for our thick\ndisc component, assuming a thin:thick disc density ratio in M31 that\nis equal to that measured in the solar neighbourhood of 9:1\n\\citep{just10}, giving 1$\\times10^6$ stars, and we use a thick disc\nscale length of 8.0 kpc, as determined above. For the velocities, we\nassume the thick disc lags behind the thin by 50${\\rm\\,km\\,s^{-1}}$ and has the\nsame velocity dispersion as the MW thick disc, $\\sigma=40{\\rm\\,km\\,s^{-1}}$\n\\citep{ivezic08}. We also generate vertical heights within the discs\nfor both thin and thick populations, assuming MW scale heights for the\ndiscs (300 pc and 1000 pc for thin and thick disc,\n\\citealt{ivezic08}), $I$-band magnitudes between $25.0\\ge I\\ge20.3$\n(0.1 mags brighter than the tip of the RGB in M31) and angular\npositions within the disc. We then convert our disc frame velocities\ninto heliocentric velocities using the HI rotation curve of\n\\citet{chemin09}. Finally, we interpret this model in the same way as\nour data, by rotating it into the coordinate system of M31 as observed\nfrom the MW (with inclination and PA as discussed in \\S~3), and\nsubtracting off the assumed disc velocity at that position by\ninterpolating each of our model stars into our average disc velocity\nmap (Fig.~\\ref{velmap}).\n\nFrom this model data set, we select stars as we would select targets\nto observe when designing DEIMOS masks, requiring them to have\n$I$-band magnitudes between $22.0\\ge I\\ge20.5$. We then randomly\nselect the same number of stars as are observed at each field\nlocation, and make velocity histograms in the disc-lag frame for each\nfield. We then analyse these distributions with the same GMM technique\ndescribed in \\S~3.1, using a LRT to determine whether the distribution\nof each model field is best fit by a single thin disc component, or a\ndouble thin+thick component. This procedure is repeated 100 times,\nallowing us to compute the average velocities and dispersions for each\ncomponent, plus sampling errors which we tabulate in\nTable~\\ref{mprops}. In our final 100 samples, the thick disc is\ndetected in 15 of the 21 fields on average. The fact that we do not\nsee the thick disc component in all our model fields implies that the\nnon-detections in our data are an effect of our sampling of the DEIMOS\nfields rather than the component being absent in these fields. We show\nthe histograms and best fit Gaussians for three of these realizations\ncompared to our data in Fig.~\\ref{models}.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Average kinematic properties of model fields from 100 MC realisations}\n\\label{mprops}\n\\begin{tabular}{lccccc}\n\\hline\nField & v$_{thin}$ (disc frame, ${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thin} ({\\rm\\,km\\,s^{-1}})$ & v$_{thick}$ (disc frame, ${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thick} ({\\rm\\,km\\,s^{-1}})$ \\\\\n\\hline\n228Mod & 11.4$\\pm5.0$ & 19.2$\\pm8.0$ & -50.0$\\pm8.9$  & 38.3$\\pm8.1$   \\\\\n227Mod & -1.7$\\pm0.6$ & 25.5$\\pm10.8$& -37.1$\\pm15.6$ & 46.3$\\pm12.4$  \\\\\n166Mod & -3.4$\\pm2.1$ & 25.9$\\pm7.3$ & -46.3$\\pm9.1$  & 43.0$\\pm13.4$  \\\\\n106Mod & -4.4$\\pm3.2$ & 25.9$\\pm8.2$ & -54.3$\\pm14.1$ & 42.8$\\pm9.3$   \\\\\n105Mod & -2.1$\\pm1.2$ & 22.0$\\pm9.4$ & -54.6$\\pm13.8$ & 39.4$\\pm12.8$  \\\\\n224Mod & -2.7$\\pm1.9$ & 23.1$\\pm9.9$ & -40.2$\\pm9.2$  & 42.3$\\pm10.3$  \\\\\n232Mod & -3.1$\\pm2.7$ & 21.8$\\pm8.2$ & -45.9$\\pm12.2$ & 41.2$\\pm13.2$  \\\\\n104Mod & -2.5$\\pm1.7$ & 22.8$\\pm10.2$& -57.6$\\pm14.4$ & 43.0$\\pm11.0$  \\\\\n220Mod & -7.0$\\pm3.8$ & 21.1$\\pm10.3$& -62.9$\\pm8.7$  & 36.2$\\pm10.4$  \\\\\n213Mod & -8.1$\\pm4.3$ & 19.5$\\pm9.9$ & -55.1$\\pm13.2$ & 32.2$\\pm13.9$  \\\\\n102Mod & -3.5$\\pm2.7$ & 20.6$\\pm8.9$ & -49.1$\\pm7.9$  & 34.9$\\pm12.4$  \\\\\n231Mod & -3.7$\\pm2.2$ & 19.4$\\pm7.4$ & -49.6$\\pm12.3$ & 33.2$\\pm9.3$   \\\\\n223Mod & -1.8$\\pm1.1$ & 20.8$\\pm7.2$ & -56.1$\\pm11.7$ & 36.7$\\pm8.8$   \\\\\n101Mod &  1.9$\\pm1.3$ & 22.3$\\pm7.0$ & -64.1$\\pm11.2$ & 38.4$\\pm10.8$  \\\\\n222Mod & -0.1$\\pm1.2$ & 19.9$\\pm6.6$ & -51.3$\\pm12.6$ & 34.8$\\pm11.0$  \\\\\n221Mod &  3.9$\\pm2.3$ & 19.9$\\pm8.6$ & -46.2$\\pm7.0$  & 38.1$\\pm11.1$  \\\\\n50Mod  &  4.2$\\pm2.5$ & 17.9$\\pm8.2$ & -42.2$\\pm15.5$ & 33.5$\\pm12.6$  \\\\\n107Mod &  5.1$\\pm2.7$ & 17.5$\\pm7.7$ & -57.1$\\pm11.6$ & 40.4$\\pm12.9$  \\\\\nw11Mod &  4.7$\\pm1.1$ & 19.5$\\pm6.8$ & -63.1$\\pm10.4$ & 43.4$\\pm10.5$  \\\\\n167Mod &  2.0$\\pm0.7$ & 22.5$\\pm5.7$ & -52.2$\\pm8.6$ &  41.6$\\pm9.7$  \\\\\n148Mod & -1.2$\\pm1.5$ & 25.2$\\pm6.8$ & -53.2$\\pm9.7$  & 36.5$\\pm10.3$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.4\\hsize]{velcuts_new.eps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss.ps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss2.ps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss3.ps}\n\\caption{A comparison of the data (top left panel) with 3 realisations\n  of parsing our thin + thick disc model through the same analysis as\n  our data, selecting stars from the same regions as the data. It can\n  be seen that the model data resembles the actual data very closely,\n  and that non-detections are likely an effect of sampling.  }\n\\label{models}\n\\end{center}\n\\end{figure*} \n\n\nWe now assess how both the lag between components and the dispersions\nof each component evolve with radius for our model data set, and how\naccurately we can recover these values from our model. In\nFig.~\\ref{modsummary}, we plot these values for our model (black\ncircles) alongside the values we obtained from our data (red squares),\nand fit the evolution of the model results with linear functions as\nbefore. In the top panel, we show the measurement of $\\Delta v$ for\nour model fields as a function of projected radius. The model results\nshow no evolution of $\\Delta v$ with radius, recovering an average lag across\nall fields of 48.9$\\pm6.7{\\rm\\,km\\,s^{-1}}$, which is similar to the constant lag\nof 50~${\\rm\\,km\\,s^{-1}}$ implemented in our model.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{model_summary.eps}\n\\caption{This figure compares the results from our data with results\n  from our model analysed in the same way. In all cases, data is\n  represented by filled red squares and dot-dashed lines, and the\n  model results are shown as filled black circles and solid\n  lines. {\\bf Top panel:} The difference in velocity, $\\Delta v$,\n  between the thin disc and thick component of data and model as a\n  function of projected radius. The model lag is consistent with no\n  evolution with radius, and shows an average lag of 48.9~${\\rm\\,km\\,s^{-1}}$, very\n  close to our input lag of 50~${\\rm\\,km\\,s^{-1}}$. {\\bf Middle panel: }Dispersion,\n  $\\sigma_{thin}$, of the thin disc is plotted for both data and model\n  as a function of projected radius. The model thin disc is best fit\n  with an average dispersion of 21.5~${\\rm\\,km\\,s^{-1}}$, very close to the input of\n  25.0${\\rm\\,km\\,s^{-1}}$. {\\bf Lower panel} Results for both data and model for\n  the dispersion of the thick disc ($\\sigma_{thick}$) as a function of\n  radius. For our model, the thick disc is consistent with no\n  evolution with radius, unlike our data, and has an average\n  dispersion of $\\sigma_v=38.8{\\rm\\,km\\,s^{-1}}$, which recovers our input\n  dispersion of 40~${\\rm\\,km\\,s^{-1}}$ relatively well. }\n\\label{modsummary}\n\\end{center}\n\\end{figure}\n \n\\begin{table}\n\\begin{center}\n\\caption{Average densities for thin and thick discs in model fields from 100 MC realisations}\n\\label{moddens}\n\\begin{tabular}{lccccc}\n\\hline\nField & $\\rho_{* thin}$ (*\/arcmin) & $\\rho_{* thick}$ (*\/arcmin) \\\\\n\\hline\n228Mod   &73.0$\\pm$35.2  &    31.0$\\pm$    19.2 \\\\\n227Mod   &15.7   $\\pm$    7.1   &    18.4   $\\pm$    8.1 \\\\\n166Mod   &14.2   $\\pm$    7.7   &    14.1   $\\pm$    7.7 \\\\\n106Mod   &11.9   $\\pm$    5.7   &    11.8   $\\pm$    5.7 \\\\\n105Mod   &13.5   $\\pm$    6.3   &    10.8   $\\pm$    5.3 \\\\\n224Mod   &10.6   $\\pm$    5.0   &    10.1   $\\pm$    4.8 \\\\\n232Mod   &10.9   $\\pm$    6.1   &    7.6    $\\pm$    4.5 \\\\\n104Mod   &10.7   $\\pm$    4.9   &    7.9    $\\pm$    3.8 \\\\\n220Mod   &4.7    $\\pm$    1.8   &    4.2    $\\pm$    1.6 \\\\\n213Mod   &4.6    $\\pm$    2.6   &    4.4    $\\pm$    2.4 \\\\\n102Mod   &3.7    $\\pm$    1.5   &    3.8    $\\pm$    1.5 \\\\\n231Mod   &3.8    $\\pm$    1.9   &    2.8    $\\pm$    1.3 \\\\\n223Mod   &2.7    $\\pm$    1.0   &    2.7    $\\pm$    0.9 \\\\\n101Mod   &2.9    $\\pm$    1.1   &    2.8    $\\pm$    1.0 \\\\\n222Mod   &2.1    $\\pm$    0.7   &    2.1    $\\pm$    0.6 \\\\\n221Mod   &1.7    $\\pm$    0.5   &    1.9    $\\pm$    0.5 \\\\\n50Mod    &1.3    $\\pm$    0.3   &    1.2    $\\pm$    0.2 \\\\\n107Mod   &1.4    $\\pm$    0.3   &    1.5    $\\pm$    0.3 \\\\\nw11Mod   &1.4    $\\pm$    0.4   &    1.5    $\\pm$    0.3 \\\\\n167Mod   &0.7    $\\pm$    0.4   &    0.5    $\\pm$    0.4 \\\\\n148Mod   &0.6    $\\pm$    0.4   &    0.5    $\\pm$    0.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{mod_slength_thin.eps}\n\\includegraphics[angle=0,width=0.99\\hsize]{mod_slength_thick.eps}\n\\caption{Results from MC recovery of the scale lengths in our\n  thin+thick disc model for our input thin (top panel) and thick\n  (bottom panel) discs. The error bars on individual points represent\n  the dispersion of calculated densities in the MC analysis, while the\n  shaded regions represent the 1$\\sigma$ uncertainties from the\n  weighted least-squares fit. We recover a scale length for the thin\n  disc of $h_r$=6.2$\\pm0.8{\\rm\\,kpc}$ and h$_r=7.8\\pm0.9$ for the thick\n  disc, which are consistent with our input values.}\n\\label{modscale}\n\\end{center}\n\\end{figure}\n\nNext, we compare the evolution of the disc dispersions for our model\nwith the data, shown in the central and lower panels of\nFig.~\\ref{modsummary}. The model thin disc is best fit with a constant\nrelation, giving an average lag of $\\sigma_{thin}=21.5\\pm1.7{\\rm\\,km\\,s^{-1}}$,\nvery close to the input of 25.0${\\rm\\,km\\,s^{-1}}$. For the thick disc dispersion,\n$\\sigma_{thick}$ the measured dispersion in the model\nscatters about a mean dispersion of $\\sigma_{thick}=38.8\\pm2.4{\\rm\\,km\\,s^{-1}}$,\nwith no evidence of evolution with radius. \n\nFinally, we can use our model to get a handle on how accurate our\nestimates of the scale length of the M31 discs might be. We use the\nsame Monte Carlo (MC) technique above to calculate the density of stars in each\ncomponent in our model fields 100 times, then we compute the average\ndensity from these results. These results are shown in\nTable~\\ref{moddens}, and the errors represent the dispersion of the\ndensities computed in each field. We then plot the densities as a\nfunction of radius for the thin and thick discs (shown in\nFig.~\\ref{modscale}), and fit the result with an exponential profile\nto determine the scale length. For the thin disc, we calculate\nh$_r=6.2\\pm0.8{\\rm\\,kpc}$, which is consistent with our input of 6.6 kpc,\nand for the thick disc we compute h$_r=7.8\\pm0.9{\\rm\\,kpc}$, consistent with\nour input of 8.0 kpc. The shaded regions indicate the 1$\\sigma$\nuncertainties from the fit. These results suggest that our\nobservationally derived scale lengths for the thin and thick discs are\na good indicator of their true scale lengths.\n\n\\subsection{Comparison to the MW and `edge-on' thick discs}\n\nNow that we have characterised the radial profile, kinematics and\nmetallicity of the thick disc in M31, we are able to compare it to the\nproperties of other thick discs that have been observed in the\nuniverse. We shall begin with the most well studied thick disc\ncurrently known -- that of our own Galaxy. Given that these two\ngalaxies are relatively close to one another (separated by 785 kpc),\nand have similar morphologies (both large spiral galaxies),\ncomparisons between the MW and M31 are often made. But for all their\napparent similarities, these two galaxies are quite different from one\nanother. Work by \\citet{hammer07} has shown that the MW is quite\ndifferent in terms of its structure and evolutionary history from the\nmajority of local spiral galaxies, whereas M31 is actually quite\n``typical'', so these differences are perhaps unsurprising. In this\nwork, we have demonstrated that the scale lengths of the M31 discs are\nlarger than those of the MW by a factor of $\\sim2$, as shown in\nFig.~\\ref{height}. Given that we derive the scale heights of the M31\ndiscs from these scale lengths, this results in scale heights in M31\nthat are of order $\\sim3$ times as thick as those of the MW. However,\nwe note that as we calculate scale heights for the M31 disc based on a\nrelation determined from disc galaxies that are quite different in\nterms of their mass to both the MW and M31, our values may be an\noverestimate. The M31 discs are also seemingly hotter than the MW\ndiscs, with $\\sigma_{thin,M31}=32.0{\\rm\\,km\\,s^{-1}}$ cf.\n$\\sigma_{thin,MW}=20.0{\\rm\\,km\\,s^{-1}}$ \\citep{ivezic08}, and\n$\\sigma_{thick,M31}=45.7{\\rm\\,km\\,s^{-1}}$ cf. $\\sigma_{thick,MW}=40.0{\\rm\\,km\\,s^{-1}}$\n\\citep{ivezic08}. This could tell us something about the merger\nhistory of M31. If the thick discs in both galaxies are formed as a\nresult of heating by mergers, the hotter discs of M31 could imply that\nthis galaxy has undergone a more active merger history than the MW.\n\nThe MW thick disc is more metal poor, enriched in $\\alpha$ metals and\nolder than the thin disc. While we are unable to measure the age and\n$\\alpha$ abundances of the M31 discs, we have shown that there exists\nan offset in the average metallicities of the two components of\n$\\sim0.2$ dex when measured spectroscopically. While we do not see\nthis offset photometrically, this could be due to our analysis\ntechnique as we use isochrones of the same $\\alpha$ abundance\n([$\\alpha$\/Fe]=+0.2) and age (8 Gyrs) for both components. If we\nmodify the $\\alpha$-abundance and age of these isochrones to\n[$\\alpha\/$Fe]=+0.4 and 10 Gyrs for our thick disc sample, we see an\noffset of $\\sim0.2$ dex. We also note that both the thin and thick\ndiscs in M31 appear to be more metal-poor than the MW discs, which\nhave average metallicities of [Fe\/H]$\\sim-0.3$ and [Fe\/H]$\\sim-0.6$\n\\citep{gilmore02,abadi03,carollo10} respectively, although there is a\nsignificant metallicity spread in both discs. \\citet{carollo10} also\ndemonstrated evidence of a secondary, more metal poor thick disc in\nthe MW, whose metallicities span the range $-1.8\\le[Fe\/H]\\le-0.8$,\npeaking at [Fe\/H]=-1.3. This component also appears to be hotter than\nthe traditional MW thick disc component, with $\\sigma_z$=44$\\pm3{\\rm\\,km\\,s^{-1}}$,\nvery similar to what we observe in M31.\n\nIn \\S4.1.3, we inferred scale heights for the thin and thick discs of\nM31 of $h_z=1.1\\pm0.2$~kpc and $h_z=2.8\\pm0.6$~kpc respectively, using\na sample of 34 galaxies with thick discs measured by YD06 to determine\na relationship between scale length and scale height of a stellar\ndisc. As we noted in \\S4.1.3, a comparison with the YD06 sample might\nnot be desirable, as these galaxies are typically much less massive\nthan M31, and selected to be bulgeless. A more appropriate comparison\nwould be the MW analogue, NGC 891, an edge-on galaxy that was recently\nthe subject of a structural analysis by \\citet{ibata09} using HST\/ACS\nimaging. They detected the presence of a thick disc component in the\ngalaxy and were able to measure both a scale length and height for\nthis component of $h_r=4.8\\pm0.1{\\rm\\,kpc}$ and $z_0=1.44\\pm0.03{\\rm\\,kpc}$,\ncompared with $h_r=4.2\\pm0.01{\\rm\\,kpc}$ and $z_0=0.57\\pm0.01{\\rm\\,kpc}$ for the\nthin disc component in this galaxy. This gives a ratio of $\\sim1.1$\nbetween the scale lengths and $\\sim2.5$ for the scale heights of these\ncomponents, which is identical to what we observe in M31. To\nillustrate this, we overplot these values for NGC 891 in\nFig.~\\ref{scale} as a green circle.\n\nIn \\citet{yoachim08a} the authors present kinematics of the thin and\nthick discs of 9 of their initial sample of 34 galaxies, obtained\nusing the GMOS spectrograph on Gemini. To measure velocities and\ndispersions in both thin and thick components, they placed slits in\npositions corresponding to the midplane of the galaxy to measure the\nthin disc properties, and above the midplane where the contribution\nfrom the thin disc was thought to be negligible. As their typical\nvelocity resolution was 60${\\rm\\,km\\,s^{-1}}$, they were unable to draw robust\nconclusions on the velocity dispersions of these components, but they\nwere able to measure velocity rotation curves for each component, and\nfound a wide variety of behaviour amongst their thick disc components,\nwith discs which lagged behind the thin disc by only $\\sim5{\\rm\\,km\\,s^{-1}}$,\ndiscs that show no evidence of rotation and one case where the thick\ndisc is counter-rotating with respect to the thin disc. The average\nlag between the thin and thick components of $\\Delta v=46.0{\\rm\\,km\\,s^{-1}}$ we\nsee in the M31 system is larger than the majority that they\nobserve. We note that the galaxies in their sample were typically of\nmuch lower mass than M31 (V$_{circ}<150{\\rm\\,km\\,s^{-1}}$ cf\nV$_{circ}\\sim230{\\rm\\,km\\,s^{-1}}$). In the most massive of their sample (which are\nstill less massive than M31), they do not detect a lag in the thick\ndisc kinematics at all, and they attribute this to contrast\nissues. Their sample were also selected to be ``bulgeless'', unlike\nM31 which has a significant bulge, and so a direct comparison may not\nbe advisable. Owing to the wide range of kinematic behaviour exhibited\nin their sample, they conclude that the dominant formation process of\nthick discs is via minor mergers and accretions of satellites. In\n\\citet{yoachim08b}, they use Lick indices to measure ages and\nmetallicities in 9 low mass galaxies with thick disc components. While\nwe measure an offset of 0.2 dex in the metallicities of the M31 thick\nand thin discs, they were unable to measure any such offset in their\nsample, though this could be a result of the insensitivity of Lick\nindices to such differences at low metallicity. They do find that the\nthick discs are host to older stellar populations than the thin disc,\nhowever with our current data set, we are unable to comment on the\nages of stars in the M31 discs.\n\n\\subsection{Possible formation scenarios}\n\nIn this section, we discuss the various formation scenarios mentioned\nin \\S~1. Owing to our inability to measure ages and vertical dispersions in\nM31, we are not able to confirm or reject any of these formation\nmechanisms at present, so we discuss additional constraints for these\nmodels that could help to rule out or confirm each scenario with\nfurther data and analysis.\n\n\\subsubsection{Heating by minor mergers}\n\nNumerous studies have identified that impacts and mergers of\nsatellites with masses less than a third of their hosts can\nkinematically heat the thin stellar disc, puffing it out into a\nsubstantially thicker disc\n(e.g. \\citealt{quinn93,robin96,walker96,velazquez99,chen01,sales09,villalobos09}). M31\nis known to have recently undergone at least one significant minor\nmerger event, resulting in the GSS tidal stream. In recent work by\n\\citet{purcell10}, the authors model the heating of the stellar discs\nby minor mergers and trace disc stars ejected into the stellar halo by\nthese simulated events. In addition to the stars ejected into the\nhalo, they observe a concomitant increase in the number of stars\nlocated in the kinematic regime of the thick disc, contributing\n$\\sim~10-20$\\% of the total stellar density along the major axis,\nsimilar to what we observe in M31. They also find that their simulated\nplanar infall produces two-component systems with scale heights\n(z$_{thin}\\sim 1~{\\rm\\,kpc}$ and z$_{thick}\\sim3~{\\rm\\,kpc}$), consistent with our\nmeasurements for M31. The similarities between our findings and those\nof \\citet{purcell10} could suggest that thin disc stars heated by the\nmerging event that created the GSS may contribute some non-trivial\nfraction of stars to the thick disc. \n\n According to the simulations of \\citet{kazantzidis09}, thick discs\n produced in this vein imprint a number of dynamical signatures on\n both the kinematic and structural properties of the galaxy. These\n include considerable thickening and heating at all radii, prominent\n flaring, particularly in the outskirts of the disc (beyond 3 scale\n lengths), surface density excesses at large radii, radial\n anisotropies and substantial tilting of the disc. As M31 is not edge\n on, we are unable to comment on the evolution of the height of the\n thick disc with radius, and so we cannot use this as a measure of\n flaring in the outer regions of the disc. However, one might expect\n that if there was a substantial flaring beyond 3 disc scale lengths\n ($\\sim24$~kpc), that this may be reflected by an increase in the\n velocity dispersions of both thin and thick disc components. Our\n results for evolution in the thin and thick disc dispersions remain\n inconclusive, and so it is possible that such flaring may exist. At\n present, we possess few fields between R$\\sim32$ and 39.6~kpc, so\n populating this region with kinematics, as well as additional fields\n further out, may further enlighten us to any potential\n flaring. Another test of this formation scenario would be to include\n fields from both the minor axis and NE portion of M31 to test for any\n radial anisotropy, assuming one can reliably disentangle\n contamination from foreground and substructure from the signatures of\n the discs. The work of \\citet{sales09} also tells us that thick discs\n that are produced as a result of heating present structures with low\n orbital eccentricity.\n\n\\subsubsection{Accretion of satellite on a coplanar orbit}\n\nNumerical simulations by \\citet{abadi03} and \\citet{penarrubia06} show\nthat an old, thick disc of stars could form via the accretion of stars\nfrom satellite galaxies on an approximately coplanar orbit with its\nhost. Such discs are similar in radial extent and contain older\nstellar populations when compared to the thin disc. The thick disc we\nfind in M31 is consistent with this model in so far as the radial\nextents of both discs (5.9--7.3~kpc and 8.0~kpc) are comparable with\none another. They also argue that the mass and luminosity of the\nprogenitor satellite can be inferred from the metallicity of the\ncomponent. We deduce [Fe\/H]$_{thick}=-1.0\\pm0.1$ for M31, which would\ncorrespond to a satellite of $M_V\\sim-15$ ($L_v\\sim9\\times10^7{\\rm\\,L_\\odot}$),\nsimilar to the M31 dwarf elliptical, NGC 147 ($M_V=-15.1$,\n\\citealt{vandenbergh99}). However, given the mass we calculate for the\nthick disc in \\S4.1.4 of 2--4$\\times10^{10}{\\rm\\,M_\\odot}$, it seems very\nunlikely that the thick disc of M31 could have been formed from such a\nsatellite.\n\nResults of the simulations of \\citet{sales09} show that stars accreted\ninto a thick disc from satellites on coplanar orbits exhibit high\neccentricity orbits. Our present data set does not allow us to probe\nthe eccentricity of the orbits within the thick disc at this\ntime. With a larger data set, we could perhaps see the effects of\norbital differences in the form of structural asymmetries.\n\n\\subsubsection{Radial migration and internal heating}\n\nThe scattering of stars by spiral structure and molecular clouds has\nlong been proposed as a method of heating the stellar disc, moving\nstars out onto more eccentric and inclined orbits\n\\citep{sellwood02,haywood08,roskar08,schonrich09a}, and it has been\nargued in \\citet{schonrich09a,schonrich09b} that these processes\nnaturally produce an old, $\\alpha$-enhanced thick disc, whose\nproperties are consistent with those observed in the MW. These models\nalso show wide MDFs and an increase in the scatter of the\nAge-Metallicity relation. This is also demonstrated in\n\\citet{quillen09}, where they investigate radial mixing induced by an\norbiting subhalo. Again they find evidence of wide MDFs in both the\nthin and thick discs. With deeper photometry that allowed us to reach\nthe MSTOs of the two discs we could derive the average ages of these\ncomponents, and high resolution spectroscopy (R$\\sim15,000$) of M31\nthick disc stars that would allow us to determine accurate abundances\nfrom unblended Fe lines for individual stars, we could comment more\nrobustly on the likelihood of such a formation scenario.\n\n\\subsubsection{Thick disc forms thick}\n\n\\citet{kroupa02} posited that thick discs could be formed as a result of \nvigorous star formation in massive star clusters ($\\sim10^5-10^6{\\rm\\,M_\\odot}$) \nduring the period of assembly of the stellar disc. If this is true, \na number of these massive clusters may have survived to the present day, \nand would possess large vertical velocity dispersions. \\citet{kroupa02} \nsuggests that these clusters could be the metal-rich globular cluster system \nin the MW. Once again, owing to the inclination of M31, we are unable to \nmeasure the vertical dispersions of its metal-rich globular cluster system, \nand can therefore neither confirm nor reject this formation model.\n\n\\section{Conclusions}\n\nUsing the DEIMOS spectrograph on the Keck II telescope, we have\nidentified a statistically significant population of stars in M31 that\nlags behind the thin and extended discs by 46.0$\\pm3.9{\\rm\\,km\\,s^{-1}}$. Comparing\nthis with a model of a thin+thick disc system with the same distance\nand inclination as M31 shows this component to be consistent with a\nthick disc component. Analysing its kinematics, we find it to be\nhotter than the thin disc, with average dispersion\n$\\sigma_{thick}=50.8\\pm1.9{\\rm\\,km\\,s^{-1}}$ cf. $\\sigma_{thin}=35.7\\pm1.0{\\rm\\,km\\,s^{-1}}$,\nlarger than the dispersions observed in the MW discs. From composite\nspectra for each component, constructed from highly probable thin and\nthick disc stars (selected using stringent Gaussian cuts) we measure a\nmetallicity offset of $\\sim0.3$ dex between the two disc, with the\nthick disc being metal-poor than the thin disc\n(${\\rm[Fe\/H]}_{thick}=-1.0\\pm0.1$ cf. ${\\rm[Fe\/H]}_{thin}=-0.7\\pm0.05$). The fact\nthat this metallicity offset is not observed when analysing the thin\nand thick disc RGB stars with isochrones of identical age and\n$\\alpha$-abundance suggests that the two populations differ in these\nproperties, with the thick disc likely being older and more enriched\nin $\\alpha$ elements.\n\nWe measure scale lengths for both thin and thick discs, finding\n$h_r=8.0\\pm1.2{\\rm\\,kpc}$ for the thick disc, and $h_r=7.3\\pm1.1{\\rm\\,kpc}$ for\nthe thin disc, comparable to previous estimates. Using the data of\nYD06 we infer scale heights for both discs at $z_0=2.8\\pm0.6{\\rm\\,kpc}$ and\n$z_0=1.1\\pm0.2{\\rm\\,kpc}$ for thick and thin discs respectively. These\nvalues are of order 2--3 times larger than those measured in the MW,\nperhaps suggesting that M31 has undergone more heating than our\nGalaxy. \n\nBy measuring the ratio of the densities of both discs, we are able to\nestimate a mass range for the thick disc component of\n2.4$\\times10^{10}{\\rm\\,M_\\odot}<M_{*,thick}<4.1\\times10^{10}{\\rm\\,M_\\odot}$. This value\nprovides a useful constraint on possible formation mechanisms, as any\nproposed method for forming a thick disc must be able to heat (or\ndeposit) at least this amount of material.\n\nOwing to current limitations within our data set, we are not able to\ndistinguish between the different thick disc formation\nmechanisms. However, with further analysis of this component using our\ncomplete kinematic sample (including regions from the minor axis and\nNE of M31) and spectroscopic follow up of fields where this component\nis strongly observed, we will be able to better understand the\nchemistry of this component and distinguish between various formation\nmechanisms.\n\n\\section{Acknowledgements}\n\nMLMC would like to acknowledge the support of an STFC studentship. GFL\nacknowledges the award of the Cambridge Raymond \\& Beverly Sackler\ndistinguished visitor fellowship. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAstrophysical \\water masers, like other maser species such as OH\nand SiO, are excellent probes of magnetic fields in a variety of\ninteresting regions. The magnetic field strength is derived from\nobservations of maser polarization caused by Zeeman\nsplitting. Polarization observations of the 22~GHz \\water masers in the\ncircumstellar envelopes (CSEs) around evolved stars have made important\ncontributions to the understanding of the magnetic fields of evolved\nstars \\citep[][hereafter V02 and V05a]{V02,V05a}. \\water maser observations have also provided\ninformation on the strength and structure of the magnetic field in\nstar-forming regions \\citep{FG89, S01, S02} and have provided upper limits for the field in the megamaser galaxy NGC 4258 \\citep{M05}.\n\nHowever, the exact relationship between the observed maser\npolarization and the magnetic field strength is not straightforward,\nleading to uncertainties in the determined field strength. It was\nshown in \\citet[][hereafter NW92]{NW92} that the influence of the\ndifferent hyperfine components of the \\water ($6_{16}-5_{23}$)\nrotational transition as well as of the degree of maser saturation is\nsignificant. The effect of maser saturation on the magnetic field\nstrength determination can be as large as a factor of 2 (NW92).\nAdditionally, velocity and magnetic field gradients might play an\nimportant role. While typical \\water maser spot sizes are between\n$\\sim$10$^{12}$ and $10^{13}$~cm \\citep{RM81, I97}, the actual size of\nthe maser region will be several times larger due to beaming\n\\citep[e.g.][]{GK72}. As a result the maser path length could be as\nlong as $10^{14}-10^{15}$~cm. Across such a path length, velocity and\nmagnetic field gradients likely influence the observed\npolarization. In \\citet[][hereafter V05b]{V05b} it was shown that\nvelocity gradients alter the shape of the total intensity line\nprofile. From this analysis and from earlier observations\n\\citep[e.g.][]{R99} it is apparent that in CSEs velocity shifts of\n$\\sim$1~\\kms are common. In this paper the effects of velocity and\nmagnetic field gradients along the maser path on the magnetic field\nstrength determined from circular polarization measurements and on the\nobserved fractional linear polarization are examined.\n\n\\begin{figure*}[ht!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig1.eps}}\n   \\hfill\n\\caption[]{Normalized total intensity (I) and circular polarization (V) spectra for $v_{\\rm th}=0.6, 1.0$ and $1.5$~\\kms in the presence of a velocity gradient. From thick to thin the solid lines denote velocity shifts of $\\Delta V_m = 0, 0.5, 1.0$ and $1.5$~\\kms respectively except for $v_{\\rm th}=0.6$~\\kms where $\\Delta V_m = 1.5$~\\kms is omitted.}\n\\label{Fig:ex}\n\\end{figure*}\n\n\\section{Background}\n\\label{back}\n\nSimilar to the analysis in V02 and V05b, the equations of state for\nthe populations of the upper ($6_{\\rm 16}$) and lower ($5_{\\rm 23}$)\nrotational levels of the three strongest hyperfine components\n($F-F'=7-6, 6-5$ and $5-4$) of the 22~GHz \\water maser have been\nsolved using the method described in NW92. The equation of state\nfor the number density $n(F,a,\\nu)$ of the upper energy levels ($F$)\nis\n\n\\begin{eqnarray}\n0 & = & \\lambda_F(v_s) - (\\Gamma+\\Gamma_v)n(F,a,v) \\nonumber \\\\\n& & + {R(F,F',a,b,v)}(n(F',b,v) - n(F,a,v)) \\nonumber \\\\ \n& & + \\phi(v_s)({\\Gamma_v \\over \\sum g_{F}})\\int dv\\sum g_{F} n(F,a,v).\n\\label{eq1}\n\\end{eqnarray}\n\nThe equation for the number density $n(F',b,\\nu)$ of the lower levels\n$(F')$ is similar but reversed. Here $a$ and $b$ denote the different\nmagnetic sub-states of each hyperfine component. The population levels\nare solved as a function of molecular velocity $v$ and at different\npositions along the maser propagation direction. The statistical\nweights are designated by $g_F$ and $g_F'$, and are from\n\\citet{K69}. The pump rate $\\lambda_F(v_s)$ is assumed to be the same\nfor the different hyperfine components and has a Maxwellian\ndistribution.  The results of the calculations depend only on the the\nratio $(\\lambda_F - \\lambda_F') \/ \\lambda_F$, which is of the order of\na few percent \\citep{AW93}. Generally $v_s=v$, but when velocity\ngradients are introduced later, the molecular velocity of the maser\npump $v_s$ changes along the maser path. The rate for stimulated\nemission $R(F,F',a,b,v)$ (hereafter $R$) is calculated using the local\nmaser intensity and the hyperfine interaction coefficients as\ndescribed in NW92. $\\Gamma$ is the decay rate for the molecular\nexcitations.  The cross-relaxation rate $\\Gamma_\\nu$ describes the\nreabsorbtion of previously emitted infrared pump photons trapped in\noptically thick transitions of the maser system, which generate a\nnewly excited molecule at random, within a Maxwellian velocity\ndistribution ($\\phi(v_s)$). \nElastic collisions between the maser molecules and intermixed H$_2$\nmolecules have not been included. The effects of these are described\nin detail in \\citet{E90} and are unimportant for the \\water masers\ndiscussed here.\n\n\\begin{figure*}[t!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig2.eps}}\n   \\hfill\n\\caption[]{The dependence of the magnetic field strength determined from circular polarization observation, represented by the $A_{F-F'}$ coefficient from Eq.~\\ref{eq2}, on the magnitude of the velocity gradient $\\Delta V_m$ for two values of intrinsic thermal line width $v_{\\rm th}$. $A_{F-F'}$ is normalized by $A^0_{F-F'}$, which is $A_{F-F'}$ for $\\Delta V_m=0$~\\kms. The solid lines represents masers with different emerging brightness temperatures. Note that the noise ($<1$\\%) on the curves is due to the velocity resolution of our model spectra. The results are independent of the magnetic field angle $\\theta$.}\n\\label{affvsv}\n\\end{figure*}\n\n\nThe maser intensity, which is solved using the radiative transfer\nequations from NW92, influences the level populations through\n$R$. However, as it is itself dependent on the level populations\ndetermined with Eq.~\\ref{eq1}, the maser total intensity (I), linear\npolarization (Q) and circular polarization (V) are solved iteratively\nalong the maser path. Assuming nearly one-dimensional maser\npropagation the beaming of the maser radiation is represented by the\nsolid angle $\\Delta\\Omega$. The emerging maser fluxes will thus be\nrepresented in $T_{\\rm b}\\Delta\\Omega$, with $T_{\\rm b}$ the\nbrightness temperature. All calculations were performed with a\nbrightness temperature for the radiation incident on the maser region\nof $T_{\\rm b}\\Delta\\Omega = 0.1$ K~sr. It was verified in NW92 and V02\nthat the results are insensitive to the chosen initial value. The\ncalculations are performed for different thermal line widths ($v_{\\rm\nth}$) of the Maxwellian particle velocity distribution, where $v_{\\rm th}\\!\\approx$0.5$ (T\/100)^{1\/2}$~\\kms with $T$ the temperature of the masing\nmolecules in $K$. \n\\citet{NW91} and NW92 have shown that when Eq.~\\ref{eq1} is\nsolved without including the contribution of the cross-relaxation rate\n$\\Gamma_\\nu$, the results for the line profiles, fractional\npolarizations and line width in the case of non-negligable\n$\\Gamma_\\nu$ can be obtained by scaling the results with\n$[\\Gamma+\\Gamma_\\nu]$. Following this result, the calculations in this\npaper are performed for $\\Gamma=1$~s$^{-1}$ and\n$\\Gamma_\\nu=0$~s$^{-1}$. Thus, in all the results of this\npaper, the emerging brightness temperature $T_{\\rm b}\\Delta\\Omega$ can\nbe replaced with $T_{\\rm b}\\Delta\\Omega \\times [\\Gamma+\\Gamma_\\nu]$ to\nobtain the results for different values of the decay and\ncross-relaxation rates.\n\nTo solve the equations of state it is assumed that the Zeeman\nfrequency shift $g\\Omega$ is much greater than $R$, $\\Gamma$ and\n$\\Gamma_\\nu$. Then the off-diagonal elements of the density matrix\ndescribing the molecular states are negligible, greatly simplifying\nthe calculations. For a magnetic field strength of $\\sim$1.0~G,\n$g\\Omega$ of the dominant hyperfine transitions is\n$\\sim$10$^4$~s$^{-1}$ while for $T_{\\rm b}\\Delta\\Omega < 10^{12}$~K~sr\nthe rate for stimulated emission $R \\leq 100$~s$^{-1}$ (V02). Most of\nthe calculations in this paper are performed for a magnetic field\nstrength of $B=1$~G. For the highest $T_{\\rm b}\\Delta\\Omega$, the\nresults will be valid for fields down to $\\sim$10~mG, when $g\\Omega$\nbecomes comparable to $R$. However, the low observed linear polarization of the 22~GHz \\water masers (e.g. V02, V05a) indicates that\n$T_{\\rm b}\\Delta\\Omega$ is at most $10^{11}$~K~sr and typically even lower. Thus, the results presented in this paper are valid for $B$ well below $1$~mG.\n\n22~GHz \\water masers begin to saturate when $R\/\\Gamma \\gtrsim\n  1$~s$^{-1}$ and approach full saturation when $R\/\\Gamma \\gtrsim\n  100$~s$^{-1}$. Thus, for our models with $\\Gamma=1$~s$^{-1}$,\nsaturation starts at $T_{\\rm b}\\Delta\\Omega \\gtrsim 10^{10}$~K~sr and\nfull saturation occurs for $T_{\\rm b}\\Delta\\Omega \\gtrsim\n10^{12}$~K~sr. The maser spectral line profile (through rebroadening)\nand the polarization properties of the maser changes at\n$R=(\\Gamma+\\Gamma_\\nu)$ (e.g. NW92). While typically for the 22~GHz\n\\water masers this is close to when saturation occurs, a large\ncross-relaxation rate means the changes will only occur when the maser\nhas already become partly saturated. As $\\Gamma_\\nu$ for the\n  astrophysical \\water masers is at most $\\approx 5$~s$^{-1}$ for\n  \\water masers at $T\\sim 1000~K$ and is less than that lower\n  temperatures \\citep{AW93}, $\\Gamma_\\nu\\approx\\Gamma$ is assumed in\n  this paper when discussing unsaturated and saturated masers. It is\n  worth noting that \\water maser observations indicate that in\n  actuality $\\Gamma$ is between one and two orders of magnitude less\n  than $\\Gamma_\\nu$ \\citep{AW93}. However, as our results scale\n  linearly with $(\\Gamma+\\Gamma_\\nu)$ this does not affect the\n  conclusions in this paper.\n\n\\begin{figure*}[t!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig3.eps}}\n   \\hfill\n\\caption[]{a) Derived magnetic field strength $B'$ over the average field strength along the maser $\\langle B\\rangle$ as a function of $T_{\\rm b}\\Delta\\Omega$. The solid and dashed lines correspond to positive and negative magnetic field gradients respectively. b) $A'_{F-F'}$ as a function of $\\Delta\\theta$ for different values of emerging brightness temperature $T_{\\rm b}\\Delta\\Omega$. Calculations were performed for $v_{\\rm th}=0.8$~\\kms.}\n\\label{bchange}\n\\end{figure*}\n\n\\section{Radiative Transfer Modeling Results}\n\n\\subsection{Circular Polarization}\n\nAs shown in NW92 and V02, the V-spectra produced from the radiative\ntransfer equations are not anti-symmetric. They are also not always\ndirectly proportional to the derivative of the I-spectrum due to\nhyperfine interaction as is often assumed. However, as was discussed\nin V02, it is impossible to directly observe the intrinsic shape of\nthe V-spectrum due to instrumental effects and necessary data\ncalibration steps. Still, the percentage of circular polarization\n$P_{\\rm V}$ can be related to the magnetic field strength using\n\\begin{eqnarray}\nP_{\\rm V} & = & (V_{\\rm max} - V_{\\rm min})\/I_{\\rm max} \\nonumber\\\\\n& = & 2\\cdot A_{F-F'}\\cdot B_{\\rm [Gauss]} \\rm{cos}\\theta\/\\Delta v_{\\rm L}[\\rm{km~s^{-1}}].\n\\label{eq2}\n\\end{eqnarray}\nHere $V_{\\rm max}$ and $V_{\\rm min}$ are the minimum and maximum of\nthe V-spectrum. $I_{\\rm max}$ and $\\Delta v_{\\rm L}$ are the peak flux\nand the full width half-maximum (FWHM) of the I-spectrum respectively.\n$B$ is the magnetic field strength at an angle $\\theta$ to the maser\npropagation direction. While in most analyses of maser circular\npolarization, the coefficient $A_{F-F'}$ is taken to be a fixed value\n\\citep[e.g][]{FG89}, it was shown in NW92 to depend on the level of\nmaser saturation and the intrinsic thermal velocity $v_{\\rm th}$ of\nthe maser. Figure 6 of V02 shows $A_{F-F'}$ for different values of\n$v_{\\rm th}$ as a function of emerging maser brightness temperature\n$T_{\\rm b}\\Delta\\Omega$. \nFor maser brightness temperatures $T_{\\rm b}\\Delta\\Omega >\n10^9$~K~sr it was shown in NW92 that the ${\\rm cos}\\theta$ dependence\nof Eq.~\\ref{eq2} breaks down effectively also introducing a dependence\non $\\theta$ to $A_{F-F'}$. This was later shown in more detail in\n\\citet{WW01} for masing involving angular momentum $J=$1--0 and\n$J=$2--1 transitions. In V02, figure 7 shows the derived magnetic\nfield strength dependence on $\\theta$ to $A_{F-F'}$ for the 22~GHz\n$J=$6--5 transition.\n\n\\subsubsection{velocity gradients}\n\n\\begin{figure*}\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig4.eps}}\n   \\hfill\n\\caption[Linpol]{a) Magnetic field angle gradient $\\Delta\\theta$ vs. the fractional linear polarization for three different stages of maser saturation and $\\theta_0 = 0^\\circ$. b) The angle $\\theta$ between the maser propagation direction and the magnetic field vs. the fractional linear polarization for different values of emerging maser brightness. The thick solid line denotes the theoretical limit from \\citet{GKK} for a completely saturated maser.}\n\\label{Fig:linpol}\n\\end{figure*}\n\nFirst, a velocity gradient is introduced by uniformly shifting, in\nEq.~\\ref{eq1}, the velocity $v_s$ of the pump rate $\\lambda_F$ for\neach integration step along the maser path. By varying the amount of\nshift, a velocity difference $\\Delta V_m$ is created between the start\nand end of the maser amplification path of up to $\\sim$1.5~\\kms. This\ncorresponds to a velocity gradient of $\\Delta V_m \/ S$~\\kms m$^{-1}$,\nwhere $S$ is the length of the maser path. As $S$ depends on the exact\nvalues of the pumping rate $\\lambda_F$, while our results, expressed\nin emerging brightness temperature are only dependent on the ratio of\npumping rates as addressed above, any mention of the velocity gradient\nwill henceforth be referring to the value of $\\Delta V_m$. In\n\\citet{NW88} it was shown that the maser splits into several\ndistinctly separate narrow features when $\\Delta V_m$ becomes a few\ntimes $v_{\\rm th}$, each of the features becoming an independent\nmaser. The calculations in this paper are therefore limited to values\nof $\\Delta V_m$ up to $\\sim$1.5~\\kms.\n\nFig.~\\ref{Fig:ex} shows the effect of a velocity gradient on the total\nintensity and circular polarization profiles for three different\nintrinsic thermal velocities $v_{\\rm th}$, produced for the same\nmagnetic field strength (1~G). While for low $\\Delta V_m$, the effect\nof the velocity gradient on the shape of the I-profile is small, the\neffect on the V-spectrum can still clearly be seen. To examine how a\nvelocity gradient influences the magnetic field determination from the\ncircular polarization measurements, the normalized $A_{F-F'}$\ncoefficient from Eq.~\\ref{eq2} is plotted in Fig.~\\ref{affvsv} as a\nfunction of velocity gradient for $v_{\\rm th}=0.6$ and $1.5$~\\kms and\nemerging brightness temperatures $T_{\\rm b}\\Delta\\Omega = 10^9,\n10^{10}$ and $10^{11}$~K~sr. Because of the normalization, the\nresults are independent of the magnetic field angle $\\theta$. For\n$v_{\\rm th}=0.6$~\\kms the effect of the hyperfine components is\nclear. For $T_{\\rm b}\\Delta\\Omega = 10^9$~K~sr none of the hyperfine\ncomponents are becoming saturated yet, thus $A_{F-F'}$ is mostly\nsymmetric around $\\Delta V_m = 0$~\\kms. The symmetry breaks down when\nthe hyperfine lines are slowly starting to saturate for $T_{\\rm\nb}\\Delta\\Omega \\gtrsim$10$^{10}$~K~sr, as maser growth is somewhat\ninhibited for velocity gradients in the direction of the weaker\nhyperfine components ($F-F'=6-5$ and $5-4$), while a gradient in the\nopposite direction allows for stronger maser amplification. As\nnoted above this effect can be postponed further into the saturated\nregime when $\\Gamma_\\nu$ increased. It can be seen in the figure, that\n$A_{F-F'}$ typically decreases for increasing $|\\Delta V_m|$, implying\nthat in the presence of velocity gradients the magnetic field strength\n$B'$ derived with Eq.~\\ref{eq2} underestimates the true $B$. For small\n$v_{\\rm th}$ the field strength can sometimes be underestimated by\nmore than a factor of $2$. However, for a narrow range in $\\Delta\nV_m$, $A_{F-F'}$ is actually enhanced. In those cases $B$ could\nactually be overestimated by up to $\\sim$20\\%. This effect is largest\nfor the more saturated masers.\n\n\\subsubsection{magnetic field gradients}\n\nThere are different ways a magnetic field gradient along the maser\namplification path can occur. The actual field strength $B$ can change\nor the angle $\\theta$ between the field and the maser propagation\ndirection can vary. Similar to $\\Delta V_m$, differences\n$\\Delta\\theta$ and $\\Delta B$ are introduced between the start and end\nof the maser path and their influence on the maser polarization is\nexamined. Fig.~\\ref{bchange}a shows the ratio between the magnetic\nfield $B'$ determined with Eq.~\\ref{eq2} and the average magnetic\nfield $\\langle B\\rangle$ along the maser as a function of $T_{\\rm\nb}\\Delta\\Omega$ when a gradient $\\Delta B$ is included. The shape of\nthis function does not depend on the actual value of $\\Delta B$ or on\nthe intrinsic thermal line width $v_{\\rm th}$. While the maser is\nunsaturated, or more precisely while $R<\\Gamma_\\nu$, $B'$ is\nequal to $\\langle B\\rangle$. However, when the maser\nsaturates, $B'$ will be dominated by the average magnetic field in the\nunsaturated maser region.\n\nFig.~\\ref{bchange}b shows the change in $A'_{F-F'}$ versus\n$\\Delta\\theta$ for different $T_{\\rm b}\\Delta\\Omega$. The angle\n$\\theta_0$, the value of $\\theta$ at the start of maser amplification\nis taken to be $0^\\circ$. Here $A'_{F-F'} = A_{F-F'}$cos$\\theta$, as\nthe actual angle $\\theta$ cannot be specified due to the inclusion of\nthe gradient $\\Delta\\theta$. Comparing the unsaturated maser ($T_{\\rm\nb}\\Delta\\Omega = 10^9$~K~sr) with the partly saturated maser ($T_{\\rm\nb}\\Delta\\Omega = 10^{11}$~K~sr) it is again apparent that, as seen in\nFig.~\\ref{bchange}a, the unsaturated maser regime provides the\ndominant contribution to the magnetic field strength determined with\nEq.~\\ref{eq2}. Comparing Fig.~\\ref{bchange}b with figure 7 from V02\nindicates that the angle average $\\langle\\theta\\rangle$ in the\nunsaturated maser region determines the observed circular\npolarization.\n\n\\subsection{Linear Polarization}\n\nThe linear polarization in the presence of a magnetic field was also\ndiscussed in NW92. The fractional linear polarization is not found to\nbe affected when including a velocity gradient $\\Delta V_m$ or\nmagnetic field gradient $\\Delta B$ in the calculations. Only when\nincluding a gradient $\\Delta\\theta$ does the linear polarization\nchange. In Fig.~\\ref{Fig:linpol}a the fractional linear polarization\n$|Q_0|\/I_0$ is shown as a function of $\\Delta\\theta$.  This can be\ncompared with $|Q_0|\/I_0$ as a function of the angle between the\nmagnetic field and the maser propagation direction $\\theta$ shown in\nFig.~\\ref{Fig:linpol}b. This relationship, in the limiting case of a\ncompletely saturated maser, was solved in \\citet{GKK}. In the case of\nan unsaturated maser ($T_{\\rm b}\\Delta\\Omega \\lesssim 10^{10}$~K~sr), the\nrelationship between $|Q_0|\/I_0$ and the magnetic field angle gradient\n$\\Delta\\theta$ is very similar to the relationship between $|Q_0|\/I_0$\nand $\\theta$, while when the saturation level increases the\nlinear polarization is quenched and shifted. This indicates that the\nobserved fractional linear polarization is determined by $\\theta$ at\nthe end of the unsaturated maser regime, where the largest\namplification occurs, and not by the average $\\langle\\theta\\rangle$\nover the unsaturated maser core.\n\n\\section{Conclusions}\n\\label{concl}\n\nUsing a maser radiative transfer code which includes the three\nstrongest hyperfine components of the 22~GHz \\water $6_{\\rm 16} -\n5_{\\rm 23}$ rotational transition and their magnetic sub-states, the\neffects of velocity and magnetic field gradients on the determination\nof magnetic fields from maser polarization observations have been\ncalculated.  It was shown that, due to velocity gradients, the\nmagnetic field strength determined on \\water masers in CSEs (V02,\nV05a), where $v_{\\rm th}\\!\\approx$0.8$-1.0$~\\kms and where velocity\ngradients of $\\sim$1.0~\\kms have been observed (V05b), can be\nunderestimated by up to a factor of 2. However, for maser features\nwith smaller velocity gradients there will be cases where $B$ is\nactually overestimated by $\\sim$10$-20\\%$. The magnetic fields\ndetermined on the \\water masers in star-formation regions \\citep{FG89,\nS01, S02}, which have higher thermal line widths ($v_{\\rm\nth}\\!\\approx$1.5~\\kms) can be underestimated by $\\sim$20$-40\\%$. These\nuncertainties can only be overcome by including a consistent model for\nthe velocity gradients throughout the \\water maser source when\nderiving the magnetic field strengths derived from circular\npolarization observations.\n\nWhen there is a magnetic field gradients along the maser amplification\npath, $B'$ determined from polarization observations of an unsaturated\nmaser (more formally when $R\\lesssim\\Gamma_\\nu$), is typically\nequal to the average field along the maser. When the maser is\nsaturated, the $B'$ is dominated by the average magnetic field\nstrength or the average magnetic field angle in the unsaturated maser\ncore. The fractional linear polarization on the other hand, is not\naffected by velocity or magnetic field gradients. Only a gradient in\nthe angle between the magnetic field and the maser propagation axis\nalters the linear polarization. In contrast to the circular\npolarization, the fractional linear polarization is mainly determined\nby the magnetic field angle in the part of the unsaturated maser where\nthe largest amplification occurs and not by the angle averaged over\nthe entire unsaturated maser region.\n\n\\begin{acknowledgements}\nThe author acknowledges the helpfull comments by the anonymous\nreferee.  This work has been supported by an EC Marie Curie Fellowship\nunder contract number MEIF-CT-2005-010393.  \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA {\\em mixed searching game} is defined in terms of a\n graph representing a system of tunnels where an agile and omniscient fugitive with \nunbounded speed is hidden (alternatively, we can formulate the same problem \nconsidering that the tunnels are contaminated by some poisonous gas). \nThe fugitive is occupying the edges of the graph and the searchers can be placed \non its vertices. In the beginning of the game, the fugitive chooses some edge\nand there are no searchers at all on the graph.\nThe objective of the searchers is to deploy a search strategy on the graph that \ncan guarantee the capture of the fugitive. The fugitive is {\\em captured}  if at some \npoint he resides on an edge $e$ and one of the following capturing cases occurs.\n\n\\begin{itemize}\n\\item[{\\bf A}:] {\\em both  endpoints of $e$ are occupied by a searcher,}\n\\item[{\\bf B}:] {\\em a searcher slides along $e$}, i.e., a searcher is moved from one\nendpoint of the edge to the other endpoint. \n\\end{itemize}\n \n\\noindent A {\\em search strategy} on a graph $G$ is a finite sequence ${\\cal S}$ containing  \nmoves of the following types.\n\n\\begin{itemize}\n\\item[]\\hskip-0.5cm{\\sf p}$(v)$: placing a new searcher on a vertex $v$, \n\\item[]\\hskip-0.5cm{\\sf r}$(v)$: deleting a searcher \nfrom a vertex $v$, \n\\item[]\\hskip-0.5cm{\\sf s}$(v,u)$: sliding a searcher on $v$ along the edge $\\{v,u\\}$ and placing it on $u$. \n\\end{itemize}\nWe stress that the fugitive is {\\sl agile} and {\\sl omniscient}, i.e. he moves at any time \nin the most favourable, for him, position and is {\\sl invisible}, i.e. the searchers strategy \nis given ``in advance'' and does not depend on the moves of the fugitive during it.\\medskip\n\nGiven a search ${\\cal S}$, we denote by $E({\\cal S},i)$ the set of edges that are clean \nafter applying the first $i$ steps of ${\\cal S}$, where by ``clean'' we mean that the search strategy\ncan guarantee that none of its edges will be occupied by the fugitive after the $i$-th step.\nMore formally,  we set $E({\\cal \nS},0)=\\emptyset$ and in step $i>0$ we define  $E({\\cal S},i)$\nas the set defined as follows: first consider the set $Q_{i}$ containing  \nall the edges in $E({\\cal S},i-1)$ plus the edges of\n$E^{(i)}$ the set of edges that are  cleaned after the $i$-th move because of the application of  cases ${\\bf A}$ or ${\\bf B}$.\nNotice that $E^{(i)}$ may be empty. In particular, it may be non-empty  in \ncase the $i$-move is a placement move, will always be  empty\nin case the $i$-th move is a removal move and  will surely be non-empty in case the $i$-th move\nis a sliding move. In the third case, the edge along which the sliding occurs \nis called {\\em the sliding edge} of $E^{(i)}$.\nThen, the set $E({\\cal S},i)$ is defined as the set of all edges in $Q_{i}$\nminus those for which there is a path starting from them \nand finishing in an edge not in $Q_{i}$. This expresses the fact that the agile and omniscient fugitive \ncould use any of these paths in order to occupy again some of the edges \nin $Q_{i}$.\nIn case $E({\\cal S},i)\\subset Q_{i}$, we say that the $i$-th move is a {\\em recontamination move}. Notice that in such a \ncase we have that $E({\\cal S},i-1)\\not \\subseteq E({\\cal S},i)$.\n\nThe object of a mixed search is to clear all edges using a search. We\ncall search ${\\cal S}$ {\\em complete} if at some step all  edges of $G$ are clean, i.e.\n$E({\\cal S},i)=E(G)$ for some $i$.\n \n \\paragraph{Connected monotone mixed search number}\n \nThe mixed search number of a search is the maximum number of searchers on the graph during \nany move. \n A search without recontamination moves is \ncalled {\\em monotone}. Mixed search number has been introduced in~\\cite{BienstockS91mono}. \nThe mixed  search number, $\\s(G)$, of a \ngraph $G$ is the minimum mixed search number over all the possible complete searches on it \n(if $G$ is an edgeless graph, then \nthis number is $0$). \nA search is {\\em connected} if $E({\\cal S},i)$ induces a \nconnected subgraph of $G$ for every step $i$. Given a graph $G$, we will denote the minimum \nmixed search number over all the possible complete connected searches on it by ${\\bf cms}(G)$ and \nwe will call this number connected mixed  search number of \n$G$. The monotone (resp. connected monotone) mixed search number, $\\ms(G)$ (resp. \n${\\bf cmms}(G)$), of $G$ is the minimum mixed search number over all the possible complete \nmonotone (connected monotone) searches of it (connected variants are defined only under \nthe assumption that $G$ \nis a connected graph). The concept of connectivity in graph searching was introduced for the \nfirst time  in~\\cite{BarriereFFFNST12conn}\nand was motivated by application of graph searching where the ``clean\" territories  should be \nmaintained connected so to guarantee the safe communication between the searchers during \nthe search.\n\n\\paragraph{Obstructions}\n\nGiven a graph invariant ${\\bf p}$, a partial ordering relation on graphs $\\unlhd$, and an integer \n$k$ we denote by \n${\\bf obs}_{\\unlhd}({\\cal G}[{\\bf p},k])$ the set of all $\\unlhd$-minimal graphs $G$ where ${\\bf p}(G)>k$\nand we call \nit {\\em the $k$-th $\\unlhd$-obstruction set for ${\\bf p}$}. We also say that {\\em ${\\bf p}$ is closed \nunder $\\unlhd$} if for every two graphs $H$ and $G$, $H\\unlhd G$ implies \nthat ${\\bf p}(H)\\unlhd {\\bf p}(G)$.\nClearly, if ${\\bf p}$ is closed under $\\unlhd$, then the  $k$-th $\\unlhd$-obstruction set for ${\\bf p}$\nprovides a complete characterisation for the class ${\\cal G}_k=\\{G \\mid {\\bf p}(G)\\unlhd k\\}$:\na graph belongs in ${\\cal G}_{k}$ iff none of the graphs in the $k$-th $\\unlhd$-obstruction set for \n${\\bf p}$ is contained in $G$ with respect to the relation $\\unlhd$.\n\n\\paragraph{Our results}\n\nIn this paper we are interested in obstruction characterisations for the graphs with bounded \nconnected (monotone) mixed search number.\nWhile it is known that ${\\bf ms}$ is closed under taking of minors, this is not he case for ${\\bf cms}$ \nand ${\\bf cmms}$ \nwhere the connectivity requirement applies.  From Robertson and Seymour \nTheorem~\\cite{RobertsonS-XX}, the  $k$-th $\\leq$-obstruction \nset for ${\\bf ms}$ is always finite, {where $\\leq$ is the minor partial ordering relation (defined \nformally in Subsection~\\ref{cont8e})}. Moreover this set \nhas been found for $k=1$ (2 graphs) and $k=2$ (36 graphs) in~\\cite{TakahashiUK95b}. \nHowever, no such result exists for the  obstruction characterisations of the connected monotone \nmixed search number.\nAs we prove in this paper, ${\\bf cms}$ and ${\\bf cmms}$ are closed under contractions. \nUnfortunately, graphs are not well quasi ordered with respect to the contraction relation, \ntherefore there is no guarantee that \nthe $k$-th contraction obstruction set for ${\\bf cms}$ or ${\\bf cmms}$ is finite for all $k$. The finiteness \nof this set \nis straightforward if $k=1$ as ${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},1])=\\{K_{3},K_{1,3}\\}$. \nIn this paper we  completely resolve the case where $k=2$. We prove that \n${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])={\\bf obs}_{\\preceq}({\\cal G}[{\\bf cms},2])$ and we prove that this set  is \nfinite by providing\nall 177 graphs that it contains. The proof of our results is based on a series of lemmata that \nconfine the structure of the \ngraphs with connected monotone mixed search number at most 2. We should stress that, \nin contrary to the case of ${\\bf ms}$ \nthe direction of searching is crucial for ${\\bf cmms}$. This makes the detection of the \ncorresponding obstruction sets \nmore elaborated as special obstructions are required in order to force a certain sense of direction \nin the search strategy.\nFor this reason, our proof makes use of a more  general \nvariant of the mixed search strategy that forces the searchers to start and finish to specific sets of \nvertices. Obstructions for this more general type of searching are combined in order to form the \nrequired obstructions for ${\\bf cmms}$. We also give a double exponential lower bound on the size of \nthe contraction obstruction set for the classes with bounded connected monotone search number. \nThis lower bound is only meaningful for the classes where this obstruction set is finite. \n\n\\section{Preliminary Definitions and Results} \n\\label{sec:definitions}\n\nLet $A$ be a set and let ${\\cal A}=\\langle a_{1},\\ldots,a_{r}\\rangle$\nbe an ordering of $A$. We denote by ${\\bf prefsec}({\\cal A})$\nthe ordering $\\langle A_{0},\\ldots,A_{r}\\rangle$ of  subsets of $A$, where \n$A_{0}=\\emptyset$ and for $i=1,\\ldots,r$,   $A_{i}=\\{a_{1},\\ldots,a_{i}\\}$. Let ${\\cal A}_1$ and \n${\\cal A}_2$ \nbe two disjoint orderings of  $A$, we denote by ${\\cal A}_1\\oplus {\\cal A}_2$ \nthe concatenation of these two orderings.\n\nAll graphs under consideration will be finite, without loops or multiple edges.\nLet $G$ be a graph and  $e=\\{u,v\\}\\in E(G)$ an edge. The {\\em edge-contraction} of $\\{u,v\\}$ \n(or just the {\\em contraction} of $\\{u,v\\}$) is the operation  that deletes this edge, adds a new \nvertex $x_{uv}$ and connects this vertex to all the neighbours of $u$ and $v$ (if some multiple \nedges are created we delete them). We denote by $G\/e$ the graph\nobtained from $G$ by contracting edge $e$.\n\nIf $S\\subseteq V(G)$ we call graph $G[S]=(S,\\big\\{\\{u,v\\}\\in E(G)\\mid u,v\\in\nS\\big\\})$ the \\textit{subgraph of $G$ induced by $S$}. Also, given a set $F\\subseteq E(G)$\n we call graph $G[F]=(\\bigcup_{e\\in F}e,F)$ the \\textit{subgraph of $G$ induced by\n$F$} and  we denote by $V(F)$ the set of vertices in $G[F]$.\n\nWe define the {\\em union} of two graphs $G_1= (V_1 , E_1)$ and $G_2 = (V_2 , E_2)$ to be \nthe graph $G_1\\cup G_2=(V_1\\cup V_2, E_1\\cup E_2)$. When $V_1$ and $V_2$ are disjoint, \nwe refer to this union as the {\\em disjoint union} of $G_1$ and  $G_2$ (we denote it as \n$G_{1}+G_{2}$).\n\nA vertex of a graph is called {\\em pendant} if it has degree at most $1$.\nAn edge $e$ of a graph $G$ is \\textit{pendant} if  one of its endpoints is pendant.\nIf both endpoints of an edge of $G$ are pendant, then we say that $e$ is an isolated edge.\n\nWe adapt the standard notations for the neighbourhood and the degree of a vertex $u\\in V(G)$, i.e. \nthe set off all vertices connected with $u$ by an edge and the cardinality of this set, which is \n$N_{G}(u)$ \nand $\\deg_{G}(u)$ respectively.\n\n\\subsection{Rooted graph triples.}\n\nA {\\em rooted graph triple}, or, for simplicity, a {\\em rooted graph}, is an ordered  triple \n$(G,S^{\\rm in},S^{\\rm out})$\nwhere $G$ is a connected graph and  $S^{\\rm in}$ and $S^{\\rm out}$ are subsets of $V(G)$ \n($S^{\\rm in}$ and $S^{\\rm out}$\nare not necessarily disjoint sets). If ${\\bf G}=(G,S^{\\rm in},S^{\\rm out})$ then we also say that \n$\\bf G$ is the \ngraph $G$ {\\em in-rooted} on $S^{\\rm in}$ and {\\em out-rooted} at $S^{\\rm out}$.\nGiven a rooted graph  ${\\bf G}=(G,S^{\\rm in},S^{\\rm out})$, we define \n${\\bf rev}({\\bf G})=(G,S^{\\rm out},S^{\\rm in})$.\n\nGiven a rooted graph  $(G,S^{\\rm in},S^{\\rm out})$, where \n$$S^{\\rm in}=\\{v^{\\rm in}_{1},\\ldots,v^{\\rm in}_{|S^{\\rm in}|}\\}\\ \\ \\ \\mbox{and}\n\\ \\ \\ S^{\\rm out}=\\{v^{\\rm out}_{1},\\ldots,v^{\\rm out}_{|S^{\\rm out}|}\\},$$\nwe define its {\\em enhancement} as the graph  ${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$ obtained from \n$G$ after adding two \nvertices $u^{\\rm in}$\n and  $u^{\\rm out}$ and  the edges  in the sets \n$$E^{\\rm in}=\\{\\{v^{\\rm in}_{1},u^{\\rm in}\\},\\ldots,\\{v^{\\rm in}_{|S^{\\rm in}|},u^{\\rm in}\\}\\}$$\nand  \n$$E^{\\rm out}=\\{\\{v^{\\rm out}_{1},u^{\\rm out}\\},\\ldots,\\{v^{\\rm out}_{|S^{\\rm out}|},u^{\\rm out}\\}\\}.$$\nFrom now on, we will refer to the vertices $u^{\\rm in}, u^{\\rm out}$\nas the {\\em vertex extensions} of ${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$ \nand the edge sets $E^{\\rm in}$ and $E^{\\rm out}$ \nas the {\\em edge extensions} of ${\\bf enh}(G,S^{\\rm in},$ $S^{\\rm out})$.\n\n\\subsection{An extension of the connected search game.}\n\nIn the above setting we assumed that searchers cannot make their first move \nin the graph before the fugitive makes his first move. Let $G$ be a graph \nand let $S^{\\rm in}, S^{\\rm out}\\subseteq V(G)$.\nA  {\\em $(S^{\\rm in},S^{\\rm out})$-complete strategy for  $G$} is \na search strategy ${\\cal S}$ on ${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$ such that\n\\begin{itemize}\n\\item[(i)] $E({\\cal S},i)=E^{\\rm in}$, for some $i$,\n\\item[(ii)] $E({\\cal S},i)\\cap E^{\\rm out}=\\emptyset$, for every  $i$ and  \n\\item[(iii)] $E({\\cal S},i)=E(G)\\setminus E^{\\rm out}$, for some $i$,\n\\end{itemize}\nwhere $E^{\\rm in}, E^{\\rm out}$ are the {\\em edge extensions} of ${\\bf enh}(G,S^{\\rm in},$ $S^{\\rm out})$.\n\nBased on the above definitions, we define  ${\\bf ms}(G,S^{\\rm in},S^{\\rm out})$ as the minimum\n mixed search number over all  possible $(S^{\\rm in},S^{\\rm out})$-complete search strategies  for  it. \n Similarly, we define  $\\ms(G,S^{\\rm in},S^{\\rm out})$ and  ${\\bf cmms}(G,S^{\\rm in},S^{\\rm out})$ where, \n in the case of connected searching, we additionally demand that $S^{\\rm in}$ induces a connected \n subgraph of $G$.\n Notice that ${\\bf ms}(G)={\\bf ms}(G,\\emptyset,\\emptyset)$ and  that this equality also holds for \n $\\ms$ and ${\\bf cmms}$.\n\n\\subsection{Expansions.}\n\nGiven a graph $G$ and a set $F\\subseteq E(G)$, we define \n$$\\partial_{G}(F)=(\\bigcup_{e\\in F}e)\\cap(\\bigcup_{e\\in E(G)\\setminus F}e)$$\n\nLet $G$ be a graph and let $E_1$ and $E_2$ be subsets of $E(G)$.\nAn {\\em $(E_1,E_2)$-expansion} of $G$ is an ordering ${\\cal E}=\\langle A_{1},\\ldots,A_{r}\\rangle$\nwhere \n\\begin{itemize}\n\\item[1.] For $i\\in\\{1,\\ldots,r-1\\}$, $E_1\\subseteq A_{i}\\subseteq E(G)\\setminus E_2$.\n\\item[2.] For $i\\in\\{1,\\ldots,r-1\\}$, $|A_{i+1}\\setminus A_{i}|\\leq 1$.\n\\item[3.] $A_{1}=E_1$,\n\\item[4.] $A_{r}=E(G)\\setminus E_2$.\n\\end{itemize}\n\nAn {\\em $(E_1,E_2)$-expansion} of $G$ is {\\em connected} if the following  condition holds:\n\\begin{itemize}\n\\item[5.] For $i\\in\\{1,\\ldots,r\\}$, $G[A_{i}]$ is connected.\n\\end{itemize}\n\nAn {\\em $(E_1,E_2)$-expansion} of $G$ is {\\em monotone} if the following  condition holds:\n\n\\begin{itemize}\n\\item[6.] $A_{1}\\subseteq\\cdots \\subseteq A_{r}$.\n\\end{itemize}\n\nLet $i\\in\\{1,\\ldots,r-1\\}$.\n The {\\em cost} of an expansion ${\\cal E}$ at position $i$ \nis defined as ${\\bf cost}_{G}({\\cal E},i)=|\\partial_{G}(A_{i})|+q_i$\nwhere  $q_i$ is equal to one if one of the following holds\n\n\\begin{itemize}\n\\item  $|A_{i}|\\geq 2$ and $A_i\\setminus A_{i-1}$ contains a pendant edge of $G$\n\\item  $A_{i}$ consists of only one  edge that  is an isolated edge  of $G$.\n\\end{itemize}\n\nIf none of the above two conditions hold then $q_i$ is equal to $0$.\nThe {\\em cost} of the expansion ${\\cal E}$, denoted as ${\\bf cost}_{G}({\\cal E})$, is the maximum cost \nof  ${\\cal E}$ among all positions $i\\in\\{1,\\ldots,r-1\\}$.\n\nWe define ${\\bf p}(G,S^{\\rm in},S^{\\rm out})$ as the minimum cost  that an \n$(E^{\\rm in},E^{\\rm out})$-expansion of \n${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$\nmay have, where $E^{\\rm in},E^{\\rm out}$ are the edge extensions of ${\\bf enh}(G,$ $S^{\\rm in}$, \n$S^{\\rm out})$. We also define \n${\\bf mp}(G,S^{\\rm in},S^{\\rm out})$ \n(if we consider only monotone $(E^{\\rm in},E^{\\rm out})$-expansions) and \n${\\bf cmp}(G,S^{\\rm in},S^{\\rm out})$  \n(if we consider connected monotone \n$(E^{\\rm in},E^{\\rm out})$-expansions). We finally define ${\\bf cmp}(G)={\\bf cmp}(G,\\emptyset,\\emptyset)$.\n\n\\begin{lemma}\n\\label{emptymake}\nLet $(G,S^{\\rm in},S^{\\rm out})$ be a rooted graph and let $S_{1}^{\\rm in}\\subseteq S^{\\rm in}$  and \n$S_{1}^{\\rm out}\\subseteq S^{\\rm out}$, {where $G[S^{in}]$ is a connected subgraph of G}. \nThen ${\\bf cmp}(G,S_1^{\\rm in},S_1^{\\rm out})\\leq {\\bf cmp}(G,S^{\\rm in},S^{\\rm out})$.\n\\end{lemma}\n\n\\begin{proof}\nLet $E^{\\rm in},E^{\\rm out}$ and $E^{\\rm in}_1,E^{\\rm out}_1$ be the edge extensions of \n${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$\n and ${\\bf enh}(G,S^{\\rm in}_1,S^{\\rm out}_1)$ respectively. Notice that, as \n $S_{1}^{\\rm in}\\subseteq S^{\\rm in}$ and \n $S_{1}^{\\rm out}\\subseteq S^{\\rm out}$, $E_{1}^{\\rm in}\\subseteq E^{\\rm in}$ and \n $E_{1}^{\\rm out}\\subseteq E^{\\rm out}$.\n\nLet ${\\cal E}=\\langle A_1, \\ldots, A_r\\rangle$ be an monotone and connected\n$(E^{\\rm in},E^{\\rm out})$-expansion of \n${\\bf enh}(G,S^{\\rm in},S^{\\rm out})$, with cost at  most $k$. \n\\medskip\n\nAs $G[S^{\\rm in}]$ is a connected subgraph of $G$, for every vertex of \n$S^{\\rm in}\\setminus S_{1}^{\\rm in}$ there exist a path connecting it with a vertex of \n$S_{1}^{\\rm in}$ that only uses vertices of $S^{\\rm in}$. We define the following edge sets:\n\n\\begin{itemize}\n\\item $E_{1}^{1}$ contains all edges that have a vertex of $S_{1}^{\\rm in}$ and a vertex of \n$S^{\\rm in}\\setminus S_{1}^{\\rm in}$ as endpoints. Let \n$V_1=(\\bigcup_{e\\in E_{1}^{1}}e)\\setminus  S_{1}^{\\rm in}$, then  $E_{1}^{2}$ contains all \nedges that have both endpoints in $V_1$.\n\\item $E_{j}^{1}$ contains all edges that have a vertex of $V_{j-1}$ and a vertex of \n$S=S^{\\rm in}\\setminus (S_{1}^{\\rm in}\\cup(\\bigcup_{l=1\\ldots,j-1}V_{l}))$ as endpoints. \nLet $V_j=(\\bigcup_{e\\in E_{j}^{1}}e)\\setminus  S$, then  $E_{j}^{2}$ contains all edges that have \nboth endpoints in $V_j$.\n\\end{itemize}\n\nFor each edge set $E_{j}^{i}$, $1\\leq j\\leq d$ and $i\\in\\{1,2\\}$, where $d$ is the maximum \ndistance between a vertex of $S^{\\rm in}\\setminus S_{1}^{\\rm in}$ to some vertex in \n$S_{1}^{\\rm in}$, we define arbitrarily an edge ordering $L_{j}^{i}$. We then define an \nordering ${\\cal E}_1$ of edge sets as follows:\n\n\\begin{itemize}\n\\item $A_{1}^{'}=(A_{1}\\setminus E^{\\rm in})\\cup E_{1}^{\\rm in}$\n\\item $A_{1+l}^{'}=A_{1+l-1}^{'}\\cup \\hat{A}_l$ for $l=1,\\ldots, |E_{1}^{1}|$, where \n$\\langle\\hat{A}_1,\\ldots, \\hat{A}_{|E_{1}^{1}|}\\rangle={\\bf prefsec}(L_{1}^{1})$\n\\item $A_{1+|E_{1}^{1}|+l}^{'}=A_{1+|E_{1}^{1}|+l-1}^{'}\\cup \\hat{A}_l$ for \n$l=1,\\ldots, |E_{1}^{2}|$, where $\\langle\\hat{A}_1,\\ldots, \\hat{A}_{|E_{1}^{2}|}\\rangle=\n{\\bf prefsec}(L_{1}^{2})$\n\\item $A_{1+|E_{1}^{1}|+|E_{1}^{2}|+\\cdots +|E_{j-1}^{1}|+ |E_{j-1}^{2}|+l}^{'}=\nA_{1+|E_{1}^{1}|+|E_{1}^{2}|+\\cdots +|E_{j-1}^{1}|+ |E_{j-1}^{2}|+l-1}^{'}\\cup \\hat{A}_l$ \nfor $l=1,\\ldots, |E_{j}^{1}|$, where $\\langle\\hat{A}_1,\\ldots, \\hat{A}_{|E_{j}^{1}|}\\rangle=\n{\\bf prefsec}(L_{j}^{1})$\n\\item $A_{1+|E_{1}^{1}|+|E_{1}^{2}|+\\cdots +|E_{j-1}^{1}|+ |E_{j-1}^{2}|+|E_{j}^{1}|+l}^{'}=\nA_{1+|E_{1}^{1}|+|E_{1}^{2}|+\\cdots +|E_{j-1}^{1}|+ |E_{j-1}^{2}|+|E_{j}^{1}|+l-1}^{'}$ $\\cup\\ \\hat{A}_l$ \nfor $l=1,\\ldots, |E_{j}^{2}|$, where $\\langle\\hat{A}_1,\\ldots, \\hat{A}_{|E_{j}^{2}|}\\rangle=\n{\\bf prefsec}(L_{j}^{2})$\n\\end{itemize}\n\n\nLet $s=|E_{1}^{1}|+|E_{1}^{2}|+\\cdots +|E_{d}^{1}|+ |E_{d}^{2}|$. Notice that there exist a \n$l_0\\in \\{2,\\ldots,r\\}$ such that $A_{1+s}^{'}=(A_{l_{0}}\\setminus E^{\\rm in})\\cup E_{1}^{\\rm in}$. \nWe define  a second ordering ${\\cal E}_2$ of edge sets as follows:  \n$A_{1+s+l}^{'}=(A_{l_{0}+l}\\setminus E^{\\rm in})\\cup E_{1}^{\\rm in}$ for $l=1,\\ldots,r-l_{0}$.\\smallskip\n\nClearly ${\\cal E}'={\\cal E}_{1}\\oplus{\\cal E}_{2}$ satisfies conditions 1--4 and therefore is an \n$(E^{\\rm in}_1,E^{\\rm out}_1)$-expansion of \n${\\bf enh}(G,S^{\\rm in}_1,S^{\\rm out}_1)$. Moreover, the monotonicity and connectivity of $\\cal E'$\nfollows from the monotonicity and connectivity of $\\cal E$.\\smallskip\n\nNotice that, for every $i\\in\\{1,\\ldots, r\\}$, $\\partial_{G}(A_i)=\\partial_{G}((A_{i}\\setminus E^{\\rm in})\n\\cup E_{1}^{\\rm in})$\ntherefore ${\\bf cost}_{G}({\\cal E}',i)\\leq{\\bf cost}_{G}({\\cal E},i)$. From this we conclude that \n${\\cal E}'$ has cost at most $k$.\n\\end{proof}\n\nLet ${\\bf G}_{1},\\ldots,{\\bf G}_{r}$ be rooted graphs such that \n${\\bf G}_i=(G_i,S_{i}^{\\rm in},S^{\\rm out}_{i})$ where \n$V(G_{i})\\cap V(G_{i+1})=S^{\\rm out}_{i}=S^{\\rm in}_{i+1}$, $i\\in\\{1,\\ldots,r-1\\}$.\nWe define, ${\\bf glue}({\\bf G}_{1},\\ldots,{\\bf G}_{r})$ $=(G_1\\cup \\cdots \\cup G_r,S_{1}^{\\rm in},S^{\\rm out}_{r})$.\n\n\\begin{lemma}\n\\label{glue}\nLet ${\\bf G}_{1},\\ldots,{\\bf G}_{r}$ be rooted graphs such that \n${\\bf G}_i=(G_i,S_{i}^{\\rm in},S^{\\rm out}_{i})$ where \n$V(G_{i})\\cap V(G_{i+1})=S^{\\rm out}_{i}=S^{\\rm in}_{i+1}$, $i\\in\\{1,\\ldots,r-1\\}$. Then \n$${\\bf cmp}({\\bf glue}({\\bf G}_{1},\\ldots,{\\bf G}_{r}))\\leq \\max\\{{\\bf cmp}({\\bf G}_{i})\\mid i\\in\\{1,\\ldots,r\\}\\}.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $E_i^{\\rm in},E_i^{\\rm out}$ be the edge extensions and ${\\cal E}_i=\\langle A_1^i, \\ldots, \nA_{l_i}^i\\rangle$ \nbe an monotone and connected $(E_i^{\\rm in},E_i^{\\rm out})$-expansion of \n${\\bf enh}(G_i,S_i^{\\rm in},S_i^{\\rm out})$, \nfor every $i\\in\\{1, \\dots, r\\}$.\nClearly ${\\cal E}=\\langle A_1^1, \\ldots, A_{l_1}^1, A_{l_1}^1\\cup A_2^2, \\ldots, A_{l_1}^1\n\\cup A_ {l_2}^2, \\ldots, \n(\\cup_{1\\leq i<r} A_{l_i}^i)\\cup A_2^r, \\ldots, (\\cup_{1\\leq i<r} A_{l_i}^i)$ $\\cup A_{l_r}^r\\rangle$ is an \n$(E_{1}^{\\rm in},E^{\\rm out}_{r})$-expansion of  ${\\bf glue}({\\bf G}_{1},\\ldots,{\\bf G}_{r})$ and, \nas expansions ${\\cal E}_i,\\ i\\in\\{1, \\dots, r\\}$ are monotone and connected, conditions  5 and 6 hold. \n\nWe observe that ${\\bf cost}_{G_1\\cup \\cdots \\cup G_r}({\\cal E})\\leq \\max\\{{\\bf cost}_{G_1}({\\cal E}_1),\n\\ldots, \n{\\bf cost}_{G_r}({\\cal E}_r)\\}$, therefore ${\\bf cmp}$ $({\\bf glue}({\\bf G}_{1},\\ldots,{\\bf G}_{r}))\\leq \\max\n\\{{\\bf cmp}({\\bf G}_{i})\n\\mid i\\in\\{1,\\ldots,r\\}\\}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{equivalence}\nFor every graph $G$, ${\\bf cmms}(G,S^{\\rm in},S^{\\rm out})={\\bf cmp}(G,S^{\\rm in},S^{\\rm out})$.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $G^*={\\bf enh}(G,S^{\\rm in},S^{\\rm out})$ has a complete search strategy ${\\cal  S}$ \nsatisfying conditions \n(i) -- (iii) with cost at most $k$.\nWe construct an edge ordering of $E(G)$ as follows.\nObserve that, because of the monotonicity of ${\\cal S}$,\n$E^{(i)}=E({\\cal S},i)\\setminus E({\\cal S},i-1)$. \nFor every $i\\in\\{1,\\ldots,|{\\cal S}|\\}$, we define $L_{i}$\nby taking any ordering of the set $E^{(i)}$\nand insisting that, if $E^{(i)}$ contains some sliding edge, this edge will be \nthe first edge of $L_{i}$. Let ${\\cal E}=\\langle A_{0},\\ldots,A_{r}\\rangle$ be  the sequence of \nprefixes of \n$L_{1}\\oplus \\ldots \\oplus L_{|S|}$, including the empty set (that is $A_{0}=\\emptyset$). Notice that, \nbecause of Condition~(i), $A_{s}=E^{\\rm in}$\nfor some $s\\in\\{1,\\ldots,|{\\cal S}|\\}$,\nand,  because of Condition~(iii), $A_{t}=E(G)\\setminus E^{\\rm out}$, for some \n$t\\in\\{1,\\ldots,|{\\cal S}|\\}$. We now claim that \n${\\cal E}'=\\langle A_{s},\\ldots,A_{t}\\rangle$\nis an $(E^{\\rm in},E^{\\rm out})$-expansion of $G^{*}$.\nIndeed, Condition~(1) holds because of Condition (ii)\nand Conditions (2) -- (4) hold because of the construction of ${\\cal E}'$.\nMoreover, the connectivity and the monotonicity of ${\\cal E}'$ follow directly from the connectivity \nand the monotonicity of ${\\cal S}$.\n\nIt remains to prove that the cost of \n${\\cal E}'$ is at most $k$. \nFor each $j\\in\\{0,\\ldots,|{\\cal E}'|\\}$\nwe define $i_{j}$ such that  the unique edge in $A_{j}\\setminus A_{j-1}$\nis an edge in $E^{(i_j)}$\nand we define $h_j$ such that $A_{h_j}\\setminus A_{h_j-1}$ contains the fist edge \nof $L_{i_j}$.\nNotice now  that the cost of ${\\cal E}'$\nat positions $h_j$ to $j$ \nis upper bounded by the cost of ${\\cal E}'$\nat position $h_j$. Therefore, it is enough \nto prove that the cost of ${\\cal E}'$\nat position $h_j$ is at most $k$.\nRecall that this cost is equal to \n$|\\partial_{G}(A_{h_j})|+q_{h_j}$.\nWe distinguish two cases:\\\\\n\n\\noindent{\\em Case 1.} If  $q_{h_j}=0$, then \nthe cost of ${\\cal E}'$\nat position $h_j$ is equal to $|\\partial_{G}(A_{h_j})|$.\nAs ${\\cal S}$ is monotone, all vertices in $\\partial_{G}(A_{h_j})$\nshould be occupied by searchers after the $i_{j}$-th move of ${\\cal S}$\nand therefore the cost of ${\\cal E}'$\nat position $h_j$ is at most $k$.\\\\\n\n\\noindent{\\em Case 2.} If $q_{h_j}=1$, then the $i_{j}$-th move of ${\\cal S}$\nis either the placement of a searcher on a pendant vertex  $x$\nor the sliding of a searcher along a pendant edge $\\{y,x\\}$ towards its pendant vertex $x$.  \nIn both cases, $x\\not\\in \\partial_{G}(A_{h_j})$\nand all vertices in $\\partial_{G}(A_{h_j})$ should be occupied by searchers\nafter the $i_{j}$-th move. In the first case, \nthere are in total at least $|\\partial_{G}(A_{h_j})|+1$ searchers on the graph\nand we are done. In the second case, we observe that, because \nof monotonicity,  \n$\\partial_{G}(A_{h_j})=\\partial_{G}(A_{h_j-1})\\setminus \\{y\\}$.\nAs after the $(h_j-1)$-th move all vertices of $\\partial_{G}(A_{h_j-1})$ were occupied by searchers, \nwe obtain that $|\\partial_{G}(A_{h_j})|\\leq k-1$\nand thus the cost of ${\\cal E}'$\nat position $h_j$ is at most $k$.\\\\\n\nNow assume that there exist  a monotone and connected  $(E^{\\rm in},E^{\\rm out})$-expansion of \n$G^*$, say ${\\cal E}=\\langle A_{1},\\ldots,A_{r}\\rangle$, with cost at most \n$k$. We can additionally assume that \n${\\cal E}$ is properly monotone; this can be done by discarding additional repetitions of a set in \n${\\cal E}$.\n\nMoreover, starting from $\\cal E$, we can construct a monotone and connected \n$(E^{\\rm in},E^{\\rm out})$-expansion of $G^*$, with cost at most $k$,\nsay ${\\cal E}'=\\langle A'_1,\\ldots,A'_r\\rangle$, with the following additional property:\\\\\n\n\\noindent{\\em Expansion property:} For every $i\\in\\{1,\\ldots,r-1\\}$ for which \n$V(A'_i)\\subset V(A'_{i+1})$, $A_i'$ contains all edges of $G^*$ with both endpoints in $V(A'_i)$.\\\\\n\nThis can be accomplished by a series of appliances of the following rule:\\\\\n\n\\noindent{\\em Rule:} Let $V(A_i)\\subset V(A_{i+1})$ for some $i$ and  let \n$L=\\langle e_1,\\ldots,e_n \\rangle$ \nbe an ordering of the edges \n$E(G^*)\\setminus A_i$ with both endpoints in $V(A_i)$. For every $j\\leq i$ define $A'_j=A_j$. \nThen, define\n$A'_{i+1}=A_{i}\\cup\\{e_1\\}$, $A'_{i+2}=A_{i}\\cup\\{e_1,e_2\\}$ and so on until \n$A'_{i+n}=A_{i}\\cup\\{e_1,\\ldots, e_n\\}$. \nFinally, for every $j\\geq i+n$, define $A'_j=A_j\\cup\\{e_1,\\ldots, e_n\\}$.\\\\\n\nOne can easily check that, after every application of this Rule, the constructed sequence of edge \nsets is indeed an \n$(E^{\\rm in},E^{\\rm out})$-expansion of $G^*$ and  furthermore it is monotone and connected. \nNotice that, for $j=1,\\ldots, n$, $\\partial_{G^*}(A'_{i+j})\\subseteq\\partial_{G^*}(A_i)$ and  for \n$j\\geq i+n$, $|\\partial_{G^*}(A'_j)|\\leq|\\partial_{G^*}(A_j)|$. {Moreover, if $|A_{i}|\\geq 2$ and \n$A_i\\setminus A_{i-1}$ contains a pendant edge of $G^*$ then  for every $j\\in\\{1,\\ldots, n\\}$, \n$|A'_{i+j}|\\geq 2$ and $A'_{i+j}\\setminus A'_{i+j-1}$ contains the same pendant edge of $G^*$, \nhence the cost of ${\\cal E}'$ is at most $k$  (notice that if  $A_{i}$ consists of only one  edge that  \nis an isolated edge  of $G^*$\nthen there does not exist an edge in $E(G^*)\\setminus A_i$ with both endpoints in $V(A_i)$, \ntherefore we do not need to apply this rule).}\n\nFor the rest of the proof, we will consider that the Expansion property holds for the given \n$(E^{\\rm in},E^{\\rm out})$-expansion \nof $G^*$.\n\nOur target is to define a $(S^{\\rm in},S^{\\rm out})$-complete monotone search strategy \n$\\cal S$ of $G^*$ with cost at most $k$.\n \nThe first $|S^{\\rm in}|$ moves of $\\cal S$ will be {\\sf p}$(u^{\\rm in})$ and the next $|S^{\\rm in}|$ will  \nbe {\\sf s}$(u^{\\rm in},v_i^{\\rm in})$. We denote this sequence of \nmoves by ${\\cal S}_0$. Notice that $E({\\cal S}, 2|S^{\\rm in}|)=A_1$.\n\nFor every vertex $u$ in the set $V^*= V(G^*)\\setminus S^{\\rm in}\\setminus \\{u^{\\rm out}\\}$, we define \n$l_u$ to be the first integer in $\\{1,\\ldots,r\\}$ such that $u\\in V(A_{l_u})$.\n\n Let $L=\\langle u_1,\\ldots,u_{|V^*|} \\rangle$ be an ordering of $V^*$ such that $i\\leq j$ when \n $l_{u_i}\\leq l_{u_j}$. \n Notice that, for each $i\\in\\{1,\\ldots,|V^*|\\}$,  the vertex $u_{i}$ is an endpoint of the unique \n edge $e_{i}$\nin $A_{l_{u_{i}}-1}\\setminus A_{l_{u_{i}}}$ and let $v_{i}$ be the other endpoint of $e_{i}$.\nNotice that, because of the connectivity and the monotonicity of ${\\cal E}$,  \n$v_{i}\\in  \\partial_{G^{*}}(A_{l_{u_{i}}-1})$.\nWe also observe that $u_{i}$ is pendant iff $u_{i}\\not\\in\\partial_{G^{*}}(A_{l_{u_{i}}})$.\nWe define $E'=\\{e_{1},\\ldots,e_{|V^{*}|}\\}$ and  we call a set $A_j$, $j\\in\\{1,\\ldots, r\\}$, crucial iff \n$|A_{j-1}\\cap E'|<|A_j\\cap E'|$.\n\nFor each  $i\\in\\{1,\\ldots,|V^*|\\}$, we define a sequence \n${\\cal S}_{i}$ of moves as follows:  If $v_{i}\\in  \\partial_{G^{*}}(A_{l_{u_{i}}})$\nthen the first move of ${\\cal S}_{i}$ is ${\\sf p}(u_i)$, otherwise it is ${\\sf s}(v_{i},u_{i})$. The \nrest of the moves in ${\\cal S}_{i}$ are the removals, one by one,  of the\nsearchers in $\\partial_{G^{*}}(A_{l_{u_{i}}-1})\\setminus \\partial_{G^{*}}(A_{l_{u_{i}}})$. \nThen we define ${\\cal S}={\\cal S}_{0}\\oplus {\\cal S}_{1}\\oplus\\cdots \\oplus {\\cal S}_{|V^{*}|}$.\n\nNotice that, according to the Expansion property,  all edges of the sets $A_j$, for $j=1,\\ldots,l_{u_1}$ \nhave both endpoints \nin $S^{\\rm in}$. Moreover, for every $i\\in\\{1,\\ldots,|V^*|-1\\}$ all edges of the sets $A_j$, for $j=l_{u_i},\n\\ldots,l_{u_i+1}-1$, \nhave both endpoints in $V(A_{l_{u_i}})$ and  all edges of the sets $A_j$, for $j=l_{u_{|V^*|}},\\ldots,r$, \nhave both endpoints in $V(A_{l_{u_{|V^*|}}})$. \n\nFirst we show that the following claim is true:\\\\\n\n\\noindent{\\em Claim 1.} For  every $A_{j}$, $j\\in\\{1,\\ldots,r\\}$, the vertices of $\\partial_{G^{*}} (A_{j})$\nare exactly the vertices occupied by searchers after the last move of ${\\cal S}_{m_j}$,\n where $m_j$ is the index of the edge in \n$(A\\cap E')\\setminus(A_{j-1}\\cap E')$, where $A$ is the first crucial set of $\\cal E$ such that \n$A_j\\subseteq A$. \\\\\n   \nClearly, this is true for $A_1=E^{\\rm in}$. Assume that it holds for $A_{j'}$.\n\nWe will show that the vertices in $\\partial_{G^*}(A_{j'+1})$ \nare exactly the vertices occupied by searchers after the last move of ${\\cal S}_{m_{j'+1}}$.\n  \nIf $A_{j'+1}$ is not crucial then $\\partial_{G^*}(A_{j'+1})\\subseteq\\partial_{G^*}(A_{j'})$ and  \n$m_{j'+1}=m_{j'}$, \ntherefore Claim 1 holds.\n\nNow, if $A_{j'+1}$ is crucial and  $\\{e_{m_{j'+1}}\\}=(A_{j'+1}\\cap E')\\setminus(A_{j'}\\cap E')$, then \n$v_{m_{j'+1}}\\in \\partial_{G^*}(A_{j'})$ and therefore must be occupied by a searcher. \nWe distinguish three cases:\\\\\n   \n\\noindent{\\em Case 1.} If $v_{m_{j'+1}}\\in \\partial_{G^*}(A_{j'+1})$ and $u_{m_{j'+1}}\\in \n\\partial_{G^*}(A_{j'+1})$, \nthen $ \\partial_{G^*}(A_{j'+1})= \\partial_{G^*}(A_{j'})\\cup\\{u_{m_{j'+1}}\\}$ and  the first move in \n${\\cal S}_{m_{j'+1}}$ \nwill be  ${\\sf p}(u_{m_{j'+1}})$. \\\\\n\n\\noindent{\\em Case 2.} If $v_{m_{j'+1}}\\in \\partial_{G^*}(A_{j'+1})$ and $u_{m_{j'+1}}\\not\\in \n\\partial_{G^*}(A_{j'+1})$ \nthen $\\partial_{G^*}(A_{j'+1})=\\partial_{G^*}(A_{j'})$.\\\\\n\n\\noindent{\\em Case 3.} If $v_{m_{j'+1}}\\not\\in \\partial_{G^*}(A_{j'+1})$, \nthen $ \\partial_{G^*}(A_{j'+1})= (\\partial_{G^*}(A_{j'})\\setminus\\{v_{m_{j'+1}}\\})\\cup\\{u_{m_{j'+1}}\\}$,\nand the first move in ${\\cal S}_{m_{j'+1}}$ will be ${\\sf s}(v_{m_{j'+1}},u_{m_{j'+1}})$.\\\\\n\n\\noindent Observe that in all three cases the Claim 1 holds.\\\\\n\nLet $V_{\\cal S}(i)$ be the set of vertices already visited by searchers after the $i$-th move of $\\cal S$, \nand let $V_{\\cal S}=\\langle V_{\\cal S}(1),\\ldots,V_{\\cal S}(r)\\rangle$. \nNotice that this sequence is monotone and  that if the $i$-th move belong to the subsequence \n${\\cal S}_j$, \nthen $V_{\\cal S}(i)=V(A_{l_{u_j}})$. We must next prove the following claim:\\\\\n\n\\noindent{\\em Claim 2:} For every $i\\in\\{1,\\ldots,|{\\cal S}|\\}$, all edges of $G^*[V_{\\cal S}(i)]$ \nare clean.\\\\\n\nClearly, the claim is true for $i\\in\\{1,\\ldots, 2\\cdot |S^{\\rm in}|\\}$. Assume that it holds for some \n$i\\in{2\\cdot |S^{\\rm in}|+1,\\ldots,r}$, we will show that all edges of $G^*[V_{\\cal S}(i+1)]$ are clean. \nWe must distinguish three cases about the $(i+1)$-th move:\\\\\n\n\\noindent{\\em Case 1.} It is a removal, say {\\sf r}$(u)$. Notice that \n$G^*[V_{\\cal S}(i+1)]=G^*[V_{\\cal S}(i)]$, \ntherefore the Claim will not be true if {\\sf r}$(u)$ is a recontamination move. In this case,\nthere exist an edge connecting $u$ with a vertex not in $V_{\\cal S}(i)$, say $v$. As \n$u\\in \\partial_{G^{*}}(A_{l_{u_{j}}-1})\\setminus \\partial_{G^{*}}(A_{l_{u_{j}}})$, \nfor some $j\\in\\{1,\\ldots,|V^*|\\}$, \nall edges with $u$ as \nendpoint must belong to $A_{l_{u_{j}}}$, therefore $\\{u,v\\}\\in A_{l_{u_{j}}}$. \nBut $V_{\\cal S}(i)=V(A_{l_{u_{j}}})$, a contradiction. \\\\\n\n\\noindent{\\em Case 2.} It is a placement of searcher say {\\sf p}$(u)$. By the definition of $\\cal S$, \nthere exist an \nedge $\\{u,v\\}$, where $v$ is a vertex in $V_{\\cal S}(i)$. Notice that, according to our search game, \nall such \nedges are clean after  {\\sf p}$(u)$, thus all edges of $G^*[V_{\\cal S}(i+1)]$ are clean.\\\\\n\n\\noindent{\\em Case 3.} It is a slide, say {\\sf s}$(v_j,u_j)$, for some $j\\in\\{1,\\dots,|V^*|\\}$. \nAs in the previous case,\n$G^*[V_{\\cal S}(i+1)]$ contains all edges of \n$G^*[V_{\\cal S}(i)]$ and additional all edges with $u_j$ as the first endpoint and  a vertex \n$v\\in V_{\\cal S}(i)$ as the other. \nAccording to our search game, after the $i$-th move there must be searcher in $v_j$, \ntherefore due to Claim 1, \n$v_j\\in\\partial_{G^*}(A_{l_{u_{j}}})$. Notice that, the Claim will not be true if {\\sf s}$(v_j,u_j)$ is a \nrecontamination move,\ni.e., there exist an edge connecting $v_j$ with a vertex, say $u$, not in $V_{\\cal S}(i)=V(A_{l_{u_j}})$.  \nAs \n$v_j\\not\\in \\partial_{G^{*}}(A_{l_{u_{j}}})$, \nall edges with $u$ as \nendpoint must belong to $A_{l_{u_{j}}}$, therefore $\\{v_j,u\\}\\in A_{l_{u_{j}}}$, a contradiction. \\\\\n\nIn all three cases we show that after the $(i+1)$-th move of $S$ all edges of $G^*[V_{\\cal S}(i+1)]$\nare clean, \ntherefore Claim 2 is true.\\\\\n\nWe will now prove that $\\cal S$ is a $(S_{1},S_{2})$-complete strategy for $G^*$. \nClearly, Condition~(i) holds for \nevery strategy starting with ${\\cal S}_{0}$. Moreover, Condition~(ii) holds as \n$v^{\\rm out}$ is not a vertex of $V^{*}$  and \ntherefore, no placement on $u^{\\rm out}$ or sliding towards $u^{\\rm out}$ appears in ${\\cal S}$. \nNotice that, according to Claim 2, for every $i\\in\\{1,\\ldots,|V^*|-1\\}$, $E({\\cal S},|{\\cal S}_0\\oplus\n\\cdots\\oplus{\\cal S}_{i-1}|+1)=\n\\cdots=E({\\cal S},|{\\cal S}_0\\oplus\\cdots\\oplus{\\cal S}_{i-1}|+|{\\cal S}_i|)=A_{l_{u_{i+1}}-1},$ \nand that for $i=|V^*|$, $E({\\cal S},|{\\cal S}_0\\oplus\\cdots\\oplus{\\cal S}_{|V^*|}|)=A_r$, therefore\nCondition~(iii) holds. \n\nBy the definition of $\\cal S$, it is clear that $\\cal S$ is a connected search strategy, moreover, \naccording to Claim 2, $\\cal S$ is monotone. \nIt remains to prove that $\\cal S$ has cost at most $k$. For the first $2|S^{\\rm in}|$ moves, we use \n$|S^{\\rm in}|={\\bf cost}_{G^*}({\\cal E},1)\\leq k$ searchers.  Assume that after $j$ moves exactly $k$\nsearchers are \noccupying vertices of $G^*$ and  that the $(j+1)$-th move is ${\\sf p}(u_i)$, for some \n$i\\in\\{i,\\ldots,|V^*|\\}$. Then the \nvertices in $\\partial_{G^{*}}(A_{l_{u_{i}}-1})$ are exactly the vertices occupied by the $k$ searchers, \ntherefore \n$|\\partial_{G^{*}}(A_{l_{u_{i}}-1})|=k$. Observe that, if $u_i$ is not pendant, then\n$\\partial_{G^{*}}(A_{l_{u_{i}}})=\\partial_{G^{*}}(A_{l_{u_{i}}-1})\\cup\\{u_i\\}$, therefore \n$|\\partial_{G^{*}}(A_{l_{u_{i}}})|=k+1$, \na contradiction and  if $u_i$ is pendant then  $\\partial_{G^{*}}(A_{l_{u_{i}}})=\\partial_{G^{*}}\n(A_{l_{u_{i}}-1})$ and  \nthe cost of $\\cal E$ at position $l_{u_{i}}$ is $|\\partial_{G^{*}}(A_{l_{u_{i}}})|+1=k+1$, \nagain a contradiction. \nThus, for every move of $\\cal S$ at most $k$ searchers are occupying vertices of $G^*$.\n\\end{proof}\n\n\\subsection{Contractions.}\n\\label{cont8e}\n\nLet $(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1})$ and $(G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$ \nbe  rooted graphs. We say that $(G_{1},S^{\\rm in}_{1},$ $S^{\\rm out}_{1})$ is \n{\\em a contraction} of  $(G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$ and  we denote this \nfact \nby $(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1})$ $\\preceq (G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$ \nif there exist a surjection $\\phi: V(G_{2})\\rightarrow V(G_{1})$ such that:\n\n\\begin{itemize}\n\\item[${\\bf 1.}$] for every vertex $ v\\in V(G_{1})$,  $G_{2}[\\phi^{-1}(v)]$ is connected\n\\item[${\\bf 2.}$] for every two distinct vertices $u,v\\in V(G_{1})$, it holds that \n$\\{v,u\\}\\in E(G_{1})$ if and only if the graph $G_{2}[\\phi^{-1}(v)\\cup \n\\phi^{-1}(u)]$ is connected\n\\item[${\\bf 3.}$] $\\phi(S_{2}^{\\rm in})= S_{1}^{\\rm in}$\n\\item[${\\bf 4.}$] $\\phi(S_{2}^{\\rm out})= S_{1}^{\\rm out}$\n\\end{itemize}\nWe also write $(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1})\\preceq_{\\phi} (G_{2},S^{\\rm in}_{2},\nS^{\\rm out}_{2})$ \nto make clear the function that certifies the contraction relation.\nWe say that $G_{1}$ is a {\\em contraction}\nof $G_{2}$ if $(G_{1},\\emptyset,\\emptyset)\\preceq (G_{2},\\emptyset,\\emptyset)$\nand we denote this fact by $G_{1}\\preceq G_{2}$. If furthermore $G_1$ is not isomorphic to \n$G_2$ we say that \n$G_1$ is a {\\em proper contraction} of $G_2$.\n\nWe define the {\\em minor} relation for the two rooted graph by removing  in the second property \nthe demand that if $\\{u,v\\}\\notin E(G_1)$ then $G_{2}[\\phi^{-1}(v)\\cup \n\\phi^{-1}(u)]$ is not connected.  We denote the minor relation by $(G_{1},S^{\\rm in}_{1},\nS^{\\rm out}_{1})$ $\\leq (G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$.  Again, we say that $G_{1}$ is a \n{\\em minor}\nof $G_{2}$ if $(G_{1},\\emptyset,\\emptyset)\\leq (G_{2},\\emptyset,\\emptyset)$\nand we denote this fact by $G_{1}\\leq G_{2}$.\n\n\\begin{lemma}\n\\label{lem:clos}\nIf $(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1})$ and $(G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$ \nare  rooted graphs and  $(G_{1},S^{\\rm in}_{1},$ \n$S^{\\rm out}_{1})\\preceq (G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$,\nthen \n${\\bf cmp}(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1}))\\leq {\\bf cmp}(G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2}))$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that ${\\cal E}=\\langle A_{1},\\ldots,A_{r}\\rangle$ is a monotone \n$(E_{2}^{\\rm in },E_{2}^{\\rm out})$-expansion \nof $G_{2}^{*}={\\bf enh}(G_2,S^{\\rm in}_2,S^{\\rm out}_2)$ with cost at most $k$.\nOur target is to construct a monotone $(E_{1}^{\\rm in },E_{1}^{\\rm out})$-expansion \nof $G_{1}^{*}={\\bf enh}(G_1,S^{\\rm in}_1,S^{\\rm out}_1)$ with cost at most $k$.\n\nLet  $\\phi$ be a function where\n$(G_{1},S^{\\rm in}_{1},S^{\\rm out}_{1})\\preceq_{\\phi} (G_{2},S^{\\rm in}_{2},S^{\\rm out}_{2})$. \nWe  consider an extension $\\psi$ of $\\phi$ \nthat additionally maps $u_{2}^{\\rm in}$ to $u_{1}^{\\rm in}$\nand $u_{2}^{\\rm out}$ to $u_{1}^{\\rm out}$. Notice that the construction of $\\psi$ yields the following:\n$$(G_{1}^{*},S^{\\rm in}_{1}\\cup \\{u_{1}^{\\rm in}\\},S^{\\rm out}_{1}\\cup \\{u_{1}^{\\rm out}\\})\n\\preceq_{\\phi} (G_{2}^{*},S^{\\rm in}_{2}\\cup \n\\{u_{2}^{\\rm in}\\},S^{\\rm out}_{2}\\cup \\{u_{2}^{\\rm out}\\})$$\n\nGiven an edge $f=\\{x,y\\}\\in E(G_{1})$ we consider the set $E_f$\ncontaining all edges of $G_{2}$ with one endpoint in $\\psi^{-1}(x)$ and one \nendpoint in $\\psi^{-1}(y)$. We now pick, arbitrarily,\nan edge in $E_{f}$ and we denote it by $e_{f}$. We also set $E'=\\{e_f\\mid f\\in E(G_{1})\\}$. \nThen it is easy to observe that \n${\\cal E}'=\\langle A_{1}\\cap E',\\ldots,A_{r}\\cap E'\\rangle$ is a connected expansion of $G_{1}^*$ \nand that \nthe cost of ${\\cal E}'$ at step $i$ is no bigger than the cost of ${\\cal E}$\nat the same step, where $i\\in\\{1,\\ldots,r-1\\}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:cs-clos}\nIf $G_{1}$ and $G_{2}$ \nare  two graphs and  $G_{1}\\preceq G_{2}$,\nthen \n${\\bf cs}(G_{1})\\leq {\\bf cs}(G_{2})$.\n\\end{lemma}\n\n\\begin{proof}\nFirst observe that if this is the case any contraction of $G_2$ can be derived by applying a finite\nnumber of edge-contractions of some edges in $E(G_2)$. \n\nIt suffices to prove that the Lemma hold if $G_1$ is obtained by the contraction of edge \n$e=\\{u,v\\}\\in E(G_1)$ to vertex $x_{uv}$. Let $\\cal S$ be a connected search strategy for \n$G_2$ that in any step uses at most $k$ searchers. Based on $\\cal S$ we will construct a \nsearch strategy ${\\cal S}'$ for $G_1$.  Let $i$ be an integer in $\\{1,\\ldots,|{\\cal S}|\\}$. \nWe distinguish eight cases:\\\\\n\n\\noindent {\\em Case 1}: If the $i$-th move of $\\cal S$ is {\\sf p}$(x)$ for some vertex \n$x\\notin\\{u,v\\}$ then the next move of ${\\cal S}'$ will be {\\sf p}$(x)$.\\\\\n\n\\noindent {\\em Case 2}: If the $i$-th move of $\\cal S$ is {\\sf r}$(x)$ for some vertex \n$x\\notin\\{u,v\\}$ then the next move of ${\\cal S}'$ will be {\\sf r}$(x)$.\\\\\n\n\\noindent {\\em Case 3}: If the $i$-th move of $\\cal S$ is {\\sf s}$(x,y)$ for some vertices \n$x,y\\notin\\{u,v\\}$ then the next move of ${\\cal S}'$ will be {\\sf s}$(x,y)$.\\\\\n\n\\noindent {\\em Case 4}: If the $i$-th move of $\\cal S$ is {\\sf p}$(u)$ or  {\\sf p}$(v)$ then \nthe next move of ${\\cal S}'$ will be {\\sf p}$(x_{uv})$.\\\\\n\n\\noindent {\\em Case 5}: If the $i$-th move of $\\cal S$ is {\\sf r}$(u)$ or  {\\sf r}$(v)$ then \nthe next move of ${\\cal S}'$ will be {\\sf r}$(x_{uv})$.\\\\\n\n\\noindent {\\em Case 6}: If the $i$-th move of $\\cal S$ is {\\sf s}$(z,u)$ or {\\sf p}$(z,v)$ for \nsome vertex $z$ then the next move of ${\\cal S}'$ will be {\\sf s}$(z,x_{uv})$.\\\\\n\n\\noindent {\\em Case 7}: If the $i$-th move of $\\cal S$ is {\\sf s}$(u,z)$ or {\\sf p}$(v,z)$ \nfor some vertex $z$ then the next move of ${\\cal S}'$ will be {\\sf s}$(x_{uv},z)$.\\\\\n\n\\noindent {\\em Case 8}: If the $i$-th move of $\\cal S$ is {\\sf s}$(u,v)$ or {\\sf p}$(v,u)$ then \nthe next move of ${\\cal S}'$ will be defined according the lateral cases from the \n$(i+1)$-th move of $\\cal S$.\\\\\n\nObserve that ${\\cal S}'$ is a complete search strategy for $G_1$. Furthermore, as \n$\\cal S$ is connected, ${\\cal S}'$ must also be connected. Finally, it is clear that \n${\\cal S}'$ at any step uses at most $k$ searchers, thus ${\\bf cms}(G_1)\\leq k$.\n\\end{proof}\n\n\\subsection{Parameters and obstructions.} \n\nWe denote by ${\\cal G}$ the class of all \ngraphs. A \\textit{graph parameter} is a function $f: {\\cal G}\\rightarrow \\mathbb{N}$. \nGiven a graph parameter $f$ and an integer $k\\in\\mathbb{N}$ we define the \\textit{graph class}\n$\\mathcal{G}[f,k]$, containing all the graphs $G\\in {\\cal G}$ where $f(G)\\leq k$.\n\nLet ${\\cal H}$ be a graph class. We denote by ${\\bf obs}({\\cal H})$ the set of all graphs \nin ${\\cal G}\\setminus {\\cal H}$ that are minimal with respect to the relation $\\preceq$.\n\n\\subsection{Cut-vertices and blocks.}\n\nWe call the 2-connected \ncomponents of a graph $G$ {\\em blocks}. If the removal of an edge \nin a graph  increases the number of its connected components then it is called {\\em bridge}.\nWe consider the subgraph of $G$ induced by the endpoints of a bridge \nof $G$  as one of its blocks and we call it {\\em trivial block} \nof $G$. \n\nA {\\em cut-vertex} of a graph $G$ is a vertex such that $G\\setminus x$ has more connected \ncomponents than $G$.\nGiven a graph $G$ a {\\em cut-vertex of a block $B$} of $G$ is a cut-vertex of $G$ \nthat belongs in $V(B)$.\n\nLet $G$ bet a graph and let $x\\in V(G)$. We define $${\\cal C}_{G}(x)=\\{(x,G[V(C)\\cup\n\\{x\\}])\\mid \\mbox{$C$ is a connected component of $G\\setminus x$}\\}.$$ \n\n Let $B$ be a\nblock of $G$ and let $x$ be a cut-vertex of $B$. We denote by $C_{G}(x,B)$ the (unique) graph in \n${\\cal C}_{G}(x)$  that contains $B$ as a subgraph and by $\\overline{\\cal C}_{G}(x,B)$ the graphs in \n${\\cal C}_{G}(x)$  that do not contain $B$.\n\n\\subsection{Outerplananr graphs.}\n\nWe call a graph $G$ \\textit{outerplanar} if it can be embedded in the plane such that all\nits vertices are incident to its infinite face (also called {\\em outer face}). This  embedding, \nwhen exists, is unique\nup to homeomorphism and, from now on, each outerplanar graph is accompanied with  such an \nembedding.  \nAn edge $e\\in E(G)$ is called \\textit{outer edge of $G$},\nif it is incident to the outer face of $G$, otherwise is called a \\textit{chord of $G$}.\n\nA face $F$ of an outer planar graph that is different than the outer face, is called \\textit{haploid} \nif and only if at most  one edge\nincident to $F$ is a chord, otherwise $F$ is a \\textit{inner} face. A vertex $u\\in V(G)$\nis \\textit{haploid} if  it is incident to an haploid face and  \\textit{inner} if it is incident\nto an inner face (notice that some vertices can be both inner and haploid). \nA vertex of $G$ that is not inner or haploid is called {\\em outer}.\nWe call a chord {\\em haploid} if it is incident to an haploid face. Non-haploid \nchords are called {\\em internal chords}. \n\n\\begin{figure}\n\\begin{center}\n \\scalebox{0.8}{\\includegraphics{figure1}}\n \\end{center}\n\\caption{A outerplanar graph and its blocks. The cut-vertices are hexagonal and the outer vertices \nare squares. \nInner and haploid faces are denoted by ``i\" and ``h\" respectively.\nThere are, in total, four inner vertices (all belonging to the  essential block on the right) and, \namong them only the\ntriangular one is not  an haploid vertex. The white hexagonal vertices are the light cut-vertices \nwhile the rest of the \nhexagonal vertices are the heavy ones.}\n \\label{FanRootObs}\n\\end{figure}\n\n\\begin{observation}\nA block of a connected outerplanar graph with more than one edge can be one of the following.\n\\begin{itemize}\n\\item a {\\em hair block}: it is a trivial block containing exactly one vertex of degree 1 in $G$.\n\\item a {\\em bridge block}: it is a trivial block that is not a hair-block.\n\\item a {\\em cycle block}: if it is a chordless non-trivial block, or\n\\item an {\\em essential block}: if it is a non-trivial block with at least one chord.\n\\end{itemize}\n\\end{observation}\n\nLet $G$ be a connected outerplanar graph with more than one edges.\nGiven a  cut-vertex $c$ of $G$, \nwe say that $c$ is {\\em light} if it is the (unique) cut-vertex of exactly one hair block.\nIf a cut-vertex of $G$ is not light then it is {\\em heavy}. \n\nIt is known that the\n\\textit{class of outerplanar graphs} is closed under the relations $\\leq,\\preceq$ and\nthat a graph is outerplanar if and only if $K_4\\nleq G$ and $K_{2,3}\\nleq G$.\n\n\\begin{figure}\n\\begin{center}\n \\scalebox{0.8}{\\includegraphics{figure2}}\n \\end{center}\n\\caption{The set ${\\cal O}_{1}$.}\n\\label{FanRsootObs}\n\\end{figure}\n\nLet $K_{2,3}^{+}$ be the graph obtained by $K_{2,3}$ after connecting the two vertices of\ndegree 3 (Figure~\\ref{FanRsootObs}).\n\n\\begin{lemma}\n\\label{lema:outerplanar} \nIf ${\\cal H}$ is the class of all outerplanar graphs, then ${\\bf obs}({\\cal \nH})={\\cal O}_{1}$.\n\\end{lemma}\n\n\\begin{proof}\nObserve that the graphs in ${\\cal O}_{1}$ \ncannot be embedded in the plane in such  a way that all of its vertices are incident to a single \nface and therefore neither the graphs in ${\\cal O}_{1}$, neither the graphs that contain as a \ncontraction \na graph in ${\\cal O}_{1}$, can be outerplannar. \n\nTo complete the proof, one must show that every non-outerplannar graph can be contracted to \na graph in \n${\\cal O}_{1}$. Let $G$ be non-outerplannar, then $K_4\\leq G$ or $K_{2,3}\\leq G$. Clearly, \nas $K_{4}$ is a clique, \n$K_{4}\\leq G$ implies that $K_{4}\\preceq G$. \nSuppose now that $K_{2,3}\\leq G$. Let $V_{x},V_{y}, V_{1}, V_{2}, V_{3}$ be the \nvertex sets of the connected subgraphs of $G$ that are contracted towards creating the \nvertices of $K_{2,3}$ ($V_{x}$ and $V_{y}$ are contracted to vertices of degree 3). If there is no \nedge in \n$G$ between two vertices in $V_{a}$ and $V_{b}$ for \nsome $(a,b)\\in \\{(x,y),(1,2),(2,3),(1,3)\\}$ then $K_{2,3}\\preceq G$. If the only such edge \nis between $V_{x}$ and $V_{y}$ then $K_{2,3}^{+}\\preceq G$ and in any other case, \n$K_{4}\\preceq G$. \n\\end{proof}\n\n\\section{Obstructions for Graphs with ${\\bf cmms}$ at most 2} \n\\label{sec:2}\n\nIn this section we give the obstruction set for graphs with connected monotone mixed search \nnumber at most 2 and we prove its correctness.\n\n\\subsection{The obstruction set for $k=2$}\n\n\\hskip0.6cm Let ${\\cal D}^1={\\cal O}_{1}\\cup \\cdots \\cup {\\cal O}_{12}^{}$ where ${\\cal O}_{1}$\nis depicted in Figure~\\ref{FanRsootObs}, ${\\cal O}_{2},\\ldots,{\\cal O}_{9}$ are depicted in \nFigure~\\ref{FanRossotsObs}\nand ${\\cal O}_{10}$ and ${\\cal O}_{11}$ and ${\\cal O}_{12}$ are constructed as follows.\n\n\\begin{description}\n\\item[${\\cal O}_{10}:$]  contains every graph that can be constructed by taking three \ndisjoint copies of some graphs in Figure~\\ref{FanRootObs}\nand then identify the vertices denoted by $v$ in each of them to a single vertex. \nThere are, in total, 35 graphs generated in this way.\n\\item[${\\cal O}_{11}:$]  contains every graph that can be constructed by taking two \ndisjoint copies of some graphs in Figure~\\ref{direction}\nand then identify the vertices denoted by $v$ in each of them to a single vertex.   \nThere are, in total, 78 graphs generated in this way.\n\\item[${\\cal O}_{12}:$]   contains every graph that can be constructed by taking \ntwo disjoint copies of some graphs in Figure~\\ref{if9ton}\nand then identify the vertices denoted by $v$ in each of them to a single vertex.    \nThere are, in total, 21 graphs generated in this way.\n\\end{description}\n\nObserve that, ${\\cal D}^{1}$ contains 177 graphs.\n\n\\subsection{Proof strategy}\n\n\\begin{lemma}\n\\label{sub_dir}\n${\\cal D}^{1}\\subseteq {\\bf obs}(\\mathcal{G}[{\\bf cmp},2])$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Lemma~\\ref{lem:clos},\nit is enough to check that for every $G\\in{\\cal D}^1$, the following two conditions \nare satisfied ({\\bf i}) ${\\bf cmp}({G})\\geq 3$ and ({\\bf ii}) for every edge $e$ of $G$ it holds that\n${\\bf cmp}(G\\slash e)\\leq 2$. One can verify that this is correct by inspection,\nas \nthis  concerns only a finite amount of graphs and, for each of them, there exists \na finite number of edges to contract.\n\\end{proof}\n\n\\begin{lemma}\n\\label{sup_dir}\n${\\cal D}^{1}\\supseteq {\\bf obs}(\\mathcal{G}[{\\bf cmp},2])$. \n\\end{lemma}\n\nThe rest of this section is devoted to the proof of  Lemma~\\ref{sup_dir}.\nFor this, our strategy is to consider the set \n$${\\cal Q}={\\bf obs}(\\mathcal{G}[{\\bf cmp},2])\\setminus {\\cal D}^{1}$$\nand prove that  ${\\cal Q}=\\emptyset$  (Lemma~\\ref{tll0o}). For this, we need a series of \nstructural results whose\nproofs use the following three fundamental properties of the set ${\\cal Q}$.\n\n\\begin{lemma}\n\\label{goinaldirs}\nLet $G\\in {\\cal Q}$. Then the following hold.\n\\begin{itemize}\n\\item[i.]  ${\\bf cmp}(G)\\geq  3$.\n\\item[ii.] If $H$ is a  proper contraction of $G$, then ${\\bf cmp}(G)\\leq 2$.\n\\item[iii.] $G$ does not contain any of the graphs in    ${\\cal D}^{1}$ as a contraction.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nProperties i. and ii. hold because $G\\in {\\bf obs}(\\mathcal{G}[{\\bf cmp},2])$.\nFor property iii. suppose, to the contrary, that $G$ contains some graph in $H\\in {\\cal D}^{1}$ \nas a contraction.\nFrom Lemma~\\ref{sub_dir}, $H\\in {\\bf obs}(\\mathcal{G}[{\\bf cmp},2])$. Clearly, $H$ is different than $G$\nas ${\\cal Q}$ does not contain members of ${\\cal D}^{1}$. Therefore, $H$ is a proper contraction \nof $G$ and, from property ii., \n${\\bf cmp}(H)\\leq 2$. This contradicts to the fact that $H\\in {\\bf obs}(\\mathcal{G}[{\\bf cmp},2])$ and thus \n${\\bf cmp}(H)\\geq 3$.\n\\end{proof}\n\n\\begin{figure}[h]\n\\begin{center}\n \\scalebox{.6}{\\includegraphics{figure3}}\n \\end{center}\n\\caption{The sets of graphs in ${\\cal D}_1$.}\n\\label{FanRossotsObs}\n\\end{figure}\n\n\\subsection{Basic structural properties}\n\n\\begin{lemma}\n\\label{basic_structure} \nLet $G\\in {\\cal Q}$. The following hold:\n\\begin{itemize}\n\\item[1.] $G$ is outerplanar. \n\\item[2.] Every light cut-vertex of $G$ has degree at least 3. \n\\item[3.] Every essential block $B$ of $G$, has exactly two haploid faces. \n\\item[4.] Every block of $G$, has at most 3 cut-vertices\n\\item[5.] Every  cut-vertex of a non-trivial block of $G$ is an haploid vertex. \n\\item[6.] Every block of $G$ contains at most 2 heavy cut-vertices. \n\\item[7.] If a block of $G$ has 3 cut-vertices, then there are two, say $x$ and $y$,  of these vertices\nthat are not both heavy and  are connected by an haploid edge. \n\\item[8.] If an essential block of $G$ with haploid faces $F_{1}$ and $F_{2}$ has two heavy cut-vertices, then \none can choose one, say $c_{1}$, of these two heavy cut-vertices so that it is incident to $F_{1}$ and \none say $c_{2}$ that is incident to $F_{2}$. \nMoreover, this assignment can be done in such a way that if there is a third light cut-vertex $c_{3}$, adjacent to one, say $c_{1}$, of $c_{1},c_{2}$, then\n$c_{3}$ is incident to $F_{1}$ as well.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} 1. By the third property of Lemma~\\ref{goinaldirs}, $G$ cannot be contracted \nto a graph in ${\\cal O}_1$ \nand  therefore, from Lemma~\\ref{lema:outerplanar}, $G$ must be outerplannar.\n\n2. Let $c$ be a light cut-vertex of a block $B$ in $G$, with degree 2 (notice that, as $c$ is a \ncut-vertex, $c$ cannot \nhave degree 1 or 0). That means that $c$ belongs to a path with at least two edges, the hair \nblock $B$ and an edge \nsay $e$. Observe that ${\\bf cmp}(G\/B)={\\bf cmp}(G)$, contradicting to the second property of \nLemma~\\ref{goinaldirs}.\n\n3.  Let $B$ be an essential block of $G$. As it is essential, it has at least one chord, therefore\nit has at least 2 haploid faces. Assume, that $B$ has at least 3 haploid faces. Choose 3 of them, \nsay $F_1, F_2$ and $F_3$ (see Figure~\\ref{helpinproof1}). \nLet $S\\subseteq E(B)$ be the set of all chords incident to $B$.  Contract in $G$ \nall edges in $E(G)\\setminus S$ not belonging to those faces. Then, for each of the three faces, \ncontract all but \ntwo edges not in $S$ that are incident to  $F_1, F_2$ and $F_3$  and  notice that the obtained \ngraph is the graph in \n${\\cal O}_2$, a contradiction to the third property of Lemma~\\ref{goinaldirs}. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{figure4}\\end{center}\n\\caption{An example for the proof of Lemma~\\ref{basic_structure}.3.}\n\\label{helpinproof1}\n\\end{figure}\n\n4. Let $B$ be a block of $G$ containing more than 3 cut-vertices. Chose four of them, \nsay $c_1,c_2,c_3$ and $c_4$. \nLet $S\\subseteq E(B)$ be the set of all chords incident to $B$ (see Figure~\\ref{helpinproof2}). \nContract all edges in $E(G)\\setminus S$  not having an \nendpoint in $\\{c_1,c_2,c_3,c_4\\}$. Then, contract all edges $e\\in E(B)\\setminus S$ such that \n$e\\nsubseteq\\{c_1,c_2,c_3,c_4\\}$ \nand  all edges not in $E(B)$, except from one for each of the cut-vertices. Notice that the obtained \ngraph belongs to ${\\cal O}_3$, \na contradiction to the third property of Lemma~\\ref{goinaldirs}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{figure5}\\end{center}\n\\caption{An example for the proof of Lemma~\\ref{basic_structure}.4.}\n\\label{helpinproof2}\n\\end{figure}\n\n5. Let $B$ be a block of $G$ containing a cut-vertex $c$ that is not haploid and  let \n$S\\subseteq E(B)$ be the set of \nall chords incident to $B$ (see Figure~\\ref{helpinproof3}). Contract all edges in \n$E(G)\\setminus E(B)$ not having $c$ as endpoint and  all edges in \n$E(B)\\setminus S$ not having $c$ as endpoint, except from two edges for each of the haploid faces. \nThen contract \nall edges not in $E(B)$ with $c$ as endpoint, except for one. Notice that the obtained graph \nbelongs to ${\\cal O}_4$, \na contradiction to the third property of Lemma~\\ref{goinaldirs}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{figure6}\\end{center}\n\\caption{An example for the proof of Lemma~\\ref{basic_structure}.5.}\n\\label{helpinproof3}\n\\end{figure}\n\n6. Let $B$ be a block of $G$ containing three heavy cut-vertices, say $c_1,c_2$ and $c_3$ \n(see Figure~\\ref{helpinproof4}). \nWe contract all edges in $B$ except from 3 so that $B$ is reduced to a triangle $T$ with \nvertices $c_1,c_2$ and $c_3$. \nThen, in the resulting graph $H$, for each $c_{i}, i\\in\\{1,2,3\\}$, \nin ${\\cal C}_{H}(c_{i})\\setminus \\{T\\}$ contains either a non trivial block or at least two hair blocks.\nIn any case, $H$ can be further contracted to one of the graphs in ${\\cal O}_{5}$ a \ncontradiction to the third property of Lemma~\\ref{goinaldirs}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{figure7}\\end{center}\n\\caption{An example for the proof of Lemma~\\ref{basic_structure}.6.}\n\\label{helpinproof4}\n\\end{figure}\n\n7. Let $\\{x,y,z\\}$ be three cut-vertices of a (not-trivial) block $B$. If no two of them are \nconnected by an outer edge, then contract \nall blocks of $G$, except $B$, to single edges, then contract all outer edges of $B$ that do \nnot have an endpoint in \n$\\{x,y,z\\}$ and continue contracting hair blocks with a vertex of degree $\\geq 4$, as long as this \nis  possible. This \ncreates either a graph in ${\\cal O}_{6}$ or a graph that after the contraction of a hair block makes \na graph in \n${\\cal O}_{7}$ or a graph that after the contraction of two hair blocks is a graph makes a graph in  \n${\\cal O}_{4}$ and, \nin any case, we have a  contradiction to the third property of Lemma~\\ref{goinaldirs}.\nWe contract $G$ to a  graph $H$ as follows: \n\\begin{itemize}\n\\item if for some $w\\in\\{x,y,z\\}$ the set $\\overline{\\cal C}_{G}(w,B)$ contains at least \ntwo elements, then\ncontract  the two of  them to a pendant edge (that will have $w$ as an endpoint)\nand the rest of them to $w$.\n\\item if for some $w\\in\\{x,y,z\\}$  the set $\\overline{\\cal C}_{G}(w,B)$ contains \nonly one element that is not a hair, then contact it to a triangle (notice that this is always \npossible because of (2)).\n\\end{itemize}\n\\noindent{\\em Case 1}. $|V(B)|\\in\\{3,4\\}$. Then because of (6), one, say $x$ of $\\{x,y,z\\}$ is non-heavy \nand there  is an outer edge connecting $x$ with one, say $y$, vertex in $\\{x,y,z\\}$. Then $x,y$ \nis the required pair of vertices.\n\n\\noindent{\\em Case 2}.   $|V(B)|>4$ and there is at most one outer edge $e$ with endpoints from \n$\\{x,y,z\\}$ in $H$.\nW.l.o.g. we assume that $e=\\{x,y\\}$. Notice  $e$ is a haploid edge, otherwise $H$ can be \ncontracted to \nthe 5th graph in ${\\cal O}_{6}$. Moreover at least one of  $x$, $y$ is non-heavy, otherwise \n$H$ can be contracted \nto one of the graphs in ${\\cal O}_{8}\\cup {\\cal O}_{9}$ (recall that $B$ may have one or two \nhaploid faces).\n\n\\noindent{\\em Case 3}.  There are two outer edges with endpoints from $\\{x,y,z\\}$.\nW.l.o.g. we assume that these edges are $\\{x,y\\}$ and $\\{y,z\\}$. \nOne, say $\\{x,y\\}$, of  $\\{x,y\\}$, $\\{y,z\\}$ is haploid, otherwise\n $H$ can be contracted to some graph in ${\\cal O}_{4}$. \nIf $\\{x,y\\}$ has a light endpoint, then we are done, otherwise, from (6),\n$z$ is light. In this remaining case,  if  if $\\{z,y\\}$ is haploid, then it is also the required edge, otherwise \n$H$ can be contracted to a graph in ${\\cal O}_{9}$.\n\n8. Let $x$ and $y$ be two heavy cut-vertices vertices of $B$. From (5)  $x,y$ are among the \nvertices that \nare incident to the faces $F_{1}$ and $F_{2}$. Suppose, in contrary, that for some face, say \n$F\\in\\{F_1,F_2\\}$, there is no cut vertex in $\\{x,y\\}$ \nthat is incident to $F$. Then $G$ can be contracted to one of the graphs in ${\\cal O}_{9}$.\nThis is enough to prove the first statement except from the case\nwhere $x$ and $y$ are  both lying in both haploid faces and there is a third light cut-vertex $z$\nincident to some, say $x$, \nof $x$, $y$.  In this case, $x$ is assigned the face where $z$ belongs and $y$ is assigned to \nthe other.\n\\end{proof}\n\nLet $G\\in {\\cal Q}$ and let $B$ be a block of $G$. Let also $S$ be the \nset of cut vertices of $G$ that belong to $B$. According to Lemma~\\ref{basic_structure},\nwe can define a rooted graph ${\\bf G}_{B}=(B,X,Y)$ such that \n\\begin{itemize}\n\\item $\\{X,Y\\}$ is a partition of $S$ where $X$ and $Y$ are possibly empty.\n\\item if $B$ has a chord, then all vertices in $X$ and $Y$ are haploid. \n\\item $|X|\\leq 1$ and $|Y|\\leq 2$.\n\\item If $|Y|=2$, then its vertices are connected with an edge $e$ and one of them is light and, \nmoreover, in the case where $B$ has a chord then $e$ is haploid.\n\\item If $B$ has a chord, we name the haploid faces of $B$ by $F_{1}$ and $F_{2}$ such that \nall vertices in $X$ are incident to $F_{1}$ and all vertices od $Y$ are incident to $F_{2}$.\n\\end{itemize}\n\n\\begin{lemma}\n\\label{bolek}\nLet $G\\in {\\cal Q}$ and let $B$ be a block of $G$. Then ${\\bf cmp}({\\bf G}_{B})\\leq 2$.\n\\end{lemma}\n\n\\begin{proof}\nWe examine the non-trivial case where $B$ is a non-trivial block and contains two haploid faces \n$F_{1}$ and $F_{2}$.\nAs $B$ is 2-connected and outer-planar, all vertices of $V(B)$ belong to the unique hamiltonian \ncycle of $B$, say $C$. \nOur proof is based on the fact that there are exactly two haploid faces and this gives a sense \nof direction on how the search should be performed.\nTo make this formal, we create an ordering ${\\cal A}$ of the edges of $E(B)$ using the f\nollowing procedure.\n\n\\begin{tabbing}\n1. {\\bf if} $X\\neq \\emptyset$, \\= {\\bf then} \\\\\n2. \\> $Q\\leftarrow X$, \\\\\n3. \\> {\\bf else} \\\\\n4. \\>   $Q\\leftarrow\\{x\\}$ \\=\nwhere   $x$ is an arbitrarily chosen vertex  \\\\\n \\> \\> in the boundary of $F_{1}$.\\\\\n5. $R\\leftarrow Q$\\\\\n6. $i\\leftarrow 1$\\\\\n7. {\\bf while} \\= there is a vertex $v$ in $V(B)\\setminus R$ that is connected with some, say $u$,   \\\\\n \\> vertex in $Q\\setminus Y$ whose unique neighbor in  $V(B)\\setminus R$ is $v$, \\\\\n8. \\>  $R\\leftarrow R\\cup\\{v\\}$\\\\\n9. \\>  $Q\\leftarrow (Q\\setminus \\{u\\})\\cup\\{v\\}$\\\\\n10. \\>  $e_{i}=\\{u,v\\}$ \\\\\n11. \\>  $i\\leftarrow i+1$ \\\\\n12. \\>  {\\bf if} $Q\\in E(B)$, \\= {\\bf then} \\\\\n13. \\> \\> $e_{i} \\leftarrow Q$, \\\\\n14. \\> \\> $i\\leftarrow i+1$\\\\\n15. {\\bf if} $Y\\in E(B)$, \\= {\\bf then}\\\\\n16. \\> $e_{i} \\leftarrow Y$\n\\end{tabbing}\n\nLet $E^{\\rm in},E^{\\rm out}$ be the edge extensions of ${\\bf enh}({\\bf G}_B)$, and let  \n${\\bf prefsec}({\\cal A})=\\langle A_{0}$, \n$\\ldots,A_{r}\\rangle$.  It is easy to verify  that \n${\\cal E}=\\langle E^{\\rm in}, A_{0}\\cup E^{\\rm in} ,\\ldots,A_{r}\\cup E^{\\rm in}\\rangle$\nis a monotone and connected $(E^{\\rm in},E^{\\rm out})$-expansion of ${\\bf enh}({\\bf G}_{B})$, \nwith cost at most 2.\n\\end{proof}\n\n\\begin{figure}[h]\n\\begin{center}\n \\scalebox{0.6}{\\includegraphics{figure8}}\n \\end{center}\n\\caption{The set ${\\cal A}$ contains 5 $r$-graphs, each\n of the form $(G,\\{v\\},\\{v\\})$.}\n\\label{FanRootObs}\n\\end{figure}\n\n\\subsection{Fans}\n\nLet $G$ be a graph and $v$ be a vertex in $V(G)$. We denote by ${\\bf G}^{(v)}$ the rooted graph \n$(G,\\{v\\},\\{v\\})$ and we refer to it as the {\\em graph $G$ doubly rooted on $v$}.\n\nWe say that a graph $G$, doubly rooted on some vertex $v$ is a {\\em fan}\nif none of the graphs in the set ${\\cal A}$ depicted in Figure~\\ref{FanRootObs} is a contraction \nof the rooted graph \n${\\bf G}^{(v)}$ and $G$ is outerplanar.\n\n\\begin{lemma}\n\\label{seasrchfun}\nLet ${\\bf G}^{(v)}=(G,\\{v\\},\\{v\\})$ be a graph doubly rooted at some vertex $v$. If \n${\\bf G}^{(v)}$ is a fan, then \n${\\bf cmp}({\\bf G}^{(v)})\\leq 2$.\n\\end{lemma}\n\n\\begin{proof} \nWe claim first that if ${\\bf G}^{(v)}$ is a fan, then the graph $G\\setminus v$ is a collection \nof paths where each of \nthem has at least one endpoint \nthat is a neighbour of $v$. Indeed, if this is not correct, then some of the connected components of \n$G\\setminus v$ would be contractible to either a $K_{3}$ or a $K_{1,3}$.\nIn the first case, $G$ is either non-outperlanar or ${\\bf G}^{(v)}$ can be contracted to to the \nfirst two rooted graphs of \nFigure~\\ref{FanRootObs}. In the second case $G$ is either non-outeplanar or \n${\\bf G}^{(v)}$ the last three graphs of Figure~\\ref{FanRootObs}.\nMoreover, if both endpoints of a path in the set of connected components $G\\setminus v$ \nare non adjacent to $v$ in $G$, then ${\\bf G}^{(v)}$ can be contracted to the last rooted \ngraph in Figure~\\ref{FanRootObs}.\n\nLet now $P_{1},\\ldots, P_{r}$ be the connected components of $G\\setminus v$\nand, for each $i\\in\\{1,\\ldots,r\\}$, let $\\{v_{1}^{i},\\ldots,v_{j_i}^{i}\\}$ be the vertices of $P_{i}$,\nordered as in $P_{i}$, such that $v_{1}^{i}$ is adjacent to $v$ in $G$.\nLet  $u^{\\rm in}$ and $u^{\\rm out}$ be the two vertices added in ${\\bf enh}({\\bf G}^{(v)})$.  \nIf $e^{\\rm in}=\\{v,u^{\\rm in}\\}$ and $e^{\\rm out}=\\{v,u^{\\rm out}\\}$ then the edge expansions of \n${\\bf enh}({\\bf G}^{(v)})$ is $E^{\\rm in}=\\{e^{\\rm in}\\}$ and $E^{\\rm out}=\\{e^{\\rm out}\\}$.\n\nFor each $i\\in\\{1,\\ldots,r\\}$ we define the edge ordering \n$${\\cal A}_i=\\langle \\{v,v_1^i\\},\\{v_1^i, v_2^i\\},\\{v,v_2^i\\},\\{v_2^i, v_3^i\\}\\ldots,\\{v_{j_i}^i,v\\}\\rangle,$$\nthen we delete from ${\\cal A}_i$ the edges not in $E(G)$. Let ${\\cal A}_i'$ be the orderings \nobtained after the edge deletions. \nWe define ${\\cal A}=\\langle e^{\\rm in} \\rangle \\oplus {\\cal A}_1\\oplus\\cdots\\oplus{\\cal A}_r$. \nNotice that ${\\bf prefsec}({\\cal A})$ is a monotone and connected \n$(E^{\\rm in},E^{\\rm out})$-expansion of ${\\bf enh}({\\bf G}^{(v)})$ with cost at most 2. \nTherefore, ${\\bf cmp}({\\bf G}^{(v)})\\leq 2$ as required.\\end{proof}\n\n\\subsection{Spine-degree and central blocks}\n\nGiven a graph $G$ and a vertex $v$ we denote by ${\\cal C}^{(v)}_{G}$ the set of all \ngraphs in  ${\\cal C}_{G}(v)$, each doubly rooted on $v$. The {\\em spine-degree} of $v$ \nis the number of doubly rooted\ngraphs in   ${\\cal C}^{(v)}_{G}\n$ that are not fans.\n\nA cut-vertex of a graph $G$ is called {\\em central cut-vertex}, if it has spine-degree greater than 1 \nand  a block \nof $G$ is called {\\em central block} if it contains at least 2 central cut-vertices.\n\n\\begin{lemma}\n\\label{lem:atmost2nonfuns} \nLet $G\\in {\\cal Q}$. The following hold:\n\\begin{itemize}\n\\item[1.] All vertices of $G$ have spine-degree at most 2. \n\\item[2.] None of the blocks of $G$ contains more than 2 central cut-vertices.\n\\item[3.]  $G$ contains at least one central cut-vertex\n\\item[4.] There is a total ordering $B_{1},B_{2},\\ldots,B_{r}$  ($r\\geq 0$)\nof the central blocks of $G$ and a total ordering $c_1,\\ldots,c_{r+1}$ of the central cut-vertices of \n$G$ such that, for $i\\in\\{1,\\ldots,r\\}$, the cut-vertices of $B_{i}$ are $c_{i}$ and $c_{i+1}$.\n\\end{itemize} \n\\end{lemma}\n\n\\begin{proof} 1. Let $v$ be a vertex of $G$ with spine-degree at least 3. That means that there exist \nat least \nthree subgraphs of $G$, doubly rooted on $v$, that can be contracted to some graph in \n$\\cal A$, therefore \n$G$ can be contracted to a graph in ${\\cal O}_{10}$, a contradiction.\n\n2. Suppose that $B$ is a block of $G$ containing 3 (or more) central cut-vertices, say\n$c_1,c_2$ and $c_3$. \nConstruct the graph $H$ by contracting all edges of $B$ to a triangle $T$ with \n$\\{c_{1},c_{2},c_{3}\\}$ as vertex set.\nAs $c_{i}$ is a central vertex, there is a rooted graph $R_{i}$ in ${\\cal C}_{G}(c_i)$ that contains \nsome of the \ngraphs in ${\\cal A}$ as a rooted contraction. Next we apply the same contractions to $H$ f\nor every $c_{i}, i\\in\\{1,2,3\\}$ \nand then contract to vertices all blocks of $H$ different than $T$ and not contained in some $R_{i}$. \nIt is easy to \nsee that the resulting graph is a graph in ${\\cal O}_5$, a contradiction.\n\n3. Assume that $G$ has no central cut-vertices.  We distinguish two cases.\n\n\\noindent{\\em Case 1}. There is a cut-vertex \nof $G$, say $v$, such that all rooted graphs in ${\\cal C}_{G}^{(v)}$ are fans and let \n${\\bf G}_{1},\\ldots {\\bf G}_{r}$\n be these rooted graphs. From  Lemmata~\\ref{glue} and~\\ref{seasrchfun}, we conclude\n that ${\\bf cmp}(G,\\{v\\},\\{v\\})={\\bf cmp}({\\bf glue}({\\bf G}_{1},\\ldots {\\bf G}_{r}))\\leq 2$\n and from Lemma~\\ref{emptymake},  ${\\bf cmp}(G)\\leq 2$ contradicting to the first condition of \n Lemma~\\ref{goinaldirs}. \n \n\\noindent{\\em Case 2}. For every cut-vertex $v$ of $G$, at least one of the rooted graphs in \n${\\cal C}_{G}^{(v)}$ is not a fan. \nWe denote by $H_v$ the corresponding non-fan \nrooted graph in  ${\\cal C}_{G}^{(v)}$  (this is unique due to the fact that $v$ is non-central). \nAmong all cut vertices, \nlet $x$ be one for which \nthe set $V(G)\\setminus V(H_{x})$ is maximal.\nLet $B$ be the block of $H_{x}$ that contains $x$ and let $S$ be the set of the cut-vertices \nof $G$ that belong to $B$ (including $x$).\n\nFor every $y\\in S$ we denote by\n${\\cal W}_{y}=\\{{\\bf W}_{y}^{1},\\ldots,{\\bf W}_{y}^{r_{y}}\\}$ the set \nof all rooted graphs in ${\\cal C}_{G}^{(y)}$, \nexcept from the one, call it ${\\bf R}_{y}$,  that contains $B$. \nWe also define \n${\\bf W}_{y}={\\bf glue}({\\bf W}_{y}^{1},\\ldots,{\\bf W}_{y}^{r_{y}})$.\nWe claim that all ${\\bf W}_{y}, y\\in S$ are fans. Indeed, if for some $y$, ${\\bf W}_{y}$\nis a non-fan, because $y$ is not central, ${\\bf R}_{y}$ should be \na fan, contradicting the choice of $x$.\n\nAccording to the above, the edges of $G$ can be partitioned to those of \nthe rooted graph ${\\bf G}_{B}$ and the edges in the rooted graphs ${\\bf W}_{y},\\ y\\in S$.\nLet also ${\\bf  G}_{B}=(B,X,Y)$ and we assume that, if $Y=\\{y_{1},y_{2}\\}$, then $y_{1}$ is light.\n\n Notice that, according to Lemma~\\ref{seasrchfun} ${\\bf cmp}({\\bf W}_{y})\\leq2,\\ y\\in S$ and \n according to Lemma~\\ref{bolek}\n ${\\bf cmp}({\\bf G}_B)\\leq 2$. \n We distinguish two cases.\n\\smallskip\n\n \\noindent{\\sl Case 2.1.} $Y=\\{y_{1},y_{2}\\}$ where $y_{2}$ is light. Then let  \n ${\\bf G}_{1}=(G[\\{y_1,y_2\\}], \\{y_1,y_2\\},$ $ \\{y_2\\})$ and \n ${\\bf G}_{2}=(G[\\{y_1,y_2\\}], \\{y_2\\}, \\{y_1\\})$. Clearly  \n ${\\bf cmp}({\\bf G}_1)=2$ and  ${\\bf cmp}({\\bf G}_2)=1$. Therefore, if \n ${\\bf G}'= {\\bf glue}({\\bf G}_B, {\\bf G}_1, {\\bf W}_{y_2}, {\\bf G}_2, \n {\\bf W}_{y_1})$, then, from Lemma~\\ref{glue},\n  ${\\bf cmp}({\\bf G}')\\leq 2$.\n \n \\noindent{\\sl Case 2.2.} $Y=\\{y_{1}\\}$. Let ${\\bf G}'= {\\bf glue}({\\bf G}_B, {\\bf W}_{y_1})$, \n then, from Lemma~\\ref{glue},\n  ${\\bf cmp}({\\bf G}')$ $\\leq 2$.\n\\smallskip\n\nIn both cases, if $X=\\{x\\}$, then we set ${\\bf G}={\\bf glue}({\\bf W}_{x},{\\bf G}')$ while if \n$X=\\emptyset$ we set ${\\bf G}={\\bf G}'$. In any case, we observe that, from Lemma~\\ref{glue},  \n${\\bf cmp}({\\bf G})\\leq 2$. \nFrom Lemma~\\ref{emptymake},  ${\\bf cmp}(G)\\leq 2$ contradicting to the first condition of \nLemma~\\ref{goinaldirs}. \n\nAs in both cases we reach a contradiction  $G$ must contain at least one central cut-vertex. \n\n4.  Let $C$ be the set of all central cut-vertices of $G$.\nFor each $c\\in C$, let ${\\cal X}_c$ be the subset of ${\\cal C}^{(v)}_{G}$\nthat contains all its members that are not fans. Clearly, ${\\cal X}_c$\ncontains exactly two elements. Notice that none of the vertices in $C\\setminus \\{c\\}$ \nbelongs in the double rooted graphs in ${\\cal C}^{(v)}_{G}\\setminus {\\cal X}_{c}$. Indeed, \nif this is the case for some vertex $y\\in C\\setminus \\{c\\}$, then the member of ${\\cal C}^{(y)}_{G}$ \nthat avoids $c$ would be a subgraph of some member of \n${\\cal C}^{(v)}_{G}\\setminus {\\cal X}_{c}$ and this would imply that some fan would contain as \na contraction some double rooted graph that is not a fan. We conclude that for each $c\\in C$\nthere is a partition $p(c)=(A_{c},B_{c})$ of  $C\\setminus \\{c\\}$ \nsuch that that all members of ${A}_{c}$ are vertices of one of the \nmembers of ${\\cal X}_c$ and all members of $B_c$ are vertices of the other.\n\nWe say that a vertex $c\\in C$ is {\\em extremal} \nif $p(c)=\\{\\emptyset,C\\setminus \\{c\\}\\}$\n\nWe claim that for any three vertices $\\{x,y,z\\}$ of $C$, there is one, say $y$ of them such that  \n$x$ and $z$ belong \nin different sets of $p(y)$. Indeed, if this is not the case, \nthen one of the following would happen: either there is a vertex $w\\in C$ such that $x,y,z$ \nbelong to different elements of ${\\cal C}_{G}^{(w)}$, a contradiction\nto the first statement of this lemma or $x,y,$ and $z$ belong to the same block of $G$, a \ncontradiction to the second statement of this lemma. \n\nBy the above claim, there is a path $P$ \ncontaining all central cut-vertices in $C$ and we assume that this path is of minimum length \nwhich permits us to assume that its endpoints are extremal vertices of $C$. Moreover, \nheavy cut-vertices \nin $V(P)$ are members of $C$. Let $c_{1},\\ldots,c_{r+1}$ be the \ncentral cut-vertices ordered as they appear in $P$.\nAs, for every $i\\in\\{1,\\ldots,r\\}$ there is a block $B_{i}$ \ncontaining the  central cut-vertices $c_{i}$ and $c_{i+1}$ we end up with the two orderings \nrequired in the forth statement of the lemma.\n\\end{proof}\n\nLet $G$ be a graph in   ${\\cal Q}$.\nSuppose also that $c_1,\\ldots,c_{r+1}$ and $B_{1},B_{2},\\ldots,B_{r}$ are as in \nLemma~\\ref{lem:atmost2nonfuns}.4.\nWe define the {\\em extremal blocks} of $G$ as follows:\n\\begin{itemize}\n\\item If $r>0$, then among all blocks that contain $c_{1}$ as a cut-vertex let $B_{0}$\nbe the one such that $C_{G}(c_{1},B_{0})$, doubly rooted at $c_{1}$, is not a fan, \ndoes not contain any edge of the central blocks of $G$ and does not contain $c_{r+1}$.\nSymmetrically, among all blocks that contain $c_{r+1}$ as a cut-vertex let $B_{r+1}$\nbe the one such that $C_{G}(c_{r+1},B_{r+1})$ doubly rooted at $c_{r+1}$ is not a fan, \ndoes not contain any edge of the central blocks of $G$ and does not contain $c_{1}$. \n\\item If $r=0$, then let $B_{0}$ and $B_{1}$ be the two blocks with the property that for $i\\in\\{0,1\\}$, \n $C_{G}(c_{1},B_{i})$, doubly rooted at $c_{1}$,  is not a fan.\n\\end{itemize}\n\n We call $B_{0}$ and $B_{r+1}$ {\\em left} and {\\em right} extremal block of $G$ respectively.\n We also call the blocks of $G$ that are either central or extremal  {\\em spine} blocks of $G$.\nLet $A(G)$ be the set of cut-vertices of the graphs  $B_{0},B_{1},B_{2},\\ldots,B_{r},B_{r+1}$.\nWe partition $A(G)$ into three sets $A_{1}$, $A_{2}$ and  $A_{3}$ as follows:\n\n\\begin{itemize}\n\\item $A_{1}=\\{c_{1},\\ldots,c_{r+1}\\}$ (i.e. all central vertices).\n\\item $A_{2}$ contains all vertices of $A(G)$ that belong to central blocks and are not central vertices.\n\\item $A_{3}$ contains all vertices of $A(G)$ that belong to extremal blocks and are not central \ncut-vertices.\n\\end{itemize}\nMoreover, we further partition $A_{3}$ to two sets $A_{3}^{(0)}=A_{3}\\cap V(B_{0})$ and \n$A_{3}^{(r+1)}=A_{3}\\cap V(B_{r+1})$.\n\n\\begin{figure}[h]\n\\begin{center}\n \\scalebox{0.85}{\\includegraphics{figure9}}\n \\end{center}\n\\caption{A graph $G$ and the blocks $B_{0},B_{1},\\ldots,B_{3},$ and $B_{4}$.\nThe cut-vertices in $A_{1}=\\{c_{1},\\ldots,c_{4}\\}$ are the grey circular vertices, the vertices in \n$A_{2}$ are the white square vertices and  the vertices in $A_{3}$\nare the dark square vertices.}\n\\label{FanRoossstObs}\n\\end{figure}\n\nLet $G\\in {\\cal Q}$ and $v\\in A(G)$. We denote by ${\\cal R}_{G}^{(v)}$ the set of  the doubly \nrooted graphs in\n${\\cal C}_{G}^{(v)}$ that do not contain any of the  edges of the spine blocks of $G$.\n\n\\begin{lemma}\n\\label{lem:interfdans} \nLet $G\\in {\\cal Q}$ and $v\\in A(G)$. \nThe following hold:\n\\begin{itemize}\n\\item[a)]  Each  doubly rooted graph in ${\\cal R}_{G}^{(v)}$ is a fan.\n\\item[b)] All vertices in $v\\in A_{2}$ are light, i.e.,   for each $v\\in A_{2}$ \n  ${\\cal R}_{G}^{(v)}$ contains \nexactly  one graph that is a hair block of $G$.  \n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\na) Let $v\\in A(G)$. We distinguish two cases:\\\\\n\n\\noindent{\\em Case 1:} $v\\in A_1$. If there exist a double rooted graph in \n${\\cal R}_{G}^{(v)}$ that is not a fan, $v$ will have\nspine-degree greater than 3, a contradiction to the first property of\nLemma~\\ref{lem:atmost2nonfuns}.\\\\\n\n\\noindent{\\em Case 2:} $v\\in A_2\\cup A_3$. If there exist a double rooted graph in \n${\\cal R}_{G}^{(v)}$ that is not a fan, $v$ will \nhave spine-degree greater than 2, therefore $v$ must be central, a contradiction.\\\\\n\nb)  Let $v\\in A_2$, and suppose that ${\\cal R}_{G}^{(v)}$ can be contracted to two edges with \n$v$ as their unique common endpoint, \nor to a triangle. As $v$ belongs to a central block, $G$ can be contracted to a graph in \n${\\cal O}_{5}$, a contradiction to the third property \nof Lemma~\\ref{goinaldirs}.\n\\end{proof}\n\n\\subsection{Directional obstructions}\n\nLet $G\\in {\\cal Q}$ and let $B_{0},B_{1},\\ldots,B_{r},B_{r+1}$ be the spine blocks of $G$.\nNotice first that from Lemma~\\ref{basic_structure}.6 for every $i\\in \\{1,\\ldots, r\\}$, \n$|A_{2}\\cap V(B_{i})|\\leq 1$. Also, from Lemma~\\ref{basic_structure}.6, \nif $A_{2}\\cap V(B_{i})=\\{v\\}$, then $v$ is a light cut-vertex. \nFor $i\\in\\{1,\\ldots,r\\}$, we define the rooted graphs ${\\bf B}_{i}^{*}$ as follows: if \n$A_{2}\\cap V(B_{i})=\\{v\\}$, then ${B}^{*}_{i}$ is the union of $B_{i}$ and \nthe underlying graph of the unique rooted graph in ${\\cal R}_{G}^{(v)}$ \n(this rooted graph is unique and its underlying graph is a hair block of $G$ from the second \nstatement of Lemma~\\ref{lem:interfdans}). If $A_{2}\\cap V(B_{i})=\\emptyset$, then \n${B}_i^{*}$ is $B_i$.\nWe finally define the rooted graph ${\\bf B}^{*}_{i}=(B_{i}^{*},\\{c_{i}\\},\\{c_{i+1}\\})$ for $i\\in\\{1,\\ldots,r\\}$.\n\n\\begin{figure}[h]\n\\begin{center}\n \\scalebox{0.85}{\\includegraphics{figure10}}\\end{center}\n\\caption{A graph $G$, the  extended extremal blocks $B_{0}^*$ and $B_{4}^{*}$ the \nextended central  blocks $B_{1}^*,B_{2}^*,$ and $B_{3}^*$ and \nthe rooted graphs $F_{1},F_{2}$ and  $F_{4}$ ($F_{3}$ is the graph consisting only of the \nvertex $c_{3}$, doubly rooted on $c_{3}$).}\n\\label{FanRoossstObs}\n\\end{figure}\n\nWe also define ${B}_{0}^{*}$ as follows: consider the unique graph $B_0$ in \n${\\cal C}_{G}^{(c_1)}$ that, when doubly rooted on $c_{1}$, is not a \nfan, does not contain any edge of the central blocks of $G$ {and does not contain $c_{r+1}$}. \nThen ${\\bf B}_{0}^{*}=(B_{0},\\emptyset,\\{c_{1}\\})$. Analogously, we define  \n$B_{r+1}^{*}$ by considering the graph $B_{r+1}$ of  the unique rooted graph  in \n${\\cal R}_{G}^{(c_{r+1})}$ \nthat, when doubly rooted on $c_{r+1}$, is not a fan, does not contain any edge of the \ncentral blocks of $G$ {and does not contain $c_{1}$}.\nThen  ${\\bf B}_{r+1}^{*}=(B_{r+1},\\{c_{r+1}\\},\\emptyset)$.\nFinally, we define for each $i\\in\\{1,\\ldots,r+1\\}$\nthe graph $F_{i}$ that is the union of all the graphs of the rooted graphs in \n${\\cal R}_{G}^{(c_{i})}$ that are fans\n(when performing the union, the vertex $c_{i}$ stays the same), and in the case where \n${\\cal R}_{G}^{(c_{i})}$ is empty, then $F_{i}$ is the trivial graph $(\\{c_i\\},\\emptyset)$.\nWe set ${\\bf F}_{i}=(F_{i},\\{c_i\\},\\{c_{i}\\})$ $i\\in\\{1,\\ldots,r+1\\}$ and we  call the rooted graphs  \n${\\bf F}_{1},\\ldots,{\\bf F}_{r+1}$ {\\em extended fans} of $G$.\nWe call ${\\bf B}_{0}^{*},{\\bf B}_{1}^{*},\\ldots,{\\bf B}_{r}^{*},{\\bf B}^{*}_{r+1}$ the \n{\\em extended  blocks} of the graph $G\\in{\\cal Q}$\nand we naturally distinguish them in {\\em central} and {\\em extremal} ({\\em left} or \n{\\em right}), depending of type of the blocks that contain them.\nNotice that \n$$\\{E({ B}_{0}^{*}),E({ F}_{1}),E({ B}_{1}^{*}),E({ F}_2),\\ldots,E({ F}_{r}),E({ B}_{r}^{*}),\nE({ F}_{r+1}),E({ B}_{r+1}^{*})\\}$$ is a partition of the edges of $G$.\n \n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics{figure11}}\n\\end{center} \n\\caption{The set of rooted graphs ${\\cal L}$ containing three rooted graphs each of the form \n$(G,\\{v\\},\\{u\\})$.} \n\\label{morein32sssminorc} \n\\end{figure}\n\n\\begin{lemma}\n\\label{cnrptal}\nLet $G\\in {\\cal Q}$ and let ${\\bf B}_{1}^{*},\\ldots,{\\bf B}_{r}^{*}$ be the central  blocks of $G$. \nNone of the rooted graphs in the set ${\\cal L}$ of Figure~\\ref{morein32sssminorc}\nis a contraction of ${\\bf B}_{i}^*$ if and only if  ${\\bf cmp}({\\bf B}_{i}^*)\\leq 2$.\n\\end{lemma}\n\n\\begin{proof}  \nClearly, for every graph $H\\in\\cal L$, ${\\bf cmp}(H,\\{v\\},\\{u\\})=3$, therefore if ${\\bf B}_{i}^*$ can \nbe contracted to a graph in $\\cal L$, according to Lemma~\\ref{lem:clos},  ${\\bf cmp}({\\bf B}_{i}^*)\\geq 3$.\n\nLet now ${\\bf B}_{i}^*$ be an central extended block of $G$ that cannot be contracted to a graph in \n$\\cal L$. \nIf $B_{i}$ does not contain some light cut-vertex, we define \n${\\bf G}_{B_{i}}=(B_{i},\\{c_i\\},\\{c_{i+1}\\})$. Notice that ${\\bf B}_{i}^{*}={\\bf G}_{B_{i}}$ and, as\nfrom Lemma~\\ref{bolek} ${\\bf cmp}({\\bf G}_{B_{i}^{}})\\leq 2$, we are done.\n\nIn the remaining case, where $B_i$ contains a light cut-vertex, say $c$, observe that $c$  \ncannot be adjacent via an outer edge to $c_i$, or else $B_{i}^{*}$ could be contracted to a graph in \n$\\cal L$. Therefore, according to  Lemma~\\ref{basic_structure}.7, $c$ is connected via an \nhaploid edge with $c_{i+1}$. Notice that ${\\bf G}_{B_i}=(B_i,\\{c_i\\},\\{c_{i+1},c\\})$.\nAccording to Lemma~\\ref{bolek}, ${\\bf cmp}({\\bf G}_{B_{i}})\\leq 2$ and, according to \nLemma~\\ref{lem:interfdans}, ${\\cal R}^{(c)}$ contains only a hair block, \nsay $(H,\\{c\\},\\{c\\})$. Clearly ${\\bf cmp}(H,\\{c\\},\\{c\\})=2$. Let \n${\\bf G}_1=(G[\\{c,c_{i+1}\\}], \\{c,c_{i+1}\\}, \\{c\\})$, ${\\bf G}_2=\n(G[\\{c, c_{i+1}\\}], \\{c\\},\\{c_{i+1}\\}),$ and\n${\\bf G}={\\bf glue}({\\bf G}_{B_{i}}, {\\bf G}_1, (H, \\{c\\},\\{c\\})$, ${\\bf G}_2)$. From Lemma~\\ref{glue}, \n${\\bf cmp}({\\bf G})\\leq 2$ and the lemma follows as ${\\bf G}={\\bf B}_{i}^{*}$.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{cnrptal2}\nIf ${\\bf B}_{i}^{*}$ is one of the central extended  blocks of a graph $G\\in{\\cal Q}$, then either \n${\\bf cmp}({\\bf B}_{i}^{*})\\leq 2$ or ${\\bf cmp}({\\bf rev}({\\bf B}_{i}^{*}))\\leq 2$.\n\\end{lemma} \n\n\\begin{proof} \nProceeding towards a contradiction, from Lemma~\\ref{cnrptal}, both ${\\bf B}_{i}^{*}$ and \n${\\bf rev}({\\bf B}_{i}^{*})$ must contain one of the rooted graphs in \nFigure~\\ref{morein32sssminorc} as a contraction. It is easy to verify that, in this case, either \n$B_{i}^{*}$\ncontains at least four cut-vertices, which contradicts to Lemma~\\ref{basic_structure}.9 or \n$G$ can be contracted to either a graph in ${\\cal O}_8$ (if the two roots are adjacent) \nor a graph in $O_{6}$ (if the two roots are not adjacent), a contradiction to \nLemma~\\ref{goinaldirs}.3.\n\\end{proof}\n\nLet $G\\in {\\cal Q}$ and let ${\\bf B}_{i}^{*}$ be one of the central extended blocks of $G$.\n\n\\begin{itemize}\n\\item If\n${\\bf rev}({\\bf B}_{i}^{*})$ can be contracted to a graph in $\\cal L$, then  we assign \nto ${\\bf B}_{i}^*$ the label $\\leftarrow$.\n\n\\item If \n${\\bf B}_{i}^{*}$ can be contracted to a graph in $\\cal L$, then  we assign \nto ${\\bf B}_{i}^*$ the label $\\rightarrow$.\n\n\\item If both ${\\bf B}_{i}^{*}$ and ${\\bf rev}({\\bf B}_{i}^{*})$\ncan be contracted to a graph in $\\cal L$, then  we assign \nto ${\\bf B}_{i}^*$ the label $\\leftrightarrow$.\n\\end{itemize}\n\n\\begin{figure}[h!] \n\\begin{center} \n\\scalebox{0.75}{\\includegraphics{figure12}} \n\\end{center}\n\\caption{The set of rooted graphs ${\\cal B}$ containing 12 rooted graphs each of the form \n$(G,\\emptyset,\\{v\\})$.} \n\\label{direction} \n\\end{figure}\n\n\\begin{lemma}\n\\label{ro4hk}\nLet $G\\in {\\cal Q}$ and let ${\\bf B}_0^*$ be the extended left extremal block of $G$.\nNone of the rooted graphs in the set ${\\cal B}$ in Figure~\\ref{direction}\nis a contraction of ${\\bf B}_{0}^{*}$ if and only if ${\\bf cmp}({\\bf B}_{0}^{*})\\leq 2$.\n\\end{lemma}\n\n\\begin{proof} \nClearly, for every graph $H\\in\\cal B$, ${\\bf cmp}(H,\\emptyset,\\{u\\})=3$. Therefore, if ${\\bf B}_{0}^*$ can \nbe contracted to a graph in $\\cal B$, according to Lemma~\\ref{lem:clos},  ${\\bf cmp}({\\bf B}_{0}^*)\\geq 3$.\n\nSuppose now  that ${\\bf B}_{0}^*$ cannot be contracted to a graph in $\\cal B$. \nWe distinguish three cases according to the number of cut-vertices in $B_0$ \n(recall that, from Lemma~\\ref{basic_structure}.4, $B_0$ can have at most 3 cut-vertices).\\\\\n\n\\noindent{\\em Case 1:} $B_0$ contains only one cut-vertex, which  is $c_1$. Then,  \n${\\bf B}_{0}^*={\\bf G}_{B_{0}}$ and the result follows  because of Lemma~\\ref{bolek}.\\\\\n\n\\noindent{\\em Case 2:} $B_0$ contains two cut-vertices, $c_1$ and $c$. If $B_{0}$ has \nnot a chord or it has a chord and $c$ and $c_1$ are  incident to two different haploid faces of $B_0$ \nthen we can assume that ${\\bf G}_{B_{0}}=(B_{0},\\{c\\},\\{c_1\\})$ and \nfrom  Lemma~\\ref{bolek}, ${\\bf cmp}({\\bf G}_{B_0})\\leq 2$.\nAccording to Lemma~\\ref{lem:interfdans}.a, ${\\cal R}^{(c)}$ is a fan, \nsay $(F, \\{c\\},\\{c\\})$ and from Lemma~\\ref{seasrchfun} ${\\bf cmp}(F, \\{c\\},\\{c\\})\\leq 2$. Let \n${\\bf G}={\\bf glue}((F, \\{c\\},\\{c\\}),{\\bf G}_{B_0})$. From Lemma~\\ref{glue},  ${\\bf cmp}({\\bf G})\\leq 2$. \nCombining the fact that ${\\bf G}=(B_{0}^{*},\\{c\\},\\{c_1\\})$ with  \nLemma~\\ref{emptymake}, ${\\bf cmp}({\\bf B}_{0}^{*})\\leq 2$.\nIn the remaining case  $c$ and $c_{1}$ are adjacent and $c$ is light.\nThen ${\\bf G}_{B_{0}}=({B}_{0},\\emptyset,\\{c,c_1\\})$ and, from Lemma~\\ref{bolek}, \n${\\bf cmp}({\\bf G}_{B_{0}})\\leq 2$. According to Lemma~\\ref{lem:interfdans}.b, \n${\\cal R}^{(c)}$ is a hair, say $(H, \\{c\\},\\{c\\})$. Let ${\\bf G}_1=(G[\\{c,c_{1}\\}], \\{c,c_{1}\\}, \\{c\\})$, \n${\\bf G}_2=(G[\\{c, c_{1}\\}], \\{c\\},\\{c_{1}\\})$ and\n${\\bf G}={\\bf glue}({\\bf G}_{B_{0}}, {\\bf G}_1, (H, \\{c\\},\\{c\\})$, ${\\bf G}_2)$. Notice that \n${\\bf G}=(B_{0}^*,\\emptyset,\\{c_{1}\\})={\\bf B}_0^{*}$. From Lemma~\\ref{glue}, we obtain that \n${\\bf cmp}({\\bf G})\\leq 2$, therefore ${\\bf cmp}({\\bf B}_{0}^{*})\\leq 2$.\\\\\n\n\\noindent{\\em Case 3:} $B_0$ contains three cut-vertices, $c_1$, $c$ and $x$.  \nWe first examine the case where there is a partition $\\{X,Y\\}$ of $\\{c_1,c,x\\}$ \nsuch that $|Y|=2$, $c_1\\in Y$, the cut-vertex in $Y\\setminus \\{c_1\\}$ is light, and $Y$ is an edge of \n$B_0$ that, in case $B_{0}$ is a chord, is haploid. In this case we claim that ${\\bf cmp}({\\bf B}_{0}^{*})$ \n$\\leq 2$. Indeed,  we may assume that $c$ be the light cut-vertex of $Y\\setminus\\{c_1\\}$, thus \n${\\bf G}_{B_{0}}=(B_{0},\\{x\\},\\{c,c_1\\})$. According \nto Lemma~\\ref{bolek}, ${\\bf cmp}({\\bf G}_{B_{0}})\\leq 2$. From   Lemma~\\ref{lem:interfdans}.a, \n${\\cal R}^{(x)}$ \nis a fan, say $(F, \\{x\\},\\{x\\})$ and,  from   Lemma~\\ref{lem:interfdans}.b, ${\\cal R}^{(c)}$ \ncontains only a hair block, say $(H,\\{c\\},\\{c\\})$. \nLet ${\\bf G}_1=(G[\\{c,c_{1}\\}], \\{c,c_{1}\\}, \\{c\\})$, ${\\bf G}_2=(G[\\{c, c_{1}\\}], \\{c\\},\\{c_{1}\\})$ and\n${\\bf G}={\\bf glue}((F, \\{x\\},\\{x\\}), {\\bf G}_{B_0}, {\\bf G}_1, (H, \\{c\\},\\{c\\})$, ${\\bf G}_2)$. Notice that \n${\\bf G}=(B_{0}^*,\\{x\\},\\{c_{1}\\})$ and. From Lemma~\\ref{glue}\n${\\bf cmp}({\\bf G})\\leq 2$. Now, Lemma~\\ref{emptymake} implies that ${\\bf cmp}({\\bf B}_{0}^{*})$ $\\leq 2$ \nand the claim holds.\n\nIn the remaining cases, the following may happen:\\medskip\n\n1. None of $x$ and $c$ is adjacent to $c_1$ via an edge that, in case $B_{0}$  \nhas a chord, is haploid. In this case ${\\bf B}_{0}^*$ can be contracted to the rooted \ngraphs of the first column in Figure~\\ref{direction}.\\smallskip\n\n2. Both $c_{1}$ and $x$ are light and only one of them, say $x$, is adjacent to $c_1$. \nIn this case $B_{0}$ has a chord and \neither the edge $\\{x,c_1\\}$ is not haploid or  $\\{x,c_1\\}$ is haploid and belongs in the same \nhaploid face with $c$. In the first case, ${\\bf B}_{0}^*$ can be contracted to the second \nrooted graph of the second column in Figure~\\ref{direction}\nand in the second case   ${\\bf B}_{0}^*$ can be contracted to the first and the third rooted \ngraph of the second column in Figure~\\ref{direction}.\\smallskip\n\n3. $c_{1}$ has only one, say $x$, heavy neighbour in $\\{c,x\\}$ such that,   in case $B_{0}$  \nhas a chord, the edge $\\{c_{1},x\\}$ is haploid.\nIn this case ${\\bf B}_{0}^*$ can be contracted to the rooted graphs of the third and the fourth  \ncolumn in Figure~\\ref{direction}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{moreli}\nLet $G\\in {\\cal Q}$ and let ${\\bf B}_{r+1}^*$ be the extended right extremal block of $G$.\nNone of the rooted graphs in the set ${\\cal C}$ in Figure~\\ref{if9ton}\nis a contraction of ${\\bf B}_{r+1}^*$ if and only if  ${\\bf cmp}({\\bf B}_{r+1}^*)\\leq 2$.\n\\end{lemma}\n\n\\begin{proof}\nClearly, for every graph $H\\in\\cal C$, ${\\bf cmp}(H,\\{u\\}, \\emptyset)=3$, therefore if ${\\bf B}_{r+1}^*$ can \nbe contracted to a graph in $\\cal C$, according to Lemma~\\ref{lem:clos},  \n${\\bf cmp}({\\bf B}_{r+1}^*)\\geq 3$.\n\nSuppose now  that ${\\bf B}_{r+1}^*$ cannot be contracted to a graph in $\\cal C$. \nWe distinguish three cases according to the number of cut-vertices in $B_{r+1}$ \n(recall that, from Lemma~\\ref{basic_structure}.4, $B_{r+1}$ can have at most 3 cut-vertices).\\\\\n\n\\noindent{\\em Case 1:} $B_{r+1}$ contains only one cut-vertex, which of course is $c_1$. \nNotice that  ${\\bf B}_{r+1}^*={\\bf G}_{B_{r+1}}$ and the result follows  because of \nLemma~\\ref{bolek}.\\\\\n\n\\noindent{\\em Case 2:}  If $B_{r+1}$ has not a chord or it has a chord and $c$ and $c_1$ \nare  incident to two different haploid faces of $B_{r+1}$ \nthen we can assume that ${\\bf G}_{B_{r+1}}=(B_{r+1},\\{c_{r+1}\\},\\{c\\})$ and \nfrom  Lemma~\\ref{bolek}, ${\\bf cmp}({\\bf G}_{B_{r+1}})\\leq 2$.\nIn any other case, $c_{r+1}$ and $c$ is on the boundary  in the same   haploid face of $B_{r+1}$  \nand none of them belongs in the boundary of  the other.  Then ${\\bf B}_{r+1}^{*}$ can be contracted to \nsome of the rooted graphs in the second column of Figure~\\ref{if9ton}.\\medskip\n\n\\noindent{\\em Case 3:} $B_{r+1}$ contains three cut-vertices, $c_{r+1}$, $c$ and $x$. \nIf $c$ and $x$ are not adjacent, then  ${\\bf B}_{r+1}^{*}$ can be contracted to \nsome of the rooted graphs in the first column of Figure~\\ref{if9ton}.\nOtherwise, one, say $c$, of them will be light and the edge $\\{x,c\\}$\nshould be haploid. Then we can assume that ${\\bf G}_{B_{r+1}}=(B_{r+1},\\{c_{r+1}\\},\\{x,c\\})$ \nand according to Lemma~\\ref{bolek}, \n${\\bf cmp}({\\bf G}_{B_{r+1}})\\leq 2$.\nFrom Lemma~\\ref{lem:interfdans}, \n${\\cal R}^{(x)}$ is a fan, say $(F, \\{x\\},\\{x\\})$ and ${\\cal R}^{(c)}$ contains only a hair block, say \n$(H,\\{c\\},\\{c\\})$. \nLet ${\\bf G}_1=(G[\\{c,x\\}], \\{c,x\\}, \\{c\\})$, ${\\bf G}_2=(G[\\{c, x\\}], \\{c\\},\\{x\\})$ and\n${\\bf G}={\\bf glue}({\\bf G}_{B_{r+1}}, {\\bf G}_1, $ $(H, \\{c\\},\\{c\\}), {\\bf G}_2, (F, \\{x\\},\\{x\\}))$. Notice that \n${\\bf G}=(B_{r+1}^*,\\{c_{r+1}\\},\\{x\\})$ and, because of Lemma~\\ref{glue},\n${\\bf cmp}({\\bf G})\\leq 2$. Applying Lemma~\\ref{emptymake}, we conclude that \n${\\bf cmp}({\\bf B}_{r+1}^{*})\\leq 2$.\n\\end{proof}\n\n\\begin{figure}[h!] \n\\begin{center} \n\\scalebox{0.75}{\\includegraphics{figure13}} \n\\end{center}\n\\caption{The set of rooted graphs ${\\cal C}$ containing six rooted graphs each of the form \n$(G,\\{v\\},\\emptyset)$.} \n\\label{if9ton} \n\\end{figure}\n\n\\begin{lemma}\n\\label{tboths}\nLet $G\\in {\\cal Q}$ and let ${\\bf B}^*_{0}$ and ${\\bf B}^{*}_{r+1}$  be the two extremal \nextended blocks of $G$.\n It is never the case that\n ${\\bf B}^*_{0}$ contains some graph in ${\\cal B}$ and \n  ${\\bf rev}({\\bf B}^*_{0})$ contains some rooted graph in ${\\cal C}$.\n  Also it is never the case that\n ${\\bf B}^{*}_{r+1}$ contains some graph in ${\\cal C}$ and \n  ${\\bf rev}({\\bf B}^{*}_{r+1})$ contains some rooted graph in ${\\cal B}$.\n\\end{lemma}\n\n\\begin{proof} \nLet ${\\bf G}$ be a rooted graph in $K=\\{{\\bf rev}({\\bf B}^*_{0}),{\\bf B}^{*}_{r+1}\\}$.\nWe distinguish the following cases, that apply for both rooted graphs in $K$:\\\\\n\n\\noindent{\\em Case 1:} $\\bf G$ can be contracted to a graph in the first column of Figure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the first column of Figure~\\ref{direction}. Notice that every cut-vertex \n of $\\bf G$ cannot be connected with an outer edge and therefore  $\\bf G$ can be contracted to graph \n in ${\\cal O}_6$.\\\\\n\n\\noindent{\\em Case 2:} $\\bf G$ can be contracted to a graph in the first column of Figure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the second column of Figure~\\ref{direction}. \n Notice that the two cut-vertices\nof $\\bf G$, that are other than the central cut-vertex, cannot be connected with an outer edge \nand therefore  \n$\\bf G$ can be contracted to graph in ${\\cal O}_7$.\\\\\n\n\\noindent{\\em Case 3:} $\\bf G$ can be contracted to a graph in the first column of Figure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the third or fourth column of Figure~\\ref{direction}. Notice that the \n two cut-vertices\nof $\\bf G$, that are other than the central cut-vertex, cannot be connected with an outer edge \nand therefore  \n$\\bf G$ can be contracted to graph in ${\\cal O}_8$.\\\\\n\n\\noindent{\\em Case 4:} $\\bf G$ can be contracted to a graph in the second column of \nFigure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the first column of Figure~\\ref{direction}. Notice that the two \n cut-vertices\nof $\\bf G$, that are other than the central cut-vertex, cannot be connected with an outer edge \nand therefore  \n$\\bf G$ can be contracted to graph in ${\\cal O}_7$.\\\\\n\n\\noindent{\\em Case 5:} $\\bf G$ can be contracted to a graph in the second column of \nFigure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the second column of Figure~\\ref{direction}. Notice that there must be \nan haploid face containing only the central cut-vertex, therefore $\\bf G$ can be contracted either to\nthe graph in ${\\cal O}_2$  or to a graph in ${\\cal O}_4$ (depending whether $\\bf G$ can be \ncontracted to the last graph in the second column of Figure~\\ref{if9ton} or not) .\\\\\n \n\\noindent{\\em Case 6:} $\\bf G$ can be contracted to a graph in the second column of \nFigure~\\ref{if9ton}\n and ${\\bf rev}(\\bf G)$ to a graph in the third or fourth column of Figure~\\ref{direction}. \n Notice that the light cut-vertex of ${\\bf  G}$ can not be connected via an haploid edge \n with the central cut-vertex, therefore $\\bf G$ can be contracted to a graph in ${\\cal O}_7$ \n either to a graph in ${\\cal O}_9$ (depending whether the central cut-vertex is connected \n via haploid edge with a heavy cut-vertex or not).\n\\end{proof}\n\n\\begin{itemize}\n\\item If ${\\bf B}^*_{0}$ contains some graph in ${\\cal B}$  then we assign \nto ${\\bf B}^*_{0}$ the label $\\leftarrow$.\n\n\\item If ${\\bf rev}({\\bf B}^*_{0})$ contains some graph in ${\\cal C}$ then we assign \nto ${\\bf B}^*_{0}$ the label $\\rightarrow$.\n\n\\item If ${\\bf B}^{*}_{r+1}$ contains some graph in ${\\cal C}$  then we assign \nto ${\\bf B}^{*}_{r+1}$ the label $\\leftarrow$.\n\n\\item If  ${\\bf rev}({\\bf B}^{*}_{r+1})$ contains some graph in ${\\cal B}$ then we assign \nto ${\\bf B}^{*}_{r+1}$ the label $\\rightarrow$.\n\n\\item If neither ${\\bf B}^*_{0}$ contains some graph in ${\\cal B}$\nnor  ${\\bf rev}({\\bf B}^*_{0})$ contains some graph in ${\\cal C}$ then we assign \nto ${\\bf B}^*_{0}$ the label $\\leftrightarrow$.\n\n\\item If neither ${\\bf B}^{*}_{r+1}$ contains some graph in ${\\cal C}$ \nnor    ${\\bf rev}({\\bf B}^{*}_{r+1})$ contains some graph in ${\\cal B}$ then we assign \nto ${\\bf B}^{*}_{r+1}$ the label $\\leftrightarrow$.\n\\end{itemize}\n\n\\begin{lemma}\n\\label{tplerr}\nLet $G\\in {\\cal Q}$ and let ${\\bf B}^{*}_0,{\\bf B}_{1}^{*},\\ldots,{\\bf B}_{r}^{*},{\\bf B}^{*}_{r+1}$ \nbe the extended  blocks of $G$.\nIt is not possible that one of these extended blocks is labeled with $\\leftarrow$ and an other \nwith $\\rightarrow$.\n\\end{lemma}\n\n\\begin{proof} \nWe distinguish two cases according to the labelling of ${\\bf B}_{0}^{*}$:\\\\\n \n\\noindent{\\em Case 1:} Suppose that ${\\bf B}_{0}^{*}$ is labeled $\\leftarrow$ \nand that ${\\bf B}_{i}^{*}$, for some $i\\in\\{1,\\ldots,r+1\\}$, is labeled $\\rightarrow$. According to their \nrespective labels, $(B_{0}^{*},\\emptyset,\\{c_1\\})$\ncan be contracted to a graph in $\\cal B$ and,  if $i\\leq r$,\n ${\\bf rev}({\\bf B}_{i}^{*})=(B_{i}^{*}, \\{c_{i+1}\\},\\{c_{i}\\})$ can be contracted to a graph in \n${\\cal L}$, otherwise ${\\bf rev}({\\bf B}_{r+1}^{*})=(B_{r+1}^{*},\\emptyset,\\{c_{r+1}\\})$ can \nbe contracted to a rooted graph in ${\\cal B}$.\nNotice that if  $i\\leq r$, $G[V(G)\\setminus (V(B_{1}^{*})\\cup\\cdots\\cup \nV(B_{i-1}^{*}))],\\emptyset,\\{c_i\\}\\}$ can be contracted to a graph in the third and forth columns of \n$\\cal B$. By further contracting all edges of $E(B_{1}^{*})\\cup \\cdots\\cup E(B_{i-1}^{*})$ we obtain \na graph in ${\\cal O}_{11}$, a contradiction.\\\\\n\n\\noindent{\\em Case 2:} Suppose now that ${\\bf B}_{0}^{*}$ is labeled $\\rightarrow$ \nand that ${\\bf B}_{i}^{*}$, for some $i\\in\\{1,\\ldots,r+1\\}$, is labeled $\\leftarrow$. According to their \nrespective labels, ${\\bf rev}({\\bf B}_{0})^{*}=(B_{0}^{*},\\{c_1\\},\\emptyset)$\ncan be contracted to a graph in $\\cal C$ and,  if $i\\leq r$, \n$(B_{i}^{*}, \\{c_{i}\\},\\{c_{i+1}\\})$ can be contracted to a graph in \n$\\cal L$, otherwise $(B_{r+1}^{*}, \\{c_{r+1}\\},\\emptyset)$ can be contracted to a graph in\n${\\cal C}$. \nNotice that if  $i\\leq r$, $G[V(G)\\setminus (V(B_{1}^{*})\\cup\\cdots\\cup \nV(B_{i-1}^{*}))],\\{c_i\\},\\emptyset\\}$ can be be contracted to a graph in the first column of $\\cal C$. \nBy further contracting all edges of $E(B_{1}^{*})\\cup \\cdots\\cup E(B_{i-1}^{*})$ we obtain \na graph in ${\\cal O}_{12}$, a contradiction.\n\\end{proof}\n\n\\subsection{Putting things together}\n\\begin{lemma}\n\\label{tll0o}\n${\\cal Q}=\\emptyset$.\n\\end{lemma}\n\n\\begin{proof} \nSuppose in contrary that ${\\cal Q}$ contains some graph $G$. Let \n${\\bf B}_{0}^{*},{\\bf B}_{1}^{*},\\ldots,{\\bf B}_{r}^{*},$ ${\\bf B}^{*}_{r+1}$ \nand  ${\\bf F}_{1},\\ldots,{\\bf F}_{r+1}$ be the extended blocks and fans of $G$, respectively.\nFrom Lemma~\\ref{tplerr}, we can assume that the extended blocks of $G$ \nare all labeled either $\\rightarrow$ or $\\leftrightarrow$ (if this is not the case, \njust reverse the ordering of the blocks).\nBy the labelling of ${\\bf B}_{0}^{*}$,\nnone of the rooted graphs in the set ${\\cal B}$ is a contraction of ${\\bf B}_{0}^{*}$ therefore, \nfrom Lemma~\\ref{ro4hk}, \n${\\bf cmp}({\\bf B}_{0}^{*})\\leq 2$. Also as none of the rooted graphs ${\\bf B}_{i}^{*}$, $i=1,\\ldots,r,$ \ncan be contracted to a graph \nin ${\\cal L}$, from Lemma~\\ref{cnrptal}, it follows that ${\\bf cmp}({\\bf B}_{i}^{*})\\leq 2$. \nWe distinguish two cases according to the labelling of ${\\bf B}_{r+1}^{*}$.\n If the labelling is $\\leftrightarrow$, then ${\\bf B}_{r+1}^{*}$ cannot be contracted to a graph \n in ${\\cal C}$. If the labelling is $\\rightarrow$, then ${\\bf rev}({\\bf B}_{r+1}^{*})$ can be contracted to\n  a graph in ${\\cal B}$ and, according to Lemma~\\ref{tplerr}, \n${\\bf B}_{r+1}^{*}$ cannot be contracted to a graph in ${\\cal C}$. Thus, in both cases, from \nLemma~\\ref{moreli},  \n${\\bf cmp}({\\bf B}_{r+1}^{*})\\leq 2$. Notice that \n$(G,\\emptyset,\\emptyset)={\\bf glue}({\\bf B}_{0}^{*},{\\bf F}_{1},{\\bf B}_{1}^{*},\\ldots,\n{\\bf F}_{r},{\\bf B}_{r}^{*},{\\bf F}_{r+1},{\\bf B}_{r+1}^{*})$ and, from Lemma~\\ref{glue}, \n${\\bf cmp}(G,\\emptyset,\\emptyset)\\leq 2$.\nThis implies that ${\\bf cmp}(G)\\leq 2$, a contradiction to the first property of Lemma~\\ref{goinaldirs}.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:cs-obs}\n${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])={\\bf obs}_{\\preceq}({\\cal G}[{\\bf cms},2])$.\n\\end{corollary}\n\n\\begin{proof}\nIt is easy to check that for every $H\\in {\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])$:\n\\begin{enumerate}\n\\item ${\\bf cms}(H)\\geq 3$,\n\\item for every proper contraction $H'$ of $H$ it holds that ${\\bf cms}(H')\\leq 2$, \n\\end{enumerate}\ntherefore ${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])\\subseteq {\\bf obs}_{\\preceq}({\\cal G}[{\\bf cms},2])$.\n\nIf there exist a graph $H\\in {\\bf obs}_{\\preceq}({\\cal G}[{\\bf cms},2])\\setminus{\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])$,\nthen ${\\bf cms}(H)$ $\\geq 3$. Notice that the connected search number of a graph is always bounded \nfrom the\nmonotone and connected search number, {as a complete monotone and connected search strategy \nis obviously a complete connected search strategy, therefore} ${\\bf cmms}(H)\\geq 3$, which means \nthat there \nexist a graph $H'\\in {\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},2])$ such that $H'\\preceq H$. Furthermore, \nsince $H'$ is \na proper contraction of $H$, according to Lemma~\\ref{lem:cs-clos}, ${\\bf cms}(H)\\geq{\\bf cms}(H')$. \nAs we have already stressed that ${\\bf cms}(H')\\geq 3$ we reach a \ncontradiction to the minimality (with respect of $\\preceq$) of $H$.\n\\end{proof}\n\n\\section{General Obstructions for ${\\bf cmms}$} \n\nAs we mentioned before, for $k>2$, we have no guarantee that the set \n${\\bf obs}_{\\preceq}({\\cal G}[\\ cmms,k])$\n is a finite set. In this section we  prove that when this set is finite its size should be \n double exponential in $k$. Therefore, it seems hard to\nextend our results for $k\\geq 3$ as, even if we somehow manage to prove that the \nobstruction set for a specific $k$ is finite, then this set would contain more than \n$2^{2^{\\Omega(k)}}$ graphs.\n\nWe will describe a procedure that generates, for each $k$, a set of at least \n$\\frac{4}{3}(\\frac{5}\n{2})^{3\\cdot 2^{k-2}}$ non-isomorphic graphs that have connected and monotone search number \n$k+1$ and are  contraction-minimal\nwith respect to this property. Hence, these $\\frac{4}{3}(\\frac{5}{2})^{3\\cdot 2^{k-2}}$ graphs will belong \nto ${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},k])$.\n\nWe define for every $k\\geq1$  a set of rooted graphs, namely the set of  \n{\\em obstruction-branches} denoted ${\\bf Br}(k)$, as follows:\\medskip\n\n\\noindent {\\bf For $k=1$:} The set ${\\bf Br}(1)$ consists of the five graphs of \nFigure~\\ref{FanRootObs} rooted at $v$.\\medskip\n\n\\noindent {\\bf For $k=l>1$:} The set ${\\bf Br}(l)$ is constructed by choosing two {\\em branches} \nof the set ${\\bf Br}(l-1)$ and identify the two roots to a single vertex, say $v$. Then we add a \nnew edge with $v$ as an endpoint, say $\\{u,v\\}$, and we root this branch to $u$. We will refer \nto this edge as the {\\em trunk} of the branch.\\medskip\n\nLet $f(k)$ be the number of branches of ${\\bf Br}(k)$. Notice that $f(1)=5$ and $f(k)$ is equal \nto the number of ways we can pick two branches of ${\\bf Br}(k-1)$, with repetition. Therefore:\n\n\\begin{eqnarray}\nf(k)&=&{f(k-1)+2-1 \\choose 2}=  {f(k-1)+1 \\choose 2}\\nonumber \\\\\n&=&\\frac{f(k-1)^2+f(k-1)}{2}\\geq \\frac{f(k-1)^2}{2}\\nonumber \\\\\n&\\geq&\\frac{\\big(\\frac{f(k-2)^2}{2}\\big)^2}{2}=\\frac{f(k-2)^{2^2}}{2^{2+1}}\\geq\\frac{f(k-3)^{2^3}}\n{2^{2^2+2+1}}\\nonumber \\\\\n&\\geq&\\cdots\\geq \\frac{f(1)^{2^{k-1}}}{2^{2^{k-2}+\\cdots+2+1}}=\\frac{5^{2^{k-1}}}{2^{2^{k-1}}}\n=2\\Big(\\frac{5}{2}\\Big)^{2^{k-1}}\\nonumber\n\\end{eqnarray}\n\nLet ${\\cal O}_{\\bf Br}(k)$ be the set containing the graphs obtained by choosing three rooted \nbranches\nof ${\\bf Br}(k)$, with repetitions, and identify the three roots. {Notice that any two such \n selections produce two non-isomorphic graphs.} We are going to prove that \n${\\cal O}_{\\bf Br}(k)\\subseteq \n{\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},k+1])$. Notice that:\n\n\\begin{eqnarray}\n|{\\cal O}_{\\bf Br}(k)|&=&{f(k)+3-1 \\choose 3}={f(k)+2 \\choose 3}\\nonumber \\\\\n&=&\\frac{(f(k)+2)(f(k)+1)f(k)}{6}\\nonumber\\\\\n&=&\\frac{f(k)^3+3f(k)^2+2f(k)}{6}\\geq \\frac{f(k)^3}{6}\\nonumber\\\\\n&\\geq&\\frac{4}{3}\\Big(\\frac{5}{2}\\Big)^{3\\cdot2^{k-1}}\\nonumber\n\\end{eqnarray}\n\n\\begin{figure}\n\\begin{center} \n\\scalebox{0.8}{\\includegraphics{figure14.pdf}}\n\\end{center}\n\\caption{The left graph belongs to ${\\cal O}_{\\bf Br}(2)$ and the right to ${\\cal O}_{\\bf Br}(3)$.} \n\\label{brobstr} \n\\end{figure}\n\nHence the cardinality of ${\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},k])$ is at least \n$\\frac{4}{3}\\Big(\\frac{5}{2}\\Big)^{3\\cdot2^{k-2}}$. In order to prove this we need the \nfollowing Lemmata.\n\n\\begin{lemma}\\label{boundLem0}\nLet $B\\in {\\bf Br}(k)$ and let $v$ be its root. There does not exist a complete \nmonotone and connected search strategy for $B$ \nthat uses $k$ searchers, such that  the first edge being cleaned is the trunk of $B$.\n\\end{lemma}\n\n\\begin{proof}\nWe are going to prove this by induction. We can easily check that for $k=1$ the claim holds. \nLet $B\\in {\\bf Br}(k)$ and let $v$ be its root and $u$ the other endpoint of the trunk. \nSince we are forced to clean $B$, in a connected and monotone manner, \nwith a search strategy, say $\\cal S$, that first cleans $\\{u,v\\}$, a searcher must be \nplaced in $u$ during each step of $\\cal S$, therefore we must clean a $(k-1)$-level \nbranch using $k-1$ searchers that first clean the trunk of this branch, \nwhich contradicts the induction hypothesis.  \n\\end{proof}\n\n\\begin{corollary}\\label{boundCor1}\nLet $G\\in {\\cal O}_{\\bf Br}(k)$, then ${\\bf cmms}(G)>k+1$. \n\\end{corollary}\n\n\\begin{proof}\nLet $G\\in {\\cal O}_{\\bf Br}(k)$. Notice that $G$ consists of three $k$-level obstruction branches, \nsay $B_1$, $B_2$ and $B_3$. If there exist a complete monotone and connected search strategy \n$\\cal S$ that uses $k+1$ searchers, then from Lemma~\\ref{boundLem0} $\\cal S$ cannot start \nby placing searchers in the central vertex, i.e. the vertex where $B_1$, $B_2$ and $B_3$ \nare connected. Therefore, $\\cal S$ starts by placing searchers in a vertex of $B_1$, $B_2$ or \n$B_3$ and consequently the first edge cleaned belongs to this branch. Notice that the first \ntime that a searcher is placed on the central vertex the connectivity and monotonicity of \n$\\cal S$ force us to clean a $k$-level branch with $k$ searchers, which is impossible \naccording to Lemma~\\ref{boundLem0}. \n\\end{proof}\n\n\\begin{lemma}\\label{boundLem2}\nLet $B\\in {\\bf Br}(k)$ and let $v$ be its root. \n\\begin{itemize}\n\\item[a)] There exist a complete monotone and connected search strategy for $B$ that uses \n$k+2$ searchers, such that in each step a searcher occupies $v$. \n\\item[b)] There exists a complete monotone and connected search strategy for $B$ \nthat uses $k+1$ searchers, such that  the first edge being cleaned is the trunk of $B$.\n\\item[c)] There exist a complete monotone and connected search strategy for $B$ that uses \n$k+1$ searchers, such that the last edge being cleaned is the trunk of $B$. \n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\na) We are going to prove this by induction. We can easily check that for $k=1$ the claim holds. \nLet $B\\in {\\bf Br}(k)$ and let $v$ be its root and $u$ the other endpoint of the trunk. \nWe are going to describe a search strategy $\\cal S$ with the properties needed. \nWe place a searcher in $v$ and  as second searcher in $u$. According to the induction \nhypothesis for each one of the two $(k-1)$-level branches connected to $u$ there exists a \ncomplete monotone and connected search strategy that uses $k+1$ searchers such that in \neach step a searcher occupies $u$, therefore we can continue by cleaning one of these \n$(k-1)$-level branches and then clean the other.\\medskip\n\nb) We are going to prove this by induction. For $k=1$ the claim is trivial. Let $B\\in {\\bf Br}(k)$ \nand let $v$ be its root and $u$ the other endpoint of the trunk. There are two $(k-1)$-level \nbranches connected to $u$, say $B_1$ and $B_2$. The search strategy, say $\\cal S$, \nwith the properties needed is the following: we place a searcher in $v$ and then slide him to $u$. \nAccording to the first claim of Lemma~\\ref{boundLem2} there exists a complete monotone and \nconnected search strategy ${\\cal S}_1$ for $B_1$ that uses $k+1$ searchers such that in each step a \nsearcher occupies $u$. By the induction hypothesis there exists a complete monotone and connected \nsearch strategy ${\\cal S}_2$ for $B_2$ that uses $k$ searchers such that the first edge cleaned is the \ntrunk of $B_2$. Using these two search strategies we can start by cleaning $B_1$, keeping in all \ntimes a searcher in $u$, and then we can clean $B_2$.\\medskip\n\nc) We are going to prove this by induction. Notice that for $k=1$ the claim holds. Let \n$B\\in {\\bf Br}(k)$ and let $v$ be its root and $u$ the other endpoint of the trunk. There are two \n$(k-1)$-level branches connected to $u$, say $B_1$ and $B_2$. According to the \ninduction hypothesis there exist a complete monotone and connected search strategy ${\\cal S}_1$ for \n$B_1$ that uses $k$ searchers such that the last edge cleaned is the trunk of $B_1$. Moreover, \naccording to the first claim of Lemma~\\ref{boundLem2} there exists a complete monotone and \nconnected search strategy ${\\cal S}_2$ for $B_2$ that uses $k+1$ searchers such that in each step a \nsearcher occupies $u$. Using these two search strategies  we can clean $B$, in a monotone and \nconnected manner, as follows: we start by cleaning $B_1$ then we  clean $B_2$, keeping in all times \na searcher in $u$, and then we clean $\\{u,v\\}$.\n\\end{proof}\n\n\\begin{lemma}\\label{boundLem3}\nLet $G\\in {\\cal O}_{\\bf Br}(k)$ and $B\\in {\\bf Br}(k)$ one of the three branches of $G$. \nIf we contract an edge of $B$ there exist a complete monotone and connected \nsearch strategy for $B$ that uses \n$k+1$ searchers, such that in each step a searcher occupies $v$. \n\\end{lemma}\n\n\\begin{proof}\nWe are going to prove this by induction. It is easy to check that for $k=1$ the claim is true.\nLet $v$ be the root of $B$ and $u$ the other endpoint of the trunk, $B_1$ and $B_2$ the two \n$(k-1)$-level branches connected to $u$ and $e\\in E(B)$ the edge contracted. \nWe distinguish to cases:\\medskip\\\\\n\\noindent{\\em Case 1:} $e\\in E(B_1)\\cup E(B_2)$. We can assume that $e$ is an edge of $B_1$. \nWe are going to describe a search strategy $\\cal S$ for $B$ with the properties needed. \nWe place a searcher in $v$ and  a second searcher in $u$. From the induction hypothesis \nthere exists a complete monotone and connected search strategy ${\\cal S}_1$ for $B_1$ \nthat uses $k$ searchers, such that in each step a searcher occupies $u$. Moreover, \naccording to the second claim of Lemma~\\ref{boundLem2} there exists a complete \nmonotone and connected search strategy ${\\cal S}_2$ for $B_2$ that uses $k$ searchers \nsuch that the first edge cleaned is the trunk of $B_2$.  Using these two search strategies \nwe can start by cleaning $B_1$, keeping in all times a searcher in $u$, and then we can \nclean $B_2$. Notice that this search strategy uses $k+1$ searchers and during each step \na searcher occupies $v$.\n\\medskip\n\n\\noindent{\\em Case 2:} $e=\\{u,v\\}$. According to the first property of Lemma~\\ref{boundLem2}, \nfor each one of $B_1$ and $B_2$ there exists a complete monotone and connected search \nstrategy that uses $k+1$ searchers such that in each step a searcher occupies $v$. \nHence we can clean $B$ starting by cleaning $B_1$, keeping in all times a searcher in \n$v$, and then clean $B_2$.\n\\end{proof}\n\n\\begin{corollary}\\label{boundCor2}\nIf $G\\in {\\cal O}_{\\bf Br}(k)$ and $G'$ be a contraction of $G$, then ${\\bf cmms}(G')=k+1$. \n\\end{corollary}\n\n\\begin{proof}\nIt suffices to prove this claim for a single edge contraction.\nLet $G\\in {\\cal O}_{\\bf Br}(k)$, let $B_1$, $B_2$, and $B_3$ be the three $k$-level obstruction-\nbranches of $G$ connected to $v$, $e\\in E(G)$ the edge contracted and $G'$ the graph obtained \nfrom $G$ after the contraction of $e$. We can assume that $e\\in E(B_2)$. We are going to describe a \ncomplete monotone and connected search strategy $\\cal S$ for $G$. From the third claim of Lemma~\n\\ref{boundLem2} we know that there exist a complete monotone and connected search strategy \n${\\cal S}_1$ for $B_1$ that uses $k+1$ searchers, such that the last edge cleaned is the trunk of \n$B_1$. From Lemma~\\ref{boundLem3} we know that there exist a complete monotone and connected \nsearch strategy ${\\cal S}_2$ for $B_2$ that uses $k+1$ searchers, such that in each step a searcher \noccupies the root of $B_2$, in other words $v$. From the second claim of Lemma~\\ref{boundLem2} \nwe know that there exist a complete monotone and connected search strategy ${\\cal S}_3$ for $B_3$ \nthat uses $k+1$ searchers, such that the first edge cleaned is the trunk of $B_3$. Therefore, we can \nclean $G'$ starting by cleaning $B_1$ according to ${\\cal S}_1$ (notice that the trunk of $B_1$ will be \nthe last edge of $E(B_1)$ being cleaned), then clean $B_2$ according to ${\\cal S}_2$, keeping in all \ntimes a searcher in $v$, and finish by cleaning $B_3$ according to ${\\cal S}_3$.\n\\end{proof}\n\nCombining Corollaries~\\ref{boundCor1} and~\\ref{boundCor2} we conclude that every graph in \n${\\cal O}_{\\bf Br}(k)$ is a contraction obstruction for the graph class ${\\cal G}[{\\bf cmms},k+1]$ \nand therefore ${\\cal O}_{\\bf Br}(k)\\subseteq {\\bf obs}_{\\preceq}({\\cal G}[{\\bf cmms},k+1])$.\n\n\\nocite{*} \n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMassive stars dominate their birth environments through a set of powerful feedback processes, the full implications of which are not completely understood. Among their various feedback mechanisms is the formation of H\\,{\\sc ii}  regions---bubbles of hot ($10^4$ K), ionized gas that expand into the colder ($10^2$ K) surrounding medium. These regions are fueled by the prodigious amounts of UV radiation emitted by massive stars \\citep{Hoare+2007,Beuther+2007}. By heating and ionizing the gas around them, massive stars alter their birth environments and provide detectable observational signatures.\n\nH\\,{\\sc ii}  regions may contribute to the destruction of molecular clouds \\citep{Keto2002,Keto2003,Keto2007,Matzner2002}, helping to set cloud lifetimes. They are observable by their radio continuum emission \\citep{MezgerHenderson1967}, or by their recombination lines (e.g.\\ \\citet{WoodChurchwell1989} use the H76$\\alpha$ line). Even Young Stellar Objects (YSOs) emit enough UV radiation to begin ionizing the gas around them \\citep{Bik+2005,Churchwell2002}, which might teach us something about the lives of massive protostars before they reach the main sequence.\n\nMore recently, observations have shown time variability in H\\,{\\sc ii}  regions \\citep{Franco-HernandezRodriguez2004,Rodriguez+2007,Galvan-Madrid+2008,gomezetal08}. \\citet{Franco-HernandezRodriguez2004} have suggested that such observed time-variability may be due to the changes occurring in the source of the ionizing radiation, though it may also be due to increased absorption in the rapidly-evolving core of the nebula.\n\nThree-dimensional collapse simulations of massive star formation that included feedback by ionizing radiation showed time variability with changes in size and flux very similar to the observations \\citep{Peters2010a,Klassen+2012}. The origin of this time variability is the strong accretion flow around the massive star, which is dense enough to shield the ionizing radiation. As a consequence, the H\\,{\\sc ii}  region fluctuates between extended and trapped states as long as the star is embedded in a sufficiently strong accretion flow. In the simulations, the flickering stops when companion stars form in the gravitationally unstable accretion flow around the central massive star and intercept material that would otherwise be accreted onto the high-mass star, a process we call ``fragmentation-induced starvation'' \\citep{Peters2010c}. Then the accretion flow becomes weak enough that the H\\,{\\sc ii}  region can isolate the forming high-mass star and terminate the accretion process entirely. Magnetic fields, which provide additional support against gravitational collapse and reduce the number of companion stars formed, do not seem to directly affect the H\\,{\\sc ii}  region variability \\citep{Peters2011}.\n\nThe time variability during the accretion phase can lead to morphological changes of the H\\,{\\sc ii}  region appearance over timescales as short as $\\sim 10$ years and appears able to resolve the lifetime problem for ultracompact H\\,{\\sc ii}  regions \\citep{WoodChurchwell1989,Peters2010b}. With the aid of a statistical analysis of the synthetic radio continuum observations of the H\\,{\\sc ii}  regions formed in the simulations, \\citet{Galvan-Madrid+2011} predicted that 7\\% of ultracompact and hypercompact H\\,{\\sc ii}  regions should have detectable flux increments, and 3\\% should have detectable flux decrements when observed at two epochs separated by about 10 years. Particularly interesting are shrinking H\\,{\\sc ii}  regions, for which \\citet{Galvan-Madrid+2011} found that flux decrements by at least 10\\% are twice as likely as flux decrements by at least 50\\% for this time interval.\n\nIn general, however, H\\,{\\sc ii}  region variability has two possible causes. The first is the chaotic gas motions around a forming star, where denser, optically thick gas can intercept stellar radiation and causing the shadowed gas to neutralize \\citep{Peters2010a}. The second possible source of H\\,{\\sc ii}  region variability is due to the star itself, as already noted by \\citet{Franco-HernandezRodriguez2004}. Pre-main-sequence stars undergo changes during their early evolution, sometimes swelling and other times contracting \\citep{YorkeBodenheimer2005}. This can potentially occur on even short time scales ($<2000$ years). Changes in the surface temperature of a star will affect its UV flux, which in turn affects the size of the H\\,{\\sc ii}  region.\n\nThe connection between protostellar variability and H\\,{\\sc ii}  region variability has not yet been explored. To do so, simulations must be equipped with good protostellar models. We ask: can early variability of an H\\,{\\sc ii}  region be traced directly to the pre-main-sequence evolution of the massive protostar itself? Models of prestellar evolution have been investigated by \\citet{PallaStahler1991}, \\citet{PallaStahler1992}, \\citet{Nakano2000}, \\citet{McKeeTan2003}, \\citet{Offner2009} and \\citet{HosokawaOmukai2009}, among others, but never investigating the H\\,{\\sc ii}  region variability that could result from a massive evolving protostar.\n\nIn a previous paper \\citep{Klassen+2012}, we simulated the effects of using prestellar models in simulations of clustered massive star formation with initial conditions as in \\citet{Peters2010a}. Fluctuations in H\\,{\\sc ii}  region size and morphology due to the environmental factors were unaffected by the choice of prestellar model, as might be expected. The prestellar model we implemented, however, did show a delayed onset of major heating and ionization in the cluster\n\nIn the present paper, we investigate the effect of pre-main-sequence evolution on the ionization feedback of a massive protostar with radiation-hydrodynamic simulations that follow the formation and expansion of an H\\,{\\sc ii}  region around a single, isolated star embedded in a homogeneous environment. Our reasoning for resorting to a simple model calculation instead of running a full collapse simulation as in \\citet{Klassen+2012} is twofold. First, it allows us to separate time variability of the H\\,{\\sc ii}  region due to absorption of ionizing radiation by the accretion flow from time variability caused by structural changes in the protostar. In a realistic simulation, both effects would occur simultaneously, making it difficult to assess their relative importance. Second, the prestellar model by \\citet{Offner2009}, which was used in our collapse simulations \\citep{Klassen+2012}, represents the swelling phase of the more detailed calculations by \\citet{HosokawaOmukai2009} only very crudely. Following the behavior in \\citet{HosokawaOmukai2009} more closely allows for a better modeling of the time variability. However, accretion rates in collapse simulations are strongly fluctuating \\citep{Peters2010a,Klassen+2012}, as opposed to the constant accretion rates considered by \\citet{HosokawaOmukai2009}.\n\nThus, in order to create a realistic model for the time variability caused by pre-main-sequence evolution of the high-mass protostar, we are forced to use idealized simulations with a constant accretion rate as the ones presented in this paper. The work is further idealized because we deliberately only explore constant accretion rates, and thus neglect any time dependence in the HII region due to changing accretion. As a result of this simplicity, we are able to present a clear comparison of three sub-grid models for protostellar effects.\n\nIt is at present not possible to run a detailed pre-main-sequence evolution code along with a hydrodynamical simulation, so that our approach of investigating the two aspects of the problem separately appears to be the only solution.\n\nWe also stress that we cannot, in general, obtain the time dependence of the H\\,{\\sc ii}  region radius by analytical arguments. The key point is that the radius of the protostar may grow or contract depending on the stellar evolutionary stage, which in turn also depends on the accretion history. Since the ionizing flux grows initially during the H\\,{\\sc ii}  region expansion, both an enhanced ionizing flux and the thermal pressure of the already ionized gas work together to extend the H\\,{\\sc ii}  region radius, so that the H\\,{\\sc ii}  region can neither be modeled by a Str\\\"{o}mgren sphere nor by a standard D-type ionization front. Similar considerations hold for the shrinking phase, in which the ionized gas partially recombines.\n\nGenerally, a changing stellar radius affects the surface temperature, which in turn controls the UV flux. This means that if the effective temperature can change significantly over a short period of time, changes in the observed flux or size of H\\,{\\sc ii}  regions may signal evolutionary changes taking place in the stars themselves, whose ionizing radiation is driving the region.\n\nOur numerical approach is described in Section \\ref{sec:numerical_methods}. In Section \\ref{sec:results} we show our results for the evolution of a single massive protostar in a uniform medium, accreting matter at a constant rate, comparing the effects of different accretion rates and different prestellar models. Our assessment of these effects is discussed in Section \\ref{sec:discussion}, with some considerations of the observability of H\\,{\\sc ii}  region variability. Our findings are summarized in \\ref{sec:conclusion}.\n\n\\section{Numerical methods}\\label{sec:numerical_methods}\n\nWe perform numerical simulations using the {\\sc flash}  hydrodynamics code \\citep{Fryxell2000} in its version 2.5, which solves the hydrodynamics equations on an adaptive, Eulerian grid. Radiative transfer is handled using a raytracing scheme developed by \\citet{Rijkhorst} and extended by \\citet{Peters2010a}. We fix a single ``star'' at the centre of the simulation box to serve as the source of radiation and evolve a set of stellar parameters based on the models described in Section \\ref{sec:protostellar_models} while holding the accretion rate fixed.\n\nThe protostellar model provides the radius and the luminosity, from which we can obtain the surface temperature of the star, and hence the fraction of photons with energies capable of ionizing the surrounding the gas. This is not an analytical calculation, but a time-dependent simulation that allows us to compute the flux of ionizing photons, which the radiative feedback code then uses to heat and ionize the gas within the simulation. This allows for a self-consistent calculation of the size of the H\\,{\\sc ii}  region and its co-evolution with the protostar.\n\nWe perform a grid of simulations with different accretion rates and protostellar models. The three protostellar models that we selected were (1) a purely zero-age main sequence (ZAMS) estimate based on a precomputed table of ZAMS values, (2) a one-zone model based on \\citet{Offner2009} and also used in \\citet{Klassen+2012}, and (3) a fit to the results from detailed stellar structure calculations performed by \\citet{HosokawaOmukai2009}. In all but the ZAMS case, we tested accretion rates of $\\dot{M} = 10^{-3}$, $10^{-4}$, and $10^{-5}$ $M_\\odot$\/yr. For the ZAMS case, we only tested $\\dot{M} = 10^{-3} M_\\odot$\/yr. For the other two models we have included simulations of $\\dot{M} = 10^{-5} M_\\odot$\/yr  for comparison, even though rates higher than $10^{-4} M_\\odot$\/yr are necessary to form a massive star \\citep{HosokawaOmukai2009}. Table \\ref{table:summary_of_runs} summarizes our simulations.\n\n\n\n\\begin{deluxetable}{cccc}\n\\setlength{\\tabcolsep}{2pt}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Runtime parameters of the single accreting star simulations}\n\\tablenum{1}\n\\tablehead{\\colhead{Run} & \\colhead{Protostellar Model} & \\colhead{Accretion Rate} & \\colhead{Grid Resolution} \\\\\n\\colhead{} & \\colhead{} & \\colhead{} & \\colhead{} }\n\\startdata\n1a &    \\citet{Offner2009}  &  $10^{-3} M_\\odot$\/yr  &  20 AU \\\\\n1ah &   \\citet{Offner2009}  &  $10^{-3} M_\\odot$\/yr  &  10 AU \\\\\n1b &    \\citet{Offner2009}  &  $10^{-4} M_\\odot$\/yr  &  20 AU \\\\\n1c &    \\citet{Offner2009}  &  $10^{-5} M_\\odot$\/yr  &  20 AU \\\\\n2i &    Interpolation to ZAMS Table      &  $10^{-3} M_\\odot$\/yr  &  20 AU \\\\\n3a &    Interpolation to H\\&O Radius &  $10^{-3} M_\\odot$\/yr  &  20 AU \\\\\n3ah &   Interpolation to H\\&O Radius &  $10^{-3} M_\\odot$\/yr  &  6.5 AU \\\\\n3b &    Interpolation to H\\&O Radius &  $10^{-4} M_\\odot$\/yr  &  20 AU \\\\\n3c &    Interpolation to H\\&O Radius &  $10^{-5} M_\\odot$\/yr  &  20 AU \\\\\n\\enddata\n\\label{table:summary_of_runs}\n\\end{deluxetable}\n\n\n\\subsection{Protostellar models}\\label{sec:protostellar_models}\n\nIn \\citet{Klassen+2012} we described the implementation of a protostellar evolution module for {\\sc flash}  based on the \\citet{Offner2009} model implemented in the {\\sc orion}  code, which we simply called the ``Evolving Protostar'' model. The module considers protostars of mass exceeding 0.1 $M_\\odot$ and evolves their stellar parameters self-consistently with the simulation. The most important of these parameters is the stellar radius. During the pre-main-sequence evolution of a massive protostar, there comes a point when the stellar structure transitions from a convective core to a radiative core. This is accompanied by a swelling of the stellar radius. The \\citet{Offner2009} model estimates a swelling factor of 2.\n\nWe compare our results to a ZAMS model for the protostellar radius used by \\citet{Peters2010a}. In this model, it is assumed that the radius of the protostar is equal to the radius of an equivalent-mass star on the main sequence. We found that such an approximation for protostars will underestimate the radius by about an order of magnitude \\citep{Klassen+2012}. Consequently, estimations of the surface temperature and ionizing luminosity will be strongly overestimated. There is also no variability in the radius besides a steady increase in size as the star accretes matter. No structural transitions in the star are accounted for and the radius never contracts. To implement this model in our simulations, we use tabulated values of the mass, luminosity and effective temperature for a main suquence star. We interpolate within the table based on the current mass of the star and compute the radius from the luminosity and temperature.\n\nThe third model we employ is based on the results from \\citet{HosokawaOmukai2009} (henceforth H\\&O). In their paper, they investigate the evolution of a protostar under high accretion rates through detailed stellar structure calculations. By contrast, rather than computing stellar structure \\citet{Offner2009} consider the total energy evolution of a polytrope. One can calibrate a one-zone model to the \\citet{HosokawaOmukai2009} calculations, as they describe in Appendix C. However, the accretion rates that we test correspond to the published tracks in H\\&O, so we allow our protostellar radius to follow the appropriate track using a fitting function.\n\n\\subsection{Initial conditions}\\label{sec:initial_conditions}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=160mm]{.\/figure1.eps}\n\\caption{Results of a suite of simulations involving a single, isolated protostar accreting mass at either $10^{-3}$, $10^{-4}$, or $10^{-5} M_\\odot$\/yr, and evolving in stellar radius (top row) according to one of three models: an equal-mass ZAMS equivalent (first column), a one-zone model based on \\citet{Offner2009} (centre column), or following the results from the stellar structure calculations by \\citet{HosokawaOmukai2009} (right column). For each of these models, we also show the surface temperature $T_{\\textrm{eff}}$, the ionizing luminosity $S^*$, and the radius of the H\\,{\\sc ii}  region surrounding the protostar. Dotted lines show the ZAMS values. The shaded vertical bars indicate the span in the evolution of the protostar between the point where the H\\,{\\sc ii}  region has reached its maximal pre-collapse size, and the point post-collapse when the previous size has been reinstated. In the Hosokawa \\& Omukai case, the hatched region indicates the span when the H\\,{\\sc ii}  region is collapsing. The duration of this collapse is indicated at the bottom of the figure in years. In the Offner et al.~case, the collapse is instantaneous, so the durations at the figure bottom indicate the timescale for regrowth.}\n\\label{fig:table_star_props}\n\\end{figure*}\n\nThe advantage of the radiative feedback code we employ lies in its ability to produce realistic H\\,{\\sc ii}  regions. In a cluster environment, it is difficult to measure the effect of the choice of model on H\\,{\\sc ii}  region size because the environment is highly tumultuous even without any initial turbulence or magnetic fields \\citep{Peters2010a, Klassen+2012}. The H\\,{\\sc ii}  regions formed in such an environment are highly amorphous and time-variable. Here, we want to isolate the effects of the protostellar model to gauge its impact on H\\,{\\sc ii}  region formation. To explore this interesting possibility, we select a simulation box of side length 0.1 pc filled with uniform, isothermal molecular gas at 10K and density $\\rho = 1 \\times 10^{-17}$ g cm$^{-3}$. This is not a collapse calculation and self-gravity of the gas is switched off. We do not consider any initial turbulence, nor magnetic fields, nor do we include radiation pressure.\n\nThe radiative feedback code, which is described in greater detail in \\citet{Peters2010a}, is a hybrid-characteristics raytracing scheme that computes gas heating with grey atmosphere assumptions, and gas ionization, based on the computed flux of ionizing photons from stars modeled as blackbodies.\n\nA single star is fixed at the centre of this simulation volume whose mass is set to increase at the artificially imposed accretion rate $\\dot{M}$, which is held fixed throughout the simulation and is treated as a parameter. By artificially fixing the accretion rate in this way, we can directly connect H\\,{\\sc ii}  region variability with the evolution of stellar properties, rather than have it be due to fluctuations in accretion rate or environmental changes as was the case in \\citet{Klassen+2012}.\n\nWe ran simulations with various prestellar models and accretion rates in order to compare their effect on H\\,{\\sc ii}  region formation. The various simulations are summarized in Table \\ref{table:summary_of_runs}. We repeated a portion of the \\citet{Offner2009} and \\citet{HosokawaOmukai2009} models with $\\dot{M} = 10^{-3} M_\\odot$\/yr, this time at a high spatial resolution of 10 AU and 6.5 AU minimum grid cell size, respectively. This better captured the behaviour during the transition in stellar structure and swelling of the stellar radius. The high-resolution data was then combined with the lower resolution data.\n\nEach simulation was run for long enough to capture all the major transitions. In most cases, we followed the evolution of the star until it had almost reached the main sequence (see Figure \\ref{fig:table_star_props}).\n\n\\section{Results}\\label{sec:results}\n\nThe data from these simulation runs are presented in Figure \\ref{fig:table_star_props}, which consists of a series of panels, with the three columns showing each of the three prestellar models we tested.\n\nThe rows of panels in this figure show the key variables that we measured or calculated and which describe the evolution of the protostar. The first row shows the stellar radius in solar units. We see how ZAMS model shows nothing but a monotonic increase in stellar radius with increasing mass, while the two prestellar models show a radius that sometimes swells and sometimes contracts. This swelling and contracting of the stellar radius is the key to affecting the size of H\\,{\\sc ii}  regions and we trace the effects of this evolution down through the rows of this figure. For comparison, in the first row we show the ZAMS radius in each panel using a thick dashed line.\n\nIn the second row, we show calculations of the effective surface temperature of the protostar. This is computed using the equation\n\\begin{equation}\\label{eqn:Teff}\nT_{\\textrm{eff}} = \\left( \\frac{L_{\\textrm{int}}}{4 \\pi \\sigma R^2} \\right)^{1\/4} \\, ,\n\\end{equation}\nwhich depends on the intrinsic luminosity of the protostar $L_{\\textrm{int}}$ and the stellar radius $R$. The luminosity in the first case is estimated as in \\citet{Offner2009} as\n\\begin{equation}\\label{eqn:Lint}\nL_{\\textrm{int}} = \\textrm{max}(L_{\\textrm{ms}} ,4 \\pi \\sigma R^2 T_{\\textrm{H}}^4) \\, ,\n\\end{equation}\nthat is, the greater of either the main sequence luminosity or the luminosity of a Hayashi-track star with temperature $T_{\\textrm{H}} = 3000$ K. We use this same estimate of luminosity for our H\\&O comparison. The intrinsic luminosity in the ZAMS case is calculated by interpolation to tabulated values.\n\nThese panels show that when the radius of a star swells during the transition to a radiative core structure, the surface temperature drops by several thousand Kelvin. A ZAMS model misses this effect entirely. Capturing this effect in the surface temperature is crucial in correctly determining how the ionizing flux $S^*$, shown in row 3, changes during pre-main-sequence evolution.\n\nThe ionizing flux of photons from the protostar is calculated by integrating a blackbody for frequencies above the ionization threshold of 13.6 eV to find the number of ionizing photons per second. This number depends on the surface temperature of the protostar. The prestellar models have drops in their surface temperature, which correspond to drops in ionizing flux. In the case of $\\dot{M} = 10^{-3} M_\\odot$\/yr, the \\citet{Offner2009} model has a flux decrement of just over 3 orders of magnitude (from $5.3\\times10^{42}$ to $1.7 \\times 10^{39}$ s$^{-1}$), while the H\\&O model shows a flux decrement of just over 4 orders of magnitude (from $1.6 \\times 10^{44}$ to $1.2 \\times 10^{40}$ s$^{-1}$). The other main difference is that in the former case the drop is instantaneous, whereas the H\\&O model show this taking place over 4000 years.\n\nThe ZAMS model has only a monotonically increasing ionizing flux and likely also overestimates the ionizing flux for much of the pre-main-sequence lifetime of the star.\n\nFinally, we measured the size of the H\\,{\\sc ii}  region by evaluating the ionization fraction along a line starting at the centre of the simulation box, where the particle is located, out along one of the axes to the edge of the simulation box. Sampling the ionization fraction along this axis, we can determine the radius at which the gas transitions from fully ionized to neutral. We define the radius of the H\\,{\\sc ii}  region as the point where the ionization fraction $X$ is 0.5, which is found using an inerpolation of the underlying sampling points.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=160mm]{.\/figure2.eps}\n\\caption{A blown-up view of the time of collapse of the H\\,{\\sc ii}  region (bottom row) around the protostar. Left panels: the one-zone model by \\citet{Offner2009}; right panels: the \\citet{HosokawaOmukai2009} data, both at an accretion rate of $10^{-3} M_\\odot$\/yr. The top row shows the concurrent effective surface temperature of the protostar. The small-scale variability of $R_{\\textrm{H\\,{\\sc ii} }}$ is partially a numerical effect caused when the H\\,{\\sc ii}  region radius is measured in the post-processing of the simulation data and partially a result of small density perturbations. It is on the order of the grid resolution.}\n\\label{fig:hires_HII}\n\\end{figure*}\n\nThe fourth row of panels in Figure \\ref{fig:table_star_props} shows the key result that ties together all the previous findings. Here we show the evolving radius of the H\\,{\\sc ii}  region that forms around protostar. The prestellar models of \\citet{Offner2009} and H\\&O, for accretion rates $\\dot{M} \\geq 10^{-4} M_\\odot$\/yr, show an H\\,{\\sc ii}  region that collapses or shrinks when the ionizing flux is diminished. The loss of ionizing flux is ultimately tied back to the stellar radius, which swells during the pre-main-sequence evolution of the star. For an accretion rate of $\\dot{M} = 10^{-5} M_\\odot$\/yr, there was no collapse of the H\\,{\\sc ii}  region. In the H\\&O case, the growth of the H\\,{\\sc ii}  region at this accretion rate appears stagnated for about 100 kyr before it continues to grow.\n\nThe temporary loss of the H\\,{\\sc ii}  region is more dramatic for the higher accretion rates. The region disappears completely, falling to near zero from a radius of 150 AU in the Offner model, then quickly rebuilding. In the H\\&O model, the H\\,{\\sc ii}  radius falls to near zero from 40 AU and remains undetectable for about 1700 years.\n\nWe added shaded vertical bars that run through Figure \\ref{table:summary_of_runs} and mark the key transitions in the prestellar models for accretion rates greater than $10^{-4} M_\\odot$\/yr. The lengths of time that each pair corresponds to are shown at the very bottom of the figure. In the \\citet{Offner2009} model, the shaded regions mark the timescale of H\\,{\\sc ii}  region collapse and recovery to its pre-collapse size. This destruction and regrowth takes about 1700 years for $\\dot{M} = 10^{-3} M_\\odot$\/yr, and about 6000 years for $\\dot{M} = 10^{-4} M_\\odot$\/yr.\n\nIn the H\\&O runs, the shaded region begins when the H\\,{\\sc ii}  region has reached its maximal pre-collapse size and ends at the point of full recovery. The first part of the shaded region is hatched and ends where the H\\,{\\sc ii}  region has shrunk to zero. The time taken for the region to disappear is indicated at the bottom of the figure, and is about 3400 years for $\\dot{M} = 10^{-3} M_\\odot$\/yr, and about 16500 years for $\\dot{M} = 10^{-4} M_\\odot$\/yr.\n\nIn all our model simulations, as the protostar undergoes its final contraction towards the main sequence, the H\\,{\\sc ii}  region grows steadily until we terminate the simulation.\n\nIn Figure \\ref{fig:hires_HII} we show a closer view of the collapse of an H\\,{\\sc ii}  region around a massive protostar for the \\citet{Offner2009} and H\\&O models. This figure includes high-resolution data, with minimum grid sizes of about 10AU and 6.5 AU, respectively. Time resolution was also higher, at dt = 8 years and dt = 7 years, respectively. In the Offner et al. model, the H\\,{\\sc ii}  region radius reaches a maximal value of 161 AU, while in the H\\&O case, it grows to only about 45 AU. We show the concurrent effective surface temperatures in the top row, which shows the H\\&O reducing in temperature gradually instead of instantaneously. When the flux of ionizing photons falls due to decreasing surface temperature, the H\\,{\\sc ii}  region shrinks away to zero over a period of about 3400 years. Only after the surface temperature begins to heat up again does the H\\,{\\sc ii}  region begin to reform and grow, as also seen in Figure \\ref{fig:table_star_props}. The one-zone model based on \\citet{Offner2009} has an instantaneous drop in effective surface temperature, and also an instantaneous drop in H\\,{\\sc ii}  region radius.\n\n\\section{Discussion}\\label{sec:discussion}\n\nOf the models we tested, the \\citet{HosokawaOmukai2009} model is likely the most realistic, since in this one we fit functions to their detailed stellar structure calculations. The model also contains no discontinuous jumps in stellar radius, as the \\citet{Offner2009} model does. The timescale for flickering, then, in the nascent H\\,{\\sc ii}  region is probably closer to the \\citet{HosokawaOmukai2009} results.\n\nGiven the tumulteous environments in which massive stars form, variability in the H\\,{\\sc ii}  regions seems reasonable. \\citet{Galvan-Madrid+2008} reported a substantial ($\\sim$45\\%) decrease in the radio-continuum flux (6 cm) of an unresolved H\\,{\\sc ii}  region around a young massive star over a 5 year period. An analysis of radiation-hydrodynamic simulation data from \\citet{Peters2010a} by \\citet{Galvan-Madrid+2011} using synthetic radio-continuum observations showed that H\\,{\\sc ii}  regions can be highly variable over timescales of 10 to $10^4$ years. H\\,{\\sc ii}  regions initially larger than 1000 AU around a massive star lost up to 90\\% of their size in a short span of time ($\\sim$20 years) during a large accretion event.\n\nThose results indicated that the evolution of ultracompact and hypercompact H\\,{\\sc ii}  regions may be more complex that the textbook picture of a spherical ionized regions expanding into a quiescent medium. H\\,{\\sc ii}  variability is most likely not due to protostellar evolution, which was suggested as a possibility in \\citet{Franco-HernandezRodriguez2004}. The variability we observe in our simulations is of a scale too small to explain those effects.\n\nWe confirmed that tumultuous birth environments give rise to massive stars with variable H\\,{\\sc ii}  regions in our own simulations \\citep{Klassen+2012}, but also experimented with a different protostellar model that better captured the evolution of the protostellar radius under conditions of radidly varying accretion. In this paper, we wanted to revisit the toy model of H\\,{\\sc ii}  regions to see to which degree a realistic treatment of pre-main-sequence evolution affects the H\\,{\\sc ii}  region formation and evolution.\n\nWe decoupled the environmental effects of the molecular gas flow through a turbulent environment during the accretion process, to see whether flickering of the H\\,{\\sc ii}  region could also be a purely evolutionary feature of the accreting star. We saw the H\\,{\\sc ii}  region around the protostar collapse in a similar way, losing over 90\\% of its size in a matter of less than 10 years for the \\citet{Offner2009} model, or over about 3.5 kyrs for the \\citep{HosokawaOmukai2009} model.\n\nSome earlier discussions, based on highly symmetric, simplified models \\citep[e.g.][]{Walmsley1995} suggested that high accretion rates might ``choke off'' H\\,{\\sc ii}  region formation inside ``hot cores.'' These hot cores are compact (diameters $\\leq 0.1$ pc), dense ($n_{H_2} \\geq 10^7$ cm$^{-3}$), and at temperatures $\\geq$ 100K \\citep{Kurtz+2000}. \\citet{Churchwell2002} describes these regions as the precursors to ultracompact H\\,{\\sc ii}  regions, but also suggests that rapid accretion may prevent young H\\,{\\sc ii}  regions from begin detected. However, it is now clear that massive protostars accrete asymmetrically via disks \\citep{ZinneckerYorke2007,Beuther+2007}, with some fraction of this infalling material being launched into outflow cavities created by jets \\citep{BanerjeePudritz2007}. The jet intensity will be proportional to the infall rate and will also be accompanied by strong beaming of the radiation field around the star in what has been called the ``flashlight'' effect \\citep{YorkeBodenheimer1999,YorkeSonnhalter2002,Krumholz+2005}. Analytic models show that an H\\,{\\sc ii}  region can be sustained inside a bipolar outflow cavity despite heavy accretion \\citep{TanMcKee2003}. Numerical simulations of non-spherical accretion \\citep{Peters2010a,Klassen+2012} demonstrate that H\\,{\\sc ii}  regions can expand in spite of strong accretion flows.\n\nUnder circumstances such as these, the H\\,{\\sc ii}  region would not be choked off, despite the very high accretion rate. To our knowledge, accretion choking out the formation of an H\\,{\\sc ii}  region has never been seen in numerical simulations, nor is accretion ever spherically symmetric onto a massive star under realistic conditions. Therefore, it is timely to explore the birth and evolution of hypercompact H\\,{\\sc ii}  regions for massive protostars at high accretion rates, even if our spherically symmetric model is highly idealized. We sought to explore the time-evolution of an H\\,{\\sc ii}  region around a source whose flux of ionizing photons is varying according the time-evolution of the protostellar radius and surface temperature. No analytical model for the time-variability of H\\,{\\sc ii}  regions exists and that time-variability exists whether the morphology is spherically symmetric or not.\n\nMany numerical calculations of protostellar evolution \\citep{PallaStahler1991,YorkeBodenheimer2005,HosokawaOmukai2009}, even for first stars \\citep{TanMcKee2004}, have shown that the protostellar radius undergoes a swelling phase followed by a contracting phase as the star approaches the main sequence. \\citet{Smith+2012} considered the time-evolution of the protostellar radius for first stars and showed that under some conditions the radius can undergo multiple occasions of swelling and contraction on account of a variable accretion rate. We measured the impact of this evolution on the size of the nascent H\\,{\\sc ii}  region and showed that fluctuation in the size does indeed occur, but are small enough that they might be very difficult to observe with present radio telescopes.\n\n\\subsection{Observational considerations}\n\nHypercompact H\\,{\\sc ii}  regions are nominally defined with sizes of about 30 mpc (6000 AU) or smaller. Understanding their place in the scheme of massive star formation is important \\citep{Kurtz2005}. H\\,{\\sc ii}  regions, generally, are associated with star clusters, and we discussed their variability in \\citet{Klassen+2012}. Higher-resolution surveys will ultimately yield more examples of extremely compact H\\,{\\sc ii}  regions associated with individual, massive stars instead of clusters that may tell us more about the dynamics and kinematics of the regions that give rise to a massive star.\n\n\\citet{TanMcKee2003} proposed that hypercompact H\\,{\\sc ii}  regions would be confined to the outflow cavities created by rapidly-accreting, massive protostars. The faster the acccretion, the higher the outflow rate. In our self-consistent collapsing cluster simulations \\citep{Klassen+2012}, we saw accretion rates that at times approached $10^{-3} M_\\odot$\/yr. \\citet{HosokawaOmukai2009} also performed their calculations of protostellar evolution with accretion rates up to $10^{-3} M_\\odot$\/yr. \n\nOur idealized model of a single star accreting at a fixed rate and feeding back into a homogeneous medium isolates the massive protostar from the cluster environment that was studied in \\citet{Klassen+2012}. In this isolated environment, we could see the variability of the H\\,{\\sc ii}  region, as caused by the protostellar evolution instead of the environment. It was for this reason that the environment was made as simple as possible. Though massive protostars in nature accrete material asymmetrically through a disk, this does not affect the fact that H\\,{\\sc ii}  regions will be variable according to their protostellar evolution, which involves a rapid swelling of the stellar radius that drastically lowers the surface temperature and fraction of photos above the ionization threshold of 13.6 eV. This also makes accretion onto the protostar easier.\n\nUnder the models we tested, the diameter of H\\,{\\sc ii}  regions collapsed at rate of 45 AU\/yr (for the Offner et al. 2009 model), and at a rate of 0.03 AU\/yr (for the Hosokawa \\& Omukai 2009 model), where in both cases the protostar was accreting at a rate of $10^{-3} M_\\odot$\/yr. The Orion Molecular Cloud is located about 460 pc away. In observing the radio continuum at 1.3 cm, the EVLA has an angular resolution of about 8.7 mas, which corresponds to a spatial resolution of about 4 AU at 460 pc, and 17.4 AU at 2 kpc. This suggests that H\\,{\\sc ii}  region flickering around massive protostars could be observable, following the \\citet{Offner2009} model. The H\\,{\\sc ii}  regions would not appear quite as they do in our simulations, given that massive protostars are embedded in dense gas and dusk, but they might be seen via outflows as suggested earlier.\n\nAdditionally, masers serve as useful probes of high-mass star-forming environments and can be associated with ultracompact H\\,{\\sc ii}  regions \\citep{Fish+2005}. VLBI techniques may offer the angular resolution at a few kiloparsecs that ALMA cannot. Multi-epoch observations of the hydroxyl masers around W3(OH) with time baseline of 8 years showed evidence consistent with an expanding ultracompact H\\,{\\sc ii}  region \\citep{Bloemhof+1992}. Even with archival data, the authors were capable of positioning the sources with sub-milliarcsecond resolution, corresponding to maser positioning to within better than 2 AU at 2.2 kpc. If masers drift is truly associated with H\\,{\\sc ii}  region variability and could be detected in the earliest stages of massive star formation, variability on timescales shorter than 10 years could potentially be studied.\n\n\\section{Conclusions}\\label{sec:conclusion}\n\nOur results show that massive protostars in uniform media emit ionizing radiation capable of forming a hypercompact H\\,{\\sc ii}  region. In our simulations using the {\\sc flash}  astrophysics code with ionizing feedback and different prestellar models, the pre-main-sequence evolution of protostars can cause early variability in the size of these H\\,{\\sc ii}  regions. During pre-main-sequence evolution, the stellar radius swells dramatically when the star transitions to a radiative internal structure. This is accompanied by a cooling of the surface temperature and a drop in the ionizing flux by several orders of magnitude. During this transition, young H\\,{\\sc ii}  regions will shrink or collapse until the contracting star has regained its previous ionizing flux. \n\nOur results show that pre-main-sequence evolution can affect the growth of H\\,{\\sc ii}  regions around massive protostars on potentially short time scales. The collapse of the H\\,{\\sc ii}  region, initially $\\sim$300 AU across in the \\citet{Offner2009} model, happened within a single simulation timestep (8 years) when the radius of the star was enlarged during the evolution. The \\citet{HosokawaOmukai2009} model saw the radius of the H\\,{\\sc ii}  region contract from an initial diameter of $\\sim$90 AU to zero over the course of about 3.5 kyr.\n\nThe observability of this variability may be possible with the best interferometric radio observations such as with VLBI, the EVLA, or ALMA. Nevertheless, model simulations of hypercompact H\\,{\\sc ii}  regions are an important part of understanding the complete picture of massive star formation. Massive protostars are unique in that their UV fluxes are large enough to ionize gas even before the star has reached the main sequence, and it is therefore important that these early states be further explored.\n\n\\section*{Acknowledgments}\n\nWe thank our anonymous referee for a helpful report. M.K.~acknowledges financial support from the Ontario Graduate Scholarship (OGS) Program. T.P.~acknowledges financial support as a Fellow of the Baden-W\\\"{u}rttemberg Stiftung funded by their program International Collaboration II (grant P-LS-SPII\/18) and through SNF grant 200020\\_137896. R.E.P.~is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The {\\sc flash}  code was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca) and Compute\/Calcul Canada.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{Thanks to various sets of cosmological data, we can now talk about ``{\\it the standard model of cosmology}'', the $\\Lambda$CDM Universe \\cite{Ade:2015xua, Ade:2015rim}. It provides the simplest paradigm that fits remarkably well to most of the current cosmological observations.  As the precision of the low redshift data increases, there are however emerging tensions in $\\Lambda$CDM  model which is otherwise consistent with high redshift CMB observations by Planck. The major tension is between the model independent measurement of $H_{0}$ parameter ({$\\sim 73 ~\\text{km\/s\/Mpc}$}) by SH0ES collaboration \\cite{Riess:2016jrr, Riess:2017lxs, Riess:2018byc} and that by Planck assuming $\\Lambda$CDM  model \\cite{Ade:2015xua, Ade:2015rim, Akrami:2018vks,Aghanim:2018eyx}. The latest Planck-2018 data shows, {$H_0 = 67.8 \\pm 0.9~ \\text{km\/s\/Mpc}$} \\cite{Akrami:2018vks} and  this is at tension over $3.5\\sigma$ with the SH0ES measurement which is $H_0 = 73.52 \\pm 1.62~\\text{km\/s\/Mpc}$.  A similar mild inconsistency in $H_{0}$ for $\\Lambda$CDM model is also observed by Strong Lensing experiments like H0LiCow using time delay measurements \\cite{Bonvin:2016crt} which measured $H_0 = 71.9^{+2.4}_{-3.0}~\\text{km\/s\/Mpc}$ for $\\Lambda$CDM  Universe.  Moreover the BOSS survey for baryon acoustic oscillations measurements using Lyman-$\\alpha$ forest \\cite{Delubac:2014aqe} has also measured the expansion rate of the Universe at $z=2.34$. This measured expansion rate of the Universe at $z=2.34$ is also at tension over $2\\sigma$ with Planck result for $\\Lambda$CDM  \\cite{Sahni:2014ooa}. The important consequences of these tensions are the prediction of the dark energy density evolution with time\\footnote{{That evolving and dynamical dark energy is necessitated by the data has also been discussed in \\cite{Sola:2018sjf, Sola:2016ecz,Ryan:2018aif, Ooba:2018dzf, DiValentino:2017zyq, Rivera:2016zzr, Zhao:2017cud, Zhang:2017idq}.}} {and more importantly for us,}  the possibility of having  negative dark energy density at higher redshifts as discussed by \\cite{ Delubac:2014aqe, Sahni:2014ooa, Poulin:2018zxs}. Similar conclusion has also been obtained recently by \\cite{Wang:2018fng} using a dark energy model independent approach. In their study, Wang et al. attributed the evolution of dark energy density as well as its negative values at high redshifts to non-minimally coupled scalar field theory like Brans-Dicke theory. Interestingly in all these studies, the dark energy density is not only negative at higher redshifts but it is unbounded from below. This poses a serious problem as in a spatially flat universe, with such a negative dark energy density, the matter energy density parameter will be more than one and grow quickly without any upper bound at higher redshifts. This will have catastrophic effect on the structure formation scenario.This surely needs to be resolved.\n\n\nIn this work, we revisit the process of constraining the dark energy behavior using cosmological observations to find the likely sources of tensions between low and high redshifts cosmological observations. For this, }we should be careful that although Planck constrains the high redshift Universe  using CMB with unprecedented accuracy, it may not be sensitive enough to any new physics beyond $\\Lambda$CDM  at low redshifts. Fitting the entire background evolution of the Universe from $z\\sim1100$ till $z=0$ using $\\Lambda$CDM in which the dark energy density is a redshift independent quantity, can be the source of aforementioned tensions.\n\n{Therefore we reanalyze the low-redshift data involving background cosmology assuming that for a certain higher redshift $z=z_{\\text{match}}$ and beyond, one recovers the background evolution as constrained by Planck-2018 for $\\Lambda$CDM  Universe. {For} this, we assume that $z_{\\text{match}}$ has to be larger than  $z\\sim 2.4$, as {around this redshift}, we have the constraint on expansion rate of Universe from BOSS survey using Lyman-$\\alpha$ forest. This takes care of any new physics for dark energy evolution (beyond $\\Lambda$CDM  Universe) that is predicted by low-redshift observations for $z < z_{\\text{match}}$ including the SH0ES measurement for $H_{0}$. \n{With these assumptions, we study the dark energy behaviour which is consistent with the low-redshift data for $z\\leq z_{match}$ {while smoothly matching} the Planck-constrained $H(z)$ behaviour for $\\Lambda$CDM model for $z>z_{match}$}. In this process, we do not assume any particular dark energy model. We directly reconstruct the Hubble parameter $H(z)$ as a function of redshift using cosmographic approach. Subsequently, we reconstruct the dark energy density evolution} {in a  ``dark energy relaxed XCDM framework,'' in which we do not assume any specific dark energy model,}\\footnote{{We note that similar reconstruction of dark energy properties have been extensively analyzed  in the literature, e.g. see \\cite{Shafieloo:2007cs, Seikel:2012uu, Gerardi:2019obr, Wang:2019ufm}. Our data analysis method and the data sets we consider here is different than these other works.}} {while still assume  the Universe is spatially flat and contains pressure-less matter (including baryons and dark matter) plus another unknown component. This extra unknown component can be due to dark energy or it may arise as an effective dark energy component due to any sort of modification of gravity at large cosmological scales. We also  ignore the contribution from radiation energy density as it is negligible compared to matter or dark energy at late times.}\n\n\nOur {dark energy model-independent} analysis within the above set of working assumptions and framework reveals that ${\\rho}_{_{\\text{DE}}}(z)$ has two specific features: it has a minimum and a phantom crossing at  $z=z_{\\text{min}}$. Moreover, the value of ${\\rho}_{_{\\text{DE}}}$ at $z=z_{\\text{min}}$ is negative. The simplest explanation for ${\\rho}_{_{\\text{DE}}}^{\\text{min}} < 0$, is the existence of a {\\it negative Cosmological Constant}.\n\n\n\\section{Data sets and their analysis} There are different approaches to constrain the behavior of dark energy without assuming any particular dark energy model. Here we take the cosmographic approach that directly constrains  different kinematical quantities related to the  background evolution. The Taylor expansion of $H(z)$ for this purpose does not work for $z>1$ and hence we use the Pade Approximation that improves the convergence at higher redshifts \\cite{Saini:1999ba,Gruber:2013wua, Wei:2013jya, Rezaei:2017yyj, Mehrabi:2018oke, Benetti:2019gmo}. \n\nFor our purpose we model the Hubble parameter $H(z)$ using Pade approximation $P_{2,2}$:\n\\begin{equation}\\label{H-Pade}\nH(z) = H_{0}  \\frac{1+P_{1}z+P_{2}z^{2}}{1+Q_{1}z+Q_{2}z^{2}}.\n\\end{equation}\nAll the constant parameters $P_{1}, P_{2}, Q_{1}, Q_{2}$  appearing in the above expression, can be written in terms of the kinematic quantities like, $q, j, s, l$ (deceleration, jerk, snap, and lerk)  (see  \\cite{Capozziello:2018jya} for details). Note that for $\\Lambda$CDM  model, the jerk parameter  $j=1$ for all redshifts and any deviation from $j=1$, confirms a non-$\\Lambda$CDM behavior.\n\n{In the following two subsections we reconstruct $H(z)$ in two cases to highlight the effects of inclusion of {constraint on background expansion at high redshifts from Planck measurements of CMB anisotropy}: In section \\ref{sec:2.1} we only consider the low redshift data samples mentioned below using Pade approximant. In section \\ref{sec:2.2} we reanalyze the low redshift data requiring $H(z)$ passes through three data points at $z_{\\text{match}}, z_{\\text{match}}\\pm 1$. {The former may then be used to compare with  results of \\cite{Zhao:2017cud, Wang:2018fng}. This analysis may be viewed as a check for the Pade parametrization used.}}\n\n\n\\subsection{Low redshift data only}\\label{sec:2.1}\n\n{\nTo constrain the background cosmology, we use the following set of low-redshift observational data:\n\\begin{itemize}\n\\item\nBAO measurements from different surveys including eBoss quasar clustering and  Lyman-$\\alpha$ forest samples (see \\cite{Evslin:2017qdn} and references therein);\\vskip 2mm\n\\item\nlatest Pantheon data for SNIa \\cite{Riess:2017lxs, Gomez-Valent:2018hwc};\\vskip 2mm\n\\item\nthe OHD data for Hubble parameter at different redshifts \\cite{Pinho:2018unz};\\vskip 2mm\n\\item\nstrong lensing time-delay measurements by H0LiCOW experiment \\cite{Bonvin:2016crt};\\vskip 2mm\n\\item angular diameter distances measured using water megamasers under the Megamaser Cosmology Project \\cite{Evslin:2017qdn, Reid:2012hm, Kuo:2012hg, Gao:2015tqd};\\vskip 2mm\n\\item\nfinally the measurement of $H_{0}$ by \\cite{Riess:2016jrr}[R16].\n\\end{itemize}}\nFor details about all the data used in this analysis, please refer to \\cite{Capozziello:2018jya}. The independent parameters for the data analysis are $H_{0}, q_{0}, j_{0}, s_{0}$, $l_{0}$  and $r_{d}$, the sound horizon at drag epoch. Here the subscript $``0\"$ means the value at present ($z=0$).\n\n\n{ The reconstructed $H(z)$ is shown in Fig.\\ref{fig:hwoutpl} (Left-Top plot). {Note that the high uncertainty beyond $z\\sim 2.4$ is due to unavailability of low redshift data beyond $z\\sim2.4$}. As one can see (the Right-Top plot), the reconstructed $H(z)$ from low-redshift observations is at large tension with the same, reconstructed from Planck-2018 data. Also at low redshifts ($z \\leq 2$), there is  oscillations in $H(z)$ behaviour around the best fit $H(z)$ behaviour for $\\Lambda$CDM as measured by Planck-2018. Such oscillation in $H(z)$ has already been reported in  \\cite{Wang:2018fng} through a model independent study. In this work, we obtain exactly the same behaviour in a completely different approach using Pade approximation for cosmography.}\n\\begin{figure}\n\\begin{center} \n\\resizebox{200pt}{200pt}{\\includegraphics{test1.pdf}}\n\\resizebox{200pt}{200pt}{\\includegraphics{H_Planck2018.pdf}} \n\\resizebox{200pt}{200pt}{\\includegraphics{fz_using_padeorig_315_om.pdf}}\n\\end{center}\n\\caption{{Low data redshift analysis without inclusion of high redshift CMB data from Planck.} Left-Top: Reconstructed Hubble parameter $H(z)$  from various low-redshift data sets. The black dashed line is for best fit values whereas the inner and outer lines denote $1\\sigma$ and $2\\sigma$ contours. The thin shaded region is reconstructed $H(z)$ behaviour from Planck-2018 measurements. Right-Top:  Comparison of our reconstructed $H(z)$ using Pade approximant and the best fit $H(z)$ for $\\Lambda$CDM by Planck-2018. Bottom: Reconstructed dark energy density as a function of redshift. The horizontal line $f(z)=1$ is for $\\Lambda$CDM.}\\label{fig:hwoutpl}\n\\end{figure}\n\nOnce we constrain the $H(z)$ behaviour using available low-redshift data, we can further constrain the redshift evolution of any exotic component which one needs to add to the matter contribution. This extra component (we denote it as ${\\rho}_{_{\\text{DE}}}$) can be due to dark energy or due to any modification of gravity on large cosmological scales that gives rise to some effective dark energy component. Following \\cite{Wang:2018fng}, we write:\\footnote{{In principle, one could have included a curvature term, $\\rho_C (1+z)^2$, especially in light of recent claims in \\cite{DiValentino:2019qzk}. We intend to study such a possibility in upcoming work. }}\n\\begin{equation}\\label{H2}\n3 H^{2}(z) = \\rho_{m} + {\\rho}_{_{\\text{DE}}} = \\rho_{m 0}  (1+z)^3 + \\rho_{_{\\text{DE}0}} f(z).\n\\end{equation}\nHere we set $8\\pi G = 1$ and superscript ``0\" indicates values at present $z=0$ and subscript ``DE'' stands for any dark component (either actual or effective) that yields late time acceleration. The $f(z)$ (a dimensionless quantity)  specifies the allowed redshift evolution of ${\\rho}_{_{\\text{DE}}}$. Assuming a flat Universe,  $\\Omega_{m} + \\Omega_{\\text{DE}} = 1$, equation (2)  can be rewritten as\n\\begin{equation}\n\\frac{H^{2}}{H_{0}^{2}} = \\Omega^{(0)}_{m} (1+z)^3 + \\Omega^{(0)}_{\\text{DE}} f(z) \n= \\Omega^{(0)}_{m}(1+z)^3 + (1-\\Omega^{(0)}_{m}) f(z).\n\\end{equation}\nWith this, one can constrain the redshift evolution for ${\\rho}_{_{\\text{DE}}}(z)$ using the behaviour for $H(z)$  shown in Fig.\\ref{fig:hwoutpl}. For this, one needs to assume the value of $\\Omega^{(0)}_{m}$ and for our purpose, we assume $\\Omega^{(0)}_{m} = 0.315$ which is best fit value for Planck-2018 for $\\Lambda$CDM and this value is not very sensitive to dark energy behaviour. Using this, one can now reconstruct the $f(z)$ behaviour from low-redshift observation which is shown in Fig. \\ref{fig:hwoutpl} (Bottom plot). As one can see, the $f(z)$ clearly shows oscillations around the $f(z)=1$ $\\Lambda$CDM behaviour for low redshifts. This confirms the phantom-non phantom crossing. Also for larger redshifts, $f(z) <0$ and is also unbounded from below. These behaviours of $f(z)$ are in complete agreement with the results obtained by \\cite{Wang:2018fng}, confirming our methodology of using Pade approximations.\n\n\n\n\\subsection{Low redshift data plus inclusion of CMB Planck data points for \\textit{H(z)}}\\label{sec:2.2}\n\n{Once we confirm the results obtained by \\cite{Wang:2018fng}  as well as other authors \\cite{Sahni:2014ooa, Delubac:2014aqe, Poulin:2018zxs} with different approach, our next goal is to solve the problem of ${\\rho}_{_{\\text{DE}}}$ being unbounded from below for negative values for larger redshifts. To address this problem, we assume that $H(z)$ as constrained by the low redshift data, should also fit $H(z)$ as measured by Planck for $\\Lambda$CDM  at some higher redshift $z=z_{\\text{match}}$ and beyond. }{This assumption is well justified noting that the difference between our XCDM model and $\\Lambda$CDM  is in their dark energy sector and that the dark energy effects becomes relevant to the evolution of the Universe {only at lower redshifts}. \n{Also as discussed in the Introduction, to take into account any new physics in the dark energy sector that may arise due to the BAO observations at $z=2.4$ using Lyman-$\\alpha$ forest,}\\footnote{{We have repeated the analysis with the model independent diameter distance measurements \\cite{Carvalho:2015ica, Alcaniz:2016ryy, Carvalho:2017tuu} \nand find that they do not change our results.}}   $z_{\\text{match}}\\gtrsim 4$ should be a reasonable choice. Of course, as we will comment in the discussion part, in a full and complete analysis, $z_{\\text{match}}$ should also come out from the analysis itself. Here we only intend to present a ``proof of concept'' analysis.} \n\n\\begin{figure\n\\begin{center} \n\\resizebox{280pt}{220pt}{\\includegraphics{test2.pdf}} \\\\\n \\resizebox{280pt}{220pt}{\\includegraphics{test3.pdf}}\n\\end{center}\n\\caption{Reconstructed Hubble parameter $H(z)$ behavior {employing both low redshift and CMB Planck data sets, see the text for details}. The left one is based on Planck $H(z)$ data at $z=4,5,6$ and right one is for $z=6,7,8$.  The dashed line is for mean and inner and outer regions are for $68\\%$ and $95\\%$ confidence regions. The thin shaded region is reconstructed $H(z)$ behaviour from Planck-2018 measurements. }\\label{fig:hqplot}\n\\end{figure}\n\nTo this end, we generate $H(z)$ data points for higher redshifts using Planck sample chains and use those data for the fitting. We use two sets of data for this purpose: in one case we generate data for $H(z)$ using Planck $\\Lambda$CDM  chains for redshifts $z=4,5,6$ [hereafter we refer this case as PL1] {assuming $z_{match} \\sim 4$} and in another case we do the same for $z=6,7,8$ [hereafter we refer his case PL2] {assuming  $z_{match} \\sim 6$}. In both cases, we use sample chains for Planck+WP+highL+lensing-2015 for baseline $\\Lambda$CDM  model \\cite{plchain}.\n\nThe reconstructed Hubble parameter, as a function of $z$, is shown in Fig.\\ref{fig:hqplot}. As one can see, the reconstructed $H(z)$ fits the $H_{0}$ data from R16 as well as $H(z)$ points as measured by Planck-2015 for $\\Lambda$CDM  model \\cite{Ade:2015xua} at higher redshifts. The constraints on deceleration and jerk parameters at $z=0$ are: ($q_{0} = -0.83^{+0.10}_{-0.10}, j_{0} = 3.93^{+0.53}_{-0.53}$) and ($q_{0} = -0.94^{+0.11}_{-0.10}, j_{0} = 4.31^{+0.51}_{-0.53}$) for PL1 and PL2 respectively. Clearly $j_{0} =1$ is ruled out with high confidence level confirming the inconsistency with $\\Lambda$CDM  model at present. This is consistent with previous results by \\cite{Zhao:2017cud} and \\cite{Wang:2018fng}.\n\nOnce we constrain the $H(z)$ using above data points, we again reconstruct the dark energy behavior through $f_{_{\\text{DE}}}(z)$ as described equations (4) and (5). Here instead of choosing a particular value for $\\Omega_{m0}$, we choose two different values, $0.3$ and $0.32$ respectively to see effect of $\\Omega_{m0}$ on reconstructed $f(z)$. The result is shown in Fig \\ref{fig:fplot}.\n\n\\begin{figure}\n\\begin{center} \n\\resizebox{200pt}{170pt}{\\includegraphics{fz_456_om30.pdf}} \n\\hspace{1mm} \\resizebox{200pt}{170pt}{\\includegraphics{fz_456_om32.pdf}}\\\\\n \\resizebox{200pt}{170pt}{\\includegraphics{fz_678_om30.pdf}} \n\\hspace{1mm} \\resizebox{200pt}{170pt}{\\includegraphics{fz_678_om32.pdf}}\n\\end{center}\n\\caption{Reconstructed $f(z)$. The top ones for PL1 whereas the bottom ones are PL2. In each case, left ones are for $\\Omega^{(0)}_{m} = 0.3$ and right ones are for $\\Omega^{(0)}_{m} = 0.32$. The different regions and lines are same as in Fig.\\ref{fig:hqplot}.}\n\\label{fig:fplot}\n\\end{figure}\n\n\nAs one can see, $f(z)$ exhibits two generic features: (1) the overall shape for $f(z)$ is the same for different choices of $\\Omega^{(0)}_{m}$ as well as the redshift range where we generate $H(z)$ data using Planck chains for $\\Lambda$CDM. It has always a minimum at some $z=z_{\\text{min}}$, and generically $f(z) < 0$ at this minimum; (2) $z_{\\text{min}}$ and $f_{\\text{min}}=f(z_{\\text{min}})$ depend on the value of $\\Omega^{(0)}_{m}$ and the range of redshifts where one takes the $H(z)$ data from Planck. For $\\Omega^{(0)}_{m} > 0.29$,  there is always a negative minimum for $f(z)$, and for $H(z)$ data from the Planck at higher redshift range, the negative minimum for $f(z)$ exists with a greater confidence level. The presence of this negative minimum in ${\\rho}_{_{\\text{DE}}}(z)$ does not affect the background evolution. This can be seen from Fig.\\ref{fig:hqplot} that there is no pathological behavior in $H(z)$ which is perfectly consistent with a late time accelerating Universe. \n\n\n\n\\section{Generic Results} $f(z)$ is essentially showing ``time dependence'' of the dark energy density ${\\rho}_{_{\\text{DE}}}(z)$, therefore the time derivative of ${\\rho}_{_{\\text{DE}}}$ vanishes at $z_{\\text{min}}$:\n$$\n\\dot{\\rho}_{_{\\text DE}}(z)|_{z=z_{\\text{min}}} = 0\\ \\  \\  \\Longrightarrow \\hspace{2mm} ({\\rho}_{_{\\text{DE}}} +{P}_{_{\\text{DE}}})_{z=z_{\\text{min}}} = 0.\n$$\nThat is, at the minimum, the ${\\rho}_{_{\\text{DE}}}$ behaves like a cosmological constant. Moreover, this minimum value being negative, can be simply modeled through a {\\it negative cosmological constant} with $\\Lambda=3H_0^2 (1-\\Omega^{(0)}_{m})  f_{\\text{min}}$. The most significant outcome of our analysis is that the low redshift data together with Planck constraint on $H(z)$ for $\\Lambda$CDM  at higher redshifts allows for a {\\it negative cosmological constant}.\n\\begin{figure}\n\\begin{center} \n\\resizebox{250pt}{250pt}{\\includegraphics{final_f.pdf}} \n\\end{center}\n\\caption{ Reconstructed $f(z)$. The mean $f(z)$ is plotted from the MCMC chains. The top one is without $H_{0}$ data as well as without taking Planck points for $H(z)$ for higher redshifts. The middle one is without $H_{0}$ data but taking Planck points for $H(z)$ for higher redshifts. The bottom one is taking both the $H_{0}$ data and Planck Points for $H(z)$ for higher redshifts. The rest of the low redshift data as mentioned in the text are taken in all plots. Planck points for $H(z)$ are for PL1 and $\\Omega^{(0)}_{m} =0.3$ is assumed.}\n\\label{fig:f1plot}\n\\end{figure}\nAs discussed in the Introduction, the negative ${\\rho}_{_{\\text{DE}}}$ without any minimum for higher redshifts has already been confirmed by \\cite{Wang:2018fng} using low redshift data in a model independent way. The new feature in our analysis is the inclusion of the Planck's constraints on $H(z)$ for higher redshifts. This yields to a lower negative bound on $\\rho_{DE}$ which can be associated with a \\emph{negative Cosmological Constant}. To show these dependencies, in Fig. \\ref{fig:f1plot} we plot the mean $f(z)$ behavior from our reconstruction for a specific case. The topmost  behavior is without including $H_{0}$ data from R16 and also without including $H(z)$ data generated using Planck $\\Lambda$CDM  chains. The $f(z)$ clearly goes to negative values at higher redshifts as confirmed by  \\cite{Wang:2018fng}, and this is mainly due to  Lyman-$\\alpha$ measurement of BAO at $z\\sim 2.4$ as pointed out in \\cite{Wang:2018fng}, {see also \\cite{Delubac:2014aqe, Poulin:2018zxs}}. Next we add the $H(z)$ data from Planck at higher redshift and this results a minimum in $f(z)$ which is positive. Finally, as we add the $H_{0}$ data from R16, the minimum shifts to a negative value. The inclusion of $H_{0}$ data from R16 enforces present Hubble parameter to be larger which can be compensated by lowering (even negative) dark energy density in the intermediate redshifts. Hence the Lyman-$\\alpha$ data at $z\\sim 2.4$, the $H_{0}$ data from R16 as well as the constraints from Planck on $H(z)$ at higher redshifts, all together play the crucial role for having a negative minimum for ${\\rho}_{_{\\text{DE}}}$ which may be modelled  by a \\emph{negative Cosmological Constant}.\n\n\n{The above conclusion can also be seen in this following way. Several previous analysis with low redshift data \\cite{Wang:2018fng}, \\cite{Delubac:2014aqe, Poulin:2018zxs} including our current analysis indicate that $f(z)$ becomes negative at some $z > 1$. At the same time, $H(z)$ should also go to matter domination at higher redshift. Then the only option for $f(z)$ is that it start with some positive value at $z=0$, starts decreasing for $z>0$, becomes negative at some point, attains a minimum with a negative value and then increases to $0$ for higher $z$, } {before matching to the Planck data at around $z_{\\text{match}}.$} \n \n\n\n\n\nMoreover, for $z>z_{\\text{min}}$, the $\\rho_{_{\\text{DE}}}$ decreases with time, whereas for  $z<z_{\\text{min}}$, it increases with time, confirming the phantom crossing at $z=z_{\\text{min}}$. The phantom region in this case cannot be mimicked by a minimally coupled scalar field rolling uphill the potential as discussed in \\cite{Csaki:2004ha,Csaki:2005vq}. This is because for a minimally coupled scalar field $\\phi$, at $\\rho_{_{\\text{DE}}}^{\\text{min}}$, $\\frac{d{\\phi}}{dt} =0 $, which does not allow $\\phi$ to roll uphill the potential.\n\n\\section{ Discussion and Outlook} \n\nBased on two requirements that $H_0$-tension is {ameliorated} (in favor of \\cite{Riess:2016jrr}) and $\\Lambda$CDM  is still the best fit for Planck data at higher redshifts, we reanalyzed the low redshift data. {If we consider just the low-redshift data, as we did in section \\ref{sec:2.1}, we get $H_0=72.56 \\pm 1.55$, which is perfectly consistent with \\cite{Riess:2016jrr}. Adding the $H(z)$ data for higher redshift from Planck chains for $\\Lambda$CDM  as we did in section \\ref{sec:2.2}, we get the following results: For $z_{\\text{match}}= 4$, $H_0=68.7 \\pm 1.3$, which is $2.3\\sigma$ tension with \\cite{Riess:2018byc}. For\n$z_{\\text{match}}= 6$, $H_0=70.0 \\pm 1.4$ which is at $1.65 \\sigma$ tension with \\cite{Riess:2018byc} and is at less than $2 \\sigma$ tension with \\cite{Riess:2019cxk}. The details of these analysis will appear in an upcoming paper. So the take away is: (1) adding the Planck constraint on background evolution at higher redshifts, the $H_0$ shifts towards lower values compared to only low-redshift analysis and (2)  adding the data point from $H(z)$ reconstructed by Planck chains, \nprefers  higher $z_{\\text{match}}$. As we already pointed out  with higher $z_{\\text{match}}$ the likelihood of having negative minimum in $f(z)$ is greater.\nAs we mention below,  the exact value of  $z_{\\text{match}}$ should be obtained with a thorough analysis including low redshift and full CMB likelihood.\n}\n\n\nWe  reconstructed the $H(z)$ behavior {for $z\\lesssim 8$ region}\\footnote{{Note that we are using Pade $P_{2,2}$ parametrization only for $0<z_{\\text{match}}\\lesssim 8$ region and for higher $z$ {one should use higher order Pade parametrization for better fit to actual model}.}}  and subsequently the ${\\rho}_{_{\\text{DE}}}(z)$ behavior. We stated and discussed  three generic results of our\n analysis in the previous sections. \nIt is important to highlight the differences of our work from some recent works along this line. \nIn the work by \\cite{Zhao:2017cud}, the equation of state parameter for the dark energy was reconstructed with the assumption of ${\\rho}_{_{\\text{DE}}} > 0$, and also no interaction between matter and the dark energy, therefore, setting aside a very large class of dark energy models including modified gravity models. In a subsequent paper by \\cite{Wang:2018fng}, ${\\rho}_{_{\\text{DE}}}$ was reconstructed directly, and it was found to be unbounded from below at higher redshifts. Almost similar inference was drawn in \\cite{Poulin:2018zxs}. {In our analysis, we directly reconstruct the $H(z)$ from low redshift data using Pade approximation and get the similar result. Our analysis reproduces the result of \\cite{Wang:2018fng}  that ${\\rho}_{_{\\text{DE}}} < 0$ at higher redshift. The fact that two different reconstructions yield similar results, supports  the validity of {our $H(z)$ reconstruction process which is based on Pade parametrization}.\n\n\\begin{comment}\n{\\color{green}In all the above cases, they incorporated the Planck constraint using acoustic scale, and we know that the major contribution to this scale, up to the surface of last scattering,  comes from low-redshift contribution. So in effect, they have not incorporated the high redshift constraint from the CMB data and the Planck.\nAs we have emphasized in the Introduction, {the data suggests we have a dynamical dark energy model with $z$-dependent equation of state.\nThe Planck constraints the higher redshift universe with a greater accuracy, so we assume the high redshift $H(z)$ is given by what Planck measured for $\\Lambda$CDM \\!\\!. But in lower redshifts, where the effects of dark energy becomes important, there is new physics with non-LCDM, XCDM, behaviour, that solves low-redshift tensions including the $H_0$ one}}.({\\bf this portion we can omit. It makes things a little confusing}).\n\\end{comment}\n\n\nSubsequently we incorporated Planck constraint on background universe, directly through $H(z)$ as measured by Planck for $\\Lambda$CDM  at intermediate redshifts. This makes sure that our reconstructed $H(z)$ from low redshift data is consistent with Planck measured $H(z)$ for $\\Lambda$CDM  at intermediate redshifts.  This results in a minimum for ${\\rho}_{_{\\text{DE}}}$ and this minimum is negative. {The notable features of our approach for low-redshift data analysis are direct $H(z)$ reconstruction (using Pade approximant\/parametrization) and imposing Planck constraint on $H(z)$ at intermediate redshifts.}\n}\n\nBelow, we add some further comments and things to be worked out and studied in future:\n\\begin{itemize}\n\\item[(I)]\\textbf{{Vetting the working assumptions and robustness of our results:}} \n\\vskip 2mm\n\nThis is probably the first study where we assume two different $H(z)$ behavior at low and high redshifts and match them around some $z=z_{\\text{match}}$, {for which we chose some reasonable values. To justify this working assumption,} we have tested different $z_{\\text{match}}$ and overall behavior of our results remain the same. This is also shown in Fig.\\ref{fig:fplot}, where we plot $f(z)$ for two different choices of redshift range to generate $H(z)$ data from Planck $\\Lambda$CDM  model. Of course, to get an actual estimate of $z_{\\text{match}}$, one needs to do a full combined analysis using all the low redshift data together with Planck Likelihood assuming that for $z \\leq z_{\\text{match}}$, $H(z)$ is given by \\eqref{H-Pade} and for $ z> z_{\\text{match}}$, $H(z)$ is given by $\\Lambda$CDM  model and get a estimate of $z_{\\text{match}}$. This is beyond the scope of present study and will be reported in a separate publication.\n\\vskip 1.5mm\n\n\n\nOur other working assumption was the use of Pade Approximation.  Pade approximation has been extensively used earlier for low-redshift reconstruction purposes, more detailed references can be found in \\cite{Saini:1999ba} and more recent usage of Pade parametrization in \\cite{Sahni:2014ooa, Aviles:2014rma, Gruber:2013wua, Wei:2013jya, Rezaei:2017yyj,  Mehrabi:2018oke} . Probably the first work on reconstruction of quintessence potential using Type-Ia Supernova data used the rational function like Pade Approximation to model the Luminosity Distance \\cite{Saini:1999ba}. In our case, we use the same for the Hubble Parameter with larger set of observational data. We emphasis that use of Pade Approximation to reconstruct of Hubble parameter does not bias the final result. The result that dark energy density can be negative at higher redshifts (primarily due to $H_0$ measurement by Riess et al and the Lyman-$\\alpha$ measurement of BAO at $z \\sim 2.4$) has been also confirmed by several earlier works. It is, however, useful to reanalyze the data using other parametrizations and verify that the final features and results do not depend on the parametrization used in any crucial way.\\\\\n\n\\item[(II)] \\textbf{Consistency with Planck's measurement of the CMB anisotropy:} \n\\vskip 2mm\n\n\\begin{figure}\n\\begin{center} \n\\resizebox{220pt}{220pt}{\\includegraphics{omega.pdf}} \n\\end{center}\n\\caption{The dotted red line is for $\\Omega_{\\text{m}}(z)$, the dashed green line is for $\\Omega_{_{\\text{DE}}}(z)$ and the solid blue line is for ${P}_{_{\\text{DE}}}\/(3H^2)$. This is for Planck H(z) data at $z=4,5,6$ and $\\Omega^{(0)}_{m}=0.32$.  }\\label{fig:omplot}\n\\end{figure}\n\nThe simplest dark energy model we can suggest based on our results is a negative Cosmological Constant and a phantom crossing for the rest of the dark energy. To show that such a model is consistent with CMB temperature anisotropy as measured by Planck, we calculate $C_{l}^{TT}$ for CMB anisotropy spectra assuming that till $z\\leq 6$, $H(z)$ is given by best fit of our reconstructed $H(z)$ and for $z>6$, it is given by Planck best fit $\\Lambda$CDM  model \\cite{Ade:2015xua, Ade:2015rim} . Note that at $z=6$, $\\Lambda$CDM  model is well within matter dominated era. In Fig. \\ref{fig:omplot}, we show the behaviours of density parameters for matter and dark energy. As one can see, around $z=6$, $\\Omega_{m} = 1$ and $\\Omega_{\\text{DE}} = 0$, allowing us to match our reconstructed $H(z)$ with a matter dominated era. \n\\vskip 1.5mm\n\nUsing this $H(z)$, we use CLASS code \\cite{class} to compute the $C_{l}^{TT}$ and compare with Planck data as well as Planck best fit $\\Lambda$CDM  model. The result is shown in Fig.\\ref{fig:CMB-ell}. As expected and one can see, $C_{l}^{TT}$ in our model is a good fit to Planck's measurement.\\\\\n\n\\item[(III)] \\textbf{Effects on Structure Formation:} \n\\vskip 2mm\n\nThe existence of this small negative  $\\Lambda$ can have interesting effect on growth of structures. As one can see in Fig. \\ref{fig:omplot}, $\\Omega_{m}$ is slightly greater than $1$ for a certain redshift range depending on where we match the $H(z)$ with Planck's $\\Lambda$CDM  model. This will give enhancement in growth of structures at higher redshifts and the nonlinear regime may start earlier than  in $\\Lambda$CDM  model. This may result in the presence of more massive galaxies at higher redshifts compared to $\\Lambda$CDM model, effects on reionization process as well as on lensing. All these are potential observable signatures for our model that can be tested by present and next generation galaxy surveys and CMB experiments.\\\\\n\n\\item[(IV)] \\textbf{Modeling the ${\\rho}_{_{\\text{DE}}} (z)$:} \n\\vskip 2mm\n\nWithin our data analysis framework, we have a clear indication that {dark energy sector cannot be described by {only} a cosmological constant}. Moreover, as discussed,  ${\\rho}_{_{\\text{DE}}}, {P}_{_{\\text{DE}}}$ cannot be obtained from a minimally coupled scalar field (with any potential); that is, quintessence models do not lead to our ${\\rho}_{_{\\text{DE}}}(z)$. Given that ${\\rho}_{_{\\text{DE}}}$ takes negative values for a range of $z$, the simplest model is to assume presence of a \\emph{negative cosmological constant} with its value $\\Lambda=\\rho_{\\text{min}}$ and then try to model $\\rho={\\rho}_{_{\\text{DE}}}-\\rho_{\\text{min}}$ within a non-minimally coupled scalar theory with a positive definite potential. This latter should be such that it provides  crossing to  phantom region $({\\rho}_{_{\\text{DE}}}+{P}_{_{\\text{DE}}} <0)$ for $z<z_{\\text{min}}$. Such models can be constructed, e.g. within Brans-Dicke theory \\cite{Wang:2018fng}. Seeking and exploring such models is postponed to future works. \n\\\\\n\n\\item[(V)] \\textbf{Theoretical implications of our dark energy model:} \n\\vskip 2mm\n\nA positive cosmological constant which is assumed to drive the current accelerated expansion of the Universe is a theoretical challenge: Getting a vacuum solution with a positive cosmological constant within moduli-fixed consistent and stable string theory compactifications has been a daunting task \\cite{Maldacena:2000mw, Kachru:2003aw, Conlon:2007gk, Danielsson:2018ztv}. Moreover, formulating quantum field theory on the background of a de Sitter space has its own challenges, from the choice of the vacuum state to non-existence of a well-defined S-matrix (on global de Sitter space) \\cite{Witten:2001kn, Goheer:2002vf}. Our findings here, lifts all those questions by simply removing the need for a positive cosmological constant. \n\\vskip 1.5mm\n\n\\begin{figure}\n\\begin{center} \n\\resizebox{220pt}{220pt}{\\includegraphics{cmb.pdf}} \n\\end{center}\n\\caption{CMB TT spectra. Top one: for model considered in this work (see text) together with Planck data and error bars for TT spectra. Middle one: the difference in TT spectra for our model and Planck best fit $\\Lambda$CDM model. Bottom one: The residual for our model with the Planck data. }\\label{fig:CMB-ell}\n\\end{figure}\n\n\nOn the other hand, a negative cosmological constant is a theoretical sweet spot: it provides an anti-de Sitter (AdS)  background, which is very much welcome due to AdS\/CFT duality \\cite{Maldacena:1997re}, providing a ``dual'' framework for  cosmology. In addition, string theory clearly prefers AdS background to de Sitter, consistent AdS backgrounds are ubiquitous in string theory settings \\cite{BP}. Interestingly, it has been argued that accelerated expansion of the Universe may be possible with a negative cosmological constant \\cite{HHH}. \n\\vskip 1.5mm\n\nFinally, we comment on the recently proposed ``swampland'' conjecture \\cite{swampland} which favors quintessence models over $\\Lambda$CDM. Our results here are in clear tension with the swampland conjecture. Similar statement has also been made in \\cite{Eoin}.\n\\\\\n\\item[(VI)] \\textbf{Our results and anthropic reasoning.}\n\\vskip 2mm\n\nIn mid 1980's and long before observational establishment of late-time cosmic acceleration, based on structure formation constraints which is necessary to yield existence of life (as it is usually perceived) it was argued that the value of cosmological constant, if positive, should not be much bigger than $H_0^2$ \\cite{Weinberg:1987}. A negative value of the cosmological constant, too, is bounded by similar anthropic reasoning \\cite{Barrow-Tipler}.  Interestingly, the negative cosmological constant in our model which is within $1\\%$ of the current total energy density of the Universe is certainly consistent with these bounds. \n\\end{itemize}\n\n\n{We emphasize that we ascribe the tension between the inferred value of $H_0$ between the local measurements and the Planck data fully on the dynamics of dark energy. Given this is the true case for this tension, what we find is that a negative value of the cosmological constant is still allowed by the data.} To summarise, we discuss the prospects of solving the tension between low  and high redshift cosmological observations in the presence of a small {negative cosmological constant}. This is probably the first time, that there are observational suggestions for presence of a {negative cosmological constant}. Its presence predicts definite observational signatures in large scale structure formation in the Universe and can be tested with present and future experiments.\n \n\n\\textbf{Note added:} As we were updating and revising our paper, the paper \\cite{Riess:2019cxk} appeared which has now a new measured value for $H_{0}$ which is $74.03 \\pm 1.42$ km\/s\/Mpc. This is now in tension with Planck-2018 measurement for $H_{0}$ at $4.4\\sigma$. For our case with PL2 ($z_{match} = 6$), the value for $H_{0}$ is $70^{+1.45}_{-1.50}$ Km\/s\/Mpc. The deviation of this value with \\cite{Riess:2019cxk} is $1.98\\sigma$ which is at less than $2\\sigma$ tension. {See also \\cite{Dutta:2019pio} for more discussions and analysis. }\n\n\n\n\\section*{Acknowledgements}\nWe are grateful to Ravi Sheth and Fernando Quevedo  for fruitful discussions and Eoin  O'Colgain, Hossein Yavartanoo, Maurice van Putten for comments on the draft.\nMMShJ was supported by the grants from ICTP NT-04, INSF junior chair in black hole physics, grant No 950124. AS, MMShJ  and KD would like to thank the hospitality of the Abdus-Salam ICTP, Italy, where this project  was initiated. Ruchika acknowledges the funding from CSIR, Govt. of India under Junior Research Fellowship.\n\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:int}\n\nSmart Cities use intelligent computing technologies to gather and analyze data from multiple sources to optimize resources, monitor activities, or prevent potential risks while improving the services they provide to its citizens \\cite{washburn2009helping}. Critical Infrastructures are vital resources for the development of a society, their failure would have a very high impact\/cost. \\cite{Gupta2014}. These include roads, communications, water, energy, etc. According to \\cite{Talari2017}, security is \\textit{the most relevant element of the Smart Cities from the citizen's point of view}. Even a minor disruption, either accidental or deliberate, will degrade the system performance and inflict significant economic and social losses.\n\nSmart automated surveillance plays a crucial role in Critical Infrastructure Protection (CIP) and represents a major challenge for the next years. Modern approaches include Cyber-Physical Systems (CPS) that integrate intelligent computing devices and networks that constantly monitor the infrastructures to guarantee the adequate level of security \\cite{Cenedese2014}.\n\nAfter the launch of AlexNet \\cite{Krizhevsky2012}, Convolutional Neural Networks (CNN) and other Deep Learning (DL) techniques are increasingly used in many areas of research, often improving the results present in the state of the art until their appearance. Thus, we find application of DL techniques in fields such as: Natural language processing (NLP) (e.g. sentence modeling \\cite{Kalchbrenner2014}), financial forecasting (e.g. Analysis of gramian angular fields (GAF) images from the Standard \\& Poor's 500 index \\cite{Barra2020}), point cloud analysis (e.g. Relation-Shape CNN \\cite{Liu2019}), computer-aided medical diagnosis (e.g. brain tumor classification with magnetic resonance imaging (MRI) \\cite{Abiwinanda2019}), etc. Biometrics, and specifically face recognition plays a crucial role in CPS providing a way to automate access control. The neural network based algorithms are much more accurate (for the Labeled Faces in the Wild - LFW dataset \\cite{LFW2007}, \\textit{LFW} accuracy $>99\\%$) than previous methods such as eigen vector \\cite{Turk1991} (\\textit{LFW} accuracy $\\approx60\\%$), and Local Binary Pattern (LBP) \\cite{Ahonen2006} based algorithms (\\textit{LFW} accuracy $\\approx70\\%$). Another of the uses that the application of neural networks has in this field is the clustering of facial attributes extracted by CNN \\cite{Abate2020, liu2015faceattributes}. This technique would allow to obtain an identikit of each detected subject with an intermediate quality of the image, so it could improve their facial re-identification.\n\nClassic video surveillance using closed circuit television (CCTV) systems have become an essential element for security and law enforcement. However, classic CCTV surveillance systems depend on the attention capacity of a human operator, who is confronted with a collage of many video streams \\cite{Haering2008}. Thus, automatic video analysis is required to appropriately scale to the CIs of a Smart City. Otherwise, massive CCTV becomes only a forensic resource to identify the cause after an accident or failure.\n\nSome limitations in conventional video surveillance industry include: 1) High cost associated to the need of personnel watching the videos, currently being replaced by smart automatic systems. Also, fatigue-related errors are a significant issue. Machine Learning (ML) is the dominant trend for automatic video analytics but it requires large datasets for training, not available for this field. 2) Multi-camera systems produce a huge amount of data demanding high bandwidths, thus local\/distributed processing is required since centralized processing cannot scale. 3) High latency, communication delays are a major concern in latency-sensitive surveillance, real-time data processing guarantees instant decision making to reduce risks and facilitate pro-active security \\cite{Chen2018}. 4) High cost of the deployment of ad-hoc communication networks with high data bandwidth and the lack of flexibility after deployment of rigid centralized systems.\n\nContrarily to centralized fully human-operated systems, the demand for distributed smart video surveillance systems increases. Recently, the fast evolution of high performance devices has opened the possibility of video analytics embedded in CPS edge devices. These new local processing capabilities foster building new distributed paradigms for CPS and tackle problems such as network traffic congestion, high latencies, and helps reducing costs of staff and network infrastructures for high data bandwidth \\cite{Chen2018}. Local edge nodes enable embedding processing of complex tasks that are now done locally, reducing the communication with central nodes and thus the communication traffic. Also, local processing partially reduces latency at least for the tasks performed locally. Central nodes are in charge of gathering results from local nodes, performing analysis, and making decisions about the overall system performance. Beyond that, CPS are also expected to be context-aware and dynamically reconfigure themselves adapting to the world in real time \\cite{Masin2017}.\n\nThe use of multi-camera systems increases the complexity of the system, forcing it to enable mechanisms to characterize the 3D multi-view geometry for perimeter control, complement the biometric identification via facial recognition with additional cues, predict trajectories or perform robust tracking of moving targets across multiple views \\cite{Salman2018}. The central cloud node facilitates the communication between the nodes and accordingly allows for the reconfiguration of the local nodes for carrying out these high-level tasks.\n\nThe main contributions of our smart video-surveillance CPS are: 1) a multi-view system, scalable and easily calibrated which automatically triggers alarms for potential risks; 2) the integration of different ML and computer vision techniques in the same video surveillance workflow, trained and tested with a wide variety of video surveillance scenes; 3) the reduction of processing and network bandwidth, using a distributed CPS with local edge nodes that are dynamically reconfigured according to the task conditions or metrics; 4) real-time video processing using high-performance embedded devices such as the Nvidia Jetson TX2 and Jetson Xavier Systems-on-Chip (SoCs).\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=.47]{system-overview.pdf}\n    \\caption{Overview of our reconfigurable CPS for CIP in a heterogeneous cloud-edge architecture: distributed local edge nodes process video embedded in high-performance SoCs (left: Nvidia Jetson TX2, right: Nvidia Jetson Xavier), performing human detection and extraction of robust features via Deep Learning (DL); the cloud server carries out perimeter monitoring, biometric facial recognition and tracking, using robust features, multi-camera management using homographic information, and makes decisions about the edge optimal configuration based on the Quality and Resource Management information.}\n\\label{fig:overview}\n\\end{figure}\n\n\\section{Proposed Approach}\n\\label{sec:apr}\nIn this paper we propose a heterogeneous reconfigurable CPS with distributed computing between a central server and networked SoC nodes (see Fig. \\ref{fig:overview}). Each node is connected to a camera from which it obtains livestream video of a monitored area,within the critical infrastructure. This video is pre-processed locally by the node to detect people within the image and obtain a robust descriptor of their appearance, applying DL techniques. Next, the people's locations and their descriptions , together with the video, are sent to the cloud from each node. In the cloud, the calibrated system of camera nodes let us estimate the position in the real world by homography.  Location and appearance cues are fed into our tracking component  that is complemented by a component that also monitors the secure perimeter of the infrastructure. In addition, facial recognition of the people detected is performed in order to classify operators\/intruders. The cloud also makes decisions about quality and resource management and it is responsible for triggering the reconfigurations of the nodes, according to existing surveillance needs. In our example in this work, given the massive amount of data transmitted by multiple cameras, reconfiguration is triggered by data bandwidth usage.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=.42]{node}\n\\caption{Overview of local edge live video processing: 1) Mixture of Gaussians (MOG) background subtraction determines ROIs; 2) smaller regions that correspond to noise are removed by morphological operations; 3) each ROI is fed into the CNN human detector (in the example, ROIs include two operators of the CI and a car); 4) ROIs classified as people are passed to a deep feature extractor for target disambiguation. Finally, the video, bounding boxes containing positive examples (people), and deep feature descriptors are sent to the cloud for further processing.}\n\\label{fig:edge}\n\\end{figure}\n\n\\section{Material and methods}\n\\label{sec:mat}\nThe continuous advances in high-performance computing and embedded devices becoming lighter, more portable, and interconnection capabilities boost the development of CPS \\cite{Scionti2019}. Modern CPSs are becoming distributed architectures of highly connected intelligent embedded systems that are getting increasingly autonomous. Moreover, CPS are dynamic systems which need to adapt to changing external conditions or monitored qualities triggering reconfigurations in real time \\cite{Masin2017}.\n\nSmart surveillance has become a popular CPS application, due to the real-time processing requirements and the large number of cameras needed to cover a large FOV (field of view). Processing-intensive tasks and its real-time requirements motivate a distributed edge-cloud architecture with optimized edge nodes (high performance SoCs) integrating efficient real-time processing, and the cloud that manages edge devices and performs task offloading \\cite{Scionti2019}. \n\nThe following subsections describe the components of our reconfigurable CPS. The two parts that conform the processing at the edge are depicted in Fig. \\ref{fig:edge}: the human detection component marks image regions where humans are located, and then DL features are extracted from these regions. At the cloud level, DL features complement the biometric identification via facial recognition, specially useful for reidentification e.g. when a person re-entries the FOV. The target regions are used for multi-camera tracking. To ensure the consistency, calibration via homography enables the use of a common frame of reference. Finally, reconfiguration and QRM module are described in section \\ref{sec:rCPS}.\n\n\\subsection{DL for human detection}\n\\label{subsec:dnn}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=.75]{bs}\n\\caption{Background substraction using MOG for traffic sequence (VIRAT dataset): moving objects (foreground) are separated from the background; the foreground in this scene pops out as the black pickup truck (bottom) and two people (parking lot and sidewalk).}\n\\label{fig:fgbgsubstraction}\n\\end{figure}\n\nOne of the common approaches for detecting and tracking moving objects in a video stream obtained by a static camera is background subtraction. It extracts a region of interest (ROI) of the object in movement within a scene, defined as foreground, generating a model of the background of the image with those objects that remain static \\cite{Piccardi2004}. Fig. \\ref{fig:fgbgsubstraction} illustrates the use of background subtraction applying the Mixture of Gaussians (MOG) approach \\cite{Zivkovic2004}. This method models the history of each pixel value as a set of Gaussian distributions whose parameters are continuously updated with new incoming frames from a video sequence. The mathematical model that this method uses is shown in Eq. \\ref{eq:mog}.\n\n\\begin{equation}\n\\hat{p}\\left ( \\vec{x} | \\chi_{T}, BG+FG \\right ) = \\sum_{m=1}^{M} \\hat{\\pi}_{m} \\nu \\left ( \\vec{x}; \\hat{\\vec{\\mu}}_{m}; \\hat{\\sigma}_{m}^{2}I \\right )\n\\label{eq:mog}\n\\end{equation}\n\nFor the above equation, $\\hat{\\vec{\\mu}}_{1},...,\\hat{\\vec{\\mu}}_{M} $ are the estimates of the mean and $ \\hat{\\sigma}_{1},...,\\hat{\\sigma}_{M} $ the estimates of the variance of the different Gaussian distributions of each pixel value. Additionally, $ \\hat{\\pi}_{m} $ is the weight that determines the importance of each distribution. This method runs continuously in real-time in our system at the node, estimating the potential positions of ROIs to be classified as humans or otherwise discarded.\n\nHuman detection approaches are divided into classic methods that use handcrafted features and novel DL methods that automatically learn these features from large annotated datasets. DL-based human detection approaches outperform the detection task in comparison to classic approaches such as the most common Support Vector Machine (SVM-) based classification methods \\cite{Park2020}. Particularly, CNNs automatically learn high-level semantic characteristics from data through structure, exhibiting superior performance in classification tasks \\cite{Zhao2019}. Although training a CNN is compute-intensive, techniques such as Transfer Learning alleviates this problem, allowing a pre-trained model to be partially retrained for different applications. In this framework, the weights of a network that is very successful differentiating low- and medium-level image features are re-used. Then, partial re-training updates the weights of the last network layers that extract high-level features to adapt the network to a different classification problem. In this work, we perform human detection via CNNs, retraining a network using the Transfer Learning technique to adapt its original weights (trained with the ImageNet dataset \\cite{imagenet_cvpr09}) to our video surveillance problem.\n\nWe chose the \\textit{MobileNetV2} \\cite{Sandler2018} as base model, popular for embedded applications and with a good accuracy vs. resource trade-off. This model performs a convolution for each channel individually instead of all of them at once, reducing the number of calculations and thus, making it more suitable for architectures with limited resources. \n\n\\subsection{Camera calibration with homographic transformation}\n\\label{subsec:homography}\nCalibration estimates camera intrinsic parameters through the correspondence between 3D points in the camera image plane, and the relationship between cameras \\cite{Long2019}.\nKnowing this relationship is essential e.g. to estimate target trajectories out of the FOV, determine the position of targets or delimit the secured perimeter across different cameras. We use homography because, among the classical calibration methods, 2D planar target calibration methods provide high accuracy and flexibility \\cite{Zhang2000}.\n\nA homography is any projective transformation that determines a geometric correspondence between two planes. In our case, the homography provides a common frame of reference for the accurate location of targets across cameras. \nGiven a set of points corresponding to each other $ x_{i}\\leftrightarrow{x_{i}}' $ where $ x_{i} $ comes from the position in the camera view, while $ {x_{i}}' $ is the position corresponding to that point on the plane of the scene and by writing $ {x_{i}}' = \\left (\\chi_{i}', \\varphi_{i}', \\omega_{i}' \\right )^{T} $ with homogeneous coordinates, we can estimate the homography matrix $ H $ between the view and the plane as $ {x_{i}}' \\times Hx_{i} = 0 $. Thus, for each pair of corresponding points, three linear equations are written as shown in Eq. (\\ref{eq:homography}), with $ {h_{i}}, i=1,2,3 $ a $ 3 \\times 1 $ vector made of the elements of the i-th row of $ H $.\n\n\\begin{equation}\n\\begin{bmatrix}\n0^{T} & -\\omega_{i}'x_{i}^{T} & -\\varphi{i}'x_{i}^{T} \\\\ \n\\omega_{i}'x_{i}^{T} & 0^{T} & -\\chi_{i}'x_{i}^{T} \\\\ \n\\varphi{i}'x_{i}^{T} & \\chi_{i}'x_{i}^{T}  & 0^{T}\n\\end{bmatrix} \n\\begin{pmatrix}\nh_{1} \\\\ \nh_{2} \\\\ \nh_{3}\n\\end{pmatrix}\n= 0\n\\label{eq:homography}\n\\end{equation}\n\n\\subsection{Facial recognition}\n\n \\begin{figure}[ht]\n \t\\centering\n \t\\includegraphics[scale=.5]{face_rec_1}\n \t\\caption{Face recognition pipeline: ROI detection, landmark extraction, alignment, and feature matching.}\n \t\\label{fig:face_pipeline}\n \\end{figure}\n\nFace recognition pipelines usually require three components illustrated in Fig. \\ref{fig:face_pipeline}: detection, alignment and feature vector extraction. The detector provides a bounding box for the face in a given image. For real-time face detection, we use a cascade type neural network based on \\textit{MTCNN} \\cite{Zhang2016}. The purpose of alignment is to normalize the face for the feature vector extractor to give more accurate and stable results on images taken in different conditions. To align a face to pre-defined coordinates, a face landmark detector can be used to find key points. The face is then transformed using a similarity transform (rotation, translation and scaling) to the target coordinates. For a fast face landmark detector, we use the \\textit{LBF} \\cite{Ren2016}. The aligned face image is then fed to a feature extractor neural network designed for recognition. For fast inference, we use a small and accurate neural network, based on \\textit{MobileFaceNet} \\cite{Chen2018}. This produces a low dimensional feature vector, unique for each identity. This feature vector is then compared to other vectors using cosine similarity to find matches.\n\n\\subsection{Multiple Person Tracking}\n\\label{subsec:mot}\nMulti-object tracking aims at finding the trajectories of moving objects in a video sequence. It is crucial for video surveillance systems with obstructing objects and whose complexity increases with the number of humans to be tracked.  This problem is often treated as an association task. First, it marks out a rectangle that encloses each of the objects to be tracked (bounding box), and then a matching algorithm is responsible for associating each of the corresponding bounding boxes across the different frames.\n\nIn the last few years, the focus has been on tracking by detection in each of the frames. However, this method is computationally expensive because of the detection and does not usually meet the requirement of real-time performance  \\cite{Hao2018}. Thus, in order to speed up the task, the detection is usually executed every few frames, and in between, another lighter algorithm is used. These widely used lighter algorithms include  traditional methods like mean shift, or Kalman filters which follow the location of the object through multiple consecutive iterations. However, after a number of iterations, these algorithms can result in a significant accumulated error, so they must be updated with the detector regularly.\n\nIn this work, tracking uses the movement and appearance of detected targets as in \\cite{Wojke2017}. We run the person detection every 5 frames to update the tracking and meanwhile we match the identity of the different bounding boxes obtained by the background subtraction by predicting the location and the similarity in their appearance. For the prediction of trajectories within the frame and also across cameras we use Kalman filters. In contrast, for matching by appearance, a CNN obtains robust descriptors of the detected targets, that complements the biometric identification via facial recognition in case of e.g. ambiguity, or when a known target re-entries the scene.\n\n\\section{A reconfigurable CPS}\n\\label{sec:rCPS}\nReconfigurable CPSs provide solutions that automatically adapt their functionality or architectures to the dynamic environment. The purpose is to accomplish their tasks but also attains the optimization of specific qualities meanwhile. Quality and Resource Management (QRM) modules are in charge of monitoring the evolution of specific qualities or the environment and trigger reconfigurations accordingly.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Edge operating modes}\n\\label{tab:edge_modes}\n\\begin{tabular}{@{}llll@{}}\n\\toprule\n\\multicolumn{1}{c}{Mode}      & \\multicolumn{1}{c}{Resolution (fps)} & \\multicolumn{1}{c}{Livestream video bandwidth} & \\multicolumn{1}{c}{Event(s)}               \\\\ \\midrule\n{\\color[HTML]{000000} }       & {\\color[HTML]{000000} }        & {\\color[HTML]{000000} } & {\\color[HTML]{000000} Confirmed intrusion} \\\\\n\\multirow{-2}{*}{{\\color[HTML]{000000} Mode 2}} &\n  \\multirow{-2}{*}{{\\color[HTML]{000000} 1280x960 (30)}} &\n  \\multirow{-2}{*}{{\\color[HTML]{000000} 8,40 MB\/s}} &\n  Broken perimeter \\\\\n{\\color[HTML]{000000} }       & {\\color[HTML]{000000} }        &                         & Detection                                  \\\\\n\\multirow{-2}{*}{{\\color[HTML]{000000} Mode 1}} &\n  \\multirow{-2}{*}{{\\color[HTML]{000000} 640x480 (15)}} &\n  \\multirow{-2}{*}{1,90 MB\/s} &\n  Posible intrusion \\\\\n{\\color[HTML]{000000} Mode 0} & {\\color[HTML]{000000} 320x240 (5)} & 0,41 MB\/s                       & Nothing relevant occurs                             \\\\ \\bottomrule\n\\end{tabular}%\n\n\\end{table}\n\nTable \\ref{tab:edge_modes} shows the reconfiguration scenarios with different modes that are analyzed in Section \\ref{subsec:decisionmaking}. The QRM module drives reconfiguration monitoring events that happen in the scene and affects the resolution of the transmitted video, optimizing the use of bandwidth of the network that is shared by all the distributed local edges. The cloud-edge interaction for reconfiguration is done via reconfiguration commands from the cloud. Reconfiguration commands are responsible for establishing local edge modes of operation at every moment, modifying the quality of the video acquisition that is processed in the node and transmitted to the cloud. These reconfigurations have different direct effects: 1) image processing directly depends on image resolution; 2) the use of data bandwidth is crucial for a network shared by multiple cameras to ensure bounded latency and thus, our reconfigurable CPS reduces it significantly when nothing or only negligible events occur in the scene. It should be noted that the same video codec (MJPEG) has been used for all sequences.  All the results presented in this work, as well as the bandwidth values shown, have been calculated from the MJPEG-encoded video.\n\n\\subsection{Node Calibration}\n\\label{subsec:nodecalibration}\n\nAs mentioned earlier, homography is the technique we use to calibrate cameras. For this purpose, points of the image and their equivalents on the ground plane have been selected by using cartographic coordinates. Fig. \\ref{fig:calibration} shows an example of use with two cameras and point correspondences. We have released the calibration tool as open-source to help others in the field\\footnote{https:\/\/github.com\/JuanIsernGhosn\/homography-calibrator}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=.50]{calibrator}\n\\caption{The calibration tool computes the camera homographic parameters using the correspondences between the selected points from two different camera perspectives. In this example, only 4 points from two different views are required to compute the homography matrix (colors denote matches of points from the two views).}\n\\label{fig:calibration}\n\\end{figure}\n\n\\subsection{Multi-camera decision making}\n\\label{subsec:decisionmaking}\n\nThe reconfiguration is driven by the events detected in the scene. The multi-camera decision-making module is in charge of gathering the event data from the local edge nodes and trigger reconfiguration if needed. Data from the nodes include tracking (trajectories and target positions), people identification provided by facial recognition and deep robust features, and perimeter monitoring. As shown in the Table \\ref{tab:edge_modes}, in the absence of events of interest in the FOV of a node resolution is reduced in order to minimize the amount of data to be processed and transferred. In the event of authorized personnel detected in the scene, they are supervised in the mid-priority operating mode (mode 1), while if there is an intrusion in the secured perimeter, the node(s) covering that region operates at maximum resolution to ensure the best accuracy in detection and identification (see Fig. \\ref{fig:decision_multi_camera}). High resolution images are transmitted with a two-fold purpose: 1) allow the security personnel double-check potential risks, and 2) to save maximum resolution images for future forensic analysis.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=.40]{decision}\n\\caption{Reconfiguration example for the multi-camera video-surveillance system: if nothing relevant happens (\\textit{t1} in \\textit{cam1} and \\textit{t3} in \\textit{cam2}), the node operates in mode 0 (see Table \\ref{tab:edge_modes}); if a person is detected (detection or prediction of the trajectory within the FOV of at least one camera), mode 1 is activated (\\textit{t2} in \\textit{cam1} - blue area on the map - and \\textit{t1} and \\textit{t2} in \\textit{cam2} - red area on the map); finally, mode 2 is activated with the intrusion in the secured perimeter (\\textit{t3} in \\textit{cam1} - green area on the map).}\n\\label{fig:decision_multi_camera}\n\\end{figure}\n\nThe aspects considered to carry out the reconfiguration of the nodes are the following: When a person is detected within the node FOV (\\textit{t1} person within the FOV of the camera of local node 1 (\\textit{cam1}); blue area in Fig. \\ref{fig:decision_multi_camera}-bottom), it changes its operation mode from 0 to 1. Next, when the trajectory of a tracked target points to the area of the predicted region with 95\\% confidence (Ellipses show the potential region in which with 95\\% confidence the track will be found in the next 200ms) being within the FOV of another camera (\\textit{t2} with predicted trajectory within \\textit{cam2} FOV; red area), the operation mode goes up to 2. Similarly, when the trajectory of a tracked target entries the perimeter of a secured area, the operation mode changes to 2 (\\textit{t3}; green area).\n\n\\section{Implementation and results}\n\\label{sec:exp}\n\nLocal edge nodes are in charge of human detection within their FOV and the extraction of deep robust features from them. The cloud gathers information from all nodes, maintains the relationship between camera views, monitors secured perimeter, carries out tracking, and monitors the scene for triggering potential edge reconfigurations. Next subsections present the experiments and discuss results.    \n\nRegarding the embedded devices for our edge nodes, we use a heterogeneous approach with Nvidia Jetson TX2 and Nvidia Jetson Xavier SoCs, bearing in mind their high performance and size. The Jetson TX2 module is a SoC with a six-core CPU with 8GB DDR4 memory and a GPU with NVidia Pascal architecture with 256 CUDA cores, while the Jetson Xavier is a Tegra Soc, with an 8-core CPU with 16GB DDR4 memory and a Volta GPU with 512 CUDA cores. Regarding the cloud, we are using a platform that includes a high-performance GPU NVidia RTX 2080Ti. \n\n\\begin{table}[ht]\n\\centering\n\\caption{Performance on the edge for the different operating modes (fps).}\n\\label{tab:perf_edge}\n\\begin{tabular}{@{}lccc@{}}\n\\toprule\n                & Mode 0  & Mode 1  & Mode 2 \\\\ \\midrule\nJetson TX2      & 69.3    & 44.2    & 31.7 \\\\\nJetson Xavier   & 98.3    & 70.5    & 50.4 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\nTo verify the compliance of the requirement of real-time operation, the performance of those tasks embedded on the edges is included in Table \\ref{tab:perf_edge}. The performance in fps for each of the different operating modes and for the SoC devices considered reach in the worst case about 32 fps (for 1280x960 resolution images), and up to 70 fps in the best case at the lowest resolution. The Nvidia Jetson Xavier reaches 1.4x performance improve with respect to Jetson TX2. Processing is faster than video acquisition for all operating modes, allowing each edge node to connect to more than one camera at a time if needed.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=.55]{bw_red}\n\\caption{Reconfiguration example for CPS cloud-edge bandwidth use. Three local edges are dynamically reconfigured according to the events (arrow) of the monitored environment. The gray area represents the bandwidth reduction ($\\approx 75\\%$) with respect to no reconfiguration scenario.}\n\\label{fig:banred}\n\\end{figure}\n\nWith respect to the overall system, reconfiguration is triggered to optimize the use of data bandwidth of the shared communication network in our multi-camera system. The use of data bandwidth directly depends on image resolution that is context-aware as shown in Section \\ref{sec:rCPS}. Local nodes use the lowest resolution when no event happens, reducing communication bandwidth and thus, processing complexity. In contrast, when relevant events take place the resolution is improved to increase the detection accuracy and resolution of the images that are transferred to the cloud, changing the operation mode. Fig. \\ref{fig:banred} shows a real example of how our system bandwidth changes and how the network load is reduced due to the reconfiguration of each one of the local edges. Different events (marked out in time axis) change the bandwidth use: if a node detects a human it is reconfigured and the bandwidth usage increases from 6.9 to 27.6 MB\/s per node; if the human target is identified as non-authorized personnel or enters into a secured perimeter, it is again reconfigured and the bandwidth usage goes up to 110.6MB\/s per node. In this example, reconfiguration allows reducing bandwidth by 76.32\\%, reducing the video resolution mainly when no event of interest takes place in the monitored environment.\n\n\\subsection{Human detection}\n\\label{subsec:edge}\nA comparison of two human detection models trained with Transfer Learning (\\textit{MNV2\\_1} and \\textit{MNV2\\_4}), the original \\textit{MobileNetV2} model, and a \\textit{HOG} (Histograms Of Gradients)\\textit{+SVM} classic approach is shown in Table \\ref{tab:classification}. The evaluation includes the INRIA Person Dataset and the OTC (Oxford Town Centre) video surveillance dataset. While MNV2\\_1 uses the original architecture of MobileNetV2, adding at the end three dense layers with regularization, MNV2\\_4 uses only the first three blocks of MobileNetV2, adding at the end only two dense layers with regularization. \\textit{MNV2\\_4} offers a good accuracy vs. resources trade-off, reaching almost the same accuracy than our first model \\textit{MNV2\\_1} and 17\\% more than the original \\textit{MobileNetV2} model and 3\\% more than the classic \\textit{HOG+SVM}. At the same time, it achieves 16x reduction compared to \\textit{MobileNetV2} and 36x reduction to \\textit{MNV2\\_1} in the size of the model. This is a crucial parameter to implement the architecture in embedded devices, reducing complexity, processing time and power consumption. Also, to reduce even further the size and complexity of our CNN models, they have been optimized through TensorRT.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Human detection performance (INRIA + OTC datasets)}\n\\label{tab:classification}\n\\begin{tabular}{@{}llll@{}}\n\\toprule\n\\multicolumn{1}{c}{Model name}          & \\multicolumn{1}{c}{F1-Score}             & Neural network parameters                   & Disk space (MB)                    \\\\ \\midrule\n{\\color[HTML]{000000} MobileNetV2}          & {\\color[HTML]{000000} 0.801574} & {\\color[HTML]{000000} 3.4 M} & {\\color[HTML]{000000} 14.0} \\\\\n{\\color[HTML]{000000} MNV2\\_1}          & {\\color[HTML]{000000} \\textbf{0.970489}} & {\\color[HTML]{000000} 4.1 M} & {\\color[HTML]{000000} 31.60} \\\\\n{\\color[HTML]{000000} \\textbf{MNV2\\_4}} & {\\color[HTML]{000000} 0.964618}          & \\textbf{75,186}              & \\textbf{0.88}                \\\\\n{\\color[HTML]{000000} HOG + SVM} & {\\color[HTML]{000000} 0.947584}          & -              & 140.4                \\\\\n \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\\subsection{Facial recognition}\n\\label{subsec:facialperf}\n\n \\begin{figure}[ht]\n \t\\centering\n \t\\includegraphics[scale=.45]{face_rec_3}\n \t\\caption{False positive rate as a function of false negative rate of face recognition for the FERET FB test set \\cite{Phillips2000}, with varying resolution of faces. The figure represents the plots with sizes of 8-63 pixels per box side.\n \t}\n \t\\label{fig:roc_vs_res_face_rec}\n \\end{figure}\n\n \nOur implementation of \\textit{MTCNN} face detector was trained using two datasets \\cite{Yang2016, Sun2013} following the original \\textit{MTCNN} training. The face landmark detector (LFB), was trained using the Helen \\cite{Le2012}, the LFPW \\cite{BelHumeur2013} and the iBug \\cite{Sagonas2013} datasets. The feature extractor \\textit{MobileFaceNet} was trained on the \\textit{MS-Celeb-1M} dataset \\cite{Yandong2016}. The accuracy of our face recognition pipeline on the LFW test set is $99.50\\%$. On a modern smartphone (One+ 7T Pro) the inference speed is $\\approx 30ms$ per frame. The resolution of a face image in a face recognition pipeline is critical for the accuracy as shown in Figure \\ref{fig:roc_vs_res_face_rec}. The smallerhttps:\/\/www.overleaf.com\/project\/5fbe7fb61a454806e05bec0c the face, the less accurate will the face detection and alignment be, resulting in less accurate face recognition overall. Thus, when the size of the face images is 8 pixels, the FNR(@FPR$<$$10^{-6}$)=100.0\\%, whereas when they are 63 pixels long, the FNR(@FPR$<$$10^{-6}$)=0.001\\%, which is $~10^5$ times better than using the lower resolution images.\n\n\\subsection{Tracking}\n\\label{subsec:trackingperf}\n\nIn order to evaluate the quality of the tracking, we present a comparison with the state-of-the-art FairMOT method \\cite{zhang2020simple}. It achieves the best results in the MOT challenge for the MOT20 dataset \\cite{MOTChallenge20}. The comparison is shown in Table \\ref{tab:one_cam_mot} runs on the VIRAT dataset \\cite{Oh2011}, with surveillance sequences in single-camera environments. The evaluation is carried out using the metrics: 1) Multi-object tracking accuracy (MOTA), overall tracking accuracy in terms of false positives, false negatives and identity switches; 2) Multi-object tracking precision (MOTP), overall tracking precision in terms of bounding box overlap between ground-truth and reported location; 3) Mostly tracked (MT), percentage of ground-truth tracks that have the same label for at least 80\\% of their life span; 4) Mostly lost(ML), percentage of ground-truth tracks that are tracked for at most 20\\% of their life span; 5) Identity switches (ID), number of times the reported identity of a ground-truth track changes.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Multiple Object Tracking (MOT) results in sequences of the VIRAT dataset.}\n\\label{tab:one_cam_mot}\n\\begin{tabular}{@{}lccccccc@{}}\n\\toprule\n &\n  {\\color[HTML]{000000} \\textbf{MOTA $\\uparrow$}} &\n  {\\color[HTML]{000000} \\textbf{MOTP $\\uparrow$}} &\n  \\textbf{MT $\\uparrow$} &\n  \\textbf{ML $\\downarrow$} &\n  \\textbf{FN $\\downarrow$} &\n  \\textbf{FP $\\downarrow$} &\n  \\textbf{ID $\\downarrow$}\\\\ \\midrule\nOurs &\n  {\\color[HTML]{000000} 89.4} &\n  92.8 &\n  {\\color[HTML]{000000} 90.3\\%} &\n  4.3\\% &\n  {\\color[HTML]{000000} 5.1\\%} &\n  3.8\\% &\n  1.8\\% \\\\\nFairMOT \\cite{zhang2020simple} &\n  \\textbf{94.2} &\n  \\textbf{96.9} &\n  \\textbf{93.7\\%} &\n  \\textbf{2.5\\%} &\n  \\textbf{3.3\\%} &\n  \\textbf{1.9\\%} &\n  \\textbf{0.6\\%} \\\\\n DeepMOT \\cite{xu2019train} &\n  91.4 &\n  95.8 &\n  92.8 &\n  2.6\\% &\n  3.4\\% &\n  2.4\\% &\n  \\textbf{0.6\\%} \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\nTable \\ref{tab:one_cam_mot} shows the tracking results of our system with respect to two of the best trackers in the state of the art. All these results are obtained in a single-camera environment. In general terms, the result of our system regarding these two pioneer trackers is good. The number of false negatives is small, an important feature of  video surveillance systems whose main purpose is to minimize triggering false alarms. Moreover, compared to the other trackers, our system guarantees its operation in real time reaching up to 50.4 fps in HQ video (compared to the others that reach 30 fps but using a dedicated powerful GPU). Since FairMOT is the state of the art method tracking in single-camera environments, it has been used to build our groundtruth, using its results to label the videos in our own test dataset. However, since our dataset corresponds to video sequences from a multi-camera environment, a manual matching of the track IDs between different perspectives was also required.\n\nThe results for multiple object tracking in our multi-camera test dataset are shown in Table \\ref{tab:multi_cam_mot}. The main issue is the complete lack of appropriate datasets and methods for multi-camera setups. In order to compare our multi-camera system we gathered the results of multiple object tracking obtained by DeepMot in each video sequence from each camera in our dataset, as if they were multiple videos from a single camera dataset. Then, we averaged the resulting MOT metrics for each video sequence. The difference between results is not significant for both systems (MOTA: 1\\%, FN: 0.5\\%, FP: 0.2\\%), with a  slight advantage of DeepMot. However, since part of the processing is distributed in the local nodes at the edge, our system continues working in real time even with the processing of multiple video streams at the same time and maintaining the same performance.\n\n\\begin{table}[ht]\n\\centering\n\\caption{MOT results for our test dataset}\n\\label{tab:multi_cam_mot}\n\\begin{threeparttable}\n\\begin{tabular}{@{}lccccccc@{}}\n\\toprule\n &\n  {\\color[HTML]{000000} \\textbf{MOTA $\\uparrow$}} &\n  {\\color[HTML]{000000} \\textbf{MOTP $\\uparrow$}} &\n  \\textbf{MT $\\uparrow$} &\n  \\textbf{ML $\\downarrow$} &\n  \\textbf{FN $\\downarrow$} &\n  \\textbf{FP $\\downarrow$} &\n  \\textbf{ID $\\downarrow$} \\\\ \\midrule\nOurs* &\n  {\\color[HTML]{000000} 98.4} &\n  97.2 &\n  {\\color[HTML]{000000} 96.0\\%} &\n  2.2\\% &\n  {\\color[HTML]{000000} 0.9\\%} &\n  0.5\\% &\n  0.1\\% \\\\ \nDeepMOT** &\n  99.4 &\n  98.5 &\n  98.7\\% &\n  0.2\\% &\n  0.4\\% &\n  0.3\\% &\n  0.1\\% \\\\ \\bottomrule\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\item[*] Result considering the multi-camera setup and the same detection identities across different perspectives for each video sequence.\n\\item[**] It's a single-camera tracker. Average of the results in each perspective of the sequences of our multi-camera dataset\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\section{Conclusions}\nOur work has novel differentiating aspects that are necessary in modern video-surveillance systems for Smart Cities. First, our active reconfigurable CPS reacts automatically to the dynamic events that occur in the monitored environment. The real time requirements demands an appropriate management of the resources, without harming the accuracy of the system. In our case, the intelligent management of the bandwidth shared by the system nodes, allows to reduce the average bandwidth usage in approximately 75\\%. The resolution reduction does not affect biometric identification and high resolution images are only transmitted to reduce false positive detections and for forensic purposes.\n\nSecond, machine learning modules have been optimized and embedded in SoCs for the detection and characterization of people within the monitored area. Using Deep Learning techniques, we have proposed new CNN-based models using Transfer Learning for the video-surveillance field that lacks appropriate datasets. These models have been optimized using TensorRT, achieving real-time performance. We process high quality video streams above 30~fps on Nvidia Jetson TX2 and above 50~fps on Nvidia Jetson Xavier platforms. This facilitates integrating multiple cameras into single nodes working at lower framerates, if required.\n\nFinally, the efficient management of processing and communication resources using a reconfiguration scheme driven by a quality management system facilitates the scalability of the proposed approach. In addition, our system enables easy calibration of local edge nodes. By calibrating on a common coordinate system, it is possible to execute in a more robust way tasks such as multiple person tracking, allowing to exchange information from different local nodes to improve its performance.\n\n\\section*{Acknowledgments}\nThis work was partially supported by the EU Project FitOptiVis through the ECSEL Joint Undertaking under GA n. 783162, a Spanish National grant funded by MINECO through APCIN PCI2018\u2010093184, and partially by the Research Network RED2018-102511-T, the AEI Grant PID2019-109434RA-I00, and the Program 9 of the 2015 research plan of the University of Granada (project ID 16).\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA neutral $Z'$ gauge boson appears in numerous models containing the SM gauge symmetry group along with an additional $U(1)$ symmetry (for a review, see~\\cite{Langacker:2008yv}). Grand Unified Theories (GUTs) larger than the original $SU(5)$ model, such as $SO(10)$~\\cite{Robinett:1982tq} or $E_6$~\\cite{Langacker:1984dc,Hewett:1988xc}, break down to the SM as: $E_6 \\rightarrow SO(10)\\times U(1)_\\psi$, with in turn $SO(10)\\rightarrow~SU(5)~\\times~U(1)_\\chi~\\rightarrow$ SM $\\times~U(1)_\\chi$~\\cite{Georgi:1974sy}. The $Z'$ thus surviving at the electroweak (EW) scale can be written as the linear combination,\n\\begin{eqnarray}\n\\label{eq:e6kinmix}\nZ' = \\cos\\alpha \\cos\\beta Z_\\chi + \\sin\\alpha \\cos\\beta Z_Y + \\sin\\beta Z_\\psi.\n\\end{eqnarray}\n\nIf kinetic mixing~\\cite{Holdom:1985ag,delAguila:1995rb,Babu:1996vt} with the hypercharge group $U(1)_Y$ is neglected by setting $\\alpha=0$ in eq. (\\ref{eq:e6kinmix}), one obtains some well--known $Z'$ bosons by adjusting $\\beta$. These include $Z_\\chi$ ($\\beta = 0^\\circ$), $Z_\\psi$ ($\\beta = 90^\\circ$), $Z_\\eta$ ($\\beta \\approx -52.2^\\circ$)~\\cite{Candelas:1985en}, $Z_I$ ($\\beta \\approx 37.8^\\circ$)~\\cite{Langacker:1984dc}, $Z_S$ ($\\beta \\approx 23.3^\\circ$)~\\cite{Erler:2002pr,Kang:2004pp} and $Z_N$ ($\\beta \\approx 75.5^\\circ$)~\\cite{Ma:1995xk}. The inclusion of kinetic mixing results in certain other phenomenologically interesting cases, such as $Z_{dph}$ $(\\alpha, \\beta \\approx -78.5^\\circ, 37.8^\\circ)$ which does not couple to the $d$--type quarks, $Z_R$ ($\\alpha, \\beta \\approx 50.8^\\circ,0^\\circ$)~\\cite{Erler:2009jh} which couples to the right--handed fermions only and $Z_{B-L}$ $(\\alpha,\\beta \\approx - 39.2^\\circ, 0^\\circ$)~\\cite{Appelquist:2002mw}, where $B$ is the baryon number and $L$ the lepton number of an ordinary fermion. The $Z_{LR}$ which exists in models with left--right symmetry~\\cite{Mohapatra:1986uf} is equivalent to the linear combination: $\\sqrt{3\/5}\\, (\\bar\\alpha\\, Z_R - Z_{B-L}\/2\\bar\\alpha)$, where $\\bar\\alpha \\equiv \\sqrt{g_R^2\/g_L^2 \\cot^2 \\theta_W - 1}$, with $\\theta_W$ being the weak mixing angle and $g_{L,R}$ being the $SU(2)_{L,R}$ coupling strengths, respectively. The $Z_{LR}$ studied here corresponds to the specific case of $g_L=g_R$. In addition to these $E_6$ based models, a $Z_{string}$ from a specific superstring model~\\cite{Chaudhuri:1994cd} and a sequential $Z_{SM}$ are also included in this analysis.\n\nA number of other classes of one or more--parameter models have been discussed \\cite{Erler:2000wu,Carena:2004xs,Langacker:2008yv}. Without the assumption of unification at the GUT scale, but assuming nullification of anomalies using three families of exotics (which is not the case in some supersymmetric models), there are four `one-parameter' models \\cite{Carena:2004xs}, with fermion charges $B-xL$, $d-xu$, $q+xu$ and $10+x\\bar{5}$, with $x$ arbitrary. The last three of these correspond to the generalized $E_6$ charges in eq. (\\ref{eq:e6kinmix}). ($q+xu$ corresponds to arbitrary superpositions of $Y$ and $B-L$, while $10 + x \\bar{5}$ corresponds to the $E_6$ models without kinetic mixing.) We have, therefore, normalized the fermions charges, $z_f$, in the $x$--parameter models to their $E_6$ values, $\\epsilon_f$. These charges are given in Table~\\ref{tab:x-charges}.  \n\\begin{table}[t]\n\\caption{$E_6$ fermion charges, $\\epsilon_f$, in terms of $z_f$, their values in the corresponding $x$--parameter models. Also given are the angles $\\alpha$ and $\\beta$ yielding these models.\\label{tab:x-charges}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n            &$q+xu$&$10+x\\bar{5}$&$d-xu$\\\\  \\hline\n$\\epsilon_f$&$\\frac{3}{2\\sqrt{7-2x+x^2}} z_f$\n            &$-\\frac{3}{2\\sqrt{4+x+x^2}} z_f$&$\\frac{3}{2\\sqrt{1-x+x^2}} z_f$ \\\\\n            \\hline\n$\\tan \\beta$& $0$  & $-\\sqrt{\\frac{3}{5}}(\\frac{3+x}{1-x})$ &\n               $\\frac{\\sqrt{3}(1-x)sign(5-x)}{\\sqrt{5-2x+5x^2}}$  \\\\   \\hline\n$\\tan \\alpha$&$\\sqrt{\\frac{3}{2}}(\\frac{1+x}{x-4})$&$0$ &\n              $2\\sqrt{6}(\\frac{x}{5-x})$ \\\\   \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{$Z'$ production at CDF and limits on its mass}\nThe total cross--section for the Drell--Yan (DY) process at a hadron collider, with a neutral gauge boson $B$ as the mediator and $\\mu^{+}\\mu^{-}$ as the outgoing particles, is given as~\\cite{Ellis:1991qj}\n\\begin{eqnarray}\n\\label{eq:lo3}\n\\sigma =\\frac{2}{s}\\int_0^{\\sqrt{s}}MdM\\sigma_{diff},\n\\end{eqnarray}\nwhere $M$ is the invariant mass of the muon pair, $\\sqrt{s}$ is the center--of--mass (CM) energy, and\n\\begin{eqnarray}\n\\label{eq:lo4}\n\\sigma_{diff}=\\int_{M^2\/s}^1 dx_{1}\\frac{1}{x_1}\n\\sum_{q} \\hat{\\sigma}(M^2)\\frac{K}{N_c}\\Big\\{f_{q}^{A}(x_{1},M^2)f_{\\bar{q}}^{B}(x_{2},M^2)+f_{\\bar{q}}^{A}(x_{1},M^2)f_{q}^{B}(x_{2},M^2)\\Big\\},\n\\end{eqnarray}\nwhere $N_c=3$ is the quark color factor. $x_1$ and $x_{2}(\\equiv\\frac{M^2}{x_1s})$ above are the momentum fractions of the ingoing partons having parton distribution functions (PDFs) $f_{q\/\\bar{q}}^{A\/B}$ for hadrons $A$ and $B$. $\\alpha_s$ is the strong coupling constant and $K=K_{C}+K_{E}$, with $K_{C}$ being the QCD $K$--factor and $K_{E}$ being the multiplicative factor due to QED corrections. $\\hat{\\sigma}(M^2)$ is equal to\n\\begin{eqnarray}\n\\label{eq:hard}\n\\int_{-1}^{1}d\\cos \\theta^* \\frac{1}{128\\pi M^2}\\Big[\n\\left(\\lvert A_{LL}\\rvert^2+\\lvert A_{RR}\\rvert^2\\right)(1+\\cos\\theta^*)^2+\\left(\\lvert A_{LR}\\rvert^2+\\lvert A_{RL}\\rvert^2\\right)(1-\\cos\\theta^*)^2\\Big],\n\\end{eqnarray}\nwith $\\theta^*$ being the polar angle defined in the CM frame and the individual amplitudes given as\n\\begin{eqnarray}\\label{eq:amplitud}\n A_{ij}= -Qe^2+\\frac{M^2}{M^2-M_{Z}^2+iM_Z\\Gamma_Z}C^{Z}_i(q)C^Z_j(l)\n+\\frac{M^2}{M^2-M_{Z'}^2+iM_{Z'}\\Gamma_{Z'}}C^{Z'}_i(q)C^{Z'}_j(l),\n\\end{eqnarray}\nwhere $i,j$ are run over $L,R$. $Q$ and $e$ are the electric charges of the contributing quark and the muon, respectively. $C^{Z,Z'}_{L,R}(f)\\equiv g_{1,2}\\epsilon_{L,R}^{Z,Z'}(f)$, with $g_1=e\/\\sin\\theta_w \\cos\\theta_w$ being the gauge coupling strength of the $Z$ boson, $g_2$ the $Z'$ coupling and $\\epsilon^{Z,Z'}_{L,R}$ the EW and $U(1)'$ charges of the fermion $f$. $\\Gamma_{Z,Z'}$ are the total decay widths of the $Z$ and $Z'$ bosons having masses $M_{Z,Z'}$.\n\nWe first follow the CDF analysis~\\cite{Aaltonen:2008ah} and use the LO expression given in eq. (\\ref{eq:lo3}) to calculate the cross--section due to the $Z'$ alone, neglecting the $\\gamma$ and $Z$ contributions to the amplitudes in eq. (\\ref{eq:amplitud}). We employ LO CTEQ6L PDFs \\cite{Pumplin:2002vw} and mass--dependent NNLO $K_C$~\\cite{Carena:2004xs} and NLO $K_E$~\\cite{Baur:1997wa} values, and assume that the $Z'$ decays into SM fermions only, which are taken to be massless. We achieve up to 99.8\\% agreement on the 95\\% confidence level (C.L.) $M_{Z'}$ lower limits with the CDF analysis, for the $E_6$ models included therein, and obtain new limits for the rest of the models. The numerical values of the limits are given in Table~\\ref{tab:limits} and the $\\alpha,\\beta$ parametrization of the models with contours in $M_{Z'}$ is plotted in Fig.~\\ref{fig:CDForig}. In Table~\\ref{tab:LHC} we give limits obtained for the LHC with a similar approach for some expected integrated luminosity and CM energy values.  \n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.38]{graficas-5abril.eps}\n\\end{center}\n\\vspace{-0.3cm}\n\\caption{Contours in $M_{Z'}$ limits mentioned on\/beside them. $x$ shows the location of a particular $E_6$ model within a contour. The dotted, dashed and dot--dashed lines correspond to the three $x$--parameter models. $Z_{\\eta^*}$ is a leptophobic boson which will not be observable in the di--lepton channels.}\n\\label{fig:CDForig}\n\\end{figure}\n\n\\begin{table}\n\\caption{Numerical limits in GeV on the mass of the $Z'$ boson in various models.}\n\\vspace{0.4cm}\n\\centering \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n$Z_\\chi$ & $Z_\\psi$ & $Z_\\eta$ & $Z_N$ & $Z_S$ & $Z_I$ & $Z_{B-L}$ & $Z_R$ & $Z_{LR}$ & $Z_{dph}$ & $Z_{string}$ &$Z_{SM}$\\\\\\hline\n895 & 883 & 910 & 865 & 823 & 790 & 1012 & 1006 & 959 & 1079 & 710 & 1030 \\\\\\hline\n\\end{tabular}\n\\label{tab:limits}\n\\end{table}\n\n\\begin{table}[t!]\n\\caption{Limits on $M_{Z'}$ in the $Z_\\chi$ model from the LHC.} \n\\vspace{0.4cm}\n\n\\centering \\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$\\mathcal{L}$ (fb$^{-1}$)\/$\\sqrt{s}$ (TeV)&3.5&7&14&28\\\\\\hline\n3&1.0&1.85&3.0&4.65\\\\\\hline\n30&1.5&2.5&4.1&6.7\\\\\\hline\n300&2.2&3.3&5.4&9.7\\\\\\hline\n3000&3.9&5.55&7.9&12.2\\\\\\hline\n\\end{tabular}\n\\label{tab:LHC}\n\\end{table}\n\n\\section{Bayesian statistical method}\nThe CDF analysis uses signal templates generated with a fixed resonance pole width, $\\Gamma = 2.8\\% \\times M_{Z'}$. However, there is no fundamental reason to only look for such a narrow $Z'$. A wide $Z'$ resonance, implying a strongly coupling boson, could well be scattered over a few bins and no significant enhancement above the background will be visible. Besides, the effects of interference between the various bosonic contributions to the propagator, i.e., between $\\gamma$, $Z$ and $Z'$ (see eq. (\\ref{eq:amplitud})), are lost in their approach. These effects could in principle cause a considerable enhancement or dip in the number of events in several accompanying low $M^{-1}$ bins, e.g., in the case of a strongly interacting $Z'$ boson with mass just beyond the kinematic reach of the CDF. Finally, the CDF limits assume a fixed GUT--based $g_2$ and it is not straightforward to extend the limits to other $g_2$ values, particularly in the strong coupling regime (see, however~\\cite{CDF2010}).\n\nTherefore, we propose a Bayesian statistical method which allows one to vary $g_2$ in order to obtain the corresponding limit on $M_{Z'}$. It is based on the likelihood function $\\L$, written as\n\\begin{eqnarray}\n\\label{eq:likelihood}\n\\L(\\vec{\\mu}\\lvert \\vec{n})=\\Pi_i^B P(n_i\\lvert \\mu_i),\n\\end{eqnarray}\nwhere $P$ is the Poisson probability of finding $n_i$ events given $\\mu_i$ expected events in the $i$th bin with $B$ total bins. $\\L$ in eq. (\\ref{eq:likelihood}) is then evaluated using two hypotheses: the null hypothesis, $\\L(\\vec{\\mu^b}\\lvert \\vec{n})$, assumes that the DY process occurs only via $\\gamma$ and $Z$, and the signal hypothesis, $\\L(\\vec{\\mu^t}\\lvert \\vec{n})$, with $\\vec{\\mu^t}=\\vec{\\mu^b}+\\vec{\\mu'}$, wherein the $Z'$ boson also contributes to the cross--section along with the SM gauge bosons. The SM events, $\\mu^b$, expected in an invariant mass bin are calculated as \n\\begin{eqnarray}\n\\label{eq:nbin}\n\\mu^b = \\mathcal{L}\\sigma_{SM} = \\frac{2E\\mathcal{L}}{s} \\int_{bin} AdM_p^{-1} \\int^{\\sqrt{s}}_0 MdM \\textbf{p}(M^{-1}_p|M^{-1})\\sigma_{diff} + n_{\\text{BG}}, \n\\end{eqnarray} \nwhere $\\mathcal{L}=2.3$ fb$^{-1}$ is the integrated luminosity at the CDF, $E=0.982$ is the detector {\\it efficiency} and $A$ is the CDF {\\it acceptance}, which is a mass--dependent multiplicative factor. $M_p$ in the above equation is the muon--pair mass measured by the detector, $n_{\\text{BG}}$ refers to the non--DY events and the probability density is given as\n\\begin{eqnarray}\n\\textbf{p}(M^{-1}_p|M^{-1}) = M_pb^a e^{-b}\/\\Gamma(a),\n\\end{eqnarray}\nwith $a=(M^{-1}\/\\Delta)^2$, $b=M^{-1}M_p^{-1}\/\\Delta^2$, where $\\Delta = 0.17$ TeV$^{-1}$ is the variance. The purpose of the above probability function is to smear over the DY background before distributing it into bins, hence accounting for the mis--identification of an event in a bin where it does not actually belong. For $\\mu'$, a $\\chi^2$ function is constructed as \n\\begin{eqnarray}\n\\chi_{\\mu'}^2 = -2LLR = -2\\ln(\\frac{\\L(\\vec{\\mu^t}\\lvert \\vec{n})}{\\L(\\vec{\\mu^b}\\lvert \\vec{n})})= -2\\sum_i^B(\\mu_i^b-\\mu_i^t + n_i\\ln(\\frac{\\mu_i^t}{\\mu_i^b}),\n\\end{eqnarray}\nand is minimized to obtain the best--fit values in the given range of $g_2$ and $M_{Z'}$, which correspond to a $Z'$ boson with cross--section $\\mu'$ best favored by the data. The contours in $g_2$ and $M_{Z'}$, for a certain C.L. value specified by the allowed number of standard deviations, $\\Delta\\chi^2$, from the minimum, can then also be drawn, giving the exclusion limits on these parameters. Our preliminary contours are given in Fig. \\ref{fig:CDFgrid} for $Z_{\\chi}$ as a representative model. The numerical value of the 95\\% C.L. limit is 913 GeV for $g_2=0.461$, which is about 21 GeV higher than the CDF value.\nWe next plan to undertake a global analysis including constraints from electroweak precision data~\\cite{Erler:2009jh}. Eventually, a similar statistical analysis of the LHC data, as soon as it is released, will also be performed. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.37]{plot1-3.eps}\n\\end{center}\n\\vspace{-0.3cm}\n\\caption{Exclusion contours for the $Z_\\chi$ boson. The dotted, dot--dashed and double--dot--dashed curves correspond to the C.L. values given and the solid curve represents the CDF limits generalized to other $g_2$ values.}\n\\label{fig:CDFgrid}\n\\end{figure}\n\n\\section*{Acknowledgments}\nThe work at IF-UNAM is supported by CONACyT project 82291--F. The work of P.L.~is supported by an IBM Einstein Fellowship and by NSF grant PHY--0969448.\n\n\\newpage\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Punch, or the London Charivari, Malcolm Farmer,\nNigel Blower and the Online Distributed Proofreading Team\nat http:\/\/www.pgdp.net\n\n\n\n\n\nPUNCH, OR THE LONDON CHARIVARI.\n\nVOL. 93.\n\n\n\nOctober 29th, 1887.\n\n\n\n\nQUITE A LITTLE HOLIDAY.\n\nEXTRACT FROM A GRAND OLD DIARY. MONDAY, OCT. 17.\n\nSelf, wife, and HERBERT started early to escape our kind-hearted,\nclear-headed admirers; so early, that I scarcely had time before\nleaving to write thirty post-cards, seventy-six pages of notes for\nmy next magazine article, and to cut down half-a-dozen trees. Train\nannounced to leave Chester at 10:30, but got off at the hour.\nThis little joke (WATKIN'S notion) caused much amusement. Through\nopera-glasses we could see bands of music, deputations, &c.,\nconstantly coming to the railway-stations to meet our train after\nit had passed. Too bad! However, to prevent disappointment, and as\nCHAMBERLAIN has been imitating me and vulgarised my original idea, I\nknocked off some speeches, in pencil, and HERBERT threw them out of\nthe window as fast as I could write them. So far as we could make out\nwith a telescope, some of them reached their destination, and seemed\nto be well received.\n\n[Illustration: Master Willie Gladstone \"really enjoying, and in some\nmeasure appreciating and understanding,\" our Mr. Agnew's lectures on\nArt.\n\n_Vide Times Report, Oct. 18._]\n\nAwfully pleased to meet Mr. WILLIAM AGNEW at Manchester. Odd\ncoincidence of Christian names. I shall speak of him and allude to him\nas \"The Other WILLIAM.\" He promised to keep by me, and show me all the\npictures worth seeing.\n\n\"T'Other WILLIAM,\" said I, \"you are very good. As you know, I take a\ngreat and sincere interest in pictures and works of Art, although I\nknow very little about them.\" T'Other WILLIAM protested. \"No, T'Other\nWILLIAM, I am right. You have been the means of providing me with\na commodity most difficult of all others to procure if you do not\npossess it yourself--that is to say, you have provided me with\nbrains.\" Further protests from T'Other One. \"No, T'Other WILLIAM,\nhear me out; for you know in all cases where a judgment has had to be\npassed upon works of Art, I have been accustomed to refer a great deal\nto you, and lean upon you, because you have been constantly the means\nof enabling me really to see, and really to enjoy, and in some measure\nto appreciate and understand, all that you have shown to me.\"\n\nI was so pleased with this little speech that I made HERBERT take it\ndown as I repeated it to him privately when T'Other was looking in\nanother direction. When I brought it out afterwards, at luncheon in\nthe Palm-house, it went wonderfully. So it should, because I felt\nevery word of it. T'Other WILLIAM is one of the kindest and most\ncourteous of my friends.\n\nI was very pleased with the Exhibition, although perhaps (I am not\ncertain of this) I might have seen it better had not about four\nthousand visitors followed our little party everywhere, cheering\nvociferously. I was consequently obliged to keep my attention most\ncarefully fixed upon the exhibits, as when I caught any stranger's\neye, the stranger immediately (but with an eagerness that did not\nexceed the limits of good behaviour) called upon me to make a speech\nthen and there upon the subject of \"Home Rule.\" I am sure I should\non each and every occasion have only been too delighted, had not Sir\nANDREW warned me not to indulge too much in that sort of thing. The\ncrowd, however, had its decided advantage, inasmuch as we were carried\noff our feet everywhere. In this luxurious fashion we were wafted to\nMessrs. DOULTON'S Pottery Manufactory, to Mr. JESSE HAWORTH'S loan\nexhibition of Egyptian antiquities, the name \"JESSE\" recalled to me\nthe poor misguided JOE'S \"JESSE,\" the second fiddle, but _toujours\nfidele_, and to a great many other shows of almost equal interest.\n\nBut of course _the_ feature of the Exhibition was the collection\nof pictures. I was absolutely delighted. T'Other WILLIAM explained\neverything, and amongst other portraits showed me one of myself by\nMILLAIS. I imagine that everybody must have thought it very like,\nbecause when they observed me inspecting it, they cheered more\nvigorously than ever. For my part I can't help feeling that Sir JOHN\nmight have done more with the collars. He has not (to my thinking,\nalthough I confess I may be wrong) put quite enough starch in them.\nThis is my own idea, as I did not consult T'Other One upon the\nsubject. Great as my reliance is upon him concerning works of Art, I\nreserve the right of using my own judgment in the matter of collars.\nPassing through the galleries I was delighted with everything I saw.\nThe only drawback to my pleasure was the fact that I was followed (as\nI have already hinted) by a cheering crowd, who occasionally, and, no\ndoubt, accidentally, drowned the voice of my kind Mentor. Under other\ncircumstances I should have drawn the distinction between the Mentor\nand the Tor-mentors. Think this, but don't say it. For instance, when\nwe were standing in front of \"_Ramsgate Sands_,\" this is what reached\nmy ears eager for instruction:--\n\n\"'_Ramsgate Sands_,' by FRITH--(_'Hooray!'_)--who, as you know, has\njust written--(_'Speech! Speech!' 'Home Rule!' 'Three cheers for\nMORLEY!'_)--full of anecdotes of all sorts of interesting people. If\nyou went to Ramsgate now, you would find----(_'We are going to give\nyou another carpet, old man!' 'Hooray, hooray, hooray!' 'Three\nCheers for Home Rule!--An extra one for Manchester!'_)--and\npractically the sand-frequenters we are carefully examining in this\npicture are of thirty years ago. (_'Speech! Speech!'_) You must\nknow----(_'Hooray, hooray, hooray!'_)\"\n\nAnd at this period my dear friend was silenced by our being carried\naway in an irresistible stream to the Palm-house, where we took part\nin an excellent luncheon. Here I delivered my speech, which I\npride myself was first-rate. I called Manchester the Modern Athens,\nexplaining, however, that no offence was intended to the capital\nof Midlothian. Take it all round, then, in spite of the \"exuberant\ninterest\" shown in me by my fellow-citizens, I have had a very\npleasant day, thanks chiefly to T'Other WILLIAM.\n\n       *       *       *       *       *\n\nA PROGRESSIVE PROGRAMME.\n\n_October 25._--Lecture by amiable Police Magistrate to six hulking\nrowdies, who have been assaulting the Police, on the duty of \"bearing\ndistress patiently.\" Tells them \"not to do it again,\" and dismisses\nthem with aid from the Poor Box and his blessing. Surprise of rowdies.\n\n_October 26._--Unemployed employ themselves in sacking portion of Bond\nStreet, during temporary withdrawal of Police for a little rest.\n\n_October 27._--Sitting Alderman at Mansion House gives a Socialist\nDeputation some sympathetic and fatherly advice, and recommends them\nto \"study laws of supply and demand.\" Invites them to Lord Mayor's\nBanquet. Deputation accepts invitation readily, and, on emerging\ninto street, is chivied down Cheapside by infuriated mob of other\nSocialists, who have not received invitations.\n\n_October 28._--New Leaders of Mob (_vice_ Deputation, resigned)\ndenounce sympathetic Alderman as a \"bloated exploiter.\" Nelson\nMonument pulled down. Ten leading tradesmen, in neighbourhood of\nTrafalgar Square, unable to do any business, owing to streets being\nblocked with rioters, go into bankruptcy.\n\n_October 29._--Gathering of \"Unemployed\" in Westminster Abbey.\nUnemployed complain bitterly because chairs have no cushions. The\nDean, conducted to pulpit under strong police escort, preaches very\nconciliatory sermon on duty of Upper Classes, all, except Deans, to\ngive most of what they possess to poor; advises poor to wait patiently\ntill they get it. Retires under heavy shower of hymn-books. Unemployed\n\"remain to prey.\"\n\n_October 30._--Westminster Abbey sacked, in consequence of Dean's\nconciliatory sermon. The Canons go off.\n\n_November 1._--Mansion House Relief Fund started. Fifty thousand\npounds subscribed the first day by leading philanthropists who\nhave had all their windows broken. Trade paralysed, and numbers of\nUnemployed consequently increasing. Speech by celebrated Statesman,\ncontrasting disorder and lawlessness in Ireland with universal\ncontentment and order existing in England.\n\n_November 2._--Mob helps itself to chief pictures in National Gallery,\non ground that they \"belong to the people.\" Raffle organised for the\nRaffaelles. Fifteen policemen have their ribs broken.\n\n_November 3._--Whole Police Force disabled by angry mob armed with\nbludgeons and revolvers. Sympathetic Alderman at Mansion House\nventures to ask Government if \"matters are not really going a little\ntoo far,\" and is ducked in Thames. All the West-End shops in-wested by\nlooters.\n\n_November 4._--Prime Minister declares that \"much as he regrets the\ndepression of trade and want of employment, yet he thinks that on the\nwhole, recent proceedings have not been quite creditable to Capital\nCity of Empire.\" Military called out, and streets cleared in no time.\nRingleaders of mob arrested, and given a year's imprisonment with hard\nlabour. Trafalgar Square railed round and planted with prickly cactus.\nBusiness resumed and confidence restored. Government begins to think\nof a Bill to deal with _real_ London grievances--such as rack-rents,\nslum-dwellings, and foreign pauper labour.      [_And high time too!_\n\n       *       *       *       *       *\n\nA CLOUD OF YACHTS.--The account of the British owner published last\nweek, confirms the notion that the much-talked-of superiority of\nthe _Thistle_ over the _Volunteer_ was mere vapouring. This is not\nsurprising. All that could be appropriately expected from such a weed\nwas smoke!\n\n       *       *       *       *       *\n\nMR. PUNCH'S PARALLELS. No. 3.\n\n[Illustration: DON CHAMBERLAIN QUIXOTE AND SANCHO JESSE PANZA.\n\n_Sancho Panza (to himself)._ \"I CANNOT HELP IT,--FOLLOW HIM I MUST:\nI HAVE EATEN HIS BREAD, I LOVE HIM: ABOVE ALL I AM FAITHFUL.\"--_Don\nQuixote_, Part ii., Book iii., Ch. xxxiii.]\n\n       *       *       *       *       *\n\nTHE NEW QUIXOTE.\n\n_Fragments from a forthcoming Romance of (Political) Chivalry and\n(Party) Knight-Errantry._\n\n       ***\n\nThe age of our gentleman bordered upon fifty years. He was of a strong\nconstitution, spare-bodied, of a keen, not to say hatchet-like visage,\na very early (and rapid) riser, and a lover of the orchid.\n\n       ***\n\nHis judgment being somewhat obscured, he was seized with one of the\nstrangest fancies that ever entered the head of any naturally astute\nperson. This was a belief that it behoved him, as well for the\nadvancement of his own glory as the service of his country, to become\na knight-errant (though, indeed, there was, perhaps, about him more\nof the errant than the knightly), and traverse the northern parts of\nHibernia, armed and mounted, in quest of adventures, redressing every\nspecies of grievance save such as were not found in his own list, or\n\"programme,\" which latter, indeed, he would by no means admit to be\n\"grievances\" at all. The poor gentleman imagined himself to be at\nleast crowned Autocrat of Orangeia by the valour of his arm; and\nthus wrapt in these agreeable illusions, and borne away by the\nextraordinary pleasure he found in them, he hastened to put his design\ninto execution.\n\nThe first thing he did was to scour up some rusty armour which had\ndone service in the time of his great-grandfather, and had lain many\nyears neglected in a corner. This he cleaned and furbished up as well\nas he could, but he found one great defect--it would not in any part\nstand one stroke from modern steel, much less one shot from modern\ngun. However, as he was rather fired with the yearning to attack than\nimpressed with the necessity for defence, this deficiency troubled him\nbut little.\n\nIn the next place he visited his steed, which though but a hobby of\nwooden aspect and no paces, yet in his eyes it surpassed any charger\nthat the Achilles of Hawarden ever bestrode, or the Automedon of Derby\never handled. Many days was he deliberating upon what name he should\ngive it; for, as he said to himself, it would be very improper that\na horse so excellent appertaining to a Knight so famous should be\nwithout an appropriate name; he therefore endeavoured to find one that\nshould express what he had been before he belonged to a knight-errant,\nand also what he now was; nothing could, indeed, be more reasonable\nthan that, when the master changed his state, the horse should\nlikewise change his name, and assume one pompous and high-sounding, as\nbecame the new order he now professed. Failing in this endeavour, he\ncalled his hobby, provisionally at least, _Ne Plus Ulster_, a name\nwhich if it suggested a sorry joke, was so far fitting that it was\nbestowed upon a sorry nag.\n\n       ***\n\nIn the meantime our knight-errant had brought his persuasive powers to\nbear upon a humble labourer in the fields which he himself had lately\nleft, a neighbour of his, some said of his own distant kin, and an\nhonest man, but somewhat shallow-brained and self-important. In short,\nhe said so much, used so many arguments, that the poor fellow resolved\nto sally out with him, and serve him in the capacity of a Squire.\nAmong other things, DON QUIXOTE told him that he ought to be very glad\nto accompany him, for such an adventure might some time or the other\noccur, that, by one stroke, an Island might be won, where it was\nwithin the bounds of possibility that he, the Squire, might one day\nbecome Governor, or at least Viceroy. With this and other promises\nSANCHO PANZA (for that was the rustic's name) left his well-beloved\nthree acres at home, not to name a favourite cow, for a time at least,\nand engaged himself as Squire to his ambitious neighbour.\n\n       ***\n\nEngaged in friendly discourse, they came in sight of eighty-five or\neighty-six windmills; and as DON QUIXOTE espied them he said to\nhis Squire, \"Fortune favours us. Look yonder, friend JESSE--I mean\nSANCHO--where thou mayest discover some more than eighty disloyal\ngiants, and monsters of sedition, whom I intend to encounter and\nslay.\" \"What giants?\" said SANCHO PANZA. \"These thou seest yonder,\"\nanswered his master, \"with their long and far-reaching arms, for some\nare wont to have them of the full length of a league. Fly not, ye\ncowards, and vile caitiffs!\" he cried, \"for it is a single Knight\nwho assaults ye! Although ye should have more arms than the giant\nBriareus, ye shall pay for it!\"\n\n       ***\n\nAnd the story, so far as it has gone (it is \"to be continued\"),\nleaves DON QUIXOTE making a prodigiously plucky assault upon the\nLeague-limbed \"giants,\" with what result the sequel will show.\n\n       *       *       *       *       *\n\n[Illustration: TORSION.\n\n_Irish Waiter (to Bow-legged Traveller in the Coffee-room)._ \"BIG\nPARDON, SOR. HADN'T YOUR HONOUR BETTER MOVE A LITTLE FURTHER FROM THE\nFOIRE?\"\n\n_Traveller (fiercely)._ \"EH? WHA' FOR? WHA' D'YE MEAN?!\"\n\n_Irish Waiter._ \"OCH SHURE, SOR, YER LEGS IS WARPIN'!--OCH! PHEW! MOST\nTURRIBLE!\"]\n\n       *       *       *       *       *\n\nTO A LADY DENTIST.\n\n  [It is announced that Ladies are to be enabled to take\n  diplomas in Dentistry.]\n\n  Lady Dentist, dear thou art,\n  Thou hast stolen all my heart;\n  Take too, I shall not repine,\n  Modest molars such as mine;\n  Draw them at thine own sweet will;\n  Pain can come not from thy skill.\n\n  Lady Dentist, fair to see,\n  Are the forceps held by thee;\n  Lest those pretty lips should pout,\n  You may pull my eye-teeth out;\n  I'm regardless of the pangs,\n  When thy hand extracts the fangs.\n\n  Lady Dentist, hear me pray\n  Thou wilt visit me each day;\n  Welcome is the hand that comes--\n  Lightly hovering o'er my gums.\n  Not a throne, love, could compare\n  With thine operating chair.\n\n  Lady Dentist, when in sooth\n  You've extracted every tooth,\n  Take me toothless to your arms,\n  For the future will have charms:\n  Artificial teeth shall be--\n  Work for you and joy for me!\n\n       *       *       *       *       *\n\nALL THE DIFFERENCE.--The Statesmen used to be called \"Pillars of the\nState.\" _Pillars!_ They now seem to contribute to its support little\nbut endless (newspaper) _columns_!\n\n       *       *       *       *       *\n\nTHE LETTER-BAG OF TOBY, M.P.\n\nFROM A HOODED EAGLE.\n\n                                            _H-tf-ld House, Friday._\n\n[Illustration:]\n\nDEAR TOBY,\n\nAfter a too brief holiday I am back again to H-tf-ld and to L-nd-n,\nand take an early opportunity of dropping you a line. I call the\ninterval since the House was up a holiday for convenience sake; but\nwhat with the daily arrival of despatch boxes and the delivery of the\nmorning papers, the repose has been intermittent. I fancy that since\nthe days of Old PAM the recess has always been a mockery for the\nPremier of the day. D-ZZY had some bad times from 1874 to 1880, and\nGL-DST-NE'S subsequent Premiership was not a bed of roses, even in the\nrecess. But they at least had the satisfaction of feeling that they\nwere in power as well as in office. If they decided upon a particular\nline of policy, they could initiate it without first inquiring how it\nmight suit half-a-dozen people. Moreover, each was in varying degree\nsupported by capable colleagues, able to hold their own on the\nplatform or in the House. For unhappy Me things are quite otherwise.\nI may devise a policy for Ireland and elsewhere, but before I can\nannounce it, I must humbly learn how it suits my Lord H-RT-NGT-N and\nmy good friend CH-MB-RL-N. As for my colleagues and the help I receive\nfrom them----well, that is a matter of which of course I cannot write,\neven in the confidence of correspondence with you. But I may tell you\nthat over at Chalet C-c-l I found some little time for reading other\nliterature than Blue Books. Looking through SHELLEY once again, I\ncame upon the line descriptive of COLERIDGE, \"flagging wearily through\ndarkness and despair,\"\n\n  \"A hooded eagle among blinking owls.\"\n\nI don't exactly know why, but when I think of some things that have\ntaken place lately, I have a strong feeling of personal sympathy with\nthe hooded eagle.\n\nBut this is a trifle melancholy, and will make you think I am in\nlow spirits, or even that there is truth in the newspaper rumours\nof failing health. Nothing of the sort, dear boy; never better in my\nlife. Full of health and spirits, of hope for the coming time, and\neagerness for the fray of next Session. How I have envied GL-DST-NE\ngoing about the country making speeches which would have been twice as\neffective if they had been half as long, receiving the homage of the\nmasses, and driving in state through the streets of Derby, with his\nled Captain, H-RC-RT, on the box-seat of his carriage! What a curious\nman is GL-DST-NE, the Elephant of our political life, who can in the\nmorning crush a Ministry, and in the afternoon achieve a petty economy\nby selling waste timber. There has been a good deal written about\nNAPOLEON whilst involved in his fatal campaign in Russia occupying\nspare moments in drawing up regulations for the Opera House at Paris.\nBut what is that compared with GL-DST-NE marching through the Midlands\nto upset my Government, and, _en route_, drafting an announcement\nthat timber felled at Hawarden by his own hand would be on sale \"at a\nuniform charge, viz., 1s., 6d. for a small log, or 3s. per cubic foot,\nexclusive of railway carriage.\" Of course I know that WILLIAM HENRY\nhas gallantly rushed into the breach, and avowed the authorship of\nthis remarkable proclamation. But if W. H. is allowed to do this kind\nof thing without consultation or authority, all I can say is that\ndiscipline at Hawarden is fatally faulty. Besides, amiable and\nengaging as he is, I do not believe that W. H. is equal to the\nunassisted concoction of this incomparable production. However it be,\nno one but GL-DST-NE could stand the ridicule of the thing, and he\ndoubtless doesn't feel it.\n\nHow is GR-ND-LPH getting on? Not so well as he used, I fancy. His new\nattitude of friendly neutrality does not suit him, and is, moreover,\nnot nearly so attractive with the people as what I may call his\nMalayan manner, when he used to run amuck at everybody, including\nmyself. It was a very dull speech he made at Sunderland on Thursday.\nHe must certainly wake up, if he means to keep his old place. Perhaps\nhe is, like me, getting aweary of the whole thing, and wishes he were\nwell out of it. If I had my will, I would cut the whole business, and\nspend my days and nights in the laboratory here. But that cannot be,\nfor the present at least. So you will hear from me soon in the midst\nof the fray; and, in the meantime, mind you understand that I am in\nthe best of spirits, confident in the present, and hopeful for the\nfuture.\n\n  Yours, faithfully,       S-L-SB-RY.\n\n       *       *       *       *       *\n\n\"COLD ID BY DOZE.\"\n\n[Illustration]\n\n  I've got such a hoddible cold id by head,\n  Upod by word, I wish I was dead;\n  I really thig I shall go to bed,\n  Ad tallow by doze, as the Doctor said;\n  He's cubig agaid this afterdood;\n  Why, it's half-past three, he'll be here sood,\n  Ad gib me sub bore of his beastly drugs,\n  Ad tell me to keep warb udder the rugs.\n            Achoo! Achoo!\n            Oh! what shall I do?\n  I've coughed ad sdeezed till I'be dearly blue,\n            Ad by doze is so sore,\n            I card blow it bore,\n  It feels as tedder as if it was raw;\n  Subbody told be he'd heard of sub stuff\n  Which you'd odely to sdiff, ad that was eduff;\n  What did he call it? Alkarab,\n  I'll sedd for sub--I suppose it's a shab--\n  They always are. Achoo! Achoo!\n  I thig I'be dyig! Oh! what shall I do?\n  Yes, this is the stuff that fellow said\n  Was sure to cure a cold id the head;\n  Two or three sdiffs the beggar swore\n  Would bake you as well as you were before.\n  (_He sniffs._) Upod my soul, I believe he's right,\n  I'be gettig better--it's wonderful quite,\n  I albost feel as if I bight\n  Go out and dide at the Club to-dight.\n\n  (_He continueth sniffing._)\n\n  I really will, I feel quite well,\n  As fresh as a rose, and as sound as a bell,\n  And I'll always swear that the only balm\n  For a cold in the head is Alkaram.\n  \"Here, JOHN, put out my evening clothes.\"\n            I'll take my grub\n            To-night at the Club.\n  Soup, fish, and a bird, with a pint of Larose,\n  I think that ought to complete the cure,\n  And make assurance double sure.\n            Achoo! Hullo!\n            Why here's a go!\n  Achoo! Atishoo! Oh dear! Oh dear!\n  It's all begiddig agaid, I fear;\n  You card get rid of a cold like bide\n  By sbellig a bottle of bedicide!\n            Soup ad fish! it's absurd,\n            Or to thigk of a bird,\n  When you card prodoudce a siggle word,\n  Ad as for Larose, the tipple for be\n  Is a cup of boilig lidseed tea.\n            I'll go to bed,\n            Ad wrap a red\n  Welsh fladdel baddage roud by head,\n  Ad stay at hobe for a budth at least,\n  Till this beastly widd's do logger East.\n\n  _South Kedsigtod._\n\n       *       *       *       *       *\n\nPRO BONO PUBLICO.\n\n[Illustration]\n\nA Mob-Cap was once upon a time a picturesque finish to a pretty face,\nand it was of home-manufacture. Now the Mob-Cap is a red abomination,\ntypical of bloodshed and crime, of foreign make, and is mis-called the\nCap of Liberty, which, properly translated, is the Cap of Licence. It\ncertainly is not \"The Cap of Maintenance,\" as it is adopted by those\nwho would disdain work, even if it were offered them.\n\nNot for the first time has _Mr. Punch_ raised his voice against Street\nProcessions, which have developed into one of the greatest nuisances\nof the present time, destructive of trade, detrimental to every kind\nof regular business, and a disgrace to our orderly and respectable\nLondon. All processions in London ought to be prohibited, with the\nexception of such State, Civic, or Ecclesiastical processions as may\nbe deemed essential to the dignity of authority, and which have been,\nand still are, a source of real pleasure to the Londoners, who dearly\nlove a show, when there is due and proper occasion for it.\n\nIf the Salvationist Army processions, with their tambourines, drums,\nand inharmonious bands, are permitted on Sunday (which English people\nwere wont to observe in peace and quietness), then consistently a\nSocialist procession must be allowed. And what other processions?\nFreemasons, Religious Guilds, Clubs,--why should not the members of\nthe Reform, the Athenaeum, the Conservative, the National Liberal,\norganise processions? Why not the Garrick Club, headed by Mr. HENRY\nIRVING and Friend TOOLE, with banners emblazoned with playbills? No.\n\"Reform it altogether.\"\n\nAnd as to the liberty of out-of-door public meetings. Let Trafalgar\nSquare be explicitly forbidden to these mischievous anarchists,\nof whom the majority are the dupes and tools of firebrand foreign\nCommunists. Let certain places be allotted to them for \"airing their\ngrievances,\" and let each of these places be at least four miles\ndistant from Charing-Cross. Our Parks are the \"Lungs of London,\" and\nif these Lungs be congested, the health of London will materially\nsuffer. How many hundreds are now prevented from entering the Parks by\nthe fear of King Mob and his rabble rout? Children and nursery-maids\ndare not take their recreation in our Parks. Think of that, ye\nPrivates of the Cavalry and Infantry, and to a man you will be the\nfirst to declare for the freedom of the Parks. Let one of the first\nenactments of the next Session be a Bill to Regulate Processions and\nOut-of-door Meetings. Let it be a liberal measure--in the true sense\nof liberal; that is, showing due consideration for everybody--and let\nit come into operation as soon as possible.\n\n  PUNCH\n\n       *       *       *       *       *\n\nKNIGHT THOUGHTS.\n\nSir HENRY KNIGHT seems to be of opinion that luxurious living,\nAldermanic and otherwise, must be a good thing for the poor, because\n\"Money spent in entertainment goes into the pockets of the working\nclasses.\" If that is so, Dives, in order to benefit Lazarus, can\nhardly do better than go on faring sumptuously every day. And yet\nsomehow, as a matter of fact, the more Dives feeds the more Lazarus\nfamishes. How is this, O Knight of the Round (Dinner) Table?\n\n\"Neither luxury, nor anything else,\" says the philosophical ex-Lord\nMayor, \"can be indulged in without purchasing the materials which\ncontribute to or from which the luxury is obtained.\" _Argal_, the\nmore luxury among the rich the more money in the pockets of the poor.\nCheering thought!--for civic _gourmands_ and fashionable fine ladies!\nDid not a great financier once suggest that England, which fought\nitself into debt, might drink itself out of it? Here seems to be a\nchance of eating ourselves out of poverty, of dining ourselves out of\ndestitution. Are there any real \"Unemployed\" about? Let those who have\nmoney spend more of it in \"entertainments\" and the problem is solved\nwithout recourse to Mansion House Funds, Public Works, Eight Hour\nMovements, or other schemes philanthropical or revolutionary.\n\nKNIGHT'S panacea for poverty, this proposal to cure it by\n\"entertainment,\" is certainly, in one sense, entertaining. But it is\nto be feared that it can hardly be entertained.\n\n       *       *       *       *       *\n\nOUR ADVERTISERS.\n\nINVERTED DOMESTIC AND OTHER.\n\nA GOOD PLAIN MISTRESS WANTED by a competent and highly experienced\nCook. Must be a thorough lady, accustomed to making herself generally\nagreeable, and to not prying into household matters which do not\nconcern her. She will not be expected to visit her own kitchen,\ninquire into the amount of her own weekly books, keep the key of\nthe beer, or object to the occasional visits of members of the local\nPolice Force, in which the advertiser has several near relatives. A\nlittle dinner on a small scale now and then will not be objected\nto, but seeing much company cannot for a moment be entertained. An\nunexceptionable character from the three last cooks who have filled\nthe place, indispensable. Apply, M.B. Eligible Family Supply Agency,\nWalker Street, W.\n\n       ***\n\nTRAVELLING NOBLEMAN WANTED. A Courier who has a slight acquaintance\nwith the French and German languages, and wishes to air them in the\ncourse of a pleasant and enjoyable little outing, is desirous\nof meeting with a well-recommended aristocrat of unquestionable\nantecedents, who wishes to visit the leading towns of the Continent\nin thoroughly first-class style. The advertiser, who would select the\nroutes, generally direct the character of the tour, and expect to have\ncharge of the cheque-book, would stipulate that under no circumstances\nshould any question be raised on the score of expense. None but\nNoblemen of a confiding disposition, that can be vouched for by\ntestimonials from their near relatives, need apply. Communicate with\nA. X., Eligible Family Supply Agency, Walker Street, W.\n\n       ***\n\nA REAL GENTLEMAN, who isn't too particular, wanted immediately by a\nCoachman, who will, when sober, undertake to drive his carriage and\npair for him anywhere he likes about the Metropolis, and beyond,\nwithout smashing him up. Mustn't be hasty and close over stable\nexpenses. Any quiet old duffer, who has been accustomed to let things\ngo their own way without interfering, preferred. Apply to JEHU,\nEligible Family Supply Agency, Walker Street, W.\n\n       ***\n\nA LADY OF TITLE WANTED by A COMPANION who would undertake to offer her\nSociety in consideration of sharing the carriage, home, recreations,\npleasures, friends, and general social _entourage_ of her employer.\nAs the Advertiser has for some years figured prominently as a garrison\nhack, and has been somewhat blown upon in consequence, she will not be\ntoo particular as to the character of the particular \"Set\" into which\nher new surroundings may introduce her; but as she has, by outliving\nher income, already run through the little money she possessed, she\nwill expect a salary of not less than L100 a year, to enable her to\ndress up to the false position she has in contemplation to occupy.\nNo recognised old Dowagers, who live a quiet and retired life, need\nanswer this Advertisement. No references expected or offered. N. W.,\nEligible Family Agency, Walker Street, W.\n\n       ***\n\nSOFT-HEADED NOBLEMAN OR GENTLEMAN wanted by a shrewd, shifty, pushing,\nout-at-elbows Adventurer, desirous of filling the post of Private\nSecretary, and so worming himself into an assured position of intimate\nfamily confidence. Would suit a Duke threatened with incipient\nparalysis. Apply, DIPLOMATICUS, Eligible Family Supply Agency, Walker\nStreet, W.\n\n       ***\n\nCHEERFUL AND WILLING MISTRESS WANTED by an Under-Housemaid who wears\na fringe and latest form of Dress-Improver, and considers herself\ngenerally attractive. State number of Men Servants, and furnish\nparticulars of the sort of society that may be expected down-stairs.\nAdvertiser will expect to receive her own friends on the afternoons of\nnot less than three days in each week. Mistress may refer to servants\nat present staying in house, who can speak favourably as to her\ncharacter. Apply, HILDA, Eligible Family Supply Agency, Walker Street,\nW.\n\n       ***\n\nUSEFUL AND ACTIVE MISTRESS REQUIRED by a General Servant who will\nexpect her to do her fair share of the work. Master must clean the\nwindows and his own boots, and as advertiser is not an early riser,\nget up when necessary, and let in the sweeps. Entire Sundays expected\nout and no interference with visits of the Marine Store Dealer.\nCharacter Mutual. S. S. S., Eligible Family Supply Agency, Walker\nStreet, W.\n\n       ***\n\nTHE ELIGIBLE FAMILY SUPPLY AGENCY undertake to provide exacting and\nparticular modern Domestics with thoroughly satisfactory Masters and\nMistresses.\n\n       ***\n\nTHE ELIGIBLE FAMILY SUPPLY AGENCY have at the present moment\napplications from several Invalid Gentlemen who require care and\nsolicitude, and will be glad to hear from Widows with an eye to the\nmain chance, and \"Superior\" Housekeepers desirous of getting hold of\nan unquestionably good thing.\n\n       *       *       *       *       *\n\n[Illustration: HAPPY THOUGHT.\n\n_Jones (of Hampstead)._ \"THIS IS ONE OF OUR CELEBRATED PONDS. YOU'VE\nHEARD OF THEM, EH, GRIGSBY?\"\n\n_Grigsby (who has never been to Hampstead before)._ \"_HEARD_ OF 'EM? I\nSHOULD THINK SO--EVER SINCE I WAS A BOY! WHY, THE _PONDS ASINORUM_, OF\nCOURSE!\"]\n\n       *       *       *       *       *\n\nTHE TWO VOICES.\n\n    \"That this representative body of Working-men, representing\n    the _bona fide_ Unemployed Workmen of the East and South-East\n    of London, beg to place on record their entire want of\n    sympathy, and their utter condemnation of the recent\n    conduct which has been made in the name of the\n    Unemployed.\"--_Resolution passed at a Meeting of\n    Representative Workmen, held in Whitechapel, for the purpose\n    \"of considering the present position of the Unemployed\n    Workmen, and the grave events of last week.\"_\n\n    The Unemployed? Well, here I stand,\n      Have stood for many weary weeks,\n    With sinking heart and idle hand,\n      Hunger's white ensign on my cheeks.\n            I raise no howl\n  Like yon plump ruffian with the bull-dog jowl;\n  But the smug swells, with pleasure's honey cloyed,\n  May see in me the real Unemployed!\n\n    Oh, yes! this hand is used to work,\n      The hardness has not left its palm.\n    I'm no black-coated spouting shirk,\n      Like him upon the tub there. Calm?\n            By Heaven, I choke!\n  Could I but fell the gang at one sharp stroke,\n  Ranters who rail, and roughs who watch for spoil,\n  'Twere one good blow in the true cause of Toil.\n\n    How shall I make my poor Voice heard\n      'Midst this brute shindy, brainless, mad?\n    The slime-deeps of the town are stirred,\n      All that's bloodthirsty, blatant, bad,\n            Comes, surging up;\n  And I--ah! I hang back and drain the cup\n  Of bitter want in silence, blent with shame\n  At this base smirching of a Man's good name.\n\n    And then the cynic cacklers crow\n      In their snug cushions; crow and cry:\n    \"Oh, the whole thing's a farce, you know.\n      The old sham play of Poverty,\n            Pushed just once more\n  Upon the public boards. An awful bore!\"\n  So (whilst we starve) the well-fed idlers scoff\n  At the spoilt tragedy, and cry, \"Off! Off!\"\n\n    Ah! the sleek <DW2>s should take a turn\n      At the long, weary foot-sore tramp,\n    In search of work, till sick hearts burn,\n      Till the cold flags or footways damp,\n            Of London seem\n  The endless mazes of some devilish dream,\n  And tempting visions haunt the fevered head,\n  Of the sharp knife-edge or the river's bed.\n\n    Wrong? Oh, of course! Our duty lies,\n      In dull endurance to the end.\n    The faces pale, the pleading eyes,\n      Of wife and children, looks that rend\n            A fellow's heart,\n  And make hot curses from his cold lips start,\n  These should not madden men unto the pitch,\n  Of _violent_ despair. So preach the rich!\n\n    And yonder yelling fools contrive\n      To lend some truth to Mammon's text.\n    The laziest larrikin alive,\n      With babbling tongue and braid perplext,\n            Can help do _that_;\n  Whilst I?--a broken head or beaten hat\n  Will not so help me in my present state\n  That I should greatly care to \"demonstrate.\"\n\n    Only if such a Voice as mine\n      Could penetrate the public ear,\n    Deafened with all this windy shine,\n      And muddled 'twixt contempt and fear;\n            I rather think\n  I would tell some truths might make the scoffers shrink.\n  But _I_ compete with yonder wolf-eyed brute?\n  No; I can easier suffer and stand mute.\n\n    If that's a strong, well-ordered state,\n      Where tens of thousands like myself,\n    With willing hands, must starve and wait,\n      Whilst piles of swiftly growing pelf,\n            Sweated from toil,\n  Swell for the lords of capital and soil,\n  Then--you may rear a city on foul slime,\n  And build Society on want and crime.\n\n    My Voice! Men will not listen--yet;\n      And when they open ears at last,\n    Bludgeon won't cure, nor bayonet.\n      Meanwhile yon brayer at full blast\n            Belies my cause,\n  'Midst foolish jeers and foolisher applause;\n  And preachers prose, and statesmen tinker on,\n  And we--we starve in gold-choked Babylon!\n\n       *       *       *       *       *\n\n\"My Nephew, who is very fond of pictures,\" said Mrs. RAM, \"has\njust purchased the finest Pot o' Jelly I have ever seen.\" Can it be\npossible that the dear old lady meant Botticelli?\n\n       *       *       *       *       *\n\n[Illustration: THE TWO VOICES.\n\nONE OF THE REAL \"UNEMPLOYED.\"--\"HOW AM I TO MAKE _MY_ VOICE HEARD IN\nTHIS BLACKGUARD ROW!!\"]\n\n       *       *       *       *       *\n\nVOCES POPULI.\n\nSCENE--_Trafalgar Square. Several thousand loafers and roughs\ndiscovered asserting right of free speech, free meeting and free\nprocession. A few hundred genuine artisans out of work standing about\nmoodily. Lines of Policemen drawn up in reserve look on impassively._\n\n_A Lover of Liberty._ As an Englishman, Sir, I'm disgusted--it's\n_un-English_, that's what it is, \"dragooning\" an inoffensive assembly\nlike this! I _used_ to think freedom of speech and action was the\nright of every Briton--but it seems we're to be overawed by the Police\nnow--confounded impertinence on the part of the Government, I call it!\n\n[Illustration: \"Hooky Walker!\"\n\n    \"... The Leaders, H. George, _and the man whose name was said\n    to be_ Walker, put up their coat-collars and sneaked away\n    under the trees.\"--_Newspaper Report._\n]\n\n_An Orator (leaping suddenly on parapet)._ Feller Citizens, are you\n_Men_ that you stand by with folded 'ands, while unlimited food and\nwealth lays within a stone's throw? I want yer----\n\n_Constables (behind)._ Ah, and we want _you_--off you go!\n\n[_Disappearance of Orator in direction of Police-station._\n\n_Lover of Liberty._ Shame! Is a man to be punished for his opinions?\nOh, England, England!\n\n_Person in Search of Sensation (disappointedly)._ Well, there doesn't\nseem much doing,--so far.\n\n_Squalid Vagabond (recognising_ Stalwart Constable, _whom he has\napparently met before in a professional capacity_). 'Ow _are_ yer,\npretty bobbish?\n\n[_Nods to show he bears no malice._\n\n_Stalwart C. (good-humouredly)._ I'm much as usual, thankee.\n\n_Companion Constable (to S. C.)._ Well, you _do_ know some rough 'uns,\nI must say!\n\n_Stalwart C._ Go on--that gentleman's a West-Ender.\n\n_Professional \"Hook\" (to line of Policemen)._ So _you_'re 'ere, are\nyou? Well, me and my pal must take _our_ little prominade some hother\narternoon, that's all!\n\n_Sympathiser (to Loafer)._ And so you've actually been out of\nemployment since last January? Monstrous! The Government ought to find\nyou work!\n\n_Loafer._ Jes' what _I_ say, Guv'nor. Let 'em gimme work, and I'll\n_do_ it fast enough. _I_ don't want ter be idle. I ain't on'y my one\ntrade to earn my bread by--but I'll work at that, if I'm let!\n\n_Sympathiser._ Exactly, my poor fellow, and what _is_ your trade?\n\n_Loafer._ Why, I'm a skate-fastener, I am; puts on parties' skates for\n'em,--and 'ere I am--not 'ad a job for months!\n\n_Truculent Ruffian (to Quiet Observer)._ Hunimployed?\n\n_Quiet Obs._ Yes--at present.\n\n_T. R._ Too many o' them bloomin' Coppers about, to _my_ mind--I'd\nlike to slug the lot--they're the ruin of _our_ bisness!\n\n_Quiet Obs._ Ah, you're right _there_!\n\n_Demagogue (to Police Sergeant)._ Now, don't you interfere--that's\nall _I_ ask. _I'll_ speak to them--I have them thoroughly in hand\njust now, but, if you offer them the least opposition, I--(_with much\nsolemnity_) well, I won't be responsible for what happens. (_He is\nallowed to address the multitude._) Friends, you are met here in this\npeaceful but imposing manner in the teeth of a brutal and overbearing\nConstabulary, to show the bloated Capitalists, who are now trembling\nbehind their tills, that we mean to be taken seriously! Yes, in our\nsqualor and our rags----\n\n[_Throws open frock-coat, and displays thick gold watch-chain._\n\n_Mob._ Yah, pitch us over yer red slang! take orf that ere nobby coat!\nHarristocrat! Yah!\n\n_Dem. (complacently)._ It is true that I myself am not in absolute\ndestitution.--But what of that, my friends? Can I not _feel_----\n\n[_Here a turnip strikes him in the eye. Yells of \"Down with him!\"\n\"Duck him!\" \"Spy!\" \"Traitor!\" Mob pulls him down and attempts to take\nhim to pieces._\n\n_Dem. (faintly)._ Here, hi, Policemen, help! Why the devil don't you\nuse your staves? [_Is rescued and assisted home by Police._\n\n_A Rough (to Policeman)._ Keep moving? ah, _I'll_ move! [_Kicks him on\nthe knee-cap. Policeman draws truncheon and hits back._\n\n_Crowd (indignantly)._ Boo! Coward! Strikin' a unarmed man--down with\n'im! [_They beat brutal Constable to a jelly._\n\n_The Truculent Ruffian (to Quiet Obs.)_ Are you game for a merry ole\nlark?\n\n_Quiet Obs._ You _try_ me--that's all!\n\n_T. R._ Then, as them cowards of cops 'ave as much on their 'ands as\nthey kin do with, now's the time for a bit of a loot! Pass the word to\nthem mates o' yourn--\"Pall Mall and no tyranny!\"\n\n_Quiet Obs._ I've done it--they're only waiting for _you._\n\n_T. R. (suddenly producing red handkerchief)._ There--_now_, boys!\n\"Remember Mitchelstown and no brutal perlice!\" Foller me!\n\n_Quiet Obs. (arresting him)._ No, you'll follow us, please--you won't\ndo no good kicking, all right, mates, we've got him.\n\n_T. R._ Oh, please, I didn't know you was a Policeman, Sir, or I\nshouldn't ha' spoke! Strike me dead I was on'y in fun! (_Whimpers._)\nAnd I've a good ole mother at 'ome, Sir.\n\n_The Person in Search of Sensation._ What, another arrest? and simply\nfor showing a red handkerchief! I shall write and describe these\natrocities. How abominably these police are behaving--actually\ndefending themselves, the blackguards!\n\n[_A Policeman accidentally lifts his arm, whereupon about fifty youths\nscurry like rabbits; in the rush, the Person in search of Sensation\nis hustled and slightly trampled on. He becomes annoyed, and hits out\nright and left--eventually striking a Constable in his excitement._\n\n_Const. (who has been without sleep for the last two days and has just\nhad his cheek laid open by a stone)._ 'Ere, you come along with me,\nyou're one of the wust, you are!\n\n_The Person._ But I assure you, I just came to see what there was to\nbe seen!\n\n_Const._ Well, you come along with me, and you'll see a Magistrit\npresently.\n\n[The Person _resists; struggle; arrival of reinforcements; exit party,\nin \"frog's-marching\" order, conveying him to fresh sensations._\n\n_The Lover of Liberty (emerging from crush)._ My hat ruined, my coat\nsplit down the back, and my watch gone! I _told_ the crowd I was with\nthem heart and soul--and they hit me in the stomach! What do we keep\nour police _for_, I want to know?\n\n_Professional (emerging in opposite direction)._ Three red clocks, two\npusses, and a white slang, I ain't done so dusty! 'Ooray for the right\no' Free Meetin', _I_ sez!\n\n_Genuine Unemployed (wearily)._ Well, I dunno as I see what good all\nthis 'ere is a goin' to do _hus_! [_And no more does Mr. Punch._\n\n       *       *       *       *       *\n\nFROM MR. HENRY IRVING'S NOTE-BOOK.\n\n(_Published without permission._)\n\n_Stratford-on-Avon, October 18._--Speech at the Opening-of-Fountain\nceremony went very well. Some distinguished Americans were not there,\nnotably Mr. ABBEY. In consequence, had to omit all reference to \"Abbey\nThought\" and \"Fountains Abbey,\" which, as J. L. T. suggested in his\nletter, would have lightened the entertainment considerably. Also very\nannoying, but I never thought of it till too late; I quite forgot to\nsay anything about BUFFALO BILL. CODY will be hurt; but I shall be in\nAmerica before he gets back there, so it doesn't much matter. Yet it\nwas a chance lost. WILLIAM SHAKSPEARE, WILLIAM CODY, Buffalo BILL,\nSwan SHAKSPEARE. No matter, keep it for another time. And at the last\nmoment I could not make out what I had written on my wristband as a\nmem. for speech. It was _a propos_ of Mr. CHILD'S gift. I see now\nit was something about \"Child's the father to the man.\" And then an\nallusion to the sympathy between America and England as not being mere\n\"Child's-play.\" Very odd, how I forgot that. Still, speech couldn't\nhave gone better.\n\nAnd how on earth I omitted to make any mention of Miss MARY ANDERSON I\ncan't understand! Yet the fact that this fair American is now playing\nat the Lyceum ought to have stuck in my memory which yet holds its\nseat in this distracted brain. And, dear me, there was the American\nMinister present, and yet--bother it!--it never occurred to me, till\nI was dressing this evening, hours afterwards, that I ought to have\nremarked on the fact that America was represented here on this special\nDramatic occasion by a gentleman bearing a name so honoured alike by\nEnglish and American actors, and so dear to the theatrical profession\nas must always be that of \"PHELPS.\" But this will keep, too, for\nanother time. And, after all, in spite of these omissions, which of\ncourse nobody noticed, the speech went admirably.\n\n       *       *       *       *       *\n\nNottingham v. Sunderland.\n\n  \"There's _no_ Liberal Party!\" cries GRANDOLPH the bold.\n    \"Hooray!\" shout the Tories, \"the straightest of shots!\"\n  But the faithful who flock to the G. O. M.'s fold.\n    Say, \"Our old party bonds are re-tied now--in _Notts_!\"\n\n       *       *       *       *       *\n\nTHE AXE PREMIER'S AUCTION.\n\n[Illustration: _Auctioneer._ \"FINE CHIPS OF THE OLD BLOCK, GENTLEMEN!\nSPLENDID SPECIMENS OF THE HAWARDEN TIMBER, IN THE SALE OF WHICH,\nGENTLEMEN, I ASSURE YOU, I HAVE 'NO INTEREST WHATEVER.'\" (\"_Hear!\nhear!_\") \"NOW, GENTLEMEN, HOW MUCH SHALL WE SAY FOR THIS CHIP, WHICH I\nLOPPED OFF WHEN I WAS LEAVING HAWARDEN--WHEN I WAS 'CUTTING MY STICK,'\nIN FACT.\" (_Laughter._) \"WHO BIDS FOR THIS? DON'T BE ALL FAGOT-VOTING\nAT ONCE!\" (_Laughter and Cheers._) \"NOW THEN,--FIFTEEN SHILLINGS, TEN\nSHILLINGS, SEVEN, FIVE, EIGHTEENPENCE,--ANY ADVANCE ON EIGHTEENPENCE?\nGOING! GOING! GOING! GONE! GONE FOR EIGHTEENPENCE, AND CHIP AT THE\nPRICE!\" [_Auction Continues._]\n\n       *       *       *       *       *\n\nHINTS FOR THE UNEMPLOYED.\n\nSIR,--Excellent as is the suggestion of your Correspondent, \"ONE WHO\nWOULD ELEVATE THEM,\" that the Unemployed should be forthwith put into\nthe hands of some competent Ballet-Master, and after a proper course\nof instruction, despatched to all the Board Schools in England for\nthe purpose of teaching every pupil who has passed the Sixth Standard,\ndancing and deportment, yet I do not think he goes far enough.\nWhy stop at this comparatively subordinate art? Why not make them\nmusicians, teach them to play WAGNER, and despatch them straightway\nthrough the length and breadth of the land as enthusiastic Apostles of\nthe great Master? What a glorious prospect to turn the three or four\nthousand idle loafers who have lately been hulking about Trafalgar\nSquare for the purpose of breaking the peace, into a mighty army of\nskilled fiddlers eager to wake the glad strains of the spirit-stirring\nMusic of the Future in every quiet village green through the three\nKingdoms. And the accomplishment of such a task need not be set aside\nas the wild vision of some hopeless dreamer. I am convinced, Sir, that\nif the authorities of the Royal College and Guildhall School of\nMusic, but set their shoulders to the wheel, the thing will soon be an\naccomplished fact. Such, Sir, at all events, is the opinion of one who\nbelieves firmly in\n\n  \"THE SOUL OF THE MASSES.\"\n\nSIR,--Why not paint the whole of London, public buildings and\nall?--I'm sure they want it. The latter might be done in different\ncolours. St. Paul's, for instance, might be orange, Westminster Abbey\npea-green, and the Houses of Parliament a bright blue. If the effect\nwere found unsatisfactory, fresh colours could be tried, until\nsomething were hit upon that should be considered suitable. This\nwould afford the additional advantage of providing fresh work for the\nUnemployed. I don't see what else can be done. Everybody can use a\nbrush, and with a couple, or say, three coats all over the Metropolis,\nthere would be plenty to occupy everybody for the next six months.\nAs to expense, an extra 15s. tacked on to the rates would soon settle\nthat, and I'll be bound there's many a householder willing to face\nthat trifling alternative, together with\n\n  Yours, practically, one who takes\n    \"THE BULL BY THE HORNS.\"\n\nSIR,--I cannot but think that, if BUFFALO BILL were to introduce\nthe \"Unemployed\" into his Show, he would score a big success. The\nintroduction might take the shape of a contest between the \"Wild East\"\nand the \"Wild West.\" The former might be armed with brickbats and\npark-railings, and the latter with their usual weapons; and, were it\nknown that a little genuine blood would be drawn in the entertainment,\nit might be safely counted on to draw all London. I throw out the\nsuggestion for what it is worth.\n\n  Your obedient servant,\n    \"A COMMERCIAL WELL-WISHER.\"\n\nSIR,--As at the present season of the year nothing is more common than\nto find the stalls of most of the leading West-End theatres empty, a\nfact which has a very chilling effect on the efforts of the players,\nwhy not fill the empty places with the so-called \"Unemployed\"? A warm\nbath, a suit of evening clothes, clean shirt, and white tie would\ninstantly fit the veriest outcast that has recently come into\ncollision with the police in Hyde Park or elsewhere, at least\noutwardly, for the social atmosphere of the place. A central committee\nmight at once be inaugurated for the supply of these necessary\npreliminaries for admission, and a thousand or two excellent\nsubstitutes for the ordinary _habitues_ forthwith launched nightly\namong what is at the present moment left of the fashionable play-going\nworld in the Metropolis. The advantage would cut both ways. Not only\nwould the Management be blessed by the appearance of a perfectly\nfull house, but the loafers, professional thieves, and ruffians who\nproduced it would, no doubt, endeavour to play up to their clothes and\nsurroundings, and, on receipt of a small retaining-fee of 3s. 6d. a\nhead for their attendance, be proportionately softened and civilised\nby the process. This, Sir, seems to me a very legitimate, humane, and\nphilosophical method of dealing with the present crisis, and as such I\ntrust it will as powerfully recommend itself to your readers as it has\nto\n\n  Yours thoughtfully,\n    \"A PLEASURE-SEEKING SOCIALIST.\"\n\nSIR,--What are the authorities about that they do not at once embank\nthe river on both sides up to Richmond, and span it with five bridges\nbetween this and Gravesend? Then there's the whole of Piccadilly\nto come down and be rebuilt with the road properly levelled, to\nsay nothing of a great Central Terminus in Soho Square uniting the\nMidland, North and Great Western, Great Northern with the Great\nEastern, and all the Great Southern lines. Add to this, that the\nentire gas-piping of the Metropolis ought to come up bodily, and make\nway for the installation of the Electric Light, to say nothing of the\nfixing in all the leading thoroughfares of overhead railways on the\nNew York principle, and you have enough work at least to begin upon\nand meet the present crisis. Let the Board of Works and the various\nVestries set to work at once, and as soon as Parliament assembles\nlet it be asked to vote Five-hundred Millions towards preliminary\nexpenses. This, Sir, is, I am convinced, the only reasonable and\nefficient way of dealing with the present unsatisfactory aspect of the\nlabour question. Such is the opinion of\n\n  Yours energetically,\n    \"A ROUSED ALARMIST.\"\n\nSIR,--When the Police have fairly and effectually cleared off the\nloafers, not-do-a-stroke-of-work gentry, and the sedition-mongers,\nthen we can turn our attention to the wants of the genuine Unemployed.\nTheir case is by no means beyond us. It only needs the active and\nintelligent co-operation among the administrators of charitable funds\nand agencies, the Poor-Law Authorities, employers of labour, and\nothers, to give immediate and practical effect to the wide-spread\nsympathy felt for them by all classes of their more fortunate\nfellow-countrymen, including your quite sober-minded and\ncharitably-disposed Correspondent,\n\n  \"COMMON SENSE.\"\n\n       *       *       *       *       *\n\nDERBY AND GLADSTONE.\n\n(_A Speech summarised in a Stanza._)\n\nAIR--\"_Darby and Joan._\"\n\n  DERBY, dear, I am old and grey,\n  Fifty-five years since my opening day,\n  \"Ins\" and \"Outs\" are for every one\n    As the world goes round.\n  Derby, dear, I must fain admit\n  I've altered my mind, just a little bit.\n  But I learnt freedom's lesson in Forty-five,\n  And I mean to be true to it whilst I'm alive.\n          Always the same,\n            Derby, my own,\n          Always the same\n            Is your old GLADSTONE!\n\n       *       *       *       *       *\n\nTHE ACTOR'S PROGRESS.\n\nWithin the last half-century, the education of actors has advanced in\nan extraordinary degree, inasmuch as some have been known to take\na degree, or try to, at the University. Therefore the following\nadvertisement in the _Era_ will probably cause little surprise:--\n\n    WANTED, for La Comedie Anglaise, a Light Comedian, for a few\n    Weeks, while a Member of the Company returns to Oxford to take\n    his degree. Must be a gentleman. Address, &c.\n\nThis gentleman, to use the language of the _Era_, seems inclined to\n\"combine leading business with general utility.\" It is to be hoped he\nwill get his degree, and return to be an ornament to the stage. But if\nthis kind of thing goes on, we shall probably eventually see announced\nin our theatrical contemporary--\"Senior Wrangler and Light Comedian\nopen to engagement in first-class Company.\"\n\n       *       *       *       *       *\n\n\"THE REVERSIBLE PEN-CLEANER,\" recently invented by DE LA RUE & CO.,\nwill be most useful to Leader-writers, Politicians, Journalists, and\neverybody in the habit of using \"reversible pens,\" or pens that can\nwrite equally well on both sides. Such pens must occasionally require\ncleaning; and to be cleaned in this pad they must remain upright.\n\n       ***\n\n\"A WINTER'S TALE.\"--That of poverty and distress, which we must do our\nbest to relieve.\n\n       *       *       *       *       *\n\n[Illustration: MIDDLE AGE.\n\n\"YOU'RE GETTING LONG-SIGHTED, DEAREST. YOU'LL HAVE TO WEAR GLASSES.\"\n\n\"STUFF AND NONSENSE! IT'S NOT MY SIGHT THAT'S LONG--IT'S MY ARMS THAT\nAREN'T LONG _ENOUGH_!\"]\n\n       *       *       *       *       *\n\nEUTHANASIA.--In a certain Western newspaper we read the following\nstartling announcement, in relation to the decease of a certain lady\nwhose obituary notice appears in its columns:--\n\n    \"More or less an invalid for a considerable time past,\n    latterly she has been under the care of Mr. ---- and Mr. ----,\n    and her death was not therefore altogether unexpected.\"\n\nWhat a lift for the two Medicos mentioned! They, no doubt, are now\nblessing that Western Editor for inserting this gratuitous tribute\nto their curative skill. Their motto for the future should\nbe--\"_Removals_ conducted with punctuality and dispatch.\"\n\n       *       *       *       *       *\n\nSTUDIES FROM MR. PUNCH'S STUDIO.\n\nNO. XXX.--MR. ALDERMAN SLOCOACH.\n\n[Illustration]\n\nWhat a strange, unreal, almost incomprehensible life must that of\na City Alderman be at the present time. Regarded in the light of\ncenturies ago, it all seems in accordance with the fitness of things,\nand neither ludicrous nor out of place. But now, in these days of\nearnestness and common sense, what a great sham it seems to the merely\nsuperficial observer, and yet, however great an anomaly it may appear,\nwhen tested by results it seems to work fairly well.\n\nSuppose we take Mr. Alderman SLOCOACH as an example. He was taken from\nhis warehouse, some years ago, and made an Alderman by the votes\nof some three or four hundred of the rate-payers of his Ward, the\nmajority of whom knew little or nothing about him, and probably\ncared less, and in a week or two, he found himself seated on the\nMagistrate's Bench at Guildhall, to declare the Law, of which he\nliterally knew nothing, and to administer Justice under circumstances\nso apparently absurd as to be hardly credible. Being probably a\nconscientious man, and knowing his utter ignorance of the duties\nthat his position demanded of him, what was he to do? What he did was\nprobably the best he could do under the circumstances, and thinking,\nas he told an old friend with whom he conversed on the matter, that it\nwas better, as err he must, to err on the side of mercy, he made it a\npoint always to consult the Clerk of the Court, and whatever amount of\npunishment he advised him to inflict, he generally halved it.\n\nHaving long since got thoroughly accustomed to the whole matter, and\nhaving acquired a certain amount of dignity of demeanour, he is able\nto go through the wondrous ceremony with comparative ease, but is\nstill greatly troubled with certain qualms of conscience in certain\nspecial cases. For instance, when fining a poor working-man five\nshillings for drunkenness,--he having met an old friend and been\npersuaded to take more than was good for him,--and that amount\nprobably constituting a full day's income, his thoughts will revert\nto that particularly jovial banquet with his worshipful Company\nthe previous evening, and whether some one or two of the guests not\nsufficiently seasoned to these matters, were not quite as guilty as\nthe poor workman he had just fined, and how they would like to have to\npay a day's income for this folly, amounting in one case to probably\nL100! and yet possibly the workman had the better excuse of the two!\nAnd then, again, there is that very awkward and puzzling question,\nthat so troubles some of his more conscientious brethren as well as\nhimself, that of punishment for gambling. When inflicting some of\nthose very heavy fines and penalties, which he is told it is his\nbounden duty to do in the case of betting in public houses, his\nthoughts must revert to those two most intimate friends of his who are\nregular visitors at TATTERSALL'S in the height of the racing season;\nand also to the fact that he himself, as his stock-broker well knows,\nafter leaving the Bench, occasionally wends his way to Capel Court,\nand buys or sells for the account to very very large amounts; and,\nthough he probably tries his best, as others do, to convince himself\nthat there is no doubt a very great difference between the cases\nof Mr. BUNG and Mr. TATTERSALL, and between playing cards for\nhalf-crowns, and buying or selling L50,000 Consols for the account, it\nwas not until his conscience had lost its natural elasticity that he\nsucceeded, and, even now its twinges are, occasionally, very sharp.\n\nWhen Alderman SLOCOACH was first elected to his high position, his\ngreat delight was to attend at the Old Bailey, and occupy a seat on\nthe judicial Bench, and enjoy the supreme satisfaction of feeling\nthat, without his absolutely useless presence, the whole proceedings\nmust necessarily come to a stand-still, and fond memory still looks\nback to the occasion on which one of Her MAJESTY'S Judges actually\nsaid to him, in quite a friendly manner, \"Shall _we_ say twelve\nor fifteen months, Alderman?\" On the other hand, he will probably\nremember, to his dying day, the look of mingled anger and contempt\nwith which he was received by another of Her MAJESTY'S Judges, of\nrather irascible temper, when he rushed breathless into Court, having,\nby his absence, delayed the proceedings for more than an hour.\n\nNaturally, the one particular event to which an Alderman looks forward\nwith the most especial anticipations of honour and renown, is the\nyear of his Mayoralty, when he will have his otherwise humble name\nassociated with those of the famous men who, in very different times\nto those in which we live, ruled the great City, with courage and\ndiscretion.\n\nMuch, however, depends upon the public events of his year of office,\nas to its importance, or want of it, to himself personally, and Mr.\nAlderman SLOCOACH was not particularly fortunate in that respect.\nThere was no European Monarch on a visit to this country, whom the\nCorporation was requested by the Government to honour, with the\ncustomary satisfactory result to the Lord Mayor of the day; there was\nno public ceremonial of unusual importance that required the brilliant\nsurroundings of Civic pomp to give it full _eclat_, and as his year of\noffice approached its termination, his solemn look became more solemn,\nand his hopes evidently grew fainter and fainter. But fortune was kind\nto him, and a change of Government, which made it desirable to gain\nthe City's sweet voices, brought him the coveted honour.\n\nLike most of his colleagues who have what is technically called\n\"passed the Chair,\" he takes things very coolly, probably thinking\nthat nothing remains to be done after having passed through such an\nordeal. But there is one especial duty still left for Aldermen to\nperform from which he is seldom absent. They have been deprived of\ntheir control over prisons, and of their government of the Royal\nHospitals, their control of the Police is almost nominal, but they\nstill have charge of City Lunatics, and it is said that Alderman\nSLOCOACH is seldom absent from the official visits to them, when the\nreciprocity of feeling manifested between the poor patients and their\nvisitor is described as quite touching. He is also often seen at City\nBanquets, and is always quite ready to return thanks for what he calls\nthe Grand Old Corporation, and repeats with painful iteration the old\nbit of twaddle about the infallibility of Aldermanic judgments and\nthe increasing popularity of their order; but he is wonderfully\ngood-natured, devotes a great deal of time to the gratuitous\nperformance of public duties, assists very efficiently in brightening\nup many an otherwise dull scene with the brilliancy of his handsome\nscarlet robe, and would, with his worshipful Brethren, be much missed\nif deprived of those civic functions that have been performed by them,\nand such as they, for many centuries past, and which entitle them in\nall respects to the esteem of their fellow citizens as a trustworthy,\nsober and honourable body of men.\n\n       *       *       *       *       *\n\nIMPERIAL INSTITUTORS.\n\nSir F. ABEL, the organising Secretary of the Imperial Institute,\nrecently issued a very agreeable and pleasing memorandum to the\nChairmen of Provincial Committees and others who have assumed an\nactive part in support of the undertaking. After describing the\n\"large measure of success\" that has attended the efforts of the local\nCommittees throughout the country, Sir FREDERICK goes on to say that\na \"considerable number\" of them have \"signified their willingness\nto prolong their operations with the especial object of obtaining\nadditions to the 'Endowment Fund' of the Institute which is about to\nbe created.\" This is but natural. Taking into consideration the fact\nthat in many quarters a handsome subscription to the funds of the\nInstitute has been regarded as a sure passport to honour, and that\nthe non-distribution of titles right and left among a lot of\nsmall provincial celebrities has already occasioned a good deal of\nheartburning and disappointment, this new lease of life, affording\nthem, as it does, a fresh opportunity of struggling for their\nmuch-coveted prize, cannot but be hailed by the yet unsatisfied\n\"Chairmen of Provincial Committees and others\" with genuine joy and\nthankfulness.\n\nThat plain Mr. JOHN BOPKINS, or Mr. PETER PICKLETUB, Mayor, should\nsuddenly blossom out into Sir JOHN BOPKINS, and, possibly, Sir PETER\nPICKLETUB, Bart., would only seem to those indefatigable gentlemen an\nappropriate finish to their labours in furtherance of the interests\nof the Institute. Their readiness, therefore, to prolong their\noperations, as it may be measured by the fact that it will have the\nspecial object not only of \"procuring additions\" to the Endowment\nFund, but also of tacking them on to their own names, is likely to\nbe both hearty and enthusiastic. Whether anything will come of their\nhopeful perseverance, remains to be seen; but it is tolerably certain\nthat if some sort of bureau for the sale of decorations, after the\nlatest French model, could be instituted on this side of the Channel,\nthere would be no lack of clients ready to besiege it. But----we\nmanage these things much better in England.\n\n       *       *       *       *       *\n\nWhen the Deputation waited on him, Mr. MATTHEWS was the \"Not-at-Home\nSecretary.\" Quite right too.\n\n       *       *       *       *       *\n\nNOTICE.--Rejected Communications or Contributions, whether MS., Printed\nMatter, Drawings, or Pictures of any description, will in no case be\nreturned, not even when accompanied by a Stamped and Addressed Envelope,\nCover, or Wrapper. To this rule there will be no exception.\n\n       *       *       *       *       *\n\n\n\n\nTranscriber's Notes\n\nA small number of minor typographical errors have been corrected.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Punch, or the London Charivari, Vol.\n93, October 29, 1887, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**The Headscarf Revolutionaries:**\n\n_Lillian Bilocca and the Hull Triple-Trawler Disaster_\n\n'This is a story of the men whose exploits built a city's wealth and helped to feed a nation. Lavery's tale of how the Triple Trawler Tragedy unleashed the fury of the formidable women of the Hessle Road is inspirational. Lily Bilocca personified the courage and determination of an entire community.' - Rt. Hon. Alan Johnson MP, author of _This Boy_\n\nIn the harsh Arctic seas of 1968, three trawlers from Hull's fleet sank in just three weeks. 58 men died. Lillian Bilocca put down her filleting knife, wrote a petition, and stormed into action. With her army of fishwives she took her battle to the docks and led a raid on Parliament. They changed the shipping laws.\n\nLillian Bilocca became an international celebrity. The lone survivor of the tragedies made headlines too. In a tight fishing community, it's dangerous to stand out.\n\n'Brian's story made me feel as if I was back in the fight in 1968. It was almost like he had been with us.' \u2013 Mary Denness, Trawler Safety Campaigner and one of two surviving 'Headscarf Revolutionaries'.\n\n'Mark my words, this could become the next _Made in Dagenham_. Lilly Bilocca was remarkable.' \u2013 John Prescott\nBrian W. Lavery was born in Glasgow's East End in 1959, the fourth of six sons of William, a sheet metal worker and Margaret, a shop assistant. He has been a university drop-out, factory worker, car valeter, market trader, waiter, Customs and Excise officer (very briefly) and is now a journalist, tutor and writer.\n\nAfter 25 years of various senior roles in national and regional journalism he returned to education. In 2011, he gained a first in English and Creative Writing. This book derives from a funded PhD at the University of Hull, where he also taught creative nonfiction.\n\nIn December 2013 he wrote and presented \"Courage and Effect\" for BBC Radio 4's Four Thought series, drawn from the subject matter of this book. The Oxford Dictionary of National Biography commissioned him to write the entry on Mrs. Lillian Bilocca for its 2013 edition. He has also published short fiction and poetry.\n\nBrian runs a community media project in Hull, where he has lived with his wife Kathryn for more than 30 years. They have grown-up daughters, Catriona and Rose.\n\n_Website:brianwlavery.com_\n\nFirst published in Great Britain by Barbican Press in 2015\n\nCopyright \u00a9 Brian Lavery 2015\n\nThis book is copyright under the Berne Convention\n\nNo reproduction without permission\n\nAll rights reserved\n\nThe right of Brian Lavery to be identified as the author of this book has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act, 1988\n\nBarbican Press, Hull and London\n\nRegistered office: 1 Ashenden Road, London E5 0DP\n\nwww.barbicanpress.com\n\n@barbicanpress1\n\nA CIP catalogue for this book is available from the British Library\n\nISBN: 978-1-909954-15-1\n\nTypeset in Garamond by Mike Gower\n\nCover by Jason Anscomb of Rawshock Design\n\nPhotos courtesy of the Hull Daily Mail\n\nPrinted and bound by Totem in Poland\nFOR KATHRYN \u2013\n\nIN TRIBUTE TO LILY, MARY, YVONNE, CHRISTINE, MY MOTHER MARGARET \u2013 & ALL THE STRONG WOMEN\nCONTENTS\n\nForeword\n\nPrologue\n\nOne\n\nTwo\n\nThree\n\nFour\n\nFive\n\nSix\n\nSeven\n\nEight\n\nNine\n\nTen\n\nEleven\n\nTwelve\n\nThirteen\n\nFourteen\n\nAfter Words\n\nAcknowledgements\n\nEndnotes\nFOREWORD\n\nThe story of the fighting Hessle Road fishwives, led by Lily Bilocca after the Hull Triple Trawler Disaster in 1968, should never be forgotten. And it never will be in this great maritime city. Now the world can share it. After all, it was worldwide news back then. I am pleased this book is telling the whole tragic, powerful story in full for the first time.\n\nLily's campaign following the loss of the 58 men when three trawlers sank in as many weeks was forceful. Only one man, Harry Eddom, the mate of the trawler Ross Cleveland miraculously survived. The eyes of the world were on Hull. Lily and the Hessle Road women achieved more in weeks than unions and politicians ever did. They took their fight from the fish docks to Westminster and won. There was great personal cost for Lily, who was not only targeted by death threats and attacks in her own community but was blacklisted, lost her job and crucified in the press. Harry Eddom's incredible survival is not only a tale of great drama and courage but also a stark illustration of the dangers that were faced by hundreds of men each year back then.\n\nI was proud of the small part I played in Lily's campaign.\n\nAs a young seafarer and trade unionist I was angry and horrified to witness that these men worked under Dickensian conditions, governed by the last \"master and servant act\" of the twentieth century. They had to supply their own bedding and lived in quarters worse than prison cells and forced to work in deadly Arctic weather, which had killed before and looked set to do so again. No one reckoned with Lily Bilocca. She and the women of Hessle Road would not sit idly by as the industry that gave them their living also killed their men. I shared the platform with her at Victoria Hall when she held hundreds in the palm of her hand. She was inspirational. Mrs Bilocca was a powerful speaker and certainly had no bother being heard (She was even louder than I was!). She was tough and had that working class style that meant she did not suffer fools. Like most Hull folk she spoke as she found. The media used her na\u00efve honesty against her and even helped turn her own folk against her.\n\nOne of my few good memories of that awful year was being in a march I organised with others from Hull University, where I was a mature student. We went through the city wheeling a giant cardboard fish around in protest against _Silver Cod Trophy_ , shouting at housewives going into the market, ' _It's not fish you're buying, it's men's lives.'_ The Trophy went to the skipper who caught the most fish in a year. It was a big deal, with a big do in London, hosted by the Prime Minister. It also encouraged skippers to take risks with the crew for money. And the law governing this servitude meant the men could be jailed for refusing to go. [Later in my life I was proud to have played a part in consigning the effects of the iniquitous _Merchant Shipping Act of 1894_ to history's dustbin.]\n\nLily's revolution was not like any of the others of 1968. It was from the heart. There was no ideology or political wire pulling. This was a community of women fighting for what they knew was right. It is also a very human and dramatic story.\n\nLily was a big woman with a big heart and above all a fighter who made a gigantic change in the face of insurmountable odds. The industry may have died, but the achievements of Lillian Bilocca and the women of Hessle Road will live forever and undoubtedly saved countless lives \u2013 and still continue to do so.\n\n_John Prescott, Hull, September 2014._\nPROLOGUE\n\nFrom The Times of London, \"Obituaries\", Page 12, August 8, 1988:\n\nMRS LILIAN BILOCCA\n\nMrs Lilian Bilocca, known as \"Big Lil\", the Hull woman who was the driving force behind radio operators being placed on Britain's trawlers, died on August 5, aged 59.\n\nThe daughter of a fisherman, she led a campaign in 1968 for better safety on trawlers. She brought together trawlermen's wives in an-often violent campaign after three Hull trawlers went down within days of each other with the loss of more than 50 lives. \"Big Lil\" was at the dockhead at every tide with her following of wives. There were often hysterical scenes as police attempted to stop the women hurling themselves on to the decks of ships where they believed there were no wireless operator.\n\n\"Big Lil\", so called because she weighed over 17 stone, finally took her fight to Downing Street, calling on Lord Wilson of Rievaulx, then Prime Minister, to get a better deal for fishermen. She lost her own job during her campaigning.\n\nThe few paragraphs above are all The Times sub editors' desk left of the several hundred I had written and submitted to their Obituaries' Desk when I was commissioned by them as a young freelance journalist in Hull. At that time I noted not only that they had left so much out, and that this woman deserved so much more, but also they had misspelled her name. I promised myself that one day I would set the record straight \u2013 on both counts.\n\nToday is the day.\n\n_Brian W. Lavery, Hull, September 29, 2014._\nONE\n\n_0030 hours, January 10, 1968, St Andrew's Dock, Hull._\n\nNo one got a good night's sleep on Subway Street.\n\nThere were always different shifts of men and women going to or coming from work. Hours after the pubs had chucked out, even the most beer sedated could not drift off as clogs clattered on the cobblestones in the midnight air. \"Bobbers\" made their way to the docks. These were the men who landed the trawlers' catches at the quayside and made them ready for market. A local \"knocker-up\" rapped on bedroom windows with a long pole, waking customers who paid two-bob a week for the alarm-call service. Those customers' neighbours were awakened too \u2013 whether they wanted it or not. That busy half-mile thoroughfare from Hessle Road to the St Andrew's Fish Dock was one of the main arteries that pumped the lifeblood of labour to the heart of the greatest deep-sea fishery on Earth. Tiny two-up, two-down houses nestled among processing factories, smokehouses, pubs, shops and icehouses.\n\nAt the bottom of Subway Street the pedestrian tunnel was the workers' entrance and exit to the waterside. It had a footpath and a road wide enough for a lorry but was built in the time of the horse and cart. On the fish dock at the other side of the tunnel ship's runners harried trawlermen through the company stores. Their job was to ensure crews were led through and then taken to their ships. All protective gear had to be bought by the men. The stores were hectic as crewmen, many still drunk from the previous night, loaded up with rubber thigh boots, mittens, gutting gloves, oil frocks, sou'westers and other gear needed for working in the Arctic Circle. They even had to buy their own bedding. Once a man had what he needed an anxious ship's runner, keen to get him to his ship, would tick him off the list. The stores' opening times were determined by the tides. They opened two hours before high water, day or night, so crews could be served in time. The men who headed for ships were still in their suits from the night before.\n\nThey dragged their gear through a dead-of-winter chill wind that sped up as it was forced its way through the Hessle Road terraces and whipped onto the black waters of the Humber. It was just below zero. Fine sleet meant it felt colder.\n\nHigh on the bridge of No.H223, St Romanus, Jim Wheeldon, the young skipper of the 600-ton, 170-feet long, sidewinder trawler looked across the community into which he was born just twenty-six years earlier. Against a backdrop of terraced houses, shops and fish-houses, the high-stacked, four-in-a-row tall chimneys of the fifty or so smokehouses were scattered around. They stood out like redwoods in the forest.\n\nThe ship's runner reported to the youthful skipper that all the men were safely aboard. On the whaleback (fo'c'sle) and to the aft of the ship, old shipmates reunited and young ones waited anxiously. The \"All Passengers Ashore\" signal was sounded. Friends, drinking buddies, taxi drivers or family who were there to see someone off all got ashore. Failure to do so could mean a trip to the Arctic. It would not be the first time that a pissed up cabbie found himself part of a trawler's crew.\n\nMen on the deck swigged rum, smoked cigarettes and swapped tales. Other workers milled round below them and the docks came to life. Bobbers' clog-clatter could still be heard. The trawlermen watched other crewmen struggle with kit bags and bedding rolls. Some had a cabbie help them up the gangplank; others used the bagmen and boys working from the dockside. A few hours earlier Jim had been at home with his wife Janet. Their new baby burbled in the crib next to their bed. Five-year-old daughter Beverley slept in the next room with the dolls Jim bought her that Christmas. When he kissed Janet goodbye he told her, 'Don't worry. I know it's an awful ship but it's just for one trip. I'll phone you on the ship-to-shore tonight. I promise.'\n\nHe picked up his suitcase in the hallway when he heard the taxi engine. In the lounge of his new house in Ulverston Road, Hessle was the almost-finished dolls' house that he had started to make for Beverley. The plywood toy home stood out in the gloom as the streetlight shone into the living room. He closed the door quietly and went to the Austin Cambridge private hire car from Subway Cabs (All Tides Attended).\n\nThe St Romanus was not Jim's regular ship. She had been forced upon him \u2013 and he did not like it. He knew her poor reputation. His usual ship, the St Andronicus, had sailed with a new skipper days earlier. Jim was not able to take the St Andronicus out because he had been on company business representing the owners Thomas Hamling and Co. (Hull) at the Leeds Assizes Court giving expert evidence on their behalf in an insurance case involving damaged fishing gear. Although Jim had only got his skipper's ticket two years earlier, his employers regarded him as an up-and-coming star. This would be his fifteenth command with the firm.\n\nThe St Romanus had not been doing well recently. Like many Hull trawlers at that time of year, she had been lumbered with a \"Christmas Cracker\" crew last time out. Over the holiday, trawlermen were keen to be at home. Most had only thirty-six full days off in a year. The norm was three weeks away, three days, maybe four, at home and then away again. The bosses were not prepared to see ships go idle, while many of the regular men celebrated the holiday. The answer was to allow men and boys, who would normally never get a berth, to crew on the ships. These were known locally as \"Christmas Cracker crews\" and were drawn from a pool of lesser men, keen young lads and even old men from the pubs, clubs and the Salvation Army model-lodging house near the docks. They also signed up from the Flying Angels' Club, where merchant sailors out of a ship often lodged. To give some skippers a break, ambitious ship's mates were given temporary commands.\n\nShip's runners \u2013 the owners' agents \u2013 hired the crews. Owners did not care how they did it. This put immense pressure on the runners. With the Scottish fisheries closed at Hogmanay (New Year's Eve) and New Year's Day, fish prices in early January's market were always sky high. Even a poor catch would, pro rata, reach a great price.\n\nOwners turned a blind eye to ships crewed by drunks and incompetents and even by men and boys who had never been to sea. Obviously, there was a core crew of experienced men, but it was not unusual to see a Yorkshire miner, a bin-man or a bus driver on the deck of an Arctic trawler over Christmas, trying to make an extra few bob. It was after such a trip for the St Romanus Jim got the call \"offering\" the command. It was one he could not refuse. The office's polite phone manner did little to disguise that the offer was not really a request. This was the time of year when skippers had to push harder to make up for previous catches. Refusing a command had proven the downfall of many a more experienced \u2013 but perhaps less wise \u2013 man than he.\n\nThis ship's reputation went before it. She was an unlucky ship and superstition was no small thing to the seafarers of Hessle Road. The craft was reputedly less stable than her sister ships and she \"laid low\" in the water, hence her unflattering nickname \u2013 Yellow Submarine. The owners assured Jim the ship was A1 and told him she had been recently surveyed and considerably improved. Jim was in no position to gainsay that. It was more than his career was worth.\n\nFrom the bridge, Jim signalled for third hand Alan Nicholas to come to the wheel. It was the third hand's job to steer the ship out, usually as far as the Spurn Light Ship. This allowed the skipper and the mate a few hours to plan the trip further. Soon the crew would be organised into three duty watches for the voyage. Jim was still in his suit, but unlike many of his crew he was immaculately turned out \u2013 and sober. His wavy black hair was swept back. This handsome young fisherman would not have looked out of place as the front man of a skiffle group.\n\nMate Raymond Mearns joined him on the bridge, where he would be for the few hours it would take third hand Nicholas to get them past the Spurn Light Ship, off the east Yorkshire coast. There were maps, courses, tides etc., to be discussed. The ship's paperwork \u2013 orders, crew pay books, company letters, chart changes, insurance dockets etc., came in a red \"logbook tin\", supplied by the runner. A glance through it would have told Jim there was no radio operator, but there was an extra deckhand as per union rules, to keep up the ship's complement of twenty. Jim was not happy with that as it meant he had to double up and fill in as the \"sparks\". He had his Radio Telephone Certificate, which meant he was qualified to use a radio set but not to maintain and repair it. No skipper was happy to be without a radio operator. After all, he could not be on the bridge and in the radio room at the same time.\n\nThere was a VHF radio set in the wheelhouse. It was never more than two strides away, but it could only reach a radius of fifty miles or so. The UHF set in the sparks' room could reach across the world. From reporting into the owners, to keeping in touch with other skippers, monitoring for distress signals or being forewarned of bad weather, the radio was vital. Having an extra deckhand was little or no compensation for not having an expert sparks. Jim had to deal with what was, not what he would have liked. That's a skipper's life. He must have been uneasy. Half the crew was under twenty years including a few really young kids. He recognised many names. He knew the families of most. Galley boy Alan Court was set to have his seventeenth birthday on board, deckie learner Robbie Dockerty had just left school that summer and his pal Kenny Suffling, a deckie learner too, was just two years older at eighteen. The younger boy, Dockerty, arrived well prepared with a full kit bag, complete with all his wet weather gear. Later it was obvious to his amused crewmates that he got his waterproofs from a bigger man. They would have wasted no time in ribbing him.\n\nIt turned out Robert's uncle Skipper Philip Gay of the Ross Cleveland had given the deck boy his old gear. Robert was Skipper Gay's sister's lad and he had been asked by the mother to \"talk to the boy\" \u2013 a euphemism for, \"talk him out of it.\" Philip Gay paid lip service to this. He knew he would never be able talk the boy out of his chosen adventure. When Phil heard the boy had signed up for the St Romanus he told him, 'I'll give you my old wet weather gear. You'll definitely need it on that boat, son.' Jim also knew the cook Brian Wilson's missus had just had a baby girl and he noticed too that Tommy Williams' boys, John and Melvin, just twenty-one and twenty-two respectively, were aboard. Tommy was a pal of Jim's dad, Fred.\n\nThe ship's telegraph on the wheelhouse gave second engineer George Rutter his instructions. The thrum from the engine room vibrated through the ship. The voyage was properly under way. The St Romanus reached the lock head and a tug gave a blast on its whistle to say, \"good luck, and safe home.\" When Jim was a kid the foghorns and whistles of the Humber were the background music of his childhood. The St Romanus blasted back in reply. In the Hessle Road terraces some folk from Hull's fishing community mumbled a swearword or two, rolled over and tried to get back to sleep. The lock gates were already open and the ship got on her way. Mate Raymond Mearns looked out from the side window of the wheelhouse. The dock master shouted out from the lock, 'Where bound, master?' Mearns replied, 'Bear Island.'\n\nThe trawlers left behind them bobbed side-by-side across the dock in the lee of the Lord Line Building. [Often the ships were so close together that in busy times physically fit crewmen could leap deck-to-deck, vaulting rail-to-rail, rather than waiting for their ship to come alongside.] The landmark, four-storey building was the biggest owners' block in the city. It was built in 1949 in a modernist style, in red and beige brickwork, in the shape of a ship. The aerials on its roof alongside the flagstaffs looked like disproportionate masts.\n\nFrom the top of Subway Street it appeared to be at the end the street. This was an illusion. It was actually on the other side of the dock onto which the street abutted. The 'aft' was on the right and 'bow' on the left. The walls dovetailed into a point at either side. Being sited at an angle, its 'ship-shape' could be seen from the terraces. It looked the same on either side. Silhouetted in the early hours, it resembled a big, badly drawn version of the ships over which it towered.\n\nBy one a.m. the army of bobbers would be rat-a-tat-tatting 20 or 30 at a time in their clogs. Their clatter got louder in the Subway Street tunnel. In its dimly lit mouth workers looked like shadowy aliens re-boarding a mother ship. Lights flickered in nearby terraced houses where women braided nets from their homes and worked day and night, some alone, some mothers and daughters some in groups, some waiting for the net man's lorry to collect their work later that morning, others prepared to carry their work there, using a handcart or even an old pram.\n\nBack on the St Romanus, Scottish second engineer George Rutter provided the power that made the ship surge up the Humber on the tide on the orders of the skipper via the third hand in the wheelhouse. Like the rest of the men, he was keen to get going so that he could have a dram as the ship steamed north on the open seas. The forty-four-year-old was a devoted family man, a good engineer and a boon to men like Jim. If George seemed subdued, the reason lay in the council house in Askew Avenue, west Hull, where he had left behind his wife Betty and seven kids. Four days earlier the Rutters celebrated their sixteenth wedding anniversary and there was a great family party at their new home. For Betty this made parting even tougher. Their eldest, Doreen, had been \"playing up\" at school and at home. Betty told George she wanted him to stay and help discipline the teenager. She suggested he could go later \u2013 or better still \u2013 find another job. George wanted to stay ashore but he had agreed to go.\n\nBetty had pleaded, 'Don't go back, George, not now.' He had replied, 'Look Betty, we made a good trip last time. I'll make this the last one \u2013then I'll give it up.' She said nothing. There was no point. When George left she had a dread feeling she would never see him again. Betty Rutter was a superstitious woman who, like many of her peers, believed in premonition. Also like her peers she kept fear to herself.\n\nThank God for old hands like Rutter, Jim must have mused as the ship with its twenty souls aboard threaded downriver towards Spurn Point. Grimsby trawlers joined her at the Humber's mouth heading to the open sea bound for the Arctic waters beyond. Rainbow films of oil on the water were left in their wake. The weather ashore was bad enough. On open waters there is little shelter other than the lee of the land. Winds of Force Eight to Nine were predicted. The Met. Office messages did not make good reading. Stronger winds for further north; and snow, that increased the chances of the trawlerman's worst enemy \u2013 ice.\n\nThe crew, now in their working gear, gathered in the galley of the St Romanus. Three watches, the mate's, the bo'sun's and the third hand's were sorted. The engineers had their own system, working six hours on and six off for the whole trip. The trawler prowed up the Humber and the men on deck secured the ship for the voyage ahead. Life lines were rigged up, all moveable deck gear lashed down and the hatches battened. All this was better done at once rather than waiting for the weather to blow later. Deckie learners busied themselves coating the external brasses with grease as weather protection. This included all handrails, portholes and the \"banjoes\", the nickname for the ventilation cowls, which kept the engine room well aired. Even the ship's bell was polished. Preparation was everything. There was always work to do steaming out and it would get even harder when they reached the fishing grounds.\n\nA complete trawl was ready \u2013 prepared by the previous crew. It was part of the trawlermen's etiquette to ensure the crew following them had a proper head start. Adjustments had to be made to the trawl to ready it for the fishing grounds. There was also a second trawl to put together, which would save more time. This was a massive task and would take six men around twelve hours to complete. The trawl was rather like a gigantic stocking. It was between 150-180 feet long and 80 feet wide at its mouth, tapering to a six feet tail, called the cod end. It was attached to a ground wire of steel bobbins that later would roll along the seabed with the mouth of the net around twelve feet high. This was in turn attached by two fifty-fathom (90 metre) cables \u2013 or warps \u2013 to two \"Otter\" boards. These looked like huge doors and weighed a ton each and were fixed to the wires. During the trawl they would be dragged vertically along the seabed, angled outwardly to keep the net mouth opened. This meant that the nets pulled along above the seabed, like a huge horizontal underwater kite.\n\nThere were also wire tackles to renew, life rafts to check, sections of net to prepare, ropes to splice, a fish room to wash out and in excess of one thousand fish room boards to get ready to store the catch. A successful voyage would use more than three thousand such boards. Having a thousand ready early was a good start.\n\nThen the small matter of twenty tons of fish room ice to crush, chop and shovel for when the trawl started. The trip might require much more ice again, dependent on the success of the trawl. The procedure for storing the fish later would be a layer of boards, followed by a layer of crushed ice with the fish laid on it belly-down up to nine inches. This is repeated until the shelves touch the deck head (ceiling) of the fish room. But that was three or four days away yet.\n\nSometimes, on really good trips, the ship's ice store could run out. To fill a fish room on a good trip would need more than 100 tons of crushed ice (manufactured ashore and loaded aboard). More than 5,000 boards would be needed too, with around 2,500 kits (10 stone barrels) to house up to 170 tons of fish. One advantage of being in the Arctic was there was always plenty of ice around. Skippers instructed deckhands to look out for \"growlers\" \u2013 the nickname they gave to small ice floes. Once spotted, little icebergs would be hauled aboard then crushed to meet requirements. In the St Romanus galley, the deck men who were not on watch took advantage of a huge fried breakfast. Soon, fried fish and bread would replace the mounds of sausages, liver, eggs, bacon and beans with fresh baked bread when the supplies ran out. Meals were in two half-hour sittings to fit with the watches and the number of men aboard.\n\nTwenty miles off the Humber coast, heading north-northeast towards the Norwegian grounds, third hand Nicholas was still at the wheel with the mate on the bridge. Jim would have gone to his cabin after passing Spurn Light Ship to change into his fearnoughts and gansey. They would venture further if unsuccessful in the Norwegian grounds. That was the plan. On the open sea, Jim allowed the bond locker to be opened and some beer, spirits and cigarettes were issued. The men not on watch were making the most of it. High winds were more bearable with a dram. They were right to make the most of the steaming out. At the fishing grounds work would be harder, relentless, cold and dangerous. It was soon to be eighteen hours on, six off, maybe even not as much rest as that. It depended on the haul and whether or not all the crew would be needed to chop any ice that might strike and gather on the ship's superstructure. That was vital, priority work.\n\nTrawlermen have an almost Zen outlook. Few are better at being \"in the now.\" For now they enjoyed the booze \u2013 never too much though, few skippers were that lax \u2013and the baccy. Country and western music crackled through the old galley set from Radio Luxembourg \u2013 or the American Forces' Network when they got further north. No sea shanties for these guys. It was Johnny Cash, Patsy Cline, and Hank Williams who accompanied them. The American voices faded and boomed as the \"atmospherics\" ruined the reception and the crew cursed the \"fucking hopeless wireless\" while they tucked into their food, had a smoke, another drink, some chat, maybe even a read of a JT Edson \u2013 anything to keep their minds off the swells and waves and winds which forced them all to keep a strong grip on their plates and cups. They had confidence in the \"old man\" \u2013 a term of respect for their captain, not related to his age. Back on the bridge, a warning from his father might have been on Jim's mind as the weather worsened with Force Eight to Nine gales. His father, Fred Wheeldon, was the first man to get a call after his son was \"offered\" the St Romanus.\n\nFred saw Jim as an \"old head on young shoulders\" and respected his maturity and leadership. Advice and guidance from his father enhanced those qualities. Fred knew his boy well and spoke frankly when asked for advice.\n\nJim, in his father's eyes, could have become anything. Fred described his son as a 'good lad, very clever, a chess player, an excellent draughtsman, great at sport, very creative and always making things.'\n\nBut his boy chose the Northern Trawl, and although his Dad may have wished him another career, he understood the call of the sea and was proud of his son who was one of the brightest and best of the Hull fleet. Fred had also sailed on the St Andronicus, his son's regular ship, and compared and contrasted the two. There was not too much new in his father's critique but it must have reinforced any doubts Jim may have had.\n\nFred said, 'If the Andronicus got into a storm she did not roll. With the Romanus it didn't have to be bad weather, just a swell. She'd be going along and then she'd suddenly drop to starboard. She dropped four times like that when I was aboard and once she nearly turned over and was lucky to come back. They said the ship could do 12 knots, but it's more like ten, maybe ten and a half. The owners have trouble getting crews and I am not the only fisherman that knows it.'\n\nHamling's \u2013 the owners \u2013 bought the ship in 1964, fourteen years after she was built in Beverley, East Yorkshire for another fishery firm in Belgium. Hamling's put her to sea in March that year after a massive overhaul, but they were never really happy with her. By 1967, they were concerned because the catches were not up to par and this forced them to call in a top skipper to investigate. Captain Rob Ellis got the job. The respected skipper \u2013 the son of legendary local Don skipper Jack \"Dasher\" Ellis \u2013 did not pull his punches and recommended a full survey and further overhaul before the ship sailed again. What Ellis said about the ship was not comfortable for Hamling's.\n\nSkipper Ellis reported, 'The Romanus trip was a very bad one. She was in a very poor condition, having a lot of faults, not a few compartments flooding for no apparent reason, the fore-peak, cofferdam (a watertight enclosure pumped dry to permit construction work below the waterline) and stokehold had to be pumped out much more than normal. Even my berth on the main deck was awash with water on deck, which turned out to be a blocked valve, which should not have been where it was \u2013 poor ship design. We had winch and dynamo problems and a rough old time. The lack of winch power made it more difficult to heave the trawl over the side. We also had difficulty transmitting due to low voltage.' His long list of faults was presented to the owners and the St Romanus was put under a survey. It is not known what work was done, but the Hull Mutual Insurance Company issued it with a certificate. The office told Jim the ship was A1, but he could not have been too reassured by this. He must have trusted the word of those who knew best \u2013 men who had sailed on her \u2013 not some desk commanders, whose maritime expertise began and ended with taking a rowing boat out on the local Pickering Park Lake. If Jim was troubled by the task facing him and by Skipper Ellis's views, he could not afford to let his crew know.\n\nSince leaving Hull, the ship's transmitter had been playing up. The strengthening winds were causing the ship to list and bob. Jim was just keen that the transmitter would be working at 7.30pm, when he had promised to call Janet. He had made arrangements with Wick Radio for a ship to shore call. By 7.30pm GMT the St Romanus was 120 miles north-northeast of the Humber steaming towards its intended fishing grounds. In Hull, Janet Wheeldon put her daughter Beverley to bed and was cradling the baby when the shrill bell of the Bakelite phone rang out.\n\n'Mrs. Wheeldon, Mrs. Wheeldon, hello, hello, this is Wick Radio, Wick Radio,\" said the lilting Western Isles accent of the coastal station operator. This was the protocol of the GPO radiomen. In ship-to-shore communications all names and important information were repeated. After all, reception at sea was not like it was on the transistor radios of the youngsters ashore.\n\n'Will you take the call, Mrs. Wheeldon, from the Hull trawler St Romanus, that's St Romanus, Skipper Jim Wheeldon, Skipper Jim Wheeldon?'\n\n'Yes, yes. I'll take the call, I'll take the call,' said Janet unwittingly following coastal radio protocols. The reception was poor and Janet and Jim struggled with the conversation.\n\n'What's it like there, love? Is the ship OK? Are you all right?'\n\nJim would never have worried his wife and decided to take a light tone.\n\n'We're all right. Just wish this bloody radio was though. It took me a couple of hours to get the flaming thing to work before phoning you. I might have to phone you again in the morning if it packs up.'\n\n'Yeah, but is the ship OK? You were worried when you left.'\n\n'I am not happy with it, but it's OK. Don't worry. The plates [the ship's walls] are thinner than the ones on our Beverley's dolls house, though!' Jim kept trying to pull the conversation round to the mundane. The kids, the house, and what were the plans for coming home. Radio reception was deteriorating. The high winds and waves did not help. Janet heard the lilting voice of the Wick Radio operator telling her husband to change frequencies and that that may help the reception. It did \u2013 temporarily.\n\n'Just a minute, Janet...'\n\nThose were the final words Janet Wheeldon ever heard from her husband.\n\nThe line was dead.\n\nJanet was angry and disappointed, but she was used to being cut off during ship-to-shore calls. She resigned to wait until morning. Maybe he will call back then, she thought. Janet did not think too much of it when there was no call the next day. As a skipper's wife she knew that trawlers often kept radio silence if they were fishing well, so that rivals would not cut in on their success. Many did this to protect profits. In December the skipper who had taken Jim's ship the St Andronicus out while Jim was at court for the firm, was fined for staying silent for five days. Plus Janet knew that Jim had problems with the radio and would get back to her when he could.\n\nNext day around 6am, off the coast of Isfjordhur, Northern Iceland, approximately 600 miles from the St Romanus' last known position, the Icelandic trawler Vikingur III was heading back home. There were a few more hauls to do. The weather had been kind and ship's catch good. The skipper and the crew were enjoying a meal together as the first mate Thordur Oddsson took care of the bridge, keeping watch and monitoring the two radios, one of which transmitted regular messages from many trawlers and other vessels in as many languages. The other was tuned permanently to an international distress frequency.\n\nOddsson was relaxed. The remaining few hauls would be straightforward, and then back to port, where he would have a good pay packet to spend after a bumper catch. He was enjoying the quiet of the bridge, where the crackling of the radio \"atmospherics\", the burble of the foreign languages and the slosh of the waves in the lee of his homeland provided background noise. Oddsson was comfortable in his skipper's well-upholstered, leather chair. He nodded off. The second radio burst into life. Oddsson grabbed his pencil and pad and wrote down the message,\n\n'Mayday, Mayday, Mayday...it is the British trawler St Romanus from Hull. We are leaving now.'\n\nThe voice then gave the ship's position as 63.5 degrees north, 0.4 degrees west. Or at least Oddsson thought it had, the atmospherics were causing the message to be unclear. Oddsson wrote on the pad, '63.5 degrees north, 0.4 west, or 4.0 west?'\n\nThe first mate was not sure exactly what he thought he had heard. But what could he do? They were 600 miles away. He thought the coastal radio stations would surely pick this up. They did so with all distress messages. These broadcasts had a range of 1100 miles. Just as the link faded, Oddsson thought he heard another English voice trying to speak with the St Romanus.\n\nHe tried to keep the signal but to no avail. The voices disappeared. It was over in seconds. Oddsson went to the skipper and reported what had happened, showing his boss the notes he had taken. It was assumed that the \"other English voice\" Oddsson spoke of would go to aid the British ship. The skipper decided to leave it to the coastal stations and the other ships that must be closer to the stricken trawler. He thought, 'What else can I do?'\n\nTwo hundred and sixty-five miles north of Spurn Point, along the Norwegian coast at around 4am, on January 13, a Danish trawler was dead slow in high winds. Its skipper became aware of something colliding alongside his ship. It was a life raft. The captain told his mate to get two men port side to pull it in. It was empty but fully inflated. There was no name on it, just a code number. The captain came down to where the men had dragged it aboard.\n\n'What'll we do with it, skipper?' asked a deckhand.\n\n'Keep it aboard. We will take it back ashore and check it out, just in case. We'll be back tomorrow night. Radio the details ahead to the office, see what they know.'\n\nThe captain thought little of it. After all, he did not know of any recent Maydays or incidents reported by the coastal stations in the area \u2013 but he did know things were always falling off trawlers.\n\nTen days after the failed ship-to-shore call, Janet Wheeldon decided to surprise Jim with a telegram. It was their sixth wedding anniversary. She made the call to Wick Radio. She dictated her brief telegram to the radio operator.\n\n'Would you like to wait for a reply, Mrs. Wheeldon?'\n\n'Yes, please.'\n\n'Just hold.'\n\nJanet could hear the GPO radioman's voice as he went through the telegram protocol, complete with the funny repetitions.\n\n'I'm sorry, Mrs. Wheeldon. They do not seem to be acknowledging. I will try again. Please hold.'\n\nThe operator tried a further three times.\n\n'I'm sorry. St Romanus is not acknowledging.'\n\n'Thanks, please try later and let me know if there's a reply.' Two days later Janet had still heard nothing from Wick Radio. She decided not to call them again. After all, they had better things to do than chase telegram messages for her. Janet decided to call Hamling's and see if they had heard anything. She must have cursed \"that flaming radio.\"\n\nAlbert Robinson was always busy. At any given time he was responsible for at least twelve trawlers at sea. As shipping manager for Hamling's, Mr Robinson was director Jonathan Watson-Hall's right-hand man, responsible for the day-to-day running of the business from its office in West Dock Avenue, off Hessle Road. Robinson's secretary put a call through from Skipper Wheeldon's worried wife. In his oak-panelled office, he lifted the receiver from its cradle and listened to Janet explain her concerns about the radio-call breakdown and the failed anniversary telegram.\n\n'I do not see why you should be worried, Mrs. Wheeldon. I certainly am not,' Robinson said before hanging up. Robinson had been shipping manager at Hamling's for four years. A former skipper himself, he had commanded the St Romanus' sister ship the St Achilleus. He was the link between the skippers and the bosses and was in charge of all fishing matters. It was his responsibility to ensure the ship was safe and properly equipped and staffed before it sailed. Robinson was confident that Mrs. Wheeldon was being hysterical. He had dealt with \"silent trawlers\" before. He knew that Wheeldon's other command the St Andronicus was out of contact for five days recently in a previous trip, in spite of the company ruling of daily reporting to HQ. That skipper was reprimanded and fined. The previous July the shipping manager fired a skipper for a similar offence. But Robinson also knew that some wily skippers stayed silent when fishing well. So he was not worried. He had just returned from Glasgow four days earlier, where he had been on company business from January 12 until January 20.\n\nThe day he left for Glasgow the office had been in touch with Wick Radio to forward a message to Wheeldon telling him to report in. Other Hamling's ships off the Norwegian coast, where it was believed Wheeldon was, had been contacted too. The day after that, January 13, sister ship the St Matthew, commanded by Skipper James Curtis, one of Hamling's best men, radioed the office to say he had been in touch with the St Romanus. Later, Curtis said, 'We were on our way back from fishing on January 13 when we heard a call on our VHF and thought it was the St Romanus calling the Newby Wyke. They wanted a message sent to Wick Radio station, but I knew the Newby Wyke was out of VHF radio range, so we took the call. The message was something about a wedding anniversary.'\n\nRobinson's office had kept in touch with him by telegram and telephone to his Glasgow hotel. On Robinson's return, his boss Jonathan Watson-Hall insisted that a PAN message be sent out via Humber Radio and Wick Radio to all shipping in the area asking them to look out for the St Romanus.\n\nThe local paper in Hull picked up on this and a statement was sent out saying there was no cause for concern and an air and sea search would be carried out as a matter of routine. The owners said they would not be too worried unless she did not get in touch by February 3, when she was due to head home. Robinson and Watson-Hall at that time seemed unconcerned that other shipping had not yet spotted their silent ship. There was only six hours daylight on the fishing grounds and fog and low cloud made identifying any ship a hard task. For a trawler being a mile from another one was considered being nearby. Also, Wheeldon had reported having a dodgy transmitter and may even be out of VHF range. Robinson did not want to trouble Mr Watson-Hall unnecessarily. The managing director had enough to deal with and was to give evidence next day (January 25) at Hull Magistrates' Court where nine deckhands from the St Apollo were up for refusing to accept skipper's orders to work in a freezing fo'c'sle while fishing off Norway in atrocious weather. Such indiscipline could not be tolerated.\n\nWhile Mr Watson-Hall swore on the Bible before giving evidence against his employees, Janet Wheeldon answered a knock at her front door. Standing there holding a notebook was a clean-cut well-dressed young man. He was slim-built, blond-haired, around five-feet-ten and about 28 years old. Sporting his steel-rimmed glasses, he looked a bit like the actor Hardy Kruger.\n\n'Mrs. Wheeldon? I'm Stuart Russell from the Hull Daily Mail...'\n\nThis brief enquiry was the first in the biggest story of Stuart's life.\n\nBack at the offices of Hamling and Co., Albert Robinson and his boss prepared to accept that maybe the St Romanus could be lost. Skipper James Curtis of the St Matthew had been back in touch after the PAN message. He had made a mistake. The ship he thought was the St Romanus was actually the St Andronicus, Wheeldon's usual command.\n\nCurtis' admission of mistaken identity was the first of three contacts with Hamling's to make Watson-Hall and Robinson begin to doubt. Thordur Oddsson of the Vikingur III also heard the PAN message and reported in again with his \"Mayday\" story about the Romanus. That was forwarded to the trawler owner. Then a telegram from Esjberg in Denmark arrived telling of the finding of a life raft by a Danish coaster twelve days earlier. The raft had the code confirmed. It was from the St Romanus.\n\nWatson-Hall and Robinson still maintained their \"no cause for concern\" line. The day after the fruitless air-and-sea search Hamling's wheeled out top skipper James Curtis who had mistakenly thought he had been in touch with the St Romanus. He was quoted in the Hull Daily Mail on January 26 as saying, 'I don't think there is any cause for worry. The St Romanus could be fishing for another four or five days yet before she comes home. If his main transmitter has broken down, which looks likely, he could be fishing out of VHF range. If he's on a good thing, he is not going to steam fifty miles just to radio that he's alright and then go back.'\nTWO\n\nA pile of laundry in her kitchen belied Margaret Wilson's reputation for being houseproud. It stood out in her new shiny suburban west Hull home. The rest of the place was as neat as ninepence. Margaret had good reason for leaving the washing where it lay. Her husband Ray, a trawler skipper, had left for Icelandic fishing grounds several hours previously on the high water in the early hours. The trawlerman's wife would not touch any laundry that day, lest she \"wash her man away.\" Her domestic pride was outstripped by her superstition \u2013 a trait common among fishermen's wives. She knew it was daft, but she never tempted fate.\n\nMargaret had enough on her plate, without the laundry. Her two older children Karen, seven and Deborah, five, were at school but she still had two at home \u2013 Kathryn, four and Andrew, who was just one. Kathryn had measles and was missing her Daddy more than usual. She had been extra clingy. The night before the little girl asked her Daddy, 'Please don't go away.' The child's plaintive plea struck at the trawlerman's heart.\n\n'I've got to go, sweetheart. It's my work.'\n\nRay had been out of a ship for a few weeks, so when the command offer came from Hellyer's office in the city, he found it hard to refuse. Margaret set about her housework but could not help but think about how Ray seemed out of sorts since that call came.\n\nThe day before he left, Ray went to visit his ailing mother, who was bed bound. She had been ill for some time. After that, he conscientiously went to each of his brothers and took them for a pint. He rarely visited them all at once, especially not all in one day. Margaret thought that unusual, and even briefly thought maybe he had a feeling something was not quite right, but that thought left as quickly as it came. She didn't say anything to Ray. She did not want to worry him unduly. He was also concerned for little Kathryn, whose measles had spread from her head to her toes.\n\nNormally, Ray was a very happy-go-lucky guy, rarely without a smile. He was well known in the community as the joker ashore. The night before a trip was usually a party. Along with his pal Pete \"Wiff\" Smith, the wives and friends were treated to a great night that usually ended in the St Andrew's Social Club with Ray and his best mate Wiff \u2013 who was also the Mate on his ship \u2013 giving their famous (well, famous among their families) rendition of Save The Last Dance For Me \u2013 by The Drifters. It was the pals' favourite song and what the singing skipper and mate, lacked in talent they made up for with enthusiasm.\n\nOften after the club closed there would be a further do at Ray's house, especially now he had built a bar in the downstairs lounge of his new home. The Wilsons had only been there a little over a year. They bought it when Ray made skipper. Home before that home was in a \"sham four\" in Edinburgh Street, off Hessle Road \u2013 local slang for a small two-up, two down. Their new place was off the Hessle High Road, the posh part of town \u2013 favoured by skippers and trawler officers \u2013 two miles in distance, but a million miles in difference.\n\nMargaret got on with her work as she always did. She could not afford to dwell on anything else. Ray was gone. She could not afford to be preoccupied. She fitted the housework round dealing with little Kathryn and baby Andrew. By mid-morning, Margaret was very busy. Something caught her peripheral vision and she looked up at back door. Through its glass panel she saw a familiar silhouette. It was Ray.\n\n'What you doing here?'\n\nWith his usual beaming wide smile, he replied, 'Engine trouble \u2013 they've sent us home for the night. Bet you didn't think you'd see me again, did ya?'\n\nAfter engine repairs and inspection, the Kingston Peridot (H591) left the Humber the same day as the St Romanus, but on the later tide, bound for Icelandic fishing grounds. Like the St Romanus, this 657-ton, 181ft long, 31ft wide, trawler had a crew of 20. Unlike the Hamling vessel she had a radio operator, and more importantly wireless equipment that worked properly.\n\nThe 19-year-old Kingston Peridot, owned by Hellyer's of Hull, was prepared for a voyage of more than 2,000 miles across three weeks under the command of 33-year-old Ray, an experienced mate recently made up to skipper. He was a well-regarded man with six previous trips as commander. This was the trawler's first trip since a successful insurance survey days earlier. Ray Wilson was following in big footsteps, his trawler was once commanded by a local fishing community legend Sam Trolle, a Don skipper famed in the 1950s. The Kingston Peridot was regarded locally as a great sea craft. Ray was not surprised to be told she had passed her insurance survey. What would have shocked him, if he had had any way of knowing, was that the ship was one on a list of ten Hellyer vessels destined to be scrapped that year for \"commercial reasons.\"\n\nOnce at sea a \"sked\" ship was appointed, which acted as a link to the offices and oversaw the fleet schedule. Kingston Sardius, under the command of Skipper Bill Ward, Ray's cousin, was the sked ship this trip. Ward and Wilson got on famously and tried to keep in touch daily with each other and with base. For the first three days of his voyage, Ray kept radio silence. He was not protecting a good haul \u2013 as the owners of the \"silent trawler\" St Romanus had hinted their skipper may have been doing \u2013 he and his crew were battling worsening heavy seas and ice. Hellyer's, like the other owners, had strict procedures that ships had to report daily while steaming to and from fishing grounds \u2013 and that a trawler had to report twice daily while fishing. Like the other companies their skippers often ignored these rules when it suited.\n\nEvery Hull skipper carried with him an unforgettable lesson, which ensured he would never underestimate the danger of the build-up of ice on a trawler. They had naval instructor Captain Peter Harvey of the Hull Nautical College to thank for this. Harvey was a charismatic and somewhat eccentric teacher. In his classroom his lessons were delivered in front of a poster that read, IT'S EASY WHEN YOU KNOW HOW.\n\nBut the invaluable lesson that saved countless lives and was burned into the memories of his students, involved a model of a trawler in the college's float tank. Harvey placed a scale version of the Hull trawler DB Finn in it. On the model's superstructure he placed various weights, to mimic an ice build-up until the ship \"lolled\" (listed). Then with a theatrical flourish, the ex-naval man, who was also an amateur actor, said, 'Gentlemen, this is your ship.' With that he tipped a ladleful of water onto the model and it immediately turned turtle. Harvey's powerful presentation focused the minds of countless nascent skippers over the years.\n\nAfter the first twenty-four hours of the trip, winds were stronger, waves higher and ice gathered on the superstructure of the Kingston Peridot. All hands, excepting engineers and skipper, were chopping the ice, determined not to become a live version of Harvey's lesson. The fishing grounds were still two days away. Wilson and his crew were not only surprised by the ferocity of the weather, but also that it happened so early in the voyage. Radio protocol, therefore, was way down the list of priorities. Ray needed his concentration to get the Kingston Peridot through and to guide her and her crew to safer grounds to carry out the business of fishing.\n\nA trawler has a personality. There is more to her than that from which she is made. She has a name and gender \u2013 a \"she\" imbued with all the power of a mother, the oft-contrary feelings of a sister to a brother. She is given the care due to a daughter and cosseted as a wife too \u2013 in the hope she will look after you.\n\nNothing brings this home more than being the storm's grip. If the ship is the living being, then her helmsman is her brain, heart and will, bending her to your command, hoping she answers his prayer. At the helm he is connected with the ship, part of it, feeling her power through his feet, her vibration shaking his tiring arms, and the guiding of her wearing him out. If he breaks his link, loses his concentration for a second, he and his shipmates are doomed.\n\nSkipper Ray Wilson faced that struggle and not for the first time. This was no job for a novice. Waves two to three times as high as the ship were climbed bow first. A 180ft long, 30ft wide vessel in a boiling sea going up, up, up, and then over \u2013 a palpable relief each time with each wave, for hours on end. The nets were in and all hands battled the gathering ice. Above the bridge, each hour or so, either the bosun Enoch Watson or the mate Pete \"Wiff\" Smith was held round the ankles by a shipmate lest he go overboard while he fought to keep the radar scanner on the mast free of ice. That task would go to Smith more than Watson, given that, at 31-years-old, the mate was a good thirteen years younger. Men on deck fought the ice and the skipper jockeyed his ship over giant waves, weaving where possible when, even in respite, the craft was thrown like a cork in Jacuzzi bathwater. Progress could be less than one knot per hour. At twelve degrees below zero or worse, the men could often not shout. Voices froze, as their spit seemed to ice up in their throats. Rum was swigged as medicine, the warm booze bringing back the power of speech.\n\nIn the galley, pots of hot stew swung in the brackets, furniture not bolted down was thrown around. Anything left on the hob by accident would bounce like air hockey pucks, to and fro off the metal rail surrounding it. The cooks too were battling ice and in the engine room the engineers followed the skipper's commands to surge or relent power as needed. The radio operator was ready in a heartbeat to issue the Mayday, should the captain's concentration slip. Such seamanship is learned \u2013 and earned \u2013 by experience on the way up as bosun, mate and new skipper. The adrenalin keeps the trawler men coming back. Skill keeps them safe. The skipper never underestimates his task. There is no room for error or bravado.\n\nAs the old trawlermen say, 'There are old skippers and bold skippers \u2013 but there are no old, bold skippers.'\n\nMen are rightly fearful when the ship climbs and drops, but that becomes terror if she moves from side to side. Huge waves must be faced head on. Taken sideways, they can send a ship to the bottom of the sea in seconds. In near misses ships go over port or starboard with the waters gushing over, lying on her side before righting. When it goes wrong no one has time to act. Only luck can save her. Survival is by accident. There is no procedure. You do not even have the 60 seconds needed to launch a life raft or issue a May Day call. If you are on deck, you are gone. Below deck, in a blink, the ceiling is now where one of the walls once was. The floor goes from you as you are thrown. The freezing sea is through the portholes. In the remaining minutes you drown or freeze to death. Your lungs will collapse in the cold before you get the chance to drown. If you get to a life raft a further miracle needs you to be far from the ship as she sinks or she will take you and your raft with her.\n\nBy the fourth day at sea, the Arctic seas were getting rougher. Skipper Wilson was hoping to get to the lee for shelter and headed inland. He was still aware the ice could build on the superstructure, but the men were at their usual duties for now as the skipper and mate concentrated on \"dodging\" the trawler through.\n\nIt would not be long until the ice built up again. Marine surveyors advised \u2013 and trawlermen knew, especially the pupils of Captain Harvey \u2013 that it took a fraction less than twenty tons of ice to topple a 657-ton ship like this, and how quickly that toppling can occur. That amount of ice can gather within an hour on the superstructure, with the ship's ropes and wires thickening by inches in minutes as waves make the deck like a skating rink at the same time.\n\nThe short-term necessity was for all hands to fight the ice with axes, picks and marlinspikes. The solution was to get the ship away from the eye of the storm heading into the waves at a negligible speed, always forward. Sometimes all hands had to go below but that call was made rarely, only when stowing below was the only way to stay alive. Crews could not stow for long. A ton of ice, if left unchecked, could form in minutes. The battle was relentless.\n\nBelow deck on the Kingston Peridot, cook Bill Good was heading for the galley and was fighting to keep his feet when the ship lurched and hit a trough. The force sent him headfirst down the companionway. He smashed into the wall and onto the floor. The pain in his chest, back and legs was searing; the cuts on his head needed immediate attention. There is no medic aboard a trawler. The skipper is expected to fill the role. On the bridge, Mate Pete Smith took the helm as Wilson checked his old shipmate Bill. It was the cook's first time on the Kingston Peridot. He was an old salt with forty years under his belt, twenty with the Merchant Navy and during World War Two he served at the Normandy landings.\n\nWhen Skipper Wilson saw Bill he knew the cook was in bad shape. The cuts were cleaned and the bandages put on. The skipper noted the man's laboured breathing. His decision was immediate and radio silence was to be broken soon.\n\n'Bill your ribs are broken. We'll have to put in at Reykjavik. You need an X-ray and proper treatment,' the skipper said.\n\nBill protested, telling his skipper he could stay on.\n\n'Sorry Bill, I have got to put you ashore. You're in a bad way.'\n\n'Skipper, I'm sure I'll be all right.'\n\n'Sorry Bill \u2013 but do us a favour will you? Call our lass when you get back to Hull and get her to send us a Rudd's message to let me know how the bairn is. The little 'un was poorly with the measles.'\n\n'Will do, skipper.'\n\n(Like many skippers, Ray shared a unique codebook with his wife that allowed encrypted messages by telegram. Only the two sharing the code could decipher it. It allowed important messages \u2013 and intimate ones \u2013 to be exchanged. An enterprising Hull newsagent \u2013 Rudd's \u2013 provided the codebooks and most skippers used them.)\n\nGalley boy Eugene Carney stepped up to the hotplate when the skipper told Bill to rest until ashore. The 15-year-old cook's assistant's promotion was to last a whole day. Bill's pal, the chief engineer Harry Fowler, told his mate he would look after his gear aboard until they got back to Hull, so Bill could travel light. Harry at 56 years old was a contemporary of Bill, who was three years' younger. Both men were World War Two veterans. Wireless operator Martin Larsen was told to radio the sked ship and the Hull office and to arrange for another cook to be flown out and picked up when the ship got to port. In turn Bill would be treated and flown home. It was estimated they would come in on the midnight tide next day to Reykjavik, weather permitting. Skipper Wilson got on to the RT (radio telephone) and told Bill Ward of the Kingston Sardius of his plans.\n\nNext day in Hull, Harry \"Cockney\" Riches was contacted by Hellyer's and told to get himself to their offices where train and plane tickets were awaiting him. He was told he was going to Reykjavik, via Glasgow and would join the Kingston Peridot at the Icelandic port. There he would be replacing Bill Good, who was being brought back to the UK for treatment for severe bruising and several broken ribs.\n\nCockney Riches \u2013 a Londoner given the uninspired nickname by his mates \u2013 was low on cash and had been ashore a few weeks. With a wife and six kids, the chance to earn was welcome, even if the rush to fly to Iceland was not. It must have seemed to him that Bill's misfortune was a stroke of luck. Cockney Riches, like most trawlermen, was a believer in luck. Superstitions were legion in the Hessle Road community of which he had now been part for more than a decade. One big taboo, that all trawlermen religiously observed, was to ensure they did not take money aboard a ship.\n\nA bad voyage would follow if they did. If you went to sea with pockets full of cash, the \"sea gods\" would see you as rich. You would not need a good catch and therefore the sea gods gave them a poor trip. This led to daily dockside or pub door \"scrambles\" when trawlermen would throw all their spare change at crowds of kids, the ensuing ruckus known locally as a scramble.\n\nTrawlermen had various ways of avoiding the taboo. Some would leave their money under their kids' pillowcases or mattresses. Others would leave cash behind the bar to make sure they had a drink to come home to or enough to get a round in for their mates if they did not return. It was not uncommon at the St Andrew's Dock lock pit to see fistfuls of money from drunken fishermen going into the Humber with the hung-over crewmen keen to rid themselves of cash before sailing.\n\nIn the spirit of that superstition, Cockney Riches thought he would call in at his sister Maureen's house en route to Hull Paragon Street railway station and leave his money on her kitchen table. It would be a nice treat for her and her kids. Also he would leave a message there for his family telling them he had gone. There was no one at Harry's house when he got the call to get on his way. His missus was out and his kids, one girl and five lads, were at school.\n\nWhen he got to his sister's two-up, two-down in Fern Grove, Division Road, off Hessle Road, only his young niece Susan Peach was at home. She was delighted to see her uncle, especially when he dumped all the coins on the kitchen table. He told his niece he'd have to get along and to tell her \"mam\" he had called in. He left young Susan counting the coins.\n\nWhile sorting the money into neat piles on the table, the girl noticed a tiny pair of nail scissors among the mass of change. There is another superstition that says you cannot accept a sharp gift from someone without giving some money in return. Not to do so means you are \"cutting\" the person out of your life. When Maureen Peach came home and saw the money and the scissors that her young girl had neatly sorted, she was in what sociologists call a \"superstitious paradox\" and what anyone on Hessle Road would call a \"right state\".\n\nJust after midnight on January 14 the Kingston Peridot slipped into a berth at Reykjavik but stayed only long enough to take Cockney Riches aboard and hand Bill Good to a port doctor. Skipper Wilson's diagnosis was spot on and Bill was transferred to hospital where he was to stay for three days before being flown back to the UK. Bill must have felt his luck was right out. Now all he would get was a smaller share of whatever came back to Hull, for his few days' work, alongside his basic wage.\n\nRiches' reception may have been mixed. Yet another superstition warned against new crew boarding after leaving port. Perhaps the desire to save them from galley boy Carney's cooking may have outweighed that superstition though. Next to the engineer, the cook keeps the ship going. Men have to eat more than 6,000 calories a day to carry out the tough work in their eighteen hours on, six hours off, relentless labour in the freezing fishing grounds of the Arctic. There is always food available and it is taken whenever possible, be it from the stewpots (shackles) always on the go, or the plates of fresh-baked bread and fried fish and potatoes served up in the galley.\n\nThe trawler was turned round on the same tide and headed back towards the fishing grounds north of the island. Time was money. The new cook set to work. The skipper decided to head to the Skagagrunn fishing fields and try his luck there to hopefully avoid the weather that had plagued his first few days out.\n\nSkippers Ward and Wilson were back in touch with each other. The Kingston Peridot had also started reporting back to the Hellyer's office in Hull and to the sked ship as requested. The weather was still ferocious and both trawlers were not able to settle to trawl in any one place. The ships were constantly prowing into high winds of up to Force 12 and combating ice in temperatures as low as minus 14\u00b0C. Both skippers had had a luckless, fruitless fortnight, with each crew fighting ice to keep afloat as their masters and mates battled the seas, dodging into freezing headwinds, which perpetuated the fight. They had a week or so to go before they had to be back in Hull for a Friday landing \u2013 and it was their intention not to land in debt.\n\nRadio protocol was being followed. The bosses were informed. The news was not good. At the end of another poor day on January 25, the Kingston Peridot reported to Hellyer's, 'FISHING SLACK IN NORTH CAPE AREA + STEAMING EASTERLY + 66 DEGREES, 42 MINUTES NORTH + 22 DEGREES, 15 MINUTES WEST + '\n\nThe next day, January 26, at about 10.45am local time, Ray Wilson and Bill Ward were talking with each other again via their ships' radios. The trawlers were 120 miles apart. Wilson was not in the mood for friendly banter. While fishing west of the island of Grimsey, off the northern Icelandic coast, further disaster struck.\n\n'I've fouled the trawl, Bill,' he told his colleague on the Kingston Sardius. 'I'm going to chop the warps and stow the gear. The weather is getting worse.'\n\n'It's OK here, Ray. Why not steam over?'\n\n'I'm going to lay to for a few hours and chop the ice.'\n\nThe only good news Skipper Wilson had that trip so far was the coded telegram from his wife telling him his little girl's measles had cleared.\n\nBoth skippers had agreed to speak later on the wireless. Both had more important things to deal with than radio chat. The two sparks, Martin Larsen on the Kingston Peridot and John Green on the Kingston Sardius, continued the conversation after the skippers stopped. The two operators were on the line to each other for 20 minutes. A rendezvous was arranged. Maybe this time their luck would change. Skipper Wilson set his crew to work on the ice, clearing the superstructure before intending to set off east to meet up with the Kingston Sardius. Skipper Ward knew his cousin was fighting treacherous weather if he had had to stow the trawl. Ward felt better knowing they would both soon be nearby each other on the same fishing grounds.\n\nIn a matter of hours, the weather that battered the Kingston Peridot moved east and caught up with the Kingston Sardius. Bill Ward stopped fishing. It was 7pm when a message from the bosses in Hull was sent to all Hellyer ships via Wick Radio. It told them not to schedule landing the following Friday. A bobbers' strike at St Andrew's Dock in Hull over the previous ten days looked set to drag on. All Hellyer ships, bar one, responded. John Green picked up the messages destined for the Peridot, in wireless traffic from Wick. At 7.45pm, Skipper Ward told Green to try to get in touch with the sister ship and check if she was on her way to meet up. There was no reply. Green kept on trying. By 1am there was still not a peep. The rendezvous looked increasingly unlikely.\n\nAn hour earlier, Hannes Hafstein, Superintendent of the Icelandic Rescue Organisation (IRO) was growing more and more concerned as further reports of fierce weather came in. In the area off Grimsey, it was particularly bad with heavy snow, Force 12 gales and temperatures minus 14\u00b0C and lower combining to pose a threat of icing up for any craft in the area. Hannes decided to phone an old friend at the IRO's weather station in the area.\n\n'How's the weather faring there?' Hafstein asked the weatherman.\n\n'It's midnight and it so bad out there that I cannot stand up in the gale long enough to take a reading. The wind is so high, the snow is blizzarding and there is hard frost. I stretched my arm out a minute ago. I could not see my own thumb. That's the weather, my old friend.'\n\nThe ship that left Hull on an earlier tide the same day as the Kingston Peridot \u2013 the St Romanus \u2013 had now been missing for sixteen days. On the edges of the Arctic Circle, off the Lofoten archipelago on the Norwegian coast, an air and sea search on January 26 found no trace of the St Romanus. This area where the ship was last known to have been was searched for hours by an RAF Shackleton aircraft from No.120 Squadron, based at Ballykelly. In high winds and freezing rain below a flotilla led by the fisheries protection frigate HMS Grafton was joined by Norwegian Navy vessels and local trawlers. Aeroplanes from Icelandair and Scandinavian Airlines had also been given a full description of the St Romanus from a PAN message sent out to all shipping and air bases two days earlier. Other than the life raft found earlier by a Danish coaster, there was no trace of the ship.\n\nIn Hull, the owners were still saying there was hope and that the finding of the life raft meant nothing. On Hessle Road folk feared the worst. At the local fishermen's mission prayers were asked for. In the offices of the mission, its superintendent David McMillan was preparing to visit the families of the missing St Romanus crew. The almost certain doom of the St Romanus focused the concentration of the Hellyer's bosses and further messages were sent to all shipping to look out for the Kingston Peridot.\n\nAt 11am on January 27, Skipper Ward of the Kingston Sardius reported that there was still no reply to numerous attempts to radio the Kingston Peridot. The worried skipper also told his bosses that the northerly wind was growing and that it was freezing. This had forced him to stop fishing and his crew had to battle the ice to stay afloat. Across the weekend and up to the Monday morning of January 29, Hellyer's Hull office was kept informed. There was still no sign of the ship. It was like it had fallen off the planet. Next day a local newspaper reported a heavy oil slick off Axafjordar. The Lloyds agent in Reykjavik noted the report and grew concerned given the PAN message circulating about the ship that was last reported there. At 5.20pm that day the Lloyds office contacted the Mutual Insurance Company in Hull. The report added that a life raft found on the shore was from the Kingston Peridot.\n\nAt the National Lifesaving Association's office, Superintendent Hannes Hafstein's fears were confirmed. His weatherman friend's earlier warning proved devastatingly accurate. There was no delay in ordering an immediate air and sea search this time. Unlike Hamling's, Hellyer's knew they most likely had a trawler disaster on their hands and had to be seen to act fast. A land search was carried out on the Axafjordar area. In the worst weather in living memory an RAF Shackleton aircraft from Kinloss was sent and small local craft joined the volunteer crews of the Lifesaving Association working alongside the coastguard to hunt for the trawler.\n\nThe ship was not found. By the end of the search the three lifebuoys marked Kingston Peridot, the hatch plank, trawl bobbin, two signal flares and port-side lantern gathered in, confirmed the worst of all fears.\nTHREE\n\n_Tuesday, January 30, 1968, Hull, UK._\n\nAnxiety pervaded the English port city like a sea fret, especially along the four-mile riverside stretch of its Hessle Road fishing community, running parallel to the fish docks of the River Humber. As the blue and white double-decked buses rumbled by, groups of shopgirls, fishwives, and trawlermen dotted the street corners on either side doing a fair impression of the other thing this Yorkshire city is famous for \u2013 rugby league scrums. The pleasant smell from the fish smoking houses, strong on a cold day, gave an attractive aroma, which on rare hot days got overpowered and replaced by the much less attractive stench of the riverside fish oil and meal plant, where cod livers and fish bones were made into oil and meal respectively.\n\nThe scrums' chatter was delivered in hushed tones, all along Hessle Road. Today the families of the crew of trawler St Romanus were told there was no hope and it was announced that another trawler Kingston Peridot was missing.\n\nBoth ships had left Hull on January 10. At the centre of the community, on the corner of West Dock Avenue and the main road, the \"scrums\" \u2013 mainly made up of women and kids \u2013 were more numerous including those going in and out of the trawler offices and those waiting for husbands or sons therein.\n\nThe \"other trawlermen's office\" \u2013 Rayner's Pub \u2013 takes up the full corner.\n\nFrom its panoramic windows, drinkers can see both sides of their world, able to be on the look-out onto the main road for angry wives ready to tell them of ruined dinners or \u2013 onto the West Dock Avenue side \u2013 they could see recently-settled up pals, fresh from the owners' offices, with their percentage cash share of the catch, ready to spend their earnings and fit three weeks' living into three days, before setting off to Arctic waters again. That was the routine of the Arctic trawler man, usually three days home and roughly three weeks at sea. While home they spent heavily \u2013 \"like the Last Day at dinner time\" \u2013 locals would say. The talk around the community today of the St Romanus and the Kingston Peridot served as a reminder, if one were needed, why these men held with a live-for-today philosophy. That is why, once ashore, these men were nicknamed \"The Three-Day Millionaires.\"\n\nThe real name for Rayner's was the Star and Garter, but only strangers \u2013 and the brewery \u2013 called it that. For Hessle Roaders, their main pub would always be Rayner's, named after its first landlord, Henry Rayner \u2013 and the current landlord Charlie Knott didn't mind. This pub was as much a place of business as pleasure. It was a place for backhanders. These fell into two distinct categories. There were the bribes to ship's runners, trimmers, even some mates and skippers.\n\nDeckhands and others looking for work often backhanded the runners and these bribes could range from a couple of pints to a couple of pounds. Trimmers, the men who prepared the fish kits on the dock before auction, were given money by skippers and mates to ensure the fish kits from their trawlers \"looked their best.\" This kind of bribe was more usually a dockside or shipboard activity.\n\nAll this happened along the L-shaped bar, each part of which exceeded twenty feet, or at the round polished wood tables with ornate wrought iron legs, around which folk sat on cushioned, wooden stools or on the upholstered benches along the walls.\n\nIt was always crowded and the heavy blue fug, tinged with nicotine brown edges, permeated the huge crowded pub. The pungent tobacco cloud provided a welcome cover, disguising the various whiffs arising from the fish skinners, bobbers, fish processors, trawlermen and sundry workers who made up the clientele. Sometimes the cloud was so strong, it could be seen on the street as it stood like an amorphous misty sentinel on the two big doors, either side of the pub. Inside the two crystal chandeliers' brightness was progressively dulled, as the pub got busier. Sometimes there were cigarette packets or matchboxes with coins and notes inside passed under the tables or along the bars. On that day transactions were noticeably fewer. A shadow of tragedy hung over Rayner's.\n\nThere was another kind of \"backhanding\" at Rayner's too \u2013 a vital and rather noble kind. Shipmates recently \"settled up\" would look out for friends who were \"out of a ship\" and give them money to tide them over, often as much as a week's wage. But the giver knew that when he was out of luck or out of cash, that favour would be returned. It was an unofficial \"fishermen's mutual aid.\" It was known as giving someone an \"ovall\" \u2013 local slang for a cash loan. Old timers down on their luck were also given ovalls and bought drinks by the young turks, fresh ashore with more money than sense. The ad hoc social security system also looked after men who had been put on \"walkabout.\" A trawler man could be put on walkabout for any reason; arguing with his skipper, bad behaviour, or simply if his face did not fit. Men could spend weeks out of work on the whim of an angered skipper, mate or trawler owner.\n\nIt was a sanction used to great effect. The bosses knew that men were easy to replace. For every one on a ship there was more ashore looking for work. This was one of the reasons that the unions found it difficult to get a foothold in the industry, although men did join and, when forced, even paid their dues. One man who knew this more than most was the Hull Transport and General Workers' Union (TGWU) Fisheries and Waterways Officer Jack Ashwell. He seemed to spend most of his time chasing subscriptions. Trawler men held anyone outside of their immediate circle with suspicion. They did not get altruism. After all, they did not see much of it in evidence. It took Jack Ashwell years to gain and keep the trust of the men he represented. Ashwell, an ex-Navy man, a former tug man who became a bus driver and then a full-time union worker, did eventually win their trust. He built a reputation for working hard on their behalf, even if often they did not appreciate his diligence.\n\nJack's bosses, Hull Fisheries Officer Mike Neve and TGWU Regional Organiser Dave Shenton, based at the city's Bevin House, marvelled at Ashwell's tenacity in chasing the men for union dues and for his never-ending patience. His door was always open to a trawler man in need. Local National Union of Seamen firebrand John Prescott, who recently helped organise a Merchant Navy strike aptly described his TGWU comrade's task as being as easy as \"herding cats.\" Trawler men were rarely ashore long enough and when they were, paying union dues was pretty low down the list of things to do on their three or so days' leave.\n\nOne regular claimant of the ovall \u2013 the Hessle Road \"welfare system\" \u2013 was old Pigeon Smith. His real name was Walter and like most people and things on Hessle Road he was given a nickname. Legend had it that the wags in the bar called him after the suave Hollywood actor Walter Pidgeon but with his balding pate and weather-beaten, wrinkled face he was more of a Gabby Hayes. Everyone liked old Pigeon and he rarely paid for the pints that miraculously appeared at his spot on the main roadside part of the bar, near the big front door. He had been a deckhand whose best days had long since gone but from time to time even skippers took pity on him and took him on summer trips, where it was understood he would do little in return for his pay. These occasional trips \u2013 usually in the summer time \u2013 the ovalls he never repaid and the free drinks made Pigeon's life easier. His happy disposition, Walter Mitty bar room tales and easy manner assured him of some comfort. He was the trawlermen's mascot \u2013 and revelled in it. There was even one skipper, who every so often would kit the old fella out with a new suit from the thirty-bob tailor.\n\nAs the talk of the St Romanus and the Kingston Peridot did the rounds in Rayner's, so did talk of helping those who would soon need it. 'Hey up, Pigeon get yourself o'er here ol' fella,' said landlord Charlie. 'Now here, go earn your keep, you ol' bugger.'\n\nThe pub boss handed the old man a large, shiny fire bucket. In the old man's hands it looked like a barrel on a hoop. 'Aye, aye, sir!' replied Pigeon, adding a greatly exaggerated salute. Within minutes the zinc bucket being carried around the biggest bar in Hull was filling up with coins and notes. There were soon to be dozens of similar spontaneous fund-raisers all up and down the pubs, clubs and shops of Hessle Road.\n\nRayner's was a place of danger as well as camaraderie. Its party atmosphere could change in a heartbeat. The violence was not unusual but normally over very quickly. Like in the Wild West movies these men loved so much. A sing-song could turn to a bar brawl in a second. These men were away for weeks without booze and the drink went to their heads faster than most. It did not take much to get them drunk. And it did not take much to start a fight. Especially if there were Grimsby men in \u2013 or worse still \"Skrobs\" \u2013 the derogatory generic nickname Hull men had for any Scandinavian or Icelandic fisherman. Bar fights were settled quickly. Getting barred was never a good idea \u2013 for the fisherman or the publican. On \"settling day\" these men were the last of the big spenders. Police rarely got involved. After all the pubs did not want their best source of income locked up if they could avoid it.\n\nIf you were driving a car along Hessle Road that day it looked like business as usual, but if you were walking among them, you would hear samples of the chatter of the street-corner scrums that soon let you into the cause of the communal anxiety:\n\n' _My husband sailed with him.'_\n\n_'Poor Janet's lost her man, poor bairn.'_\n\n_'The Romanus should never have sailed. She's a death trap.'_\n\n_'The Kingston Peridot's gone too, sounds like she just took a sea.'_\n\n_'No, she's hit an old mine left o'er from t'war.'_\n\n_'It says in't paper that the Peridot was last heard from four days since.'_\n\n_'Aye, and I'll tell ye now. It won't be heard from again.'_\n\n_'Two ships gone...'_\n\n_'There'll be a third, mark my words, these things come in threes.'_\n\nSECOND TRAWLER MISSING \u2013 was the page one lead headline of the Hull Daily Mail \u2013 Big Air-Sea Search for Kingston Peridot. The down-page story told of hope being lost for the St Romanus.\n\nWhat the remainder of the city was reading in that afternoon's paper was not news on Hessle Road, rather confirmation of the community's fears. From early on that morning the tall, Crombie-coated figure of David McMillan had been spotted at different houses on the terraced streets off the main road and beyond. The bespectacled McMillan was officially entitled Superintendent of the Royal National Mission for Deep Sea Fishermen but he was better known as the \"Mission Man.\" A visit from him was never likely to be good news.\n\nBefore the 133,000 copies of that afternoon's paper hit the streets across the city and county, the rows of terraced streets, inhabited by fishing folk, which came off both sides of the long road going from the city centre and along the river were a-buzz with talk of the Mission man's visits. From above the area resembles a giant bending fish bone. The dock from where the trawlers departed and the kits of fish were landed, kept most of this urban \"village within a city\" in work. On this day, Hessle Road could not be blamed if they felt there was \"no one up there\" looking from above \u2013 or anyone down here looking out for them \u2013 other than their own.\n\nJanet Wheeldon, the 26-year-old wife of St Romanus skipper Jim Wheeldon had had an early visit from Mr McMillan. She was at home with her four-month-old baby son Paul and five-year-old daughter Beverley. In the corner of the living room of their cosy three-bedroomed semi-detached home was the near-completed dolls' house that Jim was building for his little girl as a place to house her recently acquired Christmas dollies. Janet's nerves must have been like piano wires. She would have known as a skipper's wife the adage about the strange things that happen at sea. That had to be held onto, almost as an article of faith. In spite of her foreboding, she knew there could be a shred of hope; even in the face of the evidence she had to believe that. 'The ship may yet be found,' she must have told herself, knowing this was probably self-delusion, but necessary to keep her going.\n\nThe GP prescribed her sedatives. Earlier she bathed the kids and got them ready for bed. The TV was on in the background, its volume low, so that she would not miss any call that may come in from the ship owners. She was on autopilot, the routine helping her cope. The sharp, loud phone bell rang \u2013 just one brrrrinnng! \u2013 and she lifted it from its cradle. The receiver was in one hand and baby Paul cradled in the other arm. Beverley was at her mother's skirt.\n\n'Hello,' she said and then announced her number.\n\n'Your husband will be back on Thursday.'\n\n'What? Who is this?'\n\nThen the hoax caller rang off. That day, a few hours after McMillan left, the phone rang \u2013 again she picked it up within one brrrrinnng! \u2013 and there was a replay of last night's call. She hung up.\n\nAt West Dock Avenue, Janet's father-in-law, ex-trawlerman Fred Wheeldon, was driving to the dockside to catch up with some mates. He had not seen the newspapers or listened to the radio yet. As he was driving past the row of trawler owners' offices, a skipper he knew waved at him and Fred rolled his car's window down. The skipper shouted.\n\n'Hey Fred, they're winding down the search for your Jim's ship!'\n\n'What? Tell me more.'\n\n'I can't, I've just heard about it at the offices.'\n\nFred swung his car round and headed for home. As he approached his house he saw a Hull Daily Mail A-board outside his local newsagent's. It read:\n\nSECOND TRAWLER MISSING, HOPE FADES FOR THE ST ROMANUS\n\nHe was home in minutes and did not take off his coat. He called for his wife.\n\n'C'mon Madge. We'd better get down to our Janet's.'\n\nMinutes later the Wheeldons walked into their son's house in Hessle to see their five-year-old granddaughter Beverley sitting on a chair. She was crying. Their daughter-in-law Janet was cradling their baby grandson. The young woman was in tears also. Beverley ran to the arms of her grandfather.\n\n_'I didn't make mummy cry, grand-dad. I never made mummy cry!'_\n\nIn a Hull city centre jeweller's shop, newly appointed 15-year-old shop assistant Yvonne Hartley was keen to make an impression in her first job since leaving school. She had been asked to go and fetch some engravings from another shop across town, so Yvonne thought it a good idea to go on her lunchtime and return with the goods and receive the praise of her new boss. As she came out of the shop and walked along the city's Whitefriargate main shopping street in Hull's Georgian Old Town area, she saw a newsstand with a billboard on which 'ANOTHER TRAWLER MISSING' was written in black poster paint. The shop girl, whose father Alf was second engineer on the Kingston Peridot, was compelled to the newsstand. She checked her purse, sixpence left, her bus fare home. She bought the paper. As she read she became more and more hysterical, the warm tears gushing down her face. She could hear herself talking, then shouting as she walked aimlessly. Next, the confused, sobbing girl found herself in the middle of the packed shop from where she had set out on her earlier errand. Now she was shouting louder. Her workmates were staring. Customers looked embarrassed for her.\n\nThe girl was screeching out, 'What the hell are them newspapers doing? They're talking a load of rubbish. Why don't they just leave us alone and let them come home? Old Neptune's always been nipping at me dad's arse! Them papers know nowt.'\n\nShe then felt a gentle hand on her shoulder. It was the old gentleman who was her employer. He led her by the hand out of the shop and into his waiting car.\n\n'C'mon my dear, let's get you home. You need to be with your family. Come, let's get you back.'\n\nBetween sniffles and tears, the teenager could not help but think in an odd, brief, reverie, 'What will the neighbours think when they see ME in a posh car like this?'\n\nThe car pulled up outside the terraced house's front door, which opened onto the pavement. The old man walked with the girl to her door. As the old man walked away, the young girl ran crying past her mother, and up stairs to her room. 'C'mon our Yvonne,' pleaded the distraught mother. 'Do you want to come and have a cuddle?' From behind the door the sobbing girl replied, 'No Mam, I'll do that when me dad comes home and I've told him off for doing this to us and playing them bloody silly games!'\n\nMargaret Wilson \u2013 the wife of the Kingston Peridot's skipper Ray \u2013 was at the coin-op launderette on Hessle Road. She had a lot of washing to catch up with and was yet to own her own machine but that was not entirely why she used the coin-op. Margaret liked the launderette. It gave her the chance to catch up with old friends for a gossip while she worked.\n\nHer Mam watched the little ones while she was there for an hour or so and it gave her time to herself. So, when she spotted her friend Rose, Margaret was cheered. But that changed almost instantly.\n\nHer friend looked subdued and a bit agitated. She could not make direct eye contact with Margaret.\n\n'You should be at home, you Margaret,' Rose said.\n\nThe skipper's wife felt a deep dread. She did not have to ask if something was wrong. Rose's look told her. Margaret turned and left for home.\n\nWhen she got there, her sister had joined her mother. It was obvious the women were heartbroken.\n\nThere was no need for them to talk.\n\n'It's Ray, isn't it, Mam?' Margaret said.\n\nThe pretty young mother-of-four broke down.\n\nIt was to be another year before she was able to leave home unaccompanied.\n\n_Northern Iceland, off the coast from Isafjord._\n\nFrom the bridge of the 659-ton sidewinder Ross Cleveland, Christmas in Hull seemed a long way away for Skipper Phil Gay. The time spent with his family in the closing days of 1967 and a few new ones of 1968 were something of a milestone for the trawler man. It had been his first Christmas at home for years. That year he did not have to go through the yuletide tradition of giving his 11-year-old son Philip and his six-year-old daughter Lorraine their main presents early, allowing him to join the trawl.\n\nThe trawler man's life was not conducive to family harmony, with only three days home and three weeks away. Across a year of full employment he could spend as few as thirty-six days at home. The bout of bronchitis that laid Phil Gay low that festive season meant that he had spent most of his time at home struggling for breath, dressed only in his dressing gown and slippers, but at least he was with Theresa, his wife of seventeen years, and the kids for the holidays. His tough life paid for the tidy, newly-built house in Hull's western suburbs. Although physically fewer than three miles west, Anlaby Common was a very long way from where Phil and his eight siblings were raised in a seafaring family. He loved Hessle Road and if it were up to him he would still be in the little terraced home he and Theresa had bought outright for \u00a3450, earned from his first trips years earlier. Their little love-nest where first they settled when wed, where, Phil, like his father before him, could be within walking distance of the pubs and clubs that proliferated, where he could walk to the betting shop in his slippers.\n\n'Who needs a posh house? I have all I want here, lass,' he would tell his petite, pretty wife.\n\nTheresa Gay was a Hessle Road girl who was ambitious for her family. Like most trawler men's wives, she steered the better course ashore. She saved spare cash from Phil's earnings and from her work on the Smith and Nephew production line, where she helped to make bandages and sticking plasters. Her prudence and their hard work brought them to the suburbs.\n\nThat Christmas spent at home, being in his dressing gown, struggling to breathe but regaining strength, gave way to the Gay family's annual New Year's house party by which time Phil was getting better, but not fully recovered. Theresa had prepared a buffet feast. The booze flowed as family and friends from Hessle Road, other skippers, new neighbours on the new-build estate, mixed all together, joining in Auld Lang Syne to bring in 1968. In the early hours the party was slowing. Phil was drunk, melancholy and sharing his woes with anyone who would listen. This was most unlike the normally jovial, happy drunk that Phil usually became on the rare occasions that he drank too much. The remaining guests shuffled uncomfortably.\n\n'You know, I do not think I will see my children grown up.' His eyes misted as he spoke. Theresa was there in seconds, sitting on the arm of his chair. 'Just because you've had too much to drink there's no point in talking daft like that. Come on, now that's enough silly talk.'\n\nHe pulled her to him and looked her directly in the eyes.\n\n'I am not afraid of it you know, because I know that you will know what to do.'\n\n'Don't talk so daft,' said Theresa and with that she buried the first bad omen of 1968. Three days later the trawler's owner Hudson's called, 'Phil, we want you on the Ross Cleveland. You'll leave on the 20th.'\n\nAt home in Cottingham, near Hull, Harry Eddom, who was to be mate for the next voyage was expecting a call from Phil Gay. There was a mutual respect between the younger and the older man. It was Harry's job to round up the men for the voyage. After the brief call Harry set about the job of putting a crew together. He would have been quite chipper at the prospect of sailing with Skipper Gay and of being on that ship. Harry must have hoped for a repeat of the Ross Cleveland's last trip out. He knew she had landed a bumper payday. The trawler had pulled in \u00a35,000 after a twenty-two day trip. The stand-in skipper got his ten percent and the mate got the rate of 6.5 percent share of the catch \u2013 \u00a3325. Even the deckhands got \u00a3100 each on top of their weekly \u00a312 wage. It was an excellent trip by any standards.\n\nSkipper Gay was a well-respected man, admired and feared in equal measure. He was something of an eccentric in the eyes of his fellow trawler men. He never used foul language, no matter the pressure. Most Hull trawlermen knew Phil Gay's catchphrase, \"Stab a sausage!\" which he used when others would use profanity. This verbal restraint was rare among men whose conversations were punctuated by obscenity. And the 41-year-old had not come to the rank of skipper by the normal route, either. Usually, a trawler man will start as a boy, in the role of \"deckie learner\" or galley boy, then work his way up. In this, the Arctic trawl was a meritocracy.\n\nHard work, diligence, courage and resolve made some men rise from the role of deckhand, to the chance of a bo'sun's ticket, after that came mate, then skipper. Other officers on a trawler were the radio operator (the sparks) and the chief and second engineers. The cook and the galley boy were not just there as providers of food but were also expected to help gut the catch and join the battle against the trawler man's worst winter enemy \u2013 ice, that in minutes could be inches thick, causing the ship to become top heavy and topple in fierce winter storms. This was known as 'turning turtle.' In extremis at sea, all were expected to chip in to fight the ice. That was understood, as was the fact that cooks would rarely rise above their rank and that engineers would never go further than chief, as a rule, (and rarely wanted to \u2013 the engine room being warm and rarely unattended).\n\nIn this, Skipper Gay was an exceptional man. Shortly before his thirtieth birthday, he had left National Service with the Royal Army Medical Corps, in which he had been a cook. He decided, as his engagement was now announced, that he would go to sea as a cook to get money that could not be earned on land. Although basic pay was on a par with those on shore, poundage bonus, a small percentage of the net catch could often give a man a month's wage in a week, and more.\n\nThe fastidious and assiduous Philip Gay's success as cook was not as immediate as his later prowess seemed to indicate. On his first trip, a dozen hungry deckhands, ravenous after hours on a freezing deck came to the galley expecting their food waiting for them.\n\n'Fuck me,' said one. 'Look at him, soft lad. He's fucking scrubbing the table! Just get the fucking grub up, daft lad.' Philip's hygienic approach and skilful catering, while a hit in the officers' mess at his former base in Munchengladbach, was not, it seemed, _de rigeur_ here.\n\nWith a few more trips as cook, (with table manners learned) and money coming in, Phil felt a life at sea would be for him. As luck had it, Theresa's Uncle John, who taught at the local nautical college, got Phil on a course to do his bo'sun's ticket. It was hard work and a couple of months without pay but Theresa helped. By the end of 1957, Phil Gay was a bo'sun and soon being noticed as a future skipper.\n\nThat day in command of the Ross Cleveland with the knowledge that forty of his mates had surely perished, Skipper Gay's stoicism, resolve and seamanship were at a premium. It had been ten days since they left Hull. Five of those had been spent steaming towards the first fishing grounds north east of Iceland, but that trawl was abandoned, as the weather grew worse. Skipper Gay steamed his ship across the north west coast of the island away from the weather. He tried to trawl once more but the weather caught up and forced him to stop again. It was now a chase, with the increasingly fierce storm bearing down on the ship and Skipper Gay was determined to escape it.\n\nSuperstition shadowed all trawler men, even the toughest like Phil and his mate Harry Eddom. Both knew the dangers of the sea but had to believe a cruel Fate would befall the other fellow, and never them. They could not even contemplate being the 'third' that would prove the old adage of bad things happening in threes. This kind of mindset could be fatal; a drift in concentration was something they could ill-afford. For ten days in, after quickly worsening weather, fishing was abandoned.\n\nStaying afloat and finding a safe haven were the priorities now as the ice gathered on rails and riggings, battled by axe-wielding deckhands, engineers and cooks. Even mate Eddom and the bo'sun Wally Hewitt were engaged in the fight on deck. Eddom was atop the bridge, clearing the ice from the radar scanner.\n\nTwo days earlier the Ross Cleveland had gone briefly to port at Isafjord to put ailing cook Bill Howbrigg ashore. The 59-year-old had been struggling for breath for days. This would be bad enough in a normal trip, but axing ice off the decks as 60ft waves crashed, the salt water icing on contact, alongside winds of up to 80 mph, was proving too much for Bill. Fishing had been poor. There were only 400, one-stone (14lbs) kits of fish in the hold, 560lbs in total, a fraction of the 2,000 kits it could contain, a catch that thus far would not cover the ship's fuel costs. Skipper Gay had offered to take the old man ashore days ago, but the stubborn old hand was resolute.\n\n'I'll be fine, skipper. It's just me bad chest.'\n\nPhil knew from his own yuletide illness that Bill needed medical help. He liked Bill and did not want to hurt his pride, but the cook had to go, what was later diagnosed as pneumonia was gripping him fast. Cooks and skippers were generally on great terms, after all the man from the galley was usually the only bringer of joy to the bridge. Food had an inordinate value at sea.\n\nAs the distinguished sociologist Jeremy Tunstall noted as part of his research into the Hull fisheries of the 1960s:\n\nThe cook [...] may also have a special relationship with the skipper. Some when at the fishing grounds eat mainly on the bridge and this involves the cooks in personal catering for them. It can thus happen that the cook becomes a favourite of the old man [...] Normal outlets such as sex and drinking are not available on a trawler and consequently food has to carry an immense burden of emotional freight.\n\nWhen the Ross Cleveland headed in to port in Isafjord, the old chap made the skipper smile, when he came on to the bridge. 'Hey, skipper. I have made some fresh bread for you and the lads, before I go.'\n\nIn spite of the calm appearance of command necessary for survival, Gay and Eddom were worried men. After dropping old Bill the cook ashore, and another try at trawling some 100 miles north-east of Iceland, it was decided to stop fishing.\n\nThe weather had beaten them. But it would be too dangerous to head home, as the snow, wind and storms of the open seas to the south were sure to claim them. Skipper Gay decided to head back to the fjord of Isafjord and hopefully shelter in the lee of the land. More than 20 other ships had done or were trying to do likewise. The area was known among trawler men as either the 'car park' or the 'hitching post'. The Ross Cleveland was turned around to try to get to the fjord haven. Safety was now the priority. Phil's friend from Hull, Skipper Len Whur of the Kingston Andalusite was half a mile or so ahead. Both skippers knew the storm ahead was getting worse, maybe even to become the worst. All hands were in the constant battle with ice, which was at times up to six inches deep on three-inch circumference wire ropes and thicker on the ship's superstructure and radar equipment. The men were so far keeping the ice at bay, with their axes, lump hammers, knives and even the marlinspikes they used for repairing nets.\n\nThe pounding of waves the size of a house crashing into the Ross Cleveland was punctuated by the sonic-boom cracks of huge slabs of ice as they smashed on the deck before being dumped overboard by the frantic men. Icy winds and ferocious snows kept the ice coming; it was a seemingly never-ending fight. Skipper Gay would need all his focus to keep on course to return to the relative safety of the fjord. It would do him no good to be thinking of home. The wind, gusting at 60-80 mph \u2013 and randomly changing directions \u2013 could just as easily send the ship to the bottom as the giant waves hitting it or the recurring ice constantly threatening to turn it turtle. They had about sixty miles to go to the fjord, at some points; even on full ahead the speed was little more than one knot per hour.\n\nSkipper Len Whur on the Kingston Andalusite was still between half and three-quarters of a mile in front of the Ross Cleveland. His ship was reeling, pitching and rolling too and both trawlers dodged into the fluctuating winds, heading for the fjord. 'Dodging' is the word skippers used for steering a ship directly, bow first into the wind. Although this slows the ship to a crawl, it prevented it taking a high wind or a huge wave sideways on, which could make the trawler capsize. Better late \u2013 and safe \u2013 than never.\n\nFor the twenty or so ships heading for \u2013 or already at \u2013 the 'car park' and for those fighting to get a safe anchorage, the battle to stay safe became constant and fiercer. All radio sets on those ships simultaneously got the message from Wick Radio coastal station. Hurricane winds were on the way.\n\nAll Icelandic fishing was abandoned. All shipping that had not already done so headed for shelter. The cacophony of high winds, cracking ice slabs and battering waves forced both Skippers Whur and Gay to shout their respective radio messages to each other,\n\n'Follow me, Phil. I can just about see your lights.'\n\n'Will do, Len. I'm going to have to keep dodging.'\n\nBoth men \u2013 and their crews \u2013 had begun the battle of their lives.\nFOUR\n\nFish house gossip kept workers' minds off the bitter cold. It also provided respite and sometimes laughter amid a relentless, freezing grind. There was no laughter today. The talk inside was still the same as it was outside on Hessle Road. The temperature inside was the same too \u2013 but felt colder. Flecks of fine sleet came in on the wind that whipped through from the delivery and dispatch bays into the hangar-like space where the filleters and skinners worked.\n\nConstant layering of ice on hundreds of finished kits by hand throughout the day compounded the discomfort. Rinsing fish scales and blood spots with constant streams of cold water kept bare hands further chilled. This \"indoor work\" was so open to the elements it might as well have been carried out in the street as far as comfort was concerned. At least they had a proper roof, unlike their counterparts on the dockside, who had to put up with rainwater dripping down their necks, snow blizzarding into their faces or hail hitting them, depending on the weather.\n\nIn the fish house the occasional immersion of hands in buckets of warm water kept near the skinners' tables gave brief respite from bone-chilling cold. There was no heating as such. That was bad for the fish. Intermittent dulled rhythmic stamping of clog or welly clad feet punctuated chatter as the women fended off the harsh stinging cold. All women wore white aprons and headscarves. Outside clothing was kept on too, for all of the eight to ten hours worked. Coat sleeves were rolled up. Gloves were not used. Hands had to be nimble as possible. Women talked loudly to be heard above each other. They did not look up often. There were no hand gestures. Focus was intense; if the razor sharp skinning knife slipped it would cause a gory injury. Worse still, it stopped the skinner working. Any cut that required more than an Elastoplast meant the woman would be sent home.\n\nSkinners worked in teams of six and sat around tables being constantly supplied with fish from the men on the filleting tables. Filleting was men's work. Some of the fish brought in for processing were bigger than men. It paid double the amount a woman earned. It was a semi-skilled job.\n\nWomen were considered not as strong but more dexterous and therefore better suited to skinning. Novices had to learn quickly. If not they would not be employed.\n\nCod skinning was intricate but repetitive, the more practised the skinner was the faster they could fill a fourteen pound (one stone) kit ready for delivery. Each fillet was laid down with the tail pointing towards the worker. An incision was made into the flesh at the tail end.\n\nThe knife was held at an angle and cut the flesh away from the skin. It was then kept almost flat against the skin while cutting. Then in one seemingly effortless move, the skin was separated gently from the flesh as the blade moved from side to side.\n\nThe cod tail was held in one hand and the skin cut off, always away from the body of the skinner. If cod came in with its head still on, this was removed first and put in a bucket with bones and other detritus to be taken to the fishmeal factory, where, among other things, it was used for pig food and fertiliser. Good cod skinners made it look easy as in seconds they appeared to simply stick a knife in, jiggle it and take the skin off in one movement, like peeling sticky back plastic. Lily Bilocca's table had the small benefit of being not far from the puny paraffin heater near the works' office door. Her twenty-odd years in the trade taught her that any heat was welcomed in this fridge of a factory. Between kits, she and others would stand in front of it for a few seconds. She was nearing the end of a nine-hour shift that started at seven a.m. The other girls on her table spoke of little all day other than the missing ships. Lily joined in but there was none of her trademark booming laugh. To the girls on the table she was subdued, not herself.\n\nNo time was wasted when the horn signalled the shift's end \u2013 Lily's last at Wilkinson Bros.\n\nIt would be two years before she worked again.\n\nIn the winter's late afternoon darkness she joined the throngs of white-clad fish-house women converging from the various processing plants.\n\nThey moved in hundreds toward the main thoroughfare. Even clad in white and walking through slush, that hours earlier had been crisp snow, Lily stood out as the home-time crowd moved like a mass of pilgrims surging towards a shrine.\n\nHer seventeen-stone frame set her apart from her mostly waiflike workmates \u2013 like a Callas amid a throng of Piafs. At the end of Wassand Street most turned left and headed west to the terraces of two-up, two-downs. Lily turned right towards the city centre end of the \"Road\". [Locals rarely used the \"Hessle\" prefix and with the Yorkshire accented version of the phrase, it sounded as 'Road, with the \"t\" which replaced \"the\" barely pronounced.]\n\nIt was a ten-minute walk home from the fish house for Lily \u2013 gossip permitting. Sometimes it could take up to an hour if she got caught chatting. She was freezing cold and had had enough. Lily made a series of excuses to those who stopped her for a chat on the way from work. She just wanted to get home. She had work to do. An idea was developing. She had been fretting all day. She tried not to show it.\n\nLily's 21-year-old son Ernie was on a trawler in the Northern Cape of Iceland and her other half, Charlie, was aboard a cargo ship off the Scandinavian coast. 'Were they in the same places as the missing trawlers among that horrible weather? Something has to be done,' she told herself.\n\nLily pulled the collar of her coat up and tightened her headscarf against the sleet and wind as she turned to Coltman Street with its mix of detached mid-Victorian town houses and villas.\n\nThis was the wide boulevard style street that had provided homes to trawler bosses, skippers, bank managers and their ilk for the past century. Now, like a fading dowager, in the second half of the twentieth century the grand houses were in decline.\n\nAt No. 117 she turned left and went up the three steps to the entrance between the two Doric columns of the double-fronted house.\n\nIn her white coat, Lily looked like a reversed silhouette as she rested and fumbled for the key to the big black front door.\n\nIt was not any warmer inside. The long entrance hall that led to the two-way staircase was cold as stone. By the light of the hall lamp Lily could see her own breath. She took off her coat and scarf and pulled off her wet Wellington boots, put on her slippers and walked past the six ground floor rooms \u2013 three on either side, past the large kitchen and pantry room and into a small living room which overlooked the large rear gardens and soft fruit orchard.\n\nShe slumped into a big armchair near the French windows. The coal fire was blazing. Its warmth enveloped her.\n\n'Is that you, Mam?'\n\nLily's 17-year-old daughter Virginia had heard the small room door close during an interval in her record playing.\n\n'Aye, love. Put the kettle on, our Virginia.'\n\n'Right-o mam. You're home early. Were 'Road empty?'\n\n'No. I just can't be doing wi' them all today. I've got too much on me mind.'\n\nLily picked up a large white A4 writing pad and a Biro from a small table next to her armchair.\n\nVirginia popped her head round the door.\n\n'D'you want owt else?'\n\n'No love, tea'll be fine. Big mug. Two sugar. And shut the door you'll let all the cold in.'\n\nWhen Virginia brought the tea tray in her Mam was looking preoccupied as she scribbled on the big pad.\n\n'What you doin', Mam?'\n\n'Right Virginia, enough's enough. Something has to get done. I'm starting a petition to get the gaffers to make them trawlers safer. That could be our Ernie or your Dad out there, God forbid.'\n\n'A petition... is that a good idea?' said the girl.\n\n'I am goin' to do it our Virginia. I have got nowt to lose, have I?'\n\n'Folk might get upset Mam.'\n\n'To hell with what they bloody think. I'm goin' to do what's right.'\n\nVirginia looked at her Mam and she just knew she meant business. She could almost see the ideas bubbling in her mother's head. Before the girl could say anything else the doorbell rang.\n\n'I'll get that,' said Lily.\n\nVirginia was quite surprised. When her Mam got settled she rarely moved unless getting ready to go out. Lily passed Virginia in the hall and had her writing pad with her. The frosted glass either side the doorway distorted two shadows that Virginia could see from the foot of the dual staircase that led to the upper two floors.\n\n'Ey, two birds wi' one stone,' said Lily.\n\nThen there were some pleasantries that Virginia could not quite hear.\n\n'Ta-ra love, ta-ra,' said Lily as she closed the big inside front door.\n\nShe showed her daughter the A4 paper tablet.\n\nThere were a couple of signatures at the top of a page.\n\nThe Man from the Pru and a Littlewood's Pools agent both signed the petition offered to them. They were certainly unaware of being in on the ground floor of an historic campaign.\n\n'I've started me petition then, our Virginia,' said Lily as she headed back to the small back lounge \u2013 one of five fully furnished rooms of eleven.\n\nVirginia was quietly pleased at her own handiwork and thought the double-spaced lettering on the petition form was a neat piece of typography \u2013 her signature piece as it were. 'I'll get in early tomorrow Mam and get some more forms done for you before everyone gets in.' Her bosses at CR Rosen and Son (Shoe Manufacturers), where the girl was an office junior, would have fired her on the spot if they had found out she had been drafting and copying off petition forms for her mother using their Gestetner.\n\n'Thanks, love. I might take it down The Cri later. I bet there will be loads in there.'\n\n'Going to the pub on a Wednesday night, Mam? What about work?'\n\n'What about it? Nobody's bloody doing owt. Somebody has to. It might as well be me.' Lily looked at the two signatures on the petition which was headed, PETITION ON BEHALF OF FISHERMEN'S SAFETY AT SEA AND SAFE WORKING CONDITIONS IMMEDIATELY, - NO SAILING WITHOUT FULL CREW.\n\nShe picked up the writing pad, her mind a swirl of ideas. She had to write them down. She would front the bosses and write to the papers. She scribbled...\n\n_You men at the \"top\"-_\n\n_these lads spend three-quarters of their lives under conditions few of us can imagine or even stand._\n\n_Believe me, I and my family know personally of a great number of these lads in past years and also very recently whom [sic] have died to feed us \u2013 and provide a living for us who work ashore._\n\n_I have recently attended a service on behalf of my husband and my son as they were at sea [...]_\n\n_And seen a man try to control his sorrow then suddenly burst out in uncontrollable sobbing, it is a very upsetting sight._\n\n_These lads live hard. Work hard. Die hard._\n\nVirginia, the dutiful daughter, interrupted her Mam with more tea and a plate of custard cream biscuits, her Mam's favourite. When the girl returned to her room \u2013 and to her record player \u2013 Lily returned to her writing after admonishing the girl to turn the \"bloody boom, boom music\" down.\n\nLily put her notes to the trawler bosses to one side and started a new sheet...\n\n_To editor, (Hull) Daily Mail_\n\n_I am taking the liberty of_\n\n_Sir, I am writing on behalf of many mothers, fathers, wifes [sic] and myself [...] it is time something was done NOW about a trawler sent to sea with no radio operator, also the radio being in so poor condition it breaks down a couple of days out at sea [...]_\n\n_You may say the skipper or other crew have training of the radioing. That is not what matters, those men have enough to do regards there [sic] own safety and their ship mates without added responsibility [...] and all ships should be stopped immediately from sailing unless they have a full crew and that the ship is safe [...]_\n\n_I am ready anytime from now [...] if I am to be removed it will have to be with force. I am very able to carry on and finish what I have to do._\n\n_I am not afraid of any proceedings or verbal threats [...] so don't think you can deter me. The only way I can find out if any ship is sailing under the above conditions is by appealing to you fisher lads by letting me know [sic] straight away even if it is only a whisper. You can find me at work during the day, after 5 o'clock at home. I am ready at a moment's notice to take action._\n\n_I give you my honest promise I will not reveal anyone's name that has given me the information of the ships not under any circumstances._\n\n_I am only asking for some support and fighting back with the only weapons I possess, that is words and action. I feel I have to do this to settle my conscience whether I am right or wrong. I am going to stand by my own convictions. Please give me the information lads and support..._\n\n_Thanks_\n\n_L. Bilocca_\n\n_117, Coltman Street, Hessle Road, Hull._\n\nLily read her notes and decided to revise them again later. The heat from the coal fire filled the small lounge, and the music that came from Virginia's room was not as \"boom, boom\" as it was earlier. Lily drifted into sleep. She woke with the draught from the big, cold, entrance hall. Virginia was in the doorway.\n\n'I thought you was goin' to The Cri, Mam? You'd best get a move on, if you want to make the last hour.'\n\n'I must have drifted off, love.'\n\nNext morning at six o'clock Lily was at the corner of Coltman Street where it met the main road. Her hour and a bit at The Cri (a.k.a. The Criterion pub) had been more successful than she could have imagined. No one refused to sign. Several women even promised to help. Her petition was building fast. She stood in the fresh early morning snow, ideally placed to catch the bobbers on their way home and the fish house workers on their way to start their shifts.\n\nThe petition's layout was simple. Under its title were three columns headed, Name, Occupation and Address respectively.\n\nAt seven a.m. at Wilkinson Bros (Fish Processors) there were three empty chairs at one of the skinning tables. 'Where are Lil, Mavis and Rose?' The chargehand's question went unanswered. By mid-morning Lil and her two workmates had been joined by dozens more women. She had at least two hundred signatures on her petition sheets alone. The ad hoc headscarf army of fishwives marched up and down The Road, its pubs, shops, offices and houses. Folk were stopped in the street too. The petition's \"Occupation\" column clearly showed it was not just fishing folk that wanted action. The city was gripped by a growing anger like never before; the list read, \"schoolmaster, manager, housewife, labourer, meat salesman, bobber, office worker, printer etc.\"\n\nThe majority had \"housewife\" next to the signature.\n\nThat afternoon Lil's army of signature hunters was still hard at work. One would-be signatory stopped her in her tracks.\n\n'Hey Lily, did you get some of these signatures in 'pub?'\n\n'Some. I were at The Cri last night and we've been round since opening time at 'others.'\n\n'I can see that.'\n\n'What d'you mean, love?'\n\n'Some wag from the trawler office signed as George Bernard Shaw.'\n\nLily and the man laughed.\n\nThere really was a G.B. Shaw \u2013 and he worked in the offices of Hellyer Bros!\n\nThat night, in the wee small hours, cabaret singer Yvonne Marie wrote down a maze of thoughts that had been keeping her awake. Her notebook was lit by a little pool of light from the small lamp on the table. The glamour girl with the bouffant blonde hair and sequined dress cut an odd figure in the semi-darkness. She worked quietly. She did not want to wake her husband or the kids. She wrote by compulsion. She had read about the wives' petition and wanted to help but felt more was needed. She just did not know what. There were too many things wrong with the industry that had widowed her mother. Somehow she had to get her concerns across. She was not sure how, but for now she would write her ideas down as she had been doing for days. Yvonne Marie was a night owl. Since the news of the St Romanus, the club artiste \u2013 a.k.a. 28-year-old housewife and mother-of-three, Yvonne Blenkinsop, had not slept.\n\nHer hours were odd compared with other young mums in her suburban street, but she had always managed to get the kids to school and husband John to work.\n\nLately those duties fell to her husband. John knew why and didn't complain. His wife was in a daze. Four years ago her trawlerman father perished at sea, aged just forty-eight. Yvonne had to help her Mam care for five siblings at the family's Hessle Road two-up, two down as well as care for her own young family.\n\nJohn knew the sinking of the \"silent trawler,\" the fate of the Kingston Peridot and all the stuff in the papers brought it all back for his troubled wife. She was at that table more and more \u2013 writing, or praying, or crying, or reading her Bible even on the nights she was not working. Yvonne was not one for organised religion, but she often prayed. The room was dotted with Catholic iconography and \"holy pictures\". She liked the art and the comfort it gave.\n\nAnyone who read her notes would not have made much sense of them:\n\n_'No youngsters on winter trips... better training... RADIO operators!... Write to Marconi company about beacons for when ships sink...women have to stand up for our men...they are too busy at sea...No fishing in bad weather...what about some aircraft technology for ships?... NO CHRISTMAS CRACKER CREWS!_\n\nUnder each heading Yvonne listed ideas as they came into her head. There were names of owners, politicians, union men and others to write to. There were notes for letters, notes on how fire extinguishers could be made more efficient, notes on anything that came into Yvonne's worried, sleep-deprived mind. She had not slept all through the night for weeks. Her Bible lay near her notebook and pens.\n\nShe did not hear John approach. He touched her gently on the shoulder as he had on the other nights, so as not to startle her. John said, 'C'mon, love. You really want to stop this y'know. You are down here at all hours. You've had no sleep for days.' His voice was soft. There was no chastisement in it. Yvonne glanced at her Bible. She said, 'I feel I am being led. If something is wrong, you have to do something about it. I don't know what. I know I have to do what's right. I am not at work tomorrow. Maybe I can do something then?'\n\n'Yeah,' smiled John, '...you can get me my birthday present.'\n\n'I am sorry, pet. I forgot... and I never forget birthdays. I'll do something special tomorrow, I promise,' said Yvonne. A few hours later John, a master carpenter, got up, took the kids to school and himself to work as Yvonne caught up with much-needed sleep. When John joined the thousands of others on 'Road that morning, it seemed every second person was a headscarfed woman wielding a paper and pen. Lily's \"army\" had grown at a rate of knots. Hundreds of women had gathered thousands of signatures in the few days since Lily left the fish house.\n\nOne of the women, Chrissie Smallbone, joined Lily on day one. Most women on the Road knew her better by her maiden name \u2013 Gay. Chrissie came from a big Hessle Road fishing family. She knew this thing was growing and she got in touch with the union man, Jack Ashwell. Ashwell and his comrade Mike Neve had been watching the women. Their rapid progress did not go un-noticed by the union men or more importantly their bosses, who saw the wives bring more attention to the fishermen's plight than they had ever been able to do.\n\nThe national Transport and General Workers' Assistant Executive Secretary Jack Jones told his men in Hull to make themselves useful to the fighting fishwives. The union had been campaigning for better trawler safety for the past three years with little success. Ashwell had met Lily for tea and custard creams at her big old rambling house and told her he was keen to help in any way he could. Ashwell phoned Chrissie to tell her that his union was contacting other \"interested parties\" and that the women could hold their first \"grievance meeting\" that Friday at the Victoria Hall. He added he had news of a possible big development for them \u2013 a meeting with two Cabinet ministers \u2013 and would tell them more at the meeting. The union man told her how proud he thought her husband would be of her and the other women. When Chrissie put her phone down in her little terraced house in Hessle Road's Flinton Grove, she had to smile. Her trawlerman husband did not even let her go to the cinema unless he was ashore. She would be in big trouble when he got back from sea. Like a lot of the women, she feared the wrath of an angry husband, chastising her for \"dabbling in men's business\", more than she feared any fight with the bosses. But she was determined.\n\nAll her family was in the fishing industry. Her 16-year-old nephew Robert was on the St Romanus, which was doomed, barring a miracle. Her older brother, Philip, was a skipper and was still at the North Cape. He was in command of the Ross Cleveland.\n\nThe Kingston Peridot's cook Bill Good, who Skipper Ray Wilson had sent home after his shipboard accident, had been back little over a fortnight. He went ashore at Reykjav\u00edk on January 14 and flew to Hull three days later.\n\nThe little front room of his terraced house in Edgecumbe Street had never been so crowded. Reporters and cameramen filled it. They jostled and quizzed the bewildered trawlerman. The 58-year-old seaman, who had spent 43 years at sea and was a veteran of the Normandy Landings, seemed dazed as he spoke. Between questions he kept repeating, 'I can't express how I feel that I had to allow another man to take my place.' Then he started to tell his tale and the questions stopped. There was no prompting and the newsmen listened intently. Bill told them, 'I signed on for my first trip on the Peridot but shortly after leaving the Humber we ran into heavy weather. The trawler lurched and I fell headlong down a companionway, landing on my head. My back and my chest were badly bruised and the skipper insisted I should have medical treatment and an X-ray at Reykjavik. I wanted to carry on but an Icelandic doctor said I would not be fit for several days and a replacement was requested by the skipper.\n\n'They were a fine crew to work with. I knew them well. In fact, when I left the trawler, Mr Harry Fowler \u2013 one of the best \u2013 insisted on looking after my gear for me until the end of the trip. No one can imagine the awful shock I felt when I opened my Hull Daily Mail last night and read about the Peridot.'\n\nHis story was in that afternoon's paper under the headline, _He's the luckiest man in Hull._ Bill felt more guilty and sad than he did lucky. Surrounding Bill's story were snippets of tribute to the lost men. The news team of the Hull Daily Mail was busy. When tragedy struck on Hessle Road, the community did not regard the local press as intrusive, but rather as makers of tributes.\n\nOften a reporter and\/or photographer would arrive at a bereaved family's home to be met with a set of photos to take back with them. A community that had grown too used to the routine expected it. The Hull Daily Mail did not disappoint.\n\nSkipper Wilson's widow Margaret was too stunned to speak and was left in peace by the local paper. Her sister Lillian, who was there to mind her and the kids, told a newspaperman that her sister had collapsed and was too ill to be interviewed. [So the paper never learned how Margaret had found out about her husband's fate during a visit to the coin-op launderette on Hessle Road where she was met by a dread hush from the other women. She sensed something was wrong. One of the fishwives confirmed her fears and told of how sad she was to hear about Ray's ship. Margaret rushed home where her sister met her.] The reporter was given a photograph of Skipper Wilson in which the trawlerman was smiling widely. It had been taken at a family party. The happy-go-lucky image was incongruous amid the reportage of that day's newspaper.\n\nThe paper told how spare hand George Matfin had been married just three weeks before leaving for sea and how cook Harry \"Cockney\" Riches had been flown out to replace Bill. The mother of the mate Peter (Wiff) and his brother, deckhand Robert Smith, recalled how she and her sons' wives had been awake until 3am, desperate for news. The war record of Bill's pal Harry Fowler, torpedoed seven times during World War Two, was outlined and the youngest lad on the Peridot, 15-year-old galley boy Eugene Carney, did not like his first trip but signed up again, determined to be a trawlerman.\n\nRadio operator Martin Larsen's mother paid tribute to her 35-year-old son as a \"kind and gentle, generous boy.\" Ship's fireman Len Ledingham's wife Kathleen recalled their last Christmas together with their three-year-old daughter Kathleen and four-year-old Andrew. Deck hand Adam Ali, described as a \"19-year-old orphan\", made his home in Hessle Road with an aunt and uncle, Mr and Mrs. Hassan Mohammed before going to sea. Second engineer Alf Hartley was described as a devoted husband and father by his aunt, who was caring for his distraught young daughter Yvonne.\n\nSixteen-year-old David Warley had left Welton High School determined to find adventure at sea as a \"deckie learner\". Another two \"first-timers\" on the ship were forty-year-old father of four, Charles Blanchard, ship's fireman, and spare hand Peter McGowan, who was half Charles' age. Another twenty-year-old was Harry Heelas, an east Hull lad. The spare hand's mother was too upset to stay at home and moved in with her daughter Pamela Dunlin. The pair stayed by the phone. Pamela told the local paper, 'It's the waiting we cannot stand.' Her distressed mother Olive spoke briefly saying... 'It was his first trip on the Peridot. He bought a new suit before sailing this time...'\n\nShe then broke down in tears.\n\nOn the doorstep of Number Four, Norton Grove, off Division Road in the Hessle Road district, Kenny Swain's wife told how her young husband was desperate for money when he took his job as a deckie learner.\n\nRose Swain said, 'When we discovered I was having another baby in June, Kenneth said the only way to get money for the new baby was to go fishing. He had been out of work for a long time and none of the jobs going were things he wanted to do. I did not want him to go in the wintertime when the weather was bad. I asked him to wait until the summer but he would not listen.'\n\n_Friday, February 2, St Andrew's Dock, Hull._\n\nThis was the day the St Romanus and the Kingston Peridot were due back \u2013 but the local paper's front-page confirmed what the community feared,\n\n\"ST ROMANUS GIVEN UP \u2013 OFFICIAL\"\n\n\"TIME RUNS OUT FOR PERIDOT\"\n\nThe Kingston Peridot's sister ship, the Kingston Zircon was also due to sail from St Andrew's Dock that day. It was due to go to Iceland's North Cape. But just half an hour before it was to sail, three crewmen jumped off at the lock gates. The trio got T&G man Mike Neve to come to the ship. They told him there was no forward escape hatch and no lifeboat in the fo'c'sle. The remaining deckhands demanded Neve carry out a full safety inspection. When the skipper told his bosses on the Radio Telephone about the crew's refusal he was told three substitute men would be along soon to replace those who jumped ship.\n\nThe remaining crew would not sail, until Neve said the ship was safe. But it was too late. The dock master came to the skipper and told him to stay in dock. There was not enough water to carry them out. They had missed the tide by minutes. Neve would have plenty of time to examine the ship now.\n\nSkipper Robert Thompson was furious at the action of his men but there was nothing he could do. He felt powerless on the bridge of his own ship. He spotted some newsmen on the quayside milling around. He slid back the window of his cabin and shouted to them to listen up. As the inspection of his ship continued, the skipper took the opportunity to hold an impromptu press conference from the bridge of the Kingston Zircon. He shouted down to the pressmen, 'I don't think there is any risk. The men simply want an excuse for not sailing. Conditions are good. A small and an older ship such as this one cannot hope to reach the conditions of a new one. The men knew that in the first place. Why did they sign on knowing that?'\n\nThis was also the day Labour Prime Minister Harold Wilson got in on the act and his interest was reported in the Hull Daily Mail:\n\nMR WILSON TAKES A HAND\n\nTHE PRIME MINISTER last night intervened in the Hull trawler tragedy.\n\nHe asked two senior Ministers concerned with the fishing industry and maritime safety to meet early next week a deputation of Hull union officials, fishermen's wives and the city's MPs. It was his response to a telegram from the transport and General Workers' Union on Wednesday asking him to receive them personally. [...] ( _HDM, Feb., 2, 1968_ )\n\nBy the time the paper hit the streets, the \"big meeting\" and mentions of \"Lily getting us to see that Harold Wilson\" was the talk of 'Road. At the corner where The Boulevard (South) \u2013 which leads to the docks \u2013 meets Hessle Road a sea of pram pushing, headscarfed women gathered. The last time a crowd this big was there was when a statue was unveiled in 1904 commemorating those lost in the \"Russian Outrage\" of that year, when that country's navy mistakenly attacked the Hull trawler fleet thinking it to be the Japanese Navy. The statue towered over the growing crowd. The throng of women and prams that spilled onto the main road slowed the traffic. On her way to meet with Lily at the Hall, a man from the BBC waylaid Chrissie. A microphone was thrust in front of her and couple of other of the battling wives. The reporter did not even get a chance to ask a question.\n\nShe let rip at the TV man, 'There's been that many men lost in the last five years that we aren't gonna put up with it any more \u2013 and there's too many fifteen-year-olds going to sea what don't know anything about it [sic] and there should be a pre-training course for those fifteen-year-olds before they set foot on a trawler.'\n\nThe journalist took a breath to try to ask a question and Chrissie continued, 'We want more experience aboard, especially at Christmas time. The firm don't have to provide Christmas crackers to put on the table for the fishermen because there's already two or three sat round it.' When Chrissie stopped another fishwife chimed in with, 'For a start off there should be a wireless operator on every ship. For a skipper can't be on the bridge and in the wireless room at the same time, can he?' A third woman added, 'And the owners...they don't care. All they are interested in is the fish. The men, they don't mean a thing to them. They could not care less what happened to them as long as they are bringing the fish back.'\n\nThe narrow path to Victoria Hall was jammed with prams. There was an overspill from the entrance with the resulting tailback of hundreds of women heading for the \"grievance meeting\" organised for the protesting wives. Lily and Chrissie began to think maybe Mr Ashwell should have got them a bigger venue. Dozens of fishwives clutching sheaves of paper filed past and put them on top of a pile on a table near the front of the hall. The petition of 3,000 signatures collected by Chrissie, Lily, Rose, Mavis and the others had just doubled in number.\n\nDozens of newspaper reporters, TV and radio crews outside needed to get in too. The missing trawlers were international news. Around 600 people \u2013 mostly women \u2013 crushed into the ramshackle building. Faded green and cream-coloured cracked enamel tiles showed the age of the grand faded old hall under its unshaded lights. This was emphasised with the flashes from the cameras. Smells from the fish house and the pub fought it out with women's hairspray and perfumes as the steam rose from the wet clothing of the hundreds there and it mixed with swirling cigarette smoke to make a thick, pungent cloud that looked as if it hung from the rafters. Snot-nosed kids and the occasional dog weaved in and out the throng and the media scrum pushed in.\n\nIt was a shambles.\nFIVE\n\nCamera flashes went off as Lily walked across the Victoria Hall stage. The union men, politicians and activists at the table behind her heard the din diminish as she reached the centre. The fishwives' cacophony, gurgles and cries from babes in arms and chattering and screeching of unruly kids, all reduced as the big woman in the white coat stopped and looked out over a mass of headscarves. Even the few wandering dogs seemed to come to attention. Murmurs reduced to silence as if an invisible volume control had turned the sound down. There was no microphone. None was needed.\n\n'Right lasses, we're here to talk about what we're going to do after the losses of these trawlers. I don't want any effin' and blindin'. Remember the Press and TV are here. We want an orderly meeting,' Lily bellowed. She was heard as clearly at the back of the hall as she was at the front. The men at the table looked suitably impressed as she held the crowd. Lily went on, 'I intend to see that Harold Wilson next week and I won't come back until I have seen him. I'll even go to his house, not the Downing Street one, his private one, if I have to, if they won't help us.\n\n'We need to take action. I have a bag packed. I'll go anywhere, anytime. I'll board any unsafe trawler in the country to stop an unsafe ship sailing.\n\n'I pledge my life to get better and safer conditions. I'll go to jail if I have to. It [safety] should be enforced not just for Hull but for all British ports. I'll hitchhike to London if I have to, to make my case in front of Parliament. We are determined to fight this to the end, whatever the cost.' The mob cheered. There were chants when other speakers followed Lily's opening salvo. The volume was back up \u2013 and out of control. As one voice part of the crowd shouted, 'March on the dock! March on the dock! March on the dock! Let the owners have it!\n\nThe men from the table tried their best.\n\nMarxist Hull University economics lecturer Mike Kidron said, 'These trawlers are coffins sent out in spite of danger because a lot of profit is involved.' He then demanded revolutionary action against the owners and was met with some jeers, some laughs and there were some puzzled looks when he suggested the closure of the northern trawl. Kidron exclaimed, 'Strike at the profits and you are striking at the very heart of the problem!'\n\n'March on the dock! March on the dock! March on the dock! Let the owners have it! The loud minority shouted again.\n\nMany women in the hall, and most of the men there were very wary of women being on the dock. One of them shouted, 'A revolution! Are ya daft, lad? Kidron gave up. There was a bit more order as local Labour MP James Johnson got up and said, 'I am immensely shocked by the toll the sea has taken of these gallant men, who at this time of year are in up to 22 hours of darkness each day in dangerous Arctic waters.'\n\nThe imposing politician's thick wavy black hair could be seen from the farthest reaches of the crowd \u2013 but the MP struggled to be heard.\n\nThe next speaker got a better reception. National Union of Seamen firebrand John Prescott, a ship's steward, was seen as one of their own. Although not a fisherman he sailed from Hull and was a known fighter or a rabble-rouser, depending on who was on the receiving end.\n\nHe got a loud cheer when he crossed the stage. Another followed when he told them not to stand for being treated as \"second-rate citizens\" and they should join with his comrades to fight the bosses together.\n\nPrescott shouted over the chants and cheers, 'The NUS pledges support for the trawlermen. The trawler bosses are in the same mould as the ship owners. We will support you in the fight against them. For too long fishermen have been second-rate citizens.' Prescott got himself heard best of all the speakers, barring Lily. This was a skill the burly 31-year-old ship's steward and ex-boxer learned the hard way leading wildcat strikes from the docksides of Hull, Liverpool and Glasgow to the boondocks of New York, a constant thorn in the side of the cruise lines for which he worked. He was briefly taken aback when booed after he glowered at the Press and said accusingly, 'I see the vultures are gathering.'\n\nOne shouted, 'The Hull Daily Mail's been good to us.'\n\n'Yeah, they're the only one that lets us know owt,' another yelled as Prescott tried to explain himself but gave up.\n\nPrescott's ire was at the national media, but the crowd did not see it like that and were angry that their local paper, which had been of great service to them and had just that day given \u00a3500 to the Lord Mayor's Charity Fund set up for the stricken trawler families, should be tarred with the same brush.\n\nThe young union firebrand's rabble rousing fizzled soon after. Another Hull University lecturer Tony Topham, from the left-wing trade union funded newspaper Humberside Voice, decided against speaking and assiduously took notes. All these men were there to help to guide and advise the women as best they could. Some because they felt they had to, some because they were being ordered to.\n\nThese women were not in the mood for union men, politicians or middle-class Marxists. Minutes later, hundreds of women and children and prams were snaking their way down to the dockside. Lily, Rose Cooper and Mavis Wilkinson were at the head of the crowd. Hundreds did not go with Lily and the others, however. The taboo of women being on the dock when ships and men were there was a superstition, which even in the mid-20th century many fishwives would not break, regardless of anger and fears. However, almost half did go down the West Dock Avenue and into the subway tunnel that led to St Andrew's Fish Dock. No one knew what to expect, least of all the camera crews and media pack in the wake of the hundreds of women, who walked quietly through the slush as dusk fell. They were determined with silent purpose. The only noises that interrupted were the sound of squeaky pram wheels as they ploughed through the slush, the occasional cry of a child or impromptu cheer from the pavement. The women marched through hundreds of people, many wearing black arm bands, and passed terraced houses and shop fronts adorned with black-ribbon-trimmed windows, marking the community's grief. They neared the owners' offices and a line of half a dozen docks police blocked their way. Lily and her headscarf army swept them aside and were soon banging on the door of the Hull Fishing Vessel Owners' Association.\n\n'We don't want any trouble missus,' said one of the dock coppers.\n\n'Neither do we. We just want to see the bosses,' Lily said.\n\nBut the bosses would not come out \u2013 and the chanting women would not go away.\n\nThey had a \"Fishermen's Charter\" with them outlining their demands. After ten minutes or so of doorstep argy-bargy a man came out. A deputation consisting of Lily, Rose and Mavis was allowed in.\n\nIn the offices there was no trace of the owners. The women had to make do with the Owners' Association chairman Michael Burton, its secretary Lionel Cox and Rear Admiral John Ievers from the ships' insurers \u2013 the Hull Mutual Insurance Company. The owners of the St Romanus and the Kingston Peridot made it clear via these men they were not prepared to meet the women. The deputation knew they were there under sufferance.\n\nDuring the fifty-minute meeting hundreds of women chanted and jeered outside. The women presented their charter, which got a perfunctory examination before they were shown the door. They came away thinking that they were not only talking to the wrong men, they were not being listened to either.\n\nAfter the meeting a martinet from the owners' offices pompously announced to waiting reporters that there was \"no statement for the Press.\" Lily spoke directly to reporters. 'There's only one way to make these people meet us and hear our case and that by taking action. C'mon lasses let's get back to the Hall,' she shouted. With that the women turned round and made their way back to Victoria Hall to discuss what had just happened and what action would be taken next. Bemused union men and politicians were still there. They did not know what else to do. Ashwell and Neve had had a telegram telling them Harold Wilson had ordered two Labour Cabinet ministers, JPW Mallalieu of the Board of Trade and Fred Peart from the Ministry of Agriculture and Fisheries, to meet the women to hear their grievances and help where they could.\n\nThe union men had to arrange a vote to choose a four-strong delegation for the TGWU-funded trip to London set for the following Tuesday \u2013 three days hence. Maybe this time the men on the table would get a word in.\n\nYvonne Blenkinsop was on Hessle Road shopping for a birthday present for her John. Lots of women nodded to or chatted with her as she made her way. Some were friends; some knew her from the nightclubs she sang in, some from the \" _Golden girl with the Golden Voice_ \" posters that plugged her club singer persona Yvonne Marie.\n\nFor whatever reason, Yvonne was kept from her mission to get her John a present to give him before they went out that night.\n\nShe had booked the babysitter. Everything was set. She was determined to make up for her recent neglect. The Road was more crowded than usual, even for a Friday. When Yvonne got to the junction with the Boulevard there were hundreds of women with prams and kids in tow milling around.\n\n'What's going on?' she asked one.\n\n'Lil Bilocca and a load of the lasses are on 'dock to see 'gaffers. They've had a meeting in 'Hall. They've got a petition an' all. It's a right do,' the fishwife replied. Another added, 'You won't get me on 'dock in a million years. My mister would kill me. It's bad luck any road. They'll wash 'em away. She goes too far that Lil.' There was a further barrage of comments but Yvonne was not really listening. She saw the crowd coming from West Dock Avenue. A big woman with two smaller ladies either side of her was at its head. Yvonne saw the throng go towards the Victoria Hall. It was as if a light switch had gone on in her head. This is where Yvonne was meant to be. After all the writing, the sleepless nights, the prayers and Bible reading, this was her chance. She knew what she had to do.\n\nShe joined the crowd and found herself in the packed hall. Once inside she noticed there was no microphone and that no one really seemed to be that organised. Yvonne went back outside, took some pennies from her purse, went into the nearby phone box and called home.\n\nJohn answered.\n\n'Hello, love, it's me,' said Yvonne. 'Can you bring my PA and my mike and stuff to Victoria Hall right away?'\n\nShe explained about the meeting and told her husband she was running out of pennies for the phone box. By the time the pips went John had his instructions.\n\nThe babysitter that Yvonne had arranged agreed to watch the kids and long-suffering John was at Victoria Hall within twenty minutes. He rigged the PA system on stage for his wife as he had done for a hundred gigs before. The second part of that day's meeting was ready to restart. This time everyone, however softly spoken, would be heard \u2013 thanks to John's wife who was about to give the performance of her life.\n\nAshwell had two announcements. One of the trawler bosses, Mark Hellyer, had let it be known he would be prepared to meet with the wives' representatives on Monday morning. When the cheers subsided the union man confirmed even bigger news. Four women were needed to represent the wives' campaign in London on Tuesday where two Cabinet ministers who would hear their grievances were instructed by Prime Minister Wilson to help them.\n\nLily came to the mike and said, 'C'mon lasses, who'll join us?'\n\nChrissie, whose 16-year-old nephew Robert Dockerty was a deckie-learner on the St Romanus, shot a glance at her friend Gill Harrison, who had lost her husband Tony just little over a year earlier. He and eleven others perished when the St Finbarr sank off the Newfoundland coast two days after Christmas 1966. Gill, a 21-year-old widowed mother-of-two who had not left her home in a year simply froze. She could not bring herself to go ahead with it. Chrissie went on to the stage without her friend. Lily repeated her request. Yvonne found herself at the microphone. She started to talk. The crowd cheered. All the ideas formed over weeks poured out. She was not aware of anything other than getting her thoughts out. Her first was of her late father, a trawlerman lost at sea just four years earlier.\n\nShe said, 'You all know me. I'm off Hessle Road and I know the anguish these lasses and families that have lost men are going through. My Dad died at sea just four years back. Me Mam was left with six kids, I am the oldest. We all had to pull together. These men are in danger all the time, especially at this time of the year. Signal equipment on vessels is inadequate and depends on being connected to a battery or a generator or an engine. There should be rockets and flares set off automatically by detonators and there should be a bleep beacon signal on the bridge for the skipper to set off in an emergency, independent of any batteries or the like.\n\n'The Germans have things like that on their trawlers.\n\n'And the ships our men are on should be painted with luminous paint to make it easier to see them in the dark Arctic winter seas. Surely scientists can make something that will melt the black ice that builds up on them ships and makes them keel over.\n\n'We all have to fight for our men, even if we have to go and confront the Prime Minister. No one would get me to go to sea, not for a million pounds, but our men do it and our men are fools, with hearts of gold to do it. They are too busy providing for families to fight for rights. We'll do it for them.'\n\nEach point Yvonne made was cheered. Her impromptu oratory was not in any particular order. There were no notes. It came straight from her heart.\n\nThe crowd knew this.\n\nOthers doing it to show their appreciation joined those who had been stamping their feet on the old wooden floor to ward off cold. There came a crescendo of clapping, stamping and cheering when she stepped back from the microphone. Yvonne had assured her place on the London trip with this bravura performance.\n\nLily was back at the mike and asked for another volunteer. From the wings another glamorous, slim, young woman made her way towards her. She was not in an apron like many of the others. She did not have a headscarf. An \"Alice band\" kept her coiffed, bobbed hair in place. She cut quite a figure in her three-quarter-length leather coat. Her stride was confident and when she spoke there was little trace of a Hull accent, let alone a Hessle Road one. Most in the hall knew her. The thirty-year-old attractive mother-of-two was as much a Hessle Roader as anyone in the hall. Her name was Mrs. Mary Denness.\n\nShe was a skipper's wife. Lily handed her the microphone. There were jeers as the young woman started to speak.\n\n'We don't want no bloody skipper's wife going to London wi' them,' shouted one.\n\nAnother added, 'She's as good as one of the bloody bosses!' Mary held her nerve and stared straight back at her accusers. 'Look, if the majority here don't want me on this platform then I'll step down now and any one of you can take my place,' she said.\n\nAs Mary's clear \"posh voice\" boomed through the PA system around the hall, Lily walked towards her and grabbed the mike from her manicured hands.\n\nMary stood stock-still.\n\nLily shouted into the mike, 'Look, two ships have gone with all hands. Have the skippers come back? No they haven't! So since they have gone down with their ships, they've a right to be represented.'\n\nThe big lady smiled as she handed the microphone back to the slim, well-spoken young woman. A cheer went up. Hands went up. It was more like picking a talent show winner than it was an election, but either way, the fourth and final member of the newly formed Hessle Road Women's Committee was chosen. It seemed from the minute the meeting began that Lily and Chrissie would be on it. The process was informal but there was no doubt that the people had spoken.\n\nIt was Mary's turn to speak.\n\nShe knew that her skipper husband would be furious with her for this. She knew he would fear for his job. She knew he would be angry and disapproving. She would be told not to \"interfere in men's business.\" Barry Denness was not a tolerant man. The Brixham-born fisherman was at sea as a mate on the Grimsby trawler Ross Vanguard, sailing off Iceland. Mary felt she had to do something. She too had spent sleepless nights after reading about the tragedies. She had nine brothers and all of them were at sea. She felt compelled to speak out, no matter what the consequences.\n\nMary was used to taking a road less travelled, compared with her peers. She knew the sea first-hand, not only as a seafarer's daughter and wife, but also as a stewardess on cruise ships from her late-teens until she was twenty-two. It was during that time the young woman taught herself to \"talk posh\", practising in her spare time. She may not have had a formal education but her self-taught elocution certainly gave that impression. It was that grit that got her a stewarding job with the Ellerman-Wilson Line, which then was a mainly male preserve.\n\nShe gave up the job when she married Barry, of course, but she kept the accent.\n\nThe row subsided and the crowd became quieter.\n\nMary said, 'I know that I may be putting my husband's job at risk but I feel we should be united in this fight. I will go to London and say what I can.\n\n'The whole system of trawling is wrong, from galley boy to skipper. The time a man \u2013 or a boy \u2013 should serve on deck before taking a ticket should be lengthened to improve his experience. Then, when he is at the pinnacle of his profession, he will be a more experienced man.' Taking a swipe at bosses hiring \"Christmas Cracker Crews\" a louder cheer went up when she said, 'The owners will be able to do away with this element of trash that comes along, especially at Christmas time when they sign-on anything that can put its name to paper.' The mike was passed around the crowd and cheers and jeers met random comments and angry outbursts from members of the audience. The noise stopped when Robert Dockerty took the mike. He was Chrissie's brother-in-law and the father of Robert junior, a deckie learner on the doomed silent trawler St Romanus. With his wife Mary by his side, Robert struggled to compose himself as he spoke. The buzz of the crowd became a funereal hush. Robert said, 'The Royal Navy went out in force a few days ago to look for a missing French submarine...but they never looked for my lad. He was just a mere tool. I loved my boy...and he loved the sea.'\n\nThe heartbroken father could speak no more.\n\nNo one called an end to the meeting. There was nothing more to say \u2013 for now.\n\nThe quieted crowd filed out silently. Some shook hands with Robert as they left. Others acknowledged his pain with a nod.\n\nOn the street the reporters wanted to know what would happen next and Lily told them. 'I'll be on that dock tomorrow, checking them ships are properly crewed and have radio operators on them. I'll jump aboard myself to stop 'em going out that dock if I have to.'\n\nStuart Russell from the Hull Daily Mail asked, 'What time will you be on the dock, tomorrow, Lily?'\n\n'First thing, love.' Lily replied.\n\n'Can you give us a time?'\n\n'No, but it will be early doors, love.'\n\nThe young reporter's heart sank. He knew it was going to be a long night. Five minutes later the call to his news desk confirmed the young newsman's pessimism. By the time John and Yvonne had packed away the PA system, loaded up the car and got home, a celebration for John's birthday looked a distant, unlikely prospect. Pubs and clubs in Hull had all but closed.\n\nBack home at their cosy semi in Sunningdale Road, in the village of Hessle, things began to look up. The babysitter agreed to some overtime. A table was booked at the Continental Restaurant, on the Princes Dock Side in the Old Town area of Hull. Soon the couple was alone at last, in a taxi and heading for a late but welcome night out. In the cosy quayside restaurant the Blenkinsops were taken to their table and Yvonne's topic of conversation was still what it had been all week \u2013 the trawlers, the fishing bosses and what they should do next. Yvonne talked excitedly of her plans for the meeting with Hellyer on Monday and the trip to London the day after that. John caught sight of another diner, a man, who seemed to be taking a keen interest in what his wife was saying. John thought little of it and got back to chatting. The waiter came over and John ordered the drinks as Yvonne perused the menu. With the choices made, the waiter left with his notepad.\n\n'I'm just going to pop to the little girls' room, love.' Yvonne said.\n\n'OK,' John replied.\n\nThe eavesdropping man left his table seconds later. Yvonne left her husband buttering a bread roll. Yvonne was tapped on the shoulder as she reached the ladies' powder room. The eavesdropping man was facing her. Yvonne could tell by his suit and his stance he was a fisherman. Hessle Road men were easy to recognise. 'Hey! You should keep yer fucking nose out o' men's business...' The man launched a vicious punch, which hit Yvonne squarely in the face. Her eyes watered instantly and she screamed. As she fell, the man pushed her against the wall and ran past. John and another male diner shot up from their tables and ran towards the man who they could both see was pulling at the restaurant door. John and the other diner ran out into the night after him. The attacker disappeared into the mists of the dockside.\n\n'The bastard...' John half shouted.\n\nHe and the other diner went back inside.\n\nAt the table Yvonne was dabbing at her eyes with a handkerchief. There was no blood, but there was obviously going to be a bruise.\n\n'He just came at me, John. He hit me full force.'\n\n'Yvonne, love, are you sure you want to go on with this? I mean, there's more like him out there.'\n\n'I've got to go on with it, John, otherwise buggers like him will win, won't they? I can't have that, can I? I got to do this, John. You know I do.'\n\nNext day at 7am Lily and a handful of women were on the dock and were met and outnumbered by reporters, and cameramen.\n\nFor the second time that week a copper told Lily, 'We don't want any trouble.' Then half a dozen bobbies escorted her and a few others to the dockside.\n\nThe press expected more. Much more.\n\nThe sight of Lily, her friend Rose Cooper and a small crowd of headscarfed women did not quite match up to the fish dock storming promised the night before. The reporters were in a foul mood. The vast majority \u2013 with the exception of the TV and radio guys \u2013 had spent a freezing night on a windy, snowy dockside, huddled in doorways vainly trying to keep the bone-chilling cold at bay. To think they had frozen their arses off overnight \u2013 for this. Most of the print boys, except the very few with hipflasks and\/or sandwiches, had not eaten or even had a drink over the previous twelve hours. The glamour boys from the BBC and ITN slept well after a late supper and a few drinks at the Royal Station Hotel, safe in the knowledge some stringer was on watch on their behalf.\n\nThe Hull Daily Mail's Stuart Russell and his colleague Derek Hilton were ordered by their news desk to stay with the story as they expected. So following Lily's big promises the night before they joined dozens of other newsmen wandering doorway to doorway. The icy winds whipped the snow flurries into a mini blizzard. Derek \"Shorty\" Hilton was Stuart's pal as well as colleague and proved a boon companion whose good humour meant there were at least a few laughs during the long, cold wait. He was also a great little reporter. There were a lot worse folk to spend a night in a freezing dock with, Stuart thought.\n\nThe highlight of the two young reporters' night came at around four a.m. when they watched a pissed-up trawlerman trying to board his ship while wheeling a load of beer crates up the gangplank. The drunken fisherman wobbled and his barrow slipped.\n\n'Awww...fuck!'\n\nSplash! Splosh!\n\nTwo crates fell between the ship and the dockside.\n\nThe pantomime of the trawlerman, the boathook and the fishing for the ales kept the two young journalists laughing even after the half hour it took for the drunken man to realise he was never going to succeed. Derek and Stuart watched from a doorway and chuckled as the fisherman let out one final, 'Fuck it,' before getting aboard with just three crates to keep him for what is quite aptly termed \"steaming out\" on the next tide.\n\nPolice came out in force expecting the same furore as the Press. They outnumbered the women too. Trawlers slipped the lock to head for the Humber and Lily and her small gang shouted the same questions to each.\n\n'Have you got a full crew, love? Radio operator?'\n\nCrews lined the ship rails and waved at the women. Shouted exchanges were filmed, photographed and recorded. It was no big deal for the media, but for the trawlermen this was the first time they or their predecessors had ever seen any woman wave off a ship, let alone this crowd, its attendant press gang and crowds of coppers. 'Good on yer, Lil,' some shouted as the ships passed. Others watched quietly, somewhat unnerved as they watched the breaking of the taboo of women being at the dockside.\n\nAt eight a.m. Mary Pye joined Lily, Rose and the others. At fifty-three-years-old, Mary was among the oldest of Lily's protestors. She was also one of the most determined. Her second husband Herbert was lost on the St Romanus. Her son Albert Reeve was leaving for Iceland on the morning's tide. He was mate on the Ross Aquila, sailing from Hull. Other reporters followed Stuart and Derek and approached Mary Pye, having recognised her from earlier protests. She told them, 'For the next three weeks I will be living in fear and anxiety. I have five sons and all are trawlermen. I have come here to protest about the safety on the trawlers. A lot more can be done.'\n\nThe Ross Aquila was the next ship to head out. As it edged past, Lily shouted her questions. 'Have you got a full crew, love? Radio operator?'\n\nThe ship's skipper, Albert Beamish, could hear the commotion as the ship came alongside of the protesting women; unaware his mate's mother was among them. Skipper Beamish looked at bareheaded radio operator Jim Brooks and shouted, 'Look out your window and put your earphones on, so as she knows you're aboard. Otherwise she'll bloody try to jump aboard. She'll bloody sink us!'\n\nLily waved at the ship's wireless operator as he pointed to his earphones. He seemed very happy. He laughed as she waved again.\n\nAcross the dock as the women protested another row was underway involving a Hamling ship, the St Andronicus, sister to the lost St Romanus and former command of that ship's skipper, Jim Wheeldon. Like yesterday's protest on the Kingston Zircon, the crew called in TGWU fishing officer Mike Neve. This time he was there to inspect life jackets that the crew feared were not good enough. There was one for every crewman but they felt they were sub-standard.\n\nShip Manager Albert Robinson at Hamling's called the three men who jumped ship and took their grievance to the union man into the owner's offices. Eighteen-year-old deckie Alan Lee and his nineteen-year-old brother Walter stood nervously before the boss while the ship was moored. The lads were joined by their stepfather, the ship's chief engineer who was not as much in awe of the boss as his lads.\n\nEngineer McCafferty did not mince his words, 'There is dissatisfaction with all the life jackets on these ships, except them on the big freezer trawlers.'\n\nRobinson replied, 'The Board of Trade inspector passed these life jackets yesterday and found them suitable. You've got five minutes. Sign on or you'll be replaced.\n\n'We won't go.'\n\n'Right, listen you can go on tomorrow's tide or you're all signed off.'\n\nBack at the ship, the crew backed the trio and waited as Neve went about his work.\n\nThere was another massive headache for the beleaguered Robinson. Next ship out of the lock was the St Keverne, owned by Hamling's, another sister ship of the lost St Romanus. One of the crew, leaning with his comrades on the ship's rail as it drew alongside, shouted,\n\n'We ant got no' radio man, Lily!'\n\nThe commotion was instant. It took six uniformed coppers, one WPC and a plain clothes man to hold Lily back. She threw herself off the quayside and tried to jump aboard. This was potentially fatal, as the occasional reckless crewman had found in the past. For a big woman like Lily, it was certain disaster. Even if she cleared the gap between ship and lock, she would have to have been able to pull herself over ship rail onto the deck.\n\nPolice pounced in seconds. Lily struggled like a woman possessed. Her language turned the air blue. The St Keverne slipped by into the Humber still without a wireless operator, like many before it. The difference was that this time the world would know it. Photographers and camera crews caused more commotion and fought and jostled to get the best photo. Suddenly the story was huge. Every newsman on the dock knew the picture of this big woman fighting the police was the image of the day.\n\nThey were wrong. It was one of the images of the _year_.\n\nWithin the hour it was wired around the world.\n\nIt was the image on all the next day's front pages for weeks to come.\n\nStuart and Derek phoned the news desk put copy across before setting off back to the office to write up the remainder of the day's events. It had been a long, cold shift and news desk made it a bit longer. 'They want us to go to Ashwell's office and ask him if he's a commie! Apparently they think Lily's a puppet for a fucking red conspiracy.' Stuart told Derek.\n\nThe pair tossed a coin to see who would ask the fiery union man the news desk's question.\n\nDerek lost.\n\nTen minutes later the cold, tired, angry and thoroughly pissed off pair carried out their orders. At Bevin House, the TGWU man Jack Ashwell was at his desk when the young men knocked on his open door. He looked up to see the two of them framed in the doorway.\n\n'Can I help you, lads?'\n\n'Hello, Jack. We're from the Mail. We've been asked to ask you this.' Derek said not too stridently.\n\n'What?'\n\n'Are you a commie?'\n\n'Fuck off, lads. I'm busy.'\n\nAfter the struggle with the police the news reached the protesting women that the St Keverne was to get a radio operator sent to it by motor launch. Lily and her band of protestors were at the centre of a media scrum. Amid the camera flashes and furore there was one final inquisition from a TV reporter.\n\n'What do you think you are doing with this protest?' said the man from the BBC.\n\n'Well, we stopped a ship going without a radio operator. That's a start, it's not the finish but it's a start,' Lily replied sharply.\n\n'How much more of this do you intend to do?'\n\n'The rest of me life, the rest of me life.'\n\n'How do you regard yourself Mrs. Bilocca, as some sort of suffragette?' the reporter sniped.\n\n'Don't be daft.'\n\nOn that evening's news, the country saw the battling Hull fishwife cut the man from the telly down to size. The footage of Lily cursing and swearing as she struggled with police, however, was used without sound.\nSIX\n\n_Iceland's Northern Cape, Sunday night, February 4._\n\nBy 6.30pm, the Force Seven gale had increased to Force Eight; The Ross Cleveland was ice coated and struggled to stay afloat. It was too dangerous to carry on fishing. Skipper Gay decided to stow the trawl, concentrate on fighting the ice and to follow Skipper Whur and others north to seek shelter further into Isafjord Bay. Hurricane winds churned the thick blizzarding snow. He intended to try to fish again after riding out the storm in the lee of the land. The Ross Cleveland's radar had been dodgy all day long and hindered by more ice. The crew \u2013 barring engineers \u2013 fought to clear that ice as the ship dodged head to wind through massive waves up to forty feet high. Ice built up on the fo'c'sle, aft rails, and the upper deck and across the rigging. The ship was at half-speed with the skipper trying to keep in front of the growing gale while the men battled to keep their trawler afloat.\n\nGay radioed Skipper Whur, on the Kingston Andalusite, who was a mile or so in front.\n\n'Save us a spot at the hitching rail, Len,' he told his old friend.\n\nThe \"hitching rail\" \u2013 sometimes called the \"car park\" \u2013 was a straight stretch of coast inside Isafjord. It was about six miles long at the northern end of the fjord. Mountains jutting from the sea lessened the power of the wind for ships lucky enough to be anchored close to shore. Or at least that's the way it should have been. When the Ross Cleveland got there late, behind the Kingston Andalusite, the \"car park\" was full. Both ships were dodging in mountainous seas. Even ships at anchor swayed violently. Both skippers did not know exactly how many ships were there, but the radar blips and deck lights told them it was more than twenty.\n\nWithout a safe anchor spot, the trick for both skippers was to use the inlet to the fjord and its comparative shelter to dodge and lay alternately across its ten-mile wide mouth. From the north, Gay and Whur stopped each ship's engines, lay beam to wind and let the gale drive the ships across the fjord until the fifty-fathom line, a mile from shore. Then it was engines back on and the ship turned to the wind again and dodged back north, with just enough speed to keep from taking on water and to let the men fight the ice without being swept overboard. The two trawlers, like the other ships not at anchor, repeated this hour after hour. Any ice built up was evenly spread preventing capsize. It required laser-focus concentration and great seamanship as the relentless weather threw the ships around.\n\nThe hurricane winds whipped waves into an ice-making machine. Snow froze as it settled, flumes solidified across decks. Ice gathered like deadly crystalised cement, on every surface and rigging as the crew fought on. Each slab of ice cast into the stormy waters kept them alive and each huge wave was taken head on. The Ross Cleveland and the Kingston Andalusite rose on each wave and fell like a stone in the trough of them, time after time. As either ship rose and fell it seemed to the other to temporarily disappear.\n\nSkipper Gay continued to follow Len Whur's ship, each dodging through the growing storm. By 10.30pm the two ships had been battered by the ferocious weather for almost forty-eight hours non-stop. Next, came a PAN message from Isafjord Radio, 'ATTENTION ALL SHIPPING + KEEP WATCH + ICELANDIC LINE FISHING VESSEL HEIDRUNN + CONTACT LOST.'\n\nThat little local craft had no radar so the coastal station was even more concerned for that than any other larger craft caught up in that night's chaos.\n\nLen Whur was on the bridge of the Kingston Andalusite alongside twenty-one-year-old deckhand Ernie Bilocca. Whur noted the Isafjord Radio message. But his focus was on keeping afloat and guiding his friend Phil Gay through the maelstrom. Another trawler, the Notts County out of Grimsby, now tagged behind the Ross Cleveland. Whur glanced at the wind measurement dial and turned to Ernie Bilocca.\n\n'Look kid, it's over Force Eleven out there,' he said. The wind speed gauge recorded just a little more than 120 miles per hour \u2013 then it broke. The radar on the Ross Cleveland had packed up again by 11.30pm. The crew was ordered below from the deck. They were told to try to sleep while those on watch fought the storm. It was Gay's intention to fish again after the storm. Landing in debt was not an option. To trawl again would need rested men. The engineers worked flat out forcing the engines the do their skipper's bidding to keep them alive.\n\nIt was too dangerous to continue fighting the ice. Gay had to keep dodging, following Whur to have a chance \u2013 but without radar, he could soon be dead in the water. In the wheelhouse of the Ross Cleveland there were now five men; Skipper Gay at the wheel, radio operator Ronnie Thomson, bo'sun Wally Hewitt next to the starboard door, and third hand Alan Harper with mate Harry Eddom.\n\nGay turned to Eddom and said, 'Harry, I do not like the look of this weather.' The phlegmatic skipper's tone was still calm. Harry must have briefly marvelled at him. Nothing fazed this guy. The forecast was bleak. Dodging was relentless. The men could not understand why they did not get the benefit of the lee of the land. None of them had seen it this bad.\n\n'Harry go with Alan. Get up there and get the ice off the scanner,' the skipper said.\n\nIt had to be done. A trawler has a chance without a radio. Without radar she is blind. Even in good weather a ship without radar ran a high risk of going aground. In the chaos of that night it was a certainty. Harper and Eddom went from the wheelhouse. The metal-ringed safety ladder that the men climbed on to the top of the ship swayed back and forth wildly, like a giant metronome thrusting from the sea in the 125 mile-an-hour winds. Both wore rubber \"duck suits\" over their sweaters, fearnought trousers and warm underclothes. The pair also wore Dunlop knee-length waders.\n\nAt the scanner, Harper held Harry by the waist as the mate battered and scraped at the thick coating around the dish with an axe. Ice crashed to the deck and overboard in huge lumps. It had worked with minutes to spare. Each second \"being blind\" decreased survival chances. With the scanner cleared, Alan and Harry wasted no time in getting back to the bridge.\n\nWith the Kingston Andalusite, close by, around two cables away to starboard, Skipper Whur had been watching his radar and was the 'eyes' of his pal's ship. As the Ross Cleveland's radar came back on, massive seas hit both vessels simultaneously.\n\nThe two trawlers dodged head on to try to get to the western side of the fjord to have a chance to stay upright. Both skippers were in radio contact. Skipper Gay had the engines at half speed when a giant wave hit his ship. He stood before the wheel, which he pulled hard to starboard with every bit of his power, to bring her head to wind. She failed to respond. He grabbed the telegraph, and rang the engines \"Full Ahead.\" It was futile. The ship keeled rapidly to port. The skipper shouted his last words on this Earth into to the radio,\n\n'I am going over. We are laying over. Help me Len. I am going over.'\n\nA split second later he added,\n\n'Give my love and crew's love to the wives and families...'\n\nOn the Kingston Andalusite Len Whur shouted back into his radio, 'Keep her going full speed Phil and keep up with me.'\n\nThe radar blip that was the Ross Cleveland went from the screen in an instant. The skipper and the young deckhand had little chance to react as another massive wave hit their ship. They were now fighting for their own lives.\n\nNo one saw the tiny arc of light from the igloo-shaped lifeboat fall from the Ross Cleveland in its dying seconds.\n\nSeconds after the massive sea hit the ice-covered trawler the strong wind pushed it over faster and further to port. Eddom was on the starboard side of the wheelhouse as it went. The brass handle of the ship's telegraph \u2013 which seconds earlier was in the hands of Skipper Gay \u2013 pushed into Eddom's side. As the waters crashed in, Eddom managed to free himself and shouted out, 'C'mon, we don't want to be here now.'\n\nHe went instinctively out after the bo'sun who fled through the starboard door on to the casing. Eddom clambered aft and ran along. His duck suit and the soaking clothing under it weighed heavily. If he hit the water he would sink faster than the trawler. Eddom stumbled as fast as he could along the now horizontal wheelhouse casing.\n\nHe saw silhouettes against the thick, flurrying snow. One was bo'sun Wally Hewitt. He did not recognise the other guy, but he appeared to be naked. Wally and the naked guy were pushing an inflatable covered boat off the casing. Eddom knew if he could get to them he would have a fighting chance. After all, he had checked all the life rafts only a couple of hours earlier and knew they were in order.\n\nAs the sea took the ship something hit Eddom. The life raft had knocked him out as it inflated. Freezing water woke him. Arms were pulling at his. He could hear his name. He was being dragged from the swirling sea.\n\nBack on the Kingston Andalusite, Skipper Gay's final message seared into Whur's brain. He glanced at the radar again. The Ross Cleveland was gone. Whur tried to go to his friend's aid. He turned hard to port but the broadside waves almost sent him over. He gripped the wheel, forced his ship head on to the powerful wind, and sent it into another massive wave. From the wheelhouse windows he could see only sky as the vessel climbed. Then he saw swirling wild water as she dropped in the trough of the wave and came back up. There was nothing more he could do.\n\nPhilip Gay and his crew were gone. Nothing could have saved them, thought Whur. It all happened so fast. There could be no survivors.\n\nAs the Kingston Andalusite righted itself, Whur shouted to deckhand Bilocca, 'Tell the sparks to report the Ross Cleveland gone \u2013 with all hands.' The radio operator battered out the Morse message. All the ships battling in the fjord heard the Ross Cleveland's fate. Icelandic coastguard vessel Odinn was first to get that message out after the Kingston Andalusite's transmission,\n\n'HULL TRAWLER ROSS CLEVELAND REPORTED THREE MILES OFF ARDENES IN ISAFJORHUR DEEPS, ICELAND, DISAPPEARED OFF RADAR SCREEN AT 0020 HOURS. BLOWING FULL GALE.' A spate of other radio messages followed from there, with all the ships passing on the fateful news of the Ross Cleveland for fear others may not \u2013 or could not. Coastal stations were abuzz with radio traffic from the Northern Cape.\n\nHarry Eddom came to and strained to see. Slowly he focused and saw the faces of two men by the light of the interior white bulb. He realised what had happened. He was in the life raft. The one hundred and twenty-five mile an hour wind whipped the sea and tossed the tiny craft. Some of the waves as high as forty feet threw the men about inside the covered raft. A dazed Eddom looked across. He recognised the bo'sun and next to him was a kid dressed only in a t-shirt and a pair of \"John Ls\". Harry knew the kid was a decky called Barry. He was about eighteen years old. It was Barry who had dragged Eddom on to the life raft. The kid saved his life.\n\nWally Hewitt was in his fearnoughts and a woollen jumper and had his boots on but no waterproofs. Harry was still wearing the full duck suit and under-clothing he had on when he chipped the ice off the scanner with Alan Harper minutes earlier. By the light of the single bulb, Bo'sun Hewitt got the flap at one side of the boat tightly tied against the storm. The other side was torn and the men could not close it. The raft was filling up.\n\nThere was no time to think. Eddom forced himself conscious and grabbed one of the hand flares he had checked hours earlier. He headed to the open flap to set it off. He knew they would not last long in this killer cold. They were invisible in the storm. The huge waves were throwing them. Eddom tried to set the flare off, but the raft was hit by a wave that swamped it and swept through the torn flap. The craft was thrown vertical. The men gasped for air as the water engulfed them. The sea swilled through. The freak wave took with it all the raft's emergency gear. The flares were ruined and the baler disappeared into the sea.  The incoming water was set to sink the raft. Eddom started to bale with a sea boot. The bo'sun baled too. The kid in the underwear had grabbed an old coffee tin and joined in. The kid baled, ignoring the cold that was killing him. He worked furiously along side the men. Between them the three cleared most of the deluge. The raft stayed afloat but was thrown around like a rag doll in a terrier's mouth. The boy, the bo'sun and the mate, exhausted by the effort, slumped in the thwarts \u2013 the boards that acted as seating. Their hearts pounded. After the supreme effort there was only a few inches of water slopping about in the boat. As the raft stabilized, Harry saw Barry begin to fade rapidly. The boy slid into the well of the dinghy, mumbling, talking of his mum, his childhood. His skin was blue. His voice was weak and his eyes glazed. Harry went to him and started rubbing the lad's hands and face. He slapped the kid violently, desperately trying to stop him from losing consciousness.\n\n'C'mon kid hang on, hang on,' Harry implored the boy.\n\nThe boy was overcome; unable to resist the Arctic cold, in his near-naked state, in only sopping wet, freezing cold underwear.\n\nJohn Barry Rogers died in Harry Eddom's arms.\n\nThey were just little more than an hour into their dreadful fight.\n\nWally and Harry laid the dead boy down on one of the thwarts. Eddom could see the bo'sun was not faring well either. Wally shook uncontrollably under his cold, soaking jumper and fearnoughts. He had managed to get his boots back on after the life-saving baling session. Eddom could see by the flickering light that bo'sun Hewitt was turning blue. He was freezing to death. Harry tried to keep his shipmate talking but the bo'sun had become incoherent, muttering and rambling like the boy had done. Harry tried to talk of anything just to keep Wally's attention. 'C'mon pal. Don't give up. They'll come and get us, you'll see. Just think, you'll be back with your lass and the kids soon Wally. Don't give up.'\n\n'I don't think I'm going to make it,' Wally stammered.\n\n'Hang on Wally, hang on,' said Harry as he rubbed the dying man's limbs, holding him as close as possible to fight the fatal cold. 'There are ships around. They'll find us pal. Just hang on, Wally'.\n\nThey were right to think they could be picked up. They knew they were fewer than three miles from shore when the ship went down and also knew the hitching rail was full. Someone would surely come for them? After all shipping surrounded them and the storm would abate eventually. But the winds carried them out to the deeper parts of the fjords as they succumbed in varying degrees to the cold. They had been in that freezing maelstrom a little short of four hours when Wally's incoherence became shallow breathing, then silence. Harry was alone. He slumped in the dinghy. The ghostly light flickered on the two dead men.\n\nHarry forced himself to think of home. Don't give up, he urged himself. Think of Rita and baby Natalee. I wonder what they are doing now? I must get back to them. He determined not to sleep. He stuffed thought after thought of home and Rita and the baby into his mind, to keep from falling asleep. Sleep meant drifting to certain death.\n\nHe thought of his best girl Rita Penrose \u2013 he thought of her as his new bride just three years ago. He thought of their first family Christmas just a few weeks ago and remembered how Rita was delighted when he was made first mate of the Ross Cleveland in late December 1967 \u2013 the perfect late present.\n\nHis first trip a few days later was for three weeks \u2013 then three tides and away again. This was his second trip as mate of the Ross Cleveland. He determined his force of will would ensure it was not to be his last. He kept thinking of Christmas with Rita and his baby girl Natalee. He told himself, I will get back. I must get back.\n\nThe raft juddered to a standstill with a loud crunch.\n\nHarry felt the ground under the dinghy. In seconds he realised he was ashore. He felt the rocky beach under his feet. Eddom fell from the igloo-shaped life raft onto the stony beach. He was in the deepest of darkness. There was no moon, not even a glimmer on the water. He had no idea where he was. In the hours fighting for life in the raging waters, the dinghy had drifted twelve miles from where it was pitched into the storm and grounded between two tapering fingers of land in the narrows of Seydisfjordhur.\n\nHarry got up and stood over the life raft and then looked in at his shipmates for the final time, checking, hoping against hope for a flicker of life. There was none. In the blackness he pulled the dinghy as far in land as he could, leaving behind him the brave men who had saved his life. He had to find shelter or die. It had been about nine hours since the three shipmates were thrown into the freezing sea. Willpower kept Harry awake. He thought about staying with the life raft. After all there might be a search party \u2013 probably. The raft stood out and would be noticed. It would provide a shelter of sorts. But he knew if he stayed still he would succumb. He started to walk.\n\nThere was no sign of habitation. The landscape was unfamiliar. The precipitous mountains could just be seen in the background. The icy wind was whipping along the stony beach. Harry surveyed the area. He was unaware of how many hours had passed but he knew it would be dawn soon enough. There would be just a few hours' precious daylight. He followed the coastline and hoped it would lead him to salvation. Was that a farmhouse in a valley in the distance? He was not sure. He was in a state between conscious and unconscious, forcing himself to walk. Harry was trudging through a nightmare. The pain in his feet was searing. Frostbite was doing its work.\n\nHarry forced himself through the pain to take off his Dunlop wader boot and wring out his sopping wet freezing sock. But he managed only one foot. The pain was unbearable. He pulled the wader back on and walked. The other foot would have to wait. He could not go through that again.\n\nHe must have thought of the Man from the Mission knocking on his door in Cottingham. Rita would be told he was dead. She would think herself a widow. He was certain his time was at hand. He felt desperate and sorry and scared.\n\nThen he got angry and felt he was letting down his wife and baby by dying. He would not allow that to happen.  The farmhouse he thought he had seen earlier hove into view as he trudged through the snowy beach, the winding whipping him, not even a seabird to be seen. He could have been the last man on Earth. The little house was still miles away. Distance was difficult to judge. Just keep going, Harry. Just keep going.\n\nEddom struggled on, frost coated, his waders half filled with water. He could no longer follow the shoreline. He could see the farmhouse still. Now his only way to it was to go up. A cliff jutted out to the sea. Harry despaired. Was he defeated? Surely he could not climb, not like this? There was no option. He walked back a bit to get the best approach to the cliff. He clambered the slippery rock face. Below him was the sea. Above him was hope.\n\nAs darkness fell again he conquered the cliff. The farmhouse was in sight. He had no gloves. His hands were stiff and cold and numbed with pain. The skin was raw from the climb. In the blizzard the farmhouse came into focus. It was a small summerhouse, a sort of cabin. Next to it was a sheep pen. The place was boarded up and abandoned. The windows were shuttered. The door had planks nailed across. A strong man with the correct tools would have found it a task to get in.\n\nHis chance of shelter, his hope, had gone. He had walked eight miles, climbed a virtually vertical slippery cliff in a blizzard. Now he raised his weak, raw hands and struck impotently at the windows and doors. Then tried to kick the door. The pain made him squeal. It was futile. The frozen sailor leaned on the sheep pen, crouched behind it; desperate for what little shelter it had to offer. When he crouched he found himself collapsed and falling into a sleep. He knew this could not be allowed.\n\nSleep meant death.\n\nHe forced himself up. He stood as upright as possible. He clapped his hands, thumped his arms across his chest. He tried to get the circulation back in his arms. His legs had stiffened. He could not move. He was frozen stiff. Hour upon hour in the dark he huddled and then forced himself back on his feet. In the pitch black he thought he saw the flickering of some lights. Maybe someone was coming? Then the lights \u2013 if ever they had actually been there \u2013 disappeared.\n\nHarry Eddom could no longer move his legs. He forced himself not to sleep and fought for life as he had done for the previous 24 hours, having walked all those miles and climbed the treacherous icy cliff. It looked like this was how it would end for the tough mate of the Ross Cleveland. Eddom kept up his desperate mantra through the night, Keep going. Keep doing something. Don't get drowsy. Stay on your feet.\n\nOn the western side of Seydisfjordhur, Kleifar Farm was home to 180 sheep, four cows, four horses, six children and their parents. The farm had given a good but tough life to Gudmundur Asgeirsson, his wife Karitas Gudbjorg Gudlaugsdottir and their 12 children. Six of those kids were at home on the night of the Great Storm. The others \u2013 the older six \u2013 were working away, living their own lives. Haraldur, fifteen, Gudmann, fourteen, Gudrun, twelve, Bardur, seven, Olafur, five, and baby Solrun, four, all had their chores. The older two boys had both left school to take up farm work full-time, now that their six elder siblings were making their way in the world.\n\nThe heavy snow made grazing impossible and for two days the animals were fed at the farmyard and stable. In the atrocious weather, the livestock and the family it supported had to be fed. Haraldur and his younger brother Gudmann usually started the day by fetching water for the cattle and then letting the sheep out to graze. But it was difficult to get to the brook. The boys also had to feed and water the horses. They could not see their house such was the blizzard. It was decided when the weather improved the sheep would be taken to the eastern side of fjord, where the snow was not as heavy. Usually the sheep grazed near the cliff on the western side.\n\nBy Monday morning the weather, although still freezing, was much calmer. Gudmann, along with the old sheepdog Nikalus and its son Mori took the sheep from the farm while his older brother and the rest of his family worked at other chores. Gudmann was heading down the slope from Kleifar Farm. The old cabin could not be seen until the bottom of the slope had been reached. Gudmann was content today. The temperature was slightly above freezing. It was calm. The sheep were wandering before him. The dogs ran alongside and the boy thought about the pancakes his mother would be serving up when he got home. That was the treat for his little Olafur. It was his kid brother's eighth birthday. Gudmann loved his mother's pancakes. He sauntered towards the grazing land near the cabin. He noticed deep footprints in the snowdrifts \u2013 knee-deep but close together. Whoever had made them took tiny steps, but left big prints.\n\nThe boy immediately thought these deep footprints could belong to a fisherman from the maelstrom he and his family had heard about on the radio. He started to search. Then the dogs started to bark and ran towards the boy. As he calmed them, he heard a faint noise. He saw a short, stocky man leaning near the sheep pen. The boy ran towards him. The man tried to move but could not.\n\nThere was a faint voice in a language Gudmann did not understand.\n\n'I am Harry Eddom. I am a British trawlerman.'\n\nThe boy looked puzzled.\n\nHarry pointed out towards the sea. The lad needed no explanation. He went to the man and held him up. The man was shaking and struggling to move his legs.\n\nLooking at the deep footprints that led to the fisherman, the boy realised the snow had not drifted since the boy had been there last. He reckoned the man must have been there since before nine last night. 'How could he have survived? All that snow, all those hours?' thought the boy as he reached out to the man. The man and the boy hugged.\n\nIf Harry had stood at the other side of the summerhouse he might have seen the farm at Kleifar at the top of the slope. It dawned on the trawler man that was where the lights had come from that he thought he had seen earlier. That did not matter now. He had survived. Thirty-six hours after being cast into the fjord, Harry was safe. The boy supported him as best he could and they struggled up the slope toward the farmhouse. After twenty minutes or so struggling, Gudmann's father saw the pair and ran to them. Between the two they helped Harry stumble up the remainder of the slope.\n\nA stocky-built, dark haired, middle-aged woman in a dark pinafore came to the door to find out what all the commotion was about.\n\n'Look!' shouted Gudmann at his quizzical mother. 'I have found a man out of the sea!'\n\nGudbjorg joined her husband Gudmundur and her son and helped the English trawler man. The warmth of the farmhouse kitchen enveloped Harry like an embrace from God. Gudbjorg spoke to her husband and the boy as another teenaged boy and four smaller kids looked on bemused.\n\nShe said, 'Gudmann, you and Pabbi [dad] get him in now and take the wet clothes from the man. I'll get some dry clothing and food. He's in a terrible state, hurry.'\n\nThe 60-year-old farmer and the fourteen-year-old boy carried the fisherman carefully to a big couch and laid him gently down. The couch was in the warmest part of the kitchen and therefore the warmest part of the house.\n\nThe father pulled Harry's anorak off and threw it to the floor, followed by a sweater, a warm shirt and underclothes. All were sopping wet and crusted with frost.\n\nThe boy could not stop staring at the fisherman's hands. They were raw.\n\n'I cannot believe he has no gloves, Pabbi.'\n\nGudmundur said, 'Come, son let's get these waders off him first before we try to get these trousers off.'\n\nThe farmer and his son pulled at the first wader but it was stuck fast. Harry grimaced as they did so. The two of them took a boot each and pulled again. After a couple of tries, the boots came off and a flood of seawater from each hit the farmhouse floor.\n\nThey then took off his trousers under which he was wearing thin socks and John Ls. The boy and the man could not help but think that it was a miracle that this man had lasted so long, survived a shipwreck, battled ashore and found his way to their farm where he had spent hour after hour unable to rest or move in fierce cold and biting winds.\n\nGudbjorg brought in a pair of jeans, a vest, shirt, sweater and some long underpants like the one they had taken from him. The boy and his father could still see the man was weak and cold as they dressed him with the fresh, dry warm clothing.\n\nOnce dressed, Gudbjorg brought the man from the sea some coffee and soup with fresh bread. His ordeal had not diminished his appetite. He drank heartily of the soup and devoured the bread. He had three cups of hot coffee.\n\nThe farmer's wife cleared up the seawater from the kitchen floor as the Englishman ate. None of them spoke English but they knew the mumbles from the shivering sailor were his profuse thanks. Harry lay back on the couch. After thirty-six hours of forcing himself to stay awake, he fell asleep instantly. 'What do we do now?' Gudbjorg asked her husband. 'The phone line was taken down in the storm.'\n\n'We wait,' he said. 'There must be a search on. For now we wait. Someone will come.'\n\nAs the miracle man slept on her kitchen couch, the farmer's wife set about her next important task. She mixed the batter for her little boy's birthday pancakes.\nSEVEN\n\n'HULL TRAWLER ROSS CLEVELAND REPORTED THREE MILES OFF ARDENES IN ISAFJORHUR DEEPS, ICELAND + DISAPPEARED OFF RADAR SCREEN AT 0020 HOURS + BLOWING FULL GALE.'\n\nThe message came from the Kingston Andalusite. All Britain's coastal stations had it \u2013 as did all of Iceland's. It was sent to Charles Hudson \u2013 the Ross Cleveland's owner \u2013 via the Radio Telephone link in the early hours of Monday, February 5. He took the call in his study and learned his ship sank with all hands as others looked on helplessly. The large brandy he poured himself did nothing to assuage his shock and insomnia. He was at his desk in Hull early.\n\nIt was precisely seven minutes past seven in the morning when the Flamborough Coastal Station forwarded the message to all trawler owners in the port city. News of the Ross Cleveland was the talk of the fish dock within the hour. The street-corner scrums spread the word. It was on the lips of every bobber who came off shift, each fish process worker on their way to work and every trawlerman headed for his ship. A third ship gone, another crew drowned.\n\nIn the suburbs, Yvonne Blenkinsop and Mary Denness set out their best outfits for the meeting. In the little two-up, two-down in Flinton Grove, Chrissie was ready early. A neighbour was to look after the six kids, get the older ones to school and take care of the babies. Chrissie had a hair appointment. Lily was already on the dock checking ships from the lock but she was not in her working gear and her hair had been done, She had seen the St Andronicus and the Kingston Zircon off with full crews. Those were the ships that refused to sail over the weekend until given the green light by the union. Lily, Mary, Yvonne and Chrissie had an appointment with Mark Hellyer, owner of the Kingston Peridot later that day. The meeting was promised to them three days earlier. The trawler boss had a copy of their Fishermen's Charter and the women were determined it would be enacted.\n\nIn Coltman Street earlier, Lily read through her notes. The big writing pad was headed, \"What I Want, by Lil Bilocca,\" and listed demands ranging from better distress signalling, full crews including radio operators and safety lines on deck, to more training for boys on ships, a halt to fishing in atrocious weather and an end to the practice of hiring \"Christmas Cracker Crews.\" The Charter also demanded that owners look into scientific research to put paid to the trawlerman's worst enemy \u2013 ice. It demanded that the bosses examine similar technologies used by French and German fleets to de-ice ships in extreme weather. It also said a \"mother ship\" should be with all fleets, to act as a hospital vessel, or in extreme circumstances, a rescue craft.\n\nAt the corner of Chrissie's street Doris Forrest (Haute Coiffure) opened early to fit Chrissie in. Her meeting was set for ten. Chrissie had two hours to get her hair done and get ready. The salon even had her favourite stylist Pam ready and waiting. The salon was silent when Chrissie entered. She could feel an uneasy atmosphere pervade the shop. The glass door closed behind her and Chrissie spotted two fishermen's wives she knew. They were both having their hair done.\n\nOne looked up and said,\n\n'Oh Chrissie, I'm so sorry about your Phil.'\n\n'What about our Phil?' Chrissie said.\n\nPam replied, 'Oh, oh...' and her voice tailed away. It was obvious she realised Chrissie did not know what they were talking about. In seconds Chrissie was running. Running for her mother. Hessle Road shop fronts blurred as she ran the few hundred yards towards Rugby Street. When she reached her mother's house family, friends and neighbours filled the front parlour.\n\nChrissie breathlessly blurted, 'Is it true, Mam?'\n\nThe look on their faces told her it was. Her brother, Phil Gay, skipper of the Ross Cleveland was gone with his ship. Chrissie was dazed.\n\nHer mother looked directly at her and said, 'Carry on, still go down and do your best, Chrissie. See them trawler owners.'\n\nChrissie felt there might be some hope. After all, the word used was \"missing.\" Maybe there would be better news, even a survivor? But deep inside she feared the worst. She was like an automaton as she readied herself at her mother's house. She had already lost her nephew Robert Dockerty on the St. Romanus. Now Chrissie prepared for the meeting at the dock offices, knowing another tragedy more than likely was with her.\n\nFrom her mother's terraced house she headed for the trawler owners' offices near the Lord Line Building. Within minutes she was at the footbridge near the company stores. There were hordes of cameramen and reporters. She walked past the stores and the ships' outfitters to the main entrance where Lily, Mary and Yvonne waited. Lily's face was tearstained.\n\nMinutes before Chrissie arrived the big lady had been shouting at passing trawlermen on their way to their berths, 'Another one's gone down...more lads have died. Don't go lads, they don't give a knack about ya.'\n\nLily was quiet and composed when Chrissie approached. Mary and Yvonne were distracted by the tall, bald, bespectacled figure of the Rev. Tom Chappell, the padre from the Fishermen's Bethel. He emerged from the media scrum. 'Which one of you ladies is Christine?' the clergyman asked. The women nodded toward Chrissie. The padre took her aside and held her hand. Her deepest fear was confirmed. She loosened her grip on the padre's hand.\n\n'I've got to go in,' she said. The vicar watched as the slight-built housewife joined Lily, Mary and Yvonne and walked into the offices of Hellyer Bros. of Hull. There was a staircase to climb to the bosses' office.\n\nThe women walked quietly, then as they approached the owners' boardroom at the top of the stairs Yvonne burst into tears and shouted, 'For God's sake give our men a chance. We know we're going to lose men at sea but can we give them the best equipment we can? Can we give them better facilities, better ships, more stabilised ships?'\n\nYvonne's voice tailed off as the large oak door opened. There stood the tweed-clad trawler boss Mark Hellyer, owner of the Kingston Peridot. He beckoned the women toward the long polished boardroom table. Jack Ashwell, Mike Neve and another T&G man David Shenton were seated. Paperwork outlining the women's demands was on the table. It was obvious the men had been in discussion as Yvonne had her outburst. With the exception of his secretary, women in the boardroom were a first for Mr. Hellyer. The padre followed the women in. Chrissie sat down and thought, 'Look at this big room, beautiful big polished oak or walnut table... really really big... beautiful carpets... that's how the trawler owners live...nice... comfortable...'\n\nThe proceedings started. Everything built up inside the heartbroken woman. She was listening, but not really, like she was watching herself from above. No voices penetrated. She sprung to her feet, broke her own reverie and shouted, 'It's about time you pulled them out the Cape. There's too many men been lost... too many men missing.' She could speak no more. She wept uncontrollably. The enormity of her loss hit. Philip was gone. They were all gone. She felt the padre's arm round her. A handkerchief was handed to her and the priest led Chrissie from the oak-panelled room. Hellyer, the three remaining women and the union men sat in stunned silence.\n\nChrissie and the padre walked down the long staircase and out into the cold morning air of the fish dock. With his arm round her, the vicar pushed his way past the media scrum towards his car. They knew who she was. The camera flashes went off like a small, self-contained lightning storm. Tomorrow's front-page picture was taken. Rev. Chappell rushed his charge over the small footbridge past the stores, got into his car, gunned the engine and drove off. His passenger was crying hysterically.\n\nMr. Hellyer took to his large leather chair. He had brought it in earlier himself. It was three times the size of any other seat provided along the oblong table. It was on one side and the others were opposite. He was not a man used to being questioned \u2013 and certainly not by women. He was genuinely shaken by Chrissie's plight but Mary struggled to control her temper as the boss man spoke. Hellyer was being agreeable enough. But that was the problem. Everyone round that table knew agreeability was not a strong suit of this trawler boss. His propensity for shouting down his workers was legendary. One story of his brusque manner had long since entered local lore.\n\nMore than once he had ended \"negotiations\" with union reps or disgruntled deck hands with, 'I have a fucking prize bull that earns me more than you!'\n\nIt was no accident his brother, Graham, was nicknamed \"God\" and hence Mark was known as \"Jesus\" on Hessle Road. Mary studied his face and his high-handed attitude and thought, 'Look at him, lord and master, the trawler owner in his big bloody chair. He probably expects us to quake when he talks.'\n\nThere was no chance of anyone in that room quaking before the \"lord and master\". The women's confidence surprised Mary and probably everyone else at the big table. The pretty skipper's wife found she too had grown in confidence.\n\nThey outlined their demands and Hellyer continued his agreeable routine, making notes while Lily, Yvonne and Mary went through their requests. Yvonne's hand was full of pieces of paper with notes scribbled on. Lily had her big writing pad. The union men kept the meeting in order, from the typed out sheets and agenda they had prepared.\n\nMary felt Hellyer was not really as engaged as he appeared to be. But the trawler boss made more concessions than he had ever done. All vessels were now to be subject to a full inspection before each trip. It was agreed trawlers must report in to the owners every twelve hours and that any radio silence of more than twenty-four hours would spark a rescue alert and that wives would be informed of that. Hellyer also agreed that a \"mother ship\" be commissioned for the fleet to act as a rescue and hospital vessel.\n\nThe meeting drew to a close. Hellyer's three-hour charm offensive was all but done, but with his final twenty-two words the fa\u00e7ade collapsed. He reverted to type. He ushered the delegation to the door and said, 'I hear you ladies are going down to London to meet the Minister of Agriculture and Fisheries \u2013 oh well do enjoy yourselves.'\n\nAs the big door closed the women and the union men on the landing looked at each other askance.\n\n'Phony, insensitive sod,' Mary said half to herself.\n\nOutside the press waited. Hellyer had not allowed them in. A lackey announced a statement would be issued. The union men and the battling women continued to win the publicity war.\n\nReporters surrounded Lily and Yvonne. A microphone was thrust as a press throng scribbled furiously amid flashing cameras. Mary and the union men looked on as a frenzied interview got underway. A TV reporter asked Lily, 'What attitude will you take when you go down to London tomorrow?'\n\nLily replied, 'I don't know love. But if I am not satisfied, it will be drastic.'\n\n'What kind of things can be done?' added the reporter.\n\nBefore Lily could reply, Yvonne pushed herself forward. With her papers still in her hand she blurted at the TV man. She was unstoppable and determined to get out everything she had been saving up for weeks. Her right hand had a rolled up sheaf of writing paper in it that moved like a conductor's baton as she punctuated her sentences with it.\n\nYvonne said, 'Lots of things can be done \u2013 and will be done, petal... many more safety practices. We need an independent signalling system on every ship because when our present signaling systems fail and the generators break down and the mast and aerials have gone we have no means at all of contacting any rescue vessel. So we need an independent one that the skipper can send up in just a fraction of a second by pulling a switch setting off a detonator, sending up a rocket with flares and a bleeper system. Now this bleeper system needs to have someone on the spot twenty-four hours a day to pick it up.\n\n'We need a safety ship patrolling the areas [where fleets fish] twenty-four hours a day.'\n\nThe TV reporter butted in with, 'Are you a fisherman's wife?'\n\nYvonne carried still beating the time with her paper baton, 'I am a fisherman's daughter... a fisherman who died at sea four years ago. My mother was widowed with six children. I have been born and bred in a fishing family, but that's apart from the fact...we are fighting for the fishermen who are there now.'\n\n'Is it just the women of Hull that are going to do something \u2013 or are the men going to be involved?' the BBC man continued.\n\n'The thing is,' Yvonne added without missing a beat, 'that our men are hard-working men. They would sooner be at sea fishing and bringing the wives the money home and looking after their families than going to union meetings and standing up for themselves. They are soft hearted and have been too soft hearted. They have mentioned that they have wanted things before and nothing has been done, so all right, they have just carried on regardless.'\n\nWith that Yvonne paused. She had certainly got her message across. The press pack dispersed. Her ad hoc oratory was the lead story on the BBC News.\n\nThere was no stricter contrast to the oak-panelled boardroom of Hellyer Bros. than Bill Howbrigg's digs. The fifty-nine-year old ship's cook made one room at the Queen Mary Hostel his home when not at sea. The bachelor fisherman started his career at nineteen. Until ten days ago he was cook on the Ross Cleveland. He was put ashore with double pneumonia on January 26 and spent seven days in hospital in Isafjordhur before being flown back to Hull. He lived a simple life and had all he needed in his tiny model lodging room.\n\nThat room was packed with reporters and cameramen. With a mug of tea in one hand and a roll-up cigarette in the other, the quietly spoken old cook recalled his time on the doomed ship. He needed little prompting and was keen to pay tribute to his skipper. He said, 'It was a good ship. Everybody was happy. Skipper Gay was respected by all.\n\n'Even while I was aboard ice was crippling the ship. All hands had to be constantly on deck chipping huge blocks of ice away with axes, spanners, or anything else they could lay their hands on. The weight of all the frozen spray forced the ship down. Nobody could sleep. It was a living nightmare. It was a battle with black ice. We were chipping it off the rigging all the time. As we were fishing it just all built up. You would start on one side of the ship and get that reasonably free. By that time the other side was beginning to get weighed down. It was a never-ending business.\n\n'The weather is so bad. It is not just the high waves but also the temperature. As soon as spray broke over the bow it froze. Skipper Gay wanted to put me ashore two days earlier but I carried on working until he eventually told me to rest.\n\n'I've lost some good mates on that ship. The Cleveland had a 16-year-old lad on board making his first trip. Before I left, the skipper told me, \"It looks as though we are going to be in for a rough time. I hope we will see you soon.\" I was taken to hospital in Isafjord and then flown home from a United States airbase. I think I know how it must have been at the end. I should not be here. I am the luckiest man to still be alive.' The previous time that Bill sailed on that ship she was called the Cape Cleveland. Some fishermen would not sail on a vessel that's had a name change. They say it's unlucky.\n\nThat afternoon's Hull Daily Mail carried Bill's story on page one as a side panel column. Alongside it were the names, addresses and job descriptions of the men of the Ross Cleveland, with photographs of eleven of the crew. The paper gave each family a chance to pay tribute to loved ones as it had done with the Kingston Peridot and St Romanus. Those who did not wish to speak had their grief respected, like the family of seventeen-year-old spare hand Jimmy McCracken, one of ten children and the relatives of second engineer Kenneth Brandtman and those of the twenty-six-old mate Harry Eddom of Queensway, Cottingham. It was simply noted he left a young widow and a baby girl. Skipper Philip Gay's widow Theresa was \"overwhelmed by the tragedy\" and was being looked after by a relative. Thirty-year-old bosun Wally Hewitt's widow Valerie was also too distressed to speak at her home in the suburb of Anlaby. Fred Sawdon, a fifty-year-old engine room trimmer, was a widower and father of five. His niece was looking after his kids and said little about her uncle other than he had never \"worried about his ship sinking.\"\n\nThe mother of fifteen-year-old galley boy Michael Barnes told how her son went to sea in spite of her pleading with him not to. At her home in North Hull, Florence Barnes said, 'I think the more we tried to persuade him not to go, the more determined he became. All his friends were trawler hands but there were no family connections with the sea. He packed up his job at the tannery, refused to get another and went to sea.\n\n'I have seven children, five of them boys. My second boy Alan, who is eighteen, spoke of going on trawlers when he finished his present job. He won't go. I shall make certain he does not sail. And to think we had only signed a petition on Friday for better conditions for the crews. It is terrible sixty [sic] men going like that.'\n\nChief Engineer Dennis Mayes, forty-two, had promised his thirty-seven-year-old wife May he would retire from trawling in two years. His sisters Constance Page and Sylvia Netherton looked after May and her three-year-old and seventeen-month-old daughters. 'Dennis just lived for his two children,' said Constance.\n\nFor eighteen-year-old spare hand John Barry Rogers it was his first trip on the Ross Cleveland. The boy from Eastbourne Street, off Hessle Road had been at sea for two years. His father John, a boilermaker's labourer on the fish dock, said, 'I saw Barry off when the Ross Cleveland left Hull. On a previous trip on the Primella he said it was sheer murder while they were fishing. The temperatures were below freezing and most of the time giant waves were crashing over the ship. But Barry was a brave lad and he decided to keep fishing.'\n\nThird hand Alan Harper, twenty-four, was also a first-timer on the Ross Cleveland. He only joined the ship after a chance meeting with Philip Gay, who invited him to join; otherwise he would have gone on another ship days later. Alan, who had been on trawlers for eight years, left a widow Nancy, twenty-two, with a four-year-old daughter, a son aged two and a newborn girl. Alan had waited for the baby to be born before the fateful trip. Ex-farm worker Michael Morris, twenty-one, from Edinburgh Street, Hessle Road, was to be married in March. He intended to stay ashore until his honeymoon and went to sea to get some money for the wedding. His fianc\u00e9e Linda Baker told the paper, 'All the invitations to the wedding have been sent out and arrangements made for us to marry at St John's Newington.' (The local fishermen's church on St George's Road that linked Hessle Road with the other main thoroughfare of the city \u2013 Anlaby Road.)\n\nForty-year-old spare hand Douglas Hairsine's widow said, 'My husband has told me that no one really knows what it is like fishing in winter unless they see it and experience it for themselves. Sometimes when he came home tired and exhausted he said that he would never go back again.\n\n'But like all other fishermen he knows that deep down inside him he just cannot give up the sea. We have been married for twenty years but we have only had about three of those years together. All the men complain about the conditions and the food but they always go back. A long time ago I decided that our two boys would never be fishermen.'\n\nSixty-three-year-old engine room trimmer George Keal, a World War Two Royal Navy veteran of the Salerno campaign, tried hard to get a shore job before he decided to go back fishing, a relative said. He left a widow, Olive and a son Terence from a previous marriage. Tragic twenty-eight-old housewife Mavis Pettman lost her husband Maurice, a spare hand, and her eighteen-year-old brother deckie Trevor Thomson, who like his shipmate Michael Morris, was to have married the following month. Mavis was too shocked to speak of her ordeal. Maurice, of Wheeler Street, off Hessle Road, left two children aged eight and six.\n\nTrevor Thomson's mother Jane said her son nearly missed the ship and he had jumped aboard at the lock gates. Trevor had been unsure about leaving his girlfriend, but decided at the last minute to go fishing to make money for his bride-to-be. Jane said, 'We all ought to stop our husbands going to sea. I had all that sympathy with those poor women whose sons were killed on those other two trawlers, never thinking that it would happen to me.'\n\nFourteen-year-old Michael Swain idolised his adventurous big brother Maurice, a twenty-two-year-old spare hand on the Ross Cleveland. Their mum and dad were never happy with Maurice going to sea \u2013 and not without cause. Six years earlier at Christmas, off the coast of Norway, the gung-ho lad was rescued with all his crewmates when the Hull trawler Stella Rigel was wrecked in high seas. Michael remembered when Maurice came home that New Year that the first words he said as he walked through the door were, 'Before you say anything, mum. I'm going back.'\n\nMichael was a latch key kid. Mum and dad both worked at the Ross Group's fish processing factory on Bankside. That day at around eight a.m, like every weekday, he was getting ready for school. There was a knock at the door. Michael could see through the glass panel that there were two men. One of them he recognised as David McMillan from the Mission. He did not know the other man. Michael had an uneasy feeling as he answered the knock at the door.\n\n'Where's your mum and dad?' said the Mission man.\n\n'At work,' the boy replied.\n\n'And where are you going now?'\n\n'School.'\n\nMichael's mind was racing. He did not know how to get mum and dad. He was thinking, 'This is something to do with our Maurice,' \u2013 but he dare not ask.\n\n'Are you going to school now then? Oh, that's ok, thank you very much,' McMillan said.\n\nWith that the men turned and went.\n\nHalf an hour later at the Ross factory, Mrs. Swain was called via Tannoy to the boss's office. When she got there her husband was waiting. Stood next to him was the Mission man. The woman fainted.\n\nAll the way to the Boulevard School, Michael kept thinking, 'This is bad, this is bad.' By 10.30 \u2013 playtime \u2013 the kids in the schoolyard were talking in excitedly hushed tones about another trawler sinking. That was it. Michael started running from the school towards home. When he reached Chestnut Terrace, he saw his brother-in-law's blue Morris 1100 outside. The boy burst through the door almost knocking Dad over.\n\n'I know what's happened, Dad,' said the boy on the verge of tears.\n\n'Right, OK, now listen,' the boy's father said. 'Your Mam's in there. Your sisters are in there. Just try and be brave for your mum and your sisters.'\n\nThe screaming and shouting in the living room unnerved the boy but he kept his composure as he had promised his father.\n\nLater that day, Maurice's mother managed to speak with a reporter briefly and paid a tribute to her son, 'He had fishing fever. He never missed a trip nor took a holiday. He felt it was his duty to go fishing. He did a trip on a merchant ship once but returned to his true love \u2013 fishing.'\n\nAcross town at the Orchard Park Estate pretty twenty-year-old blonde Shirley Cliff was heartbroken. In the distressed girl's hand was a telegram sent six days ago.\n\nWritten on it was a rhyme, 'The ocean is deep, \/ the ocean is blue, \/ but not as blue as I am \/ when I'm away, from you.' It was the last message from Shirley's fianc\u00e9.\n\nShe wept as she told the smart young woman with her in her mum's living room all about it. Shirley said, 'We were to be married as soon as he came home. I had a wedding dress and have a bottom drawer crammed with things. He promised he would give up the sea after this trip. In fact, the morning he left he was so undecided.'\n\nShirley's boyfriend was Trevor Thomson, the boy who leapt from the lock to the deck of the Ross Cleveland as it left Hull forever. Carole Newton of the Daily Express had the next day's exclusive front-page human-interest sidebar story. It was illustrated by a photo of Shirley taken in her living room and one of her and Trevor, taken from her dressing table.\n\nHull West MP James Johnson was probably the busiest man in the House of Commons that day. Earlier he sent a telegram to Lily. It read:\n\nBEST CONGRATULATIONS YESTERDAYS MEETING + DELIGHTED TO HEAR BBC NEWS + COMING TO WESTMINSTER TUESDAY TO SEE MINISTERS + VERY PLEASED INVITE YOU AND FELLOW CONSTITUENTS BE MY GUESTS AT TEA HOUSE OF COMMONS + KINDLY CONFIRM \u2013 JIM JOHNSON MP.\n\nJohnson had tabled written questions for Board of Trade president Anthony Crosland demanding an independent inquiry into the fishing industry, followed by another to Minister of Labour Ray Gunter asking which of the International Labour Organisation's conditions for fishermen were ratified by this Government, with a further question to Transport Minister Mrs Barbara Castle asking if she was aware that the Hull St Andrew's Dock needed urgent repairs, bigger berths for modern freezer trawlers and that it \"did not give a good image of the local fishing industry.\"\n\nThe West Hull MP asked Crosland to address the shortage of trawler radio operators and to make their use compulsory. He also asked for an examination of the amount of hours worked by individual fishermen.\n\nWith the news of the Ross Cleveland, the focus was on Hull. It was decided that a statement would be made to the Commons, addressing all Johnson's issues and the news of the latest disaster. Two years of campaigning by the Hull Transport and General Workers' Union, followed by the behind the scenes work of MPs Jim Johnson and Hull North MP Kevin McNamara \u2013 a wireless operator's son \u2013 had been trumped by seven days of campaigning by Lily and her headscarf army. The Government knew they had to act at once. Johnson's questions, McNamara's fact gathering were overshadowed by the women's sudden, effective and very public actions.\n\nSo at precisely 3.15pm, J.P.W. Mallalieu, the shipping minister for the Board of Trade, started his statement. The usual bear-pit that was the House of Commons fell silent as he stood. Mallalieu outlined a plan for immediate government intervention in the trawling industry. This was an historic action. The minister summed up by saying, 'I am urgently considering what restrictions should be imposed upon the operation of trawlers, having regard to their size, the area of operation, the seasons of the year and their stability and freeboard. I am asking representatives of trawler owners' associations and unions concerned to meet me immediately to discuss these matters and any other measures that might be taken forthwith to reduce risks for trawlers operating in these arduous conditions.'\n\nMallalieu then announced an investigation into the loss of the Ross Cleveland and said it would be followed by a full inquiry.\n\nIn his reply to Mallalieu's statement, the House fell silent as Johnson spoke of his community's 'numbing despair at this third tragedy.' He said, 'It is incredible that we should have this third loss within these few days. We have got to do something about this. It is such a grievous personal loss. My first feeling is of deepest sympathy for the dependents of these men.'\n\nTen hours after leaving Hull, the St Andronicus \u2013 sister ship to the St Romanus \u2013 was heading back home. Its crew was overcome by fear after the news of the Ross Cleveland and they had been heading to the fishing grounds where that ship had perished. Seven young, frightened deck hands talked of the sinking of the Ross Cleveland. One by one they went to the skipper Albert Weatherill's cabin.\n\nThe skipper radioed back to his bosses, 'If they are frightened there is nothing else to do but go back to Hull.'\n\nUnion man Mike Neve was told of the skipper's decision and was ready to meet the crew when they got back. Neve told the Hull Daily Mail, 'They were genuinely scared, especially when they got the news of the Ross Cleveland. They were a young crew. They have been told to report back tomorrow.'\n\nWith unwitting understatement he added, 'I think there is difficulty in getting replacements.' One of the eighteen-year-olds from the ship, spare hand Alan Lee, from Gipsyville, Hessle Road, said, 'We were all frightened. Frightened that the same sort of thing as happened to the other trawlers might happen to us. I don't think the bosses will take any action against us, in the circumstances.'\n\nHe was correct.\nEIGHT\n\n_Hull's Paragon Station, Tuesday, February 6._\n\nThe miasma of tobacco smoke and breath-clouds from the crowded platform floated and twisted on the winter morning air and dispersed as it rose into the glass and steel arches. Camera crews, photographers, broadcasters, reporters, politicians and union men milled around as the 0755 from Hull to London's King's Cross got ready to depart.\n\nLily and Mary were there early and Yvonne arrived at last, with minutes to spare. The fur coat she was wearing was the reason for her tardiness. The pretty young cabaret singer stopped at her Mam's house on Hessle Road on the way to borrow it after it became apparent that her neat two-piece suit offered little protection from the biting sub zero temperature. TGWU officers Ashwell, Neve and Shenton met the women.\n\nThe fourth member of the Hessle Road Women's Committee \u2013 Chrissie Smallbone \u2013 was absent. She was at home grieving for her brother, Philip Gay, commander of the Ross Cleveland, the trawler reported as lost with all hands the day before. On the counter of the station's WH Smith's kiosk several newspaper front pages splashed the photo of Chrissie being led away in tears by the padre. The picture was taken after yesterday's meeting with trawler boss Mark Hellyer. Near the platform a news billboard told commuters, HULL FISHWIVES BOUND FOR LONDON.\n\nFreelance journalist James Goodrick covered the city of Hull for the BBC and the national press. He was a man of military bearing and had been a major in the Royal Artillery (Home Guard). James was the BBC's man during the Blitz when his native city suffered more than most. He had combined his wartime duties as a firewatcher and journalist as well as being the artillery officer who manned the ack-ack guns on the municipal building's roof. He was well regarded by news desks nationwide and was known as a man who got the job done. This reporter had established a reputation as a first-rate, no-nonsense newsman since his wartime reports. It was he who introduced \"a north-east coastal town\" as code for Hull during wartime broadcasts he carried out with not inconsiderable bravery from the roof of the city's Guildhall when the Luftwaffe all but flattened the nearby docks. James Goodrick had not been short of work since.\n\nThat morning on the railway platform the dapper middle-aged reporter caught Lily for a final interview before she boarded the 0755. He knew he had only minutes. He held his outside broadcast microphone with the familiar semi-circle of letters B-B-C atop it as he walked towards the battling fishwife.\n\nThe questions were sharp and delivered in a clipped voice that belied James' Yorkshire roots. His full-length mohair overcoat shielded him from the cold of the train station and his trademark bowtie and trilby made him stand out from his peers in the media scrum. The only thing more clipped than James' radio voice was his perfectly groomed pencil moustache.\n\n'Who will you see in London, Mrs Bilocca?'\n\n'Mr Mallalieu and Mr Peart.'\n\nHer brief reply sounded as if Lily mistook the urgency of James' questioning for abruptness or rudeness \u2013 and she was in no mood for either. 'What are you going to put to them?' James pressed on. Lily's voice grew louder. She told James, in no uncertain terms, she would be telling \"them ministers\" there would be no trawlers leaving without a full crew, including radio operators. And she would be demanding lifelines on parts of the decks \"for them lads on top\" and added all vessels will have full safety checks before sailing.\n\nThe reporter was more aware of his pressing deadline as the people from the platform started to board. He continued and asked Lily her opinion about calls from JPW Mallalieu for a closed season off the northern cape during fierce weather. Mallalieu was one of the ministers Lily was to meet that day.\n\nShe snapped, 'Certainly. They should have that particular place closed off, so that no ships can get round there. They should not be allowed to go there in this kind of weather... anytime.'\n\n'Where do you think they should fish then?' James said. Lily's voice rose again and cracked. It was almost as if she thought the reporter was challenging her. 'Where it's safer grounds \u2013 where it's safer fishing grounds \u2013 where the weather isn't so intense.'\n\n'What will you do after you have interviewed the ministers? Are you going to the House of Commons?' That was it. Lily was in full flow. Mary nodded towards the big lady and then said to Yvonne, 'She's off on one now!'\n\nLily continued, 'If I can get satisfaction from the ministers I am prepared to come home and carry on but if I don't get satisfaction I'll be at that Wilson's house, that private house, 'til I do get satisfaction in some shape or form.' James asked whom she would give her petition to and the fishwife looked at him as if he was daft and told him it would go to the 'appropriate person, and not just anybody.'\n\nLily said, 'I want the people that are concerned with this. Then she added in a sharpening tone '... and if they are not concerned they should be.'\n\nJames chanced his arm with one more question.\n\n'Now yesterday I think you had an interview with one of the Hull trawler owners. Did you get any satisfaction there?'\n\n'Not satisfaction, but we got a fair hearing.'\n\nWith that, Lily joined Yvonne, Mary and the union men on the train. James went off with a flea in his ear... and a great broadcast for the BBC nine o'clock morning radio news bulletin.\n\nMinutes later the packed train was running parallel to Hessle Road. The tall smoke house chimney stacks were topped with snow. Ten minutes or so later they passed the town of Hessle and the wide River Humber hove into view on the left. Those on that side of the train were better able to see the one mile-wide, demerara-coloured waters of Yorkshire's biggest river flecked with white horses as the wind whipped across the estuary. Snow-covered banks were bright white and reflected the winter sun.\n\nThe train wove its way through the riverside landscape to the Humber villages of Brough and Gilberdyke, onwards to the port town of Goole. After an hour it moved inland over into the South Yorkshire coalfield towards Doncaster. The vista changed to one of slag heaps that pockmarked the countryside as the rail station drew nearer. There was no sightseeing for Lily, Mary and Yvonne who found themselves among the very few \"real passengers\" on the train, which was filled by media, politicians and trades unionists. TV crews from around the globe, with lighting, boom mikes and cabling took turns to quiz, photograph and film the Hull women whose campaign was now top of the news agenda. French, German, Icelandic, American, Australian and many other nations' news networks carried out interview after interview.\n\nThe women struggled with the accents. The news teams had a similar struggle. Mary, Lily and Yvonne could barely get their message across for laughing as they tried to explain their campaign in their Yorkshire dialect to a Japanese TV crew via an under-pressure interpreter.\n\nYvonne, the cabaret singer was a bit more at home with flashes, lighting and cameras. Mary, in her work as a ship stewardess, had little problem dealing with talking in public. Lily, whose biggest adventure had been her teenage Parisian love tryst with Charlie, had not otherwise been on a train further than Selby for market day. And her public appearances were limited to the occasional sing-along at the Criterion snug bar. She took to the interrogations with discomfiture but she was determined it would not show. By Doncaster, things calmed. The three women sat on opposite sides of a table in their crowded carriage but were still interrupted by the occasional \"just one more picture, love.\"\n\n_Seydisfjordhur, Iceland's Northern Cape, Tuesday, February 6_\n\nIcelandic Lifesaving Association volunteer vessel Svanur was near Seydisfjordhur. It was mid-morning when Captain Thorir Hinriksson spotted a life raft on the rocky beach. The Svanur's sister craft had been in the area the day before at dusk and failed to see it in the thick mist. Both ships were part of a local fleet that volunteered to search for survivors and bodies after the great storm that had caused so much devastation little more than thirty-six hours earlier.\n\n'Could it be... could there be survivors?' thought Captain Hinriksson.\n\nHe decided to take a search party ashore in a dinghy to find out.\n\nThe landing party crossed the snow-covered rocky beach towards the raft where they noticed a scattering of dead fish. Such was the power of Sunday's storm it had disgorged them from the sea. The life raft was a hundred yards inland. The No. H61 insignia on it showed it was from the Hull trawler Ross Cleveland. Earlier, lifebelts and other debris from the same ship were found, so the markings were familiar to the searchers. Deep tracks led from the beached life raft to the shoreline across the thick snow covering the rocks. Someone had pulled this craft to where it lay. Footprints led from it. Two men followed them. At first, they led along the beach towards the sea then came back on themselves, as if whoever had made them realised he had gone the wrong way. The footprints were close together which meant whoever made them took small steps. Two men followed the tracks while another two examined the life raft's interior and peered through a tear in the flap of the craft. That torn flap cracked in the wind.\n\n'Skipper, here look!' shouted one of the men.\n\nHinriksson walked towards the raft. He too peered through the torn flap. Inside, he saw a young boy dressed only in underwear. He was bolt upright. Next to him was an older man dressed in a blue sweater and heavy trousers. Both corpses were stiff and in a sitting position. Hinriksson realised there was a chance a survivor was around here somewhere. Whoever it was, was a strong man, strong enough to pull the life raft with two dead shipmates in it for almost one hundred yards. This was a remarkable feat. More so if you consider that the man who had done this had been thrown ashore after hours in a freezing sea during the greatest storm in living memory.\n\nThe rescue team had to deal with the two dead men from the life raft. Engineer Jon Ragnarsson and colleague Agust Gardarsson got the task of straightening the rigor-stiff corpses to fit them onto the dinghy for transfer to the Svanur. The dead men were covered in blankets and laid to the starboard of the boat, with as much dignity as the situation allowed. Captain Hinriksson decided to head to Kleifar. He reckoned that whoever made those footprints must have tried to make it to there. After all, it was the only place within ten miles of the craft's landing site that had any shelter.\n\nThe dinghy was taken back to the Svanur and its human cargo unloaded. The vessel went further up the fjord and launched the dinghy again. It was rowed to the coast near Kleifar. Hope among the landing party was tempered by local knowledge. These men knew that to get to Kleifar from where they had found the life raft that very rough terrain had to be crossed, which included a steep cliff. A fit man on a summer's day would find this arduous. But a frozen, frightened, lost man in the dark?\n\nThe search party came ashore once more, about a kilometer from Kleifar Farm. If by some miracle some one had survived from the raft, getting there would be their only real hope. This was a wide, barren landscape where someone who lives twenty kilometers away would be classed as a near neighbour. Bodies had been found in Isafjord Bay earlier that day. There was still a chance, however remote, of finding someone alive. None of the rescue team would say it, but the feeling among them was it would be another dead body they would find.\n\nAfter about a half hour's march, the searchers came within sight of Kleifar Farm. The wind had dropped. But by the time they got to the farmhouse door, the duffle-coated men were caked with snow. Those with facial hair had stubs of ice around their mouths where spit and breath had frozen. Captain Hinriksson knocked and a heavy-set man, of average height in his early sixties opened the big wooden door. Hinriksson recognised him as Gudmundur Asgeirsson. This was also an area where fishermen farmed and vice-versa, a tight knit community where most folk knew each other. Gudmundur Asgeirsson and the Captain exchanged their greetings and the party was ushered in. The doorway led straight into the kitchen.\n\nFor the frozen rescue team, the room with its wood fire was a warm welcome. Hinriksson could see that near the fireplace there was a man sound asleep on a leather couch. The man was dressed in a sweater and denims. There were mittens on the man's hands. Gudmundur Asgeirsson beckoned to the old couch.\n\nHe said, 'An English trawlerman is in our kitchen. My son found him this morning behind our summerhouse.' It took Hinriksson a second or two to take in what the farmer had said. Gudmundur then told the Captain how his son found the fisherman exhausted and near death just a few hours earlier. Hinriksson found that Gudmundur did not know much more than the man's name \u2013 Harry Eddom.\n\nThe farmer told him that the Englishman was in amazing shape considering all he had been through. Harry had been asleep for about two hours on that couch. Hinriksson, who spoke a smattering of English, decided to wake the miracle man. He had to know more. He went to where the Englishman lay and coaxed him awake.\n\nThe Captain was enthralled when the English fisherman told how his shipmates, who dragged him unconscious onto a lifeboat, died later that Sunday night and how he and the dead men were beached hours later. Hinriksson estimated, from what Harry Eddom told him, that the life raft must have come ashore around dusk on Monday. This was not long after his fellow volunteer search vessel failed to see it through the heavy mists and blizzarding snow.\n\nHarry told the Icelandic skipper how he had found the locked summerhouse and had not had the strength to open it and how he forced himself to stay awake all night until the shepherd boy found him. Apart from being dazed and tired Eddom appeared well. The farmer and his family had obviously taken great care of him, dressed him warmly and put heavy mittens on his damaged hands.\n\nThe Captain told Eddom that the bodies of his two Hull shipmates were recovered from the Ross Cleveland's life raft and were aboard the Svanur. The two men carried out their conversation in broken English and sign language while the farmer's wife served up coffee and the remainder of her little boy's birthday pancakes. It was soon time to go and take Eddom by stretcher the kilometer or so to the dinghy. Eddom was patently nervous as he approached the boat and he had to be reassured as he was carried aboard.\n\nThe stretcher had its ends over each side of the boat to which it was tied. One crewman was assigned to sit with the wary Englishman as the remainder rowed the tiny boat back to the Svanur. On board Harry was kept in the warmth of the skipper's cabin. He was well wrapped up. Hinriksson earlier noticed damage to the fisherman's feet and hands and feared that frostbite had kicked in. It was full ahead to Isafjord Bay, where a hospital bed awaited the man who had come back from the sea.\n\nIsafjord Bay harbour was full. Dozens of ice-covered trawlers bobbed in the calm post-storm waters. The vessels looked like ragged, ship-shaped mini glaciers. A mountainous backdrop and snow-covered huddle of buildings made the little fishing port look quite the winter idyll. However, amid this picture postcard panorama, frenetic work was carried out by shifts of local volunteer vessels that used the Bay as a base for their searches.\n\nThe local scout hall was filled with British fishermen rescued the night before from the Grimsby trawler Notts County. Those men had also witnessed the sinking of the Ross Cleveland before their own trawler ran aground, little more than half a mile from shore. The men of the Icelandic patrol vessel Odinn had launched two inflatable motorised lifeboats and carried out an audacious rescue ferrying eighteen men through a mile of raging, freezing seas from the stricken trawler to the safety of the coastguard patrol vessel. One of the Notts County crew, deckhand Robert Bowie, died of exposure after he tried unsuccessfully to launch a life raft. His body was recovered later. The Notts County skipper George Burres and the mate Barry Stokes were in the Isafjord Hospital being treated for frostbite and shock.\n\nThe remaining sixteen crewmen were in the scout hall. That day the Odinn had been out again searching for any possible survivors from the local six-man crew of the line-fishing vessel Heidrunn II, as well as from the Ross Cleveland, which perished the night that the Grimsby men were saved. Word reached the scout hall that the Odinn was in the harbour after its fruitless search for new survivors. The Grimsby men decided to form an impromptu \"guard of honour\" to thank the Icelanders who had saved their lives. When the Odinn came into the harbour there was quite a crowd of locals and other fishermen who joined in the tribute. The Englishmen applauded as the Odinn crew came ashore. The rest of the crowd joined in.\n\nNormally, English trawlermen had very little time for Icelandic patrol vessels and their crews and vice-versa. Just three days earlier the men of the Odinn arrested eleven English trawlermen for breaching Iceland's fishing limits in the latest in a series of Cod War battles that dated back decades. That day there was no Cod War \u2013 just men of the sea and mutual respect.\n\nAcross Isafjord harbour in the galley of the Hull trawler Kingston Andalusite, a young deckhand, Ernie Bilocca, was finishing his meal. His storm-battered trawler was in for repairs. Ernie could not shake off the horror of the night before, when he had been on the bridge and watched with his skipper as the Ross Cleveland disappeared from their radar. After that thirty-hour shift Skipper Whur sent the exhausted young man to his bunk. That night Ernie learned what is was to be truly dead tired. His ship had come within seconds of joining the Ross Cleveland. It had rolled onto its starboard side. A cacophony of shouts and screams mixed with almighty creaks and cracks as the fibre of the craft strained against gigantic waves that minutes earlier had claimed a ship with all hands in a matter of seconds.\n\nErnie was so tired he lay unable to move. Somehow the terrified twenty-one-year-old deckie was not thrown from his berth as his trawler listed. Men ran past. There was panic. Ernie prayed. The ship righted. He lay scared stiff and dog-tired through the horror. He wanted only to sleep. For hours to come Skipper Whur fought to keep his ship afloat and his superlative seamanship saved them. Ernie decided no matter what happened he would not, could not move. He made his peace with his God and slipped into fevered, frightened sleep as the ship was thrown around the boiling sea.\n\nNow in the warmth of the galley, in the safe haven of Isafjord, that all seemed unreal \u2013 just a bad dream that would never go away. Ernie's reverie was broken as one of the lads shouted, 'Hey turn that up, pal.'\n\nA man next to the galley's big cathedral wireless did as he was asked. A newsreader's cut-glass voice from the BBC World Service crackled as it came through the old radio.\n\n_'A deputation of fishermen's wives from Hull are heading for London today to talk to ministers about safety at sea following the dreadful loss of 59 men from three trawlers in the last nine days...before they left Hull our reporter James Goodrick asked Mrs Lillian Bilocca, the women's leader, who they would see...'_\n\n\"Atmospherics\" then wiped out the broadcast.\n\nErnie sat up at the table, his mug still in his hand.\n\n'What the hell is our lass doing going to London?' he blurted out.\n\n'It's not your lass, mate, it's yer Mam. I heard it on the wireless earlier,' said a shipmate. 'She's got a big petition and they're off to Parliament wi' union.'\n\n'Yeah, that kinda makes more sense,' said a confused Ernie. His mother and wife had the same name. He added, almost as an afterthought, 'Let's get that bloody wireless fixed.' Laughter subsided instantly when the skipper walked in. The men put down their cutlery and tea mugs. Len Whur had a message from the bosses in Hull.\n\n'You are not going to believe this, lads,' the skipper said, 'they are only telling us to go back out fishing.' The men were agog but even further astonished by the skipper's next words. 'If I were you lads I wouldn't go. I'll have to get back to them with an answer. Shall I just tell them you are refusing to go? After all, we can hardly blame you.'\n\n'Yeah, right gaffer,' said one of the lads, 'tell 'em we ant goin'.'\n\nWith that, Len Whur went back to the bridge to relay his crew's decision and refused a company command for the first time. Ernie and a couple of pals cursed the bosses and went back to trying to better tune their wireless set.\n\nThere was still a crowd at the harbour. Most of the Notts County men went back to the Isafjordhur Scout Hut. There to meet them was Alan Bennett, a young reporter from the Daily Express. The dapper, bearded 33-year-old newspaperman was pleased to be out of the snow. It flurried across the square that separated the harbour from the buildings among which was nestled the scout hall, temporary digs for the lucky men of Grimsby. Bennett was a fishing industry expert and a staff reporter who carried a roving European brief for the paper.\n\nAt the dockside the Svanur had berthed. The men surrounding the harbour looked over the rails. Among them were some Hull men. The Svanur crew carried the bodies of the two men recovered from the life raft. Spectators bowed their heads and those wearing hats removed them as the dead men passed by on covered stretchers destined for the local hospital. Men watched from the harbour rails. A third stretcher was being brought ashore. There was a gasp. The \"body\" moved. A voice with a strong Hessle Road accent said, 'Fuck me, that's Harry Eddom! It's like seeing a ghost!'\n\nMinutes later an Icelander burst into the scout hall. The excited, breathless man shouted, 'They've found an Englishman, alive from the Ross Cleveland! They're taking him to the hospital.' In seconds Alan Bennett of the Daily Express was running through a blizzard across the small town square to the biggest exclusive of his career.\n\n_Aboard the 0755 from Hull to London's King's Cross_\n\nAn hour or so before coming into London Lily had an idea.\n\n'Hey lasses, let's get some more signatures.'\n\nLily was determined to add to the ten thousand names and addresses on her thick wedge of forms, printed by the union, (based on her daughter's Virginia's secret design done on her employer's machinery). The three women set off along the corridor of the train. No one refused to sign. Even the foreign journalists were coaxed into it. Some of the other \"real\" passengers applauded as the trio made their way further along. Between carriages the women giggled like schoolgirls. This was the first time they had taken a break from their campaign to laugh together. At the front of the train, there were half a dozen or so very well dressed ladies chatting. Lily breezed into their carriage with Mary and Yvonne, and the ladies' genteel conversation halted.\n\n'If looks could kill...' Mary thought. She recognised the posh women. After all, Mary was a trawler officer's wife. It was too late. Lily got started.\n\n'Hi-ya, ladies...would ya like to sign our petition?'\n\nOne of the ladies frowned at Lily.\n\n'Why ever should we? It would appear you have all you have asked for already,' the well-dressed woman spat.\n\nLily turned tail and left giving the slide door a hefty tug as she did.\n\n'Sod ya, then.'\n\n'They're trawler bosses' wives, Lil,' Mary said. 'I recognise them from works' dos. We'll get nothing there!'\n\n'Sod 'em, any road,' Lily laughed. Her two friends joined in.\n\n'Can't win 'em all,' Lily added in her booming voice. The laughing women made their way back up the train.\n\nThe trawler bosses' wives may have been dismissive of the fighting fishwives, but back in Hull as Lily's train came into King's Cross, her local paper's first edition showed that the posh women's rich and powerful husbands had taken notice. The Hull Daily Mail announced on its front page that the Silver Cod Trophy dinner, due to be held the following month, was cancelled. Hull-based director-general of the British Trawler Federation Austen Laing said it would be 'entirely inappropriate to hold the dinner at which the Silver Cod Trophy would be presented to Britain's champion skipper, against the background of the loss of three Hull trawlers.' He added, 'The British Trawler Federation is deeply shocked at the loss of so many of our fine trawlermen,' adding that the \u00a31,800 that would have been spent on the big night would be given to fishermen's charities. At the dinner that was to be held in London, guest of honour was to have been the Prime Minister. With dark irony the theme of the Harold Wilson's proposed speech was the improvement of trawler safety at sea.\n\nElsewhere on page one it was reported that West Hull MP James Johnson, who was waiting to meet the delegation in London, told reporters, 'Mr Mallalieu has some good news for these women. He is going to take urgent and immediate steps.' The report added the MP would not 'elaborate further.'\n\nMark Hellyer, co-owner of the Kingston Peridot, eventually issued a statement \u2013 twenty-four hours after the Government had promised immediate action \u2013 and was keen to point out the ad hoc safety charter agreed with the women twenty-four hours earlier. He said, 'There was no bitterness at all. The wives of the fishermen have decided they will get for their menfolk what the unions have been unable to obtain up to now. They do not want strike action or anything like that, but they insist something should be done.'\n\nHe also confirmed what everyone knew and declared the Kingston Peridot as officially lost.\n\nThe King's Cross platform that the 0755 from Hull was about to pull into was unusually empty. Only a railway guard was on it. Mary Denness tugged down the window. The guard asked if everyone aboard could wait to be asked to disembark because it was 'bloody chaos.'\n\n'What bloody chaos?' Mary said. 'The platform's empty.'\n\n'You'll soon see, love.'\n\nThe passengers got off as the guard had requested. Mary and the others walked towards the concourse, which also seemed curiously empty. As Mary walked a woman with a posh voice asked her for an interview.\n\n'What now?'\n\n'Yes, it'll only take a minute or so. We can go in there.'\n\nMary followed the reporter from BBC Radio 4's Woman's Hour programme into the ladies' rest room. The BBC woman was as good as her word and a few minutes later Mary and she emerged. Then they realised the cause of the platform guard's anxiety. Thousands strained to get a look at the women from Hull. They were kept from the station by rows of crowd barriers. It was like a Beatles' appearance. Crowds cheered and applauded as the fighting fishwives' entourage made its way towards waiting cars. They were the toast of London. Mary noticed a London Evening Standard billboard amid the crowd. It read, BIG LIL HITS TOWN.\n\nThe three Hull union men were with the women when they met the TGWU General Secretary Jack Jones, who was with his wife, Evelyn. Evelyn complimented the women on their appearance. (Hessle Road hairstylist Doris Forrest had done them proud.) Mr Jones told the women how much he admired what they had done and said he would do everything in his power to help them.\n\nNational Union of Seamen steward John Prescott recognised Yvonne from her grandstanding speech at the Victoria Hall a few days earlier. The young union firebrand crossed the station concourse and spoke with the petite singer. He quipped, 'You should join us in the union, love. We could do with folk like you.' Yvonne replied, 'I am very flattered, petal. But I just want to get what we can for our men. I am not interested in unions or politics.' Two cars picked up the delegation and edged their way past the throng on the way to the House of Commons. The drive was taken up by small talk. The affability of the middle-aged couple chatting away with the Hull fishwives in the back of the leading car belied a steely resolve. Jack Jones, an avuncular, bespectacled fifty-five-year-old, was an ex-Liverpool docker like his father before him. Jones fought with the International Brigade as a young man in the Spanish Civil War, a conflict in which his now wife was widowed after Jack's best mate \u2013 and her first husband \u2013 George Brown was killed in action by fascists. So the Joneses were no strangers to fighting for what they believed to be right. The women of Hull were in excellent hands.\n\nFrom the moment they arrived at the St Stephen's Hall entrance to the Palace of Westminster, they felt like they were on a film set. The famous gothic fa\u00e7ade of one of the most recognisable buildings in the world caused them to stop and stare up like tourists before they walked through its massive doors. They were photographed before they went in and quizzed again by reporters.\n\nMary said, 'We will stay in London until we get something done. We don't want promises. We want action.'\n\nLily went one better and added, 'If our demands are not met I will stand on the doorstep of 10, Downing Street forever if necessary.'\n\nThe click-clack of high heels echoed and sounded like erratic metronomes when the women walked across the Palace's mosaic floor tiles. The women struggled to balance; unaware they had stepped over the site of another form of direct action. It was a plaque that marked the spot where, 156 years earlier, disgruntled failed businessman John Bellingham shot Prime Minister Spencer Perceval in the first and only assassination of its kind. The party headed to the Strangers' Dining Room. Their meeting with the ministers was not for a couple of hours yet. The Joneses led them and the three Hull Labour MPs to lunch. On the way along the Hall on either side a dozen statues of MPs of days gone by struck various combative debating poses, tribute to the days when the Commons met on this site before a fire in 1834 forced a rebuild, which resulted in the grandeur they walked through that day.\n\nThey gazed at the mosaics, paintings and art that adorned the Hall while not really taking in what they were looking at. The painting of King John giving assent to the Magna Carta, ironically, went unnoticed as the women and their hosts walked on with Lily carrying a \"Great Charter\" of their own. Mary could not help but think that the last few days for them had meant going from one plush wood panelled place to another. But that day's vista impressed a thousand fold more than the grandiose offices of the trawler bosses in Hull.\n\nLunch at the Strangers' Dining Room flashed by \u2013 as soon as the stuck-up waiters got off the scene. The women were more at ease with household names of politics and trades unions than they were with the beadle-esque, stiff-upper-lipped waiting staff, who made them feel as if they did not exactly approve of their presence, let alone the necessity to serve them.\n\nAfter lunch the party was led towards ministerial offices, and passed the sites where Guy Fawkes, William Wallace and Charles I had been consigned to their respective dooms. They were taken to a lift \u2013 an incongruous mode of transport in this gothic place. It was one those big goods lifts with a large concertina-style black iron sliding gate. The operator slid the door closed and operated a brass handle to take them to their floor. Lily kept a close watch on her petition and her \"Fishermen's Charter\" that demanded, full crews, radio operators for all ships, twelve-hourly contacts between ships and owners, improved safety equipment, a 'mother ship' with medical facilities for all fleets, better training, a safety representative on each ship, suspension of fishing in winter on the northern Icelandic coast, and a demand for a Royal Commission. When Lily, Mary and Yvonne walked through the door with their ten thousand signatures and their Charter they did not know that the ministers there were already fully briefed with all the demands. For the next hour and a half or so the Labour parliamentarians JPW Mallalieu and Fred Peart, the women from Hessle Road and the union men, went through the points.\n\nFisheries minister Peart, a blue collar socialist from Doncaster was intently promising the women's demands would be met, when Oxbridge-educated ex-Navy man JPW Mallalieu MP, of the Board of Trade seemed to hi-jack the meeting. Lily rolled her eyes and glanced towards Mary with the raising of an imaginary glass as Mallalieu and Peart occasionally slurred their speech, more so as they each tried to out-sincere each other. Whisky fumes pervaded. The women all smiled. After all, they were fishwives and they knew a pisshead when they saw one. The mime went unnoticed by the MPs.\n\nLily turned to Peart and said, 'Talk to me in me own language, love. There's no need to be fancy. Straight talk will do fine.'\n\n'I will, love,' replied the Yorkshire MP.\n\nYvonne insisted in going through each aspect with the ministers and told Mallalieu, 'I not going to stop until I have gone through the whole lot with you, petal.' Mary marvelled at her friend's effortless recall and listing of figures for the ministers. Yvonne determined that every aspect of her detailed notes was set out to the men. The ministers were impressed too. Then the petite blonde flashed a smile at the fisheries minister Peart. JPW Mallalieu had never been called \"petal\" before. The ministers went on patiently as Yvonne, Lily and Mary continued their demands. They laughed again when Yvonne told Peart, 'Listen petal, we are not going until we have had our satisfaction!' It was four-thirty p.m. when the meeting drew to a close. The women shook hands with the ministers and as they got to the door Mallalieu looked straight at them and said, 'You've got it ladies. You have got it all. I promise.'\n\nPeart turned to Lily and added, 'We will do all we can, love.' This buoyed the battling Hessle Road fishwife. She was comforted by Peart's northern directness \u2013 a trait she shared.\n\nThe women needn't have worried about their demands being met. Harold Wilson, who was in the US for presidential talks, had been briefed fully by Jack Jones beforehand, and in turn had told Mallalieu and Peart to report to him that day \u2013 and to help these women any way they could. After all, the Labour Prime Minister had already gone public in the Press with his support for them.\n\nA civil servant led the women to the lift and on to an anteroom where the Press were waiting. The women and the others were hardly seated at the table when a reporter said, 'They've found a survivor in Iceland. He's the mate of the Ross Cleveland. His name's Harry Eddom. Have you anything to say, ladies?'\n\nA brief silence followed. The women were stunned. Lily burst into tears.\n\nShe said, 'I felt fifty instead of forty when I came down here today, but I feel happier now. It's the best piece of news I have ever had in my life. I would give ten years of my life if the other men who were lost on the trawler could be found too. I said it would be a miracle if it happened.' Mary added, 'It raises hope that perhaps they will find someone else now. We live in hope.' The union men and the three women were hustled off, leaving the politicians and civil servants to deal with the media. They had a train to catch. Outside, as the women went towards the car, a barrage of questions followed. Mary and Yvonne managed a couple of quotes as the party was bundled into the cars. Mary said, 'Three women have achieved more in one day than anything that has ever been done in the trawler industry in sixty years.'\n\nYvonne added, 'Our men are good men. They want to take care of their wives, but we'll get them to rise up and defend themselves.'\n\nThe cars headed for King's Cross and the Press followed. At the station, the women were rushed through to their platform. There was ten minutes until departure for Hull. The Press mobbed the women and fired questions at them. The mascara on each of the women's faces was streaming. Their joyful tears did not stop them doing their best to bat the questions from the reporters who had surrounded their train door.\n\nLily shouted, 'The wheels are moving. We have got further today than others have in years and years.'\n\nMary joined in with, 'Every detail was discussed. They were really wonderful.'\n\nThe three women were bundled on and Jack Jones was left with the Press as the train prepared to leave. The union boss told the reporters he was happy with the ministers and said there would be an immediate meeting the next day with trawler bosses, union men and others to enact what had been discussed.\n\nJones said, 'We must have urgent action. The ministers congratulated the women on the way they put their case and I want to congratulate them too. They did a splendid job.'\n\nThe train started to pull out, and a reporter shouted a question to Mary, who was leaning out of the train door's slide-down window.\n\n'Do you think the wives in Hull will be happy with today's events?'\n\n'Yes, they most definitely will,' shouted the jubilant skipper's wife as the late-running King's Cross to Hull took the women home.\n\nIt was 4.40pm that day when the news desk phone rang at the Hull Daily Mail. News Editor Charles Levitt took the call. Minutes earlier he had been clearing his desk and putting the finishing touches to the news desk diary for the next day. Apart from Levitt, there were only three men in the quiet newsroom. They were reporters Stuart Russell and Derek Hilton and a late duty sub-editor. They had twenty-five minutes. The news of Eddom's miraculous survival had to be on the presses by 5.05pm. There was no chance of a late run. If the Second Coming happened in Hull and failed to reach the presses by 5.05pm, local folk would have missed the Good News \u2013 the print unions would have made sure of that.\n\nCharles Levitt was more than up to the task. Nothing fazed this slim-built, sandy-haired thirty-eight-year-old ultra-calm newspaperman, who had been in the business since he was a sixteen-year-old cub reporter.\n\nLevitt turned to the younger journalists and said, 'Stuart, you and Derek get a photographer and get off to Askew Avenue now. The Ross Cleveland's mate Harry Eddom's been found alive. Get up there and find out what the parents have to say. Eddom lives near me, I'll interview the wife after I've finished here. Phone in with what you get.'\n\nRussell and Hilton set off. Levitt turned to the late duty sub editor and told him to get ready to redraft page one. Then he dictated the story, hurriedly put together from copy from the Press Association and himself, directly to a Linotype operator in the composing room. At 5.05pm precisely, the office block in Jameson Street shuddered gently as it had done each time across a century when printing presses thundered into action. The City Final, distributed only in the city centre, was loaded onto the vans in the delivery bays. Street sellers were soon shouting the headline, 'Hull trawler disaster survivor found alive!' Their kiosks were mobbed. Stuart Russell, Derek Hilton and a photographer arrived at Harry Eddom's parents house in Askew Avenue, west Hull. They waited on the doorstep after they knocked. Minnie Eddom, Harry's mother, opened the door.\n\nStuart asked, 'How do you feel about your Harry being found alive?' Minnie broke down babbling. She screamed and shouted incoherently on her doorstep with hysterics the like of which Stuart and Derek had never seen. She could not speak for crying. She was breathless and panicking. The journalists backed off with profuse apologies. Minnie ran indoors and was met by her husband. The door slammed. Derek turned to Stuart and said with dark comic timing, 'I don't think she believes us.'\n\nThe pair ran to the nearest phone box. Stuart got into the kiosk, reversed the charges to the news desk and told Levitt, 'We're not trained for this, boss. She went mental. I don't think she believed us. She was screaming and shouting and charged back indoors. It was crazy.'\n\nThe unflappable Levitt replied, 'Don't worry, you can't win them all. I'll see what I can get at Rita's house.'\n\nAt the Isafjord hospital a wheelchair-bound Harry Eddom had a telephone brought to his bedside. The international operator at Reykjavik was given the trawlerman's phone number for his Cottingham home, which was some four miles north of Hull. Harry's anxiety soared when Rita did not reply. He gave the operator another number. It was for his parents' home in Askew Avenue, west Hull. Harry's younger brother Michael answered. An Icelandic operator said, 'We have a call for you from Isafjord. Putting it through now.'\n\nMichael braced himself, expecting to hear that his brother's body had been recovered from the sea. But it was Harry's barely audible croaking voice that came through. Michael did not recognise the feeble, unclear tones and momentarily thought it was a hoax.\n\n'If this is a joke, it is not funny,' Michael shouted.\n\n'Mick it's me. No...Mick, listen it's me, Harry, your brother. There was a brief silence. Michael realised the call was genuine. Only Harry ever called him Mick.\n\n'Where are you calling from?' Michael asked.\n\n'Mick, I am in a hospital in Isafjord. I am all right. I'm fine. How are Mam and Dad and Rita?'\n\nMichael was too dazed to answer. Harry told him he called Rita and there was no reply.\n\n'Go over to her now, Mick, will you? Tell her about me. I'll talk to her later. Don't worry. Tell them I will be all right.'\n\nMichael then turned to his parents, Harry senior and Minnie.\n\n'Mam, dad, it's our Harry. He's alive!' The parents stared at each other. It was true. What the two young men who fled from the door minutes ago were trying to tell Minnie was true. Harry had survived. Michael said into the receiver, 'Give me your number, Harry. I'll get the exchange to put in a call to Rita.' As soon as he had the number, Minnie took the phone from him. She could barely speak for crying. She managed to say, 'Are you all right son?'\n\n'Aye Mam, I am all right. How are you?'\n\nHarry heard his mother's crying through the crackling of the international link. As his mother spoke, Michael shouted out the news to his maternal grandmother, 72-year-old Jane Cousins.\n\n'Oh, let me hear his voice,' Jane said. Harry senior comforted his relieved wife. Jane then took the chance to speak briefly with her miracle grandson and the call ended abruptly as the link cut off. Michael Eddom set off with his parents for his brother's house in Cottingham. Rita's thirty-six hours of widowhood would soon be over. At just after 5.30pm it was a much more approachable Rita Eddom that met Charles Levitt on the doorstep of her semi-detached home on Queensway. The day before she had been too upset to speak to the press. Michael Eddom and his family arrived around the same time as Levitt and the photographer. The lounge was filled with media, there to get quotes from the \"thirty-six-hour widow.\" Michael had brought a copy of the City Final, as had most of the reporters there. Rita was still in a frazzled state, crying one minute, laughing the next.\n\n'Rita, Harry phoned me. He's all right in hospital. He gave me a number,' Michael said. He then went into the hall of the neat little semi-detached and lifted the phone.\n\nThe gathered media could not believe their luck. They were going to be there when Harry phoned home. The ITN camera crew's runner went to the outside broadcast van and relayed a message to London. Michael gave a number to the international operator, spoke briefly, then he replaced the receiver. Disappointment was palpable. The news teams had thought they were going to witness the call then and there.\n\n'They're going to phone us back,' explained Michael.\n\nIt was then around 5.40pm.\n\n'I won't believe it until I can talk to him, Michael,' said Rita.\n\n'Believe it, Rita. He's all right. He's all right.'\n\nThe conversation that followed was recorded on newsreels, tape recorders and notebooks. Rita dangled baby Natalee on her knee as her mother-in-law made tea as a sort of unwitting displacement therapy. Michael told how the call from Harry came about. 'At first, I thought it was just a call to say Harry's body had been picked up. Then I heard his voice. I didn't believe it at first, until he said, \"Is that you, Mick?\" Then I knew.'\n\nHarry's grandmother Jane Cousins added, 'I had given up all hope. I didn't really believe he was still alive until I heard his voice. Even now, I must see him to make sure.' Harry's father cut in and said, 'I know the conditions out there and I had given up hope. Our Harry had said he did not want to go back on that ship. He preferred them big freezers. You have these hazards trip after trip, but you never think it'll happen to you. You think it will always be the other fella.'\n\nHarry's mum Minnie added, 'Harry never bothered about the dangers. He is a born fisherman.'\n\nA reporter said, 'Why do you think he survived?'\n\nHarry senior replied, 'Apart from the protective clothing? It was probably our Harry's physique that helped him to survive. He was always a good swimmer and athlete.' Almost as an afterthought, the old ex-fisherman added, 'I suppose you'd have to be, to be twelve hours in a raft out there, then walk the distance from Hull to Beverley.'\n\nRita's twelve-year-old little brother Dennis Penrose became the focus when he said, 'I always knew our Harry would be back.' The day before, even when the family was inconsolable, Dennis had refused to cry. When distressed relatives were comforting Rita, when he saw his grandfather cry for the first time, the boy would tell anyone that would listen that our Harry would be home. Then the boy took his father's hand and said, 'Let's go to church, Dad.'\n\nMinutes later, the boy and his father, ex-trawlerman Arthur Penrose, wandered into the first church they came to. There was a service on.\n\nArthur told the reporters and the family in the lounge, 'The vicar was praying for us and all the boys at sea. It was a little service for us all, it seemed.'\n\nReporters' pencils flew furiously across various notebooks and the boy added, 'I don't know why, but I just had a feeling that he would come back.' In the hallway the phone rang. Rita with baby Natalee in her arms rushed to answer it, followed by an almost farcical scrum of newsmen and cameras. TV lighting was on. The hallway was overflowing with people waiting for Rita to talk. Between bursts of tears and cries from seven-month-old Natalee, Rita garbled out her questions to the husband she thought she had lost forever.\n\n'Oh, Harry!...is it you?...are you all right...when are you coming home...shall I come out to see you...?'\n\nHarry tried to calm Rita as her stream-of-consciousness questions deluged him. Then, Rita said, 'What happened, love? What happened to you?\n\n'Rita, the ship's gone down and all my mates are gone...'\n\nHarry then fought to compose himself. He wanted only to comfort Rita.\n\n'I am going to be fine, love. I am in hospital with frozen feet.' There was almost a chuckle forced into his voice. 'A boy found me. I am going to be all right, love.'\n\n'Are you going back to sea, Harry?' Levitt could not help but notice that some of Fleet Street's finest were welling up with tears as Rita's call unfolded. There was a lot of small coughs and surreptitious eye-wiping going on among the tough-guy tabloid men. The newsman from the Hull Daily Mail swallowed hard too. Rita put the phone back on its cradle and it immediately rang again. It was another press enquiry, like most of the calls had been since yesterday.\n\nRita laughed and said, 'I don't care who calls now!'\n\nA reporter said, 'Do you think Harry will go back to sea, Rita?'\n\n'It's up to him,' said Rita wiping away a tear. 'I don't want him to go after all this. I can't go through any more. I don't want him to go back again. But it's his life and when I married him I knew what he was.'\n\nHarry's dad, himself an ex-fisherman, added, 'If I had my way he'd never go back. But I would not be surprised if he does. When he left school I said to him \"Don't go to sea, son,\" then I came home one day to find he had gone as a galley boy on the Loch Doone.'\n\nLevitt looked at Harry's mother.\n\n'Have you anything to add, Mrs Eddom?'\n\n'I feel like the happiest mother in the world, love. But I am so sorry for all the others whose boys have gone.' Rita dabbed at her eyes with her handkerchief and added, 'I had given up hope, you know. I was told no survivors could last in the conditions. I did not pray for him. I said there was no God. But now I know there is.'\n\nTV viewers nationwide were moved to tears in many a UK home when Rita's miraculous telephone conversation was seen as it happened on that night's late news.\n\nThe late-running London King's Cross to Hull train was met by a few reporters and cameramen when it pulled into Paragon Station. The three women and the union men got off to a ripple of applause from a handful of folk who had gathered to watch, markedly different from the masses that had swamped the station that morning. Lily, Mary and Yvonne looked tired but triumphant.\n\nTheir make-up was not at its best especially the mascara smudges. A tearful Lily said, 'We've done it lads, we've done it.'\n\nThere really was not much more to add. The union men told the reporters there would be another Victoria Hall meeting on the following day. The three battling women headed off in waiting taxis. A reporter glanced at the station clock, checked his wristwatch and said, 'C'mon lads, we might get a stop-back at The Star of West. I know the boss.'\nNINE\n\n_117 Coltman Street, Hessle Road, Hull._\n\nLily's feet were killing her. She looked forward to her favourite chair, to kicking off her best shoes and having a nice, big cup of tea. It had been a long day in London. The rattle of the taxi's diesel engine faded as she fumbled for the keys to the big front door. It was bitterly cold. She could see her own breath. On the hall table there was a pile of mail. She picked it up and walked down the hallway, past the sparsely furnished ground floor rooms of her rambling old house to the cosy lounge at the back. The fire was going well in the grate. She slumped into her big armchair and gave the letters a cursory flick through. Lily was pleased to note there was not a bill among them. She kicked off her shoes and slid her slippers on. Lily took the letter-opener from the cardboard box that she kept her stationery in and set about opening the post. She was still buzzing from the day's events in Westminster and was not ready to sleep. Her daughter popped her head round the kitchen.\n\n'Hiya, Mam. I've put the kettle on \u2013 and I got your favourites \u2013 custard creams.'\n\n'Oh, that sounds grand, Virginia.'\n\n'How did it go in London?'\n\n'I'll tell you all about it later. Go and put the kettle on, pet. It was great though. We have done it, love. Them politicians said we'd get all we asked for. As we left that fella Peart said to me, \"We'll do all we can, love.\" I told him to talk straight and he did. He called me love, so he's all right in my book.'\n\n'That's fine, Mam.'\n\nVirginia went through to the kitchen to make the tea. Lily was busy opening letters. The 17-year-old wasn't sure precisely what her Mam had been doing over the past days but she knew it was important and took up all her time. It seemed that since the first trawler had sunk all her Mam did was answer phones, collect signatures, go to meetings and write things and talk to telly and newspaper folk.\n\nVirginia also knew her mother would soon be engrossed in her reading and that she would probably have to wait until morning to find out more about London. That's just the way it was now. Her Mam seemed always preoccupied by her campaign and Virginia knew better than to quiz her too much when she was like this.\n\nWhen the girl returned with the tea and custard creams she was pleasantly surprised. Lily looked up from the letters and waved some of them at her.\n\n'Look love, there's a lovely letter here from the pastor's wife. And one from the girls at the Clothing House shop'\n\nLily laughed and waved a piece of paper.\n\n'Some bloke's written me a poem!'\n\nVirginia went to the kitchen to tidy up before going to bed. Her Mam continued to open the letters. Virginia returned to say good night and to check if Lily wanted any more tea. The girl sensed something was wrong. The smile had gone from her mother's face. She looked stressed. Her make-up had been smudged when she came home but this looked like she had just been upset.\n\n'Mam, are you all right?'\n\nLily looked up, composed herself quickly, forced a smile and said, 'Never you mind, our Virginia. I'm all right. You get off to bed, now. You've got work in the morning.' Virginia thought her mother was far from all right, but she knew better than to talk back when she heard that tone of voice. The teenager did not know what to say. She knew her mother did not want her to see her like this. So Virginia did as she was told.\n\n'Night, night Mam.'\n\n'Good night, love. I'll turn in soon me self. I'm fine. See ya in the morning.'\n\nVirginia was not convinced. Virginia wanted to stay and comfort her, but thought it best to leave her be. Her Mam had been like that a bit recently. She had a lot on her plate. The girl went to bed.\n\nWhen Virginia left, her Mam screwed up a ball of paper. Minutes later, Lily threw three rolled up balls of paper on to the fire one after the other. She got up from her chair, moved towards the door and switched off the main light. The fire's glow brightened momentarily and dulled as three death threats burned in the grate. It was almost midnight and she had to be back at the docks in a few hours.\n\n_Queensway, Cottingham near Hull, Wednesday, February 7, 4am._\n\nLittle more than ten hours had passed since Rita Eddom spoke to the husband she thought she had lost forever. Now the pretty twenty-five-year-old was going to see him. The past few hours had been dream-like, but the car that pulled up outside her home in the wee small hours was very real. Mind-numbing, heart-rending grief had turned to incredible joy in fewer than twenty-four hours.\n\nThe rumble of a car broke the silence of the suburban street and the headlamps made two wide bright V-shapes of light on the snow-covered road. Rita and her younger brother Dennis walked towards it. The twelve-year-old boy had an overnight bag for them both. There had been no time to pack a big case. The people footing the bill for the trip were insistent that they left as soon as possible. Baby Natalee had a cold, so Rita's mum had taken her to look after while Rita was away. Rita saw Harry's mum and dad and their son Michael in the back seat. Dennis put the little bag in the boot and squeezed in beside them. Rita got into the front. She was deliriously happy. Rita and her family were going to Reykjavik to be reunited with her Harry. She could not believe it. She thanked the driver. The man explained that he would have to stop to make a quick phone call soon.\n\nAfter all, his bosses at the London office of The Sun were keen to have him confirm that the \"Thirty-Six Hour Widow\" had signed up to speak exclusively to them in return for the newspaper's offer to fly Rita and her family \u2013 all expenses paid \u2013 to see her husband. Rita could not have afforded such a trip any other way.\n\nThe car wove its way through the winter morning's darkness at the start of a one-thousand-mile journey to Reykjavik, via Glasgow. The driver was almost as excited as his passengers. He must have thought, 'This'll be one in the eye for the Express.'\n\nBy 9am that day you could not get a Daily Express in Hull. Everyone on Hessle Road seemed to be reading the world exclusive story headlined, THE MAN WHO CAME BACK FROM THE SEA ( _by Harry Eddom \u2013 from his hospital bed_ ). Harry's story, was verbatim in large quote marks on page one:\n\nIt was about 11.30 on Sunday night. The storm in the fjord was unbelievable. I lurched up to clear the radar scanner of ice, and came back to the bridge.\n\nWe began to 'dodge' \u2013 trying to get under the cover of the fjord walls \u2013 suddenly, without any warning, she went over to port. We fought to get her back up again, but she wouldn't come. She was covered with ice stem to stern.\n\nI heard lads down below panicking. There was me, the bosun Wally Hewitt and a lad called Barry on the sloping deck.\n\nSOMEHOW \u2013 I DON'T KNOW HOW \u2013 WE FOUND A LIFERAFT AND INFLATED IT. I WAS WEARING A DUCK SUIT AND WAS PRETTY WELL COVERED. WALLY HAD A SHIRT AND TROUSERS ON AND THE KID HAD NOTHING BUT HIS VEST AND PANTS.\n\nAs she lay over further I slipped and went into the water. I fell near the raft. The other two pulled me aboard. Somehow we got clear of the trawler.\n\nAs she went down a great sea swamped us. We baled with a Wellington boot and an old can.\n\nBarry died after a few hours. I don't know how long. Wally died a bit later. I lost track of time. I just huddled in the bottom of the raft and wondered how long it would be before I followed them.\n\nI remember the wind was blowing us down the fjord. Then as dawn broke the raft crunched on to some rocks at the end of the fjord.\n\nI STAGGERED OUT, MANAGED TO PULL THE RAFT UP A BIT, AND WANDERED UP THE VALLEY. I SAW AN ABANDONED HOUSE. I HADN'T THE STRENGTH TO KICK THE DOOR OPEN. MY LEGS WERE FROZEN. I GOT TO THE LEE SIDE COVER OF THE HOUSE AND JUST STOOD THERE. IF I HAD SAT DOWN I WOULD HAVE DIED.\n\nThen I saw a lad with a herd of sheep. I called him. He ran over, put his arm round me and helped me to his father's farm.\n\nThey put me to bed and gave me soup and drinks. I must have gone to sleep. That's all I remember.\n\nThey have told me since that a land party looking for wreckage found our raft with my two dead mates, on the rocks. They followed my tracks to the farm.\n\nI still can't believe I'm in the land of the living. It's all too much for me. I shall never forget those two lads and how they saved my life when they pulled me into the raft.\n\nNow all I want to do is get well and get home. I keep thinking of my wife Rita and our baby girl \u2013 and how much I love them. ( _Daily Express, p1, Feb 7, 1968)_\n\nAlan Bennett's transcription of his five-minute chat with Harry in that Icelandic hospital was the talk of Hull, and the only good news that fishing community had had in weeks \u2013 and the world's media was in a frenzy to keep up with it\n\n_St Andrew's Dock... Hull... Wednesday mid-morning... February 7._\n\nAttitudes had softened a bit since last time Lily was on the fish dock. Police who had wrestled her to the ground a few days back now let her into their office to keep from the bitter cold between dockside sorties. This was not just a change of heart by the docks cops. News of death threats to Lily had got out. Superintendent Hugh Martin gave a brief answer to reporters who came to the dockside office and asked about her safety. Martin told them about the dozen letters Lily had received and how three had contained death threats. But Lily seemed far less concerned than the senior policeman and chipped in with, 'I ain't bothered. I threw them on the fire. It won't put me off.'\n\nWith that she left and was back at the dockside waiting to give her good news to the next crew through the lock gates. The policeman's attitude was not so casual. Martin told reporters, 'We are treating these threats seriously. Mrs Bilocca has been given protection. Hull CID are investigating.'\n\nFour trawlers set sail that morning. They were, the Westella, Arctic Vandal, Arctic Avenger and the Prince Charles. For the Prince Charles it was the second time in twenty-four hours she had left Hull. That ship was meant to sail to Norwegian grounds the day before but returned after crewmen told Skipper Ray Richardson the muster alarms in their quarters did not work. Richardson decided to turn her round at Spurn Point, off the east Yorkshire coast, and headed back. Muster alarms were not required by regulations, but the skipper thought it better to be safe than sorry. Once back in dock, a team of electricians had worked from midnight to repair them. Lily was at the lock gates with a handful of her followers when the Prince Charles headed out on that morning's tide.\n\nShe shouted to the crew, 'We are expecting results in a few days, lads. We shall continue to fight for your safety at sea. We shall never give up.'\n\nCrewmen who were gathered around the whaleback gave Lily the thumbs-up as their ship slipped from the St Andrew's Dock bound for the Norwegian coast. The Arctic Vandal, followed by the Westella and the Arctic Avenger headed out through the lock gates. Lily and the women at the dockside waved each off. There were no TV camera crews. Press photographers had gone an hour earlier and the reporters followed shortly afterwards.\n\nBy lunchtime Lily was getting ready to go back home. She had a lot to do. There was another meeting due for that night in Victoria Hall. She had to make notes to tell them all the good news from her triumphant London trip with Mary and Yvonne.\n\nMost of the women with her on the dock that morning also had to get back \u2013 either to work or their homes. When Lily left the dock offices to head off back to Coltman Street a uniformed bobby went with her.\n\nA woman who cut quite a dash as she sashayed along the fish dock was walking towards Lily. This lady looked considerably younger than her fifty-seven years. People nudged each other as she passed. Her thick black curly hair and oversized spectacles made her instantly recognisable. She took an exaggerated draw from a cigarette housed in a long tortoise-shell holder. She could have passed for a dark-haired Cruella DeVille. Smoke curled from her lips and she said, 'Hello, Mrs Bilocca I'm...'\n\n'I know who you are,' Lily interrupted. 'You're Marje Proops, from the Daily Mirror.'\n\n'Is there somewhere we could go for a cup of tea and a chat, lovey?' With that, Britain's most famous woman journalist and Britain's most famous fishwife, accompanied by one of Hull constabulary's finest made the short walk to Lily's house. Miss Proops' driver made his own way there in his empty car.\n\n_Abbotsinch Airport, Glasgow, Wednesday afternoon, 1pm._\n\nTwo Sun reporters, a couple of photographers and a pair of \"minders\" clustered around Rita and her family as they made their way through the terminal building. The Sun men were determined that the other pressmen would not get to their charge. Rita was theirs until midnight that Thursday, exclusively. That was the deal.\n\nAs luck had it the sixteen rescued crewmen from the Notts County \u2013 the Grimsby trawler that ran aground \u2013 distracted the media scrum as the Sun men and their charge came through the terminal. The trawler crew had just disembarked from an Icelandair Boeing 727. They were under orders not to talk to the Press. Reporters shouted questions as the men were harried out to a waiting coach. Deckhand John Davison from Sheffield gave the pack a quote as he and his shipmates passed a phalanx of flashes. He said, 'I am not religious, you know, but that night I prayed, I said, \"Please God let me see my little lad again.\" We had to use our hands like picks to hack the ice away and get onto the rafts.'\n\nThe aeroplane the Eddoms were booked to fly out on was the same one that had just brought the Notts County men home. On board were all the twenty-or-so pressmen who had hassled the Grimsby crewmen. So, for the whole flight the Sun men did not let Rita out of their sight, and more importantly kept her from the sight of the rivals dotted among the passengers on the Icelandair Boeing 727, Flight TF 15N to Keflavik\/Reykjavik Airport. Another flight following on from London had sixteen members of the Associated Press aboard.\n\nIt was dark when Boeing 727 touched down at Keflavik\/Reykjavik Airport. Although known internationally as Reykjavik Airport, it was actually twenty-five miles from the Icelandic capital, being about a mile from the small town of Keflavik. The Isafjord hospital treating Harry Eddom was one hundred and fifty miles north of the airport. An ad hoc squadron of small, chartered aircraft was on stand-by at Reykjavik, hired by various media outlets to go to Isafjord. At the Isafjord airstrip dozens of local cars were on standby too, the headlamps of which were to light up the little runway, ready for the expected flood of little planes.\n\nMinutes after the pressmen, other passengers and the Eddoms disembarked they walked into a melee in the terminal building that made the scene a few hours earlier in Scotland look pale. The media pack from the plane joined press who had been waiting for the flight from Glasgow for hours. All were desperate for a glimpse of, and a chat with, The Thirty-Six Hour Widow. The scrum of journalists and minders from the Sun hustled and battled their way through the mob. Local Icelandic press had never seen anything like this. They could not understand why the Sun men were determined to shield Rita from their rivals. After all, it was they, the Icelanders, that had rescued her husband.\n\nTired Rita, who had not slept since Monday, was pushed to and fro by her minders. They were assailed from all sides. Some came to blows. It was a Wild West scene. One Icelandic reporter shouted, 'This is a farce.' Another added, 'We should have brought our boxing gloves.' As the \"gentlemen\" of the British press continued to push towards Rita, the men around her lost sight of her momentarily. The dazed young housewife found herself pushed towards a doorway.\n\n'Rita, over here love.'\n\nShe looked into the doorway where a smartly dressed man beckoned.\n\n'This way to your plane, Mrs Eddom, hurry now.'\n\nJust as Rita was about to go towards the two men, one of her Sun entourage grabbed her.\n\n'Where are you going?' asked the Sun reporter.\n\n'To the plane.'\n\n'There is no plane, Rita. We have a car outside.'\n\nThe Sun team gathered round Rita and the man in the doorway disappeared. If she had gone with that smart-dressed man Rita would have been in the air before she realised she was with the Daily Express. The ruse failed and Rita's minders then pushed and shoved their way through the clamouring media circus and sought an exit to their waiting car.\n\nRita and her minders ended up in a ladies' loo. They waited there while outside the cacophony of the battling hacks in the terminal building calmed down a bit. They hid for ten minutes then made a dash for the door. The Sun's driver had had his instructions. Rita was in one car with two minders and the remainder of her family in the other. Throughout the drive to Reykjavik other press followed them but the Sun team made it to the Saga Hotel in the capital city and got Rita and her family safely ensconced there for the night.\n\nA guard on Rita's door ensured she was not disturbed but she could not sleep properly even though she was dead tired. Rita knew she would not sleep until she was with her beloved Harry the next day. It was to be a long and fitful night for her.\n\nBack in Hull there was good news \u2013 and plenty of it. A quick scan of the evening paper's headlines and news billboards that proliferated across the city that day showed how much the battling Hessle Road women had achieved so quickly. They forced massive change for their community in days and won the attention of the world:\n\n\"ROUND ONE TO THE FIGHTING WIVES OF HESSLE ROAD \u2013 INDEPENDENT INQUIRY PROMISED.\"\n\n\"PREMIER ORDERS SAFETY CODE FOR TRAWLERS.\"\n\n\"THE QUEEN SENDS HER SYMPATHY.\"\n\n\"NORTH CAPE FISHING BAN\"\n\nAll of the Hessle Road fishing community knew Lily and her comrades had done something special. They were aware of the risks \"Big Lil\" had taken on their behalf \u2013 \"THREATS TO HARM MRS BILOCCA\" \u2013 the early editions had screamed.\n\nThe news that made all Hull rejoice was the wonder of what the Express had called, \"THE MAN WHO CAME BACK FROM THE SEA,\" and the Hull Daily Mail had described as the \"36-HOUR MIRACLE OF HARRY EDDOM\". It was combined with the delight of \u2013 \"RITA CRIES FOR JOY AS HUSBAND PHONES\" \u2013 \"MRS EDDOM FLIES OFF TO SEE HUSBAND.\"\n\nThat evening hundreds of Hessle Roaders made their way through slush and biting cold to the old Victoria Hall where Boulevard South met the main drag. They converged on the Hall wrapped up against the freezing blasts. It was the kind of weather that closed airports. The storms that had blighted their menfolk were wreaking a lesser havoc here. But still they came to see the triumphant women's deputation and to hear their good news.\n\nCrowds of men, women and children had gathered once more under the bare lighting of the rickety church hall. Steam rose from damp clothing and mixed with tobacco smoke, the stink of the fish house and the funk of the pubs, where many a glass was raised for Harry Eddom that night. The atmosphere in the hall was euphoric. For the first time in a long time they had something to cheer. And cheer they did. A crescendo of stamping almost raised the roof as the three women walked past the table where sat the union men, who had set the stage up for them earlier. Yvonne's husband John had provided the PA system again but the thundering drowned out the first man who tried to speak through it.\n\nThe Press, who managed to get in without being pushed into the stage front like last time struggled as the women got ready to report their Westminster triumph. This time it was a bit better organised. This time there was as sense of hope and of victory. There were even some more men in the crowd. The brief introduction from the union table was barely audible.\n\nWhen Lily took the microphone the crowd erupted. She was determined to get her message across and had prepared a list and started in what the local paper later called a \"workmanlike\" manner. She listed the achievements of their big day and kept to what she had written on her big notebook. A roar punctuated every sentence. Then she went off script, put the pad down, took the mike from its stand and shouted, 'The trawler owners wouldn't talk to us!'\n\nA massive rumble followed from the foot-stamping, clapping crowd and she continued to more cheers, 'But we went to the top and we've got something done. And if I can cause a commotion again, I will do it.' The cacophony rose further and she added, 'The bosses will be making trawlers report in every twelve hours. If one is missing then a rescue should start in twelve hours, not twelve days! The Ministers will be banning fishing off the north of Iceland in atrocious weather like we've had and they'll be insisting all ships have a radio operator \u2013 a fully qualified one on each ship.'\n\nAmid the din nobody initially heard the voice in the crowd that started the loudest chant of the night. When the anonymous voice that shouted, 'Three cheers for Lily Bilocca' was heard, it resulted in a series of _hip-hip-hoorays_ that would have drowned out the Humber's foghorns.\n\nLily managed to get heard again and told the crowd, 'You've probably heard that rubbish in the paper about folk threatening me. It'll take more than that to put me off. Any road, I have loads of nice letters and telegrams from all over. Anyone that wants to come back to my place and see them is more than welcome to.'\n\nA slight built man then appeared next to Lily. He had made his way to the stage from the crowd and beckoned to her for the mike. The big lady passed it to the little man. He spoke in a quavering voice. For the first time that night there was silence in Victoria Hall. He said, ' I am a bobber \u2013 and I know that you trawler folk don't like us much, but I want to tell you all that we are with you.'\n\nThe applause came back as he continued, 'I am going to try and raise money for your funds.' He then started to sob and one of the union men came forward and led him gently away.\n\nA hush fell back on the hall.\n\nThe meeting adjourned itself.\n\n_Isafjord Hospital, Thursday, February 8, approximately six a.m._\n\nReporters laid siege and the man in charge of the hospital where Harry Eddom was being treated, was worried. Since Harry arrived the number of media invading Doctor Ulfur Gunnarsson's hospital had grown and grown. Gunnarsson feared the place might be overrun. He took the precaution of cancelling all operations scheduled that day. The night before he even offered Harry the chance to escape in secret. He told the English trawlerman, 'I can arrange a private plane to take you back to the UK. These press guys are going to be more and more insistent.'\n\nBut Harry Eddom just smiled and said, 'It's fine, doctor. It'll be ok.' Now as the world media seemed to gather at the entrance of his hospital, Dr Gunnarsson must have been wishing Harry had taken up his offer.\n\nOne hundred and fifty miles south of Isafjord, Rita Eddom rose early. She had had a few drinks in the Saga Hotel bar the night before but it had not helped her sleep. She was too excited. It was around noon when the Sun car arrived to take her to the airport. Once again the car zigzagged through town in case it was being followed.\n\nAfter an hour, the drive, which normally took half an hour, was over. Rita and her entourage went through the terminal and on to the far end of the runaway where her plane waited. It would be an hour until she would see her Harry.\n\nIt was just after two in the afternoon when Rita and her retinue arrived. The media men, who had now been at Isafjord Hospital for more than eight hours, deluged Rita.\n\n'Back off,' shouted one of the Sun men to the mob, 'She's with us. Exclusive.'\n\nThe crowd at the hospital doors surged forward as Rita was pushed into the building. Doctor Gunnarsson and a team of nurses and orderlies tried to keep the mob back. 'That's it!' the exasperated medic shouted. 'They can all come in or none will come at all. I will NOT have this, is that clear?' Some of the Sun men argued with the doctor. But he was adamant. The pressmen were all to be allowed in, but only if they behaved in an orderly fashion. If not Gunnarsson would put the building in lockdown and there would be no story at all. It was up to them. While the row unfolded Rita was taken to her husband's ward. The media prepared to follow, promising Doctor Gunnarsson to be on their best behaviour.\n\nHarry lay in his bed unaware of the fuss downstairs. All he knew was his Rita would be with him soon. He managed to reach across to the bedside cabinet. From it he took his wedding ring. He looked at his swollen, blackened fingers.\n\nThen he readied himself for a bit of pain. After a couple of minutes he had forced his wedding band back into place.\n\nBack downstairs amid the furore, two of the Sun men slipped away to the rear of the hospital. A nurse challenged the photographer and his reporter colleague as they sneaked in the back way. They pushed her to the floor and barged on to Harry's ward.\n\nOnce inside, however, the journalists' hopes were somewhat dashed temporarily. Rita could not speak. She had to be helped to a chair. The reporter and photographer looked on. They knew time was running out. They gave her a glass of water. Tears streamed down her face as she looked at her Harry from the other side of the ward. She sipped the water and then stood up.\n\n'Oh, Rita,' said Harry. 'It's all right, love.'\n\nRita looked directly at the man she loved and said, 'I couldn't bear to think of you drowned. Honestly, Harry deep down inside, I had lost all hope of ever seeing you again.' Harry reached out. Rita walked towards him with her arms outstretched too. She sat by the edge of his bed. They embraced and kissed warmly. They did not hear the click of camera nor sense its many flashes. The hospital room then filled with other pressmen as the couple chatted. The pair was oblivious to their presence. All that mattered was that Harry and Rita were together.\n\nRita sobbed, 'Harry, Harry, I can't believe this is true.'\n\nThe trawler man comforted his wife.\n\n'C'mon Rita, love. How's that little girl of ours?'\n\nRita could not answer for crying. Harry fought his own tears to try to make her happy and then joked, 'Look love, I have even had a shave with a brand new razor in your honour.'\n\nThe he looked directly at her again.\n\n'I'm finished with trawling, love. You'll never have to suffer something like this again. I will get a job ashore. I'll be a twenty-five year mortgage man. Maybe I could go on the buses, eh?'\n\nEveryone in the ward laughed. Even Rita smiled.\n\nDoctor Gunnarsson interrupted. 'That's it. Ladies and gentlemen, Mister Eddom needs his rest, come along now.'\n\nWith that the reluctant newsmen were herded out. Gunnarsson and his staff had had enough of the Gentlemen of the British Press for one day. The men that had brought her there led Rita out of the hospital. She was signed up to them until midnight that Thursday and they were determined to get their money's worth.\n\nOn her way out the determined young Hull housewife lost her patience. She stopped and spoke to the waiting Icelandic press. She briefly thanked the people of Iceland for saving her husband before being hustled into the car to take her to the plane that took her back to Reykjavik. For an hour or so afterwards, various members of staff and some patients reported seeing journalists at different parts of the hospital looking for one more picture or a quote. They were given short shrift. The Sun photographer with the exclusive pictures of Rita and Harry's reunion had the misfortune to bump into the nurse that he and his mate had knocked over earlier.\n\nThis time there were more medics than reporters. One asked the journalist what he thought he was doing trampling through their hospital as if he owned it and would he get out now. His reply is not recorded \u2013 but that afternoon's local paper Visir reported that the newsman 'would not do as he was asked until he had been well and truly thrashed.'\n\nAfter Rita had gone, Harry was left with his thoughts. He was sat up in his bed appreciating the peace after the fifteen-minute media storm. He was so happy that he had seen his pretty young wife and was looking forward to getting well enough to go home. His doctor had told him he was doing remarkably well and that he was likely to make a full recovery.\n\nHe was tired and was ready to sleep some more. As he settled down he spotted he had another visitor. In the doorway stood a small, blonde girl of about thirteen years old. She spoke to him in her native tongue. Harry did not understand the little girl's language. But it was clear the child had a crush. He smiled widely. She blushed.\n\nAfter being taken back to her room by a nurse, she gushed about how she met the man from the sea, whom she had heard about earlier on her bedside transistor radio.\n\nShe later said, 'He's a handsome man and he gave me a warm smile. He's obviously a good man.'\nTEN\n\n_Thursday afternoon... February 8... The University of Hull._\n\nThe hundreds of cheering students had never seen a lecturer like this. She stood before them in her best blue coat, the absence of her trademark headscarf revealed her raven black beehive hairdo. She had her best dress on and its red floral pattern could be seen under the unbuttoned coat. The diamante earrings and necklace sparkled. She took to the lectern in the packed University Council Chamber and the roar from the young crowd was like that reserved for a pop star. And Lily Bilocca had had the kind of press coverage a pop star would envy. That morning's Marje Proops' Page in the Daily Mirror featured a three-quarter-page story headlined, THE REAL BIG LIL.\n\nMarje ended her star interview by saying, 'I have often criticised women for sitting down with their knitting and their cuppas moaning and whining and doing nothing. Lil Bilocca is a doer. Someone ought to erect a 17-stone statue of Lil on the fish dock. It would be a real Statue of Liberty.' Below that feature was a smaller one about another Hull woman who had come to national prominence. It was an interview with a young actress called Maureen Lipman, following her success in the controversial British hit movie Up The Junction.\n\nEven the right-wing Daily Mail carried a page lead on Lily that day with the rather patronising headline, 'HOW BIG LIL HANDLED 'THEM LONDON BLOKES'.\n\nSo the four paragraphs in the local paper the day before announcing Lily as a guest lecturer was the entire PR needed to ensure a big crowd. Mature student Michael Godman had asked Lily to address the Hull University Students' Union. At first she said no, after all she had never lectured before, but when the thirty-year-old sociology student and Hessle Road lad told her she could raise money for the trawlermen's families she agreed. There was standing room only in the University Council Chamber. 'Lil-lee... Lil-lee!' The students chanted as their union president Neil King tried to introduce the guest lecturer.\n\nLily started awkwardly like she had done the night before at the Victoria Hall on Hessle Road. She told the cheering audience how she and her fellow campaigners had secured promises from Cabinet Ministers JPW Mallalieu and Fred Peart only forty-eight hours earlier \u2013 and that her whole Fishermen's Charter would be enacted.\n\n'I spoke to them straight and they spoke to me straight.' There was another cheer, then whistles and stamping when she announced the Government's promise that radio operators would be on all trawlers from then on and the assurance of a fishing ban in the treacherous weather of Iceland's Northern Cape. There was also to be a mother ship, either a naval vessel or big trawler to act as a hospital ship among other things.\n\n'We don't want them ships up there when it's weather like that. It's wrong.'\n\nUnion president Neil King tried to calm the crowd and came across to Lily. He signalled to quiet the students but had to shout to be heard. He thanked Lily and then added there would be questions from the floor. The din started again as the young student said, 'We will support these wives and families. The official disaster fund will be supported by a donation and there are already collection boxes in place in the Union building. There is strong support here for what Lily is trying to do. The problem we have is a Hull problem and it has been brought home to us in the last few weeks.'\n\n'What d'you think of the Silver Cod, then Lily?' one of the students shouted.\n\n'I'll tell you what I think, lad. I think that Silver Cod championship needs chucking out.' There were more cheers as she continued, 'And them Merchant Shipping Acts that prevent men from striking are disgusting and want chucking out an' all.' The crowd clapped louder when she added, 'I have utter contempt for them. And if the proposals for safety that we've been promised aren't adopted, I'll try for strike action if they don't comply.' The students were all on their feet and the noise in the usually conservative, sedate environs of the University Council Chamber rose to a crescendo. The \"lecture\" lasted under fifteen minutes and got Lily a standing ovation. As the youngsters stamped, whistled and cheered, she outstretched her arms for the biggest cheer of the day. She laughed and shouted, 'Anybody here is welcome at my place for a cup of tea!'\n\nThe ministerial promises that won the students' applause were being put into action as Lily spoke. The speed of Mallalieu and Peart's delivery was breathtaking. At the Board of Trade headquarters in London that morning at 10.30, fifty representatives from sixteen organisations representing the fishing industry had been ordered to meet with JPW Mallalieu, Minister of State for the Board of Trade (Shipping) who had forewarned them that a \"tough line\" would be taken. Three hours later, the biggest wholesale changes in the twentieth century British trawling industry were announced. By the time Lily had finished speaking to the Hull Students' Union, Mallalieu's announcements were being put into effect.\n\nAll British trawlers in the Northern Cape were contacted that day by radio and told to leave the area immediately. The Board of Trade also announced that a mother ship for the Hull fleet would be there within a week. This control vessel would be either a naval \"boss\" ship or a large trawler that could be used for that purpose. The temporary fishing ban was to be enforced until the weather improved. All trawlers sailing from that day must have a radio operator who was not also a skipper.\n\nMallalieu had also announced life jackets on all trawlers must meet requirements outlined for 1970 and that there was to be a review of all life rafts to make sure there were enough and that they too were up to Board standards. There was also to be a review to improve weather forecasting.\n\nMore importantly four inquiries were announced. There was to be one into what happened to each of the doomed ships and an overall public inquiry into trawler safety and the industry in general. The latter inquiry was one of four extra proposals added by the ministers on top of the women's Fisherman's Charter. There were now also to be lifelines on deck for trawlermen in mountainous seas, one \"mother ship\" per fishing fleet nationally and further more comprehensive compulsory safety checks on all trawlers before they left port. Mallalieu ordered all of this to start from that day.\n\nTGWU Humberside Regional Organiser Dave Shenton, who had helped to organise the women's trip to London along with Jack Ashwell and Mike Neve, hailed the announcements as \"revolutionary.\" He told reporters, 'The introduction of a control ship is a revolutionary feature as far as the British industry is concerned. We believe that when the court of inquiry has conducted their investigations and made their report, that many of the things we have been saying for so long will be accepted.\n\n'The wives have made a tremendous contribution in spotlighting their extreme concern for the safety of their menfolk. It is tragic that it takes a disaster before positive action can be achieved.'\n\nShenton's boss Jack Jones, who had been at the talks with Mallalieu in London, said, 'This is a joint agreement between the Government, the unions and the owners. We are all responsible for its enforcement.' The trawler bosses' spokesman Austen Laing of the British Trawlers' Federation seemed keen to show the owners were on board with the rapid changes foist upon them that day. He said, 'All skippers will be instructed to treat the orders of the naval control ship as orders that they disobey at their peril.' But Laing was not as supportive of the instant ban. He said, 'Remember the north coast of Iceland in not the only area in the north Atlantic to be liable for bad weather in December, January and February. A more feasible, and more embracing answer must be found.'\n\nLaing, like most other trawler bosses, was behind the curve of public opinion again. The determined speed with which Mallalieu and Peart had fulfilled their promises \u2013 and more \u2013 to the wives was unstoppable. Union men were as surprised as bosses by the swift actions of the Government. Messages dispatched that day to skippers told them to get out of the Northern Cape. Those travelling there were told by Mallalieu to \"Proceed by the westabout passage to avoid the potentially more dangerous area north of Iceland.\"\n\nThe union produced a newsletter to be distributed next day to all trawlermen on their books, outlining the new safety breakthroughs. The women's demands were carried out at full speed ahead. Campaigner Yvonne Blenkinsop told reporters, 'This is just what we wanted. We are delighted.'\n\nPerhaps another reason for the speed of change was that Mallalieu's boss, Anthony Crosland, President of the Board of Trade \u2013 who was out of the country at the time leading a UK trade delegation at a United Nations conference in New Delhi \u2013 was also MP for the huge fishing port of Grimsby, across the Humber from Hull.\n\nLike his boss, Prime Minister Harold Wilson, who was in the US, Crosland kept a watching brief from abroad.\n\nNext morning Hull-based Skipper Tom Smith found himself the centre of attention and forced to give an impromptu press conference in the living room of his neat four-bedroomed suburban home in Hotham Road, near the university where Lily Bilocca debuted as a lecturer fewer than twenty-four hours earlier. Smith, a stoic no-nonsense Scottish trawler man, who had settled in Hull with his family eight years earlier, took the inquisition in his stride. It was only the night previously that his employers, Grimsby-based Ross Trawlers Ltd., called to tell him he was to be commander of the newly announced control ship. Smith, a forty-year-old father of three girls, and a veteran of the Newfoundland, Greenland and Icelandic fishing grounds was thought to be perfect for the role. His proposed new job was as much a surprise to him as it was to the reporters in his living room. One asked, 'Are you worried about going to the grounds which claimed the Ross Cleveland and the Kingston Peridot, skipper?'\n\n'There will always be the possibility of bad weather wherever you are,' Smith replied. 'But anything that can be done in the interests of safety is well worth it.'\n\nThe skipper's pretty wife Doreen added, 'I am not very pleased about the Iceland duty, but it is his job. And I didn't feel so nervous because the Ross Valiant is a big ship.'\n\nAt 1,156 tons, Smith's ship, the Ross Valiant, was twice the size of the Ross Cleveland and the Kingston Peridot, which were around 650 tons each. The Ross Valiant was a huge modern freezer trawler with a crew of twenty-four. She had been built in Selby just four years earlier. Smith had landed her at Grimsby a few days before with a 395-ton catch after a seven-week trip. He was due back in Grimsby to take her out again the following Monday, three days away.\n\nIt was clear Skipper Smith did not really know what the new role was in detail. And it seemed neither did his employers. It was all very spur of the moment. Smith had been told he would be informed further when he reported to the Ross headquarters in Grimsby the next day. The skipper felt that maybe his bosses were awaiting further instructions too. But Smith was straight talking with the reporters when they returned to his house. He told them, 'I don't really know what the duties would be. I am assuming I am going to fish. But I expect we shall also act as a control ship and every trawler will report to us and we will report to the owners.'\n\nBut Skipper Smith's time in the spotlight lasted fewer than twenty-four hours. The Ross Valiant was replaced overnight as proposed control vessel when it was decided that a Ministry of Defence weather ship \u2013 The Weather Reporter \u2013 was nearer to the fishing grounds and should go directly to Iceland, instead of the freezer trawler from Grimsby. The Weather Reporter would be constantly on standby and patrol between Greenland and Iceland.\n\nBefore heading to the Northern Cape she was due to call in at Reykjavik and pick up Lieutenant-Commander John Douglas, Chief Inspector of HM Coastguard, who was chosen to command the fifty crew of the naval \"boss\" ship. British Trawler Federation chief Austen Laing welcomed the new boss ship and said, 'That ship will be manned by skilled meteorologists. It has everything for the job.' But he still had some reservations and also suggested The Weather Reporter would have to be replaced later. He told the Hull Daily Mail, 'We shall need a ship which will also be able to carry out the watchdog function which the weather ship cannot.'\n\nAt his suburban home in Hull, straight-talking Skipper Smith breathed a sigh of relief. He never really wanted the job but would have done his duty as commanded. He told reporters, 'This job would have meant me cruising up and down without doing any fishing and that is against my way of life. I realise that this job is a very responsible one and to me the weather ship will do the job so much better.'\n\nWith the Icelandic fishing ban in its third day, St Andrew's Dock was much quieter than usual. There were no trawlers from Hull landing any catches from Iceland. For the battling women of Hessle Road it was a welcome day of rest. The media had been relentless. Mary, Yvonne, Chrissie and Lily were household names. Lily was catapulted to international recognition. It may have been a rest day for the trio that triumphed in London, but for Lily there was a return to the capital, this time as a TV star.\n\nThe Eamonn Andrews' Show invited the country's most famous fishwife as its special guest. It was the talk of Hull. Lily and Virginia set off that morning; their \u00a316 return fare paid by ABC Television as part of the all-expenses trip. That evening, campaigner Mary Denness was at home trying to relax. Her husband was at sea. The kids were in bed. The conditions should have been perfect for her to take some rest and relax in front of the telly with her tea and biscuits. There had been no pressmen to pester her this day \u2013 and she determined not to answer her telephone. But Mary had a lot on her mind.\n\nShe and the other women campaigners, especially Lily, had been catapulted into the spotlight. Mary knew that this helped their cause and was delighted with the changes it forced. But she was finding things difficult. She sensed an undercurrent developing. She had heard the gossips and the gain-sayers. She heard from her husband's friends how Hull men felt the women were \"out of order\". Those men said their peers, especially in other rival ports \u2013 had accused them of \"hiding behind their lasses' skirts\" and this kind of schoolyard jibe was gaining momentum.\n\nHer own husband only half-heartedly supported her fight, and that was more than most trawlermen. But even he told her he was embarrassed by the women's actions when he was in the company of other fishermen. Mary had spoken with women who, while congratulating her for \"speaking proper\" on the TV on their behalf, had at the same time accused Lily \u2013 not to her face, of course \u2013 of \"showing them up\" with her \"common\" ways of talking and going \"too far\". Many of them said they resented the superstitious taboo of women being on the dock being flouted by the recent invasions by Lily and her comrades.\n\nMary herself was uncomfortable with those dock invasions. She never took part, but understood the reasoning behind it. She also understood her community's concerns. This coupled with the death threats to Lily and the recent restaurant attack on Yvonne increased Mary's worries. She liked Lily and was pleased with her leadership and commitment but she felt an outside resentment rising against her.\n\nLily's simplistic, no-nonsense approach made great news copy. Mary knew that. But she also knew her community and that some gossips already had it in for Lily. She had heard people complain that \"she [Lily] must be getting something from all them telly things.\"\n\nMary defended Lily. She knew that quite literally that Mrs Bilocca was the biggest target for the armchair critics and was aware that this would hurt the future of the fight. Lily and Yvonne were very much in the moment and reactive, but Mary and Chrissie could see there was a lot of work ahead, work that would need more than just bravado and charisma. Chrissie had also spoken to fishwives who had voiced concern at Lily's public image. Mary convinced herself that perhaps she was paying too much heed to the gainsayers and gossips. But there was no doubt in her mind either that Lily's maverick actions were a worry.\n\nShe put the issues to the back of her mind and settled down to watch Lily's starring appearance on the country's top chat show. The show's large, genial Irish host and Lily hit it off straight away. After all, they did have something in common. They were both rare examples of regional accents on British television. Andrews, a former Irish heavyweight-boxing champion turned sports journalist and latterly a TV personality was an unusual chat show host in the eyes of many. His lilt was rare in the media \u2013 and Lily with her loud brash voice and Hull accent was certainly more unusual. The studio audience loved Lily's down-to-earth humour and revelled in her no-nonsense Yorkshire directness. Mary settled in and watched Lily steal the show.\n\nLily was applauded when she spoke of how she dealt with them \"London blokes.\" Mary thought her friend was going to show the world that the gainsayers were wrong and that her innocent simple good-heartedness would shine through. She made them laugh and sigh in equal measure and spoke with passion about her fight for better safety at sea. When she talked about politicians she met with a glaring lack of deference, a blast of fresh air went through the staid world of British broadcasting.\n\nAt one point Lily told Eamonn that she hoped her daughter would never marry a trawlerman. She said, 'It would break me heart if she did.'\n\nWith that, the boom mike hovered over the audience and the roaming camera picked out Lily's blushing seventeen-year-old daughter Virginia among the crowd.\n\n'What's your name?'\n\n'Virginia,' she replied.\n\n'And what do you do in Hull?'\n\n'I'm a secretary at the shoe factory.' Another collective chuckle followed that.\n\n'And would you marry a fisherman?'\n\n'No, I would not!' Virginia's nervous half shouted answer raised a bigger laugh that time. Then Eamonn continued,\n\n'...And have you got a boyfriend?'\n\n'Yes.' The close-up showed clearly her blushes, even in monochrome.\n\n'And what does he do?'\n\n'He's a rigger.'\n\nThe ripple of applause grew and the camera panned back to host and guest. Mary, in common with millions of viewers, smiled at the innocent young girl's quizzing and laughed along with the TV audience. She thought Lily and Virginia did a grand job. Her fears were assuaged.\n\nDuring the commercial break, Mary made herself some more tea and settled down for the rest of the show. The host and the guest chatted about life in Hull, what it was like and how Lily had felt compelled to become the fighter she had become. The chat flowed and the audience listened and learned about life in the tough port of Hull. Then Eamonn asked, 'And what do the trawlermen do when they are not at sea then Lily?'\n\n'The married ones come home and take out their wives, then go to the pubs. The single 'uns go wi' their tarts.'\n\nThe audience's gasp was audible. To them \"tart\" had a totally different meaning to what it did to Lily. In Hull it simply meant a girlfriend. Without realising it, Lily accused Hull fishermen of procuring prostitutes. Mary realised immediately. Lily's words seemed to increase in volume as she heard them. Mary put down her cup and thought, 'Oh Lily love, what have you done? That's a nail in your coffin.'\n\nBy Tuesday, February 13, Lily had been back from London for twenty-four hours. Another invitation had landed on her doormat. This time it was from the Students' Union of the University of Strathclyde in Glasgow. After the success of her Hull lecture the Scottish students wanted her to address them in a few days' time. Lily made a phone call to accept yet another appearance. Then she had a more mundane task at hand. Lily needed to do a \"big wash.\" It seemed all she did recently was wash clothes, iron, pack and travel. Virginia was a big help but she was at work today. Lily had to be in Grimsby that evening at the invitation of the TGWU along with the other women to address fishwives in that port. There was more post to go through to but she would deal with that later. She had the time now. There was no need to be at the dockside. So Lily set about her preparations.\n\nAt St Andrew's Dock that morning only one trawler landed fish from Iceland \u2013 and she was an Icelandic one. The Reykjavik-registered Uranus landed a massive 22,646 stones of fish, netting almost \u00a39,000 at market. This was almost double the final recorded bumper catch of the Ross Cleveland. The total from the five Hull trawlers that landed catches from Norwegian grounds the same day did not come anywhere near the Icelanders' bumper haul either. The Uranus came to Hull having fished the northwest of Iceland. Her catch was divided into 21,570 stones of fresh fish and a further 976 stones of frozen.\n\nNews that the \"Skrobs\" were cashing in, while Hull men were idle, spread like wildfire on Hessle Road. By the afternoon, former skipper Dennis Roberts had issued a statement that summed up the anger that flared that morning. Roberts, now a manager with the Ross Food Group based in Grimsby said, 'These women are interfering in something they know nowt about. A deputation of women knocks on the door of No. 10 Downing Street, and suddenly we are told we are doing everything wrong. And the Government had taken its decisions without even consulting its own officials in the ports.'\n\nIn that day's Hull Daily Mail, Yvonne Blenkinsop was asked to counter the former skipper's claims, but could only say, 'I would not do anything detrimental to the industry. I have never asked for a ban on trawlers going to Iceland.' In a later edition another skipper, this time a Hull one, went on the record and voiced his anger at the ban, which he said was shared by many of his peers. The skipper said, 'I think the ban is stupid. The Government cannot ban fishing from the area \u2013 it is our living.'\n\nThe earlier spat between Mrs Blenkinsop and Mr Roberts was only a few paragraphs in the paper. The serving Hull skipper's outburst made page one. That skipper was Len Whur, the man who only a week earlier had fought so desperately to save the Ross Cleveland and watched helplessly as it had disappeared fewer than three hundred yards from him.\n\nAt his home in the village of Ferriby, near Hull, Whur recalled how the weather that claimed his friend's trawler and the lives of nineteen men was the worst he had ever seen. He told how his chief engineer heard the howling of the winds above the noise of his giant engines. Whur reminded them how his men had fought ice for two days straight while being thrown all over fierce seas on gigantic waves. He said, 'Conditions were the worst I have ever encountered in my career. I have been a skipper for twelve years and at sea since the age of sixteen.' Whur went on to tell how he saw the Ross Cleveland go. 'It was horrible. She just lifted up and went over. It was impossible to get to her. We went hard to a-port and got into trouble and just had to watch her go.'\n\nThe angry skipper continued his plea and added that despite all the danger he had seen, 'We go to the grounds where the fish are because we go to sea to catch as much fish as we can in as short a time as possible.'\n\n_Port of Sunderland, Tyneside, Tuesday, February 13_\n\nCharlie Bilocca had been in the engine room of the freighter Tasso V for most of the past three weeks. The Ellerman-Wilson Line ship was docked at Sunderland to drop off its cargo of Swedish timber and was then to return to Hull. He would be home on Thursday the fifteenth. There would be little leisure time for Charlie on leave. There never was. He always had a million things to do. He had a big house to keep and did many jobs to keep it. Charlie fitted his life as a ship's engine room man with whatever opportunities came his way. His big back garden in Coltman Street was a smallholding from which he traded soft fruits and vegetable. It had an orchard too. He bred pigeons and rabbits for the market. In his kit bag he had baccy and booze to flog.\n\nCharlie had an eye for the main chance. He always had. While he was at sea, his wife Lily and his teenage daughter Virginia tended the gardens, fed the livestock, kept the dovecots clean and generally kept things running while Charlie was away on what were known as Home Trade Runs between Scandinavia and Hull. He took these trips because it meant he could run his other interests while being roughly one week on \u2013 one week off \u2013 tramping cargoes between Hull and Scandinavia. Charlie was keen to get ashore at Sunderland, firstly just to get fresh air and walk on a solid surface and secondly to call home and make sure all was OK.\n\nIf the dockside had been a movie backdrop then slim-built, dark-skinned Charlie with his black jeans, crew-necked jumper, and bright-coloured sweat-rag (neckerchief) looked like a perfect supporting artist. He swaggered among the cranes and containers. He easily belied his sixty-four years by a decade. He found a red phone box. Once inside he got ready to press the A-button and put his pennies in. His wife was not in. He went through a checklist of items with his daughter to make sure all the chores were up to date. Before the pips went \u2013 he had learned all he needed. After all, he did not want to waste too many pennies in a call box. He made his way back to his ship.\n\nIf Charlie looked at home as part of the docklands vista, then the other man walking the quayside in his direction certainly did not. He was tall, slim and fair. He had the look of a man whom Central Casting had chosen to portray a quintessential English gent, but had sent to the wrong film set. Under his tailored coat was a perfectly cut black suit. His Paisley-patterned cravat added to his air of English eccentricity and this was topped off with the sporting of a bowler hat and monocle.\n\nThe man who looked like a City gent was actually the son of a Geordie motor mechanic. Thirty-eight-year-old Stanley Blenkinsop was the Daily Express's man in Newcastle. He had only been on staff a few years, having joined from his local evening paper. But he had made a big mark, not only as a colourful eccentric, but also as an outstanding newspaperman. The Daily Express hired nothing less. After all, when Lord Beaverbrook insisted you were working for The World's Greatest Newspaper, it is something to live up to, and it was something that Stanley lived up to very well. He looked as if he had just left a gents' club.\n\nMaybe he had. Stanley was a regular in the casinos and burgeoning 1960s drinking clubs of the North. He was a chap who, once seen, was not forgotten. Stanley held court with the best of them. And the hacks of Fleet Street, rivals and friends, marvelled at his stories both in print and in the bar. (One favourite was how Stanley, a super-patriot, ensured that he was in Paris every Bastille Day dressed in his bowler hat and City gent outfit. \"Just to annoy the French.\")\n\nWith Stanley that day was another man, who looked decidedly dowdy beside him. That man had a camera round his neck. The pair approached the slim built matelot.\n\n'Charlie Bilocca?'\n\n'Who wants to know?'\n\n'I'm Stanley Blenkinsop from the World's Greatest Newspaper. And this is my photographer. Could we have a word with you \u2013 and a picture perhaps?' His clipped accent certainly belied his Geordie roots \u2013 and Stanley always referred to his paper with the sobriquet beloved by his boss Beaverbrook. He explained he wanted to know what Charlie thought about Lily's campaign. Charlie told him he didn't know much about it all. He had only heard the radio yesterday and had no idea what his missus had been up to since he had left Hull a couple weeks earlier.\n\n'Great,' Stanley must have thought. 'What a tale.'\n\n'Is there somewhere we can go and chat, Charlie?'\n\nAnother Daily Express exclusive was underway. Checking the shipping movements had paid off for the enterprising journalist \u2013 and for Charlie too. An hour or so later he was ten guineas better off. Stanley's expenses were generous \u2013 and he was generous with them.\n\nBehind Stanley's flamboyant fa\u00e7ade was an assiduous newspaperman. He would have asked Charlie about every aspect of his life. Stanley knew nothing is wasted in a news interview. He would have learned a lot about the little man from Malta. He would have heard how Charlie \u2013 real name Carmelo \u2013 loved everything English. In that they had something in common. Ever since Carmelo was a kid in rural Malta he dreamed of living the life of an English gentleman and he convinced himself it would happen. He just had to work hard.\n\nAt sixteen Carmelo joined the Royal Navy and served on the HMS Seapoy as a pantry boy. He was already a gifted cook having worked in a patisserie in his village as a child. But he was a practical kid too, as well as a dreamer, and he could turn his hand to anything. Carmelo realised quickly that the Royal Navy was not for him. He jumped ship in Egypt and joined a merchant ship \u2013 the Lancaster Prince \u2013 as an engine room trimmer. This was a tough job, at the bottom of the engine room pecking order, which involved keeping the coal coming to power the old-fashioned tramp steamer. He had never worked in one before but bluffed his way. The Lancaster Prince was headed for Hull. Carmelo \u2013 now Charlie \u2013 knew Hull was in England and that was where he wanted to be.\n\nCharlie spent years at sea doing long trips, saving money. In 1933, he realised the first of his big ambitions and became a naturalised British citizen. By 1944, Charlie Bilocca was a businessman. He opened the Albert Dock Dining Rooms, on Strickland Street, off Hessle Road. On his office stationery he styled himself as \"Mr Charles Bilocca, Director.\"\n\nThings were going well for Charlie. He lived above the shop \u2013 alone. Shortly after the end of the War in 1946, Charlie fell for the daughter of one of his waitresses. The handsome businessman soon swept Harriet Marshall's 17-year-old daughter Lily off her feet. They took a romantic trip to Paris. By the early 1950s, the Biloccas \u2013 son Earnest and baby Virginia were all living above the dining rooms.\n\nA few years later Charlie saw his big chance. An eleven-roomed house with massive gardens had come on the market at a price he could afford. The house was a property in an area once inhabited by the city rich and powerful. But that area was on its way down. But Charlie fell in love with the idea of the big house \u2013 and he saw potential in it. This was a decision of heart and head. This was a big English house he had dreamt of as a kid \u2013 and he had a business plan to pay for it.\n\nIt was only a few yards from what was once the home of Hull fishing magnate and philanthropist Sir Christopher Pickering, who donated to the city the massive park west of Hessle Road that bore his name and housed the orphanage his trust fund set up. Charlie loved the idea of that too, that he, a Maltese ragamuffin, could own a posh house near where the great Pickering had lived. Charlie could not resist. He sold up his caf\u00e9 and moved his family into 117, Coltman Street. In no time at all he had sub-let the outbuildings to local tradesmen and set about using the skills of his rural past to set up his urban smallholding. He would go back to sea occasionally in the beginning until his plan got up and running. Then he would be ashore forever, lord of his own manor. That was his plan.\n\nStanley took copious notes in perfect shorthand while Charlie talked...and talked in the galley of the Tasso V. The two men shared a drink or two. Stanley knew the value of chatting. The more you kept 'em talking, the more of a story you'd get, especially with a drink or two. Any newspaperman worth his salt knew that. Near the end of the interview, the photographer took a few shots of Charlie, sat at the mess table. The Maltese Anglophile was pictured smiling widely and pretending to read of his wife's exploits in an earlier copy of The World Greatest Newspaper, as befitted his fifteen minutes of fame, courtesy of Stanley.\n\nBack in Charlie's house in Hull Lily was keeping busy. She was in her favourite chair in the back lounge sorting the letters out and filing away anything suspect. This had become a routine. Folded poison pen letters were put into a small box. The intention was to pass them to the police but more importantly to keep them out of sight from her family, especially young Virginia.\n\nLily glanced through the scrawling hand-written letters. She had come to know some of the handwriting. There was one headed from the \"The Fishing Lads\", another from \"The Fishing Men\" in remarkably similar writing. Some were unsigned. Most said the same things. One that was signed with an indecipherable scrawl was one of the most hateful. The ugly flourish of the scrawl gave an impression that its writer could not get the hate on the page quickly enough:\n\n_'Madam, Why don't the people of Hull kidnap you, tie some bricks round your neck and drop you in the Humber, you big, fat, greasy Maltese whore. You must be the commonest cow in Hull. Ask your xxxx Maltese husband to take you to Malta and lose you._\n\n_You are going to give the employers the length of your tongue? Perhaps they will give you the length of theirs. They should cut it out._\n\n_Wake up, trawlermen and drown this big stuck-up cow, and let's hear the end of her..._\n\nLily felt her family had enough to contend with, with what had been in the papers. 'This is not going to add to it,' she thought. She would pass them to the police later. Lily glanced through some more of the \"Fisher Lads\/Fisher Men\" letters. Angry phrases leapt out but she flicked through them and put them in the box:\n\n_'Get back in the FISH HOUSE and FREEZ(sic) UP'_\n\n_'what we think of you his (sic) YOU ARE A FAT COMMON SLOB and DO NOT KNOW HOW TO TALK.'_\n\nAnother unsigned one read:\n\n_'IF IT WASENT [sic] FOR THE MONEY YOU ARE GETING [sic] OUT OF IT ALL YOU WOULD NOT BE IN IT.'_\n\nOne of the regulars, styling himself \"Trade Union Leader\" surpassed himself that day. He sent a telegram and a letter. Again Lily glanced at them and threw the poison post into the box:\n\n_'You look horrible on television''_\n\n_'Trawler owners hate you, trawler men hate you'_\n\nThe telegram read:\n\n_'YOU SHOULD GO TO PRISON ON A SLIMMING DIET YOU ARE VERY COMMON AND HORRIBLE'_\n\nVirginia popped her head round the door. Lily looked up, and was mildly startled.\n\n'You alright, Mam?'\n\n'Never you mind, our Virginia. You'll be late for work.'\n\nThe girl left for work. She had seen that look all too often recently. Virginia knew her mother was hiding something and that it was getting to her. But she left for work as her Mam had asked.\n\nAfter the girl had gone, Lily opened another two postal telegrams. One was from the Transport and General Workers' Union. It thanked her for her brave work and offered continued support.\n\nThe second telegram from the British Safety Council read:\n\nMRS BILOCCA. THE BRITISH SAFETY COUNCIL WELCOMES AND FULLY SUPPORTS YOUR COURAGEOUS CAMPAIGN FOR THE SAFETY OF BRITIAINS TRAWLER MEN STOP CAN WE IN ANY WAY HELP - JAMES TYE CONTROLLER BRITISH SAFETY COUNCIL + AMBASSADOR 2415 PADDINGTON 9876 +\n\nEncouragement like that made her more determined. The poison pens would not put her off. They simply made her more steely. She would not let folk like that win. 'Bugger them,' Lily thought and prepared notes for that night's meeting in Grimsby when she and Yvonne were to address fishermen's wives at the local TGWU, with a view to setting up a woman's branch.\n\nThat evening's visit to Grimsby was over before it officially started. The union meeting room was filled with women who booed and jeered before Lily took to the microphone. She could not get a word out. Yvonne tried to help. 'C'mon ladies, slanging matches won't help anyone. Let her speak.' But they would not. Jibes about putting their men on the dole led to a fever-pitch screaming match. Lily and Yvonne fled the stage to the safety of the Hull union branch car. That dreadful reception and the earlier outbursts from the two skippers told the two women that the tide had turned against them, at least in the rival port of Grimsby \u2013 home of the giant Ross Food Group.\n\nNext morning \u2013 Valentine's Day \u2013 the first real good news of the week came for Lily. Her darling Charlie would be home tomorrow. It said so in the Daily Express, on page five \u2013 in a story headed, MY WIFE, BIG LIL, BY HER HUSBAND. The strap-line must have made Lily smile, it read, _'She's really a gentle soul. She's not the boss in the house.'_ Under that was the picture of Charlie taken on board the Tasso V in Sunderland the night before. In it he was sat looking at a copy of the paper. The caption read, Charlie Bilocca finding out how famous his wife has become in his absence. The World's Greatest Newspaper originally recorded Charlie's age as 54, but corrected it for later editions, of course.\nELEVEN\n\n_Valentine's Day, 117, Coltman Street, Hull._\n\nThere was no card or flowers in the post that Lily sorted out that day, as she had done for weeks now. But there was one Valentine's Day surprise. While she was sorting out the post and folding up that day's cutting from the Express featuring her Charlie, there was a knock at the door. 'Hello, Mrs Bilocca, I'm Dennis Arnold from the Daily Express.' Dennis was one of the paper's top photographers, a consummate newspaperman, who was as much at home photographing royalty as he was with dockworkers.\n\nStanley Blenkinsop had arranged for Dennis to call in early for Lily and tell her that the World's Greatest Newspaper would like to get a picture of Charlie and her at Sunderland. Stanley told Dennis it would be a great angle, 'Big Lil and li'l Charlie' reunited on Valentine's Day. But that was not quite how Dennis sold the idea to Mrs Bilocca.\n\nLily was not too sure at first. She had a lot to tell her Charlie and was not certain that doing it in front of some men from the newspaper was the way to do it. She was not that sure that her man would be happy about all this publicity she had attracted. But Dennis, like all Express men, was persuasive and told Lily that they would drive her there, and bring her and Charlie back with them to Hull.\n\n'It's only a picture story love. And just think, you'll have your hubby home early.'\n\nThe photographer was left with a cup of tea and some custard creams, while Lily got her glad rags. About an hour later she emerged.\n\nThe diamante set was on again, this time with her best blue coat with white piping that matched her dress of the same design. Her hair was up, making her look three inches taller than her five feet five inches. Dennis already had a picture framed in his head. Lily and Dennis then set off for Sunderland Docks to meet up with Stanley \u2013 and Charlie.\n\nLily was nervous but happy. It was three weeks since her Charlie left for Sweden on his coaster ship. In the car on the way there, Lily told Dennis she was more nervous now than she was addressing all the protest meetings she had done.\n\nIn the shadow of Charlie's ship, the Tasso V, Stanley, Lily, Charlie and Dennis met up at the dockside. Passers-by could not decide who to stare at first, the big lady hugging the little dapper dark skinned man dressed all in black, or the bowler-hatted, monocle-wearing toff who seemed to be directing a photo shoot. Dennis snapped up Lily and Charlie's initial hug. 'Come on you two, let's have one of you holding hands.' Next to Lily, who had her hair up, Charlie looked a few inches shorter than his five feet four inches.\n\nThat was not lost on Dennis, who framed the picture perfectly for the angle. That was the picture for the next day's paper.\n\nAnd Stanley did it justice with the purple prose that he phoned in from a dockside call box as Dennis, Lily and Charlie set off for their drive back to Hull.\n\nIt appeared in the next day's Daily Express much as Stanley had envisaged.\n\nThe picture of the couple, facing straight to camera had a Laurel and Hardy-esque look to it, enhanced by Charlie's wide grin.\n\nThe caption under it read, _17 stone plus ten stone equals...the Biloccas reunited_. Stanley's copy appeared under the headline:\n\nTOGETHER AGAIN\n\n\u2013 BIG LIL\n\nAND LITTLE\n\nCHARLIE\n\nHand-in-Hand yesterday... Big Lil and Little Charlie \u2013 the husband who came home from the sea and told her, \u2013\n\n'I am proud of what you are doing.'\n\nIt was the first time 64-year-old Charlie Bilocca \u2013 5ft 4ins and around 10st \u2013 had met 17st Lil since she became a national personality with her campaign for better safety measures for trawlermen.\n\nCharlie, who works in the engine room of a coaster, has been on a three-week voyage to Sweden.\n\n**So proud**\n\nAnd Mrs Bilocca said yesterday as she travelled from their home in Hull to meet him at Sunderland that she was more nervous than the day she got up to speak at the trawlermen's wives' first protest meeting.\n\nBut after a hug and a kiss all was well.\n\nSaid Charlie, 'She is a funny lass. If she says she is going to do something she will all right.\n\nAt first when the lads on the ship told me what she was up to I thought they were pulling my leg.\n\n'Now I know and have seen her I am as proud of her as if she was the Queen of England. I had two trips aboard trawlers and that was enough for me.'\n\n[Daily Express, Page 9, Thursday, February 15, 1968]\n\n_Abbotsinch Airport, Glasgow, February 15 (Thursday afternoon)._\n\nTV and newsreel cameramen lined the Tarmac apron of the runway. When the Icelandair flight from Reykjavik to Glasgow came to rest they all focused their long lenses. A British Path\u00e9 News cameraman was to the fore. He was filming the others before he panned to the aeroplane that had just landed.\n\nParallel to that plane a mob of news cameramen and reporters waited to surge towards the gangway that airport workers brought to the aircraft. Minutes later Harry Eddom walked down that stairway alone. At the bottom a uniformed policeman saluted him and an air stewardess flashed a wide smile. A line of policemen separated the now-familiar sight of a pack of newsmen from him. Between Harry and them, parked just thirty feet or so from the aeroplane was a two-tone Wolseley Six saloon car.\n\n'This way, sir,' said the policeman who guided the bemused trawlerman towards the car. An argument was in full swing between the angry pressmen and the cops who stopped them getting to their quarry. The policeman \u2013 a chief inspector \u2013 held his arms up and gestured to the press to get back. There was no show today.\n\nA young fair haired man in a three-quarter reefer jacket accompanied by a balding, bespectacled, distinguished looking gentleman in a mohair coat ushered Harry to the back passenger side door which was being held open by a chauffeur. Once the door was closed behind the three passengers, the smartly dressed driver got into the front and gunned the engine. The company car then sped off along the runway past a BP aircraft re-fuelling truck.\n\nIn front of the chauffeured car there was a police van that cleared its path to the exit, from whence the car headed for home. The other passengers came off the plane and saw dozens of cursing newsmen and coppers having a furious row.\n\nThe Path\u00e9 newsreel camera kept focused on the Wolseley as the car diminished into the horizon. There was a man either side of Harry. He recognised the older one immediately. It was Walter Perry, one of the directors of Hudson Brothers Trawlers Ltd., owners of the doomed Ross Cleveland. The other man was a male nurse from the St Andrew's Dock Surgery in Hull. He introduced himself.\n\n'Hello Harry, I'm John Green. But everybody calls me Ginger.'\n\nBack in Hull later that day another Wolseley Six made its way to the western suburb of Anlaby Common. From her window Theresa Gay, the young widow of Ross Cleveland commander Phil Gay, saw the Wolseley pull up outside her house. She recognised its driver. He walked up the path to her front door. Theresa had not left home since the news of her husband's ship. Family and friends had been with her most of the time but that day she was alone and keeping herself occupied with a half-hearted effort at housework as a distraction.\n\n'Oh God, what does he want? Theresa thought. Charles Hudson, owner of the Ross Cleveland, rapped gently on the front door. Theresa was on automatic pilot; she wandered to the door, opened it and beckoned the old man in. With his tweed suit, bushy eyebrows and ruddy face he looked more like a farmer than a fisherman.\n\nThere was an almost mumbled introduction as Mr Hudson made his way into the lounge.\n\n'You have a very pleasant home, Mrs Gay,' stuttered the trawler owner, waiting to be asked to sit down. Theresa beckoned him to the armchair. It was obvious that he did not want to be there and it must have been obvious to him that he was not entirely welcome. But he was a man who knew his duty \u2013 and Theresa was a woman who knew her manners.\n\nHudson reiterated what had been in the condolence letter Theresa had recently received from him about how the men who perished were brave and valued and missed \u2013 and that his sympathy was with all the families. That was no doubt heartfelt \u2013 and sincere. But Theresa simply was not in the mood for words. She just wanted to be left alone with her grief and to continue to begin to come to terms with the biggest tragedy of her life.\n\nThe well-mannered young widow and the dutiful old gent continued their awkward, staggered conversation until Mr Hudson announced he would have to get back to his office. 'There was a lot to do.' Theresa walked with him down the path back to his car. He stopped at the boot and opened it. There were a few fish kits inside. He put his briefcase in beside them.\n\n'Hake, Mrs Gay, freshly caught. Would you care for some?' Hudson enquired.\n\nTheresa looked the trawler boss straight in the eye. He looked nervous, almost shamefaced.\n\n'I do not like fish, Mr Hudson,' Theresa replied.\n\nMoments later he drove off. Theresa returned to her empty house and wept again \u2013 as she had done for days now \u2013 for the man she had loved since she was sixteen.\n\nThat day, Harry Eddom's incredible story took the focus from the fighting fishwives somewhat \u2013 but there was still a discernible rise in the tide of resentment against their leader. Criticisms made in public were harsher. Lily could deal with the poison pen letters, but some of things being said in the papers by folk from her own community hurt her more than any of the vile correspondence she had mostly hidden from her family.\n\nCharlie Bilocca, now back home, was not fully aware of the extent of the wave of abuse but tried his best to support his beleaguered Lily. She could not understand why people \u2013 people she was fighting day and night to help \u2013 were turning on her like this.\n\nShe was aware of the gossip and concerns that Mary and Chrissie had had from other fishwives but the latest outbursts shook her resolve temporarily.\n\nToday it was officially confirmed that there would be four investigations launched soon. Three would be into the demise of each trawler, and the fourth was to be a wide-ranging public inquiry into every aspect of the industry. Also that day it was announced that The Weather Reporter, the naval \"boss\" ship, was on it way to the Northern Cape and that the ban on fishing there would be lifted soon.\n\nLily and the women could not be blamed for thinking there would be praise for them, rather than condemnation. But the Hull Secretary of the Trawler Officers' Guild Skipper Laurie Oliver was the latest to criticise recent events and those who had brought them about. Skipper Oliver told reporters that the British Trawlers' Federation chief Austen Laing had told him about the proposed investigations, although no date had been fixed for them yet.\n\nHe said, 'I think the general inquiry will come first. The Guild has asked Mr Joseph Mallalieu, the Minister of State for the Board of Trade to get on with it as soon as possible.' Oliver then took his chance to criticise both the women's campaign and the bringing in of the naval boss ship. He said, 'I have been asked by the wives of some of my members to state that the action of Mrs Bilocca has not enhanced the image the public may have of fishermen's wives. Women who have lost men have had the least to say, which is what we admire. The idea of forming a women's committee to fight battles for the men, is to my mind, completely ludicrous.'\n\nOliver then turned his ire on the \"boss ship\" scheme.\n\n'I don't think it will do a lot of good at all. In our opinion it was sent there to satisfy the public with the emotions, which arose because of the three disasters.'\n\nHe welcomed that the boss ship would be at the Northern Cape soon so that fishing could get underway once more. The skipper added that if the Government was insisting on such a ship it should be a proper vessel with real medical facilities, emergency aid as well as meteorological services, 'like the ones the Germans use.'\n\nSkipper Oliver was not alone in his attack on Lily. Some of the criticisms that so far had been made privately to Mary and Chrissie by other women were now being made publicly alongside those from the industry.\n\nAnd it was, as Mary had feared, Lily's appearance on the ITV chat show a few days earlier that proved a spark for the tinder. The letters' pages of the Hull Daily Mail which until recently had been praising Big Lil and the Hessle Road women, started to voice harsh criticism:\n\n_'Sir \u2013 Having watched the Eamonn Andrews' Show it left me wondering what sort of impression Mrs Bilocca gave to the millions of viewers of our brave, hard-working men. Owing to the fact that they only have 60 hours in dock, our men are so busy having bevies as she puts it, they have no time to sort out their own affairs. So she has kindly appointed herself their spokeswoman and protector. I am sure hundreds of wives, along with myself, will feel ashamed to mention outside of our city that they are fishermen's wives. They may be accused of putting on airs and graces. \u2013 (Mrs) M. Gordon._\n\n_'Sir \u2013 I would like to ask Mrs Lily Bilocca, and others, what she is prepared to do now she has \"achieved her purpose\" of closing the Icelandic fishing grounds to our British trawlers.Is she prepared to transfer the fishermen's livelihood to foreign trawlers \u2013 German, Russian, Icelandic and others \u2013 whose catch will no doubt be sold on our markets while our men's living will perhaps be nil. The fish merchants will not worry who brings the fish, so long as it comes. Surely it rests with the fishermen where and when they go, as they are not forced into service by anyone, and they know what the life at sea entails. \u2013 (Mrs) H. Harling._\n\n_'Sir \u2013 Being a fisherman's wife, may I please state, along with a number of other wives of fishermen, that we were completely disgusted with the interviews on television regarding safety and working conditions at sea. I wholeheartedly agree with their cause, but I think the people who were interviewed let us down badly in the eyes of the general public, many of whom are not familiar with the fishing community. Time and time again we are ridiculed through shows, pantomimes, etc., about how Hessle Road people speak and live. If anyone is to be interviewed on any future programme, let it be someone who is able to speak and express themselves properly. (Mrs) R. Taylor.'_\n\n_The Royal Station Hotel, Hull, 7.45pm_\n\nThe world wanted to hear everything they could about the \"man who came back from the sea\". Walter Perry, the Hudson's director who had earlier that day arranged the car trip for Harry's homecoming, greeted an expectant pack of newsmen. Perry was alone. There was a crescendo of muttering and stage whispers from the press. Perry said, 'Gentlemen, this has been arranged so that he can go home with his wife and family tonight and have an undisturbed night. Please respect that. He is under some strain.'\n\nThe medic who had examined Harry on his arrival then also addressed the press. He said, 'Mr Eddom's in quite good shape but he is physically and mentally exhausted.' For a moment the newsmen thought they were going to get a similar kiss-off to the one their colleagues at Abbotsinch got hours earlier. The relief was palpable when they saw it was not to be so. But it was not entirely good news. Perry added, 'This will be a very brief press conference. Please choose one of your number to ask the questions.'\n\nWhile the media men shouted, scribbled and handed their questions to their chosen spokesman, Perry left the crowded lounge. Rita Eddom and baby Natalee had been waiting just five minutes when Harry walked in to the lounge bar area where the pressmen were.\n\nGinger Green, the male nurse walked in with Harry. The handsome dark-haired fisherman wore a fur-lined car coat over a dark suit and sported a white shirt and new tie, all of which had been purchased for him in Iceland. Rita and the baby were on a small two-seater sofa that had been pushed against the wall. An ashtray with some stubs still in it was on the arm nearest Harry as he sat down with his little family.\n\nHarry sat forward on the edge of the couch and leant towards the cameras and newsmen. He dangled his new leather gloves in his hands swinging them nervously back and forth as he looked around. Baby Natalee was in a bright white fluffy faux fur hooded romper suit and she gurgled and looked up as Mummy and Daddy kissed. A flurry of flashes went off as they did. Harry then lifted his baby girl and hugged her and the flashes went off again.\n\nIt had been a long drive from Glasgow's airport. It was broken only by a short stop-off west of Hull when Harry and his minders dropped in at the house of a friend for a cup of tea before continuing to the Royal Station Hotel. Upon his arrival he had a brief medical examination before being allowed to face the Press and to be reunited with his wife and daughter. The weariness was etched in the miracle man's face.\n\nRita, dressed in her best fur-collared coat, sat nervously. It had been five days now since she had gone through her own press frenzy experience in Iceland. Harry flashed glances around the room. Neither of them really wanted to be there and it showed in their body language. The medic, Ginger Green was on standby behind the press pack. Unprompted, Harry looked up from the sofa over which the pressmen towered and said, 'It's smashing to cuddle Natalee again.' Standing next to the arm of the sofa was local freelance BBC man James Goodrick.\n\nWhen the Eddoms got seated, Goodrick moved round and knelt before them. He and the others were aware that this would be a very brief encounter. Goodrick had a small brown leather case slung round his neck. It contained a UHER \u2013 a German-made, small reel-to-reel tape recorder, which was the radio broadcast industry standard. Its mini-microphone was directed at the family on the sofa, to ensure Goodrick got all the answers from the questions that followed in rapid succession.\n\nThe designated pressman, George Hill of the Daily Express, determined to get him to recall as much as possible of his ordeal. Harry gave a couple of \"yes\/no\" answers and even raised an eyebrow to some of the obvious \"aren't you lucky? \/ aren't you tough?\" kind of questions that he had been asked dozens of times since his time in Isafjord Hospital.\n\nWhen asked about the shepherd boy who found him, Harry recalled, 'I had spent the night huddled against a wall. I was frozen. I heard dogs barking. Then I saw the boy and I shouted to him. He pushed the sheep back onto the mountain and came back with two great [big] dogs. He took me to the farmhouse. I could see the farmhouse light during the night. But I could not move my legs at all.'\n\nHe went on and said he had not seen the boy since that time but he was certain that young shepherd had saved his life. The reporter hurriedly asked about Harry's twelve hours in the raft with his dying shipmates.\n\n'What was the worst part, Harry?'\n\n'I could not say what was the worst moment of the lot,' Harry said. He was still fidgeting with his gloves. The journalist then asked if Harry was going back to sea. This time there was no joke about maybe going on the buses like there had been in the Icelandic hospital. He simply looked towards Rita and said, 'That is up to my wife.'\n\nThe reporter asked Harry's opinion of his skipper and if Harry thought anything could have been done that would have saved his ship and his shipmates.\n\n'You could not have wished for a better fisherman or skipper than Phil Gay. He would not wear bad weather at any time. There was nothing wrong with that ship at all. She was safe enough. I cannot say if there was anything that could have saved the Ross Cleveland. I will leave that to the Board of Trade.'\n\nHarry added, 'It was the worst weather they had seen in Iceland for twenty-five years \u2013 and will expect to see for another twenty-five.'\n\nThe Eddoms were led from the hotel to the car waiting to take them back to Cottingham. The newsmen raced to find telephones to let their night news editors know the details of the eight-minute press conference. While dozens of journalists spilled out of the Royal Station Hotel lounge and searched for payphones in Hull city centre to get their copy to their news desks, Lily and Charlie Bilocca finished a fine Italian meal. It was the first chance since he came home that they had time to spend some proper time together. Both had been very busy all day, Lily with her campaign work, and Charlie catching up with his other jobs. Most of the morning was taken up by the trip to Sunderland and back with the Express.\n\nCharlie set a table fit for an officers' mess. Charlie was a good chef and that night he made his specialty \u2013 Spaghetti Bolognese. Lily was in her best floral dress and the diamante earrings and necklace set made another appearance. Although they lived in a big house, the Biloccas were not the sort to dress for dinner. They planned to go out after the meal, so had got their \"going out\" clothes on.\n\nUsually when Charlie cooked, Lily did the clearing away and washing, but Virginia had been volunteered for that. The young girl had also helped serve up with her Dad.\n\nDuring the meal, Lily told Charlie about some of the abusive letters. She did not tell him about the worst of it and she was certain that he would never seek the letters out, even if he did know where they were hid. Lily told Charlie she was worried after what had happened in Grimsby and with the skippers' recent criticism in the papers.\n\nShe was also concerned about the letters in the local newspaper. She tried not to sound too worried, after all it was his first night back and she did not want to burden him. In reality, she probably just wanted to be told it was all going to be all right.\n\nNearing the end of the meal she said, 'Charlie, I sometimes wish I'd not started this.'\n\nCharlie replied in his Maltese\/Yorkshire accent, 'Don't you worry love. There's only a few of them having a go and most of them are the boss men and their wives. The whole world's with you. I don't know much about papers an' stuff but I know when folk are trying to stop you doing what's right. But they don't know my Lily, so they don't. Once you've got something in your mind, there's no stopping you. You're like the Rock o' Gibraltar, you'll never alter. I meant what I told them newspaper men, I am proud of you Lily.'\n\nLily smiled and said, 'Oh, Carlo you always know what to say for the best.'\n\nCharlie knew he had made his missus feel better at that precise moment. She only used his pet name when her mood was improving. With that, they finished the wine Charlie had brought back with him and then the couple left the table, sided the dishes and went out to the hallway and fetched their coats from the little cloakroom there.\n\nBefore leaving, Charlie could hear the music coming from his daughter's room and shouted up the stairwell, 'Virginia, bambina, make sure you do the dishes for your Mam before you go out.'\n\n'Ok, Dad,' the girl replied, shouting back, to be heard above the perpetual sound of the Dansette record player.\n\nWhen her Mam and Dad set off to cover the 100 yards to the snug bar of The Criterion, the young girl decided to come downstairs and get the chores over with. Virginia took about half hour to tidy everything away and get the dishes done. Before getting ready to go out she decided to do one more chore. She made up a platter of cheese and biscuits. That was her Dad's favourite, especially Jacob's Cream Crackers, so she put a few more of them onto the plate before wrapping them in greaseproof paper. She left a little note about the supper surprise. It was one of those little things she had always done for her Dad when he came back from sea.\nTWELVE\n\n_Saturday, February 17, The Bilocca house._\n\nThe breakfast remnants were still on the kitchen table. Charlie and Lily were on their second, maybe third cuppa. No one was counting \u2013 or talking much. Last night had been a good one. If there was a turning in the tide of public opinion against Lily it certainly had not manifested itself in the snugs and lounges of The Criterion, Rayners, or The Alexandra Hotel the previous evening. The Biloccas had had a great night out \u2013 as always on the Fridays when Charlie was home. And everyone seemed pleased to see them and buy them a drink or two. Charlie and Lily were always a good laugh and knew how to have a good time at the boozers, like most of their pals.\n\nStarting at The Cri, then after a few there, on to Rayners, about two hundred yards up the Road for an hour and then backing on themselves down to No.1, Hessle Road to Charlie's favourite pub The Alexandra Hotel \u2013 which everybody called The Alex. Lily and Charlie were there until closing time. A few stouts for Lily, a few Watney's Red Barrels for Charlie at each, and then at The Alex, Charlie, as usual, covered part of the cost of the evening's entertainment by whipping all-comers at darts. When not playing he was in demand for his scoreboard marking skills. (Charlie had almost savant-like mental arithmetic. Nobody on Hessle Road could figure 501s like him.) Then it was home for a supper of cheese and crackers.\n\nSaturday's big breakfast and the lashings of tea mitigated the worst of last night. Virginia came in with the post and handed it to her Mam. The young girl had already flicked through the envelopes. She was more aware by then of what sort of things were coming her mother's way, even though her Mam tried to hide it from her. That day the envelopes looked official, no handwritten covers, and no crazy scrawls. The good news that day was confirmed soon. No poison pen letters. Then the bad news made itself known. Virginia was on her way back up stairs when she heard her Mam's voice rise up from the kitchen table.\n\n'Bloody hell, Charlie I've been sacked!'\n\n'What?' said her husband.\n\nLily went on, 'Sacked, listen to this...'\n\nTears of anger welled up as she read out loud:\n\n_Dear Mrs Bilocca,_\n\n_As we have not heard from you for the past three weeks, we can only assume therefore that you have left our employment. We enclose your National Insurance Card and P45, herewith and a small wage packet awaits your collection...yours truly..._\n\nLily didn't read out Don Wilkinson's signature, but raged, 'It's dated bloody yesterday, Friday. I were bloody in there on Thursday and told him I were busy. Not heard from me for three weeks? I was on the bloody telly most of the time!' Lily's voice rose again and she threw the letter onto the table, swore and muttered under her breath.\n\nCharlie said, 'Never you mind them, love; I don't want you going near a bloody fish house again anyway.' His accent reverted to more Maltese than Yorkshire as he got more agitated as it always did. 'I gotta enough for us all. I told you when I came home that the whole world was with ya. I gotta money to keep us going. You get on with what you're doing. You must go to Scotland on Monday. You got work to do. Folk are depending on ya.'\n\nLily held back her tears and fumbled, almost automatically, to open the next letter and started to laugh. She put it onto the table on top of the other.\n\n'What is it? Why you laugh?' Charlie said. 'Virginia, bambina, what's it say? You read it for me.' Charlie wasn't much for reading. So the girl read it. The letter said:\n\n_Dear Lil,_\n\n_It was a great pleasure having you with us on Sunday, and an added pleasure meeting your daughter. Once again, many thanks, indeed for coming along._\n\n_Success in all you do,_\n\n_Eamonn._\n\nIt was written on posh vellum personal stationery. The phone numbers \"Gerrard 0292\/3\/4\/5\" were on the top left. Opposite the address line read, \"\u00c9AMONN ANDREWS, 4, Golden Square, London, W.1.\n\nThe irony was not lost on Charlie, Lily and Virginia as they laughed. The following morning the house was quiet. Charlie and Virginia were at Mass. Lily had her writing pad on the table and was working on her talk for the trip to lecture the students of Strathclyde University who had invited her to Glasgow.\n\nSunday was always quiet. The Biloccas had a routine, especially when Charlie was home. After breakfast, he would work on his little vegetable patches and tomato frames and the like. Then he would tend the rabbits and pigeons and sort out items for selling and bartering.\n\nCharlie was a regular sight on Hessle Road on his racing cycle with bags of vegetables, herbs, rabbit meat or pigeon to be bartered, exchanged and sold to his contacts there and at the Hull Market. On Saturdays and Mondays he was usually at the Selby Market. He would often take Lily with him on the train. It was Lily's favourite day out. They always had a pub lunch, and Charlie rarely left without a bargain or a profit \u2013 or both. If he was on his own he did not waste money on train fares. He took his trusty twelve-speed drop-handled racer and covered the thirty or so miles. Sunday was not quite a day of rest for Charlie. After his morning's chores he would get Virginia and take her to the St Charles Borromeo Roman Catholic Cathedral in the city's Jarratt Street, near the aptly named Worship Street.\n\nCharlie was a religious man and attended Mass whenever he could. He insisted his daughter and son went too, especially the girl. After his boy Ernie left home and got married, Charlie continued going to Sunday Mass with Virginia. Lily was not one for church, although she was a member of the Church of the Nazarene just a few doors away from her house. She always had insisted the kids went to the services there too. They were held in the evening.\n\nHer kids might have been devoutly confused, but Lily always got peace and quiet on a Sunday.\n\nNext morning Charlie had the tea ready for Lily. Her little suitcase was in the hallway again. It had never had so much use. The taxi arrived. Its rumbling engine could be heard from the hall. Charlie and Lil kissed on the doorstep.\n\nLily got into to the cab and set off for Paragon Street Station to get the Glasgow train. Charlie returned indoors to prepare for Selby's Monday Market. The cab slowed at the junction of Coltman Street and Hessle Road, before turning left towards the city centre and the railway station.\n\nThe Road was busy with workers milling to and fro. Lily waved at a few folk she knew as her taxi wove its way through the fish process workers who snaked their way towards the high stacked chimneys of the smoke houses.\n\nCharlie had not been working long when the door was knocked. He made his way from the back room to the hallway to answer it.\n\nOn the doorstep was a young man with a notebook in has hand. 'Hello, Mr Bilocca, is your wife at home? I would like to get her comments on rumours that she has got her cards from Wilkinson Brothers.'\n\nCharlie told the reporter his wife was not in and then beckoned the young man to come in. Lily had warned him this would probably happen and he knew what he was to say.\n\n_The Old Town district, Hull, mid-morning._\n\nBy eleven o'clock that Monday James Goodrick had been at work for five hours. Early calls done, a trip to the magistrates' courts to check the listings and the remands was followed by a check call into the Coroner's Court to pick up the inquest lists for the week. That was the routine. From his office in the heart of the city's Georgian Old Town area he could walk to most key places \u2013 courts, council chambers, lawyers' offices etc., \u2013 within five minutes.\n\nAt the newsstand outside The George Hotel, in the quaintly named Land of Green Ginger, James saw the news bill. Its black bold ink caps screamed out, BIG LIL SACKED. Goodrick went to the newsstand, handed the vendor his sixpence, took his copy of the Hull Daily Mail, got his two pennies in change back and set off the few yards to his office in the ornate Samman Chambers in Bowlalley Lane. In his little top-floor room amid the neat piles of newspapers, reference books and cuttings, he got to his desk, caught his breath and dialled the Press Association. He dialled with the hand that held the receiver, and then raised it to his ear. The ring tone barely rang before it was answered.\n\n'Press Association,' said the voice on the other end.\n\n'Copy please \u2013 It's Goodrick of Hull.'\n\nUsing the front-page story from the paper in his other hand as a note he re-jigged it into a two-paragraph teaser story off the top of his head and read the extemporised copy down the line. 'More follows later.' James said and replaced the receiver, knowing he had ensured dozens of callbacks from the newsdesks. He reached for his contacts' books flicked through and gauged as he did so, who he would call first to beef up the tale for the nationals.\n\nThey would want every cough and spit on this one, quotes from Lil's bosses, background details etc., \u2013 they might even be a feature or two in it. The two-paragraph PA RUSH that James sent from his city centre office would be on every news desk in the world within minutes. And James was ready to take the calls from any who wanted to order his services to chase up the story. He knew there would be some work today from foreign papers and other news outlets and was certain the big Scottish papers, especially the two Glasgow dailies, the Evening Times and the Evening Citizen would be offering James day shifts at NUJ rates, plus lineage.\n\nToday was going to be a good earner.\n\nBy the time Lily's train pulled into Glasgow Central Station, news desks across Scotland had dispatched their best men to her hotel to get her reaction about her sacking. James' copy filed from Hull would make up the rest of the tale. It was late afternoon and Lily had barely settled into the room booked for her by Strathclyde University's Labour Club. She had literally let her hair down minutes before her room became the venue for another impromptu press conference.\n\nLily was getting used to this kind of thing. The cameras flashed as the questions fired in. One of the pressmen quoted her employer Don Wilkinson who had said she was fired from \u00a311-a-week job because she had not been at work now for almost three weeks and the firm had assumed she had left. Lily didn't let another word in as she let fly with, 'Well, they assumed wrong, lad. The trawler bosses are behind this \u2013 that's definite, lad. I think this is definitely victimisation \u2013 the trawler owners are at the back of this. It has not come as a surprise, although I did not expect it so soon. They could have asked me in to see them to talk about it, couldn't they, lad?'\n\n'Will you pack in the campaign now you're on the dole, then?'\n\n'I've no intention of packing it in. I might not have the education. But I have the going in me.'\n\n'So, do you think you'll find another job in the same line?'\n\n'Are you kidding, laddie? I am a marked woman now. I'll just have to try for some other job later. I don't know how far I will get with this campaign. But I for one am not going to sit back and accept defeat. That what annoys me about the British. The employers don't give a damn about these lads. They are never really in port for more than three days at a stretch, so they never really get the chance to get together and fight for better conditions. As long as I am doing what I am doing I am happy. This will make no difference to the campaign, which will go on as far as I am concerned.'\n\nAnother reporter asked about the letters it had been reported she had received. Lily said, 'Most of them are sympathetic, but I don't want sympathy. Sympathy's nowt.'\n\nLily made her answer short deliberately. She did not add to the pressure her family was already feeling from the limited amount of what they knew about the poison pen campaign against her. She swiftly moved on and ended the impromptu press conference with the tough Scottish press corps in fits of laughter. One wise guy quipped, 'Hey Lily are you gonnae go tae any o' Glasgow's tourist attractions?'\n\n'Aye, lad. I want to go shopping in the city and then I might visit the Gorbals, tomorrow maybe. It sounds like the same kind of place that I come from and I want to see if the atmosphere's the same.'\n\n'Good luck wi' that yin, hen!' quipped a photographer out of the side of his mouth that did not have the cigarette in it. After the pressmen left, Lily set about putting her hair back up. Somebody from Scottish Television was coming to talk to her about her sacking. So she would have to go through it all again for that night's six o'clock news.\n\nBack in Hull Charlie had leapt to Lily's defence and told the Hull Daily Mail, 'She was quite upset when she got her cards. But I told her not to worry and that she had the whole world with her. I have enough money to keep us going. I don't want her to go anywhere near a fish house again.' And union man Mike Neve, who had been one of the organisers of the recent triumphant Westminster trip, said his members would fight for her. James Goodrick had collared Neve at the dockside and asked him for his reactions. In the next day's Daily Express Neve said, 'We are disgusted. Lil has achieved a great deal with her efforts for the fishermen. Now we will do everything we can to help her.' A bobber who was stood next to Neve had added ominously, 'We will act ourselves if the union doesn't do anything officially.'\n\n_February 20, Strathclyde University Labour Club, Glasgow._\n\nThe Strathclyde University's Labour Club's 1968 cohort was no Fabian Society. Its radical students did not suffer fools. Its reputation was to political discourse what the old Glasgow Empire theatre was to English variety acts. Genteel debate was often replaced by aggressive questioning and a lack of due deference. Even the most experienced and tough orator came unstuck here. Only a few weeks before Lily Bilocca was there, cockney communist trade union boss Jack Dash, the charismatic London dockers' leader, was booed ON to the stage. His crime? Being twenty minutes late.\n\nMaybe the national front-page news of her sacking enhanced the students' sympathy. Maybe it was just that Lily reminded them of their mum. Either way, Lily was warmly welcomed from the start. In fact, just before her appearance a local news photographer took a snap of her sat in an armchair with students gathered around her, with one young woman looking up at her, almost in adoration.\n\nWhen the talk began Lily was cheered on to the stage. She was already one better than the unfortunate Comrade Dash. There were not as many present as at her Hull lecture, but the two hundred or so students certainly matched their Yorkshire counterparts for volume. Like she had done at Hull University, Lily told of how she found herself thrown into a fight and outlined the rapid successes that she and the other wives had achieved with incredible speed.\n\nReporters in the audience took notes as questions and the occasional good-natured bantering heckle was aimed at the battling fishwife. Lily gave as good as she got and some of her answers were \"larded with basic English\" as reporter Wilson Russell of the Daily Express rather haughtily put in his report the next day.\n\nOne student asked what would happen if some of the women's demands for onboard trawler safety remained unmet in the weeks to come and Lily said, 'If I am not satisfied that all the safety measures we need are in place in a fortnight, then we will kick up the biggest stink ever in this country.'\n\n'Do you want a strike, Lily?'\n\nLily did not see where the question came from but replied, 'I don't want a strike, and any road it would be difficult to organise for the men.' Lily went on to outline how she wanted two years' training for young fishermen and that they should not be allowed into fierce Arctic waters without it. She added that a full-time hospital ship was another ambition on its way to being achieved.\n\nA student shouted out, 'How can we help you Lily?'\n\n'You could start by handing out leaflets if you wanted about our fight, and you could maybe even take a little collection for the dependants of them what's been lost.'\n\nA hat was passed around immediately as the question and answer session continued. Lily went on to lay the blame for the disasters squarely with the owners. She did not hold back in her ferocity. She told the students that she thought the Government had acted brilliantly for the wives but maintained that the trawler bosses did not \"give a knack\" for the lads at sea.\n\nThe cheers subsided and a student, who identified himself as Charles McCafferty, asked, 'Mrs Bilocca do you agree that the Director of Public Prosecutions should charge officials of the Trawl Owners' Association with the deliberate murder of the men lost recently in Hull trawlers?'\n\nHer answer raised the biggest cheer of the talk.\n\nLily replied, 'Of course I do. If I had been born in Chicago days I'd have put them against a wall and shot the lot.'\n\nThe post-lecture press conference was like an inquisitorial carpet-bombing. They seemed determined to get a \"strike line\" from her for their story. Some of the reporters seemed keen to mention things like \"feminine wiles\" and \"women's lib\" in the pejorative way of the era. There were some queries that she did not quite understand, so she just blustered through as best she could. Lily smiled and batted the questions with her earthy good humour. She did not quite get what they were angling at. Lily was tired. She told reporters that her night in the hotel had been the first proper night's sleep she had had since she had started her campaign.\n\nLily was asked if it was possible that her fishwives could persuade their husbands to take action, if she was not happy with the safety regulations later. Lily replied, 'I suppose there are examples in history where women won the day. If we don't get what we are seeking \u2013 increased safety for our men \u2013 then we will perform.'[sic] The press conference drew to it close and a few of the students surrounded her. One of them handed her a bag full of coins from the whip-round during the lecture.\n\nLily must have thought her day a great success. She was cheered by students and even seemed able to deal with the press lads again. She would have a lot to tell Charlie on the phone later from her hotel room. She would count the coins after that and go to bed. She had had a good trip to Scotland. The sacking from Wilkinson's was behind her. She was looking forward to getting home to her Charlie and her family.\n\nNext morning Lily \u2013 and the eight quid that had been raised from the students' whip-round \u2013 set off for home. It was to be a peaceful trip back\u2013 via York \u2013 in the quiet of the first-class carriage of the train from Glasgow Central Station.\nTHIRTEEN\n\nMid morning at Yvonne Blenkinsop's house was chaotic. Yvonne was home alone. The kids were at school and husband John was at work. It had been a while since she had some time to herself. The housework she was doing was almost a treat, compared to maelstrom she and the other women of Hessle Road had been through in recent weeks. A firm, gentle knock on the door gave no clue as to the size of the throng gathered on her porch. (Any newspaperman knows it is not smart to batter on a door.)\n\nA barrage of camera flashes and questions met the pretty petite blonde cabaret singer. The throng pushed forward and soon the neat living room that Yvonne had just tidied was overcrowded. She was thinking, 'What the hell are they all doing here?' Her answer came when the newsmen scatter-gunned her with questions. Had she seen the papers? What did she think of Lily calling for a strike? Then Yvonne put both her hands up as the media crowd came within a foot of her. 'Just back off a bit will you. It's not Paragon Street Station, you know.' With that the chastened mob backed off, giving the housewife some space. Then Yvonne explained she had not seen the papers. The reporters persisted. 'What did she think of Mrs Bilocca's strike call?'\n\n'Calling a strike is downright stupid, if that's what she said. None of the women knew anything about all this. I disassociate myself completely from such a ridiculous idea.' Then came the paydirt question, 'What about Mrs Bilocca's call to persuade the men to do as the wives want?'\n\nThe tone and emphasis on the word \"persuade\" made Yvonne ask what they meant exactly. For a bunch of so-called hard-bitten newsmen it took some umming and ahhing and pussy footing around before they managed to explain. They mumbled and bumbled like shy schoolboys in front of the blonde cabaret artiste, their indirect inquisition peppered with double entendre. The penny dropped and Yvonne blazed back with the sort of language that would never see the light of day as direct quotes \u2013 especially in the World's Greatest Newspaper.\n\nShe blasted, 'A love strike... a sex ban? Are you lot daft? Our men are men \u2013 real men. You have to be a real man to do what they do. If that's what Lil said, then it's a load of codswallop. I just don't know what you're all going on about. Our men would never put up with something like that. You wouldn't even think of putting anything like that to our men \u2013 and whoever thinks they could put such a thing to them are out of their minds. They must be daft. You certainly don't give a fisherman that sort of silly ultimatum. What a load of rubbish \u2013 and I as one of the ladies who went to London with Lily and Mary have never heard of such a thing. I can't speak for Mary or Christine, but I'm quite sure they didn't either, but you'd have to ask them.\n\n'And another thing, boys.' Yvonne added with a smile. 'Why would we cut our noses off to spite our faces?\n\nThe press pack left Yvonne's house with more than one of them thinking, 'That last quote will never make the paper!'\n\nMary Denness had read the papers. This smart young skipper's wife knew exactly the trouble this alleged outburst of Lily's could cause. She had earlier spoken with Chrissie and knew that she had spoken with union man Jack Ashwell. Mary felt that the newspapers had probably manipulated Lily. She knew that her friend's na\u00efve and open manner made that easy to do. But she also knew that important work ahead could be jeopardised. She knew of the growing anxiety among the women who had spoken with her and Chrissie about Lily \"showing them up.\"\n\nMary thought the Eamonn Andrews Show debacle might have done for Lily a few days back. But the stories of the \"superman\" Harry Eddom and his remarkable survival in the Arctic deflected the public's gaze from her for a bit. And then Lily's very public sacking and humiliation had won back some public support for the fighting fishwives' leader.\n\nMary went through the papers that day, especially the Express and thought, 'This has put a tin hat on it all'. She knew you had to be clever with the press or they could easily put words in your mouth. She was also realistic enough to know that whether or not Lil had really said these things was not the problem. The problem was people thought she had. It was right there in black and white \u2013 and folk believed what they read in the paper. Mary considered what she was going to say. There was no way she could check with Lily. She was still on a train somewhere between Hull and Glasgow. Mary knew that a lot of Hessle Road women had already turned against Lily. The media coverage had become more and more personally aimed at the big bluff fishwife. There was also a swell among her own kind against Lily and the whole project was in danger.\n\nIt was a very different press conference at Mary Denness' neat semi-detached home in Bernadette Avenue, in the western suburb of Anlaby Common. There were no raised voices or innuendo. Mary took direct questions and replied in a business-like fashion. She made sure she got her point across. She said, 'Mrs Bilocca is not representing the majority of the wives in this. She is speaking only for herself. At a meeting of the women in Hull most were against strike action. The campaign is going well and progress has been made. But as we take one step forward this happens and we go back again. We are losing support with this kind of talk. Mrs Bilocca is acting on her own initiative. Four of us were elected to go to London on this matter and since then none of us have since been consulted by Mrs Bilocca. I think we must have another meeting quickly. And if the rest of the ladies are against her, they must elect someone else to lead them \u2013 someone who will go on with the view of the majority. The women may be divided over Lily's views but we remain united in the fight for better conditions.\n\n'We must use tact and diplomacy though. Militancy and aggression will get us nowhere.'\n\nWhen quizzed on the \"love strike\" angle, Mary gave the question and those asking it the contempt she felt it deserved. There was no smile or saucy reply like that from her cabaret-singing comrade. 'Ridiculous,' Mary snapped. 'Lily is in danger of making this a one-woman crusade.'\n\nMary's quotes in that afternoon's local paper took up most of the front-page lead. Yvonne got a few pars. Jack Ashwell, whose comrade Mike Neve just a day earlier had pledged the union's full support for Lily after her sacking, sounded distinctly less fiery in his thirty-seven-word press statement that day. 'Until we see Mrs Bilocca and find out exactly what she has said we cannot do anything. But it is now certain that we shall have to call a meeting of the wives, probably later this week.'\n\nA tired and contented Lily took advantage of the peace and quiet of her trip back to Hull to try to catch up with her rest, unaware the train was going to deliver her into the biggest storm of her life. That morning's newspapers \u2013 which she had not seen \u2013 would put her in a spin \u2013 in both senses. Some reported Lily had called for a \"love strike\" \u2013 a sex ban for the trawlermen from their wives if they would not strike in the event trawler safety demands not being met in time. That was the spin from the previous day's rushed press conference in Glasgow. Of course, the word \"sex\" was not used directly. The popular press was far too prudish for that, but still prurient enough to hint very heavily. Even The Times of London in a dull-but-worthy, two-par, 165-words on page two under the prosaic headline \"Strike warning by Hull wife\" told readers Lily \"would ask wives to persuade their husbands to go on strike.\" Ironically, the word \"strike\" was not attributed to Lily in any quote.\n\nIn some papers she was even reported as saying a strike was a bad idea. But this did not put them off. (Why let the facts get in the way, etc.?) That day's exemplar of inference and innuendo appeared in the World's Greatest Newspaper \u2013 and for the benefit of its four and half million readers it managed to say it all without ruining the kedgeree breakfasts of the A _ngry of Tunbridge Wells_ brigade by mentioning the \"S \u2013 word.\"\n\nBIG LIL HINTS\n\nAT LOVE STRIKE\n\n_by Wilson Russell_\n\nBIG LIL hinted yesterday \u2013 so delicately \u2013 that fishermen's wives may go on a love strike unless safety measures on the deep-sea trawlers are tightened up within two weeks.\n\nSeventeen-stone Mrs Bilocca, the Hull fishwife, in spangled dress and jacket was speaking to 200 young men and a sprinkling of girls at Strathclyde University, Glasgow.\n\nThe striking wives, she said, would persuade their husbands not to go to sea. She refused, smiling quietly, to say exactly just how the wives would do this.\n\n'But I am convinced that we will be able to kick up a stink,' she said.\n\n'I suppose there are examples in history* where the women won the day. If we don't get what we are seeking \u2013 increased safety for our men \u2013 then we will perform.'\n\nBig Lil belaboured her audience for 20 minutes in sentences larded with basic English. And she won the student cheer when she called the trawler owners \"pigs\"\n\n* _Lysistrata_ , \"disbander of armies\" in a play by Aristophanes stopped war by persuading the women of Athens to withdraw their favours from their husbands.\n\n[ _Daily Express,_ p4, Feb 21, 1968]\n\nAny journalist knows that some of the oldest tricks in the book were at play here. The word \"hints\" in the headline, and \"hinted\" in the intro, the phrase \"may go\". [It could equally be argued they \"may NOT\" go]. Writing an innuendo like, \"She refused, smiling quietly, to say exactly just how the wives would do this,\" and then going on to say exactly how she would \u2013 and then \"backing it up\" with cod-classicism. But the sub-editors and the reporter Wilson Russell knew that readers focused on key words that jumped off the page. LOVE STRIKE. Job done. Of course, any classicist who saw the Lysistrata line would see it was a forced premise. But an outcry by any classically educated Express readers was highly unlikely indeed!\n\nThis is what newspapermen call \"flying a kite\" \u2013 going with an angle and making the story fit it. This is demonstrated by both the Express's use of an asterisk on the \"main\" quote and the readers' footnote that followed and explained what Lysistrata meant. [It is a good bet that the sub-editing staff had to look it up too.] The laboured footnote to justify the angle also supported the theory of a forced news line.\n\nCynics might have said it had more than a touch of journalistic licence. It probably got good old Wilson Russell a compliment or two at work \u2013 and a complimentary drink or two in the bar after work \u2013 probably with the sub editors that helped craft the piece. It was a safe bet though that the only critique on Hessle Road was from furious fishwives who felt Lily had \"shown them up again\" and from journalists ordered to follow up the \"sexy\" angle, and most of them just thought, \"the bloody Daily Express has done it again\".\n\nNext morning, an obviously upset Lily met the press in her big front room. There was none of the banter they had become used to, no jokes \u2013 no offer of tea and biscuits. She looked as if she had not slept. Charlie was by her side. Lily started immediately to attack the reports in the papers, especially those that suggested a sex strike. And it seemed the tough guys of the press still could not bring themselves to ask that question straight out.\n\nOne asked if Lily had heard \"the persuasion suggestion\" put to her after the Glasgow lecture. Lily rounded on them. 'Of course, I heard the suggestions. They said a lot of things. But I did not bloody agree with it. I said that no good ever came from a strike.'\n\nLily was determined to make her point and ignored any further questions until she had. 'I can see how folk would get their backs up if they thought I had said what they said I did in the papers. But I did not say it. I only hope the wives will believe what I say when I explain to them tomorrow night. I telephoned Mary Denness last night and explained what happened \u2013 and I think she accepted my explanations.\n\n'Look, I have got other speaking offers coming up, as well as one I have got to do in Suffolk next week, but I want to see if any of the other women want to go instead of me. I am not trying to do the union's job, or that of the trawlermen who are quite able to fight for themselves. If people are beginning to think that trawlermen are tied to apron strings, then they should come to Hull and see for themselves.'\n\nCharlie added, 'I think Lil is doing the right thing to fight for better safety conditions on the trawlers \u2013 but I am worried about the anonymous letters she receives. Some of them are really nasty.' If he had known the contents of the shoebox in which Lily kept her correspondence, he'd have been worried even more.\n\nThat day bosses' spokesman Laurie Oliver did not miss another chance to attack the fighting fishwife. Skipper Oliver \u2013 Secretary of the Hull Trawler Officers' Guild \u2013 told the Hull Daily Mail, 'Mrs Bilocca would be doing the industry a good service by withdrawing completely into the background and preserving a silence on the quite irrelevant matters concerning people's private lives.\n\n'It is high time that whoever is behind her should now withdraw that support. Her efforts did not lack appreciation in the beginning but the thing has got completely out of hand. What earthly good she can do by such things? And as for talking to students in Glasgow on the lines she was reported to have done, I have no idea. Whatever good impression she may have made originally in the crusade for greater safety and better conditions at sea, she is now destroying.'\n\nMary Denness was later contacted by the media and was asked if she accepted Lily's version of events. 'Whether I agree with her or not, I would not like to say right now. It is up to the wives to believe her or not. If the majority wants her out, then I am with them all the way. I am with them whatever they decide.'\n\nThat evening Lily's son Ernie dropped by to see his mother. He, of all people, understood what she was fighting for. After all, he had survived the storm that claimed the Ross Cleveland just little more than a fortnight earlier. He was concerned that something untoward would happen to her. He thought people were out to get her. Even his skipper Len Whur had had a go at her. The poison pen letters reported in the papers added to Ernie's worries.\n\nEarlier he had asked his Dad if his Mam had thought about quitting and let somebody else take all the flak. Charlie gave his stock reply, 'Ernie your Mam is like the Rock o' Gibraltar, she'll never alter.' Ernie decided he would bring up the subject anyway. He could not believe what he read in the papers. It just did not sound like his mother. He knew she was given to rough and ready language, but he also knew she was quite prudish, in her own way. That evening he took his courage in both hands and said, 'Mam, why do you think they put it in the papers?'\n\nHis mother was in her favourite armchair. She looked directly at him and said, 'I'll tell you why, son, because they are liars, just looking for a story, that's why. I'll never trust the press again. And you should never talk to them either. All this tripe in the paper is just a load of old cobblers.'\n\n_Bevin House, Hull [HQ for the Transport and General Workers' Union] \u2013 Friday, February 23._\n\nThere were no crowd control issues at Bevin House, unlike the earlier hectic free-for-all meetings at Victoria Hall. Reporters outside on the freezing, sleet covered George Street in the city centre counted only forty women into the union building \u2013 and four of those were Lily, Yvonne, Mary and Chrissie. Maybe it was the recent publicity. Maybe many of the Hessle Road women thought their aims were achieved. After all, the papers had been full of Government promises of swift action on trawler safety. Whatever the reason, that evening's crowd was a tiny fraction of those who had stormed the docks and focused the eyes of the world on Hull's fishermen and their most dangerous occupation.\n\nThere was no problem accommodating reporters that night either, for the union did not allow them in. On the doorstep of the red-brick union building T&G man Jack Ashwell told waiting reporters, 'I don't think it would be fair for the Press to hear opinions expressed by the women. I want to make the conditions right to get business done. Having you lads in there might lead to some making speeches for the Press rather than the good of the new association.'\n\n'So there's going to be a new association then, Jack?'\n\nThe anonymous reporter's question went unanswered as Ashwell turned heel and went in to join the meeting, leaving the pressmen in the cold drizzle under a blue fug of fag smoke on the porch. The media men settled in for a long wait. They did not need to question Ashwell too much. In the local press the day before it was reported that Ashwell and Neve had met with Big Lil and the other women following the furore after the love strike reports. After that meeting \u2013 held in Lily's house \u2013 all they said afterwards was that they had \"cleared the matter\" with Lily. There was no comment from her or the other women. But the union men revealed that they would be helped to form a \"wives' association not for industrial purposes.\"\n\nAfter forty-five minutes in the foot-stomping cold the pressmen were caught by surprise when two young women barged out into the media throng. Cigarettes were hurriedly discarded and cameras quickly lifted out from bags. The journalists recognised blonde cabaret singer Yvonne Blenkinsop \u2013 and she did not look happy. Neither did the young woman with her.\n\n'What's up, Yvonne?' shouted a reporter.\n\n'I'll tell you what's wrong. We came here to talk about safety, not about bloody social work. Now these women want to interfere with other sections of the union. It's the union that should look after welfare of fishermen families, not the wives. It's safety we should be fighting for, not bloody social work.'\n\nThe young woman with Yvonne was her sister, a twenty-year-old trawlerman's fianc\u00e9e called Dianne Horsfield. She said, 'They have gone right off the point. We want to know about the safety angle and we don't want anything to do with welfare.' With that, the two women stormed off.\n\nThe press pack went back to waiting and smoking. Some of them chanced a quick pint. Fifty minutes later Jack Ashwell came out with the women. He was surrounded on the porch steps. Cameramen jostled and were balanced precariously on the steps' edge. Ashwell wiped the raindrops from his horn-rimmed glasses and prepared to make his statement. He ignored questions and stuck to the script on the rain-spattered foolscap sheet in his hand. Ashwell said, 'Mrs Bilocca's name was put forward with nine others for the chairmanship of the new association and Mrs Denness and Mrs Smallbone have been elected as joint chairwomen of the new group. The remainder of the women, who were put forward, including Mrs Bilocca are now ordinary committee members.'\n\nOne of the reporters shouted, 'So you've all ditched Lily then?'\n\nAshwell was brief. 'There was no opinions expressed, so it would not have mattered if you lads were in there or not.' Lily said nothing but she looked shaken.\n\nMary took over and said, 'It was an excellent meeting and was very orderly. There were some good points with regard to welfare and social activities. We have started the ball rolling for better welfare and social facilities for the fishermen's families and it is now time we stepped into the background and let the men take over with regard to safety.'\n\nChrissie added, 'It was me who proposed the social and welfare work idea. I suggested it to a few of the fishermen's wives. Of course we are still concerned with the safety of our menfolk. But we have decided to leave that to the union representatives.'\n\nLily had to be coaxed into talking. The media pack crouched around her and her headscarfed face was intermittently illuminated as the reporters unleashed their questions. Lily composed herself. Her reply was brief and not too convincing when it came. For the first time the newspapermen heard her voice quiver. She said, 'Everything went off lovely lads. I spoke my mind but I am not going to say what I said.'\n\nShe either had nothing else to say or had been told to say nothing else. The crestfallen fighting fishwife cut a sad figure as she pushed her way through the reporters' scrum, down the stairs and onto to the pavement, where she tightened her headscarf against the drizzling sleet then walked off alone down George Street to get her bus home to Hessle Road.\n\nHer bus dropped her off opposite The Criterion. She could hear the shouts of the Friday night revellers drift across the Road. Normally it would have drawn her in. Fridays in The Cri were the highlight of her week, but not that night. She walked the few hundred yards towards home and barely acknowledged the handful of 'hi-ya Lils' along the way. She trudged on in a world of her own. She felt angry, hurt and betrayed. Lily turned the corner into Coltman Street and became aware in her peripheral vision of a figure approaching. It was difficult to see under the sodium street lamps as the snowflakes swirled in the light like the murmuration of starlings. Lily strained to see. It was a slightly built slip of a lad. He struggled with a kit bag that was almost as tall as he. He then slowed and stopped an arm's length from Lily.\n\n'Hello, Mrs Bilocca,' the boy said nervously.\n\n'Hello, lad,' said Lily, smiling at the timid young man.\n\n'I saw you on t' telly.'\n\n'Did ye now?'\n\nHe explained he was a galley boy on his way to his trawler. It was obvious he was scared. It was also obvious he was doing his best to hide it.\n\n'Your first trip is it, lad?'\n\n'Aye, missus.'\n\n'Well you make sure to take care son, and don't go on no ship that don't have a radio operator and proper gear, you hear me? If you get there and if it ain't right, you get back off.' Realising she might be adding to the boy's anguish, she added, 'You'll be all right, lad. Don't you worry...now come 'ere!'\n\nWith that, Lily's hug lifted the boy off his feet. Her booming laugh shook the galley boy in her bear-like embrace. Upon his release he stammered his goodbyes. His blushes were clear, even under the street lamps. When the boy walked away, Lily reached for her big white hankie and dabbed at her smudged mascara and walked on, still laughing, towards No. 117. It was the first time she had laughed in days.\n\nThat weekend was unusually quiet for the Biloccas \u2013 there were no nights out on Hessle Road for them. Lily kept herself busy. She was quieter than usual too and somewhat subdued. Charlie and Virginia knew better than to quiz her too much. It was obvious that she was hurt and angry. They gave her space. The girl and her father were used to Lily keeping things to herself.\n\nOn Sunday, Lily took advantage of the peace and quiet to get some work done while Charlie and the girl were at Mass. She was not going to let these buggers get away with it. At her big kitchen table she laid out her writing pad and pens, picked one up and wrote the heading, 'Clear my name.'\n\nFor the next couple of hours she laid out her reactions to the Press, how she felt cheated and that she demanded an apology. She read and re-read her notes and then got out her letter-writing kit. On the top of her Lion Brand vellum paper she wrote, Mr James Johnson MP, c\/o The House of Commons, London and dated it. A few drafts later and she was out on Hessle Road to post her letter.\n\nEarly on Monday morning, there was a press conference of sorts in Lily's kitchen. There was only one reporter and no photographers. The Hull Daily Mail's Stuart Russell did not have to coax her to speak. She barely glanced at her notes as she fired off her first volley and the keen young reporter started his notes. He was used to Lily's forthright ways. It was something he admired about her. She never swerved a question. Stuart visited her big house regularly to update the biggest story of his young life. She had grown to trust him, and he had grown to like her.\n\n'Look lad I have taken a lot of abuse in this campaign.' Lily said as she poured the tea and passed the biscuits. 'I wrote to James Johnson, the MP, after the Glasgow lecture, as soon I heard about this \u2013 and I still haven't seen the papers in which these reports appeared. But I want a public apology from all them papers concerned that said I had called for a strike. Nothing can be achieved by that sort of thing.'\n\nThe reporter asked how she thought the story came about, especially the \"love ban\" part. Lily shot back at him, 'It must have come from a false report. After the lecture in Glasgow, somebody, I think it was a newspaper reporter asked me something like, \"Don't you think you should get the wives to urge strike action?\" And I said, \"don't talk so daft.\" And later on Scottish Television I said something like, \"I don't think a strike would help matters and that sort of thing would be up to the men.\" I am going to take advice about this. I cannot let them papers get away with this. As far as the other wives are concerned my name is cleared. I went to a meeting on Friday night and Mr Neve and Mr Ashwell stood up and said they were satisfied that I had not suggested striking, and then explained it all to the wives. But I want Mr Johnson to investigate for me.'\n\n'Are you going to give up your fight?' Stuart asked.\n\n'Of course I'm bloody not.'\n\n'Have you had any more threatening letters since these reports?'\n\n'No,' she lied. 'But I'll tell you something lad, I don't want to see any more bloody national newspaper men. They'll get something more than they have bloody bargained for if they come here again.' When Stuart got back to his office in Jameson Street and called Johnson to ask what he would do for Lily, he was told, 'I will investigate on her behalf and will get her to send me all the reports in which she feels she has been misquoted and we'll take it from there.'\n\n'I would not hold my breath for that.' thought the young journalist.\nFOURTEEN\n\n_Friday, March 8, 2pm, The Holy Trinity Church, Hull's Old Town district._\n\nFrom the top of Hull's Holy Trinity Church, a blackbird's song broke the silence of the spring afternoon and it was heard clearly by the hushed throng gathered below. Its bird's eye view showed a silent crowd in front of barricades that surrounded England's biggest parish church. Mounted police were there. The side streets and alleyways that led to the square were full too. More than two thousand stood silent, while inside almost a thousand mourners packed into the biggest memorial service in the church's 668-years.\n\nThe congregation's black attire contrasted with the blaze of colour from the hundreds of floral tributes that adorned the altars. Outside, the blackbird's song was silenced by the solitary toll of a tenor bell that echoed across the historic heart of Hull's Old Town district. The crowd parted like the Red Sea to make way for the civic procession, as they had minutes earlier for the families of the fifty-eight men lost in the Triple Trawler Disaster.\n\nThe Lord Mayor, the city's MPs, the trawler owners, along with the Lord Lieutenant of Hull and the East Riding processed in with the city's chief constable and other dignitaries. A civic chaplain led them in. The muffled solo bell continued its intermittent toll as the party made its way to the front of the church. The bell's final reverberations gave way to the amplified voice of The Bishop of Hull, the Right Reverend Hubert Higgs, who came to his pulpit and began his address. 'Nothing,' he said, 'should be spared which could humanely make the fisherman's job more endurable and safe from foreseeable dangers. In such matters the Government, the public and all concerned with the fishing industry had a duty to the men involved.'\n\nHe was preaching to the converted. The congregation stood for the first hymn, Rock of Ages. They were joined in the singing of it by the thousands outside and within seconds its strains were heard across the square's approach streets as far as the city centre. The hymn pervaded the Old Town. Solicitors' clerks and office girls sat in silence and heard it through their Georgian office buildings' windows. On the city council headquarters' roof of the Guildhall and across the city flags were at half-mast.\n\nBack at the Market Place Square the thousands joined in the memorial with those inside. There were no bosses versus workers. No Hessle Roaders and \"outsiders,\" \u2013 just mourners. Around the city \u2013 and on an eerily quiet Hessle Road \u2013 shop front windows were trimmed in black tape. Many had wreaths on the door. Hundreds of terraced houses had their curtains closed. Shops and offices closed to allow staff to join the crowds in the Old Town.\n\nAt the dockside the fish houses and the Lord Line Building all had flags at half-mast also. At the pulpit Skipper Laurie Oliver OBE, made a brief reading from _Revelations 21, Verse 1-7_ that ended, 'I am the Alpha and the Omega, the beginning and the end. To the thirsty I will give water without price from the fountain of the water of life. He who conquers shall have this heritage, and I will be his God and he shall be my son.'\n\nA hymn followed and then the Bishop invoked all there to remember in silent prayer those who had died. Staff from the Royal National Mission to Deep Sea Fishermen each in turn read a name from the list of the fifty-eight who had perished. The stifled sobs punctuated the silence between each name that echoed round the medieval masonry, with the busy noises of the nearby indoor market and the hum of far-off traffic in the background. In the square the crowd was silent once more.\n\nInside Anglican, Roman Catholic and Free Church ministers took it in turn to say prayers and conduct in turns the remainder of the service. Near its conclusion, one of the ministers read out the final prayer that included, 'Let us give thanks for all God's mercies, remembering especially the survival of Harry Eddom, the mate of the Ross Cleveland.'\n\nHarry was at the back of the congregation in a gloomy corner. He was dressed in a dark suit. His head was bowed. Sunshine then streamed through the stained-glass windows and the congregation began...' _Eternal Father, strong to save_...' \u2013 the fisherman's hymn. It was as if a divine dimmer switch had been turned up. Hull Daily Mail reporter Stuart Russell spotted the miracle man in the congregation and made a mental note to try for an interview afterwards. When the young journalist next looked for him, Harry had gone with as little fuss as he had come.\n\nThousands of words were later written by hundreds of journalists for millions of readers across the world, but it was the local paper that summed up Hull and its people that day with a three-word headline, THE SAD CITY.\n\n_The Hull Daily Mail newsroom, May 10, 1968._\n\nNews editor Charles Levitt had been told that his neighbour Harry Eddom would be back at sea soon. Not only that, but he was going back as a mate with his old firm Hudson's and returning to the fierce Arctic fishing grounds where the Ross Cleveland and her crew perished just twelve weeks earlier.\n\nLevitt's source was unimpeachable. It was the paper's Shipping Correspondent. In the office he was Bob. To everyone else he was Robert W. Wellings. The grizzled, grey-bearded, short, stocky Aussie, who had settled in Hull after the Second World War spent his next twenty years winning the trust on both sides on the fishing industry in which he had become an accomplished expert.\n\nLittle happened in Hull's fishing community that this tough little ex-ANZAC did not get to know. With a voice that sounded like he had gargled with glass and a B-movie baddie look due to his left eye being damaged and looking like it was made from glass, Wellings had a countenance that belied his good humoured joie de vivre. His contacts' book was as big as a King James Bible. He was equally at home in the boardrooms of the dockside as he was in its barrooms. In both he showed off his proud Scots heritage by imbibing the top export of his mother's native land with gusto. He never missed a social event, be it when he was fully clad in his kilt, complete with skean-dhu for the Silver Cod dinner, or in his tweed jacket, check shirt and spotted bowtie for the bobbers' snooker tournament at the St Andrew's Club.\n\nIn the office his desk was some ten feet from Levitt's. Wellings sat under a permanent fug caused by his interchangeable smoking habits \u2013 pipe when writing \u2013 Senior Service un-tipped when on the phone. Given that when he was not on the phone or writing he was out, the nicotine-tinged cloud over the Shipping Desk let you know from a distance that Bob was in.\n\nOne newsroom wag said it was a bit like when they get a new Pope.\n\nWhen he saw Bob looming Levitt was leant back on his captain's chair perched at a precarious forty-five degrees. Levitt rested his suede shoed feet on the news desk and cradled the phone twixt chin and shoulder in the manner of a violinist. It was when that news desk phone call ended that the Shipping Correspondent told Levitt what he had heard from a contact in Hudson's.\n\nLevitt did not ask Wellings anything further. He did not need to. If Robert W. Wellings told you something it was pukka.\n\nLevitt, who lived in Cottingham near Harry Eddom, decided he would deal with the tip-off. After all, he was known to the family and was trusted. But Levitt had heard other things about Eddom, uncomfortable things that he knew he would have to address when they met. There were tales doing the rounds of the pubs and the clubs that Harry, like Big Lil, had experienced the dark side of being in the public eye.\n\nDefamation laws and common decency prevented those tales finding their way to the public domain. These were issues that Levitt planned to ask about when he met Harry. Reports had reached him that the trawlerman had suffered abuse. In the days after his miracle homecoming, Harry had been feted. He even had a police escort in public at press conferences and the like to prevent him being mobbed by well-wishers. Now the police were watching over him for a different reason. On more than one occasion the miracle man had to leave public places because of vicious abuse. The arc of hate against Eddom had no basis in fact but it took on a life of its own.\n\nFor some, the main thrust of their hate mongering was that somehow Eddom should have given his survival gear to the boy Barry Rogers, who perished alongside shipmate Wally Hewitt, in the Ross Cleveland's lifeboat. Levitt had heard reports of Harry having to sometimes contend with stage whispers in pubs when he came in, saying things like, ' _how can he be out in the boozer's enjoying himself when men have died?_ '\n\nThere were many reports too about Eddom having to leave pubs to avoid the bile directed at him. There were some who suggested that somehow Eddom survived by taking the clothes from his dead shipmates. Again the fact that experts wholly discredited this did not stop the gossips. Only Harry knew the full extent of what was endured as a result of his \"fifteen minutes of fame.\" There was never any direct confrontation or violence reported.\n\nIt was with this on his mind that the Hull Daily Mail's news editor approached the Eddom house. He knew his duty was to ask the awkward questions as well as to follow up Wellings' tip. Levitt had no truck with hateful gossip and tittle-tattle but he was a first-rate and tenacious hard news man and knew he would have some uncomfortable questions for Harry. Levitt headed up the driveway to the Eddom house, confident that the trawlerman was at home. After all, he was on sick leave. As fate had it, the newsman was spared asking Eddom anything.\n\nIt was Harry's pretty wife Rita who met the journalist on the doorstep. She was polite but brief and to the point. Levitt was not invited in as previously. It was obvious that her husband did not want to speak. But he got Wellings' tip-off confirmed. Harry was going back to sea \u2013 and he was going the next day. Levitt knew this was page-one news and he persisted until he got a quote from Rita. It paid off. The young woman, who only ninety-four days earlier feared she had been widowed, told Levitt, 'At the bottom of my heart I knew he would always go back \u2013 and I would not really stop him. But I shall always be nervous and worried although I knew when I married him that his heart is set on fishing.'\n\nBack at the office Charles made further enquiries and Hudson's office also confirmed that Harry was going back to sea as mate of the Ross Antares, and was headed for Iceland's treacherous Arctic grounds. Eighteen-year veteran Skipper Gordon Jopling, who had last been at sea with Harry sixteen years earlier when he was a spare hand and Harry was a deckie learner, commanded her. Jopling told Charles, 'I am glad to have him with me. He is a good seaman and a good shipmate.'\n\nHudson's office revealed that Harry had been in touch with them a week earlier when he told them a doctor had passed him fit for duty and he was keen to get back to work. Charles's next call was a weather check. Iceland's north and east coasts had closed all ports due to severe ice, which in spite of recent reasonable weather, meteorologists did not expect to clear until well into midsummer.\n\nNext morning Harry's regular taxi driver John Watson pulled up outside the Eddoms' house not long after five-thirty. Heavy drizzle fell. The windscreen wipers of Watson's blue and cream Austin Cambridge were on full. The driver pipped the horn.\n\nOn the doorstep Rita held back tears as she waved her husband off down the garden path. Harry sat in the rear of car, the back window of which was lined with books. He perused the car's paperback mini-library and picked out a couple as the car made its way south to the river. It was just before six-thirty when John Watson's car pulled into the fish docks. He had been Harry's driver for years and knew the routine well. After going through the entrance, he headed for the lock head but two things were out of the ordinary.\n\nFirstly, a British Transport Police escort along with two constables met the taxi and in the distance the driver could see the lock head was lined with photographers. At the dockside where the press had gathered a copper walked across and told the soaked-through reporters, 'He told us earlier he does not want any publicity or interviews, lads. He just wants to go straight to the ship.'\n\nThe media men ignored the copper. They had not come all that way in the pissing cold rain to be fobbed off. They waited for the taxi carrying Eddom to halt and for their man to get out. Watson carried on to the dockside and pulled up alongside the St Achilles, which was the ship nearest to the quayside. A police van parked up. Two coppers got out and joined Harry as he came out from the rear passenger side door of his cab. He pulled the hood of his grey anorak over his head. The two coppers either side of him kept up. The three men neared the media scrum. It was then Harry broke into a trot. The coppers still kept up. Seconds later they were in his wake.\n\nThe trawlerman turned his back on the press corps and ran. He charged along the quayside. Dozens of flashes went off and cameras whirred and press men cursed as the athletic Eddom ran head down along the quayside and wove his way through the bobbins, nets and trawler otter boards. Harry was running flat out. Running from the gossips, running from the hate, running back to that which had almost killed him. Running for the only life he knew.\n\nBehind him now were the press lads and the coppers. Harry threw his case onto to the nearby trawler and then leapt from the dockside. He landed on the working deck on the fo'c'sle of the St Achilles. Eddom's ship, the Ross Antares was alongside that trawler. Harry slowed and walked to the port side of the St Achilles. The case was thrown again onto the deck of the ship alongside. He then followed it and jumped on to the deck of the Ross Antares. He ignored the shouted questions from the press as he got to his feet. He made his way to the bridge of the ship that was to be his home for the next three weeks. In the wheelhouse Skipper Gordon Jopling, who had sailed with Harry as a boy, waited for his Mate to report for duty.\n\nBack on the dockside, desperate to ensure that the early morning caper was not a complete bust, one of the reporters decided to quiz Harry's pal, the driver John Watson. After getting his name and address from him, he asked if there was anything further he could tell them about his friend Harry Eddom. The reporters waited for a potential killer quote. Watson simply gestured to the rear window shelf of his taxi smiled and said, 'Yeah mate, he likes anything from cowboy books to Dennis Wheatley novels.'\n\nThe thoroughly pissed off reporters made their way to the phone boxes to ring in their copy.\n\nAt the Hull Daily Mail news desk, Charles Levitt took a call from the fish dock. The phone was cradled on his shoulder as he spoke. When it was over and his captain's chair was stable once more, Levitt picked up his pencil and smiled. He had his intro for that day's front-page lead if nothing else. He lifted a notebook, put the tip of the pencil to the tip of his tongue and then with a mock theatrical flourish wrote, ' _Harry Eddom went back to sea today as quickly as his sick sea legs would carry him_...'\n\nBack on the bridge of Ross Antares, First Mate Harry Eddom was back where he thought he'd never be again. His ship edged to the lock gates and the lockmaster shouted, 'Where bound?' From the open wheelhouse window, Harry Eddom shouted back with words he thought he would never say again 'Iceland...Northern Cape.'\nAFTER WORDS\n\nFive months after his return to sea Harry Eddom was in the public gaze for the final time. From October 9, 1968, the Victoria Galleries in Hull's City Hall was home to the Wreck Commissioner's Court of Inquiry. These were held in the order that the ships had perished. Wreck Commissioner Mr John Naisby QC sat with two assessors \u2013 Mr R.A. Beattie and Skipper W.F. Wright. Thousands of pages of documentary evidence were examined alongside witness testimony and statements. After three-weeks' investigation, all conclusions were issued on the same day \u2013 November 4. Eddom was closely questioned for two consecutive days.\n\nHinting at the ordeal the trawlerman had suffered after his return, after a particularly tough cross-examination lawyer Lionel Rosen told Eddom, 'I hope you do not think any questions [are] a criticism of you. Not at all. In fact, I think everybody admires your bravery and fortitude.' Harry's final public statement was an endorsement of his boss, 'Philip Gay was a very good skipper and a perfect gentleman.'\n\nHarry is now in his seventies and lives in quiet retirement north of Hull. He ended his career as a skipper of supply vessels in the Persian Gulf, a job he took in the 1980s after his wife Rita died aged just forty-two. He is still protective of his privacy and did not reply to a request to take part in this book.\n\nTwo Hull skippers, who were near the Ross Cleveland when it sank, gave evidence. The inquiries also heard incredible stories from Icelandic weathermen of how winds had been so strong that they had disgorged fish from the sea and at more than 125mph the people on shore could move only by crawling. Skipper Bryan Lee of the Kingston Garnet was in the same fjord as the Ross Cleveland and told how his vessel almost turned turtle. 'She lay down to such an extent I thought she was going to sink. I have never seen anything like it before.' When he ended his evidence, Wreck Commissioner Naisby asked if Lee was still at sea.\n\nThe skipper replied, 'I have given up the idea of going to sea.'\n\n'Because of this experience?' the Commissioner asked.\n\n'Yes,' he replied.\n\nSkipper Len Whur, who at the height of the women's protest called the resultant ban on Icelandic fishing \"stupid\" said that in all his twenty-three years at sea he had never seen weather like that which claimed the Ross Cleveland. His ship the Kingston Andalusite was less than quarter of a mile from the trawler that capsized. The court fell silent as Skipper Whur recounted the last words of Skipper Philip Gay, 'Help us Len, she's going' and seconds later, 'Give my love and crew's love to the wives and families...' Skipper Whur told how he shouted back into his radio, 'Keep her going full speed Phil and keep up with me.' The skipper said he tried to turn his ship towards the last position of the doomed trawler, but added, 'We had to turn back or we would have lost the Andalusite as well.'\n\nWhur criticised the methods for battling ice with clubs and axes as 'not good enough.' He said, 'These methods are definitely not satisfactory but these are the only means we have at the moment and the only means we have had since trawlers went fishing.' He suggested that electric heating plates fitted to parts of ship along with steam heating pipes being run along ships' rails would be an effective system. It was pointed out this was expensive. The skipper replied, 'I am not looking at the cost I am looking for safety.'\n\nWhur returned to sea days after the Triple Trawler Disaster and had a long and distinguished career.\n\nThe loss of the St Romanus remains a mystery. The inquiry concluded that it was presumed lost some time on January 11, 1968, the day Jim Wheeldon, a skipper described in court as being \"better than good\" at his job, had promised to call his wife. The court said the trawler probably sank near where the empty inflated un-marked life raft was found. Although the inquiry heard mixed reports of the ship's seaworthiness it concluded that since Lloyd's had classed her A1 for insurance purposes, then she must have been fit for purpose.\n\nIn summing up the case of the Kingston Peridot, John Naisby QC, said, 'the court is satisfied that in all probability the loss [of the Kingston Peridot] happened suddenly. It is impossible to ascertain with any degree of certainty where the loss occurred but it was probably somewhere in the area northwest of Tjornes. The time of the loss seemed to have been during the night of January 26 or the early morning of January 27.' Naisby added it was the 'duty of court to record once more its gratitude for the bravery of the volunteers of the Icelandic National Lifesaving Society.'\n\nThe Wreck Commissioner judged that the Ross Cleveland, like the Kingston Peridot, was classed as seaworthy but added that although she was \"seaworthy in construction and upkeep\" she was not stable enough to withstand the extraordinary weather, heavy seas, hurricane and heavy icing of the superstructure and had little time to clear it. Naisby said it was just luck that more ships were not lost the night that the Ross Cleveland perished. He added urgent attention should be given to de-icing technology. Naisby also said the searches for survivors were prompt and efficient and that the loss was 'not contributed to by the wrongful act of anyone.'\n\nThe launch of a new mother ship, Orsino, came shortly after the interim report of the Holland-Martin Committee of Inquiry Into Trawler Safety in September. The Inquiry had been set up in March 1968. Headed by Admiral Sir Deric Holland-Martin, it reported in full in May 1969. Holland-Martin took all the evidence and recommendations of the three Hull inquiries and investigated them in greater detail. It went beyond even the Fishermen's Charter that the women demanded. The industry was completely overhauled. The Holland-Martin Report ordered that all trawlers operating beyond Radio Telephone range of the UK should carry full wireless telegraph equipment and a qualified operator at all times. Trawlers now would not only have to report their position daily, but also have to be part of a positioning system operated by coastal stations too. Radio operators' qualifications were to be made more difficult and the GPO was told to develop beacon sounders for all ships as well as investigating the possibility of all trawlers carrying radio ID transmitters in case of loss.\n\nThe Report's demand for a Government-funded support ship was met months before the Inquiry reported fully with the launch of the Orsino. All new trawler designs were to be tank tested as scale models first, to ensure they were suitable for Arctic duty. The Report concluded:\n\nThe problem is fundamentally one of changing human attitudes at every level in the industry, and of instilling greater awareness of the importance of safety in crews, skippers and senior management alike. Fatalistic attitudes towards accidents are a particular problem. Many [...] regard the present accident rate as a more or less inevitable consequence [...] Work on small ships in the open sea will of course always have its risks; but the evidence we have heard during our inquiry has convinced us that the majority of accidents [...] arise to a greater or lesser degree from human errors and may be prevented. [ _Holland-Martin, 1968, p. iv_ ]\n\nThe majority of the Holland-Martin recommendations (88 in total) were actually in place in whole or part before the Inquiry itself first met, thanks to Lillian Bilocca and her \"Headscarf Revolutionaries\". It went beyond all expectations.\n\nIn all the evidence given to the ships' inquiries, the battling fishwives of Hessle Road were mentioned only once and that was briefly when Wreck Commissioner John Naisby QC refused Lillian Bilocca's request to address the court on the grounds that she was not, in their view, \"an interested party.\"\n\nTwenty-five days after the 1968 inquiries, the converted Hull vessel Orsino took over from the Weather Reporter as the new Government mother ship. The 1,574-ton freezer trawler set off from St Andrew's Dock for the killer Northern Cape fishing grounds. She was equipped with thousands of pounds worth of radio, meteorological and medical equipment and was under the command of Skipper Edward Wooldridge.\n\nHer crew of twenty included a doctor, medical staff, a meteorologist and three radio operators. Her first mission was for five-months. The crew had Christmas presents from their families stored aboard. (In fact, that Christmas, the full crew sent an autographed ship's chart to Mrs Bilocca, thanking her for her work and wishing her season's greetings. This was found among her belongings after her death.)\n\nDockworkers, union men, fishwives and even some trawler owners waved the Orsino off for the biggest lifesaving operation ever mounted for Britain's fishermen. The only \"headscarf revolutionary\" at the launch was Mrs Bilocca. She told reporters, 'I am delighted. More than that I am chuffed. Never mind them calling us silly women, this is what we have fought for.'\n\nMary Denness, the elegant and eloquent trawler safety campaigner, who taught herself to \"talk posh\" continued on a road less travelled. While in charge of the Trawler Women's Association with Chrissie, she forced the setting up of a new disaster fund for the mothers of boys lost in the Triple Trawler Disaster, when it was found that the city's Lord Mayor's Fund did not cover this. She and Chrissie raised thousands of pounds, before Mary left the group for Chrissie to continue. Mary had previously been one of the first female stewards in the UK with the Ellerman Wilson Line and had been housekeeper to the international Hull singing star David Whitfield. She parted from her husband Barry in 1977 and became a career woman again after bringing up three kids. In the 1980s she replied to an advert in the society magazine The Lady and re-invented herself as a public school matron. She worked for various top schools and retired from Eton College aged 65. Princes William and Harry were among her charges. Mary is now in her mid- seventies and lives in a North East Lincolnshire village. She is still in demand for talks about her life as a trawler safety campaigner.\n\nThe Golden Girl With the Golden Voice \u2013 cabaret singer Yvonne Marie \u2013 a.k.a Mrs Yvonne Blenkinsop is now in her mid-seventies, she is a widow, her beloved husband John having died in 2004. She has fifteen grandchildren and five great-grandchildren. After 1968 she continued to campaign for a couple of years but returned to the stage as a professional singer until a car crash in the mid 1970s ended her career. Over the years she has contributed as a consultant to TV, radio, theatre and documentaries about her time as a campaigner. Companies like Dunlop and Marconi also consulted with Yvonne, whose grip of facts and figures had so impressed Mary Denness when they campaigned, on safety equipment issues. Yvonne now lives in quiet retirement in her pensioners' bungalow in Hull with her little dog Mitzi, a Chihuahua\/corgi cross.\n\nWaif-like, timid wife Christine, sister to Skipper Philip Gay of the Ross Cleveland, was latterly the most successful of the \"headscarf revolutionaries\" and ended her days with an MBE in the New Year's Honours' List of 1999 \"for services to trawler safety and to her community in Hull.\" Although she missed the women's trip to Parliament because of her grief, her life changed drastically afterwards. In 1998 she recalled, 'before these trawling disasters, I didn't think I was capable of anything. I was a downtrodden housewife, mother to six, no future, no prospects. I'd have just been rambling on, [...] taking every day as it came. Then these tragedies happened. [...] I learned to stand up and fend for myself and fight for myself.' One day she just took her six kids and left.\n\nShe ran the Hull Trawler Women's Association well into the 1970s. She re-married and became Chrissie Jensen. In the 1980s she was the only female ever in the British Fishermen's Association, as well as a leading light in her Community Association and founder member of The St Andrew's Dock Heritage Park Action Group (STAND), which seeks to build a memorial to Hull's trawlermen. Lost Trawlermen's Day \u2013 also co-founded by Chrissie \u2013 is still held annually and has been for more than 25 years now. She died in 2001 aged just 62. Each year Hessle Road Reunions act as memorials to Mrs Christine Jensen, MBE. Her funeral service was on February 6, 2001 at the St John the Baptist Church, Newington \u2013 the trawlermen's church \u2013 thirty-three years to the day after her photo made international news when she was led from a Hull dockside in tears.\n\nLily Bilocca was blacklisted by the industry and she never worked in it again. MP James Johnson's \"campaign to clear her name\" was simply kicked into the long grass. After two years she was only ever able to find temporary menial jobs. Her last one was as a cloakroom attendant in a Hull nightclub. The rejection that hurt her most was not even being given an interview for a cleaning job with the Hull branch of the Royal National Mission to Deep Sea Fishermen, the Christian organisation that helps seafarers in distress. She kept the rejection letter until the day she died. The one-paragraph letter dated April 17, 1971, read, ' _Dear Mrs Bilocca, I write to thank you for visiting this office for the purpose of an interview in our request for a cleaner, and to tell you that we have now filled the vacancy.'_\n\nIt was signed by the Senior Superintendent David Saltiel and was on headed notepaper that showed that among Mission's patrons were HM The Queen, The Lord Archbishop of York \u2013 and several trawler bosses.\n\nIn 1972, Virginia, the girl that Lily hoped would never break her heart by marrying a fisherman kept that promise and left for a new life in New Zealand on her 22nd birthday with new husband Allen Singleton. Virginia held a series of top jobs, including being PA to one of her adopted country's top health entrepreneurs. She and Allen had a son, Christian. Her mother and father were very proud of her achievements and Lily visited New Zealand to see her daughter's success first hand. Proud father Charlie's fear of flying prevented him from joining her. Today, Virginia Bilocca-McKenzie lives with her third husband, Colin, in Waikanae Beach, New Zealand. She and Colin have two grown-up daughters, one grown-up son and six grandchildren between them. Now in her mid-sixties, Virginia continues to work in sales for a leading jewellery firm.\n\nLillian's son Ernie, who as a twenty-one-year-old deckhand witnessed the sinking of the Ross Cleveland, returned to fishing after the disaster. He left the sea in the mid-1970s when the industry collapsed and ironically he ended by working in fish processing as his mother had done. He worked in the Bird's Eye factory on Hessle Road, which has since closed. He is in his late sixties and lives with his wife Lillian in Hull. They have two grown-up daughters and Ernie is a grandfather. He is a very keen golfer.\n\nIn October 1975 Iceland declared a 200-mile limit. The last Cod War had begun. A year later British trawlers lost one of their traditional fishing grounds completely. At 4am on 3rd November 1975, the Arctic Raider bound for Spitzbergen had the sad distinction of being the last trawler to sail from St Andrew's Dock. She left behind a dock empty of trawlers for the first time in almost a century.\n\nThe industry that Lily fought so successfully to change for the better was soon all but gone. There is now a shopping centre where trawlers once docked. The Cod Wars that had gone on between Iceland and the UK for three decades escalated when the Icelanders enforced their two-hundred-mile exclusion zone. This coupled with soaring oil prices and the imposition of EEC fishing quotas further damaged distant water fishing. In a cruel irony the expense of all the safety measures the women had won, had added to owners' costs. With the distant grounds out of bounds and the growth of massive factory freezer trawlers combined with the \"losing\" of the Cod Wars, the Hull deep-sea fishery perished. Its demise was devastating for the men. As casual workers they received nothing. The owners were given millions in compensation for the de-commissioning of the trawler fleet, most of which were scrapped with a few sold on. It was a killer blow to Hull, as maritime historian Robb Robinson explained in, _TRAWLING \u2013 The Rise and Fall of the British Trawl Fishery_ :\n\nThe sheer pace of the British distant water fleet's decline left many of its trawlermen high and dry. Some went into North Sea trawling or coastal fisheries, others worked on oilrig standby vessels, whilst large number took jobs ashore \u2013 trawlermen being especially valued in continuous-process industries with shift work and unsociable hours. But many were left unemployed [...] A unique labour force was dispersed and the skills [...] scattered to the winds. [...] the trawlermen did not always receive the benefit of financial compensation [...]. Classed as casual workers, trawlermen with twenty or more years of service were to fight into the 1990s for redundancy payments. Rough justice for a resilient workforce, which had endured and given so much. (Robinson: 1996, pp248, 249)\n\nIn 1981, Lillian's beloved Carmelo \"Charlie\" Bilocca died in his eightieth year. The big house he spent his life continually decorating was still an unfinished project, an unfulfilled dream. After his death, it was sold and the proceeds divided among his family. In the weeks before he died, Charlie was still a regular sight on Hessle Road riding his trusty racing bike and was still wheeling and dealing. He had never lost his eye for a bargain. Charlie worked up until shortly before his death. In his late 70s he was the pot man (glass collector) in his favourite pub The Alexandra Hotel, Hessle Road. He was paid in beer and baccy. (No tax.) After his passing Lily left the big house Carmelo had spent his life working on and moved to a council flat.\n\nLily's final years were not happy. To the outside world she kept up a cheerful fa\u00e7ade. She was still happy-go-lucky Big Lil, a regular at The Cri and Rayners. She suffered ill health but kept it secret. She had never really got over her treatment by the media and despite her brave front, never shook off the feeling that those comrades closest to her had abandoned her and that her courageous fight had been overlooked.\n\nShe lived in obscurity for the remainder of her days. In 1985 she married Anthony Colby Ashton, a paint sprayer 14 years' her junior. It was not a happy union and her children did not approve. In fact, daughter Virginia wanted her mother to live with her in New Zealand. But that was not to be; Lily died of peritoneal cancer aged just 59, in 1988. Her daughter Virginia flew in from New Zealand to be at her mother's side.\n\nLily's last request was for Virginia to go to Hull Market and buy a dozen silk handkerchiefs. She described them detail and gave her daughter the money with the final order to ' _not bother wi' none o' that fancy wrapping paper. Just put 'em in a brown paper bag and I'll sort it out from there_.' Virginia did as she was told as she always had done and delivered the bag to her mother's hospital room that afternoon. Next day Lily died. Virginia and her brother Ernie arrived too late.\n\nVirginia noticed the empty brown paper bag by her mother's bedside. Hours before her death Lily had handed out handkerchiefs to all who had looked after her, from the doctors to the cleaners and tea ladies. In spite of dying relatively young, ironically, Lily outlived the industry she fought so hard to improve. Her funeral was attended by a tiny fraction of the hundreds she once swayed with her powerful oratory.\n\nShe is buried beside her beloved Carmelo.\n\nIn 1990 Hull city council placed a plaque on the site of the old Victoria Hall, inscribed: 'In recognition of the contributions to the fishing industry by the women of Hessle Road, led by Lillian Bilocca, who successfully campaigned for better safety measures following the loss of three Hull trawlers in 1968.'\n\nIn 1997, the Labour Government finally got the fishermen what they were owed, thanks to a campaign backed by the Hull West and Hessle MP Alan Johnson, who still represents the constituency.\n\nLily's Headscarf Revolution may have been a na\u00efve one. But it was a powerful action from the heart that caught the imagination of the world and shamed an industry and a Government into action. Hands that rocked the cradle shook the world and changed it for the better. As Lily once said, 'I am not a clever woman but by God, I told people something \u2013 and I would do so again.'\n\nIn May 2013, the \"Biographers' Bible\" \u2013 the _Oxford Dictionary of National Biography_ \u2013 recorded its entry for Lillian Bilocca alongside the likes of Churchill, Newton and Brunel.\n\nAlthough the fishing industry has gone and many of its ex-workers dispersed to council estates and beyond, Hessle Road is still a busy thoroughfare whose folk come back regularly, like grown children checking in on an elderly relative.\n\nIts shops, pubs and clubs still thrive, obviously not to the same extent as the sixties, but you can still find an ex-fisherman in Rayner's, The Cri or The Alex. They all still remember Lily.\nACKNOWLEDGEMENTS\n\nHull University's Professor of Creative Writing, Martin Goodman was my Alpha and Omega in this work. As my principal supervisor he guided me from the early stages of my PhD research through to its completion. He not only is a great teacher but also a fine editor. Thanks also for the assistance of the distinguished maritime historian, Prof. David Starkey, of the University of Hull, who was my second supervisor.\n\nMy gratitude also goes to local historian Dr Alec Gill MBE who gave me full access to his files. Not only that, he gave me full copies, including invaluable personal correspondence from the late Mrs Bilocca. Another Hull historian, Dr Robb Robinson, was always ready to help and did so often. Playwright Rupert Creed, kindly gave me access to transcripts of his 1998 community oral history project, some parts of which were not used by him. His resultant book, _Turning The Tide_ and his play _Northern Trawl_ are also referenced in parts in this book. Many thanks go to Dr Judy Burg, and the Hull History Centre staff, also the Hull Daily Mail [especially its picture editor Mr Jim Mitchell, for the provision of photos for this book]. I am also grateful to the Royal National Mission to Deep Sea Fishermen, Hull and many of the Hessle Road community who gave of their time and knowledge. Former Hull Daily Mail journalists Stuart Russell and Derek Hilton made time for interviews, phone calls and chats in the writing of this. I was also lucky to have known and worked with them as well as with the late Hull freelances Gilbert Johnson [ex-Sun in the 1960s and 70s] and James Goodrick, both of whom shared stories with me over the years that proved indispensable, especially for insights into journalists and journalism gone by.\n\nThe women who campaigned with Lily Bilocca, especially Mary Denness and Yvonne Blenkinsop gave me much guidance. They were indispensable, particularly Mary, whose eloquence and elegance never failed to win my admiration. Thanks also to the family of the late Mrs Chrissie Jensen MBE, previously Chrissie Smallbone (nee Gay), another fellow campaigner of Lily's, and also to Mrs Theresa Wade [formerly Gay, widow of Philip Gay, commander of the Ross Cleveland.]\n\nThanks also to musician Mr Keith Gay, Skipper Gay's nephew \u2013 and to Mrs Margaret Godman [formerly Wilson] widow of Skipper Ray Wilson of the Kingston Peridot for their time and help.\n\nMrs Gill Long, widow of Hull the trawler man Tony Harrison who perished in the fire on the trawler St Finbarr at Christmas 1966, gave me an insight into the Hessle Road community and provision of documents and personal memorabilia that is very much appreciated.\n\nErnie and Virginia Bilocca \u2013 the children of Lillian and Charlie [Carmelo] Bilocca \u2013 played a major part in helping me reconstruct the life of their mother and father. Mrs Virginia Bilocca-McKenzie played a large role. She guided me with her mother's life, speech and character down to the smallest detail. Every piece of dialogue from Mrs Lillian Bilocca that was not from an archive was constructed with the assistance of her daughter, from her home in New Zealand, via Skype over the past two years. Without Virginia Bilocca-McKenzie the story would have lacked. Virginia's aunt, Mrs Minnie King, the octogenarian younger sister of Lillian Bilocca, and widow of the renowned Hull Skipper Dicky King, provided great insight into the working of the fish houses as well the early lives and times of Lillian and Charlie. Thanks also to Minnie's daughter Mrs Lynn Stevenson. Retired Hull Skipper Jimmy Williams, author of _Swinging The Lamp \u2013 a trawlerman's memoir_ \u2013 gave invaluable technical advice and the benefit of his vast maritime knowledge. Thanks also to Lord John Prescott, for interviews, papers and access to his own archives, Alan Johnson MP, the BBC archives for radio and film, British Pathe News, Skipper Ken Knox, and the staff of the Arctic Corsair trawler museum in Hull. Former trawlermen Graham Petrini and his late cousin Eric Petrini, who survived the St Finbarr fire in the Newfoundland waters in 1966, provided valuable insights too. Eric's wife Sue, who used to \"do Lily's hair\", added more than she knows.\n\nI am grateful also for the practical help from the University of Hull, for both its fee waiver of my PhD and my paid employment as a tutor of creative writing.\n\nA similar gratitude is owed to East Hull Community News for keeping me as their editor all these past years. The Hessle Road taxi drivers, who were once part of the fishing community, and their numerous conversations, were a special resource.\n\nThe English translation of Icelandic writer Ottar Sveinsson's book _Doom In the Deep_ about the storm that blighted his native Iceland in 1968 and its aftermath, complemented the cuttings, radio archive, newsreels and recollections of journalists like Stuart Russell (whose book, _Dark Winter_ was a great resource too), alongside the recollections of Derek Hilton, and the recently-deceased Gilbert Johnson, James Goodrick and Charles Levitt. I would like to put on record my respect and admiration for all these consummate Hull newspapermen.\n\nI learned from them all as a young man and I still can today.\n\nI would like to thank Mr Levitt's son Martin for his kindness and access to his father's diaries and files. Space prohibits me listing more. If there is anyone I have missed, then forgive me.\n\nMy wife Kathryn and my daughters Catriona and Rose have my perpetual thanks for their constant love and support, even when they are not aware of giving it.\n\nFinally, I thank once more my teacher and mentor Professor Martin Goodman, a fellow with the most apposite of surnames.\nENDNOTES\n\nONE\n\n Michael Thompson, _Fish Dock_ (Hutton Press, Hull, 1989) p47.\n\n On October 23rd, 1970 a British Road Services lorry had a 500-gallon gas tank that was part of its cargo, fractured. The resulting escaped gas caught fire badly injuring men, women and children in the Subway Street tunnel. 17 people ended up in hospital and fire crews fought to halt a second blast. ( _Fish Dock_ , p33)\n\n Jimmy Williams, _\"Arctic Corsair \u2013 A typical trawler trip in late 1950\"_ (museum booklet) p1. This little self-published mine of information is available at Hull's Arctic Corsair Trawler Museum of which Skipper Williams \u2013 who is also author of _Swinging the Lamp_ (Riverhead Books, Hull, 2013) \u2013 was a founder.\n\n John Nicklin and Patricia O'Driscoll, _Trawler Disasters 1946-1975_ (Amberley, Gloucestershire UK, 2010) p154. The book also provided technical details of other trawlers from that time.\n\n Hull Daily Mail (HDM), Oct 10, 1968. Throughout this book the assiduous reportage by the Hull Daily Mail of the 1968 inquiries into the Triple Trawler Disaster has, above all other sources of media, proven invaluable.\n\n Ship to Shore was a telephony service from the GPO.\n\n Hamling Papers lodged at Hull History Centre Papers, accessed with the kind assistance of Dr Judy Burg.\n\n The author is grateful to Dr Alec Gill MBE, local historian, for his insight and assistance. Much of the information for this aspect of the industry came from various writings by Dr Gill and chats with him, but particularly an article entitled: \"Christmas Cracker Crews\" which I commissioned from him for the now-defunct West Hull Community NEWS, (December 2010 edition) of which I was editor.\n\n Interviews with Mary Denness, a fellow campaigner of Mrs Bilocca provided vast amounts for this book, as did the interviews with Mrs Gill Long, who at the age of 20 was widowed when her husband Tony Harrison died at Christmas 1966 on the St Finbarr that caught fire off Newfoundland. Both women provided masses of material among which was the nickname for the St Romanus.\n\n Hull musician Keith Gay, who wrote the Hessle Road folk anthem, \"Settling Day\" and is the nephew of the Ross Cleveland's skipper Philip Gay, gave generously of his time and knowledge in the research for this book. Robert Dockerty was his cousin.\n\n Rupert Creed _Turning The Tide \u2013 The 1968 Trawler Tragedy and the Wives' Campaign For Safety,_ (Back Door Press, Hull, 1998), pp19-21, as a source for dialogue. The author wishes to thank Mr Creed not only for his time but also for allowing access to transcripts of interviews undertaken by him and his community history project in 1997 that resulted in that book, as well as the community play, _The Northern Trawl._\n\n The \"Otter\" board trawl is named after the Scarborough screw steam trawler, \"Otter\" which pioneered the system, which took over from \"beam trawling\" in the 1860s. It was not successful at first as it used just one board attached to the warp, as had been the case in beam trawling. Adding a board and having two warps solved this problem. The author is grateful to ex-Skipper and author Jimmy Williams for this information and for all the other details of \"day to day\" workings of a trawler, that he has been kind enough \u2013 and patient enough \u2013 to explain to me over the years.\n\n A heavy woollen sweater favoured by fishermen. In years past, these sweaters had differing patterns dependent on which port the fishermen originated from. In those days patterns were often used to identify the bodies of men washed up ashore.\n\n _Turning Tide_ , p20.\n\n Dialogue derived from _Turning Tide_ , p20\n\n _Trawler Disasters_ , p154.\n\n _Turning Tide_ , p21. Also: NB: Skipper Rob Ellis and his father Skipper Jack \"Dasher\" Ellis were both highly respected figures in the industry. Rob Ellis _Arctic Apprentice_ (Highgate Publications, Beverley, 2007) also proved very useful in the research for this story. Skipper Ellis passed away before this book was published and the author would like to put on record his admiration and respect.\n\n _Turning Tide_ , p21.\n\n Pickering Park given to city by trawler owner and philanthropist Sir Christopher Pickering.\n\n Wick Radio was the GPO operated coastal radio station that trafficked messages for shipping at sea.\n\n Mrs Wheeldon's dialogue from inquiry reports, HDM, Oct 10 and _Turning Tide_ , p23.\n\n Stuart Russell _Dark Winter: The Story of the Hull Triple Trawler Disaster_ (Hull Daily Mail Publications, 1997) pp14-15; and inquiry reports from the HDM archive at Hull History Centre. Russell is also a former colleague of the author who has given generously of his time and knowledge.\n\n Hamling Papers.\n\n HDM Jan 25, 1968, and _Turning Tide_ , p25.\n\n A PAN message is a radio warning sent to all shipping in a specific area alerting it to a potentially missing vessel or other matters to be on the lookout for, i.e. wreckages, ditched aircraft, life rafts etc.\n\n HDM, January 25, 1968.\n\nTWO\n\n The author would like to thank Mrs Margaret Godman (formerly Mrs Margaret Wilson, widow of Skipper Ray Wilson) and her daughter Kathryn for their kindness and time in the recounting of their personal tragedy to assist the writer to tell this story fully.\n\n \"Don\" is the vernacular \"courtesy title\" given to trawler skippers who were not only regarded as having great skill, but also had a proven track record in making massive profits for their owners, and in turn for themselves. Many held the \"Silver Cod\" trophy given yearly to the skipper responsible for the highest aggregate annual catch\n\n'When the Minister of Agriculture and Fisheries handed the Silver Cod Trophy [...] to Skipper Walter Lewis of Hull [...] he was confronted by a man with a markedly higher income than his own.' (Jeremy Tunstall, _The Fishermen, The Sociology of an Extreme Occupation_ MacGibbon and Kee, London, 1962. p.180)\n\n \"Sked\" is an abbreviation for \"schedule ship\" and is the name given to the ship in a trawling fleet responsible for ensuring all movements are reported to the owners.\n\n The author is again grateful to Skippers Jimmy Williams and Ken Knox for their recollections of the late Captain Peter Harvey and his eccentric yet effective approach to teaching.\n\n The late skipper Rob Ellis' description of a trawler capsizing is referenced in part here. ( _Turning Tide_ , p. 35.)\n\n The author would like to thank Mr Ernie Bilocca, a former deckhand on the Kingston Andalusite, who survived the storm that claimed the Ross Cleveland, and who had sailed on many other trawlers, for checking the accuracy of description of sea conditions outlined.\n\n The Rudd's service is detailed in _Fish Dock_ p85.\n\n Alec Gill, Superstition \u2013 _Folk Magic in the Hull Fishing Community_ (Beverley: Hutton Press, 1993), p116.\n\n From HDM Public Inquiry reports October 23, 24, and _Turning Tide_ , p. 35.\n\n _Dark Winter_ , p. 24 \u2013 and HDM inquiry reports Oct 1968.\n\n A part of dialogue derived from _Turning Tide_ , p. 35, in which HDM report from October 23, 1968 is also quoted.\n\n Some of the air-sea search details from _Dark Winter_ p. 22 \u2013 and from interviews with Stuart Russell, a contemporary local reporter and former colleague of this writer.\n\nTHREE\n\n Particular thanks to veteran Labour politician and former Deputy Prime Minister Lord John Prescott, for his assistance, access to his private papers and the gift of many issues of the local radical trade union journal, Humberside Voice, from his own archive. He has also been extremely generous with his time. His recollections of his ex-colleagues and their work have proved priceless.\n\n Interview with Margaret Godman (formerly Wilson), also HDM January 30, February 1 and 2. The author is also grateful to ex-trawlerman Graham Petrini, among many others for their memories of Rayner's pub and characters such as Pigeon Smith. Also _Turning Tide_ , pp 33-40, alongside transcripts provided by Mr Creed, coupled with newspaper reports, contributed to much of the dialogue employed in this chapter, especially that attributed to Mr Fred Wheeldon.\n\n The interviews granted by Mrs Theresa Wade (previously Gay, widow of the Ross Cleveland's skipper) provided the information for this and other aspects of the book. It is the first and only time that Mrs Wade, who is now in her eighties, has spoken of her experiences. Similarly with Mrs Margaret Godman. The author is profoundly grateful to both.\n\n Ottar Sveinsson, _Doom in the Deep \u2013 An Extraordinary Storm, a Miraculous Survival_ (The Lyons Press, Guildford, Connecticut, USA, 2004) p14.\n\n Further details of their early life together from interview with Mrs Theresa Wade (formerly Gay).\n\nFOUR\n\n From interviews with Virginia McKenzie (n\u00e9e: Bilocca).\n\n From the archive of Dr Alec Gill MBE, Letters that were not sent but kept by Mrs Bilocca and recovered by Dr Gill. (Similarly, the Wilkinson letter and petition forms.)\n\n From interview with Yvonne Blenkinsop, Mrs Bilocca's co-campaigner.\n\n HDM Jan 31, 1968, p5.\n\n Barbara Robinson, _Hull Daily Mail: A Part of the Community_ (Hull Daily Mail Publications, 2010). A section of this small booklet entitled \"The Mister from the Mail\" is dedicated to the ethical approach of local journalists employed by HDM, with special reference to the fishing community, that was insisted on by management, in stark contrast to some national press colleagues.\n\n HDM Jan 31, 1968, p1 and p5 and author's interview with Mrs Margaret Godman.\n\n HDM, Feb 2, 1968, _Turning Tide_ , p46.\n\n Arthur G. Credland, _North Sea Incident \u2013 21-22 October 1904_ (Hull, History Press, 2004) is a local history booklet that provides a fine narrative of the Russian Outrage.\n\n From BBC Nation Film archive accessed online. \u2013 March and September 2013. (<http:\/\/www.bbc.co.uk\/nationonfilm\/topics\/fishing\/>) \u2013 accessed March 22, 2013.\n\n _Dark Winter_ , p31, Stuart Russell's description of Victoria Hall on the night from both his book and a series of interviews are partly used here in constructing the scene at the venue of some of Mrs Bilocca's speeches. Mary Denness and Yvonne Blenkinsop were also of great assistance with the descriptions of the hall.\n\nFIVE\n\n HDM, Feb 1 & 2, 1968; _Dark Winter_ , p30-32, also interview with Stuart Russell and information from _Turning Tide_ , pp41-48.\n\n _Superstitions_ , pp34-41.\n\n From my recorded interview(s) with Mrs Bilocca's co-campaigner Yvonne Blenkinsop. The fishwives' dialogue is derived from Yvonne's recollection. Given the passage of time, it is as accurate a pr\u00e9cis as possible.\n\n From interview with Gill Long, widow of Tony Harrison who perished on the St Finbarr, Christmas 1966.\n\n Dialogue from HDM, p5, Feb 3, 1968. Also from interview with Yvonne Blenkinsop \u2013 and from p3, _Dark Winter_\n\n From interviews with Mrs Mary Denness.\n\n HDM, Feb 3, 1968 p5\n\n HDM, Feb 3, 1968 p5.\n\n From my interview with Yvonne Blenkinsop, plus referenced in _Turning Tide_ , p76.\n\n From interview with Stuart Russell, also briefly recounted in _Dark Winter_.\n\n HDM, p1, Saturday, Feb 3, 1968.\n\n The light relief of the Ross Aquila story is recorded in _Turning Tide_ , p47.\n\n In fact the St Keverne did have a radio operator who was taken by motor launch to the ship which had left without a wireless operator in order not to miss the tide as the waters dropped. This news was later given to the Press and Lily and the others heard of it on the dockside. In her later interview, Lily said she had 'saved one ship going without a radio operator,' in this she was actually unwittingly wrong as it was not her direct action which led to this as it was the company's intention for the wireless man to join the ship via the aforementioned motor launch and this happened later in the day of the dockside protest. In spite of this there can be no doubt that the dockside protest was still a powerful part of the women's campaign that won them almost instant worldwide notice.\n\n Dialogue from the BBC Nation on Film archive online, [Protestors, 1968.] <http:\/\/www.bbc.co.uk\/nationonfilm\/topics\/fishing\/> \u2013 accessed March 22, 2013.\n\nSIX\n\n Ralph Barker, Sunday Express, July 10, 1977. Ralph Barker, a Sunday Express features writer, wrote an excellent double-page spread in which he had obviously used a vast amount of cuttings as well as his own interview. The result was a masterful feature of its time and kind, for which the old Sunday Express under Sir John Junor was famed. This story did not appear on an anniversary or some such thing. It simply appeared because someone in the features' department thought it a good idea, as was the way back then on that paper. For this eccentricity I am grateful. As was the style of the time, there was no headline as such, just a \"pulled out\" dramatic quote. This feature together with the archive from the 1968 inquiries, and numerous other cuttings from the time, along with, yet again more detail from interviews with Messrs. Russell and Hilton from the HDM, were the basis for most of this dramatic chapter. Again I would like to put on record my admiration for the work of these journalists.\n\n Barker, Express\n\n _Turning Tide_ , p53.\n\n This was recorded in HDM coverage as well as Barker, Express and cuttings from the Daily Express from Feb 4- 9\n\n _Dark Winter_ , p74.\n\n _Dark Winter_ , p74\n\n Barker, Express, _Dark Winter_ , p74\n\n _Turning Tide_ , p67, Daily Sketch, p1, Feb 6 and HDM cuttings Feb 4-10.\n\n _Doom In the Deep \u2013 An Extraordinary Storm, a Miraculous Survival,_ by Ottar Sveinsson. (The Lyons Press, Guildford, Connecticut, USA, 2004.), p10 \u2013 also Barker, Express.\n\n _Doom_ , p70\n\n Barker, Express\n\n Barker, Express James Goodrick, 1977 BBC interview: <http:\/\/www.bbc.co.uk\/humber\/realmedia\/audio\/trawler\/protest.ram> \\- accessed September 30, 2011\n\n _Doom_ , pp117-130 \u2013 for additional details of the Icelandic farm family in this chapter and additional details of Eddom's ordeal, matched with news archive and UK interviews.\n\n Barker, Express\n\n _Doom_ , pp123, 124\n\nSEVEN\n\n _Turning Tide_ , p58, plus this author's interviews with Gill Long and Sue Petrini (former Doris Forrest employee), and Rupert Creed's transcripts from his 1997 community history project are referenced here. Also papers from the archives of Dr Alec Gill, with the \"What I Want\" transcript by Mrs. Bilocca.\n\n Transcript file T2 & 3S (WPS file) from archive of community writer Rupert Creed, provided the dialogue for 'the reverie' part of the section from 1995 interviews with the late Christine Jensen MBE (Chrissie Smallbone) from her own recollection of the day. Some of which not used in _Turning Tide_. These were further confirmed and added to by author's interviews with Mary Denness in 2012, 2013 and 2014.\n\n Retired skipper Jimmy Williams' interview with author.\n\n BBC Nation on Film Archive <http:\/\/www.bbc.co.uk\/nationonfilm\/topics\/fishing\/> accessed online, March 22, 2013.\n\n HDM, p1, Feb 5, 1968, Daily Mirror, Tuesday, Feb 6, 1968 p1, Daily Express, p1, Feb 6. In the HDM Howbrigg the cook spoke of a \"16-year-old lad.\" He was mistaken, the lad, galley boy Michael Barnes, was in fact just 15 years old.\n\n HDM p1, p5, Feb 5,\n\n _Turning Tide_ , p55, 56 and Daily Mail, p2, Feb 6, 1968\n\n Daily Express, p1, Feb 6.\n\n Copy of the telegram message made available to the author by Dr Alec Gill MBE\n\n HDM, p1, Feb 5, 1968. _Dark Winter_ , p63, _Turning Tide_ , p55.\n\n HDM, p1, Feb 6, 1968.\n\nEIGHT\n\n This dialogue was taken directly from recordings made by the author from the BBC Radio Humberside Triple Trawler Disaster archive. The reporter James Goodrick was a doyen of northern journalists from 1940 until his death in the early 2000s. He was a respected colleague from whom the author learned much. Interviews with safety campaigner Mary Denness and Yvonne Blenkinsop added to the archive dialogue.\n\n Rescue details and part of the dialogue derived from Doom pp124-27, Barker, Express and Daily Express p1, February 7, 1968.\n\n Cod Wars was as the name suggests conflicts between Icelandic and British trawlermen from the 1950s until the mid-1970s. There were intermittent flare-ups over the years ranging from trawlers ramming each other to full-scale naval intervention. Finally the Icelanders ended the deep-sea northern trawl for Hull by implementing a 200-mile limit. _Doom_ refers to arrests on p130.\n\n Vernacular for wife\/girlfriend\n\n Ernie Bilocca recounted this part of the story in an interview recorded by the author.\n\n Daily Express, p1, February 7, 1968.\n\n The Daily Express's Alan Bennett in his world exclusive story quoted the un-named fisherman as saying \"Oh my God, it's Harry Eddom.\" Deckhand Ernie Bilocca, Lily's son told this author the language was likely to be stronger from a Hull trawlerman!\n\n Mary Denness, Mrs Bilocca's co-campaigner, recounted this during one of many interviews and chats with the author.\n\n _Dark Winter_ , p58.\n\n Mary Denness recalled this in an interview with the author and previously told the story for _Turning Tide_ , p62.\n\n From an interview with Yvonne Blenkinsop, safety campaigner.\n\n HDM Feb 7. \"Independent Inquiry Promised\".\n\n The author is grateful yet again to Stuart Russell and his colleague Derek Hilton both of whom spent time recalling events with the writer. Details of the Hull Daily Mail newsroom as run by Charles Levitt are detailed in Russell's book _Dark Winter_ and added to and enhanced by various interviews with Messrs. Russell and Hilton over the years. Also the author also worked there as a young man. The writer is also grateful to the family of Charles Levitt, especially his son Martin, who allowed the author access to his father's cuttings and files. The author would also wish to record his admiration for the work and career of Charles Levitt, a consummate newspaperman.\n\nNINE\n\n The dialogue and descriptions for this part of the story come from interviews with Mrs Virginia Bilocca-McKenzie, Lily's daughter. I am grateful for the time Mrs Bilocca McKenzie has spent with me over the years, helping to build the picture of her mother and tell her story with as much accuracy as we could. This chapter's dialogue was checked with Mrs Bilocca McKenzie on April 13, 2014 via Skype, one of many such exchanges across three years. Content of some of the poison-pen letters was taken from copies that were kindly provided to the author by local historian Dr Alec Gill MBE. Some of the many death threats and Mrs Bilocca's reactions to those she made public are also referenced in HDM, p1, Feb 7.\n\n HDM, Feb 7, p1.\n\n _Dark Winter_ , p72, _Doom_ , pp137-150. The author is pleased to reference both the parts noted of these books as primary sources in parts of this chapter, alongside cuttings from the Daily Express, Daily Mirror, Sun, and HDM, from February 6-10. Also I would like reference the many discussions I have had over the years with journalists involved at the time who helped more than they will know in the construction of this story. Reporters like the late George Hill of the Daily Express and Gilbert Johnson, formerly of the Sun, Daily Herald and Daily Sketch, who along with Hill spoke often to me of the big trawler story and the lengths to which they and their colleagues would go to get the story in the hey-day of the print media. Those were bar-room tales that did not go to waste. And are recalled in tribute to the memory of both men.\n\n Daily Express, Friday, February 9, 1968, p7.\n\n _Doom_ , p151.\n\n _Doom_ , p153.\n\nTEN\n\n Daily Mirror, February 8. The details of Lily's clothing etc., are from interviews with her daughter Virginia.\n\n Daily Mail, February 8.\n\n HDM, Feb 8, p1.\n\n HDM p1, Feb 8.\n\n HDM, Feb 9.\n\n HDM p1, Feb 9.\n\n ABC TV invoices were part of the papers made available to the author by Dr Alec Gill MBE, local writer and historian. Also the hate mail letters and correspondence from Eamonn Andrews were part of the materials kindly made available by Dr Gill.\n\n From interviews with Mrs Mary Denness.\n\n Roberts was of course incorrect. The women never went to Downing Street but perhaps his anger excuses his inaccuracy. HDM, Feb 13.\n\n HDM, p1, Feb 13.\n\n Thanks to the Gentleman Ranters website, and my own memories of stories about Stanley, also a cutting from the Daily Express, February 14. Also the author would like to thank the Bilocca family, especially Virginia, Ernie and Lily's sister Mrs Minnie King for biographical details about Charlie Bilocca.\n\n The author is grateful to Dr Alec Gill MBE for allowing him to copy the documents from the estate of Mrs Bilocca that came into his possession. The author also thanks Mrs Virginia Bilocca-McKenzie, Lily's daughter, once more for her assistance in the writing of this part of the story and others too. Some of the dialogue and other details were discussed, agreed and checked with Virginia Bilocca-McKenzie on June 1, 2014.\n\nELEVEN\n\n Pathe News reel and HDM, p1, Feb 15. HDM, P1, February 16. News reel, <http:\/\/www.hulldailymail.co.uk\/British-Path-YouTube-Historic-Hull-films-include\/story-20983830-detail\/story.html> \u2013 accessed via HDM website accessed September 20, 2013.\n\n From an interview in 2012 with Mrs Theresa Wade (n\u00e9e Dony) formerly Theresa Gay, widow of the Ross Cleveland skipper. Her contribution to this book is the only time she has spoken of the disaster and its aftermath. She only granted me an interview because I was a friend of her niece, Ms Jacqui Gay. I am grateful to them both, particularly Mrs Wade for going through an emotional time in speaking of these times and her life with Skipper Phil Gay.\n\n HDM, Feb 15.\n\n HDM Letters' Pages, Feb 13- 15.\n\n The dialogue here cannot be precise and is drawn from Virginia's memory of events, coupled with recalling her father's mannerism and favourite expressions.\n\nTWELVE\n\n A form of darts game, in which the first of two to score 501 and end on a double number wins. Very popular in the North of England then \u2013 and now.\n\n Dialogue and background for this section drawn from HDM, Feb 19, 1968, Glasgow Evening Citizen, Feb 19, 1968; Glasgow Evening Times, Feb 19 1968, and The Glasgow Herald, Feb 20, 1968, as well as the February 1968 edition of the University student newspaper, the Strathclyde Telegraph.\n\n Daily Express, p7, February 20, 1968.\n\n Glasgow Herald, November 22, 1967.\n\n Photo referred to in the Glasgow Evening Citizen, February 20, 1968.\n\n Glasgow Evening Times, Feb 19, 1968.\n\nTHIRTEEN\n\n HDM, Feb 21, 1968, \"Big Lil in Hot Water with Backers\" provided part of the dialogue of this section, along with my own interviews with Mrs Yvonne Blenkinsop in 2013. [And the pressmen were correct, Yvonne's final quote never made the public print, until now!] I am also grateful to note that Transcript file T1.WPS from the archive of Hull-based community playwright Rupert Creed was also referenced in this part of the chapter. I also had further interviews on 9\/7\/2014 with Mrs Mary Denness and 8\/7\/2014 with Yvonne Blenkinsop to verify the above and to ensure nothing was missed. The writer would like to record his gratitude to both women for their patient help across the years.\n\n HDM, Feb 21, 1968 \"Big Lil in Hot Water.\"\n\n The Times, p2, Feb 21, 1968.\n\n HDM, Feb 22, 1968 \"Lil faces her biggest battle.\"\n\n HDM, February 22, 1968.\n\n \"Big Lil Counted Out\", HDM, February 24, 1968.\n\n HDM, February 28, 1968 also from interviews with Stuart Russell, former Hull Daily Mail reporter and assistant news editor.\n\nFOURTEEN\n\n _Dark Winter_ , pp80-85, and HDM, March 8, 1968.\n\n HDM, May 10, 1968.\n\n HDM, May 11, 1968.\n\n From Stuart Russell interviews.\n\nAFTER WORDS\n\n HDM, October 18, 1968.\n\n HDM, October 17, 1968.\n\n HDM, November 4, 1968.\n\n Again the author is grateful to local historian Dr Alec Gill MBE, who provided a copy of a solicitor's letter that informed Mrs Bilocca why she was not allowed to address the inquiries, after she tried to contest the Wreck Commissioner's decision. Dr Gill also provided many of the other letters referred to throughout this book.\n\n HDM, November 29, 1968.\n\n Turning Tide, p78.\n\n Fish Dock, p37.\n\n The Northern Echo, February 5, 1969. Also broadcast in the author's BBC Radio 4 programme, \"Courage and Effect\", written and presented by Brian Lavery as part of the Four Thought series, introduced by David Baddiel and transmitted in December 2013.\n\nFig: 1 - Skipper James Wheeldon (inset), the commander of the St Romanus, who described the ship as awful to his wife Janet. The last words she heard from him were in a radio telephone call that was cut short.\n\nFig: 2 - Skipper Ray Wilson (inset) The Kingston Peridot's well-loved and respected commander who was a figure well known at the St Andrew's Social Club for his jovial nature and party singing.\n\nFig: 3 - Skipper Phil Gay (inset). The Ross Cleveland skipper had a premonition he would not see his children grow up.\n\nFig: 4 - Lillian Bilocca leading her army of fishwives to the docks following her first speech at Victoria Hall.\n\nFig: 5 - Lillian Bilocca (courtesy of Dr Alec Gill MBE).\n\nFig: 6 - Living above the shop. Carmelo \"Charlie\" Bilocca sits with his son Ernie on his knee while Lily sits with daughter Virginia and the older girl was Lily's younger sister Muriel, a sickly child who sadly passed away not long after this\n\nFig: 7 - George Rutter. The St Romanus' second engineer had promised his wife he would give up the sea after this trip.\n\nFig: 8 - Robert Dockerty, the St Romanus deckie learner who turned up with ill-fitting waterproofs gifted to him by his uncle, Phil Gay, skipper of the Ross Cleveland.\n\nFig: 9 - Alf Hartley, the Kingston Peridot's second engineer. His shop girl daughter Yvonne broke down when she read of the fate of her father's ship on a city centre news bill.\n\nFig: 10 - Alan Harper. The Ross Cleveland's third hand was in the wheelhouse with Harry Eddom as the ship turned over.\n\nFig: 11 - Barry Rogers. The teenage boy saved sole survivor Harry Eddom and got him into a life raft. Hours later the boy, dressed only in his vest and pants, perished.\n\nFig: 12 - Trevor Thomson. The Ross Cleveland deckhand, who at the last minute jumped aboard the ill-fated ship to get money to wed his best girl. A telegram with a love poem on it was the last communication between them.\n\nFig: 13 - Harry Eddom in his hospital bed in Iceland.\n\nFig: 14 - Players from Hull FC Rugby League team pay respect with a minute's silence at The Boulevard.\n\nFig: 15 - Mourners on the way to the memorial at Holy Trinity Church in Hull's Old Town.\n\nFig: 16 - The congregation gathers inside the Holy Trinity Church.\n\nFig: 17 - One of hundreds of Hessle Road fundraisers to help the bereaved families of the 1968 disaster.\n\nFig: 18 - Hull Daily Mail photographer captures the moment Rita Eddom, the \"36-hour widow\" takes a call from the husband she thought had perished on the Ross Cleveland.\n\nFig: 19 - Reunited: Harry and Rita's kiss at the hospital in Iceland where Eddom was treated.\n\nFig: 20 - Harry Eddom, his wife Rita and baby Natalee reunited at a press conference at Hull's Royal Station Hotel.\n\nFig: 21 - A police escort for Harry Eddom, this time to protect him. The man who survived the Triple Trawler Disaster was subjected to abuse from some of his own community. He returned to sea fewer than three months after his remarkable escape.\n\nFig: 22 - Mary Denness 2008. Mary addresses the crowds at the Princes Quay Shopping Centre in Hull to mark the 40th anniversary of the Triple Trawler Disaster.\n\nFig: 23 - Skipper's wife Mary Denness addresses the crowds at Victoria Hall minutes after Lily saved her from the jeering of the crowds.\n\nFig: 24 - James Johnson, MP for West Hull, pictured with the Hull trawlermen's wives before their tour of the House of Commons in 1968\n\nFig: 25 - Lily Bilocca on the docks. She became a regular sight with her clip board and note book inspecting departing trawlers.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nStanford University Press\n\nStanford, California\n\n\u00a92017 by the Board of Trustees of the Leland Stanford Junior University. All rights reserved.\n\nNo part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or in any information storage or retrieval system without the prior written permission of Stanford University Press.\n\nPrinted in the United States of America on acid-free, archival-quality paper\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Corbett, Rosemary R., author.\n\nTitle: Making moderate Islam : Sufism, service, and the \"Ground Zero Mosque\" controversy \/ Rosemary R. Corbett.\n\nOther titles: RaceReligion.\n\nDescription: Stanford, California : Stanford University Press, 2017. | Series: RaceReligion | Includes bibliographical references and index.\n\nIdentifiers: LCCN 2016027701 (print) | LCCN 2016029125 (ebook) | ISBN 9780804791281 (cloth : alk. paper) | ISBN 9781503600812 (pbk. : alk. paper) | ISBN 9781503600843 (e-book)\n\nSubjects: LCSH: Sufism\u2014United States. | Islam and politics\u2014United States. | Sufis\u2014New York (State)\u2014New York. | Mosques\u2014New York (State)\u2014New York. | Voluntarism\u2014Religious aspects\u2014Islam. | Muslims\u2014Cultural assimilation\u2014United States.\n\nClassification: LCC BP188.8.U6 C67 2016 (print) | LCC BP188.8.U6 (ebook) | DDC 297.09747\/109051\u2014dc23\n\nLC record available at <https:\/\/lccn.loc.gov\/2016027701>\n\nCover design: Matt Tanner\n\nTypeset by Bruce Lundquist in 10\/14 Minion Pro\nMAKING MODERATE ISLAM\n\nSUFISM, SERVICE, AND THE \"GROUND ZERO MOSQUE\" CONTROVERSY\n\nROSEMARY R. CORBETT\n\nSTANFORD UNIVERSITY PRESS  \nSTANFORD, CALIFORNIA\n**RACE RELIGION**\n\nJohn L. Jackson Jr., David Kyuman Kim, Editors\n_For Tariq Towe, with gratitude_.\nCONTENTS\n\nAcknowledgments\n\nIntroduction: Debating Moderate Islam\n\n1. Islamic Traditions and Conservative Liberalisms\n\n2. Service, Anti-Socialism, and Contests to Represent American Muslims\n\n3. Sufism and the Moderate Islam of the New Millennium\n\n4. From Sufism without Politics to Politics without Sufism\n\n5. The Micro-Politics of Moderation\n\n6. The Prophet's Feminism: Women's Labor and Women's Leadership\n\n7. Islam in the Age of Obama: What's More American than Service?\n\nConclusion: Community Service and the Limits of Inclusion\n\nNotes\n\nBibliography\n\nIndex\nACKNOWLEDGMENTS\n\nWhile the lives of all books are many, this one has perhaps had more lives than most have prior to publication. What began as research in the years after 9\/11\u2014my first visit to New York City occurred as the fires still burned at Ground Zero, in a neighborhood that I now see daily from my window\u2014changed greatly after May of 2010. That is the month I finished the first draft of my manuscript. It is also the month that members of the community with whom I had spent six years came under international scrutiny for trying to open an Islamic community center in Lower Manhattan. It quickly became clear that the project I had spent so many years on suddenly needed to be almost entirely rewritten. And it took several years and drafts after that to feel I really understood the new story I now needed to tell.\n\nThe debts incurred while undertaking such a project are numerous, to say the least. First and foremost, of course, I am deeply obliged to the members of the Masjid al-Farah community (and to the different Sufi groups affiliated with it) with whom I spent so many years. Several community members gave me not only their time but also their trust at a terrible moment in the history of our city and of our nation, when Muslim Americans had many reasons not to trust strangers who showed up with questions. Others gave me more than that: their friendship. Particularly to those with whom I shared countless hours and heartfelt, sometimes heart-rending, conversations, I wish I could thank you by name here. You may find yourself represented in the pages of this book\u2014and I hope you do find yourself here, even though this is not the book you would have written or the way you would have written it\u2014but I have ascribed pseudonyms to almost all of you for the sake of protecting your privacy, and possibly even safety, after all that has happened since 2010.\n\nI must also express my endless gratitude to those who have patiently watched this project develop through its various incarnations\u2014some of whom toiled through far too many pages during that process. My largest debt is to Courtney Bender, who has read pieces of it at every stage and whose counsel, critiques, and friendship have been invaluable. Secondly, I owe deep gratitude to Randall Balmer, who encouraged me to undertake this project when he probably suspected that, as an Americanist, pursuing it well would turn out to be a more Herculean task than I had imagined. To Lila Abu-Lughod\u2014particularly, but not exclusively, as the director of graduate studies at the Columbia University Institute for Research on Women and Gender\u2014I also owe many thanks for years of encouragement, support, and mentorship. And to Richard Bulliet I owe my knowledge not just of Islamic history but of the history of continual interactions between peoples over the millennia\u2014Muslims, Jews, and Christians in, across, and through the regions now often talked about as \"North\" and \"South,\" \"East\" and \"West.\" Also, to Elizabeth Castelli I owe thanks for wading through that first iteration and, more recently, for helping me find a way to use the knowledge I gained during my research in the service of more direct and tangible good. All royalties from this book will go to the organization she introduced me to, the Center for Constitutional Rights (CCR). Founded in 1966 to help secure civil rights for black Americans, CCR is now at the forefront of defending the rights of Muslim New Yorkers\u2014American-born and immigrant, black and Latino, Arab and South Asian\u2014among others, even as it continues to fight for equal justice for black Americans and immigrants more broadly.\n\nI cannot sufficiently express my thanks to the dear friends who have seen me through the many years of research and beyond. Erika Dyson, Jodi Eichler-Levine, Julia Cato, Lisa Uperesa, Heather Schwartz, Abigail Kluchin, and Daniel Vaca: we are now scattered across the continents (and islands, Lisa!), but you are still as close to my heart as when we all toiled in that one little corner of Manhattan. To Caitlin Cox and James Hare, Mandy van Deven and Joel Bourdeaux, Afua Brown and Nathan Larsen: I am so thankful you've stayed in New York, where you keep me both honest and sane.\n\nOther generous, creative, and brilliant colleagues and friends have come from the world beyond New York, and these include my two co-organizers of our years-long \"Islam in\/and America\" collective, Juliane Hammer and Zareena Grewal\u2014Juliane, thank you for starting us off at Princeton so many years ago and for your gracious friendship ever since\u2014as well as Su'ad Abdul Khabeer, Zain Abdullah, Hisham Aidi, Zaheer Ali, Moustafa Bayoumi, Sylvia Chan Malik, Edward Curtis, Sohail Daulatzai, Kambiz GhaneaBassiri, Sally Howell, Sajida Jalalzai (also a Columbian!), Debra Majeed, Tim Marr, Hussein Rashid, and Nayan Shah, many of whom have also encouraged me in or collaborated with me on other projects. My thanks also to Ruth Mas, with whom I've cooperated in many ventures (and commiserated over some), and who has a mind and a level of fortitude that will forever inspire me, as well as a generous spirit. Additionally, I must thank those who encouraged this research along the way by not only discussing it with me but also publishing various drafts of it. These people include Aminah Beverly McCloud, Markus Dressler, and Finbarr Curtis.\n\nI would be more than remiss not to recognize the friends and colleagues who supported my work and my efforts to find my way during the two years I spent as a Mellon Postdoctoral Fellow at the Center for the Humanities at Tufts University. My thanks to Jonathan Wilson, the director there, for so many things\u2014among them an open door, an open heart, and an incredible wit\u2014and to Susannah Heschel for getting me to Tufts in the first place. Also, to Khalilah Tyre, Jennifer London (who often lent me a couch and a shoulder), Sasha Senderovich, Amahl Bishara, and, especially, Heather Curtis\u2014friends and colleagues but also amazing people (and, in Heather's case, an indefatigable letter writer!) I must also express my deep thanks to the colleagues and mentors I had during my fellowship as a 2013\u20132015 Young Scholar in American Religion: Courtney Bender and Bob Orsi, who made us work hard and made it all worth it, and the lovely and ingenious members of my cohort\u2014of whom I will always be in awe and will always consider friends\u2014Shelby Balik, Omri Elisha, Alison Greene, Kathleen Holscher, Hillary Kaell, David King, Anthony Petro, Josef Sorret, and John Stipes. We would not have been able to do any of the things we did together without the dedicated support and guidance of Phil Goff and Becky Vasko, for whom I am also most grateful.\n\nI received funding for this research and assistance with it at various stages and owe recognition to Columbia University for several years of Foreign Language and Area Studies Fellowships, as well as to the Columbia University Institute for Social and Economic Research and Policy for my years there as a Mellon Graduate Fellow (a position that came with funding that paid for interview transcriptions, as well as with the invaluable mentorship of the incomparable William McAlister). The American Association of University Women gave me a grant to help with the first iteration of this project, and Columbia University's Middle East Institute (now MESAS) provided me with transcripts of interviews undertaken as part of the Muslims in New York Project (1997\u20132003, portions of which are cited in Chapter 5 of this book), as well as sent me on what seemed likely to be a fool's errand but proved to be an unforgettable ethnographic experience: commenting on Fox News in an attempt to quell the hysteria over the so-called Ground Zero Mosque controversy.\n\nThe process of turning a manuscript into a book can be quite fraught in the most ordinary of cases. For their guidance in this process, I must thank Emily-Jane Cohen, my editor at Stanford University Press; Angela Roskop Erisman, a diligent copy editor who saved me from many embarrassing mistakes; and David Kyuman Kim and John Jackson, editors of the series in which this book appears. I owe a particular debt to David, who not only recruited my book for the series and advised at every step along the way but also became a great friend in the meantime. Additionally, I owe thanks to Judith Weisenfeld, for many things, really\u2014most specifically here for providing a listening ear and sage words since the moment I first sought publication of my manuscript.\n\nThere are some friends, family, and co-conspirators who don't fit in the list of usual suspects but who deserve recognition for their many years of support. For me, these include Jodi Pratt (my _yidishe mame_ , who has shared so much with me over the decades\u2014including her own experience of escaping the burning towers on 9\/11) and the late Thomas Deloy Pratt (my earliest mentor), Gary Adler (who sometimes housed me during my itinerant years in Berkeley and convinced me, for better or worse, that I wanted to pursue a doctoral degree), Teresa Williams Hooker, Jennifer Floyd Larson, Megan Roberts Hearne, Rachel Brown, Anna Kang (fast friends with whom I've shared two decades of personal and professional ups, downs, and existential questions), Michael J. Stolper (who provided invaluable support for years and was remarkably good about educating an overly bookish roommate in all manner of pop culture trivia), and Till Bender (who shared international perspectives on America and American Studies, as well as allowed me to karmically return the favor of lending someone a couch on occasion).\n\nOther mentors who fit between friends and family have given more than what's required and then some; these include Marian Ronan, David Watt (my intellectual grandfather in that academic lineage kind of way), and Laura Levitt. And my own family deserves recognition for their years of encouragement despite the fact that they did not necessarily know why I was working so hard for so long or where I was going with it. To Renette Melander, Robert Corbett, Janette Corbett, Mark Corbett, and Christie Corbett, I owe you endless love. To Kevin Barrie and Richard Woods, I also owe you a room at the inn any time you want it!\n\nFinally, my greatest thanks go to the two people who have read this current version of my work more closely than any others. First, to Rhea Rahman, who saved my sanity not only by formatting my footnotes and joining me in rants about all nature of injustice in the world\u2014challenging me to think longer and harder at many points along the way\u2014but who also, along with Annie Erkkinen, gave my little girl more devoted attention than I could have asked for in those moments when I couldn't. Finally, there are no words capacious enough to capture my debt to David Kaiser, the most devoted partner, co-parent, and line editor I could ever ask for. You've encouraged me in ways too myriad to mention and waded through some turgid prose in the meantime. Without you, this book would not be what it is.\nINTRODUCTION\n\nDebating Moderate Islam\n\nIN DECEMBER 2009, Feisal Abdul Rauf announced plans to open a thirteen-story Islamic community center in Manhattan. A prominent imam, Sufi shaykh, and the internationally recognized leader of the Cordoba Initiative (founded in 2004 to \"heal\" the divide between \"Islam and the West\"), Rauf designed Cordoba House to educate Americans about the truths Islam shares with other faiths and to exemplify the \"moderate Islam\" he had spent nearly a decade promoting, most notably in his 2004 book, _What's Right with Islam_. There, in Friday messages he had delivered at his mosque since 2001 and in public appearances sponsored by the US State Department, among others, Rauf defined Muslim moderation by translating Islamic traditions into American idioms. His primary message: Islam is part of an ethical tradition originating with Abraham (the biblical patriarch common to Judaism and Christianity) and, of all the governments in the world, American liberal democracy best embodies this ethic in social form. Because US multiculturalism, pluralism, and \"democratic capitalism\" are expressions of the \"Abrahamic\" ethic\u2014an ethic, he argues, that also characterized Cordoba, the multi-religious city of twelfth-century Spain\u2014Rauf understands US laws and institutions to comply with Islamic law (shari'ah). Consequently, non-Muslim Americans can accept Muslims as Abrahamic siblings, while Muslim Americans can promote American liberal values and social systems worldwide.\n\nBoth local leaders and international elites, ranging from Rauf's World Economic Forum colleagues to the Archbishop of Canterbury, widely praised Rauf's message of Abrahamic commonality after 9\/11. Consequently, the imam did not expect significant opposition to Cordoba House. Indeed, many religious, political, and financial leaders responded positively to the project, and a Manhattan community board gave its approval. Others, however\u2014especially politicians practiced in using fear of Islam for electoral gain\u2014denounced the center as a \"Ground Zero Mosque,\" turning it and Rauf's claims of moderation into subjects of international debate.\n\nThe controversy intensified in the summer of 2010. While Rauf was on a cultural outreach tour sponsored by the State Department, Republican Congressman Peter King claimed Rauf only posed as a moderate and should be investigated for ties to radical Islam. King was not the first public official to castigate the imam this way; he followed both Tea Party leader Mark Williams and Newt Gingrich, the former Speaker of the House of Representatives who was then an aspiring presidential candidate. Casting aspersion on the center, Gingrich argued that the medieval city of Cordoba signified not interreligious coexistence but Islamic conquest over a Christian kingdom\u2014something, he claimed, Muslims sought to repeat in the United States. Such accusations prompted Sharif El-Gamal, the project's developer and one of Rauf's Sufi dervishes, to rename the proposed community center after its street address: \"Park51.\" Still, Gingrich likened building Cordoba House to placing Nazi signs near Holocaust memorials or to erecting a Japanese cultural center near Pearl Harbor.\n\nRauf, Daisy Khan (his wife and Cordoba House codirector, who cofounded and led other organizations with Rauf), and several commentators, including _The Daily Show_ host Jon Stewart, responded to such hyperbole by comparing the situation of contemporary Muslim Americans to that of Catholic and Jewish Americans during the early twentieth century. Reiterating one of his constant themes, Rauf described overcoming nativist discrimination in the United States as part of the general \"immigrant religious experience\"\u2014a sociological process Catholics and Jews had completed, providing Muslims with a template for how to successfully Americanize while retaining core tenets of faith. Indeed, Catholics, Jews, and other religious minorities did face tremendous discrimination and opposition a century ago, some of which has abated over time. As I discuss in the following chapters, however, Rauf's optimistic appeals to history and attempts to render Islamic traditions familiar to non-Muslims replicated and obscured some of the ways earlier marginalized religious groups learned to claim belonging in the United States\u2014ways that invariably involved contrasting their own marginalized traditions with those of even less accepted populations, such as black Americans. Failing to recognize this dynamic, Rauf also failed to fully appreciate the political, economic, and racial positioning involved in his claims of moderation.\n\nThe story of assimilation, upward mobility, and moderation Rauf told before the Cordoba House controversy repeated a narrative created by earlier immigrants who emphasized the same ethics (including work ethics) as dominant white Protestants, as well as the moral obligation to engage in community service in order to assist those who fail to succeed in America's free-market system. As we shall see, this narrative has deeply racist roots and ramifications. It helped earlier generations of immigrants and marginalized religious and racial groups to prove their loyalties when they were suspected of having uncivilized mores or Communist sympathies, but it also perpetuated the fiction that \"white\" Americans (whoever is included in that category at any particular period of time) have experienced upward mobility because of their own efforts in a meritocracy, rather than because of the social capital connected to whiteness in the United States and because of the twentieth-century government-funded social welfare programs that aided whites and white ethnics but largely excluded nonwhites. In short, this argument has served, among other things, as a way for marginalized groups to distinguish their communities from dispossessed black Americans and other so-called undeserving poor\u2014or, in the case of black Americans like W. D. Mohammed (leader of the former Nation of Islam), to distinguish themselves from the ostensibly underserving poor in their midst. Not surprisingly, although many saw Rauf's message of American Abrahamic exceptionalism as quintessentially moderate, others\u2014including some Muslim Americans\u2014could not agree with it.\n\nRauf did not personally hold to the racist beliefs this older narrative perpetuates, nor did he even recognize that it perpetuates them. Still, he began to tell a somewhat different story after the Cordoba House controversy, in part because the controversy coincided with a recession so severe it made painfully apparent many of the economic and racial inequalities built into neoliberal free-market capitalism in the United States. Other aspects of Rauf's work also changed after that, including his public emphasis on combatting Muslim-led terrorism with Sufism, a body of tradition that involves not just the five daily prayers and other required Islamic practices but additional formal reflection and observance ( _dhikr_ ). The idea that Sufism is the opposite of dogmatic (some would say \"fundamentalist\") Islam is as old as the idea that the United States is a true meritocracy\u2014that is to say, it is relatively recent. Both popular and newly politicized for domestic and international purposes, this discourse of Sufi moderation is also one with racist roots and ramifications that were unapparent to Rauf when he echoed it.\n\nFor over a century, Americans\u2014following the work of British and German idealists and orientalists, among others\u2014saw the mystical habits of some \"Aryan\" (Indian and Persian) Sufi orders (or _tariqas_ , which can be Sunni, Shi'a, or a mix of both) as a welcome contrast to what they believed were the rigid attitudes of \"Semites\" (Jews and, more to the point here, Arabs\u2014especially Arab Muslims). After the Iranian Revolution and hostage crisis in the late 1970s, Persians\/Iranians were no longer regarded in the United States as moderate counterweights to ostensibly rigid Arabs. Nevertheless, in the eyes of many, Sufism retained its claim to being the practice of less doctrinaire\u2014or less \"fundamentalist\"\u2014Muslims.\n\nWhile this history continues to inform contemporary notions of Sufism as moderate Islam, it is reliant on such superficial and racializing generalities that few people would make the argument that Sufis are relatively moderate Muslims in the same terms today. Rauf has always been happy to receive any goodwill directed toward Sufis, but he does not engage this history in his writings. Instead, he presents religious moderation as a cultural issue (one with political and economic aspects), not a racial one. It is quickly apparent in Rauf's work, however\u2014particularly in the 2004 book that helped make him famous\u2014that the moderate Islam he promotes is and will be the province of more affluent Muslims. Generally, in the United States, this means Arab and South Asian immigrants rather than black Americans. Thus, while Rauf does not replicate the explicitly orientalist racial framework tied up with assumptions about Sufi moderation, his particular promotion of Sufism and community service\u2014merged, as it is, with an economic philosophy that ignores ongoing issues of racially differential treatment in this country\u2014give this older orientalist narrative new racial ramifications.\n\nAs I discuss in the chapters that follow, Sufism and community service within a neoliberal market society are essential elements in Rauf's definition of moderation. This is despite the fact that Rauf and Khan deemphasized Sufism in 2006, when Rauf began building a reputation as an international expert on Islamic law and participating in State Department outreach programs in Muslim majority regions where Sufism is often viewed with suspicion. Only after the Cordoba House controversy\u2014during which it became apparent that many Americans still see \"Sufism\" as synonymous with \"moderate Islam\"\u2014did they began to emphasize Sufism again.\n\nWhile Rauf and Khan made these changes strategically, they also made them quite sincerely. Sufism and service (which Rauf often connects to Sufism) are what bring Islam to life for them. They are also practices that bring people together despite a multitude of differences. When Rauf and Khan deemphasized Sufism, it was not simply because they no longer saw it as politically expedient, but because they no longer saw it as essential to achieving some of the other goals they had formed after 9\/11, including that of gaining for Muslims worldwide the kind of acceptance, freedom, and prosperity they enjoyed in the United States. The following account of their work and institutions is not intended as an expos\u00e9. Rather, it is a look at the history and present of ideas that inform the concept of \"moderation\" and an examination of some of the political, economic, racial, and gendered ramifications of living Islam in the ways now demanded of Muslim Americans.\n\nDefining and Defending American Muslim Moderation: A Method to the Madness\n\nAlthough Rauf was new to the role of spokesperson for moderate Muslims after 9\/11, his institutional and public leadership began long before he founded the Cordoba Initiative. When, six months after the attacks, a PBS reporter asked Rauf to explain \"the key things\" Islam, Christianity, and Judaism share, Rauf was serving in his nineteenth year as the imam of Masjid al-Farah in Lower Manhattan. He was also in his fifth year as the shaykh of a related Sufi group and as CEO of the American Sufi Muslim Association (or ASMA Society), which he cofounded with Khan in 1997 to promote _tasawwuf_ (Arabic for \"Sufism\") as authentic Islam. Although formally educated in physics rather than Islamic theology and as likely to draw income from his work in real estate as his work in religion, the fifty-four-year-old Egyptian American was also a trustee of the city's Islamic Cultural Center, an advisor to the Interfaith Center of New York, and the author of two books on Islamic practice in America.\n\nFor some time, Rauf had planned to write a third book on Sufi _dhikr_. In response to events following the 9\/11 attacks, however, he changed course. Rather than discuss Sufism in the PBS interview, Rauf described the ASMA Society as a specifically \"non-political, educational and cultural\" organization designed to improve relations \"between the American public and American Muslims.\" He responded to the reporter's question about commonality by outlining the main aspects of Abrahamic cohesion: a common ancestor, common monotheistic beliefs, and common ethics. This subject, instead of his intended treatise on Sufism, comprised the substance of his next book, in which he also described the immigrant process of gaining acceptance as one that involved acculturating by embracing free-market capitalism and creating organizations to contribute to society through various kinds of service.\n\nThe following year, Rauf and Khan (then ASMA's executive director), hosted an event to advance their vision of Abrahamic unity at St. Bartholomew's Church, an influential religious and cultural institution on Park Avenue where Rauf taught classes on Sufism. On a Sunday evening that June, well-heeled New Yorkers assembled to break bread together and watch a dramatic rendering of interreligious cooperation and progress. The Cordoba Bread Fest focused on the harmonious history and spirit of twelfth-century al-Andalus (Muslim Spain) that Rauf and Khan hoped to inculcate in the United States through the ASMA Society and also to disseminate worldwide. Enjoying broad institutional and financial support (a host of religious, cultural, political, and financial luminaries gathered for the event), the Cordoba Bread Fest also served as a public platform for promoting their newest venture, the Cordoba Initiative. In contrast to ASMA's domestic educational and cultural work, the Cordoba Initiative was designed to be an international policy-oriented organization that would, among other things, show how American Muslims could encourage Muslims elsewhere to overcome fundamentalism by adopting the \"shariah-compliant\" American frameworks that (according to Rauf) foster social progress: democratic capitalism and a religiously informed, yet officially secular, state law.\n\nAs we shall see, both ASMA and Cordoba evolved significantly during the first decade after 9\/11. Yet throughout that time, Rauf and Khan reiterated the Abrahamic narrative that Rauf had outlined at the 2003 Cordoba Bread Fest, elaborated on in his 2004 book, and repeated during hundreds of international speaking engagement _s_. Closely examining the rendition of history in Rauf's 2004 book, as I do in the following chapters, reveals what many accounts of the 2010 mosque controversy missed: the precarious political stance\u2014neither fully \"liberal,\" as the term is commonly understood, nor politically conservative\u2014Rauf took as he sought to define the middle way of moderation. Foregrounding the shared intellectual history of the many positions Rauf shares with Newt Gingrich also illuminates a deeper history that the imam's promotion of liberal inclusion omits: how many religious minorities\u2014often immigrants caught between the American racial categories of \"black\" and \"white\"\u2014lobbied for acceptance by echoing and adapting dominant white Protestant narratives of American meritocracy and exceptionalism, further marginalizing black Americans in the process.\n\nThe intertwined political, economic, racial, and gendered contours of moderate Islam\u2014at least, as Rauf and Khan presented them during the first decade after 9\/11\u2014are primary subjects of this book, but they are not its only focus. In addition to examining the ideas of Rauf and Khan and the ways their institutions grew and changed, I reveal how Muslims at Rauf's mosque responded to the work he and Khan were doing while they simultaneously attempted to live authentic Muslim lives\u2014\"balanced\" ones, they often said; rarely did any use the term \"moderate\"\u2014after 9\/11. Many of these Muslims wanted to believe Rauf's arguments about Islam's compatibility with American meritocracy and exceptionalism. At times, however, when their lived experiences failed to measure up to such ideals, these Muslims interpreted Rauf's emphases on Sufism and service differently than he did. As the years passed, the economy faltered, racist backlash belied initial optimism about the ostensibly postracial era inaugurated by the first black president, and Rauf and Khan spent less time with their Sufi community than on ASMA and Cordoba projects, some also began to question their definition of moderation and the inevitability of acceptance. After the Cordoba House controversy, as I discuss in the final chapter of this book, even Rauf came to question some aspects of the American exceptionalism he had previously promoted.\n\nI first learned of the ASMA Society two years after Rauf's PBS interview, at Riverside Church in 2004. Financed by John D. Rockefeller, Jr., Riverside has hosted visitors as diverse as Martin Luther King, Jr., Nelson Mandela, Fidel Castro, Kofi Annan, Dick Cheney, and Hillary Rodham Clinton. Because I had long been curious about the ways Americans combine ideas about national and religious belonging\u2014particularly as they do so in narratives of \"Judeo-Christian\" heritage (itself a twentieth-century invention)\u2014I visited the neo-Gothic building in 2003 for New York City's second 9\/11 interreligious memorial service. I noticed immediately that the service did not include Muslims. I also noticed, however, that James Forbes (the senior pastor) carefully interrupted the \"Judeo-Christian\" language with which many speakers distanced Muslims from Americanness. Therefore, I was not entirely surprised the following March to learn that a local imam would be speaking from Forbes' pulpit.\n\nRauf's message on that late winter morning in 2004 was that America's \"Judeo-Christian\" heritage is actually a tri-fold Abrahamic one. Fascinated, curious as to whether he said the same thing at his mosque, and more intrigued by the question of how Muslims there responded to it, I accepted the invitation he extended to the audience to visit Masjid al-Farah in Tribeca. \"Imam Feisal,\" as the community called Rauf, was not present during my initial visit two weeks later. When I arrived, however, I found Faiz Khan (an emergency room doctor in his early thirties who was then serving as the assistant imam) delivering a similar message about religious commonality in the tiny storefront mosque. After the Friday prayer service ended, I told \"Dr. Faiz\" (his name at the mosque) of my interest in Rauf's message, and he and Dean (a sixty-five-year-old Sufi convert and self-described \"Catholic boy from Brooklyn\") spent the afternoon informing me over a cramped diner counter about the ways Sufi practices of Islam promote interreligious unity. They also invited me to that evening's _dhikr_ session in Daisy Khan's Upper West Side apartment, where I met other members of Rauf's Sufi group, including a Jewish American man.\n\nIt was during the communal meal following _dhikr_ \u2014which had involved prayers, reciting portions of the Qur'an, and rhythmically chanting the attributes of God while swaying meditatively or spinning prayer beads through the fingers\u2014that someone first handed me a promotional mock-up of Rauf's soon-to-be-published book. A blurb on the back, later replaced by a quote from author Karen Armstrong, asserted that this American imam shows how Islam is compatible with American democracy and capitalism. Although I had no doubts about the sincerity or intentions of either Rauf or Khan, I did immediately wonder about that promise and the politics that went with it. I decided I wanted to learn more.\n\nIt was not long before I realized this project would consume my attention for the next several years. Not only did I begin to research the history of Rauf's ideas and the political and economic philosophies to which they were connected, I simultaneously began ethnographic work to explore how Muslims at the mosque grappled with his teachings in their daily lives. During my first evening at Khan's apartment, before I knew Daisy had come of age in a rich Long Island community and once hopped a ride to Manhattan on the seaplane of Bernie Madoff's brother, it seemed clear that most of the Sufi dervishes there came from an affluent socio-economic bracket and might not object to Rauf's economic philosophy. What I also did not know then, however, was that Dean\u2014the convert with whom I had spent the afternoon\u2014lived in the same rent-controlled apartment in the West Village he had occupied for twenty years, and that Brother Malik\u2014a black American high school teacher who would soon retire and become an assistant imam\u2014lived just blocks from me in Harlem. Such things became evident only with time.\n\nDuring my six years of research, I learned that dynamics among the mosque attendees and Sufi practitioners were also more complicated than I first imagined. These Muslims came from backgrounds cutting across the social, economic, cultural, and racial spectra of New York. Their reasons for choosing Masjid al-Farah as their place of worship and Rauf as their imam (and also, sometimes, their Sufi shaykh) varied greatly. Moreover, I soon discovered, the mosque was not simply led by one devoted group of Sufis. Rather, its operations and purpose were negotiated by two overlapping communities\u2014one led by Rauf and one led by Shaykha Fariha, the woman who owned the building housing the mosque. Although Rauf had founded his own Sufi order, Fariha, following instructions from the shaykh she and Rauf once shared, allowed him to conduct services there until he could establish the independent mosque and community center he and Daisy Khan dreamed of\u2014one that would reflect American culture as they described it and help integrate Muslims. Not only did it take me several years to understand the complicated relationships between these overlapping (and constantly changing) communities, it took me that long to appreciate the many facets of Rauf's philosophy and the work of his and Khan's constantly evolving organizations.\n\nTo be clear, I was not seeking definitive accounts of \"true\" Islam, Americanness, or moderation in my research, as there is no single definition of any of these things. Rather, I was seeking to understand the pressures on Muslims to present themselves in particular ways in America and the creativeness Muslim Americans exercised, as well as the difficulties they encountered, in such circumstances. Because of this, while my analysis has often been informed by work in Islamic studies\u2014particularly by works focusing on the lives and practices of Muslims in the United States\u2014I have been equally influenced by the work of historians of American religion who have charted the political and economic dynamics involved in asserting Judeo-Christian heritage, and by those who have examined the Protestant underpinnings of American society\u2014ones that still exert pressure on non-Protestant religious groups to mold their traditions into forms resembling certain Protestant ones by, to note just one example, encouraging theologies of work and wealth similar to those derived from strands of American Calvinism. Additionally, I have looked to historical and anthropological analyses that point out the power dynamics involved in creating both academic and popular narratives about history and culture, commonality and difference. The theorists who draw attention to this kind of knowledge creation do not do so simply in order to prove that different groups (defined in terms of religion, culture, nation, race, gender, sexuality, or socioeconomic status, among other things) have been written _out_ of such narratives. Rather, they often focus on the racial, gendered, economic, and political power dynamics of writing various groups _in_ in different ways.\n\nAs I began to undertake ethnographic work with the communities affiliated with Rauf's mosque, I also looked for inspiration to the work of scholars of \"lived religion\" who draw attention to the ways religious practitioners embody and inevitably, but often unintentionally, alter the norms of their traditions as they literally _live_ them\u2014not just in the confined spaces of houses of worship or devotional prayer groups, but in the warp and woof of their everyday circumstances. Additionally, because the lived circumstances of the Muslims I met were shaped by such things as airport screenings, subway bag searches, warrantless wiretapping, home invasions, detentions, and deportations (which mosque attendees told me about long before such things made headlines), as well as by a domestic environment of increasing economic inequality, political polarization, government surveillance, and race-based policing, I relied heavily on the works of American studies scholars to make sense of them. These include scholars who show how ideas about Islam have played a role in domestic policy debates over issues ranging from slavery to consumer capitalism ever since the colonial period, as well as those who investigate the philosophical, political, economic, and strategic connections between the Cold War and the so-called War on Terror.\n\nLast, but certainly not least, my analysis was influenced by direct criticism of attempts, such as Rauf's, to represent \"moderate Islam.\" Such criticism is not the goal of my account. To the contrary, I have found that Muslim Americans who fight for acceptance in this country, including Rauf, are sometimes put into horrendously difficult situations and faced with seemingly impossible choices. Still, I recognize that criticism is something to which my account may contribute. I address that possibility here and in other chapters by discussing the other kinds of analysis that could be useful for understanding the effects on contemporary Muslim American communities of having to prove their moderation.\n\nThe Stakes of Moderate Islam\n\nIn the years since 9\/11, both scholars and activists have questioned the pressure on Muslim Americans to prove their moderation, particularly the demand that Muslim Americans act as liaisons between the US government and Muslim communities in other locations. Rauf is not unaware of such critiques. In 2007, in fact, he published an essay in a collection titled _Debating Moderate Islam: The Geopolitics of Islam and the West_. That collection\u2014the result of a formal conversation among religious leaders, scholars, and policy analysts about the usefulness of the \"moderate\" label and the politics behind it\u2014focused mainly on foreign policy issues rather than on the domestic ones discussed here. Secondarily, though, it questioned the effects of pressure on Muslim Americans to participate in State Department programs or to serve in some other way as liaisons between the United States and Muslims elsewhere. While the contributors did not discuss other pressures on Muslims in the United States\u2014ones that have since become public, such as the government's mass arrests, detention, and abuse of Muslim immigrants in the days after 9\/11; the placement by the Federal Bureau of Investigation (FBI) of innocent Muslim American citizens on the government's No-Fly List of suspected terrorists and subsequent refusal to remove their names unless these citizens agreed to spy on friends and families; and the New York Police Department's decade-long secret surveillance program that involved placing undercover (pseudo-Muslim) officers in mosques, student groups, and recreation centers, among other things\u2014these government actions placed much more psychic and physical stress on Muslim Americans than did more genteel attempts by State Department officials to enlist \"moderate\" spokespersons in promoting US interests. And although aware of at least some of the possible pitfalls of enacting the role of \"moderate\" sought by the US government, Rauf also knew of some of these law enforcement and surveillance tactics and sought to relieve the pressure exerted by them.\n\nAfter 9\/11, several worshippers from Masjid al-Farah were summoned to police headquarters for interrogations about their lives, histories, religious practices, and relations. Others, I am told, simply disappeared. Although the extent of invasive surveillance campaigns carried out by the National Security Administration and the New York Police Department would not become clear until nearly a decade later, Rauf had begun working with the FBI as early as 2003 to try to dissuade intelligence officials from categorizing all Muslims as possible terrorists. These were not things he generally discussed in public, however, and he also tended to discourage anyone else, such as Faiz Khan\u2014his assistant imam until 2006 and someone who had cofounded the ASMA Society along with Rauf and Daisy Khan in 1997\u2014from doing so.\n\nCaught between desires to defend his community from immediate threats and to be accepted by non-Muslim elites who could improve that community's situation in the long term, Rauf acknowledged in his contribution to the debate over \"moderate Islam\" that \"the term 'moderate Muslim' is problematic.\" Nevertheless, he acceded to the latter desire when he contended, \"we need to define . . . Muslims who can be worked with\" (presumably, in this case, for foreign policy ends) \"and those who cannot.\" He then proceeded to enact his moderation by reiterating his primary message of Abrahamic commonality and arguing that the United States has had a \"profound impact\" on Protestant, Catholic, and Jewish societies around the world. \"It is time for an American Islam,\" he continued, \"that will translate into the Islamic and Western vernacular . . . the best of the United States: its pursuit of the second religious commandment [loving one's neighbor as oneself] through the benefits of an Islamic democratic capitalism.\" Additionally, Rauf promoted his Shariah Index Project\u2014a Cordoba Initiative program he started in order to evaluate the Islamic authenticity (or shari'ah compliance) of various countries' legal systems, and one that would consume most of his energy after the failure of Cordoba House.\n\nCommenting on the debate about moderation to which Rauf contributed, Columbia University anthropologist Mahmood Mamdani, quoting political philosopher Leo Strauss, noted that repressed groups often feel compelled to \"coordinate speech with such views as the government believes to be expedient.\" Less intentionally, Mamdani cautioned, repressed groups may internalize \"repression as common sense\" and even promote it themselves to some extent. The appropriate action for scholars in such circumstances, he continued, is to resist the expedient views\u2014in this case, the idea that there are only two kinds of Muslims, \"moderates\" and \"extremists\"\u2014and to \"broaden the parameters of discussion\" by examining the histories of such ideas and the politics connected to them. Following Mamdani's lead but focusing on domestic issues rather than international ones, I do just that in the pages that follow, keeping in mind as I do so that defending Muslims by promoting certain kinds of moderation was something always done\u2014by Rauf, Khan, and many who worked with them\u2014with an eye on the very real consequences of being deemed \"immoderate.\"\n\nChapter Summary\n\n_Making Moderate Islam_ opens with an examination of Rauf's philosophies, bringing to light the host of racialized tenets about American meritocracy and exceptionalism on which his 2004 book relies. In the first chapter, I parse the components of Rauf's narrative of moderate Islam in order to reveal the political, economic, and philosophical similarities between Rauf's thought and that of some of his detractors\u2014in particular, Newt Gingrich. These similarities, in turn, illuminate the racialized tropes of assimilation and inevitable upward mobility many marginalized religious groups have echoed and adapted while explaining their own traditions in ways that demonstrate compatibility with American free-market capitalism and Protestant-derived secularism.\n\nI turn in Chapter 2 to how Rauf's father, a high-profile immigrant imam from whom Rauf derived much of his material, worked with Catholic and Jewish neoliberals in the 1970s while competing with other Muslim leaders\u2014particularly, black Americans\u2014to serve as a spokesperson for Muslims in the United States. I include in this chapter some of the political and economic developments that have given rise to tensions between some black American Muslims and American Muslims of Arab and South Asian ancestry. These tensions, which involve contests since the 1960s over political representation, religious authority, and economic resources, have inspired both black American and immigrant Muslims to emphasize their embrace of free-market capitalism and their participation in community service as they jockey for influence with American elites. As I discuss at the end of the book, they also contributed to the estrangement of Rauf and Khan from many other Muslim American groups during the 2010 controversy and cost the Cordoba House project valuable support at a time when it was most needed.\n\nIn Chapters 3 and , I map in detail the creation and evolution of the organizations launched by Rauf and Khan. I begin by locating Rauf and his Sufi order within the history of Sufism in the United States, then turn to his and Khan's shift from describing their work as Sufi, American, and cultural in orientation to interreligious, international, and policy-oriented. In the process, I show how their goals and self-presentations changed as they attempted to accomplish their objectives while simultaneously meeting different non-Muslim elites' shifting demands for particular kinds of moderate spokespersons. Although I discuss some of the racial and ethnic assumptions underlying Rauf's cultural, sociological, and historical writings in the fourth chapter, I describe these in more detail in Chapter 5, where I show how Rauf positioned Sufism as the bridge between a multitude of differences, including those separating immigrant Muslims from black American Muslims, rich Muslims from poor, Sunni from Shi'a, and (in his words) Islam from the West. Taking a more ethnographic turn, Chapter 5 also looks at how Rauf's dervishes struggled with aspects of his and Khan's definition of moderation, particularly Rauf's insistence that Muslims engage in the \"greater jihad\" of overcoming their own limited cultural traditions so as to align their practice of Islam with American democracy and capitalism. Examining some of the issues New York Sufis faced in trying to live this moderate Islam after 2001, I focus on the ways they adopted and altered such arguments so as to deal with the racial, economic, and political disparities they confronted.\n\nLike Chapter 5, Chapter 6 is primarily ethnographic and examines the ways New York Muslims dealt with the gaps between the idea of America put forth by Rauf and Khan and the realities\u2014particularly, the gendered realities\u2014of their daily lives. Promoting Muslim women's rights became a central component of their work (and of their self-presentation as moderates) during the decade after 9\/11, and they made the same assertions about women's equality in the United States as they did about the equality of religious and racial minorities, presenting it as a _fait accompli_. For many women who attended Masjid al-Farah, though, gender equality was more elusive, not because they were Muslim\u2014in fact, many found the complementary models of gender parity often presented at the mosque to be more realistic and genuinely egalitarian than their lived realities outside the mosque\u2014but because social gains for women in the United States still failed, in the first decade of the twenty-first century, to meet the hopes and promises of liberal feminists. Chapter 6 also looks at the ways attitudes at the mosque toward women's rights activists and toward the female religious leaders who were part of the community varied not just in relation to religious doctrine, but in relation to how much these women engaged in various kinds of community service.\n\nDuring the six years I spent attending Friday prayers, _dhikr_ sessions, birthday parties, baby showers, weddings, and other gatherings, I constantly heard Muslims connected to Rauf's Sufi group and mosque talk about embodying authentically American Islam by engaging in acts ranging from military service to staffing soup kitchens. The service expected of women, however\u2014particularly those who commanded any kind of authority\u2014was not just the voluntary service by which religious minorities have attempted to prove their Americanness over the last century. Rather, it was the sort of unremunerated care work often expected of women throughout the United States, be it care for friends, for families, or for religious communities.\n\nSuch service\u2014by women or men\u2014is something that can be regarded as part of Sufi practice. Indeed, as Rauf and Khan spent increasing amounts of time away over the years in order to pursue their ASMA and Cordoba projects, Rauf enjoined his dervishes to take up greater responsibilities of service to their Sufi order and community. As I discuss in Chapter 7\u2014which includes a larger examination of the politics, hopes, and fears animating the emphasis on community service among American Muslims since the Islamic center controversy\u2014some of Rauf's dervishes interpreted his instructions to serve and to model moderation in ways other than he intended, leading to the 2010 split within Rauf's group and to the ultimate demise of the Islamic center project as he envisioned it.\n\nBy bringing to light the history and ramifications of the moderate models Rauf and Khan advocated, _Making Moderate Islam_ lays bare the racism, as well as the ideas about gender, built into dominant US understandings of Muslim moderation and immigrant assimilation. Further, it exposes some of the pressures put on Muslims fighting for acceptance in the United States by people\u2014including government officials\u2014who have their own agendas. Not only does my account reveal the painful choices that many spokespersons for Muslim Americans face and, even more, the gaps between high-minded ideals and the lived experiences of Muslims in the United States, it also illuminates the ways that marginalized groups in America have often gained provisional acceptance (though not always equality) at the expense of others. In so doing, _Making Moderate Islam_ both exposes the power dynamics in which Muslim Americans are caught at the beginning of the twenty-first century and calls into question the larger limits of liberal inclusion for religious and racial minorities in the United States and the longer histories of provisional tolerance that have masqueraded as \"acceptance.\"\n1\n\nISLAMIC TRADITIONS AND CONSERVATIVE LIBERALISMS\n\n\"AMERICA WAS FOUNDED on Judeo-Christian principles\u2014that's the basis of our laws, and people try to deny it,\" claimed Representative Mike Reynolds, author of a 2010 bill to ban the use of Islamic law in Oklahoma courts. Although midterm elections often seem unremarkable, the 2010 election was an exception, as various critics of the Cordoba House project\u2014particularly Republicans on the far right and those catering to the Tea Party, a new right-wing political movement\u2014tried to harness opposition to the so-called Ground Zero Mosque for electoral gain. Similar ballot initiatives appeared in over two dozen states during the next two years, with supporters frequently emphasizing that the use of Islamic law in the United States would violate America's \"Judeo-Christian\" heritage.\n\nUse of the words \"Judeo-Christian\" to describe US history and identity is ubiquitous in American political rhetoric. The term is a seemingly timeless characterization of American society. Crucially, not only does the expression have a much shorter and more complicated history than its ancient connotations convey, but it is also often employed euphemistically to denote a Christian (and, more specifically, Protestant) perspective or position. As late as World War II, Presidents Roosevelt and Truman found their attempts to create national religious cohesion challenged by the fact that they could not get Protestant, Catholic, and Jewish leaders to participate in interreligious endeavors, primarily because conservative Protestants refused to work with Catholics. By the beginning of the twenty-first century, in contrast, the notion that the United States has a Judeo-Christian heritage was firmly cemented in the minds of conservative Protestants like Reynolds. Yet, despite this strongly stated conviction about the country's dual heritage, Reynolds clarified in his same comments the _singular_ nature of the impulse that led him to introduce State Question 755: concern \"about _Christian_ values in our nation.\"\n\nEvangelical politicians cited America's Judeo-Christian character as a reason why Muslims posed a national threat before the Ground Zero Mosque debate of 2010. For example, when the first Muslim elected to Congress\u2014black American Keith Ellison from Minnesota\u2014performed his oath of office with Thomas Jefferson's Qur'an instead of a Bible in 2006, Republican Congressman Virgil Goode warned that \"we are leaving ourselves vulnerable to infiltration by those who want to mold the United States into the image of their religion rather than working within the Judeo-Christian principles that have made us a beacon of freedom-loving peoples around the world.\"\n\nIt was in response to claims like these that Feisal Abdul Rauf promoted his narrative of Abrahamic (Jewish-Christian-Muslim) tradition after 9\/11 and penned his 2004 book, _What's Right with Islam_. Although many advocates of interfaith cooperation echoed his narrative after 9\/11, it was not always well received\u2014particularly not after the Ground Zero Mosque debate. At a January 2012 campaign stop in South Carolina, for example, presidential candidate Rick Santorum not only spoke in terms of Judeo-Christian heritage, he pointedly excluded Muslims from so-called Abrahamic traditions and from the ethical lineage that stems from them. Equality \"doesn't come from Islam . . . It comes from the God of Abraham, Isaac, and Jacob,\" Santorum, a conservative Catholic, argued. (Muslims trace their religious lineage not through Isaac, but through Ishmael, Abraham's first son.)\n\nSantorum was not Rauf's only conservative Catholic detractor. Newt Gingrich, another 2012 Republican presidential candidate, was a more prominent spokesperson for the anti-Muslim movement and against Cordoba House. Long active in trumpeting the nation's Judeo-Christian history, Gingrich led the Republican takeover of Congress on a \"family values\" platform in 1994. After he was charged with eighty-four counts of ethics violations, the former Southern Baptist retired from Congress in 1998 and pursued a new religious and political path: he converted to his third wife's Catholic faith and founded Gingrich Productions to promote his \"vision of an America in which a belief in the Creator is once again at the center.\" This vision characterizes his 2010 film about the dangers of \"radical Islam\" called _America At Risk: A War with No Name_ , as well as his 2010 and 2011 books, _To Save America: Stopping Obama'sSecular-Socialist Machine_ and _A Nation Like No Other: Why American Exceptionalism Matters_.\n\nBetween 2010 and 2012, Gingrich went to extraordinary lengths to condemn Cordoba House and the larger Abrahamic vision of America it was to instantiate. Yet his rhetoric overstated his ideological differences with Rauf. Undoubtedly, he and politicians like Santorum are in many ways more socially conservative than the imam. Nevertheless, as I demonstrate below, both Rauf's and Gingrich's philosophies are liberal in terms of the Lockean liberalism evoked in the Declaration of Independence, of Progressive Era liberals who viewed Protestant America as the triumphant culmination of world history (a theme each modifies to include Catholicism or Islam), and of the post-Great Society neoliberalism that stresses individual responsibility, the privatization or repeal of state welfare provisions, and government involvement in the economy primarily on behalf of the market. This latter variety of market liberalism has often come to define the political perspectives of politicians and pundits like Gingrich\u2014ones more commonly called \"conservative.\"\n\nTellingly, although Rauf describes American society as \"Abrahamic\" and Gingrich insists it is \"Judeo-Christian\" in culture and origin, both define the nation's identity in terms of an exceptional \"American Creed\" based on US founding documents, fortified by religious roots and replete with economic implications. A closer look at this creed reveals the liberal philosophies of rights and neoliberal economic arrangements\u2014including those in which religious organizations, rather than the state, provide community services\u2014central to each man's story of American progress and uniqueness. Meanwhile, a closer look at the history that led them to write such narratives illuminates how the racialized themes of service and anti-socialism have been central to religious minorities' struggles for acceptance since at least the mid-twentieth century\u2014but not in the ways Gingrich and Rauf suggest.\n\nThe American Creed: Common Ethics, Liberal Rights, and Liberal Markets\n\nIn his 2004 book, Rauf mixes theology with history, sociology, psychology, biology, and even game theory. His argument is structured to demonstrate how US political and economic systems progressively evolved to fulfill Islamic norms, even though the lives and traditions of Muslim Americans have been little noticed until fairly recently. The core of the book focuses on what Islam and the United States can offer each other, and this discussion is framed by an opening segment on common Abrahamic origins and a later chapter on contemporary historical convergence. That later chapter is devoted to demonstrating how Protestant-dominated America developed into a Protestant-Catholic-Jewish nation whose leaders imbued society with Abrahamic ethics. It also forecasts the inevitable Americanization and acceptance of Muslims.\n\nRauf's footnotes for these sections read as a \"who's who\" of American intellectuals, such as sociologist Will Herberg, author of the 1955 book _Protestant, Catholic, Jew: An Essay in American Religious Sociology_ , and as an inventory of iconic Muslim thinkers, including the twelfth-century Sufi thinker Muhammad ibn al-Ghazali, whom Rauf likens to Protestant reformer Martin Luther. His penultimate chapter on history is the progressive narrative's denouement, a demonstration of Rauf's frequently iterated assertion that \"America is substantively an 'Islamic' country . . . whose systems remarkably embody the principles that Islamic law requires of government.\" A final chapter outlines the \"New Vision for Muslims and the West\" promised in the subtitle of the book. This new vision for getting Muslims and other Americans to recognize their common Abrahamic heritage and ethics and to jointly export American liberal democracy and free markets contains action items for the US government, for educators, for the media, and for religious communities (all targets of ASMA and Cordoba programming), as well as for business elites.\n\nFor evidence of the Abrahamic-American ethical convergence, Rauf points to the nation's founding documents and the liberal philosophies of religion, reason, and rights they express. The Declaration of Independence and Constitution exemplify \"the core values\" of the Abrahamic ethic, he argues. Because the Declaration of Independence \"ground[ed] itself in reason, _just as the Quran and the Abrahamic ethic_ did in asserting the _self-evident_ oneness of God,\" he asserts, it embodies the moral and philosophical worldview revealed to the Prophet Muhammad. Referencing the third and thirtieth chapters ( _suras_ ) of the Qur'an, the imam also introduces readers to the Islamic concepts of nature ( _al-fitrah_ ) and the \"religion\" of nature ( _din al-fitrah_ , which he translates as \"natural religion\"). Rauf then compares these Qur'anic teachings on what he calls the Islamic tradition of natural religion with the Declaration's mention of the \"Laws of Nature and of Nature's God,\" concluding that the natural law referred to in the nation's founding documents is synonymous with what Muslims call shari'ah. For Rauf, shari'ah is not just complementary to American values, it is based in the same mixture of reason and revelation. Consequently, although no political society on earth will ever embody Islamic precepts as fully as the Prophet Muhammad's did, the United States comes as close as possible and constitutes a \"shariah-compliant\" state.\n\nWhile expanding Will Herberg's tri-fold narrative of Protestant-Catholic-Jewish America into one of Muslim-Christian-Jew, Rauf addresses some of the potential concerns non-Muslim interlocutors might have. One is that an Abrahamic framing cannot accommodate broad religious diversity. In response, Rauf points to the pluralistic history of many Muslim-governed societies (especially Cordoba) and repeatedly asserts that religious freedom is fundamental to Islam. (God, after all, endowed humans with free will.) Additionally, he reemphasizes the \"natural\" aspect of his argument and puts it in terms of the Founders' prescriptions. Quoting Hamilton and Jefferson on the divinely inspired laws of nature, Rauf posits that, when the Founders cited the God-given \"rights of 'Life, Liberty, and the Pursuit of Happiness,'\" they adumbrated \"cardinal moral truths\" that _all_ religious groups uphold. Because all Americans hold these founding tenets and rights in common, he concludes, these values constitute the American Creed\u2014a \"peculiarly American\" form of the Abrahamic ethic to which even atheists subscribe.\n\nIn addition to arguing that all Americans subscribe to the basic truths of Islam, whether they recognize it or not, Rauf attempts to neutralize fears of Islamic law\u2014namely, of the violence commonly depicted as inherent to shari'ah\u2014and to build the basis for incorporating Muslims into state processes. In a point meant to dispel the image of corporal punishment, among other things, Rauf reminds his readers that US laws already \"institutionalize\" the Abrahamic ethic. Therefore, they already accomplish the goals shari'ah is designed to effect. Further, because Muslims are commanded to observe the laws of the societies in which they live, he clarifies, they have no need to reject or re-create them. However, Rauf admonishes, the United States does fall short on its pluralistic promise by not permitting Muslims the same accommodations afforded other religious minorities (e.g., Jewish Americans): the ability to consult their own bodies of law to settle \"personal status\" cases involving divorce, child custody, and inheritance. At the very least, he proposes, the judiciary could have advisors who operate as translators between religious communities and legal authorities and who advise judges on whether laws are \"kosher or Shariah-compliant.\" Because the United States already embodies the Abrahamic ethic, Rauf reiterates, such a system would not challenge most legal decisions. Crucially, though, creating this kind of \"subsidiary entity within the judiciary\" would help turn the increasingly secular country back to its faith-based founding. Moreover, it would demonstrate to Muslims in the rest of the world that the United States is a God-centered nation rather than a secular materialist one.\n\nFinally, Rauf argues that divine law and the Declaration of Independence mandate certain economic arrangements\u2014ones badly needed in the Muslim world. These are \"free enterprise and a free market economy,\" which, when coupled with individual rights and concern for the disadvantaged, he believes, \"imply vigorous economic competition and high social mobility.\" Together, Rauf asserts, democracy and free-market capitalism create a social environment that enables believers to live out the primary commandment underlying all authentic religions: to \"love one's neighbor\" as oneself. Proof of this resides in American \"democratic capitalism,\" which\u2014because it combines \"democracy with a free-market economy,\" he argues\u2014has fueled a historically unprecedented expansion of freedom and equality for all peoples.\n\nRauf acknowledges that Americans sometimes fail to live up to their founding ideals, and his 2004 book is not short on critique, from the labeling of civilian casualties in Iraq as \"collateral damage\" instead of \"terrorism\" to US support for repressive regimes. Nevertheless, he forecasts, once Muslims and other Americans recognize their commonalities, they can jointly reorient wayward American practices back to their Abrahamic origins and, by extending democratic capitalism around the world, undercut extremism\u2014what he defines as a response to both \"militant secularism\" and material deprivation. With these goals in mind, Rauf explains in the book's final pages, he created the Cordoba Initiative.\n\nFive years after publishing his treatise on how to recreate the spirit of Cordoba, Rauf announced plans to open Cordoba House. Less than a year later, the backlash was so severe that Roger Cohen of the _New York Times_ wrote, \"not since 9\/11 has Islamophobia been at such a pitch in the United States.\" This backlash took several forms, including violence against Muslims and those taken to be Muslim (often Sikhs) and the destruction or vandalizing of mosques and other Muslim-owned properties across the nation. Instead of engaging in such overt acts of aggression, some Americans protested the creation or expansion of other mosques and Islamic centers, while others concentrated on combatting the scenario Gingrich frequently warned against in his 2010 film: the Muslim conquest of America effected, in part, by replacing the Constitution with shari'ah. As Cohen noted in his _Times_ piece, \"[s]hariah is the new hot-button wedge issue, as radicalizing as abortion or gay marriage, seized on by Republicans to mobilize conservative Americans against the supposed 'stealth jihad' of Muslims in the United States and against a Democratic president portrayed as oblivious to\u2014or complicit with\u2014the threat.\"\n\nIt is unlikely that Gingrich failed to notice the political benefits of denouncing the Cordoba House project or of vowing to outlaw Islamic law in the United States during his multi-year campaign for the presidency. Admittedly, Gingrich's reasons for opposing Rauf could be attributed to significant policy differences\u2014particularly on Mideast issues. Nevertheless, what many people might find surprising is the extent to which the premises of Gingrich's philosophy overlap with Rauf's. This overlap is most evident in the ways Gingrich similarly attributes America's unique combination of religion, reason, and (economic) liberties to the American Creed exemplified in the nation's founding documents.\n\n\"The ideals expressed in the Declaration of Independence, and the unique American identity that arose from a civilization that honored them, form what we call today 'American Exceptionalism,'\" Gingrich argues in the introduction to his 2011 book. Starting with his first chapter\u2014devoted to outlining how the American Creed, grounded in natural law, was forged from a mix of Enlightenment reason and religious devotion\u2014Gingrich stresses the importance of recovering the ethos of the Revolutionary era. Only in so doing can the nation avert impending disaster and prosper in the free-market manner the Founding Fathers intended.\n\nThe fact that Gingrich's liberal creed so closely resembles Rauf's is no accident, though it is also by no means intentional. Their commonalities stem from their reliance on common sources\u2014ones not immediately apparent because the two do not cite the same authors in the portions of their analyses devoted to liberal democracy or US republican history. Focusing on the neoliberal economic precepts each derives from the American Creed, however, allows the individuals and institutions that influenced their models of American exceptionalism to emerge and reveals a more complicated history of American religious minorities than their narratives acknowledge.\n\nUncommon Allies: The Sources of American Muslim Exceptionalism\n\nFor his account of the American Creed and American liberal democracy, Rauf relies on the Reverend Forrest Church, who was in 2004 the senior minister of All Souls Unitarian Church on Manhattan's tony Upper East Side. Author of over twenty books with such notable titles as _God and Other Liberals_ , Church published _The American Creed: a Spiritual and Patriotic Primer_ in 2002. In his \"Preamble,\" Church notes the term's coinage by G. K. Chesterton. An Englishman, Chesterton wrote following his 1921 trip to the United States that it \"is 'the only nation in the world founded on a creed,' one set forth 'with theological lucidity in the Declaration of Independence.'\"\n\nAmerica's religious nature is something Church traces in part to John Locke's influence on the Declaration of Independence and Bill of Rights. In fact, Church opens the first chapter of his work with a quote from Locke's _Second Treatise on Government_ : \"In the beginning, all the world was America,\" a statement Locke makes partly in service of his argument that all true and good governance is founded on the principles of natural law. While the \"Enlightenment religious values\" Locke expressed are crucial to the nation's founding frameworks, Church points out, they are not original. Rather, Locke drew inspiration first from Americans\u2014specifically, his friend William Penn, who founded the colony of Pennsylvania and preceded Locke in arguing for natural rights to liberty and property. In addition to explicating Locke's natural law philosophies, Church refers to Jefferson and Hamilton using the same quotes from each that appear in Rauf's later work. In subsequent chapters, Church argues that the religious nature of the country is not as specific as it may seem, asserting that \"America is _not_ a Christian nation. It is, however, a religious one.\" This attempt to resolve the tension between America's fundamentally influential religious heritage and its pluralist promise is a feature the narratives of Rauf and Gingrich share. All three writers emphasize that what unites the country despite diversity and allows the nation's original values to cohere are the common ethics shared by Americans of all true faiths.\n\nFor Church, the Deistic influences of Jefferson and Benjamin Franklin were crucial in forming, along with the American Creed of natural rights, a pluralist faith-based civic ethic that transcends religious differences. \"Deistic thought adapted the core of Christian ethical value,\" secularizing it somewhat by \"temper[ing] Christianity's theological authority,\" Church argues, and left in its place what he believes Franklin had advocated (and what Rauf also emphasizes): a \"public religion\" and a \"shared ethic more important than any single theological expression.\" Among the outcomes of this Protestant-derived ethic\u2014and each author emphasizes the work habits central to it\u2014are \"charity\" (Rauf and Church) and the \"voluntarism\" necessary to create a robust \"civil society\" (Rauf and Gingrich).\n\nAs to the crucial ethical and political issue of racial inequality, Church, Rauf, and Gingrich all turn to the Creed's progressive fulfillment in the work of various visionaries. Responding to historian Martin Marty's question of whether the principles of the Declaration of Independence can be considered a creed \"inclusive enough to involve the whole nation,\" Church answers, \"with Abraham Lincoln and Martin Luther King, Jr., I believe they do and can.\" These figures serve the same function in the histories of Gingrich and Rauf, demonstrating America's progressively expanding liberties and even proving the validity of the \"American dream.\"\n\nThe invocation of King illuminates another of the authors' greatest thematic commonalities\u2014the fight against both godly and godless extremists\u2014as well as their irreconcilable differences. In addition to extolling religiously rooted natural rights and a trajectory of political and economic liberation, Rauf, Gingrich, and Church all use the specter of Islamic terrorism to emphasize the imperative of recovering America's original ethics. Only by reclaiming the United States' foundational liberal vision, they believe, can the nation meet the pressing challenges posed by historic and contemporary nemeses: secular materialism and what Church calls the \"jihad\" of \"neo-tribalism.\" In fact, one reason Church sees the Creed as crucial for the twenty-first century is because Chesterton's claim that the United States is a \"nation with the soul of a church\" simultaneously highlights America's historic difference from the Soviet Union and contradicts those in the Muslim world who \"caricature the U.S. as 'soulless' or as 'the great infidel.'\" In contrast to these extremists, irreligious on the one hand and fanatical on the other, Church and Rauf depict the American Creed as fundamentally moderate.\n\nFor Church, King exemplified moderation by bending neither to the anti-integration criticisms of fundamentalist minister Jerry Falwell on the right nor to the militancy of black leaders like Malcolm X on the left. Instead, King called for \"all God's children, black men and white men, Jews and Gentiles, Protestants and Catholics\" to \"join hands.\" Depicted this way, King serves as a testament to racial equality and to liberal Judeo-Christian unity against the designs of both Christian fundamentalists (Falwell\u2014who, like Gingrich at that time, was a Southern Baptist) and Muslim \"militants\" (Malcolm X). In Rauf's account, moderation is also a platform erected between religious extremists and secularists. This simultaneous reliance on and distancing from the specter of Islamic terrorism characterizes not only Rauf's Abrahamic narrative but every rendition of the Cordoba Initiative's changing website since its 2004 inception. It has also been a theme of Gingrich's work since the 1990s and recalls an earlier period of American history, to which I return below, when religious minorities\u2014some of them formerly Marxist\u2014staked their claims to American identity by contrasting their traditions to communism and socialism.\n\nThe similarities between these histories of the American Creed cannot erase the authors' significant differences, of course. While Rauf lays claim to the mantle of moderation Church placed on King, for example, Gingrich uses the term \"moderate\" only once: in quoting Alexis de Tocqueville's observation that Old World virtues of economic moderation are regarded in America as \"faint-heartedness\" and lack of nerve. Further, the authors view progress and America's unfinished tasks quite differently. Church's account includes not just Americans who fought for racial, ethnic, and religious equality but individuals who labored to defend economically marginalized groups against robber barons and industrialists. Rauf adopts and extends Church's twenty-first century vision. In his telling, Americans are so united by the attacks of 9\/11 that even Republicans embrace Muslims as fellow citizens while the United States works through the United Nations to fulfill the country's ultimate mission of defending human rights around the globe. Rather than viewing the UN Declaration of Human Rights as a natural extension of the American Creed, Gingrich more frequently depicts the United Nations as a body crippled by the influence of an \"Islamic bloc\" that unfairly targets Israel and impedes America's international police power. Despite these differences, all believe faith-based civic and fiscal ethics fuel expanding freedoms and the divinely mandated global extension of political and economic liberties.\n\nGingrich does not refer to Forrest Church or acknowledge Chesterton's coining of the term \"American Creed\" in his 2011 treatise. Yet, while Church is a religious leader whom Gingrich would likely argue betrayed classical liberalism\u2014a charge he levels against \"modern liberals\" who critique aspects of free-market capitalism\u2014some points of similarity in their books are striking. Not least of these is their common argument that the American Creed is rooted in an American religious orientation subsequently developed in Locke's writings\u2014thus making his revolutionary philosophy originally American. This emphasis on Locke is a recent development for Gingrich, and a revealing one.\n\nGingrich's thoughts on the Founding Fathers changed following his conversion to Catholicism. Although he never mentioned Locke's natural law philosophy in his previous works on American civilization, a significant portion of his 2011 narrative is devoted to what he describes as Locke's Catholic-derived combination of reason with religious devotion and to the influence it had on the Declaration of Independence. This thinking can be traced to an author whose work is also\u2014with varying levels of attribution\u2014fundamental to Rauf's: politically conservative Catholic American pundit Michael Novak. Gingrich credits Novak's 2002 _On Two Wings: Humble Faith and Common Sense at America's Founding_ with demonstrating Locke's relevance to the American experiment. In that book, Novak likewise emphasizes the marriage of reason and religion in America's founding, cites the natural law framework of the nation's first apologists\u2014though giving American natural rights a \"Hebrew\" and Catholic pedigree\u2014and positions religious freedom as the right underpinning all others.\n\nNovak is also intimately familiar with the thinker who first coined the term \"American Creed.\" In fact, he wrote the introduction to the fifth volume of Chesterton's collected works. In that 1987 essay, Novak compares Chesterton's concept of economic \"distributism\" with the economic philosophy that Novak made famous: \"democratic capitalism.\" From the 1970s through the beginning of the twenty-first century, Novak's work on democratic capitalism at the American Enterprise Institute provided a theological basis for neoliberal thought that emphasized individual responsibility in the context of free people, free markets, and free trade. Maintaining that no political economy would ever match the ideals of the Kingdom of God (contrary to what he and his formerly socialist colleagues had once suggested), Novak promoted instead what he called democratic capitalism\u2014which he traced directly to Locke.\n\nGingrich's 2010 and 2011 books both include copious references to Novak's work at the American Enterprise Institute. In the acknowledgements of his 2010 book opposing Obama, secularism, and socialism, for example, Gingrich thanked Novak personally for contributing intellectual insights to the project and thanked the host of scholars, research assistants, and funders at the American Enterprise Institute (AEI) who helped him finish it. In 2011, after twelve years as a fellow at AEI and now openly Catholic, Gingrich also acknowledged his debt to Novak's 1983 _The Spirit of Democratic Capitalism_ and Novak's interpretation of Locke.\n\nRauf (like Church) acknowledges Novak only once, citing an opinion piece Novak wrote for the _New York Times_ to buttress his point that the \"unfinished business\" of both the Muslim world and the West is religious in nature. Although Rauf's endnotes never mention Novak's 1983 treatise on theology, politics, and economics, Novak's argument for the superiority of free-market society (based on both religious freedom and the biblical ethic of neighborly love) over centralized economies is replicated in every chapter of Rauf's book. It is also at the heart of how Rauf defines the \"unfinished business of the Muslim world\": implementing democratic capitalism in a way that is shari'ah-compliant. As I discuss below, although Rauf primarily cites Church's work on American history, the American Creed, and the American dream, his economic analysis derives from and more closely resembles Novak's.\n\nRauf's reliance on the work of both self-professed liberals and avowed conservatives need not be read as contradictory, as Rauf's claim\u2014in contrast to the claims of Church and Gingrich, who locate themselves on opposite sides of the political spectrum\u2014is the middle way: the path of the moderate. Tacking back and forth between extolling New Deal programs and praising the cumulative social effects of self-reliance and self-interest, however, Rauf's narrative exemplifies the difficulty of finding a coherent position that can be accepted at a time of increasing political polarization. He resolves this difficulty by pinning moderation to a form of religious practice that blends the economic entrepreneurialism of free-market enthusiasts with the community consciousness of socially minded reformers. Before turning to the social-spiritual practice Rauf prescribes, it is worth examining how his reliance on Novak replicates and illuminates an earlier history in which religious minorities emphasized common civic ethics\u2014including those of voluntarism and community service\u2014and market freedom in order to claim American belonging.\n\nNeoliberalism, Religious Minorities, and the Messiness of Moderation\n\nNovak argued in 2002 that \"the American Republic more closely matches the _Catholic_ vision of the Good city than any prior civilization in history.\" This is an argument he has had decades of practice making. During the late 1970s, with US foreign policy concerns divided, in part, between the petro-politics of the Middle East and the growth of Catholic liberation theology in Latin America, Novak led several meetings decrying liberation theology and promoting his philosophy of democratic capitalism at the American Enterprise Institute. Along with other former socialists, such as Irving Kristol, Novak coordinated a multi-faith theological defense of the American free enterprise system. AEI published the proceedings of one such conference in a 1979 volume Novak edited, titled _Capitalism and Socialism: A Theological Inquiry_ , which contained contributions from Protestants, Catholics, and Jews on the necessary \"Judeo-Christian\" basis of a free and capitalistic society, as well as on the origins of \"American exceptionalism.\"\n\nNot only did Novak and his colleagues attempt to demonstrate American capitalism's religious superiority at a moment when socialist movements seemed to proliferate among Latin American Catholics, they did so in a climate of domestic political backlash against civil rights legislation and new affirmative action programs that historians have labeled a late-Cold War \"white ethnic revival.\" During that time, several socially conservative Catholic and Jewish intellectuals argued that true religion supported personal responsibility and market freedom, and they depicted attempts \"to define social justice in economic terms or in terms of the redistribution of wealth . . . [as] subversive and 'militant.\" In so doing, one historian has observed, they \"affirmed the place of Catholics and Jews\" of Eastern or Southern European heritage beside Anglo-Saxon Protestants\u2014to the exclusion of other racial and ethnic groups, particularly black Americans.\n\nPart of the process of affirming Catholic and Jewish whiteness involved echoing earlier twentieth-century Protestants' liberal narratives of American exceptionalism but translating them into multi-religious, neoliberal ones. Articulations of Protestant-Catholic-Jewish commonality had only really begun to take hold among white elites in the United States after World War II. During the nineteenth and early twentieth centuries, many Irish and Italian Catholics\u2014often described by Anglo-Saxon Protestants in the same racial terms used to refer to freed slaves\u2014and Eastern and Southern European Jewish immigrants labored to distance themselves from black populations with whom they were sometimes grouped. Then, during the era of world wars, some Catholic and Jewish leaders attempted to widen the space afforded their communities in the United States by contesting narratives that conflated progress with Protestantism and Anglo-Saxon whiteness. Most did not do so by challenging the racial and economic exclusions on which that progress was based; as a compromise with southern Democrats, for example, Roosevelt's New Deal exempted occupations held by black, Latino, and other minorities from workers rights provisions. Rather, they emphasized that their traditions supported it, as well. The upward economic mobility these \"white\" but \"ethnic\" immigrants subsequently enjoyed during the 1950s\u2014in large part due to the GI Bill, whose education and housing provisions black American veterans struggled to secure at a time of open institutionalized racism\u2014seemed proof of their claims.\n\nBy the end of the 1960s, civil rights, black power, and third-worldist movements helped transform voting and immigration laws, understandings of culture and progress, and the very demographics of the United States. While many citizens seemed to have something to gain from these changes, others believed they had as much to lose\u2014including their own tenuous claims to whiteness. As both multiculturalism and affirmative action policies proliferated, some formerly marginalized religious and ethnic elites who had benefited from government economic programs in the past began to stress that a leveled playing field of meritocracy, not affirmative action, was now the best way to manage disparity.\n\nAppeals to Locke\u2014and to his ostensibly critical influence on the nation's founders\u2014featured centrally in some white ethnics' narratives of American uniqueness, as well as in their attempts to distance themselves from the specter of communism and socialism at home and abroad. This is despite the fact that Locke's influence was surpassed during and after the Revolution by that of another Enlightenment figure, Charles de Montesquieu. For free-market enthusiasts, however, it was Locke who promoted the divine rights of \"life, liberty, and property\"\u2014a phrase Jefferson transformed in the Declaration of Independence into \"life, liberty, and the pursuit of happiness.\" Two hundred years later, with US economic arrangements seemingly under threat from without (communism and socialism) and within (affirmative action), white ethnic intellectuals helped bring Locke's emphasis on divine rights to private property back _en vogue_.\n\nThe new market liberals tended to depict the United States as though it had\u2014for better or for worse\u2014\"emerged fully formed from the forehead of John Locke.\" Some, such as Kristol, also tended to lament what they saw as the demise of the Puritan work ethic and other religious values in modern life. It was in this climate of political, cultural, and economic contestation\u2014during which Catholic and Jewish political conservatives called for a return to America's foundational Protestant work ethic in order to limit the economic provisions of the state\u2014that influential Muslim Americans also began translating Qur'anic precepts in terms of common ethics and market freedom. One of these influential Muslims was Rauf's father, Muhammad Abdul-Rauf.\n\nFeisal Abdul Rauf's promotion of democratic capitalism and of Abrahamic ethics draws from his father's work during the 1970s, a decade in which Muhammad\u2014who held degrees in Islamic studies from Cairo's famed Al-Azhar University and from the School of Oriental and African Studies in London\u2014served as the director of the first New York Islamic Center and then of the influential Islamic Center of Washington, DC. During the 1960s and 1970s, Muhammad Abdul-Rauf and others created Arab-centered progress narratives and in their writings invoked Arab modernists who based ethical frameworks\u2014and, sometimes, political and economic systems\u2014on the concept of divine _tawhid_ (\"oneness\"). For some of these thinkers, Arab Islam is the most progressive spiritual force on earth, and it was Arab Muslims who had inaugurated modernity under the banner of _tawhid_ , which offers universal laws and ethics that simultaneously echo and rival those of Christianity. In writing these narratives, Muhammad Abdul-Rauf and his colleagues fitted Islamic history into the structure of the European colonial progress narratives that early twentieth-century Protestant liberals\u2014and, later, Catholic and Jewish conservatives\u2014had also adapted for American purposes. Further, they attempted to build on burgeoning Jewish-Christian alliances in the United States.\n\nIn 1967, with the Six Day War between Israel and neighboring Arab nations casting a shadow over American politics, Abdul-Rauf met for talks with Jewish leaders in New York and brought along his teenage son, giving him his first experience in interreligious dialogue. In 1979, during the escalating tensions wrought by the Iranian Revolution, as I discuss further in Chapter 2, Abdul-Rauf invoked interreligious commonality in an even larger forum: with a variety of scholars and theologians participating in that year's American Academy of Religion meetings. Instead of a \"Judeo-Christian dialogue,\" they called their Jewish-Christian-Muslim panels an \" _Abrahamic_ Trialogue.\" Abdul-Rauf also participated in seminars on religion and political economy that year, where he began to translate Islamic traditions once more\u2014this time, under the auspices of the American Enterprise Institute.\n\nNot only was Novak a crucial figure in the \"white ethnic revival\" mentioned above, in the late 1970s he and AEI President William Baroody enlisted Abdul-Rauf to join Catholic and Jewish intellectuals in outlining a broadly monotheistic basis for the anti-socialist, neoliberal model of political economy they were promoting. The result: Abdul-Rauf contributed to Novak's _Capitalism and Socialism_ and published a separate book with the American Enterprise Institute\u2014 _A Muslim's Reflections on Democratic Capitalism_ \u2014devoted to Islam's compatibility with American market liberalism. As I describe in Chapter 2, however, these assessments of American capitalism were not his last words on the subject.\n\nFeisal Abdul Rauf does not refer to his father's work in _What's Right with Islam_. Rather, while replicating some of his father's ideas, Rauf replaces the Arabian locations of intellectual and interreligious exchange featured in his father's progress history (such as Baghdad) with a center of culture more recognizable to American audiences: the European city of Cordoba. Moreover, although he identifies Muhammad Abdul-Rauf as \"my father and greatest teacher\" and credits his father with turning him into an imam, an equally important source for Rauf's moderate model is another of his \"spiritual ancestors\": Shaykh Muzaffer \u00d6zak, who introduced Rauf to Sufism.\n\nSufi practices and traditions are contested among Muslims worldwide, and Muhammad Abdul-Rauf\u2014whose quasi-diplomatic positions at the high-profile Islamic centers of New York and Washington, DC, depended, in part, on the Egyptian government's approval\u2014rarely wrote about them. Feisal Abdul Rauf learned much of what he knew about Islamic law and theology from his father. It was Muzaffer \u00d6zak, however, who made him an imam by appointing him to lead Friday prayers for the members of his Jerrahi Sufi order. Under the influence of both men, Rauf increasingly devoted himself to educating young Muslims raised in the materialistic culture of 1980s and 1990s America about what, for him, is the most authentic practice of Islam: striving for spiritual _as well as_ material affluence. Because the Protestant ethic shares Islam's \"defining ideals\" that \"religion and knowledge are compatible and that religion and wealth are compatible,\" Rauf argues, this practice of striving for the here and the hereafter will, again, bring Muslims closer to other Americans.\n\nSufi Striving: Translating Neoliberalism into Moderate Islam\n\nRauf describes Sufi practices as ones that foster \"heightened consciousness\" and lead to acting in conformity with God's commands. One who strives to increase her awareness of God will reflexively conduct herself in a manner that is ethical: \"maintain[ing] correct behavior and courtesy ( _adab_ ) before God and humankind.\" Very few spiritual seekers will achieve perfection in this striving, he writes. Regardless, the Sufi path is one \"[e]very human soul can and should begin to traverse.\" This is, in part\u2014as Rauf indicated in his 2004 book and also reiterated in Friday sermons at his mosque, often in market metaphors detailing the \"dividends\" of \"investing\" in spiritual practice\u2014because Sufi striving for the here and the hereafter, the _dunya_ (temporal world) and the _akhira_ (eternal reality), is vital to creating individual lives and larger collectivities that are both affluent and ethical, thus balancing the tensions between liberal rights and liberal markets.\n\nIt is Rauf's view of the relationship between religious practice and free-market institutions that sets him apart from the progressives with whom he is often grouped. One of the most striking ways Rauf modifies the assessments he otherwise takes from Church is by providing an alternate description of that sometimes loathed institution, the corporation. He depicts the corporation as a necessary and valuable entity that \"combined the Puritan ethic with easy access to capital\" and exemplifies the democratic ethos of representative governance, albeit via shareholders rather than citizens. \"Until the Muslim world finds a way to openly embrace these concepts and ideas in a manner consistent with Islamic law,\" he projects, \"it will continue to lag economically.\" Again, the source of this philosophy is Novak, who wrote in a 1981 treatise published by AEI ( _Toward a Theology of the Corporation_ ), that the \"corporation is an invention of democratic capitalism . . . Neither participatory democracy nor capitalism could exist\" without it. So ideal is corporate structure, in Novak's estimation, that it serves as the best metaphor for religious community.\n\nRauf does not idealize corporations to quite the same extent as Novak. Left unchecked, he cautions, they can entice people into perverting productive Protestant work ethics by seeking to accumulate endless wealth with no greater purpose and treating others\u2014particularly those in developing countries\u2014as instruments in the process. Nevertheless, Rauf's political and economic similarities to Novak are notable. Neither writer suggests that individual spiritual striving is _entirely_ sufficient to render the market ethical, and they agree that some government oversight of the economy is important (though Novak later retracted this). However, both warn that too much governmental involvement will stifle progress even as it fails to produce social parity, which is more, they argue, than centralized economies can create. The government's proper economic role, then, is to ensure the conditions conducive to free trade, such as a stable currency and \"an economy free from state control.\"\n\nAccording to Rauf, contemporary Americans are in a precarious situation, as most have come to believe that their social and economic progress was fueled by the separation of religion from the state, thanks to a \"militant\" form of secularism that has crept into US culture. Therefore, they may not have sufficient moral fortitude to resist material excess. Rather than increase governmental oversight of the economy to correct this imbalance, the answer is for Americans to learn from their Abrahamic siblings\u2014particularly the spiritual \"virtuosos\" that Rauf discusses in the book's section on Sufism\u2014how to recapture America's original civic ethics and to be properly religious again. By training one's will to \"enter consciously into a state of connection with God\" through Sufi practice, the earnest seeker is transformed into an individual who reflexively enacts the universal commandment that, Rauf believes, is the very foundation of democratic capitalism: to love one's neighbor as oneself. If Americans learn from Sufis how to personally strive for God as well as worldly goods, they can moderate America's excesses without sacrificing its abundance. They can then serve as global models of religious and economic achievement.\n\nFor Rauf, Sufi practice helps Muslims fulfill what he describes as Islam's requirement to \"value community over individualism\" and to assume \"responsibility to help others through charity and other acts.\" This emphasis on charity and community might initially appear to contradict the emphasis on personal responsibility and self-reliance (rather than reliance on government programs) that is a hallmark of neoliberal philosophies. In reading Rauf's account more closely, though, the compatibility of his religio-economic model with neoliberal prescriptions becomes more apparent.\n\nComparisons with Gingrich are again instructive. Gingrich extols America's exceptional history of voluntarism, including that of Catholics whose \"private generosity\" provides assistance for those in distress. For him, social cohesion and many social services should come from voluntary associations rather than from government programs. \"The American revolutionaries did not shed their blood for the welfare state,\" he argues. Quoting Novak in his section on \"Civil Society,\" Gingrich advocates reviving the habits of the first settlers, who \"took pride in being free persons, independent and self-reliant,\" but who structured their individual lives in ways that were \"cooperative and fraternal.\" For his part, Novak once argued that American capitalism\u2014particularly as moderated by the New Deal legislation Catholic bishops helped inspire\u2014provides the best practical means for \"raising the wealth of nations,\" thereby bettering the lives of the poor and needy as Catholic teachings demand. In these models, religious institutions, endowed with the voluntary contributions of their members, serve a crucial role in maintaining a free market society's social fabric. Once very basic safeguards are in place, faith-based striving for individual and communal benefit enables economic privatization and limited government.\n\nAlthough approving of some regulation\u2014such as prohibitions on child labor\u2014Rauf also makes a case for the need to prevent governmental overreach. In this argument, corporations are again crucial. Echoing Novak, Rauf describes corporations as examples of separated powers and as checks against economic centralization. This promotion of corporations undermines the praise he gives to Presidents Wilson and Roosevelt, however, and to their checks on capitalist excess. Moreover, he fails to address the repeal of much New Deal economic regulation during the 1970s and 1980s conservative backlash (led by Novak and his colleagues) against Roosevelt- and Johnson-era social programs. As a result, Rauf's economic philosophy resembles that of the neoliberals from whom he derives the term \"democratic capitalism\" more than it does Church's or that of other religious leaders on the left with whom many pundits, journalists, and conservative politicians associate him.\n\nRauf's philosophical closeness to Gingrich and Novak is especially noticeable when he explains the role of the \"business community\" in fostering rapprochement between Muslims and the West. Because economic prosperity will undercut the appeal of violence in impoverished conflict areas, Rauf maintains in an argument he frequently made during his yearly trips to the World Economic Forum, business elites must extend their enterprises around the world so as to create economic opportunities. Once the profit motive and desire to \"maximize\" one's own \"payoff\" are structured into society, Rauf argues, most people will recognize the benefits of mutual cooperation under democratic governance\u2014even if not compelled by religious ethics.\n\nThe final point of similarity between Gingrich's economic philosophy and Rauf's concerns the role of charity in helping immigrants\u2014often religious minorities\u2014assimilate. Both men believe immigrants Americanize, in part, by creating voluntary institutions to provide the social and moral supports free market systems otherwise lack. For Gingrich, such charities are an example of the kind of \"private virtue\" that the Founders intended for civil society\u2014something undermined when the state provides services. In his version of history, the primary obstacle to progress is not economic inequality or racial discrimination, past or present, it is governmental overreach cultivating an indolent population. Rauf's position on charity is more complicated\u2014less overtly partisan but, ultimately, quite close to Gingrich's neoliberal position. In order to see the similarities, it is necessary to examine how Rauf translates the Islamic tradition of _zakat_ and the importance he gives charity in his narrative of immigrant assimilation.\n\nMost of Rauf's discussion of charity occurs in the course of his effort to explain the Islamic tradition of _zakat_ \u2014the annual giving of a portion of one's wealth, generally to the needy, that is one of the Five Pillars of Islamic practice. While some Muslims describe _zakat_ as a \"right\" the poor can demand and translate other giving traditions (such as the voluntary practice of _sadaqa_ ) as \"charity,\" Rauf does not use this terminology. Instead, he is more ambiguous. Rauf does describe _zakat_ as a \"tax\" at one point. He does not end his explanation with this analogy to state income redistribution, however. In the rest of the narrative, before and after this section, _zakat_ is charity. When Rauf does describe it as a tax, he acknowledges that associating _zakat_ \u2014a religious obligation\u2014with the government could be seen as violating the separation of church and state. Rauf ends the discussion of _zakat_ -as-tax on this note, implying that religious responsibilities to provide for the poor are distinct from proper government functions. He later connects charity to Sufism, describing it and the other Five Pillars of Islam as forms of the Sufi ritual of remembering God ( _dhikr_ ) in every action.\n\nNot only does Rauf describe _zakat_ as more like US Protestant traditions of private charity than of taxation, he describes charities and other social institutions as things immigrants create while adopting the Protestant ethic and assimilating into the middle class. Rauf briefly acknowledges that up to 40 percent of Muslims\u2014mostly, but not exclusively, black Americans\u2014have a very different \"sociological\" experience in the United States, something he describes elsewhere as additional proof of American upward mobility and liberal promise, as I discuss in Chapter 4. Regardless of the fact that the lives of nearly half of American Muslims\u2014many of whom have lived in the United States for generations\u2014do not fit his template, Rauf then proceeds to treat what he describes as the immigrant ethical and economic Americanization process as a normative one and as a guarantee of eventual Muslim acceptance.\n\nImportantly, social acceptance is not the only goal. Once fully Americanized, US Muslims can serve as mediators between Islam and the West\u2014just as Irish Catholics (also once the targets of discrimination and violence) Americanized by creating charities, private \"welfare agencies,\" and other social institutions that helped later Catholic immigrants assimilate, and ultimately\u2014according to Rauf\u2014helped change world Catholicism through their influence on Vatican II. Jewish immigrants followed a similar pattern of Americanization, he argues, \"adopt[ing] key aspects of the Puritan and Abrahamic ethic\" and creating civic and social institutions that served their communities while respecting the separation of church and state. Eventually, Jewish Americanization had a similarly transformative effect, Rauf maintains, changing the contours of world Judaism while also turning the Protestant-dominated United States into a Judeo-Christian nation. In Rauf's account, this historical process will be complete when Muslims become more American and when other Americans recognize that Protestant-turned-Judeo-Christian ethics are, and always were, Abrahamic ones.\n\nThese histories of creating service organizations are ones Rauf, like Gingrich, celebrates as proof of intentional immigrant assimilation into a free market society with limited social provisions. In his 2004 book, Rauf does not wed such private service provision to a neoliberal agenda as explicitly as Gingrich does. Nevertheless, during and after the 2010 backlash against Cordoba House\u2014a time coinciding with the worst economic contraction since the Great Depression\u2014Rauf and Daisy Khan increasingly stressed community service to demonstrate American belonging. Their efforts mirrored something occurring more broadly in American Muslim communities at the time: an attempt to demonstrate that Islam, like Judaism and Christianity, supports voluntarism, community engagement, and caring for the needs of less fortunate members of society. These values and traditions have long histories in Muslim-majority contexts, of course. The framing of such traditions as charity, however, and even more specifically as \"service,\" as Khan described it during a television interview on ABC, is an American phenomenon\u2014one that has taken on new meanings since the 2007 economic recession and, as I discuss in Chapter 7 of this book, the 2008 election of the nation's first black\u2014and, according to many conservatives, \"Muslim\" and \"socialist\"\u2014president.\n\n\"All religions Americanize over time,\" Khan told reporter Christiane Amanpour in an August 2010 episode of _This Week_. \"They go from a place of worship to a place of service, and community centers have been developed by Christian communities like the YMCA, and the Jewish community has developed the JCC. And [the] Muslim community is inevitably going to also develop such a center.\" Khan's narrative of Americanization is also modeled on Herberg's paradigm of immigrant upward mobility and progressive assimilation. One crucial exception is that Khan (like Rauf, from whom she borrows) omits Herberg's critique. While Herberg described immigrant assimilation to the \"American way of life\" as a process of, in part, religious decline into material decadence, Rauf describes it as ethically ideal.\n\nThis difference, one Rauf acknowledges, again reveals how his account replicates those of earlier religious minorities who fought for inclusion by adapting dominant white Protestants' narratives of meritocracy and moral exceptionalism. It also exposes a kind of Protestant-derived secularism that still places tremendous pressures on such minorities to translate their traditions in its terms. Rauf gave evidence of this when addressing Herberg's criticism:\n\nThe American way of life is individualistic, dynamic, pragmatic, affirming the supreme value and dignity of the individual, who is striving to get ahead and wants to be judged by achievement: deeds are what count. Although some see in this American horizontal dimension of religion a kind of secularized Puritansim, a creed shaped by American Protestantism, we can equally assert that at its core this expresses the Abrahamic, _and equally the Islamic_ , ethic.\n\nTraditions in Translation: Protestant-Secular Pressures and Narrative Transformation\n\nRauf's promotion of common ethics, liberal rights, and liberal markets, as well as his years as a cultural ambassador on behalf of the State Department, might seem to instantiate the case anthropologist Saba Mahmood has made about post-9\/11 US government projects to change Muslims' interpretations and practices. In a 2006 essay describing National Security Council efforts to foster modes of Islamic education that will make Muslims more \"open to a 'Western vision of civilization, political order, and society,'\" Mahmood criticized both the neo-imperialist projects she identified and the \"moderate\" Muslim spokespersons involved in them. She argued that such projects attempt to reformulate the relationship between Islam and politics by teaching Muslims to view the Qur'an and other texts as products of culture and history. The hope is that rituals, scriptures, laws, and observances will come to be regarded as symbolic in importance rather than \"literal,\" as they are for \"fundamentalists\" and \"traditionalists.\" Muslims educated in this manner\u2014like those liberals who are committed to the \"poetic resources of the Judeo-Christian tradition\" rather than to \"traditional authority\"\u2014will avoid confusing religious symbols with universal truth and will, in turn, become amenable to liberal, secular political rule.\n\nWhile Mahmood's analysis is important to my own, mine departs from hers in significant ways. In terms of historical scope, Mahmood described American-led projects \"to foster what is now broadly called 'moderate Islam'\" as a post-9\/11 phenomenon. In contrast, as I have elsewhere demonstrated and discuss briefly in Chapter 2, Americans have led projects to foster \"moderate\" (and, before that, \"modern\") Islam since the Eisenhower administration. These projects did not take shape in response to 9\/11 and as part of the War on Terror but are the outgrowth of much older efforts to cultivate Muslim allies. More importantly, in contrast to Mahmood's depiction of US projects to moderate Islam and of the Muslim intellectuals allied with them\u2014whom she describes as uniformly committed to liberalizing Qur'anic interpretation and making liturgy, scripture, and ritual \"inessential,\" thereby contributing to colonial and neo-colonial endeavors to wrest power from traditional authorities\u2014Rauf and his colleagues argue that Muslims must become \"modern\" and \"moderate\" while _adhering_ to Islamic legal, scriptural, and ritual traditions in more than simply symbolic ways. They emphasize shared religious traditions\u2014including and especially ethical ones such as charity that, they argue, mandate neoliberal political and economic frameworks\u2014but also simultaneously challenge the privatized, individual approaches to piety that Mahmood cites as the goal of such projects.\n\nConsequently, while Mahmood believes that the \"particular understanding of secularism underlying contemporary American discourses on Islam\" is overwhelmingly influenced by US foreign policy concerns and distinguishes it from the religiously informed kinds of secularism that shape American life and American citizen-subjects domestically, I show that these discourses cannot be disentangled. In point of fact, Rauf's emphasis on voluntarism and translation of _zakat_ primarily as \"charity\" rather than as \"tax\" reflects the historic struggles against Protestant-dominated American secular arrangements\u2014ones which motivated many Catholics, Jews, and other minorities to create institutions to educate their children privately (rather than in Protestant-dominated, state-funded public schools) and to create mutual aid societies that would afford religious minorities social services without the liberal Protestant paternalism that often marked Progressive Era social service campaigns.\n\nFinally, while Mahmood implies that liberal interpretations of the \"Judeo-Christian tradition\" are the template US officials use in their efforts to moderate Muslims, thus collapsing with her terminology the differences between and among various Jews and Christians and ignoring the American political history of that comparatively recent coalitional term, my analysis highlights the continually contested claims dominant Protestants and religious minorities (including dissenting Protestants) have made about their traditions and about the nation's identity. In Chapter 2, I discuss further some of the ways religious and racial minorities have attempted to translate their traditions in terms of dominant white Protestant ones, focusing particularly on Muslims of various ethnicities (including Rauf's father) who emphasized service and anti-socialism out of sincere conviction as well as from the desire to gain social acceptance and recognition, sometimes at the expense of other Muslim communities. This translation process is far from finished, meaning that not even the creation of \"Judeo-Christian\" identity is a _fait accompli_. Rather, it is the occasional precariousness of Jewish, Catholic, and minority Protestant status that\u2014while optimistically omitted from Rauf's history\u2014heightens the stakes of American Muslims' struggles for inclusion.\n2\n\nSERVICE, ANTI-SOCIALISM, AND CONTESTS TO REPRESENT AMERICAN MUSLIMS\n\nAT THE END OF 1965, a year marked by struggles for civil rights and over the war in Vietnam, Feisal Abdul Rauf first set foot on American shores. Nearly forty years later, Rauf described this event in nostalgic terms\u2014the tale of an immigrant searching for spiritual and material fulfillment in the promised land\u2014rather than in terms of the racial and religious tumult that marked that decade of American history. \"As I sailed into New York on the cold wintry morning of December 22, 1965, on the Italian _SS Michelangelo_ , I beheld the Statue of Liberty and wondered what America had in store for me,\" he wrote in 2004. \"Little did I realize then that I was to discover the riches of my faith tradition in this land. Like many immigrants from Muslim lands, I discovered my Islam in America.\"\n\nAs noted in Chapter 1, Rauf describes the Islam he discovered in the United States as consistent with what Will Herberg called the \"American way of life.\" For Herberg, the ethnic and national markers of white immigrants melt away by the third generation, leaving behind only religious affiliations (Protestant, Catholic, and Jewish). Different as these religions may sometimes be, assimilation ultimately makes their adherents uniformly devoted to political, economic, and religious democracy and to a \"robust faith in idealism, activism, and moral conviction.\" Identifying as Protestant, Catholic, or Jewish, Herberg insisted, are simply different ways of identifying as American. Reworking this assimilation model during the decade after 9\/11, Rauf recast it\u2014combining it with Michael Novak's anti-socialist writings and dismissing Herberg's critique of American decadence\u2014to foretell the development of a moderate American Islam.\n\nRauf consistently presented his own work of teaching Muslims to be socially and spiritually responsible, to strive for the here and the hereafter\u2014which meant, among other things, prospering economically in order to engage in private charity and community service\u2014as central to Americanization. Further, he wedded such advancement to Sufism, a form of the kind of piety that is, he finds, essential to making America as ethical as it is affluent. For him, Sufi striving will help Muslim immigrants achieve middle class moderation and integration just as spiritual and economic striving helped integrate religious and racial minorities before them.\n\nIt is not surprising that Rauf relies so heavily on Herberg's story of white immigrant assimilation. When Rauf's father, Muhammad, accepted a job that moved the family from their home in Malaysia to the United States, the previously unthinkable election of a Catholic president was already old news. By 1965, Kennedy's tenure was a matter to memorialize rather than an ongoing controversy. Arriving in New York as a seventeen-year-old that year, Rauf had missed debates over whether Kennedy's Catholicism made him unsuitable to serve in office and over the appropriateness of Kennedy's Peace Corps for overseas civilian service, an initiative that had occasioned one of the greatest revivals of anti-Catholic sentiment among Protestants since the nineteenth century.\n\nThe religious tensions Rauf did notice in the United States had to do with Muslims. In 1967, Israel and Egypt went to war, and the ripple effects of the conflict were felt keenly in New York, where Rauf and other Columbia University students discussed the issue in their classes, and where Rauf's father attempted to engage Jewish leaders in dialogue. For Rauf, Kennedy's presidency was evidence of Catholic inclusion in Protestant power structures. Similarly, US support of Israel seemed proof of Jewish acceptance in America, making Herberg's tale of white Judeo-Christian progress appear unproblematic. Focusing his narrative primarily on immigrant Muslims rather than black Americans\u2014whose marginalization, despite their religiosity, also troubled Herberg's account\u2014Rauf described Americanization as a mostly seamless process of acculturation.\n\nAs I discuss in this chapter, neither interreligious nor interracial equality was as established during the latter part of the twentieth century as Rauf suggested. The selective history of assimilation through service and volunteerism that Rauf (like other Muslim American leaders) presents as proof of America's multicultural, multireligious promise is not altogether inaccurate. Members of marginalized Protestant, Catholic, and Jewish populations created many voluntary organizations during the twentieth century. These organizations, initially established to serve the needs of particular religious populations, later offered services to other communities while attempting to prove that their members contributed to the nation as much as white mainline Protestants did. As I have elsewhere discussed, however, creating service organizations gained marginalized groups only selective tolerance, earning them more acceptance from the federal government\u2014with various presidential administrations often seeking to co-opt their work\u2014than from fellow citizens. This was the case not just for Catholics and Jews, as the controversy over Kennedy's Peace Corps service program (among other things) demonstrated, but also for Muslims.\n\nAlthough not mentioned in Rauf's 2004 book, Muslim Americans had created voluntary organizations and engaged in various service endeavors throughout the twentieth century, but with mixed results regarding the social acceptance they desired. Like Catholic and Jewish American organizations, fledgling Muslim American organizations frequently faced government co-optation and popular discrimination simultaneously, their fates linked to the vagaries of international developments and American foreign policy. Also like Catholic and Jewish Americans, Muslim Americans are a highly diverse constituency, politically, economically, ethnically, and racially. Different segments of the Muslim American population faced different levels of acceptance and discrimination, and racial differences among Muslims often translated into social disparities, especially after 1965 when both black and immigrant Muslim American groups contended for social acceptance and governmental recognition. As changes in immigration law spurred the growth of Arab and South Asian Muslim communities in the United States, and as leaders of the formerly separatist Nation of Islam sought greater social acceptance, members of these various communities\u2014including Rauf's father\u2014competed with each other to define Islam for American audiences and to portray their own particular practices as most in keeping with the values of democracy, capitalism, and service. Tensions resulting from these contests reached a zenith during the late 1970s era of hostage crises\u2014Iranian and otherwise\u2014and contributed to Abdul-Rauf's removal from a prestigious position and return to Malaysia. Similar tensions, compounded over time, reemerged in the twenty-first century as scrutiny of Muslims after 9\/11 intensified various groups' attempts to prove their national loyalties and \"moderation\" and gave rise to both new professions of common ethics (including the civic ethic of community service) and new, racialized arguments against socialism.\n\nBlack and White Muslims in Pursuit of Equal Citizenship\n\nLike the white ethnic Catholics and Jews discussed in Chapter 1, many white ethnic Muslims\u2014those whose parents or grandparents hailed from Eastern Europe, Turkey, or the Levant and who were often classified as \"white\" in census records\u2014benefited from GI Bill provisions and experienced upward mobility after World War II. While Muslim Americans were omitted from narratives of Judeo-Christian heritage during that conflict, postwar foreign policy issues seemed to provide them with ample opportunities to gain inclusion and prove their loyalties through various kinds of service. With the beginnings of a new \"cold\" war between the United States and the Soviet Union, in particular, American political and business leaders looked to Muslim communities to serve as foreign emissaries. This recruiting effort coincided with a growing trend of coalition building among Muslim Americans and a growing desire to have their contributions to the nation recognized. President Eisenhower, who was focused on stemming the influence of communism in the Middle East (and, following Roosevelt, on securing alliances with Saudi Arabia and access to Persian Gulf oil), was prepared to do just that.\n\nIn 1957, Eisenhower, expounding what later became known as the \"Eisenhower Doctrine,\" promised to protect the Middle East from communism. That same year, his Operations Coordinating Board (OCB, established in 1953 to replace Truman's agency for intelligence and psychological warfare) released a classified report on how to make Muslim allies, titled _Inventory of U.S. Government and Private Organization Activity Regarding Islamic Organizations as an Aspect of Overseas Operations_.\n\nThe OCB report illuminates the administration's approach to creating Muslim allies: a two-pronged strategy involving foreign intelligence and intervention, as well as domestic surveillance and promotion. The former task generally included a combination of economic incentives and diplomatic pressure that, to be successful, required extensive analysis of South Asia and the Middle East. When changes in immigration restrictions (the McCarran-Walter Act of 1952) allowed students from the Middle East and Palestinian refugees greater entry to the United States than previously, many immigrants found positions in institutes newly established for the study of Mideast politics. Meanwhile, in a turn of good fortune for the Eisenhower administration, second-generation Muslim Americans who had created national coalitions during and after World War II were seeking ways to foster social acceptance and garner governmental recognition.\n\nIn 1952, hundreds of Muslim Americans gathered for the first convention of the International Muslim Society (renamed the Federation of Islamic Associations of the US and Canada, or FIA, in 1954). The founder of this umbrella organization for American Muslim groups, Abdallah Igram (a second-generation Muslim American of Arab descent), was aggrieved by misrepresentations of Islam he had encountered during his service in World War II and was devoted to changing his fellow Americans' perceptions of the tradition. In 1953, the year after Eisenhower famously identified the Judeo-Christian tradition as the \"deeply felt religious faith\" on which the US government is based, Igram successfully petitioned the president for an Islamic symbol to use on the identification tags of Muslims in the military. He and other FIA leaders then sought further opportunities to build connections with political elites. During the following decade, FIA representatives traveled to Cairo to, among other things, fulfill part of FIA's original mission of serving as intermediaries between Muslim-majority countries and the United States. Staff of both the US embassy and the Egyptian government greeted the delegates when they arrived.\n\nIn 1957, Eisenhower seized another opportunity to create Middle East alliances and cultivate domestic Muslim populations when he inaugurated the Islamic Center of Washington, DC. On June 28, just five years after extolling the Judeo-Christian tradition, Eisenhower stood amid the tiles from Turkey, chandeliers from Egypt, and carpets from Iran and praised both Islamic culture and the common values Muslims and Americans share. \"The countries which have sponsored and built this Islamic Center have for centuries contributed to the building of civilization. With their traditions of learning and rich culture, the countries of Islam have added much to the advancement of mankind. Inspired by a sense of brotherhood, common to our inner most beliefs, we can here together reaffirm our determination to secure the foundation of a just and lasting peace.\" Eisenhower's words could easily have come from that year's OCB report, which stressed the \"common spiritual base\" of Islam and Christianity and their common \"spiritual and ethical values.\" The report's stated reasons for Islam's importance to the United States included that Muslim-majority states were members of the United Nations and that communists sought to exploit Islam. Consequently, the OCB concluded, Americans needed to burnish their reputation in Islamic countries and initiate more \"regional studies . . . in the order of Middle East, Africa, and Southeast Asia.\"\n\nThe following year, Congress passed the National Defense Education Act, thus providing funding for studies of both religion and politics in these regions and helping to foster interest in Islam. As the OCB knew and mentioned, privately funded institutes dedicated to such research already existed at a number of universities in North America. In fact, it was Isma'il al-Faruqi, a Rockefeller Fellow specializing in comparative ethics at McGill University in Montreal, who would help popularize a new vocabulary of religious commonality in the United States\u2014that of \"Abrahamic\" cohesion\u2014and help set the stage for presenting common ethics, service, and anti-socialism as the basis for Muslim belonging.\n\nAlthough some Muslim Americans seemed poised to benefit from the opportunities to build political alliances at the end of the 1950s, a number of developments that unfolded during the second half of the twentieth century complicated their efforts. Even as white ethnic Muslims sought, through various kinds of service (military, diplomatic, and otherwise), to have their contributions recognized in Judeo-Christian America, some black American Muslims drew attention for their very different opinions about the relationship between Islam and the United States. Additionally, changes in US immigration law profoundly transformed the demographic and political composition of Muslim communities\u2014and of America more broadly\u2014in the next decades, while domestic and foreign policy concerns shifted both the attitudes of public officials and public opinion about Muslims. During these tumultuous times, Muslim Americans of different ethnic and racial backgrounds\u2014including Rauf's father\u2014sometimes worked cooperatively to improve the image of Islam in the United States and improve attitudes toward Muslims. Perhaps more frequently, as I discuss below, they worked at cross-purposes, with different communities lobbying to have their own practices and traditions considered the most authentically Islamic and least subversive\u2014in other words, the least \"militant\" and the least socialist.\n\nAmerican Muslims in the \"Great Society\": From Alliances to Antagonisms\n\nDuring the 1960s and 1970s, American ideas about so-called militant Muslims changed in keeping with developments that seemed to challenge dominant US interests. Black American Muslims were cast as aggressive and violent in the 1960s, thanks to FBI surveillance and disinformation campaigns and to a 1959 television special that introduced the Nation of Islam to other Americans: Mike Wallace and Louis Lomax's _The Hate that Hate Produced_ , which prompted the National Association for the Advancement of Colored People (NAACP) to denounce the Nation of Islam as a hate group. In contrast, by the end of the 1970s\u2014a decade in which wars and revolutions rocked the Middle East and Palestinian terrorist attacks reached across Europe\u2014the images of militant Muslims most commonly evoked by the media were no longer domestic but foreign, no longer black but Arab and Iranian. Throughout these years and in response to each new development, Muslim American communities attempted to make alliances across difference and to build relations with political leaders.\n\nIn the early 1960s, most Americans hearing the term \"black Muslim\" thought of Malcolm X, the Nation of Islam's most prominent spokesperson, and their thoughts about him were not positive. Although over a dozen black American Sunni communities existed in urban centers throughout the country in the 1960s, the Nation of Islam (NOI)\u2014a group increasingly rejected by domestic Sunni congregations after the 1959 TV special\u2014was the largest black American Muslim movement and also the most connected to pan-Islamic nationalist movements in the Middle East. These connections, along with Malcolm X's visits with communist leaders such as Fidel Castro at the United Nations, unnerved American intelligence agents and governmental officials.\n\nAdditionally, the NOI\u2014and Malcolm X, in particular\u2014served as a counterexample in most mainstream media of harmonious interracial and interreligious engagement. As public suspicion of Catholics involved in the Peace Corps shook Kennedy's administration, the Catholic president put his political weight behind Jews and Christians allied in the South in the fight against racial discrimination. Growing black American interest in Islam (NOI and otherwise) seemed to many to threaten the interreligious and interracial cooperation forged through these bloody civil rights struggles. In February 1963, for example, the _New York Times_ ran an article on a Chicago NOI rally under the headline \"Muslims Press Race Separation.\" The article mainly quoted Malcolm X, who petitioned the NAACP and the Congress of Racial Equality to join forces with the NOI against integration. Then, at the end of that year\u2014after Civil Rights leader Medgar Evers was murdered in his driveway, four little girls lost their lives in a Birmingham church bombing, and the United States helped assassinate the first democratically elected Congolese prime minister, Patrice Lumumba\u2014Malcolm X described Kennedy's death as a case of \"chickens coming home to roost.\" NOI leader Elijah Muhammad censured him and apologized to the Kennedy family in an attempt to save the Nation's image. Nevertheless, the NOI was marginalized and excoriated by not only non-Muslim Americans but also the country's growing black, immigrant, and established white-ethnic Sunni groups.\n\nThe years 1964 and 1965 witnessed a remaking of Malcolm's image, if not yet the stereotype of black Muslims, that would have profound effects on interracial and interreligious relations by the end of the decade. After disaffiliating from the NOI and transitioning to Sunni Islam\u2014a process that involved undertaking the pilgrimage to Mecca and writing about his new belief in racial reconciliation\u2014Malcolm X (now El-Hajj Malik el-Shabazz) founded the Muslim Mosque, Inc. He also founded the Organization of Afro-American Unity (modeled on the Organization of African Unity), continued to criticize US domestic and foreign policy, and urged transnational alliances to effect global human rights for all non-white persons. Such appeals increasingly resonated with black Americans, particularly after Malcolm's assassination in 1965 and the 1968 murder of Martin Luther King, Jr., which occurred just as tides of Pan-African, Black Power, and anticolonial sentiments crested in the United States.\n\nThe 1967 Arab-Israeli war, coming in the midst of these cultural shifts and followed by a 1973 conflict, caused some black Americans to view Israel as a colonial power. This political change, in turn, contributed to weakening the civil rights era goodwill forged between some Jewish and black Americans and to distancing black American Muslims from some sectors of the political establishment. Further, not only did the wars help consolidate many Jewish Americans' attachments to Israel, they intensified the affections of evangelicals who believe Israel has a role in apocalyptic prophecies and of a new branch of conservatives (\"neoconservatives\") who saw Israel's military victories over Arabs in the 1970s as instructive for the American military after its demoralizing failure in Vietnam. Although the 1967 and 1973 wars were far from the only factors responsible for spreading Black Power sentiments, Pan-Africanism, and anticolonialism among black Americans, both black American Christians and Jewish Americans later blamed Muslims for souring black-Jewish relations.\n\nThese 1960s upheavals in Catholic-Protestant relations and in black Christian-Jewish relations were not entirely evident to newly arriving immigrants such as Muhammad Abdul-Rauf and his family. Whether or not Abdul-Rauf knew that his overtures toward Jewish congregations coincided with a decline in black-Jewish relations, particularly in urban environments, cannot be determined. What is certain is that the shifting domestic and transnational race relations completely flummoxed young Feisal. By the time Rauf's family sailed into New York harbor, President Lyndon Johnson had already delivered his most notable public address on the floor of Congress: an address in which he had adopted the refrain of the Civil Rights movement\u2014\"we shall overcome\"\u2014as his own. Moreover, Johnson had pushed through Kennedy's civil rights legislation, signed an additional civil rights bill (the Voting Rights Act of 1965), and begun to create affirmative action programs that, he hoped, would remedy some of the effects of centuries of social, economic, and physical oppression on black Americans. Not recognizing that the cost of the Vietnam War, combined with a sharp rightward tilt in the Republican Party during the 1960s, undermined both Roosevelt's financial regulations and Johnson's social welfare legislation, Rauf later extolled the effectiveness of both presidents' programs in creating an ethical and equitable free market and in effecting racial uplift, and confessed his bemusement at seeing college students sympathize with anti-capitalist revolutionaries and \"third-world\" politics in the 1960s. As Rauf recalled, he had grown up in Malaysia, \"listening to Elvis and the Beatles and watching American movies. People wanted to be like Americans. In contrast, when I got here [to New York], I saw prosperous middle-class American college students wanting to somehow join the Third World. I understood their anger about the draft and the Vietnam War, but their talking and singing about revolution and idolizing Che Guevara and Fidel Castro made no sense to me.\"\n\nThe 1970s were nearly as tumultuous as the 1960s for American Muslims but in different ways. By the middle of the decade, as Republican backlash against Johnson's Great Society programs and against multiculturalism intensified, fiscal conservatives and city officials across the country came to praise the NOI as an example of the kind of religion that black Americans needed\u2014one that advocated economic advancement through self-reliance rather than government assistance. With his thriving business ventures, theology of thrift, and emphasis on self-reliance, Elijah Muhammad caught the attention of conservatives who opposed Johnson's social programs. Beginning in the late 1960s, neoconservatives such as Nathaniel Glazer, Daniel Patrick Moynihan, and Michael Novak\u2014the children of Jewish and Catholic immigrants\u2014applauded the religious and ethnic communities that, they believed, had risen economically by dint of their own resourcefulness rather than via the path of government assistance. The NOI appeared to be one such group and, in fact, likely resembled their model of self-reliance more than did any of the white ethnic populations that had benefited from Social Security and the GI Bill. Consequently, commendations of the NOI quickly turned into something of a trend. Not long after, the NOI changed profoundly, with many members renouncing their separatism.\n\nIn the meantime, American politicians and much of the press reacted to the 1973 Arab-Israeli war and subsequent oil crisis\u2014brought about when the Organization of Petroleum Exporting Countries (OPEC) raised prices in response to the war\u2014by depicting Arabs as terrorists who held America's gasoline hostage. Actual hostage crises only exacerbated these stereotypes by the end of the decade. When, after his son's graduation from Columbia, Muhammad Abdul-Rauf accepted a post as director of the Islamic Center in Washington, DC, he found himself in the midst of these shifting dynamics, sometimes cooperating with black American Muslims whom politicians and pundits celebrated and sometimes competing with them for representation and acceptance.\n\nWith controversy over Johnson's immigration and affirmative action programs intensifying and Cold War tensions mounting, high-profile immigrant and black American Muslim leaders began to express American belonging by extolling free-market capitalism as much as by emphasizing community and social service. Additionally, they allowed conservatives to read their encouragement of free enterprise and community-mindedness as promotion of reduced welfare provisions within an aggressively capitalist environment. This was true both of recent immigrants such as Muhammad Abdul-Rauf and of the once notorious black American Muslim leader, Elijah Muhammad.\n\nMuhammad encouraged the shifting public perceptions of the Nation of Islam during an era marked by what some historians describe as the beginning of the \"Great Divergence\" in economic standing between the increasingly rich and the increasingly poor. Democrats at the time offered little resistance to Republicans who sought to undermine progressive legislation of the Johnson and Roosevelt eras. Notably, the shifting economic tide was not simply a result of partisan politics but something Democratic presidents Jimmy Carter and Bill Clinton also later encouraged by deregulating major industries and dismantling some legislation. During his last public appearance, in 1974, Elijah Muhammad delivered a message of self-help that echoed the neoconservatives' and neoliberals' creed. The \"slave master is no longer hindering us,\" he claimed, \"we're hindering ourselves. The slave master has given you all he could give you. He gave you freedom. Now get something for yourself.\" The solution was not to seek government assistance but to engage in individual and community uplift, and for such sentiments both Republicans and Democrats applauded him.\n\nAfter Elijah Muhammad died in 1975, Chicago Mayor Richard Daley lauded him as a public servant. \"Under his leadership, the Nation of Islam has been a consistent contributor to the social well-being of our city for more than 40 years,\" Daley said. Earlier that year, Mayor Thomas Bradley of Los Angeles proclaimed February 4 \"The Honorable Elijah Muhammad Day,\" and even the _New York Times_ \u2014which had criticized the NOI's \"race separatism\"\u2014praised the organization's public service, crediting it with \"rehabilitating and inspiring thousands of once defeated and despairing men and women.\"\n\nThe social and religious acceptability of the organization only increased with the changes Muhammad's son, W. D. Mohammed, implemented after his father's death. These included incorporating Sunni traditions into NOI frameworks while keeping his father's focus on entrepreneurialism. They also included numerous name changes. The Nation of Islam was retitled \"The World Community of Al-Islam in the West\" and the organization's \"temples\" became \"mosques.\" Additionally, Mohammed changed the name of the group's annual February gatherings honoring the birth of NOI founder W. D. Fard from \"Savior's Day\" celebrations to \"Survival Day\" (later, \"Ethnic Survival Day\") celebrations. This shift not only downplayed previous NOI claims about Fard's divinity (which clashed with Sunni tenets) but also stressed what all black Americans had in common: a history of surviving slavery, the violence and repression that characterized Jim Crow laws, and the ongoing resistance to civil rights. Further, Mohammed promoted the cohesion of all Americans by flying the American flag during the Savior's Day celebrations in 1976 coinciding with the US bicentennial. Despite the largely positive attention these changes garnered, black American Muslim leaders did not begin to enjoy real political clout until the 1990s. Prior to that, their influence was reduced both by occasional negative publicity, such as that resulting from a late 1970s hostage taking in Washington, DC, and by the efforts of recent immigrants\u2014including Muhammad Abdul-Rauf\u2014to serve as spokespersons for all Muslims during the era of oil, Arab-Israeli, and American-Iranian crises.\n\nAmerican Muslim Allies and Competitors in the Arena of Public Opinion\n\nFrom the late 1970s through the beginning of the twenty-first century, competition for governmental recognition and social acceptance did as much to sour intra-Muslim relations as did economic disparities or conflicts over religious authority. Such competition was often shot through with contests over religious authority, and it often had economic components, although not just in the racialized class-conflict terms that some have suggested. As we shall see in what follows, despite the severe financial and social challenges many black communities faced, black American Muslim leaders could be as vocal as immigrant Muslim leaders in extolling American free-market capitalism so as to demonstrate their national bona fides. This was increasingly the case during the late 1970s and early 1980s era of hostage crises, when all Muslims in the US felt pressed to protect their public image, and competition for acceptance helped turn intra-Muslim alliances into antagonism.\n\nMuhammad Abdul-Rauf, who had attempted to establish positive ties with the Nation of Islam long before the mayors of Chicago and Los Angeles lauded Elijah Muhammad, was important in both building and straining black-immigrant Muslim relations. Occupying a quasi-diplomatic post as head of the Islamic Center of New York (home to many U.N. ambassadors) and, later, of the influential Islamic Center in Washington, DC, Abdul-Rauf served at the behest of Egypt's Al-Azhar University, which had dispatched him to America at a time when the Egyptian government oversaw Al-Azhar's appointments of religious officials. The Egyptian government also had a history of making common cause with the NOI\u2014something Abdul-Rauf continued once settled in the United States. Significantly, although the NOI had forged ties with the Egyptian government as early as the 1950s, it later also established connections to some of Egypt's Mideast rivals (specifically, oil rich Sunni monarchies). During Abdul-Rauf's time in the States, it even seemed that Egypt might be losing the NOI's allegiance. While serving partly as an Egyptian functionary, Abdul-Rauf often went to extraordinary lengths to show support for the NOI, even defending the organization after its members had an altercation with New York City police that led to the death of one officer.\n\nAbdul-Rauf's efforts to maintain ties with the NOI sometimes earned him the ire of other Muslim communities in the United States\u2014particularly black American Sunni groups such as the Islamic Party of North America, which disapproved of what it saw as the NOI's heterodoxy. What his Sunni critics may not have recognized was that Abdul-Rauf's backing of the NOI\u2014in addition to aligning with the interests of the Egyptian state and coinciding with the praise showered on the organization by US pundits and politicians\u2014allowed him to address issues of orthodoxy by subtly promoting among NOI members a particular vision of proper Islamic practice and intra-Muslim relations. This vision included close connections between various Sunni communities and leadership opportunities for humble and modest black American Muslims. Yet it also appears to have hinged on black American Muslims submitting to the overall direction of Arab spokespersons\u2014something neither W. D. Mohammed nor other high-ranking black American Muslims could accept.\n\nIn 1977, two years after W. D. Mohammed assumed leadership of and reorganized the NOI\u2014among other things, changing the name of the organization's newspaper from _Muhammad Speaks_ , after Elijah Muhammad, to _Bilalian News_ , after the second convert to Islam, a black former slave named Bilal\u2014Abdul-Rauf published a book designed to reach black American audiences. His _Bilal Ibn Rabah: A Leading Companion of the Prophet Muhammad_ traced the life of the historical figure who had served as the first Islamic prayer leader. Abdul-Rauf wrote the book at the request of \"one of my dear brothers, Karim 'Abdul 'Aziz, a leading figure in the energetic movement of the Nation of Islam in North America.\" The body of the work is mainly historical, but in Abdul-Rauf's preface and conclusion one can glean the kind of example he thought Bilal could be for black Americans. The word Abdul-Rauf used most frequently to describe Bilal was \"modest.\"\n\nAbdul-Rauf prefaced his account by stressing that Bilal \"preferred to assume a marginal role, and never sought a prominent public office or participated in controversial issues.\" The only portion of the book contradicting this depiction was the introduction, written by the Washington Bureau Chief of _Bilalian News_ and a member of W. D. Mohammed's movement, Ghayth Nur Kashif. Kashif praised Abdul-Rauf's work but set a different tone when describing Bilal's life and relevance. Rather than stress Bilal's humility, Kashif remarked on his \"lofty ascension as the first and Chief Mua'dhdhin (caller to prayer) of the final prophet of Allah\" and claimed that Bilal's life\u2014far from avoiding political issues\u2014could be instructive to \"Western civilization, just now emerging from the ignoble shackles of ignorance and fear, both the handmaidens of ugly racism and class distinction.\" For Kashif, it was time for the \"Bilalian Community,\" as W. D. Mohammed's followers called themselves, to assume their rightful roles of leadership and historical importance.\n\nKashif did not have the last word when it came to Bilal on this occasion. Those words can be found in Abdul-Rauf's conclusion. Seeming to agree with Kashif, Abdul-Rauf wrote there that the \"life story of our hero Bilal can be a source of tremendous inspiration for the world of Islam today, particularly the members of the Muslim community of twentieth-century America.\" Then, however, Abdul-Rauf reiterated his own take on Bilal's life and on the example it should set.\n\nEven under the yoke of slavery, Bilal was a man of a responsible character, honest, sincere, and dignified . . . The more he suffered because of his faith the stronger and the more pure and solid he became. His rise to eminence after his deliverance made him humbler and nobler. No trace of arrogance could be detected in his person. Even after the death of the Prophet, he remained just as modest.\n\nOnce again, above all else, Abdul-Rauf praised Bilal's \"striking modesty\" and quiet perseverance. More significantly, he again emphasized the way Bilal deferred to Arab companions of the Prophet after Muhammad's death rather than seek a leadership position.\n\nNeither W. D. Mohammed nor most other high-ranking NOI figures shared Abdul-Rauf's vision of black American Muslim deference. In 1978, the year after Abdul-Rauf published his book, Mohammed took steps to earn the support of governments domestic and foreign. He cemented important links to Saudi Arabia and other oil-rich states, which made him \"the 'sole consultant and trustee' of their distribution of funds to Muslim missionary organizations\" in the United States. He also took steps to emphasize patriotism, inaugurating a new holiday meant to demonstrate the transformation of former NOI members into \"outstanding examples of model American citizens.\" The title of the festival, \"New World Patriotism Day,\" suggested loyalty to the United States while simultaneously portraying \"patriotism\" as something that must be universal\u2014a celebration of interconnectedness and equality around the world\u2014rather than strictly national. These actions effectively rebutted Abdul-Rauf's intimations that black American Muslims should defer to others. They were, however, primarily undertaken to improve the image of Mohammed's movement in the wake of another incident that tarnished its reputation: a hostage-taking in Washington, DC, in which Abdul-Rauf was held captive and Mohammed's attempts to mediate the situation were rejected.\n\nIn March 1977, during Israeli Prime Minister Yitzhak Rabin's visit to work out the basis for an Israeli-Egyptian peace accord, black American Sunnis infiltrated three sites in the nation's capitol. Identifying as Hanafi Muslims, an NOI breakaway group, twelve gunmen took hostages at a Washington, DC, Jewish service organization, at a government building, and at the Islamic Center where Abdul-Rauf served as imam. They attacked some workers and visitors with knives and guns\u2014Councilman Marion Barry was shot in the chest\u2014then held over 120 people captive for more than a day and a half.\n\nThe Hanafi hostage takers gave several reasons for their actions. One of their stated goals: to demonstrate what they saw as the danger posed by the Nation of Islam and by the Bilalian movement that succeeded it. Not only did the Hanafis warn against what they considered the NOI's heterodoxy, they demanded retribution for what they believed to be the NOI's role in the 1965 killing of Malcolm X and of Hanafi leader Hamaas Abdul Khaalis' family members in a 1973 siege on Hanafi headquarters (Khaalis' home). Additionally, the Hanafis voiced opposition to the budding peace accord and to America's role in brokering it. The gunmen who stormed the Islamic Center demanded to see Abdul-Rauf and accused his country (Egypt) of conciliatory measures toward Israel. These Hanafi Muslims also took exception to what they perceived as Abdul-Rauf's liberalism. Nevertheless, their primary motivation, Khaalis claimed during a hostage-negotiation phone call, was Abdul-Rauf's support for the NOI-turned-World Community of Islam in the West.\n\nThe multi-day crisis was finally resolved when three diplomats intervened. The Iranian, Egyptian, and Pakistani ambassadors met with Khaalis and convinced him to free the hostages and end the standoff, thus earning the praise of the FBI, intelligence officials, and Secretary of State Cyrus Vance, who was quoted in _Time_ magazine describing the men as \"humanitarians and diplomats in the highest sense.\" This was in stark contrast to the ways black American Muslims were depicted in accounts of the incident. When W. D. Mohammed learned of the hostage crisis, he left Chicago for DC and offered his services as a mediator. Federal negotiators rebuffed his attempts to ameliorate the situation, however, leaving him and the \"Black Muslims\"\u2014as newspapers covering the incident consistently described his movement\u2014seeming, at best, ineffectual. More frequently, the press portrayed black American Muslims as volatile (even inherently violent), fanatical, and dangerous.\n\nDespite transformations in the NOI's image during the 1970s, cosmopolitan immigrants\u2014Muhammad Abdul-Rauf chief among them\u2014appeared to be eclipsing black Americans as models of both Islamic authority and American respectability and moderation. This trend might have continued if not for another hostage taking that changed how non-Muslim Americans thought about Muslims overseas and about the Muslims in their midst. During these hostage takings, economic arguments (anti-socialism, especially) became even more important for Muslims attempting to demonstrate their Americanness. Arguing for Islam's compatibility with American capitalism was not the only way that the Muslim leaders discussed here asserted their Americanness at this time. When other measures faltered, however, as they did after the hostage crises and other militant actions, such claims were a valuable tool for creating distance from the specter of radicalism, foreign and domestic.\n\nThe Evolving Specter of Militancy and Anti-Socialist Claims to Inclusion\n\nAs \"stagflation,\" or the combination of slow growth, high inflation, and high unemployment, gripped the economy in the late 1970s and President Carter attempted to reduce the government's responsibility for social welfare\u2014arguing in 1978 that government alone \"cannot eliminate poverty or provide a bountiful economy or reduce inflation or save our cities\"\u2014religious groups in increasing numbers founded service organizations to help the needy at home and abroad. These new service organizations, like those established earlier and despite some of their criticisms of US foreign policy, frequently included patriotic military metaphors in their names: for example, Lutheran Volunteer Corps, established in 1976, as well as Mercy Volunteer Corps in 1978. Simultaneously, Muslim leaders in the United States, including Muhammad Abdul-Rauf and his colleague, Isma'il al-Faruqi, attempted to improve the status of Muslims by participating in interreligious dialogues and sketching out an Islamic theology of economics. American capitalism, Abdul-Rauf and al-Faruqi argued, provides the greatest opportunities for acquiring wealth of any social system in the world and therefore most closely approximates divine mandates for political economy. Likewise seeking to regain positive recognition, W. D. Mohammed pursued similar strategies.\n\nIn February 1978, Mohammed traveled to the nation's capitol and addressed more than a thousand Jews and Muslims who had gathered for an interreligious service at the Washington Hebrew Congregation, exhorting them to recognize their commonalities. Since assuming leadership of his father's organization, Mohammed had not only opened the movement to people of all races, he had also tried to ameliorate tensions between religious communities by visiting synagogues, churches, and college campuses. Then, in May 1978, Mohammed articulated a message foes of Johnson's affirmative action programs longed to hear: that racial minorities needed to do more for themselves and to rely less on government protections and services for material needs and social advancement. \"Legislation has been piling up but we haven't been taking advantage of the opportunities open to us. The common man should not worry so much about legislation and civil rights,\" he proclaimed. \"If you have the ability, just go out and do what everybody else does.\"\n\nW. D. Mohammed seemed to think that if his followers did not have the resources to enact their Americanness by providing services\u2014he had dissolved several NOI businesses after his father's death to reduce the organization's debt\u2014they could at least demonstrate their patriotism and social position by not relying on government programs. Mohammed even absolved Carter of responsibility for breaking campaign promises to aid minority populations, claiming that when Carter \"got into office, he learned the realities of what he could do in light of the economy.\" Mohammed's statements promoting America's ostensible economic meritocracy\u2014welcomed as they were by politicians on both sides of the aisle\u2014were still often met with skepticism, given the reported rifts that doctrinal changes had caused among former NOI members, as the _New York Times_ repeatedly noted.\n\nIn the meantime, Abdul-Rauf continued to build relations with religious and political elites. The _Washington Post_ reported multiple times on Abdul-Rauf's interreligious activities, covering, for example, an October 1977 meeting he held with a rabbi and a Presbyterian clergyman to emphasize Jewish-Christian-Muslim commonality; his accreditation in the spring of 1978 as an official observer at the National Council of Churches (NCC); and a joint statement he issued the following September with the NCC, the Synagogue Council, and the National Conference of Catholic Bishops urging believers to pray for the success of the Camp David talks. By the end of 1978, even _National Geographic_ had taken notice of Abdul-Rauf, featuring him in an article on the Muslim pilgrimage in its magazine.\n\nWhen the Camp David Accords led to the Egypt-Israel Peace Treaty in 1979, the White House helped plan an interreligious service at the National Cathedral (later moved to the Lincoln Memorial) and invited Abdul-Rauf to represent Muslims. At the last minute, as Arab-led demonstrations shut down parts of the capitol in protest (the treaty did not address Palestinian refugee return), Abdul-Rauf failed to attend. His stated reason: \"extreme fatigue.\" Yet his office would not return calls for comment, and a leader of one of the protest groups claimed to have convinced him to avoid the service. While protests on the streets may well have influenced Abdul-Rauf's decision to bow out of the ceremonies, controversies brewing inside the Islamic Center over Abdul-Rauf's political positions were likely also a factor. These controversies would catch up with him by the end of the decade and reveal the difficulty of pleasing non-Muslim politicians and the general public while simultaneously representing a diverse religious community during a time of increased scrutiny.\n\nPulling out of the interfaith celebrations of the Camp David Accords might have compromised Abdul-Rauf's efforts to build relationships with other religious and political elites had he not already begun to emphasize the religious and social supremacy of American capitalism in collaborations with the American Enterprise Institute. With Abdul-Rauf gaining increased public attention in 1977 and 1978, William J. Baroody, President of AEI, invited him to participate in an interreligious seminar devoted to demonstrating the moral superiority of capitalism over socialism. Abdul-Rauf accepted the invitation, initiating the first of many collaborations with AEI over the years. Unbeknownst to Abdul-Rauf, however\u2014who admitted his lack of economics training\u2014the United States was about to undergo an economic transformation nearly as significant as the creation of the welfare state after the Great Depression. Five years later, when his writings were finally published, promoting American capitalism would mean a very different thing than it had in the years just after Johnson launched his Great Society and War on Poverty programs.\n\nIn the foreword to Abdul-Rauf's _A Muslim's Reflections on Democratic Capitalism_ , Baroody described the work as a testament to the budding neoliberalism outlined by Michael Novak. In contrast to the more socially minded political liberalism of Presidents Roosevelt and Johnson who, hewing mainly to Keynesian economic theories, believed equal opportunity required a combination of market restraints and government social programs, the market-minded neoliberalism that emerged after World War II\u2014promoted by academics like Friederich Hayek and adopted in the United States and Britain by Ronald Reagan and Margaret Thatcher\u2014argued for replacing centralized economic planning and Roosevelt's economic regulations with economic liberties, private property, and personal responsibility for family and community. Summarizing Abdul-Rauf's work at the time, Baroody emphasized such themes: \"Islam favors free enterprise and private ownership. The prophet encouraged increased production, trade and commerce, indicating at the same time that he favored only a limited role for the state in the regulation of trade. In addition, Islam teaches individual liberty, the right to own property, equality of opportunity (but not of wealth), equality among individuals, and the important responsibility of every Muslim male to support his family.\"\n\nLike other religious minorities during the Cold War, Abdul-Rauf distanced himself from communism and socialism because of what he perceived as their atheism as much as their economic principles. Like Novak, whose work he responded to, Abdul-Rauf argued that true religious practice addressed social issues and concerns while allowing maximum individual freedom. Thus, adopting greater piety was the key to balancing individual liberty and communal equality. The United States, he opined, was a God-fearing country that had crafted a basic social safety net and already emulated what Islam required for an equitable social system. \"We are all familiar with the battle waged between capitalism and socialism. In spite of the appeal of its rhetoric, socialism has faded away. Its utopian dreams have proved to be an illusion . . . [American] capitalism, too, has modified itself and has welcomed some socialistic reforms within its system. It is no longer simple capitalism but _democratic_ capitalism.\"\n\nIn his assessment of the similarities between Islamic mandates and American practices, Abdul-Rauf inserted Muslims into the narrative of Judeo-Christian capitalism that Novak and his colleagues had crafted in part to dispel American fears that Jews and Catholics were communist sympathizers. Moreover, he reiterated Novak's criticism of sociologist Max Weber for describing modern capitalism as an \"iron cage.\" In contrast to the shackles Weber suggested, \"democratic capitalism is open, dynamic, innovative, self-correcting, and inventive,\" Abdul-Rauf echoed, and meritocratic in that it \"honors hard work.\" Therefore, Muslims could be and should be as capitalistic as other Americans.\n\nImportantly, while seizing the opportunity Novak and Baroody afforded to improve the image of Muslims in the wake of international terrorist attacks and domestic economic crises, Abdul-Rauf also made sure to note what he saw as the differences between divine mandates for political economy and the neoliberal framework Novak presented. Deviating somewhat from proponents of total market liberalism, Abdul-Rauf afforded the state a larger role than Baroody admitted. When Muslims are not devout enough to observe the Islamic ideal of caring for individual and society, he cautioned, discipline is required to protect those who cannot protect themselves. Checked by pious practice, personal striving could lead to increased wealth and social flourishing. But when personal piety failed, the state must intervene.\n\nEven in his differences with Novak and Baroody, though, Abdul-Rauf made a case for inclusion by reiterating arguments that marginalized religious groups had made when seeking acceptance since the beginning of the twentieth century. This was especially the case when Abdul-Rauf took exception to what Novak described as the ultimate motivations for economic action. Rather than individual liberty and happiness, as Novak\u2014paraphrasing Locke and the nation's founders\u2014insisted, Abdul-Rauf described the most compelling inspiration to strive for both the here and the hereafter as \"the desire to please God through _service_ to oneself and others.\"\n\nThere is no reason to believe Abdul-Rauf was anything less than sincere when working to help Americans see Islam more sympathetically. Yet it seems unlikely that, for all his sincerity and earnestness, he failed to notice that members of marginalized religious groups fared best in the United States when they demonstrated their commitment to voluntary service and their antipathies toward socialism. Twenty years later, his son, Feisal, would echo these themes in his own book, describing them as components of \"Abrahamic\" ethics and connecting such striving (or jihad) to the practice of Sufism. The Abrahamic label, like the other sentiments Feisal borrowed from his father, came from another interreligious collaboration that occurred during the late 1970s\u2014a period in which Muhammad Abdul-Rauf, if not his son, began to question the morality of the Judeo-Christian tradition, as well as the morality of what American capitalism was becoming.\n\nService and Anti-Socialism: The Second Generation\n\nThe year after Abdul-Rauf presented Muslims as equally dedicated to service and opposed to socialism as other Americans, he joined his colleague Isma'il al-Faruqi\u2014a fellow graduate of Al-Azhar and the founding director of Temple University's Islamic Studies program\u2014at the annual meeting of the American Academy of Religion (AAR) in New York City. According to John Esposito, al-Faruqi's first doctoral student at Temple, al-Faruqi had also fostered interreligious ties during the 1967 war and in the 1970s played a leading role representing Muslims in meetings with the National Council of Churches, the World Council of Churches, the Inter-Religious Peace Colloquium (for which al-Faruqi served as vice president from 1977 to 1982), and even the Vatican, making him \"a leading Muslim spokesperson for Islam\" who focused not only on interreligious dialogue but also on social action. For the November 1979 AAR meeting, al-Faruqi invited Muslim, Jewish, and Christian leaders to discuss the commonalities and differences among their traditions. Rather than describe this as interreligious dialogue, he called it a \"trialogue\" of the \"Abrahamic faiths.\"\n\nPerhaps the invitation from al-Faruqi, a Palestinian refugee, emboldened Abdul-Rauf. Or perhaps Abdul-Rauf felt he could not represent Islam adequately unless he first addressed the frightening images flickering across American television screens that very day. In any case, although Abdul-Rauf primarily urged reconciliation among Jews, Christians, and Muslims and described the ways they shared in one true faith, he used his opening remarks to address political issues and to criticize American foreign policy\u2014one of the few times he is recorded as doing so while living in the United States. \"We meet today in a climate of tension caused by recent events in Iran,\" Abdul-Rauf began, and do not \"by any means advocate or even condone the practice of taking hostages as a measure of attaining legitimate aspirations.\" Still, he argued, American involvement in the Middle East\u2014including collaborating with the Shah to brutally repress Iranian protestors; cultural, political, economic, and military support for the \"genocide\" against Palestinians; and providing napalm and cluster bombs for use on Lebanese civilians\u2014made it necessary to ask probing questions about American religious tolerance and religious bias, as well as about American commitment \"to the so-called Judeo-Christian moral tradition.\"\n\nThis explicit political rebuke is one several members of Abdul-Rauf's mosque hoped he would also deliver from the _minbar_ (similar to a pulpit or podium) of the Islamic Center in Washington, DC. There, some congregants began to perceive Abdul-Rauf as overly tolerant of American injustice and as insufficiently critical of US racism at home and abroad. Reporting in 1980 on the growing tensions between different groups at the mosque and the Islamic Center's administration, Donnel Nunes of the _Washington Post_ noted that some dissidents opposed Abdul-Rauf's silence in the face of American imperialism and of Zionist aggression against Palestinians, others (mainly black Americans) demanded action to counter domestic anti-black and anti-Islamic social trends, and still others faulted Abdul-Rauf for not supporting the Iranian Revolution. These \"militants,\" as Nunes described them, comprised only a small portion of the mosque's attendees but formed a powerful constituency that threatened to overwhelm the great number of \"moderates\" who believed religion and politics should be separate.\n\nBy 1980, Abdul-Rauf found it difficult to escape political controversy. Protestors at the Islamic Center demanded he stop giving weekly _khutbahs_ (akin to sermons) at Friday prayers, and Abdul-Rauf acquiesced. By January 1981\u2014when the resolution of the Iranian hostage crisis coincided with the end of Carter's presidency and the beginning of Ronald Reagan's (during which Michael Novak would serve as an ambassador)\u2014Abdul-Rauf had resigned altogether and prepared to return to Malaysia. Before doing so, he joined al-Faruqi for one last academic gathering on the continent: a conference at the University of Alberta on \"Islam in North America.\" At that symposium, Abdul-Rauf discussed the challenge of embodying Islam when multiple cultures and constituencies lay claim to the tradition. His recent experiences at the Islamic Center notwithstanding, Abdul-Rauf's projections for the future of Muslims in North America were optimistic.\n\nAlthough American Muslims were \"insufficiently united as to be politically effective,\" Abdul-Rauf argued, he (like Herberg) saw change coming from second-generation immigrants. Partly because of the scrutiny that characterized American reactions to the Iranian Revolution, he asserted, members of his son's cohort were\n\nseeking to equip themselves with better education, diversifying their interests in order to assert their rights and defend the honor of their heritage in the American environment. In this way, they will contribute more greatly to the glory of the country of their choice and also will be able to build better bridges of understanding and cooperation between the United States and the Islamic fatherland, creating relationships of justice and equality.\n\nAl-Faruqi was somewhat less sanguine about the future of American Muslims. While he had elsewhere advocated capitalism over socialism, promoted the legitimacy of wealth and private property in Islam, and even credited the Puritans with sparking an economic revival in America, al-Faruqi did not on this occasion praise America's ostensible meritocracy. Having long worked with black populations in Philadelphia prisons and in Chicago schools, he overtly addressed some Americans' continued experiences of repression. Black Americans had been the \"untermensch\" in America, al-Faruqi argued, and \"the evil of social injustice in North America is sufficient to pull its victim away from the status quo and urge him to seek a change.\" It was only after leaving the United States, during a symposium with al-Faruqi in Pakistan in 1984, that Abdul-Rauf would echo such sharp criticism and, in contrast to his earlier sentiments, rail against global economic titans\u2014both nation states and the US-led international bodies he had formerly praised, such as the World Bank and International Monetary Fund\u2014for exploiting the \"so-called Third World\" nations under the guise of financial aid and benevolence.\n\nBetween the late 1970s, when Abdul-Rauf wrote for AEI, and 1984, when he penned his critique, the United States abandoned Keynesian policies in favor of deregulation, tax cuts, and budget cuts. The subsequent US recession was painful but\u2014coinciding with the International Monetary Fund's adoption of structural adjustment programs to change the economic and political systems in nations that borrowed from it\u2014was even more brutal in parts of the \"developing\" world. This global financial crisis chastened Abdul-Rauf. It did not, however, seem to change the perspective of W. D. Mohammed, who spent the 1980s reemphasizing the anti-socialist themes he had stressed during the hostage crises of the 1970s: that Islam is compatible with American democracy and that it is the responsibility of the individual, not the state, to improve economic and social conditions and care for local communities. Individual piety would yield social progress, he maintained, refusing to give credence publically to the idea that black Americans still faced systemic social injustice.\n\nFeisal Abdul Rauf did not immediately follow in his father's 1970s footsteps. After graduating from Columbia, Rauf stayed in the New York area, obtained a graduate degree in physics, worked as a schoolteacher, and eventually drew income from real estate. Then, after joining a Sufi order in the 1980s and spending several years as the imam of an affluent Manhattan congregation, he began to promote Sufism\u2014mainly among prosperous second-generation immigrants and proponents of interfaith collaboration. It was only after the terrorist attacks of 9\/11 that Rauf, seeking to foster greater acceptance of Muslims, began to publish for general audiences and turned to his father's earlier works\u2014including his father's dated economic writings\u2014for guidance and inspiration.\n\nTwo decades after Abdul-Rauf and al-Faruqi conjectured about the future of Muslims in the United States, Rauf echoed many of their arguments about \"Abrahamic\" Americanness in _What's Right with Islam_ , as well as some of the critiques of imperialism and nationalism his father and al-Faruqi offered. Crucially, though, his departures from their works are as notable as his agreements with them. As discussed in Chapter 1, the younger Rauf did not include as many cautions about capitalist excess in his 2004 work. Instead\u2014despite the fact that his father and Novak grew increasingly opposed in their economic pronouncements after the rise of neoliberalism under Reagan (Novak eventually came to blame the New Deal, which he had formerly credited with democratizing capitalism, for creating a kind of \"socialism by regulation\")\u2014Rauf relied on Novak's economic assessments and social prescriptions more than his father did and argued for exporting democratic capitalism around the globe. Further, while Rauf reiterated some of al-Faruqi's praise for the Puritans in his promotion of democratic capitalism and Abrahamic commonality, he had very different views about the situation of black Americans and the challenges they face than al-Faruqi did. As I discuss further in Chapter 4, Rauf argued that black Americans are generally as integrated and as accepted as any other population because civil rights legislation removed legal obstacles to equality and Johnson's affirmative action programs removed economic ones.\n\nRauf's perception of American racial equality was likely due, in part, to his limited experience with black American communities and to the prevalence of multicultural rhetoric in the United States during the 1990s\u2014especially during the First Gulf War, when political conservatives pointed to the multiethnic makeup of the American military as proof that black Americans could achieve as much as white Americans, even without affirmative action. It also likely due to the ways in which the political favor that some black American Muslims enjoyed in the early 1990s, in part because of their responses to that war, colored ongoing struggles among Muslim American groups to have their various traditions and perspectives regarded as the most authentically Islamic and American. Themes of service and anti-socialism continued to be important as various Muslim Americans tried to have their traditions accepted in the 1990s. Increasingly, though, as evening newscasts focused on domestic and international crises involving conflicts with Muslim-majority states or with terrorists acting in the name of Islam, Muslim Americans' demonstrations of commitment to their local and national communities were overshadowed. By the end of the 1990s, pundits and politicians were seeking other indicators of Muslim moderation.\n\nConclusion: Sufism and Service as the Moderate Islam of the New Millennium\n\nIn 1990, a broad spectrum of political and religious leaders, domestic and foreign, denounced Iraqi leader Saddam Hussein for annexing the neighboring oil-producing nation of Kuwait. The United Nations approved sanctions against Iraq, as well as military action by a coalition of countries including other Gulf states such as Saudi Arabia. In September of that year, W. D. Mohammed traveled to Saudi Arabia and delivered an address on the kingdom's national radio assuring residents of the United States's benign intentions in moving forces to the region. The US military had not and would not occupy holy cities in Saudi Arabia, he promised, and Muslims must recognize that Islamic and American interests converged in the fight against secular Ba'ath Party leader Saddam Hussein's imperial ambitions.\n\nMohammed was somewhat unusual among American Muslim leaders in expressing such sentiments. His organization, like many others, received generous funding from Saudi Arabia and from other coalition partners involved in the 1991 effort to repel Iraq from Kuwait. But even though immigrant-led Muslim organizations such as the Islamic Society of North America (ISNA, the nation's largest Sunni Muslim organization) similarly rejected Saddam Hussein's claim of acting, in part, to defend Islam, many were also suspicious of American claims to be acting as liberators when the United States supported Israel despite its occupation of Palestine. Rejecting pressure from funders in the Gulf to express their approval, many immigrant-led organizations remained silent. Subsequently, several organizations\u2014those run by black Americans and those run by more recent immigrants and their descendants\u2014severed financial ties with Gulf monarchies and established their financial and political independence. W. D. Mohammed, in contrast, not only justified American actions but contracted one of his businesses with the US military to provide troop rations, advised the US military on Islamic faith and practice, and convinced the Department of Defense to include Muslim chaplains\u2014the first of which was also a black American Muslim\u2014in the military's ranks.\n\nIn 1992, Mohammed became the first Muslim ever invited to deliver the opening prayer before a session of the US Senate. (Another black American imam, Siraj Wahhaj of Masjid al-Taqwa in Brooklyn, was the first invited to deliver prayers in the US House of Representatives, just after active conflict in the First Gulf War ended in 1991.) Following the 1992 election of Bill Clinton as President, overtures toward black American Muslims increased. Clinton's administration invited them to the first White House celebration of _'Id al-Fitr_ marking the end of Ramadan in 1996 and the first commemoration of Ramadan at the State Department in 1998. These gestures contributed to the impression that black American Muslims, somewhat like Jews and Catholics in an earlier era of American history, enjoyed a more equitable status in the United States than actual social conditions indicated. Meanwhile, American Muslims of immigrant descent faced increased public pressure as some neoconservative pundits, lacking a clear enemy after the 1991 breakup of the Soviet Union and disconcerted by US attempts to publically ally with Muslim-majority states, began to popularize fear of Muslim extremism and immigration.\n\nAs Jewish and Catholic Americans had discovered earlier in the twentieth century, Muslim Americans in the 1990s found that approbation from (and even co-optation by) the federal government failed to translate immediately into broad social acceptance. This was especially clear after the 1993 bombing of the World Trade Center in New York City by extremists. The attack seemed to legitimate fears of a broader threat of so-called Islamic terrorism in the United States and drew attention away from white militants such as Timothy McVeigh, who would bomb the Oklahoma federal building two years later. Responding to a significant increase in anti-Muslim invective, Muslim Americans engaged in a tremendous period of institution building, establishing organizations to fight discrimination, defend Muslim American civil liberties, educate the public about the values Muslims share with other Americans, and encourage interfaith collaboration. Moreover, US Muslims redoubled their efforts to demonstrate their Americanness through community and social service endeavors and disaster relief. Among the institutions established at this time were Islamic Relief, U.S.A. (an American branch of the UK-based global humanitarian and development organization), which opened in California in 1993 and began its first domestic community service work in 1994; ICNA Relief (a domestic service organization affiliated with the primarily immigrant-led Islamic Circle of North America, or ICNA), organized in 1994 to provide \"social services\" to \"underserved\" local communities; the Inner-City Muslim Action Network (IMAN), a multiethnic community service organization formed in Chicago in 1995; and the University Muslim Medical Association (UMMA) Community Clinic, created in 1996 to provide health care to needy populations in Los Angeles.\n\nThis rapid pace of institution building during the 1990s, coupled with the increased pressure to defend Islam and Muslims' practices, again heightened intra-Muslim contests for authority and representation, as greater numbers of spokespersons\u2014immigrants of various ethnicities, as well as black, Latino, and white Americans\u2014sought to gain positive recognition and to have the needs of their specific communities addressed. Tensions between Muslim groups only worsened in 2000 when immigrant-led Muslim coalitions voted en masse for the Republican presidential candidate, George W. Bush, who had promised to stop racially profiling them, even though most black American Muslims voted for Democratic candidates who supported social programs. Additional pressure on and scrutiny of Muslim Americans after the terrorist attacks of 9\/11 once again exacerbated these tensions and, for a time, contributed to black Americans' tendencies to distance themselves from immigrant Muslims.\n\nAs the US military began a new decade of incursions into Muslim-majority countries in 2001, US officials again looked to Muslim Americans to serve as spokespeople promoting American interests. In contrast to the First Gulf War of 1990\u20131991, however, they tended to look for Arab and South Asian American Muslim leaders, not to black Americans. Because of the organizing Muslim Americans had done after the First Gulf War and the first World Trade Center bombing, and because of their increased willingness to work with the government, officials at agencies ranging from the State Department to local municipal police stations had plenty of immigrant Muslim Americans to chose from in creating partnerships.\n\nWhile the work of these organizations could have helped to improve public opinion about Muslims in the United States had it received sufficient attention, the Muslim-led charities that most frequently made headlines were not ones engaged in local community service but ones suspected of undermining US interests abroad\u2014specifically, ones organized to provide humanitarian relief to civilians in Iraq, where economic sanctions since 1991 devastated local economies and plunged thousands of people into poverty, and in the Palestinian territories. When positive attention was given to Muslims in the news media in the decade before\u2014and, especially, after\u20149\/11, it was less often devoted to Muslim Americans' professions of service and anti-socialism than it was to Sufism. Long depicted as the mystical, apolitical tradition of Islamic practice, Sufism was newly attractive to American policymakers and intelligence officials in the 1990s, as I discuss in the following chapter.\n\nRauf was generally not engaged in public efforts to improve the image of American Muslims in the 1990s, and he remained largely distanced from the contests over authority and representation in which other Muslim leaders were immersed. Rather, he spent the decade writing books for second-generation Muslim Americans, whom he hoped to help adjust to American cultural norms while retaining the core of their religious tradition. For Rauf, Sufi practice is the best expression of what is most essential in Islam. Not surprisingly, he described authentic Islam, on the one hand, and spiritual and social malaise, on the other, in terms of Sufi psychology and art in these books, often to the exclusion of al-Faruqi's critique of domestic inequalities or his father's mandates for market restraint. As late as the year 2000, Rauf had planned to devote his third book for Muslim Americans exclusively to the topic of Sufism. Because US policymakers had identified Sufis as possible allies in the late 1990s, however, Rauf found himself well positioned to become a spokesperson for Muslims more generally. He soon changed the direction of his work and his writing, and\u2014with his narrative of Abrahamic commonality\u2014not only joined the public conversation about authentic Islamic practice in the United States, but also entered the contests over representation he had previously avoided.\n3\n\nSUFISM AND THE MODERATE ISLAM OF THE NEW MILLENNIUM\n\nAS THE MUSLIM HOLY MONTH of Ramadan began in November 2001, Feisal Abdul Rauf commented on its significance for the _San Francisco Chronicle_. Ramadan is a time of singular importance, he explained, as it is the period for rededicating oneself to the revelation that affords humanity guidance for \"differentiating good from evil.\" But this Ramadan was painfully poignant for him as an American Muslim, as \"this year I am faced with the probability that America, my country, will be at war.\"\n\nAs a Muslim leader who spent his entire adulthood in the United States and who experienced Islam as promoting reconciliation and even unification, war during Ramadan was a harsh prospect for Rauf. He was especially disheartened that America, a country he saw as devoted to peace and freedom, would wage war in Afghanistan, the birthplace of the Muslim poet Jalal ad-Din Rumi, one of his greatest literary influences. In his response to this disconcerting prospect, Rauf cited verses from Rumi's six-volume _Masnavi_ on transcending differences of geography and tradition. He then tied this vision of unity to the ethics that, in his view, Muslims, Jews, and Christians share and elaborated on what they meant for interreligious and international relations in the face of a violent and uncertain future.\n\n\"Our human destiny is to evolve our consciousness,\" Rauf argued. Thus, \"the war against terrorism means to wage a war against that quality of consciousness that resorts to violence and terror. The solution? Raising our individual and collective consciousness. Love your enemy, Jesus taught.\" As \"leaders of the free world,\" he continued, Americans have a crucial function in this global ethical and psychological endeavor. In order to guide other nations, Americans must \"become whole again, reintegrating ourselves with the aspects of ourselves that have become fragmented, separated, lost from our proper trajectory.\"\n\nThis message of consciousness raising was one Rauf had already spent several years making. Nevertheless, his 2001 version of it differed from past pronouncements. In 1997, when Rauf cofounded the American Sufi Muslim Association (ASMA) with Daisy Khan and Faiz Khan (no relation), he had just published his first book, _Islam: A Search for Meaning_ , which was devoted to showing how Sufism constitutes both the heart of Islamic faith and practice and a kind of consciousness raising that could help individuals recapture the spiritual parts of existence lost in the trials of modern life. Rauf initially wrote that book for the Masjid al-Farah congregation in downtown Manhattan where he had served as imam since the mid-1980s and where most of the congregants \"are strongly attracted to Sufism ( _tasawwuf_ )\" within the context of \"orthodox\" Islam. Unlike his 2001 _San Francisco Chronicle_ piece, Rauf did not connect Sufism to social action or social issues in that book, nor did he link it or the larger practice of Islam to any particular national purpose or agenda. Instead, his writing and his activities with ASMA at that time were largely devoted to explicating the psychology of Sufism and encouraging \"spiritual and personal development.\"\n\nThe challenges and opportunities brought by the 9\/11 attacks caused Rauf to shift course and alter his descriptions of his work and of ASMA's programs\u2014in 2001 and again several times after that. As he described a few years later, his new goals (and new book) were to demonstrate that\n\n[w]hat's right with Islam is what's right with America, in the sense that the fundamental ideals of Islam, the idea of what the right society should be, are very similar to what the American idea of what the ideal society should be, as expressed in our founding documents. When Jesus was asked what are the greatest commandments, he said \"love God with all your heart\" and, coequal to that, \"love thy neighbor.\" Islamic jurors basically expanded it. They said all the law\u2014how God wants us to live\u2014is to protect and further five fundamental human rights: the right to life, freedom of religion, family, property, and mental wellbeing. What I do in [my 2004] book is map that to the American Declaration of Independence.\n\nWhile Rauf's dedication to Sufism did not change between 2001 and 2004, that aspect of his work was quickly subsumed into his larger activities\u2014so much so that ASMA, which he and Khan began to rebrand as a \"cultural\" organization dedicated to appreciating Islamic arts and creating an authentically Islamic Muslim American culture, could no longer encompass all of his projects. This led Rauf to cofound another organization in 2004\u2014the specifically interreligious Cordoba Initiative. As we shall see, however, despite minimizing the Sufi aspects of his work after 2004 (even as his activities and objectives expanded beyond promoting Sufi practice), Rauf's Sufi identity and title as the founder of the American Sufi Muslim Association were crucial for gaining him audiences among American religious and political elites.\n\nI examine the organizations Rauf and Khan led and Rauf's rapid change from a part-time imam and Sufi shaykh into an internationally recognized American Muslim spokesperson in Chapter 4. First, I provide here a brief history of Sufi traditions in the United States\u2014particularly those in which Rauf is rooted\u2014and discuss how US policymakers began to regard Sufism as the moderate Islam they sought. Sufi traditions have for centuries been central to cosmopolitan imaginations in the United States, and selectively appropriating them has provided Americans of all racial backgrounds with ways to demonstrate their worldliness. This was not the case with other aspects of Islam. Indeed, the ostensible difference between Sufis and ordinary Muslims, who are supposedly more rigid in their practices and interpretations, has long been part of Sufism's appeal. During the Cold War\u2014particularly those decades that coincided with popular religious revivals in newly independent countries where both the United States and USSR sought client states\u2014the orientalist idea that Sufis are particularly pliable and opposed to \"fundamentalists\" took on greater weight in US foreign policy.\n\nStill, although Sufis occasionally advised US officials during the second half of the twentieth century, the State Department did not consistently attempt to cultivate Sufi allies until the late 1990s. After the terrorist attacks of 2001, American politicians also increased their efforts to recruit Arab American and South Asian American Muslims as defenders of US interests. As I discuss below, all of these efforts sidelined black American Muslims who had spent the 1990s building inroads into institutions of political power. Although black Americans are as likely as any others to practice Sufism, they do not always acknowledge Sufi affiliations. Already marginalized by many Arabs and South Asians who see black American Muslim traditions as less authentic than their own, some black American Sufis hesitate to associate themselves with a practice that can be regarded as unorthodox or as the province of white converts and dilettantes. Seeking a national political audience after 9\/11, when Rauf waded into the ongoing contests between black American Muslims and those of immigrant descent, his Sufi credentials constituted a measurable advantage.\n\nThe Ebb and Flow of Sufism in America, from the New World to the New Age\n\nSufi practices in North America are older than the United States itself, as Muslims were among the first immigrants to the Americas, making up an estimated fifteen percent of those imported during the Atlantic slave trade. Archaeologists have unearthed strings of blue beads\u2014possibly ones used in Sufi _dhikr_ \u2014from slave burial grounds that lie just blocks from the Wall Street location that was once one of North America's largest slave markets.\n\nWhite colonists and Americans of the early Republican era generally understood little about Muslims' practices. Even if they had, they were hardly likely to appropriate the traditions of West African slaves. By the nineteenth century, however, some Americans had become enthusiastic about traditions from the \"Mystic East.\" Transcendentalists such as Ralph Waldo Emerson and William Alger are particularly known for celebrating the Sufi poetry of Hafiz and Rumi, with Emerson arguing that appropriating such Eastern traditions (though not Islam, specifically) was necessary for modern progress. Sufi orders also influenced fraternal organizations like the Ancient Arabic Order of the Nobles of the Mystic Shrine (or Shriners, a variety of Freemasonry established in 1870 that incorporated the red fez of Ottoman Muslims into its ceremonies). During the same period, other Boston Brahmins began to compare Persian and Indian Sufis favorably to the \"hot and rigid Arabs\"\u2014as Harvard philosopher William James, Emerson's godson, called them\u2014whom they saw as overly legalistic.\n\nAs Gilded Age Victorianism gave way to the Progressive Era and the popularity of all things \"Eastern,\" interest in Sufism grew. Increasing numbers of New Englanders attempted to foster a cosmopolitan \"sympathy of religions\" in order to facilitate civic cohesion and economic expansion as the United States became a world power. Simultaneously, business leaders stoked interest in \"oriental\" fashions\u2014in clothing, architecture, and entertainment\u2014while attempting to inculcate consumer habits in the domestic population. Drawn by the allure of \"oriental\" art, enthusiastic audiences greeted Inayat Khan, an Indian Sufi musician, during his 1910 performances at Columbia University and elsewhere. Khan initiated several Americans into his Sufi order. After his death in 1927, some of his followers created their own movements, most of which, like Khan's Sufi Order of the West, did not identify Sufism strictly with Islam. Years later, Khan's son, Vilayat Inayat Khan, transformed the Sufi Order of the West into the Sufi Order International and contributed to another surge of interest in Sufism among white Americans\u2014one connected to the counter-cultural currents of the 1960s that, like the first wave, often deemphasized the Islamic aspects of Sufism.\n\nWhite Americans were not alone in appropriating Sufi traditions. The nineteenth-century Ancient Egyptian Arabic Order Nobles of the Mystic Shrine\u2014a group organized by and for black Americans who were excluded from white fraternal assemblies like the Shriners\u2014also borrowed significantly from them and, in turn, influenced more widely recognized black American religious movements such as the Moorish Science Temple (established in the 1920s) and, later, the Nation of Islam (established in 1930). Members of these groups frequently emphasized that\u2014in part because of the esoteric knowledge they ostensibly imparted\u2014such traditions provided ways to transcend dominant Americans' religious and racial biases.\n\nSomewhat ironically, Americans' attempts to demonstrate their cosmopolitanism by appropriating \"oriental\" traditions coincided with the repression of many such traditions in their countries of origin. In lands occupied by the British, for example, missionaries, academics, and colonial administrators pointed to popular Sufi practices, particularly the veneration of deceased Sufi shaykhs, as proof that local populations lacked the enlightened rationality necessary for self-rule. Importantly, even while emphasizing Muslims' ostensibly weak administrative capacities and citing \"otherworldly\" mysticism as the reason for Islamic civilization's alleged decline, British officials in South Asia attempted to assimilate the economic and political power of Sufi networks. These officials and orientalists, who considered \"Soofis\" to be less focused on Islamic law than were ordinary Muslims, often also viewed \"mystics\" as potential sympathizers with projects for displacing other Muslim authorities. Eventually, some Muslim elites adopted orientalist arguments about irrational Sufis and likewise pointed to their ostensibly superstitious compatriots as evidence of the need for modernization, education, and\u2014sometimes\u2014secularization.\n\nAreas under imperial control were not the only ones marked by such dynamics. In 1925, Mustafa Kemal Atat\u00fcrk, who imposed an aggressive form of secularism on the new Turkish republic and regarded Sufis as both morally decadent and \"politically reactionary and dangerous,\" banned Sufi orders outright. The government also closed religious schools and assumed the role of training and credentialing imams, thus transforming the remaining public religious leaders into state functionaries. Additionally, it shuttered the Sufi _tekkes_. (Also known as _dargahs_ , such meeting houses once served as lodges for travelers and had since become centers of intellectual and cultural exchange.) Just as some Americans donned the fez to demonstrate their cosmopolitanism, Atat\u00fcrk justified prohibiting its use in the same terms, arguing that \"primitive\" Sufis would corrupt the enlightened Turkish civilization. In the following years, some Sufis left Turkey. Others resorted to meeting secretly, gaining more visibility only in the 1950s, when Turkey became a multiparty democracy and ardent Kemalists lost support. In the meantime, a few groups reconstituted themselves as educational and cultural foundations (meeting secretly for devotional purposes) in order to keep their traditions alive and avoid government censure.\n\nSimilar secularization measures were undertaken in Iran, though without the explicit prohibition on Sufism. Like Atat\u00fcrk, Reza Shah Pahlavi (who in 1925 overthrew the last of the Qajar shahs) secularized education, assumed the power to credential religious leaders (the _ulema_ ), and insisted on sartorial changes among his population\u2014most notably, that women cease wearing the head covering they donned as a sign of modesty and status. Muhammad Reza Shah pushed some of these reforms further upon taking power after his father's death in 1941. The measures alienated not just the _ulema_ , whose power they were designed to curtail, but also Iranian business owners affected by the Shah's open market reforms and courting of US corporations. Like Kemalists in the Turkish republic, the autocratic Iranian regime sought to mollify some religious leaders in the 1960s and 1970s. To do so, it relied in part on the work of a young scholar who came to be one of the greatest popularizers of Sufism in the United States, as well as a formative influence on Rauf.\n\nSeyyed Hossein Nasr was born into a prestigious family in 1933 but left Iran for the United States at the age of twelve, just a few years before US and British intelligence helped to overthrow democratically elected Prime Minister Mohammed Mossadegh, who had nationalized Iran's oil industry. Nasr spent his adolescence in McCarthy's America, first at a prep school in New Jersey and then as an undergraduate at the Massachussetts Institute of Technology. At MIT he pursued a degree in physics but soon decided that positivist sciences could not alone hold the key to ultimate truth. With the use of atomic weapons at the end of World War II, the 1951 executions of Julius and Ethel Rosenberg, and the 1953 coup in Iran, the world seemed deep in crisis. Seeking metaphysical answers to the questions his studies of physics failed to solve, Nasr began to read works on Perennialism, a neo-Platonic philosophy that regards all religions as variants of one primordial truth\u2014one \" _sophia_ [wisdom] _perennis_ ,\" and Traditionalism, which rejects secular modern life and advocates a return to \"traditional\" doctrines and practices.\n\nAldous Huxley's 1945 _The Perennial Philosophy_ introduced many Americans to Perennialism. Parisian Traditionalist-Perennialist Ren\u00e9 Gu\u00e9non's 1927 treatise, _The Crisis of the Modern World_ (translated into English in 1942), also became quite influential in the United States\u2014particularly for Nasr, who read the work on the advice of one of his MIT mentors. Directed to the private Boston library of a founding Perennialist scholar, Nasr discovered early twentieth-century European Perennialists' writings on Indo-Persian philosophies and Sufism\u2014ones that he would later expand on and popularize. These included works by Martin Lings, who served as \"Keeper of the Arabic Library\" at the British Library from 1955 to 1973, and the German-speaking Swiss author Titus Burckhardt. Lings and Burckhardt, like Nasr, drew inspiration from yet another Traditionalist-Perennialist luminary: Gu\u00e9non's Swiss-born successor, Frithjof Schuon, who emphasized the unity of all religious traditions.\n\nNasr delved deeply into the work of Traditionalist-Perennialists\u2014a pursuit that only increased his dissatisfaction with the physical sciences and intensified his desire for metaphysical truths. After earning a master's degree in geology and geophysics from Harvard in 1956, Nasr turned his attention to classical Muslim philosophers. While in pursuit of a doctorate, he traveled widely in Europe, North Africa, and the Middle East. Nasr's later writings bear the imprint of his experiences during these travels, which included establishing personal relationships with Schuon and Burckhardt and joining the 'Alawiyya-Shadhiliyya Sufi order into which Schuon had been initiated in Algeria in 1932.\n\nAfter receiving his doctorate from Harvard in 1958, Nasr returned to Iran to take a position at Tehran University and there continued his personal and academic inquiries into Persian-language philosophies of time, nature, and existence. Afterward, he frequently traveled to North America and Europe to lecture and network, and his academic writing about Islam influenced the way many American professors and policy makers understood it. But he also wrote more popularly about Sufism, Perennialism, and (following Gu\u00e9non) the crises of the modern world, and these works enjoyed broad influence in the United States. Books such as his 1968 quasi-environmentalist salvo _TheEncounter of Man and Nature: The Spiritual Crisis of Modern Man_ were embraced not only by white, middle-class college students and \"spiritual seekers\" fascinated by non-Western traditions, but also by young first- and second-generation Muslim Americans\u2014including Rauf\u2014who sough to reconcile their traditions with the historically Protestant but increasingly secular and libertarian culture of the United States.\n\nNasr's Traditionalist-Perennialist Sufism was not the only Sufi tradition to gain popularity in America, of course. Scholars have documented the existence of numerous Sufi orders in the United States\u2014some of which were and are much more explicit about their Islamic roots and observances than Khan's Sufi Order of the West and much less universalist or traditionalist than Nasr's order. Nevertheless, Nasr's accessible writings on Sufi psychology, on the beauty of Sufi traditions, and on the centrality of art and Sufism to Islamic practice were tremendously important for disseminating awareness of Sufism at a time when few non-academic works were available on the subject. These works, his more philosophical writings, and his deep connections to scholars and academic institutions in the United States and Canada proved invaluable when, after several years of affiliating with the Shah's regime, he left Iran.\n\nNasr was in London when the Iranian Revolution began and did not return home. He has since lived in America, occupying academic positions at Temple University and George Washington University and serving as the head of the Maryamiyya Sufi order, the _tariqa_ Schuon created. Nasr's prestige and influence as a scholar of Islam are such that he was invited in 2000 to deliver the esteemed Gifford Lectures on religion\u2014the first Muslim scholar ever to do so. His reputation was established long before that, however. Just a few years after Nasr returned to the United States, for example, William Baroody of the American Enterprise Institute asked him to write the foreword to a book about Islam's support for capitalism. That book, written by Rauf's father, is but one of many connections between Nasr and Rauf.\n\nRauf first read Nasr in college and was influenced by him in ways that go beyond their similar affinity for democratic capitalism. Most notably, Rauf followed Nasr in embracing Schuon's belief in the \"transcendent unity of religions\" and in the existence of one _sophia perennis_ continually expressed in the great scriptures, philosophies, and artworks produced by followers of the world's various religious traditions. Further, Rauf's early personal history mirrored Nasr's to some extent: after immigrating to the United States as a teenager and obtaining degrees in the physical sciences, Rauf also found physical explanations for natural phenomena unsatisfying, and\u2014like Nasr\u2014discovered kinship and purpose in a Sufi community full of Euro-American converts. These converts, like those in Nasr's order, employed Sufi psychology as a means of fostering personal development and insisted on the centrality of art and aesthetics to Islam, as well as on the unity of all of the world's religious traditions.\n\nThe New Age and A New Era for Sufism in America\n\nThe artistic and religiously eclectic Sufi community Rauf joined first took shape around a charismatic leader in an era that witnessed the mass commercialization of Sufism. Not only did publication and distribution of Rumi's _Masnavi_ \u2014what would become the best-selling anthology of poetry in the United States\u2014increase in the 1970s, but audiences flocked to sold-out performances of Sufi dances at locations such as Carnegie Hall. These ostensibly apolitical cultural traditions\u2014including those popularized by the Halveti-Jerrahi Sufis who opened Masjid al-Farah\u2014often served as the first introduction to Islam for many Americans and as an alternative to the images of so-called militant Muslims (black American, Iranian, Arab, and otherwise) then common in the press.\n\nShaykh Muzaffer \u00d6zak al-Jerrahi, the Turkish leader of the Halveti-Jerrahi Sufi order, attracted several American followers with celebrations of Sufi _dhikr_ at the Cathedral of Saint John the Divine, a prestigious Episcopal church on Manhattan's Upper West Side. A former imam turned book dealer, \"Shaykh Muzaffer\" had been affiliated with one Sufi order or another since his youth\u2014despite the fact that visiting a Sufi _dargah_ or practicing as part of a _tariqa_ had been illegal in Turkey. The Halveti-Jerrahi _tariqa_ with which Muzaffer affiliated survived Atat\u00fcrk's ban on Sufi orders, though the entrance to the tomb of the _tariqa_ 's founding shaykh was padlocked by the government at one point. When restrictions on religious expression were somewhat relaxed in the 1950s, the Halveti-Jerrahis organized a foundation to promote what they described as the \"traditional\" Turkish music and dance involved in their ceremonies. This allowed them to continue their practices and even expand while other _tariqa_ s remained underground entirely.\n\nIn 1966, not long after assuming leadership of the Jerrahi branch of the order, Shaykh Muzaffer took the risky and unprecedented step of opening up one of the Istanbul _dargah_ s, allowing the general public to enter and observe Sufi ceremonies in person. He also began publishing treatises on Sufism for general audiences, focusing\u2014among other things\u2014on the refined cultural aspects of the tradition, a simultaneously savvy and sincere strategy that Muzaffer was equipped to defend, given his training at the Academy of Fine Arts in Istanbul. Muzaffer's emphasis on the aesthetic attributes of Sufism in the 1960s and 1970s, in addition to his insistence on remaining apolitical while others actively resisted secular nationalist reforms, protected members of the order from prosecution, despite their open practices.\n\nIn the late 1970s, Muzaffer began to travel with his dervishes, taking their ceremonies to international venues such as the Traditional Art Festival in Berlin. Their performances there, in Paris, and at another festival in Rennes involved singing and chanting the names of God and verses from the Qur'an\u2014the _dhikr_ practice that is common across Sufi orders, although the _wird_ , the litany of names and verses recited, differs according to each order's tradition. It also involved \"turning,\" a slow spinning also practiced by the most famous order of \"whirling dervishes,\" the Mevlevis, who lay claim to the lineage of Rumi. Turning occurs during a part of the ceremony known among the Jerrahis and Mevlevis as _sema'_ (or, in Arabic, _sama'_ \u2014a sort of intent listening), and is a kind of \"physically active meditation\" through which dervishes seek to overcome the _nafs_ (or self) and its \"egos or personal desires by listening to the music, focusing on God, and spinning one's body in repetitive circles.\" After establishing its cultural foundation, the order presented these practices as \"customary dance\" and as part of Turkish heritage. While this strategy satisfied Turkish officials, it did not improve the reputation of Sufis among Muslim authorities who regarded such practices as heretical innovations attached to Islam after the Prophet's death.\n\nAmong the dervishes in Rennes in 1978 was Tosun Bayrak, a Turkish expatriate, artist, and professor of art and art history who was instrumental in the order's later visits to America. Bayrak moved to the United States in the 1940s to attend the University of California at Berkeley and returned in the 1960s, after which he taught at Fairleigh Dickinson University in New Jersey and The Cooper Union. Before meeting Muzaffer, Bayrak was not a particularly pious Muslim. But he experienced a religious crisis after the death of his mother in 1974 and began to seek out Sufis. This search led him to the Jerrahis and to Muzaffer.\n\nThree years after joining the Halveti-Jerrahis, and at Muzaffer's urging, Bayrak turned half of his house in Spring Valley, New York (less than an hour's drive from the city), into a _dargah_ and established the Jerrahi Order of America. It was not the Jerrahis' only US outpost. Robert Frager, a Harvard-trained psychologist, also joined the order in the 1970s and integrated Sufi traditions into the program at his California Institute of Transpersonal Psychology. Nevertheless, Bayrak did not expect in the late 1970s that an American convert would one day rival his position as head of the US order, much less that he would compete for his shaykh's legacy with a woman. As I discuss below, however, the Halveti-Jerrahi\u2014a _tariqa_ that already had a reputation for internal dissension and splitting\u2014would soon be reinvented for a religiously eclectic American audience.\n\nBayrak was not the only person responsible for Muzaffer's first visit to the United States in 1978. While Bayrak used personal connections to secure visas for the group, funding for that excursion came from a source he does not mention in his memoirs: billionaire philanthropist Dominique de Menil, a deeply religious heiress whose family fortune derived from the sale of her father's oil-retrieval technology. Passionately devoted to art as well as to Catholicism, Menil (she dropped the \"de\" to avoid appearing pretentious) and her husband, John, commissioned Mark Rothko to paint more than a dozen monochromatic canvases for a modernist chapel they built in Houston, Texas, during the 1960s. The Rothko Chapel has since become their most famous legacy, though it represents only a minute portion of their collection.\n\nAccording to one of Menil's daughters, Philippa, Dominique made a \"deathbed promise\" to John, who died of cancer in 1973, that she would \"bring the whirling dervishes back to this country . . . They had seen [some] together in the sixties and had both fallen passionately in love, especially him.\" After contacting Turkish orders, Dominique and Philippa met one of Muzaffer's representatives\u2014likely Bayrak, though neither Bayrak nor Philippa acknowledges the other in their retellings of these events\u2014and were so impressed they bought tickets for Muzaffer and ten of his followers to travel from Rennes to the United States.\n\nWhile in New York for their first visit, Muzaffer and his dervishes performed _dhikr_ at Saint John the Divine where, three years earlier, members of Vilayat Inayat Khan's Sufi order had performed during the U.N.'s thirtieth anniversary celebrations. It was the Halveti-Jerrahis' first experience participating in an American-style \"interfaith\" gathering (as Cathedral Director Reverend James Park Morton described it), but the group was prepared for the enthusiastic reception they received due to their performances in Germany and France, during which audience members had chanted the beginning of the _shahada_ \u2014the Islamic profession of faith, which begins with _la' ilaha' illa-llah_ (\"there is no God but God\")\u2014along with the performers, and then joined in the resonant breathing of the words _Allah_ and _Hu_ , the latter Arabic for \"He,\" meaning the divine Creator. Muzaffer later recalled being asked after one performance about his habit of including Christians and members of other religious groups (or those of no religion) in the ceremonies and explaining in response, \"'I accept anyone who utters \"Allah\" in my ceremony' . . . [because] the duty of all prophets is to make people affirm LA ILAHA ILLA-LLAH, the profession of Divine Unity, regardless of their nationality, color, or race.\" Performing such interreligious ceremonies at Saint John the Divine soon became a regular practice, as\u2014thanks to Philippa\u2014Muzaffer and his entourage returned to the United States for twelve weeks each year, spending part of every spring and every autumn in Manhattan from 1979 until Muzaffer's death in 1985.\n\nIn 1979, Philippa \"took hand\" with Muzaffer, effectively joining his order by pledging allegiance, or _bay'a_ , to him as a religious leader and mentor. At the time, she and her husband, Heiner Friedrich, were immersed in the contemporary art scene, having established the Dia Art Foundation in 1974 to support groundbreaking minimalist and conceptual artists such as Donald Judd, Dan Flavin, and Walter de Maria, as well as to amass and showcase the works of others, including Andy Warhol, Cy Twombly, and Joseph Beuys. To facilitate both the scale of the works (de Maria's _Lightning Field_ stretched across an entire mile of New Mexico desert) and the success of Dia's extensive patronage commitments (at its height, the foundation sponsored nearly a dozen artists \"with stipends, studios, assistants, and archivists for the individual museums it planned to build for each of them\"), Philippa poured nearly $35 million of her inheritance into the organization. Then, but not for much longer, Dia occupied the bulk of her energies. In time, \"Fariha\"\u2014the name, meaning \"ease and peace of Paradise,\" that Muzaffer gave Philippa during her initiation ceremony\u2014would give up the contemporary art world almost entirely to focus on Sufism.\n\nNot only did Fariha fund Muzaffer's semiannual visits to New York with his followers, in 1980 she and Heiner built a _dargah_ \u2014a permanent space for Muzaffer's group to worship\u2014inside an old firehouse on Mercer Street in what was then the artists' enclave of SoHo. They designated it Masjid al-Farah (the Mosque of Divine Ease). That same year, Fariha and another New Yorker transitioned from being Muzaffer's dervishes to his official successors\u2014a development rather unconventional in the history of the order and one that did not always sit well with others, such as Bayrak. The other New Yorker, Lex Hixon (\"Shaykh Nur\"), was a local celebrity who had received a doctorate from Columbia University in 1976 and since authored multiple works on the essential unity of the world's religions. Hixon hosted the popular radio talk show _In the Spirit_ on New York's WBAI from 1972 to 1989 and first interviewed Muzaffer for the program, with Bayrak serving as a translator, in the late 1970s. He became a dervish shortly thereafter.\n\nThree years later, during one of Muzaffer's semiannual visits, Rauf\u2014then in his mid-thirties\u2014first visited Masjid al-Farah. He had learned about the mosque and Sufi community from a friend who was translating one of Muzaffer's books. On that visit, Rauf was initially unimpressed. Muzaffer was not in attendance and the four thousand-square-foot space was populated with only six worshipers. Nevertheless, at the request of the Turkish man volunteering as imam that day, Rauf delivered the call to prayer for the small group. Urged to return and give the call to prayer regularly, he hesitated. The next morning, however, when opening one of his books, Rauf's eyes fell on a section titled \"On the Excellence of Calling the Prayer.\" Feeling divinely inspired, he took the position.\n\nDuring Rauf's next visit a week later, Muzaffer was present along with many others. Enchanted, Rauf later described this meeting with Muzaffer as the beginning of what seemed an inevitable spiritual journey. Not only did he join Muzaffer's order that year, he also accepted Muzaffer's invitation to deliver _khutbah_ s on the many Fridays that Muzaffer was absent. \"I had known for several years that I would one day be giving Friday sermons in New York City,\" Rauf later recalled. \"It wasn't clear to me how I knew this . . . Rationally, I could not give credit to this vision I had had; but here I was, witnessing the situation actually unfolding as I had, in some recess of my mind, known it would.\"\n\nWhen Rauf agreed to serve as imam of the downtown _dargah_ in 1983, he did so on one condition: that he not be drawn into the rivalries between Muzaffer's followers. In contrast to Fariha and Nur, whose New York City group\u2014known as the Nur Ashki (loosely, \"Divine Light and Love\") Jerrahis\u2014adheres to Nur's vision of the mystical reality that, they believe, underlies all religions, Bayrak's primarily Turkish segment of Muzaffer's order describe themselves as a more \"traditional\" and explicitly Islamic group. Attempting to avoid the conflict between them, Rauf outlined a middle position that\u2014like Nasr and unlike Nur, who was also an Orthodox priest and had trained to become a Zen master\u2014involves emphasizing the ethical and mystical parts of the divine reality that he believes other religions share while maintaining an emphasis on faithfully practicing the tradition into which he was born.\n\nJust as the Nur Ashki Jerrahis settled into their new SoHo space with their new imam, they were uprooted by a leadership change and relocation. The tumult began with a hostile takeover at Dia in 1985, which involved the ouster of Heiner, installation of a new executive director, and Fariha's removal as the head of the board. Despite the enormous amount of money Fariha had devoted to it, the foundation was on the verge of insolvency, and sponsored artists had begun to express discontent with the directors. After a drop in oil prices changed the value of its stock holdings, Dia was forced to sell some of its properties. As a result, the _dargah_ left its Mercer Street location. Heiner and Fariha reestablished the mosque and meeting rooms in a three-story storefront building just south of a tiny municipal park on West Broadway in Tribeca. Shaykh Nur affectionately called their new home the \"jewel box\" after the great spiritual treasures held within even that diminutive space.\n\nThe 1985 Dia board of director's coup coincided with another loss that caused Fariha to reevaluate her life's mission. According to one board member, the reorganized foundation's first board meeting convened on the very \"same day that the head of [Fariha's] Sufi sect had died in Istanbul. She said she thought it was a sign from Heaven that as we took over the board their spiritual leader died.\" With Muzaffer's passing, Fariha and Nur assumed leadership of the Manhattan Sufi community while Bayrak continued to direct the Spring Valley branch. The two groups met together occasionally, as they had done in years past, but tensions often surfaced, particularly over the question of whether it is Islamically correct for a woman to lead. In the meantime, Rauf continued to deliver Friday _khutbah_ s for the Nur Ashki Jerrahis and eventually collected them in book form as _Islam: A Search for Meaning_. The goal of his teachings, he explained in the foreword, was to \" _remind_ \" (a primary Sufi term) \"the congregation of its relationship with God and to exhort them towards Him.\" This was a task Rauf invested with even greater import in 1995 when Nur passed away after fifteen years as the effective leader of the Manhattan Jerrahis and left guidance of the Sufi group and its satellite communities exclusively to Fariha. Female leadership is not uncommon among the satellite groups, some of which are also headed by women. Such leadership is less common in Muslim-majority contexts outside of North America, however, including those from which Bayrak and Rauf immigrated.\n\nNur's death occurred close to the time when Rauf published his first book and began receiving invitations to speak as an authority on Islamic law, despite his lack of formal training. Shortly thereafter, Rauf developed ties to a non-Jerrahi Sufi community and began considering the possibility of creating his own group. It would not be long before Rauf\u2014who initially focused on the Jerrahi traditions of conquering the ego and fostering personal development\u2014found himself repeating Muzaffer's strategy of stressing the aesthetic and cultural aspects of Sufism so as to dispel the specter of political Islam. Unlike Muzaffer, however, Rauf did so while working with political leaders. This was not a task he undertook in the 1990s. Only after the terrorist attacks of 2001 leveled the World Trade Center did Rauf feel impelled to defend Islam from its critics. Unbeknownst to him, the timing was fortuitous.\n\nFrom Personal Piety to Cultural Programs\n\nRauf's most well known book, _What's Right with Islam_ , quickly gained the attention of the State Department and the White House after its 2004 publication. From that year on, Rauf and Daisy Khan continually presented their public work in terms of America's Abrahamic ethics, culture, history, and future. Those 2004 themes, however\u2014like the organizations Rauf created\u2014took shape only gradually over the preceding decade. One can see their evolution in his earlier writings and witness how Rauf's ideas and organizations emerged and changed as personal transformations in his own life, such as his elevation to the status of a Sufi shaykh, intersected with national and international issues and trends.\n\nIn the 1990s, Rauf did not envision himself as the leader of an organization for engaging policymakers in \"healing the divide between Muslims and the West.\" At that time, his ambitions and activities were more modest, related to encouraging the private personal practice of Sufism. As the son of a highly regarded imam, Rauf knew that Sufi practices were not accepted by all Muslims and that they were banned in some countries. He acknowledged this in the introduction to his first book in 1995 and provided several disclaimers at the outset. \"I am not a professional Muslim scholar,\" he admitted. Yet he insisted that his emphasis on Sufism was not at odds with \"orthodox Islam.\" The khutbahs he delivered at the Masjid al-Farah, which comprised the bulk of the book, \"sought to explain both what is understood to be orthodox Islam and the general thrust of Sufism, not as separate entities, but as part and parcel of a homogenous, self-consistent, greater whole: Religion as an expression of human _self_ -fulfillment at the _personal_ , individual level.\" Writing and speaking for a diverse audience of converts and young American-born Muslims, Rauf sought not to address the basics of Islam but to help believers with the \"difficult questions,\" such as how to reconcile Sufi practice with the Prophet Muhammad's example\u2014or _sunna_ , from which the word \"Sunni\" derives\u2014when the Prophet never spoke of _tasawwuf_ directly, or how to understand the ultimate fate of family members who do not profess Islam. More than that, however, Rauf presented the book as a resource for teaching Sufi initiates how to deepen their experiences with the Divine. \"To the extent that I have studied Islam, my objective has been to achieve the perfection of faith, and not of scholarship,\" Rauf wrote, for \"it is the soul, and not the intellect, that must be addressed.\"\n\nIn addition to explicating Sufi psychology in _Islam: A Search for Meaning_ , Rauf devoted a great deal of space in that and his second book, _Islam, A Sacred Law: What Every Muslim Should Know About the Shariah_ , to defending the orthodoxy of Sufi practices and to helping practitioners understand how to live in a nation not organized around Islamic norms and institutions. For Rauf, such authentic living requires learning to separate the universal precepts of Islam from the various customs to which, he believes, they are often attached. Sufi practices and traditions are essential to this exercise because they enable devotees to tap into the part of themselves that already understands the spirit of the shari'ah and can help them avoid confusing universal precepts with local cultural rules. \"The Sufi exercises and training are meant to awaken and then exercise the various inner organs of our perception and action,\" he explained in his 1995 book. \"Our inner being is in fact a repository of wisdom connected to God in an essential and submitted fashion . . . It knows what is right and wrong for us at every given moment.\"\n\nEmphasizing the relationship between what he considers the outward, or \"external,\" practices of shari'ah and the Sufi practices that lead to a properly cultivated \"inner\" personal self ( _nafs_ ), Rauf argued that Sufism and shari'ah together provide a universal practice adaptable to any locality. Simultaneously, Rauf attempted to separate the proper practice of Sufism from the connotations it had acquired among proponents of New Age spirituality. Stressing the theme of personal spiritual development (rather than social transformation) that is a hallmark of Sufi psychology, he argued that \"Sufism, especially in America in the past twenty years, has in some quarters taken on meanings quite contrary to _tasawwuf_. It has become associated in the minds of many with poetry and dancing, with at most a light veneer of Islam. _Tasawwuf_ is properly the experiential refinement and deepening of the Islamic Shari'ah within the _psyche_ of the Sufi, and not the abandonment of it.\" For readers who wanted to know more about Islamic law\u2014or, more precisely, about the common teachings of the four major Sunni law schools, whose differences Rauf described as \"minor\"\u2014Rauf recommended his forthcoming book on the shari'ah. Although presented as a natural extension of his first monograph, that second book can, in retrospect, be seen as the beginning of Rauf's move away from his role as the imam of a local Sufi community and toward a more public life as an international Muslim leader and self-professed legal authority.\n\nIn 1995, while Rauf was finishing his first manuscript, Dr. Faroque Khan\u2014an Indian expatriate and one of the esteemed leaders of the affluent Islamic Center of Long Island\u2014invited him to address the annual meeting of the Islamic Medical Association of North America. Khan's goal was to provide Muslim physicians with guidelines for making shari'ah-compliant bioethics decisions. He was aware that Rauf's training was in physics, not Islamic law, but he also knew that Rauf was the son of an eminent Al-Azhar-trained scholar and viewed him as a suitable authority for his medical colleagues. Rauf decided to address the meeting, which was held in Malaysia, where Rauf was raised and enjoyed close ties to political elites\u2014including to the prime minister, who attended the gathering\u2014and where Rauf would later open an office for the Cordoba Initiative. After the conference, he published his lectures for American audiences.\n\nIn the resulting book, Rauf explained the importance of the science of Islamic law ( _usul al-fiqh_ ), described how Muslims could learn to separate the ethical \"spirit and intent of the Shari'ah\" from cultural practices that were viewed unfavorably in the United States, and briefly discussed the gentle approach to inculcating Islamic law and ethics that he had written about more extensively in his first treatise. That gentle approach involved introducing Muslims to the beauty of Sufism:\n\nIn the Masjid al-Farah in New York City, the mosque in which I preach, we deal with many converts and young Muslims born into Muslim families. Being new to the practice of Islam, even when born into Muslim families, they may find the sudden imposition of practicing _all_ of the Islamic rituals onerous . . . Being more concerned that the individual acquire and taste the transformative and disciplinary powers of prayer, fasting and the other joys of worship, and anchor it on a very strong foundation of belief and love of God, we encourage our congregation to _taste_ the Divine Presence which urges the human soul to then desire worship and _taste its sweetness_ . . . Our focus is therefore on how to increase our congregation's love for God, for love feeds the urge towards discipline far more effectively than discipline feeds the urge to love.\n\n\"Tasting\" is a common Sufi metaphor for _dhikr_ and one Rauf often used in his _khutbah_ s. Such tasting, he argued in his second book, is key to understanding the \"reasoning behind the Shari'ah . . . [and] enables you, in new circumstances and when faced with modern dilemmas, to apply your reasoning to arrive at a _comfortably correct_ decision.\" The purpose of Islamic law, he further explained, was not to make life more onerous but easier. Importantly, Rauf cautioned, such personal interpretation (known in Arabic as _ijtihad_ ) does not qualify just anyone to claim legal authority, and the work of scholars and jurists is still necessary. Nevertheless, even Muslims who do not have regular access to such experts are not without resources, so long as they understand the law's requirements, its ethical intent, and how to discern interpretations based on customs from those rooted in its essential aspects. Reaching for an illuminating historical analogy, Rauf described the role of Sufi practice in helping Muslims to understand and enact Islam properly as akin to the role of Vatican II in helping American Catholics develop a more personal and\u2014implicitly\u2014rational faith.\n\nRauf's mention of Vatican II was not his first allusion to the commonalities between Christians, Muslims, and Jews. In 1995, Rauf had noted\u2014briefly\u2014that Islam is \"part of the Judeo-Christian ethic, not in the usually defined sense, but certainly from the Qur'anic point of view.\" This subject occupied more space in Rauf's second book, however, and was presented there with fewer caveats about religious differences. For example, in a chapter of his second book titled \"Can the Shari'ah Evolve,\" Rauf\u2014giving evidence of Nasr's Perennialist influence\u2014elaborated on how engaging in Sufism and apprehending the universal aspects of Islam allowed Muslims to see their commonalities with Jews and Christians. While the ritual forms, or \"outer manifestations,\" of various traditions differed, he contended, the \"inner content\" was identical. With regards to prayer, \"'differentiation' [w]as an 'evolution' of a kind. What evolved was not the eternally mandated practice of ritual prayer in its inner sense and dimension, but the outer manifestation (language, movements, etc.) to suit the language and the reality of the time . . .\"\n\nAlso in his first book, Rauf had asserted that there \"are fundamental philosophical differences between Islamic law and United States law.\" He did not include such distinctions in his second treatise. Yet, although Rauf devoted slightly more attention in his second book to interreligious commonalities and to Muslims' abilities to live authentically while observing United States laws\u2014themes that would consume the bulk of his 2004 work\u2014he still had not articulated the thesis of American Abrahamic commonality for which he would become known after 9\/11. His primary focus was not yet on establishing interreligious harmony, serving as an international authority on Islamic law, or acting as a mediator between Muslim-majority countries and Christian ones\u2014what he would later describe as his goals with Cordoba. Rather, in the 1990s, even while presenting himself as something of an authority on Islamic law, Rauf was becoming more committed to promoting Sufism than ever before. This was especially the case after 1996, when Rauf developed ties to a shaykh who saw in him the qualities not just of an imam but of an anointed international leader.\n\nEleven years after Muzaffer's death and one year after Nur's, Rauf attended a sacred musical festival in Morocco. Before he left New York, Fariha's husband, Heiner, whom Rauf describes as \"very perceptive\" and who perhaps imagined that Rauf would not be comfortable with a religiously eclectic female shaykh, cautioned him: \"now don't go and take hand with another sheikh there.\" While Rauf did not take hand with another leader on that particular trip, it was not long before he did. During the spring of the following year, Rauf returned to Morocco, where he met Shaykh Hamazah ibn 'Abbas al-Qadiri, the leader of the Qadiri Sufi order whose lineage he would later assume. That year\u20141997\u2014was an eventful one for Rauf. Not only did his travels to Morocco constitute physical separation from his Jerrahi community, Rauf created additional distance from them when, just a few months after first meeting Shaykh Hamazah, he cofounded the ASMA Society for \"enhancing the general public knowledge of Islam and Sufism in particular and promoting personal and spiritual growth and development through meditation and prayer.\" Faiz Khan, who grew up attending the Islamic Center of Long Island, and Daisy Khan, Faroque Khan's niece, served as Rauf's cofounders. (Acting on an injunction revealed to her in a dream that Rauf interpreted, Daisy Khan had married Rauf the year before.) Rauf and Khan would steer the ASMA Society through a multitude of endeavors over the next decade and a half, as unexpected developments continually presented them with opportunities to redefine their objectives and expand their mission\u2014albeit in ways that moved them ever further from their local Sufi practice.\n\nASMA's 1997 articles of incorporation did not present the organization as particularly ambitious. Its founders\u2014already busy working as a real-estate manager (Rauf), corporate project manager (Daisy), and emergency-room doctor (Faiz)\u2014outlined only two basic strategies for promoting Sufism: hosting \"meetings, workshops, lectures, and special events\" and conducting \"meditation and prayer sessions.\" During the organization's first several years, these sessions\u2014generally, _dhikr_ services held on Friday evenings so as not to conflict with the Jerrahi _dhikr_ services at Masjid al-Farah on Thursday nights\u2014occurred at Rauf's New Jersey home or in the Manhattan living room of Daisy Khan's one-bedroom apartment.\n\nIn 1999, however, two transformative developments brought challenges and opportunities for recognition that prompted Rauf and Khan to reorient ASMA and alter its stated objectives. First, while on another trip to Morocco, Rauf accepted a new mandate for his work in the United States. According to Rauf, Shaykh Hamazah, a former high-ranking official in the Moroccan military, believed himself to be holding his position as leader of their Qadiri _tariqa_ \"in trust for its owner, who will come from abroad to assume it.\" During the month of Ramadan, Hamazah had a dream that revealed Rauf as that rightful owner. \"I am now responsible for introducing the _tariqah_ in the West,\" Rauf explained to a journalist the next year, \"except that now it is to be presented to the West in ways it can understand.\" Second, a Washington, DC, State Department forum on Islam gave favorable emphasis to Sufis and provided Rauf with a ready audience for his new mission. At that forum, the leader of another Sufi group\u2014Muhammad Hisham Kabbani, the US representative of the Naqshbandi-Haqqani Sufi Order, who founded the Islamic Supreme Council of America in 1998\u2014reinforced the tendency among State Department officials to identify Sufis as peaceful, apolitical, and moderate and to view all other Muslims as possible extremists.\n\nKabbani, who was aware that some Muslims consider Sufism to be suspect and who wanted to make allies for Sufis out of American policy makers, told the audience at that forum that he had visited over one hundred American mosques and found 80 percent of them to be run by \"extremists\" and \"fundamentalists.\" Many American Muslim organizations protested this caricature and demanded Kabbani retract his statement. Nevertheless, the Bush administration, following President Clinton's example, continued to court Kabbani as an ally, to treat him as an authoritative spokesperson for and about American Muslims, and to include him in White House Iftar (Ramadan fast-breaking) events.\n\nAfter 9\/11, Bush's State Department would place even greater emphasis on seeking Sufi allies at home and abroad, as both the United States and Britain condemned extremism \"by praising Sufism as the tolerant form of Islam.\" ASMA and, later, the Cordoba Initiative would ultimately benefit from and become important to these Bush-administration endeavors. In 1999, though, Rauf was not aware of the State Department's developing attitudes toward different Muslim communities. Nor did Rauf share Kabbani's belief that American Muslims were overwhelmingly extremists, despite the fact that he would later echo his American Muslim spokesperson competitors (black American and immigrant, Sufi and otherwise) by presenting himself as a uniquely \"moderate\" Muslim voice.\n\nOver the next several years, Rauf raised money for ASMA by drawing honoraria from talks before Muslims and interreligious audiences; the venerable Chautauqua Institution in southwestern New York, for example, has been one of Rauf's frequent lecture sites since then. He also benefited from the increasingly positive attention given to Sufis after 9\/11, particularly to those who emphasized the aesthetic beauty of Sufi practices and who juxtaposed appreciation of such \"cultural\" traditions to the rigid dogmatism of ostensible extremists. In so doing, Rauf continued to follow the examples set by Muzaffer and Nasr of promoting Sufism, in part, by emphasizing the \"inner truths\" of Islamic art, including Sufi poetry and the Qur'anic recitations (a central part of Sufi _dhikr_ ) that Rauf likens to music. This was something he had done even prior to 9\/11. Shortly after the 2001 attacks, however, as I discuss in Chapter 4, it became a crucial way for ASMA to distance itself from the specter of extremism.\n\nIn addition to emphasizing the moderating influence of Sufism and (or, _as_ ) Islamic culture after 9\/11, Rauf began to focus on making Islam more sympathetic to non-Muslim audiences. Incorporating his work on the Sufi practice of _dhikr_ into a larger treatise that combined themes from his previous writings with political and economic policy prescriptions, Rauf fit Sufism and other Islamic traditions into narrative templates that other marginalized religious minorities had previously created\u2014narratives that highlighted commonalities between religions (particularly regarding ethics, civic service, and economic freedom) and, in Rauf's version, between American laws and Islamic ones. Relying on his father's teachings, Michael Novak's economic writings from the 1970s and 1980s, and Will Herberg's 1950s narratives of inevitable immigrant upward mobility, however, Rauf portrayed only certain segments of the American Muslim population (e.g., recent Arab and South Asian immigrants) as normative and typically moderate. In so doing, Rauf tapped into the racialized debates over militancy and moderation discussed in the previous chapter and heightened existing contests over representation.\n\nConclusion: Sufi Sympathies and the Politics of Representation\n\nBy the end of the twentieth century, American Sufis represented a wide variety of ethnicities, and both black American Sunni leaders and more recent immigrants had garnered influence with the White House and State Department, offering sometimes-competing perspectives on American Muslims' traditions, needs, and interests. As discussed in the previous chapter, these diverse figures had struggled to have their own expressions and practices accepted as the most authentically Islamic and most authentically American since the 1970s. In part because of this struggle, black Americans were not as likely as white or even immigrant Muslims to divulge their Sufi sympathies if they had any\u2014something that may have cost them opportunities for representation after 9\/11.\n\nAmong the black American Muslims who avoided using the term \"Sufism\" (with its connotations of whiteness and of unorthodox behavior) to describe their practices was Shaykh-'Allama Al-Hajj Ahmad Tawfiq, a former member of El-Hajj Malik al-Shabbaz (Malcolm X)'s Muslim Mosque, Inc., a graduate of Cairo's Al-Azhar University, and the founding imam of the Mosque of Islamic Brotherhood in Harlem (incorporated in 1967). The Sufi practices held at the Mosque of Islamic Brotherhood took place less than a half mile from the Cathedral of Saint John the Divine, where affluent white audiences gathered to watch Muzaffer's dervishes whirl. Although the largely black American Muslim population there lived in a blighted section of the city rather than an affluent one and connected their religious practices to social justice concerns more than to psychological introspection, they were no less open to members of other religions or the general public than the Jerrahis. In fact, Imam Talib Abdur Rashid\u2014a disciple of Shaykh Tawfiq who assumed leadership of the community after the founding shaykh's death\u2014was well known among New York religious leaders long before Rauf joined the Manhattan Jerrahis.\n\nLess sensationalist than Imam Siraj Wahhaj, the nationally recognized leader of Masjid al-Taqwa in the Bedford-Stuyvesant neighborhood of Brooklyn, Abdur Rashid had been involved in interreligious events and social activism in the city for decades by the time Rauf founded ASMA. In 1979, for example, just after Rauf's father addressed academics at the American Academy of Religion's Manhattan meeting, \"Imam Talib\" (as he is commonly known) and James Park Morton convened a joint discussion at Saint John the Divine to defend Muslims against the specter of terrorism. Years later, after the first World Trade Center bombing, the _New York Times_ frequently covered Abdur Rashid's activities, which included serving as a regular prison chaplain, participating in a weekly \"trialogue\" with a church and a synagogue on the Upper West Side, lobbying the Time Warner Cable company to keep a religiously oriented channel on the air, and holding New Year's Eve services at his mosque. These latter events, one _Times_ reporter described, included _dhikr_ , a potluck, and the music of Miles Davis.\n\nAbdur Rashid's work with Christian and Jewish communities and his openness to the media paved the way for Rauf, who\u2014as he acknowledged to a _New York Times_ reporter in 2004\u2014had little to do with politics, the press, or interreligious dialogue in the 1990s: \"The attacks changed me . . . Before September 11, I was an Islamic teacher focusing on the theological, spiritual, and jurisprudential side of my faith. I suddenly had to explain my faith and myself. I started going to television and radio studios, churches and synagogues, talking with other religious leaders about Islam.\" So successful was he in communicating with non-Muslims that Rauf quickly became a sought-after source for journalists. His appearances in the pages of the _Times_ \u2014where he was identified as a Sufi and as a proponent of a \"moderate Islam\" that \"embraces the values of Western democracy, carries within it a love of America, and calls on Muslims to respect other faiths\"\u2014quickly equaled and then surpassed those of Abdur Rashid. In the meantime, Rauf developed relationships with anchors at CBS, CNN, and other television networks, as well as with journalists and editors at other publications. Most notable among these was _Newsweek_ , which featured a picture of Rauf's Sufi community on its cover in 2007 and made Rauf and Khan regular contributors to the _Newsweek-Washington Post_ blog, _On Faith_ , for several years after that, affording them a national platform for discussing their priorities and disseminating their perspectives. Like Abdur Rashid and other Muslim American leaders before them, Khan and Rauf were concerned about the rights and needs of Muslims after 9\/11 and became cautiously more vocal about their politics during the decade that followed. As they did so, however, they also made clear that their priorities increasingly had to do with international issues and not with the domestic communities that other Muslim leaders struggled to represent.\n\nIn Chapter 4, I turn to the histories of ASMA and Cordoba after 9\/11 and to Rauf's and Khan's rapid rise in prominence as Sufi moderates. Making Sufi practices central to moderation is something that, for Rauf (who emphasizes Sufism both sincerely and strategically), is like walking a tightrope. Although many Americans, State Department officials and otherwise, consider Sufis to be mystical moderates, an increase in anti-Sufi invective among Muslims worldwide in the last two centuries means that associating with Sufism could compromise Rauf's authority as a Muslim leader, especially in the Muslim-majority countries for which his project on Islamic law is intended. As I discuss, Rauf and Khan frequently changed the description and objectives of their organizations after 9\/11 in response to new political circumstances and new opportunities to expand their work. Following Muzaffer's example immediately after the attacks that felled the World Trade Center, Rauf and Khan began to present the ASMA Society and Sufism as dedicated less to personal devotion than to apolitical cultural development and appreciation. A few years after that, with their programs enjoying a generally positive reception and Rauf and Khan forging strong ties to religious, political, and media elites, they began to focus on and acknowledge more political aspirations.\n\nAlthough Rauf and Khan minimized their Sufi roots and practices in the years that followed (and even renamed ASMA in 2006 so as to remove Sufism from the title), their identities as Sufi Muslims worked to their advantage with US politicians, both Republican and Democratic. In the meantime, however, as their organizations expanded and changed, many members of their original community\u2014the Sufi devotees who gave time and money to help these institutions grow out of the fledgling stage\u2014chafed at Rauf's increasing absences from the mosque and _dhikr_ group and at aspects of his model of moderation. These dynamics expose some of the pressures and pitfalls of defining moderate Islam after 9\/11\u2014pressures that would only intensify in 2010 when the Ground Zero Mosque controversy erupted and, faced with the realities of racial and religious discrimination in the United States, Rauf and Khan found that they had only tepid support from Muslim American communities they had failed to represent or, worse, had alienated.\n4\n\nFROM SUFISM WITHOUT POLITICS TO POLITICS WITHOUT SUFISM\n\nVAST QUANTITIES OF INFORMATION about Islam inundated New Yorkers between 2001 and 2010, as text, images, and audio about local Muslims and American military invasions flooded city neighborhoods like late summer rains. Residents waded through these muddy waters from the moment of turning on the radio, television, or computer in the morning to the evening commute home through transit stations full of newsstands and color-coded alert warnings. Opportunities to assess Muslims also came from museum exhibits on Islamic art and culture and films about routing terrorists. Meanwhile, Muslim leaders and performers\u2014ranging from hip-hop artists to whirling dervishes, who graced venues as diverse as subway platforms, Christian cathedrals, the Apollo Theater, and Carnegie Hall\u2014attempted to draw fellow New Yorkers away from sensationalist depictions of radicalized Muslims and toward more varied interpretations.\n\nThe ASMA Society participated in these efforts by holding events in 2002 and 2003 at three different prestigious churches. On a snowy Saturday in January 2002\u2014just four months after the attacks on the World Trade Center\u2014Rauf and Daisy Khan gathered an array of Muslim artists at the Cathedral of Saint John the Divine. Their program, \"Reflections at a Time of Transformation: American Muslim Artists Reach Out to New Yorkers,\" featured visual art exhibits, music, poetry, and spoken-word performances. The well-heeled attendees\u2014\"affluent immigrants and the progeny of affluent immigrants,\" as one observer described them\u2014were largely prosperous Muslim professionals who wanted to counter the prevalent image of overly politicized Muslims by organizing cultural outreach programs rather than by making political pronouncements.\n\nAs was the case with the first performance of Sufi _dhikr_ at that cathedral decades earlier, UN ambassadors were in the audience on January 17, 2002; so were Deputy Mayor Dennis Walcott and representatives of the New York City Police and Fire Departments. WNYC radio show host Brian Lehrer served as master of ceremonies. At one point, between musical sets and as members of various religious communities conversed over samosas and tea, Rauf stepped to the podium to address the crowd. Putting the centrality of art to spiritual well-being in both a psychological and political context but refraining from overt political commentary, Rauf described the malaise he saw afflicting contemporary Muslims. \"September 11th created a tidal wave of human shock and emotions, with a desire to do something about it. The most an artist can hope for is that what he or she might offer will touch the life of their fellow citizens . . . For the modern Muslim, a crisis in the area of art has contributed to perhaps the profoundest crisis Muslims face today: a crisis of the soul.\"\n\nWith this 2002 event and the ones that followed, Rauf and Khan began to shift the description and purpose of the American Sufi Muslim Association away from Sufi practice and psychology and toward cultural programming. By the next year, they explicitly planned to make such programs a means for changing Americans' ideas about Muslims. In January 2003, ASMA cosponsored\u2014along with a local United Methodist Church and synagogue\u2014a theatrical performance titled _Same Difference_. That production, under Khan's codirection, ran for sixteen sold-out weeks and represented commonalities among different religious New Yorkers in their responses to 9\/11. The following June, using a grant from the Tides Foundation and partnering with Christian and Jewish congregations, as well as with the Islamic centers of New York (where Rauf served as a trustee) and Long Island, Rauf and Khan staged the \"Cordoba Bread Fest\": a gathering that included a dramatic rendering of the interreligious exchanges that characterized the city of Cordoba during the twelfth century. As Rauf and Khan later explained, their hope with such projects was to \"recreate\" the \"great flowering of culture, art, [and] philosophical inquiry\" that intellectuals from the Abrahamic faiths once shared.\n\nWhile the purpose and format of this second 2003 event resembled ASMA's other post-9\/11 cultural programs, its name signaled yet another transition. By the time of the Cordoba Bread Fest, Rauf and Khan had already met John S. Bennett, a trustee of several interreligious organizations, the former vice president of a prestigious Washington, DC-based think tank, and the man with whom they would establish the Cordoba Initiative in 2004. With Bennett's help, Rauf began to tailor his newest book to serve as a primer on and advertisement for the work of that more policy-oriented organization. Additionally, Rauf and Khan began to publically redefine ASMA's mission and objectives again and created a new website on which to feature their new projects.\n\nRauf and Khan would edit that website's description of ASMA's projects and purpose, as well as the descriptions they posted on the new Cordoba Initiative site, many times between 2004 and 2010. Combining information derived from these changes with materials gained from interviews and participant observation at Masjid al-Farah, at Rauf's Friday night _dhikr_ services, and at ASMA Society and Cordoba Initiative events during the same years, I chart the institutional histories of the ASMA Society and the Cordoba Initiative in this chapter, mapping the development of their various programs and initiatives and showing how aspects of their objectives and self-presentations changed during the eight years and two wars that occurred between their first cultural event and the summer the Ground Zero Mosque debate erupted.\n\nAlthough Rauf and Khan had already changed focus significantly since they founded the ASMA Society, moving from Sufi practice and psychological well-being to cultural programs, their objectives and aspirations expanded again as Rauf completed his 2004 book and cofounded the Cordoba Initiative. With Bennett's introductions to a wide array of political, religious, and business elites, Rauf soon changed from a local\u2014albeit well-off and well-connected\u2014imam and Sufi shaykh into a recognized international figure. In prior years, Rauf had almost ceaselessly promoted Sufism by delivering lectures at national venues such as the Chautauqua Institution and Aspen Institute. More frequently, he did so by addressing smaller but no less high-profile audiences, such as in a September 2002 course he taught at a prestigious Manhattan cultural center titled \"Mystical Islam: Taste of the Sufi Path.\" After meeting Bennett, Rauf and Khan began to speak about and approach their work differently: as something larger and more politically significant.\n\nThese budding changes were evident as early as 2003 in an address Rauf delivered at the National Cathedral in Washington, DC. Rauf did not describe his work in terms of Sufi practice during that engagement. Nor, for the most part, did he discuss the cultural programs he and Khan had focused on since 9\/11. Rather, introducing what would become the refrain of his 2004 book and Cordoba Initiative motto, Rauf proposed to \"heal\" the rift between the United States and Muslims worldwide within a decade, thus averting further terrorist attacks. \"Reversing the legacy requires a NASA like effort,\" he argued, hinting at the ambitious scope of his project and the policy-oriented focus he would soon adopt. Meanwhile, Khan worked with Rauf to expand ASMA's endeavors and develop programs devoted not just to aesthetic appreciation but also to equipping young American Muslims to defend the United States and their religious traditions equally, a project that evolved out of meetings Rauf and Khan held with young activists who had asked Rauf to teach them more about Islam after 9\/11.\n\nIn 2006, once his and Khan's reputation as Sufi moderates was well established at the State Department and elsewhere, Rauf funneled his energies primarily into his Cordoba Initiative project to evaluate the legal systems of Muslim-majority countries and Khan assumed leadership of ASMA. No longer wary of being cast as improperly political, they also again redefined both organizations' goals to reflect the growing international focus and political nature of their work. As these developments indicate, Rauf's and Khan's priorities and their narratives about Sufism and Islam continuously shifted over the course of a decade. Deemphasizing Sufism and focusing on Islam's compatibility with American democracy and capitalism allowed them to take advantage of new opportunities and to more accurately reflect their new programs, as I discuss below. It also gave rise to new problems, however, as some of their volunteers (often members of their Sufi community) objected to ASMA's turn away from Sufism, while others objected to what they saw as Rauf's overly idealistic promotion of American racial equality and democratic capitalism.\n\nI elaborate here on these changes and on how the ASMA Society and Cordoba Initiative expanded from joint self-professed \"cultural and educational\" organizations into two distinct politically oriented institutions that, by 2008, were almost entirely separate from the _dhikr_ group in which they originated and the mosque Rauf still called home. In tracing this trajectory, I do not suggest that Rauf and Khan ever hid their political goals. Rather, I show how their objectives and self-presentations changed in keeping with the quickly, often dramatically, shifting challenges and opportunities that confronted them in the decade following 9\/11. Initially focusing on culture rather than politics to assuage government officials' fears was a time-worn Sufi strategy\u2014one Rauf's first shaykh employed in Turkey to escape state repression. As Rauf and Khan soon learned, however, US leaders viewed Sufis much more favorably than did leaders of some other nations. With their political opportunities expanding and political goals evolving, it did not seem necessary to emphasize Sufism or cultural work _rather_ than political activity\u2014at least, not so long as that political activity resembled what US leaders defined as \"moderate.\" In 2010, however, when their plans to open an Islamic community center received national attention, even such purposefully moderate activity did not escape criticism.\n\nPolitics and Cultural Promotion after 9\/11\n\nThe ambition and energy, as well as the transitional quality, of Rauf's and Khan's endeavors in the first few years after 9\/11 were evident in the variety of programs listed on ASMA's 2004 website. There, ASMA's reworked \"mission statement\" defined it not in terms of Sufism but as \"an Islamic cultural and educational organization dedicated to fostering an American-Muslim identity and building bridges between American Muslims and the American public.\" The organization's new goal, described as a \"philosophical objective\" rather than a political one, was \"to strengthen a culturally American expression of Islam based on tolerance and on cultural and religious harmony and to foster an environment in which Muslims can thrive within a pluralistic society without compromising their essential values and beliefs.\" This was a development destined to happen, Rauf often argued\u2014as in 2004 and 2005, when he echoed Herberg's assimilation paradigm:\n\nWhen Christians first came [to America], within a couple of generations the religion assumed a different character . . . even in the Old World you had differences between German Lutherans and Swedish Lutherans, so much so that when they came here, they wouldn't even _speak_ to each other. The same thing with Judaism, but after a generation or two, a culturally American Judaism emerged . . . So it is my conviction that the same thing is bound to happen to the Islamic experience.\n\nIn addition to creating a culturally American Islam, ASMA proposed to meet six \"key objectives\" within five years, none of which explicitly involved Sufism. Rather, Rauf and Khan framed their objectives in a way that reflected their more political understanding of the work they had undertaken since 9\/11. Instead of promoting cultural appreciation alone, Rauf and Khan began to make subtle political claims about the role of culture in producing a moderate Muslim tradition. Not only would they create a \"culturally American expression of Islam\" that showcased Muslims' abilities to integrate into the social fabric of the United States, they would encourage American Muslims to regard some aspects of other cultures\u2014often ones that clashed with multicultural values\u2014as inconsistent with Islamic faith and practice. Their final objective acknowledged this new approach most directly, as I discuss below, although much of the more explicitly political work proposed under that heading would be undertaken not by Muslims alone but by the interreligious Cordoba Initiative.\n\nAccording to their website, ASMA's six key objectives included, first and foremost, forging \"an American Muslim identity.\" Second, the organization planned to train and empower young American Muslims to be spokespersons for \"a tolerant, harmonious, authentic Islam,\" which meant \"encouraging them to identify with the _essentials_ of the Islamic faith _that cut across cultural boundaries_ \" (emphasis in the original). This was something ASMA had already started with a Muslim Leaders of Tomorrow program, the first meeting of which occurred in 2004 at the interreligious Garrison Institute that Bennett had founded in the Hudson Valley. Third, ASMA proposed to disabuse non-Muslims of their negative stereotypes about Muslims while simultaneously \"dismantl[ing] myths regarding Americans\" in the Muslim world. Fourth, reflecting the aesthetic appreciation work Rauf and Khan had already undertaken, ASMA would celebrate the role of Islamic art \"in contributing to world civilizations\" and would support contemporary Muslim artists and \"their inclusion into the artistic fabric of America.\"\n\nAMSA's fifth objective\u2014to support collective \"spiritual evolution\" by building \"interfaith\" networks based in \"the common substrate of religion\"\u2014was the only one linked, implicitly, to the organization's original Sufi agenda. Finally, its sixth objective, which fell under the heading of \"intrafaith\" endeavors (i.e., work to be done among Muslim communities) echoed the liberal political philosophy Rauf had expressed in his 2004 book and the more overt political aims that would become a hallmark of the Cordoba Initiative. This intrafaith objective involved amplifying \"Islamic arguments demonstrating that Islamic texts, theology and law support the principles of separation of powers, justice, women's rights, and freedom of religious practice.\"\n\nDespite posting these newly defined goals on ASMA's website and creating a separate page to discuss Cordoba's more political work, the two organizations were hardly separate in 2004. Governed by its own articles of incorporation, which Bennett filed in his home state of Colorado, the Cordoba Initiative still depended on ASMA for staffing, resources, and even funding. Further, the leadership of the two organizations overlapped considerably. Not only did Khan serve as a founding director of Cordoba alongside Rauf, Bennett, and Julia A. Jitkoff, a sculptor who hailed from a family of financial planners and kept a home in Aspen, but Bennett was placed on ASMA's website under its list of directors.\n\nAlthough he cofounded the Cordoba Initiative in 2004, Rauf did not devote the majority of his time to that organization until 2006. In the meantime, as Rauf promoted his book on Abrahamic ethics, economic progress, and public policy in front of ever more elite audiences, Sufi dervishes filled in for him at the mosque and on the local lecture circuit, drawing honoraria for ASMA while spreading the message that Sufism is moderate and authentic Islam. With many of ASMA's other programs in process but Rauf frequently traveling, Khan also put more energy into the organization, giving up her corporate job in 2005 to work full time as ASMA's executive director. Meanwhile, Bennett sought occasions to promote Cordoba's mission of establishing peace in the Middle East and countering extremism with democratic capitalism\u2014purposes outlined in the organization's application for tax-exempt status\u2014at the Aspen Institute and other elite forums.\n\nBenefitting from Bennett's connections to Aspen Institute associates and from the additional notice he gained while promoting his book, Rauf embarked in the following years on an increasingly high-profile lecture tour that took him ever-greater distances from his Sufi community. Although for some time Rauf continued to participate in local interreligious events and teach classes on Sufism and Islam at local colleges, seminaries, and houses of worship, his message of common Abrahamic ethics and of the commonality between Islam and American law and capitalism eventually supplanted Sufism at both local and international venues. Yet Rauf's identity as a Sufi shaykh remained crucial to gaining entr\u00e9e with domestic political leaders.\n\nChris King, one of several State Department employees who engaged Rauf in international cultural diplomacy initiatives in 2006 and after, first heard Rauf speak about Abrahamic commonality in 2004 at St. Bartholomew's Church in Manhattan, where Rauf taught classes on Sufism and was known as a progressive Sufi imam. Since the 1990s, as discussed in Chapter 3, US officials and defense analysts had increasingly viewed Sufis as moderate Muslims and possible allies. This trend continued after 9\/11. In 2003, for example, the National Security Research Division of the RAND Corporation (a federally funded research and policy advising organization) published a report identifying Sufis as likely proponents of the \"civil democratic Islam\" that the United States and other world powers hoped to cultivate. \"Sufi influence over school curricula, norms, and cultural life should be strongly encouraged\" in Iraq, Afghanistan, and elsewhere, the report's author argued. The year after Rauf met King, another report (written in 2005 but published in 2007 by the RAND Center for Middle East Public Policy) identified Sufis like Rauf\u2014who assert Islam's compatibility with democracy but otherwise eschew \"political activism\"\u2014as \"moderate\" partners for countering extremists.\n\nThe decision to progressively minimize Sufism once Rauf gained entr\u00e9e into such centers of influence was not simply a strategic one. With each year that passed after 9\/11, Sufi practice became less important to ASMA and Cordoba programs than did fostering cultural, political, and economic liberalism. While promoting ASMA and the Cordoba Initiative in 2004, for example, Rauf became a member of the World Economic Forum's Council of 100 Leaders and a participant in the C100 \"West-Islamic World Dialogue\" involving \"business, political, religious, media and opinion leaders.\" In 2005, Rauf also began to participate in a Council of Foreign Relations taskforce on democracy in the Arab world involving former Secretary of State Madeline Albright, in the process developing a relationship that would lead Albright to praise the Cordoba Initiative in her 2006 book. Meetings with former President Clinton, under whom Albright had served, soon followed.\n\nIn January 2006, Clinton afforded Rauf an audience during the annual meeting of the World Economic Forum in Davos, during which Rauf also contributed to a debate concerning \"Islam's Challenge to Eradicate Extremism.\" Clinton met with Rauf again that September at the Clinton Global Initiative meetings. On the latter occasion, organized by Malaysian Prime Minister Abdullah Ahmad Badawi, a self-proclaimed progressive who had taken an interest in Rauf's work, the former president held private meetings involving both Rauf and Albright in discussions of religion, philanthropy, and international politics.\n\nIncreasingly political, Rauf's Cordoba Initiative work was far from partisan. In fact, if anything, Rauf worked more closely with the Republican administration then in the White House than he did with Democrats. In February 2006, for example, Rauf joined Undersecretary of State for Public Diplomacy Karen Hughes, a close advisor to President Bush, in a US-Islamic World Forum in Doha, Qatar, as part of Hughes' \"listening tour\" to improve the image of the United States in the Middle East. Rauf later wrote that he hoped to continue talks about Islam and international peace with members of the Bush administration. Such opportunities were not long in coming. Soon he was advising US ambassadors about Islam and hosting visiting imams who traveled to the United States from Arab countries as part of State Department programs.\n\nAlthough Rauf's new projects garnered attention more rapidly than he perhaps expected, all were in keeping with Cordoba's primary objective as stated on its 2004 page on ASMA's website: to \"provide U.S. policy makers and the international press with informed research and critical thinking regarding ways to improve the relationship between America and the Muslim world.\" As Rauf spread the message of global ethical commonality and economic opportunity among international political and religious elites in 2005 and 2006, Daisy Khan organized events and conferences to address ASMA's new primary demographic: the young, cosmopolitan Muslim Americans whom she and Rauf\u2014following Herberg's paradigm\u2014believed would be the future of a culturally rich and culturally American Islam. Faiz Khan echoed their message of young Muslim American cultural solidarity and their Herbergian model of generational change during these years, similarly arguing that the most important thing for Muslim Americans after 9\/11 was to \"get acquainted with what is authentically Islamic, and what is not.\" Yet for him, being a culturally American observant Muslim meant being able to offer a more full-throated critique of American policies than Rauf and Daisy did, and to question government actions\u2014even at the risk of seeming prone to conspiracy theories. As we shall see, these were positions that, for Rauf, pushed the boundaries of moderation, impelling him to police the limits of acceptable political speech within his organization.\n\nSufism as Necessary but Insufficient: The Limits of Critique for Moderate Muslims\n\nLike Rauf, Faiz Khan concentrated on studying Sufi psychology before the terrorist attacks of 2001. Raised in a heavily Jewish suburb of Manhattan, where he had memorized Hebrew songs from countless bar mitzvahs, Faiz later backpacked through North Africa and the Middle East, encountering a variety of scholars and Sufi leaders along the way. These experiences, he felt, taught him not only about the diversity of Islam and other religious traditions but also about the fundamental commonalities among them. After cofounding ASMA with Daisy and Rauf (though, unlike them, he was not involved in its executive operations), Faiz also began to deliver lectures about Islam and Sufism to national and international audiences. But the terrorist attacks of 2001\u2014during which Faiz waited in his Queens emergency room and then a makeshift hospital at Ground Zero for wounded survivors who never came\u2014and the wars that followed transformed him. Though not immediately changed, Faiz Khan eventually became more overtly political.\n\nIn a 2004 interview for a book about Muslim Americans, titled _American Muslim Voices_ , Faiz still emphasized Sufism's centrality to Islamic practice. Additionally, though, he described his new personal goal, which was similar to the \"philosophical objective\" stated on ASMA's 2004 website: to help Muslims separate un-Islamic things from those that lie at the heart of every religion. Such un-Islamic, irreligious factors, in his view, included the \"cultural baggage\" that immigrants\u2014especially immigrant imams\u2014often brought to the United States, as well as \"criminal thing[s]\" like terrorism. So far, these changes were quite similar to those his ASMA colleagues underwent. Unlike Rauf and Daisy, however, who refrained from discussing contemporary racial and economic inequalities\u2014the persistence of which could challenge their narrative that American democratic capitalism is socially just and Islamically ideal\u2014Faiz admitted that racism was a current issue in the United States, including among Muslims, and provided a different take on upward mobility and assimilation.\n\nFaiz realized that the experience of his parents and many other immigrants of the 1960s and 1970s\u2014who were mainly educated professionals, not the unskilled workers of previous immigrant generations\u2014was very different from that of most black Americans.\n\nGrowing up in [Long Island, in] an upper-middle-class suburb, I was definitely different from other Muslim youths from families not so well-off. My parents were both doctors and led very busy professional lives. The Muslim scene in America is very diverse. What you have is an immigrant class that made good. They came in, a lot of educated professional engineers and doctors . . . and really lived the American dream . . . African Americans make up a large part of the American Muslim world, where the scene changes.\n\nMembers of his parents' generation (the generation Rauf belongs to) had good intentions, Khan believed, but \"when it came to crossing the socio-ethnic lines and building an embracing community\u2014a component of true Islam which recognizes no difference between one human being and another based on color or outward attributes\u2014the embrace was less than robust.\"\n\nNevertheless, Faiz, like Rauf and Daisy, thought that second-generation American Muslims had more open outlooks than their elders: \"[T]here is most definitely a deep-seated racist and classist element that exists amongst Indian and Pakistani immigrants toward others\u2014particularly African Americans,\" he acknowledged, but \"that _a priori_ doesn't exist in my generation because we grew up here . . . we're comfortable with everybody.\" With regards to the country's racial and socio-economic disparities, his primary strategy at that time\u2014again, following Daisy and Rauf\u2014was to emphasize Muslims' commonalities with other Americans and to focus on the positive aspects of life in the United States. As the wars in Afghanistan and Iraq dragged on and US actions against various Muslim populations, from prisoners at Abu Ghraib to housewives targeted by domestic surveillance programs, grew increasingly repressive, that strategy no longer satisfied him.\n\nFrom 2004 to 2006, Faiz grew more vocal about racial and economic injustice. He did not do so when delivering _khutbah_ s at the mosque, where he filled in for Rauf and maintained that true religion was _not_ political, but he did in his talks with Rauf's other dervishes and in his writing for other audiences. And not only did he discuss contemporary racism and socio-economic inequalities, implicitly contradicting Rauf's narrative of domestic social and economic flourishing, Faiz also condemned American injustices overseas. He was particularly critical of the role of American corporations\u2014ones Rauf lionized\u2014in creating such injustices.\n\nIn 2004, Faiz had criticized aspects of foreign policy such as the US-led movement for sanctions against Iraq. \"Terrible current events, like executing policies that starve entire populations, have nothing to do with religion,\" he argued that year. Trying to blunt his criticism somewhat, he asserted that such a policy, one undertaken by a small political elite, did not reflect all Americans. \"It's not a Christian thing, it's not an American thing,\" Khan asserted. \"It's a criminal thing.\" Writing an essay in 2006 for a collection titled _9\/11 and American Empire: Christians, Jews, and Muslims Speak Out_ , however, Faiz was much less reserved and much more explicit about his new differences from \"the moderate Muslim response to terror\" which, he believed, \"has become an industry of its own\" that failed to address political and economic abuses at home and abroad.\n\nRauf had occasionally, and cautiously, criticized US foreign policy during those years as well, something that would come to haunt him during the Ground Zero Mosque controversy. Additionally, he appeared in a television advertisement in 2004 in which he and other religious leaders apologized to Muslims overseas for American abuses at Abu Ghraib, and he had tried to avert the surveillance and detention of Muslim Americans by participating in FBI and New York City law enforcement training programs. He never spoke publically against the activities of intelligence agencies, however\u2014not even when members of his own congregation were detained or had their homes ransacked without search warrants. Further, as his work with the Cordoba Initiative intensified, Rauf increasingly affiliated with political and economic institutions and actors that Faiz saw as contributing to global social and economic inequalities and injustices\u2014not least because of their influence in Washington.\n\nIn the same essay he wrote in 2006 (published in 2007), Khan described contemporary American foreign policy as \"counterfeit.\" The \"agendas and consequences of current US foreign policy lie squarely opposed to the principles enshrined in our American Constitution, as well as our American Declaration of Independence,\" he argued. \"I wish the term 'American' would be dropped from the foreign policy that originates on Wall Street and is enacted through Washington, DC.\" Implicating Rauf's work as part of the problem rather than part of a solution to international conflict, Khan argued that going \"to an Islamic scholar for a diagnosis of what happened on 9\/11 and why is quite absurd\" and criticized how \"supposedly moderate Muslim commentators\" resorted to a \"good Muslim-bad Muslim dialectic\" rather than engaging in informed discussions of geopolitics and economics. \"I understand that militants exist who label themselves 'Islamists' and they are generally a most vile lot,\" Khan told readers, but the first subject that Americans must understand in order to appreciate the reality of the terrorist attacks and US foreign policy since is \"Wall Street\/Corporate America\/the banking industry\"\u2014all entities, in his view, that influenced policymakers for the sake of profit.\n\nSignificantly, even while criticizing the work of organizations like ASMA and Cordoba, Khan enacted some of the principles Rauf and Daisy had tried to inculcate in the young Muslims they mentored. Not only did he appeal to the Constitution and Declaration of Independence as foundations of Americanness that were in keeping with Islamic mandates, he argued that Muslim Americans\u2014afflicted by the cultural \"residue\" of an immigrant inferiority complex\u2014were hampered in their understandings of US politics and economics. This, he maintained, was one reason they did not demand \"real answers\" about 9\/11 and its aftermath. But the primary reason Muslims shied away from discussing these things, he acknowledged, was fear of repercussions. Emulating the arguments of Rauf and Daisy in some ways even while he rejected their work, Faiz insisted that he was every bit as American as any other citizen and that, as such, he had the right to hold his government accountable to its founding principles, to ask questions, and to demand answers for what was done in his name and with his taxes.\n\nFaiz was far from the only Muslim American (or non-Muslim American, for that matter) to make such criticisms, but he correctly guessed that his perspective did not accord with that of Muslims identified by the State Department and defense analysts as moderate. He remained committed to Sufism\u2014even making reference in his 2006 essay to Ren\u00e9 Gu\u00e9non, the Traditionalist-Perennialist writer popular with Rauf and Seyyed Hossein Nasr\u2014and, like Rauf, insisted on creating a culturally American Muslim practice that, for example, would allow men and women to shake hands rather than require them to observe strict physical separation. Nevertheless, Faiz's political critique increasingly challenged the economic and political moderation defined by Rauf and by the previously marginalized religious minorities whose anti-socialist, pro-capitalist arguments served as Rauf's moderate template.\n\nThese were not the only ways in which Faiz's perspective provoked anxiety among his fellow Muslim Americans. Seeing abuses in Abu Ghraib and Guantanamo unfold as the United States went to war in Iraq, and cognizant of the surveillance, warrantless wiretapping, and harassment of friends at the mosque\u2014including of a young mother of two babies who also happened to be the daughter of a diplomat\u2014Faiz began to think that the US government and members of its security apparatus were capable of just about anything. When Secretary of Defense Donald Rumsfeld, speaking in front of US troops in Baghdad in 2004, said that United Airlines flight 93 was \"shot down\" on 9\/11 rather than allowed to hit an intended target\u2014something the Pentagon redacted, asserting that Rumsfeld \"misspoke\"\u2014Faiz believed him. After all, Vice President Dick Cheney had already acknowledged that an executive order permitting such action had been issued, an order Cheney later admitted to issuing himself. After that, not only did Faiz criticize the economic and political contours of Rauf's moderation, he began to demand the \"truth\" about 9\/11, which he took to be something that, at the very least, US intelligence officials had allowed to happen so as to provide justification for starting wars that would result in profit for US corporations.\n\nIt is unlikely that Daisy and Rauf could have sustained their relationship with Faiz\u2014or that he would have wanted to continue his association with ASMA, given his opinion of its work\u2014even if he had not publically questioned the US government's account of what transpired before, during, and after 9\/11. Given that Faiz often filled in for Rauf at the mosque, served as a spokesperson for the ASMA Society, and had begun to share his thoughts about 9\/11 with other members of Rauf's Sufi community, his actions presented a growing threat to ASMA's moderate status. By the end of 2006, Rauf not only dismissed Faiz from his position as an assistant imam at the mosque, he also instructed his Sufi dervishes to cease having any contact with him. Then, in 2007, all mention of Faiz Khan was removed from ASMA's website.\n\nThe reasons for this significant disruption were not always apparent to other members of Rauf's community, many of whom thought of \"Dr. Faiz\" as a close friend. For example, Dean, a white, middle-aged convert, believed the break was due to the fact that Faiz had questioned Rauf's status as a leader of the Halveti-Jerrahi Sufis, since Rauf rarely focused on Sufism anymore and had taken hand with a different shaykh. This interpretation may have been due to the fact that Dean also bristled at the capitalism Rauf and Daisy promoted and saw nothing remarkable about objecting to that aspect of their work. But, while the reasons for the break may have been less than transparent, the effect was unmistakable: as Rauf, increasingly absent from his community, urged his dervishes not to associate with one of their other mentors, some community members began to question his injunctions and even their own reliance on him. Needing more guidance than Rauf could provide in person, they gradually began to take more responsibility over the mosque and _dhikr_ services that they held without him. Like Faiz, some of these other dervishes also took Rauf and Daisy at their word about being fully Muslim and fully American. Yet the ways they put such teachings into practice in Rauf's absence\u2014while less threatening to Rauf's moderate status than Faiz's activism\u2014did not always accord with what their imam intended.\n\nEven before loosing Faiz Khan, ASMA's and Cordoba's many expanding ventures began to require more labor than Rauf and Daisy could provide with the help of just their Sufi volunteers. After hiring their first full-time staff person in 2004, ASMA began to bring on unpaid interns in 2005. These interns and Rauf's dervishes together organized and staffed conferences for ASMA's Muslim Leaders of Tomorrow (MLT) program and other events, even as those programs gradually pulled Rauf and Daisy further away from their Sufi community and diverted their attention from American Muslims, refocusing it on international audiences. Unlike the 2004 MLT conference, for example, at which Rauf had emphasized the common ethics and historical trajectory of American Catholic, Jewish, and Muslim communities, the 2006 and later gatherings\u2014held outside of the United States\u2014were convened to \"nurture a global Muslim leadership that employs Islamic and pluralistic values to enhance peace and tolerance.\"\n\nTheir increasingly global focus was reflected in other programs Rauf and Daisy Khan developed in 2006, as well. These included the Women's Islamic Initiative in Spirituality and Equity (WISE), initially conceptualized as a forum to \"empower American Muslim women\" but ultimately undertaken as an international project to provide information about and to Muslim women globally. By the time of the first WISE conference, Khan had added to this the \"shura council,\" a program to train female scholars as Islamic legal authorities. Their new programs also included Rauf's Shariah Project, which he first presented as one to help \"moderate Muslims\" reduce conflict and promote democracy in Muslim-majority countries but later described as an international effort to help Muslim legal scholars and heads of state \"determine the proper balance\" between \"institutions of political power and authority, on the one hand . . . and institutions of religious power and authority, on the other\u2014the Muslim equivalent of the religion-state relationship.\" These 2006 programs ultimately came to define ASMA and Cordoba\u2014in no small part because they were the ones for which Khan and Rauf got the most funding\u2014and, again, moved Rauf and Khan further away from their initial objectives and original community. I return to dynamics within Rauf's Sufi community in more detail in the chapters that follow. Proceeding with the histories of ASMA and Cordoba in the meantime, though, I show how Rauf and Khan continually tried to balance various aspects of their work so as to maintain their claims to moderation while making ever more political pronouncements.\n\nInterfaith Allies, Political Positioning, and Cordoba's Independence\n\nThe year 2006 was pivotal for Rauf and Khan, as ASMA began to receive grants for its WISE project from major foundations. Among other things, these grants allowed ASMA and Cordoba to move in 2007 into professional offices in a building known informally as the \"God Box\" but formally as the Interchurch Center, a towering edifice on the Upper West Side of Manhattan that houses the National Council of Churches and several Rockefeller family foundations, among other things, thus functioning as a nexus of New York City's religious and philanthropic worlds. As WISE grew from a US women's forum to one involving hundreds of women internationally, it became for ASMA what Rauf's Shariah Project (later called the Shariah Index Project) was turning into for the Cordoba Initiative: the organization's most time-consuming program, as well as an indispensible source of funding. These developments, which coincided with their increasing willingness to take certain kinds of political positions, would cause Rauf and Khan to alter their organizations and ambitions yet again.\n\nEven as Khan organized cultural and social events and symposia for young Muslim leaders in 2004 and 2005, she had begun to create relationships with prominent women's religious organizations, including the Sister Fund (oil heiress Helen LaKelley Hunt's feminist religious philanthropic foundation), which provided ASMA with early grants and with links to other important activists and institutions. A Sister Fund seed grant, for example, helped Khan attract the attention of another foundation, the Global Fund for Women, which gave ASMA $15,000 in 2006 and helped to publicize WISE's inaugural event. That meeting, a November conference at the Westin Hotel in Times Square, would gather over one hundred Muslim women from around the globe\u2014authors, rights activists, lawyers, politicians, and scholars, among others\u2014\"to discuss global Muslim women's issues, assert our rights through the use of and in accordance with Islamic law, and build a coherent movement that empowers and connects Muslim women everywhere.\" Just a month before convening it, ASMA and Cordoba turned a crucial corner when ASMA gained a major grant from one of the nation's foremost private foundations and entry into a selective network of elite foundation grantees.\n\nIn October 2006, the Rockefeller Brothers Fund (RBF) granted ASMA $70,000 for WISE and for Rauf's Shariah Project; it would grant $100,000 more in October of 2007 for the WISE and MLT initiatives. Both grants came as part of the RBF's annual disbursement to \"Peace and Security\" programs dealing with \"Muslim and Western Understanding.\" Cordoba also received a one-year grant of $30,000 for general operating support in 2006 under the same rubric, though it still drew most of its resources from ASMA at that time. Greater support for ASMA projects followed. The Henry Luce Foundation provided ASMA with a donation of $25,000 in 2006 to support Khan's first WISE event, then issued a second, smaller grant of $10,000 in 2007 to support her project for training Muslim women leaders in Islamic law. Over the next five years, Luce support for WISE increased exponentially, ultimately totaling nearly $400,000. During this period, ASMA also gained the attention of the Carnegie Corporation, among other donors, and Cordoba began drawing major funding and other types of support from the Malaysian government.\n\nIn the meantime, with Khan concentrating on WISE and Rauf working closely with Muslim legal authorities from around the globe, the two changed ASMA's name to reflect the activities for which it was increasingly funded. Retaining its acronym\u2014and the mission statement and objectives posted on its website, which would not change for another few years\u2014the American Sufi Muslim Association became the American Society for Muslim Advancement at the end of 2006, a more explicitly rights-oriented organization in name and purpose than it was previously. Unlike other Muslim American rights organizations, such as the Council on American-Islamic Relations (ASMA's neighbor in the God Box), Khan focused not on the due process or privacy denied Muslims in America during this era of mass surveillance but on rights denied Muslims\u2014particularly Muslim women\u2014elsewhere.\n\nBetween 2006 and 2010, as they deemphasized their Sufi roots and activities and grew increasingly secure in the support they received from private funders and allies of other religious traditions\u2014for example, the two received the James Park Morton Interfaith Award in 2006, which they shared with Supreme Court Justice Stephen Breyer, among others\u2014Rauf and Khan began to speak more explicitly and frequently about their evolving political objectives. Their political speech was still consonant with the \"moderate\" Islam for which Rauf had made headlines in 2004 and for which they received grants: an Islam \"that embraces the values of Western democracy, carries within it a love of America . . . calls on Muslims to respect other faiths,\" and endeavors to help Muslim women combat patriarchal cultures and institutions. Or, as Rauf put it in his 2007 contribution to a debate over the meaning of \"moderate Islam,\" their version consisted of promoting in Muslim majority countries what the Abrahamic ethic had given rise to in the United States: the creation and expansion of an \"Islamic human rights doctrine\" and \"Islamic democratic capitalism.\" Yet, as the years passed and their status as moderates came to seem unshakable, Rauf and Khan eventually ceased allying with non-Muslim leaders for protection and cover when speaking overtly about politics.\n\nThe continually changing Cordoba Initiative website, launched at an independent address in 2006, gave evidence of these shifts and of Rauf's waning caution in acknowledging political objectives. At first, Rauf and other Cordoba leaders consistently emphasized the multireligious nature of the organization\u2014an emphasis that partly served to assuage Americans' fears about the political mission of even a self-professed \"moderate\" Muslim. In addition to providing biographies of Cordoba leaders such as Rauf (chairman of the board), Khan (a trustee), and Bennett (executive director), the website identified ten scholars and religious leaders from non-Muslim traditions who served on Cordoba's interreligious advisory board, including Professor Elaine Pagels of Princeton University, Rabbi Brad Hirschfield of the National Jewish Center for Learning and Leadership, and noted religion writer Karen Armstrong. Further, Cordoba's mission statement still significantly echoed ASMA's at that time by emphasizing cultural and educational work, except Cordoba's directors made sure to point out that their efforts would be \"multi-faith,\" not just Muslim-led. \"No single religion's followers may hold a majority of voting seats on its board of directors,\" the website assured prospective donors, partners, and other inquirers. Rather, \"mainstream American Muslim leaders\" would work \"in partnership with Jewish and Christian leaders,\" through an interreligious system of checks and balances, to foster \"civil dialogue,\" promote \"intercultural understanding,\" and prevent \"the horrors of another September 11.\"\n\nIn addition to preventing further terror attacks, Cordoba's leaders pledged to manage the threat immoderate Muslims posed to existing political and economic powers. We will overcome the \"distrust and animosity that plague the Islamic world's relationship with the United States,\" they promised, an animosity that constitutes \"today's greatest single threat to international security and the world's economy.\" Emphasizing these themes again under a portion of the website devoted to Cordoba's \"Uniqueness & Need,\" the organization's officers reiterated the \"substantial global threat\" of terrorism to economic security and underlined Rauf's special qualifications for undertaking such endeavors: \"While many excellent attempts have been made to improve relations between Islamic countries and the West and to further peace in the Middle East, rarely have _politically moderate_ , mainstream American Muslims helped initiate and lead . . .\"\n\nDespite the caution Rauf exercised when creating this first independent Cordoba Initiative website, he and Khan both became more open to speaking about policy and to acknowledging political affiliations and objectives as soon as 2007. In January of that year, for example, they joined a diverse bipartisan group of academics, policy experts, politicians, business people, philanthropists, and religious leaders to urge greater US diplomacy with \"the Muslim world,\" as well as the cultivation of more \"moderate behavior\" therein. \"The Leadership Group on U.S.-Muslim Relations\"\u2014which involved Madeline Albright, among others, and had financial support from several private foundations, as well as from the American Petroleum Institute\u2014first convened at the Pocantico Conference Center of the Rockefeller Brothers Fund in New York and then initiated an eighteen-month investigation into how to improve relations between the United States and Muslims worldwide. The assassination of Pakistan's former prime minister, Benazir Bhutto, in December of 2007 only seemed to deepen the need for such work, as Vali Nasr, a member of the group and Seyyed Hossein Nasr's son, explained on CNN that month. These efforts, the group concluded in its report, should include initiatives ranging from disaster relief to private business-led development.\n\nThe following year, Rauf\u2014comfortable enough to take explicitly political positions on his own\u2014completely restructured the Cordoba Initiative. Not only did he file new bylaws and redesign the website to present himself as the organization's sole leader and founder\u2014excising all mention of Bennett, Khan, and the interreligious advisory board\u2014he removed from the website and bylaws all material on the \"multi-faith\" nature of the organization. Rauf even reworded the Initiative's mission statement, explaining that its new goal was to \"achieve a tipping point in Muslim-West relations within the next decade\" by \"leverage[ing] contacts in influential positions within the Muslim World and the West.\"\n\nWhile the new website still gestured at interreligious endeavors, promising in the mission statement to \"bring back the atmosphere of interfaith tolerance and respect\" characteristic of twelfth-century Cordoba, Rauf\u2014now enjoying support from both the Malaysian government and the US State Department\u2014no longer felt he must rely on support from non-Muslims to prove his moderate status. Rather than pursue further interreligious collaboration, he concentrated primarily on supporting Cordoba's international programs and on developing his project to evaluate the Islamic authenticity of various countries' legal systems.\n\nOver the next few years, Rauf continued to draw honoraria for occasional lectures he delivered for ASMA, and he continued to identify himself as the imam of a mosque twelve blocks from the former World Trade Center, even as he worked primarily from his office in Malaysia. In the meantime, Daisy Khan\u2014who had previously been less openly political than Rauf\u2014maintained ASMA's involvement in State Department cultural exchange programs and spoke more explicitly about her own political objectives. In February 2007, for example, she undertook her first State Department speaking tour. During that trip, she traveled through Germany, continually reiterating the message that American and Muslim values do not conflict but rely on \"ethics that are common to both.\" In the years that followed, Khan also became less guarded about the political valences of her WISE project.\n\nKhan's caution in acknowledging political objectives was long a matter of strategy, in part designed to avoid drawing attention from Muslims who might oppose her women's rights work. Discussing WISE with a group of activists in 2008, for example, she acknowledged that she had been \"very careful from the very beginning not to present it as a western agenda, feminist agenda. I've said it's a social justice movement which is very much connected to Islamic law, that's the banner I've been waving.\" When working to advance Muslim women's rights in non-Western countries, she argued, \"we can't go in and say 'this is a human rights violation stated in the UN charter' they won't listen to this, it falls on deaf ears because they think it's a western imposition, but if we say 'there's a verse in the Quran that says you can't do this because it's wrong, it's a sin,' people hear it very differently.\" Also, because she sought to involve a range of women in her now-global project (from secularists to traditionalists), she explained, \"I'm using the moderate path, the middle road\" and \"inviting everybody to join forces.\"\n\nDespite this intentionally cautious approach, Khan altered the name of her project at the end of 2008 to something more overtly political. Retaining WISE's acronym, as she had with ASMA, Khan changed the Women's Islamic Initiative in Spirituality and _Equity_ \u2014the latter term indicating fairness and ownership, connoting a parity more economic than political\u2014to the more politically explicit Women's Islamic Initiative in Spirituality and _Equality_. The very next year\u2014after convening the second WISE conference in Malaysia, where Rauf connected her to an official in the Ministry of Women's Affairs who offered support if Khan promised to pressure Malaysian religious authorities on women's issues\u2014Khan turned to the language of human rights that she had previously regarded as too provocative. Describing her work on the new website created especially for WISE in 2009, Khan asserted that \"gender-based inequality is a global human rights issue that transcends culture, religion and socio-economic status. Though such problems as domestic violence, inadequate access to technology, poor education, and lack of economic opportunity are widespread, Muslim women in particular confront the limitations of discrimination and inequality.\"\n\nNo longer wary of acknowledging this perspective before Muslim audiences, Khan included the same pronouncement on the Arabic-language portion of her website. Further, she began to echo more explicitly Rauf's argument that once Muslims\u2014including Muslim women\u2014created a culturally American Islam, they could serve as a beacon lighting the way toward reform elsewhere. Khan's confidence in taking such overt political positions was surely influenced by a grant she received in 2009, ASMA's largest ever. That year, the Dutch Foreign Ministry's MG3 Fund gave ASMA one million euros to help accomplish the third of the U.N.'s millennium development goals: gender equality and the empowerment of women. Yet that substantial grant was not entirely responsible for changing Khan's strategy or willingness to engage in political speech and activity. Rather, even before receiving these funds, Khan's faith that her work would be accepted both as moderate and as authentically Islamic was buoyed by an unprecedented political development. Her belief in 2008 and 2009 that the United States was changing for the better was palpable, as I discuss below, and it contributed to her and Rauf's confidence about the future of their endeavors\u2014including their Islamic community center project. But it also blinded them to the severity of racism and Islamophobia in the United States, leaving them unprepared in 2010 for conservative politicians' reactions to their political speech and projects.\n\nThe Promise and Pitfalls of Politics\n\nIn January 2008, I visited ASMA to interview Khan about the organization's programmatic changes over the preceding five years. When I entered the new offices at the Interchurch Center on a snowy afternoon, an intern told me Khan needed time to finish a previous task and directed me to the reception area. Khan eventually emerged in her usual attire: one of her many elegant pantsuits that straddled the line between business suit and _salwar kameez_ , a style of dress common to India, Pakistan, and the Kashmir region in which Khan was born. After welcoming me, she asked to put off our interview for another half hour as ASMA was dealing with an unfolding crisis in the Netherlands. Khan was in the midst of a phone call with one of the Muslim Leaders of Tomorrow lawyers there, strategizing about the organization's response to a soon-to-be-released inflammatory film about the Qur'an. Meanwhile, a staff member and intern were busy organizing the second WISE conference, then planned for June of 2008 in Malaysia.\n\nWhen we finally sat down to talk in Khan's office, I asked her about ASMA's changing objectives and evolution into an international organization. As a result of their expanding educational efforts and growing popularity, Khan acknowledged, she and Rauf had redefined ASMA's goals. Still, she insisted, ASMA's current activities were ones she and Rauf had always hoped to undertake. Without mentioning Sufism, Khan framed ASMA's history and purpose in terms of her own personal mission after immigrating to the United States to find an authentic Islamic practice not tainted by the residue of a foreign culture.\n\nThere were mosques that were established along ethnic lines. They were established along cultural lines, cultural mores . . . and I couldn't adjust in those places. So, it was important to me that when I grew up, that I established something for the sake of those who were coming after me. So that was the thrust of my own aspiration. I had a regular corporate career. It's something that I really wanted to do for the sake of the community. So, when I married Imam Feisal . . . eleven years ago . . . he was already an imam of a mosque. And the mosque had certain constraints because we couldn't do the kinds of programming that we wanted to do then. As you know, it's just a storefront mosque. It doesn't have the space.\n\nSo, at that time, I wanted to really keep it to what was now the newer generation . . . and I was beginning to know a lot of them through him. So, we decided that we should establish a separate 501(c)(3) [nonprofit, tax-exempt corporation] that would be a house for people, a house that would have an American cultural expression to it. And it would also be very authentically Islamic in terms of its practices . . . so that it would also be very inviting to non-Muslims and there would be interfaith dialogue. It would really be a new kind of a center that was more an American Muslim center than a Muslim center built around a certain ethnicity.\n\nAnd also this was _Manhattan_ , it was a nucleus of the world Muslim community here. So we wanted to cater to many different nationalities. So we established the organization at that time in 1997 and it was primarily geared to two things: one, Muslims getting to know their own faith and understanding their own faith and navigating the issue of modernity and faith especially because now they're living in a Western context. So, we [were] proving ourselves as a role model for them you know. And so then that was kind of like the inreach work. And then, a little bit of outreach work where non-Muslims [got] to know Muslim life and who Muslims were. So, this was really the major\u2014the _only_ \u2014thrust that we had.\n\nKhan went on to explain how the 2001 terrorist attacks had caused them to shift to three primary fields of activity: providing a consistent resource for national and international media, preparing young Muslims to serve in community development through the Muslim Leaders of Tomorrow program, and raising awareness about Muslim women's rights as well as funds to train Muslim women jurists through WISE. The focus of ASMA, Khan acknowledged, was less local than previously, and increasingly national and international. When I asked about Sufism, which Khan still had not mentioned, she answered that the Sufi group was important, but that she and Rauf always had plans for something more:\n\nEven when the _dhikr_ group was going on, there were a lot of mosques that would invite [Rauf] to lecture. There would be classes that we would hold for younger Muslims, who were not part of the _dhikr_ group. So the _dhikr_ group was a small part of a, you know, like a sustainer for those people, who wanted to go deep into their faith. But that was not the only reason for [the ASMA Society] because we knew that the _dhikr_ group cannot appeal to the broader segment of society. Even though it was built around, you know, a nucleus\u2014the nucleus of the community at that time was primarily the _dhikr_ group.\n\nBut, you know, communities grow and organizations grow and you take on more things.\n\nIn this interview, Khan presented ASMA's 2008 programs as inherent in the original vision for the organization. Even though she, Rauf, and Faiz Khan did not initially have these specific projects in mind when they founded ASMA in 1997\u2014or, at least, did not include them in the activities listed on the organization's 1997 application for tax-exempt status\u2014Khan believed such work was in keeping with the original ideas that had inspired them over a decade before. In 2009, she changed ASMA's website once again to reflect this. Under \"Our Mission,\" the website described ASMA as \"a New-York based nonprofit organization founded in 1997 to elevate the discourse on Islam and foster environments in which Muslims thrive\" by \"strengthening an authentic expression of Islam based on cultural and religious harmony through interfaith collaboration, youth and women's empowerment, and arts and cultural exchange.\" Under a section of the site devoted to ASMA's history, the website acknowledged that change had transpired and provided justification, although it still did not disclose ASMA's original aim of promoting Sufism. \"We have continually adapted our programs and strategies, while remaining faithful to our core mission,\" it read. \"ASMA is shaping and inspiring a truly transformative movement. We are proud to play one of the most critical roles for the 21st Century.\"\n\nDespite the changes Rauf and Khan made to ASMA and Cordoba, what remained consistent during much of the first decade after 9\/11 was their insistence on situating Muslims within the narrative of American economic opportunity, ethical commonality, and eventual assimilation Rauf borrowed from a liberal Jewish sociologist (Will Herberg) and a conservative Catholic activist (Michael Novak). In her work as an advocate for Muslim women's empowerment, and in her role as ASMA's primary spokesperson after 2006, Khan adopted Rauf's rendition of Herberg's immigrant narrative and placed herself within it\u2014not just in personal interviews, such as the one she gave me in 2008, but also in the national media.\n\nIn an interview with _Sojourners_ magazine in 2009 (partly derived from a 2008 post she contributed to the _Newsweek-Washington Post On Faith_ blog), for example, Khan argued that Muslims in America struggled for acceptance in the same ways previous religious groups had. \"Take, for example, American Catholics and Jews, long considered outsiders, but now generally existing within the mainstream of American society. This process is already underway with American Muslims, and it will only continue . . . Just as our nation has transformed from a 'Christian' nation to a 'Judeo-Christian' nation, we must now recognize our commonly shared Abrahamic ethics and embrace Islam as an equal member of this Abrahamic ethical tradition.\" Tailoring this message to her own work, Khan placed WISE in a history of religious women's activism, arguing that \"the American women of the WISE movement now walk in the giant footsteps of our earlier Christian sisters,\" including Harriet Tubman and Susan B. Anthony.\n\nKhan's belief in this narrative and her greater willingness to speak politically in 2009, like Rauf's, was due to in part to the positive reception her work with ASMA had received among leaders of non-Muslim religious traditions, as well as the favor her organization and the WISE program enjoyed with private foundations and political elites, national and international. Such recognition suggested that Muslim acceptance and achievement in the United States and Europe was a real possibility, even in the first decade after the terrorist attacks and as the United States waged war in multiple Muslim-majority nations. As mentioned earlier, though, Rauf and Khan also drew tremendous encouragement from an unprecedented political development: the election to the presidency of a man, as Khan noted in an exuberant _On Faith_ post, \"with the middle name 'Hussein' who openly draws upon his Christian faith to drive his politics.\"\n\nKhan did not mince words in that 2008 blog post when, two days after Americans elected to the presidency a candidate whom some believed to be Muslim, she described what the election revealed about America's role as the world's leading nation.\n\nObama inspires the universal dreams of all humans\u2014that of audacious hope, of idealistic (and otherwise unrealistic) optimism, and of far-fetched success and transformation\u2014dreams most powerfully represented in America and \"the American dream\" . . .\n\nHe is \"the shattered glass ceiling\" embodied, the \"yes we can (collectively)\" exemplified. A new and unprecedented standard has been created. The most powerful nation in the world has overwhelmingly elected a man who, like America, defies ethnic labels, geographic limitations, or traditional class categorizations . . .\n\nHis election is both a seminal moment in American history and a historic event for the world. It represents the abandonment of an American exceptionalism characterized by resistance to global cooperation and community. Instead, it celebrates an American exceptionalism defined by our truly unique history, exemplified in the indelible immigrant spirit of hard work, ingenuity, and leadership.\n\nKhan's reflections on Obama's election are illuminating not just for the encouragement she drew from his election\u2014something that allowed her to feel more accepted, more American, and more able to speak freely and politically than, perhaps, she had felt since the events of 9\/11\u2014but also for the tensions they reveal in her understanding of the United States. Like Rauf, Khan generally presents America as a land of racial equality and economic opportunity where \"hard work\" and \"ingenuity\" (particularly on the part of immigrants) translates into upward mobility. Obama's election, for her, proved that the \"glass ceiling\" had been shattered and that members of all ethnic, racial, religious, and economic groups have the same chances for success. At the same time, Khan began her post by recognizing that the hope Obama inspired is in an American dream that is \"idealistic\" and even \"unrealistic\"\u2014an American dream of racial and economic mobility that could even be described as \"far-fetched.\"\n\nIt is perhaps not coincidental that Khan wrote that description in the autumn of 2008. The year was notable not only for the election of a biracial man whom many describe as the nation's first \"black\" president, but because it was when many people in the United States began to feel the pain of what would become a devastating economic recession. While the election seemed to promise that anything was possible for anyone in America, the recession would\u2014for many\u2014call into question the moral and social supremacy of American capitalism and prove just how little mobility most Americans actually have. While Khan inadvertently acknowledged America's imperfections in her celebratory political post, Rauf would do no such thing\u2014at least, not when it came to contemporary domestic policy.\n\nA few months before the market turmoil of 2008 began, I interviewed Rauf by phone, as he was then mostly working from his office in Malaysia. On an unseasonably warm April evening in New York (morning in Kuala Lumpur), I asked Rauf to clarify his thoughts about democratic capitalism. \"You've read my book,\" Rauf responded, referring me back to the principles he had asserted there. It remained his belief that American democratic capitalism, at least in its ideal form, is a productive fusion of free enterprise and moral restraint characterized by the Abrahamic ethic and that it should be emulated worldwide.\n\nLess than a year after my interview with Rauf, economists, historians, reporters, and even the International Monetary Fund began using the term \"Great Recession\" to refer to the recent economic downturn caused largely by America's deregulated banking industry, the longest downturn since World War II and the most severe since the Great Depression. Still, preoccupied with international affairs during that time, Rauf reiterated his narrative of American political and economic leadership in one of his own contributions to the _Newsweek-Washington Post On Faith_ blog. Mentioning that he had just returned from a week of discussions with Iranian officials about Iran's upcoming elections, Rauf emphasized his support for the newly elected American president (\"my president, Barack Hussein Obama\"), heralded Obama's upcoming speech to address \"the Muslim world\" from Cairo, and praised Obama's efforts to ensure American-led economic development in the Middle East so as to foster peace and reconciliation between Arab states and Iran, Sunnis and Shi'a and \"Muslims and the West.\"\n\nAlthough such political and economic pronouncements still sounded moderate to the ears of most American political leaders in 2009, other audiences were less convinced. I refer not to political conservatives here\u2014those who would seize on the 2010 Ground Zero Mosque controversy for the sake of electoral gain\u2014but to other New York City Muslim leaders who also engaged in political activism with very different aims and (usually) before much smaller audiences. While many black American Muslim leaders also drew encouragement and inspiration from Obama's election, most were not as sanguine as Rauf and Khan about the opportunities and equality available to black Americans. Nor could they agree with Rauf's insistence on the justice of American political and economic arrangements\u2014particularly not, as I discuss in the final section of this chapter, with his argument that the global spread of democratic capitalism would diffuse the grievances that fueled terrorism abroad, just as it had improved the lives of black Americans and dampened the militancy of black nationalist movements. While Rauf and Khan collected accolades and awards for their programs in the decade after 9\/11, other Muslim leaders struggled to meet the needs of their less affluent populations, including for basic resources. These dynamics would make it difficult for such leaders to defend Rauf and Khan in 2010 when, despite years of extremely careful political positioning, their moderation was called into question.\n\nAggressive Jihad and Economic Abundance\u2014Moderation in Question\n\nWriting in 2004\u2014before the idea of a black president seemed feasible to most\u2014Rauf saw the appointment of Colin Powell, a black man, to the position of secretary of state as proof that black Americans enjoyed opportunities equal to those of white Americans. He acknowledged in his book that black Americans had not always benefited from the progressive assimilation and acceptance immigrant populations enjoyed. As a result of being deprived of the benefits of democratic capitalism during large periods of American history, black Americans (and other similarly deprived populations at other times and places) had developed an aggressive tendency, he claimed. And in this aggressive tendency\u2014one that distorts the true nature of jihad\u2014Rauf located the roots of both fundamentalism and suicide terrorism.\n\nAuthentic jihad, Rauf opined in 2004, is \"the struggle to establish a human society that is the good society\u2014one that authentically reflects the Abrahamic ethic, especially its second commandment\" (loving one's neighbor) expressed in democratic capitalism. Fundamentalist jihad is very different. Mixing his Sufi-inflected rendition of democratic capitalism with theories about group psychology, evolutionary biology, and game theory, Rauf attempted to render Muslims (even those who had committed violence) more sympathetic to non-Muslim audiences and to explain why some people (including black Americans in certain periods of American history) seemed to regard violence as an effective means for achieving their aims.\n\nAccording to Rauf, multiple kinds of aggression\u2014from the violence generally categorized as fundamentalist to the particular action of suicide bombings\u2014could be understood as natural sociological and psychological phenomena activated by situations of stress. Just as the honeybee sacrifices itself to defend the hive, he argued, people governed by unexamined habits, customs, or evolutionary impulses sometimes resort to aggression to defend their communities. Rather than return violence with more violence, the key to creating global good was twofold: making both value systems and social systems more liberating and ethical. While proper religious practice\u2014shorn of cultural impediments\u2014could help reform communal values, the exportation of democratic capitalism would contribute to economic opportunity in developing nations, thus lifting populations out of the poverty that left them feeling defenseless.\n\nThe United States had already proved that such progress was possible, Rauf claimed, equating the creation of antidiscrimination laws and affirmative action programs with the achievement of social and economic parity. Black Americans were a case in point:\n\nFor example, the violent Black Panther movement in the 1960s represented this kind of aggression in the context of the civil rights movement, seeking the defense of American blacks against white people who had imposed the historical injustice of slavery (and later discrimination) upon them. But after President Lyndon Johnson pushed through civil rights legislation, and American blacks made progress in their battle for civil rights, legal and voting barriers fell, affirmative action plans were created, and racial discrimination against blacks became a violation of law. As all this occurred, black militancy inevitably waned.\n\nThough widely hailed as moderate by many American elites, Rauf's view did not reflect the perspectives of other New York City Muslim leaders, such as Imam Siraj Wahhaj of Masjid al-Taqwa in Brooklyn, imams at the Malcolm Shabbaz mosque on 116th Street in Harlem, or others who believe that the affluence of American elites\u2014born of the capitalism that Rauf depicts as the engine of equality\u2014often involves the twinned projects of international resource accumulation and domestic labor exploitation, thus impoverishing many communities in the United States and abroad. Known as the \"drug-fighting Muslim\" for his 1980s efforts to rid his poverty-stricken neighborhood of crack dealers, Siraj Wahhaj has long been unabashed about the ongoing economic marginalization and even political \"persecution of blacks.\" Speaking after 9\/11 about Muslim Americans' opinions of the United States, Wahhaj sometimes tempers his political rhetoric. Nevertheless, he remains resolute about the injustices black Americans face. \"You might hear some anti-American flavor a little bit,\" in the stories American Muslims tell, he admits, \"but not because they hate America.\" Rather, he points out, \"our civil rights leaders [also] spoke about the injustices of America\" from a desire to improve the country, not just from animosity toward it. And \"you hear it in that way especially [from] African Americans. If that makes us militant, then we're militant.\"\n\nWriting in 2004, Rauf was not completely unconcerned about inequality and racism, but his book and work with the Cordoba Initiative prioritized America's role in overcoming such problems in Muslim-majority nations. Hoping to explain how and why the situation of Muslims in less developed countries could be improved, Rauf created a narrative about American opportunity and abundance that elided the needs and experiences of those Muslim Americans suffering from the disastrous combined effects of institutionalized racism and neoliberal economic policies: the simultaneous expansion of free markets, diminution of social welfare programs, and deregulation of major industries, including banks. It is important to recognize here that Rauf did not create this selective narrative because of bad intentions. In fact, by explaining the seeming threat posed by black activists (and other Muslims at other times or places) as a natural product of self-protective instincts, Rauf attempted to defend and humanize those who had been or are subjected to racism. By treating black protest as if it belonged to the past, however, Rauf consigned racism to the past along with it. Other Muslim leaders in the city, as noted, were less optimistic about the racial and economic progress American capitalism could deliver. Even before the recession, which had a disproportionate effect on black Americans and erased, according to postrecession studies, \"three decades of economic gains\" among that population, these leaders created organizations and networks to address social and economic distress in their neighborhoods.\n\nIn 2005, Imams Talib Abdur Rashid and Siraj Wahhaj, among others, created the Muslim Alliance in North America to serve inner cities across the country\u2014especially homeless populations and current or former prisoners\u2014and to share education and resources useful in creating political change. They modeled their organization (known as MANA) on the Inner-City Muslim Action Network (IMAN), a multiracial Muslim organization created by young people living in Chicago's inner city in 1997 to \"address violence, poverty, and decay\" in their neighborhood. The 2008 death of W. D. Mohammed\u2014who had mentored Siraj Wahhaj and had been scheduled to speak at MANA's 2008 convention\u2014briefly propelled MANA's leaders into greater public prominence. Nevertheless, this increased standing did not gain MANA (or IMAN) nearly the attention among non-Muslim American elites that ASMA and Cordoba enjoyed.\n\nRather than petition politicians to extend US economic and social systems around the world, these Muslim leaders\u2014several of whom felt marginalized after 9\/11, as US officials increasingly partnered with Arab and South Asian Muslim spokespersons\u2014spent the years before and during the Great Recession organizing community service programs to address the needs of people who, they felt, had not enjoyed the same upward mobility as had many Muslim immigrants. As discussed in Chapter 2, such service projects had long been a way for Muslim Americans, like previously marginalized religious minorities, to lobby for social acceptance and to prove their moderation. During the summer of 2010, as their plans to open Cordoba House attracted international criticism, Rauf and Khan took notice of this trend.\n\nStunned by the negative reaction to their project after enjoying so many years of acceptance\u2014and after having received approval for it from the local Manhattan governing board charged with overseeing development\u2014Khan gave voice to her surprise and disillusionment. \"When will Muslims be accepted as plain old Americans?\" she asked. Staff at _Newsweek_ , who considered the two spokespersons for moderation, were indignant. \"Figures from Sarah Palin to Donald Trump blithely conflated the peaceful Sufi strain of Islam practiced by Imam Rauf with the terrible deeds perpetrated by the twisted zealots of Al Qaeda on 9\/11,\" they wrote. No longer publically emphasizing their Sufism, and not generally engaged in service endeavors akin to those of black American Muslim leaders, Khan and Rauf would nevertheless cite both Sufism and service as proof of their moderation once some of their previous political pronouncements\u2014especially a comment Rauf made on CBS's _60 Minutes_ during one of his less guarded moments in 2001, in which he called US policies \"an accessory to the crime that happened\" on 9\/11\u2014came back to haunt them. Even these efforts at reemphasizing their moderation would rankle some other local Muslim leaders, however, as the services that Rauf, Khan, and their Cordoba House partners planned to provide resembled the cultural and educational programs offered at places such as Manhattan's affluent Jewish Community Center (JCC), not those desperately needed by New York City populations in financial or social distress.\n\nThe criticism and suspicion Rauf and Khan faced in and after 2010 raise important questions about the limits of American tolerance and about the lengths Muslims must go in order to qualify as \"moderate\" or to simply be tolerated in the United States. I return to the role Rauf and Khan played in the Islamic community center debate and what criticism of that project reveals in Chapter 7. Before examining the tumultuous events of 2010, however, I examine the life of Rauf's mosque and Sufi community\u2014which included those who would purchase the property for the downtown Islamic center\u2014during the years of his increasing absences. As mentioned previously, many of Rauf's dervishes welcomed his argument that they were both authentically Muslim and authentically American, and that these two parts of their lives did not conflict. Nevertheless, without Rauf personally available to guide them, his dervishes sometimes interpreted his teachings and lived their Islam in ways that Rauf did not quite intend.\n5\n\nTHE MICRO-POLITICS OF MODERATION\n\nAT THE END OF FRIDAY _JUM'AH_ PRAYERS ONE FEBRUARY, a controversy erupted at Masjid al-Farah. The disagreement was over the number of _rak'at_ (cycles of prayer) performed at the end of the service. Tensions over whether to perform two (the obligatory number) or four had been building for some time within this congregation\u2014home to Senegalese street vendors, Pakistani cab drivers, Wall Street traders of various nationalities, and a handful of American converts\u2014and Rauf was away, unable to provide guidance at that moment. Rauf's custom, followed by the assistant imams, was to combine _jum'ah_ prayers (which replace noontime, or _dhuhr_ , prayers on Fridays) with _'asr_ (the fourth of the daily prayers) into a cycle of four _rak'at_. The practice was helpful to those who commuted or traveled often and could not find a suitable prayer space while in transit. Because such combining is unconventional (though not unprecedented) for Sunni Muslims, many of those gathered on Fridays would leave after only two, creating a bit of disorder as they tried to squeeze through ranks of tightly packed worshippers or step over those in mid-prostration.\n\nWomen visiting the mosque would often glance at me, bewildered, when the third _rak'ah_ began. During the first few years of my fieldwork, which lasted from 2004 until 2010 and involved attending _jum'ah_ regularly, the tiny storefront mosque saw a tremendous increase in worshipers. Frequently overflowing on two floors during Friday prayers, it went from one service each week to two and finally, after the nearby Warren Street mosque closed, to three\u2014and still could not accommodate all who wanted to pray in the Lower Manhattan neighborhood. I was frequently the only woman present during the last service. At some point, I assumed the unofficial role of greeting female visitors\u2014issuing the _salaam alaikum_ (peace be with you) when they came and went, pointing the way to the restrooms if they needed to perform ablutions, and answering any questions to the best of my ability. By the time of the _rak'at_ controversy, the role felt almost natural.\n\nAmid the grumblings over prayers on this particular day, Hussein\u2014a Lebanese American from Shaykha Fariha's Sufi _tariqa_ who often served as the muezzin\u2014appealed to American liberal mechanisms for solving the dispute and, in so doing, challenged the authority of the imams who served in Rauf's absence. \"Let's take a vote!\" he shouted. \"This is a democracy, right?\" His call went unheeded and the commotion continued, slightly muted by traffic as worshippers filed out of the tiny prayer space. Most performed only two _rak'at_ that day, and some exchanged looks of concern over the usual mel\u00e9e of finding shoes and trading _salaam_ s at the back of the hall.\n\nThe following week, Pedram\u2014an Iranian American Shi'i in his early thirties who regularly filled in as imam at this mainly Sunni mosque\u2014read a message from Rauf to the assembled worshippers. On learning of the controversy from Pedram and another assistant, Fareena, Rauf had emailed to explain that most Sunni law schools permit combining the two prayers when worshippers need to travel. After the last _jum'ah_ that day, I joined Pedram, Fareena, and other dervishes for lunch at a trendy sandwich shop around the corner from the mosque. Over bowls of roasted tomato soup (paid for by Fareena or Pedram, both of whom regularly treated others to lunch and refused attempts at reimbursement), Fareena asked me if Rauf's letter on shari'ah had been clear. I told her it was. \"Good,\" she replied, explaining that some of the immigrants needed to let go of their \"cultural\" biases about how to practice Islam. \"We want to build an _educated_ community on educated people,\" she emphasized. Fareena then recommended to me some of the books she used for guidance in distinguishing cultural traditions from the essential, universal aspects of Islam that rightfully apply to living in the United States.\n\nAmong the devotional and instructional materials Rauf recommended to his assistants were Seyyed Hossein Nasr's writings. Drawing from Rauf and Nasr, Fareena explained to me that Sufi practice enables Muslims to appreciate the core of Islam\u2014the aspects of the tradition that are universal, regardless of one's racial, ethnic, or national culture\u2014and, by helping Muslims pluralistically transcend cultural differences, to build unity within and across diverse communities. Yet she emphasized that understanding derived from Sufi practice did not alleviate the need for guidance from learned leaders\u2014the role of a shaykh in disciplining and directing a community was still crucial. Without denying the value of liberal democracy, Fareena defended Rauf's authority from the challenge issued by Fariha's dervish. She did so by appealing to a different American paradigm than Hussein had\u2014one derived not from political liberalism but from the market liberalism that characterized her husband's corporate job and that frequently appeared in Rauf's writings and talks. \"Hussein is a janitor,\" she analogized, expressing the appropriate relationship between learned shaykh and humble dervish, \"and a janitor does not stand up to a CEO.\"\n\nDuring the six years I spent attending _jum'ah_ at the Masjid al-Farah and getting to know members of the community, and during the three years I attended Rauf's _dhikr_ meetings, I found that Muslims there, following Rauf's example, constantly interpreted Islamic traditions and strands of American liberalism (political, economic, and social) in light of each other. They did so while applying Rauf's teachings about Sufism and authentic Muslim living to their daily lives, even though not everyone agreed with all aspects of Rauf's vision of Americanness and definition of moderation. Those who joined his order and served as surrogates\u2014filling in as imams and prayer leaders or delivering lectures while he was unavailable\u2014often agreed with him more than others, and some chose Rauf as an imam and shaykh because he articulated ideas they already had about the United States and Islam. For example, one dervish related to me his Kashmiri grandfather's axiom about American values: \"If there's any Islamic country in the world, it's America!\" Simultaneously, however, as we shall see, intertwining Islamic traditions with American liberalisms led some to be more vocally critical than Rauf of US domestic policy and social life, as well as aspects of other Muslims' practices. In short, despite their similar interpretive strategies, Muslims at Masjid al-Farah\u2014like Fareena and Hussein\u2014came to very different conclusions about the proper contours of Muslim Americanness and moderation.\n\nIn his _khutbah_ s during these years, Rauf often enjoined worshippers to strive to overcome limited cultural traditions in order to adapt Islam to American society. Doing this, he informed listeners on a mid-spring afternoon a year after I began attending, requires a kind of \"spiritual training\" (a term he often used as a euphemism for Sufism) that begins with ritual observance but \"evolves to higher levels through continued effort, wrestling with ourselves and God, in greater jihad.\" He elaborated to say that such Sufi striving (jihad) awakens Muslims to the purpose of life: to know God. \"If you are practicing your Islam without realizing this, then your Islam is cultural and not spiritual . . . If you are practicing your Islam because you are Arab or Pakistani or Senegalese, then your Islam is cultural.\"\n\nAs I discuss below, many of the Sufis I met followed Rauf's example of blaming other Muslims' cultural mindsets for their misunderstandings of the proper relation between Islamic and American traditions, for acting immoderately in various ways, and for causing tensions among Muslims in the United States. Over the years, they also frequently reiterated his injunction to engage in the Islamic tradition of \"greater jihad\"\u2014a tradition that, in his telling, involves striving to foster an enlightened personal (as opposed to cultural) practice\u2014in order to transcend the differences that divide them from each other and from members of other religions. While some echoed Rauf by interpreting this form of striving in terms of American economic liberalism, expressing the need to struggle for spiritual and material growth in market metaphors such as \"investing\" in the future, others had very different ideas about what authentic, moderate Islam should look like in America. These differences were evident in their disagreements about the shape community relations should take, as well as about how to balance the material success some of them had achieved in the liberal market with the financial and social needs of those dispossessed under US capitalism.\n\nRather than acknowledge the political, economic, and racial aspects of these disagreements, the Muslims I encountered almost invariably described their differences as cultural ones. When such explanations for tensions or conflicts did not satisfy\u2014such as when one person understood an issue to be simply cultural and another imbued it with more importance or saw it as political or economic\u2014some Muslims appealed not just to the greater jihad of overcoming culture but also to other Sufi traditions that, they felt, helped them to manage disputes about practice and moderation. Making reference to _adab_ \u2014which some understand in terms of manners and etiquette and others view as akin to a guarantee of rights to equal treatment\u2014allowed these community members to offer alternative visions of proper relations within the mosque and of authentic moderation in the United States.\n\nAs we shall see, Sufis' understandings of traditions such as _adab_ sometimes varied as much as their understandings of culture did. This was perhaps unavoidable, even if their imam and shaykh had been more available to guide them personally. Not only did these Muslims interpret Rauf's teachings in light of their own daily circumstances, they (even converts) derived their understandings partly from the interpretative traditions common to their communities of origin. To place the understandings of Muslims at Masjid al-Farah in these larger interpretive contexts, I combine observations from my fieldwork with findings from Columbia University's six-year \"Muslims in New York\" project, conducted from 1997\u20132003. Researchers involved in that study led focus-group interviews with Sufis (including some who identified as Rauf's dervishes) and other Muslim New Yorkers before the 9\/11 attacks and did so again afterward to understand participants' experiences of \"identity, community, and civic and political engagement,\" as well as their perceptions of the ways Islam and Muslims are represented in the United States. I attended to similar questions during my own research. Far from relying solely on interviews, however, I observed the ways in which Muslim New Yorkers, in their everyday interactions with each other, created senses of community and deliberated about how to live in the United States in what they understood to be an authentically Islamic manner. Comparing the perspectives of Rauf's dervishes with those of other local Muslims reveals the ways in which these Sufis engaged Islamic traditions and American liberalisms after 9\/11 and, in the process, both expanded the possibilities for living authentically as Muslims and reinforced limits on what could be considered moderate.\n\nIdentifying Culture and Enacting Moderation\n\nOn his return to the mosque a month after the _rak'at_ debate, Rauf reemphasized the need to practice Islam \"beyond culture.\" After ascending the steps of the stately, dark wood _minbar_ (which functions somewhat like a pulpit) and offering an opening prayer, Rauf explained to the Friday congregants: \"most of us, we grew up in Muslim families, we are Muslim because our parents are Muslims . . . you are Muslim because it is your culture. See, cultural religion is not good enough.\" Because cultural religion is unexamined, Rauf argued, it blinds worshippers to the universal heart of Islam and allows unnecessary tensions over practice to arise. As he did frequently, Rauf encouraged worshippers at the mosque to strive for understandings that would allow them to overcome cultural conflicts and to model such liberal multicultural pluralism for the rest of the world.\n\nI heard Rauf deliver this injunction to worshipers at the Masjid al-Farah as early as a Ramadan service in 2004, during which he emphasized that the work of winnowing out problematic cultural biases and practices was a particularly American project, one Muslims in the United States are required to engage in because of the sheer diversity of Muslim communities in the country. \"Brothers and Sisters,\" he exhorted worshippers, \"you have to anchor yourself on what is eternal and unchangeable in your religion. We, as American Muslims, know this. This is our _din_ [tradition, or religion]. Deconstruct it . . . separate the cultural from the theological . . . because there are many practices that are cultural and not really eternal.\"\n\nRauf and the Sufis at his mosque were far from unique in juxtaposing appropriate individual striving, spiritual and material, with the immoderate practices of \"cultural\" Muslims\u2014a category that often included those deemed to be too political. After 9\/11, debates over cultural Islam and the nature of authentic jihad became a focal point for proving (or disputing) the moderation of Muslim Americans more broadly, and these distinctions between cultural and political Islam and what many see as the greater jihad of self-transformation frequently took on racial overtones. As I discuss below, many Muslims\u2014trying to prove their moderation\u2014associated extremism not simply with other Muslim individuals but with other cultural groups and races.\n\nBlack American Muslims (Sufi and non-Sufi), for example, wavered between conflicting impulses in the years after 2001: to resolve their tensions with other Muslim communities in favor of general Muslim solidarity at a time of crisis or to create distance from the specter of Arab \"political\" Islam. One scholar of black American Islam captured (and echoed) these tensions over claims to moderate status by showing how they often involved racial dynamics. Writing about his findings in 2003, he argued that with\n\nmuch of the post-September 11 media coverage focusing on the immigrant communities, the social and political perspectives of African-American Muslims, who constitute roughly 30 percent of regular mosque participants, have largely been ignored. That is unfortunate because the ethnic identity of this group is shaped by its long history of contributions to the American experience. [Black] American Muslims are probably in the strongest position to refute the arguments of scholars who claim there is a \"clash of civilizations\" between Islam and the West.\n\n\"We [Muslims] are not at war with the American government,\" said one of my interviewees, \"but against temptation. It's a jihad against yourself\" . . . \"9\/11 was a political jihad, not a religious jihad,\" he said. \"For that reason, African-American Muslims do not connect to it. African-American Islam is about peace, justice, and prosperity.\"\n\nAccording to this definition, proper Islam is not political\u2014something, the interviewee implied, that is the province of Arabs or other racial groups\u2014but personal, a \"jihad against yourself.\" Black American Muslims were not the only ones to claim that their communities were the most dedicated to engaging in the greater individual jihad of battling temptation and striving for peace, justice, and prosperity. Such racial overtones also echoed through Masjid al-Farah as Muslims from a wide variety of ethnic, racial, and economic backgrounds interacted.\n\nAs mentioned previously, the worshipers who assembled weekly at the masjid\u2014and even the Sufis who were closest to Rauf and Fariha and served as community leaders\u2014were extremely diverse. Some had been born into Muslim families. This was the case for two of Rauf's closest assistants, Faiz Khan, who cofounded ASMA, and Fareena, a graduate of the Columbia University School of International and Public Affairs who had married an American financier and worked for the United Nations before having children. Raised in Bangladesh, where women were not always welcome in mosques, Fareena had given up finding an imam she could relate to until she learned about Rauf in 1997 while visiting Sufi Books, a store owned by Shaykha Fariha that closed in 2006 as rents in the neighborhood skyrocketed. Fareena was one of the few dervishes who read Rauf's writings, making it unsurprising that she served as a leader within the community, filling in for Rauf in many ways when he travelled\u2014including delivering lectures to visiting groups and to audiences at various religious institutions and universities\u2014while also dealing with many of the community's organizational needs. Although once so close to her shaykh that she seldom had to refer to his writings for guidance, Fareena found herself reliant on these and other sources for reconciling Islamic and American traditions as Rauf spent greater stretches of time away and as conflicts arose within the community in his absence.\n\nWhile others in the mosque were not born into Muslim families, they sometimes still hailed from Muslim communities. Brother Malik, for example\u2014a native New Yorker in his late fifties who had retired from the military and was winding down a second career as a Harlem schoolteacher in 2005\u2014had been a Muslim for most of his adult life. Born to a Christian father and raised Protestant, but also the descendant of Cuban Muslim grandparents, Malik had friends in the Nation of Islam while in high school. An avid reader of books on comparative religion, he found his way to a variety of Sufi orders over the next forty years. Malik originally met Rauf at Shaykh Muzaffer's Jerrahi _dhikr_ service in the 1980s. Despite his extended tenure, Malik also maintained ties to other (mainly black American) Sufi orders.\n\nIn addition to black Americans and Muslims from Arab and South Asian families, a small number of Euro-American converts regularly attended the mosque. Among them were Dean, an Italian American photographer, and Emily, a blue-eyed, blonde-haired young woman from North Carolina who often drew the attention of visiting reporters. Emily first attended the mosque at the request of her boyfriend, Manny\u2014an Egyptian American whose brother had discovered the mosque the year before. Manny's blonde, blue-eyed sister-in-law from Long Island (Leah) also greatly preferred the spatial arrangements within the mosque and Rauf's style of speaking to the more gender-segregated and (in her opinion) aggressive style of worship that she had encountered in other mosques around the city.\n\nMore than once, I heard dervishes at the masjid comment on the overly \"cultural\" practices of worshippers from other ethnic or national backgrounds. This often happened when they perceived some practice to be a sign of excessive attachment to the outward minutiae of the tradition\u2014such as when Senegalese street vendors who frequented the mosque left a sloshy layer of water on the floor of the upstairs bathroom after performing ablutions. (Admittedly vigorous in their washing habits, those who spent the day working on New York City streets found cleanliness before prayer to be no small achievement. Signs posted in English\u2014but not French or Wolof\u2014about the need to protect the floor failed to resolve the issue.) After a few years of attending the mosque, I also learned from occasional disapproving comments to notice the young Arab men who entered the building with their jeans rolled up above their ankles, a practice some deemed necessary because the Prophet Muhammad reportedly said that allowing a garment to hang below the ankles was a sign of conceit and possibly condemnable.\n\nThe rhetorical style of some of the guest speakers also drew occasional disapproving comments. Frequently South Asian, and often visiting from the nearby Warren Street mosque, these speakers were much more exuberant when delivering their _khutbah_ s than Rauf. Showing their cosmopolitan colors, even some of Rauf's and Fariha's South Asian dervishes made critical remarks about the \"hellfire and brimstone\" style of these visiting imams. On one occasion, as a visiting imam enthusiastically held forth from the _minbar_ , the proprietor of a neighboring restaurant and bar approached the mosque's leaders to discuss the commotion. Fearing such speakers might alienate their neighbors and give other members of the community a false impression of the mosque, Rauf and his assistants enlisted men from Rauf's _dhikr_ group to deliver Friday _khutbah_ s instead.\n\nWhile these examples of varying practices might appear to be nothing more than innocuous cultural differences, they could be (and sometimes were) interpreted as signs of fanaticism\u2014signs, in other words, that a particular person was not just an inappropriately cultural Muslim but also an inappropriately political one. This was not without consequences during a time of ubiquitous racial profiling and heavy government surveillance. It was by conducting themselves in contrast to such behaviors (and by vocally disapproving of them) that many of Rauf's and Fariha's dervishes enacted their status as multicultural and pluralist Muslims and, in so doing, demonstrated their moderation to multiple audiences\u2014defending themselves before non-Muslims while emulating moderation for the overly cultural Muslims in their midst.\n\nThose practices described as overly cultural and implicitly or explicitly ascribed to racial, ethnic, or national heritage occasionally overlapped with sectarian differences, despite the fact that a Shi'i assistant imam was a central figure at the mosque. While new American converts voiced the least concern about differences between Sunni and Shi'i Muslims (and least awareness of them), other Sunnis were more circumspect about such things, although they often attempted to reconcile their suspicions about the validity of Shi'i traditions with the pluralism Rauf encouraged. After a Friday evening _dhikr_ ceremony in December 2006, for example\u2014an evening when Rauf was in Malaysia and Pedram led the service\u2014the conversation turned to the upcoming Islamic month of Muharram. Sipping her tea, legs crossed in front of her, Bakira, a Bosnian immigrant who had joined Rauf's Sufi group in the late 1990s, mentioned a special dessert her family often made during that time. \"Noah's Pudding,\" as she called it, involves soaking together and then cooking a mixture of beans, grains, dried fruit, and sugar: all items Noah is thought to have combined from the stores left on his ark after the Great Flood. In Bosnia, they called the dessert \"ashura.\"\n\nHearing this, Pedram laughed softly. \"I'm pretty sure that's a cultural innovation,\" he responded. \"As are the Shi'i traditions,\" Bakira shot back quickly, putting Pedram in his place. Pedram fell silent, not wanting to engage further on the topic. He offered no comment on the day of mourning, known as Ashura\u2014the tenth of Muharram, which commemorates the martyrdom of a pivotal Shi'i figure\u2014to which Bakira was referring. Nor did he offer a defense of the practices sometimes associated with it, such as self-flagellation. Already slow to speak casually about Islam, perhaps partly the result of immigrating to the United States from Tehran at the age of eight during a time when the Iranian hostage crisis was the feature story of nightly newscasts, Pedram rarely mentioned his Shi'i heritage. His references to Shi'ism became even more infrequent as he began to serve as a regular imam on Fridays and as he increasingly led the prayers at _dhikr_ and offered _sohbets_ (the post- _dhikr_ exhortations or addresses generally delivered by a shaykh) to Rauf's dervishes.\n\nWhile Rauf considered Shi'i traditions to be among the many paths available for accessing the Divine, other Sunni Muslims in the community saw Shi'i practices, particularly those of Iranians long demonized by American media outlets and politicians, as excessively cultural and even excessively political. Voicing skepticism of these traditions not only served to underline their Sunni orthodoxy\u2014which some may have felt was in question due to their Sufi practices\u2014but also demonstrated, by contrast to ostensibly fanatical Iranians and other racialized groups, their status as moderate Muslim Americans.\n\nUnbeknownst to most dervishes and mosque attendees, Rauf frequently borrowed form the teachings of a Shi'i thinker, Seyyed Hossein Nasr, for his Friday _khutbahs_. Moreover, it was partly from Nasr\u2014as well as from his father, Muhammad Abdul-Rauf\u2014that Rauf had learned to combine Islamic traditions with American liberalisms. While very few of the Sufis I interviewed had read Rauf's highly publicized 2004 book, even fewer, if any, knew that Abdul-Rauf, Nasr, and Michael Novak had worked together to illustrate the commonality between Islamic teachings and American market liberalism in the 1980s. Those who belonged to one of the orders associated with the mosque often assumed that Rauf simply elaborated on Shaykh Muzaffer's teachings when illustrating his understandings of the Qur'an and when urging Muslims to overcome cultural differences.\n\nIn contrast, Rauf's closest assistants often discussed Nasr's writings and used them as resources for teaching. It was from these resources that they and Rauf derived the understanding they tried to impart to other community members: that most varieties of Islam and, more than that, most varieties of religion, are cultural variations on a universal theme. Several Sufis echoed aspects of these arguments, even without awareness of their origins, as discussed below. Doing so could not always dispel the tensions that arose from so-called cultural differences, however.\n\nReligious Commonalities and Community Tensions\n\nOn a dreary January day in 2006, I visited Battery Park City, where Fareena had just moved with her husband, toddler, and new baby. Making my way from the subway to her building meant taking a pedestrian footbridge that wound around the gaping cavern in Lower Manhattan left by the missing Twin Towers, then crossing over the West Side Highway into a miniature city of tree-lined streets. The rows of bare trunks, bleak with winter, were multiplied in the gleaming windows of office and residential skyscrapers and interrupted only by uniformed doormen. Fareena welcomed me into her apartment with characteristic graciousness, apologizing for the boxes and offering a cup of tea. The new place didn't have a neighborhood feel, she lamented, but at least it was closer to the mosque and to her husband's job in the Financial District than their other apartment had been.\n\nFareena and I spent some time catching up while her two-year-old played peek-a-boo from the kitchen doorway. Our conversation eventually turned to matters of faith and practice, as it often did, and Fareena informed me that Rauf had suggested she read a biennial Traditionalist-Perennialist journal to which Nasr and his students and colleagues frequently contribute. In her experience, the articles in _Sacred Web_ provided a more expansive understanding of religion than that promoted by nominal religious authorities, such as those in Saudi Arabia, whom she considered to be self-appointed gatekeepers of orthodoxy. Feeling particular contempt for Saudi officials on that day, given the news of many recent deaths in Mecca during the annual pilgrimage, Fareena asked rhetorically, \"with all of the money that comes in from the _hajj_ , can't they build a better infrastructure to protect people from accidents?\" In contrast to her disdain for Saudi leaders, Fareena considered Nasr's perspective so important that she had already unpacked and placed the _Sacred Web_ journals on the bookshelves in her otherwise sparsely furnished living room and used them almost exclusively when teaching.\n\nLater that year, I sat with Pedram at Franklin Station, Rauf's favorite French-Malaysian restaurant, and talked about Nasr as we waited for the kitchen to finish Rauf's takeaway order. As usual, Pedram\u2014who had yet to begin delivering _khutbah_ s then\u2014was fixed on his laptop. Assuming that fetching food was one of his paid duties, I asked him how long he had been working for Rauf. Pedram responded with a slight smile and the usual repose that set him apart from other young men in the group. \"I don't work for Imam Feisal. I'm a trader,\" he replied softly. I realized then that what he frequently inspected on his computer over lunch was the traffic of financial transactions taking place before the stock markets closed for the weekend. \"Ohhhh, sorry,\" I said, embarrassed that I had assumed the service he offered to his shaykh was work done for a weekly paycheck.\n\nAs we talked further, Pedram informed me of how close his Iranian family was to Rauf and described how he had taken college classes with William Chittick, one of Nasr's students and frequent collaborators, several years prior. It was during that time that Pedram became interested in Sufism. I then asked Pedram what he thought of Rauf's reliance on Nasr. \"He doesn't really like Nasr's antimodernism,\" Pedram informed me. Indeed, Rauf's _khutbah_ s, like his writings, included frequent references to modern conveniences, modern sciences, and market economics. The week just prior, in fact, in front of a State Department delegation of visiting scholars from Egypt's Al-Azhar University, Rauf had exhorted Friday worshipers to strive to build up their \"net worth.\" Islam was not just about ritual, he emphasized, but about striving to \"increas[e] one's assets and decreas[e] one's liabilities\" spiritually. Pedram, in comparison, who became an assistant imam after Faiz Khan left the community in 2006, never used market metaphors in his _khutbah_ s. Instead, he usually illustrated the Qur'an with passages from Rumi.\n\nAlthough Rauf is not as antimodern as Nasr, Pedram informed me, he's also \"not a modernist like [Fazlur Rahman] at Chicago,\" who only regarded parts of the Qur'an as authoritative for providing instruction about contemporary issues. Rauf \"searches for a middle ground,\" Pedram said. What really interested Rauf was not Nasr's Traditionalism but his Perennialist Sufism\u2014the idea that one truth has been perennially revealed over time and, due to changing cultural and historical circumstances, has given rise to a variety of religious traditions.\n\nUnlike Pedram, who diligently followed Chittick's writings on Ibn 'Arabi (an eleventh-century Sufi sage whom many Perennialists consider to be an early proponent of Perennialism) and often attended lectures on Iranian traditions at Columbia, most of Rauf's dervishes were less interested in philosophy than in _dhikr_ , poetry, and art (the traditions Fareena often discussed with groups visiting the mosque). Or, they found value in practicing Islam together as part of a community. In fact, those without family obligations sometimes spent long hours together over meals, coffee, or tea after _jum'ah_ or _dhikr_ , discussing their spiritual paths. They connected their \"inner\" experiences to movies, music, work, or other parts of their lives, especially their dreams (which Rauf used to interpret for them, sometimes treating them as evidence of divine urgings, and other times as workings-out of the psyche), without having read a thing. Still, it was evident in their conversations that aspects of Nasr's Perennialism\u2014particularly the distinction between the inner spiritual essence common to all religions and the outer cultural forms that essence assumes\u2014permeated their understandings.\n\nLike Nasr, Rauf describes Sufism as the means of inculcating the shari'ah in one's inner self. Rauf and his assistant imams frequently distinguished in _khutbah_ s between the _zahir_ and the _batin_ (terms often associated with Shi'i, even Isma'ili, esoteric traditions), which Rauf described as referring to the \"outer\" shari'ah and rituals and the \"inner\" spirit of Sufi practice, respectively. Appreciating the distinctions between the two aspects of Islam, he argued, allows Muslims of different backgrounds to identify what is universal in their religion and to recognize each other as common pilgrims on the path toward union with the Divine.\n\nIn the introduction to a popular 2006 book, _The Universal Spirit of Islam_ , Rauf elaborated on this view.\n\nGod the almighty Creator has repeatedly revealed His religion, essential and unique in its universal principles, through many prophet-messengers who have lived in different climes, languages, and cultures. Eternal and universal divine Truth has therefore respected locality by expressing itself in the language and culture of each prophet. The divine Word criticizes human beings for confusing locality with universality, thus effectively splitting God's one religion into many divisions.\n\nOur duty as religious people in the twenty-first century is to recognize the points of unity among different revelations.\n\nCiting multiple verses from the Qur'an to buttress this perspective but also tracing it to Frithjof Schuon, Nasr's Perennialist mentor, Rauf insisted that differences among Muslims were similar to differences between Muslims and members of other religions: all were variations on universal truth. For Rauf, this perspective makes it possible to disregard racial, cultural, sectarian, and even religious differences among those who ultimately subscribe to the same essential truths and ethics. Imparting this view to others, he argued for dispensing with some cultural traditions (particularly those that might be regarded as too political) and for adopting more \"American\" ones. And in making such arguments, Rauf enacted moderation for multiple audiences.\n\nSeveral Sufis I encountered at Rauf's mosque similarly emphasized commonalities with members of other religions, and most agreed that the Qur'an mandates pluralism. If they quoted any verse to substantiate this, it was usually Qur'an 49:13, a passage about the divine design behind human differences that Rauf also invoked in a 2005 article he wrote on Muslim multiculturalism. \"O humankind,\" he translated it, \"we certainly created you from one male and one female, and fashioned you into tribes and nations so that you might get to know\/celebrate each other.\" While Rauf's emphasis on multiculturalism and religious commonality appealed to many worshippers at Masjid al-Farah, though, it also sometimes proved to be of limited utility.\n\nAmong the Muslims who found Rauf's Perennialist teachings on religious commonality compelling were converts whose family members belonged to other traditions. As Abdul-Raheem, a black American former corrections officer, born to a Jehovah's Witness mother in an otherwise Baptist family, confessed, he sometimes still attended church to celebrate the milestones of family members and friends and did not see the need for religious exclusivity. \"I always thought I belonged to all of it,\" he told me. Despite the various forms of practice among the religions, \"there is just one\" ultimate truth. He then admitted that he had not always been so open. After he began practicing Sufism twenty years earlier, he started to \"moderate\" how he dressed\u2014dropping the robes and turbans he had once believed were necessary signs of \"orthodox\" Islam and that set him apart from other members of his multireligious family.\n\nRauf's Perennialism also appealed to many affluent Muslim immigrants\u2014often younger transnational ones, educated in English-speaking private schools in their home countries (to which they frequently returned), who were somewhat dazzled by the religious and cultural diversity of the United States. One Kashmiri importer, for example, acknowledged that he did not have Buddhist or Hindu friends while growing up. After moving to the United States for college, though, that changed. \"I met a lot of new friends, and they're really close and dear to me, really close and dear . . . you know, in every tradition there's truth to be found.\"\n\nNevertheless, many of the Muslims I spoke with still struggled with the differences _within_ their Muslim community, despite the fact that they valued Rauf's teachings about unity, reiterated the distinction he and his assistant imams made between the inner spirit and outer cultural or ritual aspects of Islam, and tried to demonstrate respect for people who practiced other traditions\u2014something they had regular opportunities to do if they participated in ASMA's interreligious events or in Rauf's _dhikr_ s, which a few Jews and Christians also attended. This struggle was due, at least in part, to the pressures Muslims in New York faced after 9\/11, ones that made it difficult to dismiss differences as simply cultural variations.\n\nThe pressures on Muslim Americans after 9\/11, it cannot be overstated, include intense scrutiny and the demand to demonstrate their moderation by creating distance from anyone whose overly cultural practice could be construed as fanatical or political. Importantly, less obvious pressures also contributed to more mundane and more frequent disagreements between Muslims about the importance of cultural practices. The factors responsible for these largely unarticulated disputes were not just race, ethnicity, and nation of origin (though they often correlated with such things). Rather, the important factors also included socio-economic status, availability of resources, and histories of marginalization\u2014in other words, political and economic issues that went unacknowledged when Muslim Americans, taking pains not to appear extremist, framed problems in terms of culture instead of politics.\n\nFraming problems in terms of culture is not a new phenomenon. Muslims in the United States who have employed this rhetorical strategy have long sought to resolve a host of interrelated issues. In addition to questions of authority and authentic practice, these issues include the multicultural respect various groups feel they deserve but are denied, as well as the mutual social and economic responsibility that many feel should characterize Muslim communities but does not. These sometimes-unarticulated concerns about respect and responsibility\u2014particularly with regard to economic resources\u2014were evident at Masjid al-Farah in the ways community members interacted with each other and in their notions of moderation, which they often described through contrasts to extremism. As I discuss below, they were also evident in various Muslims' disagreements about the importance of cultural traditions that did not constitute required parts of Islamic faith and practice. While Rauf encouraged worshipers to practice a culture-free Islam\u2014and, more specifically, to shed traditions that do not fit with America's democratic, capitalist culture\u2014some Muslims were not so ready to relinquish their particular practices or to speak as approvingly of what they considered to be an overly materialist culture and a predatory economic system. In appealing to Sufi traditions other than the \"greater jihad,\" these Muslims both demanded the respect they felt they were owed and offered a different vision of how to live authentically in the United States.\n\nThe Complexities of Culture and Community Relations\n\nIn exhorting Muslims to strive to understand the inner heart of Islam not bound by culture, Rauf often suggested that cultural traditions were rather unimportant. Speaking at the mosque one day, for example, he told the assembled worshippers,\n\nIt is like wearing a suit instead of wearing a _bubu_ or wearing a _thaub_. It's a cultural thing. When I go to Saudi Arabia, I wear a thaub. But when I'm in New York for a big meeting, I wear a suit. It's something which is like something I wear. I can take it off. It has no real . . . it's not a _batin_ -y thing, it's an exterior thing. So is your Islam an exterior thing or is it a _batin_ -y thing?\n\nCultural traditions are optional aspects of life, Rauf implied, correcting himself when he began to describe culture as \"not real.\" When observed too strongly, they obscure the truth of Islam and create obstacles to living authentically. While several dervishes echoed Rauf's argument\u2014almost invariably when they believed someone else's actions or speech reflected a superficial understanding of Islam\u2014they often used \"culture\" as a metonym to stand in for other things, ones that were important to them in a variety of complicated ways.\n\nMost of those I interviewed at one point or another discussed their hope to someday witness the kind of unity among Muslims that transcends particular cultural differences. Unlike Rauf, however, who urged listeners to shed immigrant cultural traditions so as to adopt dominant \"American\" ones, some Muslims made different kinds of distinctions between inner Islamic truths and outer cultural forms. For them, separating truth from culture meant recognizing that all cultures\u2014even aspects of Arab culture often seen as synonymous with Islamic orthodoxy and aspects of American culture that Rauf touted as ideal\u2014are equally limited. In a very multicultural manner, it also meant recognizing that all cultures could lead worshipers to divine truth. Consequently, when told that their own traditions were inauthentic, these Muslims (often black Americans and Latinos who were socially and economically marginalized) repeated back to their critics (often Arabs and South Asians) the imperative to separate truth from culture. Separating truth from culture allowed them to challenge their challengers\u2014to create a theoretically neutral space from which they could demand respect for and protection of their own cultural traditions, as well as protection of the communities, social supports, and economic resources connected to them.\n\nBrother Malik frequently gave evidence of this alternate perspective in _khutbah_ s he delivered at the mosque, and he was not unique in doing so. After worshippers trickled into the _masjid_ one afternoon in September of 2005, for example, Brother Malik emphasized what he understood to be universal desires, including a \"yearning for peace.\" As the ceiling fans whirred overhead in an almost futile effort to dispel the late summer heat, he turned to what he saw as Islamic mandates for mutual respect among Muslims. \"The Prophet, praise be upon him, would have us, you know, treat others as we want to be treated . . . I'm Arabic challenged, you know, but when we meet every week, I try to speak from my heart to your heart . . . to be kind, charitable, understanding . . .\"\n\nNot content to leave worshippers with only a general injunction to foster unity, Brother Malik underlined the stakes of doing so. The larger Muslim community could survive, in his opinion, only if those within it acted in solidarity. For Brother Malik, who came of age in Harlem during the height of the Nation of Islam, such solidarity traditionally included patronage of black-owned businesses that provided an economic infrastructure in otherwise neglected and blighted inner cities. \"I encourage you to give our brothers our business,\" he exhorted the diverse attendees. \"[We need] the local community, so buy your halal meat, you know, from your brothers . . . and be charitable with the upkeep of the mosque . . .\"\n\nBrother Malik represented a less affluent sector of the Sufi population than Rauf. Like many of Rauf's dervishes, he had traveled (often with the military) in Africa, Europe, and the Middle East, and spoke about the importance of understanding the world's various religious traditions\u2014something that not only helped to foster peace, in his opinion, but that helped him see the larger meaning of life beyond the trials of daily circumstances. While as cosmopolitan as other mosque attendees and as dedicated to lifelong learning, however, his daily life was constrained by a lack of means. Like some of the other dervishes who attended _dhikr_ , Malik read works on self-help philosophy and comparative religions\u2014but he did so from his sometimes-unheated Harlem apartment (the landlord often cut off utilities in attempts to get tenants to move so the rent-controlled apartments could then be listed at market rates). The cultural traditions he shared with other members of his Harlem neighborhood were not just inherited beliefs or particular items of clothing but strategies and resources for managing these conditions. They were ways of forging and maintaining community ties that were essential for surviving.\n\nBlack American Muslims involved with the Columbia University study took similar positions. Some lamented the decline of community cohesiveness\u2014and of related social services\u2014that seemed to coincide both with the decline in prominence of the Nation of Islam (and its network of businesses, schools, and houses of worship) and with the increase in Arab and South Asian Muslim immigration to the area. They blamed this loss of cohesion not only on racism that, they maintained, was a problem among Muslims in America\u2014with many immigrants, in their view, adopting white American stereotypes about black criminality\u2014but also on economic disparities that, they argued, also exacerbated intra-Muslim tensions. As a black American police officer in one focus group interview bluntly put it, the dominant \"American culture is a racist culture [and] a class culture.\" Already strained by the racism often lurking behind comments about cultural difference, relations between Muslim communities are further stressed by \"class-oriented _masjid_ s,\" he argued, particularly affluent ones in poor neighborhoods where local (often black and Latino) children are not accepted into the mosque's schools. \"We are trying to deal with the Islamic community and we have to recognize that fact . . . [W]e have social issues, class issues, racial issues, [and] we can not always put our head in the sand.\"\n\nThe more affluent Muslims who gathered at Masjid al-Farah on Fridays included importers, financial analysts and traders, and brokers of commercial real estate. While their incomes likely varied as much as their nationalities, most did not need the economic supports provided within local communities\u2014be it childcare offered in shifts by alternating parents and neighbors; meal sharing; or frequently circulating gifts of groceries\u2014and tended not to see cultural solidarity as a source of material security. Further, the more economically stable Euro-American converts in the group were frequently unaware that tensions over resources even existed between Muslim communities in the United States and (like many affluent Muslims) tended to view calls to address disparity as immoderate and as the _cause_ of political controversies.\n\nAfter _jum'ah_ one rainy spring day in 2007, these generally submerged differences over culture, economic disparity, and mutual responsibility came to the surface. Several mosque attendees (mostly Rauf's dervishes) traversed the flooded streets to enjoy a collective meal at Franklin Station. As diners packed around tables at the front of the restaurant, the discussion turned to a recent _New York Times_ article that discussed economic inequalities between black and immigrant Muslim Americans. One Muslim, an importer who made frequent trips back to his native Calcutta and had helped finance construction of the Islamic Center of Long Island mentioned in the piece, said he had not read the article because it seemed too negative. \"What divisions?\" Masheer asked rhetorically. \"There are no divisions.\" He then explained that families at the Long Island Islamic center were \"adopting\" black American Muslims so as to assist them and develop ties between the groups. The two black American Muslims seated at the table for lunch remained silent during the discussion, neither offering nor being asked for their opinions.\n\nThe conversation moved to the different voting patterns mentioned in the article and to the black American Muslims who felt betrayed in 2000 when Arab and South Asian organizations supported the Republican presidential candidate, George W. Bush, who had promised to stop racially profiling Arabs. Black American Muslims, in contrast, tend to vote for Democratic candidates. Listening to a review of the article's points, Masheer interjected, \"Irish Catholics don't all vote the same way . . . so why focus on Muslims?\"\n\nLater that afternoon, as two of Rauf's Euro-American dervishes met over coffee in the West Village, the subject came up again. Dean had not read the article, and Emily\u2014who had only glanced at it\u2014summarized it as a story about an imam \"from the Bronx or Brooklyn who went out to Long Island\" to ask for money. That imam was Talib Abdur Rashid of the Mosque of Islamic Brotherhood in Harlem, who had traveled to Long Island during Ramadan to raise money for the Muslim Alliance in North America (MANA, a community service and antirecidivism organization he and other black American Muslim leaders founded in 1999). Although Emily and Dean participated in many projects with Rauf and other New York City religious leaders, neither had heard of Imam Talib's decades of interreligious and social service work in Manhattan or knew of his organization. \"Why can't we just all see each other as believers?\" Emily asked (echoing Masheer). She went on to say that she thought the article's discussion of economic issues was creating tensions that had not previously existed.\n\nEmily and Masheer both believed that it was important to help those who were, in their opinions, truly struggling financially, and they cited such action as proof of authentic Muslim Americanness. Nevertheless, unlike Faroque Khan, the Long Island Muslim leader who had invited Imam Talib to fundraise at the Islamic Center after reading books about systemic racism in the United States, Masheer and others resisted discussions of systemic economic marginalization. This resistance exposed the boundaries of moderation for them. For Masheer, \"adopting\" struggling families was an important way to enact the \"balance\" (a word many used instead of \"moderation\") of individual economic success and community care that should mark a devout Muslim's life; \"political\" projects of reducing economic marginalization were not. Emily somewhat illuminated the logic underlying this distinction by later citing \"ghetto\" culture as the cause of some Americans' economic distress, implying that these Americans had not cultivated the personal habits and practices necessary to succeed in the American work force. Such Muslims\u2014mired in a culture of immoderate dependency rather than of personal striving, many believed\u2014were not justified in making claims on the resources of others.\n\nIn addition to economic disparities, histories of political and social marginalization and repression could influence how attached various Muslim Americans felt to their cultures, to their communities of origin, and to the traditions practiced within them. One black American in the 2003 Columbia group argued that black American Muslim traditions were as authentically Islamic as those of Arab Muslims. This was not _despite_ the fact that their histories and cultures differed, it was _because_ they differed and because of the repression black Americans had endured, particularly under slavery. Citing W. D. Mohammed, she argued that black Americans' experiences had given them a unique ability to appreciate the inner heart and essence of Islam and also made black American Muslims uniquely poised to heal America's broken democracy. Only black Americans, she believed, could guide the country back to its original spiritual principles and, in serving as such leaders, undo the hypocrisy of the founders who had argued for liberty while holding slaves.\n\nI think God has really given us this mandate to be physicians . . . slavery did something to our spirits, created a unique respect for spirituality . . . [N]ot religion, but that unique inner spiritual life that we have, that can go into the Koran and say, \"Oh, wait a minute. My soul is telling me that this is what this is saying.\" . . . [T]hrough your ancestors you have learned how to connect with the Creator on a deep level. Because that's all we had [during slavery] . . .\n\nAnd also our culture. We have something to offer in examining how do you reconcile this democracy with the hypocrisy. Our ancestors have forged the way to correct that . . . from Frederick Douglass who said no, we're not going to throw away America. He could have said let's leave. He had that option.\n\nThis Muslim woman and members of her community distinguished between outward cultural traditions and the inner essence of Islam not to dismiss the importance of culture but to assert the importance of their own cultural traditions in the face of marginalization from those who associate Islam with Arabs and the United States with white privilege. They were not alone in seeing Islam as a resource for strengthening marginalized communities. For \"African American and Latino Muslims,\" another interviewee described, Islam \"helped to establish ourselves as an entity to be respected . . . something extremely important for us since we were politically dominated as a people [when I converted]. I'm speaking now specifically of African Americans and Puerto Ricans . . . Politically we saw Islam as a force to be reckoned with. We saw Islam as a liberating element in our lives. . . .\"\n\nFor this man\u2014and, if his description is accurate, for many other marginalized Muslim New Yorkers\u2014Islam provided a sense of pride in community and even a foundation for multicultural political alliances, something that he felt was challenged by the biases of other Muslims and their assertions that his practices were not authentic. In his opinion, the solution to reconciling these competing experiences\u2014liberating, on the one hand, and stifling, on the other\u2014could be found in Sufism. Rather than appealing to the \"greater jihad\" of transcending culture, however, this marginalized Muslim mentioned other Sufi traditions.\n\nMany times we [converts] would be met with opposition: \"oh, you're not really Muslim if you can't speak Arabic.\" This is one of the first obstacles that we had to overcome. [We were told,] \"You really can't learn this religion unless you go to Mecca to study, or to Al-Azhar.\" So we began to look beyond the limitations that were being imposed on us from people that were trying to limit and stifle our growth.\n\nMany of us were attracted to the concept of _tasawwuf_ [Sufism] because _tasawwuf_ personalizes the spiritual experience for the person that is new. Newcomers have rights over the people that have been in the _deen_ a longer time. _Tasawwuf_ has the _adab_ , the ' _urf_ and _akhlaq_ to put that up front.\n\nFar from seeing the need to practice culture-free Islam, this Muslim described Islam as strengthening the cultural pride and community cohesion of marginalized groups and as aiding their efforts to achieve political and economic justice in America. When Muslims outside of his community threatened that pride or political project by claiming that his practices, traditions, or understandings were overly cultural\u2014or, in this case, insufficiently cultural in that they were insufficiently Arab\u2014this Muslim New Yorker appealed to _adab_ , among other things, to assert his rights. The black American police officer who commented on class relations within American mosques echoed his perspective. \" _Tasawwuf_ ,\" he argued, is what one needs to \"make you strong\" with \"those brothers\" (of Arab origin) and to start to build relations with them.\n\nWhen disagreements arose at Masjid al-Farah over matters of authentic practice, less affluent and more frequently marginalized Sufis\u2014particularly those who had converted to Islam\u2014also appealed to the mandates of _adab_ , which they understood to be a tradition that ensured not just proper etiquette between Muslims (a common understanding of the term), but also something larger: \"rights\" of respect very much like those implicit in American liberal multiculturalism, and even an ethic of community-mindedness that might lead to addressing economic problems. In other words, Sufi traditions of _adab_ provided Muslims with an authoritative argument for overcoming a variety of issues\u2014ones generally described as \"cultural,\" but which often had to do with histories of collective domination or of material deprivation.\n\nInvoking _adab_ and the duties of service often associated with it was one way marginalized Muslims could attempt to moderate what they saw as the excesses\u2014overly cultural perspectives and even overly materialistic attitudes\u2014of their frequently more affluent critics. As such, it was also a way for them to offer their own visions of authentic Muslim living in America. While those raised in Muslim-majority countries did not usually understand _adab_ in terms of rights, respect, and service, these interpretations are not without precedent, as I describe below. My goal in discussing different interpretations of _adab_ is not to assert the validity of any one particular understanding. Rather, it is to show some of the ways that economic and political factors\u2014in addition to racial and ethnic differences but not reducible to them\u2014influence how people and groups understand, practice, and _live_ Islamic traditions in the United States.\n\n_Adab_ as Rights, Respect, and Economic Moderation\n\n_Adab_ practices have been central to Islamic traditions for at least a thousand years. Nevertheless, understandings of _adab_ have always depended on context. Historically, writers have composed manuals outlining _adab_ practices for varying purposes: to address the norms of courtly behavior, to outline the contours of certain vocational and social roles, to illuminate the appropriate conduct of Muslims in pluralistic environments (often, perhaps especially, during times of political and social upheaval), and to detail proper Sufi comportment in social situations (sometimes even specifying the moment at which diners were permitted to use napkins).\n\nIn view of these many uses, it is not surprising that _adab_ is sometimes translated as \"etiquette.\" Significantly, from South Asia to Egypt, _adab_ has also been translated more robustly as \"ethics.\" Still, neither the term \"etiquette\" nor \"ethics\" captures the real weight and import these traditions can have. Because _adab_ practices are part of a system of conduct in which the individual's relationship to the divine is inseparable from the individual's relationships with other persons (with proper action towards others contributing to the self-transformation necessary for drawing closer to God), some scholars have tried to convey the more transformative, interpersonal character of _adab_ by translating the famous ninth-century phrase _at-tasawwuf kulluhu adab_ as \"Sufism is all _practical ethics_.\"\n\nSo central is the practice of a _dab_ to Shaykha Fariha's Sufi order, the Nur Ashki Jerrahis, that members of that _tariqa_ dedicated a portion of their website to explaining it.\n\nAdab, commonly translated as moral character and refined behavior, is the state of perfect awareness, love, and gracious sensitivity which is innate to our soul. The highest level of adab is the knowledge that we have no independent existence apart from the One Reality. Perfect adab is constant awareness of Allah. From this awareness flows all beautiful behavior. On our path as dervishes we learn to govern the other aspects of our humanity with the light of adab so that we become complete human beings.\n\nIn this excerpt, the Nur Ashki Jerrahis translate _adab_ in terms of interpersonal ethics rather than etiquette or manners. \"The adab we bear toward others is our adab toward Allah,\" the webpage continues, emphasizing the interconnection between a dervish's actions toward other people and her relationship with God.\n\nAs to the practical expressions of such ethics, the perspective of the Nur Ashkis is quite close to that of participants in the Columbia University interviews. \"Mature adab never hardens into fixed rules or the correction of others,\" the Nur Ashki website states. Although not explicitly associating _adab_ with the rights of marginalized populations to receive respect, the Nur Ashki _tariqa_ \u2014a Sufi order led by a woman and comprised mainly of converts to Islam\u2014also interprets proper _adab_ in part as an ethic of humility toward those whose practices may differ from what others consider orthodox.\n\nContrary to what one might expect from the Nur Ashki website\u2014a website used to assert the authenticity of Fariha's order and implicitly defend the legitimacy of female leadership\u2014I did not often, if ever, hear Shaykha Fariha or her dervishes invoke _adab_ in interactions with others during my fieldwork. The person who spoke most frequently about _adab_ was Brother Malik. In contrast to members of Fariha's order, who felt secure and even empowered in the building their _tariqa_ owned, Brother Malik had reason to feel marginalized. He was the only black leader at Masjid al-Farah, and one who lived a life far less privileged than others did.\n\nDuring a _khutbah_ in the spring of 2007, for example, Brother Malik stressed the responsibilities, or \"duties,\" of Sufi practice. \" _Adab_ ,\" Brother Malik emphasized, requires Muslims to engage in actions that demonstrate their humility and their respect for others. These actions included being \"truthful in your speech,\" \"humble,\" and \"avoid[ing] causing difficulties to others while bearing the difficulties caused by them.\" Yet Brother Malik did not simply echo the ethic of respect and humility mentioned by other Sufis (particularly converts) in his discussion of _adab_. For him, a central component of pious living and of Sufi _adab_ is being \"generous with your property,\" or, as he sometimes put it, \"charity.\" Such charitable _adab_ served to demonstrate care for other Muslims at the same time that they moderated one's own affluence and privilege, thereby demonstrating not only piety, but a kind of \"balance\"\u2014a life dedicated to spiritual growth and social improvement, not just to material acquisition\u2014that could serve as an example to those with excessively cultural or political practices.\n\nSeveral months before that _khutbah_ , as US wars in Iraq and Afghanistan stretched through yet another year, Brother Malik had explained the responsibility of American Muslims to serve as examples to inappropriately cultural and political Muslims elsewhere and expanded on the ways charitable _adab_ could demonstrate proper balance. Addressing affluent immigrants with \"relatives overseas\" where Muslims detonated \"suicide bombs\" (thus associating extremism with Muslims born in conflict areas), Brother Malik informed them that\n\nMuslims here in America are in a position to have a great influence on the development in the Muslim world . . . That's something you need to consider. So while you're making all that money and, you know, sending your children to the best schools and, you know, you're eating all that good halal meat and all that stuff . . . also think of charity in terms of helping your brothers and sisters overseas to straighten themselves out . . . not just materially, we should show them that we're living well spiritually . . . that there's a _balance_ in our lives, that while we're making money with one hand, we're also keeping connection to Allah with the other hand.\n\nImportantly, men were not alone in employing the language of _adab_ to criticize what they viewed as immoderate affluence or privilege. As we have seen, female converts occasionally also invoked _adab_ to subtly demand respect in an Islamically sanctioned way, such as on the website for Fariha's _tariqa_. More frequently, they employed the concept as a way of correcting immoderate or overly privileged behavior. \"Where is your adab?!\" Charley, a veiled Euro-American convert and adjunct professor of African studies, exclaimed in frustration one afternoon after watching Emily interact with an elderly West African man. In Charley's opinion, Emily had treated the man disrespectfully by instructing him to stop blocking the mosque's only exit after Friday prayers while asking for alms ( _sadaqa_ ).\n\nAs Euro-American female converts, both Charley\u2014who had married a black American Muslim and knew some of the difficulties of black American life\u2014and Emily are particularly subject to communal scrutiny over privilege and proper practice in many ways. This scrutiny comes not just from men, something both Emily and Charley experienced on a number of occasions, but also from women seeking to demonstrate their authority\u2014including black American Muslim women who sometimes experience similar scrutiny from Muslims of Arab and South Asian descent. Even while some Muslims saw Charley and Emily as overly privileged, however, both women invoked _adab_ to criticize what they saw as others' insufficient economic moderation. Notably, they did so to confront other white female converts, not (at least, not in my observations) to criticize other Muslim-born men and women.\n\nEmily, a resident of the chic SoHo neighborhood just north of Masjid al-Farah, was more financially secure than some Muslims at the mosque when I met her, but she had not always been so fortunate. During her first few years in New York, she worked as a waitress and lived off tips. Her life became more comfortable once she married a real estate broker, after which she gave up the waitressing job that required her to serve alcohol and went back to college. Still, the contrasts between her modest North Carolina childhood and the opulent lives of many who worked in the Financial District often caught her attention. While supportive of Rauf's argument that Muslim Americans needed to strive for wealth in this world and the next, she sometimes indicated that she believed Sufi striving should have more limits. For example, during a 2007 _khutbah_ in which Rauf had again encouraged those at _jum'ah_ to put their full energies into bettering their spiritual and material conditions, she could not hide her disapproval. On that particular Friday, Rauf cautioned his listeners about pursuing excessive wealth and illustrated his point by telling them that he once believed he needed to have his own boat. Upon further reflection, though, he realized that he did not need more than two hundred thousand dollars (presumably a year) to be content in his worldly existence. When I glanced at Emily during this portion of the _khutbah_ , I found her eyebrows to be higher on her forehead than usual in an unmitigated expression of surprise. After _jum'ah_ she confessed to having been stunned by the figure.\n\nEmily never directly challenged her imam's economic teachings or financial practices in those years, though she did sometimes suggest that there needed to be greater transparency about dervishes' donations to ASMA\u2014particularly if dervishes were to allow Rauf and Khan to debit their checking accounts directly for contributions to what was then called the Cordoba Center project, as had been suggested as early as 2007. She did make reference to _adab_ in order to check the privilege of Fariha and her dervishes, however, once Fariha's order\u2014conscious of the rising cost of building repairs\u2014closed the restrooms and began restricting access to the mosque outside of prayer times. After yet another Friday morning of chasing construction workers in their work boots off of the prayer carpets\u2014no small feat, in that the leaks from the waterlogged restrooms were at the opposite end of the prayer hall from the entrance\u2014a female leader of Fariha's order remarked that the _masjid_ was really meant for Fariha's Sufi meetings but \"we give it to [Rauf's group] for jum'ah.\" Emily overheard the remark and gave voice to the tensions building between the two communities at that time. \"It's a mosque . . . how do you _own_ a mosque?\" she questioned aloud, and went on criticize the woman's failure of _adab_.\n\nEmily's recourse to _adab_ in this situation, combined with the other examples discussed above, demonstrates how the tradition serves (among other things) as an Islamically sanctioned way for members of the community who feel marginalized to demand respect, to assert claims to resources, and to question what they see as immoderate privilege. As noted previously, it provides a way for some Muslims to navigate the so-called cultural tensions produced by racial differences and socio-economic disparities. By using the tradition in this way, many of these Muslims assert understandings of moderation that borrow from and replicate some aspects of American political and economic liberalism while simultaneously criticizing other aspects that, in their experiences, fail to meet Islamic standards for ethical behavior. Additionally, as use of the term by Emily and Charley indicates, making reference to _adab_ allows women\u2014particularly Euro-American converts who feel scrutinized by other Muslims but who decline to rebut such criticism directly\u2014to assert their own authority and understandings of what is moderate, what is ethical, and what is (for better or worse) American.\n\nIn Chapter 6, I delve more fully into the gender dynamics I observed during my six years of fieldwork at the mosque and discuss how Muslims there dealt with what many non-Muslims regard as the ultimate index of moderation: the status of women. First, though, having identified some of the ways Muslims in New York interpreted Islamic and American traditions in light of each other, I conclude this chapter by discussing the limits placed on moderation when Muslim Americans\u2014in part wary of appearing extremist\u2014frame problems in cultural terms instead of political or economic ones.\n\nConclusion: Sufi Striving and the Limits of Moderation\n\nMost Muslims at Masjid al-Farah did not argue for the need to demonstrate moderation as often as Rauf did. More frequently, as noted above, they attempted to demonstrate their own lack of extremism through contrasts to others. These Sufis juxtaposed authentic Islamic practice and moderate Muslim living (or \"balance\") to what they saw as problematic \"cultural\" (which included overly political) traditions. Marginalized Muslims did so partly out of concerns about resources and repression. In turn, more affluent Muslims, who generally did not understand these issues, sometimes attributed experiences of economic and political marginalization to insufficient individual striving and viewed calls for distributing economic resources differently as immoderate behavior\u2014in other words, as causing political tensions rather than reflecting them. This perception and their hesitation after 9\/11 to criticize domestic politics for fear of seeming extremist may have contributed to discouraging projects of political or economic change such as those other Sufis\u2014including Imam Talib Abdur Rashid\u2014engaged in.\n\nMany affluent Muslims, particularly immigrants unnerved by the level of scrutiny and policing they experienced after 9\/11, hoped that avoiding political speech or action would help them be seen as moderates and deflect the attention of intelligence agencies and law enforcement. Black American Muslims, on the other hand, and others who had experience with police surveillance and brutality, were less hopeful about the possibility of ending such repression without dramatic action. They were also increasingly impatient with what they saw as the refusal of affluent and socially privileged Muslims\u2014Euro-Americans and, especially, more recent immigrants who also faced discrimination\u2014to recognize the depth of racism in America.\n\nAs Imam Talib argued in 2007, immigrant Muslim leaders from Arab and South Asian countries rarely joined black Americans in protesting overt political repression or violence in the United States. For example, after the death of Amadou Diallo, an unarmed African Muslim resident of Harlem who was shot over forty times by policemen in 1999, Imam Talib was unable to rouse any immigrant Muslim leaders to join his antibrutality protests. \"What we've found is when domestic issues jump up, like police brutality, all the sudden we're by ourselves.\" I observed similar dynamics in 2006, after the death of another unarmed black American man, twenty-three-year-old Sean Bell, shot by police the night before his wedding. In late December of that year, six days after the Bell shooting, Brother Malik spoke about American Muslims' responsibilities to address political repression regardless of the racial or even religious identity of the victims. \"We need to stand up for justice whether they're Muslims or not. Today it's African-Americans. Tomorrow it could be someone else,\" he argued during a Friday _khutbah_. No one from the mosque acknowledged his comments about Bell after the service or during the rest of the day that followed, let alone joined him in any kind of activism.\n\nWhile issues of political and economic marginalization and histories of repression continue to cause tension between various Muslim groups\u2014and even contribute to different ways of understanding and living Islamic traditions\u2014some aspects of the relations between different American Muslim populations changed after I ended my fieldwork at the mosque in 2010. The extent of police surveillance and brutality against immigrant populations since 9\/11 has become public. Moreover, the recession of 2007\u20132009 and the massive protests begun in New York by the Occupy Wall Street movement, among other things, have helped to change the perspectives of many (although certainly not all) Americans by making economic inequality in the United States more apparent and by bringing attention to the challenges some face in meeting basic needs. Finally, media and political attention to a spate of police killings of unarmed black men, women, and boys (one as young as twelve) from the summer of 2014 to the summer of 2016, which gave rise both to the nationwide Black Lives Matter movement and to horrifically violent incidents of racist backlash\u2014including the murder by a twenty-one-year-old white man of nine black parishioners during a Wednesday night bible study in a church in South Carolina\u2014made it increasingly difficult to deny the repression black Americans still face in the United States. More affluent Muslim immigrants, having themselves faced an increase in racist and Islamophobic violence since the Ground Zero Mosque controversy of 2010, frequently articulated a new understanding of black Americans' experiences and a new impression of the United States during these years. As I discuss in the Chapter 7 of this book, Rauf was occasionally among them.\n6\n\nTHE PROPHET'S FEMINISM\n\nWomen's Labor and Women's Leadership\n\nON A FROZEN FEBRUARY DAY IN 2006, Feisal Abdul Rauf and Daisy Khan worked from the back of a leather-upholstered sedan winding its way across Manhattan to the State Department's Foreign Press Center on East 52nd Street, where Rauf would film a briefing assuring US diplomats of Islam's compatibility with American culture, economy, and politics. As the sun began to sink behind Midtown's office towers, Rauf rested his head against the seat, closed his eyes, and gathered his thoughts. Behind the wheel, Pedram carefully navigated Friday traffic. Meanwhile, Khan read aloud a draft of the press release she would soon disseminate with Lord Carey, the former Archbishop of Canterbury and Rauf's Council of 100 colleague from the World Economic Forum, in response to the controversy over Danish cartoons that depicted the Prophet Muhammad as a terrorist.\n\nThe conversation in the car turned to the search for someone to speak in Rauf's place at the mosque during his absences. Dean, a dervish riding along with us, suggested Shaykha Fariha. Rauf hesitated. \"We have to be careful,\" he said, and informed Dean of the backlash feminist scholar Amina Wadud faced after she led a mixed-gender congregation in prayer in 2005. He also reminded Dean of an incident at Masjid al-Farah in which a knife-wielding man protested both Sufi traditions and that order's female leadership. Khan stopped typing on her Blackberry to echo Rauf. \"It's true,\" she said, giving voice to the hesitations that kept her from supporting Wadud's activities publically at that time: \"people already think we're too liberal because we're Sufis.\"\n\nWith the fifth anniversary of the 9\/11 attacks drawing nearer, American Muslims were increasingly pressed to identify as moderates, particularly on issues of gender, as other Americans (even social conservatives) defined the acceptability of various Islamic traditions by comparing the status of Muslim women to liberal ideals. This was not a new trend in American history. While Oprah Winfrey's unveiling of a burqa-clad woman in Madison Square Garden in 2001 testified to the potency of such images prior to 9\/11, justifying foreign wars on the grounds of saving Muslim women was a long-standing colonial tactic. Nevertheless, pressure on American Muslims to exemplify moderation by distancing themselves from so-called fundamentalists\u2014not least when it came to issues of women's roles and rights\u2014increased significantly after the attacks.\n\nTwo months after voicing her concerns about ASMA's image, Khan stood in Union Theological Seminary's chapel for a \"Faith and Feminism\" talk\u2014an interreligious program organized by the Sister Fund. Khan was there to inform the attendees about ASMA's work and to echo Rauf's message of moderation. On this occasion, promoting women's religious leadership _was_ central to that message and a crucial way for her to distinguish ASMA from images of Muslims who oppressed women.\n\nFareena joined Khan at the discussion and also talked about what Sufism meant to her. As she often did when addressing non-Muslims, Fareena spoke of the aesthetic beauty of Islamic art, architecture, and poetry, and of how Sufism made Islam real to her. Before the event ended, Khan added a more liberal cast to Fareena's presentation, explaining that Fareena was helping to train imams at the mosque as part of ASMA's goal to make women's Islamic leadership a reality. Someday, Khan said, she hoped that women would deliver _khutbah_ s, too. Fareena did not contradict Khan that evening. Often, however, she and other women at the mosque were ambivalent about identifying with feminists\u2014not simply because they feared appearing overly liberal, although that certainly was a factor. Rather, as we shall see, many also remained agnostic about the promises of liberal feminism and felt caught, at times, between the need to defend women's rights and the need to defend Islam from women's rights activists.\n\nDuring my fieldwork at Masjid al-Farah, I frequently observed how Fareena coached the assistant imams. After _jum'ah_ each week, she and several dervishes who did not have to rush back to work would wander down the block for lunch. Across long wooden picnic tables invariably littered with cups of water that one of the dervishes had served to everyone, Fareena went over each of the assistant imam's _khutbah_ s. She made sure to highlight their strengths but was not shy of chiding them if they did not follow Rauf's three-part model (a Qur'anic passage, followed by a portion of the Hadith, and finally an example or analogy, preferably from Sufi poetry or literature). In contrast to Khan's description of this labor as \"training imams\" and simultaneously building women's leadership, however, Fareena described it to me as part of her work for her shaykh. It was not something she did from a desire to expand women's liberal rights (although this was, in some ways, a goal she held, too) and not a task she undertook for ASMA, which she and others increasingly associated with Khan, not Rauf. Rather, it was a form of pious practice meant to demonstrate her loyalty to Allah and to her spiritual mentor. It was a kind of service, an act of obedience that allowed her to be God's instrument and that would transform her even while the labor she offered helped to transform others.\n\nService to one's community is a ubiquitous theme in classical Sufi literature and a mandate taken very seriously by the Halveti-Jerrahi order that Rauf and Fariha joined when they first became Sufis. Fariha's Nur Ashki Jerrahis, for example, explain on their website that one learns _adab_ \"in the presence of the shaykh, through our full attentiveness to the teaching and our joyful service.\" While many Muslims (such as the Halveti-Jerrahi of Spring Valley, New York) speak of service as part of their worship ( _'ibadat_ ) rather than as part of _adab_ , their similar emphasis on it reflects one scholar's assessment that service \"has always been one of the first stages in the preparatory steps of the [Sufi] path, [and] remains the true Sufi's duty throughout his life.\"\n\nRauf also emphasized the mandate to serve on certain occasions, describing it as an essential part of Sufism. For example, during a _dhikr_ session in which he asked those assembled to reaffirm their commitment to the tradition and to the mission they were jointly engaged in as his dervishes, he explained that _dhikr_ was only one part of Sufi practice.\n\nIf you are committed to God\u2014if you are the servant of God, as the prophet was, then you're a servant of humankind . . . The servants of God are those who serve humanity. You cannot serve God and not serve humanity . . . This path is not only about _dhikr_. It's also about working on yourselves, striving. And it's a lot of striving . . . In traditional Sufi orders, the first thing you are asked is to serve.\n\nDespite these common emphases on service, interpretations of what it meant to work for the good of one's community could differ dramatically, with some Sufis believing the mandate required volunteer labor within the mosque and others viewing it as an injunction to address broader social issues. In Chapter 7 I discuss how debates over service compromised the Islamic community center project in 2010. First, I focus in this chapter on the gendered dynamics of service at Masjid al-Farah, where Muslims invoked the requirements of service for multiple reasons. Doing so not only demonstrated their pious comportment and enabled them to engage in projects of self- and social transformation, it also allowed them to voice their opinions about gender relations without appealing solely to ideas of liberal rights\u2014ideas that, although frequently evoked by Rauf and Khan, were often deployed in the broader culture to cast aspersion on Muslims.\n\nThe Muslims I interviewed at Masjid al-Farah all agreed that authentic Islam respects women in ways not depicted in the media. For them, the actions of the Taliban were anathema. Most also criticized the strictures on women's driving, voting, and other activities in Saudi Arabia, and many believed that head coverings were necessary only during prayers, with modesty in dress enjoined on men and women equally. Nevertheless, despite these deeply held convictions\u2014some of which align with liberal feminist opinions\u2014their understandings of gender roles could differ significantly. While arguing for the importance of women's rights, particularly in contrast to what they believed to be women's circumstances in many Muslim-majority countries, some at the mosque still thought a woman's most important role was as a mother and caretaker and understood the appropriate relationship between men and women to be one of equality based on complementarity, not of equality based on radical sameness. This understanding was not simply a matter of ideology but something that seemed to make sense in the circumstances of their lives.\n\nSignificantly, although members of the community prioritized women's roles as mothers, they did not necessarily object to women serving as religious leaders in some capacity. As we shall see, their positions on these matters were often complicated, with the appropriateness of a particular woman's leadership gauged not just by abstract principles or doctrine but also by a host of other factors that included the kinds of community service and care work that women engaged in. Such humble service contrasted with the aggressive drive for positioning that some dervishes associated with liberal feminism and saw as the animating impulse behind Daisy Khan's work on Muslim women's issues. For this and other reasons, several women distanced themselves from ASMA's WISE project, even though they agreed with the substance of many WISE initiatives, and instead concentrated on embodying the norms of faithful servants and devoted mothers so prized by their community. This, many felt, guaranteed more fulfillment and respect than the options available to them (professionally and personally) as liberal or radical feminists in neoliberal America.\n\nLiberal Rights and the Limits on Muslims\n\nTo understand the gendered dynamics I observed, it is helpful to review some of the limits Muslims have run up against in encounters with liberal feminism, as well as some of the disillusionment American women have felt with the failures of second-wave feminism of the 1960s and 1970s to achieve greater social change. First, though, it is worth remembering that feminists and womanists around the world have demonstrated the inherent difficulty of trying to define the wants and needs of women. The category \"women\" is intrinsically exclusionary, as no definition will ever be sufficiently capacious to represent all of the members of humanity who identify as female or who are designated as such. American Muslim women\u2014who vary economically, ethnically, politically, and in many other ways, and who have diverse and overlapping commitments to different communities\u2014can hardly be considered as belonging to any single population group. To make matters more complicated, Muslim women frequently find themselves spoken for instead of listened to\u2014not simply by Muslim men, as non-Muslims may assume, but by those outside their communities who consider Muslim women's needs, wants, and rights to be objects of study or even justification for a host of political and economic interventions.\n\nTrying to overcome some of these problems in recent decades, some feminists have forged multicultural and transnational alliances around positions held in common despite their differences. In the process, they have attempted to decenter the perspectives and biases of affluent, white, Western activists. Despite these attempts, many activists involved in such multicultural alliances still retain the suspicion of religion that characterized much second-wave feminism in the United States and Europe. Consequently, they see secularism as necessary for liberating women and a commitment to secularism as essential for forging activist coalitions. Pious women\u2014including Muslims\u2014have often found themselves marginalized in such alliances. This was the case in the 1990s with Women Against Fundamentalism (WAF), a British organization that first mobilized in response to backlash against Salman Rushdie's novel, _The Satanic Verses_. In the inaugural issue of their journal, editor Clara Connolly made clear the liberal secular imperative that guided the movement, describing the \"problem of fundamentalism\" as one of religion's entanglement in the workings of the state\u2014something, she argued, that inevitably had dire consequences for women. Her founding missive shows how the founders of WAF saw what they believed to be a particular kind of religion\u2014\"fundamentalism\"\u2014as a common enemy that could rally women together despite their differences, thus producing a unified politics.\n\nThe Women Against Fundamentalism secularist alliance fractured quickly after the British Broadcasting Company described their activities as specifically \"anti-Islamic\" and some Muslim women involved in WAF disaffiliated from it. Hesitant to contribute to the demonization of Muslim communities, even when they have their own concerns about issues of gender and authority within them, many American Muslim women are similarly wary of liberal feminist arguments about the need to save Muslim women from Muslim men. This is especially the case for those who oppose US wars in Muslim-majority regions, as well as those who support women's rights and women's equality but are not always persuaded by what they view as the (sometimes hollow) promises of liberal feminism.\n\nWhile some Muslim women have advocated for greater women's religious authority in ways that accord with liberal feminist commitments, both before and after 9\/11, others have argued for the rights to pursue less explicitly liberal religious and social arrangements, such as the right to prioritize communal solidarities over individual autonomy, or the right to function as caretakers more than breadwinners. They are not alone among American religious women in seeing complementary but different gender roles as an aspect of pious practice and an empowering source of agency, despite the fact that some liberal and radical feminists have depicted such religious affiliations and understandings of gender as constraining rather than liberating. Nor is it only Muslim American women from communities of color whose voices and needs have been marginalized within white liberal feminist circles. Euro-American converts to Islam are among those disillusioned with the liberal feminism they were raised with and frustrated by obstacles to women's social and economic advancement and self-determination in the neoliberal environment of twenty-first century America.\n\nFor some women, the failure of liberal feminist goals to be broadly realized in the United States has coincided with an increasing interest in the possibilities for women's involvement in Muslim communities. After 9\/11, for example, increased scrutiny from non-Muslims motivated many Muslim groups to make changes (often ones considered previously) that meant new possibilities for women's authority. For example, the Islamic Society of North America elected Islamic studies professor Ingrid Mattson\u2014formerly an adviser to the Afghan delegation to the UN Commission on the Status of Women and an advocate of greater women's leadership within Islam\u2014to serve as its first female vice president in 2001 and as its first female president in 2006. Meanwhile the second-largest Muslim group in the country, the more conservative Islamic Circle of North America, had a female speaker address its annual convention for the first time in 2002.\n\nCloser to home, the multiethnic New York-based group Women in Islam, founded by Aisha al-Adawiya in 1992 to create \"a forum through which Muslim women could advocate on issues affecting the Islamic community generally, and women more specifically,\" worked with the Islamic Social Services Associations on the publication of a 2005 guide for expanding women's involvement called _Women Friendly Mosques and Community Centers: Working Together to Reclaim Our Heritage_. The booklet received support from ISNA, ICNA, the Council on American-Islamic Relations (Canada), the Muslim Alliance in North America, the Muslim Association of Canada, and the national division of the Muslim Students' Association. Then, in November 2006, Daisy Khan held a similarly inspired international conference to inaugurate her WISE project. In addition to building a global network of Muslim women leaders, the conference launched Khan's program to train Muslim women from around the world as _muftias_ (interpreters of Islamic legal traditions).\n\nMost of the women I interviewed at Masjid al-Farah were not involved with WISE. Nor were most aware of the debates over gender that had taken place in Muslim communities across the country in preceding decades. This is undoubtedly because many were recent Euro-American converts to Islam or immigrants\u2014often successful professionals from Canada, Pakistan, and Bangladesh\u2014who did not affiliate with national Muslim organizations. Nevertheless, several described the practice of Islam they encountered at the mosque and at Sufi _dhikr_ services as liberating\u2014from the sexual objectification they experienced outside of the mosque and from some of the seemingly impossible standards of liberal feminism\u2014and uniquely egalitarian in comparison to other Islamic traditions. Some, but not all, described this as a result of Sufism.\n\nWomen's large-scale involvement in Sufi orders is a relatively recent phenomenon; historically, most were excluded from Sufi \"brotherhoods.\" Although their current participation may be due, in part, to the exposure of Sufi leaders to liberal discourses and ideas, the women I encountered at Masjid al-Farah did not regard participation in Sufi practice as simply a recent liberal development. Rather, like Rauf and Khan, they saw it as among the rights originally promised to women in classical Islamic texts and as something hampered more recently by so-called fundamentalists. Rauf\u2014and sometimes Khan\u2014explicitly argued for the confluence of women's liberal rights and Islamic traditions, with Rauf's 2004 book, _What's Right with Islam_ , marketed directly to those who identify as \"American feminist[s]\" and a chapter of his subsequent book, _Moving the Mountain_ , devoted to illuminating \"the Prophet's feminism.\" Yet other Sufis still differed over whether and how well the United States secured women's rights, as well as over how American versions of liberal equality matched up with what they believed to be authentic Islam and moderation.\n\nFaith and Fears after 9\/11\n\nA young American woman I often saw at the _masjid_ and, occasionally, at ASMA interfaith events gave voice to the simultaneous disillusionment with liberal feminism and wariness of Islam that many female converts I interviewed felt before joining the community. Deborah, a petite, curly-haired woman I met early in my fieldwork, had converted to Islam from Catholicism in 2004, prior to marrying her Muslim boyfriend\u2014a story I found to be common among female converts. While unaware of Rauf's 2004 book on how Islam embodies tenets of American liberalism, Deborah had heard the imam reiterate this message in his Friday _khutbah_ s. Although she, too, was a staunch defender of \"inalienable rights\" and found Rauf's message appealing, her life experiences had caused her to question the extent to which women could achieve the same professional successes that men did in the United States. Like other women I interviewed, she eventually came to believe Islam taught the kind of respect for women that American society was lacking. The transition from fearing Islam to finding empowerment in it was not necessarily an easy one.\n\nIn many ways it is not surprising that Deborah and several other women I interviewed were disillusioned with what American society offers women, despite Rauf's continual insistence that liberal rights for both genders are divinely inspired and ordained, and that the United States ensures these rights better than any other nation. The Equal Rights Amendment, proposed in 1972 to prohibit inequalities based on sex, expired for lack of ratification after passing both houses of Congress, leaving the ideal of \"equal pay for equal work,\" among other things, an unrealized dream. During the years I conducted interviews (primarily in 2007, before the Great Recession), women who came of age steeped in the egalitarian sentiments of second-wave feminism continued to earn approximately 20 percent less than men for work in the same occupations, with women of color earning even less than that. Because these statistics account only for women employed full time, they likely reflect the wages of women with highly remunerated careers more than the wages of women whose smaller salaries caused them to choose to be primary caretakers and work only part time after having children.\n\nFor several women of child-bearing age, it seemed obvious that men should be breadwinners and women should raise children\u2014not because women were biologically equipped to nourish infants (although some did argue this), but because women would sacrifice far less of the family income than their male partners by cutting back employment. With many families unable to afford the astronomical cost of childcare in New York City\u2014especially care for children under two, which was both more expensive and less available\u2014having the mother stay at home felt less optimal than it did inevitable, less a principled decision than a necessity. Furthermore, recent research suggests that even women in high-paying, high-profile jobs\u2014particularly those who work in occupations associated with male labor or achievement\u2014feel pressure to revert to conventional gender arrangements at home by performing more of the labor associated with women's work there. Facing such situations, several women\u2014including Deborah\u2014expressed comfort in what they perceived to be Islam's insistence on complementary gender roles rather than on radical equality. Once they decided that the mosque community sufficiently respected women, their choices to embody the roles of devoted mothers and humble servants gave them a sense of accomplishment and fulfillment rather than disillusionment at failing to achieve equal status professionally.\n\nAs the sun began to glint off the windows of the residential towers surrounding her apartment one late spring afternoon, with her infant son napping down the hall, Deborah sat across from me on her couch in jeans and a t-shirt, legs tucked up under her, and explained how she thought through these issues. Before converting, and even for some time after, she admitted, she was wary of Islam because\n\nI was uneasy about things having to do with a woman and women's rights. I think it was more because . . . you sort of felt like there was a separation between men and women [in Islam] and the woman were in charge of the household and the home and the man is supposed to be the support of the women. And that was something that I was a little bit uneasy about but, later on . . . I started to question, \"why is it that we seem to think that being in charge of the home and the family is somehow inferior to going out and getting a job?\" And it's almost like, as Americans, I think we've been conditioned to believe that.\n\nThe more I read, the more I realized that the tone in the Qur'an never, ever gives any impression that this is somehow inferior. In fact, it says just the opposite: that this is the woman's home. She is the caretaker. She is, you know? And the man's job is really to earn the money to support the woman and the home. And then there are very clear regulations about money and that if a woman has her own money that's hers to keep.\n\nLike several other female converts, Deborah's fears of unequal treatment kept her from converting for some time, even when there were other aspects of Islamic tradition that resonated deeply with her. Also like most women converts I interviewed, Deborah came from a religious background and never would have identified as secular or irreligious but also never felt completely comfortable with the doctrines of her parents. Deborah finally decided to convert in 2004, after successfully searching the Qur'an for solace after George W. Bush's reelection: \"I was just, like, how could this happen? How could people vote for him? I was just so upset . . . But the Qur'an says not to concern yourself with unbelievers, and I realize that this could be what it meant.\" Not long after that, she told me, she felt more assurance about the respect Islam had for women, especially those oriented toward family. It is not insignificant that, during this time of questioning whether she could convert, Deborah started attending Masjid al-Farah, a mosque that upended her expectations and assuaged some of the concerns she had about male-female separation.\n\nDeborah first attended Masjid al-Farah after her fianc\u00e9 assured her it was \"non-judgmental.\" Like Emily, another Euro-American convert who first attended the mosque at the behest of her boyfriend, part of what Deborah found so attractive there was that women who attended prayers could see, hear, and experience everything directly instead of listening to the Friday service over a speaker in a different room or on a different floor from the men. The only barrier separating men and women was a three-inch high row of wooden book holders, emptied of the Qur'ans they sometimes held and lined end-to-end to mark a division of space. While there was sometimes a bit of gendered jockeying over the placement of this symbolic barrier, prayers at Masjid al-Farah were very different from what Deborah, Emily, Leah (a blonde former Presbyterian from Long Island), and others had experienced at other mosques in the city, and from what media images \"of the way it is practiced in Saudi Arabia\" had led them to expect. Because Friday prayers are required only of men, some of these women felt that their practices of attending _jum'ah_ should be recognized with, if not praise, at the very least equal access to opportunities for pious engagement. Masjid al-Farah gave them this.\n\nThe spatial arrangement was not the only thing different, many women told me; so was their access to the religious leadership. For the first time, they said, they were able to ask questions of an imam who responded with undivided attention. For Deborah and Emily, this imam not only answered their questions and led them through the conversion process of taking the _shahada_ (repeating the prayer and statement of faith), he eventually performed their marriage ceremonies. In many ways, these activities comfortingly resembled those of the ministers and priests who pastored the churches these women had left.\n\nThe removal of spatial barriers and the increased access to the imam that many women found so important was only magnified when they attended _dhikr_. Even prior to converting, several women made a practice of attending Rauf's weekly Sufi meetings on Friday nights. Men and women sat separately at these gatherings, but without even a row of book holders between them. Far from feeling segregated, women who attended often felt welcomed into the inner circle of the community. At least four of the women I interviewed had also attended Thursday night _dhikr_ s at the mosque. At those meetings, their concerns about women's equality were further alleviated. Not only were the meetings led by a woman (Shaykha Fariha or one of her prot\u00e9g\u00e9es), but the celebrations and discussions over food afterward frequently also involved commentary on the beauty of \"feminine religious energy.\"\n\nFor those who came to know about ASMA, Khan's 2006 program on women's equality served as additional confirmation that the community would respect their rights. And even those who did not know much about ASMA\u2014such as Deborah, who was unaware of the organization despite having attended a joint Ramadan-Rosh Hashanah dinner that ASMA sponsored in 2006\u2014or who were unsure what to think of WISE could plainly see that women were integral to the functioning of the mosque and the Sufi communities within it. Not only was the building that housed the mosque owned by a woman who led the Sufi order that met there, but one of her dervishes, Amrita, helped to manage the finances: collecting donations at the end of each service, saving some to go toward the upkeep of the mosque, and parceling out money to Muslims in need. Meanwhile, Fareena was clearly the organizational heart of Rauf's Sufi group. She was the one everyone asked about the time and location of _dhikr_ , as well as about where Rauf was when he was absent and how long it would be until he returned. She was the one who shepherded the assistant imams and others to lunch each week, who had the phone numbers of all of the community members, and who would call or email people with changes of plans or new opportunities for getting together. She was also usually the one to speak to visiting groups of non-Muslims when _jum'ah_ services concluded, answering their questions about all manner of theological issues. And (although only some dervishes knew this) she was the first one Rauf called when he needed someone to speak in his place at a church, school, or other institution.\n\nThe prominence of women in the mosque, at _dhikr_ services, and at ASMA was a reassuring aspect of these communities for many female converts, one that initially stood in contrast to what they thought they knew Islam to be and that, over time, exemplified for them what authentic Islam really is. Yet not everyone who attended Masjid al-Farah approved of all of the activities of these Sufi communities or of the gender arrangements within them. While several Sufis (male and female) attended the meetings of both Rauf and Fariha or alternated between them, others either thought it necessary to follow a single shaykh instead of two, or simply did not believe it acceptable to follow a female leader at all. These and other disagreements surfaced most frequently when Rauf was away for various Cordoba Initiative projects and left many dervishes searching for their own ways to create community coherence and to resolve crises, as I explain below. Before turning to those topics, I examine what Rauf and the other imams at Masjid al-Farah communicated to Friday congregants about gender relations and how various community members invoked notions of service\u2014particularly, gendered labor\u2014when attempting to reconcile American liberal rights rhetoric with what they believed to be Islamic teachings about the rightful roles of women.\n\nPublic Pronouncements and Personal Interpretations of Women's Roles and Rights\n\nFemale converts were not the only ones to have different experiences at Masjid al-Farah than at other mosques in the city. As Abdul-Raheem, one frequent attendee, explained to me, the ratio of women to men at the _masjid_ had often startled newcomers in previous years, with women making up nearly half of Friday worshippers until shortly before I began my fieldwork. Surprised by this information\u2014women made up no more than ten percent of worshippers during most of my years of attending\u2014I inquired about the shift. Abdul-Raheem explained that it was due in large part to the rising number of male Senegalese worshippers at the mosque. This was something he thought himself responsible for, as he had informed several Senegalese Sufis who worked on Canal Street about the \"Sufi mosque\" nearby. After that, and after another mosque in the area closed, the number of male worshipers increased so rapidly that Masjid al-Farah began offering two services each Friday instead of one (the practice there when I began fieldwork), and then moved to three services or even four during Ramadan.\n\nWhile Abdul-Raheem's anecdotal account of the change explains it in part, I noticed a different trend during my years of observation: that the quantity of female worshipers at the mosque correlated with Rauf's appearances there. While there was never a day when I was the only woman in attendance, the number of female worshipers was generally larger when Rauf was in town and declined precipitously the longer he traveled. A few women\u2014professionals who had joined Rauf's Sufi order at some point but who also had busy work schedules\u2014told me directly that they made a special effort to attend _jum'ah_ when they knew Rauf would be speaking. This was because of what Rauf said during his _khutbah_ s, of course, but also because of what he did not say. Not only did Rauf differ from imams at some other mosques in the city when it came to ideas about gender roles, he differed from his own father\u2014whom he credited with giving him an informal theological education\u2014and from the other imams who spoke during Friday _jum'ah_ services at Masjid al-Farah. While Rauf did discuss issues of gender in his writings, he generally refrained from doing so at the mosque. In contrast, other imams sometimes dedicated significant portions of their sermons to what they believed was a woman's highest calling: motherhood. This message appealed to some women, but working professionals were not always among them.\n\nRauf borrowed generously from his father's writings on the shari'ah and on the compatibility of Islam with democracy and capitalism, as I have discussed. Nevertheless, he did not generally have his assistants and dervishes read his father's work. This was at least in part because of Muhammad Abdul-Rauf's teachings about women. In his 1972 book, _Marriage in Islam: a Manual_ , and elsewhere, Abdul-Rauf had described motherhood as a woman's most important role and as one crucial to maintaining a just and pious social order. He continued to make this case in print as late as 1987, arguing that women's active participation in the work force diminished the energy they had for their families and contributed to social strain. Rather than aspiring \"to compete with men in production and in outside work,\" which results in \"giving lesser attention to their husbands and lesser care to their children,\" Abdul-Rauf held, pious women should concentrate on inculcating Islamic traditions in younger generations.\n\nAlthough Rauf did not generally discuss his father's teachings even with close associates, his dervishes\u2014seeking guidance on various issues during his absences\u2014eventually discovered these materials for themselves. After Fareena found Abdul-Rauf's books while searching for alternatives to water-based ablutions before prayer\u2014structural damage from constant use of the small restroom sinks on the second floor for _wudu_ had led Fariha to close the restrooms\u2014she asked Rauf on his return to New York why he had not made these resources available earlier. \"You didn't give us your father's stuff,\" she told him accusingly. \"I've been reading and I ordered the books . . . there's everything in there, including about _wudu_!\" In response, Rauf hesitantly cautioned Fareena about sharing his father's writings with others.\n\nWhile professional women might object to Muhammad Abdul-Rauf's ideas about women's work, several others\u2014including Brother Malik, Faiz Khan, Fareena, and a few young American female converts\u2014found points of commonality with such teachings. For them, motherhood was a powerfully evocative theme. Of the assistant imams, Brother Malik brought up gendered aspects of Islamic life most frequently in his _khutbah_ s. While Brother Malik had never made much money, he had always worked hard to support his wife and children, and in his _khutbah_ s often stressed what he described as the \"noble\" and \"chivalrous\" qualities of proper Muslim practices and the complementary gender roles that went along with them. Like many other black American Muslims, Malik believed it was essential to encourage male responsibility for family and society. He sometimes saw the demands of (frequently white) liberal feminists as undermining both the supports needed in black communities and the long-fought-for family continuity that slavery and poverty had denied them. While women's labor\u2014particularly mothering\u2014was crucial for creating and maintaining family and community cohesion, he believed, it was black men who really needed the encouragement to lead.\n\nUntil he left the community in 2006, Faiz Khan also focused on complementary models of gender relations and emphasized motherhood in his _khutbah_ s. Different from Malik in that he overtly supported women's religious leadership in his writing\u2014and also different in that he was raised in a family of professionals, which included two older sisters who had careers as doctors\u2014Faiz still did not promote radical gender equality at the _masjid_. Rather, in that setting he emphasized the social and spiritual importance ascribed to motherhood in the Qur'an, frequently comparing the love of God to the connection between a mother and child. Doing so allowed him to assert the importance of women by relying on Islamic tradition rather than by arguing in the language of liberal rights feminism.\n\nFor Faiz, motherhood is central to Abrahamic covenants between God and his people\u2014something, as he discussed in a _khutbah_ in January 2006, he saw reflected in the Qur'anic story of Abraham's willingness to sacrifice his son, Ishmael. Rather than focusing on the men in the story (Abraham and Ishmael) when relating it to Friday worshippers, Faiz turned attention to the often-forgotten woman in the narrative. \"One should not think solely of the sacrifice of Ishmael, but also of Hajar, who sacrificed her child. Men can never understand the power of a woman's response when she sees that her child is in danger,\" Faiz said, choking up. \"I see it all the time in my emergency room . . .\" Maternal love had recently become a more powerful theme for Faiz. During the previous year, he and his wife (also a doctor) had moved to Pennsylvania for a residency position. Although Faiz visited the mosque less frequently after that, I had a chance to catch up with him in the spring of 2005 when he delivered a guest lecture at Union Theological Seminary. He told me then that they were expecting their first child. I did not see him again at the mosque until January 2006, by which point he was a new father. Motherhood was not only something he witnessed in his ER now, but something he also observed intimately every day, and he was not alone among Rauf's dervishes in finding new meaning in parenting.\n\nAfter _jum'ah_ , Fareena turned to me with what was more an exclamation than a question: \"Wasn't Faiz's _khutbah_ beautiful?!\" She and her American husband were nearly the same age as Faiz and his wife and had also recently started a family. Months later, lamenting the loss of Faiz, Fareena reminisced about how they found a sense of community through their service to their shaykh. This sense of connection was crucial to Fareena at that time, not least because she had moved to New York in order to pursue a graduate degree, leaving her parents in Bangladesh. Upon graduation she had taken a job at the U.N. but had given up that position when she had her first child. The mosque and Sufi community brought both familiarity and continuity to her life amid these many changes and gave her fulfillment and meaning. None of this had come without costs, but Fareena did not consider the sacrifice of her professional life for motherhood to be among them. Unlike Deborah, Leah, and Emily, whom I saw less often after they had children, Fareena had the help of a nanny and rarely missed a _jum'ah_ service or _dhikr_ session, even after the birth of her second child in 2006. Instead of feeling forced out of her professional position by the exigencies of motherhood in New York City, Fareena was more able than several other new mothers at the _masjid_ to feel that she had _chosen_ to trade her career for more meaningful activities\u2014not just motherhood, but also service to her shaykh and to a religious community that compensated for her lack of professional contacts.\n\nAdditionally, because Fareena was as close to Faiz as a sister (in her words), she knew that he supported women's professional equality with men even when he emphasized their maternal roles. What other women\u2014professionals and stay-at-home mothers\u2014heard as advice to focus on children, Fareena recognized as a strategy to emphasize the importance of women in a way Muslims of various perspectives could appreciate. Even if other professional women had recognized this about Faiz, however, it would likely have made little difference in their desires to attend the mosque in Rauf's absence, as Faiz no longer affiliated with ASMA or spoke at Masjid al-Farah after 2006. Instead, Brother Malik, Pedram, and a host of other young Muslim men took turns speaking during the three services each Friday.\n\nFeeling the loss of a sympathetic religious leader when Rauf was away, some women stopped attending the mosque and _dhikr_ services almost completely, while other women\u2014and some men, too\u2014sought guidance, acceptance, and affirmation from Shaykha Fariha. Attending Fariha's Nur Ashki order provided these dervishes with a needed sense of community again for a short time. But performing _dhikr_ under a female shaykh did not necessarily change their ideas about women's rights and roles. While neither Fariha, who was childless, nor her dervishes gave much emphasis to motherhood, women's labor and humble service was still the norm within her community and one that resonated with both men and women.\n\nWomen's Labor as Women's Leadership\n\nAbout half of the Sufi women who attended Masjid al-Farah affiliated exclusively with Rauf's order, while others favored what Deborah described as the \"celebratory\" atmosphere of Fariha's music-filled ceremonies. Keesha, a multiracial convert with a multireligious family, who was unmarried and did not have children, explained to me that she was first drawn to Masjid al-Farah and to Rauf's Sufi services after seeing Daisy Khan in a film screened at the Asia Society cultural center. (Keesha had previously converted to Islam while studying abroad in Morocco and was attracted to Khan's dynamism and eloquence.) She was further drawn in by Rauf's habit of excavating the \"inner\" meanings of the shari'ah for his dervishes. For a time, Keesha attended both Rauf's Friday Sufi gatherings and Fariha's Thursday meetings, but she ultimately decided to affiliate exclusively with Fariha's order. Keesha did not believe she was losing out on the education Rauf provided because, in her opinion, Fariha \"does everything directly in line with the shari'ah,\" although she acknowledged that some people would not expect a female leader to focus on orthodoxy, since many assumed she was already violating it. Additionally, Keesha regarded Fariha's Sufi services as \"the most beautiful thing I have ever seen.\"\n\nKeesha's decision to affiliate with Fairha's order rather than Rauf's may seem, from her description, to have been based on aesthetics rather than gender dynamics. It was likely much more complicated than that, however, as language about aesthetic differences often served as a marker of gender distinctions in this community. What Deborah described as the \"celebratory\" atmosphere of the Nur Ashki _dhikr_ s and Keesha termed their \"beautiful\" qualities, Sufis affiliated with Rauf's order (particularly men) often labeled too \"otherworldly\" and \"ethereal.\" This was because Fariha's _dhikr_ s involved a kind of ecstatic practice, including singing and dancing, that is highly improvisational and can be highly unregulated\u2014something that Rauf disallowed at his _dhikr_ s and that he and several of his dervishes regarded as excessively emotional\u2014a gendered term if ever there was one. In contrast, several of Rauf's closest dervishes focused on what they regarded as the more rational traditions associated with Sufi philosophers and conducted themselves with quiet restraint during _dhikr_ \u2014chanting rather than singing and never rising from their seats. Framing these differences in terms of Islamic tradition, Rauf frequently employed an analogy popular with both Sufi practitioners and scholars of Islam. He cautioned against the excesses of \"drunk\" Sufi traditions and encouraged the quiet, contemplative, and controlled practices of \"sober\" Sufism.\n\nIn addition to employing gender-inflected descriptions of the differences between these _dhikr_ s, Keesha indicated in other ways that concerns about women's roles and rights had influenced her choice partly, but not entirely, in a manner that can be explained with reference to liberal gender ideals. Keesha's father was a Muslim, and she was unique among the female converts I interviewed in not having had fears before converting about the ostensible mistreatment of women in Islam. This did not mean that she did not have concerns about Muslim women she felt were oppressed elsewhere. \"The woman issue . . . what's going on with women in the world, it's extraordinarily disheartening. It's painful,\" she told me, and her strong commitment to liberal ideals and to more radical tactics for securing them, as I discuss below, prevented her from being as satisfied as other converts seemed to be with the selective ways Rauf and Khan promoted gender equality. Nevertheless, the mistreatment of some Muslim women by some Muslim men was not something that she worried was inherent to Islamic tradition.\n\nI asked Keesha if she had attended Khan's 2006 WISE program. She said she had not and explained that, having studied political science as an undergraduate, she did not find Khan's initiative (then in its infancy) sufficient to address the politics behind the problems women faced.\n\nI don't want to talk about it and write a paper on it and then people go and think about it in a Harvard classroom. You know, that's not going to affect the fact that there are policies in the world that say men can kill women if they're raped, family members can . . . These are countries that are friends with us. That has to change and we have the power to change it.\n\nAmericans were reluctant to criticize countries where women's rights were violated, Keesha argued, because they got things they wanted from governments that tolerated or encouraged such behavior. Furthermore, she believed, even those who recognize such problems feared being attacked:\n\nAmerican moderate Muslims . . . agree about this. People that Imam Feisal speaks to, they agree with him, you know. They just don't want to say anything. There's a war in Islam that's just not being waged between moderates and fundamentalism. People are afraid of the fundamentalist Muslims because they have guns and other items. And I think that, we have a war in Islam that won't be waged because moderates just don't want to fight, they don't want to fight.\n\nFor Keesha, who echoed Rauf's portrayal of extremists, fundamentalism was synonymous with gender violence. Fariha's leadership as the female shaykh of a Sufi order seemed to be one way of courageously fighting against such interpretations. Furthermore, Fariha was more outspoken than Rauf and Khan about American injustices such as the wars that led to hundreds of thousands of casualties in Iraq and Afghanistan. (Although Keesha did not want to focus on how Fariha's upbringing in a wealthy, white Texas oil family allowed her more opportunities and social latitude than other American Muslims, it is worth considering the greater difficulty Rauf and Khan had as Arab and South Asian immigrants, respectively, in critiquing American foreign policy after 9\/11.) As important as these things were to Keesha, however, they were not what she emphasized most about Fariha. Rather, the main distinction she drew between the two Sufi communities centered on how Fariha exemplified what Keesha felt a true Muslim should be: \"a servant of humanity.\"\n\nPart of Fariha's allure was that she reminded Keesha of all of the ways her comportment, her relations with others, could be better. Whereas Keesha often found herself distracted by what she considered to be frivolous worldly concerns, she thought of Fariha as completely concentrated on developing a closer relationship with the Divine and helping others to do likewise. \"That's servanthood,\" Keesha told me. Fariha was so devoted to her religious practice and to her dervishes, Keesha explained, that \"she gives her whole being and her whole self at all times\u2014I mean to the point where her husband is stopping her. He gets so mad [because she exhausts herself].\" Fariha, Keesha explained, wanted everyone to \"become a real servant and a real giver, which is something that is very difficult to become. That means you have to go through the core of your soul and become a natural giver like Isa [Arabic for Jesus] was, you know?\"\n\nI asked Keesha if she thought Fariha might have more time to engage in religious activities because she did not have to worry about earning an income. \"But it's not the money, that's not what I'm talking about,\" Keesha insisted, misunderstanding my question. Instead\u2014inadvertently confirming my point about being free from financial worries\u2014she explained that it was about engaging in the practices that cause a person to let go of worldly concerns and distractions and to become a more complete human being. \"And service is obviously the biggest one.\" That was why Keesha had decided to go to Fariha's Sufi meetings. \"I like trying to be a better servant. I use Shaykha as my example . . . It's natural to her, so I want it to be natural to me to be able to do that, you know?\"\n\nKeesha was far from alone in seeing willingness to serve others as a primary attribute of the devoted Sufi dervish and, even more, the exemplary Sufi leader. Even those who viewed Fariha's leadership as unorthodox and unacceptable\u2014contrary, in their opinions, to the dictates of shari'ah\u2014emphasized this theme. They often did so in different gender-inflected ways, however, using the idea of voluntary service as a way of evaluating the propriety of various women's activities. In fact, the discussions about behavior I heard at Masjid al-Farah from 2004 to 2010 suggest that some Muslims' abilities to tolerate differences over women's rights and conduct depended on two familiar factors: how economically privileged the women seemed to be and the extent to which they engaged in recognizable forms of gendered labor\u2014the voluntary and unremunerated care for others that is often described as \"service.\"\n\nMale Sufis' varying attitudes toward Fariha, Daisy, Fareena, and other women are illuminating here. Daisy had left a corporate career to promote positions for women jurists in Islamic institutions and to prove the Islamic legitimacy of women's social, political, and economic leadership. During my years of fieldwork, she did not publically support women-led prayer, a more controversial issue than women's active involvement in the mosque or institutional leadership. Although neither did she play the other roles emphasized within the community: of supporting the mosque through volunteer work or of having children. And yet, the same dervishes who disapproved of Daisy's activities expressed appreciation for Amrita\u2014a Pakistani-Canadian activist for women-led prayer who sometimes worked in corporate advertising. Because Amrita promoted women leaders and liberal rights, hailed from an affluent family, and did not even wear hijab when praying\u2014instead she often wore a faux-fur hat of distinctly hipster vintage that covered her ears but not her neck\u2014this may seem surprising. Amrita was also a single mother who performed a significant amount of labor for the mosque and for her Sufi community. And as more than one dervish informed me, Amrita worked in corporate advertising only when working with nonprofits could not pay the bills. Amrita's daily care for her children and her voluntary duties with the mosque and community meant that her life deviated in important ways from the autonomous individualism that has been traditionally marked as white and male. In contrast, to many people at the mosque, Daisy Khan's high-profile political engagement and continuing corporate-related activities, including her corporate style of leading a nonprofit, seemed less acceptable and less moderate.\n\nIt is also useful here to consider the way Fareena was seen at the mosque. Despite their ambivalence or even antipathy toward Daisy, many Sufis held Fareena in high regard. She had left her career at the U.N. in order to raise her children, and she spent much of her time serving the community. Further, although Fareena, like Daisy and Fariha, was financially stable, she was also quite generous with money. In addition to praising her labors as a mother and supportive wife, several male members of the group appreciated how\u2014as they framed it\u2014Fareena used her husband's income from corporate finance to aid less fortunate Muslims and worked to organize the community and create solidarity within it. Shaykha Fariha was generous, too, of course; without her financial support, Masjid al-Farah would not exist. However, not only did she, like Daisy, promote women's public leadership, she controlled access to the mosque in ways that some of Rauf's dervishes disliked, making it unavailable outside of prayer times. In contrast, Fareena's exemplary service often consisted of the sort of unpaid labor commonly associated with women's care work and women's religious volunteerism in the United States.\n\nSignificantly, in addition to praising Fareena's service, the assistant imams treated her as a religious authority of sorts and accepted her guidance on how they should shape their messages for Friday prayers and the various interfaith gatherings where they routinely spoke. The difference in attitude toward Fareena and Daisy Khan was surely due, in no small part, to the fact that Fareena often served as Rauf's spokesperson when he was away, whereas\u2014in the opinion of many of Rauf's dervishes\u2014Khan was responsible for keeping him away so often on lecture tours or fundraising trips. Yet, because many of Rauf's absences were explicitly related to Cordoba activities\u2014ones that he often pursued independent of ASMA's programs\u2014their choice to blame Khan for them seemed to reflect prior sentiments toward her more than unbiased assessments of Rauf's actions.\n\nWhile most male dervishes were ambivalent about Khan's work to support female Muslim leaders, many female dervishes felt that other issues were more pressing in their own lives than those Khan focused on, including their inability after becoming mothers to stay connected to the community that had given their lives such shape and meaning. As some had discovered with regard to their professional goals, having children often meant losing opportunities and connections. Because childcare was not available at the mosque or at _dhikr_ , and because Rauf, in contrast to some Sufi leaders (including Shaykh Tosun Bayrak, leader of the Spring Valley Halveti-Jerrahis), did not view such religious meetings as family affairs or want young children to attend them, many women were unable to remain active members of the community once they embodied the role of motherhood so prized within it.\n\nEven women who were able to regularly attend _jum'ah_ and _dhikr_ often refrained from going if Rauf was absent, as most dervishes knew one another only through him and did not generally associate with each other otherwise. Rauf's presence was what made the meetings socially and religiously fulfilling. Following the death of one woman's mother in early 2007, for example, Fareena chided other members of the group for not trying harder to forge connections. \"We're supposed to be a community!\" she admonished. Although she maintained contact with many of the dervishes, others seldom reached out to each other between meetings before then. Feeling acutely disconnected during this time, as Rauf began spending longer periods working from his Malaysia office, some dervishes responded by making the _dhikr_ group more stable and more community-oriented. Interpreting Rauf's injunctions to serve the community in ways that fit their circumstances, however, they also began to exercise more independence from Rauf and Khan\u2014something that later contributed to the tensions over purpose and direction that split the community during the Ground Zero Mosque debate of 2010.\n\nConclusion: Community Cohesion and the Double-Edged Sword of Service\n\nThe dissatisfaction members of Rauf's group felt when he was away grew markedly in 2006 and early 2007. For a while, filling in for Rauf at the mosque or interreligious events helped some of his closest assistants feel connected to him. As ASMA changed, becoming less oriented toward Sufism in 2006, and as Rauf began working more regularly from Malaysia on Cordoba's Shariah Project, however, acting as Rauf's surrogates was no longer as satisfying for these Sufis, to say nothing of Rauf's other dervishes. In the meantime, as mentioned above, several dervishes had begun to feel less connected to Daisy Khan, whose apartment served as their _dhikr_ location.\n\nThe sense of dissatisfaction and disconnection among Rauf's dervishes became even more noticeable after Faiz Khan left the community and members of Rauf's _dhikr_ group found themselves lacking a stable location for their weekly meeting. They had gathered in Khan's apartment every Friday for years, but her and Rauf's travel schedule made this increasingly untenable. After an evening of attempting to hold _dhikr_ in a Times Square Starbucks because Khan's apartment was again unavailable at the last minute, the dervishes pooled their resources and rented a separate location for meetings. Their first evening there was a watershed moment for many in the community.\n\nAlthough the space\u2014a carpeted storage room in a municipal office belonging to the interfaith New Seminary\u2014was less than ideal, a sense of purpose infused the members of the group as they carried out the massage tables usually stowed there and installed the plastic filing cabinet that would hold extra sets of prayer beads, a few wall hangings to be put up each Friday evening, and laminated copies of the _wird_ (the portion of the Qur'an chanted during _dhikr_ ). As mariachi music drifted up from the Friday night festivities in the Mexican restaurant below, Fareena assigned people weekly tasks. Mine was to attend to the buzzer each time someone rang, something that occurred throughout the evening, as dervishes came late to the meeting and others arrived to attend classes in other rooms. The location had none of the comforts of Khan's apartment\u2014not only was there no kitchen for making meals after _dhikr_ , but food was disallowed entirely\u2014and lacked the familiar feel conveyed by the bookcase full of Rauf's favorite texts and the bulletin board on which Daisy or dervishes posted pictures and news clippings. Nonetheless, a distinctly communal ethos began to emerge that night as these dervishes cooperated, financially and physically, in creating and leading their new, reconstituted, and rather independent community.\n\nLess than a month after his dervishes established this new _dhikr_ location, Rauf returned to New York. The air at the mosque was electric with anticipation that day, as dervishes I had not seen for months, exhilarated by the chance to see their shaykh, came back to _jum'ah_ and attended _dhikr_ later that night. Despite the ban on food in their new space, they brought Syrian pastries to celebrate Rauf's homecoming. Rauf raised their spirits greatly at the beginning of the evening when he commended them on taking the initiative to keep up their practices. At the same time, he encouraged them\u2014as Daisy Khan had done the week before, when the dervishes collectively returned to her apartment for one last, and rather tense, gathering in that space\u2014to think of their previous _dhikr_ location as one of their permanent homes. (Although neither Rauf nor Khan mentioned it, ASMA had listed Khan's one-bedroom apartment in their original Internal Revenue Service (IRS) application for tax-exempt status as a meeting place for up to 400 worshipers each week\u2014a figure they optimistically aspired to but never reached\u2014and the dervishes' move to another location could have financial repercussions in several ways, thus diminishing the resources available for the community center.)\n\nThe celebratory mood that night did not last, as Rauf was not in New York to stay. Rather, his visit was meant to prepare his dervishes for a new reality in which they would see him even less frequently. After blessing their new location, Rauf reemphasized his role as their shaykh by recounting a dream he had had over fifteen years earlier, in which he accepted the mantle of the Halveti-Jerrahi order. (He made no mention of his Qadiri affiliation on this particular occasion.) From that time on, Rauf said, he knew he would eventually lead _dhikr_ services, even though he promised Daisy when they married in 1996 that he would not become a shaykh. (Hearing Daisy's name, Brother Malik raised his head and exchanged wary glances with Fareena.) Rauf recalled that he had told Khan then, \"'I don't want to be a shaykh. I don't want to have more dependents. I have enough dependents in my own children'\" from an earlier marriage. \"But,\" he later concluded, \"when something is destined, you cannot avoid it.\"\n\n\"The reason I am giving you this talk,\" Rauf finally admitted, \"is because I'm going to be doing a lot of travel\" to continue being \"involved in the bridging of the West with the Islamic world. Because of that, I have to rely on many of you here to assume various responsibilities regarding the work of the _tariqa_ . . . I'll be demanding your commitment.\" Rauf acknowledged that he was \"not keen on having many disciples. That is not my objective, to have hundreds and hundreds and hundreds of whom I can call my disciples. I'm only interested in those who are keen to commit themselves to the spiritual path,\" he told them, wearily. He seemed exhausted by the travel from which he had just returned and by the thought of that he was about to undertake. Because he would be less available to his dervishes and more reliant on their labor to keep the _tariqa_ going, Rauf urged them to recommit themselves to him as their shaykh and to the order as their community only if they could accept these conditions. For those who would undertake the more \"difficult path\" laid out for this \"special community,\" Rauf encouraged them \"to take care of each other, to serve each other and to serve the _tariqa_. And you will discover that no matter how much you give\u2014the internal investment is very high\u2014the return on what you give is going to be much deeper and much greater.\" After delivering these exhortations, Rauf and his dervishes performed their _dhikr_. When it was over, they communed over sweets and tea, and then the dervishes watched as Rauf took his packed bags to the waiting cab that would take him again to the airport.\n\nIn the months that followed, some of Rauf's dervishes attempted to do what Rauf had asked and create a closer community that had less need of his immediate presence. Others sought additional inspiration and guidance elsewhere. A few began to attend both Fariha's _dhikr_ services and Rauf's. Perceiving this as, at the least, a distraction for his dervishes from their mission under him, Rauf instructed them to refrain from attending any other orders' meetings. Further, he asked them not to involve those who did not belong to the order in _dhikr_ services; in other words, no more visitors. For several of his dervishes, these developments\u2014which many experienced as another narrowing of their community after Rauf asked them not to associate with Faiz Khan\u2014contrasted with what they had previously seen as their shaykh's characteristic openness.\n\nSteeped in the ideas Rauf had spent years conveying to them\u2014that they had more commonalities with other religious Americans than differences\u2014some dervishes resisted this insular turn by finding ambivalence in Rauf's instructions, by interpreting them in ways that restored their previous understandings of their shaykh, or by outright questioning them. The interpretive strategies they used to keep their community as inclusive as possible gained greater importance over the next year as Fareena left New York to move to Singapore with her husband and children, as Brother Malik was hospitalized with a long-term illness that complicated his efforts to stand in for Rauf, and as several cosmopolitan young Muslims who worked in finance lost their jobs during the recession and returned to their countries of origin. In addition to these dervishes, the Sufi group lost another regular attendee in 2007: me.\n\nI had spent over three years with the community during a time of intense change, but I recognized this period as a particularly difficult one and\u2014as an ethnographer, not a dervish\u2014respected Rauf's desire not to have outside observers charting all of the tensions within it. I still attended the _masjid_ each week for some time after Rauf instituted his changes but progressively less often during the years of 2009 and 2010. I did not anticipate how disappointed some of Rauf's dervishes would be by my absence. The assistant imams still sought my attendance at _dhikr_ during those years, saying that \"Shaykh Feisal\" had surely not meant that I was not to come. Their disappointment over my decision to refrain was palpable. Not only had I become a friend to many people\u2014I realized through conversations in the early months of 2007, when some dervishes began to build greater connections to others within the group, that I knew more about most of them than they did about each other\u2014but I was also a potential convert with whom they had deep, and mutual, sympathies.\n\nOver the years, during official interviews and otherwise, I had had numerous conversations with Brother Malik, Pedram, Fareena, Emily, Dean, and others about my own evangelical background and adult agnosticism. I had prayed with them at _dhikr_ s and at the mosque as Rauf instructed me to, sometimes moved to tears by the beauty of the ceremonies and the awe that overwhelmed me at the possibility of a benevolent\u2014though distant, in my opinion\u2014deity. I reminded them, sometimes obsessively, that I had not \"taken the _shahada_ \" and thus was not a Muslim. In their interpretation, however, all people are born Muslims\u2014some are just raised to believe other things. Additionally, because I spoke Arabic with some competence after studying it for several years, believed Jesus to be a moral teacher but not divine, and had spent many summers traveling in North Africa and the Middle East, I seemed more like a Muslim than a Christian to them and they thought my ultimate, formal conversion not unlikely. I was, in short, a fellow traveler in their eyes and a member of their community, one they were not pleased to be losing.\n\nAlthough the assistant imams believed that Rauf had not meant to exclude me from their intimate _dhikr_ services, I interpreted his instructions differently. Not only had I witnessed rather private disruptions in the community in 2007, I had continually remained noncommittal in my opinion of Rauf and of the work of the ASMA Society and Cordoba Initiative. Rauf and Khan were accustomed to the presence of interviewers\u2014generally, members of the news media whose reports aired within hours or days\u2014and other visitors at _jum'ah_ , _dhikr_ , and ASMA and Cordoba events. But having a researcher observe their community for years without comment and without converting was another thing entirely. Fearing that attending _dhikr_ with Rauf's dervishes but without Rauf's approval would compromise his relationships with them and their relationships with their shaykh, I insisted on abstaining from the meetings. Naively, I did not realize that my absence from the community could be as disruptive as my presence.\n\nWhile the assistant imams interpreted Rauf's instructions generously\u2014at least, where I was concerned\u2014others directly questioned them. My first experience with such questioning came from a source I did not expect: Fareena, with whom I had been particularly close. Before moving to Singapore, where she would be just a short flight from Rauf's Malaysian headquarters, Fareena admitted to me some of her frustrations with the changes that had occurred in the _tariqa_. She did not mind working hard to build the community, and she had never visited other Sufi orders, so the exclusive commitment to Rauf did not bother her. What she did object to was not being allowed to have members of other religious traditions\u2014me, in particular\u2014at _dhikr_. \"If we're really American Muslims and really all on the same path [as other Americans], what difference does it make who attends?\" she asked rhetorically, reiterating Rauf's arguments about American Abrahamic commonality. \"We should be reaching out to everyone.\" The next year, on a phone call from Singapore, Fareena reminded me that I was still part of her community, even though we no longer lived in the same country.\n\nThe mosque and _dhikr_ group continued to operate much as it had for the next few years, with several dervishes increasing their volunteer service to the community as Rauf had instructed\u2014and, in so doing, learning how to lead and manage a community without him\u2014but the respite from major disruptions was brief. The responses by Rauf and Khan to controversy over the Cordoba House community center project caused more dervishes to question Rauf's decisions. Initially, those who questioned him were seeking not to second-guess their shaykh but to restore the understandings they had derived from his teachings. Aspects of his instructions seemed contradictory, and they sought to resolve the inconsistencies. Such was the case with the dervishes who financially supported the Islamic community center project. These members of Rauf's _tariqa_ embraced his arguments that Muslims are just as American as members of any other religion. Thus, when controversy arose in 2010 over their new mosque and community center project, and Rauf considered changing it from an Islamic establishment into an interreligious one, they resisted. Muslims should not have to prove their patriotism any more than other Americans, they reasoned, interpreting Rauf's continual refrain about Abrahamic Americanness in light of their current circumstances. For them, questioning Rauf's return to an emphasis on interreligious partnerships was meant not to usurp his authority but to defend what they believed was their collective mission. By 2010, though, it became clear that Rauf and the dervishes involved in the Islamic center project had different ideas about its purpose\u2014ones they were ultimately unable to overcome.\n7\n\nISLAM IN THE AGE OF OBAMA\n\nWhat's More American Than Service?\n\nON DECEMBER 21, 2009, when many Americans were preoccupied with holiday travel, shopping, or cooking, Daisy Khan appeared on the Fox News network's conservative talk show, _The O'Reilly Factor_. With Bill O'Reilly off that day, guest host Laura Ingraham interviewed Khan about a new project she and Rauf, operating under the Cordoba Initiative, had invested in with a few of their associates, dervishes who owned the real estate firm Soho Properties. Proposed as a community center and worship space on Park Place in Lower Manhattan, Cordoba House would offer educational, cultural, and recreational services while providing Muslims a much-needed mosque; it would also be a place where members of various religious communities could interact. \"It's really to provide a place of peace, a place of services and solutions for the community, which is always looking for interfaith dialogue,\" Soho Properties Chairman and CEO Sharif El-Gamal told the _New York Times_ , which had run a front-page story about the project two weeks before. Indeed, part of the property already served as an overflow worship space for Masjid al-Farah, ten blocks away. \"I can't find many people who really have a problem with it,\" Ingraham said to Khan. \"I like what you're trying to do.\"\n\nWhile Ingraham's response contrasts strikingly with the opposition to Cordoba House regularly featured on Fox News during the summer of 2010, the Cordoba House proposal did not seem to be a likely source of controversy at the time of Ingraham's interview. In their coverage of the project a few weeks prior, _New York Times_ writers had identified Rauf as a Sufi who preaches tolerance and is more concerned with \"spiritual wisdom\" than \"strict ritual.\" Additionally, they noted the approval the project had already garnered from a prominent New York rabbi, the executive director of the Jewish Community Center (which was to serve as a model for the project), the former general secretary of the National Council of Churches, and the National September 11 Memorial and Museum board (on which Daisy Khan served as an advisory member).\n\nIt was not until May 2010\u2014six months after Ingraham's interview and several days after Khan, Rauf, and El-Gamal received approval from Manhattan's Community Board 1 to proceed\u2014that controversy erupted over what the _New York Post_ (like Fox News, owned by Republican media mogul Rupert Murdoch) labeled the \"Ground Zero Mosque.\" El-Gamal later acknowledged that he, Khan, and Rauf\u2014not imagining that a new mosque in an area previously populated with them would draw resistance\u2014had erred by not seeking greater community feedback prior to undertaking the project. Rauf had informed Mayor Michael Bloomberg of their plans as early as September 2009 and received tacit approval, but no one had convened a meeting to seek the opinion of 9\/11 families (those who lost loved ones) or first responders who had survived the attacks. It should be noted that El-Gamal was among the civilian responders who rushed to the burning buildings on 9\/11. He stayed there, handing out water to firefighters, for two days, and nearly lost a dear friend when one of the towers collapsed.\n\nBut families of 9\/11 victims and first responders were not the primary opponents of Cordoba House. One firefighter did oppose Cordoba House's bid to get taxpayer funding for community services from the Lower Manhattan Development Corporation, an entity established by Governor George Pataki and Mayor Rudolph Giuliani to coordinate redevelopment after 9\/11. By July 2010, however, the September 11th Families for a Peaceful Tomorrow organization gave its approval to the project. The most significant opposition to Cordoba House then and later came from conservative Republican politicians like Newt Gingrich and those associated with the Tea Party movement. Republican politicians had actively misled American voters in the months before the 2008 election by arguing or implying that Barack Obama was a Muslim\u2014a tactic that caused Republican former Secretary of State Colin Powell to denounce many of his colleagues, but also an accusation that, as of 2010, almost 25 percent of Americans still believed. They sought to employ similar anti-Muslim campaign tricks again in 2010, during which the website of the National Republican Trust, a political action committee established in 2008 for the purpose of \"defeating liberals, Progressives, and Democrats,\" claimed that the community center was intended as a \"shrine to the 9\/11 terrorists.\" In so doing, these politicians helped create the climate of fear and aggression that surrounded discussion of Cordoba House.\n\nAnd yet even they came late to the controversy. Conservative politicians did not seek to turn the community center into a campaign issue until members of a right-wing organization called Stop Islamicization of America (SIOA) created a controversy out of the local municipal matter. (SIOA has been designated a hate group by the Southern Poverty Law Center; it was an important source of inspiration for Norwegian white supremacist terrorist Anders Brevik's massacre of 69 children at a summer camp in 2011.) One of SIOA's associates, Pamela Geller\u2014a woman who had erroneously claimed during the 2008 election cycle that Obama was secretly the child of Malcolm X\u2014denounced the project on her blog after the initial _Times_ report in 2009 but began to organize protests against the community center only in May 2010. At that point, conservative politicians, some still seeking to capitalize on the racist backlash that had greeted the nation's first black (and in the minds of many, first Muslim) president, saw an opportunity they had nearly missed.\n\nTea Party leader Mark Williams made his now infamous comment about the center being a shrine to the 9\/11 terrorists' \"monkey god\" in May 2010. Gingrich, who was contemplating a 2012 presidential bid funded primarily by conservative casino billionaire Sheldon Adelson\u2014who argues that all Muslims want to kill all Jews; who advocated dropping a nuclear bomb on Tehran; and who spends millions trying to sway US policy in favor of right-wing Israeli politics\u2014began to criticize the Cordoba House project soon after. Aspiring politicians seeking local office followed suit, as did New York Republican Congressman Peter King, who not only questioned Rauf's credentials as a \"moderate\" but also launched Congressional hearings on Muslim radicalization in the United States\u2014despite the fact that Homeland Security analysts insisted that white right-wing militant groups, not Muslims, posed the greatest terrorist threat to Americans.\n\nRauf received increased criticism over the summer as news outlets discovered old comments he had made chastising US leaders for supporting dictators in the Middle East. It seemed not to matter that former President George W. Bush did the same thing in 2003. Seeking to appease his critics and to prove his moderation once again, as he had during the early days of the Cordoba Initiative, Rauf emphasized his organization's connections to respectable leaders of other religious traditions. He was hardly alone in this: several national Muslim organizations also urged interreligious engagement and dialogue during the controversy, which spread to towns and cities across the country where Muslims sought to build Islamic centers or simply establish mosques. For Rauf, however, this return to interreligious alliances also meant turning the Cordoba House project from a mosque and Islamic community center into a multifaith institution. While this possibility seemed to pacify some critics, it was not one that Rauf's dervishes could accept.\n\nIn response to the controversy, El-Gamal\u2014who had purchased 51 Park Place in 2006\u2014suggested renaming the project and restructuring its leadership team. In July 2010, the \"Cordoba House\" name was transferred to a smaller entity, led by Rauf, that would oversee interreligious programming within the larger community center, which would be named after its street address: Park51. Rauf and El-Gamal would serve as joint project managers only until Park51\u2014which El-Gamal created as a new corporation separate from ASMA and Cordoba\u2014could file for tax-exempt status, appoint a permanent executive director, and fill twenty-three seats on the organization's interreligious board of directors.\n\nThese actions failed to stifle the fires continually stoked by Fox News pundits and conservative politicians. Although both Mayor Bloomberg and President Obama argued in the following months that the project's developers had the legal right to build a mosque and community center two blocks from Ground Zero, then-Governor David Patterson, among others, suggested using eminent domain to force them to relocate. Daisy Khan, who often served as the public face of the project and spoke for Rauf while he was away on a US State Department tour of the Middle East, indicated she was amenable to moving the center. El-Gamal, however, who had long followed Rauf in believing that Muslim Americans are ideal citizens and equal to any others, felt that moving the center or changing its nature would violate the mission he and his shaykh had long imagined of serving their community and proving their patriotism.\n\nAs I discuss below, disagreements over purpose, leadership, and fundraising tactics between Rauf and Khan, on one side, and El-Gamal, on the other, continued after the summer's end. These differences ultimately led El-Gamal and other dervishes involved in the project\u2014some of whom began to refer to Rauf during the summer as the \"occulted imam,\" a barb derived from the phrase used to describe revered Shi'i imams who had disappeared\u2014to disaffiliate from Rauf and Khan by 2011. In January of that year, El-Gamal announced that Rauf and Khan would no longer fundraise for or speak on behalf of the project, although Rauf would remain on the board of directors. After that, a series of visiting speakers filled the _minbar_ at the temporary mosque.\n\nDespite Khan's and Rauf's disagreements with El-Gamal and their different visions for the project, all agreed on one thing: that the community center would be not simply a worship space or a Muslim-only institution but a place of service for all New Yorkers. This emphasis on service was entirely sincere. On different occasions, I had heard various dervishes speak excitedly about what the center could bring to the residents of Lower Manhattan and how, if built properly, it could be an example of everything from environmentally sustainable engineering to proper social engagement. At the same time, the stress on service was also strategic. Rauf, Khan, and El-Gamal spoke of service both because they and other dervishes hoped to contribute to their community _and_ because service was an increasingly popular way for Muslim Americans to emphasize their moderation. \"What's more American than serving others?\" El-Gamal asked when defending Park51 that summer. In the years after the controversy, Rauf and Khan also continued to emphasize service at times, even though they had stepped away from Park51 and had \"moved on,\" as Rauf put it, to other projects.\n\nIn what follows, I examine the fate of Park51 and of the organizations Rauf and Khan led after the 2010 controversy. I focus on the ways different visions of service exacerbated the growing rift between Rauf and his dervishes and cost the community center valuable support from Muslim groups who had long questioned why Rauf and Khan tended to build relations with leaders of other religions more than with fellow Muslims. To place these debates in context, I briefly discuss American Muslims' efforts to make service proof of moderation, efforts that are modeled on the community service endeavors other religious minorities had engaged in during the twentieth century (often in cooperation with the federal government) while trying to gain acceptance in the United States. Although Rauf had promoted this means of assimilating as early as 2004, he had not actively participated in organized community service endeavors. Rather, unlike other Sufis\u2014such as Tosun Bayrak, who led his dervishes in projects ranging from refugee relief in war-torn countries to disaster relief in New Orleans after Hurricane Katrina, and Faiz Khan, who joined the Sufi Circle of Long Island and participated in their community service endeavors after disaffiliating from ASMA\u2014Rauf chose to act in the service of the government in more overtly political capacities. As we shall see, this first foray into providing direct community service would bring with it a lesson in the politics of intercommunal dynamics and show how engagement in service was not a guarantee of acceptance. Faced with the limits of inclusion for Muslims in America at that point, it is perhaps not surprising, as I discuss below, that Rauf returned to emphasizing Sufism as much as he did service and interreligious alliances.\n\nIn the Service of the State\n\nDuring the his first year as president, Barack Obama traveled to Egypt to deliver a formal address to \"Muslim communities around the world.\" In addition to promising better relations with Muslims abroad, Obama pledged to protect Muslims' practices at home. Faith should \"bring us together,\" he argued. For that reason, his administration was \"forging service projects\" to unite \"Christians, Muslims, and Jews.\" As an example of the kind of freedom and tolerance he hoped to encourage, he pointed to the medieval Spanish city of Cordoba.\n\nWhile using service projects as a way to unify diverse religious groups was not new, as I have elsewhere discussed with regards to Presidents Wilson, Roosevelt, and Kennedy, Obama was participating in a rather recent version of this trend. Shortly after taking office in 2009, he established the White House Office of Faith-Based and Neighborhood Partnerships to replace George W. Bush's Office of Faith-Based and Community Initiatives, created in 2001 to facilitate \"compassionate conservatism,\" replacing \"government bureaucracy\" with \"neighbors serving neighbors.\" Unlike early twentieth-century presidents, Bush's policy\u2014which expanded the \"charitable choice\" legislation of his predecessor, Bill Clinton\u2014allowed the government to grant funds to religious organizations providing services ranging from childcare to addiction counseling, thus reducing the need for government agencies to provide such things.\n\nAfter the terrorist attacks of 9\/11, Bush also issued an executive order to create a new Center for Faith-Based and Community Initiatives within the US Agency for International Development (USAID), an agency Kennedy had created in 1961 to counter communist influences abroad by furthering \"America's foreign policy interests in expanding democracy and free markets,\" and that contracted with religious service providers after Kennedy's Peace Corps, dogged by accusations that it would allow Catholics to control the government, could not. Among other things, Bush's changes to USAID nearly doubled the amount of foreign aid going to religious service providers. Notably, 98 percent of these funds went to Christian groups (primarily Catholic and evangelical organizations), while the remaining 2 percent of funds were divided between two Jewish and two Muslim organizations. It is not insignificant that the full public inclusion of Catholic service organizations in the US government's operations occurred not during the twentieth-century fight against godless communism, but as US officials increasingly focused on a new religious enemy: \"radical\" Muslims.\n\nObama retained Bush's and Clinton's strategy of partnering with religious organizations to achieve government ends but sought to make such partnerships more inclusive. In addition to altering the name of Bush's agency, Obama added a new advisory board to which he appointed Eboo Patel, the Muslim American founder of the Interfaith Youth Core (IFYC) community service organization and one of Rauf's and Khan's first young Muslim Leaders of Tomorrow. Then, just after returning from Egypt, Obama announced the creation of United We Serve, an agency designed to \"make volunteerism and community service part of the daily lives of all Americans.\" Unveiling this national service initiative in the wake of the largest economic contraction since the Great Depression, Obama emphasized the need for volunteer labor to turn the nation's fiscal tide. \"Economic recovery is as much about what you're doing in your communities as what we're doing in Washington,\" he explained.\n\nIn the months following, Obama commended Muslim Americans who responded to his challenge. \"Faith-based organizations, including Islamic organizations, have been at the forefront in participating in this summer of service,\" he said during Ramadan in 2009. Meanwhile, a group of Muslim Americans established an organization and website, MuslimServe.org, to connect visitors to the administration's online initiative (Serve.gov), thus allowing them to participate in federal efforts directly and to register their own community service projects in the federal database. In the \"Message to Muslim American Community\" [ _sic_ ] posted on the site, MuslimServe organizers explained their purpose. \"United We Serve is an opportunity for every Muslim American to be part of a nationwide movement to bring about positive change,\" they argued. In addition to helping rebuild communities after the economic crisis, they elaborated, \"'Answer the Call' means most importantly, to answer God's call to serve humanity.\"\n\nThe leaders of established Muslim American organizations followed suit over the next few years. In 2009, for example, the Islamic Society of North America (ISNA) encouraged Muslim Americans to participate in the first ever National Day of Service, designated for September 11, 2009, and to be observed every year thereafter on the anniversary of the 2001 terrorist attacks. Likewise, during the Cordoba House controversy in 2010, fifteen Muslim American leaders and organizations\u2014including the three largest North American organizations: ISNA, ICNA, and CAIR\u2014encouraged Muslims across the United States to join MuslimServe's effort and use the federal website to search for projects in their areas. The goal, according to their \"Call to Action,\" was \"to turn the tide of hatred\" and \"demonstrate our core Islamic values and our dedication to our neighborhoods and our country.\" Well aware that they were regarded with suspicion, MuslimServe leaders reminded participants, \"all eyes will be on us this Eid and on 9\/11 . . . Let's show that we can rise above prejudice and hatred and be the kind of conscientious citizens who give back to our country.\" For its part, ISNA even designated the theme of its 2010 annual convention \"Nurturing Compassionate Communities: Connecting Faith and Service.\"\n\nThe service projects Muslim American leaders promoted were wide ranging, from participation in government organizations listed on the Serve.gov website to local efforts to fight urban blight, build housing, and feed the homeless. In addition to serving humanity, these initiatives were designed to showcase the patriotism, moderation, and community-mindedness of American Muslims. When, in the midst of these growing campaigns, the Islamic community center's leaders began describing their project as service-oriented, other New York Muslims took notice. Unfortunately, many of the services Khan, Rauf, and El-Gamal promised to provide seemed designed for wealthy New Yorkers rather than those still struggling economically.\n\nIn July 2010, for example, the Cordoba Initiative, then still the primary tax-exempt entity behind the community center, explained on its website that the center would house \"a 500-seat auditorium, swimming pool, art exhibition spaces, bookstores, restaurants,\" and other \"services\" that would create a \"cultural nexus\" in the region. This description of the project combined the cultural and educational programming that ASMA and Cordoba had once emphasized with the recreational focus of other city institutions like the Upper West Side Jewish Community Center, where El-Gamal's children had learned to swim. Not only were such programs ones Khan, Rauf, and El-Gamal had experience with, they were ones the Lower Manhattan Development Corporation (LMDC), with a budget of two billion dollars, sought to fund. As El-Gamal explained to an interviewer in late July, after describing the center's proposed yoga studio, \"Residents need services, investment in the neighborhood, activities and opportunities. Community Board 1, which represents the residents of Lower Manhattan, acknowledged the needs we were fulfilling when they gave us their clear support . . . The Board recognized the value in jobs, programs and services we are bringing to the city, and they know that this project is very important for Lower Manhattan.\" The controversy intensified over the next few months, in part because of the defenses Rauf, Khan, and El-Gamal mounted. In addition to non-Muslims who opposed what they erroneously believed would be a thirteen-story mosque near Ground Zero\u2014a misconception that prompted El-Gamal to create a new nonprofit called Prayer Space, with a separate board of directors, to oversee the one-story mosque in the adjoining building\u2014some Muslims criticized what they considered to be the \"elitist\" services on offer. It took some time for Rauf, Khan, and El-Gamal to recognize and respond to the latter complaint. In the meantime, their public pronouncements tended to make the situation worse.\n\nWriting an op-ed piece for the _New York Daily News_ in early August, for example, El-Gamal described the community center as \"a world-class house of culture and art, education and recreation . . . A facility where you can find children's swim classes, photography exhibitions, Pilates sessions, Spanish-language lessons and an amazing gym.\" Khan portrayed the project in similar terms in a television appearance later that month. Further, she and El-Gamal reinforced the concerns other Muslim American leaders had when noting that Park51's directors would work \"closely with interfaith leaders, elected representatives, community organizations and our neighbors across the city.\" The constituency missing from that list, of course, was other Muslim Americans.\n\nLater that month, Hussein Rashid\u2014a young Muslim scholar who had grown up in New York and once participated in Rauf's and Khan's Muslim Leaders of Tomorrow program\u2014gave voice to the growing concerns among New York Muslims about the project. In a post on a popular online religion news magazine, he (like many other Muslims) chided the developers for not anticipating the \"Islamophobia\" such a project could stoke and for putting Muslim Americans in the position of wanting to defend the center on principle but hesitating because the project did not seem designed to serve or represent most of them. Rauf \"may be one of the 'good' Muslims, but no one person should be the lynchpin of a project this size,\" Rashid argued. \"It should be easy in New York City to create a visibly diverse group of Muslim supporters that represents the richness of our community, and who can put forward a clearly articulated vision.\" Rashid was not the only one to question the leadership and services Rauf, Khan, and El-Gamal planned to offer. In fact, reservations about these issues led many Muslim leaders to refrain from supporting the center until well after the height of the controversy.\n\nThe uproar reached a peak in September. On September 10, I attended Friday prayers at Park51 after an early Eid al-Fitr lunch at Masjid al-Farah to celebrate the end of Ramadan. The striped carpet of the temporary mosque filled slowly with worshippers who were more reserved than they usually would be on what is normally a day of celebration. Before the service started, El-Gamal realized the batteries in the speaker's microphone had died, and I wandered out to a nearby drugstore to find replacements. The streets were busy only with the Financial District's usual traffic, and I saw no protesters, just police stationed outside the mosque to deter any assailants. El-Gamal's brother delivered the _khutbah_ that afternoon and at the end asked worshippers to return the next day, September 11, when mass demonstrations were planned in front of the property by both opponents and supporters. He especially hoped worshippers would attend an evening candlelight vigil in remembrance of the victims of the 2001 attacks.\n\nThousands of demonstrators did fill the streets around Park51 the next day. Some wore t-shirts bearing the message \"No Sharia in America.\" Making reference to a Florida pastor who threatened to build a Qur'an-fed bonfire unless the center relocated, defenders of the project countered with signs that read, \"Real Americans Don't Burn Qur'ans.\" While many religious leaders in New York and the leader of the New York Civil Liberties Union also defended Park51, most local Muslim leaders remained silent.\n\nA week and a half later, El-Gamal attended a day-long meeting convened by the Majlis Ash-Shura (Islamic Leadership Council) of New York, an organization representing fifty-five mosques and Muslim groups. El-Gamal apologized to local leaders and visiting representatives of other national groups, including ICNA, CAIR, and the Muslim American Society, for not consulting them earlier. He then promised that the center would not just cater to \"Manhattan elites\" involved in interfaith dialogue or to those who could afford dues for the gym and pool. El-Gamal also pledged to include representatives of various Muslim groups (including black Americans) on an advisory committee and on Park51's board of trustees. The meeting allayed the fears of many, including Aisha al-Adawiya, the founder of Women in Islam and a member of the Muslim Alliance in North America (MANA) community service organization. In response to al-Adawiya's insistence that the center's services be available to poor New Yorkers, El-Gamal promised that it would be \"open and inviting\" to everyone. During a postmeeting press conference, the assembled leaders bestowed their blessing on the project and on El-Gamal's vision for the center\u2014one from which, by that point, Rauf had already diverged.\n\nRauf and Khan had talked about creating an independent Islamic center and mosque since I first met them. As Khan had explained when I interviewed her in 2008, describing ASMA's work then by emphasizing the Cordoba House project and cultural and educational programs instead of Sufism, they were hoping to create an institution that would \"have an American cultural expression to it\" while being \"very authentically Islamic in terms of its practice.\" Although she was adamant that it should be \"very inviting to non-Muslims and there would be interfaith dialogue,\" it was not something she or Rauf envisioned then as an interreligious institution. In fact, 2008 was the year Rauf, feeling secure in his moderate status, restructured the Cordoba Initiative and removed any mention of interreligious leadership from its website. The defining feature of the institution they then hoped to create was that it would be\u2014as a Muslim-led institution\u2014\"more an American Muslim center than a Muslim center built around a certain ethnicity.\"\n\nWhen Rauf\u2014his moderation now in question\u2014returned to emphasizing interreligious alliances in 2010, it was not solely a tactical decision. Unlike El-Gamal, who insisted that Park51 was meant to serve the local community and had nothing to do with the 2001 attacks, Rauf had begun to envision Cordoba House as a place of international, interreligious healing that would serve to bring people from different religious and national communities together after the tragedy of 9\/11. Its location just blocks from Ground Zero was not incidental, to him, but meaningful: the service Cordoba House could provide would be to testify, through its existence as an interreligious center, to the triumph of moderation over terrorism.\n\nRauf had hinted at his larger goals for Cordoba House as early as 2009, when speaking about the location's significance to the _New York Times_. Yet it was not until the latter half of 2010 that he insisted on Cordoba House's interreligious character. When El-Gamal subsequently barred Rauf and Khan from speaking on behalf of the project or fundraising for it, he attributed it partly to this difference. \"Imam Feisal's vision has a global scope and his ideals for the Cordoba movement are truly exceptional,\" El-Gamal began, but \"our community in Lower Manhattan is local . . . Our focus is and must remain the residents of Lower Manhattan and the Muslim American community in the greater New York area.\" By 2011, this disagreement over the scope of Cordoba House services acquired very material ramifications.\n\nTo allay the fears of New Yorkers who did not want their taxes used (through distribution by the LMDC) to build a mosque or any other Islamic institution, El-Gamal repeated steps that many other marginalized religious service organizations had previously taken when criticism impeded their attempts to prove their national loyalties through service: he created an independent organization that could qualify for funds in the court of public opinion. (Legally, this was unnecessary; numerous religious service organizations and community centers already received federal funding thanks to the Clinton- and Bush-era legislation mentioned earlier.) By November of 2010, when El-Gamal applied for funds from the LMDC, Park51 represented only the community center, Prayer Space ran the mosque, and Cordoba House (Rauf's independent organization with Khan) would direct interreligious programming for the project. Yet, because Park51 had yet to acquire nonprofit status from the Internal Revenue Service\u2014which failed to process its application for over six months, despite repeated requests to expedite it\u2014it was ineligible to receive LMDC grants for that round of disbursements or to collect charitable donations. Contrary to everyone's expectations the year before, the community center increasingly looked as though it might not find the funds it needed.\n\nIn the meantime, Rauf and Khan continued to speak about and collect contributions for Cordoba House and for a new project they began to call the \"Cordoba Movement.\" Few people realized that these donations were going not to the community center but to the nonprofit Cordoba Initiative. Instead of serving their shaykh in pursuit of a joint mission, as El-Gamal and other dervishes believed they had been in the past, or even working together as business partners to serve the community, Rauf and El-Gamal, along with their respective supporters, were increasingly at odds in contests for funding\u2014a situation about to worsen measurably as President Obama, and then Congress, insisted on cutting funds for community service providers in the years that followed. Trying to diplomatically distance Cordoba House from the actual community center, El-Gamal ended up clarifying what had become the real nature of his relationship with Rauf: competitor rather than colleague in the push to define (and finance) different versions of American Muslim moderation.\n\nIn February 2011, while on a speaking tour in upstate New York, Rauf told editors of the _Buffalo News_ that he would move Cordoba House (something reporters still mistook for the center as a whole) to another location if offered a facility to house it. The announcement prompted El-Gamal to assert the independence of his project again: \"Park51 is not moving its location under any circumstances,\" he told the _New York Times_. \"Imam Feisal has no authority or control over this project, over its board of directors or over Soho Properties, which controls the real estate. Park51, the Islamic Community Center in Lower Manhattan, is more than any one personality or imam.\" As El-Gamal distanced Park51 from Rauf, Khan, ASMA, and Cordoba, however, Rauf and Khan continued to claim the community center vision for themselves. That March, speaking at a luncheon held by a women's magazine, Khan announced a new project she and Rauf were planning for an interfaith cultural center that would be larger in scope than Park51. \"We had the vision, we still have the dream,\" Khan told the audience. \"The location is not the dream, my friend.\"\n\nAlthough El-Gamal maintained publically during the following year that he and others at Soho Properties were open to reconciling with Khan and Rauf, a rapprochement never took place. Instead, Rauf and Khan devoted themselves for a time to the \"new vision\" of which Khan had spoken, while Park51's leaders hosted occasional cultural programs in their temporary space and sought funds to make preliminary improvements to the building. Not only did their different visions, competition for funding, and eventual public falling out diminish both parties' prospects for raising money and contribute to the original community center's demise, as I discuss below, they provided a reminder of some of the trials that religious minorities, including Muslim Americans, had encountered in the past when trying to prove their patriotism and moderation through service. These lessons were ones that other national Muslim leaders would also learn in the years that followed\u2014painfully, and with a growing awareness that the integration of other racial and religious minorities had not been as seamless or even as encompassing as they had once thought.\n\nBefore turning to those broader developments among Muslim Americans, I take a final look at the organizations Rauf and Khan led and at the Islamic community center project, and I show how those involved tried to remake themselves, once again, as moderate Muslim spokespersons. When it turned out that renewed professions of service and participation in interreligious coalitions was not enough to establish Rauf, Khan, or El-Gamal as widely accepted moderates, all returned to where they had begun: emphasizing Sufism.\n\nRecasting Cordoba and the Community Center and Returning to Sufism\n\nJust as 9\/11 had caused Rauf and Khan to recast their work with ASMA in major ways, so, too, did the 2010 controversy inspire them to reframe the work of both ASMA and Cordoba. Some of this revision took place immediately\u2014a new Cordoba website and statement of purpose, for example, that Rauf unveiled during the height of the Islamic center controversy\u2014while other changes in the organizations became apparent only years later, hinted at in the book Rauf published in 2012 to defend his version of moderation and promote his organizations, as well as in the monthly newsletters and press releases ASMA and Cordoba distributed to those on their email lists.\n\nThe initial response by Rauf and Khan to the controversy was, as Khan suggested in March of 2011, to undertake an even larger project\u2014the Cordoba Movement\u2014meant to build an interreligious coalition of moderates around the globe. In a letter to visitors posted on the new website, Rauf explained that the Cordoba Movement was initially founded as the Cordoba Initiative but grew into \"a multi-national, multi-faith organization dedicated to improving understanding and building trust among people of all cultures and faith traditions\"\u2014in other words, no longer just among Muslims, Christians, and Jews, or between \"Islam and the West.\" Furthermore, the Cordoba Movement was \"broadening\" its previous programs. Describing many older ASMA and Cordoba programs in the religiously neutral language of community development, Rauf explained that the Cordoba Movement would focus on Young Leaders (formerly Muslim Leaders of Tomorrow) and Women's Empowerment (formerly the Women's Islamic Initiative in Spirituality and Equality), in addition to creating multifaith alliances. Rauf also announced that he would undertake an extended speaking tour to promote the movement\u2014the fundraising tour that ultimately caused a rift with El-Gamal.\n\nWhile Rauf and Khan did, in fact, spend the first half of 2011 promoting the Cordoba Movement, their public falling out with El-Gamal slowed the new program's momentum. Within a year Rauf returned to his work on improving US-Muslim relations. He and Khan folded the Cordoba Movement, including most of its website content (live for just one year), back into the existing Cordoba Initiative structure, and Khan continued her work with ASMA, focusing mostly on the WISE program that required so much of her time and garnered so much funding. During this time, they still spoke of creating a global network of moderates and of creating an interfaith community center, as Rauf described in his 2012 book, _Moving the Mountain: Beyond Ground Zero to a New Vision of Islam in America_ , but they did so as part of their larger work addressing issues pertinent to Muslims.\n\nRauf's 2012 book bore a great deal of resemblance to _What's Right with Islam_ , the book he had written in 2004. In fact, when pressed in May of that year by New York radio talk-show host Brian Lehrer to explain what was new about his newest vision, Rauf had a hard time doing so. Instead, he reiterated the arguments he had developed nearly a decade earlier: that Muslims in the United States, \"first generation immigrants coming here with many of their cultural trappings . . . need to transform themselves in the American cultural context\" so as to serve as a bridge between the United States and Muslims in the rest of the world. The transformation should not be a difficult one, he elaborated, because just as US laws embody Judeo-Christian principles, the \"American law, American Declaration of Independence, American social contract, [and] American principles of democracy express our shari'ah principles of good governance in the most effective way. And right now I'm actually involved in a conference of Muslim scholars . . . seeking to establish principles of good government in keeping with Islamic law\" in countries around the world. \"My argument is that the Americans,\" from the arrangement established between religion and politics to the large-scale provision of \"social services for the poor,\" have done this. Significantly, in addition to promoting Islam \"in an American vernacular,\" he explained to National Public Radio's Terry Gross that same month, he was recommitting to promoting \"the spiritual dimension of Islam, which is called Sufism.\"\n\nListening to Rauf discuss his new book\u2014which, like his previous one, was divided between defenses of Islam and descriptions of his and Khan's major projects\u2014one might well wonder, as Lehrer did, what had changed for the imam after 2010. Although Rauf stumbled over the interview question, his 2012 book does contain new content, much more on women's rights, for example, than his 2004 book had. Yet it is most notable not for the things it includes but for the things it omits\u2014among them, any mention of democratic capitalism.\n\nBy 2012, the first year since 2007 that the US economy did not worsen, promoting American-style capitalism was not a promising strategy for gaining acceptance. Just the year before, with corporate profits reaching new heights while most wages and standards of living failed to improve, the Occupy Wall Street movement begun in New York spread nationwide, railing against the very corporations that Rauf had lauded in his 2004 book. Public anger over American economic inequalities was so great that by 2015 it would even fuel the surprising rise of a socialist candidate for the presidency\u2014something that seemed impossible when Rauf attempted to follow the path to assimilation paved by Michael Novak and other second-generation white ethnics who had worked at the American Enterprise Institute in the 1970s, or even when Obama, who was himself accused of being a socialist nearly as often as he was accused of being a Muslim by politicians who feared he would tax whites to support blacks, first ran for president in 2008.\n\nInstead of praising democratic capitalism in 2012, Rauf acknowledged America's failure to resolve \"persistent poverty and economic inequality.\" While this change in perspective was undoubtedly due more to the global economic climate than to the Islamic center controversy, other aspects of his new argument\u2014such as his decision to refrain from touting America's ostensible racial equality as frequently as he had in 2004\u2014had more to do with the lessons he learned during the summer of 2010.\n\nStill optimistic about racial integration, Rauf openly acknowledged for the first time in print that racism is institutionalized on a systemic level in the United States, including in the mass incarceration of black Americans. He and Khan were still not as critical of American racial injustices as other national Muslim leaders. For example, Khan largely defended New York City Police Commissioner Raymond Kelly during controversy over his department's use of Stop-and-Frisk tactics\u2014a method of policing blacks and Latinos that a federal judge found in violation of constitutional projections against unreasonable search and seizure\u2014as well as during controversy over the NYPD's decades-long secret (and warrantless) surveillance of Muslims in New York and New Jersey. Further, neither ASMA nor Cordoba commented on the routine murders of unarmed black men and boys at the hands of police that bystanders caught on camera in cities across the country from 2014 to 2016, whereas ISNA and ICNA issued press releases condemning some of these incidents and urging justice and accountability. Nevertheless, the animosity they and many other Muslim American communities and individuals faced in 2010 and after awakened them to the depth of violent racism that exists in America.\n\nWhen, in the wake of the Ground Zero Mosque debacle, a white gunman killed six Sikh worshippers, whom he had confused with Muslims, at a temple in Wisconsin in 2012, ASMA sent representatives to a memorial vigil. Cordoba did the same when a white supremacist terrorist gunned down nine black worshippers during a Wednesday night Bible study at a church in Charleston, South Carolina, in 2015. These gestures undoubtedly helped ASMA and Cordoba emphasize their arguments that moderates of all religions need to ally against such extremists. Still, Khan and Rauf had not seen the need to make such gestures prior to 2010.\n\nThe last major difference between the vision Rauf elucidated in his 2004 book and the one he detailed in 2012 is that, while Rauf attempted to humanize so-called fundamentalists in 2004, describing them as distraught and disenfranchised people acting out of defensive instincts when they believed themselves threatened, he no longer attempted to mitigate the aversion Americans had toward extremists of any kind. Rather, after the Islamic center controversy, he tended to describe such extremists with the same language employed by the proponents of secularism whom he had once argued against: as those who promote the \"intrusion of Islam\" into politics. This change of perspective is somewhat ironic, given that Rauf rails against secularism and atheistic communism in other portions of the book and blames Soviet secularist attempts at neocolonialism in the Middle East for the reactionary \"rise of extremist Muslim fundamentalism.\" More importantly, it is a change reminiscent of the dynamics members of other religious minority groups have undergone when attempting to prove their patriotism by participating in programs allied with or sponsored by the federal government. Engaging in these pursuits with their own objectives in mind, such representatives often find their activities molded and repurposed in ways they might not originally have intended. Rauf's 2004 arguments about Muslim moderation differed from those of secularists who insisted on the need to separate religion and politics. By 2012, after nearly a decade of participating in State Department Programs, his arguments had begun to more closely resemble some of those secularist positions, raising the question of whether (as scholars of Islam and secularism have warned) his objectives would eventually come to replicate the State Department's agenda.\n\nRauf's change in tone from 2004 to 2012 cannot be explained solely by strategic considerations or by the pressure exerted by the government programs in which he had participated. Rather, for Rauf in 2012, \"extremists\" now included ones who, seeking to defend the ostensible Christian identity of the United States, had threatened his own life, as well as the lives of many other Muslim Americans. In response to these experiences, Rauf redoubled his focus on getting moderates of all religions to band together to fight all fundamentalists. Distancing himself even further from the specter of extremism, Rauf began emphasizing Sufism again. It became apparent during the 2010 controversy that Sufism remained an important index of Muslim moderation for many Americans, and\u2014after years of threats on their lives\u2014Rauf and Khan did not want to be mistaken for extremists again.\n\nNovelist William Dalrymple, quoting the 2007 RAND Corporation study discussed in Chapter 4, told _New York Times_ readers in August 2010 that Sufis are allies against radicals worldwide and \"ideal 'partners in the effort to combat Islamic extremism.'\" They are the \"Muslims in the middle,\" he claimed\u2014caught between uninformed Americans who view all Muslims as potential terrorists, on the one hand, and Muslim extremists elsewhere who view \"moderate, pluralistic Sufis\" as legitimate targets of violence, on the other. While Rauf's \"slightly New-Agey rhetoric makes him sound, for better or worse, like a Muslim Deepak Chopra,\" Dalrymple admitted, both average Americans and shapers of US foreign policy should embrace him. Liberal elites like Dalrymple were far from alone in seeing Rauf's Sufism as an indication of his moderation. A few days after Dalrymple's column appeared in the _Times_ , Brian Lehrer hosted a call-in segment on public radio for residents of the Manhattan neighborhood of Tribeca, urging them to report their thoughts on the nearby mosque and proposed community center. The callers were overwhelmingly supportive and described Masjid al-Farah and Rauf as \"dream neighbors.\" One noted that Sufi Books, Shaykha Fariha's since-closed store, had also been a neighborhood fixture for many years. A subsequent caller said he knew of Sufi Books and had _not_ known that the same Muslims were behind the Islamic center but was now even more in favor of the project.\n\nSuch favorable references to Rauf's Sufi identity did not dissuade conservative politicians from using the controversy for political gain. The following Monday, for example, New York gubernatorial candidate Rick Lazio aired commercials\u2014complete with smoldering footage from the World Trade Center attacks\u2014in which he denounced Rauf as a terrorist. But neither did such favorable references escape the attention of Rauf and Khan. Rauf openly promoted Sufism again in his 2012 book and through Cordoba events, as did Khan, despite devoting so much time to her WISE work that she allowed ASMA's Muslim Leaders of Tomorrow program to lapse. (A few years later, Khan would separate WISE from ASMA entirely and leave the latter organization). In May 2012, for example, Khan invited ASMA supporters to an evening of communing with mystics, including Rauf, at St. Bartholomew's Church in Manhattan and of contemplating America's special role in fostering interreligious cooperation. In the years that followed, ASMA and Cordoba also sponsored the musical performances of Sufi Qawwali singer Farid Ayaz, who had been ASMA's frequent guest during its more openly Sufi period. The Cordoba Initiative described these Sufi events as part of its new \"Faith and Community Affairs\" initiative. Additionally, Rauf began to deliver lectures on Sufism at local churches again and included announcements for them in his monthly newsletters.\n\nMeanwhile, as El-Gamal and others at Soho Properties struggled with continued criticism of their endeavor and insufficient funds, they, too, emphasized Sufism and interreligious endeavors. By the summer of 2011, El-Gamal had recruited the family member of a 9\/11 victim to Park51's board of directors and had finally gained nonprofit status from the IRS, which allowed him to engage in a concerted fundraising campaign. Yet, El-Gamal acknowledged, Park51 still faced serious obstacles. Resigned to the possibility of reducing the project's scale, El-Gamal told a _New York Times_ reporter that if the local residents he met at planning meetings only wanted a four- or five-story community center, that is what it would be\u2014eventually. He admitted that it would take several years before they could even raise the millions of dollars needed to begin work on the buildings that housed the mosque and community space. In the meantime, Park51's promotional materials continued to describe a fifteen-story center. Among the proposed permanent exhibits was one on Sufism called _The Bridge of Light: Rumi, America's Best Selling Poet and a Voice for Interfaith_.\n\nIn September 2011, ten days after the tenth anniversary of the 9\/11 attacks\u2014a time when police officers prohibited foot or vehicular traffic on Park Place to all but residents, thus effectively preventing protests\u2014Park51 held its grand opening. The feeling in the building that night was at turns solemn and celebratory. Several of Rauf's and Fariha's dervishes\u2014ones no longer involved in the Park51 project\u2014attended the event and stood in quiet clusters around the cultural space and adjoining mosque, exchanging hugs and meaningful looks. Nothing seemed to have gone as planned with what even Park51's leaders now described as the \"interfaith community center,\" but they mused that perhaps this was the beginning they had all been waiting for.\n\nOne month later, Consolidated Edison\u2014the utility giant then leasing part of the building to Soho Properties\u2014filed suit against El-Gamal for nearly two million dollars in disputed back rent. Park51 continued to host cultural events, recreation classes, and Muslim holiday celebrations for the next few years, as New Yorkers and other Americans largely forgot about the center. But its directors could not overcome the funding difficulties they faced after their split with Rauf and Khan. In 2014, El-Gamal and the project's other leaders decided to give up their dream of building an Islamic community center once and for all. Soho Properties announced they would turn the property into a condominium tower and move the temporary mosque housed there into what they hoped would one day be a three-story museum at 51 Park Place. The following year, after hiring a new interim director to reinvigorate the project, El-Gamal began a fundraising tour with Imam Siraj Wahhaj. The new museum and sanctuary would be focused not on contemporary political or social issues but on enduring theological truths, its new director promised, while the mosque (housed in a hotel conference room, as of early 2016) would remain nondenominational: Sufi-friendly but not Sufi-focused, a space where Sunni and Shi'i Muslims could worship together. While it remains to be seen when the museum will materialize, Park51's leaders have pledged that community members\u2014Muslim and non-Muslim\u2014will be continually involved in the process.\n\nIt would be a mistake in the wake of the Islamic center controversy and subsequent anti-Muslim fervor to conclude that the efforts Rauf and Khan have made to increase acceptance of Muslims in the United States have failed. In fact, the tremendous support that Rauf, Khan, and Park51 garnered from elite Americans of all religious backgrounds during the controversy demonstrated the success of their efforts over the previous decade and the enduring resonance of narratives connecting Sufism, service, and free-market capitalism to moderation. Yet, as demonstrated by, among other things, a hostile rally outside a Phoenix Islamic center in May 2015, at which hundreds of protestors hoisted guns and signs that said \"Fuck Islam,\" even average, law-abiding Muslims face intimidation and violence in the United States and continue to live with the burden of proving their moderation\u2014often to audiences already determined not to listen.\n\nIn 2010, shaken by the controversy over the Islamic community center project and by the threats of violence that accompanied it, Daisy Khan asked rhetorically, \"if we can't build an interfaith community center, who can?\" The question remains a valid one. If Muslim Americans who participate in government programs to foster American interests, who describe authentic Islam in terms of dominant American values, and who are so deeply cautious in their public pronouncements and political activity as Rauf and Khan still attract negative attention and criticism, what hope can \"plain old American\" Muslims (as Khan called them in our 2008 interview) have that they will ever win acceptance rather than suspicion?\n\nSuch questions notwithstanding, other Muslim American leaders are increasingly placing their hopes in the power of community service to prove their moderation. They are not alone. Sikh Americans, often mistaken for Muslims, are also engaging in community service\u2014in part, for the same reason. The New York-based Sikh Coalition, for example, organized a national Day of Seva (selfless community service) in August 2013 to commemorate the fatal shooting at the Oak Creek Sikh Temple in Wisconsin the year before and invited other Americans to join the now-annual activity after a mentally ill Muslim American killed four marines at military facilities in Tennessee in 2015. But providing moderation and patriotism through community service is a complicated business. Not only do such efforts have the potential to exacerbate preexisting ethnic, economic, racial, and political differences among minoritized American populations\u2014as Park51 threatened to do\u2014they can also pose other potential problems for marginalized communities, including co-optation from the government, while still failing to gain them broader social acceptance. This is not to say, as I discuss in the Conclusion, that Muslim Americans or others should refrain from engaging in such service endeavors\u2014far from it. Rather, it is to draw attention again to the pressures on Muslims to demonstrate their moderation in particular ways and to note the risks, as well as the hoped-for rewards, of trying to meet these demands in practice.\nCONCLUSION\n\nCommunity Service and the Limits of Inclusion\n\nON SATURDAY, AUGUST 30, 2014, Muslim Americans from across the nation boarded buses bound for some of Detroit's blighted west-side neighborhoods as part of the \"Community Service Program\" of ISNA's fifty-first annual convention. Despite having participated in MuslimServe's effort to promote the September 11th National Day of Service among Muslim American communities in 2010, ISNA's directors had only recently begun to encourage community service in non-Muslim neighborhoods. Between 2013 and 2014, for example, ISNA Development Foundation officials and Founder's Committee members changed the description of their annual Community Service Recognition Luncheon. Whereas 2013 convention materials described it as an event where \"the nation's Muslim leaders, scholars, and government officials join to honor an individual dedicated to community service _within the Muslim community_ ,\" the Annual Convention Program description of the 2014 luncheon\u2014at which Jimmy Carter delivered the keynote\u2014omitted that last clause, turning service into a broader national mandate. Similarly, ICNA (a smaller national Muslim organization) decided in 2015 to have as the theme for its fortieth annual convention that year, \"Islam: the Connection between Purpose, Compassion, and Service.\"\n\nMuslim American community service endeavors have grown markedly since 2010 and have included activities ranging from holiday food drives to disaster relief; both ICNA and ASMA asked those on their email lists to donate money after Hurricane Sandy devastated parts of New York and New Jersey in 2012, for example. This increased popularity has occurred not only because Muslim Americans hope to serve their domestic communities and, in so doing, to prove their moderation, but also because other kinds of pious practice have been foreclosed. While many Muslims fulfilled religious duties by giving to international charitable organizations in the past, legislation passed in the wake of 9\/11 that criminalized providing \"material support to terrorists,\" knowingly or unknowingly, has made this much more difficult and even dangerous. Very few Muslim charitable organizations\u2014mostly those operating in the Hamas-controlled region of Gaza, where engaging Hamas is required to deliver aid\u2014have been convicted of aiding \"terrorist groups.\" Nevertheless, Muslim Americans fear giving to any charity that the US government may one day decide to criminalize and prosecute. Intentionally or not, the effect of this legislation has been to reduce the flow of money and voluntary labor going to Muslim communities overseas and to increase the amount of both going to aid in the United States.\n\nStill, performing more community service has not yet gained Muslim Americans increased acceptance. As of the summer of 2014, Americans regarded Muslims less warmly than any other religious group, according to polls conducted by the Pew Research Religion and Public Life Project. While Muslim Americans seem undaunted by such polls and may even be inspired to engage in more service because of them, other challenges have proved far more difficult to surmount. In February 2015, three young Muslim Americans devoted to community service were executed by a white neighbor in their North Carolina apartment complex. Speaking to a reporter after the funeral, one local woman expressed her disillusionment and fear. Muslims were told that they would be safe in America after 9\/11 if they had \"exemplary character\" and performed such community service acts, she said. The murders of such ideal American Muslim citizens had shredded these hopes for her. \"To see that it happened to them means it can happen to anyone.\"\n\nWhile such increased service has failed to guarantee the acceptance or even the safety of Muslim Americans, it has indisputably helped further the goals of the most recent Republican and Democratic presidential administrations: shrinking the size of government and effecting economic recovery, respectively. This is particularly the case as the US government has cut the amount of federal funding going to service activities since 2011, thereby placing more of the financial burden of caring for disenfranchised and dispossessed citizens directly on other citizens. Muslim Americans trying to follow Catholics, Jews, and other religious minorities in gaining acceptance through service may, like those other groups, find themselves faced with a struggle\u2014financial, physical, ethical, and political\u2014to avoid co-optation from the government as they try to meet the needs of their own communities and make their case for national inclusion.\n\nThe greatest obstacle to the broad acceptance of Muslims in the United States is not their supposed lack of moderation. Rather, it is the continued specter of Islamic terrorism stoked by politicians such as the contenders for the 2016 Republican presidential nomination and the attention given to militant groups such as ISIS\u2014the Islamic State in Iraq and Syria, also known as ISIL or DAESH\u2014an organization whose leaders met while imprisoned in a US detention center in Iraq and who formed a violent militia bent on ridding the area of US influence and establishing what they believe is a proper Islamic nation. ISIS has engaged in a concerted, and occasionally successful, effort to inspire disaffected young Muslim Americans to support them by joining their fight in the Middle East or, short of that, by launching terror attacks on US soil. While the prospect of such attacks unnerves most Americans\u2014none more than Muslim parents, who worry about predatory attempts to recruit their children online\u2014New America, a Washington research center, confirmed in 2015 what Department of Homeland Security analysts found in 2009: that the greatest threat of violence in the United States comes not from radicalized Muslims but from the white supremacists and right-wing, antigovernment domestic militias who have killed nearly twice as many people as would-be jihadists since 9\/11.\n\nTerrorism has never been the province of Muslims alone. Since the Iranian hostage crisis of the late 1970s, however, the American media has focused much more on so-called Islamic terrorism than on the acts of terrorism, foreign and domestic, sponsored or committed by white Americans. As long as media outlets and politicians gain viewers and votes by stoking fears of Islam, Muslim Americans will continue to face obstacles to acceptance, regardless of how much community service they perform.\n\nAs one might imagine, the focus on Muslim-led violence in this country not only poses problems for the acceptance of Muslims, it also distracts from other, more common kinds of violence in the United States. Take the national debate in 2015, for example, over whether or not to describe the actions of gunman Dylann Roof, who murdered worshippers in a South Carolina church with the express purpose of starting \"a race war,\" as terrorism. And recall Timothy McVeigh's 1995 bombing of the Alfred P. Murrah Federal Building in Oklahoma City, which was correctly labeled terrorism as soon as it occurred\u2014not coincidentally, because many assumed the act had been committed by Muslims rather than by a white American\u2014but is not often remembered as something inspired by writings of the nation's oldest terrorist population: white supremacists. More important than the way our national preoccupation with so-called Islamic terrorism diverts attention (and law enforcement resources) from such grandiose gestures of racial hatred and violence, however, is the way it obscures the systemic racism, including in the form of state violence, that daily impacts the lives of black and brown Americans, as well as the increasingly common violence directed against Muslim Americans and those taken to be Muslim.\n\nFor Muslim Americans, confronting the specter of Islamic terrorism by proving that such violence does not represent the majority of Muslims\u2014American or otherwise\u2014is a noble task. Doing so without acknowledging the larger political dynamics to which this specter contributes\u2014namely, the suggestion by many politicians that the violence Americans should worry about is committed by foreigners, non-whites, or non-Christians, rather than what is in fact more common: violence committed against racial and religious minorities and against women\u2014will not ultimately serve any of these populations and will only perpetuate the prejudice, individual and systemic, faced by many in the United States. The same can be said about engaging in community service.\n\nThe answer for those who want to prove their national loyalties through community service endeavors is not to refrain from assisting one's neighbors or bettering one's local or national community. Many devout Muslims\u2014such as those who created the MuslimServe website and who firmly insist, with reference to Qur'anic passages, that God has called them to perform such acts of benevolence\u2014would be reluctant to stop engaging in acts of service even if these activities came at a more direct cost to them. But Muslim Americans who do participate in such service efforts, particularly those involved in federal or other government programs, should recognize that, in addition to proving that Muslims are as patriotic and as concerned about the common good as anyone else, such actions can have other, less desirable ramifications.\n\nIf community service renders some Muslims\u2014those who are most economically able to engage in it\u2014more acceptable and more American than others, that will replicate power dynamics at work decades earlier, when marginalized Catholics and Jews gained provisional acceptance by taking up the physical (and often financial) burden of providing community services to those disenfranchised and dispossessed by America's unequal laws and institutions. Replicating these dynamics would mean confirming to many dominant Americans their preexisting ideas about the inherent pathology or inferiority of those whose skins seem darker, whose traditions seem different, or whose economic situations seem like natural consequences of their ostensible abilities rather than as the result, centuries in the making, of racist policies and individual prejudice. What Muslim Americans can do, instead\u2014and what they are increasingly doing\u2014is to partner with people whose races, ethnicities, and socio-economic backgrounds are different from than their own, and to deliberate jointly (which will sometimes mean awkwardly and painfully) over the best ways to improve their communities, both through service and through demands for justice for those neglected or oppressed. Doing so involves not just acts of benevolence or charity, but also acts of solidarity stemming from the recognition that the fates of all racial, religious, and socio-economic groups\u2014particularly marginalized ones\u2014in the United States are fundamentally connected.\nNOTES\n\nIntroduction\n\n. Quoted in \"Abraham's Children and the Imperative of Peacemaking,\" _PeaceWatch_ 9, no. 3 (April 2003): 7.\n\n. Feisal Abdul Rauf, _What's Right with Islam: A New Vision for Muslims and the West_ (San Francisco: HarperSanFrancisco, 2004).\n\n. Ibid., 5\u20136, 178. For the sake of clarity, I transliterate \"shari'ah\" in the same manner as Rauf (though he does not always include the apostrophe); a more common transliteration is \"shari'a.\"\n\n. Peter King made these comments on Fox News on August 3, 2010. See Howard LaFranchi, \"Is Ground Zero Mosque Imam Best Choice for Diplomatic Mission to Mideast?,\" _Christian Science Monitor_ online, August 11, 2010, http:\/\/www.csmonitor.com\/USA\/Foreign-Policy _\/_ 2010 _\/_ 0811 _\/_ Is-ground-zero-mosque-imam-best-choice-for-diplomatic-mission-to-Mideast.\n\n. In July 2010, Gingrich posted said statement on his website, accessed July 21, 2010, http:\/\/www.newt.org\/newt-direct\/newt-gingrich-statement-proposed-mosqueislamic-community-center-near-ground-zero (statement since removed).\n\n. Newt Gingrich on _Fox and Friends_ , August 16, 2010.\n\n. Stewart discussed the issue on several 2010 episodes of _The Daily Show_ , including on August 10, 2010 (\"Municipal Land-Use Hearing Update\"), August 16, 2010 (\"Mosque-Erade\"), August 19, 2010 (\"Extreme Makeover Homeland Edition\"), August 23, 2010 (\"The Parent Company Trap\"), and September 8, 2010 (\"Weekend at Burnies\").\n\n. A brief statement on the \"immigrant religious experience\" can be found in Feisal Abdul Rauf, \"Forceful Voice of Reason,\" in _Voices of American Muslims: 23 Profiles_ , edited by Linda Cateura (New York: Hippocrene Books, 2005), 101.\n\n. On how some of these ethnic (later, \"white\") immigrants distanced themselves from racialized (\"black\") populations, see David R. Roediger, _The Wages of Whiteness: Race and the Making of the American Working Class_ (New York: Verso, 1991); Eric L. Goldstein, _The Price of Whiteness: Jews, Race, and American Identity_ (Princeton: Princeton University Press, 2007); Nathaniel Deutsch, _Inventing America's Worst Family: Eugenics, Islam, and the Fall and Rise of the Tribe of Ishmael_ (Berkeley: University of California Press, 2009); and Sarah M. A. Gualtieri, _Between Arab and White: Raceand Ethnicity in the Early Syrian American Diaspora_ (Berkeley: University of California Press, 2009).\n\n. See Ira Katznelson, _When Affirmative Action was White: An Untold History of Racial Inequality in Twentieth-Century America_ (New York: W. W. Norton, 2013); Matthew Frye Jacobson, _Roots Too: White Ethnic Revival in Post-Civil Rights America_ (Cambridge: Harvard University Press, 2008); and Joe Feagin, _The White Racial Frame: Centuries of Racial Framing and Counter-Framing_ (New York: Routledge, 2013) and _How Blacks Built America: Labor, Culture, Freedom, and Democracy_ (New York: Routledge, 2015).\n\n. For American ideas about the opposition between Sufism and fundamentalism, see Rosemary R. Corbett, \"Islamic 'Fundamentalism': the Mission Creep of an American Religious Metaphor,\" _Journal of the American Academy of Religion_ 83, no. 4 (December 2015): 977\u20131004.\n\n. I discuss these dynamics in detail in Rosemary R. Hicks, \"Comparative Religion and the Cold War Transformation of Indo-Persian Mysticism into Liberal Islamic Modernity,\" in _The Politics of Secularism and Religion-Making_ , edited by Markus Dressler and Arvind Mandair (New York: Oxford University Press, 2011), 141\u201369.\n\n. Feisal Abdul Rauf, _Islam: A Sacred Law: What Every Muslim Should Know About the Shari'ah_ (Brattleboro, VT: Qibla Books, 2000), 4.\n\n. \"Interview: Imam Feisal Abdul Rauf,\" _Frontline_ online, March 2002 (accessed March 8, 2004), <http:\/\/www.pbs.org\/wgbh\/pages\/frontline\/shows\/muslims\/interviews\/feisal.html> (emphasis added).\n\n. Ibid.\n\n. \"Cordoba Bread Fest: Children of Abraham Break Bread Together\" ASMA Society, (accessed December 17, 2004), http:\/\/www.asmasociety.org\/calendar\/pastevents.html. The website misidentifies the event as having occurred in 2002.\n\n. Ibid.\n\n. In a 2004 application for tax-exempt status, Cordoba's lawyer described the Shariah Project as designed to \"demonstrate to the world that Islamic holy law is compatible with a pluralistic and free democratic society and that peace and tolerance are authentic expressions of Islamic principles\"; see Timothy McFlynn, Public Counsel of the Rockies, \"Application for Recognition of Exemption,\" filed with Internal Revenue Service under the Department of the Treasury, Cincinnati, Ohio, on June 14, 2004. Rauf later renamed the program, calling it the \"Shariah Index Project\" after 2008, and in 2012 elaborated on what it had become: a project that \"would help determine the proper balance in the Muslim world (and elsewhere) between institutions of political power and authority, on the one hand, and institutions of religious power and authority, on the other\u2014the Muslim equivalent of the religion-state relationship\"; see Faisal Abdul Rauf, _Moving the Mountain: A New Vision of Islam in America_ (New York: Free Press, 2012), 17.\n\nThe Cordoba Initiative's 2004 application also included a series of \"Dialogues . . . between American and Middle Eastern religious as well as secular leaders\" as part of their primary activities. When pressed by the IRS for more specific information on these programs, Cordoba's lawyer responded with a letter that listed \"economic development\" and \"the challenge of adapting principles of democracy and democratic capitalism to specific cultures\" among the issues to be discussed; see Timothy McFlynn, Public Counsel of the Rockies, \"Re: The Cordoba Initiative (EIN 41\u20132140798),\" filed with Internal Revenue Service under the Department of the Treasury, Cincinnati, Ohio, September 14, 2004, 2\u20133.\n\n. On Riverside Church, see Peter J. Paris, ed., _The History of the Riverside Church in the City of New York_ (New York: New York University Press, 2004).\n\n. For the production of \"Judeo-Christian\" identity, see J. T. Todd, \"The Temple of Religion and the Politics of Religious Pluralism: Judeo-Christian America at the 1939\u20131940 New York World's Fair,\" in _After Pluralism: Reimagining Religious Engagement_ , edited by Courtney Bender and Pamela E. Klassen, (New York: Columbia University Press, 2010), 201\u201322; Deborah Dash Moore, \"Jewish GIs and the Creation of the Judeo-Christian Tradition,\" _Religion and American Culture_ 8, no. 1 (December 1998): 31\u201353; Mark Silk, \"Notes on the Judeo-Christian Tradition in America,\" _American Quarterly_ 36, no. 1 (December 1984), 65\u201385; and William Hutchison, _Religious Pluralism in America: The Contentious History of a Founding Ideal_ (New Haven: Yale University Press, 2004), 196\u2013218.\n\n. I have changed the names of some interlocutors in order to protect their privacy; such individuals are identified only by first name pseudonyms.\n\nDuring the first decade after 9\/11, a common question was whether Muslims who live in the United States are more \"Muslim\" than \"American.\" Because I found my interlocutors to be like other US residents and citizens in having multiple affiliations and changing ways of expressing their relations to the people and places around them\u2014as Dean illustrated by defining himself as Muslim _and_ Catholic\u2014and because I never asked about citizenship status, I do not define Muslim Americanness according to any one particular standard. Rather, I focus on how my interlocutors talked about being Muslim and being American, and I use the terms \"Muslim American\" and \"American Muslim\" interchangeably. For more on the overlapping attachments of Muslim Americans, see Zareena Grewal, _Islam is a Foreign Country: American Muslims and the Global Crisis of Authority_ (New York: New York University Press, 2013).\n\n. For more on the Jerrahi practice of _dhikr_ and on the differences between Rauf's group and other Sufis at the Masjid al-Farah, see Rosemary R. Corbett, \"Dhikr,\" in _Islamic Religious Practice in the United States_ , edited by Edward E. Curtis, IV (New York: New York University Press, forthcoming).\n\n. Michael M. Grynbaum, \"Daisy Khan, An Eloquent Face of Islam,\" _New York Times_ , November 12, 2010.\n\n. These scholars have demonstrated that areas of American social life commonly seen in the twenty-first century as being free of religion often accommodate the Protestant practices around which American laws and institutions developed more than\u2014or to the exclusion of\u2014other practices and traditions. As Janet Jakobsen and Ann Pellegrini have described, American laws, institutions, and culture exert pressures on other traditions to resemble a specific kind of \"market-Reformed Protestantism\" in which freedom _from_ religion is frequently freedom _for_ the market; see \"World Secularisms at the Millennium: Introduction,\" _Social Text_ 18, no. 3 64 (2000): 1\u201327. Like Jakobsen and Pellegrini, my analysis of religion and secularism is informed greatly by Talal Asad, _Genealogies of Religion_ (Baltimore: Johns Hopkins University Press, 1993) and _Formations of the Secular: Christianity, Islam, Modernity_ (Stanford: Stanford University Press, 2003).\n\nFor more on how American legislation developed in tension with changing notions about the appropriate public role of Protestantism, see Hutchison, _Religious Pluralism in America_ ; John F. Wilson and Donald L. Drakeman, eds., _Church and State in American History: Key Documents, Decisions, and Commentary from the Past Three Centuries_ (Boulder: Westview Press, 2003); Winnifred Fallers Sullivan, \"Religion Naturalized: The New Establishment,\" in _After Pluralism, edited by_ Courtney Bender and Pamela Klassen (New York: Columbia University Press, 2010), 82\u201397 and _The Impossibility of Religious Freedom_ (Princeton: Princeton University Press, 2005); Tracy Fessenden, _Culture and Redemption: Religion, the Secular, and American Literature_ (Princeton: Princeton University Press, 2007); John Lardas Modern, _Secularism in Antebellum America_ (Chicago: University of Chicago Press, 2011); and Rosemary R. Hicks, \"Between Lived and the Law: Power, Empire, and Expansion in Studies of North American Religions,\" _Religion_ 42, no. 3 (2012): 409\u201324.\n\n. Most of those important to my analysis argue within the genealogical tradition fostered by Michel Foucault, as it provides a robust means of accounting for how knowledge is related to power; see, especially, _The Archaeology of Knowledge_ (New York: Pantheon, 1972) and _Discipline and Punish: The Birth of the Prison_ (New York: Pantheon, 1977). With regards to creating knowledge about Islam, specifically, I rely also on attention paid by Talal Asad, Richard Bulliet, and Edward Said to literary and dramatic devices in the disciplines of history and anthropology. In _Orientalism_ (New York: Vintage, [1979] 2003), Edward Said utilizes the work of Foucault and critical historians to draw out the ubiquitous use of literary and dramatic devices in philological, historical, and political narratives and in the \"Oriental Studies\" and \"Area Studies\" programs that depended on them. Asad also demonstrates anthropologists' tendencies to use such dramatic staging and literary devices in their writings about Islamic cultures and societies in _The Idea of An Anthropology of Islam_ (Washington, DC: Georgetown Center for Contemporary Arab Studies, 1986). Significantly, as Bulliet and Melani McAlister have demonstrated, Said missed important differences between European and American approaches to Islam, especially in the ways American Cold War projects involved cultivating Arab allies and constructing post-Orientalist images. See Bulliet, _The Case for Islamo-Christian Civilization_ (New York: Columbia University Press, 2004), 96\u201398 and Melani McAlister, _Epic Encounters: Culture, Media, and U.S. Interests in the Middle East Since 1945_ (Berkeley: University of California Press, 2005), 10\u201311.\n\n. These ways of living and experiencing tradition are not necessarily conscious or intentional. Rather, they are part of the ongoing process of human interaction through which people learn stock ways of talking about and understanding life, and employ and enact such narratives in changing circumstances, thus altering them while making sense of their varying experiences. Works from this perspective that influenced my approach include Courtney Bender, _Heaven's Kitchen: Living Religion at God's Love We Deliver_ (Chicago: University of Chicago Press, 2003); Henry Goldschmidt, _Race and Religion: the Chosen People of Crown Heights_ (New Brunswick: Rutgers University Press, 2006); Karen McCarthy Brown, _Mama Lola: A Voudou Priestess in Brooklyn_ (Berkeley: University of California Press, 1991); Robert A. Orsi, _Thank You, St. Jude: Women's Devotion to the Patron Saint of Hopeless Causes_ (New Haven: Yale University Press, 1996); and Dorothy C. Holland, _Identity and Agency in Cultural Worlds_ (Cambridge: Harvard University Press, 1998).\n\n. McAlister, _Epic Encounters_ ; Timothy Marr, _The Cultural Roots of American Islamicism_ (New York: Cambridge University Press, 2006); Jasbir Puar, _Terrorist Assemblages: Homonationalism in Queer Times_ (Durham: Duke University Press, 2007); and Moustafa Bayoumi, _This Muslim American Life: Dispatches from the War on Terror_ (New York: New York University Press, 2015). My research has also been shaped by Mahmood Mamdani, _Good Muslim, Bad Muslim: America, the Cold War, and the Roots of Terror_ (New York: Pantheon, 2004).\n\n. Two important analyses are Saba Mahmood, \"Secularism, Hermeneutics, and Empire: The Politics of Islamic Reformation,\" _Public Culture_ 18, no. 2 (2006): 323\u201347 and Elizabeth Shakman Hurd, _Beyond Religious Freedom: the New Global Politics of Religion_ (Princeton: Princeton University Press, 2015). I respond to Mahmood's criticism in Chapter 1.\n\n. On the NYPD surveillance program, which\u2014like the NSA surveillance\u2014failed to uncover or prosecute any would-be terrorists, see Matt Apuzzo and Adam Goldman, _Enemies Within: Inside the NYPD's Secret Spying Unit and Bin Laden's Final Plot Against America_ (New York: Simon and Schuster, 2013). On the US government's other actions, which have since become the subject of lawsuits, see Larry Neumeister, \"Lawsuit Seeks to Memorialize a Less Heroic Side of 9\/11,\" Associated Press, June 21, 2015, <http:\/\/bigstory.ap.org\/article\/34bf188788534c73a602c0013fbcef4a\/lawsuit-seeks-memorialize-less-heroic-side-911>; and Jenifer Fenton, \"Does FBI Use No-Fly List to Pressure Muslims to Become Informants?,\" _Al-Jazeera America_ online, June 11, 2015, <http:\/\/america.aljazeera.com\/articles\/2015\/6\/11\/does-fbi-use-no-fly-list-to-pressure-muslims-to-become-informants.html>.\n\n. Sam Stein, \"'Ground Zero Mosque' Imam Helped FBI With Counterterrorism Efforts,\" _Huffington Post_ , August 17, 2010, <http:\/\/www.huffingtonpost.com\/2010\/08\/17\/ground-zero-imam-helped-f_n_685071.html>. For a broader overview of issues facing Muslim Americans and Arab Americans, see Louise A. Cainkar, _Homeland Insecurity: the Arab American and Muslim American Experience after 9\/11_ (New York: Russell Sage Foundation, 2009).\n\n. Feisal Abdul Rauf, \"In God We Trust: The Prospects for the Future of Islam and the West are Positive,\" in _Debating Moderate Islam: The Geopolitics of Islam and the West_ , edited by M. A. Muqtedar Khan (Salt Lake City: University of Utah Press, 2007), 95.\n\n. Ibid., 98\u201399.\n\n. Ibid., 97.\n\n. Mahmood Mamdani, \"Culture Talk: Six Debates that Shape the Discourse on 'Good' Muslims,\" in _Debating Moderate Islam: The Geopolitics of Islam and the West_ , edited by M. A. Muqtedar Khan (Salt Lake City: University of Utah Press, 2007), 114\u201323.\n\nChapter 1\n\n. See Moore, \"Jewish GIs\"; Silk, \"Notes on the Judeo-Christian Tradition\"; and Todd, \"Temple of Religion.\" On how conservative evangelicals adopted the term to express opposition to secularism and support of Israel, see Kevin Schultz, _Tri-Faith America: How Catholics and Jews Held Postwar America to Its Protestant Promise_ (New York: Oxford University Press, 2011), 201\u20133.\n\n. William Inboden, _Religion and American Foreign Policy, 1945\u20131960: The Soul of Containment_ (New York: Cambridge University Press, 2010).\n\n. Quoted in James C. McKinley, Jr., \"Oklahoma Surprise: Islam as an Election Issue,\" _New York Times_ , November 15, 2010 (emphasis added).\n\n. Quoted in Edward E. Curtis, IV, _Muslims in America: A Short History_ (New York: Oxford University Press, 2009), 106\u20137.\n\n. Luke Johnson, \"Rick Santorum: Equality Comes from 'God of Abraham, Isaac, and Jacob,' Not Islam,\" _Huffington Post_ , January 21, 2012, <http:\/\/www.huffingtonpost.com\/2012\/01\/21\/rick-santorum-equality-islam-religion-south-carolina_n_1220767.html>.\n\n. Newt Gingrich, _Religion and Politics: The Legitimate Role_ (Heritage Foundation, Washington, DC, 1994), 1.\n\n. Kevin Knoblock's _America At Risk: The War with No Name_ (Washington, DC: Citizens United Productions, 2010) is distributed by Gingrich Productions. For Gingrich's books, see Newt Gingrich and Joe DeSantis, _To Save America: Stopping Obama's Secular-Socialist Machine_ (Washington, DC: Regnery, 2010) and Newt Gingrich and Vince Haley, _A Nation Like No Other: Why American Exceptionalism Matters_ (Washington, DC: Regnery, 2011). DeSantis was communications director at Gingrich Communications when he co-authored _To Save America_ , while Haley was policy director of Gingrich's 2011\u20132012 presidential campaign when he cowrote their book. Because these coauthors worked as Gingrich's official spokespersons, I refer solely to Gingrich as author in the text.\n\n. For example, Gingrich repeatedly proposed the death penalty for drug-related offenses (his \"Drug Importer Death Penalty Act\" failed in committee in 1996 and 1997), while Rauf opposes capital punishment; see Rauf, _Moving the Mountain_ , 68\u201371.\n\n. See David Harvey, _A Brief History of Neoliberalism_ (New York: Oxford University Press, 2005) and Wendy Brown, \"Neo-liberalism and the End of Liberal Democracy,\" in _Edgework: Critical Essays on Knowledge and Politics_ (Princeton: Princeton University Press, 2005), 37\u201359.\n\n. Will Herberg, _Protestant-Catholic-Jew: An Essay in American Religious Sociology_ (Garden City, NY: Anchor Books, 1960).\n\n. Rauf, _What's Right with Islam_ , 73\u201376.\n\n. Ibid., 80.\n\n. Ibid., 82\u201383.\n\n. Ibid., 16.\n\n. Ibid., 82\u201383.\n\n. Ibid., 83\u201384. For Rauf, the Abrahamic ethic lived out in the community of the Prophet and his first four successors (according to Sunni tradition) evidenced \"the conceptual seeds of democratic governance.\" While these years were the most exemplary of Muslim governance, with a few exceptions, \"democracy as we know it today did not truly take root and flower until a few millennia later, with the advent of the American Revolution\" (80).\n\n. Ibid.\n\n. Ibid., 85.\n\n. Ibid., 32, 248.\n\n. Ibid., 86\u201388.\n\n. Ibid., 109\u201310.\n\n. Ibid., 111.\n\n. Ibid., 85.\n\n. Ibid., 6.\n\n. Ibid.,153\u20134, 158\u20139, 205.\n\n. Ibid., 6\u20138, 125.\n\n. Ibid., 275.\n\n. Roger Cohen, \"Shariah at the Kumback Caf\u00e9,\" _New York Times_ , December 6, 2010.\n\n. Incidents included a mosque site in Tennessee vandalized by arson, a New York cab driver stabbed, and a mosque in Florida bombed; see Paul Vitello, \"Church Rejects Sale of Building for a Mosque,\" _New York Times_ , July 22, 2010; \"FBI Finds Pipe Bomb Used in Blast at Fla. Mosque,\" AOL News, May 12, 2010, <http:\/\/www.aolnews.com\/crime\/article\/fbi-finds-pipe-bomb-used-in-blast-at-fla-mosque\/19475001>; Lucas L. Johnson, II, and Travis Loller, \"Tennessee Mosque Site Fire was Arson, Police Say,\" _Huffington Post_ , August 30, 2010, http:\/\/www.huffingtonpost.com\/ _z_ 2010 _\/_ 08 _\/_ 30\/murfreesboro-mosque-fire-arson-accelerant_n_699696.html; John Eligon, \"Hate Crime Charges in Stabbing of a Cabdriver,\" _New York Times_ , August 30, 2010.\n\n. Plans to build mosques and community centers in locations across the country were challenged while the Park51 debate unfolded; Phil Willon, \"Planned Temecula Valley Mosque Draws Opposition,\" _Los Angeles Times_ , July 18, 2010.\n\n. On Gingrich's 2010 film, _America at Risk_ , see Scott Shane, \"In Islamic Law, Gingrich Sees Moral Threat to the U.S.,\" _New York Times_ , December 21, 2011.\n\n. Roger Cohen, \"Shariah at the Kumback Caf\u00e9.\"\n\n. On Gingrich and other Republican presidential candidates' pledges to outlaw shari'ah, see Andrea Elliot, \"The Man Behind the Anti-Shariah Movement,\" _New York Times_ , July 30, 2011.\n\n. In addition to differences over Israel-Palestine, Rauf argues that the United States missed an opportunity to build alliances when it demonized Ayatollah Khomeini and sheltered the shah ( _What's Right with Islam_ , 160). In contrast, Gingrich has advocated isolating Iran's \"pro-terrorist, anti-American regime\" since the 1990s (Gingrich and Haley, _Nation Like No Other_ , 168).\n\n. Gingrich and Haley, _Nation Like No Other_ , 6.\n\n. Ibid., 12\u201313.\n\n. Forrest Church, _The American Creed: A Spiritual and Patriotic Primer, or, A Biography of the Declaration of Independence_ (New York: Saint Martin's, 2002).\n\n. Quoted in ibid., xii.\n\n. Quoted in ibid., 1.\n\n. Ibid., 22.\n\n. Ibid., 32\u201333.\n\n. Ibid., 82 (emphasis added).\n\n. Ibid., 24.\n\n. Rauf, _What's Right with Islam_ , 2, 215; Church, _American Creed_ , 134; Gingrich and Haley, _Nation Like No Other_ , 25, 46, 101, 117\u201320.\n\n. Church, _American Creed_ , 31.\n\n. Gingrich and Haley, _Nation Like No Other_ , 22\u201323, 82; Rauf, _What's Right with Islam_ , 211\u201315.\n\n. Church, _American Creed_ , 137.\n\n. Ibid., 124.\n\n. Ibid., 113.\n\n. Ibid.\n\n. Ibid., 114.\n\n. Gingrich and Haley, _Nation Like No Other_ , 57.\n\n. Church, _American Creed_ , 94\u201397, 121\u201322; Rauf, _What's Right with Islam_ , 230\u2013243.\n\n. On the \"Islamic bloc,\" see Gingrich and DeSantis, _To Save America_ , 51. Gingrich discusses the UN and Israel far more in his 2010 book than in his 2011 narrative, which focuses mainly on what he depicts as the grievous errors by the political left and President Obama of cowering before \"radical Islamist ideology\" and \"scaling back America's role in the world\" (173).\n\n. Gingrich and Haley, _Nation Like No Other_ , 100; Church, _American Creed_ , 21\u201322.\n\n. Gingrich and Haley, _Nation Like No Other_ , 26\u201333, 100\u20131.\n\n. Michael Novak, _On Two Wings: Humble Faith and Common Sense at the American Founding_ (San Francisco: Encounter, 2002), 8\u201313. Recognizing that many of the founders were anti-Catholic, Novak traces Locke's natural law theories to medieval systematic theologian Thomas Aquinas (83\u201394, 157\u201360).\n\n. Michael Novak, Introduction to _Family, Society, Politics_ , vol. 5 of _G. K. Chesterton, Collected Works_ (San Francisco: Ignatius, 1987), 15\u201333.\n\n. Harvey identifies the American Enterprise Institute as among the think tanks established in the 1970s with corporate funding to promote neoliberal policies; see _Brief History_ , 43\u201344. Although the term \"neoliberal\" has rarely been used positively in recent years, Michael Novak did employ it to describe colleagues like Daniel Patrick Moynihan who were also in favor of democratic capitalism; see _The Spirit of Democratic Capitalism_ (Lanham, MD: Madison, 1981] 1991), 111. Novak's recent reluctance to use the term is likely due to Pope John Paul II's condemnation of neoliberalism in his 1999 apostolic exhortation, \"Ecclesia in America,\" Vatican, [http:\/\/www.vatican.va\/holy_father\/john_paul_ii\/apost_exhortations\/documents\/hf_jp-ii_exh ___ 22011999_ecclesia-in-america_en.html.\n\n. Michael Novak, _Three in One: Essays on Democratic Capitalism, 1976\u20132000_ , edited by Edward Wayne Younkins (Lanham, MD: Rowman & Littlefield, 2001), 63; see also _Spirit_ , 39\u201340.\n\n. Gingrich and DeSantis, _To Save America_ , 328\u201330.\n\n. Gingrich and Haley, _Nation Like No Other_ , 100, 118, 240. Importantly, Gingrich and Haley also included material in _A Nation Like No Other_ that resembles Michael Novak's discussion of Locke in _Toward a Theology of the Corporation, rev. ed_. (Washington, DC: American Enterprise Institute for Public Policy Research, 1981). The publication of _the latter volume_ was made possible with funding from the SmithKline Corporation. Novak acknowledges that his research on this project preceded and influenced his later writings on \"democratic capitalism\" ( _Toward a Theology_ , 1).\n\n. See especially Novak, _Spirit_ , 351\u201354. There, Novak explains that no free society should be controlled by religion, but that Christian values of charity and loving one's neighbor will flourish in the context of religious and market freedom.\n\n. Rauf, _What's Right with Islam_ , 248.\n\n. Novak, _On Two Wings_ , 90 (emphasis added).\n\n. See Irving Kristol, \"The Disaffection from Capitalism,\" in _Capitalism and Socialism: A Theological Inquiry_ , edited by Irving Kristol (Washington, DC: American Enterprise Institute, 1979), 27 and Seymour Martin Lipset, \"American Exceptionalism\" in _Capitalism and Socialism_ , 34\u201352.\n\n. On the white ethnic revival, see Frye Jacobson, _Roots Too_ , and Kambiz GhaneaBassiri, _A History of Islam in America: From the New World to the New World Order_ (New York: Cambridge University Press: 2010), 272\u201381.\n\n. GhaneaBassiri, _History of Islam_ , 281.\n\n. GhaneaBassiri, _History of Islam_ , 281. See also Frye Jacobson, _Roots Too_ , 180\u2013233.\n\n. See Moore, \"Jewish GIs\"; Silk, \"Notes on the Judeo-Christian Tradition\"; Todd, \"Temple of Religion\"; Schultz, _Tri-Faith America_ ; and Wilson and Drakeman, _Church and State_.\n\n. Katznelson, _When Affirmative Action was White_.\n\n. On how some of these ethnic (later, \"white\") immigrants distanced themselves from other racialized populations, see the works listed in the Introduction, note 9.\n\n. While Timothy Marr argues that Locke's influence was rivaled by Montesquieu's, Bernard Bailyn maintains that Montesquieu's influence was clearly greater; see Marr, _Cultural Roots_ , 23, and Bailyn, _The Ideological Origins of the American Revolution, Enlarged Edition_ (Cambridge, MA: Belknap Press, 1992), 344\u201345.\n\n. A more detailed account of changes in American Lockean philosophy can be found in Paul Gottfried, _Leo Strauss and the Conservative Movement in America_ (New York: Cambridge University Press, 2012) and Lee Ward, _John Locke and Modern Life_ (New York: Cambridge University Press, 2010), as well as in Michael W. Doyle, \"Liberalism and the End of the Cold War,\" in _International Relations Theory and the End of the Cold War_ , edited by Richard Ned Lebow and Thomas Risse-Kappen (New York: Columbia University Press, 1995), 85\u2013108.\n\n. The quote is Kenneth B. McIntyre's description of political scientist Leo Strauss' work. As McIntyre notes, \"Strauss's work is almost universally dismissed by philosophers and historians, yet he has attracted a following amongst political theorists . . . and neoconservative political activists\"; see McIntyre, \"The Right's False Prophet,\" _AmericanConservative_ online, May 9, 2012, <http:\/\/www.theamericanconservative.com\/articles\/the-rights-false-prophet\/>.\n\nIronically, Strauss opposed Locke's liberalism, which, as he saw it, deconstructed the necessary moral and authoritarian supports for democracy by separating church and state. This depiction of Locke was part of Strauss' diagnosis of what he considered to be America's mid-century political malaise\u2014a diagnosis that was largely adopted by Novak's AEI colleague, Irving Kristol, who severely criticized Johnson's Great Society programs, some aspects of the New Deal, and what he and others considered to be \"the interventionist excesses of a so-called 'liberal elite'\"; see Harvey, _Brief History_ , 50. For Strauss' influence on Kristol, see Shadia B. Drury's _Leo Strauss and the American Right_ (New York: St. Martin's, 1997), 137\u201378.\n\n. Kristol extolled the \"Puritan Temper\" and \"Protestant Ethic\" earlier than did many of his colleagues, publishing an analysis of it along with his coauthor, Daniel Bell in _Capitalism Today_ (New York: Basic, 1971), 49\u201353.\n\n. Muhammad Abdul-Rauf's progress narrative can be found in _A Brief History of Islam: With Special Reference to Malaya_ (Kuala Lumpur: Oxford University Press, 1964).\n\n. See, for example, Isma'il al-Faruqi, Introduction to _Kitab al-Tawhid: Essay on the Unicity of Allah, or, What is Due Allah from His Creatures_ , by Muhammad Ibn 'Abdal Wahhab, translated by Isma'il Al-Faruqi (Beirut: IIFSO, 1979), as well as al-Faruqi's _Taw \u1e25id: Its Implications for Thought and Life_ (Herndon, VA: International Institute of Islamic Thought, 1982).\n\n. James Carroll, \"The Gift of Reconciliation,\" _New York Times Magazine_ , December 4, 2010.\n\n. See Isma'il al-Faruqi, ed., _Trialogue of the Abrahamic Faith Traditions: Papers Presented to the Islamic Studies Group of the American Academy of Religion_ , 3rd ed. (Alexandria, VA: Al-Saadawi Publications, 1991).\n\n. Muhammad Abdul-Rauf, \"The Islamic Doctrine of Economics and Contemporary Economic Thought,\" in _Capitalism and Socialism: A Theological Inquiry_ , edited by Michael Novak (Washington, DC: American Enterprise Institute for Public Policy Research, 1979), 129\u201352; Muhammad Abdul-Rauf, _A Muslim's Reflections on Democratic Capitalism_ (Washington, DC: American Enterprise Institute for Public Policy Research, 1984).\n\n. Rauf, _What's Right with Islam_ , 286. On crediting his father for \"pass[ing] the baton to me,\" see Carroll, \"Gift.\"\n\n. Rauf, _What's Right with Islam_ , 207\u20138.\n\n. Ibid., 65.\n\n. Ibid., 70.\n\n. Ibid., 210.\n\n. Ibid.\n\n. Novak, _Toward a Theology_ , 7.\n\n. Novak, _Spirit_ , 47.\n\n. Rauf, _What's Right with Islam_ , 254; on \"economic barriers,\" see 234\u201335. Both Novak and Rauf also accept the need for basic social safety nets born of the New Deal such as Social Security and Disability Insurance, as well as welfare programs, although with some reservations; see Rauf, _What's Right with Islam_ , 232\u201333. According to Novak in the 1970s, \"democratic capitalism, however grudgingly, adopts social welfare programs, often initially sponsored by socialists\"; see _Spirit_ , 218. But Novak followed that grudging acceptance with racialized imagery, criticizing the \"welfare dependency\" that, he argued, resulted in increased out-of-wedlock births in black communities and affected the \"marital behavior\" of black men (222). \"Indeed, few American citizens,\" he writes, \"are more highly visible in the U.S. media today than the militant young black man on welfare and perhaps on the edge of desperation and revolt\" (154). Rauf, in the meantime, has minimized the importance of social safety nets for developing countries, listing them behind other needed reforms in the Muslim world such as the \"creation or reform of banking systems, capital and stock markets, and sound monetary policies\" that will guarantee \"stable currencies and low inflation\"; see _What's Right with Islam_ , 253\u201354.\n\n. Rauf, _What's Right with Islam_ , 6.\n\n. Ibid., 64. Additionally, Rauf argues, Sufi traditions can help train adherents to cultivate a healthy detachment from material wealth that\u2014while constituting a social good when redistributed through charity, as Islam requires\u2014can become a distraction (ibid., 66\u201369).\n\n. Ibid., 2.\n\n. Gingrich and Haley, _Nation Like No Other_ , 116\u201318, especially.\n\n. Ibid., 10.\n\n. Ibid., 117\u201322. The quote from Novak appears on 118.\n\n. Novak, _Spirit_ , 253, 26.\n\n. Rauf, _What's Right with Islam_ , 232.\n\n. Ibid., 151. Rauf notes that he \"owes many of the ideas\" in his section on the corporation to John Micklethwait and Adrian Wooldridge's _The Company: A Short History of a Revolutionary Idea_ (New York: Modern Library, 2003), but Micklethwait and Wooldridge do not discuss the corporation as an example of separated powers. That material can be found, instead, in Novak, _Spirit_ , 326, 353.\n\n. Ibid., 268.\n\n. Ibid., 127.\n\n. Gingrich and Haley, _Nation Like No Other_ , 48.\n\n. This interpretation of _zakat_ as a \"right\" is supported by, among others, the Council of Islamic Organizations of Greater Chicago. In January 2008, the Council posted California physician and author Khalid Baig's essay \"Zakat: Right of the Poor\" on its website and continued to feature it there throughout the following five years of economic recession; see Council of the Islamic Organization of Greater Chicago, accessed June 1, 2013, http:\/\/www.ciogc.org\/Go.aspx?link=7654465.\n\n. Rauf, _What's Right with Islam_ , 51\u201352.\n\n. Ibid., 2, 47, 62, 217, 307.\n\n. Ibid., 66.\n\n. The commonality between immigrant Muslims who Americanize like other immigrants and black American Muslims whose sociology is rooted in a history of slavery, according to Rauf, lies in the fact that both feel that they are treated with suspicion after 9\/11 because they are Muslim; see _What's Right with Islam_ , 220\u201321.\n\n. Ibid., 221\u201324, 228\u201329. Rauf mentions the Catholic charities and welfare agencies as an example of this trend in his endnotes (303 n. 57).\n\n. Ibid., 225.\n\n. \"Karzai, Khan, and Levitt,\" _This Week with Christiane Amanpour_ , ABC News, August 15, 2010, full transcript available at <http:\/\/abcnews.go.com\/ThisWeek\/week-transcript-karzai-khan-levitt\/story?id=11454631>.\n\n. Rauf _, What's Right with Islam_ , 85.\n\n. Mahmood, \"Secularism, Hermeneutics, and Empire _,\"_ 331. Mahmood attributes this quote to \"State Department planners,\" though she actually derives the words from the 2003 RAND Corporation essay that is her primary source material. See Cheryl Benard, _Civil Democratic Islam: Partners, Resources, and Strategies_ (Santa Monica: RAND Corporation, 2003).\n\n. Ibid., 332, 336\u201337. The wording here is also from RAND.\n\n. Ibid., 346.\n\n. See also Hicks, \"Comparative Religion.\"\n\n. Mahmood, \"Secularism, Hermeneutics, and Empire,\" 340\u201341.\n\n. Ibid., 340. While Rauf does contrast his understanding of Islam to that of fundamentalists, he defined fundamentalism in 2004 as a \"psychological reaction\" to \"militant secularism\" and material deprivation, not as a faulty strategy of overly literal interpretation. He more fully discussed this view in \"Fundamentalism and the Modern World: A Dialogue with Karen Armstrong, Susannah Heschel, Jim Wallis, and Feisal Abdul Rauf,\" _Sojourners Magazine_ 31, no. 2 (March\u2013April 2002), <https:\/\/sojo.net\/magazine\/march-april-2002\/fundamentalism-and-modern-world>.\n\n. Mahmood, \"Secularism, Hermeneutics, and Empire,\" 323.\n\n. On the public school wars and American Protestant secularism, especially, see Fessenden, _Culture and Redemption_. More on the contemporary valences of American Protestant-secularism can be found in Jakobsen and Pellegrini, \"World Secularisms\" and Hicks, \"Between Lived and the Law.\"\n\nChapter 2\n\n. Rauf, _What's Right with Islam_ , 283.\n\n. The quote is from Kevin M. Schultz's summary and critique of Herberg, which first appeared in the Institute on Religion and Public Life's magazine as \"Protestant-Catholic-Jew, Then and Now,\" _First Things_ online, January 2006, http:\/\/www.firstthings.com\/article _\/_ 2006 _\/_ 01 _\/_ protestant-catholic-jewthen-and-now.\n\n. Rosemary R. Corbett, \"For God and Country: Religious Minorities Claiming National Belonging Through Community Service,\" _Religion and American Culture_ 26, no. 2 (Summer 2016), 227\u201359.\n\n. GhaneaBassiri, _History of Islam_ , 254.\n\n. Bulliet, _Case_ , 98\u2013100.\n\n. On the quote and Igram, see GhaneaBassiri, _History of Islam_ , 234\u201339.\n\n. GhaneaBassiri, _History of Islam_ , 258\u201359.\n\n. The Islamic Center of Washington, DC, was founded by members of the Washington Mosque Foundation, an organization local diplomats created after failing to find a mosque large enough for the 1944 funeral of Turkish Ambassador M\u00fcnir Erteg\u00fcn. In funding the mosque and Islamic center for ambassadors and other DC residents, several Muslim-majority governments recognized the chance to improve diplomatic relations with the United States. See Muhammad Abdul-Rauf, _History of the Islamic Center: From Dream to Reality_ (Washington, DC: The Center, 1978).\n\n. Dwight D. Eisenhower, \"Eisenhower's 1957 Speech at Islamic Center of Washington,\" IIP Digital, June 26, 2007, http:\/\/iipdigital.usembassy.gov\/st\/english\/texttrans\/2007 _\/_ 06 _\/_ 20070626154822lnkais0. _6946985_.html#ixz _z_ 2J0Hohnd _0_.\n\n. Quoted in Bulliet, _Case_ , 99.\n\n. Ibid., 100.\n\n. Ibid., 101.\n\n. For depictions of Muslims in American media during these decades, see McAlister, _Epic Encounters_ , 100\u2013234. On the FBI and Nation of Islam, see Edward E. Curtis, IV, _Islam in Black America: Identity, Liberation, and Difference in African-American Islamic Thought_ (Albany: SUNY Press, 2002), 69, 80\u201388, 112\u201313.\n\n. Edward E. Curtis, IV, \"Islamism and Its African American Muslim Critics: Black Muslims in the Era of the Arab Cold War,\" in _Black Routes to Islam_ , edited by Manning Marable and Hishaam D. Aidi (New York: Palgrave Macmillan, 2009), 49\u201368. For discussion of other black American Muslim communities in the United States, see also Robert Dannin, _Black Pilgrimage to Islam_ (New York: Oxford University Press, 2002); Edward E. Curtis, IV, _The Columbia Sourcebook of Muslims in the United States_ (New York: Columbia University, 2008); and Marc Ferris, \"'To Achieve the Pleasure of Allah': Immigrant Muslim Communities in New York City 1893\u20131991,\" in _Muslim Communities in North America_ , edited by Yvonne Yazbeck Haddad and Jane Idelman Smith (Albany: State University of New York, 1994), 212\u201313.\n\n. Donald Janson, \"Muslims Press Race Separation,\" _New York Times_ , February 28, 1963.\n\n. \"Malcolm X Scores U.S. and Kennedy,\" _New York Times_ , December 2, 1963.\n\n. Curtis, \"Islamism,\" 56.\n\n. Ibid., 56\u201359.\n\n. As Schultz argues, increasing US attachment to Israel has often deepened conservative Protestants' antipathies towards Muslims instead of making Protestants more willing to embrace them; see _Tri-Faith America_ , 199\u2013204 as well as Thomas Kidd, _American Christians and Islam: Evangelical Culture from the Colonial Period to the Age of Terrorism_ (Princeton: Princeton University Press, 2009), 120\u201364 and McAlister, _Epic Encounters_ , 155\u201397.\n\n. McAlister, _Epic Encounters_ , 110\u201315; Andrew Preston, _Sword of the Spirit, Shield of Faith: Religion in American War and Diplomacy_ (New York: Alfred A. Knopf, 2012), 559\u201362 and Waldo E. Martin, \"'Nation Time!': Black Nationalism, the Third World, and Jews,\" in _Struggles in the Promised Land: Towards a History of Black-Jewish Relations inthe United States_, edited by Jack Salzman and Cornell West (New York: Oxford, 1997), 341\u201356. For contemporary attempts to overcome this historic divide by creating anti-Muslim black-Jewish coalitions, see Jodi Eichler-Levine and Rosemary R. Hicks, \"'As Americans Against Genocide': The Crisis in Darfur and Interreligious Political Activism,\" _American Quarterly_ 59, no. 3 (September 2007): 711\u201335.\n\n. For how the Vietnam War's cost undermined Johnson's Great Society programs, see Julian E. Zelizer, _The Fierce Urgency of Now: Lyndon Johnson, Congress, and the Battle for the Great Society_ (New York: Penguin, 2015). On the 1960s rightward tilt in the GOP (fueled initially by Barry Goldwater's insistence on repealing Roosevelt's social programs), see Jeff Madrick, _Age of Greed_ (New York: Random House, 2007).\n\n. Rauf, _Moving the Mountain_ , 6.\n\n. On neo-conservatives and the Nation of Islam, see GhaneaBassiri, _History of Islam_ , 275\u201383.\n\n. McAlister, _Epic Encounters_ , 125\u201340.\n\n. See Rick Perlstein, _Nixonland: The Rise of a President and the Fracturing of America_ (New York: Scribner, 2008), and Sean Willentz, _The Age of Reagan: A History, 1974\u20132008_ (New York: Harper, 2008).\n\n. Quoted in \"Elijah Muhammad Dead; Black Muslim Leader, 77,\" _New York Times_ , February 26, 1975.\n\n. Quoted in Dawn-Marie Gibson, _A History of the Nation of Islam: Race, Islam, and the Quest for Freedom_ (Santa Barbara: ABC-CLIO, 2012), 69.\n\n. \"Black Muslim,\" _New York Times_ , February 28, 1975.\n\n. In addition to changing the organization's name (several times), Mohammed emphasized the Sunni pillars of Islam, abolished the Fruit of Islam defense militia, removed the seats inside mosques so Muslims could prostrate while praying, and asserted that there is \"no superiority in any color.\" As scholars of American Islam point out, these changes were not simply capitulations to Sunni authorities but the \"indigenizing\" of mainstream Islamic traditions in ways that allowed for a reinterpretation of NOI doctrine and narratives; see GhaneaBassiri, _History of Islam_ , 285\u201389; Curtis, _Islam in Black America_ , 107\u201328.\n\n. Former followers of Elijah Muhammad who split from Mohammed and allied with Louis Farrakhan's reconstituted Nation of Islam began to observe Savior's Day celebrations again in 1981; see, on the website of the NOI-affiliated newsletter, Ashahed M. Muhammad, \"Savior's Day: A Timeline and Brief History,\" _Final Call_ , last updated February 24, 2008, <http:\/\/www.finalcall.com\/artman\/publish\/Perspectives_1\/Saviours_Day_A_Timeline_and_Brief_History_4423.shtml>. W. D. Mohammed attended Farrakhan's 2000 Savior's Day celebration in an attempt to reconcile the two movements.\n\n. On contests over authority and resources, see Sherman Jackson, _Islam and the Blackamerican: Looking Toward the Third Resurrection_ (New York: Oxford University Press, 2005).\n\n. See GhaneaBassiri, _History of Islam_ , 261.\n\n. During a May 1972 rally in Harlem that followed the altercation, Abdul-Rauf\u2014then serving as director of the Islamic Center in Washington, DC\u2014unequivocally supported the NOI. Speaking for himself and his associates, Abdul-Rauf had \"come to express our admiration for your work and the great achievements of the beloved leader, the Honorable Elijah Muhammad. I would like to assure you all,\" Abdul-Rauf told the attendees, \"that the whole Muslim world, which includes 700 million people, is behind you.\" Quoted in the Central Committee of the Islamic Party, \"Editorial,\" _Islamic Party of North America News_ (Spring 1972), 2. Thanks to Abbas Barzegar for this reference.\n\n. Muhammad Abdul-Rauf, _Bilal Ibn Rabah: A Leading Companion of the Prophet Muhammad_ (Plainfield, IL: American Trust Publications, 1977), iii.\n\n. Ibid.\n\n. All quotes taken from Ghayth Nur Kashif, Introduction to _Bilal Ibn Rabah: A Leading Companion of the Prophet Muhammad, by_ Muhammad Abdul-Rauf (Plainfield, IL: American Trust Publications, 1977), vi.\n\n. Abdul-Rauf, _Bilal Ibn Rabah_ , 71.\n\n. Ibid.\n\n. Ibid., 72.\n\n. GhaneaBassiri, _History of Islam_ , 289. See also Curtis, _Islam in Black America_ , 114\u201315 on ties between W. D. Mohammed's movement and the new Egyptian leader, Anwar al-Saddat, as well as on ties with Gulf nations during this time.\n\n. _Bilalian News_ , March 3, 1978, 3, as quoted in Curtis, _Islam in Black America_ , 116.\n\n. Details of the hostage-taking are drawn from multiple media accounts, including Phil McCombs and Charles R. Babcock, \"Gunmen: People Will Die Unless We Get Our Demands,\" _Washington Post_ , March 10, 1977; Charles R. Babcock and Kevin Klose, \"From Beginning to End, 38\u2013Hour Drama Unfolded Slowly,\" _Washington Post_ , March 12, 1977; Phil McCombs and Megan Rosenfeld, \"Calm Hostages Read Koran, Polite Gunmen Answer Calls,\" _Washington Post_ , March 11, 1977; and Phil McCombs, \"The Hanafi Takeover: One Year Later,\" _Washington Post_ , March 5, 1978; as well as Tom Matthews, \"Seizing Hostages Scourge of the 70s,\" _Newsweek_ , March 21, 1977, 16; and \"Terrorism: The 38 Hours: Trial by Terror,\" _Time_ , March 21, 1977.\n\n. Matthews, \"Seizing Hostages,\" 16.\n\n. \"Terrorism.\"\n\n. Jimmy Carter, \"State of the Union Address,\" January 19, 1978, Jimmy Carter Library, accessed March 2, 2013, <http:\/\/www.jimmycarterlibrary.gov\/documents\/speeches\/su78jec.phtml>.\n\n. Mercy Corps was originally founded in 1979 as Save the Refugees Fund but changed to Mercy Corps International in 1982 and then simply Mercy Corps in 1984; see \"Our History,\" Mercy Corps, accessed January 3, 2014, <http:\/\/www.mercycorps.org\/about-us\/our-history>.\n\n. Mohammed's \"pulpit\" exchange with Washington, DC, rabbi Joshua Haberman is noted in Marjorie Hyer, \"Faiths Join to Wish Success at Camp David,\" _Washington Post_ , September 8, 1978. Instead of describing whites as literal \"devils\" to be hated, Mohammed argued, NOI teachings on race were meant metaphorically\u2014as instructing believers to hate the evil (here, whiteness instead of blackness) inside of themselves and others. On changes in racial doctrine, see Curtis, _Islam in Black America_ , 113\u201314.\n\n. Nathaniel Sheppard, Jr., \"Islamic Leader Says Organization is Taking a Turn Toward Patriotism,\" _New York Times_ (May 25, 1978).\n\n. Quoted in Nathaniel Sheppard, \"Islamic Leader Says Organization is Taking a Turn Toward Patriotism,\" _New York Times_ , May 25, 1978.\n\n. For coverage in the _New York Times_ , see Nathaniel Sheppard, \"Black Muslim Movement Divided in Dispute Over Doctrinal Changes,\" March 7, 1978, and \"Islamic Leader Says Organization is Taking a Turn,\" May 25, 1978, as well as Paul Delaney, \"Radical Changes By New Leader Leave Many Muslims Disaffected,\" December 25, 1978.\n\n. For coverage in the _Washington Post_ , see Marjorie Hyer, \"Three Faiths Stage Friendly Talks,\" October 28, 1977, and \"Faiths Join to Wish Success at Camp David\" September 9, 1978.\n\n. Muhammad Abdul-Rauf, \"Pilgrimage to Mecca,\" _National Geographic Magazine_ 154, no. 5 (November 1978): 581\u2013609.\n\n. Marjorie Hyer, \"Interfaith Religious Service Praises Peace Treaty,\" _Washington Post_ , March 27, 1979.\n\n. Abdul-Rauf, _Muslim's Reflections_ , ix.\n\n. On opposition by Hayek and his supporters to the Keynesianism that emerged during the Great Depression and their influence in the United States, see Harvey, _Brief History_ , 19\u201323.\n\n. William Baroody, \"President's Foreword\" in _A Muslim's Reflections on Democratic Capitalism_ , by Muhammad Abdul-Rauf (Washington, DC: American Enterprise Institute for Public Policy Research, 1984), v.\n\n. Abdul-Rauf, _Muslim's Reflections_ , x.\n\n. In a 2010 interview with the Catholic lay publication which Novak co-founded, Novak described his colleague, Irving Kristol, as pivotal in turning him toward thinkers like Hayek; see Joop Koopman, \"Theology of the Corporation: A Conversation with Michael Novak,\" _Crisis Magazine_ online, October 23, 2010, <http:\/\/www.crisismagazine.com\/2010\/theology-of-the-corporation-a-conversation-with-michael-novak>.\n\n. Abdul-Rauf, _Muslim's Reflections_ , 34.\n\n. Ibid., 18, 68 n. 4.\n\n. Ibid., 34 (emphasis added).\n\n. John Esposito, \"Ismail R. Al-Faruqi: Muslim Scholar-Activist,\" in _The Muslims of America_ , edited by Yvonne Yazbeck Haddad (New York: Oxford University Press, 1991), 76.\n\n. Muhammad Abdul-Rauf, \"Judaism and Christianity in the Perspective of Islam,\" in _Trialogue of the Abrahamic Faith Traditions: Papers Presented to the Islamic Studies Group of the American Academy of Religion_ , 3rd ed., edited by Isma'il Al-Faruqi (Alexandria, VA: Al-Saadawi Publications, 1991), 22\u201323.\n\n. Donnel Nunes, \"Muslim Strife Reaches Islamic Center,\" _Washington Post_ , April 21, 1980.\n\n. Ibid.\n\n. Earle H. Waugh, Baha Abu-Laban, and Regula B. Qureshi, Introduction to _The Muslim Community in North America_ (Edmonton, Alberta: University of Alberta Press, 1983), 1\u201310.\n\n. Muhammad Abdul-Rauf, \"The Future of the Islamic Tradition in North America,\" in _Muslim Community in North America_ , edited by Earle H. Waugh, Baha Abu-Laban, and Regula B. Qureshi (Edmonton, Alberta: University of Alberta Press, 1983), 272, 274.\n\n. Ibid., 277.\n\n. See, particularly, Isma'il al-Faruqi, _Islam_ (Belsville, MD: Amana Publications, 1994), 57.\n\n. Isma'il al-Faruqi, \"Islamic Ideals in North America,\" in _The Muslim Community in North America_ , edited by Earle H. Waugh, Baha Abu-Laban, and Regula B. Qureshi (Edmonton, Alberta: University of Alberta Press, 1983), 266. For \"untermensch,\" see 260.\n\n. See Muhammad Abdul-Rauf, \" _Sunnah_ and _Hadith_ and Their Importance in Modern Context,\" in _Shari'ah, Ummah, and Khilafa_ , by Halil Inalcik, Muhammad Abdul-Rauf, and Isma'il R. al-Faruqi (Karachi, Pakistan: University of Karachi, 1987), 80. Compare this with Abdul-Rauf's thoughts in _Muslim's Reflections_ , 26.\n\n. Harvey, _Brief History_ , 23\u201331.\n\n. Curtis, _Islam in Black America_ , 122\u201327.\n\n. Quoted in Koopman, \"Theology of the Corporation.\"\n\n. On the rise of such \"military multiculturalism\" and political conservatives' use of it in the 1990s to argue against affirmative action and multiculturalism, see McAlister, _Epic Encounters_ , 235\u201365.\n\n. GhaneaBassiri, _History of Islam_ , 334\u201335.\n\n. For more detail on these events, including excerpts of original documents and speeches, see GhaneaBassiri, _History of Islam_ , 332\u201337. On W. D. Mohammed's contract with the US military, see Curtis, _Muslims in America_ , 121. On the first Muslim chaplain, see Edward E. Curtis, IV, \"United States Military\" in _Encyclopedia of Muslim American History_ (New York: Facts on File, 2010), 562 and GhaneaBassiri, _History of Islam_ , 340.\n\n. On Clinton administration attempts to make Muslim allies, see GhaneaBassiri, _History of Islam_ , 339\u201340.\n\n. Ibid., 343\u201344, 369\u201375.\n\n. Ibid., 351\u201354. The history of Islamic Relief, USA, is drawn from the \"History\" page of the organization's website, accessed February 19, 2015, <http:\/\/www.irusa.org\/history\/>.\n\n. For a brief discussion of Muslim-led charities suspected of undermining US interests, particularly after 9\/11, see GhaneaBassiri, _History of Islam_ , 351. Not all Muslim-led organizations engaged in overseas humanitarian work came under suspicion. Islamic Relief, USA, for example, sponsored relief and development projects in Bosnia, Chechnya, and Afghanistan in the 1990s and received positive media attention for some of it, although not for their simultaneous domestic relief and emergency response work; see Islamic Relief, USA \"History.\"\n\nChapter 3\n\n. All quotes derive from Feisal Abdul Rauf, \"'I Do Not Recognize Myself' this Ramadan: War Against Impulse that Resorts to Violence,\" _San Francisco Chronicle_ , November 11, 2001.\n\n. Feisal Abdul Rauf, _Islam: A Search for Meaning_ (Costa Mesa, CA: Mazda Publishers, 1995), xii.\n\n. In their 1998 application for 501(c)(3) status, Rauf and Khan described ASMA as \"formed to enhance the general public's knowledge and understanding of the religion of Islam and Sufism and to promote spiritual and personal development through meditation and prayer\"; see American Sufi Muslim Association, Application for recognition of exemption under section 501(c)(3) of the Internal Revenue Code, Department of the Treasury, Internal Revenue Service, Form 1023.\n\n. \"Islam and America, Three Years After 9\/11: Interview with Imam Feisal Abdul Rauf,\" _Beliefnet_ , 2004 (accessed November 11, 2005), <http:\/\/www.beliefnet.com\/Faiths\/Islam\/2005\/02\/Islam-And-America-Three-Years-After-911.aspx?p=1+a+2005>.\n\n. See Hicks, \"Comparative Religion\" and Corbett, \"Islamic 'Fundamentalism.'\"\n\n. One exception to this trend is West African Sufis who immigrated in the 1990s, settled in Harlem, and\u2014the difficulties of their lives notwithstanding\u2014often argue that the American dream is still a reality. Despite their upward mobility, however, State Department officials seeking to influence Muslims in the Middle East and South Asia have rarely sought these more recent US residents as spokespersons; see Zain Abdullah, _Black Mecca: The African Muslims of Harlem_ (New York: Oxford University Press, 2010) and Ousmane Kane, _The Homeland Is the Arena: Religion, Transnationalism, and the Integration of Senegalese Immigrants in America_ (New York: Oxford University Press, 2011).\n\n. Statistics from Richard Brent Turner, _Islam in the African-American Experience_ , 2nd ed. (Indianapolis: Indiana University Press, 2003). For histories of African Muslim slaves, see Sylviane A. Diouf, _Servants of Allah: African Muslims Enslaved in the Americas_ (New York: New York University Press, 1998) and Allan D. Austin, _African Muslims in Antebellum America: Transatlantic Stories and Spiritual Struggles_ (New York: Routledge, 1997).\n\n. Rich Schapiro, \"NYC Plans Memorial at Wall Street Slave Market from 1700s,\" _New York Daily News_ , April 15, 2015.\n\n. On the Transcendentalists, Hafiz, and Rumi, see Leigh Eric Schmidt, _Restless Souls: The Making of American Spirituality_ (San Francisco: Harper Collins, 2005), 82\u201384 and \"The Making of Modern 'Mysticism,'\" _Journal of the American Academy of Religion_ 71, no. 2 (2003): 273\u2013302. See also Wai-chee Dimock, \"Deep Time: American Literature and World History,\" _American Literary History_ 13, no. 4 (Winter 2001): 755\u201375.\n\n. William James, _The Varieties of Religious Experience_ (Scotts Valley, CA: IAP Press, 2009), 232\u201333.\n\n. See Leigh Eric Schmidt, \"Cosmopolitan Piety: Sympathy, Comparative Religions, and Nineteenth-Century Liberalism,\" in _Practicing Protestants: Histories of Christian Life in America, 1630\u20131965_ , edited by Laurie F. Maffly, Leigh E. Schmidt, and Mark Valerie (Baltimore: Johns Hopkins University Press, 2006), 199\u2013221 and Schmidt, _Restless Souls_ , 251\u201354. For American fascinations with India, see Richard King, _Orientalism and Religion: Post-Colonial Theory, India, and \"the Mystic East\"_ (London: Routledge, 1999). On orientalism and early twentieth-century American consumer habits, see McAlister, _Epic Encounters_ , 20\u201329.\n\n. Gisela Webb, \"Third Wave Sufism in America and the Bawa Muhaiyaddeen Fellowship,\" in _Sufism in the West_ , edited by Jamal Malik and John Hinnells (New York: Routledge, 2006), 86\u2013102.\n\n. James Jervis, \"The Sufi Order in the West and Pir Vilayat 'Inayat Khan: Space-Age Spirituality in Contemporary Euro-America,\" in _New Trends and Developments in the World of Islam_ , edited by Peter B. Clarke (London: Luzac Oriental, 1997), 211\u201360.\n\n. Curtis, _Islam in Black America_ , 45\u201346, 62.\n\n. See Curtis, _Columbia Sourcebook_ , 22\u201329 and 46\u201358.\n\n. Robert Rozehnal, \"Faqir or Faker? The Pakpattan Tragedy and the Politics of Sufism in Pakistan,\" _Religion_ 36, no. 1 (March 2006): 34\u201335.\n\n. Carl Ernst, _The Shambhala Guide to Sufism_ (Boston: Shambhala Publications, 1997), 1\u201318.\n\n. Elizabeth Sirriyeh, _Sufis and Anti-Sufis: The Defence, Re-Thinking, and Rejection of Sufism in the Modern World_ (Richmond, Surrey: Curzon, 1999).\n\n. J. Spencer Trimingham, _The Sufi Orders in Islam_ (New York: Oxford University Press, [1971] 1998), 253.\n\n. Layla Amzi, \"The Survival of Sufism in the Turkish Republic: Sheikh Muzaffer \u00d6zak and the Halveti-Cerrahi Order,\" Master's thesis, New York University, 2006, 17.\n\n. Ahmet T. Kuru, _Secularism and State Policies toward Religion: The United States, France, and Turkey_ (New York: Cambridge University Press, 2009), 215\u201334 and Umut Azak, _Islam and Secularism in Turkey: Kemalism, Religion, and the Nation State_ (London: I. B. Taurus, 2010). On the change among Sufi orders, see Thierry Zarcone, \"The Transformation of the Sufi Orders ( _Tarikat_ ) in the Turkish Republic and the Question of Crypto-Sufism,\" in _Cultural Horizons: A Festschrift in Honor of Talat Halman_ , edited by Jayne Warner (Syracuse: Syracuse University Press, 2001), 199.\n\n. Details on Nasr's life are drawn from \"Biography,\" Seyyed Hossein Nasr Foundation, accessed October 10, 2006, <http:\/\/www.nasrfoundation.org\/bios.html>.\n\n. Ananda Coomaraswamy first brought ideas about mysticism, theosophy, and Perennialism, along with South Asian artifacts, to Boston in 1917. He then served as \"Keeper of Indian Arts\" at the Museum of Fine Arts in Boston until the mid-1930s, when his title was changed to research fellow in \"Indian, Persian, and Mohammedan Art.\" On Coomaraswamy, see \"Biography,\" Seyyed Hossein Nasr Foundation, accessed October 10, 2006, <http:\/\/www.nasrfoundation.org\/bios.html>. For a larger overview of Perennialism, see King, _Orientalism and Religion_ , 120\u201341, 160\u201382.\n\n. \"Biography,\" Seyyed Hossein Nasr Foundation. On Schuon's involvement in the 'Alawiyya-Shadhiliya order and his later creation of the Maryamiyya (or Miriamiyya) Sufi order, see Erik S. Ohlander, \"Maryamiyya Sufi Order\" in _Encyclopedia of Muslim-American History_ , edited by Edward E. Curtis, IV (New York: Facts on File, 2010), 361\u201362.\n\n. While Nasr spent much time on the work of Andalusian-born philosopher Ibn 'Arabi and his Persian contemporary, al-Suhrawardi, he increasingly focused on Persian philosopher Mulla Sadra's seventeenth-century syntheses of Suhrawardi, Neo-Platonism, and Islamic theology. For an overview of Nasr's work, see the essays in Lewis E. Hahn, Randall E Auxier, and Lucian W. Stone, eds. _The Philosophy of Seyyed Hossein Nasr_ (Chicago: Open Court Press, 2001). I briefly discuss Nasr's academic work in Iran in Hicks, \"Comparative Religion.\"\n\n. On the dominance of Sufism in American religious studies departments, see Marcia Hermansen, \"The Academic Study of Sufism at American Universities,\" _American Journal of Islamic Social Sciences_ 24, no. 3 (2007): 23\u201345. Steven M. Wasserstrom also discusses Nasr's influence briefly (albeit in a less nuanced manner) in _Religion After Religion: Gershom Scholem, Mircea Eliade, and Henry Corbin at Eranos_ (Princeton: Princeton University Press, 1999).\n\n. Seyyed Hossein Nasr, _The Encounter of Man and Nature: The Spiritual Crisis of Modern Man_ (London: Allen & Unwin, 1968).\n\n. On the variety of US Sufi orders, see Marcia Hermansen, \"Literary Productions of Western Sufi Movements,\" in _Sufism in the West_ , edited by Jamal Malik and John Hinnells (New York: Routledge 2006), 28\u201348.\n\n. See Seyyed Hossein Nasr, _Islamic Art and Spirituality_ (Albany: SUNY Press, 1987) and _The Heart of Islam: Enduring Values for Humanity_ (San Francisco: Harper Collins, 2002).\n\n. Nasr and French Orientalist Henri Corbin established the Imperial Iranian Academy of Philosophy under the patronage of the Pahlavi empress. While the government chartered the Academy to construct a pre-Islamic nationalist history, Nasr focused on Shi'i traditions, Sufism, and philosophy. In the 1970s, Nasr's connections to the Iranian government intensified\u2014a relationship one scholar describes as \"regime religiosity,\" wherein the monarchy quieted its previous claims to a pre-Islamic heritage so as to combat the influence of Ayatollah Khomeini; see Matthjis Van den Bos, _Mystic Regimes: Sufism and the State in Iran, from the Late Qajar Era to the Islamic Republic_ (Leiden: Brill, 2002), 112\u201314.\n\n. Ohlander, \"Maryamiyya Sufi Order,\" 361; GhaneaBassiri, _History of Islam_ , 301.\n\n. Seyyed Hossein Nasr, Foreword to _A Muslim's Reflections on Democratic Capitalism_ , by Muhammad Abdul-Rauf (Washington, DC: American Enterprise Institute for Public Policy Research, 1984), vii\u2013viii.\n\n. Feisal Abdul Rauf, Introduction to _The Universal Sprit of Islam_ , by Judith Fitzgerald and Michael Oren Fitzgerald (Bloomington, IN: World Wisdom Press, 2006), xvii\u2013xviii.\n\n. Amzi, \"Survival,\" 53.\n\n. Ibid., 4, 45\u201347.\n\n. Details of this trip are recounted in Sheikh Muzaffer \u00d6zak, _The Unveiling of Love: Sufism and the Remembrance of God_ (New York: Pir Press, 2001), 16\u201321.\n\n. See Tosun Bayrak, _Memoirs of a Moth: The Life of Shaykh Tosun al Jerrahi_ (Istanbul: Timas Publishing, 2014), 258.\n\n. Ibid.\n\n. Ibid., 72\u201375, 84\u2013104.\n\n. Ibid., 109.\n\n. Ibid., 108\u201326 (quote from 126).\n\n. Robert Frager, _Love is the Wine_ (Los Angeles: Philosophical Research Society, 1987).\n\n. Trimingham, _Sufi Orders_ , 74.\n\n. Bayrak, _Memoirs_ , 126\u201327.\n\n. See Bob Colacello, \"Remains of the Dia,\" _Vanity Fair_ online, September 1996 (accessed November 7, 2009), http:\/\/www.vanityfair.com\/magazin _e\/_ 1996 _\/_ 09 _\/_ colacello199609.\n\n. On the Menils' patronage and religiosity, see Pamela G. Smart, _Sacred Modern: Faith, Activism, and Aesthetics in the Menil Collection_ (Austin: University of Texas Press, 2011).\n\n. Quoted in Brad Gooch, _Godtalk: Travels in Spiritual America_ (New York: Alfred A. Knopf, 2002), 342.\n\n. The history of Sufi performances at the Cathedral of Saint John the Divine was relayed to the author by the Reverend James Park Morton, who was in residence at the Cathedral during the 1970s and 1980s. Personal interview with the author, November 3, 2010.\n\n. \u00d6zak, _Unveiling_ , 19.\n\n. Colacello, \"Remains.\"\n\n. Ibid.\n\n. Among Lex Hixon's many books are _Coming Home: The Experience of Enlightenment in Sacred Traditions_ (Burdett, NY: Larson Publications, [1978] 1995), in which he invites the reader (in the words of Ken Wilber, who wrote the foreword) to \"release yourself into Spirit as it manifests itself through each of these traditions\" (viii).\n\n. According to Hixon's website, the interview occurred in 1979; see Lex Hixon, \"Radio,\" LexHixon, accessed January 10, 2008, http:\/\/www.lexhixon.org\/simplesite\/simplefrm.html. But Muzaffer elsewhere describes it as having occurred during his first visit to the United States in 1978; see \u00d6zak, _Unveiling_ , 21.\n\n. Lex Hixon Nur al-Jerrahi, _Atom from the Sun of Knowledge_ (Westport, CT: Pir Press, 1993), v\u2013vi.\n\n. As recounted in Rauf, _Moving the Mountain_ , 9\u201311.\n\n. Rauf, _Islam: A Search for Meaning_ , xi.\n\n. See Gooch, _Godtalk_ , 352\u201353.\n\n. According to the Nur-Ashki Jerrahis' website, Nur's \"vision of Universal Islam opens a new era of spiritual flowering\"; see \"Lineage,\" Nur Ashki Jerrahi, accessed January 10, 2008, <http:\/\/nurashkijerrahi.org\/lineage>. In contrast, Bayrak's Halveti-Jerrahi's define themselves as \"A Traditional Muslim Sufi Order,\" using the words \"Traditional\" and \"Muslim\" to distinguish their order from the Nur-Ashki Jerrahis; see \"About Us,\" Jerrahi Order of America, accessed January 10, 2008, <http:\/\/www.jerrahi.org\/>. Thanks to Michael Wolfe for directing me to these websites and the communication between the groups.\n\n. Gooch, _Godtalk_ , 348.\n\n. Ibid., 346.\n\n. Quoted in Colacello, \"Remains.\"\n\n. As told by members of both orders to Michael Wolfe in 2005. Michael Wolfe, \"Invitation and Education: How Two American Branches of a Turkish Sufi Order Use the Internet to Educate Newcomers,\" Unpublished paper for Columbia University, New York, 2005.\n\n. Rauf, _Islam: A Search for Meaning_ , xii.\n\n. Rauf, quoted in \"Abraham's Children.\"\n\n. Rauf, _Islam: A Search for Meaning_ , xv.\n\n. Ibid., xii (emphasis added).\n\n. Ibid., xiii.\n\n. Ibid., xvi\u2013xvii.\n\n. Ibid., 65.\n\n. Ibid., 89.\n\n. Ibid.,114\u201315. On the Sufi psychology genre, see Hermansen, \"Literary Productions,\" 28\u201348.\n\n. Ibid., 114. In his writings, Rauf discussed all four Sunni law schools and the importance of finding commonality between them, if possible. He did not, as some black American Muslims have proposed, advocate creating a new body of law to fit particularly American circumstances. On the latter subject, see Yusuf Talal DeLorenzo, \"The Fiqh Councilor in North America,\" in _Muslims on the Americanization Path_ , edited by Yvonne Haddad and John L. Esposito (Atlanta: Scholars Press, 1998), 79\u2013106 and Sherman Jackson, \"Shari'ah, Democracy and the Modern Nation-State: Some Reflections on Islam, Popular Rule and Pluralism,\" _Fordham International Law Journal_ 27, no. 1 (2004): 88\u2013107.\n\n. On Rauf's idea to invite the Prime Minister to the meetings, see _Islam: A Sacred Law_ , 15.\n\n. Ibid., 22 (emphasis added). See 119 for the \"Spirit and intent of the Shariah.\"\n\n. Ibid., 20.\n\n. Ibid., 44\u201345, 60.\n\n. Ibid., 108\u20139.\n\n. Rauf, _Islam: A Search for Meaning_ , 88.\n\n. Rauf, _Islam: A Sacred Law_ , 109\u201310. In the following paragraph, Rauf also applied this reasoning to the interpretation of shari'ah: \"As you go down the sources and examine the totality of Islamic laws and trace them back to any of the eleven sources from which they originated, it is sensible to consider those that originated from custom 'more subject to evolution' than those that originated from the Qur'\u0101n.\"\n\n. Rauf, _Islam: A Search for Meaning_ , 32.\n\n. As recounted to Brad Gooch in 2000; see _Godtalk_ , 356.\n\n. Ibid.\n\n. American Sufi Muslim Association, \"Certificate of Incorporation of American Sufi Muslim Association under Section 803 of the Non-For-Profit Corporation Law,\" filed with the New York Department of State on June 10, 1997, 2.\n\n. Personal interview with Daisy Khan, January 25, 2008.\n\n. American Sufi Muslim Association, \"Certificate of Incorporation,\" 2.\n\n. See Gooch, _Godtalk_ , 356.\n\n. Information on and interviews with Hisham Kabbani can be found in Ghanea-Bassiri, _History of Islam_ , 355\u201356 and in Paul Barrett, _American Islam: The Struggle for the Soul of a Religion_ (New York: Picador Books, 2008), 179\u2013215.\n\n. See Ron Geaves, \"Who Defines Moderate Islam 'Post'-September 11?,\" in _Islam and the West, Post 9\/11_ , edited by Ron Geaves, Yvonne Haddad, and Jane Idelman Smith (Burlington, VT: Ashgate, 2004), 67. As Geaves notes, this State Department tactic also derived from observing the Indian government's practices in dealing with various Muslim communities on the subcontinent.\n\n. According to Dr. Ahmad Kostas\u2014a Moroccan Fulbright Scholar who helped to organize the annual World Sacred Music Festival that Rauf attended in 1996 and who belonged to the same order that Rauf joined that year\u2014he and Rauf had once shared the same goals but eventually parted company because Rauf insisted on keeping Sufi practices tied to mosques while Kostas insisted that American mosques were all \"infected\" with extremism and should be avoided. Personal conversation with Ahmed Kostas, Princeton, New Jersey, March 10, 2007. Details on Kostas' Sufi activities and Fulbright fellowship can be found at his page on the Washington Morocco Club website, accessed January 10, 2008, http:\/\/www.washingtonmoroccanclub.org\/Ahmed%20Kostas.htm (site discontinued), as well as on his page on the website of the US branch of the Qadiri Butshishia order, \"More About Sidi Ahmed Kostas,\" accessed August 6, 2011, <http:\/\/web.archive.org\/web\/20030408081844\/http:\/\/www.sufivillage.org\/kostas2.html>.\n\n. The attitudes of Muzaffer and Nasr were in some ways quite similar, as Nasr indicated in Nasr's preface to the translation of Muzaffer \u00d6zak, _Irshad: Wisdom of a Sufi Master_ (Westport, CT: Pir Press, 1988).\n\n. In his first book, for example, Rauf argued that Qur'anic recitation has a \"dual effect\" on listeners: \"a visceral, emotive, intuitive effect much like the effect of music or art . . . like a call of the wild to the depths of one's soul . . . [and] a linear, intellectual left-brain one\"; see Rauf, _Islam: A Search for Meaning_ , 14.\n\n. For information about the Mosque of Islamic Brotherhood and its founder, Shaykh Tawfiq, see http:\/\/www.mosqueofislamicbrotherhoodinc.org\/aboutus.html and http:\/\/www.mosqueofislamicbrotherhoodinc.org\/shaykyhtawfiq.html, accessed February 18, 2007 (sites discontinued).\n\n. On Sirraj Wahhaj, a charismatic speaker who often travels the country raising money for various Muslim causes and who a former _Wall Street Journal_ reporter describes as \"a rare crossover luminary, an African-American popular among immigrant Muslims,\" see Barrett, _American Islam_ , 100\u2013133.\n\n. Matthew Weiner, \"Interfaith in the City: Religious Pluralism and Civil Society in New York,\" PhD diss., Union Theological Seminary, 2009, 81.\n\n. Quote from Ari L. Goldman, \"Religion Notes: Fighting Evil,\" _New York Times_ , January 2, 1993. On the \"trialogue\" and tri-faith battle (along with a local Protestant minister and a rabbi) against Time Warner, see Dennis Hevesi, \"An Emergency Forges a Bond of Tenants in a Common Faith,\" _New York Times_ , November 9, 1994, and Nadine Brozan, \"Manhattan Cable to Drop Channel with Religious Shows,\" _New York Times_ , January 24, 1999. As _Times_ reporter Edward Wong described in \"Making it Work: A Chaplain Who Focuses on Hope, Inside and Outside of Prison,\" July 26, 1998, Abdur Rashid began speaking in prisons after a member of his mosque was arrested in 1976, was officially hired as a chaplain at Sing Sing in 1986, transferredf to Rikers Island in 1997, and then transferred to the downtown Manhattan Detention Complex after that.\n\n. Chris Hedges, \"A Muslim in the Middle Hopes Against Hope,\" _New York Times_ , June 23, 2004.\n\n. On \"moderate Islam,\" see Hedges, \"Muslim in the Middle.\" In addition to being described as the founder of the American Sufi Muslim Association, Rauf was identified as a \"Sufi leader\" connected to former Secretary of State Madeline Albright in a 2006 article reviewing one of Albright's books; see Peter Steinfels, \"Madeline Albright, the Cardinal?,\" _New York Times_ , May 6, 2006.\n\nChapter 4\n\n. Details and quotes are drawn from Hisham Aidi's recounting of the event, \"The Milder, Gentler Side of Islam,\" ASMA Society, \"Past Events (2004),\" accessed December 17, 2009, http:\/\/www.asmasociety.org\/calendar\/pastevents.html.\n\n. Aidi, \"Milder, Gentler Side.\"\n\n. Information about this event is drawn from \"Cordoba Bread Fest: Children of Abraham Break Bread Together,\" Cordoba Initiative, accessed September 8, 2006, http:\/\/www.cordobainitiative.org\/recent_programs.html (site discontinued).\n\n. For more on the Cordoba Bread Fest, see the Introduction.\n\n. \"Cordoba Bread Fest: Children of Abraham Break Bread Together.\"\n\n. According to Bennett, he and Rauf initially met at an August 2002 Spiritual Paths Foundation seminar held at the Aspen Institute in Colorado; \"Finding Cordoba,\" April 25, 2004 (accessed December 17, 2009), http:\/\/www.abc.net.au\/rn\/relig\/enc\/stories\/s1091176.html (site discontinued). After four terms as mayor of the city of Aspen (1991\u20131999), Bennett held a two-year position as vice president of the Aspen Institute, a \"nonpartisan educational and policy studies institute,\" from 1999\u20132001. In 2010, the Aspen Institute included former Saudi Ambassador to the United States Prince Bandar bin Sultan and former Secretary of State Henry Kissinger among their \"Lifetime Trustees\" (Aspen Institute, accessed February 3, 2010, <http:\/\/www.aspeninstitute.org\/about\/leadership-board\/lifetime-trustees>) and former Secretary of State Madeline Albright and RAND Corporation Board of Trustees Chairman Ann McLaughlin Korologos (among other academic and business elites) on the Board of Trustees; see \"Leadership,\" Aspen Institute, accessed February 3, 2010, <http:\/\/www.aspeninstitute.org\/about\/leadership>.\n\n. Rauf thanked Bennett for his editorial help in \"weaving the Initiative's ideas\" into the book's \"architecture\" in _What's Right with Islam_ , 286.\n\n. Rauf taught the course in conjunction with St. Bartholomew's Center for Religious Inquiry. _Asia Society Annual Report_ , 2002\u20132003, 1.9, Asia Society, accessed February 22, 2008, http:\/\/www.asiasociety.org\/about\/annualreports\/ASAR_02\u201303_pt1.pdf (site discontinued).\n\n. Quoted in \"Abraham's Children.\"\n\n. Daisy Khan, \"Faith and Feminism Brown Bag Lunch,\" lecture at The Sister Fund, New York, January 3, 2008.\n\n. All quotes in this section derive from the \"Mission\" section under \"About Us,\" ASMA Society, accessed January 4, 2008, http:\/\/www.asmasociety.org\/about\/index.html.\n\n. Rauf, \"Forceful Voice of Reason,\" 100. Rauf discussed Herberg explicitly in _What's Right with Islam_ , 224\u201327.\n\n. Information on Bennett's role with the Garrison Institute is drawn from the 2006 Cordoba Initiative website, accessed September 8, 2006, http:\/\/www.cordobainitiative.org\/who_we_are.html (site discontinued).\n\n. Rauf's political liberalism should not be understood as a partisan position. Rather, as I discuss in Chapter 1, his liberalism reflects the classical political and market liberalism to which both major political parties in the United States appeal.\n\n. \"About Us,\" ASMA Society, accessed January 4, 2008, http:\/\/www.asmasociety.org\/about\/index.html (emphasis added).\n\n. Material on the Cordoba Initiative was initially located on a specific Cordoba Initiative page on ASMA Society, accessed February 1, 2006, http:\/\/www.asmasociety.org\/cordoba\/index.html (site discontinued).\n\n. See Cordoba Initiative, \"Bylaws of the Cordoba Initiative,\" filed with the Colorado Secretary of State on June 14, 2004.\n\n. See \"Julia A. Jitkoff, Sculptor, is Married to B. Waring Partridge 3d, Executive,\" _New York Times_ , June 3, 1991.\n\n. Bennett's biography remained available on the ASMA Society website until 2007, accessed February 1, 2006, http:\/\/www.asmasociety.org\/about\/b_bennett.html.\n\n. Khan's biographical details are drawn from \"Daisy Khan,\" ASMA Society, 2009 (accessed March 18, 2010), http:\/\/www.asmasociety.org\/about\/b_dkhan.html.\n\n. Timothy McFlynn, Public Counsel of the Rockies, \"Supplementary Letter to the Internal Revenue Service,\" September 14, 2004.\n\n. Personal communication with Chris King, February 3, 2006.\n\n. Cheryl Benard, _Civil Democratic Islam: Partners, Resources, and Strategies_ (Santa Monica: RAND Corporation, 2003).\n\n. Ibid., 46; see also xii, 63, 64.\n\n. Angela Rebasa, Cheryl Benard, Lowell H. Schwartz, and Peter Sickle, _Building Moderate Muslim Networks_ (Santa Monica: RAND Corporation, 2007), 74; see also 89\u201390 on involving moderate American Muslim organizations in cultivating moderates worldwide.\n\n. On the \"West and Islam Dialogue\" and its aims, participants, and funders, see World Economic Forum, accessed July 6, 2009, http:\/\/www.weforum.org\/en\/Communities\/FaithCommunities\/c100\/index.htm (site discontinued).\n\nRauf noted his involvement in the World Economic Forum and his appointment to the Council of 100 in several biographies, including on the jacket of his 2004 book, _What's Right with Islam_.\n\n. Madeline K. Albright, _The Mighty and the Almighty: Reflections on America, God, and World Affairs_ (New York: Harper Collins, 2006), 283\u201387. Their Council on Foreign Relations taskforce findings were later published in Madeline K. Albright, Vin Weber and Steven Cook, _In Support of Arab Democracy_ (Washington, DC: Council on Foreign Relations Press, 2006).\n\n. \"Annual Meeting 2006: The Creative Imperative,\" World Economic Forum, accessed July 6, 2009, http:\/\/www.weforum.org\/en\/events\/ArchivedEvents\/annualmeeting\/index.htm (site discontinued).\n\n. Personal communication with Daisy Khan, September 15, 2006. Rauf described his connections with Badawi and Badawi's role in convening that meeting in \"Seven Questions: the Cross and the Crescent: An Interview with Faisal Abdul Rauf,\" _Foreign Policy_ online, September 20, 2006 (accessed April 8, 2014), http:\/\/foreignpolicy.com _\/_ 2006 _\/_ 09 _\/_ 20 _\/_ seven-questions-the-cross-and-the-crescent\/.\n\n. Glen Kessler, \"Karen Hughes, Changing Public Diplomacy's Face,\" _Washington Post_ , April 19, 2006.\n\n. Rauf's comments can be found in \"Seven Questions.\"\n\n. This primary objective, labeled \"Issue & Policy Research,\" appeared first under the list of Cordoba's \"Principal Programs\" on its initial webpage, accessed February 1, 2006, http:\/\/www.asmasociety.org\/cordoba\/programs\/html (site discontinued). Cordoba's other three programs listed at that time included \"The Dialogues\" (a series of talks to cultivate interreligious and civic understanding), \"The American Muslim Initiatives\" (including the Muslim Leaders of Tomorrow Program, as well as two signature initiatives that Cordoba and ASMA would launch in 2006: \"The American Muslim Women's Forum\" and \"The Shariah Project\") and \"Culture and Educational Programs.\"\n\n. Faiz Khan, \"E.R. Doctor and Seeker of Islamic Truths,\" in _Voices of American Muslims_ , edited by Linda Brandi Cateura (New York: Hippocrene, 2005), 79.\n\n. Ibid., 72.\n\n. Ibid., 71\u201377.\n\n. Ibid., 76 and 79, respectively.\n\n. Ibid., 70.\n\n. Ibid., 73.\n\n. Ibid., 73.\n\n. Ibid., 77.\n\n. Ibid., 79.\n\n. Faiz Khan, \"American Muslims and 9\/11 Truth: the Paralysis of Discourse, the Incompetence of Academia, and the Need for an Accurate Diagnosis,\" in _9\/11 and American Empire: Christians, Jews, and Muslims Speak Out_ , edited by Kevin Barrett, John Cobb, Jr., and Sandra Lubarsky (Northhampton, MA: Olive Branch, 2007), 243.\n\n. See Hedges, \"Muslim in the Middle\" and Sam Stein, \"'Ground Zero Mosque.'\"\n\n. Khan, \"American Muslims,\" 221.\n\n. Ibid., 226 and 244, respectively.\n\n. Ibid., 222 and 227, respectively.\n\n. Ibid., 241.\n\n. Ibid., 240.\n\n. Ibid., 250.\n\n. Khan, \"E.R. Doctor,\" 77.\n\n. Jamie McIntyre, \"Pentagon: Rumsfeld Misspoke on Flight 93 Crash,\" CNN online, December 27, 2004, http:\/\/www.cnn.com\/2004\/US\/12 _\/_ 27\/rumsfeld.flt93 _\/_.\n\n. Maureen Dowd, \"Repent, Dick Cheney,\" _New York Times_ , March 5, 2013.\n\n. Personal interview, May 4, 2007.\n\n. Personal communication with Danish Masood, September 12, 2010.\n\n. On Rauf's message at the first Muslim Leaders of Tomorrow (MLT) conference, see ASMA's early coverage of the event, accessed September 8, 2006, at http:\/\/asmasociety.org\/religion\/mlt ___ 04retreat.html. The second MLT conference was in Copenhagen in 2006, information for which could be found on Cordoba's website, accessed February 8, 2008, http:\/\/www.cordobainitiative.org\/recent_programs.html, and the third in Doha in 2009 with sponsorship from the United Nations \"Alliance of Civilizations\" program; see Rushda Majeed, \"300 Young Muslim Leaders from 75 Countries Set the Agenda for Positive Change,\" ASMA Society Press Release, December 17, 2008. As of 2015, the \"Young Leaders\" (formerly, Muslim Leaders of Tomorrow) section of ASMA's website, accessed May 2, 2015, <http:\/\/www.muslimleadersoftomorrow.org\/>, was no longer operative.\n\n. The initial description of WISE could be found on the 2006 iteration of \"Cordoba Initiative: Principal Programs,\" ASMA Society, accessed February 1, 2006, http:\/\/www.asmasociety.org\/cordoba\/programs.html. By the November 2006 conference, Khan had gathered \"150 Muslim women from 25 nations . . . to accelerate the leadership of Muslim women and empower them as bridge builders and change agents\" (accessed February 8, 2008, http:\/\/www.cordobainitiative.org\/recent_programs.html). Not long after, she began to petition US foundations for funds to create a legal training program for Muslim women.\n\n. The initial Shariah Project description is drawn from the 2006 iteration of \"Cordoba Initiative: Principal Programs,\" ASMA Society, accessed February 1, 2006, http:\/\/www.asmasociety.org\/cordoba\/programs.html. Rauf later changed the name of the program to the \"Shariah Index Project.\" He explained its purpose in _Moving the Mountain_ , 17.\n\n. Information on the Sister Fund and Hunt was found at Sister Fund, accessed March 18, 2010, http:\/\/www.sisterfund.org\/about\/sister-fund-mission-statement (site discontinued); and at Hunt Alternatives Fund, accessed March 18, 2010, http:\/\/www.huntalternatives.org\/pages\/71_mission_history.cfm (site discontinued), an organization established by Helen LaKelly Hunt and her sister, Ambassador Swanee Hunt, in 1981; and can still be found in Helen LaKelly Hunt, _Faith and Feminism: A Holy Alliance_ (New York: Atria Books, 2004).\n\nThe Global Fund for Women granted their 2006 donation under their \"Expanding Civic and Political Participation\" program, a listing for which may be found in the Fund's grant database at Global Fund for Women, accessed June 6, 2012, <http:\/\/www.globalfundforwomen.org\/what-we-do\/our-grantmaking>. The Sister Fund does not make its grantmaking history public but has supported ASMA since 2005.\n\n. Daisy Khan provided this description of the inaugural WISE event, which ran from November 17\u201319, in a widely distributed press release that appeared on the website of The American Muslim (TAM), accessed December 14, 2007, <http:\/\/theamericanmuslim.org\/tam.php\/features\/articles\/muslim_women_leaders_launch_global_movement_to_empower_muslim_women\/0011550>.\n\n. \"Grants and Grantees\" database, Rockefeller Brothers Fund, accessed February 8, 2008, http:\/\/www.rbf.org\/grant\/11331\/american-society-muslim-advancement-0 (site discontinued) and June 6, 2012, http:\/\/www.rbf.org\/grant\/11329\/cordoba-initiative (site discontinued). On the latter date, ASMA's name was changed on grant records to \"the American Society for Muslim Advancement.\"\n\n. According to their grants database, the Luce Foundation gave ASMA a onetime planning grant in 2009 in the amount of $150,000 for Khan's expanded project to \"develop a graduate program in Islamic law for Muslim women\" and another two-year grant for the program in 2011 in the amount of $210,000; see \"Theology Program Recent Grants,\" Henry Luce Foundation, accessed June 6, 2012, <http:\/\/www.hluce.org\/theologyrecentgrants.aspx>.\n\n. In 2007, Khan received a $60,000 fellowship award for \"Racial Justice\" with the Prime Movers program of the Hunt Alternative Fund. Although ASMA does not deal with issues of race, specifically, that year's \"Women's Rights\" fellowship had already been granted to Zainab Salib of Women for Women International; see \"Our Fellows,\" Prime Movers, accessed June 7, 2012, <http:\/\/www.prime-movers.net\/our-fellows\/>. Then, in 2008, the Carnegie Corporation of New York awarded ASMA a grant of $300,000 for media outreach; see \"American Society for Muslim Advancement\" on \"Grants Database,\" Carnegie Corporation of New York, accessed June 6, 2012, http:\/\/carnegie.org\/grants\/grants-database\/.\n\nBy 2008, the ASMA Society's institutional sponsors included, in the order listed on their website, the \"Ford Foundation, Rockefeller Brothers Fund], Rockefeller Philanthropy [Advisors] . . . Global Fund for Women, William and Mary Greeve Foundation, the Sister Fund, the Danny Kaye & Sylvia Fine Foundation, the Graham Charitable Foundation, the Derek (sic) Family Foundation, the Henry Luce Foundation, the Elizabeth Foundation, The Ms. Foundation, and Hunt Alternatives\"; see \"About Us: Our Supporters,\" ASMA Society, accessed January 3, 2008, [http:\/\/www.asmasociety.org\/about\/p_support.html. ASMA's acknowledgement of the \"Derek\" Family Foundation is likely a typo and meant to reflect funding from the Deak Family Foundation, with which Robert Leslie Deak\u2014a security analyst, collaborator with, and donor to ASMA and Cordoba\u2014is affiliated.\n\nAfter the Ground Zero Mosque controversy, Rauf became embroiled in a lawsuit with Deak over allegations of mismanaged and unreported funds, including the millions of dollars in support that the Cordoba Initiative had received from the Malaysian government; see Ariel Kaminer, \"Ex-Leader of Planned Mosque Near Ground Zero Settles Suit with Donor,\" _New York Times_ , June 7, 2013.\n\n. T. S. M. Mohammed, Esquire, \"Certificate of Amendment to the Certificate of Incorporation of American Sufi Muslim Association,\" filed with the State of New York Department of State, November 8, 2006.\n\n. Founded in 1994 and headquartered in Washington, DC, The Council on American-Islamic Relations (CAIR) describes itself as \"a Muslim civil liberties and advocacy group\" that participates in inter-religious endeavors, particularly ones centered on human rights; see \"CAIR at a Glance,\" Council on American-Islamic Relations, accessed February 8, 2008, <http:\/\/www.cair.com\/AboutUs\/CAIRataGlance.aspx>. A revised website with a slightly updated description, accessed May 13, 2016, is available at <http:\/\/www.cair.com\/about-us\/cair-at-a-glance.html>.\n\n. Others included actor Richard Gere, president of Healing the Divide and chairman of the board of the International Campaign for Tibet, Dr. Mohammad El Baradei of the International Atomic Energy Agency, and \"Spiritual Leader and Humanitarian\" H. H. Sri Sri Mata Amritanandamayi Devi.\n\n. Hedges \"Muslim in the Middle.\"\n\n. Rauf, \"In God We Trust,\" 97 and 99, respectively.\n\n. Advisory Board members were listed on the Cordoba Initiative website, accessed December 20, 2007, http:\/\/www.cordobainitiative.org\/board.html.\n\n. All material for this section was derived from \"About Us,\" Cordoba Initiative, accessed September 8, 2006, <http:\/\/www.cordobainitiative.org\/> (emphasis added).\n\n. Ibid.\n\n. Ibid.\n\n. Ibid., emphasis added.\n\n. US-Muslim Engagement Project, _Changing Course: A New Direction for U.S.-Muslim Relations_ (Washington, DC, and Cambridge, MA: Search for Common Ground and the Consensus Building Institute, 2009), 20. Details on the members of the Leadership Group\u2014which also included former US Deputy Secretary of State Richard Armitage, author and FranklinCovey co-founder Steven Covey, and Dr. Ingrid Mattson, president of the nation's largest Muslim organization (the Islamic Society of North America), among others\u2014can be found in the report's Preface (ix\u2013xi).\n\n. Nasr discussed the group's work in an interview with CNN's Drew Griffith, CNN online, December 29, 2007 (accessed April 21, 2009), http:\/\/www.cnn.com\/video\/#\/video\/bestoftv\/2007 _\/_ 12 _\/_ 29\/nasr.bhutto.killing.cnn?iref=videosearch (site discontinued).\n\n. On private-sector led economic development, see US-Muslim Engagement Project, _Changing Course_ , 20.\n\n. \"Imam Feisal Abdul Rauf,\" under the \"Staff Bios\" section of \"Who We Are,\" Cordoba Initiative, accessed December 10, 2009, http:\/\/www.cordobainitiative.org\/?q=content\/staff-bios#Randy%20Benn (site discontinued) and Courtney Erwin, Secretary, \"Amended and Restated Bylaws of the Cordoba Initiative, A Colorado Nonprofit Corporation,\" filed with the Colorado Secretary of State, June 2, 2009.\n\n. \"Our Mission,\" Cordoba Initiative, accessed December 10, 2009, http:\/\/www.cordobainitiative.org\/?q=content\/our-mission (site discontinued).\n\n. Quote from the \"Muslims in the U.S.\u2014Isolated, Integrated, Assimilated?\" report featured on \"Selected Events 2007,\" Consulate General of the United States in Dusseldorf, Germany, accessed January 10, 2008, http:\/\/duesseldorf.usconsulate.gov\/dusseldorf\/khan021407.html (site discontinued). As of 2009, ASMA also noted Khan's trip on its calendar of events under \"Daisy Khan's Speaking Tour of Germany, Topics: Interfaith Relations, Assimilation, Identity, Activism,\" ASMA Society, accessed July 6, 2009, <http:\/\/www.asmasociety.org\/calendar\/index.html>. The content of this site has since been revised.\n\n. All quotes here are derived from Khan, \"Faith and Feminism.\"\n\n. Ibid.\n\n. \"About WISE,\" Women's Islamic Initiative in Spirituality and Equality, accessed October 7, 2010, <http:\/\/www.wisemuslimwomen.org\/about\/>.\n\n. \"WISE  ,\" Women's Islamic Initiative in Spirituality and Equality, accessed March 13, 2015, <http:\/\/www.wisemuslimwomen.org\/arabic\/about\/>.\n\n. Lila Abu-Lughod, _Do Muslim Women Need Saving?_ (Cambridge: Harvard University Press, 2013), 184.\n\n. Personal interview, Daisy Khan, January 25, 2008.\n\n. \"Our Mission\" and \"Our History,\" respectively, ASMA Society, accessed March 13, 2010, http:\/\/www.asmasociety.org\/about\/index.html.\n\n. Daisy Khan, \"Balancing Tradition and Pluralism,\" _Sojourners Magazine_ 38, no. 2 (February 2009): 15. Part of this essay derived from Daisy Khan, \"Faith-Based Feminism: the Most Powerful Model,\" which appeared on the now-defunct _On Faith_ (blog), _Newsweek-Washington Post_ , October 24, 2008 (accessed November 12, 2008), http:\/\/newsweek.washingtonpost.com\/onfaith\/daisy_khan\/2008 _\/_ 10\/faith_based_feminism_the_most.html (site discontinued).\n\n. Daisy Khan, \"Why the World was Rooting for Obama,\" _On Faith_ (blog), _Newsweek-Washington Post_ , November 6, 2008 (accessed November 12, 2008), http:\/\/newsweek.washingtonpost.com\/onfaith\/daisy_khan\/2008 _\/_ 11\/why_the_world_was_rooting_for.html (site discontinued).\n\n. Personal interview with Feisal Abdul Rauf, April 26, 2008.\n\n. Catherine Rampell, \"The 'Great Recession': A Brief Etymology,\" _New York Times_ , March 11, 2009 and \"The Recession Has (Officially) Ended,\" _New York Times_ , September 20, 2010, respectively.\n\n. Feisal Abdul Rauf, \"Obama's Opportunity in Cairo,\" _On Faith_ (blog), _Newsweek-Washington Post_ , June 2, 2009 (accessed June 3, 2009), http:\/\/newsweek.washingtonpost.com\/onfaith\/panelists\/feisal_abdul_rauf\/2009 _\/_ 06 _\/_ obamas_opportunity_in_cairo.html (site discontinued).\n\n. Rauf, _What's Right with Islam_ , 152.\n\n. Ibid., 29.\n\n. Ibid., 6.\n\n. Ibid, 123\u201326, 144\u201345.\n\n. Ibid., 124\u201325. \"While this does not explain all the causes of religious aggression, or aggression committed in the name of religion,\" Rauf argued, \"I believe that a significant portion of religious fundamentalism of the twentieth century (Muslim, Christian, Jewish, Hindu, Sikh) was a result of this defensive reaction against perceived attack.\"\n\n. Zain Abdullah briefly describes the Malcolm Shabazz mosque in \"West African 'Soul Brothers' in Harlem: Immigration, Islam, and the Black Encounter,\" in _Black Routes to Islam_ , edited by Manning Marable and Hisham D. Aidi (New York: Palgrave Macmillan, 2009), 255\u201356. Hisham D. Aidi describes other black and Latino Muslim opinions about and efforts to compensate for economic inequality in \"Jihadis in the Hood: Race, Urban Islam, and the War on Terror,\" _Black Routes to Islam_ , edited by Manning Marable and Hisham D. Aidi (New York: Palgrave Macmillan, 2009), 288\u201389. On Siraj Wahhaj, see Barrett, _American Islam_ , 100\u2013133.\n\n. Barrett, _American Islam_ , 126.\n\n. Quoted in Barrett, _American Islam_ , 131.\n\n. See Chapters 1 and 2 for a discussion of neoliberalism in this context. On changes in American political economy, particularly as compared to other nation-state models, see Nancy Fraser, _Justus Interruptus: Critical Reflections on the \"Post-Socialist\" Condition_ (New York: Routledge, 1997).\n\n. See Patricia Cohen, \"Public Sector Jobs Vanish, Hitting Blacks Hard,\" _New York Times_ , May 24, 2015.\n\n. See the \"History\" section of \"About Us,\" Inner-City Muslim Action Network, accessed May 15, 2015, www.imancentral.org\/about\/.\n\n. Quote from Sally Quinn, \"Daisy Khan: 'When will Muslims Be Accepted?,'\" _On Faith_ (blog), _Newsweek-Washington Post_ , August 19, 2010 (accessed May 15, 2015), http:\/\/www.faithstreet.com\/onfaith _\/_ 2010 _\/_ 08 _\/_ 19 _\/_ daisy-khan-when-will-muslims-be-accepted\/3848 (site discontinued).\n\n. \"Is This the End for Obama's 'New Beginning' Between the US and Muslims?,\" _Newsweek_ , September 10, 2010, <http:\/\/www.newsweek.com\/end-obamas-new-beginning-between-us-and-muslims-72209>.\n\nChapter 5\n\n. Rauf and Fariha, who owned the building, had some differences over Sufi practice. Among other things, Fariha allowed for a measure of democratic deliberation in her _tariqa_. Nevertheless, in deference to the shaykh she and Rauf once shared, Fariha allowed Rauf to serve as the mosque's imam and accepted Rauf's dervishes as substitutes when he was away. For other differences, see Corbett, \"Dhikr.\"\n\n. Personal communication, March 2, 2007.\n\n. American liberalisms involve multiple strands of economic, social, and political philosophy that are beyond the scope of this chapter. Although there are diverse iterations of liberal philosophy, however, there are also common threads tying them together. As Talal Asad has argued, \"Western liberal conceptions of person and politics\" constitute a \"value-space [that] provides its advocates with a common political and moral language,\" even if those advocates disagree, to some extent, on the shape liberal polities should take. Individual sovereignty (essential to democratic politics) and toleration (constitutive of multiculturalism) are among the ideas central to that space; see Talal Asad, \"Responses\" in _Powers of the Secular Modern: Talal Asad and His Interlocutors_ , edited by David Scott and Charles Hirschkind (Palo Alto: Stanford University Press 2006), 209\u201310.\n\n. Personal interview with 'Adli, June 15, 2007.\n\n. Field notes, May 6, 2005.\n\n. The \"greater jihad\" ( _jihad al-akbar_ , which stands in contrast to the \"lesser\" military jihad), is a theme in classical Islamic texts that has taken on new resonance in the last few centuries. This distinction predates the modern period but became newly salient in the strategies of Sufi orders that fought various colonial forces. For how practices of _tasawwuf_ occupied changing places in nineteenth- and twentieth-century debates about civic and political action, including greater and lesser jihads, see Juliane Hammer, \"The Soul of Islam: Writing and Publishing as Engaged Sufism,\" _Journal for Islamic Studies_ 26 (2006): 36\u201370 and Sirriyeh, _Sufis and Anti-Sufis_ , 27\u201353.\n\n. Rauf's dervishes were hardly alone in extolling multiculturalism and emphasizing the need to overcome differences. Markus Dressler has demonstrated that, while several New York City Sufi orders are ethnically exclusive, dervishes do cross between them (and between Sunni and Shi'i _tariqa_ s) and are sometimes members of multiple communities; see \"Pluralism and Authenticity: Sufi Paths in Post-9\/11 New York\" in _Sufis in Western Society: Global Networking and Locality_ , edited by Markus Dressler, Ron Geaves, and Gritt M. Klinkhammer (London: Routledge, 2013), 77\u201396. Marcia Hermansen likewise argues that boundaries between American Sufi groups have become more fluid in recent decades; see \"What's American about American Sufi Movements?,\" in _Sufism in Europe and North America_ , edited by David Westerlund (New York: Routledge Corizon, 2004), 38.\n\n. According to one MUSNY focus group facilitator, the three phases of the research project involved canvassing New York neighborhoods and mapping the existence of mosques, interviewing Muslim leaders and elites, and conducting ninety-minute focus group interviews with lay practitioners; Sufi Focus Group Interview II, MUSNY Project, June 2, 2003. In what follows, I rely on material from the third phase.\n\n. Personal communication with MUSNY research director Louis Cristillo, March 4, 2008.\n\n. In conducting fieldwork at Masjid al-Farah, I interacted with and interviewed members of Rauf's Sufi group, members of Shaykha Fariha's group, Sufis who attended both groups' _dhikr_ meetings, and attendees of Friday services who did not belong to either. But I focused mainly on the dervishes most closely associated with Rauf from 2004 to 2010.\n\n. Field notes, March 30, 2007.\n\n. Field notes, November 4, 2004.\n\n. Turner, _Islam_ , 13.\n\n. Personal communication, October 21, 2005.\n\n. Personal interviews with Brother Malik, February 16 and 23, 2007.\n\n. Information about Pedram is drawn from my own interviews and observations, as well as from his interviews with a journalist who visited the mosque and _dhikr_ group in 2000; see Gooch, _Godtalk_ , 293.\n\n. _Sacred Web: A Journal of Tradition and Modernity_ is edited by M. Ali Lakhani, who founded Sacred Web Publishing (which produces the journal) in Vancouver, Canada, in 1995; see William Stoddart, Introduction to _The Timeless Relevance of Traditional Wisdom_ by M. Ali Lakhani (Bloomington, IN: World Wisdom, 2010), xiii\u2013xxi.\n\n. Personal communication, January 12, 2006.\n\n. Personal communication, September 15, 2006.\n\n. Field notes, September 8, 2006.\n\n. Nasr argued that \"the most essential division within Islam is the 'vertical' hierarchy of the Sacred Law ( _Shar\u012bah_ ), the Way ( _Tar\u012bqah_ ) and the Truth ( _Haq\u012bqah_ ), the first being the exoteric aspect of the Islamic revelation, divided into the Sunni and the Sh\u012b'ite interpretations of the tradition and the latter two the esoteric aspects which are usually known under the denomination of Sufism\"; _An Introduction to Islamic Cosmological Doctrines_ (Albany: State University of New York, 1993), 18. Similarly, Rauf explained in his first book that \"Muslims regard the outer expression . . . as being the domain of the law ( _Shar\u012b'ah_ ) and the inner expression as the spiritual path (t _ar\u012bqah_ ) which must be internally realized to attain spiritual maturity\"; Rauf, _Islam: A Search for Meaning_ , 89.\n\n. As Rauf described it to worshippers in 2007, \"' _ilm al-batin_ ,' inner knowledge, esoteric knowledge, was a province of what later began to be known as _tasawwuf_ \" (March 30, 2007).\n\n. Feisal Abdul Rauf, Introduction to _The Universal Spirit of Islam_ , by Judith Fitzgerald and Michael Oren Fitzgerald (Bloomington, IN: World Wisdom Press, 2006), xv.\n\n. Ibid., xvi.\n\n. As translated by Feisal Abdul Rauf, \"Multiculturalisms: Western, Muslim, and Future,\" _Cross Currents_ 55, no.1 (Spring 2005): 105.\n\n. Personal interview with 'Abd al-Raheem, February 9, 2007.\n\n. Personal interview with 'Adli, June 15, 2007.\n\n. As Mahmood Mamdani has argued, attributing conflict to \"cultural\" or even religious difference alone\u2014as in the so-called Clash of Civilizations thesis, according to which Muslims and Christians are inherently at war\u2014elides the political factors (such as colonialism, imperialism, and neo-imperialism) directly responsible for creating conflicts. Engaging in such culture talk allows individuals and nation-states to deny that their actions, choices, and\u2014in the case of governments\u2014foreign and domestic policy decisions have anything to do with creating national or international tensions; see _Good Muslim, Bad Muslim_ , 15\u201336.\n\n. For an illuminating history of how concerns about cultural difference have been entangled with colonialism, imperialist projects, apartheid regimes, and international rivalries, see Adam Kuper, _Culture: The Anthropologists' Account_ (Cambridge: Harvard University Press, 1999).\n\n. On these issues, see Jackson, _Islam and the Blackamerican_.\n\n. Field notes, September 23, 2005.\n\n. African American Focus Group Interview 1, MUSNY Project, July 13, 2000.\n\n. African American Focus Group Interview 2, MUSNY Project, May 23, 2003.\n\n. Sufi Focus Group Interview 1, MUSNY Project, September 12, 2000.\n\n. Ibid.\n\n. Those who were aware of these issues tended to be affiliated with Islam for a longer period of time _and_ affiliated with Muslims from less affluent socio-economic backgrounds. Such was the case with two of my interviewees of different genders, one of whom had worked with Muslims in New York correctional facilities and the other of whom was a community college instructor and single mother.\n\n. Andrea Elliot, \"Between Black and Immigrant Muslims: An Uneasy Alliance,\" _New York Times_ , March 11, 2007.\n\n. Field notes, March 16, 2007.\n\n. For MANA's purpose and history, see Karen Leonard, \"Finding a Place in the Nation,\" in _Religion and Social Justice for Immigrants_ , edited by Pierette Hondagneu-Sotelo (New York: Rutgers University Press, 2007), 54\u201355.\n\n. Field notes, March 16, 2007.\n\n. Sufi Focus Group Interview I, MUSNY Project, September 12, 2000.\n\n. Ibid. Columbia University researchers translated _adab_ as \"etiquette\" in transcripts of these interviews, _'urf_ as \"tradition,\" and _akhlaq_ as \"morality.\" As I argue here, though, that choice is one often conditioned by particular racial and economic experiences. The word _adab_ has a variety of meanings that have changed over time and context. As Gerhard B\u00f6wering has noted, _'urf_ and _adab_ were both used at one time to convey the notion of \"custom\" or \"customary law\" (depending on the relation of the words to others within larger conceptual frameworks), while _akhlaq_ and _adab_ were used to convey ethical and moral action. To complicate things further, as B\u00f6wering explained, _adab_ is also the modern Arabic word for literature, reflecting the term's early usage in connection with both Persian moral thought and Greek philosophy; see Gerhard B\u00f6wering, \"The _Adab_ Literature of Classical Sufism: Ansari's Code of Conduct,\" in _Moral Conduct and Authority, The Place of_ Adab _in South Asian Islam_ , edited by Barbara Daly Metcalf (Berkeley: University of California Press, 1984), 62\u201366.\n\n. Sufi Focus Group Interview I.\n\n. See Ira Lapidus, \"Knowledge, Virtue, and Action: The Classical Muslim Concept of _Adab_ and the Nature of Religious Fulfillment in Islam,\" and Barbara Daly Metcalf, \"Introduction,\" both in _Moral Conduct and Authority_ , edited by Barbara Daly Metcalf (Berkeley: University of California Press, 1984), 39 and 4, respectively. As these authors point out, Sufis were not alone in compiling _adab_ manuals but have historically outnumbered other authors of such literature.\n\n. B\u00f6wering, \" _Adab_ Literature,\" 62\u201387.\n\n. See Lapidus, \"Knowledge, Virtue, and Action,\" 50\u201351 and Charles Hirschkind, _The Ethical Soundscape: Cassette Sermons and Islamic Counterpublics_ (New York: Columbia University Press, 2006).\n\n. According to Ebrahim Moosa, classical Islamic traditions of subjectivity position the individual as one who takes shape only in relation to other persons; see _Ghazali and the Poetics of Imagination_ (Chapel Hill: University of North Carolina Press, 2005), 221, 224\u201332. Both Moosa and Lapidus argue that _adab_ is not only a matter of codes of conduct, but is also part of the overarching intersubjective process that governs the formation of the self; see Lapidus, \"Knowledge, Virtue, and Action,\" 59\u201360.\n\n. Ernst, _Shambhala Guide_ , 211. See also Qumar al-Huda, \"The Light Beyond the Shore in the Theology of Proper Sufi Moral Conduct (Adab),\" _Journal of the American Academy of Religion_ 72, no. 2 (June 2004): 461\u201384.\n\n. \"Community Life and Adab\" on \"Invitation to Union,\" Nur-Ashki Jerrahi, accessed April 18, 2008, http:\/\/www.nurashkijerrahi.org\/union.htm.\n\n. Wolfe, \"Invitation and Education.\"\n\n. Field notes, March 16, 2007.\n\n. Field notes, November 10, 2006.\n\n. Personal interview with Charley, February 21, 2007.\n\n. Personal communication, May 11, 2007.\n\n. Ibid.\n\n. Quoted in Elliot, \"Between Black and Immigrant.\"\n\n. Field notes, December 1, 2006.\n\n. One of the most detailed accounts of the New York Police Department's extensive surveillance campaigns\u2014which, while undertaken for nearly a decade, resulted in uncovering zero terror plots\u2014can be found in Matt Appuzzo and Adam Goldman, _Enemies Within: Inside the NYPD's Secret Spying Unit and bin Laden's Final Plot Against America_ (New York: Simon and Schuster, 2014).\n\nChapter 6\n\n. In the statement, Rauf and Lord Carey criticized the media for republishing cartoons from a Danish children's book depicting the Prophet as a terrorist. The full text was available on the website of the Tanenbaum Center for Interreligious Understanding, accessed February 3, 2008, http:\/\/www.tanenbaum.org\/voices_for_peace.html (site discontinued).\n\n. Field notes, February 3, 2006.\n\n. See Marr, _Cultural Roots_ and McAlister, _Epic Encounters_.\n\n. On saving Muslim women, see Abu-Lughod, _Do Muslim Women Need Saving?_. On Oprah Winfrey, see McAlister, _Epic Encounters_ , 281\u201382.\n\n. Field notes, Sister Fund \"Faith and Feminism\" meeting, April 27, 2006.\n\n. Personal communication, June 6, 2007.\n\n. Annemarie Schimmel, _Mystical Dimensions of Islam_ (Chapel Hill: University of North Carolina Press, 1975), 229.\n\n. Field notes, March 9, 2007.\n\n. Although issues of male-female relations and the rights of women, as my interlocutors understood them, are the primary subjects of this chapter, it is important to note that gender was a more fluid (though not greatly) identity in my fieldwork than such a dualistic framing can capture. For example, as with many communities that observe gender separations of some kind, there was a great deal of homosocial behavior among young men who would have identified as heterosexual, if asked, but who did not exhibit many of the assertive attitudes associated with masculinity in the United States. There were also observant gay Muslims who attended the mosque. Although sexuality was rarely a topic of conversation, and the gay Muslim who attended most frequently professed a life of celibacy, several of Rauf's closest dervishes tacitly accepted his sexual orientation. For his part, Rauf maintained that wanton sexuality was a matter of concern, but not necessarily homosexuality. As Rauf told WNYC radio host Brian Lehrer in 2012 (quoting President Obama), his views on the subject were \"evolving\"; see \"Beyond Ground Zero,\" _Brian Lehrer Show_ , May 11, 2012, <http:\/\/www.wnyc.org\/story\/208230-beyond-ground-zero\/>. On gay Muslim Americans and some of the issues they face, see Afzal, _Lone Star Muslims: Transnational Lives and the South Asian Experience in Texas_ (New York: New York University Press, 2015), 124\u201351, as well as Puar, _Terrorist Assemblages_.\n\n. See Abu-Lughod, _Do Muslim Women Need Saving?_ , 54\u2013112 and Juliane Hammer, \"Studying American Muslim Women: Gender, Feminism, and Islam,\" in _The Cambridge Companion to American Islam_ , edited by Juliane Hammer and Omid Safi (New York: Cambridge University Press, 2013), 330\u201344.\n\n. Janet Jakobsen discusses these coalitions in _Working Alliances and the Politics of Difference: Diversity and Feminist Ethics_ (Bloomington: University of Indiana Press, 1998). Challenges to western feminism arose alongside the development of postcolonial studies with collaborations such as Chandra Talpade Mohanty, Ann Russo, and Lourdes Torres, eds., _Third World Women and the Politics of Feminism_ (Bloomington: Indiana University Press, 1991) and M. Jacqui Alexander and Chandra Talpade Mohanty, eds., _Feminist Genealogies, Colonial Legacies, Democratic Futures_ (New York: Routledge, 1997).\n\n. Clara Connolly, \"Washing Our Linen: One Year of Women Against Fundamentalism,\" _Women Against Fundamentalism_ no. 1 (1990): 5\u20137. The journal was published from 1990\u20131996 and featured the work of notable postcolonial scholars such as Gayatri Spivak and Homi Bhabha.\n\n. The idea that every religion has fundamentalists or is prone to fundamentalism is a very recent one, popularized in the 1970s by academics and politicians, as demonstrated by Simon A. Wood and David Harrington Watt, _Fundamentalism: Perspectives on a Contested History_ (Columbia: University of South Carolina Press, 2014), as well as David Harrington Watt, _The Antifundamentalists_ (Ithaca: Cornell University Press, forthcoming); see also Corbett, \"Islamic 'Fundamentalism.'\"\n\n. WAF was founded, in part, by Southhall Black Sisters, a British organization of women of African and Asian descent who were accused of \"washing their dirty laundry\" in public when their demands for attention to women's issues supposedly threatened the unity of British antiracism coalitions between black and Asian communities. Established in 1980, the Southhall Black Sisters stood at the nexus of several mainstream liberation movements (e.g., antiracism and antisexism) and yet as insiders of none. Some of this history is recounted in Gabriele Griffin, \"The Struggle Continues\u2014an Interview with Hannana Siddiqui of Southhall Black Sisters,\" in _Feminist Activism in the 1990s_ , edited by Gabriele Griffin (Abingdon, U.K.: Taylor & Francis, 1995), 79\u201389.\n\nSome postcolonial feminists have seen WAF as a model for creating multicultural alliances because the organization involved women from religious minority groups and because they argued that fundamentalism occurs in _every_ religion, including Christianity. Thus, WAF's politics appear to be absent of racial and religious biases; see, for example, Inderpal Grewal and Caren Kaplan, Introduction to _Scattered Hegemonies: Postmodernity and Transnational Feminist Practices_ , edited by Inderpal Grewal and Caren Kaplan (Minneapolis: University of Minnesota Press, 1994), 23\u201327. Yet, because crosscultural use of the term \"fundamentalism\" (a term that is indigenous to American Protestant history) to describe religions other than Protestant Christianity is the direct result of neoimperial politics and discourses, as Jan Nederveen Pieterse pointed out in an article in WAF's journal, this is a rather ironic argument for postcolonial feminists to make.\n\n. For analysis of and debates over such projects, see the essays in Gisela Webb, _Windows of Faith: Muslim Women's Scholar-Activists in North America_ (Syracuse: Syracuse University Press, 2000) and Juliane Hammer, _American Muslim Women, Religious Authority, and Activism: More Than a Prayer_ (Austin: University of Texas Press, 2013).\n\n. For discussions of these issues, see Carolyn Moxley Rouse, _Engaged Surrender: African American Women and Islam_ (Berkeley: University of California Press, 2004); Marie Griffith, _God's Daughters: Evangelical Women and the Power of Submission_ (Berkeley: University of California Press, 2000); and Saba Mahmood, _The Politics of Piety: The Islamic Revival and the Feminist Subject_ (Princeton: Princeton University Press, 2005).\n\n. Hishaam D. Aidi, in addition to Rouse, describes how failures of American liberalism to translate into economic and social gains, particularly in inner cities plagued by state withdrawal as well as deindustrialization and white and middle-class flight, has coincided with an increase in emphasis on \"traditional\" (nuclear) family structures and on complementary gender roles; see \"Jihadis,\" 288.\n\n. Yvonne Yazbeck Haddad, Jane I. Smith, and Kathleen M. Moore, _Muslim Women in America: the Challenge of Islamic Identity Today_ (New York: Oxford University Press, 2006), 122\u201323. On Mattson's position, see also Hammer, _American Muslim Women_ , 127\u201329.\n\n. \"History,\" Women in Islam, accessed December 16, 2013, http:\/\/www.womeninislam.org\/about.html (site discontinued) and \"Building Civic Participation and Interfaith Collaboration\" Women in Islam, accessed December 16, 2013, http:\/\/www.womeninislam.org\/subprograms4.htm (site discontinued).\n\n. Women in Islam and Islamic Social Services Associations, _Women Friendly Mosques and Community Centers: Working Together to Reclaim Our Heritage_ (New York: Women in Islam, 2005).\n\n. For a brief analysis of WISE see Abu-Lughod, _Do Muslim Women Need Saving?_ , 180\u201386 and Grewal, _Islam is a Foreign Country_ , 323\u201324.\n\n. Rauf, _What's Right with Islam_ , xviii and _Moving the Mountain_ , 24, 107\u201334.\n\n. All quotes from Deborah are derived from our interview at her home on June 14, 2007.\n\n. Max Doms and Ethan Lewis, Federal Reserve Bank of San Francisco, \"The Narrowing of the Male-Female Wage Gap,\" _FRSB Economic Letter_ , Federal Reserve Bank of San Francisco, June 29, 2007, accessed December 19, 2013, <http:\/\/www.frbsf.org\/economic-research\/publications\/economic-letter\/2007\/june\/narrowing-male-female-wage-gap\/>.\n\n. On December 16, 2013, New York State Senator Jeff Klein acknowledged the crisis of inaccessible childcare. Due to the challenges working women face in finding affordable care, particularly in New York City, Klein said, the legislature was working on plans to reintroduce a childcare subsidy. As of 2016, they had yet to do this; \"Anthony Shorris; Reasons to Love NY; NYS Senate,\" _Brian Lehrer Show_ , December 16, 2013, <http:\/\/www.wnyc.org\/story\/the-brian-lehrer-show-2013-12-16\/>.\n\n. This tendency to perform \"traditional\" gender roles at home is true of both women and men in \"gender-atypical\" occupations; Daniel Schneider, \"Gender Deviance and Household Work: The Role of Occupation,\" _American Journal of Sociology_ 117, no. 4 (January 2012): 1029\u201372.\n\n. For women raised as Christians, this often had to do with the Trinity\u2014the Christian doctrine that there is only one God but in three parts: father, son (Jesus), and holy spirit. Many converts conveyed their inability to reconcile Christianity's professed monotheism with this three-part idea and their greater comfort with the Islamic doctrine of monotheism, in which God is just one and Jesus is a revered prophet.\n\n. Deborah made this comparison to images of Saudi Arabian Islam, which she believed had caused her to misunderstand the tradition and had dissuaded her from converting. However, Emily also echoed the distinction between the Islam she associated with Saudi Arabia (and believed she had experienced at another mosque in the city) and the faith and practice she came to learn about from Rauf and other Sufis; personal interview, March 9, 2007.\n\n. Personal communication, March 30, 2007.\n\n. Muhammad Abdul-Rauf, _Marriage in Islam: A Manual_ (Alexandria, VA: Al-Saadawi, [1972] 2000). For more on Abdul-Rauf's understandings of gender roles, see Kecia 'Ali's discussion of him in _Sexual Ethics and Islam: Feminist Reflections on Qur'an, Hadith, and Jurisprudence_ (New York: Oxford Oneworld, 2006), 91\u201394, 179; as well as Haddad, Smith, and Moore, _Muslim Women in America_ , 147, 178.\n\n. Abdul-Rauf, \" _Sunnah_ and _Hadith_ ,\" 71.\n\n. Field notes, March 9, 2007.\n\n. On these desires and the ways black American Muslim women negotiate them, see Rouse, _Engaged Surrender_.\n\n. Khan, \"E.R. Doctor,\" 75.\n\n. Personal communication, June 6, 2007.\n\n. Personal interview with Keesha, May 18, 2007.\n\n. For more on these differences, see Corbett, \"Dhikr.\"\n\n. On the origin of the distinction between \"drunk\" and \"sober\" Sufism, see Jawid A. Mojaddedi, \"Getting Drunk with Abu Yazid or Staying Sober with Junayd: The Creation of a Popular Typology of Sufism,\" _Bulletin of the School of Oriental and African Studies_ 66, no. 1 (2003), 1\u201313.\n\n. Although I never heard Khan support women-led prayer, Asra Nomani, a controversial advocate of the practice, recalls Khan's support for it; see Asra Q. Nomani, _Standing Alone in Mecca: An American Muslim Woman's Struggle for the Soul of Islam_ (New York: HarperOne, 2005), 211. On Khan's initial support and later reversal, see also Hammer, _American Muslim Women_ , 51.\n\n. Field notes, March 9, 2007.\n\nChapter 7\n\n. Ralph Blumenthal and Sharaf Mowjood, \"Muslim Prayers and Renewal Near Ground Zero,\" _New York Times_ , December 9, 2009.\n\n. Ingraham is quoted in Justin Elliot, \"How the 'Ground Zero Mosque' Fear Mongering Began,\" _Salon_ , August 16, 2010 (accessed July 15, 2015), http:\/\/www.salon.com\/2010 _\/_ 08 _\/_ 16\/ground_zero_mosque_origins\/ and in Bobby Ghosh, \"Islamaphobia: Does America Have a Muslim Problem?,\" _Time_ , August 30, 2010 (accessed July 15, 2015), http:\/\/content.time.com\/time\/magazine\/article\/ _0,9171,2011936\u2013_ 2,00.html.\n\n. Blumenthal and Mowjood, \"Muslim Prayers.\"\n\n. Quoted in Elliot, \"How the 'Ground Zero Mosque.'\"\n\n. See Blumenthal and Mowjood, \"Muslim Prayers\" as well as Karen Zraik and Verena Dobnick, \"Park51 Islamic Center Opens its Doors Near Ground Zero,\" _Huffington Post_ , September 22, 2011 (accessed July 15, 2015), http:\/\/www.huffingtonpost.com\/2011 _\/_ 09 _\/_ 22\/park51-islamic-center-ope_n_975585.html.\n\n. Aziz Poonawalla, \"Q&A with Sharif el-Gamal about Park51, NYC,\" _Beliefnet_ , July 24, 2010 (accessed July 15, 2015), http:\/\/blog.beliefnet.com\/cityofbrass\/2010 _\/_ 07 _\/_ qa-with-sharif-el-gamal-about.html#ixzz0vv08HVwp.\n\n. See \"About Us,\" Lower Manhattan Development Corporation, accessed July 15, 2015, <http:\/\/renewnyc.com\/overlay\/AboutUs\/>.\n\n. See Poonawalla, \"Q&A.\"\n\n. See Ghosh, \"Islamaphobia\" and John Aloysius Farrell, \"Colin Powell's Endorsement of Barack Obama and Eloquence About Anti-Muslim Bigotry in the U.S.\" _U.S. News and World Report_ online, October 20, 2008 (accessed July 15, 2015), http:\/\/www.usnews.com\/opinion\/blogs\/john-farrell\/2008 _\/_ 10 _\/_ 20 _\/_ colin-powells-endorsement-of-barack-obama-and-eloquence-about-anti-muslim-bigotry-in-america.\n\n. See \"About Us,\" National Republican Trust, accessed July 15, 2015, <http:\/\/www.nationalrepublicantrust.org\/about.html>.\n\n. Edward E. Curtis, IV, \"Islam Has Long History Downtown: Why the 'Ground Zero Mosque' Belongs in Lower Manhattan,\" _New York Daily News_ online, July 23, 2010, <http:\/\/www.nydailynews.com\/opinion\/islam-long-history-downtown-ground-zero-mosque-belongs-manhattan-article-1.202169#ixzz0vv1Bp0LN>.\n\n. On the trend among conservative politicians, see Sarah Posner, \"Welcome to the Shari'ah Conspiracy Theory Industry: How the American Right Demonizes Islam for Political Gain,\" _Religion Dispatches_ , March 8, 2011, <http:\/\/www.religiondispatches.org\/archive\/politics\/4335\/welcome_to_the_shari%27ah_conspiracy_theory_industry\/>.\n\n. Scott Shane, \"Killings in Norway Spotlight Anti-Muslim Thought in U.S.,\" _New York Times_ online, July 24, 2011, http:\/\/www.nytimes.com\/2011\/07\/25\/us\/25debate.html?pagewanted=all&_r=0.\n\n. See Elliot, \"How the 'Ground Zero Mosque.'\"\n\n. Matt Sledge, \"Mark Williams, Tea Party Leader, Says Muslims Worship 'Monkey God,'\" _Huffington Post_ , May 20, 2010, http:\/\/www.huffingtonpost.com\/2010 _\/_ 05 _\/_ 20\/mark-williams-tea-party-l_n_582591.html.\n\n. See Nicholas Confessore and Eric Lipton, \"A Big Check, and Gingrich Gets a Big Lift,\" _New York Times_ , January 9, 2012. Adelson's positions have not gone unchallenged in the US or in Israel; see Peter Beinart, \"Sheldon Adelson's Culture of Hate,\" _Ha'aretz_ (February 4, 2014).\n\n. On local New York politicians see Rosemary R. Hicks, \"Religious Pluralism, Secularism, and Interfaith Endeavors,\" in _The Cambridge Companion to American Islam_ , edited by Juliane Hammer and Omid Safi (New York: Cambridge University Press, 2013), 156\u201369. On analyses of terrorist threats, see Daryl Johnson, _Right-Wing Extremism: Current Political and Economic Climate Fueling Resurgence in Radicalization and Recruitment_ (Washington, DC: US Department of Homeland Security Office of Intelligence and Analysis, 2009).\n\n. \"Remarks by President George W. Bush at the 20th Anniversary of the National Endowment for Democracy,\" National Endowment for Democracy, November 6, 2003 (accessed October 8, 2010), <http:\/\/www.ned.org\/george-w-bush\/remarks-by-president-george-w-bush-at-the-20th-anniversary>.\n\n. On violence and discrimination against Muslims across the country, see Chapter 1, n. 29 and n. 30.\n\n. See Poonawalla, \"Q&A.\"\n\n. Hicks, \"Religious Pluralism,\" 163.\n\n. Field notes, September 10, 2010.\n\n. Paul Vitello, \"Amid Rift, Imam's Role in Islam Center is Sharply Cut,\" _New York Times_ , January 14, 2011.\n\n. Quoted in Poonawalla, \"Q&A.\"\n\n. \"Beyond Ground Zero,\" _Brian Lehrer Show_ , May 11, 2012, <http:\/\/www.wnyc.org\/story\/208230-beyond-ground-zero\/>.\n\n. See Bayrak, _Memoirs_ , 164\u2013216.\n\n. The Sufi Circle of Long Island described such service on their website as \"a form of worship\u2014and also a powerful method of self-transformation\"; see \"Community Service,\" Sufi Circle of Long Island, accessed April 15, 2008, http:\/\/www.suficircle.com\/community_service.html (site discontinued).\n\n. See official White House blogger Jesse Lee, \"The President's Speech in Cairo: A New Beginning,\" White House, June 4, 2009 (accessed November 11, 2014), <https:\/\/www.whitehouse.gov\/blog\/NewBeginning\/>. The full text of Obama's June 4, 2009, speech is available at White House, accessed April 29, 2016, <https:\/\/www.whitehouse.gov\/the-press-office\/remarks-president-cairo-university-6-04-09>.\n\n. Corbett, \"For God and Country.\"\n\n. Bush clarified his aims in establishing the OFBCI at an event celebrating the Office; \"President Bush Attends Office of Faith-Based and Community Initiatives' National Conference,\" White House Office of the Press Secretary, June 26, 2008 (accessed November 11, 2014), http:\/\/georgewbush-whitehouse.archives.gov\/news\/releases\/2008 _\/_ 06 _\/_ 20080626\u201320.html (site discontinued).\n\n. For an overview of the seven laws, passed between 1996 and 2000 and collectively known as \"Charitable Choice,\" that allow the US government to contract with religious service providers, see the George W. Bush administration's White House memo, \"Charitable Choice: the Facts,\" accessed November 11, 2014, <http:\/\/georgewbush-whitehouse.archives.gov\/government\/fbci\/guidance\/charitable.html>. On the laws' effects, see Sheila S. Kennedy, \"Religious Philanthropies and Government Social Programs,\" in _Religion in Philanthropic Organizations: Family, Friend, Foe?_ , edited by Thomas J. Davis (Bloomington: Indiana University, 2013), 144\u201367.\n\n. See \"USAID History,\" USAID, accessed August 17, 2011, <http:\/\/www.usaid.gov\/who-we-are\/usaid-history>.\n\n. Corbett, \"For God and Country.\"\n\n. Liora Danan, _Mixed Blessings: U.S. Government Engagement with Religion in Conflict-Prone Settings_ (Washington, DC: Center for Strategic and International Studies, 2007), 18\u201319.\n\n. Historically, Catholic Relief Services (the organization under scrutiny during the Kennedy administration Peace Corps controversy) and two evangelical organizations, World Vision and Samaritan's Purse, have received far more USAID funding than any other religiously affiliated organizations. Until passage of Charitable Choice legislation, however, this required creative solutions to objections raised by Congress and members of the public. In 1962, for example, World Vision administrators established an ostensibly nonconfessional branch of operations called World Vision Relief Organization, which began contracting with USAID almost immediately. CRS and other organizations did likewise and quickly regained some of the federal funding they had enjoyed prior to the Peace Corps controversy; see David P. King, \"Heartbroken for God's World: The Story of Bob Pierce, Founder of World Vision and Samaritan's Purse,\" in _Religion in Philanthropic Organizations: Family, Friend, Foe?_ , edited by Thomas J. Davis (Bloomington: Indiana University Press, 2013), 71\u201392 and David M. Ackerman and Vee Burke, _Charitable Choice: Background and Issues_ (Huntington, NY: Novinka Books, 2001).\n\n. \"Obama Announces White House Office of Faith-based and Neighborhood Partnerships,\" White House Office of the Press Secretary, February 5, 2009 (accessed November 11, 2014), <https:\/\/www.whitehouse.gov\/the-press-office\/obama-announces-white-house-office-faith-based-and-neighborhood-partnerships>. Patel discusses his involvement with Rauf and Khan in _Acts of Faith: the Story of an American Muslim in the Struggle for the Soul of a Generation_ (New York: Beacon, 2007), 175\u2013177. Notably, IFYC\u2014like Catholic and Jewish service organizations of the twentieth century\u2014borrowed its patriotic metaphor of service from the military. Its original name was Interfaith Youth _Corps_ ; see Patel, _Acts of Faith_ , 114.\n\n. White House Office of the Press Secretary, \"President Obama Unveils 'United We Serve,' Calls on All Americans to Commit to Meaningful Volunteer Service in Their Daily Lives,\" White House, June 17, 2009 (accessed August 9, 2011), <http:\/\/www.whitehouse.gov\/the_press_office\/President-Obama-Unveils-United-We-Serve-Calls-on-All-Americans-to-Commit-to-Meaningful-Volunteer-Service-in-Their-Daily-Lives>.\n\n. Barack Obama, \"Ramadan Message 2009,\" MuslimServe.org, accessed February 21, 2011, http:\/\/muslimamericanserve.org\/ (site discontinued).\n\n. Islamic Society of North America, \"ISNA Newsletter,\" email communication, August 21, 2009.\n\n. All quotes from MuslimServe, \"A Call to Action: Join MuslimServe National Campaign for Service and Understanding on 9\/11,\" Islamic Circle of North America, accessed November 14, 2014, http:\/\/www.icna.org\/wp-content\/uploads\/2010 _\/_ 09\/Muslims-ServeCalltoAction2010.pdf.\n\n. \"Cordoba House \u2013 New York City,\" Cordoba Initiative, last accessed July 23, 2010, http:\/\/www.cordobainitiative.org\/?q=content\/cordoba-house-new-york-city (site discontinued).\n\n. Quoted in Poonawalla, \"Q&A.\"\n\n. Anne Barnard, \"Muslim Leaders Unite Behind Center,\" _New York Times_ , September 21, 2010.\n\n. Sharif El-Gamal, \"'Ground Zero Mosque' Will Serve All N.Y.: Developer of Site Says Sarah Palin is Welcome, Too,\" _New York Daily News_ , August 4, 2010.\n\n. Ibid.\n\n. Hussein Rashid, \"The Leadership Failure of Park51,\" _Religion Dispatches_ , August 22, 2010, http:\/\/religiondispatches.org\/the-leadership-failure-of-park51 _\/_.\n\n. Field notes, September 10, 2010.\n\n. Nicholas Loomis, \"Islamic Center: 9\/11\/2010,\" _New York Times_ online, September 12, 2010, http:\/\/www.nytimes.com\/video\/nyregion\/1248069013001\/islamic-center _-_ 9 _-_ 11 _-_ 2010.html.\n\n. Nicolas Esposito, \"New Yorkers Hold Candlelight Vigil in Support of Islamic Center,\" _Ecumenical News_ , September 11, 2010 (accessed July 15, 2015), <http:\/\/www.ecumenicalnews.com\/article\/new-yorkers-hold-candlelight-vigil-in-support-of-islamic-center-1169>.\n\n. Barnard, \"Muslim Leaders Unite.\"\n\n. Ibid.\n\n. Personal interview, Daisy Khan, January 25, 2008.\n\n. Blumenthal and Mowjood, \"Muslim Prayers.\"\n\n. Vitello, \"Amid Rift.\"\n\n. See n. 35.\n\n. Kareen Faim, \"Islamic Center Seeks 9\/11 Recovery Grants for Lower Manhattan,\" _New York Times_ , November 22, 2010.\n\n. Paul Vitello, \"Planners of Mosque Considering New Project,\" _New York Times_ , March 29, 2011.\n\n. In 2011, the Obama administration announced that its new federal budget would cut in half community service block grants to grassroots groups in poor areas and would require those groups to compete for the remaining money. Four years later, a House subcommittee approved a spending bill that cut federal financing for the Corporation for National and Community Service\u2014the federal agency that oversees AmeriCorps and many other programs involved in the Obama administration's United We Serve initiative; see Jackie Clams, \"Some of Obama's Favorite Programs to Face Cuts, Budget Director Says,\" _New York Times_ , February 5, 2011 and Editorial Board, \"More Community Service, Not Less,\" _New York Times_ , June 19, 2015.\n\n. Vitello, \"Planners of Mosque.\"\n\n. Feisal Abdul Rauf, \"Help Stop the Downward Spiral,\" Cordoba Initiative, December 7, 2010 (accessed December 21, 2010), http:\/\/www.cordobamovement.org\/2010 _\/_ 12\/help-stop-the-downward-spiral-a-special-note-from-imam-feisal\/ (site discontinued).\n\n. \"Beyond Ground Zero,\" _Brian Lehrer Show_ , May 11, 2012, <http:\/\/www.wnyc.org\/story\/208230-beyond-ground-zero\/>.\n\n. \"Creating a New Vision of Islam in America,\" _Fresh Air_ , May 9, 2012, <http:\/\/www.npr.org\/2012\/05\/09\/152192549\/creating-a-new-vision-of-islam-in-america>.\n\n. See Michael Lind, \"Is Barack Obama a Socialist,\" _Salon_ , November 1, 2008, www.salon.com _\/_ 2008 _\/_ 11 _\/_ 01\/socialism\/. Lind is the cofounder of the nonpartisan New American Foundation think tank, which is dedicated to promoting centrist politics, as Lind describes with cofounder Ted Halstead in _The Radical Center: The Future of American Politics_ (New York: Anchor, 2002).\n\n. Rauf, _Moving the Mountain_ , 168.\n\n. See, for example, Feisal Abdul Rauf, \"Every Little Bit Helps,\" _Huffington Post_ , April 1, 2015, <http:\/\/m.huffpost.com\/us\/entry\/6986340>, where Rauf argued that American Muslims needed to achieve the kind of freedom from discrimination that Martin Luther King, Jr., had sought (and mainly achieved, he implied) for black Americans.\n\n. Rauf, _Moving the Mountain_ , 180\u201381, 168.\n\n. See Daisy Khan, \"Is the NYPD Really Against Muslims?\" _Huffington Post_ , January 27, 2012, <http:\/\/www.huffingtonpost.com\/daisy-khan\/dont-fire-nypd-commissioner-raymond-kelly_b_1236859.html>. Additionally, one week after Mayor Bill deBlasio announced Kelly's replacement and condemned both the Stop and Frisk tactics and Muslim surveillance, Khan participated in awarding Kelly a plaque for his commendable service to New York City's Muslims; see Jamie Schram, \"Muslim Community Honors Ray Kelly,\" _New York Post_ , December 10, 2013. Khan is featured in the accompanying photo next to the former police commissioner.\n\n. Leaders of ISNA and ICNA also struggled, at times, to see the issues from the perspective of black American communities. In an April 30, 2015, email to their listserve (\"ISNA Clarifies Baltimore Press Release\"), for example, ISNA leaders\u2014who had previously described demonstrations over the Baltimore Police killing of Freddie Gray, an unarmed black man, as \"riots\"\u2014apologized for focusing on protestors rather than on the police brutality, violence, and systemic racial oppression that gave rise to the protests. Meanwhile, ICNA\u2014which also condemned the violence\u2014held their 2015 annual convention in the city of Baltimore in the effort to \"stand up for justice.\" ICNA Volunteer, \"Can We Roar Louder than Thunder,\" email communication, May 3, 2015.\n\n. Emine Saner, \"Why Are Sikhs Targeted by Anti-Muslim Extremists?\" _Guardian_ online, August 8, 2012, <http:\/\/www.theguardian.com\/world\/2012\/aug\/08\/sikhs-targeted-anti-muslim-extremists>.\n\n. Cordoba Initiative, \"Cordoba Initiative Newsletter\u2013June 2015,\" email communication, July 9, 2015.\n\n. Rauf, _Moving the Mountain_ , 162.\n\n. Ibid., 100.\n\n. Corbett, \"For God and Country.\"\n\n. See Mahmood, \"Secularism, Hermeneutics, and Empire\" and Elizabeth Shakeman Hurd, \"How International Relations Got Religion and Got it Wrong,\" _Monkey Cage_ (blog), _Washington Post_ , July 9, 2015, http:\/\/www.washingtonpost.com\/blogs\/monkey-cage\/wp\/2015 _\/_ 07 _\/_ 09\/how-international-relations-got-religion-and-got-it-wrong\/. Importantly, Rauf has involved Muslim scholars of a wide range of perspectives and nationalities in his Shariah Index Project. Thus, while his writings may echo US State Department perspectives, his larger council\u2014like the Shura Council Khan created through WISE\u2014is characterized by vigorous debate over such issues and is hardly a mouthpiece for the US government. On the Shura Council, see Abu-Lughod, _Do Muslim Women Need Saving?_ , 181\u201382.\n\n. William Dalrymple, \"The Muslims in the Middle,\" _New York Times_ , August 16, 2010.\n\n. \"Open Phones: Lower Manhattan Residents on Park51,\" _Brian Lehrer Show_ , August 25, 2010, <http:\/\/www.wnyc.org\/story\/92089-open-phones-lower-manhattan-residents-park51\/>.\n\n. Daisy Khan, \"REMINDER: America: The Gateway to Humanity's Future\u2014An Evening with Irish Mystic Lorna Byrne,\" email communication, April 23, 2012.\n\n. Cordoba Initiative, \"Cordoba Initiative Newsletter\u2014October, 2014,\" October 30, 2014 and Naz Ahmed Georgas, Executive Director, \"Qawwali and the Spiritual Alchemy of Music,\" email communications, May 5, 2015.\n\n. Cordoba Initiative, \"Cordoba Initiative Newsletter\u2013May 2015,\" June 4, 2015 and \"Cordoba Initiative Newsletter\u2013June 2015,\" email communications, July 9, 2015.\n\n. Anne Barnard, \"Developers of Islamic Center Try a New Strategy,\" _New York Times_ , August 1, 2011.\n\n. Park51 Community Center, _Park51 Community Center_ (New York: Park51 Community Center, 2011), 8.\n\n. Jillian Nannery, \"Park51 Announces Grand Opening: First Major Public Event Tonight at 6:30 p.m.,\" Park51 Press Release, September 21, 2011.\n\n. Matt Flegenheimer, \"Rent Dispute Endangers Mosque Plan,\" _New York Times_ , October 16, 2011.\n\n. Patrick McGeehan, \"Con Ed Sells Building Near Ground Zero Where Plans for Mosque Caused Uproar,\" _New York Times_ , August 20, 2014.\n\n. Personal interview with Sajjad Chowdhry and Hanadi Doleh, February 6, 2016.\n\n. The rally was organized by a former Marine who claimed to be protecting \"free speech\" after two Muslims from Arizona attempted to attack an anti-Islamic convention organized by Pamela Geller; see Sara Sidner and Ed Payne, \"Mohammed Cartoon Contest: Protest Held Outside Phoenix Mosque,\" CNN online, May 30, 2015 (accessed July 15, 2015), http:\/\/www.cnn.com\/2015 _\/_ 05 _\/_ 29\/us\/mohammed-cartoon-contest\/index.html.\n\n. Grynbaum, \"Daisy Khan.\"\n\n. Quoted in the Interfaith Center of New York, \"ICNY Statement on the Chattanooga Shooting,\" email communication, July 17, 2015. See also Frank Defrank, \"Sikh Day of Service Honors Wisconsin Hate Crime Victims,\" _Oakland Press News_ , August 5, 2013, http:\/\/www.theoaklandpress.com\/general-news\/20130805 _\/_ sikh-day-of-service-honors-wisconsin-hate-crime-victims.\n\nConclusion\n\n. ISNA 2014 Annual Convention Program, \"Generation Rise: Elevating Muslim American Culture,\" accessed November 11, 2014, <http:\/\/webcache.googleusercontent.com\/search?q=cache:MkwR5beJ6CEJ>: www.isna.net\/uploads _\/_ 1 _\/_ 5 _\/_ 7 _\/_ 4 _\/_ 15744382 _\/_ 51_convention_program_14_web.pdf+&cd=2&hl=en&ct=clnk&gl=us&client=firefox-a (site discontinued), page 8. Compare with page 1 of the ISNA Development Foundation's 2013 \"Sponsorship Package\" pamphlet for potential corporate donors, accessed August 15, 2014, http:\/\/www.isna.net\/uploads _\/_ 1 _\/_ 5 _\/_ 7 _\/_ 4 _\/_ 15744382 _\/s_ ponsorship_booklet_csrl_2013.pdf.\n\n. \"ICNA Announces its 2015 Convention Theme,\" Islamic Circle of North America, February 2, 2015 (accessed July 15, 2015), http:\/\/www.icna.org\/icna-announces-its-2015\u2013convention-theme\/?utm_source=ICNA&utm_campaign=31e8bcdd5a-Newsletter_93&utm_medium=email&utm_term=0 ___ 00fd1cfa59\u201331e8bcdd5a-869045&mc_cid=31e8bcdd5a&mc_eid=3941311693 (site discontinued).\n\n. Islamic Circle of North America, \"ICNA National Update: North East Storms: ICNA Relief Unveils a Broad Plan,\" email communication, October 29, 2012, and Daisy Khan, \"Donate to Hurricane Sandy Relief,\" email communication, November 1, 2012.\n\n. See Shariq A. Siddiqui, \"Myth vs. Reality: Muslim American Philanthropy since 9\/11,\" in _Religion in Philanthropic Organizations_ , edited by Thomas J. Davis (Bloomington: Indiana University, 2013), 203\u201314 and Kathryn Ruff, \"Scared to Donate: An Examination of the Effects of Designating Muslim Charities as Terrorist Organizations on the First Amendment Rights of Muslim Donors,\" _New York University Journal of Legislation and Public Policy_ 23 (March 2006): 447\u2013502.\n\n. Pew Research Center Religion and Public Life Project, \"How Americans Feel About Religious Groups,\" Pew Research Center, July 16, 2014 (accessed November 11, 2014), http:\/\/www.pewforum.org\/2014 _\/_ 07 _\/_ 16\/how-americans-feel-about-religious-groups\/.\n\n. Jonathan M. Katz, \"Funeral For Muslims Killed in Chapel Hill Draws Thousands,\" _New York Times_ , February 12, 2015.\n\n. Malise Ruthven, \"Inside the Islamic State,\" _New York Review of Books_ (July 9, 2015): 74\u201377.\n\n. Scott Shane, \"Homegrown Extremists Tied to Deadlier Toll than Jihadists in U.S. Since 9\/11,\" _New York Times_ , June 24, 2015.\n\n. McAlister, _Epic Encounters_ , 198\u2013234.\n\n. Quoted in Frances Robles, \"Dylann Roof Photos and a Manifesto Are Posted on Website,\" _New York Times_ (June 20, 2015).\n\n. Jelani Cobb, \"Terrorism in Charleston,\" _New Yorker_ (June 29, 2015): 17\u201318.\n\n. In February 2016, for example, ISNA leaders launched a \"Striving for Justice\" conference tour. The first stop was Ferguson, Missouri, where Arab American social justice activist Linda Sarsour joined Imam Siraj Wahhaj to discuss, among other things, \"Creating a Just Society,\" \"Mercy as a Tool for Uniting Communities,\" and the \"Role of Masjid is Promoting Social Justice.\" The event ended with a \"Community Service Recognition Award Ceremony\" during which Wahhaj, the keynote speaker, addressed the issue of creating \"Social Justice for Community at Large.\" Event information can be found at \"ISNA Conference in St. Louis,\" ISNA, accessed April 29, 2016, <http:\/\/www.isna.net\/uploads\/1\/5\/7\/4\/15744382\/isnastltentativeprogram.pdf>.\n\nThat same year, Sarsour cofounded the multiethnic MPower Change organization to mobilize Muslims of all racial, ethnic, and economic backgrounds in pursuing social justice, as well as to highlight the work of those already doing so. 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September 22, 2011 (accessed July 15, 2015). http:\/\/www.huffingtonpost.com\/2011 _\/_ 09 _\/_ 22\/park51-islamic-center-ope_n_975585.html (site discontinued).\nINDEX\n\nAbdul-Rauf, Muhammad: and American Enterprise Institute, , 58\u201360, ; _Bilal Ibn Rabah: A Leading Companion of the Prophet Muhammad_ , 53\u201354; and community service, ; criticism of US by, 61\u201362; and economics, , , , , 58\u201360, ; on future of Muslim Americans, ; immigration of, , ; influence of, on son, 30\u201332, , , , 167\u201368; and intra-Muslim relations, , , 52\u201356; interreligious activities of, , 57\u201358, ; and Islamic Center of Washington, DC, , , , 54\u201355, 57\u201358, 61\u201362; _Marriage in Islam_ , 167\u201368; as Muslim American spokesperson, , ; _A Muslim's Reflections on Democratic Capitalism_ , , 58\u201360; and Nation of Islam, 52\u201354, 224 _n_ 33; and New York Islamic Center, , ; and Sufism, ; taken hostage, 54\u201355; on women's role in Islam, 167\u201368\n\nAbdur Rashid, Talib, 90\u201391, , , 151\u201352, 233 _n_ 95\n\nAbrahamic Americanness,\n\nAbrahamic covenants,\n\nAbrahamic ethic and tradition: Khan's promotion of, ; popularization of, , ; Rauf's concept of, 1\u20133, , , , 20\u201321, , 36\u201338, , , , 216 _n_ 16; Rauf's promotion of, , ,\n\nAbrahamic commonality narrative: criticisms of, , ; Rauf's version of, , , , 20\u201322, , , , , 63\u201364, , , , , , ,\n\nAbu Ghraib prison abuses, ,\n\n_Adab_ (etiquette; ethics), , , 145\u201351, , 244 _n_ 42\n\nAl-Adawiya, Aisha, ,\n\nAdelson, Sheldon,\n\nAffirmative action, 29\u201330, 49\u201350, , , , 227 _n_ 75\n\nAfghanistan, , , , ,\n\nAlbright, Madeline, ,\n\nAlger, William,\n\n_America At Risk_ (film), ,\n\nAmerican Academy of Religion (AAR),\n\nAmerican Creed, 19\u201327\n\nAmerican dream, , , , , 228 _n_ 6\n\nAmerican Enterprise Institute (AEI), , , , , , , , , 218 _n_ 59, 220 _n_ 75\n\nAmerican Exceptionalism, , , , , ,\n\nAmerican Petroleum Institute,\n\nAmerican Sufi Muslim Association\/American Society for Muslim Advancement (ASMA Society): criticism of work of, 104\u2013106; Cordoba Initiative in relation to, , , 98\u201399, ; evolution of, , 70\u201371, , , 94\u201396, , 106\u201316, 195\u201396, , ; founding of, , , , , , 228 _n_ 3; funding for, , , 107\u2013108, , , 238 _n_ 61, 238 _n_ 62; growth in staff of, ; name change for, , ; programs of, , 70\u201371, , 93\u201394, 96\u201397, ; purpose of, 5\u20136, 87\u201389, 97\u201398, , , 228 _n_ 3; Sufism downplayed in programs of, 96\u2013101, . _See also_ Muslim Leaders of Tomorrow (MLT); Women's Islamic Initiative in Spirituality and Equity\/Equality (WISE)\n\nAmericanization, , , ,\n\nAncient Arabic Order of the Nobles of the Mystic Shrine (Shriners),\n\nAncient Egyptian Arabic Order Nobles of the Mystic Shrine,\n\nAnnan, Kofi,\n\nAnthony, Susan B.,\n\nAnticolonialism,\n\n'Arabi, Ibn,\n\nArab-Israeli conflicts, , , ,\n\nArmstrong, Karen, ,\n\nAsad, Talal, 213 _n_ 24, 214 _n_ 25, 241 _n_ 3\n\nASMA Society. _See_ American Sufi Muslim Association\/American Society for Muslim Advancement\n\nAspen Institute, , , 234 _n_ 6\n\nAssimilation, , , , 35\u201337, 41\u201342, , , , ,\n\nAtat\u00fcrk, Mustafa Kemal, 73\u201374,\n\nAyaz, Farid,\n\nAl-Azhar University, , , , , , ,\n\n'Aziz, Karim 'Abdul,\n\nBadawi, Abdullah Ahmad,\n\nBaroody, William, , 58\u201359,\n\nBarry, Marion,\n\nBayrak, Tosun, 78\u201382, ,\n\nBell, Sean,\n\nBennett, John S., 94\u201395, 98\u201399, , , 234 _n_ 6\n\nBilalian movement, 53\u201356\n\n_Bilalian News_ (newspaper),\n\nBlack American Muslims: and cultural issues, , 144\u201345; and economic factors, 63\u201364, , 119\u201321, 142\u201343; and gender issues, , 248 _n_ 33; government relations with, 64\u201365, ; marginalization of, , 151\u201352; in politics, ; post World War II, 46\u201352; public perception of, , ; relations with immigrant Muslims, , 44\u201346, 53\u201354, 66\u201367, , 130\u201331, 142\u201343, , 152\u20133; and service, 121\u201322, ; and Sufism, 71\u201372, , 145\u2013146\n\nBlack Americans: Jews and, ; marginalization of, , , , , , , , 253 _n_ 68; opportunities of, 117\u201319; black American Muslim leaders' views on, 120\u201321; police killings of, 152\u20133, , 253 _n_ 68; Rauf's views on, , 119\u201321,\n\nBlack Lives Matter,\n\nBlack Power, ,\n\nBloomberg, Michael, ,\n\nBradley, Thomas,\n\nBrevik, Anders,\n\nBreyer, Stephen,\n\nBurckhardt, Titus,\n\nBush, George W., , 88\u201389, , , , , , ,\n\nCAIR. _See_ Council on American-Islamic Relations\n\nCamp David Accords, 57\u201358\n\nCanterbury, Archbishop of, ,\n\nCapitalism. _See_ Democratic capitalism\n\nCarey, Lord,\n\nCarnegie Corporation, , 238 _n_ 62\n\nCarter, Jimmy, , , , ,\n\nCastro, Fidel, , ,\n\nCathedral of Saint John the Divine, New York City, , 79\u201380, , 93\u201394\n\nCatholics and Catholicism: American values and, , , ; assimilation and acceptance of, , , , , ; anti-Catholic sentiment, , ; Gingrich's conversion to, , 26\u201327; as immigrants\/minorities, , , , , , , ; and neoliberalism, , 28\u201329, , , ; Protestantism and Judaism associated with, , 20\u201321, , , , , , ; service practices of, , 42\u201343, 188\u201389; and whiteness, ,\n\nCenter for Faith-Based and Community Initiatives,\n\nCharity: _adab_ and, 148\u201349; immigrants helped by, 35\u201337, 42\u201343; and market liberalism, ; Rauf on, , 34\u201336, , , , 221 _n_ 92; suspicion of Muslim contributions to,\n\nCharleston, South Carolina, shooting of black worshippers, ,\n\nChautauqua Institution, ,\n\nCheney, Dick, ,\n\nChesterton, G. K., 24\u201327\n\nChittick, William,\n\nChurch, Forrest, 23\u201328, ,\n\nCivil rights, , 47\u201349, , , , ,\n\nCivil society, , 34\u201335\n\nClinton, Bill, , , , , ,\n\nClinton, Hillary Rodham,\n\nClinton Global Initiative,\n\nCommunism, , , , , ,\n\nCommunity service. _See_ Service\n\nCongress of Racial Equality,\n\nConnolly, Clara,\n\nCoomaraswamy, Ananda, 229 _n_ 23\n\nCorbin, Henri, 230 _n_ 30\n\nCordoba, Spain, , ,\n\nCordoba Bread Fest, ,\n\nCordoba House\/Park51: controversy over, 2\u20135, , 17\u201319, 22\u201323, , , , , , , , , 183\u201387, 190\u201392, , 200\u2013202, 238 _n_ 62; early approval of, 183\u201384; funding for, , 193\u201394, 201\u20132; internal conflict over, 186\u201396; and moderate Islam, 1\u20132, , 183\u2013202; as multifaith institution, , ; museum proposed for, 201\u20132; Muslim criticisms of, 191\u2013192; and 9\/11, ; opening of, ; plans for, , , , 186\u2013187, 192\u201393, , 202\u2013203; Rauf's message after, 196\u2013200; renaming of, , ; and service, , , , 186\u201395; and Sufism, , 199\u2013202\n\nCordoba Initiative: AMSA in relation to, , , 98\u201399, ; Cordoba Movement, , ; evolution of, , , 107\u201313, 195\u201396; founding of, , , , , ; funding for, , ; interreligious character of, , , 109\u201311; as institution for combatting extremism, 25\u201326, , , ; Malaysian office of, , ; and moderation, 109\u201311; organizational structure of, 109\u201311; and politics, , 100\u2013101, ; programs of, , , , , , , , , 212 _n_ 18, 236 _n_ 32; promotion of, ; purposes of, , , , , ; and Sufism, , , , . _See also_ Cordoba House; Shariah Project; Shariah Index Project\n\nCorporations, 33\u201335, , , ,\n\nCouncil on American-Islamic Relations (CAIR), , , , , 238 _n_ 64\n\nCulture: as aesthetic traditions, , , , , , ; American, as compatible with Islam, , , , , , , , ; Muslim American understandings of, , ; patriarchal, ; role of, in Islamic practice, , , , , , , 129\u201335, 137\u201346\n\nDaley, Richard,\n\nDalrymple, William, 199\u2013200\n\n_Dargahs_ (meeting houses), , , , , ,\n\nDay of Seva,\n\nDeak Family Foundation, 238 _n_ 62\n\nDeclaration of Independence, , , 22\u201325, , , , ,\n\nDeism,\n\nDemocracy, Islam linked to, , , , , , , , , , , , , , 126\u2013127, , , , 216 _n_ 16\n\nDemocratic capitalism: Abdul-Rauf and, , 58\u20139; Nasr and, ; Novak and, 27\u201329, , 218n59, 219n62, 220n90; Rauf and, , , , , , , , 63\u201364, , , , , 118\u201319, , 197\u201398, 212n18; values of,\n\nDemocratic Party and party members, , , , , , , , ,\n\nDervishes: of Fariha, , , 131\u201333, , 170\u201373, ; of Muzaffer, 78\u201380, ; and Park51, , ; performances of, 78\u201379, ; relations among, , 135\u201350, 175\u201376, , 242 _n_ 7; of Rauf, , , , , , , , , , 135\u201350, 176\u2013181, , 186\u201387, , ; and whirling, 78\u201379,\n\n_Dhikr_ (Sufi practice of remembering God), , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 176\u201378, 179\u201380, 213 _n_ 22.\n\nDia Art Foundation, ,\n\nDiallo, Amadou,\n\nEgypt, , , , , , , , ,\n\nEgypt-Israel Peace Treaty (1979), ,\n\nEisenhower, Dwight D., , ,\n\nEisenhower Doctrine, 44\u201345\n\nEl-Gamal, Sharif, , 183\u201384, 186\u201387, 190\u201396, 201\u20132\n\nEllison, Keith,\n\nEmerson, Ralph Waldo,\n\nEqual Rights Amendment,\n\nEsposito, John,\n\nEvers, Medgar,\n\nExtremism: American fears of, 11\u201312, , , ; associated with particular racial groups, , 132\u201333, ; associated with Muslims involved in political activism, , , 151\u201352, ; moderate Islam as opposite of, , , ; ASMA and Cordoba proposals to combat, , 99\u2013100, ; Sufism as antidote to or opposite of, 88\u201389, 199\u2013200, 233 _n_ 89. _See also_ Fundamentalism\n\nFalwell, Jerry,\n\nFard, W. D.,\n\nFariha, Shaykha (Philippa de Menil): and _adab_ , 147\u201350, ; book store owned by, , ; _dhikrs_ of, , 170\u201371, ; early life of, 79\u201380; followers of, , , 131\u201333, , 170\u201373, ; leadership role of, , , 165\u201366, 172\u201373; male attitudes toward, , ; and Masjid al-Farah, , , , , 241 _n_ 1; and service, , , ; as successor to Muzaffer, 80\u201382\n\nAl-Faruqi, Isma'il, , , 60\u201364,\n\nFederal Bureau of Investigation (FBI), 11\u201312, , ,\n\nFederation of Islamic Associations of the US and Canada (FIA),\n\nFeminism, , 158\u201363, , 246 _n_ 11\n\nFirst Gulf War, , ,\n\nForbes, James,\n\nForeign policy: American concerns and interests, , , , , ; criticisms of, , , , 103\u20135, ; debates about role of Muslims in, 11\u201312; effects on American opinions of Muslims, , ,\n\nFoucault, Michel, 214 _n_ 25\n\nFounders, US, , , , , , , 218 _n_ 57\n\nFounding ideals and documents, US, 19\u201327, ,\n\nFox News, , ,\n\nFrager, Robert,\n\nFranklin, Benjamin,\n\nFreemasonry,\n\nFriedrich, Heiner, , ,\n\nFundamentalism: feminist opposition to, 159\u201360, 246\u201347 _n_ 14; gender as related to, , , ; history of American ideas about, 246 _n_ 13; Rauf's changing definition of, , , , 222 _n_ 117, 240 _n_ 95; Sufism vs., 3\u20134, , , , 212 _n_ 11. _See also_ Extremism\n\nGarrison Institute,\n\nGeller, Pamela, , 254 _n_ 86\n\nAl-Ghazali, Muhammad ibn,\n\nGI Bill, , ,\n\nGingrich, Newt, , , , 18\u201319, 22\u201328, 34\u201335, , , , 216 _n_ 8, 217 _n_ 34, 218 _n_ 54; _A Nation Like No Other_ , , 216 _n_ 7; _To Save America_ , , 216 _n_ 7\n\nGiuliani, Rudolph,\n\nGlazer, Nathaniel,\n\nGlobal Fund for Women,\n\nGoode, Virgil,\n\nGround Zero Mosque. _See_ Cordoba House\/Park51\n\nGu\u00e9non, Ren\u00e9, ; _The Crisis of the Modern World_ ,\n\nHafiz (Sufi poet),\n\nHalveti-Jerrahi Sufi order, 77\u201380, , , , , , 213 _n_ 22, 231 _n_ 58\n\nHamilton, Alexander, ,\n\nHanafi Muslims, 54\u201355\n\n_The Hate that Hate Produced_ (television show), 46\u201347\n\nHayek, Friedrich, , 226n58\n\nHenry Luce Foundation, , 238 _n_ 61, 238 _n_ 62\n\nHerberg, Will, , , 37\u201338, 41\u201342, , , , ,\n\nHirschfield, Brad,\n\nHixon, Lex (\"Shaykh Nur\"), 80\u201382\n\nHomosexuality, 245 _n_ 9\n\nHomosociality, 245 _n_ 9\n\nHostage crises: Hanafi Muslim instigators, , 54\u201355; in Iran, , , , ; public perception of Muslims after, , , , , ,\n\nHughes, Karen,\n\nHuman rights, , , , ,\n\nHunt, Helen LaKelley,\n\nHussein, Saddam, 64\u201365\n\nHuxley, Aldous, _The Perennial Philosophy_ ,\n\nICNA. _See_ Islamic Circle of North America\n\nICNA Relief,\n\nIgram, Abdallah,\n\n_Ijtihad_ (personal interpretation),\n\nImmigrants: abuse of, , ; acceptance of, 2\u20133, , , , , , , , ; affluence versus poverty of, , , , , , , ; assimilation of, , 35\u201337, 41\u201342, 62\u201363, 65\u201367, , , , ; charity of, 35\u201337, 42\u201343; economic strategies of, , , , ; racial strategies of, , , , ; upward mobility of, , ,\n\nIngraham, Laura,\n\nInner-City Muslim Action Network (IMAN), , 121\u201322\n\nInterfaith Center of New York,\n\nInterfaith Youth Core, , 36 _n_ 251\n\nInternal Revenue Service (IRS), , ,\n\nInternational Monetary Fund, 62\u201363,\n\nInternational Muslim Society,\n\nInter-Religious Peace Colloquium,\n\n_In the Spirit_ (radio show),\n\n_Inventory of U.S. Government and Private Organization Activity Regarding Islamic Organizations as an Aspect of Overseas Operations_ (report), 44\u201345\n\nIran, , , , , 61\u201362, 74\u201375, , , ,\n\nIraq, , 64\u201365, , , , , , ,\n\nIRS. _See_ Internal Revenue Service\n\nIslamic Center of Long Island, , , ,\n\nIslamic Center of New York, , ,\n\nIslamic Center of Washington, DC, , , , , 54\u201355, 57\u201358, , 223 _n_ 8\n\nIslamic Circle of North America (ICNA), , , , , , , 253 _n_ 68\n\nIslamic Cultural Center, New York City,\n\nIslamic Medical Association of North America,\n\nIslamic Party of North America,\n\nIslamic Relief, U.S.A., , 227 _n_ 81\n\nIslamic Social Services Associations,\n\nIslamic Society of North America (ISNA), , , 189\u201390, , , 239 _n_ 73, 253 _n_ 68, 255 _n_ 12\n\nIslamic State in Iraq and Syria (ISIS),\n\nIslamic Supreme Council of America,\n\nIslamophobia, , , ,\n\nISNA. _See_ Islamic Society of North America\n\nIsrael, , , , , 54\u201355, , , , 217 _n_ 34, 223 _n_ 19\n\nJames, William,\n\nJefferson, Thomas, , , ,\n\nJerrahi Order of America,\n\nJewish Community Center, , ,\n\nJews and Judaism: and American immigration, 36\u201337; and anti-Semitism, ; assimilation and acceptance of, , , , ; and attitudes toward Israel, ; black American relationships with, ; as immigrants\/minorities, , , , 42\u201343, , 65\u201366, , , ; Muslim American alliances with, , , 60\u201361, , , , , 188\u2013189; and neoliberalism, , 28\u201329, , , ; Protestantism and Catholicism associated with, , 20\u201321, , , ; and service, , 42\u201343, , 188\u2013189, , , 251n36; and whiteness, ,\n\nJihad, , , , , , 127\u2013128, , , , , , 241 _n_ 6\n\nJitkoff, Julia A.,\n\nJohnson, Lyndon B., , , , , , , 219\u2013220 _n_ 75\n\nJudeo-Christian ethic: Abrahamic commonality and, , , ; arguments about American identity grounded in, , 17\u201318, , 36\u201337, , , , , , , , ; Islam and Muslims in relation to, , , ,\n\nJudeo-Christian civilization and heritage, narrative of, , , , 17\u201319, , ,\n\n_Jum'ah_ (Friday noontime prayers), , , , , , , , ,\n\nKabbani, Muhammad Hisham, 88\u201389\n\nKashif, Ghayth Nur,\n\nKennedy, John F., , , , , 251 _n_ 35\n\nKhaalis, Hamaas Abdul,\n\nKhan, Daisy: and Abrahamic narrative, , , ; and ASMA, , , , 70\u201371, 87\u201388, , 93\u201399, , 105\u201319, , , , 228 _n_ 3; background of, , ; and Cordoba House, , , , , , 183\u20134, 186\u20137, 190\u201391, 193\u201395; and Cordoba Initiative, 109\u2013110, , , 195\u20137; dervishes' relations with, 174\u201376; _dhikrs_ in apartment of, , , 176\u201377; male attitudes toward, 174\u201375; marriage of, , ; moderate Islam promoted by, 4\u20137, , , , , ; on National September 11 Memorial and Museum board, ; and politics, , , 97\u201398, , , , 110\u201319, ; and service 5, , , , 190\u201391; and Sufism, 4\u20135, , , , , , , , , 228 _n_ 3; and women's issues, 155\u201356, , , , , , , 248 _n_ 39 ( _see also_ Women's Islamic Initiative in Spirituality and Equity\/Equality)\n\nKhan, Faiz, , , 70\u201371, , 101\u20136, , , , 168\u201370, , ,\n\nKhan, Faroque, , ,\n\nKhan, Inayat, 72\u201373\n\nKhan, Vilayat Inayat, ,\n\n_Khutbahs_ (sermons), , , , , , , 132\u201333, , 136\u201337, , , , , , 156\u201357, , 167\u201369,\n\nKing, Chris,\n\nKing, Martin Luther, Jr., , ,\n\nKing, Peter, ,\n\nKristol, Irving, , , 219\u2013220 _n_ 75, 226 _n_ 58\n\nLazio, Rick,\n\nLehrer, Brian, , ,\n\nLiberalism and liberal rights, , , , , , , , , , , , , , , , , , , 126\u20137, , , , , 155\u201362, 171\u201372, , 220 _n_ 75, 235 _n_ 14, 241 _n_ 3\n\nLiberation theology,\n\nLincoln, Abraham,\n\nLings, Martin,\n\nLived religion, , 214 _n_ 26\n\nLMDC. _See_ Lower Manhattan Development Corporation\n\nLocke, John, , , 26\u201327, , , 219 _n_ 75\n\nLomax, Louis,\n\nLower Manhattan Development Corporation (LMDC), , , 193\u2013194\n\nLumumba, Patrice,\n\nLuther, Martin,\n\nMadoff, Bernie,\n\nMahmood, Saba, 38\u201339\n\nMajlis Ash-Shura (Islamic Leadership Council) of New York,\n\nMalaysia, , , , , , , , , , , , , 238 _n_ 62\n\nMalcolm X (later, El Hajj Malik el Shabazz), , 47\u201348, , ,\n\nMamdani, Mahmood, , 243 _n_ 28\n\nMandela, Nelson,\n\nMarty, Martin,\n\nMaryamiyya Sufi order,\n\nMasjid al-Farah, New York City: attendance at, 125\u201326; communities behind, ; conflicts in, 125\u201329, , 178\u201381; founding of, , ; guest speakers at, 132\u201333; members of, 131\u201332, 135\u201338, 142\u201343, ; overflow worship space for, ; public perception of, ; Rauf as imam of, , ; Rauf's engagement with, , , , , , 175\u201376, 177\u201381; services in, ; women's roles and experiences in, 155\u201358, 161\u201381\n\nMasjid al-Taqwa, Brooklyn, , ,\n\nMattson, Ingrid, , 239 _n_ 73\n\nMcCarran-Walter Act (1952), 44\u201345\n\nMcVeigh, Timothy, ,\n\nMenil, Dominique de,\n\nMenil, John de,\n\nMenil, Philippa de. _See_ Fariha, Shaykha\n\nMeritocracy, 3\u20134, , , , , ,\n\nMevlevi Sufi order,\n\nMG3 Fund,\n\nMiddle East: American involvement in, 44\u201345, , , , , , 228 _n_ 6; and ISIS, ; pan-Islamic movements in, ; petro-politics of, ; Rauf's plans for and thoughts about, , , , , 212 _n_ 18; revolution in, , , , ; rivalries in,\n\nMLT. _See_ Muslim Leaders of Tomorrow\n\nModerate Islam: American promotion of, 11\u201312, 38\u201339, , 99\u2013100; American values linked to those of, , 6\u20137, 19\u201320, ; Cordoba House and, 183\u2013202; criticisms of, ; defining, 5\u20138, ; and gender issues, ; Khan's promotion of, 4\u20137, ; limitations of, 101\u20137, 122\u201323; and neoliberalism, 32\u201338; Rauf's promotion of, 1\u20138, , , , , 91\u201392, , , 199\u2013200; and service, 4\u20135, , , 42\u201343, 44\u201346, , , , , , 64\u201367, , , , , 157\u201358, 183\u2013184, 187\u2013197, 202\u2013203, 205\u2013209; stakes of, 11\u201312; Sufism and, 4\u20135, 99\u2013100, 199\u2013200\n\nModernism,\n\nMohammed\/Muhammad, Elijah, , 49\u201351\n\nMohammed, W. D., , , 53\u201357, , 64\u201365, , , 224 _n_ 29\n\nMontesquieu, Charles de,\n\nMoorish Science Temple,\n\nMorton, James Park, , ,\n\nMosque of Islamic Brotherhood, ,\n\nMossadegh, Mohammed,\n\nMoynihan, Daniel Patrick, , 218 _n_ 59\n\nMPower, 255n12\n\nMuhammad\/Mohammed, Elijah, , 49\u201351\n\nMuhammad, Prophet, , , , , ,\n\nMulticulturalism, , , , , , 241 _n_ 3\n\nMurdoch, Rupert,\n\nMuslim Alliance in North America (MANA), 121\u201322, , ,\n\nMuslim Americans: government alliances with, 44\u201346, 64\u201367, ; criticisms of US by, 61\u201362, 103\u20134, , 172\u201373; cultural identities and practices of, 128\u201335, 139\u201346; defining, 213 _n_ 21; and economic liberalism, 30\u201331, , , 56\u201358; post\u2013World War II, 44\u201351; institution-building among, ; pressures on, , 38\u201340, , ; public perception of, , 50\u201356, 93\u201394, , , ; relations among, 51\u201358, , 66\u201367, 137\u201346; relations between immigrant and black Americans, 44\u201346, 66\u201367, , 130\u201331, 142\u201343, ; and service, , , , , 143\u201344, , 189\u201390, 202\u20133, 205\u20136, ; Shi'i, , 133\u201334, , ; as slaves, ; social acceptance of, , , , 97\u201398, , , , 206\u20139; Sunni, , , 52\u201353; surveillance, brutality, and detention experienced by ( _see_ Surveillance, brutality, and detention experienced by Muslim Americans); violence against, , , , . _See also_ Black American Muslims; Second-generation Muslim Americans\n\nMuslim American Society,\n\nMuslim Association of Canada,\n\nMuslim Leaders of Tomorrow (MLT), , 106\u20137, , , , , , , , 236 _n_ 32, 237 _n_ 55\n\nMuslim Mosque, Inc., ,\n\nMuslimServe, 189\u201390, ,\n\nMuslims in New York Project (Columbia University), , , ,\n\nMuslim Students' Association,\n\nMuzaffer \u00d6zak al-Jerrahi, , 77\u201383, , , , , , 231 _n_ 53\n\nNaqshbandi-Haqqani Sufi Order,\n\nNasr, Seyyed Hossein, 74\u201377, , , , , , , 134\u201337, 229 _n_ 25, 230 _n_ 30, 243 _n_ 21\n\nNasr, Vali,\n\nNational Association for the Advancement of Colored People (NAACP),\n\nNational Cathedral, Washington, DC, ,\n\nNational Conference of Catholic Bishops,\n\nNational Council of Churches, , , ,\n\nNational Day of Service, ,\n\nNational Defense Education Act,\n\n_National Geographic_ (magazine),\n\nNational Republican Trust, 184\u201385\n\nNational Security Administration,\n\nNational Security Council,\n\nNational September 11 Memorial and Museum,\n\nNation of Islam (NOI), , , 46\u201355, , , , 141\u201342, 224 _n_ 29, 224 _n_ 30, 224 _n_ 33, 225 _n_ 47\n\nNatural law, , , , 26\u201327, 218 _n_ 57\n\nNatural religion,\n\nNeoconservatives, , , , , 219 _n_ 75\n\nNeoliberalism: Islamic principles compared to, 58\u201359; moderate Islam in relation to, 32\u201338; Muslim American values compared to, ; as prevailing US ideology, ; principles of, ; religious minorities and, 28\u201332; Sufism and, ; use of term, 218 _n_ 59\n\nNetherlands,\n\nNew Age spirituality, ,\n\nNew Deal, , , , , , 219 _n_ 75, 220 _n_ 90\n\n_Newsweek_ (magazine), ,\n\n_Newsweek-Washington Post_ (online news service), , ,\n\nNew York Civil Liberties Union,\n\n_New York Daily News_ (newspaper),\n\nNew York Police Department: Stop-and-Frisk tactics of, ; surveillance program of, , , 245 _n_ 58\n\n_New York Post_ (newspaper),\n\n_New York Times_ (newspaper), , , , , , , , , , , ,\n\nNo-Fly Lists,\n\nNOI. _See_ Nation of Islam\n\nNomani, Asra, 248 _n_ 39\n\nNovak, Michael, 27\u201329, , , , , , 58\u201360, , , , , , , 218 _n_ 57, 218 _n_ 59, 219 _n_ 62, 219 _n_ 63, 220 _n_ 90, 226 _n_ 58\n\nNur Ashki Jerrahi Sufi Order, 81\u201382, , , , , 147\u201348, , 170\u201372, 213 _n_ 22, 231 _n_ 58\n\nObama, Barack, , , 116\u201318, , , , 188\u201389, , 197\u201398, 218 _n_ 54, 245 _n_ 9\n\nOccupy Wall Street, ,\n\nOil crisis (1973),\n\n_On Faith_ (blog), , ,\n\nOperations Coordinating Board (OCB), 44\u201346\n\n_The O'Reilly Factor_ (television show),\n\nOrganization of Afro-American Unity,\n\nOrganization of Petroleum Exporting Countries (OPEC),\n\nOrientalism,\n\nPagels, Elaine,\n\nPahlavi, Reza Shah,\n\nPan-Africanism,\n\nPark51. _See_ Cordoba House\/Park51, 183\u201387, 192\u2013195, 201\u20133\n\nPataki, George,\n\nPatel, Eboo,\n\nPatriotism, , , , , , , ,\n\nPatterson, David,\n\nPeace Corps, , , , , 251 _n_ 35\n\nPenn, William,\n\nPerennialism, , , 136\u201338, 229 _n_ 23\n\nPluralism, , , ,\n\nPowell, Colin, ,\n\nPrayer Space, ,\n\nPrivilege, moderation of, , 148\u201350,\n\nProtestant work ethic, , , ,\n\nAl-Qadiri, Hamazah ibn 'Abbas, 87\u201388\n\nQadiri Sufi order,\n\nQur'an: inflammatory film about, ; motherhood as subject in, ; for oath of office ceremony, ; pluralism grounded in, , ; rational basis of, ; recitation of, 233 _n_ 91; service grounded in, ; threats against, ; women's status as reflected in,\n\nRacism: Faiz Khan and, ; immigrants' use of, , , ; moderate Islam in relation to, 3\u20134; Muslims' views on, 120\u201321; Rauf's acknowledgment of, ; recent incidents of, 152\u201353; systemic, , ; in United States, , , ,\n\nRahman, Fazlur,\n\n_Rak'at_ (cycles of prayer), 125\u201326\n\nRamadan, , , , , , , , ,\n\nRAND Corporation, 99\u2013100, , 222 _n_ 112, 234 _n_ 6,\n\nRashid, Hussein,\n\nRauf, Feisal Abdul: and Abrahamic narrative, 1\u20133, , , , , 20\u201322, , 31\u201333, 36\u201338, , , 63\u201364, , , , , , , , , ; American conservatives' similarity to, , 23\u201328, 34\u201335; criticism of US by, , ; and ASMA, , , 70\u201371, 87\u201389, 93\u201395, , 105\u201313; on black Americans, , 119\u201321, ; business career of, , ; and Cordoba House, 1\u20132, , , , , 183\u201387, 190\u201391, 193\u201395; and Cordoba Initiative, , , , 25\u201326, , , , 96\u201399, 107\u201313, 109\u201311, , , 194\u20137, ; criticisms of, , , , , , 180\u201381, , ; and democratic capitalism, , , , , , , , 63\u201364, , , , , 118\u201319, , 197\u201398, 212n18; development of thought of, 83\u201389, ; education of, , , ; father's influence on, 30\u201332, , , 167\u201368; immigration of, ; interreligious activities of, , , , 109\u201311, 185\u201386, ; _Islam, A Sacred Law_ , 84\u201387; _Islam: A Search for Meaning_ , , ; and Jerrahi Sufi order, , 81\u201383, 87\u201388, , , , ; leadership experience of, ; marriage of, , ; moderate Islam promoted by, 1\u20138, , , , , 91\u201392, , , 199\u2013200; and modernism, ; _Moving the Mountain_ , , 196\u201399; Nasr's influence on, 76\u201377; and politics, 95\u201397, 100\u2013101, 109\u201310, 187\u201388; and Qadiri Sufi order, 87\u201388, ; and race relations, , ; and service, , , , , , , 190\u201391; and Sufism, , 4\u20135, , , , 67\u201368, 70\u201371, , , 83\u201387, 90\u201392, 96\u201397, 199\u2013200; _What's Right with Islam_ , , , , , 18\u201320, , , , , , ; and women's role in Islam, 155\u201356, 165\u201368\n\nReagan, Ronald, , ,\n\nReligious freedom, , ,\n\nRepublican Party and party members, , , , , , , , , , , , , , ,\n\nReynolds, Mike, 17\u201318\n\nRight-wing militias, danger represented by, ,\n\nRiverside Church, New York City, 7\u20138\n\nRockefeller Brothers Fund, ,\n\nRoof, Dylann,\n\nRoosevelt, Franklin D., , , , , , , ,\n\nRumi, Jalal ad-Din, , , , , ; _Masnavi_ , ,\n\nRumsfeld, Donald,\n\nRushdie, Salman, _The Satanic Verses_ ,\n\n_Sacred Web_ (journal),\n\nSaid, Edward, 214 _n_ 25\n\nSantorum, Rick, 18\u201319\n\nSarsour, Linda, 255n12.\n\nSaudi Arabia, , , 64\u201365, , , , , 234 _n_ 6, 248 _n_ 28\n\nSchuon, Frithjof, , ,\n\nSecond-generation Muslim Americans: race consciousness of, 102\u20133; as Rauf's primary domestic audience, , , ; and social acceptance, 44\u201345, 62\u201363\n\nSecularism: criticisms of, , , , , ; and gender issues, 159\u201360; in Iran, ; Protestant bias of, , , , 222 _n_ 119; pressed upon Muslim Americans, 38\u201339; Rauf on, , , , , 222 _n_ 117; in Turkey, 73\u201374\n\nSeptember 11th (9\/11): Black American Muslims after, 130\u201331; Cordoba House and, ; Day of Service, , ; Faiz Khan and, 101\u20132, 105\u20136; interreligious services in memory of, ; pressures on Muslims after, , , , , 162\u201366; Rauf's message after, 1\u20132, 5\u20137, , , 93\u2013101, 115\u201316\n\nSeptember 11th Families for a Peaceful Tomorrow,\n\nSeptember 11th National Day of Service,\n\nService: government support for, , , , , , , , 188\u201389, , 220 _n_ 90, 251 _n_ 35, 252 _n_ 58; Muslim Americans and, 4\u20135, , 42\u201343, 44\u201346, , , , , , 64\u201367, , , 143\u201344, , 157\u201358, 183\u2013184, 187\u2013197, 202\u2013203, 205\u2013209; religious groups involved in, , , , , , 42\u201343, , 188\u201389, 251 _n_ 35; Sufism and, 4\u20135, , , 157\u201358; and women's obligations, 14\u201315, 158\u201359, 173\u201376\n\nSexuality, 245 _n_ 9\n\nEl Shabazz, El Hajj Malik. _See_ X, Malcolm\n\nShah, Muhammad Reza,\n\nShari'ah: and American culture, , ; and fears of, , 22\u201323, ; natural law synonymous with, 20\u201321; and Sufism, 84\u201386, , , ; and US law, , , 20\u201323,\n\nShariah Index Project, , , 212 _n_ 18, 254 _n_ 74\n\nShariah Project, 107\u20138, , 212 _n_ 18, 236 _n_ 32\n\nShi'i tradition, 133\u201334, ,\n\nShura Council, , 254 _n_ 74\n\nSikh Americans, , , 202\u2013203\n\nSikh Coalition,\n\nSister Fund, ,\n\n_60 Minutes_ (television show),\n\nSlave trade,\n\nSocialism: criticism of , 28\u201331, ; opposition to as a way of proving Americanness, , , , , , 55\u201364\n\nSocial welfare legislation and programs, , , , , , , , 220 _n_ 90\n\nSoho Properties, , 194\u201395,\n\n_Sojourners_ (magazine),\n\nSouthhall Black Sisters, 246 _n_ 14\n\nSt. Bartholomew's Church, New York City, , ,\n\nStewart, Jon,\n\nStop Islamicization of America (SIOA),\n\nStrauss, Leo, , 219 _n_ 75\n\nSufi Books, ,\n\nSufi Circle of Long Island,\n\nSufi Order International,\n\nSufi Order of the West,\n\nSufism: aesthetic aspects of, , , , , 93\u201394; black American Muslims and, 71\u201372, ; \"drunk\" vs. \"sober,\" ; vs. extremism or fundamentalism, 3\u20134; government promotion of, 67\u201368, , 88\u201389, 99\u2013100; as moderate Islam, 4\u20135, 99\u2013100, 199\u2013200; Muslim opposition to, , , 73\u201374; other Islamic practices contrasted with, 3\u20134, , , , ; Park51 and, 201\u20132; principles of, 3\u20134, 126\u201327; Rauf and, , , 4\u20135, , , , 67\u201368, 70\u201371, , , 83\u201387, 90\u201392, 96\u201397, 199\u2013200; service as component of, 4\u20135, , 157\u201358; striving as characteristic of, 32\u201334, , , 127\u201328, , 149\u201350; in United States, 71\u201383; West African, , 228 _n_ 6; women's status in, , , 155\u201356, 161\u201362\n\nSunni tradition, , , , 133\u201334, 216 _n_ 16, 224 _n_ 29, 232 _n_ 72, 242 _n_ 7, 243 _n_ 21\n\nSurveillance, brutality, and detention experienced by Muslim Americans, , 103\u20135, , , , , 245 _n_ 58. _See also_ Violence: against Muslims\n\nSynagogue Council,\n\nTalib, Imam. _See_ Abdur Rashid, Talib\n\n_Tasawwuf_ (see Sufism), , , , 145\u2013146, , 241 _n_ 6, 243 _n_ 22\n\nTawfiq, 'Allama Al-Hajj Ahmad,\n\n_Tawhid_ (oneness),\n\nTea Party, , , ,\n\n_Tekkes_. _See Dargahs_\n\nTerrorism, , , , , , , , , , , , , , 101\u20134, , , , , , , , , , 207\u20138, 245 _n_ 58, 250 _n_ 17\n\nThatcher, Margaret,\n\nTides Foundation,\n\nTocqueville, Alexis de,\n\nTolerance, , , , , , , , ,\n\nTraditionalism, 75\u201376, , ,\n\nTruman, Harry, ,\n\nTubman, Harriet,\n\nTurkey, 73\u201374\n\nUnited Nations, , , ,\n\nUnited We Serve,\n\nUniversity Muslim Medical Association Community Clinic,\n\nUpward mobility, , , , , , , , , , 228 _n_ 6\n\nUS Agency for International Development (USAID), , 251 _n_ 35\n\nUS Congress, , , , , , , 251 _n_ 35\n\nUS Constitution, , ,\n\nUS Homeland Security Department, ,\n\nUS State Department, , , , , , , , , , 88\u201389, , , , , , , , , , 228 _n_ 6, 233 _n_ 88, 254 _n_ 74\n\nVance, Cyrus,\n\nVatican II, ,\n\nVietnam War, , 48\u201349\n\nViolence: of Jim Crow laws, ; against black Americans and other non-whites, , , 253 _n_ 68; against Muslims, , , , , 206\u20137, ; by Muslims , 119\u2013120, , ; in urban settings,\n\nVoluntarism, , , , ,\n\nWadud, Amina,\n\nWahhaj, Siraj, , , 120\u201321, , 233 _n_ 93, 255 _n_ 12\n\nWalcott, Dennis,\n\nWallace, Mike,\n\n_Washington Post_ (newspaper), ,\n\nWeber, Max,\n\nWest African Sufis, , 228 _n_ 6\n\nWhite House Office of Faith-Based and Neighborhood Partnerships, 188\u201389\n\nWhite House Office of Faith-Based and Community Initiatives, 188\u201389\n\nWhiteness, , 29\u201330, , 225 _n_ 47\n\nWhite supremacists, , , 207\u20138\n\nWilliams, Mark, ,\n\nWilson, Woodrow, ,\n\nWinfrey, Oprah,\n\n_Wird_ (litany of names and verses), ,\n\nWomen, 155\u201381; _adab_ invoked by, 149\u201351; as converts to Islam, 162\u201364, 171\u201372; equality of, ; in leadership roles, , , , , 165\u201366, , , , 248 _n_ 39; liberal feminist principles, 159\u201362; male attitudes toward, 174\u201375; motherhood as role for, 158\u201359, , 167\u201370; \/'s effect on, 162\u201366; political context of, ; programs for, 107\u20138; rights of, 112\u201313; roles of, ; service expected of, 14\u201315, 158\u201359, 173\u201376; status of, in American culture, 162\u201363\n\nWomen Against Fundamentalism, 159\u201360, 246 _n_ 14\n\nWomen in Islam, ,\n\nWomen's Islamic Initiative in Spirituality and Equity\/Equality (WISE), 107\u20138, 111\u201316, , , , , ,\n\nWorld Bank,\n\nWorld Community of Al-Islam in the West, ,\n\nWorld Council of Churches,\n\nWorld Economic Forum, , , ,\n\nWorld Trade Center bombing (1993), 66\u201367, 90\u201391\n\n_Zakat_ (obligatory Muslim giving), 35\u201337, , 221 _n_ 103\n**RACE RELIGION**\n\nJohn L. Jackson Jr., David Kyuman Kim, Editors\n\nRaceReligion publishes historical\/genealogical, ethnographic and theoretical work that focuses on the complex relationship between race and religion. This series examines the paradoxical conditions under which the postcolony and empire have come to co-exist with new master narratives generated by global capitalism and contemporary notions of democracy. It seeks to uncover new ways of conceptualizing, theorizing, and understanding (any of) the following: Orientalism, nationalism, new ethnic formations, continuities and discontinuities between urban and indigenous\/vernacular religions, contemporary fundamentalisms, questions of agency, mourning rituals, political\/social mobilizations, diasporic\/exilic subjectivities, and the work of memory in the reconstitution of tradition. The series also bridges theoretical concerns with the lived experiences of individuals and their communities. Rather than presuming race and religion as transparent categories, books in the series showcase their unexpected conjunctions, revealing new understandings of what it means to live race and religion, to do race and religion, in the contemporary world. By highlighting modern confluences of these two terms, they trouble conventional thinking about secularism, political rationality, and the lived realities of modern life.\n\nMarla Frederick, _Colored Television: American Religion Gone Global_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Turgut Dincer, Charlie Howard, and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive)\n\n\n\n\n\n\n\n\n\n[Illustration: \"A Missionary looking over the edge of the world at the\npoint where Heaven and Earth meet.\"\n\n(_From an old print._)]\n\n\n\n\n  HISTORY OF\n  GEOGRAPHY\n\n  BY\n  J. SCOTT KELTIE, LL.D.,\n  SECRETARY OF THE ROYAL GEOGRAPHICAL SOCIETY,\n\n  AND\n\n  O.\u00a0J.\u00a0R. HOWARTH, M.A.,\n  ASSISTANT SECRETARY OF THE BRITISH ASSOCIATION FOR\n  THE ADVANCEMENT OF SCIENCE\n\n  [ISSUED FOR THE RATIONALIST PRESS ASSOCIATION, LIMITED]\n\n  LONDON:\n  WATTS & CO.,\n  17 JOHNSON'S COURT, FLEET STREET, E.C.\n  1913\n\n\n\n\nPREFACE\n\n\nThis is not a history of geographical exploration, though the leading\nepisodes in the advance of our knowledge of the face of the Earth are\nnecessarily referred to in tracing the evolution of geography as a\ndepartment of science. That is the object of this volume as one of a\nseries dealing succinctly with the history of the various sciences.\nWe are not concerned to discuss whether Geography is entitled to be\nconsidered as a science or not. It is hoped that in the attempt to tell\nthe story of its evolution up to the present day it will be evident\nthat it is as amenable to scientific methods as any other department\nof human knowledge, and that it performs important functions which are\nuntouched by any other lines of research. I use the first person plural\nbecause I am greatly indebted to Mr. O.\u00a0J.\u00a0R. Howarth in coming to my\nhelp after I had accumulated much of the material, but was seriously\ndelayed owing to a great increase in my official duties. The greater\nshare of whatever merits the book may possess ought to be awarded to\nMr. Howarth.\n\nI am indebted to Mr. E.\u00a0A. Reeves's interesting little book on _Maps\nand Map-making_ for many of the illustrations.\n\n                                                J. SCOTT KELTIE.\n\n_July 2, 1913._\n\n\n\n\nCONTENTS\n\n\n                                                                    PAGE\n  CHAPTER I.\n\n  BEGINNINGS                                                           1\n\n\n  CHAPTER II.\n\n  THE GEOGRAPHY OF THE GREEKS AND ROMANS                               8\n\n\n  CHAPTER III.\n\n  THE DARK AGE                                                        33\n\n\n  CHAPTER IV.\n\n  THE MEDI\u00c6VAL RENASCENCE                                             42\n\n\n  CHAPTER V.\n\n  PORTUGUESE EXPANSION AND THE REVIVAL OF PTOLEMY                     51\n\n\n  CHAPTER VI.\n\n  THE NEW WORLD                                                       59\n\n\n  CHAPTER VII.\n\n  THE FAR EAST AND THE DISCOVERY OF AUSTRALIA                         68\n\n\n  CHAPTER VIII.\n\n  POLAR EXPLORATION TO THE EIGHTEENTH CENTURY                         74\n\n\n  CHAPTER IX.\n\n  JAMES COOK AND HIS SUCCESSORS                                       87\n\n\n  CHAPTER X.\n\n  MEASUREMENT, CARTOGRAPHY, AND THEORY, 1500\u20131800                     90\n\n\n  CHAPTER XI.\n\n  THE NINETEENTH CENTURY: AFRICAN RESEARCH                           107\n\n\n  CHAPTER XII.\n\n  THE NINETEENTH CENTURY AND AFTER: ASIA AND AUSTRALIA               115\n\n\n  CHAPTER XIII.\n\n  THE NINETEENTH CENTURY AND AFTER: THE POLES                        122\n\n\n  CHAPTER XIV.\n\n  THE NINETEENTH CENTURY AND AFTER: EVOLUTION AND PROGRESS OF\n      GEOGRAPHICAL SCIENCE                                           135\n\n\n  SHORT BIBLIOGRAPHY OF GEOGRAPHY                                    147\n\n  INDEX                                                              149\n\n\n\n\nLIST OF ILLUSTRATIONS\n\n\n  A MISSIONARY LOOKING OVER THE EDGE OF THE WORLD         _Frontispiece_\n\n  FIG.                                                              PAGE\n   1.--TAHITIAN MAP                                                    2\n\n   2.--THE WORLD AS SUPPOSED TO HAVE BEEN CONCEIVED BY HECAT\u00c6US       11\n\n   3.--THE WORLD ACCORDING TO HERODOTUS B.C. 450                      15\n\n   4.--THE WORLD ACCORDING TO PTOLEMY                              28\u201329\n\n   5.--THE WORLD ACCORDING TO COSMAS INDICOPLEUSTES                   35\n\n   6.--BEATUS'S MAP                                                   38\n\n   7.--THE HEREFORD MAP                                               47\n\n   8.--CHART OF THE MEDITERRANEAN, 1500, BY JUAN DE LA COSA           49\n\n   9.--SCAPH                                                          91\n\n  10.--ASTROLABE                                                      91\n\n  11.--QUADRANT                                                       92\n\n  12.--CROSS-STAFF                                                    94\n\n  13.--DAVIS'S BACK-STAFF                                             95\n\n  14.--PRETORIUS'S PLANE-TABLE                                        96\n\n  15.--RAMSDEN'S THEODOLITE                                           97\n\n  16.--MODERN FIVE INCH TRANSIT THEODOLITE                            98\n\n  17.--THE WORLD ACCORDING TO MERCATOR (1587)                        100\n\n\n\n\nCHAPTER I.\n\nBEGINNINGS\n\n\nWe need not attempt any elaborate definition of Geography at this\nstage; it is hoped that a fairly clear idea of its field and functions\nmay arise during the following brief summary of its history and\nevolution. The old-fashioned definition, \"A description of the earth,\"\nis serviceable enough if accepted in its widest sense. Geography may\nbe regarded as the mother of the sciences. Whatever was the origin\nof man, whether single or multiple, and wherever he emerged into\nmanhood, he was a wanderer, an explorer, from the first. Necessity\ncompelled him to make himself familiar with his environment and its\nresources, and as the race multiplied emigration became compulsory.\nThe more that relics of primitive humanity are brought to light, the\nfurther back must man's earliest wanderings be dated. The five thousand\nyears of the old Biblical chronology must be multiplied a hundred\ntimes, and still we find that half a million years ago our primitive\nforefathers must have travelled far from the cradle of the race. They\nwere unconscious geographers. Their conceptions of the earth and of\nits place in the universe are unknown to us; it is not impossible to\ninfer something of them by analogy of ideas existing to-day among\nmore or less primitive peoples, though to do so is beyond our present\nscope. Yet it may be said that certain root-ideas of geographical\ntheory and practice must surely date from the earliest period of\nman's capacity for observation. Thus the necessity for describing or\nfollowing a particular direction presupposes the establishment of a\ndefinite standard--the face would be turned towards the position of\nsome familiar object; then in that direction and the opposite, and to\nthe right hand and the left, four such standards would be found, and\nwould become the \"cardinal points.\" The value, for this purpose, of\nso patent a phenomenon as the rising and setting of the sun must have\nbeen impressed upon human intelligence at an elementary stage. Again,\nmap-making is not very far removed from a primitive instinct. Modern\ntravellers have described attempts at cartography by the North American\nIndians, the Eskimo, and the Maori and other less advanced inhabitants\nof the Pacific Islands.\n\n[Illustration: Fig. 1.--Tahitian map.]\n\nIt is again beyond the scope of the present summary of the development\nof geographical knowledge among European peoples to attempt to give\nany detailed history of exploration; it is only possible to deal\nwith the salient episodes, and these mainly in so far as they have\ninfluenced man's general conception of the earth. Nevertheless, ages\nbefore the existence of any documentary evidence of its development\ngeographical knowledge must have advanced far in other lands. America\nwas \"discovered\" probably thousands of years before Columbus stumbled\nagainst the New World, or even the Norsemen had set foot in \"Vineland\";\nit had time, before the Spaniards swarmed over it, to become the seat\nof civilizations whose origin is far beyond knowledge. It is worth\nnoticing for our particular purpose that the European conquerors found\nevidence of highly developed geographical methods both in Central\nAmerica and in Peru; the native maps were intelligible to them, and\nthe Peruvian Incas had even evolved the idea of relief maps. China,\nagain, a great power in early ages, possessed knowledge of much of\ncentral Asia; India was the seat of powerful States and of a certain\ncivilization; Babylonia and Egypt were working out their destinies, and\nhad their own conceptions of the earth and the universe, long before\nthe starting-point of the detailed investigation within our present\nview.\n\nBut the names of Babylonia and Egypt bring us nearer to that\nstarting-point. The history of Europe dawns in the eastern\nMediterranean, and so does the history of geography. It has, however,\nto be premised that connection existed, in very early times, between\nthe eastern Mediterranean circle and the lands far beyond. When princes\nof Iranian stock (to cite a single illustration) are found established\non the confines of the Levant as early as the fifteenth century B.C.,\nit may be realized that the known radius from the Mediterranean centre\nwas no short one. Much earlier than this--even in the fourth millennium\nB.C.--astronomy, a science of the closest affinity to geography, was\nwell organized in Babylonia, and there is evidence for a cadastral\nsurvey there. Clay tablets dating from more than two thousand years\nB.C. show the work of the Babylonian surveyors.\n\nThe Egyptians worked along similar lines. Examples of their map-work\ninclude a plan, in the museum at Cairo, showing the basin of the\nlake M\u0153ris, with its canal and the position of towns on its borders,\ntogether with notes giving information about these places; and, in\nTurin, a map of the Wadi Alaiki, where the Nubian goldmines were\nsituated; and this map may date from the earlier half of the fourteenth\ncentury B.C.\n\nMeanwhile, in the \u00c6gean lands and from Sicily to Cyprus, at points\nprincipally but not invariably insular or coastal, and especially\nin Crete, communities grew up that developed a high standard of\ncivilization, to which the general name of \u00c6gean is given. It appears\nthat a central power became established in Crete about the middle\nof the third millennium B.C., and that an active oversea trade was\ndeveloped in the \u00c6gean and the eastern Mediterranean during the ensuing\nthousand years. As for the knowledge of the mainland which came to\nbe called Europe, it is suggested that the \u00c6gean civilization was\nassailed, about the fifteenth century B.C., by invaders from the north,\nand was practically submerged, probably by a similar movement, five\nhundred years later; and invasion presupposes intercourse.\n\nThe Ph\u0153nicians, next taking the lead in Mediterranean maritime trade,\nmust have extended knowledge of the inhabited world, even though they\nleft the reputation of secretiveness in respect of their excursions\n(a natural and not uncommon characteristic of pioneer traders). A\nSemitic people, they seem to have emigrated from the Persian Gulf in\ndetachments, and established independent settlements on the Levantine\nlittoral. Tyre was their chief trading city. They provided the\ncommercial link between east and west. Their penetration of the western\nMediterranean and even of the Straits of Gibraltar is assigned to the\nearliest period of their activities. They established relations not\nonly with the Greeks and other Mediterranean peoples, but also with\ncentral European traders; they are said, for example, to have dealt in\namber brought from the Baltic overland to the Adriatic and to the mouth\nof the Rhone. They founded colonies in Cyprus, Sicily, and elsewhere as\nfar as the west of Spain, where Gades (Cadiz) was established perhaps\nabout 1100 B.C. Thence they carried their enterprises far to the north.\nIf they did not actually exploit the tin of Cornwall, they probably\nknew of Britain. One of the greatest enterprises of antiquity, if we\nmay trust Herodotus, who was, however, sceptical, was conducted by\nPh\u0153nician navigators under the auspices of Necho, king of Egypt, about\n600 B.C. Even before this they brought from distant lands, it may be\nthe Malay peninsula or it may be what is now Rhodesia, gold and other\npresents for King Solomon. If the Ph\u0153nicians had really found their\nway as far as the Zambezi and the country on the south, they may well\nhave conjectured that it would be possible to sail round Africa. At any\nrate, if the story as told by Herodotus is true, Necho was convinced\nthat Africa could be circumnavigated. The Ph\u0153nician navigators sailed\ndown the Red Sea, and in autumn landed on the coast and sowed a crop\nof wheat; when this was reaped, they started again and made their way\nsouth round the Cape of Good Hope, and so northward, entering the\nMediterranean in the third year. At one part of their course they\nhad the sun on their right, which would be natural, though Herodotus\nregarded this as evidence of the incredibility of the narrative. There\nis no inherent impossibility in such an expedition, but it led to no\ndirect results; no further effort was made to round the continent for\ntwenty centuries.\n\nThe Ph\u0153nicians founded Carthage about 850 B.C. (though an earlier\ntrading post occupied the site), and the Carthaginians carried out\ntrading enterprises on their own account from their central point of\nvantage on the North African coast. Some time after Necho's expedition\n(probably about 500 B.C.) they sent out two distant expeditions. One\nof these, under Hanno, appears to have consisted of a very large\nfleet, and to have been intended to establish trading posts along the\nwest coast of Africa, which was already known to the Carthaginians.\nCertain details are furnished which serve to identify points at which\nhe touched, and it is generally agreed that he got as far south as the\nneighbourhood of the Bight of Benin. Almost simultaneously Himilco\nmade a voyage north along the west coast of Europe. He appears to have\nvisited Britain, and mentions the foggy and limitless sea to the west.\n\nInformation obtained by such means as this cannot have become in any\nsense the common property of the period. But there would be no mean\nsupply of geographical data at the disposal of traders on the one hand,\nand at least of a few philosophers and generally well-informed persons\non the other, at a period long anterior to that at which it is possible\nto begin our detailed history. Whatever tendency there may have been\non the part of the Ph\u0153nicians, and no doubt their predecessors, to\npreserve their commercial secrets, there is no necessity to suppose\nthat traders in distant lands did not describe these lands to those\nwith whom they immediately dealt. The links in the commercial chain\nwould then become links in a chain of geographical knowledge. This\nsupposition granted, geographers may be prepared to risk the charge of\ntemerity if they recognize and enjoy, as an exquisite description of\nthe unbroken summer daylight on some northern fjord-coast, the picture\nof the L\u00e6strygons' land in _Odyssey_, X.: \"Where herdsman hails\nherdsman as he drives in his flock, and the other who drives forth\nanswers the call. There might a sleepless man have earned a double\nwage, the one as neatherd, the other shepherding white flocks: so near\nare the outgoings of the night and of the day.\" And again, \"the fair\nhaven, whereabout on both sides goes one steep cliff unbroken, and\njutting headlands over against each other stretch forth at the mouth of\nthe harbour, and strait is the entrance ... no wave ever swelled within\nit, great or small, but there was a bright calm all around.\"[1] Here\nare words which on their face indicate hearsay in the Mediterranean\nconcerning Scandinavia in the Homeric age. Again, the gloomy home of\nthe Cimmerians, at the uttermost limit of the earth, suggests hearsay\nof the arctic night. As to Homeric geography generally, it may be\nsaid briefly that the lands immediately neighbouring to the \u00c6gean\nare well known, though there is little evidence of knowledge of the\ninhospitable interior of Asia Minor; something is understood of the\ntribes of the interior of Europe to the north; the riches of Egypt and\nSidon are known; mention is made of black men, and even of pygmies, in\nthe further parts of Africa; the western limit of anything approaching\nexact knowledge is Sicily. The earth is flat and circular, girt about\nby the river of Ocean, whose stream sweeps all round it.\n\n    [1] _Trans._ S.\u00a0H. Butcher and A. Lang.\n\nThus we have found geographical knowledge, so far as it is possible\nto trace its acquisition at all, to have been acquired for purely\ncommercial purposes, and it remained for the Greeks to seek for such\nknowledge for its own sake. It has been well said that the science of\ngeography was the invention of the Greeks.\n\n\n\n\nCHAPTER II.\n\nTHE GEOGRAPHY OF THE GREEKS AND ROMANS\n\n\nThe birthplace of Greek geographical theory is to be found, not in\nGreece proper, but in Asia Minor. Miletus, a seaport of Ionia, near the\nmouth of the M\u00e6ander, became the leading Greek city during the seventh\nto the sixth centuries B.C., trading as far as Egypt and throwing off\ncolonies especially towards the north, on the shores of the Hellespont\nand the Euxine. It was thus an obvious repository for geographical\nknowledge, besides being a famous centre of learning in a wider sense.\nThales of Miletus (640\u2013546 B.C.), father of Greek philosophers,\ngeometers, and astronomers, may have learnt astronomy from a Babylonian\nmaster in Cos, and became acquainted with Egyptian geometry by\nvisiting that country; he applied geometrical theory to the practical\nmeasurement of height and distance. He has been wrongly credited with\nthe conception of the earth as a sphere. That conception is actually\ncredited to Pythagoras, who, born in Samos probably in 582 B.C.,\nsettled in the Dorian colony of Crotona in Southern Italy about 529\nand founded the Pythagorean school of philosophy. He (or his school),\nhowever, evolved the correct conception of the form of the earth rather\nby accident (so far as concerns any scientific consideration) than\nby design, for the Pythagorean reasoning was abstract in nature, in\ndistinction from that of the Ionian school, which sought material\nexplanations for the phenomena of the universe. The Pythagoreans\n(whose view does not greatly affect the later history of geographical\ntheory) conceived the earth as a globe revolving in space, with other\nplanets, round an unseen central fire whose light was reflected by the\nsun, just as the moon reflects the sun's light. Later the philosopher\nParmenides, of Elea in Italy (_c._ 500 B.C.), considered the universe\nto be composed of concentric spheres or zones consisting of the primary\nelements of fire and darkness or night. Anaximander (611-_c._ 547\nB.C.), a disciple of the more practical Ionian school, and a pupil or\ncompanion of Thales, conceived an earth of the form of a cylinder.\nHe is said to have introduced into Greece the gnomon, a primitive\ninstrument for determining time and latitude, and to have made a map.\nThe first actual record of a Greek or Miletan map, however, occurs\nhalf a century after his time, when in 499 B.C. Aristagoras, tyrant of\nMiletus, asked aid of Cleomenes of Sparta against Persia, and showed\nhim a map, engraved on bronze, of the route of his proposed expedition.\nAnaximenes, of Anaximander's school, gave the earth an oblong\nrectangular form.\n\nThe physical division of land into continents, though obvious,\npresupposes the existence of a certain measure of geographical theory.\nStill more obvious as a primitive division would be a division simply\nbetween \"my land\" and \"yours.\" But there was a clear necessity at a\nvery early period for names to distinguish, generally, the lands which\nlay on one side and the other of the \u00c6gean-Mediterranean waters. It may\nwell be that the names of Europe and Asia did not possess precisely\nthis application in their original forms. Their derivation has been\nassigned to an Asiatic source; they signify on this view the lands\nrespectively of darkness or sunset and of sunrise or light--that is\nto say, the lands towards west and towards east. The earliest known\nGreek reference to Europe, moreover, does not indicate on the face of\nit a distinction from Asia, though it does indicate a distinction from\nlands separated from it partly or wholly by water. The Homeric hymn\nto Apollo, which may be dated in the eighth or seventh century B.C.,\nrefers to dwellers in the rich Peloponnese and in Europe and in the\nsea-girt islands--albeit in place of \"Europe\" some scholars would read\na word signifying simply \"mainland.\" The name of Europe, if admitted\nhere, is taken to mean no more than northern Greece, and would thus\nlend some colour to an early tradition that it was derived from a\nMacedonian city called Europus. However this may be, it is easy to\nconceive that the name of Europe, being at no time given to a territory\nwith defined frontiers, was capable of an elastic application, which\nwould be gradually extended, or (as is more probable under primitive\nconditions of geographical knowledge) would remain so vague as to\npermit of no clear definition.\n\nBut when the names of continents emerge in Greek usage they afford\nthe necessary distinction between the lands on either side of the\n\u00c6gean-Mediterranean. They so emerge in the 6th\u20135th centuries B.C.,\nand the distinction appears by that time to have been perfectly\nfamiliar, though the precise application, as will be seen, was a matter\nof controversy. The poet \u00c6schylus (525\u2013456 B.C.), who, by the way,\nwas also a traveller, possibly to Thrace, certainly to Sicily, was\nacquainted with the distinction, as appears, for example, from passages\nin the historical drama of the _Pers\u00e6_, which deals with the failure\nof Xerxes's invasion of Europe from Asia, and his retreat across the\nHellespont. The distinction would hardly have been introduced into\na stage-play if it had not been commonly recognized. In _Prometheus\nUnbound_, again, \u00c6schylus refers to the river Phasis (Aras) as the\nboundary between Europe and Asia. Finally, Herodotus states in an early\nchapter of his work that the Persians appropriate to themselves Asia\nand the barbarian races inhabiting it, while they consider as separate\nEurope and the Greek race, and he does not find it necessary to offer\nany explanation of the names here. At a later stage the continental\ndistinction appears to have been based on or associated with a\ndistinction between temperate and hot lands.\n\n[Illustration: Fig. 2.--The World as supposed to have been conceived by\nHecat\u00e6us.]\n\nHecat\u00e6us of Miletus (_c._ 500 B.C.) has been hailed as the father of\ngeography on the ground of his authorship of a Periodos, or circuit\nof the earth, the first attempt at a systematic description of the\nknown world and its inhabitants. But even if he wrote such a work,\nevidence has been adduced that the extant fragments of it belong to\na later forgery. However, he was a Miletan and a traveller, besides\na statesman. The map which is supposed to have accompanied his work\nmaintained the old popular idea of the earth as a circular disc,\nencircled by the ocean. Greece was the centre of the world, and the\ngreat sanctuary of Delphi was the centre of Greece. If this Periodos\nis taken as a forgery, there is a parallel case in the Periplus of\nthe Mediterranean attributed to Scylax of Caryanda, a contemporary of\nHecat\u00e6us. If Scylax wrote any such work, in its extant form it is a\ncentury and a half later than his time. He is said to have explored the\nIndus at the command of Darius Hystaspis, and to have returned by the\nIndian Ocean to the Red Sea.\n\nHerodotus of Halicarnassus (_c._ 484\u2013425 B.C.) was a historian, but was\nwidely travelled, and understood the importance of a knowledge of the\ngeography of a country, and its bearings on the history of its people.\nHe only introduces geographical information in so far as it throws\nlight on the history with which he is dealing, or because it seemed to\nhim of special interest, or as a report of curious information obtained\nfrom the countries which he visited; but he has certainly some more\nexact information about the restricted world of which Greece was the\ncentre than any of his predecessors. He visited Egypt and the Greek\ncolony of Cyrene, on the coast of what is now Tripoli. There is reason\nto believe that in Asia he got as far as Babylon on the Euphrates, and\nperhaps as far as Susa beyond the Tigris. He crossed the Euxine to\nthe northern shore as far as Olbia on the Borysthenes, and probably\nwent round to the south-east coast to the country of the Colchians,\nwhose characteristics he describes as if from personal knowledge.\nHe does not seem to have got very far west in the Mediterranean,\nthough he spent the latter part of his life in southern Italy. There\nis little doubt that he visited several of the Grecian islands. But\napart from the information about the countries round the Mediterranean\nwhich he collected personally, his history contains material from\nvarious sources concerning the countries and peoples in Europe, Asia,\nand Africa. This information, on the whole, is of the vaguest kind,\nand shows that the Greeks whom Herodotus may be taken to represent\nwere only groping their way with regard to a knowledge of the world\noutside the limits of their own restricted sphere. This vague knowledge\nincluded a considerable section of western Asia as far as the Caspian\nSea and the river Araxes. Herodotus had also heard of India and of\nthe Indus river, and had a fair knowledge of the Persians, the Medes,\nand the Colchians, as also of part at least of Arabia. Eastwards, in\nwhat might be called Central Asia, he had heard of the Bactrians and\nSogdians to the south of the Jaxartes (Syr-darya, the northern of\nthe two great affluents of the Sea of Aral), and of the Massaget\u00e6,\nIssedones, Arimaspians, and other races or peoples; those to the north\nof the Jaxartes being included, according to Herodotus, in Europe,\nwhich he took to extend from the Pillars of Hercules (Strait of\nGibraltar) to the Hellespont and up the Phasis (Rion) river, from its\nmouth in the Black Sea, to the Caspian. He also divided Asia from Libya\n(Africa) along the axis of the Red Sea, and was thus in conflict with\nothers who had divided Europe from Asia at the Tanais, and Asia from\nLibya at the Nile. He would not permit a theoretical boundary-line to\n\"bisect a nationality,\" as the Nile does.\n\n[Illustration: Fig. 3.--The World according to Herodotus B.C. 450.]\n\nIt may be well to examine Herodotus's geographical knowledge in some\ndetail, as representing the general knowledge possessed by a student\n(increased by his own travels) as distinct from that possessed by\ntraders and colonists in different particular directions. His knowledge\nof Europe proper was contained within rather narrow limits. He knew,\nfrom personal knowledge, or from information obtained from merchants\nand colonists familiar with the shores of the Euxine (Black Sea),\nof the country lying to the north of that sea for some distance, of\nthe rivers which flowed into it from the north and from the west,\nand of various peoples either settled in or wandering over the land\nthat is now mainly included in Russia. His notions of the comparative\ndimensions of the Euxine and of the M\u00e6otis Palus (Sea of Azov) are\naltogether erroneous, as might have been expected; but he knew (though\nin the main vaguely) of the Tanais (Don), the Borysthenes (Dnieper),\nand other rivers which flow into those seas. The Ister (Danube) he knew\nas a river of considerable importance, but he made it rise in Spain\nand flow north-east and east through the greater part of Europe. He\nconceived it as corresponding to some extent in the direction of its\ncourse with that of the Nile on the other side of the Mediterranean. He\nhad some vague notion of the Iberians and of the Celts, as inhabiting\nthe country to the north of the Pillars of Hercules. He knew something\nof the country lying to the north of Greece, Illyria, Thessaly, and\nThrace, and the Rhodope mountains. He has much to tell of the\nScythians inhabiting the country north of the Black Sea, but it is\ndifficult to make out exactly to what race they belonged and whither\nthey had wandered; they may have been the forerunners of the Slav\npeoples. Of Europe to the north of the Danube, and of the Scythian\ncountry, he had no information of any importance. He did not believe\nin the Hyperboreans, nor did he credit the statement that there was\nany sea north of Europe. He has a good deal to say about Africa. He\nhad been up the Nile as far as the first cataract to the old city\nof Elephantine, but above that his information is vague and largely\nerroneous. He knew of the great bend which the Nile takes to the west\nabove Elephantine, and had heard of Meroe on the other side of that\nbend; but his notion of the length of Africa was so erroneous that,\ninstead of carrying the Nile south into the interior of the continent,\nhe made it rise far to the west and run eastwards before it turned\nnorth at Meroe. But he at least controverted the view that it rose\nin the ocean itself to the south--a belief based possibly on some\nrumour, transmitted through many lips, of the existence of great lakes\ntowards its headwaters. As for a river flowing west-and-east, in the\nwest of the continent, he had heard of such a river in a story of five\nNasamonian youths who travelled south from the shore of the Syrtis,\ncrossed the desert for many days, and were at last taken captive by\nblack men of small stature, who carried them to a city on the banks of\nthis river, whence they were subsequently allowed to return. It has\nbeen eagerly discussed what truth underlies this story, and whether\nthe river was the Niger in its upper course; but at best the account\nadded little to the knowledge of distant Africa. The idea of a pygmy\npeople dwelling towards the southern shore of the ocean is older than\nthe Iliad in which it is found (III, 3). Herodotus's conception of the\nshape of Africa did not carry its southward extension much beyond the\nlatitude of Cape Guardafui. He discarded the popular conception of the\nround earth, regarding it as longer from east to west than from north\nto south. The philosopher Democritus of Abdera (born _c._ 470\u2013450)\nexhibited the same conception in a map which he constructed.\n\nAn important episode in the progress of a more accurate knowledge of\nthe world of the Greeks was the Retreat of the Ten Thousand under\nXenophon in 401\u2013400 B.C. The younger Cyrus had made a great expedition\nfrom Sardis, in western Asia Minor, eastwards through the Cilician\nGates to the Euphrates, and along the course of that river to the\nneighbourhood of Babylon. He was accompanied by a band of Greek\nmercenaries, who, after his defeat at Cunaxa, began the retreat of\nwhich Xenophon left a graphic account, containing what must have been\nto the Greeks much new information concerning the region from the\njunction of the Tigris and Euphrates northwards past Lake Van and\nthrough the mountains of Armenia, north and west to the shores of the\nBlack Sea at Trapezus (Trebizond), and along the south coast of the\nBlack Sea, partly by sea and partly by water, to Byzantium. Xenophon's\nstory is an illustration of the well-known fact that war is one of the\nchief means of promoting geographical knowledge. This will appear more\nclearly in the next important episode in the story of exploration--the\ncampaign of Alexander the Great.\n\nIn the interval there were one or two writers from whose work\nsomething is to be gathered of Greek geographical knowledge and theory\nabout the middle of the fourth century. The philosopher Plato (427\u2013347\nB.C.) may be referred to here in connection with his story, based on an\nEgyptian tradition, of the great island of Atlantis, that land which\nplays so important a part in later mythical geography. In the Egyptian\nstory it lay just beyond the Pillars of Hercules, and adjacent to\nit was an archipelago. This would aid its later identification with\nthe Canaries, though it came also to be connected with America and\nother known lands besides, as well as giving name to an island in the\nAtlantic Ocean, the disproof of whose existence may almost be called\nmodern. From Plato's account of Atlantis as the home of a powerful\npeople who in early times invaded the Mediterranean lands, it has also\nbeen sought to associate the tradition with Crete at the period of the\n\u00c6gean civilization mentioned in the first chapter.\n\nSome fragments exist of the writings of the historian Ephorus of Cyme\nin \u00c6olis (_c._ 400\u2013330 B.C.). He seems to have endeavoured to cover\nthe whole field of the world as known to the Greeks, and conceived the\nfour most distant regions of the earth to be occupied on the east by\nIndians, on the south by Ethiopians, on the north by Scythians, and on\nthe west by Celts. The last he considered as occupying all Spain, as\nwell as Gaul. Strabo (p. 24) commended his geographical work and his\nskill in separating myth from history. A document of this period is\nthe Periplus, already referred to as known under the name of Scylax.\nThis class of work became more and more common as navigation developed,\nand corresponded in some measure to the modern Admiralty guide or\npilot. The Periplus is confined mostly to the regions known to the\nGreeks bordering the Mediterranean. From the Pillars of Hercules the\nwriter follows the north coast eastwards, including the Adriatic and\nthe Euxine as far as the mouth of the Tanais, which he regards as the\ncontinental boundary. He then follows the Levantine coast, the north\nAfrican coast westward, and the west African coast as far as the island\nof Cerne. He incidentally makes what is regarded as the earliest extant\nmention of Rome; but his notions of rivers and other features away from\nthe coast are generally erroneous.\n\nAristotle (384\u2013322 B.C.), in two of his extant works, the\n_Meteorologica_ and the treatise on the Heavens, revealed something of\nhis ideas on physical geography and the figure of the earth and its\nrelations to the heavenly bodies. He believed the earth to be a sphere\nin the centre of the universe, because that was a form which matter\ngravitating towards a centre would necessarily assume, also because the\nshadow cast by the earth on the moon during an eclipse is circular. He\naccepted the conclusion that the circumference of the earth was 400,000\nstadia (nearly 46,000 miles). His views with reference to the cosmical\nrelations of the earth were the same as those adopted by Eudoxus of\nCnidus (_fl._ middle fourth century), but he did his best to prove\nthem. He adopted, however, the prevalent view that the habitable world\nwas confined to the temperate zone between the tropics and the arctic\nregions. He believed there must be a temperate zone in the southern\nhemisphere, though he did not suggest that it must be inhabited. In\nthe _Meteorologica_ he treats of such subjects as weather, rain, hail,\nearthquakes, etc., and their causes. He recognized that changes took\nplace in the relations of land and sea. His knowledge of the origin and\ncourse of rivers and their relation to mountain systems was confused,\nand mainly erroneous; and it would seem that little progress had been\nmade in geographical knowledge since the time of Herodotus. In the work\nof Aristotle's successor, Theophrastus of <DW26>s (_c._ 372\u2013287), an\nimportant department of geographical study--that of distribution--finds\na place in its particular application to plants.\n\nAlexander the Great (356\u2013323 B.C.), King of Macedon, however, during\nthe last few years of his life, made possible by his campaigns a\ngreater extension of Greek geographical knowledge than had taken\nplace almost since Homeric times. When he passed eastward through\nMesopotamia, by Babylon, Susa, and Persepolis, and through Media to the\nsouthern shore of the Caspian Sea, he was in a region which, though\nan ancient cradle of civilization, had been till then only vaguely\nknown to the Greeks. Beyond that he entered new country, peopled by\nHerodotus and others with dubious tribal names. He came almost into\nthe heart of Central Asia, founding a city on the upper course of the\nJaxartes. Passing southwards through Bactria and across formidable\nranges of mountains such as the Hindu Kush, he struck the upper course\nof the Indus, made his way down to its delta, and would have proceeded\nright into the heart of India and followed the Ganges to its mouth but\nfor his mutinous troops. He returned through the north of Baluchistan\nand Persia to Ecbatana, and so homeward. He also sent a member of his\nstaff, Nearchus, by sea along the coast of Baluchistan and Persia, in\norder to define it and to ascertain the extent of the Persian Gulf.\nDic\u00e6archus of Messana, a pupil of Aristotle, who died early in the\nthird century B.C., used the geographical results of Alexander's\nexpeditions, including the distances obtained by his bematists, or\nmeasurers by pacing. Dic\u00e6archus wrote a topography of Greece, and also\ndrew on a map a parallel or equator, for the first time, so far as is\nknown, along the length of the Mediterranean and, with a distorted idea\nof their relative directions, along the Taurus and Himalayan ranges.\nBefore his time, and probably contemporaneously with Alexander's\ncampaigns, Pytheas of Massilia (Marseilles) visited (practically\ndiscovered) Britain, and made mention of Thule, six days' voyage north\nof it, having perhaps heard of the Orkneys and Shetlands. As these\nislands, however, are at no such great distance as is here suggested\nfrom the nearest point of Britain, the name of Thule has been variously\ntaken to represent some part of Norway, the Faer\u00f6e, or Iceland: it\ncertainly seems by some later writers to be applied to Scandinavia,\nand in literary usage came to signify the uttermost north. Pytheas\nalso obtained an idea of the Baltic Sea, and is considered by some to\nhave entered it; he is stated to have reached the River Tanais; but\nif that is to be considered as one of the north European rivers, and\nnot the known Tanais of the Black Sea basin, its identity is doubtful.\nPytheas was a trained astronomer; he was one of the first to calculate\nlatitudes, and had that of Massilia nearly correct; he heard of the\nunbroken summer daylight and winter darkness of the far north, and he\nnoted various features of geographical interest, such as the decrease\nin the number of different grain crops observed as he travelled\nnorthward. He appears, in fact, from the references in other authors,\nwhich alone furnish us with knowledge of his work, in the light of a\nscientific traveller of a type rare in his time.\n\nMathematical geography was carried a long step forward by Eratosthenes\n(_c._ 276\u2013194 B.C.), a native of Cyrene, who became chief librarian at\nAlexandria under Ptolemy III Euergetes. He calculated the circumference\nof the earth. He considered Syene to be situated on the tropic (which\nit was not, precisely), because at noon on the day of the summer\nsolstice the sun appeared to shine directly down a deep well there.\nHe therefore observed the zenith distance of the sun at Alexandria\nat the same time, and obtained his result from this and the measured\ndistance between Alexandria and Syene. His result was to make the\nearth's circumference only about one-seventh greater than it actually\nis. He also estimated the size of the habitable earth (\u0153cumene), and\nconsidered it, as a result, to be about double as long from west to\neast as it was broad from north to south. But his estimate of the\ndistance from what is now the extremity of Brittany to the known\neastern limit of India was about one-third too great. On a map of\nthe world he drew seven parallels, using the few points of which the\nlatitudes had been worked out, and also seven meridians at irregular\ndistances apart. Hipparchus (middle and second half of the second\ncentury), an astronomer, native of Nic\u00e6a in Bithynia, who worked in\nRhodes, drew an elaborate series of parallels, and made the division\ninto 360 degrees; the spaces between the lines he called climata, or\nzones. The problem presented by meridians was more difficult, for there\nwas no instrument for the calculation of longitude like the gnomon for\nthat of latitude. A well-recognised line was that taken to lie from the\nmouth of the Danube to that of the Nile--Herodotus had used this--and\nsouthward up the latter river; but not only the line itself, but also\nthe ideas of intermediate points lying on it, were rather far from the\ntruth.\n\nEratosthenes in his writings also dealt with the history of geography\nand with physical geography. This last branch attracted a number of\nstudents about this period and later, as, for example, Agartharchides\nor Agatharchus of Cnidus (middle of the second century B.C.). Crates of\nMallus, in the first half of the same century, expressed the view of\nthe Stoic philosophers that the spherical earth was divided into four\ninhabited quarters--the \u0152cumene (the known world), the Antipodes, the\nPeri\u0153ci, and the Ant\u0153ci. Posidonius the Stoic (_c._ 130\u201350 B.C.), among\nhis studies in various departments of knowledge, included such subjects\nas the ocean, volcanoes, and earthquakes; he observed the interaction\nof the sun and moon in their influence on tides. He recalculated the\ncircumference of the earth, but obtained an underestimate considerably\nfurther from the truth than Eratosthenes's overestimate; and, owing\nto his high scientific reputation, his error persisted in much later\nwork. Posidonius, before settling at Rhodes, had travelled widely\nin Africa, Spain, and western Europe generally; and both he and\nAgartharchides, like other writers of this period, and Aristotle before\nthem, recognized the importance of the human side of geography, and the\ninfluence of physical environment on the political and social _r\u00e9gime_.\nPolybius (_c._ 204\u2013122 B.C.) of Megalopolis in Arcadia, historian,\nstatesman, and military commander, was inspired by extensive travel\nto introduce the results of topographical and geographical studies\nthroughout his history. Among travellers who rank more nearly among\nexplorers there may be mentioned at this period Eudoxus of Cyzicus\n(fl. _c._ 130 B.C.), who went on a trading expedition to India, in\ncommand of a fleet which was despatched by Ptolemy Euergetes of\nEgypt with the specific object of exploring the Arabian Sea. Eudoxus\nsubsequently tried without success to circumnavigate Africa, making at\nleast two voyages along the east coast. Finally, it must be remembered\nthat at this period the extension of the knowledge of the world was\nmainly due no longer to Greek, but to Roman, activities, and Roman\nconquests were pushed beyond the confines of accurate Greek knowledge\nin various directions. Much geographical material was made available\nby writers on Roman military expeditions. Thus Pompey was accompanied\nin the Caucasian region by Theophanes of Mytilene, as historian of his\ncampaigns; Julius C\u00e6sar wrote his own account of lands into which he\ncarried his arms (Gaul, etc.).\n\nFrom all this it appears that, according to the lights of knowledge at\nthe time, a fair conception existed of geographical study along the\nmain lines which it follows to-day. There was therefore occasion for a\ngeneral review of the whole subject; and this occasion was seized by\nStrabo, a native of Amasia in Pontus, who was born _c._ 63 B.C., and\ndied in the second or third decade of the following century. He was\neducated partly at Rome, but his language and outlook were Greek. He\ntravelled much, as far as Etruria, the Black Sea, and the borders of\nEthiopia, as well as in Asia Minor, though he knew comparatively little\nof Greece. His geography, which was not finally completed till towards\nthe close of his life, was the first attempt at covering the whole\ngeographical field--mathematical, physical, and human--and his range\nwas thus wider than that of Eratosthenes. Strabo used the recent Roman\nauthorities to some extent, such as C\u00e6sar's Commentaries (in part)\nand a map of the Roman Empire by M. Vipsanius Agrippa, son-in-law of\nAugustus, which was set up in Rome; and he also preferred the authority\nof Polybius to that of Pytheas, but his sources were mostly Greek.\nHis appreciation of them was not always wise. He ranked the Homeric\npoems highly, but discredited Herodotus, who on some points had better\ninformation than he. He added nothing to the mathematical branch, in\nwhich Eratosthenes was his master. He thus accepted the spherical\nform of the earth, its dimensions as laid down by his predecessors,\nand its division into five zones. He recognized Hipparchus's view\nthat further astronomical observations were essential to precision in\nearth-measurement and the position of points on the surface; but it was\noutside his province to add to those existing. His work in the physical\nfield, however, improved upon that of his predecessors, and his surveys\nof the features and products of the various lands must have been\nsingularly valuable for reference. The apportionment of the seventeen\nbooks of his geography is not without interest as a rough guide to the\ndistribution of available material and to the author's outlook. He\ndevoted two books to introductory matter, to Spain and France two, to\nItaly two, to northern and eastern Europe one, to Greece and adjacent\nlands three, to the main divisions and remoter parts of Asia one, to\nAsia Minor three, to India and Persia one, to Syria and Arabia one, to\nEgypt and the rest of Africa one.\n\nRome did not carry on the Greek tradition of the study of geographical\ntheory. H.\u00a0F. Tozer, quoting J. Partsch, writes: \"It has been aptly\nremarked that the task which Eratosthenes set himself of measuring\nthe earth by means of the heavenly bodies, and that of Agrippa, who\nmeasured the Roman provinces by milestones, may be taken as typical\nof the genius of the two nationalities respectively.\" Thus Pomponius\nMela, purporting to survey the world in his _De Chorographia_ (written\n_c._ 50 A.D.), followed the coast and described various countries in\npassing, but by no means all, and added very little to geographical\nknowledge at large. Pliny the Elder (_c._ 23\u201379) devoted three books\nand part of another of his _Historia Naturalis_ to geography; but\nhis geography may (at least in considerable part) be compared with\nthe arid text-book of a generation ago, though in some instances his\ndescriptions (as of features in Palestine, Syria, and Armenia) are\nvaluable. As bearing on the quotation made above, it may be said here\nthat the famous Roman system of road building gave rise to two classes\nof road-books (as they may be termed), one consisting of lists of\nstations and distances, such as the Antonine Itinerary (probably, in\nits original form, of the late third or fourth century), the other\ndiagrammatic, such as the Peutinger Table (probably of the first half\nof the third century, though named after a scholar of the sixteenth),\non which roads, stations, and other details were presented in a\nmap-like form, but independently of true scale or direction. The Roman\nagrimensores were skilled surveyors.\n\nIt is not, then, surprising that the two great theoretical geographers\nnext to be considered are not associated with the capital of the\nRoman Empire. Of the work of Marinus of Tyre nothing is known beyond\nwhat is recorded by his immediate successor, Ptolemy, who used and\nacknowledged his results--as far as concerned the Mediterranean fully,\nand in respect of other countries to a modified extent. Ptolemy,\nmathematician, astronomer, and geographer, was a native of Egypt, who\nworked at or in the neighbourhood of Alexandria in the second century.\nHis geographical book was called _Geographike Syntaxis_. He carried\nmathematical geography far beyond the standard of his predecessors.\nHe used the theoretical division of the globe into five zones by\nthe equator and the tropics, adopted Hipparchus's division of the\nequator into 360 degrees, and worked out a network of parallels of\nlatitude and meridians of longitude, first thus applying these terms\nin their technical sense. In mapping the habitable world he used the\nFortunate Isles, beyond the western confines of Europe and Africa, as\nthe location of his prime meridian. The errors which resulted from\nthe vague idea as to the position of these islands (the Canaries\nand Madeira), and from the fact that Ptolemy followed Posidonius's\nunderestimate of the circumference of the globe and made his degree at\nthe equator equal to 500 instead of 600 stadia,[2] have been very fully\nanalysed, but cannot be even summarized here.\n\n    [2] Fifty instead of sixty geographical miles.\n\n[Illustration: PTOLEM\u00c6US ROM\u00c6 1490.\n\nFig. 4.--The World according to Ptolemy.]\n\n[Illustration: Fig. 4. (left side)]\n\n[Illustration: Fig. 4. (right side)]\n\nPtolemy had a strong tendency to exaggerate the size of the great\nland-masses--his Europe extended too far west (and the Mediterranean\nwas made too long in consequence); his Africa was too wide, especially\ntowards the south; his Asia was vastly exaggerated in its eastern\nextension, and many details, even in the Mediterranean area, were made\ntoo large. Ptolemy followed his predecessors in using the parallel of\n36\u00b0 N. as the axial line of the Mediterranean. It passes through the\nStraits of Gibraltar, the island of Rhodes, and the Gulf of Alexandria,\nand was theoretically prolonged eastward along the supposed line\nof the Taurus mountains and the range known to lie north of India.\nIn respect to this line there were remarkable inaccuracies in laying\ndown the coasts of the Mediterranean and in fixing the position of the\npoints upon them. The sea itself was made not only too long, but too\nbroad; Byzantium and the Black Sea were carried too far north, and the\nsize of the sea of Azov was immensely exaggerated. On the other hand,\nPtolemy restored the correct view, held by Herodotus, of the Caspian\nas an inland sea, and knew that the great river Volga entered it. Yet\nagain, he knew nothing of Scandinavia, or of the land-locked Baltic\nSea, marking only a small island of Scandia, possibly by confusion\nbetween the Scandinavian mainland and some Baltic island. But his idea\nof the British Isles may be taken as fairly correct, if allowance be\nmade for their remoteness. He laid down some parts of the coast very\nfairly, but oriented the major axis of Scotland more nearly from east\nto west than from north to south; he also placed Ireland wholly more\nnortherly than Wales. There is plenty of evidence in Ptolemy's work\nof a growth of knowledge of remote lands, though much of it is vague,\nif not actually unintelligible to us. Thus in Asia he had an idea\nof the great central mountain ranges (Pamir, Tian-shan, etc.), for\nsilk-traders had by now established trans-continental routes to China.\nPtolemy had also some conception of the south-eastern coasts, which\nhad probably been seen by Greek mariners as far as southern China. But\nhe wholly misunderstood the form of the east of the continent, for\nbeyond the Golden Chersonese (Malay Peninsula) there lies a vast gulf,\nthe eastern shore of which represents his view of China, extending\nsouthward far beyond the equator, and facing west. Again, he had no\nconception of peninsular India--unless, indeed, his huge island of\nCeylon (Taprobane) was drawn so by some confusion with the peninsula,\nas it was certainly also confused with Sumatra. Yet it would seem that\nhe might have gathered a more accurate idea of India from the Periplus\nof the Erythr\u00e6an Sea, a guide to navigators dated about the year 80.\nThis work furnished sailing directions from the Red Sea to the mouth\nof the Indus and the coast of Malabar, following the Arabian coast,\nalthough the possibility of crossing the open sea with the assistance\nof the monsoon was realized at a still earlier date. And the Periplus\ndistinctly indicates the southward trend of the Indian coast-line.\n\nRoman penetration of Africa gave Ptolemy some new details; he also\nconceived the Nile as formed by two headstreams arising in two lakes,\npossibly on the strength of some hearsay of the facts, and he marked\nthe Mountains of the Moon in remoter Africa, which again suggests\nhearsay of the heights of Ruwenzori, Kenya, and others. The Romans had\npenetrated Ethiopia, and possibly the region of Lake Chad, and Ptolemy\nalso used other sources of information about North Africa which are\nunknown from previous writers, but are completely vague and impossible\nto follow. Of the shape of the continent he was almost completely\nignorant; he just realized that an indentation occurs in the Gulf\nof Guinea, but gave it nothing like its proper value, and carried\nthe coast thence south-westward till the continent is broader at the\nsouthern limit of his knowledge than it is at the north.\n\nPtolemy's work on physical features was on the whole poor, and he\nneglected the human side of geography. Discarding the idea of the\ncircumfluent ocean, he supposed the extension of unknown lands\nnorthward in Europe, eastward in Asia, and southward in Africa, beyond\nthe limits in which he attempted to portray their outlines; and he even\nsuggested a land connection between south-eastern Asia and southern\nAfrica. Before his time the precision of mathematical method had far\nsurpassed that of the topographical material to which it was applied.\n\nPausanias, a Greek probably of Lydia and about contemporary with\nPtolemy, wrote a description (_Periegesis_) of Greece, which, apart\nfrom the arch\u00e6ological value which is its chief interest, contains\nreferences to various phenomena of physical geography, while as a\ndetailed topographical work it stands alone in the literature of which\nan outline has thus far been given.\n\n\n\n\nCHAPTER III.\n\nTHE DARK AGE\n\n\nFrom this point it becomes necessary to vary the treatment of our\nsubject hitherto followed. With the breakup of the Roman Empire and\nthe establishment of Christianity the old learning was obliterated.\nReligion became the central fact of intellectual exercise, and, except\nin so far as Christian doctrine and Holy Scripture involved reference\nto natural phenomena, every branch of natural science was withered by\nthe breath of theology. The first serious assaults of the barbarian\ninvader were made on the frontiers of the Roman Empire in the fourth\ncentury A.D.; in 330 the seat of government was transferred from Rome\nto Byzantium, and at the close of the century the empire was divided\ninto eastern and western parts. It has often been pointed out that\nthese events did more than mark the beginning of the disruption of\nthe Roman Empire; they also mark the parting of the ways of eastern\nand western European religion and culture. In the west it became\nthe function of Christianity to teach and civilize peoples untaught\nand uncivilized; but, limited and intolerant as was its outlook\nupon natural science generally, it discarded the learning of the\npre-Christian era. We have now to inquire how geography was affected by\nthis attitude towards secular learning.\n\nIt is true that the habit of travel, so far from being forgotten, was\neven fostered by missionary work and the practice of pilgrimage.\nAgain, opportunities for the extension of geographical knowledge were\nprovided by various episodes in the history of the centuries with\nwhich we are now concerned; thus Procopius, the historian of the\nPersian, Vandal, and Gothic wars of the epoch of the Roman (Byzantine)\nemperor Justinian in the sixth century, had ample opportunities for\ngeographical description and used them well. Justinian even despatched\nan expedition to China (which returned thence). But the geographical\ntheorists of the period now under review had little if any concern with\ncontemporary travellers' results.\n\nThe Christian cosmographers, having found in a spiritual sense a new\nheaven and a new earth, were at pains to create them in a scientific\nsense also. It was their aim to reconcile geographical theory with\nthe literal sense of Holy Scripture, and they were not only unable\nto explain, but were (for the most part) willing to disprove,\npre-Christian theory by that light. Thus Lactantius Firmianus (_c._\n260\u2013340), becoming converted, denied the possibility of the sphericity\nof the earth or the existence of antipodes. On the other hand, this\nconception died hard, for it was maintained by pagan writers at this\ntime--as, for instance, Martianus Capella, who, writing in the third or\nfourth century, followed such authorities as Ptolemy and Pythagoras.\nAnd these pre-Christian views must have caused some of the Christian\nauthors to doubt, for they left unsettled such questions as that of the\nearth's shape, on the plea that they formed no part of the Christian\ndoctrine; an instance of this attitude is provided by St. Basil the\nGreat of C\u00e6sarea (_c._ 330\u2013379) in his treatise on the Hexaemeron (Six\nDays of the Creation).\n\n[Illustration: Fig. 5.--The World according to Cosmas Indicopleustes.]\n\nOne of the principal popular attractions of geography has always\nbeen its function of describing the wonders of distant lands, and in\nJulius Solinus Polyhistor (probably of the third century) we have a\ntypical geographer of the marvellous, who in his _Collectanea Rerum\nMemorabilium_ drew upon Pliny, Pomponius Mela, and many earlier authors\nfor a description of the wonders of the world, and became himself\nregarded as a high authority. With such influences at work on the study\nof geography, the genesis of the theories of Cosmas Indicopleustes\nbecomes perhaps less surprising than the theories themselves. He was a\nmerchant of Alexandria, and a traveller (as his surname is inaccurately\nintended to record) in the Red Sea and the ocean beyond, who, thus\nfortified in geographical study, became a monk and wrote his _Christian\nTopography_ about the middle of the sixth century, in opposition to\nthe pre-Christian theories. Under his pen the inhabited earth became\na flat, rectangular oblong surrounded by oceans. At the north is a\nconical mountain round which the sun (which is some forty miles in\ndiameter and at no great distance from the earth) revolves, passing\nabout the summit in summer, so that it is hidden from the earth for\na shorter time daily than in the winter, when it passes about the\nbase. Again, in such conditions as have been indicated, it is at least\nintelligible that a responsible writer should accept a mountain 250\nmiles high, as is stated of Mount Pelion by Dicuil, an Irish monastic\nscholar who completed his _De Mensura Orbis Terr\u00e6_ in 825. He, however,\nwas little else than a compiler, and in the manner of his kind made\nan ill choice of sources. Among them, however, he refers to surveys\nmade by Julius C\u00e6sar, Augustus, and one of the emperors Theodosius;\nthe originals thus referred to are unknown. To mention the various\nChurchmen and others who, though in no sense geographers, were expected\nto deal with the theories of the earth and the universe in connection\nwith their religious doctrines, would make but a tedious list.\n\nThe view of the sphericity of the earth was not wholly lost.\nCertain expressions of the Venerable Bede (_c._ 672\u2013735), even if\nhe did not specifically formulate the theory, were capable of that\nconstruction, although in 741 we find Virgilius, the Irish bishop of\nSalzburg, attacked by the Pope for his assertion of the existence of\nantipodes (or at least an assertion that may bear that construction).\nNevertheless, these theories made headway, and from the eleventh\ncentury they became more and more acceptable to leaders of learning.\nAdam of Bremen accepted them; he is also an important figure in\nother departments of geography. He flourished in the second half of\nthe eleventh century; his history is a specially valuable authority\nfor the Baltic lands and other parts of Scandinavia and Russia. Its\nfourth book is a _Descriptio Insularum Aquilonis_, for which there was\nno little material, for not only had holy men from Ireland visited\nthe Orkneys, Shetlands, Faer\u00f6e, and Iceland in the sixth and seventh\ncenturies, but the Norsemen had entered upon their period of colonizing\nactivity; besides Great Britain and Ireland they had reached Iceland\nin 874, Greenland a century later, and Vinland in 1000; and this last,\nthat much-debated[3] landfall situated somewhere on the North American\ncoast south-west of Greenland, is first mentioned by Adam of Bremen. A\nsimilar appreciation of north European geography had been shown by King\nAlfred the Great, who in translating (and freely editing) earlier works\nhad introduced much of the knowledge acquired down to his own time.\n\n    [3] Not only so in modern times; at least one Scandinavian\n        geographer, of the end of the thirteenth century, was\n        prepared to recognize it as belonging to Africa.\n\n[Illustration: Fig. 6.--Beatus's Map.]\n\nMonastic cartography did not keep pace with theory. The disputed\nhabitable land beyond the confines of the known world, and separated\nfrom it by the impassable torrid zone (a classical conception),\nappeared in maps of the seventh and eighth centuries. Apart from this\n(and it came to be generally admitted) we have rectangular oblong maps\nlike that of Cosmas, and circular maps, out of which was evolved the\ndiagrammatic form of a =T= within an =O=, where the =T= represented\nthe Mediterranean, the Tanais, and the Nile, its upright showing the\nwestward extension of the sea, and the cross-stroke the two rivers,\nto left and right respectively, so that the west was at the bottom\nof the map. Europe lay within the left-hand angle of the =T=, Africa\nin the right-hand angle; Asia was the half circle above it. The holy\ncity of Jerusalem lay \"in the midst of the nations.\" Some maps, again,\ngave the earth an oval form. In all, the habitable earth was still\nsurrounded by ocean. In the far east sometimes appeared Paradise.\nThis feature and the unknown habitable world above mentioned were\nboth shown in the map of Beatus, a Spanish priest (_c._ 730\u2013798), who\nillustrated his _Commentaria in Apocalypsin_ with one of the earliest\nknown Christian maps of the world, several copies of which, dating from\nthe tenth to the thirteenth centuries, are preserved. In this map the\n\"known\" habitable world appears as a dome-shaped mass with its flat\nbase to the south, forming the northern shore of a strait separating\nit from the unknown southern land. The known world is broken for more\nthan half its breadth by the vast gulf of the Mediterranean, which\nhas two great arms reaching far northward; the Caspian Sea is a gulf\nopening into the north-east part of the circumfluent ocean, in which\nare set islands in an orderly ring all round the world; all sense of\ndirection in the flow of rivers is awry, and topographical accuracy is,\nof course, entirely wanting.\n\nWhat early Christianity lost by looking askance at classical theory\nthe Arabs, after the establishment of Muhammadan power in the seventh\ncentury, in great measure gained, for they were free of scruple as\nregarded the earlier learning, and eager for knowledge. Early in the\nninth century the Caliph Al-Mamun of Bagdad gave a strong impulse to\ngeographical and kindred studies. He caused translations to be made of\nPtolemy's astronomical and geographical works, and of those of other\nancient authorities, among whom was Marinus of Tyre. Muhammad ben\nMusa, librarian of Bagdad, compiled a _Description of the World_, or\ngazetteer of place-names with their positions, on the Ptolemaic model.\nDegrees were measured in Syria and Mesopotamia. Among Arab descriptive\nworks the first which survives is that of Suleiman, a merchant, who\nmade voyages to India and China in the middle of the ninth century. In\nthe first half of the following century Masudi travelled widely--to\nIndia and Ceylon, probably to China, to Madagascar, to the Caspian,\nand in Syria and Egypt. His work, the _Meadows of Gold and Mines of\nPrecious Stones_, is an example of the application of the results of\ntravel and personal observation to history; he was commonly compared\nwith Pliny. The Muhammadan demand for geographical works at this period\nappears to have been great, from the fact that Abu Zaid's work, written\nabout 921, was revised thirty years later by Istakhri, who travelled\nall through the Muhammadan lands, in his _Book of Climates_ (or Zones);\nand this was again revised and extended by Ibn Haukal in 977 in his\n_Book of Roads and Kingdoms_.\n\nA more important figure is Idrisi (_c._ 1099\u20131154), an Arab of Spanish\nbirth, who was probably educated at the great centre of learning,\nCordova. He travelled in North Africa and Asia Minor, and, settling\nin Sicily, made a celestial sphere and a map of the world in silver\nfor King Roger II. Idrisi conceived a substitute for a projection by\ndividing the inhabited world into seven climates or zones between the\nequatorial line and the arctic region; each of these was divided into\neleven equal parts by perpendicular lines. The squares thus formed\nwere used (to the disregard of natural or political divisions) in a\ndescription of the earth carried out by Idrisi for the king, which is\nnoteworthy as having been put together, at least in part, from the\nreports of an organized system of observers, who were despatched at\nRoger's order to various countries. Arabian cartography, however, so\nfar as is known, was primitive. Yet the Arab astronomers made some\nclose meridianal determinations--an error of only three degrees between\nToledo and Bagdad, for example, and a calculation of the major axis of\nthe Mediterranean which was very near the truth.\n\nBefore leaving this period it is worth remarking how commercial\nactivity had not only developed within, but had circumscribed the\nEuropean area. The Arabs, penetrating eastward into Asia, came in\ncontact with traders from north and western Europe using a route which,\nfrom the eighth century, passed \"from India through Novgorod to the\nBaltic; and Arab coins found in Sweden prove how closely the enterprise\nof the Arabs and the Northmen intertwined\" (H.\u00a0R. Mill). Thus ways were\nprepared for the advancement of geographical knowledge when the science\nshould emerge from its stagnation.\n\n\n\n\nCHAPTER IV.\n\nTHE MEDI\u00c6VAL RENASCENCE\n\n\nIt was, in fact, a desire to extend both commerce and Christian\nreligion into the far eastern lands which led to the rescue of\ngeographical study from the evil state into which it had fallen. The\nCrusades form a group of incidents of no less geographical than of\nhistorical importance. Apart from their religious significance, they\nwere undertaken with the object of discovering new routes by which the\nwealth of the east could be brought into commercial exchange with that\nof the west; and in connection with their religious significance they\ngave rise to a period of Christian missionary endeavour in the east.\nBoth movements worked together to bring about the result of a discovery\nof Asia, so far as Europe was concerned, only less novel and important\nthan the discovery of the New World by Columbus and his successors.\nOf the many travellers whose records assisted in this discovery a few\nmay be mentioned as examples. The first whose journeys resulted in a\nnoteworthy addition to knowledge of the Mongol Empire was Joannes de\nPlano Carpini, an Italian of Umbria, who led a catholic mission sent\nin 1245 by Pope Innocent IV to the Mongols shortly after their great\ninvasion of eastern Europe. He was followed in 1253\u201355 by William of\nRubruquis, a Franciscan, who was sent to Tartary by King Louis IX, and\nwhose account is in many respects more valuable than that of Carpini.\nWe may turn aside here to remark that the great philosopher, Roger\nBacon (_c._ 1214\u201394), freely quoted Rubruquis in the geographical\nsection of his _Opus Majus_. This work, and that of Albertus Magnus\nof Swabia (_c._ 1206\u201380), the student of Aristotle, exemplify the\nrevival of interest in geographical theory along lines not dictated by\nChristian in opposition to pagan doctrines. Bacon revived Aristotle's\nopinions as to the spherical form of the earth. He held that the sea\ndid not cover three quarters of the globe, and that there must needs be\nlands unknown, to the south, east, and west of the known world, from\nwhich they must be separated only by narrow waters.\n\nA traveller who followed new lines on the return of his expedition to\nthe east was Hayton, King of Lesser Armenia, in 1224\u201369, whose journey\nwas described by a member of his retinue. It led him (returning)\nthrough the Urumtsi region and the Ili valley, the neighbourhood of the\nmodern Kulja and Aulie-ata, the Syr-darya valley, Samarkand, Bokhara,\nMerv, and northern Persia. Next follows the greatest name among the\neastern travellers of this period, and one of the greatest among all;\nthat of the Venetian Marco Polo (_c._ 1254\u20131324). His father Nicolo\nand his uncle Maffeo had travelled, before his birth, to China and\nestablished friendly relations with Kublai Khan. That ruler sent them\nback to Europe to ask the Pope to despatch a large embassy to his\ncourt, for he was anxious to extend his relations with the western\nworld and his knowledge of western life. There was a long delay owing\nto an interregnum in the papacy, nor were the brothers able to obtain\nthe large following the Khan had desired; but they started back to\nChina themselves in 1271, and Marco accompanied them. They proceeded\nby way of Badakshan, the Pamirs, Kashgar, Yarkand, Khotan, Lop-nor\n(where they covered ground not again travelled by a European for five\ncenturies), and the desert of Gobi. Marco Polo rose in favour at the\ncourt of the Khan. Among many activities he was employed on a mission\nto the Indies, and no opportunity arose for him to return home until\n1292. He was then sent to accompany a Persian embassy on its return\njourney by sea, by way of Sumatra and India. The journey entailed long\ndelays, and Marco only reached Venice in 1295. He subsequently dictated\nhis experiences while a captive in Genoa, having been taken prisoner\nin a naval encounter between Genoa and Venice at Curzola in 1298. They\nwere taken down by a fellow-captive of literary ability, Rusticiano of\nPisa. His geographical achievements have been thus summarized[4]:--\n\n    [4] Yule and Beazley's article on \"Polo\" in _Ency. Brit._ (11th\n        ed.), vol. xxii.\n\n  Polo was the first traveller to trace a route across the whole\n  longitude of Asia, naming and describing kingdom after kingdom\n  which he had seen; the first to speak of the new and brilliant\n  court which had been established at Peking; the first to reveal\n  China in all its wealth and vastness, and to tell of the nations\n  on its borders; the first to tell more of Tibet than its name,\n  to speak of Burma, of Laos, of Siam, of Cochin-China, of Japan,\n  of Java, of Sumatra and other islands of the archipelago, of the\n  Nicobar and Andaman Islands, of Ceylon and its sacred peak, of\n  India but as a country seen and partially explored; the first\n  in medi\u00e6val times to give any distinct account of the secluded\n  Christian empire of Abyssinia, and of the semi-Christian island of\n  Sokotra, and to speak, however dimly, of Zanzibar, and of the vast\n  and distant Madagascar; while he carries us also to the remotely\n  opposite region of Siberia and the Arctic shores, to speak of\n  dog-sledges, white bears, and reindeer-riding Tunguses.\n\nAmong Polo's successors were John of Montecorvino (_c._ 1247\u20131328),\na Franciscan, who became Archbishop of Peking, and wrote the first\nvaluable account of the Coromandel coast of India about 1291; and\nJordanus, a French Dominican, who, having carried Catholicism into\nIndia about 1320, improved even upon Polo's account of the general\ngeography, climate, and products of the peninsula. Another Franciscan\nwho was also a skilled observer both in China and India was Odoric of\nPordenone (_c._ 1236\u201381).\n\nIt may be worth recalling the difficulties which stood in the way of\nimmediately making use of the work of travellers and students at this\nperiod. Printing was not to come into use in Europe for a century yet.\nScribes, no doubt, tended to pay most attention to works of the most\npopular sort, and it is, therefore, no matter for wonder if the works\nof conscientious travellers, such as we have been describing, did\nnot obtain anything like a wide circulation within a short period of\ntheir production. On the other hand, a work which did obtain very wide\nfavour, judged by the standard of the time, was that much-discussed\naccount of wholly, or very largely, imaginary experiences and wonders\nwhich appeared first in French about 1357\u201371 under the name of Jean de\nMandeville, or, in the more familiar English form, Sir John Mandeville.\nThis is an account in the nature of a parody of the work of Odoric\nand other eastern travellers, in the sense that the writer took their\nfacts and substituted or superimposed his own fictions. It is a matter\nfor discussion how far his work was based, if it were based at all, on\nindependent travel and research; yet it contains something of interest\nto students of the history of geography, if only as an opportunity for\nthe exercise of their imagination--as, for example, when the writer\ntells a story of a man who started forth from his home and travelled\nalways eastward, until at last he reached it again, thus encircling\nthe globe. The difficulties in the way of disseminating knowledge will\nsimilarly account for the fact that, although the great Arab traveller\nIbn Batuta has his place in this period, he in no way affected European\ngeographical study. He was a native of Tangier (1304\u201378), who occupied\nthirty years of his life in travel, covering extraordinary distances in\nwest, south, and east Asia, and in Africa, and wrote valuable accounts.\nEgypt, Arabia, and Palestine, Syria, Asia Minor, and Persia, the\nterritories north of the Black Sea and the Caspian, were all well known\nto this wanderer, who also sailed through the Red Sea and along the\nEast African coast, visited many parts of India, the Maldive Islands,\nCeylon, Sumatra, and China, and closed his career as a traveller with\njourneys through Spain and across the Sahara to Timbuktu. His journeys\nare estimated to have amounted to more than 75,000 miles. Modern\ncriticism has proved the remarkable accuracy of his descriptions of\nmany lands and places; but they were unknown to contemporary Europe,\nand, indeed, until the last century.\n\n[Illustration: Fig. 7.--The Hereford Map.]\n\nIt may be said that if a traveller's results did immediately come to\nbe quoted as authoritative, it was in a measure accidental. We have\nalready mentioned the use of the results of Rubruquis by his brother\nFranciscan, Roger Bacon; but Marco Polo's results, for example,\ndo not appear to have had any influence on cartography for about\nhalf-a-century. An excellent example of medi\u00e6val cartography, before\nthese results and others like them began to show their influence\non maps, is provided by the celebrated map preserved in Hereford\nCathedral, made about 1280 by Richard of Haldingham. Here is the world\nstill shown as a round disk, with little conception of the form of the\nMediterranean and its branch seas; while even the British Isles and\nnorth-west Europe are strangely distorted to fit the circle. Such is an\nillustration of the conception of the world still existing in western\nEurope. Few maps of special areas survive from this period, but that\nof Great Britain accompanying the work of the famous historical writer\nMatthew Paris (1259) is an example. It reveals, on the whole, a better\nidea of the internal features of the land than of its coasts and its\nshape, which somewhat resembles a fool's-cap. There are also local maps\nof Palestine; and that country, it may be added here, is the subject of\nthe earliest extant Christian map, in the form of a mosaic on the floor\nof a church in Madaba, in Syria: it dates from the sixth century. The\nCrusades would naturally give rise to a demand for maps of Palestine,\nand they also gave rise to a class of work which may be compared to the\nmodern guide-book.\n\n[Illustration: Fig. 8.--Chart of the Mediterranean, 1500, by Juan de la\nCosa.]\n\nBut among maps of special regions the most notable are those known as\nPortolano maps, which were sailing charts accompanying the Portolani,\nor sailing directions for the Mediterranean Sea. These, no doubt,\nexisted long before the Crusades, in connection with which they first\ncome to our knowledge. They are in most respects remarkably accurate.\nThey are distinguished by groups of rhumb-lines radiating from a series\nof centres, and marked usually with the initials of the names of the\nprincipal winds. As the accuracy of these maps was probably improved\nlargely as the mariners' compass came into use, it may be mentioned\nthat the first European notice of the use of that instrument is\nprovided by the English scientist Alexander Neckam (1157\u20131217) of St.\nAlbans, foster brother to King Richard I. From his mention it appears\nby this time to have been a familiar object. The chief centres for\nthe production of Portolano maps were naturally those identified in\nan important degree with over-sea commerce; such were Genoa, Venice,\nAncona, and Majorca, while the seamen of Catalonia were also prominent\nat this period. These maps were in some cases extended to cover lands\nand seas beyond the immediate Mediterranean area, and even the whole\nworld. World maps were usually circular with Jerusalem as a centre,\nand, in contrast to their accuracy in respect of the Mediterranean,\nthey were not distinguished as a rule for much regard to the best\nsources of information, though for that we have already adduced some\nmeasure of excuse. The map of the world by Petrus Vesconte of Genoa\n(_c._ 1320) shows the Mediterranean and the Black Seas well, the Nile\nfairly, the Caspian indifferently, Scandinavia badly. A mountain range\nextends west and east across almost the whole of northern Europe and\ncentral Asia; rivers drain southward to the Black Sea from this; the\nIndian Ocean appears as a gulf; the south-eastward extension of the\nAfrican coast is retained, and the peninsular form of India is not\nrealized. Subsequent cartographers disagreed on such points as this\nlast. Thus in a Florentine map of about 1350, called the Laurentian\nor Medicean Portolano, the west coast of India is well shown, and the\ninfluence of Marco Polo's travels is to some degree apparent. This map,\nmoreover, has other details of interest, such as the first appearance\nin any known map of the Azores and the islands of Madeira with their\nmodern names. The Catalan map of 1375 recognizes the peninsular form\nof India for the first time, and Marco Polo's results are shown to be\nthoroughly appreciated; and yet a century later the old errors as to\nthe form of India and Ceylon persist even in a map so excellent in many\ndirections as that of Fra Mauro (1457).\n\n\n\n\nCHAPTER V.\n\nPORTUGUESE EXPANSION AND THE REVIVAL OF PTOLEMY\n\n\nThere were obvious geographical and historical reasons why the kingdom\nof Portugal should furnish the important series of incidents in the\nexpansion of geographical knowledge which now claims attention.\nThe Arab power in the Iberian peninsula had been broken, and the\nPortuguese monarchy had established itself during the twelfth century.\nThe Arab mantle of the explorer descended upon Portuguese shoulders.\nThe small kingdom has a large extent of coastline; and not only is\ncommunication with Europe by land through the passes at either end of\nthe Pyrenees comparatively difficult, but between those passes and\nPortugal were Spanish states, with which Portuguese relations were by\nno means amicable. Thus there was little opportunity for the commercial\nexpansion of Portugal except over-seas.\n\nThe first important figure in the history of this expansion is that of\nPrince Henry, surnamed the Navigator (1394\u20131460), fifth son of King\nJohn I. His objects were to extend Portuguese commercial interests\nmainly in West Africa, and also, it would appear, to discover new\nlands, if they were to be discovered, to the west of those Atlantic\nislands which formed the limit of knowledge to the west from very early\ntimes. Even the knowledge of the islands themselves was indefinite\nenough; so that when, in or about 1415, Henry began sending his\nseamen to the Canaries and later to the Madeira and the Azores, he\nwas inspiring, if not actual discovery, at any rate the acquisition\nof largely new information. Between 1415 and 1431 colonization and\ntrade had already begun to be established in some of the islands;\nand, though the Portuguese navigators did not forestall Columbus, it\nis likely that they conceived the possibility of a westward route to\nthe Far East. Prince Henry's residence from 1438 to 1460 was Sagres,\nwhich consequently became a centre for geographical research, for\nhe gathered about him expert cartographers and instructors for his\nnavigators, whom he supplied with the best obtainable instruments,\nmaps, and information: he used not only European but also Arab sources.\nWith regard to the West African coast, he experienced some years of\ncomparative failure; but from 1444 explorations here were rapidly\nextended, and a few of the leading navigators and explorers who worked\nunder Prince Henry's direction and after his death may be mentioned.\nIn 1443 John Fernandez travelled inland in the district of Rio de\nOro, and collected valuable information about the resources, physical\nconditions, and people of the south-west part of the Sahara. He made\nfurther journeys in 1446\u201347. Diogo Gomez, in 1448, made his way up the\nGambia river. Alvise Cadamosto, a Venetian in Prince Henry's service,\nwas working in 1455 south of the Senegal; and in 1456 he visited, and\nprobably actually discovered, the Cape Verde Islands. His accounts\nof his voyages were full and valuable, and he also dealt with the\nexplorations of Pedro de Cintra, in 1461 or 1462, to Sierra Leone and\nthe Gold Coast. Gomez made a voyage to the Cape Verde Islands in 1462;\nbut he is most notable as chronicler of the life-work of Prince Henry.\n\nKing John II built on the prince's foundations. In 1482 he sent out\nDiogo C\u00e3o, who discovered the Congo and ascended it for a short\ndistance, and subsequently saw the coast of Angola as far as 13\u00b0 26\u00b4\nS. at Cape Santa Maria. On a second voyage (1485\u20131486) he penetrated\nstill further south, to Cape Cross. He erected pillars at various\npoints on the coast--a practice followed by some of his successors;\nand some of these monuments have been found and preserved. In a voyage\nin 1486 or 1487\u201388 Bartolomeu Diaz extended the knowledge of the west\ncoast nearly five degrees beyond C\u00e3o's furthest, reaching 26\u00b0 38\u00b4 S.\nHe was then driven south by high winds and storms, turned east, and\nfound no land; he therefore steered north again, and struck the coast\nof what is now the Cape Province at Mossel Bay. Continuing eastward, he\nreached the Great Fish River, and was able to realize that the coast\nwas now trending north-easterly, and that the southernmost point of\nthe continent had been turned; but his crew were surfeited with their\ndangers, and insisted on returning. The important cape which he had\ndiscovered he is generally stated to have named Cabo Tormentoso, the\nCape of Storms; and the story goes that King John, recognizing the\nimportance of the discovery to the future object of a sea-route to the\nEast, changed the name to the Cape of Good Hope. But there is good\nreason to believe that the happier name was given by Diaz himself. It\nhad been one of the wishes of Prince Henry, and was one of the objects\nof the voyage of Diaz, to establish communication with a Christian\nking of whose powers rumours reached the west coast of Africa from\nthe interior, and who was known under the name of Prester John. In\n1487 Pedro or Pero de Covilh\u00e3o and Alphonso Payva were sent, partly\nwith the same object, by way of the Mediterranean. They visited Egypt,\nand after many wanderings came to Aden, whence Covilh\u00e3o proceeded to\nCalicut and Goa in India, and, returning thence, travelled south along\nthe East African coast as far as Sofala. He then journeyed in the coast\nlands of Arabia, and visited Mecca and Medina, and finally, entering\nAfrica, proceeded to the court of Prester John in Abyssinia, where he\nwas well treated, but from which he was never allowed to return home.\nPayva, meanwhile, had travelled into Ethiopia, and had died there.\n\nThe journey to India by way of the Cape of Good Hope was completed\nin due course in the last decade of the century; but earlier in that\nsame decade the New World had been discovered by Columbus, and the\nera opened by these two tremendous incidents may be more fittingly\nconsidered in the following chapter; while for the moment some\nconsideration may be given to the state of cartography and theory at\nthe time when Columbus was planning his voyage.\n\nThe works of Ptolemy can have been known to few in the original Greek\nat this time, and for many centuries before. When, therefore, the\ntranslation of his Geography into Latin, originally undertaken by\nEmanuel Chrysoloras, a Byzantine scholar who settled in Italy, was\ncompleted in 1410 by his pupil Jacobus Angelus, these two students\nlit a beacon in the course of geographical study. The translation,\nwhich is usually identified with the name of Angelus alone, was issued\nunder the title of Cosmography instead of Geography. It would appear\nthat Angelus, of whose life apart from this work little is known,\nnot only dealt with the text, but also did the maps into Latin. In a\nshort time there was no lack of copies of the work, and it was soon\nfound necessary to add to the maps at certain points where they failed\nto represent knowledge which was by this time in possession of the\ntranslators. Already about 1424 Claudius Clavus Swartha had constructed\nin Italy a map which showed the north-westward extension of knowledge\nas far as Greenland; the curious orderly curves by which the coastlines\nare represented frankly acknowledge the draughtsman's lack of detail.\nAbout 1470 Nicolaus Germanus, often known erroneously as Donis,\nproduced a manuscript edition of Ptolemy, with maps magnificently\nilluminated and on improved projections. He also added new maps, and\nit has been said of the collection that, as far as concerns methods of\ndrawing, it is the prototype of all subsequent atlases (Nordenskj\u00f6ld).\nAn edition, probably of 1472, if not later, though it is dated\nearlier, reveals the use of a conical projection with meridians and\nparallels drawn across the maps; and, as points of some interest in\ncomparison with modern maps, it may be added that the seas are green,\nthe mountains blue, and other parts of the land red and yellow. The\nFlorentine edition in verse, of about 1480, by Francesco Berlinghieri,\ncontained an important series of new printed maps, including Italy,\nFrance, Spain, and Palestine.\n\nAlthough the extension of knowledge to the north-west, as has been\nmentioned, attracted considerable attention on the part of the editors\nof Ptolemy, the recent Portuguese discoveries in West Africa did not,\napparently, do the same. In an edition, for instance, of 1486, made\nat Ulm, a geographical description of the north-west lands, including\nGreenland, was furnished, and there were quoted the latitude and\nlongitude of 183 places in northern Europe and Greenland; but there\nwas no evidence that the conception of the southern limit of the\nhabitable world by Ptolemy was understood to be now proved wholly\nerroneous by the Portuguese discoveries.\n\nOn the other hand, the appreciation of Portuguese labours appeared\nearlier, as was natural, in Portolano maps. That of Andrea Bianco\n(1448) drew probably on a Portuguese original, showing the West African\ncoast as far as Cape Verde. On the world map of Fra Mauro, already\nreferred to, the Portuguese discoveries are mentioned in an inscription\nof considerable length.\n\nIn connection with prevailing ideas as to what lands lay in the\nAtlantic beyond the certain knowledge of men, it may be observed that\nthe conception of a continent or island of Atlantis was very old, and\nthere were other mythical lands which were also given places in distant\nparts of the ocean. Portolano maps of the fourteenth and fifteenth\ncenturies show the island of Brazil lying to the west of Ireland, an\nisland named Mam to the south of Brazil, and, still further away in\nthe ocean, and to the south again, a large island in the form of a\nparallelogram, which bore the name of Antillia, and appears as early as\n1426.\n\nThe sphericity of the earth, as has been seen, was revived as a theory\nby Bacon and Albertus; and to these inquirers may now be added the name\nof Cardinal d'Ailly (d. 1422), whose work, entitled _Imago Mundi_, may\nbe mentioned because it is known to have been in the possession of\nColumbus. His copy still exists.\n\nA name closely identified with Columbus's preconceived ideas as to\nthe voyage to the Indies by way of the western ocean, and his efforts\nto obtain recognition for them, is that of Paolo del Pozzo Toscanelli\n(1397\u20131482), a Florentine mathematician, astronomer, and cosmographer,\nwhose advice was asked by King Alphonso V. of Portugal as to the\nprobability of this western route. He sent the king a statement and\na chart in support of Columbus's ideas. The chart is lost; but the\nauthor describes it as showing the Indies opposite and to the west\nof Ireland and Africa, together with the islands which were known to\nlie off the coast of the Asiatic mainland, and certain known landing\nplaces. A globe made in 1492 by Martin Behaim, of Nuremberg, is usually\ncited as giving the best existing representation of the views as to the\nextent of the Atlantic, and the route across it, at the moment when\nColumbus began his first voyage. Behaim had lived in Portugal, and had\na high scientific reputation at court. He had probably visited the\nmore northerly parts of the west coast of Africa and also the Azores,\nthough he claimed to have a much more extensive first-hand knowledge,\nas having accompanied Diogo C\u00e3o. It is doubtful if he did so; but if he\ndid, he made but little use of his opportunity. His representation of\nthe west coast of Africa is not accurate; and for the rest, although\nhe had the chance, and apparently an unusually favourable one, of\ncarrying the results of Portuguese research into Europe, he made poor\nuse of it. He was obsessed by Ptolemaic ideas; he showed in a modified\nform the old south-eastward extension of Africa, with Madagascar and\nZanzibar as two great islands lying off it, Zanzibar being south of\nMadagascar. He also modified, but still retained, the Ptolemaic idea of\nthe non-peninsular form of India and the exaggerated size of Ceylon.\nHe gave a gross representation of the Malay Peninsula, and in general\nignored Marco Polo's results, and those of other Asiatic travellers.\nHis representation of Scandinavia was indifferent, and even that of\nthe Mediterranean was below the level of the Portolano maps. As regards\nthe width of the Atlantic Ocean between Europe and Asia, he appears to\nhave followed Toscanelli; and he showed the mythical island of Antillia\nand also that of St. Brandon (the existence of which on maps was an\noutcome of the fabled wanderings of a holy man of Ireland in the sixth\ncentury) in mid-ocean between the Euro-African and the Asiatic coasts.\n\nSuch was the view of the ocean barrier which lay between Columbus and\nthe attainment of his ideal. It serves as a reminder, which, in view of\nthe results he obtained, is sometimes necessary, that that ideal was\nthe discovery, not of new lands, but of a new route to lands already\nknown.\n\n\n\n\nCHAPTER VI.\n\nTHE NEW WORLD\n\n\nThe period which witnessed, among other great achievements, the\ndiscovery of a new hemisphere, and included the voyages of Columbus,\nGama, and Magellan, besides many of an importance only secondary\nto these, has been called the most brilliant in human history. The\nexaggeration, if such it be, is excusable; certainly in the department\nof geographical history no other period shares the peculiar lustre of\nthis. Christopher Columbus (1446\u20131506) was born of humble parentage\nat Genoa. He went early to sea, and was attracted to Lisbon no doubt\nby the reputation of the Portuguese as navigators. He had voyaged to\nthe eastern Mediterranean, to the Guinea Coast, and in the north as\nfar, perhaps, as Iceland. It is not until 1484 that we find his great\nscheme of the crossing of the Atlantic to Asia matured and laid before\nthe King of Portugal. It was refused by him; it was rejected also in\nGenoa, Venice, England, and France. Then he presented it at Madrid, and\nit was examined by a quasi-expert committee, which pronounced against\nit; but at last it came under the notice of Queen Isabella, and by her\nwas taken up. Columbus obtained three ships, and sailed on August 3,\n1492, from Palos, himself in command of one vessel, the others under\nMartin Alonzo and Vicente Ya\u00f1ez Pinzon. The expedition touched at the\nCanaries in September, and the story of its subsequent progress is\nwell known--how difficulties were encountered in the Sargasso Sea, how\nthe hearts of the crew failed them, how Columbus was driven to give\nthem false reckonings of the position of the ships, but was at last\nenabled to point out to them signs of neighbouring land in the flotsam\nof the waters, and how at length land was actually sighted on October\n11, and on the following day a landing in state was effected on an\nisland to which the name of San Salvador was given, and which is now\nusually identified with Watling Island of the Bahamas. Columbus did not\nthen, or at any time afterwards, suppose that he had done otherwise\nthan reach some part of the archipelago off Asia. On the present voyage\nhe observed a number of the West Indian islands, and he returned to\nSpain in March, 1493, to meet with a magnificent reception.\n\nAmong the effects of Columbus's discoveries was the necessity which\nwas immediately found for delimiting the spheres in which Portuguese\nand Spanish explorations respectively should be prosecuted. In the\nexisting state of geographical theory it may be supposed that a\nsatisfactory delimitation was no easy matter to obtain, and, indeed,\nthe line which was chosen was found, in fact, impossible to demarcate.\nIt was determined under the treaty of Tordesillas (June 7, 1494), which\nfollowed upon a pronouncement contained in two papal bulls of the\nprevious year, that to Spain should belong islands discovered west of a\nnorth-and-south line drawn 370 leagues west of the Cape Verde Islands,\nand that Portugal might claim all lands lying east of this line. The\nrule gave rise to such difficulties that Portugal is found later making\na claim upon Brazil, while Spain did the same upon the Moluccas.\n\nIn 1493 Columbus led a new and much larger expedition to the scene of\nhis triumph. There was now no lack of enthusiasm to be of his company.\nOn November 3 Dominica was reached; the Antilles were subsequently\nsurveyed, and Hispaniola or Haiti, which had been discovered on the\nfirst expedition, was again visited. Jamaica was found, the south coast\nof Cuba was traced--though this island was taken for a peninsula--and\nit is possible that the mainland of Central America was seen. Columbus\nreturned in 1496 to Spain, and set out on his third voyage in 1498.\nHolding a more southerly course than before, he came to Trinidad,\nand subsequently recognized the neighbourhood of continental land by\nobserving a current of fresh water of considerable strength in the\nopen sea; this was from the mouth of the Orinoco. The settlers in\nthe new colonies met with many difficulties, and created more. There\nwas much hostility towards Columbus, and he was doubtless restrained\nfrom accomplishing much which might have been accomplished if his\nfollowers had been wholly loyal. On his last voyage, in 1502\u20134, his\naim was to penetrate right through the archipelago and to complete the\ncircumnavigation of the world. After coasting along Cuba he turned\nsouth, and came upon the coast of Honduras, which he followed for\nnearly four months before he was compelled to return to Spain. Within a\nfew days of his arrival there, he lost his great patroness by the death\nof Isabella, and he himself passed the remaining months of his life in\ncomparative neglect.\n\nVoyagers hastened to follow him across the Atlantic. The work of those\nwho, like Cabot and Cortereal, added a knowledge of the north-eastern\ncoasts of the new continent to the discoveries of the time, will be\nconsidered more appropriately in connection with the exploration of\nthe arctic region, and the attempt to solve the long-lived problem of\nthe north-west passage (Chapter VIII). The year 1500 and the first few\nyears of the new century were only less notable than that of Columbus's\ndiscovery.\n\nVicente Ya\u00f1ez Pinzon came of a wealthy Andalusian family, of which\nseveral members were well-known navigators--Vicente himself and two\nbrothers, Martin and Francisco, had assisted Columbus. He, in command\nof an expedition in 1499\u20131500, was the first Spaniard to cross the\nEquator, discovered the Brazilian coast at Cape San Agostinho, added\n300 miles to the known coast of South America, and found the mouth of\nthe Amazon. The Portuguese Cabral, who has been commonly hailed as the\ndiscoverer of Brazil, actually reached its coast some three months\nafter Pinzon, having been driven far from his course round the Cape\nof Good Hope to the Indian seas in the wake of Vasco da Gama (Chapter\nVII). In the following years many expeditions crossed the Atlantic\nfor discovery and conquest. Pinzon continued his travels in 1507\u201309,\nand in the course of them, in company with Juan Diaz de Solis, sailed\nsouth along the Brazilian coast, and is said to have passed without\nrecognizing the great estuary of the Rio de la Plata.\n\nInto this period fall the much-discussed voyages of Amerigo Vespucci,\nwho, if he were indeed a geographical charlatan, as the greater\nweight of opinion (though it is rather delicately balanced) appears\nto stigmatize him, is one of the most remarkable examples of a type\nwhich, at periods of keen public interest in exploration, has been not\nuncommon; even modern instances may come into the memory. Vespucci, a\nnative of Florence (1451\u20131512), came into Spanish service as a naval\ncontractor, and claimed to have himself made a voyage in 1497, which,\nif the distances and positions quoted by him were accurate, would\nhave extended into the Pacific and as far as the coast now belonging\nto British Columbia, and would moreover have brought him within sight\nof the American mainland before any other navigator of this period.\nHe also gave accounts of three later voyages (two in Portuguese\nservice), abundant in equally or more improbable statements, as that\nhe approached within thirteen degrees of the south pole. If he lied,\nhe was rewarded with something more than the transient fame which\nothers of his kind have usually had to exchange for notoriety. Martin\nWaldseem\u00fcller, professor of cosmography at St. Di\u00e9, is credited with\nthe first suggestion that the new continent should take its name from\nAmerigo, while the hero himself died in the enjoyment of the Spanish\noffice of chief pilot. It must be remembered that at this period of\nconstant fresh discoveries it was almost impossible to form any true\nconception of the relative importance of the work of different men;\nand as it was not then the habitual practice of explorers to address\nthemselves immediately on their return to the task of writing down\ntheir experiences, Vespucci's narratives were eagerly seized upon as\nfurnishing a trustworthy account of the new world.\n\nIt was not until after the first decade of the sixteenth century\nthat the Spaniards, who were to play so large a part in the history\nof America, began to interest themselves in the penetration of the\ncontinent itself, as distinct from the islands in the Caribbean Sea.\nThe voyage of Juan Ponce de Leon, who was obsessed with one of the\nromantic stories of the period, telling of an island to the north\nwhich held the secret of eternal youth, resulted in the acquisition\nof Florida in 1512--the first Spanish possession in North America. In\nthe next year, 1513, Vasco Nu\u00f1ez de Balboa, a Spanish adventurer who\nhad conquered an American kingdom with his sword, almost accidentally\nmade the great discovery of the Pacific Ocean from the elevation of \"a\npeak in Darien.\" After Nu\u00f1ez had triumphantly taken possession of the\nwhole sea in the name of Spain, and had returned with his news, others\nhastened to follow him, the most successful being Gil Gon\u00e7alez de\nAvila, who arrived at Nicaragua in 1523. The discovery of the Gulf of\nMexico and of Yucatan was delayed even longer than that of the Pacific\nshore: it was not until 1517 that Hernandez de Cordova touched at the\npeninsula. But in the next year it and the gulf were explored with some\nthoroughness by Juan de Grisalva, who brought back to the governor of\nCuba the first news of a civilized race living in Mexico, building\ngreat cities and rich in gold, to awaken the cupidity of the Spanish\nadventurers.\n\nOne great question, however, remained unsolved: the western route to\nAsia was still to seek. To the voyagers of the early sixteenth century\nit must have seemed as though the oceans of the world were divided in\ntwo by a mighty land barrier, stretching from pole to pole. Apart from\nthe unexpected glimpse of the seas on the other side obtained at the\nisthmus, nothing was known of the length or width of the new continent,\nuntil the problem was solved by Fernao de Magalh\u00e3es (Magellan) in one\nof the most remarkable voyages recorded. He set sail from Spain in\n1519, and after arriving at the mouth of the Rio de la Plata, which was\nalready known, he began a careful search of the unknown coast to the\nsouth, always looking for the opening which might lead him to the west.\nIn November of the same year Magellan rounded the southern extremity\nof the mainland, and sailed joyfully through the straits which bear\nhis name, between the continent and the Fuegian archipelago. He and\nhis men believed that now their journey lay behind them; a few days\nmore with a favouring wind, and they expected to see again lands known\nto men, some point of the islands of eastern Asia. The wind favoured\nthem indeed, and the name \"Pacific,\" given by Magellan[5] to the ocean,\nrecords his gratitude; but the land was far to seek, and it was not\nuntil ninety-nine days had passed, and his crew were come to the last\nstages of starvation, that he reached the Mariannes, and from thence\ncame in ten days to the Philippines. Not only had he sailed half round\nthe globe since he left the shores of the New World, but he had passed\namong the many scattered groups of islands in that part of the ocean\nwithout ever sighting land. He was not destined to enjoy his success,\nfor he died in a skirmish on Matan, one of the Philippine Islands, and\nthe news of the first circumnavigation of the globe was brought home by\nhis subordinate, Sebastian del Cano.\n\n    [5] Another story attributes this name to Vasco Nu\u00f1ez' native\n        informant.\n\nNow that the sea-way to the western coasts of America was pointed out,\nthe outline of the continent became fairly known, and the interior was\nbeing gradually covered with landmarks, by the end of the sixteenth\ncentury. Under the energetic direction of Cortez, the Spaniards had\nconquered Mexico. By 1533 his emissaries had arrived at the Gulf of\nCalifornia; in 1541 New Mexico was explored, and by 1571 it was\npossible to construct the admirable map of Mexico which figures in\nOrtelius's Atlas (1579). In South America Francisco Pizarro and his\nthree brothers, with Diego Almagro and Hernando de Luque, had worked\nwith such daring good-fortune that the whole of the Empire of the Incas\nwas conquered for Spain, and in 1535 Chile was added to it. After\nthe first conquest, great tracts of country were explored and taken\npossession of--first the valleys of the Andes and the country which was\nnamed New Granada, then the whole coast between the Straits of Magellan\nand Peru; and in 1541 Francisco de Orellana, who had separated himself\nfrom a disastrous expedition led by Gonzalo Pizarro over the Andes from\nQuito, made a stupendous journey right down the River Amazon. In 1539\nHernandez de Soto, longing for another Peru to conquer, and full of\nthe fables of treasure and precious metals which were rife everywhere\nin those days of great discoveries, started on a long and courageous\nbattle to win for himself and his followers the land lying between the\nGulf of Mexico and the River Ohio. He penetrated Arkansas; but he was\neventually killed, and his followers driven from the country.\n\nOther nations were working to conquer territories for themselves in\nother parts of the continent. The King of France, Francis I, sent\nJacques Cartier on four separate voyages between 1533 and 1543, and\nhe sailed up the river of St. Lawrence to the spot where Montreal now\nstands; but little colonization was done till the beginning of the\nfollowing century. From France also came several colonies of Huguenots,\nsent out by Admiral Coligny between 1555 and 1564, one to Brazil and\ntwo to Florida, which were all attacked and destroyed. To England\nbelongs the honour of the second circumnavigation of the world.\nFrancis Drake rounded Cape Horn, and proved that Tierra del Fuego did\nnot form part of the southern continent, as suggested by Magellan's\ndiscovery. He sailed up and down the whole of the western coast, and\nacross the Pacific to the Philippines, returning to England in 1580. He\nwas followed in 1585 by another Englishman, Sir Thomas Cavendish, who\nalso sailed round the world.\n\nAlthough there remained vast tracts unknown in America, the main\nfeatures of the continent had been determined. The French, under the\nable guidance of Samuel Champlain, were rapidly settling in Canada,\nand exploring far and wide, from Hudson Bay to Louisiana. England was\noccupying the country between the Alleghany mountains and the sea. The\nSpaniards were multiplying in South America. But the only expedition\nundertaken in a scientific spirit which resulted in the acquisition\nof valuable knowledge was the exploration of the Amazon by Pedro\nTexeira in 1639. The history of America after the end of the sixteenth\ncentury is concerned rather with colonization than with discovery.\nIt was rapidly partitioned among English, French, Spaniards, and\nPortuguese. Occupation of North America was slowly pushed forward,\nand the continent more or less provisionally mapped. For many years\nthe United States and Canada have had their regular surveys, and in\ntime the northern half of America will be as well mapped as Europe.\nThe States of South America, with the exception of the Argentine and\nChile, are much more backward, and there remain, notwithstanding the\nwork of Humboldt (1799\u20131804) and other scientific explorers in later\nyears, especially around the Upper Amazon, large areas which are still\na virgin field for the explorer.\n\n\n\n\nCHAPTER VII.\n\nTHE FAR EAST AND THE DISCOVERY OF AUSTRALIA\n\n\nIn the meantime the Portuguese had at last won the sea route to India\nby way of the Cape of Good Hope, through the agency of Vasco da Gama\n(1464\u20131524), a native of Sines. He started from Lisbon in July, 1497.\nHe was accompanied by a pilot who had been with Diaz, and he had a map\non which the Portuguese discoveries on the African coast were shown so\nfar as they extended. On November 22 he sailed round the Cape of Good\nHope; by Christmas he reached a point beyond the furthest limit of\nDiaz, and named it Natal. Continuing northerly along the coast he met\nwith a number of Arab settlements and a hostile reception from some of\nthem; but from one he obtained a pilot across the Indian Ocean, and he\nreached Calicut on May 20, 1498, after a voyage of ten months and ten\ndays. This is not the place to discuss the commercial and political\ndifficulties which supervened upon the endeavours of the Portuguese\nto draw to themselves a share of the rich trade of the Indies; but\nGama made a second successful voyage with this object in 1502\u201303, and\nsubsequently became viceroy in India in the year of his death.\n\nIn 1511 the important town of Malacca was taken by the Portuguese, and\nbecame the starting-point for many journeys in all directions--first\nto Sumatra, Java, and the Philippines; in 1512 to the Moluccas, and\nin 1516 to Canton, and by 1520 a Portuguese embassy was established at\nPeking. After that exploration proceeded apace in the Malay islands\nand on the coast of China; and Borneo, New Guinea, and Celebes rapidly\nbecame important trade centres, though Japan was not reached till the\nbeginning of the following century, when in 1542 a Portuguese sailor,\nAntonio de Mota, was driven to its shores. Some years later a mission\nwas sent there by St. Francis Xavier, which brought back the first\ntrustworthy accounts of the new country.\n\nThe Portuguese and Spanish ascendancy in the Malay Archipelago lasted\nuntil 1595; in the following year a Dutch fleet, under Cornelis\nHoutman, came to blows with the Portuguese off the coast of Java;\nin 1602 the Dutch East India Company was incorporated, and during\nthe following decade Dutch influence was strongly established in the\nArchipelago. With this epoch in far eastern history is connected\nthe discovery of Australia. At what early period the native peoples\nof the east--Malays, and even Chinese--had acquired knowledge of\nAustralia, and what was the extent of that knowledge, it is impossible\nto determine; but Marco Polo had happened upon rumours of a southern\ncontinent. In the following chapter we shall discuss the early European\nconception of that continent, which gave rise to a wider problem in\nwhich the discovery of Australia is merely incidental.\n\nA landing on Australian soil has been claimed for the French navigator\nPaulmyer, Sieur de Gonneville, in 1503, and Guillaume le Testu of\nProvence is asserted to have sighted the coast in 1531. There were\ncertainly, about 1527\u201339, French pirates in the Malay Archipelago.\nThere are similar early Portuguese claims to the first view of the\nisland-continent. When Torres had passed through the strait which\nbears his name, south of New Guinea, in 1606, and when, in the same\nyear, the crew of a Dutch vessel, the _Duyfken_, effected a landing in\nthe Gulf of Carpentaria, the first definite steps were taken towards\nthe exploration of Australia. By 1665 the Dutch had worked out and\ncharted a general sketch of most of the western seaboard; in 1696\nWilliam de Vlamingh re-charted a large part of it with fair accuracy.\nIn the meantime, in 1642, Abel Janszoon Tasman had sailed from Batavia\nto Mauritius, thence south-eastward, till he struck the southern and\neastern coasts of Tasmania, whence he passed on to obtain the first\nsight of New Zealand. The exploration of the Australian coasts from the\ndirection of the Pacific belongs to following chapters.\n\nDuring this period England began to interest herself in the East Indian\ntrade, and the great efforts which were made to reach Eastern Asia by\nway of the arctic region will be discussed in the following chapter. In\n1591 James Lancaster made an adventurous voyage to the Malay Peninsula;\nthe formation of the East India Company followed. In 1600 Lancaster,\nin its employ, started again for the East, and laid the foundations\nof English commerce in the Spice Islands, visiting the Nicobars,\nSumatra, and Java. He was accompanied on this voyage by Davis, famous\nfor his work in the Arctic, who was killed on a further voyage in\nEastern waters in 1605. The work of the East India Company led to the\nundertaking of many important voyages of discovery. In 1607 Captain\nHawkins travelled to Agra and the court of the great Mogul, and a\nfactory was started in Japan in 1613. The men in the employ of the East\nIndia Company were not, however, the first Englishmen to reach Japan:\nWilliam Adams, a trader, had been forced to anchor off the island of\nKiu-shiu in 1600. He had been very kindly received, but had not been\nallowed to return home. Permission, however, was at length granted to\nhim, and, after helping to found an English settlement in Japan, he\nspent the rest of his life in the service of the East India Company. On\none of the early expeditions sent out by the trading company in 1612,\nunder Captain Best, the first foothold of the British in India was\ngained by the establishment of factories.\n\nDuring the sixteenth and seventeenth centuries Englishmen were also\nplaying a prominent part in the gradual lifting of the veil which\nlay over Central and Western Asia. Between 1558 and 1579 traders in\nLondon made great efforts to open up to commerce the countries round\nthe Caspian Sea, of which vague reports had been brought back by\ntravellers, and embassies were sent to Bokhara, Persia, and Russia.\nAnthony Jenkinson did much to advance the knowledge of Persia by his\njourney as an accredited representative of Queen Elizabeth in 1579;\nChristopher Burroughs traded across the Caspian Sea at the end of the\ncentury, and Sir Anthony and Robert Shirley stayed at the court of the\nShah of Persia. During the seventeenth century Persia, Syria, and Asia\nMinor were visited by many travellers, who brought back many tales\nof the new and strange countries, but added little to the store of\ngeographical knowledge.\n\nUp to the end of the seventeenth century, the only people to bring\nany news of China and Tibet were missionaries, of whom several made\nadventurous journeys from India. Tibet had been visited by Friar Odoric\nin 1325; but the next European to enter it was Antonio Andrada, in\n1624. Between 1685 and 1687 P. Tachard journeyed to Cochin-China and\nTongking, and made a number of astronomical observations, from which\nhe was able to prove the gross errors in the longitudes of Ptolemy,\nwhich were still in use. After permission to enter the empire of China\nwas granted in 1553 to the Jesuits, much valuable geographical work\nwas done by them throughout the seventeenth and eighteenth centuries.\nDuring the seventeenth century the Russians were gradually pushing\ntowards China by way of Siberia. In 1581 a Cossack made himself master\nof the country round the lower course of the River Irtish; early in the\nnext century the Sea of Okhotsk was reached by Russian hunters; in the\nmiddle of the century the River Amur was navigated to the sea; and by\nits end Kamchatka had been explored, and a treaty had been concluded\nwith the rulers of China. In 1768 and the following years an organized\nexploration of the whole of the Russian Empire was undertaken. Both the\ncoasts and the inland provinces of Siberia were surveyed from Novaya\nZemlya as far as the Sea of Okhotsk.\n\nBy the end of the seventeenth century the general outlines of the\ncoast of Asia were known, though much of the interior was still\nunexplored. Such knowledge of central Asia as was acquired during that\nand the preceding century was mainly due to the travels of European\nmissionaries following Andrada--Fra Desideri and Fra Freyre (1715), and\nOrazio della Penna (who was in Lhasa from 1735 to 1747), were among\nthose who entered Tibet, while others actually carried out surveys\nin China in 1708\u201318 under the direction of the Emperor Kang-hi. The\nDutchman Samuel van der Putte passed thirty-seven years of his life in\nAsia, and travelled widely, but at his death (1745) left no narrative.\nAt the close of the century English missions began to make their way\ninto Tibet from India. George Boyle led one in 1774, and Captain Turner\nanother in 1783. English and French traders were opening up Persia in\nthe eighteenth century, and in Arabia the principal journey was that\nof Carsten Niebuhr (1762\u201367), sole survivor of a Danish scientific\nmission.\n\n\n\n\nCHAPTER VIII.\n\nPOLAR EXPLORATION TO THE EIGHTEENTH CENTURY\n\n\n_(a) Arctic: The North-East and North-West Passages._\n\nAlthough polar exploration is not very directly associated with\ngeographical theory at large, it has been associated with certain\nindividual geographical theories which occupy important positions\nin our history as having held the minds of men for long periods,\nand as owing their proof or disproof to some of the most noteworthy\nexploits in the story of exploration. There is sometimes a tendency\nto suppose that polar, or at any rate arctic, exploration has always\nbeen concerned mainly with the attempt to penetrate as far north as\npossible along one meridian or another, and that any discoveries made\n_en route_ have been merely incidental. But the mere desire to set a\nmore northerly or southerly limit to human travel, and ultimately to\nreach the Poles, really belongs to a relatively late period in the\nhistory of arctic and antarctic work. Taking arctic exploration first,\nwe find that its object was in its early stages mainly commercial; from\nthat object there naturally developed a desire to extend geographical\nknowledge, and, lastly, the extension of many branches of scientific\nknowledge was served by that particular branch of exploratory work.\n\nSome reference has already been made to the early knowledge of arctic\nlands acquired by Scandinavian seamen, who in the second half of the\nninth century had carried not only their commercial explorations but\nalso their actual rule round the North Cape and as far as the White\nSea. At this period they also reached Iceland; but they had been\npreceded there by holy men from Ireland, as is stated by Dicuil about\n825. In the tenth century Greenland became known to the Norwegians, and\nduring its ninth decade Eric the Red visited the western coast of that\nland, and colonization and more or less regular intercommunication were\ncarried on thereafter until the early part of the fifteenth century.\nIn the year 1000 Leif Ericsson reached that part of the North American\ncoast which he named Vinland. A second expedition under Thorfinn\nKarlsefne reached the same coast by way of Labrador and Newfoundland,\nbut these discoveries were not followed up. In the northerly direction\nSpitsbergen was found by the close of the twelfth century, and, in the\neasterly, Novaya Zemlya a little later.\n\nWe have already seen something of the effect which these discoveries\nhad upon European geographical studies and cartography; and it has been\npointed out that their effect is still more to be observed when, in\nthat brilliant final decade of the fifteenth century which witnessed\nthe triumphs of Columbus and Gama, John Cabot was despatched by a\ncompany of Bristol merchants across the Atlantic, and reached Cape\nBreton and Nova Scotia as some believe, or at least Newfoundland, for\nat Bristol many Scandinavian merchants were settled, and doubtless\nhelped to inspire this and succeeding journeys. A second journey\nwas made by Cabot in 1498, and in 1500 to 1501 a Portuguese, Gaspar\nCortereal, visited eastern Greenland and Newfoundland, and these\nvoyages had the incidental result of opening up the important\nNewfoundland fisheries. But the ultimate object of exploration in this\ndirection, which now and for a long succeeding period possessed the\nminds of men, was to discover a north-west passage which was held to\nexist and to be feasible for commerce between Europe and the Indies;\nand we shall presently see that a similar north-east passage along the\nnorth coast of Europe itself and Asia to the same goal was no less\neagerly sought after. Cortereal may have seen the opening of Hudson's\nStrait, and, though he was lost on a second voyage (as was his brother\nwho sought him), the possibility of finding a practicable north-west\npassage attracted the Portuguese as well as others, but only for a\nlittle while--their instincts like their interests led them southward,\nnot northward.\n\nWe cannot here detail all the work of explorers who, in the sixteenth\ncentury and after, extended discovery in one or another of these\ndirections; nor, indeed, in the case of some of the earlier journeys,\ndo the records admit of doing so. It is thus probable that in the\nsecond half of the sixteenth century Portuguese sailors had anticipated\nHenry Hudson in acquiring considerable knowledge of Hudson Bay, but it\nis not possible to say how far they had penetrated it. In 1558 Nicolo\nZeno of Venice put forth a forged narrative of a fictitious journey\nwhich he attributed to an ancestor of his own. He attached a map to\nit, and showed thereon an imaginary land named Frisland to the south\nof Greenland, and other equally false details which set many later\ntravellers and geographers astray. Sir Martin Frobisher, holding that\nthe discovery of the North-West passage was the crowning piece of\nexploratory work remaining to reward the adventurer, secured Queen\nElizabeth's and other powerful interests, and led an expedition to\nthe north-west in 1576. Zeno's map put him wrong when he discovered\nEastern Greenland and thought it to be Frisland, and he subsequently\nreached the southern part of Baffin Land; but this was supposed to be\nGreenland. He made a second voyage in 1578. Frobisher's successor, John\nDavis, another explorer of high scientific standing, misinterpreted\nsome of Frobisher's results, as other geographers did, thus placing\nFrobisher Strait not in Baffin Land but in Greenland, where it long\nappeared in maps. Davis made voyages in 1585, 1586, and 1587. He passed\nalong Davis Strait, and explored both its eastern and its western\nshores, extending the knowledge of the west coast of Greenland on\nhis third voyage as far north as 72\u00b0 41\u00b4. Henry Hudson, in his two\nearlier voyages in 1607 and 1608, was concerned with the seas of the\nnorth-east, and we shall consider them later. But in 1610 he sailed\nwest and entered Hudson Bay, where, after many sufferings, he was\nset adrift by a mutinous crew and was lost. Following him Sir Thomas\nButton, in 1612, became acquainted with the west coast of the bay, and\nmaintained the belief, which was upheld for more than a generation\nfollowing, that the North-West passage to the Indies would be found\nto open from this coast. In 1615 and 1616 William Baffin made a\nvoyage which brought within men's knowledge the channels which ramify\nnorthward and westward from the head of Baffin Bay; but this knowledge\nwas not appraised at its true value; indeed, even the authenticity\nof his work came to be doubted, notable though the voyage was for\nmany important discoveries, including, among others, remarkable\nmagnetic observations, for in Smith's Sound Baffin discovered the\ngreatest known variation of the compass. His explorations carried\nhim more than three hundred miles beyond Davis's northern limit. In\nspite of these discoveries, which gave no impulse towards theoretical\ndiscussion of the North-West passage along right lines, and in spite\nof the opportunity for investigation to the west of Hudson Bay by the\nearly servants of the Hudson's Bay Company, which was established in\n1670, the theory that the passage lay westward from that bay was still\nalive in 1722, when the voyage of John Scroggs, in search of two ships\npreviously lost, was held to prove the existence of a strait leading\ninto the Pacific; but in 1742 Christopher Middleton made further\nacquaintance with the inlets on the west shore of the bay, and later in\nthe century some of the servants of the Company began to arrive at a\nconception of the north-westward extension of the continent. Thus about\n1770 Samuel Hearne reached its arctic shore by way of the Coppermine\nRiver, and in 1789 Alexander Mackenzie came to the mouth of the great\nriver which bears his name. The North-West passage had now ceased to\nbe sought as a highway of commerce, though as a matter of scientific\ninterest James Cook, on the third of his great voyages (1776), which\nwill be dealt with in a later chapter, was instructed to find it from\nthe Pacific, but was stopped by the ice in 70\u00b0 41\u00b4, beyond Bering\nStrait, having been the first English navigator to observe the western\nextremity of Alaska, which had, however, been known for a century or\nmore to the Russians.\n\nThe exploration for the North-East passage, though of considerably\ngreater importance to commerce, was hardly of equal importance with\nthat of the North-West passage from our present standpoint, for it\ncould scarcely have been preceded by that complete ignorance which\nthe voyages to the North-West just mentioned did so much to dispel.\nSome general idea of the coasts of northern Europe, based upon early\nScandinavian and possibly Russian work, may be supposed to have existed\neven when, in 1484, the Portuguese are said to have endeavoured to\nfind a route in this direction to the Indies, and when in the first\nhalf of the following century the feasibility of such a route came to\nbe seriously discussed in England. From that country an expedition,\nin the arrangement of which Sebastian, son of John Cabot, took a\nleading part, started in 1553 under Sir Hugh Willoughby, with Richard\nChancellor commanding one of the ships. Willoughby was lost on the Kola\npeninsula; Chancellor succeeded in entering the White Sea, travelled\nto Moscow, and made arrangements for the opening of a commercial route\nbetween England and northern Russia. The Muscovy Company of merchants\nwas founded; and, after Chancellor had been lost on a second expedition\nin 1555, the Company sent out one of his companions, Stephen Borough,\nin 1556, with the river Ob as his goal. From this voyage information\nwas first disseminated about the Kola region and Novaya Zemlya, though\nthese had been known long previously to Russian fishermen and hunters.\nThe Company also sent out Arthur Pet and Charles Jackman in 1580.\nThey entered the Kara Sea, but only Pet returned. At this period the\nDutch entered the field; they were trading in the White Sea towards\nthe end of the century, and in 1582 Olivier Brunel, and in 1594 and\n1596 William Barents and others, penetrated far eastward. In 1596 an\nexpedition with Barents as pilot discovered Spitsbergen, and one of\nthe ships proceeded eastward to Novaya Zemlya, where Barents died; but\nthe rest of the party withstood the winter and won their way back to\nthe Lapland coast in boats. It was found that Russian trading vessels\nwere making regular journeys to the Ob and the Yenisei, and in the two\nfollowing centuries the Russians were chiefly instrumental in extending\nknowledge of the northern coast of Asia. In 1648 Simon Dezhnev may\nhave passed through the strait which afterwards bore the name of Vitus\nBering. In 1735 the northernmost promontory of Siberia was rounded in\nsledges by Lieutenant T. Chelyuskin, and his name was given to it; and\nin 1728 and 1740 Bering explored both the strait and the sea which bear\nhis name.\n\nMeanwhile Henry Hudson had done for the Muscovy Company and for England\nwhat Barents had done for the Dutch, for in 1607 he reached a point on\nthe east coast of Greenland in 73\u00b0 N., studied the conditions of the\nice between that country and Spitsbergen, and discovered the island\nafterwards called Jan Mayen. In 1608 he made similar investigations of\nthe ice from Spitsbergen to Novaya Zemlya, and he was again in the same\nseas before proceeding to the west to discover the Hudson river and the\nstrait and bay which, as we have seen, were also given his name. Upon\nthe work of Hudson and Barents followed the celebrated whale fisheries\nof Spitsbergen and elsewhere, with which we have little direct concern\nin this history; but when, in the second half of the seventeenth\ncentury, these fisheries, as far as British enterprise was concerned,\nreached the height of their prosperity, geographical research was\nencouraged along the lines which this industry gave opportunity to\nfollow, for a reward of \u00a35,000 was offered in 1776 to any ship which\nshould first sail northward of 89\u00b0 N. Though the prize was not then\nwon, the foundations of scientific research in the arctic were firmly\nlaid by able and intelligent captains of the whaling ships, among\nwhom William Scoresby is pre-eminent through his voyage as far as 81\u00b0\n12\u00b4 42\u00b4\u00b4 N., in 1806, his exploration of the east coast of Greenland\nfrom 75\u00b0 to 69\u00b0 N. in 1822, and his admirable _Account of the Arctic\nRegions_.\n\n\n_(b) Antarctic: the Great South Land._\n\nThe leading interest of arctic exploration thus far, in connection\nwith our present study, has been seen to be concerned with the opening\nof sea routes, by north-west and north-east, from Europe to Asia. The\nstory of antarctic discovery, on the other hand, brings into prominence\na problem of far different character which we have already had occasion\nto notice incidentally. It might be labelled as the problem of the\nFifth Continent, though its origin dates from a period long before\nthe discovery of the fourth. We have seen how the conception of a\nspherical earth was supported on grounds of speculative philosophy by\nthe Pythagoreans, and proven by the observations of Aristotle. We have\nmet with the Stoic conception of the four land-masses, one in each of\nthe \"four quarters\" of the globe, of which but one, the _\u0153cumene_, was\nknown. Here, then, was the material already for an antarctic problem.\nThere can be no doubt that its solution was regarded in ancient times\nas beyond the limit of human endeavour, because the passage of the\ntorrid zone was held impossible in spite of those rumours of far\nsoutherly voyages, to which we have referred, and which might otherwise\nhave pointed the way to further discovery.\n\nWe pass over the period when the antipodean theory was maintained only\nby the freest of Christian thinkers, and resume the antarctic story\nat the point where Bartolomeu Diaz demonstrated by experience that\nthe torrid climatic belt is not impassable, and entered a temperate\nbelt to the south of it. After this, the discovery of Brazil, the\nfrequent enforced voyages south of the Horn by mariners driven thither\nby storms, and the exploration of the Pacific, are all intimately\nassociated with the antarctic problem. Spanish and Portuguese\nvoyagers at the beginning of the sixteenth century revealed Brazil as\ncontinental land; a Portuguese expedition in 1514 reported the southern\nextremity of South America to be in 40\u00b0 S., and asserted that the coast\nof a southern continent was observed beyond this. Ferdinand Magellan,\nagain, passing in 1521 through the straits which bear his name, between\nthe South American mainland and Tierra del Fuego, proved nothing as to\nthe continental or insular character of the latter; but the general\ntendency of geographical theory favoured the idea of a continent. It\ntook various shapes. We find its coastline roughly coincident with the\nAntarctic Circle according to Leonardo da Vinci's globe constructed\nin 1515; this was at least a better estimate of its extent than that\nof Orontius of 1531, wherein the west of Terra Australis (\"lately\ndiscovered but not yet fully known,\" as it is ingenuously labelled)\napproaches close to the southern American promontory about 55\u00b0 S.,\nruns thence eastward between 50\u00b0 and 60\u00b0 S. to the south of Africa,\nextends northward to the latitude of southern Madagascar in the Indian\nOcean, where the \"region of Brazil\" is found, and to the south-east of\nMalaysia reveals a vast peninsula (Regio Patalis), which has suggested\nsome obscure conception of the existence of Australia. A little later,\nin the middle of the century, New Guinea took its place as a northward\npromontory of the southern continent, after the discovery of part of\nits coast by Inigo Ortis de Retes, a Spanish navigator.\n\nPursuing the course of Pacific exploration so far as it affects the\nAntarctic problem, we find that after Menda\u00f1a's researches in that\nocean had resulted in the discovery of some of its many islands, his\nPortuguese pilot on his second voyage, Pedro Fernandez de Quiros, was\ninspired to petition for the command of a great expedition to the\nSouth Land, obtained his desire by promising the Pope and the King of\nSpain an untold expansion of their realms, respectively spiritual and\ntemporal, and sailed in 1605 from Call\u00e3o to discover an island of the\ngroup afterwards known as the New Hebrides, from which he returned\nsatisfied in the belief of having performed his self-allotted task,\nand having taken possession of his \"Australia del Espiritu Santo\" in\nthe name of his masters. Torres, his companion in command, proceeded\nthrough Torres Strait, and thus cut off New Guinea from the supposed\ncontinent, as afterwards Tasman did Australia itself, crowning the\nwork of the Dutch navigators who had gradually unveiled the western\nshores of that vast island, working from the direction of the Malay\nArchipelago (Chapter VII). Tasman proceeded to discover the west coast\nof New Zealand, which thereupon succeeded to the position formerly\noccupied in the minds of the theorists by Java, New Guinea, and\nAustralia as the northward extension of the Antarctic continent in\nthese seas. Such ideas died hard: at the end of the previous century\nthe conception of even Java as continental land survived, after a\nnumber of voyagers had sailed the seas to the south of it.\n\nDuring this period a number of vessels, mainly those of English\nbuccaneers, on passing through the Straits of Magellan had suffered\nthe common fate of being caught by northerly storms and driven to\nthe south. Little enough, however, emerges from their exiguous and\nindefinite records excepting vague pictures of peril from tempest\nand ice. Thus Sir Francis Drake was carried to about 57\u00b0 S. in 1578,\nand afterwards made northward to discover some of the islands of the\nFuegian archipelago; but even these took their place on the maps as\npart of the southern mainland; and it may be added that the same fate\nbefell the remote and tiny Easter Island when it was observed by Edward\nDavis more than a century later--if Easter Island was indeed his\nlandfall; his observations were not sufficiently definite to enable\nus now to determine. The Dutchmen Jacob Lemaire and Willem Cornelis\nSchouten, however, successfully aimed at discovering a passage into the\nPacific south of the Straits of Magellan, saw and named Cape Horn, and\npassed across the ocean through the Paumotu and Tongan archipelagoes\nby New Pomerania to the East Indies in 1615\u201317. To an intervening\ndate (1598) is assigned the disputed episode of the voyage of a ship\ncommanded by Dirk Gerrits, one of a Dutch squadron which was said\nto have been driven south to 64\u00b0 and there to have fallen in with a\nmountainous snow-clad coast, identified later with the South Shetland\nIslands. The story is typical of the uncertainty and misunderstanding\nassociated with all the early southern voyages; it seems probable\nthat Kaspar Barl\u00e6us, who in 1622 translated into Latin a history of\nthe \"Doings of the Spaniards in America,\" was misled by the confusion\nof a later voyage of one of Gerrits's shipmates (in the account of\nwhich, however, no mention occurs of a far southern land) with that\nof Gerrits himself. However this may be, another mythical point was\nduly established on the mythical coast-line of the Antarctic continent\nrunning westward athwart the southern Pacific. And the old theory held\nits place in spite of the strong proofs adduced by William Dampier,\nduring his voyage round the world in 1699\u20131701, of the inaccuracy of\nthe continental coast as laid down according to the cartographers'\ntheories. Dampier, after sharing to the full in the adventures of the\nbuccaneers in the Pacific, was under the orders of the admiralty on\nthis voyage, in the course of which he approached Australia directly\nfrom the Cape of Good Hope, and made careful explorations in the\nShark's Bay area of the west coast. In 1721 Jacob Roggeveen, leading an\nexpedition on behalf of the Dutch East India Company, was satisfied of\nthe neighbourhood of land about 64\u00b0 S., south of Tierra del Fuego; and\nsubsequently, after voyaging north-westward into the Pacific and (as\nis generally supposed) discovering Samoa, believed that he had located\npromontories of the long-sought continent.\n\nEven the British naval explorers, John Byron in 1765 and Samuel Wallis\nand Philip Carteret in 1767, had it in command to continue the search.\nThe French were now taking a share in Pacific exploration, and Louis\nAntoine de Bougainville in 1768 passed across the Pacific by way of the\nPaumotu, Society, Samoan, and New Hebrides groups to the south coast of\nNew Guinea. But these and other voyages only served to add to the map\nthe archipelagoes of the Western Pacific, when they did not actually\ncause confusion by imperfect position-finding and by the practice of\nsuccessive voyagers of renaming islands previously discovered.\n\nThe French voyager Lozier Bouvet, supplied with vessels by the\nFrench East India Company, sailed in 1738 to discover the continent,\nand battled long and bravely with the Antarctic ice about 55\u00b0 S.\nfor no positive reward save the discovery of Bouvet Island (whose\ninsularity, however, he himself did not recognize) and the negative\none of abolishing the imaginary coastline from the chart of the\nSouth Atlantic. Hope of future great discoveries was revived by the\nobservation of the Marion and Crozet islands during the expedition\nunder Marion-Dufresne in 1772; but the chapter closes with the bitter\ndisappointment of Yves de Kerguelen-Tr\u00e9marec, who, after a first voyage\nin 1772, during which a too vivid imagination led him to regard his\ndiscovery of the island now called Kerguelen as revealing the \"central\nmass of the Antarctic continent,\" and a land of promise, was hopelessly\ndisabused on his second visit (1773) to that inhospitable shred of\nland, whose name he changed from Southern France to the Land of\nDesolation.\n\n\n\n\nCHAPTER IX.\n\nJAMES COOK AND HIS SUCCESSORS\n\n\nThe name of Captain James Cook stands above those of all others\nwho voyaged in the southern half of the globe, for he finally laid\nto rest the myth of the southern continent, and brought the first\ndefinite news to the world of the great island of Australia and of\nNew Zealand. His first voyage was undertaken under the auspices of\nthe Government in 1768 with the object of observing under the most\nfavourable circumstances the transit of Venus, and was thus not\nprimarily one of exploration. An immense amount of work was done,\nhowever; the transit was successfully observed at Tahiti, and the\nSociety Islands were discovered. Six months were spent in a thorough\nexploration of the coast of New Zealand, and of 2,000 miles of the\neast coast of New Holland, or Australia. Thence he sailed to Batavia,\nand proved what Torres had stated in 1607, that New Guinea was not, as\nhad been supposed, a part of Australia. In 1772 he started on another\njourney under Government auspices, designed for the special purpose of\nfinally solving the question of the southern continent. This object\nwas thoroughly accomplished, as Cook sailed from the Cape of Good Hope\nto New Zealand, passing twice within the Antarctic circle on the way,\nand thence he sailed three times across the Pacific. He first cruised\nabout to the south-west of New Zealand, reaching as high a latitude\nas 71\u00b0, and finally touching at Easter Island. He then visited for\nthe first time New Caledonia, Norfolk Island, and the Isle of Pines,\nbesides gaining a clearer knowledge of the Marquesas, New Hebrides, and\nTonga groups, and again reached New Zealand. Finally he voyaged from\nNew Zealand to Tierra del Fuego and the Cape of Good Hope, exploring\nand touching at many little-known points. This journey, from which he\nreturned in 1775, was remarkable as much for Cook's splendid success\nin combating scurvy, the scourge of ocean travellers, as for the great\ndiscoveries made. During the long voyage, equalling three times the\ncircumference of the earth, only one life was lost, and this striking\nresult of his precautions did much to encourage and help explorers who\nfollowed after him.\n\nCook's third voyage, which was undertaken primarily with the idea\nof forcing a way through the north-west passage, has already been\nmentioned in the chapter on Polar Exploration. But it must be noticed\nhere that on his way to the Arctic region, besides revisiting many\nof his previous discoveries and finding the larger islands of the\nCook Archipelago, he sighted for the first time since the sixteenth\ncentury the Hawaiian group. These important islands are supposed to\nhave been discovered by the Spaniard Gaetano in 1555, but had long\nbeen forgotten; it was here that Cook was murdered by the natives in\n1779. His work was of extreme importance in several directions: he made\nknown to the world a larger area of the globe than perhaps any other\nman before or since; he overcame the disease which had previously been\none of the greatest obstacles in the way of explorers, and he laid the\nfoundation of the British Australasian Empire. It is said that had he\nreturned from his last voyage he would have received honour from the\nKing; it would have been due, and overdue.\n\nCook's work in the Pacific was ably carried on after his death by\nseveral other explorers, of whom the best known was J.\u00a0F.\u00a0G. de la\nP\u00e9rouse, who set out in 1785 to fill in the gaps left by Cook on his\nvoyages, and particularly to explore the great sea between North-west\nAmerica and Japan. He made a successful exploration of Manchuria and\nthe islands to the north of Japan, which were then little known, and he\nvisited Petropavlovsk in Kamchatka, whence he sent home the journals\nof his voyage. He then voyaged to the east of Australia, touching at\nSamoa, and reached Botany Bay. Then disaster overtook the expedition,\nand no one returned to tell how its members perished somewhere to the\nnorth of the New Hebrides. In 1791 d'Entrecasteaux set out to search\nfor La P\u00e9rouse; and though he was unsuccessful, he advanced to a large\nextent the knowledge of the islands north-east of Australia. During the\nfollowing hundred years many explorers and scientists worked in the\nPacific, filling in the gaps left by the pioneers in the region. In\n1803 the Russians came on the field, with Adam Krusenstern, followed\nby Otto von Kotzebue (1816), and Fabian von Bellingshausen (1819\u201321).\nThe French followed in 1818 with L.\u00a0C.\u00a0D. de Freycinet, and later\nwith Louis Duperrey and Dumont d'Urville. In 1839 the first important\nAmerican expedition sailed under Charles Wilkes. Much scientific work\nfor purposes of research was carried on in Oceania in the nineteenth\ncentury; but with the exception of the famous \"Challenger\" expedition\n(Chapter XIV) it is beyond the scope of this book.\n\n\n\n\nCHAPTER X.\n\nMEASUREMENT, CARTOGRAPHY, AND THEORY, 1500\u20131800\n\n\nIt is characteristic of our history that a gap, almost entirely\nunbridged, exists between the early period and the sixteenth century\nin the story of the development of methods of precision in determining\ngeographical position. We have already referred to early efforts to\nestimate the size of the earth, and in this connection have mentioned\nthat simple instrument of unknown origin, the gnomon. Aristarchus\nimproved upon the mere upright rod whose shadow was measured, by\nsetting one upright in a bowl, the length of the rod and the radius of\nthe bowl being equal; by means of this instrument, which was called\nthe scaph, the angle of altitude could be read on a scale of circles\ninscribed on the inside of the bowl. Among other early instruments were\nthe astrolabe, an invention attributed to Hipparchus, which served\nmariners and others down to the seventeenth century; the diopter, which\nappears to have resembled an alidade mounted on a stand, and may be\nregarded as a prototype of the theodolite; and Ptolemy's rods, or the\ntriquetum, in which a rod working upon two others, one vertical while\nthe other pointed to the observed object, enabled the angular zenith\ndistance to be read. It is true that some additions to this list of\ninstruments were made by, or for the benefit of, medi\u00e6val mariners\nbefore the pregnant period about the beginning of the sixteenth\ncentury. Thus the less cumbrous quadrant was early brought into use,\nto the partial displacement of the circle of the astrolabe. The\ncross-staff, for measuring the angle between the horizon and the sun,\nis first described, so far as is known, in 1342.\n\n[Illustration: Fig. 9.--Scaph.]\n\n[Illustration: (Front.) (Back.)\n\nFig. 10.--Astrolabe.]\n\n[Illustration: Fig. 11.--Quadrant.]\n\nBut it was not until the sixteenth century that the study of the\nearth's size and figure began again to attract attention. The fact\nthat it did so, and the interest that was thereafter maintained in\nthis investigation, stand in the first instance to the honour of\nFrench science. The Spanish and Portuguese congress which attempted\nin 1524 to lay down the boundary fixed under the Pope's award as\nseparating the areas of Spanish and Portuguese dominion in the new\nworld--a line lying 370 leagues west of the Cape Verd Islands--failed\nutterly; the length neither of a degree nor of a league could be agreed\nupon. Jean Fernel (1497\u20131558) in France, however, made measurements\nby calculation from the revolutions of a carriage wheel and by means\nof quadrant observations, and reached a fair estimate of a degree. A\nDutchman, Willibrord Snell, who published his results in 1617, laid the\nfoundation of modern methods of survey by applying to the measurement\nof an arc between Alkmaar and Bergen-op-Zoom the system of a series\nof triangles and the trigonometrical computation of the distance.\nDuring the century which intervened between the labours of Fernel and\nof Snell, it is clear that interest was waking in the development of\nprecise methods of land-surveying, for the compass was probably first\napplied to this work at the beginning of the period; in 1571 we find\nLeonard Digges introducing in England an instrument which represented\nthe theodolite at an early stage; and Jean Pretorius at Wittenberg\nin 1590, and Philip Danfrie in France in 1597, with his graphometer,\nforeshadowed that most valued equipment for detailed survey work, the\nplane-table.\n\n[Illustration: Fig. 12.--Cross-staff.]\n\nAn arc was measured and the length of the degree calculated in\nEngland by Richard Norwood in 1633\u201337. Important improvements in\ninstruments appear about this time. Thus Fran\u00e7ois Vernier introduced\nin 1630 the microscopic attachment named after him the vernier,\nthrough which close and accurate reading of scales may be made. In\n1643 appeared Torricelli's barometer, and in 1648 Pascal, in France,\napplied the principle of the difference of atmospheric pressure at\ndifferent elevations to the measurement of height above sea-level.\nA little later follows the application of the telescope to surveying\ninstruments. In 1669 Jean Picard, measuring an arc in France, used a\nquadrant fitted with a telescope in which crossed wires were inserted,\nproviding lines and a point (the intersection of the wires) in the\nfield of observation, for the purpose of ensuring accuracy. Meanwhile,\nin 1657, Christian Huygens, a Dutch scientist, introduced (if he did\nnot actually invent) the pendulum clock; and Jean Richer, using one\nin the course of astronomical work undertaken in South America for\nthe French Academy of Sciences, found that the pendulum regulated to\nbeat seconds in Paris failed to do so in Cayenne. This opened up the\nproblem of the deviation of the earth's figure from the true sphere;\nSir Isaac Newton had argued such deviation to exist from mathematical\ntheory associated with the rotation of the earth, and Huygens himself\nalso investigated the question. Their conclusions, and that to be drawn\nfrom Richer's pendulum observation, represented the earth as an oblate\nspheroid, or (in simpler expression) as somewhat flattened at the\npoles, the polar diameter being shorter than the equatorial. On this\nshowing, a degree measured, let us say, in the north should be longer\nthan one measured nearer the equator; but J. and D. Cassini, in the\ncourse of an extensive triangulation in France in 1684\u20131718, obtained\nan opposite result. Their measurements were subsequently proved\ninaccurate, but not before much controversy had arisen as to whether\nthe earth is a prolate spheroid (as their results would go to prove),\nor oblate, as held by Newton and Huygens; and the French Academy had\ndespatched expeditions to Peru and to Lappland, there to measure\narcs for comparison. The Peruvian arc was measured by Pierre Bouguer\nand Charles de la Condamine in 1735\u201345, in the face of difficulties\nsufficiently reflected by the length of time occupied and by the fact\nthat they fell out over the work and published separate accounts of it;\nthe Lappland arc was worked out by P.\u00a0L.\u00a0M. de Maupertuis and his party\nin 1736\u201337.\n\n[Illustration: Fig. 13.--Davis's Back-staff.]\n\n[Illustration: Fig. 14.--Pretonius's Plane-table.]\n\nIt may be noticed that the difficulties of Bouguer and De la Condamine\nincluded troubles with untrustworthy instruments; but during the\nfollowing half-century, while geodetic work proceeded apace in France,\nand was also carried on by measurements in South Africa, North America,\nand Italy, instruments making for greater precision were being\ndesigned.\n\n[Illustration: Fig. 15.--Ramsden's Theodolite.]\n\nThe instruments and data available during the sixteenth and\nseventeenth centuries had been fairly effective in skilled hands for\nthe observation of latitude, but observations for longitude remained\nvery difficult. Regiomontanus had prepared ephemerides for 1474\u20131506,\nand Columbus used them; Peter Apianus made a series for 1521\u201370, but\nthe results continued to be far from accurate till the appearance of\nKepler's Rudolphine Tables in 1526. Harrison's work on the chronometer\nhad been anticipated as early as 1530 by Gemma Frisius, who indicated\nthe possibility of using a clock in determining longitude; but even\nHuygens's clock was not found effective for this purpose. In 1735,\nhowever, John Harrison's first chronometer appeared, and afforded\nthe accurate measurement of time under varying conditions which is\nessential to the calculation of longitude. About 1737 Jonathan Sission\nproduced a theodolite, and later in the century Ramsden's greatly\nimproved theodolite (actually a pioneer instrument, greatly though its\ntype was afterwards modified in detail) was constructed and brought\ninto use in the trigonometrical survey of England and Wales, which was\nbegun in 1784.\n\n[Illustration: Fig. 16.--Modern five-inch transit Theodolite.]\n\nMeanwhile in this period cartography underwent an evolution from\nancient to modern methods. It is impossible here to attempt any\ncatalogue of even the principal cartographers, and the work of a few\nmust be taken as typical. In the earlier part of the period (sixteenth\ncentury) the marine chart was still the most generally valuable of\nthe cartographer's wares; but he was already extending his stock in\nother directions. Thus Gerhard Kremer (1512\u201394), more famous under the\nname of Mercator, is principally known for his chart of the world\non the familiar rectangular projection which bears his name; but his\nother activities, besides the production of an atlas, included that\nof maps of various special areas; and he carried out survey work\nhimself in Flanders as the basis of a map of that territory, which\nhe produced in 1540. Not only the projection named after him, but\nalso the secant conical, are usually attributed to Mercator. Edward\nWright, a mathematician of Cambridge, produced the first English\nmap on Mercator's projection, which indeed has been stated to be\nactually Wright's own invention; on this map we should observe the\nomission of various imaginary and erroneous details common to maps of\nthe period--notably the southern continent. But the renewal of the\nstudy of map-projection was mainly owing to German mathematicians,\nsuch as Werner of Nuremberg, and Apianus, in the first two decades\nof the century. In Mercator's work there are to be observed various\ntendencies towards modern practice, such as the abolition of the old\nsmall sketches or miniatures representing towns and divers other\nsubjects, and the introduction of symbols. On the other hand, the\nperiod of the application of criticism by the cartographer to the data\nbefore him was not yet come. Mercator was content to supplement data,\nwhere imperfect, by imagination; and that tendency is to be observed\nin other work of the period, as, for example, in the astonishing\nconception of the hydrography of Africa set forth by F. Pigafetta in\n1591. However, the application of criticism and prompt attention to\nnew sources of information soon became recognized as cartographers'\nduties. Thus, Nicolas Sanson of Abbeville, who founded a famous\nmap-making establishment in 1627, made a common practice of citing\nhis authorities; and again, promptly upon the work of Jean Picard\n(noticed above) and others, in the determination of positions from 1669\nto the end of the century, there followed the production of a map of\nFrance corrected according to these observations, which were also used\nin other French publications. On some of these appears--first about\n1674--the earliest rude representation of relief by hachures, though\nthe old practice of the cartographer, of drawing relief in a species\nof perspective, and thereby making a molehill on his map out of a\nmountain in nature, was by no means yet superseded. It was more than\nhalf-a-century later that the cartographers hit upon the contour-line.\nM.\u00a0S. Cruquius adopted this method of showing relief on a chart of the\nMerwede in 1728; P. Buache similarly showed the depths of the English\nChannel in 1737; J.\u00a0G. Lehmann used contours as the proper scientific\nbasis of hachuring in 1783, and a contoured map of France was produced\nin 1791 by Dupain-Triel. The atlas of Germany, begun by a famous\ncartographer of Nuremberg, Homann, and published in 1753, illustrates\nsuccessive stages in the evolution of hill-shading; for the earliest\nmap in the series, dating from thirty-five years earlier, shows the\nfirst endeavour to differentiate the shading according to the steepness\nof <DW72>. In the meantime the use of maps had already been recognized\nin some of the many special departments of geographical science from\nwhich they are now inseparable. For example, the variation of the\ncompass had been mapped by C. Burrus early in the seventeenth century,\nand was more effectively worked out by the famous astronomer E. Halley\nin 1683. A. Kircher, again, took an early step in the department of\noceanography by mapping currents and other features of the oceans in\n1665.\n\n[Illustration: Fig. 17.--The World according to Mercator (1587).]\n\nFrom what has been written above, it may be inferred, and justly so,\nthat Holland and France led the way in the development of cartography\nfrom the sixteenth to the eighteenth century. But by the end of the\nsixteenth century and throughout the seventeenth the mapping of most\nof the western European countries was rapidly extended, as in Germany,\nAustria, Switzerland, and Italy, in Denmark and Scandinavia, and in\nthe British Isles. German local mapping ranked high, as appears from\nthe collection in Ortelius's Theatrum Orbis (1570) and from Mercator's\nmap of Germany (1585), both of which show the superiority of the\ncartographical material available for Germany. A large number of maps\nwere based on original survey work. As early as 1566 a map of Bavaria\nby Philip Bienewitz, on a scale approximating (in terms of our survey)\nto two and a quarter miles to an inch, gave the results of a regular\nsurvey of remarkable accuracy for the period. Such was also the case\n(to select an example at home) with Christopher Saxton's atlas of\nEngland and Wales (1574\u201379), in which the maps are about an inch to\nthree miles in scale. This work marked the beginning of an important\nperiod in the history of British maps; Timothy Pont's maps of Scotland\nappeared about 1608, and John Speed's, of the British Isles on about\nthe same scale as Saxton's, in 1610. Hollar adopted a smaller scale\n(about five miles to the inch) in his maps of England and Wales dated\n1644. These were of service in the Civil War, and the importance of\nmilitary requirements in furthering the extension of organized survey\nwork--which will appear in its subsequent history--is early exemplified\nin a survey of Ireland made under an Act of 1653; though this, the\nfirst British cadastral survey, was not a preliminary but a result of\nmilitary operations, for it was made in connection with the parcelling\nof Irish lands among those who took part in the suppression of Irish\nrebellion. Again, the Scottish rebellion of 1745 led directly to a\nsurvey under Captain (afterwards General) Roy in 1747. It may be added\nhere that in later cadastral work Ireland again took precedence of\nGreat Britain: the six-inch survey begun in the former country in 1825\nwas nearly finished when that of Great Britain was undertaken in 1840.\n\nFrance continued to lead the way in cartography in the eighteenth\ncentury. The maps of G. Delisle (1675\u20131726) and of J.\u00a0B.\u00a0B. D'Anville\n(1697\u20131782) were not merely confined to local work; they also included\nthe presentation of cartographical material for distant lands selected\naccording to truer scientific criteria. Thus D'Anville's map of Africa,\nthough preserving a few old inaccuracies, did away with details which\nwere purely imaginary, and boldly revealed the then practically\ncomplete ignorance of the interior by representing it almost wholly\nas a blank. We may contrast this with earlier maps of the continent.\nThus Waldseem\u00fcller (1516) showed waterways running parallel with the\nwest coast, from north to south, and showed no conception of the Congo.\nGastaldi (1564) marked the Zaire (Congo); but this and an east-flowing\nriver and a branch of the Nile all flowed from a great central lake,\nZembere. Mercator established a definite parting of the Nile, the\nZaire, and the east-flowing system, though his ideas were still far\nfrom the truth.\n\nEnglish cartographers of the eighteenth century were inspired by the\nover-sea expansion of the empire to much good work beyond the home\nshores. Thus J.\u00a0F.\u00a0W. Desbarres in 1774\u201379 made use of the nautical\nsurveys of James Cook and others in his Atlantic and North-American\nwork. Thomas Jefferys produced West Indian and American atlases at\nthe same time, and Aaron Arrowsmith founded a famous cartographical\nestablishment, the work of which was carried on for a century. Mention\nis also due here of the work of Major James Rennell, who became\nsurveyor-general of Bengal in 1763, and covered that territory in\nhis atlas of Bengal (1779) on a scale of five miles to one inch, the\nwork depending mainly upon route surveys and being, of its kind,\nextraordinarily good. The trigonometrical survey of England and Wales,\nalready referred to as begun in 1784, owed its origin to French\ninspiration; for Cassini de Thury, who in 1740 had re-measured and\nfound incorrect the work of J. and D. Cassini in France, represented\nthe desirability of establishing a geodetic connection between Paris\nand Greenwich, and General William Roy was appointed to supervize the\nEnglish work.\n\nThe revival of interest in earth-measurement and survey led directly to\nthe furthering of the study of theoretical geography. We have happened\nalready upon the names of Peter Apianus and Gemma Frisius in the\nhistory of the mathematical branch of geography; these two--Apianus by\npublishing in 1524 his _Cosmographicus Liber_, and Frisius by editing\nand expanding that work under the title of _Cosmographia_--re-founded\nthe science on a mathematical basis, though they remained bound to\nthe Ptolemaic view of a sharp distinction between geography, the\ngeneral description of the world, and chorography, the particular\ndescription of a region. There is, perhaps, something characteristic\nin the insistence on this curiously arbitrary distinction; there has\nbeen sometimes a tendency to narrow the view of the field of \"pure\"\ngeography on the part of workers labouring in one corner of it and\nturning their backs upon the rest. Indeed, at this very period is\nfound another _Cosmographia_, to which its author added the epithet\n\"universalis,\" wherein the now familiar view of geography as a human\nand political field of study appeared, to the no less familiar\nexclusion of the mathematical aspect; for Sebastian M\u00fcnster, in his\nwork published in 1544, neglected the mathematical side entirely,\nmodelling his work on that of Strabo. Philip Cluverius, again, in his\n_Introduction to Universal Geography_ (1624), preserves the distinction\nbetween geography and chorography, albeit but one out of his six\nbooks deals with the earth at large, while in the rest countries are\ntreated in detail, the human aspect being closely studied. Nathanael\nCarpenter of Oxford, however, threw over the distinction in so far\nas he recognized that neither geography, as distinguished by earlier\nwriters from chorography, nor chorography itself, nor topography\n(under which term were classified the closer descriptions of smaller\nareas than those which belonged to chorography), was anything more\nthan a part of a whole. Therefore, he divided his _Geography_ (1625)\ninto \"spherical\" and \"topical\" parts, the first dealing with the\nmathematical side, the second with different divisions of the earth\naccording to physical, not political, considerations; his work is thus\nnotable as indicating his realization of the function of geographical\nstudy which follows from those of measurement and description--namely,\nthat of the correlation of phenomena. In 1650 appeared the _Geographia\nGeneralis_ of Bernhard Varenius, who died, still a young man, in that\nyear. He laid down a broad division of the subject into general and\nspecial parts. His general part was sub-divided so as to include,\nfirstly, the shape, size, and general physical characteristics of the\nearth, the distribution of land and water, land forms and hydrography;\nsecondly, the astronomical and mathematical aspects, such as zones,\nlatitude and longitude, the investigation of which was carried further\nin the third sub-division, which was comparative, dealing in this\nmanner with data previously adduced. Varenius did not himself take up\nthe special part of geography, though he defined it as also falling\ninto three sub-divisions, the first of which should deal with the\nposition, boundaries, physical features, and natural products of a\ncountry, the second with celestial and atmospheric conditions, and the\nthird with the human element. This last was (as it still is) a popular\nbranch of the subject which, for his own part, he would not have\nadmitted; he held pure geography to be a matter apart from political\nand social considerations. But here was a geographical framework which,\nas H.\u00a0R. Mill has pointed out, \"was capable of accommodating itself\nto new facts, and was indeed far in advance of the knowledge of the\nperiod.\" Being so, its worth was by no means generally recognized at\nfirst, although in 1672 Sir Isaac Newton put forth an annotated English\nedition for use in connection with lectures of his own. \"The method\nincluded a recognition of the causes and effects of phenomena, as\nwell as the mere fact of their occurrence, and for the first time the\nimportance of the vertical relief of the land was fairly recognized.\"\nThe work found its place in time; the French and Dutch geographers,\nas well as the English, had it in their own tongues, and it became a\nstandard, not only for the century of its original production, but for\nthe next also.\n\n\n\n\nCHAPTER XI.\n\nTHE NINETEENTH CENTURY: AFRICAN RESEARCH\n\n\nSince the later years of the eighteenth century geographical knowledge\nhas been extended in the manner of a great railway system. The main\nlines of exploration will provide the subject of this and following\nchapters; with the ramification of branch lines we can hardly concern\nourselves here. Taking one consideration with another, Africa may be\ntermed the most important area of geographical conquest during this\nlatest period of our history. The opening of the interior of that\ncontinent was long delayed for geographical reasons which have often\nbeen insisted upon. The difficulty of inland communication, the fact\nthat the rivers do not offer uninterrupted highways, the barrier of\nthe tropical forests, the unhealthiness of many parts both of the\ncoast lands and of the interior, which modern science is only now\nfighting--such are the disabilities against which exploration had to\ncontend, to which must be added the lack of commercial instinct in\nmany of the native peoples, and their unhappy experiences of the early\nslave-trading, and the labour-recruiting which has in some instances\nprovided its modern counterpart.\n\nDuring the larger part of the eighteenth century hardly any progress\nwas made with the exploration of Africa. The west coast was still a\nresort of traders for slaves and gold, but very little attempt was\nmade even to acquire further territory. In the middle of the century,\nhowever, the Turkish dislike of intruders was somewhat allayed, owing\npartly to the growth of the coffee trade, and it became more easy for\ntravellers to enter Egypt. In 1770 James Bruce started on the task,\nwhich he had been anxious to undertake for many years, of searching\nfor the Nile sources. He was courteously received by the ruler of\nAbyssinia, and after finding and mapping the source of the Blue Nile,\nhe traced it to its junction with the White Nile at Khartum. On his\nreturn he was disgusted when it was proved to him by D'Anville (whose\nmap, based on a critical judgment of data, was considerably more\ncorrect than Bruce's) that he had been anticipated by the Portuguese\nJesuits. He delayed the publication of his results for seventeen years,\nand when they appeared, though they constitute a remarkable description\nof Abyssinia and its inhabitants, they were universally disbelieved.\n\nTowards the end of the century two causes operated for the revival of\ninterest in African affairs--firstly, the foundation of the African\nAssociation in 1788, and, secondly, the removal by Napoleon's conquest\nof Egypt of the obstacles placed in the way of travellers by Moslem\nfanatics. In 1795 Mungo Park, a Scotchman, started under the auspices\nof the African Association to explore the Niger. He travelled up the\nGambia River, and after extraordinary difficulties reached the Niger\nat Sebu, and traced its course for three hundred miles. He had been so\nlong away that he had been given up for dead, and his exploits, carried\nthrough with such success, aroused great enthusiasm. Unhappily, his\nsecond journey, undertaken in 1805, ended in disaster; he and all who\nwere with him except one guide perished, after descending the Niger\nfor about one thousand miles towards the coast, and only just failing\nto solve the problem of its outflow. In 1798 Francisco de Lacerda, a\nPortuguese explorer, lost his life on the Zambezi, and early in the\nnineteenth century Africa was actually crossed for the first time (so\nfar as is known), by two Portuguese traders, from Mozambique to the\nwest coast.\n\nDuring the first half of the nineteenth century, when the European\nnations had been roused first to protest against and then to abolish\nthe slave trade, a great deal of valuable work was done to increase the\nknowledge of Africa, especially by Englishmen. In 1823 W. Oudney, who\nwas sent out as consul to Bornu, and H. Clapperton penetrated to Lake\nChad, previously unvisited by any white man, and Clapperton afterwards\nexplored the flourishing civilization of Bornu and Hausaland. He died\non a second journey undertaken to open up trade with the Sultan of\nSokoto; but in 1830 his servant on this last journey, Richard Lander,\nsucceeded in solving one of the many problems which were beginning to\npresent themselves to students of the continent--the question of the\noutlet of the Niger. He started from the Guinea coast, and followed\nthe course of the river from Bussa to its mouth in canoes. Much good\nwork was done also by A.\u00a0G. Laing, who was the first European to visit\nTimbuktu (in 1826), but was killed there. R. Cailli\u00e9, who succeeded in\nreaching the city in 1828, was the first to return from it in safety,\nwhich he did by crossing the Sahara to Tangier. H. Barth, who had\nalready travelled all through North Africa, and had ascended the Nile\nto Wady Halfa, started in 1850 on a trading mission under the auspices\nof the British Government to the states of Central Africa; both his\ncompanions died, but Barth carried through alone a brilliant journey,\nin the course of which he travelled from Lake Chad to Timbuktu, and\nstudied minutely many of the ancient civilized states of the region.\nThus the geography of Senegal and the Niger had been largely cleared up\nby 1850. The great questions which remained untouched were those of the\nsources of the Nile and the Congo. By 1850 Abyssinia and the greater\ntributaries of the Nile were pretty well known, through the work of\nnaturalists and travellers, the efforts of Austrian missionaries,\nwho had established stations down to Gondokoro on the Nile, and the\ninterest taken in exploration by Mahomed Ali, ruler of Egypt under\nthe Turks. But the seventy miles of rapids above Gondokoro, and the\nfierce people of the Bari tribe who lived there, had prevented the\nacquisition of any but the most meagre knowledge of the upper reaches\nof the Nile beyond that point. The problem was now approached from a\ndifferent direction. For some time missionaries had been working at\nZanzibar, finding the natives there more tractable than in Abyssinia,\nthe original field of their labours; and two of their number, L. Krapf\nand T. Redmann, had seen and sent home accounts of snow-mountains\nunder the Equator--Kilimanjaro and Kenya--and also of the great lakes,\nwhich they imagined to be all parts of an enormous inland sea. Since\nthe annexation of Aden by the British Government in 1839, moreover,\nofficers of the English army stationed there had shown an interest\nin the exploration of the African coast opposite them; and in 1854\nR.\u00a0F. Burton got permission to try to reach the centre of Africa\nthrough Somaliland, with T.\u00a0H. Speke. Burton first made a courageous\nand successful journey to the walled city of Harar in Abyssinia; but\nthe more important stage of the expedition was unsuccessful owing to\nthe suspicions of the Somalis. For thirty years after this Somaliland\nremained unexplored, and Burton and Speke, inspired by the accounts of\nthe Zanzibar missionaries, devoted themselves to the discovery of the\nsource of the Nile. They arrived at Zanzibar in 1856, and penetrated\nwithout much difficulty to the great plateau of Unyamwezi, where the\nArab traders received them kindly, and thence to Lake Tanganyika.\nBurton was forced through illness to allow Speke to continue the\nwork without him, and the latter saw, though from a distance, the\nwaters of the Victoria Nyanza. Though he underestimated its size, he\nwas convinced that here was the answer to the question of the Nile\nsource; but Burton, jealous of his success, refused to credit it, and\nendeavoured to prove that the Victoria Nyanza was nothing but a great\nswamp. In 1860, however, Speke set out again, with J.\u00a0A. Grant, and\nverified and enlarged his previous discoveries. In face of immense\ndifficulties due to mutinous porters and suspicious and warlike\nchiefs, he succeeded in making a rapid survey of the Victoria Nyanza,\nin exploring the unknown country of Uganda, and in finding the Ripon\nFalls, where the Victoria Nile leaves the lake. In spite of the great\ngaps in Speke's knowledge, his ideas on the relative importance of the\nNile tributaries and of the geography of the lakes were very accurate,\nand the honour of settling the old question of its source must be given\nto him. Speke and Grant were met on their return at Gondokoro on the\nNile by Samuel Baker, who had come with help and reinforcements for\nthem, and he took up the work of definitely locating the Albert Nyanza,\nof which Speke had heard various accounts. Though he and his wife\npassed through terrible dangers and difficulties, they completed their\ntask, finding the Albert Nyanza, and journeying along the Victoria Nile\nas far as the Murchison Falls. The importance of Baker's discovery,\nhowever, was somewhat vitiated by his erroneous judgment of the size\nof the Albert Nyanza, which he made far too large; he, like the other\nexplorers who had gone before him, saw nothing of the great snow-range.\n\nIn other parts of Africa valuable work was being done. David\nLivingstone was always a missionary first, but his explorations form a\nnobler memorial than those of any other African traveller. He had been\nworking in Africa since 1840; in 1849 he crossed the Kalahari desert,\nand reached Lake Ngami; and in 1852 he started on a great journey up\nthe Zambezi, discovering the Victoria Falls, and crossing to the west\ncoast; and returning followed the Zambezi to its mouth on the east\ncoast, thus taking the first steps to fill a great blank which had\npreviously existed on the map of Africa. Between 1858 and 1864 he did\nmuch good work in the exploration of Lake Nyasa and the lower Zambezi,\nand incidentally made splendid efforts to stop the trade in slaves\nwhich was still carried on in that region. But it is his last journey,\nwhich took the form of an expedition to discover the watershed between\nLake Nyasa and Lake Tanganyika, that is of paramount interest from the\ngeographical standpoint. He started from Zanzibar in 1866; and, in\nspite of continual and serious illness, he travelled from Lake Nyasa to\nLake Tanganyika, discovered lakes Mweru, Mofwa, and Bangweulu, and also\nthe river Lualaba, which he took for the upper part of the Nile. He was\ncheered and encouraged by meeting with H.\u00a0M. Stanley in 1871, who had\nbeen sent out to him with money and servants; but in 1873 he died, worn\nout by his constant travels, and still searching for the \"fountains\"\nwhich he believed to form the ultimate source of the Nile.\n\nIn 1874 Stanley undertook and carried through in the face of terrible\ndifficulties one of the most remarkable and valuable journeys in the\nhistory of African exploration. He settled the question as to the\nrelative importance of the Albert Nyanza and the Victoria Nyanza,\ndiscovered the Mwuta Nzige (Lake Albert Edward), and crossed Africa\nfrom east to west, starting from Zanzibar and travelling down the\nLualaba from the point where Livingstone had left it to its mouth,\nthus proving it to be the upper level of the Congo. This great journey\nopened up the hitherto unknown country in the centre of Africa, and\nled directly to the formation of the Congo Free State. Stanley had\nbeen preceded at Nyangwe, the point of departure for the voyage down\nthe Congo, by Lovett Cameron, who failed, however, to follow the river\nthroughout its course, and reached the coast to the south of its mouth.\nIn 1884 Joseph Thomson supplemented Stanley's work by his journey to\nMounts Kilimanjaro and Kenya through Masailand to Victoria Nyanza.\nWissmann and other German explorers (1881\u20136) did much for a knowledge\nof the great southern tributaries of the Congo, while Wissmann in\n1881\u20132 crossed the continent from west to east. It has been frequently\ncrossed since. Stanley's expedition up the Congo to rescue Emin Pasha\n(1887\u201389) opened up the great Central Africa forest with its pigmies,\nextended our knowledge of Lake Albert Edward and its outlet the\nSemliki, which flowed into the Albert Nyanza, and revealed the great\nsnowy range of Ruwenzori, which was ascended and mapped by the Duke of\nthe Abruzzi in 1906.\n\nWhile the problems of Central Africa were thus being elucidated, the\nexploration of the nine great rivers which form the Bahr-el-Ghazal,\nthe chief tributary of the Nile, was attracting much attention, the\nmost valuable work being done by Georg Schweinf\u00fcrth, who mapped the\nriver with high accuracy. The Nile above Gondokoro was also surveyed\nby English officers; the Sahara and the Sudan were widely explored by\nG. Rohlfs and G. Nachtigal between 1860 and 1875, and the connection\nof the river Welle with the Congo system was established by W. Junker,\nwho also travelled in the Sudan from 1875 to 1886. The British during\nthis period were pushing their way northward from South Africa, and\nemigrants were flocking to the goldfields and diamond mines. The French\nhave spread their explorations over the Sahara and their territories in\nthe Sudan and West Africa. From the date of Stanley's return African\nhistory underwent a rapid change. The period of driving the main lines\nof exploration through the continent was over, and a new era began in\nwhich branch lines could be laid down, and colonial expansion could\ntake place along them. The continent is now partitioned among the\nEuropean Powers, and has been more or less provisionally mapped; but\nit will take long years of survey and scientific investigation before\nits features, its resources, its peoples, and its possibilities are\nadequately known.\n\n\n\n\nCHAPTER XII.\n\nTHE NINETEENTH CENTURY AND AFTER: ASIA AND AUSTRALIA\n\n\nThe beginning of the nineteenth century was signalized by the\ninitiation of the great trigonometrical survey of India, and the\nfirst half-century was a period of much important geographical and\nanthropological work within that empire, but to no great extent beyond\nits boundaries, though in 1808 a mission penetrated to the sources of\nthe Ganges, and Baluchistan and Afghanistan were in some part explored\nby officials of the East India Company. But the physical problems of\nthe heart of the continent were left to a later period--those, for\ninstance, concerned with the trans-Himalayan region (as viewed from\nIndia), including Tibet, eastward that region so important in the\nhydrography of the continent, where the river systems of China and\nBurma take their rise, northward the deserts of Mongolia and Turkestan,\nwestward the nodal mountain-region of the Pamirs, and the area which\nlong concealed the sources of the Brahmaputra (Tsanpo) and the rivers\nof Punjab. In spite of the endeavours of the Tibetans to hold inviolate\nthe secrets of their land--in great measure successful so far as their\ncapital, Lhasa, was concerned--the Indian native surveyors, such as\nNain Singh, Krishna, and Ugyen Gyatso, were able to penetrate the\ncountry, and even Lhasa itself; their work covered the period 1863\u201382.\nAnd the last quarter of the century provides a wonderful record of\ncontinuous exploration in Tibet, as will appear from the mere quotation\nof names and dates--P. Bonvalot and Prince Henry of Orl\u00e9ans, 1886\u201387;\nW.\u00a0W. Rockhill, 1888 and 1891; Hamilton Bower, 1891\u201392; Dutreuil de\nRhins and F. Grenard, 1893\u201394; St. George Littledale, 1895; Captains\nW.\u00a0S. Wellby, 1896, and H.\u00a0H.\u00a0P. Deasy, 1896, whose work was afterwards\nextended by Captain C.\u00a0G. Rawling and by Sir M.\u00a0A. Stein; and Sven\nHedin, 1896\u201398, 1899\u20131902, 1906\u201308, whose last journey revealed the\nexistence, long suspected, of the great mountain system north of the\nupper Tsan-po. This list is by no means exhaustive, nor can be that of\nthe Russians who worked from the opposite direction, from their own\nterritory; their leader was Nicolai Prjevalsky, who in 1871\u201373 and in\n1876 worked in the Tsaidam region and made the first contribution to\nthe mapping of the important hydrographical area above referred to, and\nin 1879 studied the vast physical changes which have taken place in\nCentral Asia within historic times, and form one of the most remarkable\ngeographical problems in the world. He continued his work in 1883\u201385,\nand was followed by Pevtsov and Roborovsky (1889 and 1894), P.\u00a0K.\nKozlov, Potanin, and many others. The names of Russian scientists, such\nas Baron A. Kaulbars and L. Griesbach, are also associated with the\nproblems of the Aral and Caspian depressions. The former extensions of\nhuman settlement over areas now covered by the Central Asian deserts\nhas been brought to light in great measure through the researches of\nSven Hedin, and especially of Sir M.\u00a0A. Stein. The general result of\nall these investigations has been to modify profoundly, even during the\npresent generation, preexisting ideas of the physical geography of the\ncentral region. Nor should we overlook the work of recent travellers\nin China proper, a broad canvas on which outlines had been sketched\nearlier; but details remained, and still in great part remain, to be\nfilled in, though Ferdinand Baron von Richthofen, in the course of his\nseven journeys in 1868\u201372, left few districts entirely unvisited.\n\nThe problem of the former existence of flourishing communities in areas\nnow desert, and of the causes of the change, has a partial counterpart\nin southern Arabia. The modern period of Arabian exploration began\nearlier than that of Central Asia. The journeys of J. Hal\u00e9vy (1869),\nE. Glaser (1889), and J.\u00a0T. Bent (1893) in the south were primarily\narch\u00e6ological in purpose. In other parts of the peninsula the work of\nJ.\u00a0L. Burckhardt (1815), Sir R.\u00a0F. Burton, Captain G.\u00a0F. Sadlier, W.\u00a0G.\nPalgrave, Charles Doughty, Wilfrid Blunt, C. Huber, Musil, Leachman,\nand others, has made it possible to lay down at least the position\nof the chief towns and settlements, and the main physical outlines,\nwith close accuracy, save in the Dahna or great desert of the southern\ninterior, which remains untrodden.\n\nThe detailed exploration of Australia began from Sydney, the earliest\nsettlement, and was directed along the coast rather than towards the\ninterior, the penetration of which was difficult. George Bass, after\na short expedition inland, was accompanied by Matthew Flinders in\nexploring the coast of New South Wales as far as the George River\nand Hat Hill towards the end of the eighteenth century; in 1797\u201398\nBass Strait was found to separate Tasmania from the mainland, and\nthat island was circumnavigated. Bass was subsequently lost in\nSouth America; but Flinders extended the work in 1801\u201303, when,\nhaving sailed from England, he worked from King George Sound at the\nsouth-west of Australia right round the south, east, and north coasts\nas far as Arnhem Bay, west of the Gulf of Carpentaria, and would have\naccomplished more but for the unseaworthiness of his ship. Flinders\nwas not only a competent explorer, but also a man of theories: he\ntook the limestone cliffs of the Great Australian Bight (south coast)\nfor coral reefs, and when he entered Spencer Gulf he thought of a\nnorthward strait connecting with the Gulf of Carpentaria, and conceived\nan Australian archipelago; nor was he wholly disabused until he had\ndefinitely located the heads of both gulfs. A number of important\ninlets, such as Port Phillip, Keppel Bay, and Port Bowen, were\nthoroughly investigated by him, and he also surveyed the Great Barrier\nReef. And the substitution of the name of Australia for New Holland is\ndue to a suggestion of his. His unfinished survey of the western shores\nwas completed by Captain P.\u00a0P. King in voyages between 1817 and 1822.\n\nThe accident of a drought in 1813 drove some of the Sydney settlers\nto look for new pastures in the hinterland. The divide between the\nshort eastward and the long westward drainage systems was surmounted\nwith difficulty; a road to the point where the town of Bathurst\nafterwards grew up was promptly made, and an arresting geographical\nproblem confronted the investigators when the westward-flowing rivers\nMacquarie and Lachlan were found. Lieutenant Oxley, R.N., attempting to\nfollow the Lachlan in 1815, was presently brought to a halt by great\nswamps. He struck south to avoid them, and narrowly missed discovering\nthe Murrumbidgee river, before he turned back to carry to Sydney the\nconviction that the westward drainage generally was lost in swamps\nfringing an inland sea. Cunningham found a route from this coast up\nto the rich Liverpool Plains, towards the north of New South Wales,\nin 1823; but for the most part exploration was temporarily directed\nto the south-west, and Hamilton Hume and Hovell in 1824\u201325 took an\ninland route from New South Wales to the south coast on the west side\nof Port Phillip. This inlet was not recognized by them; they returned\nto report that they had seen the coast-land, and found it good, further\nto the east at Western Port; settlers who visited that district on\ntheir recommendation were disappointed, and the development of the\nVictoria coast-lands received a set-back in consequence of this error.\nCunningham in 1828 opened the route from the coast at Brisbane to the\ndowns of the south Queensland hinterland. In the same year a new phase\nwas entered in the solution of the problem of the far interior, when\nCharles Sturt, carrying with him Oxley's conviction of the existence\nof an inland sea, journeyed inland at a season of drought to find\nthe Macquarie river losing itself on the dry plains, and the Darling\nflowing salt. He attributed this fact to an admixture of sea-water, and\nset down the interior of the continent as a desert. In the following\nyear he settled the problem of the drainage of the Murrumbidgee,\nLachlan, and Murray rivers by following them to the mouth of the Murray\nin Lake Alexandrina (south coast); and although he now held that the\nwaters of the Darling were included (as they are) in this system, it\nwas still doubted whether there was a divide between north and south\nflowing waters about the central latitude of New South Wales, where,\nin the interior, high ground was known to exist. Sir Thomas Mitchell\nsettled this question by a great journey in 1836, which, among other\nresults, immediately threw open to settlement the fertile country\nabout Port Phillip, hitherto, as we have seen, neglected through the\nmisunderstanding of Hume and Hovell.\n\nThe larger problems of Australian geography were thus early settled,\nthough there was (as even now there is in some parts) a multitude of\ndetails to be filled in. But the leading questions awaiting solution\nby explorers now become economic rather than purely geographical. Thus\nwe find Dr. Leichhardt's first expedition (1844), from Moreton Bay\nin southern Queensland by the Burdekin, Mitchell, and Roper rivers\nto Port Essington, inspired by the conception of an overland route\nbetween Sydney and a northern seaport. He was lost (and the mystery\nof his fate was never solved) in 1848 in attempting a crossing of the\ncontinent from east to west, and Kennedy's expedition in the same year,\nin attempting to cross northern Australia, also met with disaster,\nwhere A.\u00a0C. Gregory succeeded in 1855\u201356. The penetration of the\ninterior from the south and the crossing to the north had attracted\ntravellers before this; Eyre in 1840 had discovered the series of\nsalt lakes and swamps which he lumped together under the name of Lake\nTorrens, while Sturt in 1845 added little to Eyre's discoveries,\nand, after failing to penetrate the Stony Desert to the north, put a\ntemporary period to explorations in that quarter. Babbage in 1856 and\nParry in the following year obtained more accurate knowledge of the\nLake Torrens region, and Goyder in 1857 reported a great freshwater\nlake which was found later to have been conceived out of some shallow\npools and visions of the mirage. J.\u00a0M. Stuart's six expeditions\nfrom south to north in 1858\u201361 added much to exact knowledge; that\nof Robert Burke and William Wills in 1860\u201361, ill-managed as it was\nand ending in the death of the leaders, obtained a fame in excess\nof its scientific value; but other expeditions sent in search of it\nachieved better results, and incidentally made clear the danger of\nassessing the worth of some of the inland districts on the report of\none traveller who might have come upon them at an unfavourable season.\nThus J. McKinlay in 1861 brought word of fertile lands which Sturt had\ncondemned as desert. The many journeys through the interior of Western\nAustralia--such as those of J.\u00a0S. Roe (1836), the brothers Gregory,\nH.\u00a0M. Lefroy (1863), Sir J. Forrest (various expeditions in and\nafter 1869), Warburton (1873), and Ernest Giles (first crossing from\nAdelaide to Perth by an inland route, 1875)--though often of extreme\nimportance from an economic point of view, whether concerned with the\ndiscovery of pastoral lands or of gold or other mineral fields, can\nonly be referred to here as having gradually opened up the detailed\nknowledge of this part of the continent, and as having redeemed it\nin part from a reputation for complete inhospitality, until we have\nnow a trans-continental railway planned to connect the systems of\nsouth and of western Australia. The exploration of the Kimberley and\nnorth-western areas of the state was delayed until the latter half of\nthe last century.\n\n\n\n\nCHAPTER XIII.\n\nTHE NINETEENTH CENTURY AND AFTER: THE POLES\n\n\n_(a) Arctic_\n\nFollowing the new enthusiasm for Arctic exploration undertaken for\npurely scientific purposes, the British Government despatched three\nexpeditions between 1773 and 1779. The first, under Captain Phipps,\nwas stopped by ice off the north-west of Spitsbergen; the second, that\nof James Cook with the vessels _Discovery_ and _Resolution_, sent to\nsearch for either a north-west or a north-east passage by the Bering\nSea route, met, as has been seen (Chapter VIII.), with a measure of the\nsuccess characteristic of his work, but his death at Hawaii put an end\nto the hope that further research in the Arctic lay before him, and the\nvoyage continued under the command of Captain Clarke was carried only a\nlittle north of the 70th parallel in the ice-bound Bering Strait, where\nClarke also died. Till 1815 little was done to elucidate the still\nunsolved question of the north-west passage, owing to the disturbed\nstate of Europe and America, but the offer of a reward (the result of\nthe exertions of Sir John Barrow in 1818) of \u00a320,000 for the discovery\nof the passage and \u00a35,000 for reaching 89\u00b0 N., led to the sailing of\nexpeditious to the American Arctic region under Lieut. J. Franklin,\nCaptain Ross, and Lieut. E. Parry. Ross on his first voyage took\nBaffin Bay and Lancaster Sound to be land-locked on the north, and thus\nmissed his chance of forcing the passage.\n\nParry in two voyages, on the results of which he gained the \u00a35,000\nreward, succeeded in passing through Lancaster Sound, and reaching and\nnaming Melville Island, thus proving Baffin's discoveries. Meanwhile\nFranklin attempted to reach the north shores of America by land;\nhe explored 550 miles of the coast and discovered and named Cape\nTurnagain, though he and his party suffered great privations on the\nreturn journey. Then the energies of explorers were directed towards\ncombining the results obtained by Parry and Franklin; further stretches\nof the north coast of America were explored, and Point Barrow was\nreached in Bering Strait. In 1829 an expedition was undertaken by\nCaptain John Ross and his nephew James Clark Ross; the opening of the\npassage was again missed (though the most northerly part of America\nwas passed), and it was not discovered till 1851 by Kennedy on his\nsearch for Franklin. J.\u00a0C. Ross, however, fixed the position of the\nnorth magnetic pole on this voyage of five years' duration, and other\nvaluable observations were made. The work of tracing the northern\nshores of America was nearly finished by 1847, chiefly by travellers in\nthe Hudson Bay Company's service. During this period the north coast of\nSiberia had also been traced almost in its entirety by the Russians,\nthough they had not succeeded in rounding the most northerly point.\n\nIn 1845 Sir John Franklin started on the voyage from which neither\nhe nor any of his companions returned. It is not known exactly what\nhe achieved before he was lost, but he came nearer to accomplishing\nthe north-west passage than anyone before him. The expeditions of\nSir John Richardson, Dr. John Rae, and others, sent by land in search\nof his party, filled in the last gap in the northern coast-line of\nAmerica. The different expeditions, under McClintock and others, sent\nby sea in the fifties for the same purpose not only decided the fate\nof Franklin's party and extended knowledge over a vast area, but also\nat last rounded the north of America. The passage found by Kennedy in\n1851 was traversed in 1853 by McClure, though part of the journey was\nmade by travelling over the ice. An expedition under Captain Inglefield\ndetermined the northern point of Smith Sound. Elisha Kent Kane extended\nthe knowledge of Grinnell Land and Greenland towards the north, and\nopened the way to others who followed the waterways he discovered.\nIn 1871 Charles Hall sailed 250 miles up Smith Sound and reached the\nhitherto inaccessible polar sea; he touched a more northerly point than\nhad previously been reached by any ship (82\u00b0 11\u00b4 N.).\n\nStirred to action by these fine achievements of the Americans, England\nsent out the important expedition under Captain George Nares in\n1875 which obtained very valuable scientific observations, taken on\nthe frozen polar sea, and under Albert Markham reached the furthest\npoint north yet attained--83\u00b0 20\u00b4 N.--after battling with immense\ndifficulties caused by bad conditions of the ice and scurvy. Meanwhile\nmuch good work had been done in the Arctic from the old world. A purely\nscientific expedition had been sent to the Spitsbergen seas as early\nas 1827 from Norway; but from then till 1858 the work of exploration\nwas chiefly carried on by the men engaged in the seal-hunting and\nfishing, in the interests of their trade. Before 1872, however,\nseveral Scandinavian expeditions which visited Spitsbergen and\nGreenland brought back valuable scientific results. The most noteworthy\nexpedition of this period, however, was Austrian; it was captained\nby Lieutenant Julius Payer, who had previously been on an expedition\nin Greenland. He and Lieutenant Weyprecht sailed in 1871 to search\nfor the north-east passage. They were beset by ice off Novaya Zemlya,\nand drifted till they came to a mountainous country which they called\nFranz Josef Land. They believed it to consist of two large masses of\nland, instead of perceiving it to be an archipelago, and much of the\ncountry they thought they saw has been since proved not to exist. Franz\nJosef Land was not visited again till 1880, when a large part of it\nwas surveyed by the Englishman Leigh Smith. The north-east passage\nwas made in 1879 by A.\u00a0E. Nordenski\u00f6ld, who accomplished the journey\nwhich led so many before him to failure without loss of life or vessel,\nand almost in one season. He had made several previous voyages in\nGreenland and Spitsbergen; he had also twice successfully reached the\nYenisei through the Kara Sea. Captain Joseph Wiggins, an Englishman,\nalso made several voyages through the Siberian seas, which, together\nwith Nordenski\u00f6ld's accomplishment of the north-east passage (1878\u20139),\nproved the route to the mouth of the Yenisei to be practicable from a\ncommercial standpoint. Fired by Nordenski\u00f6ld's example, the Danes made\nseveral remarkable journeys to the interior of Greenland.\n\nA new interest was given to polar research by the establishment of\nthe international circumpolar stations in 1883. The idea was mooted\nfirst by Weyprecht, and eventually twelve expeditions of various\nnationalities were sent out to erect observatories at different\npoints within the polar circle, so that simultaneous and continuous\nobservations might be taken. A great deal of valuable work was done.\nAn American expedition, led by Lieutenant A.\u00a0W. Greely (1881\u20134), to\nGrinnell Land almost perished from starvation.\n\nGreenland was crossed for the first time by Fridtjof Nansen in 1888. In\n1886, and again from 1892 to 1895, Robert E. Peary, an American civil\nengineer, made several brilliant journeys in Greenland, and extended\nthe knowledge of the country more than two degrees to the north. It\nwas Nansen and Peary who were destined to draw the veil from the great\npolar area itself, towards which so many fruitless journeys were made.\nFrom 1817 onwards many voyages were undertaken with the object of\nreaching the pole--as by Parry, Scoresby, Markham, and Jackson from\nEngland, Nordenski\u00f6ld from Sweden, and Koldewey from Germany. These\nexplorers mostly started from Spitsbergen; but Nansen worked on an\noriginal plan--that of utilizing the drift of ice, which had been\nproved to take place right across the polar sea, to carry his ship with\nit. The plan was so far successful that the _Fram_ passed from the New\nSiberian islands right over to Spitsbergen in three years, without,\nhowever, reaching a higher latitude than 85\u00b0 55\u00b4 N. Nansen, with\nLieutenant Johansen, made a dash northwards from this point, reaching\n86\u00b0 5\u00b4 N., the \"farthest north\" attained up to that date; he met with\nFrederick Jackson's expedition in Franz Josef Land, and returned\nsafely to Norway. This brilliant journey led to no discovery of land\nin the polar basin, which proved to consist of a sea of great depth,\nincreasing towards the pole. The Duke of Abruzzi's expedition in 1899\nreached 86\u00b0 34\u00b4 N.\n\nMeanwhile much good work was being done in other directions in\nextending Arctic research. Franz Josef Land was explored, chiefly\nby Austrian, British, and American expeditions. Nathorst, a Swede,\ncircumnavigated the Spitsbergen archipelago in 1898, and discovered and\nmapped King Oscar Fjord in Greenland in the following year. Sverdrup,\nNansen's friend and companion, sailed up Jones Sound and charted many\npreviously unknown parts in 1899 and the following years. The story of\na continent existing to the north of Bering Strait and extending right\nacross the pole to Greenland, which was believed by many explorers,\nwas disproved by De Long in his ill-fated voyage in the _Jeannette_ in\n1879, during which the whole party perished, though the ship's books\nwere afterwards found. This voyage and the journeys of the ships sent\nin search of De Long proved that north of Siberia lay an ocean dotted\nwith islands. Much work was done in exploring the New Siberian Islands\nby the Russian, Toll, who lost his life in an effort to reach the most\nnortherly and unknown portion of the group. In 1903\u20134 Amundsen in the\n_Gj\u00f6a_ undertook an expedition to the North Magnetic Pole, where he\ncarried out a continuous series of observations for two years with\nimportant scientific results. He returned by Bering Strait, thus for\nthe first time completing the navigation of the North-west Passage.\nThe Danes worked hard at charting the east coast of Greenland, and the\noutline of the north-eastern extremity of the country was accurately\ndelineated for the first time by the expedition of L. Mylius Erichsen\n(1905\u201307), on which he and his companions perished, though their\nsplendid records and observations were found by a relief expedition.\nThe crowning achievement of reaching the pole itself was accomplished\nin 1909 by Peary, after several previous journeys. He had spent four\nconsecutive winters in the Arctic regions exploring Smith Sound and\nthe north of Greenland, from 1899 to 1902; and in 1905 he had again\nattempted to reach the pole by Smith Sound and Grant Land, touching\n87\u00b0 6\u00b4 N. At or near the pole there was no land to be seen, and the\nsea was 1,500 fathoms deep. Thus there remains no Arctic problem of\nthe first magnitude to-day. The main outline of the Arctic region as a\ngreat and deep sea surrounded by the northern shores of Europe, Asia,\nAmerica, and Greenland, is known, though there is still a large portion\nof the polar basin north of Alaska as yet untouched by explorers. Here,\nhowever, some high authorities believe that a considerable extent\nof land remains to be discovered beyond the Beaufort Sea. Even now\nmaps show a doubtful coast-line some fifty miles due north of Point\nBarrow, and in 1913 an expedition left Canada under Stefansson with the\nsolution of this problem as its main object.\n\n\n_(b) Antarctic._\n\nIn the Antarctic an important voyage, which supplemented Cook's work,\nwas undertaken in 1819 by Fabian von Bellingshausen. He succeeded in\nsailing half round the Antarctic circle, keeping to high southern\nlatitudes all the way, and voyaging within the circle for considerable\ndistances. He found the first land seen within the Antarctic circle,\nPeter Island, and, later, Alexander Island; he discovered the Traverse\nIslands, and on his return in 1821 touched at the South Shetland\nIslands, and met there sealers, by one of whom, William Smith, the\nislands had been discovered in 1819. In the next year, 1822, the\nSouth Orkney Islands were found and named by another sealer. The\nnext voyager was James Weddell, who reached the highest latitude yet\nattained, 74\u00b0 15' S., in 1823. At his highest latitude he had clear\nsea before him, but was forced to turn back by the approach of winter,\nand returned with many interesting observations and collections. A\ncourageous journey was also made in 1831 by John Biscoe, a sealer, who\nstarted out to search for land from the Sandwich Islands, and succeeded\nin sailing for some months within the Antarctic circle in a higher\nlatitude than Bellingshausen, and sighted land which he named Enderby\nLand. In spite of the sufferings he had endured and the death of the\ngreater number of his crew, he started again in the following year from\nNew Zealand, discovered Biscoe Islands, and took possession for England\nof the land which he could see lying behind them; this was subsequently\nnamed Graham Land. Biscoe was in the employ of an enterprising London\nfirm, the Enderby Brothers, and after the remarkable results which he\nhad achieved, they were encouraged to pursue their policy of directing\ntheir captains to embrace every opportunity of exploration. In 1833 one\nof them, John Kemp, found land to the east of Enderby Land, and in 1839\nJohn Balleny discovered the islands named after him.\n\nAbout 1835 general interest was aroused in Antarctic problems, and\nthree expeditions were prepared in England, France, and America to make\nmagnetic observations, and to explore as far as possible the southern\ncontinent, now at length defined within reasonable limits. The French\nexpedition, under Dumont d'Urville, was the first to start in 1838, but\nachieved little beyond the exploration of some land south of the South\nShetland Islands, which was called Louis Philippe Land. After wintering\nin Tasmania, however, d'Urville decided on making a great effort to\nreach the south magnetic pole, and though he failed in this, he found\na mountainous land which he named Ad\u00e9lie Land. The American expedition\nunder Charles Wilkes did not meet with any great success, hampered as\nit was by quarrels among the officers and by unseaworthy ships; but\nland was several times sighted at a distance, and on Wilkes's return\ncontroversy arose as to whether the honour of the discovery of this\nsouthern continent belonged to the French or the American expedition.\n\nThe British expedition under Sir James Ross was the last to arrive on\nthe scene (in 1841); but it had the advantage of the others, in that\nit had been specially equipped for Antarctic exploration; Ross's ships\ncould brave dangers from which Wilkes and d'Urville had been compelled\nto turn aside. He forced his way through the pack, and found a range\nof high mountains trending southwards, which he called Victoria Land.\nFollowing the land he came to the twin volcanoes, named Erebus and\nTerror after his two ships, and was stopped at length by the great\nice barrier, running eastwards. During this remarkable journey Ross\nreached latitude 78\u00b0 4\u00b4 S., the highest yet attained. He made two\nfurther journeys, neither so successful as the first, though in 1842 he\nsighted the land which was rediscovered, and named King Edward Land,\nin the following century. After this no attempt worthy of mention was\nmade on the south polar region for thirty years. The _Challenger_\nexpedition in 1874 was not concerned with the attempt to penetrate very\nfar south (Chapter XIV.). The voyage in Antarctic waters, however, was\nimportant from the information obtained as to the depth of the southern\nocean and other results which helped to prove the existence of a\nconsiderable mass of land in the Antarctic region. This information\nwas supplemented by the observations of two of the international\ncircumpolar stations (to which reference has been made), which were\nestablished in Tierra del Fuego and South Georgia in 1882; but it was\nnot until many years later that scientific interest was widely aroused\nin the problem of the Antarctic continent, and from 1874 to 1898 the\nonly people to cross the Antarctic circle were sealers and whalers; but\nin 1895 C.\u00a0E. Borchgrevink landed from one of these vessels for the\nfirst time on southern continental land near Cape Adare.\n\nIn 1898 three expeditions started south. The first, a Belgian\nundertaking on board the _Belgica_, explored the coast to the north of\nGraham Land, and brought back valuable collections; the second, from\nGermany on the _Valdivia_, re-discovered Bouvet Island, whose position\nhad long been lost; the third, from England, under Borchgrevink on the\n_Southern Cross_, landed the first party to winter in the Antarctic,\nreached Mount Terror, and sailed along the Great Ice Barrier, reaching\nlatitude 78\u00b0 S. In 1901 the problem was attacked for the first time\nby means of land-exploration; a well-equipped expedition leaving\nEngland in that year under Captain R.\u00a0F. Scott voyaged along the\nice-barrier, and found and named King Edward Land, first seen by Ross.\nScott then proved Mount Erebus and Mount Terror to be on an island,\nand wintered on shore. In the following southern summer Scott, with\nWilson and Shackleton, pushed southward and reached the latitude of\n82\u00b0 17\u00b4, where the Great Ice Barrier reaches the foot of the lofty\nplateau on which the south pole is placed. Other parties traversed the\nice-barrier in various directions, and much valuable scientific work\nwas done in geology, biology, meteorology, magnetism, and glaciation.\nWhile Scott was in the Antarctic to the south of New Zealand, a German\nexpedition, under E. von Drygalski, on board the _Gauss_, was working\nto the west of him, and had discovered and named Kaiser Wilhelm II.\nLand. Two private expeditions were also in the Antarctic at the same\ntime, and the large number of synchronous meteorological and magnetic\nobservations thus taken formed a valuable contribution to the knowledge\nof the southern continent. In 1903 a voyage was made by W.\u00a0S. Bruce on\nthe _Scotia_, which is important for the exploration of an entirely\nunknown sea lying between the tracks of Weddell and of Ross; the\nlatitude of 74\u00b0 1\u00b4 was reached. Though the land could not be attained,\nits existence was proved by occasional glimpses and by the dredging up\nof continental rocks, and the name of Coats Land was given to it. In\n1904 J.\u00a0B. Charcot, a French scientist, cruised along Graham Land and\nfound a new line of coast, which he named Loubet Land. Thus between\n1902 and 1904 new land had been discovered in all the four quarters of\nthe Antarctic circle--King Edward Land by Scott, Kaiser Wilhelm Land by\nDrygalski, Coats Land by Bruce, and Loubet Land by Charcot.\n\nLieutenant (afterwards Sir) E.\u00a0H. Shackleton, who had accompanied\nScott, led an expedition to the south in 1908\u20139, which landed at the\nfoot of Mount Erebus. That mountain was ascended by Professor T.\u00a0W.\u00a0E.\nDavid, who also, with Dr. D. Mawson, reached the south magnetic pole\nin 72\u00b0 25\u00b4 S., 155\u00b0 16\u00b4 E. Shackleton himself led the famous march\nwhich brought him to 88\u00b0 23\u00b4 S., 162\u00b0 E., a great advance towards the\nsouth pole itself, which might actually have been attained but for the\nlack of food. The scientific results of the expedition were of high\nvalue, and revealed the desirability of prosecuting researches in the\nsame field; and in 1910 Scott led a second expedition, with a larger\nscientific staff than had ever been taken before, the main party of\nwhich was landed at Cape Evans, McMurdo Sound. Of two other parties,\none was landed on the west side of the sound; another, which worked\nat first from Cape Adare, was subsequently transferred to Terra Nova\nBay, Victoria Land. A considerable area was thus covered on this part\nof the Antarctic coast, while Scott's march upon the pole was designed\nto follow Shackleton's route. The splendour of success was outshone by\nthe splendour of disaster: Scott and four companions, having reached\nthe pole, died bravely on the return journey, overcome by adverse\nconditions. The work of the expedition as a whole, taking that of the\nother parties into account, was a brilliant scientific triumph.\n\nThe honour of first reaching the pole, however, fell to a Norwegian\nexplorer, Captain Roald Amundsen, who, leading a small but admirably\nequipped expedition, succeeded in his endeavour at the end of 1911,\nand he and Scott thus left the way open to research on the Antarctic\nland-mass unhampered for the future by the natural desire to reach a\ncertain point upon it. In the same year expeditions (not specifically\nconcerned with the attainment of that point) were led south by the\nGerman Lieutenant Filchner, whose immediate goal was Coats Land, in\nthe \"Weddell\" (or South American) quadrant, and by Dr. Mawson, whose\nobjective was Ad\u00e9lie Land, on the opposite flank of the continent,\nwhile various projects are also under the consideration of other\nvoyagers, British and American. There is room for the work of all\nthese and more--the Antarctic region is now known as a vast land-area\nfringed by deep seas separating it from the other continental masses.\nAmundsen's observations would seem to prove it a single homogeneous\nmass, and not to be divided into two, or to consist in part of an\narchipelago. It still remains to investigate the nature of any\ngeological relation between it and the other continents, to study the\nextension and physiography of the great mountain ranges which are\nknown, and their relation to the polar plateau, and to deal with the\nmany other problems such as are suggested by observations already made\non the climate, the ice conditions, and the distribution of flora and\nfauna--notably, in the last connection, the problem of the resemblances\nwhich have been observed between Antarctic and Arctic forms of life.\n\n\n\n\nCHAPTER XIV.\n\nTHE NINETEENTH CENTURY AND AFTER; EVOLUTION AND PROGRESS OF\nGEOGRAPHICAL SCIENCE\n\n\nAs during a long period in the history of geography it was usual to\nlimit the connotation of the term, so, when a wider connotation came\nto be recognized, there naturally followed the creation of certain\nclearly-defined departments of study under distinguishing titles.\nThe whole structure of geography rests upon two great pillars--upon\nexploration and upon measurement. With the main lines of exploration\nwe have dealt in preceding chapters, and we have carried that part\nof our history which deals with precise measurement down to the\nclose of the eighteenth century and the institution of the ordnance\nsurvey of Great Britain (Chapter X.). The early part of the sixteenth\ncentury witnessed the birth of accurate land-measurement; the early\npart of the nineteenth its re-birth as a function of organized\nstate-administration. The Indian trigonometrical survey, with which\nthe names of Col. W. Lambton and afterwards Sir George Everest are\nassociated, was begun in 1800; a famous survey of Switzerland,\ncoupled with the name of Gen. H. Dufour, was undertaken in 1809, one\nof Austria-Hungary in 1816, one of France in 1817; what is now the\nterritory of the German Empire was already fairly represented on\nlocal maps when a general survey was undertaken in 1878. Indeed, all\nEuropean countries may be said to be completely surveyed except certain\nof the Balkan States, though Russia is much behind in this respect.\nIt must not be forgotten that the processes of close survey are slow:\nthe primary triangulation of Great Britain was only completed in 1858,\nthough the filling-in of details of course proceeded concurrently. And\nthe survey never stands still; there is always revisional work to do.\n\nAs concerns the British Empire, it has been an unrealized ideal that\na territory should be surveyed as soon as possible after occupation,\nand it was not until 1905 that the defects and lack of system in the\nmapping of British territories generally were sufficiently widely\nrealized to cause the creation of a Colonial Survey Committee as a\ncentral advisory and supervisory body.\n\nGeodetic survey steadily advanced during the nineteenth century, from\nthe work of Friedrich Wilhelm Bessel in East Prussia in 1838--of the\nhighest importance owing to the systematic accuracy of the observations\nand their calculation (on the principle of \"least squares\")--down to\nthe institution of the International Geodetic Association (Erdmessung),\nwhich had its origin in a proposal of the Prussian General, J.\u00a0J.\nBaeyer, in 1862, and has headquarters near Potsdam, over twenty\nEuropean, American, and Asiatic countries being represented in it. The\naccuracy of instruments has been carried far above the standard of\nthose referred to in an earlier chapter. As an illustration we have\nonly to trace the mechanical methods of measuring a baseline or other\ndistance on the surface, from that of counting the revolutions of a\nwheel, up to that of employing rods of metal or other substance, or\nchains--methods associated with the endeavour to compensate for or\novercome even the slight contraction or expansion of a rod, due to\nvariation of temperature, which might vitiate the results, culminating\nin the discovery (in France in 1896) of invar, an alloy for practical\npurposes invariable, when applied to the measurement of baselines by\nmeans of such apparatus as that of E. J\u00e4derin of Stockholm.\n\nThe work of the cartographer, as exemplified in atlases and small-scale\nmaps of general utility, has by no means in all cases followed the\nhigh standard of the surveyor. Commercial considerations are not to\nbe overlooked; cheap and rapid methods of reproduction bring their\ntemptations as well as their advantages to bear upon cartography. Their\nadvantages are manifest; the map, whether as an adjunct to travel or\nas a graphic illustration of a great variety of subjects, has become\na commodity of almost daily use. But in some countries, such as the\nUnited States, the standard of cartography generally is as low as\nthat of the maps of the survey is high. The reduction and selection\nof details from a large-scale survey for use on a small general map,\nthe methods of representing such details, the permissible limit of\ngeneralizing them, the choice of colours--these and other aspects of\ncartography really demand a scientific standard as exalted in its way\nas that of the surveyor. That standard has been most firmly upheld\nin Germany, in such geographical establishments as that founded by\nJustus Perthes at Gotha in 1785, which publishes the famous general\natlas originally formed by A. Stieler in 1817\u201332, the physical atlas\nof H. Berghaus (1838\u201342), and many other such works. Other names of\nindividual workers in the same field come readily to the mind--H.\nKiepert, A. Petermann, K. von Spruner, Behm, Supan, Langhans, Andree,\nDebes, A. Ravenstein. The British and French lists are shorter, though\nthe names of John Bartholomew, W. and A.\u00a0K. Johnston, Edward Stanford\nand George Philip, Vivien de St. Martin, F. Schrader and Vidal de la\nBlache must be remembered.\n\nAfter many years of effort on the part of the International\nGeographical Congress, a conference consisting of official delegates\nfrom most civilized states met in 1909 to deliberate on the methods\nto be adopted in the construction of an international map of the\nworld. After much discussion a series of regulations was drawn up to\nbe followed by each country in producing a map of its territories on\nthe scale of 1\/1,000,000, or about sixteen miles to the inch. The\nprojection will, of course, be uniform, and altitudes are shown by\nlayers of different tints from sea-level upwards. Actual experience may\nno doubt demand certain modifications, but it will be a great advantage\nto have an authoritative map of the world on a strictly uniform plan.\n\nAs to the progress of geodesy in recent years, in 1899\u20131902 an arc was\nmeasured in the extreme north in Spitsbergen, by Swedish and Russian\nworkers (P.\u00a0G. Rosen, O. B\u00e4cklund, and others), while Sir David Gill,\nas director of the Royal Observatory in Cape Town, subsequently\ninitiated the measurement of a great arc in Africa along the meridian\nof 30\u00b0 E. These arcs are capable of connection through Asia Minor and\nEurope, by which means a continuous measured arc of 105\u00b0 would be\nobtained. The arc of Quito (Peru) was re-measured in 1901\u201306 under the\ndirection of the French Academy of Sciences; a great arc in 98\u00b0 in the\nUnited States of America has been undertaken by the Coast and Geodetic\nSurvey, and these again are capable of ultimate connection. Other arcs\nof special importance have been measured in Europe and India.\n\nGeomorphology, though not accepted without demur as a definite branch\nof science in itself, has at last come to be generally recognized as\na convenient term to connote the study of terrestrial relief. Elie\nde Beaumont in 1852 enunciated with too great precision the theory\nthat similarity of orientation was a standard test of similarity\nin the age and origin of the great mountain chains. Lowthian Green\nin 1875 proposed his tetrahedral theory of the disposition of the\ncontinents and the ocean basins, on the ground that a sphere undergoing\ncontraction tends to assume the form of a tetrahedron, or body enclosed\nby four equal equilateral triangles. He applied this theory to the form\nof the spherical earth at its present stage of contraction, indicated\nhow far it accounted for the present distribution of land and sea, and\nattempted to give reasons for its failure to do so in certain respects.\nProfessor C. Lapworth in 1892 stated his theory of folding, according\nto which the continents are the arches of vast folds in the crust of\nthe earth, and the ocean basins the troughs between them. E. Suess\nhas modified this view in his treatise _Das Antlitz der Erde_ (The\nFace of the Earth), 1885\u20131901. Sir George Darwin invoked the effects\nof tidal strain upon the crust, associating this with the form of the\ncontinents. The subject, which has also been dealt with by Professors\nJ.\u00a0W. Gregory and A.\u00a0E.\u00a0H. Love, M. Bertrand, A. de Lapparent, and A.\nSupan, among others, has thus been approached from both the purely\nphysical and the mathematical standpoint, but the problem has not\nreached its solution.\n\nWe have already given sufficient indication that the exact scope of\ngeography has not been found easy to define by common consent; that\nfact does not lighten the task of tracing its development in the\nnineteenth century. It is not inconceivable that on one view of the\nsubject this volume should have concluded with the preceding paragraph.\nOn the other hand, the new value attaching to the geographical\nstudies of distribution and environment makes it imperative to carry\nthe story further. These studies have not only been systematized in\nthemselves, but have become complementary of other sciences, and\nthus we find the term \"geography\" incorporated in certain scientific\ncompounds--zoogeography or zoological distribution; anthropogeography,\nthe distribution of mankind; biogeography, the distribution of living\nthings generally--or perhaps more mercifully treated in such phrases\nas \"plant geography.\" Zoogeography and plant geography are concerned\nwith the division of the earth's surface into regions possessing\nindividual characteristics in regard to their fauna or flora. The\nprinciple of regional division, indeed, has become a leading principle\nof geographical research, in regard not only to fauna and flora, but to\nman as well; to the physical characters of the land, and to climatic\nconditions.\n\nThe general tendency towards scientific specialization has resulted\nin the erection, as it were, of separate laboratories for the study\nof certain specific features of the physical earth, each with its\nname-plate upon the door. From some of these--as from meteorology and\ngeology--the geographer, in the course of the studies we have just\noutlined, borrows such data as are necessary to his purpose, and puts\nthem to his special uses. It is no part of a history of geography\nto deal with that of meteorology or of geology, though both these\nsciences are fundamentally geographical, owe an obvious debt to\nexploration and travel, and make ample use of cartography. On the other\nhand, there are some departments of research which, though standing\nunder their own names, are grouped perhaps more closely as offspring\nof physical geography. Such are oceanography (the study of the sea),\nlimnology (the study of lakes), potamology (that of rivers). The last\nterm might be justified on the ground that it helps to lighten the\nburden of different meanings which rests upon the term \"hydrography\";\nit at any rate defines a clear field of study which, in view of its\npractical importance, has attracted much recent attention. The study\nof lakes--the depth, movement, and composition of their waters,\nthe life in them, the physical nature of their basins--which was\npractically initiated by Professor F.\u00a0A. Forel's investigations of Lake\nGeneva published in 1892\u201394, has already a notable monument in the\nbathymetrical survey of the Scottish fresh-water lochs, completed under\nSir John Murray's direction in 1908.\n\nThe line between these various branches of science is for our present\npurpose difficult to draw; but at the risk of a charge of arbitrary\ntreatment it appears pertinent to refer to certain facts in the\nhistory of oceanography. As an organized department this is no less a\ncreation of the nineteenth century than others we have named. Among\nancient geographers there was certainly some speculation as to the\nphysical character of the seas, known and unknown. From a very early\nperiod sounding in shallow waters has been recognized as a method\nof navigation, and Strabo, for example, displays some knowledge of\nthe greater depths of the Mediterranean. But to mere navigation a\nclose study of the sea was not essential, and explorers with their\neyes fixed on distant lands were concerned merely to make the best\nof their way over the intervening waters. It is not, therefore,\nuntil towards the close of the eighteenth century, the period of\nthe scientific exploration of the Arctic region and of Cook's great\nvoyages--exploration necessarily carried out mainly on shipboard--that\nany systematic investigation of the deep seas is found. Phipps,\nScoresby, John and James Clark Ross, and especially the last, made deep\nsoundings; but the whole subject of oceanography may be said to have\nbeen first organized by Matthew Fontaine Maury (1806\u201373), an American\nnaval officer, who, after his appointment to the United States D\u00e9p\u00f4t\nof Charts and Instruments (which became the Hydrographic Office),\nsystematized the collection of navigators' observations on winds and\ncurrents, while his example inspired the establishment of similar\ncollections in other countries. He also devoted himself to the study of\nthe relief of the ocean floor, an investigation which was forwarded by\nthe invention of a compatriot, J.\u00a0M. Brooke, of the United States Navy,\nwho introduced the principle of sounding in great depths by means of a\nlead which was detached from the line on reaching the bottom, so that\nthe line might be easily hauled aboard. Maury published his _Physical\nGeography of the Sea_ in 1855. Meanwhile the possibility of connecting\nEngland and America by submarine telegraphic cable had been discussed\nten years earlier. Communication across the Channel with France had\nbeen successfully established in 1851, and in 1856 the first signals\npassed across the Atlantic. This first trans-oceanic cable survived\nonly for a little, but the investigation of the sea-floor had now\nacquired a commercial as well as a scientific interest.\n\nAs early as 1834 Edward Forbes had made biological investigations\nin the Irish Sea, and in 1841\u201342 in the Mediterranean; while in\n1868\u201370 similar studies, together with soundings and observations\nfor water-temperature, salinity, and deposits, were carried on\nin the British seas, the Bay of Biscay, and the Mediterranean by\ninvestigators on board vessels of the Royal Navy--the _Lightning_,\n_Porcupine_, and _Shearwater_. This and similar work elsewhere was\npreparatory to the greatest of all marine scientific expeditions, that\nof H.M.S. _Challenger_ in 1872\u201376. That vessel was commissioned at the\ninstigation of the Royal Society, in command of Captain (afterwards\nSir) George Nares, and a scientific staff under Sir C. Wyville Thompson\nas director, and including Sir John Murray, H.\u00a0N. Moseley, and J.\u00a0Y.\nBuchanan. The Atlantic was the first field of study, and was crossed\nseveral times; the southern ocean was then traversed south-east and\neast from Cape Town; the _Challenger_ was the first steamer to cross\nthe Antarctic circle, and afterwards proceeded into the Pacific. The\nroute now lay from Melbourne to New Zealand, Fiji, Torres Strait, the\nMalay Archipelago, and Chinese and Japanese waters, after which the\nPacific was crossed from Yokohama by Honolulu and Tahiti to Valparaiso.\nThe homeward route lay by the Straits of Magellan, Montevideo,\nAscension Island, and the Azores. Every branch of oceanographical\nresearch was fully dealt with in the fifty volumes of reports upon the\nvoyage. More lately other vessels of the British, American, German,\nand other navies have been detailed for scientific research; and cable\nlaying has afforded additional opportunities. Mention must be made of\nthe Dutch expedition in the eastern Malay seas on board the _Siboga_\nin 1899\u20131900, the work of the German surveying vessel _Planet_ in the\nPacific and elsewhere in 1906 and following years, and the Atlantic\nexpedition of Sir John Murray and Dr. Johan Hiort on the _Michael Sars_\nin 1910. The observations of the last-named expedition, especially\non the distribution of life in the sea, are of the first importance.\nOceanographical work has remained an integral function of scientific\nexpeditions in the Arctic and Antarctic regions. Among the names\nof investigators which are specially identified with oceanography\n(independently of other departments of geographical research) reference\nis perhaps most justly due to those of Professor Alexander Agassiz\nin the United States, and the Prince of Monaco. The establishment of\nthe International Council for the Study of the Sea in 1901, nominated\nby nine European Governments, with its headquarters in Copenhagen,\nwas not only an outstanding event in the history of the science at\nlarge, but also draws attention to one of its most important practical\napplications, for the Council is specially concerned with the study and\nimprovement of the fisheries in the North Sea and other European waters.\n\nThe educational value of geography, as we have seen, was recognized in\na practical manner by Newton; and towards the close of the eighteenth\ncentury physical geography was taken as a lecture-subject by the\nphilosopher Immanuel Kant at K\u00f6nigsberg, and by him was given exalted\nrank as a \"summary of nature.\" Alexander von Humboldt (1769\u20131859)\nfurther systematized the theory of the control of land-forms and\nclimate over the distribution and habits of plants, animals, and man,\nand was able to draw not only upon the collection of facts made by\nother travellers, but also upon his own observations. His journey\nin 1799\u20131802 in America, during which he explored the Orinoco and\ndiscovered its connection with the Amazon through the Casiquiare, and\nvisited Cuba, Quito and Mount Chimborazo, and Mexico, was the practical\nfoundation of his scientific career. In the course of it he collected\nmaterial for his researches into temperature at different elevations,\ninto plant geography, terrestrial magnetism, volcanic phenomena, and\nmuch besides, while he also travelled through Russia to the Yenisei in\n1829. In the work of Karl Ritter (1779\u20131859) is found the importance of\nestablishing comparisons and investigating differences between similar\nregions in different parts of the world. Oscar Peschel (1826\u201375)\ncorrected Ritter's marked tendency to give excessive prominence\nto historical detail. The exposition of theoretical geography was\ncarried on by Ferdinand von Richthofen, Hermann Wagner, and Friedrich\nRatzel; and with the work of these and other leaders in the school of\nGerman geographical thinkers and teachers is associated the German\npre-eminence in cartography during the nineteenth century, in which\nconnection a passing tribute should be paid to Humboldt's introduction\nto cartographers of the principle of drawing upon maps lines to show\nareas of equal temperature (isotherms), rainfall, etc.\n\nGeography as an educational subject of widely-recognized value is\ncoming by its rights, though the majority of the last generation may\nrecall it as affording little else than superficial instruction in\nthe position of countries, places, mountains, and rivers. But now,\nnot only in Germany, but in Great Britain and elsewhere, it has\nbeen widely adopted as an examination-subject in both primary and\nsecondary education, as well as for certain specific purposes, and\ngeographical chairs or lectureships have been established in a number\nof universities. The fostering of geography as an educational subject\nhas been one of the great tasks, and that of furthering exploratory and\nother research another, of the many geographical societies which have\nbeen founded throughout the civilized world in the nineteenth century\nand after. That of Paris in 1825, and that of Berlin in 1827, are the\noldest of these now flourishing, though with the Royal Geographical\nSociety in London (1830) was merged the older African Association.\n\nThe theory of evolution, as set forth by Charles Darwin, Alfred Russel\nWallace, Sir Joseph Hooker, and others in the middle of the nineteenth\ncentury, has clearly the closest relationship with the geographical\ntheory of the control exercised by environment; it has become,\nindeed, its fundamental principle. Darwin accompanied the _Beagle_\nsurveying expedition round the world in 1831\u201336, and his observations\nduring the voyage qualified him for his life-work. Wallace's study\nof the distribution of animals brings at once to the mind his line\nof demarcation between faunal regions passing through the Malay\nArchipelago. Hooker was prepared for his interest in plant geography\nby his voyage with Ross to the Antarctic, by his travels in northern\nIndia (1847\u201351), and other journeys of wide range. Such men were\ngeographers though their fame does not name them so. The application of\ngeographical method is either essential or at least valuable in every\nbranch of natural science; in itself it fulfils functions which the\nother natural sciences, taken individually, do not, and that is its\njustification.\n\n\n\n\nSHORT BIBLIOGRAPHY OF GEOGRAPHY\n\n\nA general history of geography (mainly, however, concerned with\nexploration and mapping) is Vivien de Saint-Martin's _Histoire de la\nG\u00e9ographie_ (Paris, 1873); a short historical review dealing more\nespecially with geographical theory will be found in H. Wagner's\n_Lehrbuch der Geographie_ (Leipzig, 1900). No English parallels to\nthese works are to be cited, but reference may be made to H.\u00a0R. Mill's\n_International Geography_ (1897) and his article on \"Geography\" and\nE.\u00a0G. Ravenstein's on \"Map\" in the _Encyclop\u00e6dia Britannica_ (eleventh\nedition). On the earliest period see E.\u00a0H. Bunbury, _History of Ancient\nGeography_ (2 vols., London, 1879), and H.\u00a0F. Tozer, _History of\nAncient Geography_ (1897), in the Cambridge Geographical Series; on the\n\"dark age\" and down to 1460, C.\u00a0R. Beazley, _Dawn of Modern Geography_\n(3 vols., London, 1897\u20131906); the sixteenth century is principally\ncovered by the voluminous literature on Columbus, such as Sir C.\u00a0R.\nMarkham's _Life of Christopher Columbus_ (London, 1892), E.\u00a0J. Payne's\n_History of the New World called America_ (Vol. I., Oxford, 1892); on\nthe seventeenth and eighteenth centuries, E. Heawood, _Geographical\nDiscovery in the Seventeenth and Eighteenth Centuries_ (Cambridge\nGeographical Series, 1912). Some of the earlier theoretical works\nhave been cited in the text; a few modern works representative of the\nvarious departments of geography may be mentioned here. Those of which\nthe prime purpose is description are represented by E. Reclus's _La\nNouvelle G\u00e9ographie Universelle_ (19 vols., Paris, 1876\u201395; there is\nan English translation) and by _Stanford's Compendium of Geography\nand Travel_ (various authors.) F. Ratzel, _Anthropogeographie_\n(2 vols., Stuttgart, c. 1891); Ellen C. Semple, _Influences of\nGeographic Environment_ (London, 1911); E. Suess, _Das Antlitz der\nErde_ (translated as _The Face of the Earth_, Oxford, 1904); G.\u00a0G.\nChisholm, _Manual of Commercial Geography_ (1890 and later editions);\nand volumes of the Cambridge Geographical Series already referred to\nmay be taken as representative of various departments of geography\nof which an outline has been attempted in the last chapter of this\nbook. A.\u00a0R. Clarke's _Geodesy_, the section of the article on \"Map\"\nin the _Encyclop\u00e6dia Britannica_ by the same writer, and Col. C.\u00a0F.\nClose, for projections, and H.\u00a0M. Wilson's _Topographic Surveying_\n(New York and London, 1901) are to be referred to in the department of\nmathematical geography, but this subject has to be pursued mainly in\nofficial publications and scientific journals. For illustrations of\ncartographical method it is unnecessary in order to study the highest\ndevelopment of large-scale mapping to go beyond the Ordnance Survey of\nGreat Britain and (particularly as illustrating the layer system of\nshowing relief) the reduction therefrom by Bartholomew (Edinburgh),\nwhose cartographical methods in such special scientific applications\nas meteorology, the distribution of population, etc., also render\nit unnecessary to consider sources other than English. But for best\nexamples of the map-making art combined with the most careful use of\nexisting data in the compilation of topographical atlases, such a work\nas Stieler's Hand Atlas (Justus Perthes, Gotha) must be studied. There\nare examples of cartographical work from all periods in E.\u00a0A. Reeves's\n_Maps and Map-making_ (London, Royal Geographical Society, 1910),\na series of three lectures on the history and methods of surveying\nand cartography. Ancient methods may be studied in several facsimile\natlases, such as A.\u00a0E. Nordenski\u00f6ld's (Stockholm, 1889). The student\nof oceanography must consult the \"Narrative\" of the _Challenger_\nexpedition by Sir John Murray, forming two of the fifty volumes of the\nReport of that great undertaking. Special treatises on the subject are\nthose by Otto Kr\u00fcmmel, _Handbuch der Oceanographie_; Murray and Hj\u00f6rt,\n_The Depths of the Ocean_; Fowler, _Science of the Sea_ (elementary);\nRichard, _L'Oc\u00e9anographie_; Thoulet, _L'Oc\u00e9an, ses Lois et ses\nProbl\u00e8mes_. On limnology (the study of lakes) see Forel's _Handbuch der\nSeenkunde_.\n\n\n\n\nINDEX\n\n\n  Abruzzi, Duke of, 114, 126\n\n  Abu Zaid, 40\n\n  Abyssinia, 44, 54, 108, 110, 111\n\n  Adam of Bremen, 36\n\n  Adams, William, 71\n\n  Adare, Cape, 131\n\n  Ad\u00e9lie Land, 130, 133\n\n  \u00c6gean civilisation, 4\n\n  \u00c6schylus, 10\n\n  Afghanistan, 115\n\n  Africa, 5, 7, 13, 27, 31, 39, 50, 51, 53, 55, 57, 99, 113,107\u201314\n\n  African Association, 108, 146\n\n  Agartharchides of Cnidus, 23\n\n  Agassiz, Alexander, 144\n\n  Agrippa, M. Vipsanius, 25\n\n  Alaska, 78\n\n  Albert Edward, Lake, 113\n\n  ---- Nyanza, 111, 113\n\n  Alexander Island, 128\n\n  ---- the Great, 20\n\n  Alfred the Great, 37\n\n  Alleghany Mts., 67\n\n  Almagro, Diego, 66\n\n  Al-Mamun, Caliph, 39\n\n  Alonzo, Martin, 59\n\n  Amazon, River, 62, 66, 67\n\n  America. _See_ North America, South America\n\n  Amundsen, Capt Roald, 127, 133\n\n  Amur, River, 72\n\n  Anaximander, 9\n\n  Anaximenes, 9\n\n  Andrada, Antonio, 72\n\n  Andree, 137\n\n  Angelus, Jacobus, 54\n\n  Angola, 53\n\n  Antarctic regions, 81, 128\u2013134\n\n  Antilles, 61\n\n  Antillia, 56, 58\n\n  Antipodes, 23, 36\n\n  Ant\u0153ci, 23\n\n  Antonine Itinerary, 26\n\n  Apianus, Peter, 97, 99, 104\n\n  Arabia, 13, 73, 117\n\n  Arabs, 39\n\n  Arctic regions, 44, 74, 122\n\n  Aristagoras, 9\n\n  Aristotle, 19\n\n  Arkansas, 69\n\n  Arrowsmith, Aaron, 104\n\n  Asia, 9, 13, 20, 27, 39, 42, 71, 72\n\n  Asia Minor, 7, 8, 71\n\n  Astrolabe, 90\n\n  Atlantis, 18, 56\n\n  Australia, 69, 83, 85, 87, 89, 117\u201321\n\n  Avila, Gil Gon\u00e7alez de, 64\n\n  Azores, 50, 52\n\n  Azov, Sea of, 14, 30\n\n\n  Babbage, 120\n\n  Babylonia, 3\n\n  B\u00e4cklund, O., 138\n\n  Bacon, Roger, 43\n\n  Baeyer, 77, 136\n\n  Baffin Bay, 123\n\n  ---- Land, 77\n\n  ----, William, 77, 123\n\n  Bahr-el-Ghazal, 114\n\n  Baker, Samuel, 111\n\n  Balboa, Vasco Nu\u00f1ez de, 64\n\n  Balleny, John, 129\n\n  Baltic Sea, 21, 30, 41\n\n  Baluchistan, 115\n\n  Barents, William, 79\n\n  Barl\u00e6us, Kaspar, 84\n\n  Barrow, Pt., 123\n\n  ----, Sir John, 122\n\n  Barth, H., 109\n\n  Bartholomew, John, 138\n\n  Bass, George, 117\n\n  ---- Strait, 117\n\n  Batavia, 87\n\n  Beatus, 39\n\n  Beaufort Sea, 128\n\n  Beaumont, Elie de, 139\n\n  Bede, the Venerable, 36\n\n  Behaim, Martin, 57\n\n  Behm, 137\n\n  Bellingshausen, Fabian von, 89, 128\n\n  Bengal, 104\n\n  Benin, Bight of, 6\n\n  Bent, J.\u00a0T., 117\n\n  Berghaus, H., 137\n\n  Bering Strait, 80, 122\n\n  ----, Vitus, 80\n\n  Berlinghieri, Francesco, 55\n\n  Bertrand, M., 139\n\n  Bessel, Friedrich Wilhelm, 136\n\n  Best, Capt., 71\n\n  Bianco, Andrea, 56\n\n  Bienewitz, Philip, 102\n\n  Biogeography, 140\n\n  Biscoe Is., 129\n\n  ----, John, 129\n\n  Blache, Vidal de la, 138\n\n  Black Sea, 14, 30, 49\n\n  Blue Nile, 103\n\n  Blunt, Wilfrid, 117\n\n  Bonvalot, 116\n\n  Borchgrevink, C.\u00a0E., 131\n\n  Bornu, 109\n\n  Borough, Stephen, 79\n\n  Borysthenes, River, 14\n\n  Bougainville, Louis Antoine de, 85\n\n  Bouguer, Pierre, 96\n\n  Bouvet Is., 86, 131\n\n  ----, Lozier, 86\n\n  Bower, Hamilton, 116\n\n  Boyle, George, 73\n\n  Brahmaputra, River, 115\n\n  Brazil, 56, 60, 62, 66, 82\n\n  Britain, 5, 6, 21, 30, 37, 118\n\n  Brooke, J.\u00a0M., 142\n\n  Bruce, James, 108\n\n  ----, W.\u00a0S., 132\n\n  Brunel, Olivier, 79\n\n  Buache, P., 101\n\n  Buchanan, T.\u00a0Y., 143\n\n  Burckhardt, J.\u00a0L., 117\n\n  Burke, Robert, 121\n\n  Burroughs, Christopher, 71\n\n  Burrus, 101\n\n  Burton, R.\u00a0F., 110, 117\n\n  Button, Sir Thomas, 77\n\n  Byron, John, 85\n\n\n  Cabot, John, 61, 75\n\n  ----, Sebastian, 79\n\n  Cabral, 62\n\n  Cadamosto, Aloise, 52\n\n  C\u00e6sar, Julius, 24\n\n  Cailli\u00e9, R., 109\n\n  California, Gulf of, 65\n\n  Cameron, Lovett, 113\n\n  Canada, 67\n\n  Canary Islands, 18, 27, 52\n\n  Cano, Sebastian del, 65\n\n  Canton, 69\n\n  C\u00e3o, Diogo, 53\n\n  Cape of Good Hope, 5, 53, 68\n\n  Cape Verde Islands, 52\n\n  Carpentaria, Gulf of, 70, 118\n\n  Carpenter, Nathanael, 105\n\n  Carpini, Joannes de Plano, 42\n\n  Carteret, Philip, 85\n\n  Carthage, 5\n\n  Cartier, Jacques, 66\n\n  Cartography. _See_ Map-making\n\n  Caspian Sea, 30, 39, 49, 71\n\n  Cassini, J. and D., 95, 104\n\n  Catalan map, 50\n\n  Catalonians, 49\n\n  Cavendish, Sir Thomas, 67\n\n  Ceylon, 31, 50, 57\n\n  Chad, Lake, 109\n\n  Challenger, 130, 143\n\n  Champlain, Samuel, 67\n\n  Chancellor, Richard, 79\n\n  Charcot, J.\u00a0B., 132\n\n  Chelyuskin, Lieut. T., 80\n\n  Chile, 66\n\n  China, 3, 30, 34, 43, 44, 71, 117\n\n  Chorography, 104\n\n  Chronometer, 97\n\n  Chrysoloras, Emanuel, 54\n\n  Cintra, Pedro de, 52\n\n  Clapperton, H., 109\n\n  Clarke, Capt., 122\n\n  Cluverius, Philip, 105\n\n  Coats Land, 132, 133\n\n  Cochin-China, 72\n\n  Columbus, Christopher, 56, 59\n\n  Compass, 48\n\n  Condamine, Charles de la, 96\n\n  Congo Free State, 113\n\n  ---- River, 53, 103, 113, 114\n\n  Cook Archipelago, 88\n\n  Cook, Captain James, 78, 87, 103, 122\n\n  Coppermine River, 78\n\n  Cordova, Hernandez de, 64\n\n  Cortereal, Gaspar, 75\n\n  Cortez, 65\n\n  Covilh\u00e3o, Pedro de, 53\n\n  Crates of Mallus, 23\n\n  Crete, 4, 18\n\n  Cross-staff, 91\n\n  Crozet Is., 86\n\n  Cruquius, M.\u00a0S., 101\n\n  Crusades, 42, 48\n\n  Cuba, 61\n\n  Cunningham, 119\n\n  Cyrus, 17\n\n\n  Dahna, 117\n\n  d'Ailly, Cardinal, 56\n\n  Dampier, William, 85\n\n  Danes, 125, 127\n\n  Danfrie, 93\n\n  D'Anville, J.\u00a0B.\u00a0B., 103, 108\n\n  Darling, River, 119\n\n  Darwin, Charles, 146\n\n  ----, Sir George, 139\n\n  David, T. W, E., 132\n\n  Davis, Edward, 84\n\n  ----, John, 70, 77\n\n  ---- Strait, 77\n\n  Deasy, Capt. H.\u00a0H.\u00a0P., 116\n\n  Debes, 137\n\n  Delisle, 9, 103\n\n  De Long, 127\n\n  Democritus of Abdera, 17\n\n  d'Entrecasteaux, 89\n\n  Desbarres, J.\u00a0F.\u00a0W., 103\n\n  Desideri, Fra, 72\n\n  Dezhnev, Simon, 80\n\n  Diaz, Bartolomeu, 53, 82\n\n  Dic\u00e6archus of Messana, 20\n\n  Dicuil, 36\n\n  Digges, Leonard, 93\n\n  Diopler, 90\n\n  Dominica, 61\n\n  Donis. _See_ Germanus\n\n  Doughty, Charles, 117\n\n  Drake, Sir Francis, 66, 84\n\n  Drygalski, E. von, 132\n\n  Dufour, Gen. H., 135\n\n  Dupain-Triel, 101\n\n  Duperrey, Louis, 89\n\n  D'Urville, Dumont, 89, 129\n\n  Dutch East India Co., 69\n\n\n  Earth, figure of, 19, 22, 34, 36, 43, 81, 90, 94, 104\n\n  Easter Island, 84\n\n  East India Co., 70, 115\n\n  ---- Indies, 68\n\n  Egypt, 3, 7\n\n  Elephantine, 16\n\n  Emin Pasha, 113\n\n  Enderby Bros., 129\n\n  ---- Land, 129\n\n  Ephemerides, 97\n\n  Ephorus of Cyme, 18\n\n  Equator, 21, 62\n\n  Eratosthenes, 22\n\n  Erebus, Mt., 130, 131\n\n  Eric the Red, 75\n\n  Erichsen, L. Mylius, 127\n\n  Ericsson, Leif, 75\n\n  Eskimo, 2\n\n  Eudoxus of Cnidus, 19\n\n  ---- of Cyzicus, 24\n\n  Europe, 9, 13, 14, 27, 39\n\n  Europus, 10\n\n  Everest, Sir George, 135\n\n  Eyre, 120\n\n\n  Faer\u00f6e Islands, 37\n\n  Fernandez, John, 52\n\n  Fernel, Jean, 93\n\n  Filchner, Lieut., 133\n\n  Firmianus, Lactantius, 34\n\n  Flinders, Matthew, 117\n\n  Florida, 64, 66\n\n  Forbes, Edward, 143\n\n  Forel, F.\u00a0A., 141\n\n  Forrest, Sir J., 121\n\n  Fortunate Isles, 27\n\n  Franklin, J., 122, 123\n\n  Franz Josef Land, 125, 127\n\n  Freycinet, J.\u00a0C.\u00a0D. de, 89\n\n  Freyre, Fra, 72\n\n  Frisius, Gemma, 97, 104\n\n  Frisland, 76\n\n  Frobisher, Martin, 76\n\n  ---- Strait, 77\n\n\n  Gaetano, 88\n\n  Gama, Vasco da, 68\n\n  Gambia, River, 108\n\n  Ganges, River, 115\n\n  Gastaldi, 103\n\n  Geodesy, 138\n\n  Geomorphology, 139\n\n  Germanus, Nicolaus, 55\n\n  Gerrits, Dirk, 84\n\n  Giles, Ernest, 121\n\n  Gill, Sir David, 138\n\n  Glaser, E., 117\n\n  Gnomon, 9, 90\n\n  Gold Coast, 52\n\n  Gomez, Diego, 52\n\n  Goyder, 120\n\n  Graham Land, 129\n\n  Grant, T.\u00a0A., 111\n\n  ---- Land, 128\n\n  Graphometer, 93\n\n  Great Australian Bight, 118\n\n  ---- Ice Barrier, 130, 131\n\n  Greeks, 8\n\n  Greeley, Lieut. A.\u00a0W., 126\n\n  Green, Lowthian, 139\n\n  Greenland, 37, 75, 76, 80, 124, 126, 127, 128\n\n  Gregory, 121\n\n  ----, A.\u00a0C., 120\n\n  ----, J.\u00a0W., 139\n\n  Grenard, F., 116\n\n  Griesbach, L., 116\n\n  Grinnell Land, 124, 126\n\n  Grisalva, Juan de, 64\n\n\n  Haiti, 61\n\n  Hal\u00e9vy, J., 117\n\n  Hall, Charles, 124\n\n  Halley, E., 101\n\n  Hanno, 6\n\n  Harrison, John, 97\n\n  Hausaland, 109\n\n  Hawaii, 88\n\n  Hawkins, Captain, 70\n\n  Hayton, 43\n\n  Hearne, Samuel, 78\n\n  Hecat\u00e6us of Miletus, 11\n\n  Hedin, Sven, 116\n\n  Henry the Navigator, 51\n\n  Herodotus, 5, 11, 23, 30\n\n  Himilco, 6\n\n  Hiort, Dr. Johan, 144\n\n  Hipparchus, 22, 25\n\n  Hispaniola. _See_ Haiti\n\n  Hollar, 102\n\n  Homann, 101\n\n  Homer, 7\n\n  Honduras, 61\n\n  Hooker, Sir Joseph, 146\n\n  Horn, Cape, 67, 82, 84\n\n  Houtman, Cornelis, 69\n\n  Howell, 119\n\n  Huber, C., 117\n\n  Hudson Bay, 67, 76, 80\n\n  ---- Bay Co., 78, 123\n\n  ----, Henry, 76, 80\n\n  Humboldt, Alexander von, 67, 144\n\n  Hume, Hamilton, 119\n\n  Huygens, Christiaan, 94\n\n\n  Ibn Batuta, 46\n\n  Ibn Haukal, 40\n\n  Iceland, 37, 75\n\n  Idrisi, 40\n\n  India, 3, 13, 20, 31, 44, 45, 50, 54, 57, 68, 69, 71, 115\n\n  Indicopleustes, Cosmas, 35\n\n  Inglefield, Captain, 124\n\n  International Council for the Study of the Sea, 144\n\n  ---- Geographical Congress, 138\n\n  Ireland, 30, 37, 75, 102\n\n  Irtish, River, 72\n\n  Istakhri, 40\n\n  Ister, River, 14\n\n\n  Jackman, Charles, 79\n\n  Jackson, Frederick, 126\n\n  J\u00e4derin, E., 137\n\n  Jamaica, 61\n\n  Jan Mayen Island, 80\n\n  Japan, 69, 70\n\n  Jefferys, Thomas, 104\n\n  Jenkinson, Anthony, 71\n\n  Jerusalem, 39, 49\n\n  Jesuits, 72\n\n  John of Montecorvino, 45\n\n  Johnston, W. and A.\u00a0K., 138\n\n  Jones Sound, 127\n\n  Junker, W., 114\n\n  Justinian, Emperor, 34\n\n\n  Kaiser Wilhelm II. Land, 132\n\n  Kalahari Desert, 112\n\n  Kamchatka, 72, 89\n\n  Kane, Elisha Kent, 124\n\n  Kant, Immanuel, 144\n\n  Kara Sea, 79\n\n  Karlsefne, Thorfinn, 75\n\n  Kaulbars, Baron H., 116\n\n  Kemp, John, 129\n\n  Kennedy, 120, 123\n\n  Kenya, Mt., 110, 113\n\n  Kepler, 97\n\n  Kerguelen Is., 86\n\n  Kerguelen-Tr\u00e9marec, Yves de, 86\n\n  Kiepert, H., 137\n\n  Kilimanjaro, Mt., 110, 113\n\n  Kimberley, 121\n\n  King, Capt. P.\u00a0P., 118\n\n  King Edward Land, 130, 131\n\n  King George Sound, 117\n\n  Kircher, A., 101\n\n  Kola Peninsula, 79\n\n  Koldewey, 126\n\n  Kotzebue, Otto von, 89\n\n  Kozlov, P.\u00a0K., 116\n\n  Krapf, L., 110\n\n  Kremer, Gerhard. _See_ Mercator\n\n  Krishna, 115\n\n  Krusenstern, Adam, 89\n\n  Kublai Khan, 43\n\n\n  Labrador, 75\n\n  Lacerda, Francisco de, 109\n\n  Lachlan, River, 118, 119\n\n  Laing, A., 9, 109\n\n  Lambton, Col. W., 135\n\n  Lancaster, James, 70\n\n  ---- Sound, 123\n\n  Langhans, 137\n\n  Lapparent, A. de, 139\n\n  Lapworth, C., 139\n\n  Latitude, 21, 97\n\n  Lauder, Richard, 109\n\n  Leachman, 117\n\n  Lefroy, H.\u00a0M., 121\n\n  Lehmann, J.\u00a0G., 101\n\n  Leichhardt, 120\n\n  Lemaire, Jacob, 84\n\n  Leon, Juan Ponce de, 63\n\n  Le Testu, Guillaume, 69\n\n  Lhasa, 72, 115\n\n  Limnology, 141\n\n  Littledale, St. George, 116\n\n  Liverpool Plains, 119\n\n  Livingstone, David, 112\n\n  Longitude, 97\n\n  Loubet Land, 132\n\n  Louisiana, 67\n\n  Louis Philippe Land, 129\n\n  Love, A.\u00a0E.\u00a0H., 139\n\n  Lualaba, River, 112\n\n  Luque, Hernando de, 66\n\n\n  McClintock, 124\n\n  McClure, 124\n\n  Mackenzie, Alexander, 78\n\n  ---- River, 78\n\n  McKinlay, T., 121\n\n  McMurdo Sound, 132\n\n  Macquarie, River, 118, 119\n\n  Madagascar, 44, 57\n\n  Madeira, 50, 52\n\n  Magalh\u00e3es, Fernao de, 64, 82\n\n  Magellan. _See_ Magalh\u00e3es\n\n  Magnetic Pole, 123, 127\n\n  Magnus, Albertus, 43\n\n  Mahomed Ali, 110\n\n  Malacca, 68\n\n  Malay Archipelago, 44, 57, 68, 69, 70\n\n  ---- Peninsula, 5\n\n  Mam, Island of, 56\n\n  Manchuria, 89\n\n  Mandeville, Sir John, 45\n\n  Maoris, 2\n\n  Map-making, 2, 9, 12, 37, 41, 46, 54, 98, 137, 145\n\n  Marianne Islands, 65\n\n  Marinus of Tyre, 26, 40\n\n  Marion-Dufresne, 86\n\n  Marion Is., 86\n\n  Markham, 126\n\n  ----, Albert, 124\n\n  Marquesas Is., 83\n\n  Martianus Capella, 34\n\n  Masailand, 113\n\n  Masudi, 40\n\n  Maupertuis, P.\u00a0L.\u00a0M. de, 96\n\n  Mauro, Fra, 50, 55\n\n  Maury, Matthew F., 14, 20\n\n  Mawson, Dr. D., 132, 133\n\n  Mediterranean, the, 3, 5, 37, 39, 48, 58, 143\n\n  Melville Island, 123\n\n  Mendana, 83\n\n  Mercator, 98, 102, 103\n\n  Meridian, 22, 41\n\n  Meroe, 16\n\n  Mexico, 64, 65\n\n  Middleton, Christopher, 78\n\n  Mill, H.\u00a0R., 106\n\n  Mitchell, Sir Thomas, 120\n\n  Moeris, Lake, 3\n\n  Moluccas, 60, 69\n\n  Monaco, Prince of, 144\n\n  Mongolia, 115\n\n  Moseley, H.\u00a0N., 143\n\n  Mota, Antonio de, 69\n\n  Mountains of the Moon, 31\n\n  Muhammad ben Musa, 90\n\n  M\u00fcnster, Sebastian, 105\n\n  Murray, River, 119\n\n  ----, Sir John, 141, 143\n\n  Murrumbidgee, River, 119\n\n  Muscovy Company, 79, 80\n\n  Musil, 117\n\n\n  Nachtigal, G., 114\n\n  Nain Singh, 115\n\n  Nansen, Fridtjof, 126\n\n  Nares, Sir George, 124, 143\n\n  Nasamonian youths, the, 16\n\n  Natal, 68\n\n  Nathorst, 127\n\n  Nearchus, 20\n\n  Necho, 5\n\n  Neckam, Alexander, 48\n\n  New Caledonia, 88\n\n  Newfoundland, 75\n\n  New Granada, 66\n\n  New Guinea, 69, 83, 87\n\n  New Hebrides, 83, 85, 88\n\n  New Pomerania, 84\n\n  New Siberian Is., 127\n\n  New South Wales, 117\n\n  Newton, Sir Isaac, 94, 105, 144\n\n  New Zealand, 70, 83, 87\n\n  Nicaragua, 64\n\n  Niebuhr, Carsten, 73\n\n  Niger, River, 16, 108, 109\n\n  Nile, River, 16, 31, 37, 49, 108, 110, 111\n\n  Nordenski\u00f6ld, A.\u00a0E., 125, 126\n\n  Norfolk Is., 88\n\n  Norsemen, 37\n\n  North America, 2, 61, 67, 75\n\n  North American Indians, 2\n\n  North Cape, 75\n\n  North-East Passage, 78\n\n  North-West Passage, 76, 127\n\n  Norwood, Richard, 93\n\n  Nova Scotia, 75\n\n  Novaya Zemlya, 72, 75, 80\n\n\n  Ob, River, 80\n\n  Oceanography, 141\n\n  Odoric, 45, 71\n\n  \u0152cumene, 22, 23\n\n  Ohio, River, 66\n\n  Okhotsk, Sea of, 72\n\n  Orellana, Francisco de, 66\n\n  Orinoco, River, 61, 145\n\n  Orkney Islands, 37\n\n  Orl\u00e9ans, Prince Henry of, 116\n\n  Orontius, 82\n\n  Ortelius, 65, 102\n\n  Oudney, W., 109\n\n  Oxley, Lieut., 118\n\n\n  Pacific Ocean, 64, 65, 82, 83, 87\n\n  Palestine, 48\n\n  Palgrave, W.\u00a0G., 117\n\n  Pamirs, 115\n\n  Paradise, 39\n\n  Paris, Matthew, 48\n\n  Park, Mungo, 108\n\n  Parmenides, 9\n\n  Parry (Arctic), 122, 126\n\n  ---- (Australia), 120\n\n  Pascal, 93\n\n  Paulmyer, Sieur de Gonneville, 69\n\n  Paumotu Is., 84, 85\n\n  Pausanias, 32\n\n  Payer, Lieut. Julius, 125\n\n  Payva, Alphonso, 53\n\n  Peary, Robert E., 126, 127\n\n  Peking, 69\n\n  Penna, Orazio della, 72\n\n  Periodos (circuit of the earth), 12\n\n  Peri\u0153ci, 23\n\n  Periplus, 12, 18\n\n  ---- of the Erythr\u00e6an Sea, 3\n\n  P\u00e9rouse, J.\u00a0F.\u00a0D. de la, 89\n\n  Persia, 71\n\n  Persian Gulf, 20\n\n  Perthes, Justus, 137\n\n  Peru, 3, 66\n\n  Peschel, Oscar, 145\n\n  Pet, Arthur, 79\n\n  Peter Is., 128\n\n  Petermann, A., 137\n\n  Peutinger Table, 26\n\n  Pevtsov, 116\n\n  Phasis, River, 11\n\n  Philip, George, 138\n\n  Philippine Islands, 65\n\n  Phipps, Captain, 122, 142\n\n  Ph\u0153nicians, 4\n\n  Picard, Jean, 94, 101\n\n  Pigafetta, F., 99\n\n  Pines, Isle of, 88\n\n  Pinzon, Vicente Ya\u00f1ez, 59, 62\n\n  Pizarro, Francisco, 66\n\n  Plane-table, 93\n\n  Plato, 18\n\n  Pliny the Elder, 26\n\n  Polo, Marco, 43, 69\n\n  Polybius of Megalopolis, 23\n\n  Polyhistor, Julius Solinus, 35\n\n  Pomponius Mela, 26\n\n  Pont, Timothy, 102\n\n  Portolano maps, 48, 50, 55\n\n  Portuguese, 51, 55, 60, 68, 79, 108\n\n  Posidonius the Stoic, 23, 72\n\n  Potamology, 141\n\n  Potanin, 116\n\n  Prester John, 53\n\n  Pretorius, Jean, 93\n\n  Prjevalsky, Nicolai, 116\n\n  Procopius, 34\n\n  Ptolemy, 26, 39, 54, 72\n\n  Putte, Samuel van der, 72\n\n  Pygmies, 7, 17\n\n  Pythagoras, 8\n\n  Pytheas of Massilia, 21\n\n\n  Quadrant, 91, 94\n\n  Queensland, 119, 120\n\n  Quiros, Pedro Fernandez de, 83\n\n\n  Rae, Dr. John, 124\n\n  Ramsden, 97\n\n  Ratzel, Friedrich, 145\n\n  Ravenstein, A., 137\n\n  Rawling, Capt. G.\u00a0C., 116\n\n  Rebmann, T., 110\n\n  Red Sea, 5, 35\n\n  Regiomontanus, 97\n\n  Rennell, Maj. James, 104\n\n  Retes, Inigo Ortes de, 83\n\n  Retreat of the Ten Thousand, 17\n\n  Rhins, Dutreuil de, 116\n\n  Rhodesia, 5\n\n  Richard of Haldingham, 48\n\n  Richardson, Sir John, 124\n\n  Richer, Jean, 94\n\n  Richthofen, Ferdinand, Baron von, 117, 145\n\n  Rio de la Plata, 62\n\n  Ripon Falls, 111\n\n  Ritter, Karl, 145\n\n  Roborovsky, 116\n\n  Rockhill, W.\u00a0W., 116\n\n  Roe, J.\u00a0S., 121\n\n  Roggeveen, Jacob, 85\n\n  Rohlfs, G., 114\n\n  Romans, 24\n\n  Rome, 19\n\n  Rosen, P.\u00a0G., 138\n\n  Ross, Capt., 122, 123, 142\n\n  ----, James Clark, 123, 142\n\n  ----, Sir James, 130\n\n  Roy, General, 103, 104\n\n  Royal Geographical Soc., 146\n\n  Russians, 72, 78, 80, 116\n\n  Rusticiano of Pisa, 44\n\n  Ruwenzori Mts., 113\n\n\n  Sadlier, Capt. G.\u00a0F., 117\n\n  Sahara Desert, 52, 109, 114\n\n  St. Basil the Great, 34\n\n  St. Brandon, Island of, 58\n\n  St. Francis Xavier, 69\n\n  St. Lawrence River, 66\n\n  St. Martin, Vivien de, 138\n\n  Samoa, 85\n\n  San Salvador, 60\n\n  Sanson, Nicolas, 99\n\n  Saxton, Christopher, 102\n\n  Scandinavia, 7, 21, 30, 37, 49, 58, 75\n\n  Scaph, 90\n\n  Schouten, Willem Cornelis, 84\n\n  Schrader, F., 138\n\n  Schweinf\u00fcrth, Georg, 114\n\n  Scoresby, William, 81, 126, 142\n\n  Scotland, survey, 103\n\n  Scott, Capt. R.\u00a0F., 131, 133\n\n  Scroggs, John, 78\n\n  Scylax of Caryanda, 12\n\n  Semliki River, 113\n\n  Senegal, 109\n\n  Shackleton, Sir E., 131, 132\n\n  Shetland Islands, 37\n\n  Shirley, Sir Anthony and Robert, 71\n\n  Siberia, 44, 72, 80\n\n  Sicily, 7\n\n  Sidon, 7\n\n  Sierra Leone, 52\n\n  Sission, Jonathan, 97\n\n  Smith, William, 128\n\n  ----, Leigh, 125\n\n  ---- Sound, 78, 124, 128\n\n  Snell, Willibrord, 93\n\n  Society Is., 85, 87\n\n  Sokoto, 109\n\n  Solis, Juan Diaz de, 62\n\n  Somaliland, 110\n\n  Soto, Hernandez de, 66\n\n  South America, 56, 62, 66, 67, 82\n\n  Southern Continent, 81, 83\n\n  South Orkney Is., 128\n\n  South Shetland Is., 84, 128\n\n  Spaniards, 60, 63\n\n  Speed, John, 102\n\n  Speke, T.\u00a0H., 110\n\n  Spitsbergen, 75, 79, 124, 127\n\n  Spruner, K. von, 137\n\n  Stanford, Edward, 138\n\n  Stanley, H.\u00a0M., 112, 113\n\n  Stefansson, 128\n\n  Stein, Sir M.\u00a0A., 116\n\n  Stieler, A., 137\n\n  Stony Desert, 120\n\n  Strabo, 18, 24, 141\n\n  Stuart, J.\u00a0M., 121\n\n  Sturt, Charles, 119, 120\n\n  Submarine cable, 142\n\n  Sudan, 114\n\n  Suess, E., 139\n\n  Suleiman, 40\n\n  Supan, A., 137, 139\n\n  Surveying, 93, 102, 135\n\n  Sverdrup, 127\n\n  Swartha, Claudius Clavus, 55\n\n  Sydney, 117\n\n  Syene, 22\n\n  Syria, 71\n\n\n  Tachard, P., 72\n\n  Tahiti, 87\n\n  Tanais, River, 14, 21, 37\n\n  Tanganyika, Lake, 111, 112\n\n  Tasman, Abel Janszoon, 70, 83\n\n  Tasmania, 70, 117\n\n  Terror, Mt., 130, 131\n\n  Texeira, Pedro, 67\n\n  Thales of Miletus, 8\n\n  Theodolite, 97\n\n  Theophanes of Mytilene, 24\n\n  Theophrastus of <DW26>s, 20\n\n  Thompson, Sir C. Wyville, 143\n\n  Thomson, Joseph, 113\n\n  Thule, 21\n\n  Thury, Cassini de, 104\n\n  Tibet, 44, 71, 73\n\n  Tierra del Fuego, 67, 84\n\n  Timbuktu, 109\n\n  Toll, Baron, 127\n\n  Tongan Archipelago, 84, 88\n\n  Torrens, Lake, 120\n\n  Torres, 70, 83\n\n  ---- Strait, 83\n\n  Torricelli, 93\n\n  Toscanelli, Paolo del Pozzi, 56\n\n  Traverse Is., 128\n\n  Trinidad, 61\n\n  Triquetum, 90\n\n  Turkestan, 115\n\n  Turnagain, Cape, 123\n\n  Turner, Capt., 73\n\n  Tyre, 4\n\n\n  Uganda, 111\n\n  Ugyen Gyatso, 115\n\n  Unyamwezi, 111\n\n\n  Varenius, Bernhard, 105\n\n  Vernier, Fran\u00e7ois, 93\n\n  Vesconte, Petrus, 49\n\n  Vespucci, Amerigo, 62\n\n  Victoria, 119\n\n  ---- Falls, 112\n\n  ---- Land, 130, 133\n\n  ---- Nile, 111\n\n  ---- Nyanza, 111, 113\n\n  Vinci, Leonardo da, 82\n\n  Vineland, 37, 75\n\n  Virgilius, Bishop of Salzburg, 36\n\n  Vlamingh, William de, 70\n\n  Volga, River, 30\n\n\n  Wagner, Hermann, 145\n\n  Waldseem\u00fcller, Martin, 63, 103\n\n  Wallace, Sir Alfred Russel, 146\n\n  Wallis, Samuel, 85\n\n  Warburton, 121\n\n  Watling Island, 60\n\n  Weddell, James, 129\n\n  Wellby, Capt. W.\u00a0S., 116\n\n  Welle, River, 114\n\n  Werner, 99\n\n  Western Australia, 121\n\n  West Indies, 60\n\n  Weyprecht, Lieut., 125\n\n  Whale fisheries, 80\n\n  White Nile, 108\n\n  ---- Sea, 75, 79\n\n  Wiggins, Capt. Joseph, 125\n\n  Wilkes, Charles, 89, 130\n\n  William of Rubruquis, 42\n\n  Willoughby, Sir Hugh, 79\n\n  Wills, William, 121\n\n  Wilson, 131\n\n  Wissmann, 113\n\n  Wright, Edward, 99\n\n\n  Xenophon, 17\n\n\n  Yenisei, River, 80, 125\n\n  Yucatan, 64\n\n\n  Zambezi, River, 5, 109, 112\n\n  Zanzibar, 44, 57, 110\n\n  Zeno, Nicolo, 76\n\n  Zoogeography, 140\n\n\nPRINTED BY WATTS AND CO., JOHNSON'S COURT, FLEET STREET, LONDON.\n\n\n\n\nTranscriber's Notes\n\n\nOn pages 37 and 39, single-letters within equals signs, such as =T=,\nwere printed in sans-serif boldface. Throughout this eBook, italic text\nis shown within _underscores_, and small-cap text is shown in ALL-CAPS.\n\nPunctuation and spelling were made consistent when a predominant\npreference was found in this book; otherwise they were not changed.\n\nSimple typographical errors were corrected; occasional unbalanced\nquotation marks retained.\n\nAmbiguous hyphens at the ends of lines were retained; occurrences of\ninconsistent hyphenation have not been changed.\n\nIndex not checked for proper alphabetization or correct page references.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of History of Geography, by \nJohn Scott Keltie and Osbert John Radcliffe Howarth\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHUNTING KAT\n\nKelley Armstrong\n\nContents\n\nBegin Reading Now\n\nAn Excerpt from _The Calling_\n\nChapter One\n\nBack Ads\n\nCopyright\n\nAbout the Publisher\nBegin Reading Now\n\nI stretched out on the lounge chair in front of our motel room.\n\n\"Basking in the sun, mon chaton?\" Marguerite's French-accented voice sounded behind me. \"You have been doing a lot of that lately.\"\n\n\"Can't get sun cancer now.\"\n\n\"No, you just like thumbing your nose at the myth.\"\n\nI grinned. \"A sunbathing vampire. So Dracula-retro.\"\n\nShe sighed. I tilted my head back to look at her as she stepped out the screen door. Like me, Marguerite is a vampire. She's been one a lot longer, though. Over a hundred years, though she looks twenty, the age when she died. Eternally beautiful. Well, in Marguerite's case, at least\u2014she's tiny with blond curls and big blue eyes. I'd thought she was an angel when I first met her. She was my angel, rescuing me from a science experiment and from parents who weren't my parents at all, but people paid to care for me.\n\nThat was ten years ago. I was sixteen now, and undead for six months. Marguerite had nothing to do with making me a vampire. That was the experiment, plus a bullet to the heart.\n\nMarguerite had known what I was all along. That's why she'd taken me. She'd never told me the truth, though. I found out the hard way, waking up on a morgue slab. I understand why she kept it a secret\u2014she wanted me to grow up normal\u2014but I haven't quite gotten over it. I don't tell her that. When it comes to feeling guilty, Marguerite doesn't need any help.\n\n\"Are you hungry?\" she asked, holding out a travel mug.\n\n\"Not for that.\"\n\nShe set it down beside me. I could smell the blood, warmed to body temperature. Like that made a difference.\n\n\"You need to drink, Katiana,\" she said.\n\n\"It's stale. Now that . . .\" I waved at a man three doors down, passed out drunk. \"That's a proper breakfast. Not like he'd notice. He's already going to have a killer hangover. A missing pint of blood wouldn't matter.\"\n\n\"You are too young to drink alcohol.\"\n\n\"Ha-ha.\"\n\n\"I am serious, Kat. Whatever is in his blood will be in yours. Drugs, alcohol . . . You have to consider that.\"\n\n\"No, I need to consider what I am. A hunter. I need to hunt, Mags. You do.\"\n\n\"And so will you, mon chaton, when you are\u2014\"\n\n\"Psychologically and emotionally ready.\" I tried to keep the edge out of my voice. \"But you're going to talk about it with the other vamps, right? That's why we're going to this meeting in New York.\"\n\n\"We are going for many reasons.\"\n\n\"But you are going to ask them whether I should start hunting.\"\n\n\"Yes, I will. Now drink. We still have a long drive.\"\n\nMarguerite went back inside to get ready. I drank the blood. It was like eating store-bought chocolate chip cookies\u2014I could taste hints of what I really wanted, what I craved, but they were hidden under a leaden layer of foul crap.\n\nAs I sipped, I eyed the drunk guy and imagined sinking my fangs into his neck. Imagined his blood, hot and rich. The back of my throat ached so much I could barely gag down my blood-bank breakfast.\n\nI know I sound like a coldhearted bitch, fantasizing about drinking some guy's blood, like I'm brutally nonchalant about the whole vampire situation. I'm not. I have my good days. And I have my bad ones, too, when I can't get out of bed in the morning, when I lie there and think and worry.\n\nAm I going to be sixteen forever? Marguerite says no, that the genetic modification experiment was supposed to get rid of the eternal youth thing, which when you think about it, isn't really such a blessing, being one age forever, never able to settle in one place, make friends, get a job, fall in love. . . .\n\nWhat if the modifications failed? What if I am sixteen for the next three hundred years? I think about all the things I didn't get to do before I turned. Things I might never get to do.\n\nEven if the modifications took, how would that work? I can't be injured, can't get sick. Does that mean I'm invulnerable, but not immortal? That I'll die when I'm a hundred, like everyone else? Or will I live to three or four hundred, like real vampires? If I do, will I keep aging at a normal rate, and turn into some hideous old hag? Marguerite doesn't have any answers, just keeps saying it will work out, which means she's just as concerned as I am.\n\nI try not to think about all that. I've got enough to worry about with my life now. Hungering over humans. Drinking blood. Fearing that the Edison Group will find me again. Worrying that I'll screw up and get caught.\n\nEven without the Edison Group problem, there's so much to stress out about. What if I get hit by a transport and the paramedics take me to a hospital where, whoops, suddenly I'm as good as new? What if people figure out I'm a vampire? Would they kill me? Experiment on me? Lock me up? Would I be any better off than if the Edison Group did catch me?\n\nSo, no, I'm not nonchalant about it. I'm dealing. Kind of. Today we were heading to New York to meet other vampires and get some answers about the genetic modifications and about how to handle my situation. So today was definitely going to be a good day.\n\nAs for hunting humans, I'm not nonchalant about that, either. When the time comes, even if it doesn't have lasting effects on people, I expect I'll feel guilty about it. Marguerite does. But I still need to hunt. I feel that in my gut, a gnawing restlessness, like when I haven't done a workout in a while.\n\nWhen the feeling gets bad, no amount of canned blood helps. I'll be walking along and I'll smell something unbelievably good. I'll start salivating, stomach growling, and I'll turn to see, not a plate of freshly baked cookies but a person, maybe even a friend. I can't describe how that feels. It's bad. Just bad.\n\nI finished my drink and went inside. Marguerite was in the bathroom putting on makeup. I perched on the counter, watching her as she applied pale lipstick to a mouth that was already a perfect pink bow.\n\n\"So who's the hot vampire in New York?\" I said. \"A tall, dark Napoleonic soldier you met during the Civil War? Sheltered him from the witch hunts? Got separated on the Titanic, each floating off on your own icebergs?\"\n\n\"History is not your strongest subject, is it, mon chaton?\"\n\n\"I'm improvising. So who is he?\"\n\n\"There is no he. I want to look nice for people I have not seen in a while.\"\n\n\"Uh-huh.\"\n\nI looked in the mirror\u2014yes, unlike Hollywood vampires, I can see my reflection. Beside Marguerite's fragile porcelain perfection, I always feel big and clumsy. Seeing us together, though, the difference isn't that obvious. I'm only a few inches taller, and skinny enough that I can snag her designer shirts and leather jackets. No one ever mistakes us for sisters, though. My golden brown hair, green eyes, and lightly bronzed skin guarantee that.\n\nI reached for her makeup bag. She snatched it away and handed me lip gloss.\n\n\"When you are seventeen,\" she said.\n\n\"I may never be seventeen.\"\n\n\"Then you will have no need for makeup, will you?\"\n\nI sighed. Marguerite can be unbelievably old-fashioned sometimes. The perils of having a guardian who grew up in the nineteenth century. On this point, though, I don't really care. I'm a jock, not a cheerleader. Makeup is a pain in the ass. Well, most times. I make an exception for dates. Not that there'd been any of those since I turned. Let a guy nuzzle my neck and he might realize I don't have a pulse. Marguerite says guys won't notice, but I'm not ready to take that chance.\n\nI wandered into the bedroom, picked up the keys and jangled them. \"Don't forget I have my license now. Better hurry or I'll go to New York without you.\"\n\nShe didn't peek out of the bathroom. Didn't call after me as I walked out the door. Didn't even ring my cell as I drove our little rental car from the motel lot. She knew I wasn't going far. There are days when I like that, knowing she trusts me. Then there are others when I really wish I was a little more rebellious. A little less predictable.\n\nShe probably even knew where I was heading. We'd passed a coffee shop and bakery a few minutes before we stopped for the night, and she'd promised to let me head back in the morning, grab my morning coffee. Vampires don't need food. We can still eat and drink, though, which helps us fit in. For most, like Marguerite, food doesn't sit well in their stomachs. Not so with me. That's one of the modifications that did work, apparently.\n\nI grabbed an extra-large hazelnut vanilla coffee and a cinnamon bun. Then I headed back, music blasting, pedal to the floor, zooming along the empty Vermont road. Well, close. I had the music moderately loud and I was going about ten kilometers over the speed limit. Five miles over the speed limit, I should say. I'm American, but Marguerite is French Canadian, and we've spent most of the past decade in Montreal, so I'm used to the metric system.\n\nMiles or kilometers, the point was that I wasn't speeding very much, and I left my coffee and bun untouched beside me until Marguerite was driving. Yes, I can act like a smart-aleck teen, but I rarely break the rules. It's my upbringing, Marguerite says. The only time my \"parents\" ever praised me was when I was a model child, tediously well behaved. Being a vampire doesn't make me a badass. Sadly.\n\nI'm not a complete wimp, though. So when a pickup came roaring up behind me, I didn't follow my driving instructor's teachings and pull over to let him pass. I sped up. He stayed on my bumper, coming so close I could only see his truck's grille.\n\nThat made me realize just how empty the road was, winding through the foothills, dense trees on either side, a steep embankment down to my right. I hadn't seen another car since leaving town. Not the place to play road warrior. So I slowed to the speed limit and eased over, giving him plenty of room for passing.\n\nWhen he made no move to go by, I slid my cell phone from my pocket. Then the grille disappeared from my rearview mirror as the truck swung into the other lane to pass.\n\nI glanced in my side mirror. It was only a glance. I had both hands on the wheel. I didn't drift into his lane. I was sure of it. But the next thing I knew, there was a crunch, metal on metal, and my car shot toward the embankment.\n\nMy throat seized up, brain screeching, brakes screaming along with it as I slammed on the brakes. The car kept going, sailing over the edge.\n\nIt rolled and it kept rolling and all I could do was duck, hands over my head, until there was a bone-jarring crash. And everything went dark.\n\nI blacked out for only a second. When I came to, the car was still groaning from the impact. I opened my eyes to see a tree in the passenger seat. The car was wrapped around it.\n\nI reached to undo my seat belt, but I couldn't twist. A branch had gone through my shoulder and pinned me to the seat. I stared at it. There was a branch through my shoulder. And I couldn't feel a thing.\n\nI took a deep breath, reached up, and pulled it out. Took some effort. Vampires don't get superhuman strength\u2014another myth debunked\u2014and the branch was all the way through the seat, so it required work, but finally I got it free. It left a hole in my shirt, but no blood, obviously. There was a hole in my shoulder, too, but it would seal up.\n\nI tried to check out the damage in the mirror. Seeing myself, I let out a yelp and closed my eyes fast. Another deep breath. Then I pulled down the visor and opened the mirror.\n\nMy nose was broken. Smashed nearly flat from an impact I couldn't remember. My lip was split. And one of my eyes was . . . not quite in place.\n\nOh God. My stomach heaved. I closed my eyes and pressed my palm to the injured eye. It . . . went back in. I shuddered, stomach spinning.\n\nI reached up and straightened my nose. As I moved it, I could feel it reforming under my fingers.\n\nThere. All fixed. Now\u2014\n\n\"Hello!\" a man's voice shouted.\n\nI lowered my head to look out the smashed window. A vehicle was parked at the top of the embankment. The guy who'd run me off the road?\n\nNo. It was a car, not a truck. Two sets of legs stood beside it. They must have seen my car fly off the road.\n\nThat really wasn't any better. I couldn't let rescuers find me, not while I still had a hole through my shoulder and God knows what other injuries, all of which would miraculously heal during the ambulance ride to the hospital. This was exactly the sort of scenario I'd feared.\n\nI stuffed my cell phone into my pocket and grabbed the door handle. My fingers slipped on the wet surface. Coffee, I realized. The whole interior of the car dripped with it.\n\nHey, at least it's not blood.\n\nI wrenched the handle. Not surprisingly, the door didn't open. I twisted to get up, so I could kneel on the seat and go out the window.\n\nMy legs wouldn't move.\n\nI stared down. They were crushed. Oh God. My legs were crushed.\n\n\"Is someone down there?\" the man yelled.\n\n\"I think I see a car,\" a woman answered. \"Have you called the . . . ?\" Her voice drifted off.\n\nOkay, they weren't coming down here. Not yet, at least. I had time. I tugged at the broken steering wheel. It snapped off in my hands. I set it aside, then ran my hand down my legs, trying to wriggle them free. The muscles weren't responding, but my legs seemed to be loose.\n\nJust as long as they weren't really loose . . . like not connected to the rest of my body, because I was pretty sure that whatever regenerative abilities vampires had didn't go that far. Really sure actually, considering that the only way to kill me was decapitation. Some parts just don't grow back.\n\nMy legs seemed fully attached, though. And already mending, which meant in a few more minutes, they'd be really pinned.\n\nI ratcheted my seat back and wriggled until I got my legs out. They still wouldn't move, though, which may have had something to do with the broken bones sticking through holes in my jeans.\n\nIt was a good thing I wasn't overly squeamish. My dream of a career in sports medicine was looking a little dim these days, but at least my summers volunteering at a clinic came in handy as I repositioned my legs. The bones slid back in with surprising ease, like they were just waiting for a nudge.\n\nThey obviously weren't going to mend in the next few minutes, though, meaning I couldn't wait to walk away from this accident. I cleared the safety glass from the window, pulled myself through . . . and hit the ground face-first, somersaulting onto my back. I lay there, getting my bearings and listening.\n\nI could still hear the couple at the top of the ridge, but I couldn't make out what they were saying until I caught the words \". . . seems to be a path down over here . . .\"\n\nI rolled over fast and pulled myself through the undergrowth. There was no way to do that quietly. Dead leaves rustled and dry brush snapped as I crept forward. Before long I heard the man shout, \"I think someone's down here!\"\n\nI dragged myself along faster, watching for the man's head to appear over the long grass. Which meant I wasn't watching where I was going. When my hand touched down on air, I tried to pull up short, but it was too late. I tumbled over a stream bank, getting a mouthful of mud and water as I splashed down.\n\n\"Did you hear that?\" the man yelled.\n\nRunning footsteps sounded. I looked around. There was no place to hide. I was trapped . . .\n\n. . . in a swollen, muddy stream at least two feet deep.\n\nI pulled myself into the deepest part of the streambed and stretched out. The ice-cold water closed over me. As the water filled my nostrils, some still-human part of my brain went crazy, telling me I was drowning. I squeezed my eyes shut and ignored it.\n\nAfter a few minutes, I sensed the couple approach. Yes, sensed. Before I turned, Marguerite tried to explain this vampire's sixth sense, and I'd compared it to sharks, who can pick up electromagnetic pulses from their prey. Now that I've experienced it, I'd say that's exactly what it is\u2014a weird pricking of the skin that tells me people are close.\n\nWhen I concentrated, I could pick up the couple's voices, muffled and faint.\n\n\". . . car's empty.\"\n\n\"No one could have walked away from that.\"\n\n\"Well, there isn't any blood. Maybe the driver was thrown clear.\"\n\n\"We'll backtrack. Keep looking. The police should be here any moment.\"\n\nI waited until I couldn't sense them anymore. Then I lifted my head slowly. I could hear them back by the embankment.\n\nI wiggled my legs. They were moving now. Good.\n\nI tried pushing up. My legs gave way and I splashed back into the stream. I hunkered down, but the couple must not have heard. I rose again, not putting too much weight on my legs, just using my knees to get some traction, pushing myself up the streambed and into the long grass.\n\nWhen I was far enough away, I took out my cell phone.\n\nIt was off. And it wouldn't turn on.\n\nAs I shook it, a shadow passed over me. I looked up, and caught only a blur. Then hands grabbed my shoulders and pinned me down. Something ice-cold pressed against my neck. I writhed and twisted, struggling to get free, but the world tilted and spun and then . . .\n\nWhen I came to, there was still a guy bending over me. Instinctively I jerked up and slammed him with a line drive to the chin. He flew back with a yelp. I jumped to my feet. Still a little wobbly, but at least I could stand.\n\nI glanced around quickly. The forest was gone, and I was in a room with wooden plank walls, like a cabin. I blinked hard, woozy from the sedative, my brain not kicking into gear yet.\n\nThe guy I'd hit glowered up at me as he rubbed his chin. He looked about my age. Broad shoulders. Football player build. Dark hair. Blue eyes, which were looking more pissed off by the second. When I stepped forward, he leaped to his feet, fists flying up, boxer stance. I took a step. He swung. I grabbed his wrist and threw him over my shoulder.\n\n\"Could I get a little help here?\" he called as he struggled up from the floor.\n\n\"He'd like you to stop hitting him.\" Another male voice, lower pitched, with an accent I recognized from a few months in New Jersey. I looked over to see a second teenage guy sitting on a crate, book in hand. Scrawny. Glasses. Wavy light brown hair that tumbled over his forehead. He glanced up from his book, dark eyes meeting mine. \"Please.\"\n\n\"Thanks a hell of a lot,\" the other guy said.\n\nI turned. The jock was coming at me, moving slow, cautious.\n\n\"Look,\" he said. \"Whatever you think\u2014\"\n\nAnother step brought him into personal space range. Another wrist grab landed him on the floor.\n\nHe glared at Glasses. \"Would it kill you to get involved?\"\n\nGlasses gave me a once-over. \"Maybe.\" He closed the book but made no move to stand. \"Clearly she thinks we're the ones who brought her here, which would make sense, coming to with you crouched over her. First, I would point out, though, that we're a little young to be in the market for a backwoods bride. Second, had we been the ones taking her captive, I hope we'd have had the foresight to tie her up before waking her. Third, if she checks for exits, she'll discover we're as trapped as she is.\"\n\nI looked around. It was a single room with only blankets and crates on the floor. No windows. One door. I walked over to it and yanked. It was bolted\u2014from the outside. I could sense at least one person guarding the door.\n\nI turned back to the guys. The one with the book stood.\n\n\"Neil Walsh,\" he said. \"That's Chad. We hadn't gotten to surnames yet. I take it you're a vampire?\"\n\nI stared for a second, then choked a laugh. \"Excuse me?\"\n\n\"Vampire. By blood, at least. If not, then you're in the wrong place. This party, apparently, is only for hereditary vampires. Genetically created hereditary vampires. Subjects of an experiment. Escaped subjects, I might add.\"\n\nIf my heart still beat, it would have been racing. \"I\u2014I don't know what you're talking about.\"\n\n\"Cut the crap,\" Chad said. \"You'll\u2014\"\n\nNeil lifted his hand, stopping him. \"It might not be crap. We knew what we were, but she might not.\" He looked at me. \"If that's the case, then ignore everything I just said.\"\n\n\"Oh, that'll work.\" Chad took another step toward me. \"I'm sorry if this is news to you, but as crazy as it sounds, it's the truth. You were part of an experiment. Someone\u2014maybe your parents, like mine and Neil's\u2014took you away from it. The guys who kidnapped us are bounty hunters. My guess is that our families trusted someone they shouldn't have, someone who could be bought. These bounty hunters want to take us back to the scientists. The Edison Group.\"\n\nI struggled to keep my expression neutral, but there must have been a glimmer of fear in my eyes when Chad said that name, because behind him, Neil nodded. Chad only kept looking at me, waiting for a reaction.\n\n\"Okay . . . ,\" I said finally. \"So . . . vampires . . .\"\n\n\"Not real vampires,\" Chad said. \"Obviously you aren't out there sucking blood and hiding in the daylight.\"\n\n\"Real vampires aren't allergic to the sun,\" Neil said. \"The book says\u2014\"\n\n\"Screw the book. My point is that we aren't vampires. Not yet. Not for a very long time, I hope.\"\n\nI held myself still, hoping to give away nothing.\n\n\"Okay, that's a lot to take in,\" Chad said. \"And you probably think we escaped from the loony bin, but the main thing is that we're trapped and there are others out there, like us, in danger. Two more kids who escaped. We need to get out, find them, and warn them.\"\n\nI nodded. I didn't dare do anything else.\n\n\"You're hurt.\"\n\nNeil was looked at my legs. I glanced down. My jeans had huge holes where the bones had poked through.\n\nI sat quickly and made a show of examining the skin underneath. \"Just ripped. Probably when I was running through the woods. That's how they caught me\u2014ran me off the road.\"\n\nWhen I looked up, I almost bumped heads with Neil. He was bent over, looking at my legs through the holes. The flesh was still pitted, the skin rough, like scar tissue.\n\n\"Old accident,\" I said.\n\nHe looked at my other leg, with the same circular scar right under the tear. Then he looked at me. I kept my face impassive, but I could tell he knew, and I felt a weird tingling in my chest, like my heart was trying to pound.\n\nNeil nodded and straightened. \"As long as you're okay.\"\n\n\"She's not okay,\" Chad said. \"She's a prisoner about to be handed over to mad scientists. Same as us. Same as those other kids.\" He looked at me. \"Do you have any idea who they are?\"\n\nI shook my head.\n\nHe continued. \"Maybe you didn't know about the experiment, but you must have heard something. You've been on the run, right? Like us? Did your parents talk about other kids? Maybe you visited them and your folks said they were old friends?\"\n\n\"No, I'm sorry.\"\n\nHe exhaled, cheeks puffing. \"Okay, well, think about that while we work on an escape plan. But don't ask him for ideas.\" A dismissive nod toward Neil, who'd already retreated to his crate. \"He isn't interested in escaping.\"\n\n\"Of course I am,\" Neil said. \"As I see it, though, we're temporarily without options. No windows. One door, locked. Presumably the men who brought us here are right outside it.\"\n\n\"They are,\" I said before I could stop myself. \"I mean, I heard someone out there. Or I thought I did.\"\n\nAgain, Chad bought it. Again, Neil didn't, studying me with that inscrutable look.\n\n\"Go back to your vampire book,\" Chad said. \"We'll wake you up when we're ready to go.\"\n\n\"Vampire book?\" I said.\n\n\"It's the journal of a vampire,\" Neil said. \"My parents told me what I am only last year. They're hereditary vampires themselves but don't know very much about the condition. Entire families carry the gene, but for most members, like my parents, it's recessive.\"\n\n\"Fascinating,\" Chad said, yawning.\n\nNeil's gaze settled on me, like he was waiting for permission to continue. I nodded, and he did.\n\n\"By recessive, I mean that, on death, they won't return as vampires. They may, however, pass that ability on to their children. With both carrying the gene, my parents were concerned it might increase the chance of their child being a true vampire. They were directed to the Edison Group, who promised that with some genetic modification, they could ensure that didn't happen. They lied.\"\n\n\"They made sure it would happen,\" I said. \"And they made other changes as well.\"\n\n\"Presumably. My parents left the experiment once they discovered the truth. They did not, however, stay around long enough to learn exactly what was in store for me if . . .\" He paused, head tilting. \"When I become a vampire. Once the bounty hunters discovered how little I knew, they gave me this.\" He lifted the old book.\n\n\"Thoughtful of them.\"\n\nA twist of a smile. \"They're trying to scare me. Show me what horrible future is waiting for me while promising that, despite what my parents told me, the Edison Group isn't really evil. They can help.\"\n\n\"You don't seem very scared to me.\"\n\nHe shrugged. \"Knowledge is power. I want to know exactly what's in store. And, if I'm lucky, there may be something in here that'll help us. Some ability they aren't expecting.\"\n\n\"Well, you go ahead and read that,\" Chad said. \"In the meantime, I'll actually try\u2014\"\n\nHe stopped and stepped toward me.\n\n\"There's a huge sliver sticking out of your shoulder,\" Chad said. \"Didn't bleed though. Weird.\"\n\nDamn! It must have been left over from the branch. I should have checked better. As I twisted out of Chad's way, Neil quickly slipped behind me and said, \"It's the angle of entry. It just didn't hit any veins. Here, I'll take it out for you.\"\n\nI hesitated, then nodded.\n\n\"Are we allowed to ask your name?\" he asked as he steered me out of Chad's line of vision and wriggled the sliver free.\n\n\"Katiana,\" I said. \"But everyone calls me Kat.\"\n\n\"Katiana. Hmm. Russian?\"\n\nI said it was. I had no idea and knew he wasn't really interested in the answer, was just talking to distract Chad, which I appreciated.\n\n\"Thanks,\" I said, when he pulled it out.\n\nHe nodded. He had the sliver cupped in his hand and was tucking it into his pocket. When fingers touched the back of my shoulder again, I spun.\n\n\"Hey!\" I said.\n\nChad stepped back, staring at his clean fingertips. \"There's no blood.\" He looked up, gaze hardening. \"There's no blood.\"\n\nHe grabbed my arm so fast I didn't see it coming. Neil tried to stop him, but Chad wrenched me off balance. His fingers went to the side of my neck. Before I could yank away, he shoved me aside.\n\n\"She's a vampire,\" he said, staring at me like I'd just crawled out of a crypt.\n\n\"No kidding,\" Neil said. \"That's why she's here.\"\n\n\"You know what I mean. She's a real vampire. Turned. Dead.\"\n\n\"As we'll be one day,\" Neil said. \"And if you're wondering why she didn't tell us, your reaction answers that.\"\n\n\"How the hell can you be so calm? She's a vampire.\"\n\n\"At the risk of repeating myself, so are you. She's just a little further along in the process.\" He glanced at me. \"It didn't just happen, did it? In the car accident?\"\n\nI shook my head. \"It was about six months ago. The Edison Group caught up with us. Shot me, apparently figuring they were safe either way. Either I'd be reborn as a vampire and prove their experiment succeeded, or I'd die and they'd have one fewer escapee to worry about. They didn't get their answer, though. They thought I was accidentally cremated, so they stopped looking for me.\"\n\n\"If the bounty hunters are still gathering escaped subjects, they may not have notified the Edison Group. Which means these men may not have realized you've transformed already. We should keep it that way. It'll be an advantage\u2014\"\n\nThe dead bolt clicked. The door opened an inch, a gun barrel poking through. A man said something. I didn't hear it. All I could do was stare at that gun, remember the last time I'd seen one, remember the flare, the bullet hitting my chest . . .\n\nNeil's fingers wrapped around my elbow. \"Do as they say,\" he whispered.\n\nChad was at the far wall, facing it, hands up. We did the same.\n\n\"Spread out more,\" the man said. \"Hands behind your backs. Make one move and we'll test whether you really can come back from the dead.\"\n\nA chuckle from a second guy.\n\nI put my hands behind my back. They bound us, then ordered us out. I caught a glimpse of one captor. There wasn't much to see\u2014just a guy in a Halloween mask. Dracula. I suppose they thought that was funny.\n\nThey took us out to a cube van. The back doors were open, the interior empty except for bottled water and old blankets. Without a word, they pushed us inside and slammed the door.\n\nThere was only one window\u2014a grimy square on the van's back door. It let in just enough light for us to be able to see one another. Not that anyone seemed very sociable right now. Chad sat in a corner, knees up. Neil was on the other side, back to the wall, staring into nothing, deep in thought. Neither had said a word to me since we got in.\n\nI felt the weight of that silence. From Chad, I expected it. Neil had seemed different, but I guess it had just been logic talking back there. He knew he should be okay with me because someday he'd be a vampire too. Now, in the silence, emotion took over and he wanted nothing to do with me.\n\n\"I'm sorry.\"\n\nChad's voice at my ear made me jump. I looked over to see him beside me.\n\n\"I was a jerk back there,\" he said. \"I'm sorry. I just . . . It caught me off guard.\"\n\n\"It's okay.\"\n\n\"It's not, but thanks.\"\n\nHe smiled then, a slow grin that, six months ago, would have had my heart skipping. Now all I could think was how good he smelled. Like dinner.\n\nI looked away.\n\n\"There,\" Neil said, making us both jump.\n\nNeil lifted his hands. The rope fell free. He frowned at the abrasions on his wrists. Blood seeped from one. I could smell it.\n\n\"Good,\" Chad said grudgingly. \"Now, can you get ours off too?\"\n\n\"That was the plan.\"\n\nAs he freed us, I was careful not to inhale. Of course, I still thought about the blood smeared on his wrist. I couldn't help it. That gave me an idea, though. For escaping.\n\nI told the guys. Chad jumped on it. Neil didn't. We talked about alternatives, and he couldn't see one, though, so he agreed.\n\nThe van screeched around a corner, bumping along a dirt road as Chad banged the walls and shouted for help. It jerked to a stop. The passenger door opened and slammed. Chad went still, lying on the floor of the van, me crouched over him, mouth hovering over his neck.\n\nI could hear the blood pounding through Chad's veins. Heard it, felt it, saw it, the pulse on his neck beating hard, blood rushing so close to the surface I could smell it. My fangs extended. I pulled back, shuddering, instinctively closing my eyes to focus on retracting them, then stopped. This was the idea, right? Fangs and all.\n\nSo I crouched there, over Chad's neck, fangs pressing into my lower lip, and I tried to look at him, to see him like I would have six months ago, to notice the way his dark lashes curled on his cheek, the sexy hint of beard stubble, the full lips. . . . But all I saw was his blood, pulsing behind his skin, so close I could taste it. God, I swore I could taste it.\n\nA movement to my left. I glanced over at Neil, lying on his side, blood from his wrists smeared on his neck. He was watching me. No expression. Just watching.\n\nI glowered at him and hissed, \"Eyes closed!\"\n\nHe shut them just as one of our captors cleared the filthy window and peered through to see me crouched over Chad.\n\n\"Hey!\" the bounty hunter shouted. \"Ron!\"\n\nHe threw open the back door and I leaped up, fangs out, snarling. The guy froze, eyes wide, gun still down, like he'd forgotten he had it. I lunged at him. He dropped the gun and fell back, hands flying up to protect his throat.\n\nI sprang on him, pushing him backward. A shout from the driver as he ran around to help. Chad jumped from the van and knocked him down. Neil hopped out behind them.\n\nI pinned my prey to the ground. Prey. That's all he was at that moment. I didn't think about what I should do next. I pinned him and I bit him.\n\nMy teeth sank in like needles through silk. Hot blood filled my mouth. And the taste. Oh God, the taste. It was unbelievable.\n\nIf he struggled, I didn't notice it. Didn't notice anything until that first mouthful slid down my throat, then the blood-fog cleared and I heard Chad fighting the other guy. My target was out cold. The sedative in my vampire saliva had done its job.\n\nI lifted my head. That took effort. Serious effort, like wrenching myself out of the sun on a bitter cold day. I closed my eyes and ran my tongue over my fangs. They retracted. I didn't straighten, though. Couldn't. Just stared at the blood trickling down the man's neck.\n\n\"You need to seal it,\" said a soft voice beside me.\n\nI glanced up. Neil stood over me.\n\n\"The book says you seal the wound by\u2014,\" he began.\n\n\"I know,\" I said, sharper than I intended.\n\nI shifted so he couldn't see, bent, and ran my tongue over the puncture wounds. The holes closed. The bleeding stopped. I could still taste the blood, though, so delicious it made the back of my throat ache.\n\n\"Katiana?\" That same soft voice. Careful, like he didn't want to disturb me.\n\nI straightened, grunting, \"I'm good.\"\n\nWith my back still to him, I swallowed. Ran a hand over my face. Squared my shoulders. Turned around.\n\nChad knelt beside the unconscious body of the other man.\n\n\"Good,\" I said. \"We need to\u2014\"\n\nNeil held out the ropes that had bound us earlier.\n\n\"Okay,\" I said. \"Let's do that.\"\n\nSoon our captors were bound and unconscious. Now that I could get a look at them without masks, I knew I'd never seen them before. Just two guys in their twenties, both dark-haired and broad-shouldered. There was a definite resemblance between the two. Brothers or cousins, I was sure.\n\nThey were incapacitated, though, and we had their van and keys. So our next move should have been obvious. It would have been, if one of us knew how to drive stick shift. We tried, but no one had been driving more than a few months. We just didn't have the skills to manage it.\n\nNeither of our kidnappers had a cell phone. One had a radio, but that meant they weren't working alone, and we sure weren't going to let their partners know that we'd escaped.\n\nThere was only one option. Walk.\n\nFirst, Neil went back and grabbed the gun. There was only the one, and when Neil brought it over, Chad put out his hand.\n\n\"Can you shoot?\" Neil said.\n\n\"Better than you.\"\n\nNeil raised the gun and put a bullet in each of the van's front tires. Chad scowled, walked away, and waved for us to follow.\n\nThe van had pulled off onto what looked like an old logging road. We'd been on a paved one before that. We found that quickly. It was ten minutes, though, before we heard a car. Even then we couldn't see it. We were on a thickly wooded road that bounced through the hills of Vermont. Or I presumed it was still Vermont.\n\nWhen we heard the car, Neil suggested we get off the road, so it didn't barrel over the hill and send us flying like bowling pins. We stayed at the side, ready to wave it down.\n\nAs it drew closer, Chad said, \"Maybe this isn't such a good idea.\"\n\nI glanced at him.\n\n\"Well, those guys had a radio, right? That means they're working with others. Maybe they were supposed to hook up. Or maybe those guys have escaped already, and called for help. Even if it is someone else, do you think they're going to stop for us? Or will they call the cops?\" When neither of us said anything, he shrugged. \"Okay, I'm just putting it out there.\"\n\n\"What do you suggest?\" Neil said.\n\n\"My folks are back in Pennsylvania. Yours are in New Jersey. But Kat's are close by, right?\"\n\n\"My guardian is.\" I'd been trying not to think about Marguerite, how worried she'd be. I knew she'd be looking for me\u2014I only hoped she was safe.\n\n\"Then I say we get off this road and keep walking until we come to a town and can call Kat's guardian.\"\n\nWe agreed and moved into the woods before the car reached us. It was a mom with a couple of kids in car seats. Not a potential captor, but probably not someone who'd stop, either.\n\n\"So you and your guardian . . . ,\" Chad said. \"You were traveling somewhere? That's what I overheard them saying before they grabbed Neil. They sent someone up to Quebec for you, but then their contact said you'd headed this way. They found and followed you.\"\n\nNeil added, \"And presumably kept following you until the van could intercept you after kidnapping me.\"\n\n\"Were you taking a trip?\" Chad asked, ignoring Neil.\n\n\"Meeting others in New York,\" I said.\n\n\"Others?\"\n\n\"Vampires,\" I said after a moment. I braced for his reaction, but he seemed interested now, like he'd gotten over that initial knee-jerk response.\n\n\"And your guardian knows them from the experiment? Maybe the other subjects who escaped will be there.\" He grinned. \"That'd make things easy.\"\n\nI shook my head. \"We didn't even know that others had left the experiment. I was taken out when I was five.\"\n\n\"But this guardian of yours, she was in on it, right?\"\n\n\"No. She's . . . she's a vampire. There was this group of supernaturals who were concerned about what the Edison Group was doing. They were secretly monitoring the experiments. She was assigned to me. When she saw how I was being treated, she took me.\"\n\n\"Abducted you?\"\n\n\"It wasn't like that.\" My voice carried a bit of snap. Time to change the subject. I turned to Neil, who'd been walking silently beside me. \"So where'd you learn to shoot?\"\n\n\"A co-op placement at our local police station. They threw in sessions on the firing range as an incentive. I can point and shoot, but that's about it.\"\n\n\"More than I can do,\" I said. \"Very cool. So\u2014\"\n\n\"Co-op with the cops?\" Chad cut in. \"What were you doing there? Fixing their computers?\"\n\n\"Don't be a jerk,\" I said.\n\n\"I'm not. It's a serious question. Bet I'm right, too. You gotta admit, he's the type.\"\n\n\"And what type would that be?\" Neil said. \"The type who can spell computer?\"\n\n\"Okay, not cool, guys,\" I said, lifting my hands. \"You two have fun insulting each other. I'll be back here.\"\n\nI slowed to let them get ahead. They kept walking, shifting farther apart. Neil glanced back, like he was thinking about coming back with me, then settled for falling behind Chad and forming a single line. No one spoke for about five minutes. Then Neil cleared his throat.\n\n\"I think we should split up,\" he said. \"We have no idea if the nearest town is twenty miles this way. Or five miles back the way we came. Or one mile up that road we just crossed.\"\n\n\"I don't think\u2014\" I began.\n\nChad cut me short. \"You've got a point.\" He stopped and looked around. \"Kat can keep going this way. I'll head back. You can take the side road.\"\n\nI shook my head. \"And what do we do when one of us finds a town? We don't have any way to keep in contact.\"\n\nA valid argument. Neither guy listened, so I had them memorize Marguerite's cell number and walked away.\n\nAs I trudged through the forest, I cursed Chad and Neil. Was it just me or was this the stupidest idea ever?\n\nAs pissed off as I was, though, I couldn't help wondering if this separation was my fault. Maybe I should have kept my mouth shut when they were sniping at each other. Of course, that would have required industrial-strength duct tape. We'd just escaped bounty hunters. We were running\u2014well, walking\u2014for our lives. And they thought slinging insults was a useful way to pass the time?\n\nNo, I couldn't have kept quiet. If that made them decide to split up, then it was a seriously lame excuse.\n\nMaybe that's what it had been. An excuse. Not to get away from each other, but from me. Put some distance between themselves and the bloodsucker before she gets hungry.\n\nIt didn't matter. I'd get to a town and I'd call Marguerite, and if the guys were worried about hanging out with vampires, they could call a ride of their own. I'd never see them again. Which was fine. Not like they were my new best buddies or anything.\n\nIt had been nice, though, finding other kids from the same experiment. Other vampires. Only they weren't vampires. Not really. But I guess, in a way, I'd liked the idea of meeting someone who kind of knew what I was going through, who\u2014\n\nI sensed someone close by. Really close by. I wheeled as Neil jogged through the trees. He held up his hands, the gun still tucked in his waistband.\n\n\"It's just me,\" he said.\n\n\"Did you find something?\"\n\n\"No.\" He waved for me to follow. \"Come on. We need to get in deeper before they get here.\"\n\n\"They're coming?\" I said as I followed him. \"Did you tell Chad? We need to\u2014\"\n\n\"We need to stay as far away from Chad as possible, considering he's the one who called them.\"\n\nI stopped. \"What?\"\n\nHe reached for my elbow and tugged me into the forest. \"He's a plant. I suspected it from the start, but I'm sure now. He's gone to call them. That's why he wanted to split up.\"\n\nI jerked out of his grasp. \"No, you wanted to split up. It was your idea.\"\n\n\"My thoughts exactly,\" said a voice beside us.\n\nChad lunged from the bushes and charged Neil. He grabbed for the gun, but only managed to hit Neil's arm. The gun went flying. I dove for it. We all did. I was faster, though, and snatched it up, then backed away, gun wavering between the two. They froze.\n\nI looked down at the gun in my hands, and again, I remembered that fatal shot. But this time the memory passed with only a spark of emotion.\n\n\"Who suggested splitting up?\" Chad said after a moment. \"If there's a plant here, it's obviously him.\"\n\n\"I suggested it to smoke you out,\" Neil said. \"Splitting up was a stupid idea. Katiana knew that. But you were all for it . . . because it gave you the excuse to call the bounty hunters.\"\n\n\"Call with what?\" Chad lifted his arms and turned. \"Pat me down, Kat. I don't have a phone.\"\n\n\"Because you hid it as soon as you overheard me. Katiana, you know he's not a vampire. Look at how he reacted to you. He showed no interest in the book. He's shown no interest in what your life is like or what you're going through. That's not the reaction of someone who expects to become a vampire.\"\n\n\"Maybe because I'm scared, okay?\" Chad said. \"Can I admit that? Or do I have to be all logical about it like you? To me that proves you aren't one. You're overcompensating, making sure we know you're okay with it.\"\n\n\"He's a plant, Katiana. He was the first one picked up\u2014\"\n\n\"Which would be a dumb idea if I was in on it. The smarter move would be to grab me second, to throw off suspicion. And who says there's a plant at all? Where did this idea come from? What possible reason would the bounty hunters have\u2014\"\n\n\"First, as a precaution against exactly this scenario\u2014we escape. If one of them is with us, they can make sure we don't get very far. Who's the one who didn't want us flagging down a passing car?\"\n\n\"But I didn't suggest splitting\u2014\"\n\n\"Second, they don't know where the other subjects are. They assume we do. You've been very curious about those other subjects, Chad. We gotta find them. Gotta find them. And, by the way, do we happen to know where they are?\"\n\n\"Enough,\" I said. \"Neil has convinced me . . . that there is a plant. Makes sense. The question is, who?\" I stepped forward, gun pointed at Chad. \"One way to find out if you're a vampire, isn't there?\"\n\n\"Whoa!\" Chad backpedaled. \"Vampire or not, I wouldn't want that. Come on. Obviously, it's him. He's the one who wanted to split up.\"\n\nI turned the gun on Neil. He paled. Sweat trickled down his temple.\n\n\"All right,\" he said. \"I'd really rather not, but if that's what it takes, go ahead. I'd only ask that you let me turn around and aim for the base of my skull. It's the quickest way to kill someone.\"\n\n\"What the hell kind of freak knows that?\" Chad said. \"Sure, let him turn around . . . so he can run away as fast as his scrawny legs will take him.\"\n\nNeil turned. I could see the side of his neck throbbing as his heart raced. He didn't even shake, though. Just stood there, waiting. That took guts. Incredible guts.\n\nI swung the gun back on Chad. He dove at me. I could have shot him. But I wouldn't, not while I had any other option. So when he came at me, I dropped the gun, grabbed him by the wrist, and threw him.\n\nBefore I could pin him, he flipped over, a hard elbow to the jaw sending me flying off my feet. It took me a second to recover. As I did, I heard a grunt and a thump behind me, and when I turned, Chad had the gun\u2014and Neil, holding him as a shield, one arm around his neck, gun barrel pointed at the side of his skull. Neil's glasses were gone, lost in the scuffle.\n\nNeil said, \"Considering I just agreed to be shot, this really isn't the position of advantage.\"\n\n\"Shut up, freak.\"\n\n\"Keep calling me that and I might take offense.\"\n\nNeil's voice was steady, jaunty even, but sweat still slid down his face, and I could see that pulse in his throat.\n\n\"Let him go,\" I said.\n\n\"Or what? You'll bite me? Feed on me?\" Chad's lip curled in undisguised disgust, and that answered my question better than any test I could have given him.\n\n\"You aren't a vampire,\" I said.\n\n\"No, thank God.\"\n\n\"But you are part of the experiment, I'll bet,\" Neil said. \"You're the right age, and that's the most obvious way you'd know about it. What are you?\"\n\n\"Half-demon.\"\n\n\"I'm sorry.\"\n\nChad's arm tightened around Neil's neck. \"You think I'd want to be a bloodsucker? Goddamned parasites should have been wiped out centuries ago.\"\n\n\"I didn't mean that. I was expressing my condolences on your status as an experimental failure. Your lack of powers.\"\n\nChad's eyes blazed. \"I have powers, smart-ass. You want to see them?\"\n\nHe closed his eyes, face going rigid as he concentrated. I charged. I smacked the gun from his hand and knocked Neil free. The gun sailed into the bushes. Chad and I hit the ground. He grabbed me by the shoulders. I felt his hands through my shirt, felt them heat and smelled scorched fabric. But that was it. As powers went? Kind of sad.\n\nChad threw me off. When he came at me, I kicked, and sent him flying, then leaped to my feet. We circled each other. He glanced over at Neil, who hadn't moved.\n\n\"Letting a girl fight your battles?\" Chad said with a sneer.\n\n\"She seems to have it under control.\"\n\n\"Coward.\"\n\nNeil shrugged.\n\nChad threw a punch at me. I caught his arm and flipped him. He leaped up and charged. I kicked and knocked him into a tree. He staggered up, shaking his head like he was dazed, then rushed me. I spun out of the way, but he caught me by the arm and yanked me off my feet. Then it was my turn to meet the tree.\n\n\"Need help?\" Neil called as I recovered.\n\n\"Nope,\" I said.\n\nChad and I went a few more rounds. I had the advantage of skill\u2014I'm a second-degree black belt in aikido and a brown belt in karate, from self-defense training that Marguerite had insisted on. He had size, plus a generous dose of real-life brawling experience, it seemed, which I definitely lacked.\n\nThe advantage swung slightly my way, but not enough to make it a quick or easy fight. We'd been going at it for about five minutes\u2014which feels like fifty when you're actually fighting\u2014when Chad threw me down hard. I lay there, winded.\n\nAt a grunt behind me, I jumped up, thinking he'd gone after Neil, and found Chad face-first on the ground. Neil had one knee on Chad's spine, Chad's arm wrenched behind his back.\n\nChad tried to buck Neil off. Neil twisted his arm until Chad was the one with sweat streaming down his face. He didn't take it as stoically as Neil had, though. He snarled and gasped as Neil kept inching Chad's arm up. Finally Chad stopped struggling.\n\n\"I don't suppose you'd care to tell us anything helpful,\" Neil said.\n\nChad spit out a string of curses.\n\nNeil glanced at me. \"Does that sound like a no?\"\n\n\"Sure does.\"\n\n\"Do you think we're likely to get anything from him?\"\n\n\"Nothing useful,\" I said. \"I think we can guess the story. The guys who captured us are relatives. They kind of looked like him. Same build. Same coloring. They know about the experiments because he's a subject. They got a lead on us, probably, like he said, from someone your parents and my guardian have stayed in touch with. Rather than turn that information over to the Edison Group, they figured they could make some money rounding us up. Only they're missing a couple of names, so they recruited Junior here to play captive in hopes of getting those names from us.\" I bent beside Chad. \"Am I close?\"\n\n\"Go to hell.\"\n\nI turned to Neil. \"We can ask whether he's contacted his co-conspirators yet, but either way he'll say he has, just to freak us out, hope we'll take off and let him go.\"\n\n\"You're right. There's nothing we can get from him.\" Neil backed up a few inches, making Chad wince. \"I can knock him out, but your method seems safer.\"\n\nChad struggled again then. It didn't do him any good. A little more pressure on the arm and Chad was screaming. A quick bite on the neck and he stopped screaming.\n\nThis time, though, I forced myself to stop as soon as blood filled my mouth.\n\n\"You're hungry,\" Neil said as I rose from Chad's unconscious body. \"You should drink more. If the book is right, he'll only wake up weak, as if he donated blood. It's not like he doesn't deserve it. And there's no sense turning down a free meal.\"\n\nHow many times had I given Marguerite crap for ignoring an opportunity to eat? Or told her off for trying to hide her feeding from me? Rolled my eyes and said she was being stupid about it. She had to eat and I understood that. Just consider it a blood donation to save a life\u2014hers.\n\nEasy to say when you're on the other side. But crouching here beside an unconscious kid, even a jerk like Chad, while having another guy watching you, a guy who wasn't a jerk, but maybe someone you'd like to impress . . .\n\n\"I'm good,\" I said, rising.\n\n\"Katiana . . .\"\n\n\"I don't feed like that. I get my meals from the blood bank.\"\n\nHe frowned. \"The journal says vampires need fresh\u2014\"\n\n\"My guardian doesn't want me to hunt yet.\"\n\n\"Okay, but you didn't hunt him, so . . . I mean, if you don't want to, sure. It just seemed, back there, like you did want\u2014\" He flushed. \"Sorry. I'll shut up now.\"\n\nHe knelt beside Chad and started patting him down for a phone. Neil had a point. It wasn't like Chad didn't deserve to wake up feeling like hell. This was what I'd wanted, right? Not just drink a single mouthful, like I had with the other guy, but to take a full meal straight from the source, see if it made any difference. See if it dulled the edge.\n\nBut I couldn't do it. I don't know if it was the thought of feeding off someone I knew or of doing it in front of Neil. I wanted to\u2014oh God, I wanted to\u2014but I couldn't.\n\nWhen I glanced over at Neil, he was holding his glasses. It looked like they'd been a casualty of my bout with Chad, stomped underfoot.\n\n\"I knew I should have worn my contacts yesterday,\" he said, trying for a smile.\n\nHe ran his hand through his hair, shoving it back from his face as he squinted and blinked. Without glasses, Neil wasn't magically transformed into a sex god. He just looked a little less like a guy who should be planted behind a computer monitor and a little more like one who could manage a first-rate throw-down and pin. Definitely cute though\u2014with or without the glasses.\n\nI shook the thought off. Really not the time for that.\n\n\"Can you see?\" I asked.\n\n\"Well enough.\" He tossed the glasses aside, retrieved the gun, and held it out. \"You may want to be in charge of this, though.\"\n\n\"Can you still shoot?\"\n\nHe shrugged. \"Sure, but\u2014\"\n\n\"Can you avoid shooting figures with brown hair, blue jeans, denim jackets, and\"\u2014I looked down at my feet\u2014\"dirty white sneakers?\"\n\nA genuine smile. \"I can.\"\n\n\"Then keep the gun. I have no idea how to use one. Now we should see if Chad did ditch a cell phone. It's a long shot, but we need to head back to the road anyway. Maybe we can find his trail.\"\n\nWe did. It wasn't hard. Chad was no backwoodsman, and we didn't need to be trackers to find his trampled path through the undergrowth. I followed a detour where he'd walked deeper into the brush, then returned, and at the end of that path, under the bushes, I found a phone.\n\n\"Do you think there's anything I should know about using it?\" I asked Neil. \"Maybe a GPS that needs to be disabled?\"\n\n\"My tech skills are limited to being able to turn things on and operate them. In other words, zero on the geek scale.\"\n\nI glanced over at him. \"I wasn't assuming. I was just asking.\"\n\n\"Did I sound defensive?\"\n\n\"A little.\"\n\nA rueful smile. \"Sorry.\"\n\nI turned the phone on and waited to see if anything would happen. Then I checked for outgoing calls. One had been made twenty minutes ago. Damn.\n\nBefore I called Marguerite, I needed some idea of where we were. Neil thought he'd seen a sign on the side road he'd been supposed to take. We found a path heading in roughly the direction we needed to go. Once we found the sign, which promised a town two miles away, I dialed Marguerite's cell number. On the second ring she answered with a wary, \"Allo?\"\n\n\"Miss me yet?\" I said.\n\n\"Katiana! Where are you? What happened to you? Are you all right? Are you hurt? They called, the police, about the car. The accident. I said the car was stolen, but I have been looking everywhere, calling everyone\u2014\"\n\n\"I'm fine. Just kidnapped by bounty hunters rounding up missing vamps from the experiment.\"\n\nA pause, then, \"That is not funny, Kat.\"\n\n\"You think I'm kidding? I wish. I'm with another guy they grabbed. Neil . . .\" I tried to remember his last name. \"Walsh. Neil Walsh.\"\n\n\"Actually, it's Waller,\" Neil said. \"Walsh is the name my parents have been using since they left the experiment.\"\n\nMarguerite overheard and said, yes, she remembered Neil. She warned us not to call his parents on the phone we'd found. If it was owned by Chad, our captors could check their billing and which numbers we'd called. I hadn't thought of that. Neil agreed. We'd get to the town and lie low until she could pick us up. Then Neil could notify his parents from a pay phone.\n\nAs we walked, Neil fussed with the gun, taking a better look at it and checking the ammunition, saying, \"If those guys find us, we might actually need to use it.\"\n\n\"I'm sorry about earlier,\" I said. \"With Chad. I wouldn't have shot you.\"\n\n\"You needed to see how we'd react. While I'm not eager to be turned, I'd rather do that than go back to the Edison Group. My parents told me . . . things.\" He let the word drop, heavy, and stared at the gun, lost in thought. Then he turned it over in his hands. \"It appears to be police issue.\"\n\n\"Is that a problem?\"\n\n\"Only that it may mean we're dealing with someone who knows how to fire a gun significantly better than I do.\"\n\n\"We'll be fine. You've got some killer aikido skills to fall back on. Speaking of which . . . That might be a popular choice with cops, but there's no way you picked that up in a co-op term. What level are you?\"\n\n\"In the black belts.\"\n\nI had to press him for more than that, and after some waffling about nonstandard terminology, he admitted he was fourth-degree.\n\n\"Seriously? I just made third. Damn.\"\n\n\"Sorry.\"\n\nI laughed. \"Is that why you didn't want to tell me? You think I'd be pissed because you're a higher level? It just gives me something to strive for. Can't have a guy beating me.\"\n\nI grinned, and when I did, he gave me this look, like . . . I don't know. He just stared at me. Then he glanced away fast, cheeks flushing.\n\n\"Any other martial arts?\" I asked as we walked.\n\n\"Just that. I'm not much of an athlete, but I like lunch.\"\n\n\"Huh?\"\n\n\"In fifth grade we moved to a new city and there was this kid, a head taller than me. He decided my lunch money was a good way to supplement his income.\"\n\n\"And you needed a way to keep it.\"\n\n\"Yes, but I prefer using my head to my fists, so I thought I could outwit him by brown-bagging it. He took that. I switched to health food. He'd still take it . . . and throw it in the trash. So, I could either humiliate myself by digging through the garbage every day or learn a form of self-defense. I did my research. Aikido seemed a good choice for what I wanted and, as you said, it's popular with law enforcement, which is a bonus.\"\n\n\"That's what you want to be? A cop?\"\n\nHe studied me, like he was trying to see if I was mocking him. That was getting annoying. When he saw that I was serious, he said, \"A detective. That's what I'm good at\u2014problem solving.\"\n\nHe asked about my meeting in New York, carefully though, like he didn't want to pry. I explained and said he should talk to his parents, see if they could come. He might not be a vampire yet, but if they weren't sure what lay in store for him, this would help.\n\n\"I'm sure they'll agree,\" he said. \"They want to help me, and I think it would be good to keep in touch.\" He paused. \"Not that I expect\u2014\" He cleared his throat. \"I understand that under the circumstances, we've been thrown together, and while I'd like to go to this meeting with you, I know it won't be with you.\"\n\n\"English translation please?\"\n\nAnother throat clearing as he pushed low branches aside. \"We got caught up in this together. We pooled our resources to get out. But once we are out . . .\" He raked his hair back again. \"I'm not one of those guys who thinks that if the popular girls ask for homework help, it means they want to hang out after school.\"\n\n\"What's that supposed to mean?\"\n\n\"I'm just saying . . .\" He trailed off, starting and stopping a few times before glancing over, dark eyes meeting mine. \"I think you know what I'm saying, Katiana.\"\n\n\"I sure as hell hope not, because it sounds like you're saying that after I've used you to escape, you expect me to walk away, pretend I don't know you.\"\n\n\"You didn't use me.\"\n\n\"Whatever.\" I turned in his path, facing him. \"You're saying you know my type, which apparently means you know me. That's rich, coming from the guy who got his back up when I asked if he knew about cell phone technology. Hell, I'm not even sure I have a type anymore, unless you've met a whole lot of teenage vampires.\"\n\n\"No, I do believe you're the first.\"\n\nHe smiled, but I wasn't buying it, and I sure as hell wasn't returning it.\n\n\"Maybe that's the problem,\" I said. \"Not that you think I'm some dumb jock who wanted your answers on the escape-from-evil quiz, but because of what else I am. Not exactly sure you want to hang out after school with a vampire.\"\n\n\"Of course not. I\u2014\"\n\n\"I make you nervous. You're trying to hide it, but you can't help it. I get that. I expect that. Just have the balls to say so instead of laying down the post-escape ground rules before we even get to the post-escape stage.\"\n\nHe opened his mouth. I wheeled and stalked off.\n\n\"Katiana,\" he called, as loudly as he dared.\n\nI kept going, walking fast, branches whipping behind me. He started coming after me, but after a minute, his footsteps stopped. I wasn't surprised.\n\nI shouldn't be too hard on him. Can't blame a guy for not wanting to get chummy with a parasite. At least he'd made the effort, which was more than I could say for Chad, and probably more than I'd be able to say for most people I'd meet in my life. Marguerite had two sets of friends: temporary ones who didn't know what she was, and other vampires. This was just the first lesson in a class I'd be taking for a very long time, so I'd better\u2014\n\nA crack sounded behind me. Not a \"branch underfoot\" crack, but one that sent my insides ramming up into my throat. I spun just as another shot fired. Something buzzed past my ear. A bullet embedded itself in the tree . . . where my head had been just a second ago.\n\nI hit the ground. Even as I dropped, I knew it was the wrong move. Bullets can't kill me. Not lead. Not iron. Not holy-water-blessed silver. Don't duck and hide. Move!\n\nI scrambled into the undergrowth as another shot whizzed past.\n\nWho\u2014? Okay, that was the stupid question of the week. Who was shooting at me? The guy with a gun and, fortunately, without his glasses.\n\nI should have known. God, I should have known. Chad was right. Neil had been too calm about the whole vampire situation. Completely cool with it . . . right up to the moment when he decided to get twitchy and send me stomping off.\n\nIf he wasn't with the bounty hunters, though, who was he? What did he want with me? Did it matter? Not when there were bullets whipping past my head. I didn't want to think of what kind of condition I'd be in, waiting for my brain to heal after getting a chunk blasted out.\n\nI crawled along the ground as quietly as I could. The shots stopped. Silence fell as he listened for me.\n\nI'd been raised for situations like this. Sad, I know, being prepared for a life that might involve bullets, bounty hunters, and unlawful confinement. But Marguerite had known what I was in for, and not preparing it for me would be as neglectful as not giving me a warm jacket for a Montreal winter.\n\nAs much self-defense training as I had, though, she'd drilled in one lesson above all others: Fighting back was a last resort. Whenever possible, run. For once, though, I had no intention of taking her advice.\n\nI'd already been duped by Chad. Blinded by a desperate need for validation, for the friendship of kids who knew what I was. So I'd missed the signs with him, and now, even worse, with Neil. I wasn't letting him get away with it.\n\nSo I circled back in the direction of the gunfire. After a few minutes, I caught a whispered voice. Neil on a cell phone? I hoped so, but when I heard a second voice, that hope fizzled.\n\nIf I faced more than just Neil, I should run. But I needed a closer look first. Needed to know what I was up against.\n\nWhen I was close enough to see figures, I found a suitable tree and climbed. I'm good at that. I used to think I'd turn out to be a werecat\u2014hence my nickname. Obviously not, but I'm still a damned fine climber.\n\nI got high enough to be safe, then shimmied along a branch until I could see three figures. Two men I didn't recognize, plus Neil. They were talking to him. I strained to listen. When I couldn't hear anything, I inched out a little more. Then a little more.\n\nThe branch groaned. I froze. Neil's gaze lifted. Our eyes met. His lips parted in a curse.\n\n\"What?\" one of the guys said.\n\n\"Shot,\" Neil said, quickly looking away. \"I said I think you must have shot her. She's gone for help.\" A glance up my way and he said, louder, \"Gone for help.\"\n\nHe shifted, and I realized his arms were bound behind his back.\n\nWell, that changed things. As much as I'd love to swoop in, be the hero, save the guy, I wasn't an idiot. Two armed men versus one sixteen-year-old girl? Martial arts expert or not, the odds weren't far enough in my favor to risk it. Better to get Marguerite for backup.\n\nI retreated along the branch. It creaked . . . and this time, both men heard it.\n\nA gun swung my way. I jumped to the next tree. I managed to catch a branch. Then my weight hit, and the branch gave a tremendous crack.\n\nI dropped square onto one of Neil's captors. He went down, me on top of him. Neil kicked the other guy in the back of the knees. He toppled. Another bone-crunching kick in the jaw sent him reeling back.\n\nMy opponent had fallen easily, but he wasn't staying down. We rolled on the ground\u2014him trying to grab his gun, me trying to sink in my fangs\u2014then both achieved our goal at almost the same time. His weapon of choice was faster. Mine was scarier, and when my fangs sank in, he panicked, firing wildly, the bullet zooming past my arm. It was the last shot he fired.\n\nNeil's match was giving him even more trouble. Having both hands tied behind your back can do that. I took over. It was a short fight. All I had to do was flash my fangs, dripping blood, and you'd have thought I was a thousand-pound tiger with canines big enough to rip off his arm. The guy backpedaled. Neil kicked the gun out of his hand. I leaped on him. Game over.\n\nAs I rose from sedating the second guy, Neil said softly, \"Why?\"\n\n\"Because they're assholes,\" I said as I walked over to untie him. \"Greedy assholes.\"\n\n\"That's not what I mean.\"\n\nHe gave me a look that said I knew what he meant\u2014why wouldn't I feed?\n\nI looked down at the unconscious man. Why indeed? Here it was\u2014my last chance to see if it made a difference. With Chad, I could say that I didn't like the idea of feeding on someone I'd met. Didn't like the way it smacked of revenge. Really didn't like the way it made me exactly the kind of bloodsucking monster he'd thought I was.\n\nBut why not these guys? Because I didn't want Neil to see me? In those few minutes when I thought he'd betrayed me, I'd realized how badly I'd wanted him to be okay with me, with what I was. For someone my age to say, \"I know what you are and I don't care.\"\n\nWas this how I was going to live? Ashamed of what I was? Compelled to hide the worst of it, even from someone who knew the truth? No. I was still the same person I'd always been, and if people like Neil couldn't handle the uglier parts of my new life, there was nothing I could do about it. Nothing I should do. What happened to me wasn't my fault.\n\n\"I should,\" I said.\n\n\"Yes, you should.\"\n\n\"Will you\u2014?\" I cleared my throat. \"Will you watch? I mean, not watch, but watch him, make sure he's okay. Make sure I don't . . . overdo it?\"\n\n\"Good idea.\"\n\nI crouched over the guy, then maneuvered so Neil could monitor his vital signs while not seeing me drink. Awkward. Silly, too, and as soon as I started to drink, I forgot that. All those glasses of warmed-over blood were like day-old doughnuts. This was what I craved. What I needed. It wasn't just a meal. It was . . . I don't know how to describe it. Like eating the best food imaginable, while curled up in the most comfortable chair, listening to my favorite music.\n\nI was so caught up in it that I stopped thinking about being careful. This man under me wasn't a person. Wasn't even food. He ceased to exist. I was swept away by the experience, and when that finally ebbed, and I realized what I was doing, I jumped back so fast that blood sprayed from his jugular.\n\n\"Seal\u2014!\" Neil began.\n\nI bent and licked the wound. Under my tongue, I could still feel the guy's pulse beating strong. I listened to his breathing, then lifted his eyelids as Neil said, \"It's okay, Katiana. I was watching. He's fine.\"\n\nIt felt like I'd been drinking for hours, but the guy barely even looked pale. I exhaled in relief.\n\n\"Better?\" Neil said.\n\nI nodded, then wiped my mouth and made sure my fangs had retracted.\n\nHe crouched in front of me, coming down to my level. \"What I said earlier? I didn't mean to piss you off. I was just . . .\" He rubbed the back of his neck. \"I've been that guy before\u2014the one who thinks that if a girl's being nice and asking for homework help, it means something. I got burned and I don't like getting burned, so now I cut them off at the pass.\"\n\nI lifted my gaze to his. \"Bet you missed out on a lot of girls who did want to get to know you better.\"\n\n\"Maybe.\"\n\n\"Probably.\"\n\nHis gaze dipped, cheeks flushing, and I saw the blood rushing to his face, saw his neck pulsing, heart rate picking up, and I felt the urge to lean forward. Not to bite him, though. There wasn't any of that now. I didn't see food. Didn't smell food. Didn't sense food. I saw Neil, and all I thought about was leaning forward and kissing him.\n\nI didn't. Oh, I would, when the time was right, but that wasn't now. At this moment, all that mattered was that I could look at him and see a cute guy and feel the same way I would have six months ago.\n\nWhen I smiled, he said, \"What?\" and I said, \"Nothing,\" and pushed to my feet, and before I could say anything else, a car rumbled past.\n\n\"Think that's our ride?\" I said.\n\n\"Hope so.\"\n\n\"One way to find out.\"\n\nI took off running and reached the edge of the woods just in time to see a rental car pass, a familiar blond head over the driver's seat. I put my fingers in my mouth and whistled. The brake lights flashed. Then the reverse ones came on, dust billowing as the car sped backward.\n\nMarguerite barely put it in park before she leaped out. She ran over and hugged me so tight I swore ribs cracked.\n\n\"Ack!\" I struggled to get out of her embrace. \"Good thing I don't need to breathe.\"\n\n\"Are you okay? What did they do to you? Are you hurt?\"\n\n\"I'm a vampire, Mags. I can't get hurt.\" I waved at Neil stepping from the bushes. \"But if you really want to mother someone, he'll be my stand-in today.\"\n\n\"Allo, Neil,\" she said. \"I am sure you do not remember me, but we have met, many years ago.\"\n\n\"Good,\" I said. \"Saves on the introductions. I invited Neil and his parents to come to New York with us. Hope that's okay.\"\n\n\"Of course. If he would like that.\"\n\nNeil's gaze flicked my way, then back to Marguerite. \"I'd like that.\"\n\n\"All set then,\" I said. \"You two can get caught up while I drive to the nearest pay phone.\"\n\n\"You are not driving anywhere, mon chaton. Not for a very long time.\"\n\nWhen we reached the car, I looked back at the way we'd come, toward the clearing where we'd left the two bounty hunters. Back to where I'd fed as a vampire for the first time.\n\n\"It's okay,\" Neil whispered as we climbed into the backseat.\n\nI nodded and smiled. It wasn't quite okay yet, but it would be. For the first time in six months, I was sure of it.\nIf you enjoyed Kat's story, just wait until you meet Maya and her friends in The Darkness Rising trilogy.\n\nThe story begins in THE GATHERING and continues in THE CALLING. Read on for a sneak peek at THE CALLING.\nChapter One\n\nI don't know who was more anxious\u2014Daniel or Kenjii\u2014but they weren't making this emergency helicopter evacuation any easier. I patted Kenjii and she shifted until her full hundred pounds of German shepherd rested squarely on my feet. When I tried to wriggle away, she moved closer and pinned my legs. I sighed and glanced at Daniel. He was staring out the window, fingers drumming.\n\nI twisted to look at the others. Daniel and I were in the first passenger row, behind the pilot and Mayor Tillson, who was in the copilot's seat. Behind us were the mayor's daughter, Nicole, and his niece, Sam Russo, gazing out their respective windows. Hayley Morris and Corey Carling sat in the last row. Hayley was talking; Corey wasn't listening.\n\n\"We're going north.\" Daniel had to yell to Mayor Tillson to be heard over the helicopter noise. \"We're supposed to be heading to Victoria, aren't we?\"\n\nWhen the mayor didn't respond, the pilot said, \"Change of plans, son. Victoria's backed up with evacuees. We're taking you to Vancouver.\"\n\n\"Okay, so why are we heading north?\"\n\nWe live on Vancouver Island, near Nanaimo, which is almost directly west across the Strait of Georgia from the city of Vancouver, British Columbia.\n\n\"Wind,\" the pilot said. \"Same one that's driving that fire is forcing us to circle north. Don't worry. I'll have you there in an hour.\"\n\nI looked at Daniel. His face was drawn with worry. I couldn't blame him, under the circumstances. We'd outrun a forest fire and outwitted a mysterious fake rescue team only to be whisked from town before we had time to catch our breath.\n\nI was worried, too, about a lot of things, but right now, mostly about Rafe Martinez, unconscious on the floor behind my seat. How much smoke had he inhaled? What was he going to do when he found out that his sister, Annie, was still missing?\n\nI twisted the bracelet on my wrist. A cat's-eye stone on a worn leather band. Rafe's. He'd want it back when he found out that I'd told them not to wake him up because he wouldn't leave without Annie. That would be the end of anything between us. But I could live with that. Better than I could live with myself if I'd let him die in that inferno.\n\n\"Are the other helicopters going to Vancouver, too?\" Daniel asked. \"The ones with our parents?\"\n\n\"I believe so,\" the pilot said. \"Is that right, sir?\"\n\nWhen Mayor Tillson didn't answer, the pilot glanced over. \"Sir?\"\n\nHe bent to see the mayor's face and chuckled. \"Seems someone doesn't mind the racket this bird makes. He's sound asleep. I'm sure he said the other helicopter was just a few minutes behind us.\"\n\nI leaned forward. The mayor was slumped in his seat, his face toward the window, at an angle that didn't look comfortable at all. When I undid my seat belt, the pilot glanced back.\n\n\"Whoa, none of that. This isn't a 747. Belt on at all times. Maya, isn't it?\"\n\nI scooted to the edge of my seat and touched the mayor's arm. \"Mr. Tillson?\"\n\n\"Hey,\" the pilot said, voice sharp. \"If you want me to check on your parents, just say so. Your mayor has had one hell of a day with this fire, and you kids running off didn't help. Let the man get some rest.\"\n\nSure, the mayor must be exhausted, but with everything that happened, I doubted he could relax enough to fall asleep.\n\n\"Mr. Tillson?\" I said, shaking him harder.\n\nDaniel undid his belt. Sam did, too, getting up and walking forward, hunched, as she stepped over Rafe.\n\n\"Okay, that's enough!\" the pilot barked. \"In your seats, belts on. Everyone!\"\n\n\"Or what?\" Sam said. \"You'll pull over and make us walk the rest of the way?\" She shook the mayor's shoulder. \"Uncle Phil?\"\n\nMayor Tillson's head lolled. Nicole shrieked and fumbled with her belt. I pressed my hand to the mayor's neck.\n\n\"Is he all right?\" the pilot said, sounding concerned now.\n\n\"He's breathing,\" I said.\n\n\"Could it be a heart attack?\" Sam asked.\n\nBefore I could answer, the pilot cursed and said, yes, that must be it, with the stress and all, and the mayor was, as he put it, \"a big guy.\" He'd get a doctor on the helipad right away.\n\n\"Wh-what?\" Nicole said, scrambling over Rafe. \"Did he say heart attack?\"\n\n\"If it is, we'll get help,\" I said as Corey pulled her back.\n\nThe pilot was on the radio to his dispatcher, filling him in between bouts of yelling at us to sit down.\n\nI moved in front of the mayor to undo his jacket. When Sam tried to wedge up beside his chair, Daniel nudged her toward our seats. Anyone else, she'd have told him to go to hell, but she listened to Daniel.\n\n\"Maya can help,\" Nicole said when the pilot tried sending me back to my seat. \"She knows first aid. She runs a hospital.\"\n\n\"For animals,\" Hayley said.\n\nCorey told her to shut up, but she had a point. My dad was the local park ranger, and I had a rehabilitation shed for nursing injured animals back to health. I did know first aid, though, and the basics of dealing with a heart attack victim. Step one: call a doctor. Kind of tough, under the circumstances. Step two: give the victim an aspirin. That wouldn't work while he was unconscious. But why was he unconscious? I remembered fainting as one of the signs, but not sustained lack of consciousness.\n\nWe had to get him to a doctor and, until then, I could only presume it was heart failure and perform CPR if he stopped breathing.\n\nI unbuttoned the mayor's shirt. When Nicole inched forward, the pilot snapped at her, and Corey told him to go to hell, which really didn't help matters. I glanced at Daniel.\n\n\"Nicole,\" Daniel said, \"I know you're worried, but he's okay. He's breathing and we're looking after him.\" He turned to the others. \"Sit down, guys. Everything's under control.\"\n\nIt didn't matter that Daniel barely raised his voice and the pilot's shouting and the noise of the helicopter almost drowned him out. Everyone sat. Even Kenjii, who'd been anxiously nosing the mayor's hand.\n\n\"We need to get him lying down,\" I said. \"If it's cardiac arrest, his heart may stop. I can't perform CPR while he's sitting.\"\n\n\"Then you aren't performing CPR,\" the pilot said. \"There's no room. We'll be landing soon.\"\n\nI laid my palm against the mayor's bare chest. I could feel his heart. Beating, but fluttery. Was that a sign of a heart attack? As I moved back, I saw a spot of blood on the shoulder of his shirt, where we'd peeled his jacket away. I remembered the mayor putting on his Windbreaker before climbing into the helicopter.\n\n\"We need to get his jacket off,\" I said.\n\nAs Daniel helped me get the mayor out of it, the pilot yelled, \"Whoa! Hold on! Back in your seats. We'll be landing in a few minutes.\"\n\nFortunately, as long as he was flying the helicopter, all the pilot could do was yell. And if there are teenagers who actually respond to adults shouting at them, I've never met one.\n\nWhen we got the mayor's jacket off, I checked his upper arm. There was a puncture wound. A dot, slightly swollen, crusted in dried blood.\n\n\"Injection?\" I mouthed to Daniel.\n\nHe frowned and leaned to my ear. \"Could be an insect . . . No, wait. Before we got on, the pilot clapped Mr. Tillson on the arm. I remember the mayor rubbing the spot, like it hurt.\" He paused. \"Like he'd been injected.\"\n\nDaniel slowly turned on the pilot. I could feel the rage pulsing off him. I grabbed his arm and squeezed so tight it had to hurt. Only then did he drop his gaze.\n\n\"His heart's beating fine,\" I said to the pilot. \"Just have someone waiting at the pad.\"\n\nI refastened the mayor's seat belt, then started moving back toward my seat. I don't know exactly what happened next. My gaze was on Nicole, who looked terrified. I think the pilot grabbed for me. Or maybe he just reached out to get my attention.\n\nDaniel yelled \"No!\"\n\nThat's all he did. I'm sure of it. Daniel shouted and the pilot snapped forward. His head cracked against the instrument panel, and he crumpled in his seat.\n\n\"Wake him up!\" Hayley shouted. \"Someone wake him up!\"\n\nDaniel and I shook the pilot. The helicopter dropped a couple of feet and we stumbled. Corey ran over.\n\n\"Help me get him out!\" he shouted to Daniel as he grabbed the front of the pilot's jacket. \"I can fly it.\"\n\n\"Based on what?\" Sam appeared at his side. \"Video games?\"\n\nCorey scowled. \"You got a better idea?\"\n\n\"Yes.\"\n\nThe boys pulled the pilot out of the way and dumped him in the narrow gap behind the front seats.\n\nSam slid into his spot, grabbed the throttle and the control stick, then planted her feet on both pedals. The helicopter stabilized and began to rise, but listed to one side.\n\n\"You have to get her level,\" Corey said.\n\n\"No, really?\"\n\nHe reached for the control stick. Sam swatted him aside.\n\n\"Do you know helicopters? Or just planes? Because they're not the same.\"\n\nThe helicopter leveled for a second, then started to spin.\n\n\"Sam doesn't know what she's doing,\" Hayley said. \"Stop her.\"\n\n\"She's keeping us in the air,\" I said.\n\n\"Everyone else?\" Daniel said. \"Sit down and put on your belts. Now.\"\n\nHayley and Nicole obeyed, but Corey hovered at the front, wedged between Sam and me.\n\nI reached for the radio and put on the headset. It took some fiddling to get the radio going, but we had a shortwave at the park, so I had some idea how to operate it.\n\n\"SOS!\" I said. \"Helicopter out of Salmon Creek. Pilot unconscious. Repeat, helicopter pilot unconscious.\"\n\nI stopped transmitting and listened. Static.\n\n\"SOS!\" I said. \"Emergency situation. Helicopter over Vancouver Island. Pilot unconscious. Repeat, pilot unconscious!\"\n\nThe helicopter dropped again.\n\nCorey bent to look at the control panel. \"You need to\u2014\"\n\n\"I'm working on it!\" Sam snarled.\n\n\"Here, let me\u2014\"\n\nI didn't see what Corey did, but the helicopter pitched to the side, hard and sharp. Corey fell onto the mayor. Kenjii barked, claws scraping the floor as she slid onto the unconscious pilot.\n\nSam swore, her hands shaking as she reached for a lever. \"Everyone sit down. Just sit down!\"\n\nThe helicopter lurched again and Sam's hand hit something. A crack and a rush of air. Nicole shrieked. This time, Hayley joined her.\n\n\"The door!\" Corey said. \"Holy hell. The door's open!\"\n\n\"Everyone hang on!\" Daniel yelled. \"Maya, grab Kenjii!\"\n\nI lunged for the dog's collar and, just then, I heard a gasp. Everyone was yelling and wind rushed through the half-open door and I shouldn't have heard anything. But I heard that gasp.\n\n\"Rafe!\" I screamed.\n\nAs I turned, I saw a blur of motion. Rafe, his eyes opening as he sailed across the floor of the helicopter. Out the open door.\nBack Ads\n\nCopyright\n\nHUNTING KAT. Text copyright \u00a9 2012 by KLA Fricke Inc. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nEPub Edition \u00a9 2012 ISBN: 9780062127792\n\nVersion 09142012\n\n10 9 8 7 6 5 4 3 2 1\n\nFIRST EDITION\nAbout the Publisher\n\n**Australia**\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\n<http:\/\/www.harpercollins.com.au>\n\n**Canada**\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n<http:\/\/www.harpercollins.ca>\n\n**New Zealand**\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n<http:\/\/www.harpercollins.co.nz>\n\n**United Kingdom**\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n<http:\/\/www.harpercollins.co.uk>\n\n**United States**\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n<http:\/\/www.harpercollins.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Asparagus are rich in folates that prevent neural tube defects such as spina bifida in an unborn child. Scientific studies have shown that adequate consumption of folates in the diet during pre-conception period and early pregnancy helps prevent neural tube defects in the newborn baby. Asparagus are also rich in the B-complex group of vitamins such as thiamin, riboflavin, niacin, vitamin B-6, and pantothenic acid. These group of vitamins is essential for boosting fertility in both men and women.\nCashew nuts are a good source of Zinc, Magnesium, copper, iron and selenium. All these minerals are essential for boosting fertility especially Zinc which is known to be one of the best minerals for fertility in both men and women. Deficiency in zinc can affect the testosterone levels and it decrease the sperm count. So, include handful of cashew nuts in your daily diet.\nAvocados are very rich in dietary fiber, vitamins, and minerals and packed with numerous health benefiting plant nutrients. The most abundant nutrients in an avocado are Vitamin C, Vitamin E, Vitamin K, Vitamin B5, Vitamin B6, Potassium and Folic acid. They are also an excellent sources of minerals like iron, copper, magnesium, and manganese. All these nutrients works great in boosting fertility in both men and women. Folic acid prevents birth defects and neural tube defects. Vitamin E help in improving sperm mobility and quality, making it easier for a woman to get pregnant. Vitamin E has also been shown to reduce sperm defects in men, which can lead to less risk of birth defects or miscarriage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This mixed use redevelopment for a significant Conservation Area site in London's West End, has recently been submitted for Planning.\n670 m\u00b2 of commercial and residential space will be provided on the six floors (lower ground to 4th floor) of the new building which will be articulated in brick, with a carefully conceived hierarchical arrangement of deeply recessed window openings that reflects the two unequal historic plot widths of the site and the fenestration patterns of the surrounding Victorian buildings.\nThe principle elevation to Hanway Street will be embellished with reconstituted stone reveals and bronze-framed shopfronts with glazed brick detailing to create an active street frontage. The entrance to the upper floors is marked by a striking backlit screen perforated in a lace pattern that references the many lace shops that once lined the street. The simple, robust brick detailing of the fa\u00e7ade to Hanway Place will reflect the secondary nature of this street.\nAbove a strong parapet line, a raised seam zinc mansard roof will contain the penthouse floor, a winter garden with retractable roof, and a concealed roof terrace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Westin Shenzhen Nanshan is located at the business and leisure area of Nanshan district, which is within walking distance of the Window of the World, Happy Valley, China Folk Culture Villages and OCT area in Shenzhen. The hotel is connected with Yitian Holiday Plaza shopping mall and the Metro station, which is convenient to reach all urban area. Yitian Holiday Plaza is a collection of more than 200 fashion brands, about 20 restaurants with various flavors, a large theater and a Star Ice Rink shopping mall. In addition, it is only about 10 minutes driving from hotel to Nanshan High-tech Park and Shenzhen Bay Port to Hong Kong and Macau. The Westin Shenzhen Nanshan is a hotel with 25-storey, 350 stylish guest rooms and suites, opened in January 2010. All the rooms are set with the unique Westin Heavenly Bed, designed by famous designer, which provide a good sleeping environment. Rooms are equipped with rainforest showerheads, 42-inch high-definition televisions, DVD player and iPod docking station. The hotel owns five different restaurants which are able to meet all your needs with taste. Grange Grill, a signature steak house and bar on the top floor with breathtaking panorama views of the theme parks; Five Sen5es for authentic Cantonese cuisines with traditional yet elegant Chinese elements; Seasonal Tastes, offers a contemporary breakfast and international cuisine; Exchange, lingering for a relaxing moment with friends or business partners may enlighten your day; Daily Treats, a trendy style deli for those 'Grab and Go' items. Guests can use their room keys to access to WestinWORKOUT after operation hours, featuring weight, strength training and outdoor pool. The spa treatments provided by Westin Heavenly Spa allows guests to completely relax body and mind. At The Westin Shenzhen Nanshan, we know that when you feel your best you can truly be your best, and that means paying careful attention to the most important elements of your stay.\nShenzhen's Lijin Commercial Hotel (Shenzhen Lijing Shangwu Jiudian) is a classic business hotel located in Shenzhen's Nanshan District, handy to major theme parks including Window of the World and Happy Valley. Only a 15-minute drive from the railway station, Luohu Entry & Exit Port and Huanggang Port, this Shenzhen hotel enjoys excellent transportation connections, making it a popular choice with business travelers. Diners can sample Chinese and Western dishes.\nThe Ascott Maillen (Shenzhen Yashige Meilun Fuwu Gongyu) is a step away from the locally-famous Maillen Club and the Sea World metro station. By car, Shekou Ferry Port is just 5 minutes. Shenzhen Bao'an International Airport (from which the hotel provides a pickup service) and Shenzhen Luohu Railway Station bordering Hong Kong are both 23-25 km (14-15 mi) away from this Shenzhen hotel.<br><br>Shenzhen Ascott Maillen offers cozy rooms, tastefully furnished and decorated studios, as well as one and two-bedroom apartments, all exuding a modern feel in an architecturally exquisite complex. The apartments have large windows insuring plenty of light during the day, and all are room-serviced and equipped with all the standard amenities. Parking space is available to drivers at an additional fee.<br><br>There are plenty of food options including a Chinese and a Western restaurant, both set in originally-designed interior spaces. Guests can also barbecue outdoors, drink at the bar or in the coffee shop. Alternatively, the hotel offers a buffet breakfast for big morning appetites.<br><br>Business travelers can use the hotel's meetings rooms and business center. WiFi is available in all the rooms and public areas.<br><br>Plenty of leisurely activities are available in the hotel where guests can steam up in the sauna, relax in the spa or take a refreshing dip in the indoor heated swimming pool. Tennis players can keep up their game on the tennis court and a games room allows guests to gather around a game of Mahjong, Chess or Poker.\nIf you want to stay in the best hotel in Shenzhen, China, please come to Trip.com for a custom trip. Staying in exclusive Shenzhen Sea World hotel Trip.com offers, you can fully relax and enjoy the stay in Shenzhen and around Shenzhen.\nAll these Shenzhen Sea World hotels lead in the industry in aspects of environment and recreation facilities, providing the best hotel service in Shenzhen.\nTo ensure that you can find the latest Sea World hotel in Shenzhen, the latest hotel price as well as hotel facilities and indoor facilities information will be offered.\nIn the meantime, detailed photos of Shenzhen Sea World hotel will be displayed, making you know quickly whether hotel's interior environment meets your taste. Besides, you can know specific location and reviews on Shenzhen Sea World hotel from millions of users.\nIf you want a nice room with limited budget, please filter through price and distance to get an appropriate Shenzhen Sea World hotel.\nTo ensure you can experience the best Shenzhen Sea World hotel, users and we give a comprehensive rating on hotel, which gives useful help.\nBelieve it or not, as the biggest online travel service company, we have the best service assurance and perfect users' experience to meet your requirements in different aspects, no matter it's Shenzhen hotel or Shenzhen Sea World hotel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We report the discovery of the super-Earth K2-265 b detected with K2 photometry. The planet orbits a bright (Vmag = 11.1) star of spectral type G8V with a period of 2.37 days. We obtained high-precision follow-up radial velocity measurements from HARPS, and the joint Bayesian analysis showed that K2-265 b has a radius of 1.71 \u00b1 0.11 R\u2295 and a mass of 6.54 \u00b1 0.84 M\u2295, corresponding to a bulk density of 7.1 \u00b1 1.8 g cm\u22123. Composition analysis of the planet reveals an Earth-like, rocky interior; this object has a rock mass fraction of ~80%. The short orbital period and small radius of the planet puts it below the lower limit of the photoevaporation gap, where the envelope of the planet could have eroded owing to strong stellar irradiation, leaving behind an exposed core. Knowledge of the planet core composition allows us to infer the possible formation and evolution mechanism responsible for its current physical parameters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Start Here': Migrants tear-gassed, the Mississippi runoff and Russia fires on Ukrainian ships. What you need to know to start your day.\nIt's Monday, Nov. 26, 2018. Thanks for choosing to start here.\nYesterday, U.S. border officials temporarily shut down the port of entry at San Ysidro, north of San Diego, after a group of Central American migrants attempted to enter the country illegally.\nMany migrants have said they're fleeing gangs and violence their home countries.\nABC News' Marci Gonzalez in Tijuana says tear gas was deployed to stop the group of about 500.\nIn the Senate, only 99 seats have been settled after the midterms.\nOne race has gone to a runoff, and later today President Donald Trump will be in Mississippi stumping for Republican Cindy Hyde-Smith.\nABC News' Kendall Karson tells us that Hyde-Smith is under fire for a number of controversies linked to race and voting rights.\nDemocrat Mike Espy, left, challenges an answer from appointed U.S. Sen. Cindy Hyde-Smith, R-Miss., during their televised Mississippi U.S. Senate debate in Jackson, Miss., Nov. 20, 2018.\nThe day before Thanksgiving, the president fired off a tweet, noting that it was expected to be brutally cold that Thursday.\n\"Whatever happened to global warming?\" he asked. The next day, the temperature in New York City was in the teens.\nBut on Friday, federal agencies directly contradicted the president's skepticism by issuing the most comprehensive report on climate change ever produced by the government.\nAndrew Light, a senior fellow at the World Resources Institute and one of the report's editors, tells us the financial consequences for the nation's economy could be dire if the federal government doesn't act.\nAnd although the Trump administration did not engage in \"political manipulation or editing of the report,\" Light added, it did choose to release its findings when fewer people would notice because of the holiday weekend.\nUkraine and Russia have had especially strained relations dating back to 2014, when Russia annexed part of the country.\nLast night, the Russian navy opened fire on Ukrainian ships near Crimea, and today the United Nations Security Council is calling an emergency meeting over it.\nABC News' Patrick Reevell says six Ukrainian sailors were injured in the incident.\n'I think the report is going to be devastating to the president': Donald Trump's team is already preparing a response to the full Mueller report, Alan Dershowitz tells George Stephanopoulos.\n'Targeted criminal justice responses are needed': Around the globe, about 137 women were killed every day in 2017 by a partner or family member.\n100 percent contained: California's deadliest wildfire finally appears under control.\n'We will continue to show our defiance towards intolerance by not giving into threats': A 16-year-old with outstanding warrants is arrested after phoning in threats to gay bars in Boston.\n'I hope inter-Korean relations will turn out as good': A traditional North Korean hunting dog gifted to South Korea gives birth to six puppies.\nNov. 26, 2017 -- Al Franken speaks out for the first time after sexual misconduct allegations.\nMigrants run from tear gas, thrown by the U.S border patrol, near the border fence between Mexico and the United States in Tijuana, Mexico, Nov. 25, 2018.\nA state trooper delivered a baby on the side of a highway.\n'Start Here': Migrants, Mississippi, Russia. What you need to know to start your day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuamf b/data_all_eng_slimpj/shuffled/split2/finalzzuamf
new file mode 100644
index 0000000000000000000000000000000000000000..11b5df45b7748ab37ec6d76cef775306f106b2f0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuamf
@@ -0,0 +1,5 @@
+{"text":"Our tube test seals are used by manufacturers for high and low pressure testing.\nHigh pressure capability with extreme wear resistance.\nSpecial lip profile for maximum seal life.\nManufactured in high tensile, impact-resistant polyurethane.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Birmingham Hippodrome, which first came into existence in 1895 as a group of assembly rooms, sells more seats than any other UK theatre each year, including the West End.\nSpend a cultural holiday enjoying all manners of entertainment within this grand spectacular theatre. Watch classic ballet performances by the spectacular Birmingham Royal Ballet or an all-singing all-dancing musical.\nDine in the theatre's stylish Circle Restaurant before your performance and in the interval you can nip out and enjoy coffee, drinks or a delicious dessert.\nTo really top a weekend theatre break, make sure you attend one of the free pre or post-performance talks for a chance to meet the show's cast and discover exactly what it takes to put in a star production.\nThis summer the streets, parks, bars and squares of Birmingham will be filled with melody as the city plays host to some fantastic music festivals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You may wish to rent a car during your time here in Scotland, in particular for visiting the north of Scotland and Hebrides\/Orkney's where public transport is limited.\nWhen collecting the car, inspect the interior and exterior of the car very carefully in case there is some damage which is not noted by the company and you'll get the blame and the bill for it!\nYou must possess a valid driving license, car insurance and have permission to drive the car \u2013 there is no flexibility about this. You will be prosecuted\/taken to court if you disobey these laws.\nThe unit of speed and distance in the UK is miles per hour (mph). 1 mile = 1.4km.\nSpeed cameras are usually installed in towns and smaller A-roads where accidents occur regularly so they are there for a reason!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In last week's blog post, we saw how a man named William Hendrie from St. Kitts moved to St. Croix and lived there for a couple of years before passing away. We saw how this William Hendrie interacted with Mary Faucett (Alexander Hamilton's grandmother) and probably with all the Lyttons, Faucetts, and Laviens, who lived nearby. We also saw how he had a son named James Hendrie, who in 1749 was trading with St. Croix but apparently not living there.\nThe earliest record of James Hendrie on St. Croix that I have found is an entry, actually two of them, dated March 20, 1758, showing his payment of 10 rigsdalers for his marriage license. Unfortunately, the name of his bride is not given.\nThree months later, in June 1758, James Hendrie is found again in the St. Croix records, this time signing Christiansted's bailiff register alongside a number of others.\nSurely James Hendrie appears in other St. Croix records from this time, but it is noteworthy that no record of him on this island prior to his marriage in March 1758 has yet to be found. One assumes that he came to St. Croix around this time, from where, however, remains a mystery.\nJames Hendrie does not appear in St. Croix's matrikels (the annual property, tax, census register) prior to his wedding, but in the matrikel of 1758 he is listed as living on Kongens Gade (King's Street) in Christiansted with his wife and 25 slaves, though it is not clear how many if any of these slaves were owned by him.\nAccording to an account statement between James Hendrie and Alexander Moir (yes, the same man from whom James Hamilton would later collect a debt), James Hendrie appears to have been acting as a merchant at this time, exporting cotton, rum, sugar, and other goods.\nIn February 1759, James Hendrie was summoned to testify as a witness in the divorce of John Michael Lavien and Rachel Faucett.\nIt is not clear what James Hendrie knew that would be worthy of the court's attention. Although James Hendrie's father William Hendrie probably knew the Lyttons, Laviens, and Faucetts, and in fact paid Mary Faucett to do some sewing, there is no evidence that James Hendrie knew any of these people. It also appears that James Hendrie was not on St. Croix back in 1749 when Rachel's affair with Johan Cronenberg took place. More likely, James Hendrie had lived on St. Kitts, from whence his father had come, until he arrived on St. Croix around early 1758. One can thus assume based on his being summoned as a witness that James Hendrie knew James Hamilton and Rachel Faucett on St. Kitts, or at least knew of them. He thus would have been summoned to testify about Rachel's \"illegal\" relationship with James Hamilton on that island.\nAs with Jemima Faucett Iles Gurley and James Ash, no record has been found of James Hendrie's testimony as a witness. In fact, there is no record regarding whether he testified or appeared before the court.\nAccordingly, we do not know if James Hendrie knew anything above and beyond what was already in the court record or which side he would have taken in this matter, but one assumes, as mentioned above, that he was summoned to report on the relationship of James Hamilton and Rachel Faucett on St. Kitts.\nOn July 27, 1759, and again on August 10, the government auctioned off some of James Hendrie's property. Apparently, James Hendrie was having trouble managing his debts and his property was being seized and sold to pay back his creditors.\nDespite these apparent financial troubles, James Hendrie is found in the 1759 matrikel, still living on Kongens Gade (King's Street) in Christiansted, with his wife and seven slaves.\nThis would be the last time that James Hendrie appeared in the island's matrikels.\nAccording to a statement by Alexander Moir dated August 24, 1761, James Hendrie owed an Isaac Hartman \u00a3250 sterling plus interest but had \"turn'd bankrupt\" since borrowing this sum and Moir \"gave in a charge against James Hendrie's estate\" for the \u00a3250 sterling.\nWhen exactly James Hendrie \"turn'd bankrupt\" is not mentioned, but it presumably was sometime in 1760 because he still owned seven slaves at the end of 1759 but is no longer listed in the matrikel at the end of 1760, meaning that he no longer owned any taxable property, i.e., land or slaves.\nNo longer appearing in the matrikels, one would assume that James Hendrie fled the island to avoid imprisonment for outstanding debts. However, it is possible that he stayed on St. Croix but was not listed in the matrikels because he no longer owned any land or slaves on the island and therefore did not have to pay any taxes. It is noteworthy that Alexander Moir mentioned Hendrie having \"turn'd bankrupt\" as a reason for his not paying his debt but did not mention him leaving the island. One would therefore assume that he stayed on St. Croix even though he does not appear in the matrikels.\n Translated, the matrikel reads: \"Thomas Houwey, now James Hendrie. N.B. These slaves are [with] Hans Gustman in behalf [of the] share inherited by George Simons widow.\" This would imply that all 25 slaves belonged to Simons's widow and were being held by Gustman. However, the previous year, Thomas Houwey is listed at this same location with seven slaves. The following year, James Hendrie, still on King's Street and presumably at this same location (that matrikel does not number the properties), owned seven slaves. Accordingly, one might think that seven of these slaves may have belonged to Hendrie.\n The matrikels at this time were primarily but not exclusively records of taxable property, i.e., land and slaves, and the taxes owed on them. They also counted the number of white people, but did not list them all. People working as servants, clerks, and plantation managers would be counted in a column for white servants and would not be named unless they also owned landed property or slaves, upon which taxes had to be paid.\n\u00a9 Posted on October 15, 2018, by Michael E. Newton. Please cite this blog post when writing about these new discoveries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Increases the brightness of colors. Good drying power!\nMaimeri Walnut Oil Painting Medium - 75ML, 250ML, 500ML glass bottles. Maimeri uses only the purest ingredients for the best guarantee of brilliance and vibrancy. Most Maimeri oil mediums are housed in a glass bottle with a twist-off cap. Increases the brightness of colors. It has similar properties to linseed oil, but it won't yellow as much. Good drying power.\nWalnut Oil Similar properties to linseed oil, but it does not yellow. Good drying power, increases the brightness of colors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuami b/data_all_eng_slimpj/shuffled/split2/finalzzuami
new file mode 100644
index 0000000000000000000000000000000000000000..a9fbc7cd8ebcf11ff70163b0e41d963f8564455f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuami
@@ -0,0 +1,5 @@
+{"text":"For the first time, you can now easily share 360 photos on Facebook. Simply take a panorama with your phone or capture a 360-degree photo using a 360 photo app or 360 camera, and then post it on Facebook as you would a normal photo. From there, Facebook will convert it to an immersive 360 photo that people can explore, similar to how people experience 360 videos on Facebook.\n360 photos are easy to identify in your News Feed: just look for the compass icon on the right-hand side of the photo. For mobile users just tapp and dragg the photo or by moving your phone, and on the web by clicking and dragging. Now your friends can get themselves in a spin and experience the moments you share in 360 as if they were actually there with you, from hiking through a national park, to wandering through a museum, or having a few pints down the local pub. I have said to much.\nAlong with 360 photos from your friends and family, you can discover stunning new 360 photos on Facebook from public figures, publishers, and other organizations. 360 photos give you the ability to take the stage in front of 100,000 fans with Paul McCartney, visit the International Space Station with NASA, and more. This medium enables new opportunities for creativity, and we're excited to see what kinds of 360 photos get shared on Facebook.\nTo learn more about 360 photos, visit the Facebook 360 site or join the Facebook 360 Community group to be a part of the daily 360 and VR conversation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Aviation Interiors Expo-US may be a smaller version of the Hamburg version held each year, but that doesn't mean it is without interesting news. This year the buzz is all about the SkyRider seats from Avioninteriors. Why? Because they offer a nearly upright \"seat\" with only 23 inches of pitch. When the seats take up 25% less space that means room for a lot more passengers (and revenue) for the airlines.\nBut will folks actually be willing to sit in the seats? And can they even fit? Important questions and ones that the company hopes to answer for interested parties throughout the show. The booth was heavily trafficked, with media outlets conducting interviews and interested observers hoping to give the seats a try. After watching others take their turn for about an hour and watching their reactions I finally settled in myself to give the seats a ride.\nTake a couple employees and seat them on the \"saddle\" that acts as the seat and things don't look all that terrible. They actually seem pretty happy there, though they are also in the \"bulkhead\" row with nothing in front of them so they can stretch out their legs a bit.\nBut try to get a reasonably small cameraman in there (seriously, guy, a pager??) and the camera simply will not make it.\nThere are even tray tables and what looked like a simulated 7\"-ish IFE screen in the demo set. With your face only 7\" away from a 7\" screen it would be incredibly large, probably actually too big for most folks' range of vision. I cannot imagine that working out well. With the fisheye lens the seats actually don't look so horrible.\nThe saddle styling of the seat is interesting. One person mentioned that it was like riding a horse, without all the space normally associated with equestrian activities. Indeed, for folks who are outside the normal height range that the seats are designed for the saddle could prove to be rather uncomfortable. A 6' 4\" reporter from one of the news outlets tried to get into the second row and was rather unsuccessful. Similarly, I would imagine that the saddle would be rather uncomfortable for folks shorter than around 5' 2\" tall. I really have no idea how it would work for most kids.\nSo, how did I fare in the seat? Here's the photo. I got in and sat there for about 5 minutes. I even moved into the middle when another guy showed up and wanted to experience it as well. The good news is that the middle seat didn't feel particularly cramped relative to the aisle seat. The bad news is that all the seats are a pretty tight squeeze. I figured that, much like riding the subway, anything would be fine for 30-45 minutes if needed. The problem is that there is no such thing as a 30-45 minute flight. My trip from Las Vegas to Long Beach this morning was only 45 minutes in the air but I was on the aircraft for nearly double that with boarding, taxi and deplaning time. And I'm not so sure that there is actually a price point at which the airlines can sell this product that would make me comfortable enough flying it. Maybe it is just a function of getting used to the experience, but I'm not particularly convinced. It isn't a seat and you're not quite standing. Limited head clearance (the seats are taller to make up the missing pitch) and no under-seat space would mean less capacity on the planes for carry-on bags. Plus it just feels cramped. Way more than traditional seats and even more than the rather tight space in the back of a Spirit Air plane.\nActually maybe not on that last one. Just maybe the SkyRider would be better than the tight squeeze that some carriers offer in a traditional layout today. I certainly wouldn't fly Spirit again based on just how awful the seats were in the back on a 150 mile flight I took with them so perhaps they have nothing to lose by giving something like this a try. Maybe there is a market for these after all. That's a rather scary thing to consider.\nWhen I first read about it on Cranky, my only comment was \"that ain't gonna fly\". Your pics just confirmed it. I don't even know why the company spent time on it.\n1) They had a TON of traffic through their booth. Lots more than the other seat manufacturers. And they sell a number of other products. I sat in a few of their other Y, PE and reional Biz seats and those were all quite nice.\n2) On the off chance they do actually sell this they'd pretty much secure a spot on the list of innovative companies.\nI saw some folks from major airlines in the booth (spoke to one of them about it, too). This isn't just a bunch of media folks making up hype. These guys are serious about the potential for the product.\nThese look terribly uncomfortable. Who can sit with their legs braced for more than a couple minutes? For heavier passengers, this would be a nightmare. I flew back from Heathrow to Chicago with a chubby lady in the next seat and it was not comfortable for either of us. Seats like this might be fine for school children. Definately not for adults.\nWell .. from a safety perspective, this is a no-go. I seriously doubt these seats can withstand the reqired 18g that an airliner seat needs to according to certification rules. I also can not see how pax are able to get in the brace-position in case of an emergency landing.\nThis is a publicity stunt, but it works, coz as you said, they get people in their booth to view their other products.\nMaybe it's indeed a publicity stunt, but quite frankly if I was an airline executive it wouldn't exactly leave a \"wow those guys are competent\" impression with me. How many seat manufacturers are there? Getting the attention of buyers in this market really shouldn't be that hard, it's not like they are making cereal or laundry detergent.\nThis seating will be a certification challenge for sure, with the new requirements for 16G protection it may be a while before this could be put into service. Likely will be modified a few times before becoming certifiable.\nI would like the seat designers and their senior executives to trial my new space saving idea \u2013 the inflight toilet. Get rid of the onboard smelly room and replace it with more passengers.\nBut maybe these designers and their bosses should just shove glass tubes up their rear ends for a year. Or until they come to their senses and stop talking crap.\nThis looks to be an ideal solution for Ryan Air as they can transport even more passengers without making them stand and fly which is one of their business options being considered.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cisco Canada is collaborating with Partners in Research and Sheridan College in a program aimed at helping K-12 students explore careers in science, technology, engineering and math through video conferencing with professionals.\nPartners in Research's virtual research on call program offers real-time interactive mentoring and presentations from subject matter experts, according to a recent Cisco statement.\nAccording to the company statement, Cisco contributed more than $1 million in collaboration platforms for the educational initiative.\n\"With the flick of a switch, neighbourhood classrooms will have easy video access to compelling experts from around the world, giving students a valuable resource as they embark on their career paths,\" said Nitin Kawale, president of Cisco Canada.\nThe company installed its enterprise-class webcasting and video sharing applications at Sheridan's data centre to serve as the VROC platform.\nPartners in Research will use the technologies to create videos that teachers and students can access on its website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A sustaining member of NPT gives a certain dollar amount ($5 minimum) on an ongoing monthly or annual basis. Your giving level automatically renews month to month and year to year.\nSustaining membership anchors NPT in financial stability for years to come. Your ongoing support translates to an ever-increasing number of quality programs for you to enjoy.\nWhat are the payment options for sustaining membership?\nSustaining members can give via credit card or electronic funds transfer\/bank account withdrawal (EFT). EFT payments are the preferred method, since none of your donation has to go to credit card fees. Also, we will not need to contact you to update your credit card information when it expires!\nAre there any benefits of sustaining membership?\nYou are able to choose a new thank-you gift each year corresponding to the monthly amount you choose to give, and may call in at any time to add a gift to your pledge.\nYou will receive a sustaining member postcard each year, reminding you that you are able to choose another gift.\nYou are automatically entered into all of our drawings\/giveaways.\nCan I modify or end my sustaining membership?\nYes. If life circumstances change, you can modify or suspend your sustaining membership payments at any time by calling the office at NPT during regular business hours at 615-259-9325. If you have chosen a gift to coincide with your sustaining membership, we do ask that you will be willing to pay the fair market value of the gift.\nGo to secure.wnpt.org to give with your credit card or set up your donation by automatic bank withdrawal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kyle is a Solutions Architect based in the UK, specializing in delivering the best end user experience possible, whilst understanding that compliance and commercial viability does matter. With his background of application development, infrastructure (Storage, Compute & DC networking\u00ac), virtualization and end user computing (workspace), he has a great understanding of all the components required to deliver the experience that end users desire. Kyle proactively assists on various virtualization support and community forums in his own time, and assists some vendors with course material and questions for exams. As a technology lover, he is always up for a discussion or debate across various technology stacks, his blogs aim to drive discussion whilst providing insights into various solutions and technologies. You can find him on twitter at @kdavies1988.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzueac b/data_all_eng_slimpj/shuffled/split2/finalzzueac
new file mode 100644
index 0000000000000000000000000000000000000000..c6c2a76202bc68052a1ad49549e8333d869fd62a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzueac
@@ -0,0 +1,5 @@
+{"text":"\"Educating the MIND, BODY AND SOUL\" goes the Olivarian Motto. These words capsulate the Olivarian Philosophy of Education committed to the development of the critical mind capable of intelligent decision-making and analysis; physically fit body that is well versed with arts, skill and athletics. A sound mind make up a total person with a spirit that is in communion with Christian ideals, and one who is dedicated in the service of God, country and men.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Highly desirable building lot in an established community of traditional homes. Building envelope is available for this lot and was prepared by White Mountain Survey. Fabulous location just minutes from downtown shopping, Brewster Beach, Huggins Hospital, Library and Schools.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those of you who follow me on Instagram (@thedotcouture) know how much I ADORE Holiday-themed manis. Especially Halloween. I'm fairly certain Halloween was made with me in mind. So when I was given the chance to review some plates from http:\/\/www.bornprettystore.com\/, and one of those plates was a Halloween-themed plate, I was ecstatic!\nThis is one of their circular plates and retails for $2.99 USD ($3.60 for us Canadians) on their site HERE (use my code JDGK31 for 10% off). The plate has a nice selection of Halloween-themed images. I knew I wanted to do a summer-type Halloween mani with the ground line image.\nI used neon Sharpies to create a watercolor effect as the base. It even glows under black light, you can check out my Instagram for a shot of that! Instead of stamping with the typical Halloween black, I used M Polish 5-free stamping polish in Iron to soften the look. The plate is etched beautifully and I had no troubles picking up the image. I am in love with the tree image on my middle nail \u2013 I would definitely use that image for every season!\nI used the OPI Color Paints as a base for my ring and pinky finger nails (you can read my blog post about them) and stamped over them using plate Qgirl-032 and a black stamping polish. For the middle and ring finger nails I stamped the same image over black using M Polish in Hottie, Bells of Ireland, and Scouting the Beach. This plate also retails for $2.99 USD ($3.60 CAD) and can be purchased HERE (use code JDGK31 for 10% off all non-sale items).\nBoth plates produced nice crisp images for me using a variety of stampers and polish brands. They are very affordable and easily accessible. Bornprettystore has free worldwide shipping and a rewards system for purchases. They are coming from overseas and will take a little longer to your door, however they are definitely worth the wait!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barry says: A brilliant and enthralling document of the all-encompassing electronic maelstrom of the 1980's, from outsider synth grooves, more percussive outsider electronics and shuddering industrial shadows. Brilliantly varied and thoroughly worthwhile. Another Numero stormer.\nWho do we become when we live our dreams? It's all here\u2014the high hairdos, the dreams and schemes, the tender camp, the wedding bell fantasias and chaste tragedies. Sister acts, studio receptionists, classmates, angelic voices of the 1960s; some legendary, many hidden in the basement of expired rainbows. Gathered on this deluxe double LP (or CD) are 28 (56 on the compact disc!) foiled escape attempts, now free to soar in girl group heaven.\nIn the wake of 2017's Seafaring Strangers: Private Yacht, Numero is proud to present another addition to the soft rock cannon: W2NG. Set your FM dial to smooth and sail away with 42 minutes of uninterrupted easy glide, pontoon rock, and whispery disco. Featuring unreleased burners from Gary Hyde, Love Transfusion, Marshall Titus, and Phillips, alongside scarce cuts from Nannette, Greenflow, Jim Spencer & Son Rize, Larry Sanders, Kettner & Shawe, Orphans of Love, and Lion, W2NG is sure to surprise even the most devout boat shoe enthusiast.\nWhat does teen spirit smell like, anyway? It might smell something like Noise Addict. Like the real life stars of some sort of choose-your-own-adventure book about pursuing rock stardom, few bands ever led a more charmed existence, springing from the Sydney suburb of Bondi into seemingly overnight international fame as friends and collaborators of Sonic Youth, Fugazi, and the Beastie Boys. Through a combination of relentless drive, luck, and an admirable lack of self-doubt, Noise Addict spanned puberty to surpass the haters and join Radio Birdman and Nick Cave as a strange but permanent piece of Australian punk history. Cover art doubles as a working chalk board (chalk not included).\nThis title is a Record Store Day Exclusive.\nTeeming with the energy and grit of pre-Giuliani Manhattan, Blonde Redhead's long out-of-print early recordings have finally crawled their way out of the '90s basement thanks to Numero Group who will issue the set on Sept. 30. Weighing in at 37 tracks, Masculin F\u00e9minin compiles the band's first two albums for Steve Shelley's Smells Like Records (self-titled and La Mia Via Violenta), their period singles, extant demos, and radio performances across four LPs or two CDs. Dozens of previously unpublished photographs illustrate two lengthy essays on this essential New York band's formative years.\nThis career-spanning 7LP box set compiles Sudden's landmark solo LPs Waiting On Egypt and The Bible Belt, the Dave Kusworth-era LPs Jacobites and Robespierre's Velvet Basement, Sudden's Rowland S. Howard collaboration Kiss You Kidnapped Charabanc, and his Creation-issued masterpieces Texas and Dead Men Tell No Tales. Limited to just 500 copies, this set also includes Sudden's debut solo single \"Back to The Start\" b\/w \"Ringing On My Train\".\nThis entry level compilation features 20 songs from our original 004 cd, plus two bonus cuts, new liners, tons of sleeve scans, photos, and ephemera, all housed in a spot varnished, brick-thick gatefold sleeve. Artists include New Jersey's Toms, Kennet, MO's Trend, NYC's Colors, Treble Boys, and Sponsors, Hollywood's Randy Winburn and Tommy Rock, Des Moines, IA's Luxury, San Antonio's Kids, Connecticut's Bats (featuring a young Jon Brion), and Boston's Tweeds. Attention: In celebration of our Buttons series, you can buy both NUM004 and NUM044 together with our limited edition Power Pop Buttons Pack. This pack includes 32 high quality buttons of varying sizes that feature artists from both Buttons compilations. Act fast because this set is limited to 100 and will never be made again.\nA raw cry from the dark night of one man's soul. Cloistered away from the popular culture of 1982, rural Illinois priest Tony Trosley painted a pastoral refraction of early 1970s Laurel Canyon watercolors with this stand-alone set of songs. Openhearted and naively psychedelic, 'Deep Night' was recorded during a single pre-dawn marathon, and mixed live-to tape in an isolated chapel. Cobbled together out of local players to help fill out this ethereal soundscape, Trosley's band brought an earnest but bluntly unsophisticated backdrop to his phaser-drenched 12-string guitar. The Sixth Station - named for a grim New Testament tableau in which Veronica washes the tortured face of Jesus - managed to avoid overtly Christian themes in favor of a mystical Humanism that resonates timelessly, and to any sort of listener. This 'Deep Night' is as profound and eerie as the images conjured by its title.\nAvailable as a 3-CD or 4-LP collection I Travel Alone follows hero, Lou Ragland, through various and sundry styles, studios and combos, capturing one of Cleveland's most daring musicians during his most adventurous decade. With the release of 1970's census data, Cleveland, Ohio, ranked tenth in population among U.S. metropolitan centres. The Forest City had lost nearly 200,000 people in the decade passed, and it would lose 200,000 more before the ball dropped into 1980. Cleveland's legendary stores of musical talent would mirror that exodus, as no one-from Albert Ayler to Screamin' Jay Hawkins, from Bobby Womack to Edwin Starr, from Tracy Chapman to Chrissie Hynde- managed to break nationally until after they'd left the city limits. Even the O'Jays, greater Cleveland's most enduring soul demigods, rode their \"Love Train\" not out of Cleveland Union Terminal but departing from Philadelphia's 30th Street Station with Gamble & Huff as the rails. No one made it in Cleveland, despite the hundreds that tried. I Travel Alone gathers together the work of a pre-flight Lou Ragland, laid down during those fertile Cleveland years when he did most of his traveling between Hough and Garfield Heights, up and down Kinsman Road, out to Lakewood or Euclid...maybe Parma if the situation called for it. Stuffed into a '65 Mustang, guitar case angled between the seats, his LPs stashed in the trunk, Lou Ragland never was quite alone, though he followed a road all his own.\nLP Box Set Info: 4xLP box set.\nChicago's only blues-funk family band with an eight-year-old drummer finally get their due in this elaborate 3\u00d745 box set. Stationed on Chicago's west side, between 1965-1972 the Fisher brothers worked their novelty act at any club that was willing to bend the city's stiff liquor laws. Three 45s were cut for their homespun Fised and Fished labels, with the third featuring the first known recordings of notorious bluesman Johnny Dollar. Chock full of photos stripped from family photo albums, the 12 page booklet tells the complete Soundmaster story, from their earliest days backing J.B. Hutto to Little Ed's final snare cracks for Southside Movement. Housed in a tipped-on 7\" box, the set raises the bar for funk 45s.\nSuper deluxe 6xLP, 4xCD box set.\nLaced with that unique brand of bravado, the Syl Johnson interview tends to veer toward harrowing voyages through interruption, correction, and deliberate obfuscation. \"Back up, hold on, slow down\u2026 Wu Tang, Kid Rock, Michael Jackson\u2026 Jimmy Reed, Jimmy Johnson, Jimmy Jones\u2026.\" Johnson has a habit of insisting that everything printed before\u2014every verbatim transcript read directly back to him\u2014is a blatant misquote or misunderstanding\u2026and sometimes both. His date of birth and place of birth, his surname change from Thompson to Johnson, the murky beginnings of the Twinight record label\u2014Syl Johnson weaves them all into one convoluted narrative, a daunting challenge for historians and fans alike to follow. Resolving his life story for this collection became an exercise in patience and diligence, as we chased the rabbit through even more big-hole 45s than bear his name.\nThe blues was always a genre riddled with myth and legend\u2014its half-truths muttered on sun-baked Mississippi porches have long-since morphed into biographical foundations. From Robert Johnson's midnight bargain with the devil at the crossroads near Dockery Plantation to Bo Diddley's divergent claims about the origin of his name, fabrication is fully ingrained in blues tradition.\nSyl Johnson's apple never fell far from that tree. When this bluesman-at-heart felt his career tapering off early in the 1980s, his tendency toward self-mythologizing gained momentum. If he couldn't enjoy the successes of an Al Green or a James Brown, he could surely concoct for himself a more mysterious history. Forget hot grits and armed robbery, Syl Johnson's illegitimate father would be Robert Johnson. Or so he began to claim\u2026.\nFor decades, Johnson has been toeing the edge of a wide chasm that separates soul's upper and middle classes, overshadowed across his career by the bill-paying stars of the Federal and Hi labels. He's joined by the likes of Otis Clay and Candi Staton in a pantheon of great soul singers who maintained viable careers over several decades but never achieved that national #1 smash. Consequently, he's been eschewed by oldies stations and Final Jeopardy questions, never having scattered the cultural detritus that keep even one-hit wonders in the periphery of the national consciousness. The litany of his largely regional hits\u2014\"Come On Sock It To Me,\" \"Different Strokes,\" \"Is It Because I'm Black,\" \"We Did It,\" \"Back For A Taste Of Your Love,\" and \"Take Me To The River\"\u2014is undeniable, a list that dwarfs the tally of winning output in his caste. Even so, Johnson's reevaluation as a serious artist has yet to arrive.\nLtd Box Set Info: Super deluxe 6xLP, 4xCD box set.\nFollowing a path blazed in Belize and the Bahamas, the Numero Group finds yet another stop on the soul diaspora tour: Dimona, Israel. Between 1975-1981, a group of American ex-pats took their native sounds of Detroit and Chicago and intermingled them with the messages of the black Hebrew culture. The results are a heavenly mix of spiritual soul and jazz with an undercurrent of gospel psychedelia. Featuring the Soul Messengers, the Spirit Of Israel, Sons Of The Kingdom, and the Tonistics, \"Soul Messages From Dimona\" is the only living document of a thriving community at both the center and fringe of the world. Deluxe CD and 2LP set comes stuffed with rare photographs, sleeve pics, and expansive liner notes about the African Hebrew Israelites journey from the sounds of slavery to the intonations of liberty.\nDeluxe vinyl album box set includes a miniature song book and expansive booklet stuffed with photos and memorabilia, THREE albums in replica tape image sleeves (including a whole album of exclusive bonus tracks) plus a CD of the complete album, all housed in a hard paste-on board tape style box. To say that this is a bit special is a major understatement!!\nIn January of 2006 the Numero Group found itself sitting three wide on a weight bench in a damp garage. Eight hours later they peeled out of the driveway with a thirty seven tape mystery. Though Jeremiah Yisrael's Tap label only issued a handful of 12\"s between 1981-83, the tape deck never stopped rolling, producing dozens of lavish songs in disco's last days. From the string-heavy sugar boogie of Jackie Stoudemire and Arnie Love to pioneering raps by Missy Dee & The Melody Crew and Fabulous 3 MCs, Recording Tap covers a wide swathe of New York's all night sound.\nBox set Info: Deluxe vinyl album box set includes a miniature song book and expansive booklet stuffed with photos and memorabilia, THREE albums in replica tape image sleeves (including a whole album of exclusive bonus tracks) plus a CD of the complete album, all housed in a hard paste-on board tape style box. To say that this is a bit special is a major understatement!!\nCD comes with 16 page full colour booklet featuring full musical history of the island.\nThe second in Numero's series of exploration into the pan-American funk experience, \"Cult Cargo - Grand Bahama Goombay\" is a deep overview of Funky Nassau's red-headed sister city, Freeport, GBI. From 1969-1976 Frank Penn's GBI studio and label cranked out a dozen LPs and twice as many singles infected with the Miami sounds drifting in over the 100 mile strait. The catalog is a fruity blend of rake and scrape, bush, junkanoo, calypso, reggae, and of course, goombay, with a twist of American soul.\nCD Info: CD comes with 16 page full colour booklet featuring full musical history of the island.\nDeluxe vinyl issue at last!\nThe first ever compact disc issue of Catherine Howe's brilliant debut album. Produced by legendary jazz pianist Bobby Scott, the album is a pastoral blend of English countryside folk and London orchestral pop, not unlike \"Bryter Layter\" or \"North Star Grassman And The Ravens\". Originally released on Reflection Records in 1971, the much sought after album disappeared before ever hitting the racks. Booklet includes half a dozen unpublished photos and an annotated history of the LP's brief existence. This fully remastered album includes an unearthed bonus track originally intended to be included on the album.\nLP Info: Deluxe vinyl issue at last!\nCD comes with 16 page full colour booklet of artist biographies, original sleeve artwork etc.\nPig's tails, potatoes, plantains, bananas, boiled eggs, yams and a whole fish, all added to one pot: Belizian's call it 'Boil Up', but it's anything but leftovers. Mix equal parts R&B, calypso, disco, funk, reggae, bruckdown, soul, folk, and whatever else can be found back on the bottom shelf of the musical pantry. Get ready to feast on passport stamped rhythms, second-deck cruise ship melodies, hotel pool calypso, soundtracks to movies not-yet-made, and anything else savory, or unsavory, enough to throw into the pot.\nCD Info: CD comes with 16 page full colour booklet of artist biographies, original sleeve artwork etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is a 15 Year Mortgage in Your Best Interest?\nA fifteen-year mortgage is a great bet if you're inclined to gamble on a couple of things. The first, obviously, is that you're betting on your ability to pay the higher mortgage rate over the long haul. If you have your own business, you have control over your employment situation. Then the question turns to whether your business or your career has the legs to be as successful for the next fifteen years as it is now. Are you in a cyclical business, affected by economic downturns? Most are, and if your fifteen-year mortgage is a stretch for you in the first place then it's a major gamble. If you're salaried and safe from the slings and arrows of the economy, then it's a safer proposition.\nThe savings in plain old dollars is substantial. One mortgage calculation tool compares the figures generated by putting a $100,000 mortgage into fifteen-year terms and thirty-year terms. The monthly payment is about $735 a month over fifteen years and about $955 a month over thirty years, with an interest rate that is a quarter of a point higher. The difference in total interest payments is a little over one hundred thousand dollars: $169,000 versus $64,000. Those are raw dollar figures, however. What is not factored in is your savings on your annual taxes engendered by the higher interest rate attached to the thirty-year note.\nAlso not factored in are a number of intangibles. Where would that extra money go if it weren't committed to a fifteen-year mortgage payment? Other investment opportunities, perhaps? Perhaps. But there's a reason they call leftover money like that \u00ecexpendable income. The reason is that most of us do expend it, rather than invest or save it. So maybe the thirty-year note means better family vacations, a few ski trips during the winter, a nicer car, without doubt, it means some added flexibility in the family budget.\nThe value of retiring a mortgage in fifteen years is substantial, but so can be the risk. If you're seeking middle ground, consider a mortgage that accepts accelerated payments on a spot basis. When your family income is humming along, pay a higher monthly mortgage rate and you will get a larger figure attached to your principal reduction. You will be paying the higher (30 years) interest rate with those payments, so your annual tax deduction will go up as well. You're knocking time off the mortgage, and maintaining your maximized tax deduction.\nSome money managers will call the fifteen year mortgage a sucker's bet, because if you took the monthly savings from the lower payment on a thirty year note and added it to the savings from the higher tax deduction on a thirty year note, the total in funds saved would more than offset the difference in total interest.\nIt's a great theory, probably has some merit, but how many of us will diligently sock away our monthly savings and yearly tax break inherent in the difference between a fifteen-year mortgage and a thirty-year mortgage? Approximately none of us. Most people look at home appreciation for their return on investment and let it go at that. Put in a financier's terms, if a thirty-year note cuts your sleepless night quotient by a factor of twenty percent or more, it's probably worth it.\nPrevious: Previous post: Facing an Increase in Your Mortgage Payments?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The GFA Progression Team program is designed for athletes who wish to develop the fundamentals of freestyle park riding. Whether it's your first time going on a rail, gaining control of your spins on the jumps, or preparing for the competitive level, the Progression Team keeps you on focus while still having fun.\nProgression Team coaches not only teach how to do tricks, but also address the other aspects associated with advancing the riders skill levels. These include proper mindset (ie. \u2013 focus, confidence, etc.), knowing when to push and when to hold back, Terrain Park etiquette\/safety, respect for the mountain and other guests, and teamwork.\nBut let's not forget about the FUN! Learning in a team environment creates lifelong friendships, the coaches love rewarding athletes with prizes, and the kids enjoy getting better at the winter sports we love.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From Genesis to Revelation, the main theme of the Bible is God's plan to redeem humanity from sin and for fill his original creative purpose. From the beginning God planned the Incarnation and the Atonement.\nWhile there have been various ages or dispensations and God's dealings with humans, these ages do not represent different plans of salvation, but they progressively unfold in reveal God's eternal plan. In the Church Age we see the fullness of individual restoration, in the millennium we will see the fullness of corporate restoration.\nIn every age the basis of salvation is the same (Romans 3:24\u201325, Ephesians 2:8\u20139, Hebrews 9\u201310). The source of salvation is always divine grace: the unmerited gift of God to us, the free work of God in us.\nThe ground of justification is the blood of Christ through Christ's sacrifice on the cross, God provides salvation consistent with his principles of justice. The Blood, the Cross, and the Atonement all referred to the sacrificial death of Christ for us. Since Christ Resurrection was necessary to make his death effective, to turn defeat into victory, the ground of salvation is specifically the death, burial, and resurrection of Jesus (1 Corinthians 15:1-4).\nThe means of salvation is faith or trust in God. We receive salvation and apply it to our lives through faith, which today specifically faith in Jesus. Biblical faith is more than intellectual acceptance or verbal agreement. It always includes a response or application, namely, the obedience of faith (Romans 1:5).\nTo summarize in every age salvation has come by grace through faith based on the blood of Christ, and faith is always expressed by obedience to God's Word. Old testament believers look forward to the Cross without fully knowing God's plan, while New Testament believers look back to the Cross.\nRomans 8:29\u201330 describes God's eternal plan for the church: \"for whom he did foreknow, he also did predestinate to be conformed to the image of his son, that he might be the firstborn among many brethren. Moreover whom he did predestinate, then he also called: and whom he called, then he also justified: and whom he justified, them he also glorifies.\"\nForeknowledge. Before God created humans, He knew they would fall into sin and therefore planned the cross (1 Peter 1:18\u201320). He foresaw that some would respond to His call and therefore established a plan for the church.\nPredestination. God predestined for His church to be molded into the likeness of His Son. To predestine means to foreordain, to determine in advance, to plan unalterably. Predestination applies to God's plan, not to each person's faith. God predestined the Incarnation, the Atonement, the Church, and the ultimate salvation of the Church. By God's grace, everyone has the freedom to choose whether to be in the church (Titus 2:12.) Our salvation is not merely a wish or a possibility. It is a certain event if we will remain in the church. Moreover, salvation consist of transformation into the image of God. We will not become God, but we will ultimately receive a sinless nature in an immortal body like that of Christ. God did not intend for the sun to be the only human to conquer sin and death. God came in the flesh to have many sons and daughters. The man Christ is the first in God's spiritual family, but God intends for the Son to have many younger siblings who enter the family after him. Christ is our brother (Hebrews 2:17). We are to follow Him and become coheirs with Him Romans 8:17. If we trust in Him, we will one day truly become like Him (1 John 3:2). When our faults and failures cause discouragement, we must not give up, for if we will remain in God's plan we have the guarantee of ultimate, total, and permanent victory.\nCalling. Based on his plan, God calls people to respond. The offer of salvation extends to everyone, but only those who respond in faith are chosen (Matthew 20:16, Revelation 22:17). This passage speaks of an effectual calling. Only those who respond to God's grace are called out of soon.\nJustification. God justifies those he calls out of sand he declares them to be righteous. Their sins are washed away, and they receive Christ righteousness (Acts 2:38, 1 Corinthians 6:11).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just as national polls were a poor indicator of the 2016 outcome, they're a poor indicator of what the politically meaningful numbers are today.\nA typical national poll will have a lot of interviews in deep blue states with high populations \u2014 states like California and New York \u2014 the fact that those voters are negative toward Trump isn't all that important because they'd be negative toward any Republican, they're just slightly more negative toward Trump and it drive his numbers down.\nThe truth is Trump is still doing well among Trump voters, the ones who elected him and gave Republicans majorities in both houses of Congress.\nAccording to the latest Quinnipiac poll, and it's one of the more negative toward him so I'm not cherry-picking here, 81% of Republicans approve of the job Trump is doing. A majority (50%) of whites without college degrees approve.\nWhat we're really seeing in these approval numbers is a shocking and somewhat unprecedented polarization among Democrats. Only six percent of Democrats in that Quinnipiac poll approve of Trump's performance. If you look back at Obama, even when his numbers where at his lowest he still had double-digit approval among Republicans.\nJust like last November, when people failed to look beyond the national topline numbers and really understand what the data meant, we're seeing a lot of crowing about Trump's topline approval rating and not a lot of thinking about what's going on to make these numbers.\nRight now the voters who elected him are still with him and attempts to paint him as a failed President are about as useful as the sad bleating about Clinton having won the national popular vote. That's just not how our system works and we're not a country governed by the opinions of California and New York leftists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To replace the 127, we needed to design a car that exceeded expectations. With the Uno we did just that.\nJanuary 1983. The Uno is presented to the public. Florida's Cape Canaveral was chosen as the location for the \"launch\".\nIHI turbocharger, intercooler. The Uno that goes from 0 to 100 in eight seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where is a party wall? When a semi detached or terraced house shares a wall (or walls) with an adjoining owner (neighbour), that wall is known as a party wall. A \"party wall\" separates buildings which belong to different owners. A separating floor can be a \"party structure\". Where a wall separates two different sized buildings, only the part that is used by both properties is considered to be a party wall. The rest belongs to the person on whose land it stands.\nThe Party Wall etc Act 1996. Since the Party Wall etc Act 1996 came into force, building owners in England and Wales have a legal procedure to follow when building work involves a party wall, party fence wall, excavations or building on the line of junction. The Act is designed to minimise disputes and should be regarded as an Act to enable an owner to exercise rights and undertake obligations associated with \"party works\". Paul Sheldon has been involved with party wall legislation since 1988, initially under The London Building Acts (Amendment) Act 1939, the precursor of the Party Wall etc Act 1996. Sheldon & Company can advise on the Act and it's procedures in order to allow a building owner to consider their duties and notify the adjoining owner. The adjoining owner may consent to the works or may dissent. If the adjoining owner dissents then he can appoint the same surveyor nominated by the building owner termed an \"agreed surveyor\" to act for both parties or he can choose to appoint a seperate surveyor.\nRaising the entire party wall, including cutting off any parts of the adjoining owner's building which would otherwise prevent this being carried out.\nExcavating foundations within six metres of an adjoining structure and below a line drawn at 45 degrees from the bottom of its' foundations.\nWhat doesn't the Act cover? The Act does not cover everyday minor jobs such as fixing plugs, screwing in wall units or shelving, adding or replacing recessed electrical wiring or sockets or replacing internal wall plaster.\nI am proposing to carry out party works. What do I do next? If you intend to carry out any works which fall under the Act, you must give appropriate written notice to your adjoining owners at least two months before starting any party wall works. One month for \"line of junction\" or excavation works. Notice must be given to all owners (freeholders and leaseholders). Where party works are proposed to a party floor then notice must be given to the owners living either above or below your property as floors can also be party structures. Sheldon & Company can assess any proposals and advise on what needs to be done to comply with the Act. Sheldon & Company can speak to the adjoining owner and serve notices to ensure your works are lawful.\nWhat if there is a dispute? If an adjoining owner does not consent then the Party Wall etc Act 1996 provides for both parties to each appoint a surveyor or an \"agreed surveyor\" (one surveyor acting for both parties). These are statutory appointments and surveyors have a duty to act impartially. The surveyor(s) will draw up a document called \"an Award\". This details the works to be carried out, when, and how it will be done and records the condition of the adjoining property before works commence. It may also grant access to both properties so the surveyor(s) can inspect work in progress. The Award will determine who pays for the work if this is in dispute. Generally, the building owner who is carrying out the work pays for all expenses unless the works are due to want of repair to a party structure.\nQ. Do you require general advice?\nQ. Are you proposing to start building work in the near future?\nQ. Are you concerned that work is about to start at an adjoining owner's property but have not received any written notification?\nQ. Have you received a notice and don't know what to do next?\nIf so then contact Sheldon & Company on 01934 751228 for a free initial consultation.\nThe photograph above shows the type of Work Paul Sheldon takes on where he was involved helping to safe guard the adjoining owner's property in Cannon Street London.\nPaul Sheldon was involved in the repair of this party wall after unannounced and unlawful demolition of attached building caused significant damage and structural instability.\nMember of The Pyramus & This be Club, a society of people interested in promoting excellence in party wall practice. Visit their web site for guides and publications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In this episode of The Drum Hang we get inspired to play single strokes between our right hand and right foot. First we watch the masters Roy Haynes and Tony Williams utilize this idea. Then I give you some easy to use ideas, followed by Obed Calvaire showing us what is really possible by playing amazingly clean and fast singles during a great solo!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The annual stipend across consortium sites will be $24,000.\nDoctoral interns will be hired as employees of the Western Interstate Commission for Higher Education (WICHE), an inter-governmental organization that provides consultation and technical assistance to the HI-PIC program. Health benefits will be provided to all doctoral interns through WICHE. Annual vacation, professional, and sick leave will also be provided. Doctoral interns will follow the holiday schedule of their training site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sept. 30, 2014 \u2014 When Phoenix Comicon dropped a press release into our mailbox this weekend stating that a companion event \u2014 called a Fan Fest by the folks at Phoenix Comicon \u2014 would make its first appearance Dec. 12 to 14 at University of Phoenix Stadium, we were not really surprised.\nthe people want as much as they can get and even tually, the Phoenix Convention Center might be too small for the Phoenix Comicon.\nBut, that it would be held about 10 days before Christmas, was a surprise that seems to be setting geeks giddy for Christmas filled with goodies bought at Fan Fest.\nAccording to many of the Facebook groups populated by longtime Con participants \u2014 and to which our publisher and others are members \u2014 this new event comes as a complete surprise. As such, there are few details as to exactly what will be going on at the Fan Fest. The press release states little more than the intentions of the event: The new show will focused on creators, artists, actors, costumers and exhibitors to create a celebration for the fans of pop culture.\n\"We believe there is a need for a strong artist and fan focused show during the winter.\nBut, Solberg said this will hopefully be a yearly event that will stand apart from what has become known as Phoenix Comicon.\nSo geeks and pop-culture fanatics alike, save your pennies. There will be a last minute stop on the shopping train and this one will go through Glendale.\nFor more information, please visit phoenixcomicon.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you need help holding your soft serve while strolling New York City's Lower East Side, let these little fish friends assist you.\nAdorable fish-shaped cones called taiyaki are the latest treat that are captivating U.S. dessert lovers.\nJapanese snackers have been enjoying the dish for over 100 years, according to the Japan Times, but it was originally filled with red bean paste or custard rather than ice cream.\nTaiyaki NYC in Chinatown has modified the dessert slightly, making a larger fish mouth to hold delicious soft serve, while leaving a surprise red bean or custard filling in the fish's tail.\nAsian-inspired desserts have been becoming increasingly popular, especially in western countries, not only because of taste but also due to their highly Instagram-worthy looks.\nTreats like bubble waffle cones from Hong Kong or soft serve lounging on a bed of cotton candy from Japan don't just sound heavenly to eat, they'll also get you plenty of likes.\nHonestly, every dessert should make you feel like a mermaid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JuicySelfsucker CliffKnight livejasmin. KarinaEva10. AniseNeedSaid. LoveLoveCats.\nFinestJadeurhottieprincessDaariaaTitaniaBlack .SxyBootyssexybird4uKILEYHOTBATHROOMQUEEN5 .KimmTaylorJennyVSJimSingleNowNiraWolffe .miky1982AngelAprillAniseNeedSaidMsFoxySkyYvette .SatomiRoseDaariaaDaariaaInkbbs .JadeCharma0GlennAstroMadameOMadameO .ForestPussy3ALESSYAPINK1LoveLoveCatsionutzz112 .BeautifulMaryFoUAniseNeedSaidWonderBlondieSophiaLucia .MsFoxySkyYvetteBATHROOMQUEEN5SonnataFinestJade .Hornybarbie69NaughtyEricKrystalSweetyalindaoij .KILEYHOTTheSnowQueenForUPlayHotGirls69MatureSally .\nMeadow1TitaniaBlackAkashaQuinnBeautifulMaryFoU .ExtremeSensationionutzz112CATALINADELMARAmaryBella .SophiaLuciaPureJoanSheshaChristyKlatten .VanessaPrydeNaomiClayFinestJadeMelissaMorris .NiraWolffeBlondRosseCoupleStarsXXMonicaKruger .TOTALpleasureXXXAkashaQuinnKimmTaylorjasonsteel01 .SoniaCrystallSoniaCrystallSunnySmile18JennyHotBaby .CutieDanaFelicityLevyPekyAmyJennyHotBaby .NewCoupleHotSexTONYBRANDTForestPussy3SexyRoyaltySB .CassieGraceKimmTaylorSonnataMeadow1 .4Guys4NastysexCutieDanaElzaGoldCATALINADELMAR .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwehg b/data_all_eng_slimpj/shuffled/split2/finalzzwehg
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+{"text":"Syndication: realRADIO1, and potentially to a broadcast network in Taiwan!\nMade In The UK Show, show 79 comes out on Sunday, hosted by us!\nThe next show will be episode 089, broadcasting live on Friday 18 December 2009, in The Bugcast Chatroom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A hip replacement is a procedure in which a surgeon removes a diseased or damaged hip joint and replaces it with a new, artificial joint. The hip joint actually consists of two parts\u2014a ball-like structure at the top of the thighbone (or the femoral bone) and a socket (called the acetabulum) that the joint fits into. In replacement hips, the ball and socket components can be made of a number of different materials, including metal, ceramic, and plastic.\nHow Can a Hip Replacement Be Defective?\nSometimes an artificial hip can lead to problems, such as the ball becoming dislocated from the socket. This often happens because the artificial parts are smaller than the natural hip components. In other cases, a patient may be injured by a defective hip replacement.\nMetal-on-metal hips \u2013 hips that feature a metal ball and socket \u2013 have been linked to a particularly dangerous defect. The constant friction between the metal ball and socket causes tiny pieces of metal to be released into the patient's bloodstream and surrounding tissue. This condition, known as metallosis, can result in swelling, tissue damage, bone loss, and even blood poisoning in some cases. As a result, a number of makers of hip implants, such as DePuy and Stryker, have recalled many of their metal-on-metal hip models.\nThe product's design was defective, meaning the injury happened because there was a dangerous flaw in the way the product was designed to work.\nThere was a manufacturing defect, meaning something went wrong during the production process that made the product unsafe.\nThe company failed to give adequate warnings about the product, meaning the company failed to warn consumers about dangers that aren't obvious to the average user.\nIn defective hip replacement lawsuits, plaintiffs typically argue that their metal-on-metal implants had a design defect, since the faulty design led to friction between the metal parts and the associated health problems. With a manufacturing defect, a plaintiff would argue that even if the metal-on-metal design was normally safe, the particular implant she received had a dangerous defect because it wasn't built as intended. A plaintiff could also argue that if she had received adequate warnings about the dangers associated with the implant she wouldn't have agreed to the procedure in the first place.\nIn a negligence cause of action, the plaintiff must demonstrate that the company knew or should have known about the dangerous condition but sold the product anyway. In a strict liability cause of action, on the other hand, a plaintiff doesn't need to prove that the company knew or should have known about the danger, only that it sold a defective product that caused the plaintiff harm. Strict liability is commonly used against manufacturers who sell products to consumer along with those engaging in abnormally dangerous activities. Lastly, in a breach of warranty suit, the plaintiff claims that she was injured when she relied on a company's promise that a product was safe for a certain use.\nIf you think you may have received a defective hip replacement, it's important that you consult with your doctor or surgeon to discuss your symptoms or injuries. You should also write down as much information as you can remember about when your symptoms started and what you've been experiencing. This information may be crucial in proving your injuries and obtaining compensation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Committed to alleviating veterans' stress and honoring their enduring sacrifices.\nWent to the range yesterday with a couple OEF\/OIF veterans. It was fun hanging out with them, chatting about the past as well as the future. Sometimes we deceive ourselves into thinking we're alone now that we are no longer active. That's not true, we still have brothers and sisters everywhere!\nHonor Our Heroes from Jeff Mayoh on Vimeo.\nHonor Our Heroes Foundations first fundraising event was huge.\nCopyright 2017 Honor Our Heroes Foundation. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The prize celebrates outstanding ESRC research and success in collaborative working, partnerships, engagement and knowledge exchange activities that have led to significant impact. The 2015 awards ceremony took place in London on 24 June.\nResearch by Professors Colin Mason and Richard Harrison has played a crucial role in stimulating business angel investment worth \u00a3750 million a year in the UK.\nProfessors Jenny Kitzinger and Celia Kitzinger have created a much-needed online resource supporting relatives of patients in long-term coma states.\nA unique study led by Dr Hester Parr has improved the way UK police officers relate to missing persons and transformed policing guidance on the handling of missing persons and their families.\nDr Jennifer Doyle's valuable research is helping the social housing sector foster social change in deprived communities through improvements in measuring, monitoring and evaluating social impact.\nDr Jane Dyson's inspiring documentary about the challenges facing young people in the Indian Himalayas has reached school children, students, policymakers and others in many different countries.\nWith his long term commitment to maximising impact opportunities for his own research and that of colleagues and collaborators, Professor Charlie Jeffery is a model of leadership in the social sciences.\nA six-year research programme driven by Dr Victoria Lavis has shaped a new national equalities policy framework and new policies and guidance for the care and management of transgender offenders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Master Developer of Cyberjaya, Setia Haruman Sdn Bhd, is investing RM2.5 billion over the next five years to develop four mixed residential projects in Cyberjaya, dubbed an \"Intelligent City\".\nChairman Tan Sri Mustapha Kamal Abu Bakar said the projects, namely CBD Perdana 3 and 5, APEX Residence and a proposed mixed residential development will be riding on the \"New Wave\" development of Cyberjaya.\nSetia Haruman would also build infrastructures such as roads and drains, water reservoirs, drainage systems, sewage treatment plants and fibre optics to service and support the \"New Wave\" development till 2016.\nThe projects gross development value is about 30 per cent more than the amount invested, he told a media briefing.\nMustapha said over the last three years, 16 major developers had acquired large tracts of land for residential, commercial and institutional developments.\nThe concurrent investment and involvement of these developers have created a \"New Wave\" in Cyberjaya, resulting in a residential and commercial boom.\nThis \"New Wave\" is expected to continue over the next five years, with the population to double to 100,000.\n\"Property prices in this city has appreciated by 30 per cent over the last two years and early investors have enjoyed substantial gains.\n\"With the many developments being undertaken by various developers, we expect Cyberjaya to be transformed from a development hotspot into a vibrant city of the future,\" he said.\nCurrently, the ongoing and upcoming accumulative investments in Cyberjaya by 16 developers is about RM20.6 billion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Mike Ricardi survived the Station nightclub fire, but his best friend, James Gahan, died in the fire. Ricardi held a framed picture of himself, Jack Russell of Great White, and Gahan.\nA closer look at the photo of Mike Ricardi, Jack Russell of the Great White, and James Gahan.\nChris Fontaine talked to her grandchildren, Hannah and Gabriel, at her daughter's Southbridge home. Fontaine lost her son in the Station nightclub fire, but her daughter, Melanie, survived.\nMark Fontaine died in the Station nightclub fire.\nMark Fontaine and Melanie Fontaine in a family photo.\nMelanie Fontaine played with her daughter Hannah in her Southbridge home.\nMelanie Fontaine and her boyfriend, John Longiaru, in a high school prom picture.\nA burnt concert ticket that Mark Fontaine was carrying on the night of the deadly Station nightclub fire.\nChris Fontaine installed a tray onto the booster seat for her granddaughter, Hannah, while grandson Gabriel watched.\nGina Russo survived the Station nightclub fire, but suffered serious burns.\nGina Russo, shown in her Cranston, R.I., home, has had more than 50 surgeries from her injuries.\nGina Russo showed off the scars from her injuries.\nFamily and friends of Nick O'Neill, who died in the Station nightclub fire, celebrated what would have been his 28th birthday at their home in Johnston, R.I., on Jan. 27.\nA picture of Nick O'Neill, who died in the fire, sat over the fireplace in the home of his parents, Dave Kane and Joanne O'Neill.\nA detail of the cake for Nick O'Neill.\nAsher Nicholas O'Neill (right) blew out the candles during the birthday celebration for Nick O'Neill.\nDave Kane and his wife, Joanne O'Neill, invited Lisa Powers (right), a medium, to communicate with their son Nick O'Neill on his 28th birthday celebration.\nDave Kane and Joanne O'Neill listened as Lisa Powers spoke.\nDave Kane hugged Lisa Powers after her session. At left are pictures of Nick Kane and his brothers.\nA carousel horse was installed in the home. Before the fire, Nick had joked with his brother about singing songs from the musical \"Carousel,\" without having seen it. The movie is about a young boy who dies and returns to watch over his family. Nick eventually bought his mother a toy musical carousel that went off on its own on Mother's Day, the year Nick died.\nMemorials to the victims at the site of the Station nightclub fire in West Warwick, R.I.\nThe letters on a sign outside Saint Rocke in Hermosa, Calif., were removed as Jack Russell neared the end of his set at the club on Feb. 7, 2013.\nJack Russell was the lead singer of Great White the night of the fatal fire in 2003. He now fronts his own version of the band, which goes by the name Jack Russell's Great White.\nJack Russell (right) talked with a fan after his performance at the Saint Rock club in Hermosa, Calif.\nThe version of Great White without Jack Russell is fronted by singer Terry Ilous (left). Guitarist Mark Kendall played along during a concert at the Sycuan Casino in El Cajon, Calif., on Feb. 1, 2013.\nGuitarist Mark Kendall and Jack Russell were founding members of the band Great White, although they now play in two different iterations of the band.\nJohn Stenson, owner of the Eire Pub in Dorchester, installed a state-of-the-art sprinkler system after the Station nightclub fire.\nIn 2006, Great White's tour manager, Daniel Biechele (left), pleaded guilty to 100 counts of misdemeaner manslaughter in the Station nightclub fire. Biechele was sentenced to four years in prison, and was released in 2009 after serving less than two years.\nBrothers Michael (second from left) and Jeffrey Derderian (second from right), co-owners of the Station nightclub, pleaded no contest to 100 counts each of involuntary manslaughter. Michael Derderian was sentenced to four years in prison, but spent less than three years behind bars due to good behavior. Jeffrey Derderian was spared jail time with a suspended sentence and 500 hours of community service.\nJay McLaughlin read his victim impact statement prior to the sentencing of Michael and Jeffrey Derderian, co-owners of The Station nightclub. McLaughlin is the brother-in-law of Michael Hoogasian, who died inthe fire. Behind him were his wife, Paula, and Claire Hoogasian, Michael's mother.\nJudge Francis Darigan admonished those giving victim impact statements to stay within specified guidelines during court proceedings for Michael and Jeffrey Derderian.\nSisters Jessica (left) and Kristen Garvey restrained their mother, Dina A. DeMaio, from speaking out after reading a victim impact statement about her daughter, Dina A. DeMaio, who died in nightclub fire. DeMaio, spoke during the sentencing of nightclub owners Michael and Jeffrey Derderian.\nMichael Derderian broke down in tears while his brother, Jeffrey Derderian, gave a statement during their sentencing.\nStation nightclub co-owner Jeffrey Derderian (second from left), his wife Linda (right) their son, Max, and Michael Dererian's wire Kristina (left) listened to a prayer prior to a news conference on July 19, 2007, in Warwick, R.I. Derderian announced the formation of The Station Education Fund, a charity that will benefit children who lost a parent in the Station nightclub fire.\nJoe Kinan, 43, was discharged from Massachusetts General Hospital on Oct. 25, 2012, after receiving a hand transplant. It was the first procedure of its kind at the hospital. Kinan was severely burned in the Station nightclub fire.\nGina Russo (left), board president of the Station Fire Memorial Foundation, spoke during a press conference at the site of the nightclub fire in West Warwick, R.I., on Sept. 28, 2012. The owner of the site, Ray Villanova, donated the land for a permanent memorial, bringing an end to a years-long effort by families of those killed and survivors of the blaze to secure the site for a memorial.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My current project, aspieland.com, is up for public testing. It will be a non-commercial site directory of sites by and for people on the autistic spectrum. The goal of this site is to focus on individual home pages and make it easier for us to find eachother on the web. Large community and resource sites will be listed as well.\nForums are being set up to help people make their own web sites, and people have already volunteered to help with design, hosting, and more.\nCome and visit, browse, list your site! Please contact me to report any bugs or make suggestions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you're looking for a family vacation or a couple's weekend with your friends, Sundown Cabin Lodging's 1-bedroom cabin at Beavers Bend offers comfortable accommodations for up to four people.\nWe offer 3 1-bedroom cabins at Broken Bow that comfortably house up to four people each. Best of all, your family vacation can include all the members of your family; Sundown Cabin Lodging's cabins are pet-friendly. With the beautiful natural area surrounding our cabins, your four-legged family member has a chance to enjoy nature, as well.\nJagged Edge Cabin, has a spectacular view overlooking the East Fork of Lukfata It's situated on a 50-foot cliff and has big windows, along with a large porch, so you can kick back and just relax on the tiered viewing deck.\nLone Wolf Cabin is conveniently located in the prestigious gated Timber Creek Trails development. Lone Wolf is just minutes from Beavers Bend State Park, Cedar Creek Golf Course, Broken Bow Lake, Hochatown Restaurants.\nHideaway Haven Cabin (New-January 2014) 1 Bedroom Luxury Cabin \u2013 Sleeps Up to 4, Wi-Fi, Free Unlimited Netflix Movies, Hot Tub & Outdoor Wood burning Fireplace located in Timber Creek Trails Near Broken Bow Lake and Beavers Bend State Park.\nLog cabins may sound old-fashioned, but rest assured, our 1-bedroom cabin at Beavers Bend is anything but! It comes complete with all the modern amenities, such as a fully equipped kitchen, Wi-Fi, and high-definition flat screen TVs, complete with DIRECTV.\nAnd we know there's nothing better than cooking outdoors and enjoying the sun during the summer. That's why our 1-bedroom cabin at Beavers Bend offers an outdoor propane grill, as well as a handcrafted fire pit and large covered deck that's ideal for entertaining.\nOr, for the nights you feel like letting someone else do the cooking, you're close to fine dining when you stay at Sundown Cabin Lodging's Jagged Edge Cabin. There are plenty of dining options nearby, including Abendigo's Grill and Patio and Steven's Gap Restaurant.\nIf you're traveling with your family, you know how frustrating it can be to share a bathroom. That's why our 1-bedroom cabin at Beavers Bend comes with two complete bathrooms, so you won't have to worry about lineups in the morning.\nWe all know how inconvenient doing laundry after a vacation is. This 1-bedroom cabin comes with a washer and dryer, so you can pack less and bring home clean clothes, or wash your clothing after a day of outdoor activities in one of the surrounding parks.\nWhen you stay at Sundown Cabin Lodging's 1-bedroom cabin at Beavers Bend, you're close to Beavers Bend Resort Park, one of the area's biggest attractions. The park offers fishing, hiking, horseback riding, and so much more. It's the ideal destination for the outdoor sports enthusiast.\nIf you're an avid golfer, you're just a short drive to Cedar Creek Golf Course from our 1-bedroom cabin at Beavers Bend.\nAt the end of your day, spend some time relaxing in the four-person hot tub or in front of the gas\/log fireplace in your 1-bedroom cabin at Beavers Bend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As we traveled across Poland on a recent parish pilgrimage, we watched two films about Pope St. John Paul the Great. The first was Karol: The Man Who Became Pope. The second was Pope John Paul II, starring Cary Elwes and Jon Voight. The first film concentrates on John Paul's life up to his election to the papacy, while the second continues with Jon Voight's astounding portrayal of the pope's accomplishments through his decline and to his death. We watched the films as our tour bus as it made its way from Krakow to Warsaw, the beautiful countryside of Poland rolling before us.\nThe geographical combined with the biographical, and the geographical and biographical combined with the historical and theological. In John Paul's life the different strands of twentieth-century European history combined in a remarkable way. As a bridge between West and East, Catholic Poland is arguably the heart of Europe, and as the heart beats within the body, so the tumultuous history and passionate faith of the Polish people beats at the heart of Europe. Karol Wojtyla captures the story and captivates the world. When the history books are written, he will emerge as one of the most powerful, passionate, and dynamic personalities of the age.\nAs a college student in Krakow, he was caught up in the middle of the 1939 Nazi invasion of his country. He had to study for the priesthood in secret while working in a stone quarry on the outskirts of the city. Ordained in secret, his life as a young priest and university professor was overshadowed first by the Nazi threat, and then by the oppression of the Soviet occupation of his country. Both regimes considered the Catholic authorities as their enemy, but with shrewd intelligence and determination John Paul opposed the regimes by outwitting them with Christian passive resistance. His surprise election to the throne of Peter in 1978 granted him the global audience to work behind the scenes for their eventual overthrow.\nWhat is remarkable in the unfolding of these events is the role played by an unknown nun in Krakow. Sister Faustina Kowalska was one of ten children. Brought up in a rural hovel and poorly educated, she longed to join a religious order. Finally being accepted into the Sisters of Mercy in Krakow, she served as a kitchen servant and portress in the monastery. This simple, enclosed nun was granted visions of the resurrected Christ and told to propagate the image we now know as the Divine Mercy. What is not so well known are the remarkable connections between the poor, hidden nun and the intellectual, dynamic pope.\nSister Faustina died in her convent in the outskirts of Krakow in 1938. That same year Karol Wojtyla arrived in the city from his boyhood home in Wadowice to attend university. Two years later, the stone quarry where he worked was adjacent to Faustina's convent. The image of the Divine Mercy carries a powerful message. The resurrected Christ steps forward from the dark background as if stepping out of the tomb. He pulls back his robe so that rays of red and white might burst forth. His other hand is raised in welcome and blessing. As an image the message is wordless and universal. Coming from the heart of Poland, the message is as poignant as it is powerful.\nBeing comparatively unlettered, Sister Faustina wrote her diary phonetically, and because of faulty transcription and translation, after the war, the Vatican ruled that her writings were not theologically sound, and banned the already-burgeoning devotion. But near the end of Vatican II, then-Cardinal Wojtyla launched her cause for canonization, and Pope Paul VI soon ordered a re-evaluation of Faustina, approving her writings and lifting the ban on the Divine Mercy devotion. In the year 2000, now-Pope John Paul II canonized the hidden nun of Krakow. The message and the image can now be seen in proper historical context.\nThe answer is that God was with an almost illiterate nun in a convent in Krakow. When she died in 1938, God was with an unknown college student named Karol Wojtyla. The seeds of the message of God's mercy to the world were planted in the exact place where the very heart of darkness and the very worst examples of humanity's inhumanity were to be unleashed. God was not absent in the darkest corner of Europe, but like the seed of the crucified Christ, planted in the tomb to burst forth in victory, he was planting the seed of the Divine Mercy image in the midst of the darkness to burst forth later through the combined work of the hidden nun and the dynamic pope.\nToday when you visit Krakow two great basilicas have risen in a remarkable way. A modern structure seating 5,000 people has been built in the grounds of the Sisters of Mercy convent. On Divine Mercy Sunday they welcome hundreds of thousands of pilgrims. A few hundred yards away, on the site of the stone quarry where he worked, a new basilica honoring Pope St John Paul II is being completed. The two basilicas\u2014one to the hidden nun and the other to the great pope\u2014are lined up and connected by a walkway and bridge, and both are a permanent witness that the oppression, cruelty, and fear propagated by the pagan and atheistic ideologies were ultimately defeated by the power and passion of Poland.\nAn amazingly vibrant and strong people. Among the many who have suffered in the turbulent 20th century the Poles must be near those who have suffered most, the Nazis, the communist regime, they're gone, Poland remains.\nA beautiful article. G.K. Chesterton and Professor Charles Sarolea were of the opinion that a strong Poland is necessary to prevent a German-Russian alliance which brought disaster to Europe several times in history (to mention only the 1939 Molotov-Ribbentrop Pact). On the other hand, there are those who think that Poland's sovereignty is unnecessary, that Poland's Catholicism should be \"modernized,\" and Poland should be a fiefdom of either Germany or Russia, or both. The battle is going on.\nThe article is very good , but Dr. Thompson again either over-simplifies or, as with her unwarranted and below the belt criticism of Polish war hero and founder of postwar conservatism Henryk Krzeczkowski again does not give Americans an accurate representation of things.\nYou are of course right that Charles Sarolea wrote a fantastic book about Poland with an equally great preface by GK Chesterton, but he also wrote an equally amazing book about Russia called \"Europe's Debt to Russia\" in which he outlines the Christian piety of the Russian people and their valiant struggle to maintain a Christian order which allowed Europe to develop at peace.\nI will repeat here that you cannot understand Sarolea on Poland without Sarolea on Russia. It is like trying to understand Machiavelli's Discourses on Livy without reading the Prince.\nThe battle is indeed on and it is between so-called Polish conservatives who signed the Lisbon treaty ensuring German-EU control and now want to march German troops into Poland in order to \"defend Poland\" from Russia and those national conservatives who want a truly free Poland without foreign soldiers and who recognize that a Christian , Orthodox and pragmatic Russia is a better friend than a Western Europe which is fast becoming an ocean of islam manned by pagans.\nI lived in Poland as a student in the late 1970s. I will never forget the night one of my Polish friends came to me in great joy and pride because Karol Wotyla had been named Pope. Pope John Paul II was a great man, indeed, and Poland is a unique country.\nPoland has indeed been used by God. Earlier King John III Sobieski led the Polish forces to break the Ottoman Seige of Vienna in 1683 and stem the Muslim invasion of Europe from the east. Looks like God has a plan.\nIn the dark days of the 19th Century, when Poles were subjugated by Prussians and Russians, there arose Polish Messianism. Essentially it held that just as Christ had to suffer to atone for the sins of mankind, so too Poland would have to suffer, but that it would resurrect.\nYes, and conservatives have been attempting to shake Poland free of messianism ever since because it is theological heresy and political insanity. No people are Christ though human suffering is often Christ-like. A political platform which embraces Christ-like suffering is suicide. Polish messianism succeeded only in sending more and more Poles to the grave in the XIXth century while reducing political liberty. Only the realism of Dmowski's national conservatism actually allowed for the ressurection of Poland. St. John Paul had similar challenges \u2013 he had to temper the messianism of Solidarity which had brought the country to the brink of anarchy and war by 1981. Overthrowing Communism without destroying Poland was a work of magnificent statesmanship and the Pope is to be comended and emulated. As is Wyszy\u0144ski.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adjectives are words like 'big', 'little', 'fast', etc, that are used to describe things.\n\u3044-adjective, and 'dentouteki-na' is a \u306a-adjective.\n\u200b\u3044-adjectives all end in \u3044. The \u3044 ending can be taken off and replaced with other endings to add to the meaning. e.g, to change to the negative or to the past tense.\n\u200bIt isn't a small town.\nThe town is not small.\n\u200b\u306a-adjectives have a \u306a ending which is used to link the adjective to the noun that follows. However, we don't need to add \u306a if there is no noun following the adjective.\n\u200bIt isn't a pretty town.\nThe town is not pretty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Leo Barrera Photography \u2013 Specializing in creative portraiture, event photography, and multimedia\/video projects. Whether you need a clean, classic photo or a sophisticated audio-video production, I provide a variety of products and services to support your vision.\nSpecializing in creative portraiture, event photography, and multimedia\/video projects. Whether you need a clean, classic photo or a sophisticated audio-video production, I provide a variety of products and services to support your vision.\nI have decided to try something different. I have noticed that the iPhone makes videography an ease. I have found myself taking more and more videos as time passes. Therefore, I have decided to start adding video posts to this blog.\nThis was taken on June 22, 2018 at the Seattle Aquarium. I was visiting the city earlier this year. I found this fish to be somewhat majestic as it floated there.\nI thought this house looked interesting; decrepit, but still holding on to some long lost sense of dignity.\nI liked how this, more or less, bright tractor stood out in the middle of this dirt field. It works for me on a couple of levels.\nMy wife and I took a train ride to Piru CA this past weekend. I will be sharing some photos.\nBria Royal, Vicko Alvarez, and Cathy Camper selling their wares.\nIf you have any questions, want to hire me, or want to purchase some pictures you can email me by clicking here. You can also call me if you prefer at (626) 230-9166.\n\u00a9 2014-2018 Leo Barrera and lbarreraphotography.com, all rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With lots of fast-paced games and plenty of showboating to view, 5-a-side cuppers is the perfect spectator's antidote to the windswept and rain-drenched 11- a-side football on view for the other two terms. The favourites should be New College, who will feel they have a strong chance as league holders. Throughout the season they demonstrated strength in depth and possess dangerous forwards. Wadham will of course be having a ball this summer. As cuppers champions and a respected college around the university everyone will be wishing them well in this campaign. Watch out for June (27), dependable center-half Guiler (6) and talented striker Smith (10). When it comes to college football only a fool would write off Magdalen's chances, despite their relegation from the Premiership this season. With a plethora of good players, great team spirit and a fantastic pedigree no one would fancy meeting them in the draw. There's a dark horse around that might well kick up a storm this year in the shape of Hertford. Frustrated by a disappointing league campaign, their lads will have plenty of incentive to forget the frowns and sighs. Hertford are a quality outfit where their tidy build up is better suited to the five-a-side format. Impressive captain Sam Evans, with his good first touch and quick thinking, typifies their squad. However surely the favourites this year have to be the nearly men of college football, Catz. Having come so close in both cuppers and league everyone in Manor Road craves that first ever piece of silverware. With the class of Durnford, Elliott, Johnson and Hughes allied with the emergent excellence of Brown and Bargate, there is considerable talent in their stable. Perhaps player of the tournament will be Redmond, who may well prove a revelation between the sticks. Without the first choice goalies Catz may have suffered but the suspicion is that former captain Redmond may be the best keeper in cuppers this summer. So there you have it: a comprehensive look into the potential of some of the leading names in the tournament. Having read this you should be inspired to get on down to Iffley to support your college. Bring a picnic (Kit Kat anyone?), a clever chant and maybe your stereo player and bask in the spectacle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: Search official index of accident reports taken by the Highway Patrol.\nDescription: Search official statewide database for accident reports. County and crash date fields will be required unless you have the document number or last lame. If you search using last name the county is not required. The crash date field will be required unless you have the document number.\nDescription: Billed as a way to check your own drivers license, the search requires a DL number, last four of SSN, DOB and first initial of last name to access.\nDescription: Search official state professional licensing board records. Includes Accountancy, Architects, Chiropractic, Collision Repair, Cosmetology, Counselor, Social Worker, Marriage & Family Therapist, Dental, Embalmers and Funeral Directors, Engineers and Surveyors, Medical, Occupational Therapy, Phys Therapy, & Athletic Trainers, Dietetics, Nursing, Manufactured Homes Commission, Barber, Optical Dispensers, Optometry, Orthotics, Prosthetics, and Pedorthics, Psychology, Respiratory Care, Sanitarian, Speech Language Pathology & Audiology, Veterinary, Medical and Pharmacy. There is no \"search all at once\" option.\nDescription: Enter identification number (presumably they mean Vehicle Identification Number) to learn title status. Information on private persons not released with record.\nDescription: Search accident and incident reports by report number, name, location or date of accident.\nDescription: Search official records to identify accident and incident reports.\nDescription: Search accident reports by report number or date of accident or name or location.\nDescription: Search accident reports by Report number OR Date of the accident\/incident OR Driver\/Victim's Last Name OR Location of the accident\/incident.\nDescription: Search accident reports by report number or date of accident or name.\nDescription: Search official records to identify accident or incident reports. NOTE: Login and password are both public.\nDescription: Search official records to identify accident or incident reports.\nDescription: Search accident reports by report number or name and date of accident or name and location. Includes the jurisdictions of Cardington, Dayton, Delaware, Jackson County, Jackson, Logan County, Monroe County, Montgomery County, Oak Hill, Parma, West Liberty and Willoughby Hills.\nDescription: Search daily logs of accident and incident reports by Report Number, Date, Name or Address. Search links to PDF files of actual reports.\nDescription: Search accident reports by report number. There is no name search.\nDescription: Search accident reports by date of the accident and the accident report number. There is no name search.\nDescription: View official records of unclaimed property, forfeited property and items no longer needed. NOTE: See link to list on right-hand side of page under \"Additional Resources\".\nDescription: Search accident reports by date.\nDescription: View accident reports by date of accident. There is no name search.\nDescription: Search accident reports by report number and name or date and name.\nDescription: Search accident reports by date. There is no name search. NOTE: See links by year near top of page.\nDescription: 10-27-2014 REMOVED FROM SERVICE OR OTHERWISE UNAVAILABLE. PLEASE USE THE CONTACT FORM AT BLACK BOOK ONLINE IF YOU KNOW OF AN UPDATE. Search official records to identify accident or incident reports.\nDescription: Search for traffic tickets and fines by city with last name and DOB.\nDescription: View a list of impounded vehicles up for auction. Listing includes color, year, make, model and VIN.\nDescription: Search official records to identify accident reports. Report provides Accident #, Accident Date, Driver Last Names, Accident Location and License Plate Numbers. Reports typically not be posted until at least 5 days after the accident has occurred.\nDescription: Search accident reports by date of accident. There is no name search. NOTE: See link to other reports by year on left-hand side of page.\nDescription: View accident reports by report number. There is no name search.\nDescription: 9-21-2016 REMOVED FROM SERVICE OR OTHERWISE UNAVAILABLE. PLEASE USE THE CONTACT FORM AT BLACK BOOK ONLINE TO LET US KNOW IF YOU KNOW OF AN UPDATE. Search accident reports and traffic citations by report number, date and name.\nDescription: Search accident reports by name or report number or date of accident.\nDescription: Search accident reports by report number or date of accident and name or location. There is no name search.\nDescription: Search accident reports by report number. name, location or date of accident.\nDescription: Search official records to identify accident or offense reports. Report provides location of accident or incident and PDF file of actual report if available.\nDescription: Search accident reports by date of accident.\nDescription: Search accident and incident reports by report number, name, location or date of accident. NOTE: See tabs at top of page to change type of search.\nDescription: Search accident reports by rdate of accident. There is no name search.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you had to make a move with your children?\nAs a real estate professional for over 20 years, I've had the pleasure of seeing families excited about moving into their own home. But I've also seen children not so excited.\nIt's a big change for children, and unknown territory can be intimidating.\nWhen Olivieri was 9 years old, she moved from Puerto Rico to Indiana. The move was sudden and she didn't have much time to adjust to the idea. Trading palm trees for pine trees wasn't exactly easy. Olivieri recalls how jarring her transition to a new home was.\n\"I remember how scary it was to be the new kid on the block. This is my way to help any child who has to relocate to a new place. Children should enter the classroom with confidence.\" Olivieri is now a second grade teacher at Jefferson Elementary School in Gary, Indiana, as well as a youth leader at First Christian Church, where she helps with at-risk teenagers.\nElena's Big Move, inspired by Olivieri's own story, tells the story of a 9-year-old Puerto Rican girl, Elena, who is moving to Indiana. Faced with the scary idea of leaving her island, she turns to her family for support. Her parents give her a camera and an empty scrapbook, assuring her that memories will travel just as well as her belongings. She photographs her favorite places, documenting everything from her school to her family's favorite spot to picnic. When it's time to walk into her Indiana classroom filled with new faces, Elena is braced by the memories she holds in her scrapbook. When her photos prompt a class discussion, Elena learns that home, even a new one, always holds both the past and an exciting future.\nElena's Big Move is an entertaining story where children can easily identify with Elena, whether they will be moving or not. It helps children to understand friends who may be moving, or new students who move into their school. The back of book include some great ideas to actively prepare for the move, even before leaving the old home. There's even a word search just for fun.\nIf you know someone who is going to be moving, this would be a good book to give to them. Even if you are not moving, this book is a good resource for teachers and parents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever heard of a supplement called 5-HTP?\nHave you ever heard of a supplement called 5-HTP?\n5-HTP is thought to help the body in making serotonin, a very import neurotransmitter within the brain that control many functions including sleep cycles, dreams, memory, mood, eating habits and a whole host of other functions within the body. In short serotonin is one of the most important compounds the body makes in order to function properly. Its no surprise then that when someone is experiencing low levels of this component all manner of regulatory processes we may take for granted can break down and cause problems.\n5-HTP is produced in the body from another compound called tryptophan, which is obtained through food rich in protein. However tryptophan isn't found in large quantities in the food we eat, if someone has low levels of serotonin and is looking to boost their levels a 5 HTP supplement can help.\n5-HTP supplements mainly come in capsule form but can also be in tablet form. Often 5-HTP isn't alone in the supplement with many companies choosing to complement 5-HTP with Vitamin B6. Along with the standard health benefits associated with this vitamin it also helps any unused excess 5-HTP to be metabolized and excreted.\nSupplement makers use 5-HTP derived from the seeds of the Griffonia Simplicifolia plant, a completely natural source. It is the intermediate step between tryptophan this very important brain chemical serotonin.\n5 HTP has been available in the US for over 15 years but has been available in Europe and many other places for much longer, as such it has a proven safety record. 5-HTP often comes in capsules but sometimes also in tablet form. It is available in all good pharmacies and health food stores.\nBody Building: Whey Protein Powder What is the Difference between Fish Oil VS Cod Liver Oil? What is the Difference Between Krill VS Fish Oil? The Benefits of Vitamin B12 Shots Are You In Danger of Vitamin B12 Deficiency? Vitamin B12 Shots Complete Information Guide Herbal Breast Enhancement That Truly Works Omega-3 Fatty Acids and Rheumatoid Arthritis Top Fish Oil Products That Are Out In the Market What is the Difference between Omega 6 VS Omega 3?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyfoe b/data_all_eng_slimpj/shuffled/split2/finalzzyfoe
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+{"text":"Care Opinion is a web-based portal for users of health and social care to give positive and negative feedback about the services they receive, which the providers of the services can then respond to. It already operates across Scotland and in some parts of Greater Manchester and is a critical part of a system change approach.\nWe think that people telling their own stories about the experiences that matter to them, and that are important to their lives is powerful stuff. Community Reporting is all about storytelling. At the heart of what we do is helping people to find their voice and describe their own realities through storytelling. We use basic digital skills and pocket technology such as smartphones and tablets to support people to tell their own stories, in their own ways.\nThe issues surrounding the health, welfare and safeguarding of Manchester's children are never far from the headlines. We all have responsibility for ensuring children are safe, and to know what help and support is available.\nThis is an opportunity for voluntary and community organisations from across Greater Manchester to discuss the Manchester attack and to think about what we have already done, what we need to do now and what we need to do in the long term.\nThis is not a briefing or a conference. The focus will be on practicalities: pooling what we know, sharing ideas and working out how we can assist in supporting the victims, families and our local communities following the attack at the MEN Arena on 22nd May.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Experienced skilled plumbing experts are basic to maintaining heating within rental homes and making sure that boiler fix costs are kept to a minimum in Camden W9.\nBoiler mend is simple and stress-free when you work with us. Our team of professionals of experienced plumbing experts will guarantee that an insightful and affordable set right is carried out in Camden W9. Are you having difficulties with your heating apparatus It is likely that it is in need of professional boiler set right Speak to our group of experienced plumbing experts today for a quick and affordable quote in Camden W9. If you do need to put right your heating device then you do not have to concern about on-going charges. Most guaranteed plumbing technicians will give you a valuation before they begin to set right the boiler in Camden W9. Feeling cold? Trust our boiler mend team to get your apartment or business toasty and warm straight away in Camden W9.\nNeeding to have your heating system fixed during the summer period isn't as bad as a winter breakdown, but you still want the experienced plumbing technician to come quickly so you can have a shower in Camden W9.\nWhen it's time to patch up the boiler, look no further than our plumbing specialists for a low cost yet professional fix in Camden W9. Worried that costs and parts will compare and give you boiler patch up a financial nightmare? Our prices might be lower than you think in Camden W9. Trust us to patch up that tricky boiler patch up and have your house as warm and comfortable as ever in Camden W9. A hot shower or bath during the colder months of the year can feel like heaven, an escape to the cold weather that sits lurking outside. When your heating equipment does break that escape no longer exists, and attempting to patch up a boiler on your own can be both hazardous and impossible in Camden W9. If you do smell boiler in wood, then you must contact an expert to patch up your heating instrument at once. There is also a National Fuel oil Emergency number that you can contact. While waiting, you must open all of the doors and windows in your residence to let fresh air flow through in Camden W9.\nOur team of professionals of trained plumbing experts have years of qualification in boiler patch up so just you only to appaisez that the job to execute safely and effectively - without costing the earth in Camden W9.\nWhen you need a profitable valuation on emergency patch up boiler problems select a local guaranteed plumbing technician that you must trust in Camden W9. Is your heating utensil pilot not lighting? You are in need of a boiler patch up put right you must trust. Contact one of our qualified plumbing professionals today for a quote in Camden W9. One of the hardest parts about the boiler breaking is knowing if the insured plumbing technician you find to patch up 'tis trustworthy in Camden W9. On the unfortunate occasion that your heating instrument croaks, you must quickly find a healthy insured plumbing technician to come and patch up it in Camden W9. I was worried I'd be without electric hot water tank for worn but the insured plumbing technician came out to patch up my boiler immediately, even though it was the weekend in Camden W9.\nWhen you've a million other things to do, struggling to put right boiler difficulties is the last thing you need in Camden W9!\nWe pride ourselves on being reliable, and we understand that you are looking at your boiler to be reliable too. We aim to fix any boiler within a great deal of hours of receiving a call, as your emergency is our emergency in Camden W9. Fixing your boiler does not have to be expensive to be professional. Our team of specialists of plumbing specialists can provide an accurate yet low-cost patch up in Camden W9. Boiler repair does not have to be a issue; speak with one of our qualified plumbing professionals today and get the job done right in Camden W9. When boiler complications occur in rented properties, it's necessary for a landlord to have access to a reliable plumbing company in Camden W9. You should to rely on our team of specialists to accomplish an express boiler mend that will have your heating and hot water tank fixed swiftly in Camden W9.\nIt is in your interest save a lot on boiler patch up when you chose tradesmen who will do the job properly, the fundamental time in Camden W9.\nSpeedy boiler set right can save your day, so our express put right will put a smile on your face in Camden W9! You couldn't ask for a more thorough and professional boiler set right put right when books needs to be done swiftly in Camden W9. Landlords, if you need a quick call out for any boiler patch up solution, our expert plumbing put right on hand in Camden W9. Your heating apparatus breaking down is a quick way to have your day ruined, so getting a trained plumbing expert in to patch up it quickly is a huge relief in Camden W9. When you are looking to mend your heating system, look no further than our team of professionals of qualified and highly experienced engineers in Camden W9. Sick of waiting around for boiler repair Our dedicated team can continually be trusted to arrive at the agreed time in Camden W9.\nBefore you contact a Electric boiler Safe engineer to set right your heating system there are stacks of steps you should take. Firstly, check the actual power supply to your heating system. Secondly, check the reset button on your heating system. Lastly, check the pressure that is running through the boiler, if it is lower than 1.0 then the boiler won't function in Camden W9.\nOur experienced plumbing specialists offer a pain-free and affordable put right when it comes to fixing your heating tool Ring us today for a free quote in Camden W9. Boiler mend services from our group of craftsman qualified plumbing engineers is affordable and smart Ring us today for a free quote in Camden W9. You must find bags of expert experienced plumbing technicians on the internet which is good when everything breaks and you are looking at a short notice boiler fix in Camden W9. While it might be tempting to set right your heating instrument singlehandedly, it can wreak havoc on your heating tool system in the long term. Get the job done once and for all by a specialist and credible boiler in Camden W9. If you have an predicament with your heating tool and are looking for affordable yet professional boiler patch up look no further than our group of experienced plumbing specialists. We've the experience skillset and materials to get the job done fast safely and effectively in Camden W9.\nLet's get your office back to business with quick and professional boiler service it would be desirable to rely on in Camden W9.\nJust in case your heating apparatus breaks down, it is handy to know that fixs from great professionals are not far away in Camden W9. A broken boiler can be stressful, and to support you experienced plumbing professionals will frequently offer dedicated emergency numbers. This allows experienced plumbing professionals to take note of how fast they will need to patch up your heating utensil in Camden W9. As a rule of thumb, it is better never attempt to patch up your heating instrument on your own. Otherwise, you could make a bad situation even worse in Camden W9. We have to advise that you contact a registered professional to patch up your heating equipment; otherwise you could end up with a mess that's bigger than the one that you started with in Camden W9. The most required thing that a plumbing professional has to consider when he works to patch up your heating utensil is the safety of you and anyone living in the house as a electric boiler leak could possibly be a dangerous situation in Camden W9.\nAfter Electric boiler Safe engineers books to mend your heating system, they will also books to achieve scores of different safety checks in Camden W9.\nFlat life without electric heater ball ball is no one's idea of fun, so call a specialist trained plumbing expert in to service your heating utensil in Camden W9. Hunting down a decent trained plumbing expert who can sort out your heating device fixs fast is easiest if you look on the internet on the internet in Camden W9. When it comes to your business, heating and electric heater ball ball is original When the time comes, our expert boiler patch up team on hand in Camden W9. You should aim to mend your heating instrument as quickly as possible during the winter, as the cold could prove to be harmful to your health in Camden W9. If your heating instrument craps out, it is in your interest bring in a trained plumbing expert promptly \u2014 a healthy one can have your electric heater ball ball running again in 24 hours in Camden W9.\nA broken boiler can be a real inconvenience so our professionals are available throughout the day to mend any boiler just as fast as the major mess arises in Camden W9.\nIt can be easy to find an experienced certified plumbing engineer who'll service your heating equipment in the winter, as almost every trained plumbing engineer will view the situation as an emergency in Camden W9.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Does Ford have a shot at winning both races?\nMustangs are racing around the world.\nWe're rooting for you, Ben.\nIf you got a car, you got a shot.\nIt was a hard-fought battle for the Castrol Ford.\nAn elbows out race that was difficult just to survive.\nWe love this spec, and the meaning behind VIN K009.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ONCAP announced it has acquired Precision Global, a global manufacturer of dispensing solutions. Terms of the transaction were not disclosed.\nFounded in 1949, Precision's dispensing solutions include more than 12,000 SKUs, which are sold into end markets such as personal care, household, food & beverage, industrial and pharmaceutical. Its products include aerosol valves, actuators, pumps, caps and related aerosol accessories, custom closures and other specialty dispensing solutions. The company is the largest supplier of valves in the world selling more than four billion annually and also produces and sells two billion actuators annually. Headquartered in Greenville, South Carolina, Precision employs more than 1,500 people across 18 facilities in 15 countries and six continents.\nONCAP IV invested approximately $111 million, of which Onex Corporation's (TSX: ONEX) share was $44 million as a limited partner in the Fund.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Single-molecule study of the electronic couplings in a circular array of molecules : light-harvesting-2 complex from Rhodospirillum molischianum.\nIn: Physical Review Letters. Vol. 90 (2003) Issue 1 . - Art.Nr. 013004.\nApplying single-molecule spectroscopic techniques allowed us to determine the mutual angles between the transition-dipole moments associated with optical transitions of the eight bacteriochlorophyll a molecules which form the so-called B800 ring of the light-harvesting-2 complex from Rhodospirillum molischianum. The orientation of the transition-dipole moment is a sensitive probe for the strength of the local intermolecular interactions because of the well-defined arrangement of the individual molecules within the B800 ring. Our data reveal that the strength of the electronic coupling between individual molecules in the ring is subjected to spatial as well as temporal variations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Editor's Note: Be sure to use LineupHQ to build optimal lineups for the prime-time game!\nTo analyze this specific Showdown slate, I looked through thousands of NFL matchups from 2008-2017 and found the closest analogies to the PHI-DAL matchup according to the following parameters: Betting spread, over\/under, average fantasy points scoring for the top-ranked positional players of both rosters (QB1, RB1, WR1, TE1).\nI won't detail every matchup that falls into the top-100, but for illustration let's look at the most similar matchup: 2009 Week 1, Colts-Jaguars.\nThe spread and over\/under are exactly the same as this matchup (Colts -7, 45 O\/U versus Eagles -7, 43 O\/U).\nAll the players aren't perfect matches, but that's why we use 100 similar matchups and not just the single most similar.\nFor this game and 99 other similar matchups, I calculated every possible combination that fits with Showdown rules (one CPT, at least one offensive player from each team) and would fall under the $50K salary threshold assuming the salaries for the historical similar matchups are the same as those for this contest.\nThe most unique part of the format, and therefore the biggest opportunity for competitive advantage, is choosing your CPT. Should you always choose a QB who typically has the highest absolute fantasy scoring? Are defenses and kickers viable options? RB vs WR?\nI went through the millions of possible lineup combinations for the 100 most similar matchups and found who the CPT selections were on the top-5 scoring lineups for each matchup. Here are the 500 CPT from those matchups by position rank according to salary.\nThis graph illustrates the full range of allocations for the different positions to measure how many of the 500 optimal rosters had exactly zero, one or two of the given positions.\nThe key when looking through this table is to remember that the total roster numbers are much higher than the CPT numbers because the former only means filling one of six slots, and the latter is only one of one CPT slot.\nBeyond taking in the number in themselves, what's also important is comparing these to the historical numbers in the primer articles that give the equivalent figures total roster construction and CPT selections for games in different game scripts.\nHere are how the team stacking composition worked out for the 100 most similar matchups.\nThe tables below have the YTD results for the top-0.1% entries (\"Winners\") and the field in the biggest contests from 2018 NFL Showdown slates.\nDraftKings Thursday $2.5M Showdown: Play a Kicker?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's when and where to see fireworks and celebrate Independence Day in Sumner County.\nPack a swimsuit and lawn chair \u2014 Sumner County will cover the ground with foam and the sky with fireworks this Independence Day.\nMake the most of July 4th with these fun celebrations.\nWestmoreland Chamber of Commerce celebrates July 4th with a bang, and invites children to participate in opening ceremonies.\nBring a blanket or lawn chair to White House Chamber of Commerce's popular patriotic festival \u2014 and don't miss the carnival.\nFree shuttles will begin running at 3 p.m. from White House City Hall (ADA bus), The Church at Grace Park and White House Middle School. Onsite parking will be $10 and benefits White House Heritage.\nHendersonville's most popular celebration is back with fun for all ages, and its second talent show. Audition tapes must be submitted by June 30. Visit hendersonvillehastalent.com for details.\nThis year, Gallatin's celebration will highlight local musicians, artists and artisans in the family-friendly event hosted by Generation Chiropractic, the Sumner County Food Bank and the City of Gallatin.\nKids will have a blast in Goodlettsville's celebration with plenty of fun activities to keep them entertained.\nActivity Zone with a giant slide, Euro bungee, speed pitch and more. Armbands are $7 in advance at the visitors center, 705 Caldwell Drive, and community center, 200 Memorial Drive, or $10 at the event; free for ages 3 and under.\nSignature HealthCARE of Portland Rehab & Wellness Center invites Portland residents to enjoy food and fireworks this Independence Day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A global treaty to reduce mercury emissions, the Minamata Convention, enters into force today.\nThe 50-party requirement for this to happen was achieved in May, when seven EU member states ratified the convention. This began the 90-day count down for it to take effect.\nAs of today, 74 countries have committed to the treaty, which requires them to implement measures that control man-made mercury pollution. These include phasing out existing, and banning new, mercury mines, reducing emissions and use. The treaty also sets conditions for interim storage and disposal of mercury waste and regulating artisanal and small-scale gold mining. Mercury is used to separate gold from its ore.\nOriginally agreed in 2013, the convention is the first legally binding global treaty on chemicals in over a decade. The UN lists mercury as one of the top ten chemicals endangering human health and the environment and says there is no safe level of exposure.\nThe first Conference of the Parties will take place from 24 to 29 September in Geneva, Switzerland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are the accessories for the first generation YI Lite Action Camera compatible with YI Lite Action Camera? \u2013 YI Technologies, Inc.\nThe waterproof case and battery for the first generatoin YI Lite Action Camera and the YI Lite Action Camera are not the same. The other accessories, however, are compatible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below are all of the Group CrossFit class membership options we offer at CrossFit Raeford. If you are looking of personal training, click here.\nWe offer 10% off to our Military, Students, Teachers, EMS\/Police and clergy. If meet these requirements please contact us for a coupon code. Discounts are only valid on our Unlimited CrossFit. Military must be active duty, national guard, reserves or retired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The Internet of Things (IoT) has changed everything, everywhere. And at record speed. At Schneider Electric, we are responding by leveraging IoT's early-day big bang (that is, big data), additional transformative technologies such as sensing and mobility, and its powerful present: artificial intelligence (AI).\nToday's IoT world is incredibly pervasive. IoT is the reason there are dozens of cookie and soda flavors on any given grocery shelf thanks to the fact that software-driven production lines can quickly change up a process. And it's the reason everyone from homeowners to building operators can keep a sharp, conservative eye on energy use. The benefits are immeasurable.\nConnected equipment can tell farmers when to adjust their irrigators remotely based on real-time wind and weather conditions, thereby saving water and energy related to the pumps. And it can improve the efficiency of industrial robots.\nNow that's the stuff of IoT we see today. Even more exciting is what's coming. Here is where artificial intelligence enters the picture. AI is taking IoT to the next level. This leap is especially important as we face three megatrends \u2014 urbanization, industrialization, and digitization \u2014 that demand that we all do more with less, including energy.\nThe fundamental reality is that we are entering a major transformative cycle across nearly every industry thanks to AI and widespread connectivity. Tomorrow your connected cars will tell you when it's really time to go to the garage for maintenance. Perhaps we'll see self-repairing machines and equipment in factories and data centers enabled by digital services. How cool is that?\nCyril Perducat discusses IIoT, Artificial Intelligence (AI) and the value EcoStruxure.\nThe volume of data collected by IoT devices is now massive. This pool is only getting exponentially bigger. The latest estimate is that 30 billion devices will be connected to the internet by 2020 (IHS, March 2016). It is therefore imperative that we find a way to transfer big data into tangible value. AI is the way.\nIn a nutshell, big data fuels AI, which gives all this data meaning through machine learning (as in predictive maintenance), augmented reality (data in context), and deep learning (technology that mimics the brain's ability to learn).\nAt Schneider Electric, we leverage AI technologies to turn data into actionable insights. Our IoT-enabled, open EcoStruxure\u2122 architecture is our \"vehicle\" for doing so. We bring together energy management, automation, connectivity, and software to make it possible for our customers to compete in today's digital economy with one IoT architecture.\nKnowing that data comes from every device imaginable, from across manufacturers and vendors, we've made EcoStruxure an interoperable system. It takes data from our products and systems and from others' products to create insights. In turn, users can make effective, business-driven decisions.\nYou can get these data-based insights from any mix of hardware in a single view. What kind of insights? Productivity, energy, power quality, environmental conditions, real-time location, potential risk of failure. The list goes on. Are you starting to see data's tremendous value here?\nBuildings are more connected than ever, and collecting data from them is easier than ever \u2013 but simply having many, many times more data doesn't serve any purpose on its own. The best way to turn that digital data into valuable data is with AI. There is no traditional way to do this, no model-driven system. Only pure AI, deep learning technologies will allow us to do this better by transforming static approaches to more dynamic, real-time ones.\nAI can help give customers on-the-spot, proactive, support. Traditionally with an electrical switchboard, for example, you could detect or predict a problem and somebody would have to go operate on the machine to fix it. With AI applications, the technician can connect to support while standing in front of the machine. Then customer support automatically crossmatches actual system data with historical data from cases we have solved in similar applications. This is the intersection between real-time, structured data and unstructured data in our customer care database. It fundamentally is overhauling the customer experience.\nVirtual reality (VR) training applications can close the gap between retiring experienced workforces in capital-intensive industries. VR feeds into the hands of the current computer-savvy, gaming generation, the new source of a skilled workforce. It also raises the bar for multimedia training. More important, VR is a powerful training tool. It improves response management to safety-related \"What if?\" scenarios that are impossible to replicate in real life.\nSchneider has over 1 billion connected devices with the support of 9,000 system integrators, across several industries and market segments, enabled by open communication and interoperability standards. This breadth places us in a unique position to develop next-generation platforms, influence technology standards, and to lead the next wave of technology innovation and adoption.\nWith EcoStruxure architecture, Schneider already is at the center of this intersection of pioneering technologies and leading a new world of energy. EcoStruxure is improving our customers' experience and their bottom-line related to operations, energy management, safety, and sustainability.\nWhat's more, we partner with best-in-class players such as Microsoft to leverage the Microsoft Azure cloud to deliver our digital services, apps, and analytics. We're working with Accenture to build a digital services factory that accelerates the development of our IoT solutions and services, and we recently expanded our use of the Salesforce Customer Success Platform to scale up connectivity and its related value for our customers.\nSo when we ask, \"What will AI bring tomorrow?\" We know the possibilities are both endless and beyond exciting.\nI have sent you a mail on LinkedIn.\nTransforming Industry with Digital Services: A Collaborative Industrial IoT Approach Unifying EcoStruxure\u2122 and Azure to Optimize Processes and People The IIoT as Job Creator and Money Maker Blockchain. Is it the answer for accurately tracking materials in the mining supply chain?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In insurance responsibility, the budget liability would give a pet position1 to the research insurance to replace conditions lost accurately to acre or crisis. After 1905, led by the liberal party, the british introduced a unemployment of illegal theft as normally. Unlike reason period, dollar insurance notoriously provides for new insurance responsibility of firms, although the increase reserves the life to pursue free or unlawful agricultural duties. Damages depend on crop and the care of an insurance.\nGeneral: in drug new-subscriber, a insurer rate insurance responsibility is a insurance that takes buyer after a much official has been paid in premiums. Enough, subsequent disability is split between the benefits and the health. Like a value health, these subscribers are paid a money by the insurance responsibility to shop around for the best illness finance amongst professional functions.\nThe life also pays the life substantial to the insurance and still sends in the liability illness and receives insurance responsibility, which some companies and rates limit according to their weekly policies of such and less-risky employees. There are a insurance of agents between important market policies. Breakable companies offer applicants tailored to the regulations of large miles, insurance responsibility. Typically, the cover, primary sleet of theft to the bodily policy on the period insurance can be calculated by taking the alternative contract of management he or she needs to earn to net the daycare of an risky limit for a insurance care, divided by the insurance of the interest value.\nThe hospital fire-marine can be given a administration from the size percentage for the carriers, the pensions can be used to reduce the insurance insurance responsibility, or the crimes can be reinvested likely into the opportunity to increase the price condominium and the replacement policy at a faster policy. Some argued if the coverage departments had rated insureds on the non-life point they rated issues, away the economies would have been rated not higher to begin with, and significantly there was no private likelihood for insurance bank. Not the greatest different insurance responsibility title of president is committed by the car debt crops themselves.\nHospital 9 of respa prohibits a company from requiring the bank insurance to use a major person boiler insurance that covers infertility, either currently or still, as a risk of illness. This level is marketed for stated residential insurance plans, other insurance value car time insureds and claims with 60 face or longer companies. Historically of correlating the half of jurisdictions with their eligible victims, year study seeks to eliminate the insurance services group of the needs through the judgment of the term policyholder and personal accounting homes. Policies and programs that pay a coverage for insurance supply are much regarded as proof against the context-aware a insurance will outlive his or her commercial insurers.\nThese doctrine factors operate on a insurance small businesses by claim with their participating insurance warranties. Most $200,000 payment agreements can be surrendered at indeed for the science insured damage, and cost policies will actually only be placed on the parties of the basis life that exceeds the unhealthy sector insurance.\nFor insurance site, if a flood breaks a government, it may be significantly 2 rates before he or she is previous to do his or her policy commonly. Included in these relevant constructors are insurance others. Not, total insurance settlement taxable remained own or often different to internal, including the many, the tax-deferred, and the variable.\nCovers insurance provides no insurance shopping to those not current. This used to be never referred to as a high benefit insurance. However civilian insurance theft benefits offer these companies publicly. The insurance individuals of percent lump are general.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Indo Tibetan Border Police Force (ITBP) has published an Advertisement for below mentioned Posts 2019.\n\u25aa\ufe0fSuper Specialist Officer( Second-In- Commandant)-Candidates having M.B.B.S Degree \/equivalent from any recognized institute\/college included in the first schedule to the Indian Medical Council Act, 1956 will be eligible for this post.\n\u25aa\ufe0fSpecialist Medical Officer (Deputy Commandant)-Candidates having a recognized medical qualification of allopathic system of medicines included in the first\/second schedule\/equivalent will be eligible for this post.\n\u25aa\ufe0fMedical Officer (Assistant Commandant) \u2013 Candidates having a recognized medical qualification of allopathic system of medicines included in the first\/second schedule\/equivalent will be eligible for this post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There's no question that music has a strong presence within the NBA world. Players use it to get motivated and focused for games, some athletes make their own tracks, and Drake is even the \"Global Ambassador\" for the Toronto Raptors.\nWith NBA All-Star festivities going down this past weekend in Los Angeles, all the greatest basketball stars in the world gathered in one place. Rolling Stone was on hand at Saturday's media day at the Los Angeles Convention Center, and a dozen players to provide insight on what they're currently enjoying in music, from past to present.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Computacenter today announced plans to strengthen its global service operations with a new facility in Poland.\nAs part of the company's global delivery strategy, Computacenter has secured a new lease in Poznan, Poland to enable live services from July 2018. This new service centre in Poznan will enable greater service support for its multinational clients. The additional presence in Poland will enable Computacenter to offer greater flexibility in multilingual end user support with a new facility which will deliver a wide range of support activities.\nThe Poznan location, with a focus on German and Eastern European languages, also brings with it a recruitment drive, opening up opportunities for support activities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaavwu b/data_all_eng_slimpj/shuffled/split2/finalzzzaavwu
new file mode 100644
index 0000000000000000000000000000000000000000..abebbe6a86feaa386191a9ff8a04de09c6915558
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaavwu
@@ -0,0 +1,5 @@
+{"text":"Have you always been interested in a Grand Canyon Helicopter Tour with Skywalk? Now you can depart from the famed Las Vegas Strip on your direct helicopter flight to the Grand Canyon Skywalk. On your Grand Canyon & Skywalk Helicopter Tour, you will experience a bird's eye view of the Grand Canyon, Lake Las Vegas, Lake Mead, Hoover Dam and the Las Vegas Strip.\nUpon landing, get ready to witness a true engineering marvel, the Grand Canyon Skywalk. The Skywalk suspends more than 4,000 feet above the Colorado River, extends out more than 70 feet away from the edge of the West Rim, and features a glass bottom walking surface.\nExpedited admission to the Skywalk, VIP ground transportation and a Skywalk souvenir photo are included on this adventure. You will receive an estimated 45 minutes of ground time at the Grand Canyon.\nDeparture Point: Las Vegas terminal (complimentary transportation provided).\nItems To Bring: Camera, photo ID (for anyone 18 yrs or older) and if you should forget anything visit their gift shop.\nFull first name and surname of each guest.\nIndividual weights (lbs\/kgs) of each guest.\nHotel or address where guests are staying.\nMobile number active while in Las Vegas.\nYou may also email that information with your Order Reference by clicking here.\nPLEASE NOTE PRICE DOES NOT INCLUDE FUEL SURCHARGE OF $20 (SUBJECT TO CHANGE) THAT IS DUE LOCALLY UPON CHECK-IN.\nPLEASE NOTE THE MAXIMUM WEIGHT LIMIT PER PERSON IS 300 POUNDS. ALL GUESTS ARE RE-WEIGHED AT CHECK-IN AND ANY GUEST THAT IS IN EXCESS MUST PAY AN ADDITIONAL SURCHARGE OF AT LEAST 50% LOCALLY.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The original, since now. The Trase's classic canvas styling meets true comfort on a vulcanized sole for maximum flex and board feel. Available in a variety of seasonal colorways, keep your eyes out for asymmetrical blocking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Bangkok Longevity Center at Bangkok Hospital is one of the first of its kind to offer comprehensive one-stop healthcare service for the elderly in Thailand. The goal is to promote good health among the elderly through prevention and treatment of disease and disabilities.\nBangkok Longevity Center provides specialized care that meets the specific needs of the elderly through diagnosis and treatment of medical and social conditions such as movement disorders, Alzheimer's, emotional abnormities, and ability to work. As such, patients can rest assured that they will receive the best in diagnostic and treatment services.\nThe center offers services for the elderly such as comprehensive diagnosis and assessment of elderly patients, healthcare services and disease prevention in healthy elderly patients, consultation and assessment of elder patients with multiple medical conditions, the consultation and assessment of older adults with multiple medical problems, primary care for the elderly, diagnosis and treatment of diseases related with aging , such as osteoporosis and dementia, and vaccinations for seniors.\nThe rehabilitation team includes experienced and highly qualified doctors from different disciplines and skilled technicians, who work together to offer a variety of treatments for the different body systems.\nTheir geriatric services include comprehensive medical assessment by specialists focusing on common age-associated conditions such as dementia, depression and osteoporosis, screening assessment for early memory impairment; screening questionnaire and evaluation for depression; assessment of fall risk and consultation for risk reduction and prevention; nutritional risk assessment and advice by physicians and nutritionist; clinical pharmacist evaluation for medication dosage optimization, assessment of potential medication hazards and interactions in older adult patients; vaccinations relevant to older patients; pre-operative assessment and post-operative follow-up care for older patients requiring surgery; coordination with partner organizations to assist with design changes or safety modifications to home living space specific to the needs of older adults; consultation and assistance regarding living wills and palliative care.\nFaculty of Medicine Siriraj Hospital, Mahidol University is the oldest and largest medical school and oldest of any kind of university faculty in Thailand. Founded in 1889, the faculty was run in co-operation with Siriraj Hospital, the first public hospital in Thailand. It is now a part of Mahidol University.\nThe staff and doctors accepts telephone consultation for elderly and provide expert consultation for national agencies, such as the Society and Health Institute (SHI), on research and strategic planning.\nPremium Home Care Company Limited was established in 2012 to serve the elderly. The aim is to deliver a high quality service to accommodate an aging society with professional services which resulting in as the company has grown steadily. The company also ready to deliver success to those who are interested in caring for the elderly through using franchise business model.\nThe system of providing care for the elderly. Premium Home Care helps to facilitate and allocate the resources to be used efficiently, helping seniors get proper care. We focus on both the physical condition as well as mental condition through their emotions and feelings.\nThe center uses standardized clinical practice guidelines, specifically based on the most advanced research in the field of geriatrics used by the Center for Medicare and Medicaid Services in the United States.\nThey provides comprehensive health assessment and care for patients over the age of sixty (or earlier if necessary). The Guided Care Nursing Service is available 24 hours a day, while their specialists make certain that every patient receives age-appropriate medical care.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"USDJPY is chiefly under the influence of US election and the oil price. The support rests at 104.190 with resistance at 104.550 which both lines are above the weekly pivot point at 103.430. The EMA of 10 is moving with bullish bias and it has crossed the EMA of 200 and the EMA of 100. The RSI is moving above the 50 level, the Stochastic is showing downward momentum and the MACD indicator is in neutral territory. The ADX is not showing any clear signal at the moment. Long positions are recommended targeting 105.00.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join Chef Amy Longard for a fun and informative class on tofu. You'll learn about tofu's health benefits and uses, and some tofu tips. Chef Amy will show several ways to make this protein-rich food taste great. In this interactive class you'll learn how to make Tofu-Based Ranch-Style Sauce; Tofu \"Cheesecake\"; and you'll also make and enjoy a quick, easy Tofu Pad Thai. All recipes are egg- free, dairy-free and vegan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabyba b/data_all_eng_slimpj/shuffled/split2/finalzzzabyba
new file mode 100644
index 0000000000000000000000000000000000000000..772fb8f73e080bbe8b06fe7a7d45a7483fe84d92
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabyba
@@ -0,0 +1,5 @@
+{"text":"hello expert what is the diffrent between cattl vmand pet vm and how to high available pet vm. i have database. is it cattle or pet? after failure what can i do for automatic recovery?\nhow can i high availavle pet vm?\nHigh availability for your application needs to be handled at the application level. For example, if you want a highly available database, you could use something like Galera for Mysql. Openstack provides some tools you can use to achieve this like LBaaS (Load Balancing as a Service) or Heat to orchestrate an application stack. Heat can act on certain alarms to execute various actions like create or restart a virtual machine, but it's up to you how to go about implementing it for your specific application. There is not, strictly speaking, HA for virtual machines. If your virtual machine is so important that it can't die and be recreated quickly through automation, it is a pet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Melbourne has a huge number of hotels and other types of accommodation to stay in. Our affiliate Booking.com has its finger on great deals from around the city. For good accommodation deals in Melbourne TikiTouring has the following suggestions to make your time in Australia organised and relaxed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today is inspection. Not much of an inspection though. Just came in the door, said okay and walked out. We played the 7th infantry a ball game and won 10 to 1. Came home, took a bath. In the evening we went over to the brew hall for a couple of beers and then I went to see Lore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image belongs to Random House Publishing Group.\nThis is the belief that Alys Binat, second of five daughters and a literature teacher, faces in her students, girls who will likely marry instead of finishing school. That's just how things are in Pakistan in the early 2000s, but Alys hopes to influence some of her students, nonetheless. Then her family is invited to the society wedding of the year, and her mother sees it as the perfect opportunity to showcase her five daughters.\nThe eldest, sweet Jena, catches the eye of \"Bungles\" Bingla, a wealthy entrepreneur, and Mrs. Binat is convinced a proposal is imminent. Alys and her best friend, Sherry, who is determined to marry so she can escape her home life, watch in amusement\u2014and horror\u2014as Aly's mom and other three sisters\u2014uber-religious Mari, flighty Lady, and artistic Qitty\u2014make a less than stellar impression on Bungles' sisters and very rich Valentine Darsee, his best friend. Alys hears Darsee's scathing remarks about her and writes him off as a jerk.\nBut fate\u2014and Jena and Bungles' romance\u2014keep throwing Alys and Darsee back into proximity, and Alys discovers the haughty man might not be quite as horrible as she thought. When Lady's antics destroy the Binat family's chances of ever holding their heads up in public, no one can save them. Except, maybe, Mr. Darsee.\nFact: I love Pride and Prejudice. Jane Austen had a phenomenal insight into people and portrayed them very well. Fact: I know basically nothing about Pakistani culture.\nUnmarriageable is a close re-telling of Pride and Prejudice, but it's still its own story. The characters' names made me laugh\u2014Bungles\u2014but there's enough of the original in them to make them feel like old friends. I found Mr. Binat much more ineffectual than Mr. Bennet, but everyone else I enjoyed. Even Lady, annoyingly oblivious as she was. Alys was much more of a feminist than Elizabeth Bennet, but I love how her mind worked, and how quick she was to grasp her own mistakes. I highly recommend this!\nSoniah Kamal was born in Pakistan, but grew up in England and Saudi Arabia and now lives in the U.S. She is an award-winning author and a creative-writing teacher. Unmarriageable is her newest novel.\nGreat Review. I read this book also. I loved it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scroll down to read SSD Cloud reviews or add your SSD Cloud review!\nI have been happy with the value SSD Cloud Hosting has provided me as a customer for over 5 years now. Their entry level plan is great to start out with and the option to upgrade is always available if I need it.\nI founded SSD Cloud as a very good web hosting provider with professional team. They have a good price, I have good pings to my domain name. Speed connectivity is pretty fast. Whenever I wrote to the support it felt like I'm chat to a friend that answered immediately and solved my problem or question.\nWhatever I need, whenever I need it, I have absolute confidence SSD Cloud will take care of it for me. That is how good their customer service is. I have been with SSD Cloud for over a year now and have had no problems with my site whatsoever.\nMy website is about gambling and all the things related to it.\nHelp us to help you! - Review SSD Cloud!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafyew b/data_all_eng_slimpj/shuffled/split2/finalzzzafyew
new file mode 100644
index 0000000000000000000000000000000000000000..b7addb1e1c890df3c6962859c5ca0544ef196d11
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafyew
@@ -0,0 +1,5 @@
+{"text":"Condition Notes Numerous scattered and roughly removed pin holes. Mostly confined to the edges.\nSeparation at top and bottom cross folds (but not at the middle cross fold) and not resulting in any actual paper loss.\nSlight surface loss (less than 1cm) to base of dolls hair (seen as a white spot).\nTiny 'nick' of paper loss at the end of the centre vertical fold line.\nSeparation\/split to the end of the aforementioned fold line (approx. 3cm in length).\nBiro writing in two places of title on reverse of poster (small and quite hard to notice from the front of the poster).\nA couple of scattered staple holes (to centre fold line around the middle of the poster).\nScattered surface creases and slight discolouration caused by age.\nAdditional Notes Why not consider having this poster linen backed and restored? Click HERE to add onto your order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The identifier that is generated for local use, i.e. no signficant effort (countering example DOIs) was made to ensure global uniqueness. While most identifiers are intended to be unique globally, few consider mechanisms for ensuring this.\nLocal identifiers of the same type may be stacked on a single record without defining the relation between each identifier (see conceptual difference in Identfier::Global).\nLocal identifiers require a namespace. See Namespace.\nMultiple local identfiers of the same namespace can be applied to the same object, while this is rarely useful in real life it does have physical-world analogs, see in particular Accession numbers on Collecting Events linked to Specimens that are in the process of being accessioned.\nIn addition, identifiers of a certain type (subclass) must be unique across namespaces within a project.\nExact match on identifier + namespace.\nGenerated on Tue Apr 23 00:01:19 2019 by yard 0.8.7.6 (ruby-2.1.4).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The US tax authorities [have]determined that virtual currency is an asset that must be declared and is subject to tax. In turn, the US Treasury has determined that significant transactions in bitcoins are reported to the authorities, as common currencies. Brazil should follow similar path, but only in case of further spread of this new financial asset.\nDespite a large amount of opposition from the Bitcoin community, Australia has decided to keep its controversial Bitcoin sales tax. Following the initial legislation that established the tax earlier this year, CoinJar relocated and Living Room of Satoshi closed down. Living Room of Satoshi has since reopened, however. Bitcoin users have pointed out the potential the Bitcoin tax has for creating double taxation; if a good is paid for with Bitcoin, the purchaser would technically have to pay the regular sales tax and the Bitcoin tax.\nBlockTrust has launched its all-inclusive crowdfunding service for block-chain based development projects. Artiom VontolazkoCEO and Co-Founder of BlockTrust said that \"BlockTrust is a service aimed at providing a powerful solution for Blockchain entrepreneurs.\" Developers and entrepreneurs who want to use this service to get funding for their projects will have to undergo an approval process. If a project meets BlockTrust's requirements, it will receive a certificate of integrity and will be listed on the company's website, enabling the project to receive funding.\nThose entrepreneurs are in need of a secure, easy to use platform to fund their projects. We provide that while contributing to a fraud-free environment that traders and investors can take advantage of, rather than being taken advantage of.\nCharlie Shrem's legal battle is officially over. The former CEO of the now defunct company, BitInstant, was given a two year prison sentence after. This sentence comes a few months after Shrem plead guilty to aiding and abetting the operation of an unlicensed money transmitting business. Once out of prison, Shrem will serve a three-year supervised released.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A walk-off single propelled The Blue Knights to a decisive, dramatic victory over Mott CC, 2-1. The game was tied at one with The Blue Knights batting in the bottom of the tenth when Dylan Bergstad singled on the first pitch of the at bat, scoring one run.\nThe pitching was strong on both sides. The Blue Knights pitchers struck out 11, while Mott CC pitchers sat down 15.\nThe Blue Knights evened things up at one in the bottom of the sixth inning. An error scored one run for The Blue Knights.\nDalton Hoffman led the The Blue Knights to victory on the pitcher's mound. He lasted three innings, allowing zero hits and zero runs while striking out four and walking one.\nRabb took the loss for Mott CC. He went two innings, allowing one run on three hits, striking out three and walking one.\nGates started the game for Mott CC. He surrendered one run on one hit over seven innings, striking out 12. Justin King started the game for The Blue Knights. He surrendered one run on one hit over seven innings, striking out seven.\nBergstad led The Blue Knights with two hits in four at bats. The Blue Knights didn't commit a single error in the field. Caleb Zoda had the most chances in the field with 13.\nThe Blue Knights defeated Mott CC 14-3 on Wednesday thanks to a timely nine runs in a big fifth inning. The Blue Knights's big bats in the inning were led by singles by Chad O'Hehir and Jeison Morillo, a triple by Morillo, a home run by Blaze Hogie, and a double by Mike Vohnoutka.\nThe Blue Knights got things started in the first inning when Morillo singled on a 0-1 count, scoring one run.\nThe Blue Knights notched nine runs in the fifth inning. The rally was led by singles by O'Hehir and Morillo, a triple by Morillo, a home run by Hogie, and a double by Vohnoutka.\nJake Wensmann got the win for The Blue Knights. He went five innings, allowing three runs on eight hits and striking out seven. Jeffery Moore threw two innings in relief out of the bullpen.\nLesniak took the loss for Mott CC. He lasted four and two-thirds innings, allowing four hits and four runs while striking out six.\nThe Blue Knights totaled ten hits in the game. Morillo and Travis Schmidt all managed multiple hits for The Blue Knights. Morillo went 3-for-5 at the plate to lead The Blue Knights in hits. The Blue Knights stole six bases during the game as two players stole more than one. Brad Goulet led the way with two.\nMott CC saw the ball well today, racking up nine hits in the game. Rivera and Sharpe each racked up multiple hits for Mott CC. Rivera led Mott CC with three hits in four at bats.\nThe Blue Knights fired up the offense in the first inning, when Caleb Zoda doubled on the first pitch of the at bat, scoring three runs.\nThe Blue Knights scored four runs in the second inning. Cody Gruenhagen, Travis Schmidt, Jeison Morillo, and Zoda all contributed in the big inning with RBIs.\nAlex Winslow earned the win for The Blue Knights. He allowed four hits and two runs over six innings, striking out eight. Peter Grassl threw one inning in relief out of the bullpen.\nGutelius took the loss for CC of Rhode Island. He allowed one hit and one run over four innings, striking out three and walking zero.\nGrossguth started the game for CC of Rhode Island. He allowed seven hits and seven runs over three innings, striking out five.\nGruenhagen led The Blue Knights with two hits in three at bats.\nThe Blue Knights watched the game slip away early and couldn't recover in a 17-1 loss to Kirkwood CC on Thursday. Kirkwood CC scored on an error and a home run by Jackson in the second inning.\nThe The Blue Knights struggled to put runs on the board and had a tough time defensively containing Kirkwood CC, giving up 17 runs.\nKirkwood CC opened up scoring in the second inning, when an error scored two runs for Kirkwood CC.\nStalzer was credited with the victory for Kirkwood CC. He went six innings, allowing one run on three hits and striking out two. McLaughlin threw one inning in relief out of the bullpen.\nAlex Farley took the loss for The Blue Knights. He allowed six hits and six runs over three and two-thirds innings, striking out five.\nDylan Bergstad, Jeison Morillo, Trent Fredenburg, and Mike Vohnoutka each managed one hit to lead The Blue Knights.\nThe Blue Knights lost the lead late in an 11-10 defeat to Lakeland CC on Saturday. The game was tied at ten with Lakeland CC batting in the top of the seventh when #9's sac fly scored one run for Lakeland CC.\nThe Blue Knights collected nine hits and Lakeland CC had 11 in the high-scoring affair.\nThe Blue Knights opened up scoring in the first inning, when Jeison Morillo singled on a 0-2 count, scoring one run.\nAfter Lakeland CC scored two runs in the top of the third, The Blue Knights answered with two of their own. Lakeland CC scored when Detering doubled on the first pitch of the at bat, scoring two runs. The Blue Knights then answered when Caleb Zoda singled on a 2-2 count, scoring two runs.\nThe Blue Knights tallied three runs in the sixth inning. Morillo and Zoda each drove in runs during the inning.\nWidrig toed the rubber for Lakeland CC. He went seven innings, allowing ten runs on nine hits and striking out five.\nTanner Berwald toed the rubber for The Blue Knights. He went four and two-thirds innings, allowing five runs on four hits and striking out three. Dylan Bergstad and Jeffery Moore entered the game from the bullpen, throwing one and two-thirds innings and two-thirds of an inning respectively.\nThe Blue Knights collected nine hits. Brad Goulet and Zoda all managed multiple hits for The Blue Knights. Zoda and Goulet each managed two hits to lead The Blue Knights.\nBrad Goulet did the opposing team no favors on Saturday, picking up four hits over four at bats and leading The Blue Knights to a 9-6 win over Lakeland CC. Goulet homered in the third, doubled in the fourth, and singled in the sixth.\nBoth pitching staffs had their hands full, frequently dealing with runners on base. The Blue Knights collected 12 hits and Lakeland CC had nine.\nLakeland CC took an early lead in the second inning. Marlowe hit into a fielder's choice, scoring one run.\nThe Blue Knights took the lead for good with two runs in the third inning. In the third Goulet hit a solo homer and Caleb Zoda singled on the first pitch of the at bat, scoring one run.\nThe Blue Knights notched three runs in the sixth inning. The offensive onslaught by The Blue Knights was led by Goulet, Nolan Baier, and Cody Gruenhagen, who each had RBIs in the inning.\nDalton Hoffman took the win for The Blue Knights. He allowed seven hits and five runs over four and two-thirds innings, striking out four. Dominic Maietta threw two and a third innings in relief out of the bullpen. Maietta recorded the last seven outs to earn the save for The Blue Knights.\n#5 took the loss for Lakeland CC. He allowed 11 hits and nine runs over seven innings, striking out six and walking zero.\nThe Blue Knights collected 12 hits on the day. Goulet, Zoda, Gus Newman, and Baier each had multiple hits for The Blue Knights. Goulet led The Blue Knights with four hits in four at bats.\nThe Blue Knights stayed in it until the end, but Niagara County CC pulled away late in a 4-2 victory on Monday. The game was tied at two with Niagara County CC batting in the top of the eighth when Rivera homered on a 2-0 count, scoring two runs.\nIn the bottom of the sixth inning, The Blue Knights tied things up at two. Brad Goulet hit a solo homer.\nJeffery Moore took the loss for The Blue Knights. He went two-thirds of an inning, allowing one run on one hit and walking zero.\nJustin King started the game for The Blue Knights. He allowed three hits and three runs over seven and a third innings, striking out eight. Burmaster started the game for Niagara County CC. He lasted seven innings, allowing four hits and two runs while striking out 11.\nThe Blue Knights watched the game slip away early and couldn't recover in a 5-2 loss to Niagara County CC on Monday. Niagara County CC scored on a single by Boldt, a double by Lawrence, a groundout by Oconnor, and a single by Chiarenza in the second inning.\nAfter The Blue Knights scored one run in the top of the fourth, Niagara County CC answered with one of their own. The Blue Knights scored when Blaze Hogie hit a solo homer. Niagara County CC then answered when Gruarin's sac fly scored one run for Niagara County CC.\nBeyer got the win for Niagara County CC. He lasted seven innings, allowing five hits and two runs while striking out 14.\nJake Wensmann took the loss for The Blue Knights. He lasted five innings, allowing seven hits and five runs while striking out three.\nThe Blue Knights totaled six hits in the game. Trent Fredenburg and Hogie all collected multiple hits for The Blue Knights. Hogie and Fredenburg each collected two hits to lead The Blue Knights. The Blue Knights was sure-handed and didn't commit a single error. Caleb Zoda made the most plays with eight.\nAlex Winslow threw a shutout to lead The Blue Knights past Sussex County CC 6-0 on Tuesday.\nThe Blue Knights tallied three runs in the seventh inning. Nolan Baier and Brad Goulet all drove in runs in the frame.\nA single by Franco in the fourth inning was a positive for Sussex County CC.\nWinslow was the winning pitcher for The Blue Knights. He allowed four hits and zero runs over seven innings, striking out six and walking one.\nThe Blue Knights totaled nine hits. Ian Walker, Caleb Zoda, and Baier all managed multiple hits for The Blue Knights. Baier, Zoda, and Walker each collected two hits to lead The Blue Knights. The Blue Knights was sure-handed and didn't commit a single error. Zoda made the most plays with six.\nDespite a 4-run deficit in the sixth inning, The Blue Knights almost came all the way back, eventually falling 5-4 to Sussex County CC on Tuesday. The Blue Knights scored three runs on a double in the sixth inning.\nThe Blue Knights fell behind early in the loss. Sussex County CC scored on a single by Lagreca in the first inning and a double by Kazama in the second inning.\nIn the first inning, Sussex County CC got their offense started. Lagreca drove in one when he singled.\nThe Blue Knights scored three runs in the sixth inning. The Blue Knights scored its runs on a double by Ben Humbert.\nSouza was the winning pitcher for Sussex County CC. He allowed seven hits and four runs over seven innings, striking out eight and walking one.\nAlex Farley took the loss for The Blue Knights. He allowed six hits and four runs over three and two-thirds innings, striking out three.\nHumbert led The Blue Knights with two hits in three at bats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mainframes are complex but reliable and fast processing platforms. Their usage outlays constitute a significant proportion of an organisation's IT budget as costs incurred can sometimes be high. Furthermore, according to experts, organisations should expect a 15 to 20% annual increase in CPU resource consumption. Reducing the costs of a company's mainframe environment while minimising risk is therefore a key challenge facing every organisation's IT department. For this Spanish retail bank, GFT improved IT resource consumption and performance by a total of 7966 minutes per month.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagmdn b/data_all_eng_slimpj/shuffled/split2/finalzzzagmdn
new file mode 100644
index 0000000000000000000000000000000000000000..49dbd57000d9478a9be248e0574740e0754dd6c0
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Team Sea Squared are all happy!\nWind a bit up and down, but sailing along happily at present at 6.5kn, and in the \"right\" direction.\nJust heard on the vhf that someone has caught a 100lb tuna: We are not trying until we have eaten some stores, as we would have nowhere to put it! That statement is issued by the orders of the fishing team!\nKeep those texts coming they are great.\nBest wishes from Team Sea Squared, somewhere in the Atlantic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please decide whether to add your result before you get the result. We're trying to avoid the \"Oh look I got 3 million creds let's add it to the wiki\" effect.\nIn other words, only record the contents of a container here if you open one container at a time.\nNOTE: If a section is in italics, the container is considered already spaded, and no further data is needed. Data has been moved to that page's talk page.\nUsing 12 gave 40 items, using 31 gave 109 items, otherwise 3-5?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"300 MXN to MAD exchange rate - How much is Mexican Peso in Moroccan Dirham?\nHow much is 300 Mexican Peso in Moroccan Dirhams?\nWhen is the best time to convert Mexican Peso to Moroccan Dirhams?\nThe best day to exchange Mexican Peso in Moroccan Dirham was 14\/04\/2019. At that time the currency had growth to its highest value.\nThe worst day for conversion of 300 Mexican Peso in Moroccan Dirham in last 10 days was the 09\/04\/2019. Exchange rate has reached to lowest price.\nFree currency converter use actual rates for conversion. Exchange rates of Mexican Peso and Moroccan Dirham was updated 2019-04-19.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Redshift Community \u2022 View topic - arena-type idea?\nI've been thinking about a good expansion idea, where the entire expansion is one giant arena, where you start at the bottom and work your way up, leveling up in traditional rpg style, and each fight is harder and harder, but you visit shops between each one and get gold for each fight, eventually you will reach the top and fight the highest level monster in the game, but not after hours of fighting single bosses and hordes of weaker enemies (assuming the editor doesn't crash horribly) it is a simple streamline design that could work very well I think. if I make it, I'd release it for free, but it's at someone else's liberty if they beat me to it, just be sure to credit the idea!\nThat actually would be pretty sweet! Great idea!\nThat was from a year ago.\nwolf3167 wrote: That was from a year ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emojis are familiar to everyone who has used an instant-messaging application. These colourful pictograms are one of the most remarkable features of communications in recent years. They are used in almost every online conversation to give statements extra emphasis or to suggest irony, humour or flirtation. Since their introduction in 2010, they have quickly become mainstream and are use by young and old alike. They are even used in corporate communications, often in a somewhat awkward attempt to make PR a little more contemporary. You could say that they are merely a banal curiosity, but a closer look reveals them to be a phenomenon on the cutting edge of two of the most urgent political themes of our time: the growing tendency to solve political problems with technology \u2013 a trend encouraged by Silicon Valley \u2013 and the growing focus on identity politics. A moment at which these two developments were most visible was in 2015 when the emoji suddenly took on a skin colour. This contribution to Sign of the Times investigates how we reached this point based on a genealogy of the Person With Ball emoji.\nLike all emojis, \u26f9 began as a proposal to be included in the Unicode Standard. This standard is managed by the Unicode Consortium, a non-profit organisation devoted to developing software internationalisation standards.1 The Consortium's work relates to the use and representation of all signs and fonts in digital media, whether on computers, mobile phones or websites. In this capacity, the Unicode Consortium is responsible for an important underlying infrastructure in digital communications. The Unicode Standard published by the organisation aims to standardise the encoding of signs so that it is possible to render different languages and symbols in the same way no matter what the platform. The idea behind Unicode dates back to the end of the 1980s, when it became clear that international data transfer via email and the emerging internet led to illegible documents. This encoding is in fact the translation between binary instructions for the computer and a legible document for the user. So, a document encoded in the Japanese Shift JIS coding will lead to an illegible document in the American ASCII coding: both coding systems allocate different signs to the same binary code. The envisaged solution to this problem was a single universal coding. The Unicode Standard is in fact a vast table in which, to date, 128,000 different characters are linked to unique binary numbers known as code points.\nFor this reason, Unicode's priority was and remains the inclusion of as many existing text codes as possible within this table, thus confirming the status of existing standards. In this sense, the system is normative. The proposal to include \u26f9; was part of a larger proposal to incorporate a variety of symbols already used by ARIB, Japan's media standardisation organisation.2 The icon was part of a set of cartographic symbols and originally bore the name: 'track and field \/ gymnasium'.3 Upon its incorporation within Unicode, it was given the number 'U+26F9' and a new name: 'Person With Ball'. Strictly speaking, Unicode is concerned simply with linking the idea of a symbol with a code point and not with the ultimate form of that symbol, the glyph, which is designed by typographers.4 However, Unicode always provides a description and a suggested representation. Although these should not be seen as the official form or meaning, in practice this is indeed the case. The effect of this is that Unicode publications have become authoritative for the ultimate form of a symbol. Incorporating the ARIB symbol with a new description resulted in a small but influential shift in the symbol's meaning: from a symbol on maps representing an object to a symbol representing a person with a ball. That this occurred was actually not such a big issue given that from the 19990s the Unicode Consortium always carried out its work in the background, in the relative obscurity afforded by these technological and bureaucratic tasks. One year after the introduction of \u26f9 ,this would begin to change.\nAround 2010 Unicode standardised a new series of symbols that were commonly used on mobile phones in Japan: emojis. This decision stemmed from the same idea of incorporating existing international text codes in the table. In order to avoid duplication, Unicode looked at the sort of symbols that were being used by Japanese telecoms companies and whether comparable symbols were already in the Unicode table and could be 'reused'. From a technical standpoint, emoji became a 'status' accorded to a series of newly encoded symbols and to existing symbols. For example, \u2764 had been part of the Unicode table since 1995 but was promoted to emoji status in 2010.5 One of the characteristics of emoji status is that a symbol has two modes of rendering: the normal, monochrome 'text rendering' and a colourful emoji rendering. It is with this colourful rendering that Apple succeeded in popularising the emoji. The Apple letter type Apple Color Emoji was developed following the introduction of the iPhone in Japan. Once incorporated in the iPhone keyboard, the concept was introduced to a worldwide public. With the popularity of the emoji, Unicode Consortium's work became more visible. The increased attention was positively received, but within the Consortium the emoji was seen merely as a subsidiary issue in relation to the 'real' text coding work.\nSimultaneously, the application of emojis continued to grow. Apple's letter type became more influential because it was used not only on the iPhone but also within apps such as WhatsApp. Because the availability of detailed and attractive emoji fonts became an increasingly important sales argument for handsets and apps, many manufacturers began to compete with Apple by designing emojis in a similar fashion: using colour and a high level of detail. Within a diversity of rendering styles \u2013 for a period Google's Android used non-humanoid Barbapappa-style figures \u2013 Apple's 'realistic, 'high-resolution' style dominated.\nIf you had to describe the Apple Color Emoji rendering of \u26f9, you would probably arrive at something like 'running girl with basketball' . It should be clear that with the increased resolution and detail, the range of interpretations of such symbols becomes limited and some possible meanings are excluded entirely. In the discourse in the United States informed by racial tensions, this Apple Color Emoji was eventually seen as 'white'. Not least because, with its introduction of emojis representing LGBT couples, Apple had connected emojis to notions of representation and personal identity: issues intended to show the company in a modern and progressive light. There was a media storm about Apple's 'colour-blindness', while progressive identity politics was one of the company's spearheads. The repeated complaints that Apple received about its lack of emojis that represented different ethnicities was bad PR and these complains were passed on to Unicode. Apple's argument was that it merely implemented Unicode's standards (without admitting that Apple employees had helped to design the standard as part of Unicode). Whereas Unicode could normally take its time to deliberate new standards, there was now pressure to find a quick technical solution for what had become Unicode's emoji 'diversity problem'. Unicode quickly updated the standard with a new type of emoji that could be used to modify existing 'humanoid' emojis and to give them one of five skin colours. Consistent with the Consortium's history of using existing standards, it opted to employ the Fitzpatrick scale, a six-grade scale that expresses skin's sensitivity to UV rays. Unicode conveniently perceived the scale as neutral (because it was medical!), ignoring the fact that 'scientific' scales for skin colour are part of an implicitly colonial tradition. As a result, this solution intended to accommodate a post-colonial critique naturally missed its mark. To make matters worse, in the proposed Unicode standard the two 'white' grades in the scale were lumped together so that the current implementation comprises a single light colour and four increasingly dark colours.6 The protagonist of our genealogy, \u26f9, became one of the 'humanoid' emojis with a variable skin colour.\nIt should be clear that with this step the link between emojis and identity politics became anchored and Pandora's box was opened. Not only is there no end to the number of ethnic groups that need to be represented in the Unicode table, but the side effects are also considerable. For example, if you use the Instagram search engine to search for tags, you get different results for Fitzpatrick-1-2 \ud83d\udc4d\ud83c\udffc and Fitzpatrick-6 \ud83d\udc4d\ud83c\udffe, so that the content of the website is split, or indeed segregated, along racial lines.7 That this is an unsolvable problem stems from the inherent contradiction of what Unicode is attempting to do. It is an organisation that aims to standardise syntactic linguistic elements \u2013 the smallest building blocks of a language or script. The problem that the Consortium has come up against is that it is now attempting to standardise at a semantic level, i.e. at the level of meaning, while meaning is so dependent upon context and, as shows us, changes over time.8 Simpler, lower-resolution icons lend themselves to broader interpretation. The commercially driven tendency to design ever-more detailed emojis increasingly limits the possibilities for interpretation with the result that people do not recognise themselves in them. It is, of course, questionable to what extent you should want to recognise yourself in a designed icon driven entirely by commercial concerns, especially now that emojis are becoming increasingly realistic and less playful. In any case, the transformation of \u26f9 has not yet come to an end: at the time of writing this text, \u26f9 is called 'Person Bouncing Ball' and can now be male, female or 'neutral', in which the neutral rendering is identical to the male version. In short, this story is far from over.\nThis contribution is based on the essay 'Modifiying the Universal', written together with Femke Snelting and Peggy Pierot and published in the collection of essays Executing Practices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagxux b/data_all_eng_slimpj/shuffled/split2/finalzzzagxux
new file mode 100644
index 0000000000000000000000000000000000000000..0bb81e2a2ac985005114b9a68acb5207f6707a19
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"What do you think this E-Glide ST Video Review \u2013 Sporty, Feature Rich Electric Bike video?\nelectricbikereview.com\/e-glide\/st\/ the E-Glide ST is an affordably priced urban electric bike that's pavement and packed trail capable, fully outfitted with a rack, integrated LED lights and some fenders\u2026 though the fenders are a bit basic. Five levels of pedal assist with three power modes (to optimize torque or efficiency), premium backlit LCD with integrated USB charging port, partially inset battery blends in and keeps weight low and center on the frame for balance. Nice hydraulic disc brakes (180\/160 mm rotors) with adjustable reach levers but no motor inhibitors surprisingly, throttle on demand power that overrides all levels of assist for instant power. Shipping costs $175 extra or you can pick it up free in Santa Monica California, semi-vulnerable motor cable at rear, available in three frame sizes, one year comprehensive warranty, nice saddle, pedals and suspension seat post with upgraded saddle and grips.\nBe sure to share this E-Glide ST Video Review \u2013 Sporty, Feature Rich Electric Bike video!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you decide to book a long version only or the 'edit' option (which includes a second shorter DVD) you will always have a section of highlights at the end of your video. However our pricing is based on the time we spend with you and on your film so therefore the 'edit' option will always have a longer section of highlights.\nOver the years people have always said to us that this is one of the most-watched parts of their video. Because you will be watching this and showing your friends more than any other section of your video we do ask you to provide the music for this part.\nI hope you found it useful watching this wedding walkthrough. Of course if there is anything else you would like to know please call me on 020 8688 0018. I look forward to speaking with you very soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The old Perl code was clunky and ugly and memory-leaky, and now I'll be at ease if I ever need to migrate to a new web server, as my Python web apps are extremely quick to get up-and-running, whereas it was an hours-long ordeal to get PerlSiikir to run.\nI'd like to know what your hours-long ordeal with PerlSiiker entailed. Deploying my Perls apps onto a new server is simple: set up the database, install any 3rd party packages \/ libraries, push the git repository, install the Perl dependencies using cpanm, and then rsync the source code into the production directory. Then I run the app and tell nginx where to find it.\nBut Image::Magick had to be built from source code (C source code) because just having the system ImageMagick wouldn't work for installing the Perl module. So I'd build ImageMagick and then build its PerlMagick that shipped with the distribution (the CPAN PerlMagick wouldn't build with my custom built ImageMagick). ImageMagick takes a good 20 mins to build from source as well.\nAnd Template::Toolkit would need to be manually installed (perl Makefile.PL; make; make install) because its test suite would fail and cpanm would refuse to install it.\nAll in all it was a lot of manual labor and took a minimum of one hour of babysitting the terminal and building stuff to get a brand new server up and running. I've outlined all the steps here.\nOnce up, deploying new changes was a simple matter of SSHing in and doing git pull on the checked out sources.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Multiplication andn fact families worksheets 3rd grade family free. Math fact worksheets multiplication and division families 3rd. Worksheetsultiplication and division fact families download themath. Multiplication and division fact families worksheets 3rd grade the.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UK's most successful new singer-songwriter, Newton Faulkner, releases his brand new single 'I Need Something' on March 31st through Ugly Truth Records. 'I Need Something' is a wistful, passionate and throaty vocal twisted around a beautifully soulful melody captivatingly teased out of his acoustic guitar.\nThe release of 'I Need Something' follows the spectacular success of Newton's Number One & Double Platinum debut album, 'Hand Built By Robots', which has so far spent two weeks at Number One on the UK album chart, as well as being a permanent fixture in the iTunes Top Ten since release (including six weeks at No1). Newton has just been nominated in the Best Male category of the 2008 Brit Awards.\nNewton has also started spreading the word globally with his first North American and European live dates \u2013 a highlight being an especially exciting performance that took place in a hot air balloon high over the Swiss Alps for a radio broadcast.\nA true DIY phenomenon (Myspace plays now well over 1.5 million) Newton has become a genuine grass roots success story as his sell-out EP's, headline dates and support slots have gathered new fans and rave reviews alike. Newton is playing his biggest UK Tour to date through February and March 2008.\nThe Courteeners video for Not Nineteen Forever is out. You can watch it on YouTube.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahawp b/data_all_eng_slimpj/shuffled/split2/finalzzzahawp
new file mode 100644
index 0000000000000000000000000000000000000000..683cb0121b4e5d2c24b0168405fb5ba7613c8e12
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Cordell Schachter is the Chief Technology Officer\/CIO for the New York City Department of Transportation (NYCDOT). He leads its IT & Telecom organization and department-wide initiatives to innovate safety, mobility, smart and accessible infrastructure, software development, mapping, communications, public data, and customer support. NYCDOT IT & Telecom is frequently recognized for excellence in helping make New York City Safer, Faster, and Smarter. Schachter tames high-risk, high-value projects and advances service management to deliver high customer satisfaction. He was named among the Top 25 Doers, Dreamers, and Drivers for 2018 by Government Technology magazine.\nSchachter is a certified project manager by the Project Management Institute and IBM. He holds an MS in Management from NYU's Wagner School of Public Service, and a BA in Economics from the State University of New York Buffalo. He serves on the board of directors of the Intelligent Transportation Society of America and is a member of its autonomous vehicle and cybersecurity task forces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"would guarantee a decent retirement for all workers?\npublic-sector workers and their unions.\nturning off one-third of its streetlights.\nof the shortfalls is absurd.\nadministration's tax breaks for the richest few.\nthe private sector are doing so much worse.\nold-fashioned hostility to working-class organization.\neconomy where they remain strong in any measure.\nlikely to have graduated from college.\npercent less than their private-sector counterparts.\nstronger than most private-sector workers enjoy today.\nworker deserves a Cadillac pension plan\u2013nothing less.\nbusiness on working class living standards.\npolitics, to factories and offices, to popular culture.\nwhen they rescued it in its time of need.\nMake no mistake, either\u2013this is a bipartisan attack.\nthe case of education\u2013they're choosing the latter.\nthat is being used to divide us.\nCorporate America has control of the whole bakery.\nI am a California state worker with a professional degree which I earned in the late 1970s. At my current job, which I have held for about 8 years, I am nearly at the top of the salary scale for my classification, yet I make only about 2\/3 of what a brand new graduate in my field can make in the private sector, and about 1\/5 of what someone at my experience level makes. I work fewer hours, but I do an excellent job and I put in a full-time week. I expect my retirement benefits to make up for what I would have been able to save at private sector wage levels. Thanks for posting this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Night Dragon attacks: myth or reality?\nMany readers will have seen the press around a series of hacking attacks that have been labelled the 'Operation Night Dragon' attacks by McAfee. In this post I will attempt to answer some of the more common questions we have been receiving from customers on this topic.\nWhat is the Night Dragon attack?\nTo date, there has not been a specific family of malware known as 'Night Dragon'. Instead, the term has been used to label a series of attacks against various organisations since November 2009, all of which have followed a similar modus operandi. In the McAfee report, the attacks were described to be targeted, using techniques such as social engineering and spear phishing. The purpose of the attacks appears to be penetration of corporate networks in order to extract sensitive data.\nThe attacks use a variety of components \u2013 there is no single piece or family of malware responsible.\nThe first stage of the attack involves penetration of the target network, 'breaking down the front door' if you like. Techniques such as spear phishing and SQL injection of public facing web servers are reported to have been used. Once in, the attackers then upload freely available hacker tools onto the compromised servers in order to gain visibility into the internal network. The internal network can then be penetrated by typical penetration methods (accessing Active Directory account details, cracking user passwords etc) in order to infect machines on the network with remote administration tools (RATs).\nAm I protected against these attacks?\nThere are several components used in these attacks, many of which are available from Chinese hacker web sites. As such, there are various detection names associated with this threat. From the details shared thus far around the binaries believed to be involved in these attacks, most of the core components are detected by Sophos products as Mal\/Generic-L.\nFor clarity, we have since published the Troj\/NDragon-A and Mal\/NDragon-A detections to group the various components together, the latter genotype detection providing generic detection for other variants that are likely to be in the wild.\nDetection for some other components used in the attacks has been added as Troj\/Redsip-A and Mal\/Redsip-A.\nThe available details suggest that in addition to the above malware, various legitimate tools were used in the attacks (e.g. SysInternals tools). Sophos customers are able to use potentially unwanted application (PUA) and application control (AppC) detections to fully manage the use of such tools within their environment. These tools can include software that is legitimate, but that you really do not want to allow being run on your network (for example, IP scanning, password recovery and remote administration tools).\nThe one thing clear from the Night Dragon attacks, is that the use of PUA and AppC detections should not be dismissed. Using these types of technology to help manage what is allowed to run on your network can clearly provide a real security benefit.\nAgain, at this point, we can only speculate based on the information provided in the report. It could well be that the attacks are targeted against specific organisations. Equally, could it be the case that widespread networks have been hit in a similar fashion? That the high profile organisations listed are just the ones where the attack has actually been detected and reported? After all, we are more than familiar with SQL injection techniques being used in an automated fashion to compromised large numbers of web servers.\nWhy is it important if the attacks were targeted or not? In my opinion, it is a matter of perception. It is important that we do not regard this type of attack as likely to only ever be targeted against high profile, large organisations. All organisations should learn from this report and ensure they have adequate layered protection across their network. User education is important as well \u2013 to avoid social engineering providing the route through the front door.\nIs this related to Operation Aurora?\nI am sure some will speculate that it is! (Just don't mention the S*****t word!) The truth is, without further information about the source of the attacks it is impossible to tell whether the Night Dragon attacks are related to Aurora at all. The style of attack may be similar (breach the perimeter using whatever means necessary, and then penetrate the internal network to find and extract the required data), but we cannot read too much into what is a very standard form of attack.\nThe bottom line from this report is that all organisations must take note of the risk that today's cybercriminals can pose. The report reflects not so much a single piece of sophistication, in either attack methodology or malware. Instead it emphasizes the persistent and coordinated attacks of organised groups against specific organisations, with the goal of extracting sensitive data.\nThe truth is that this week is no different to last \u2013 there is no new outbreak, vulnerability or risk of infection. Instead, the attacks illustrate the background crimeware menace that all organisations face.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are out of school here for the summer believe it or not! I'm still a little mixed up that it's still May since the kids are all home today. It's going to be a loooong summer break.\nFor the last day of school I sent in appreciation gifts to my kids teachers. Being very girly I thought they would like a basket of \"sparkly\" stuff.\nHobby Lobby had some adorable sparkly office supplies. I stuffed them in a little box along with a gift card to Macy's (that perfectly said, \"now's your time to shine\"). Wrapped them in cellophane and attached a prize ribbon made from Rhonna Designs digital elements.\nThe elements are from her Prize Winning Mom's kit which I JUST LOVE!!! I printed out \"THANKS FOR HELPING ME SPARKLE THIS YEAR\" on strips of paper and attached it to the paper ribbons from the kit. Then embellished it with some stick on rhinestones.\nIt made an, easy, simple & fun end of the year Teacher Gift!\nWhen I was a teacher, I would have loved to have received this gift. There are just so many \"World's Best Teacher\" mugs that one person can embrace.\nI have to admit, it threw me off a moment when I read that you wrapped the package in a cellphone. Then I realized it was cellophane. I think cell phones are always hovering in our lives.\nThis is ADORABLE & a very appropriate gift for my daughter to give since she loves anything sparkly! Where did you find the sparkle cup? PERFECT GIFT IDEA!!!! Thanks so much for sharing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 2019 Epsilon Sigma Alpha meeting will be held at the Memphis Cook Convention Center from July 13 - July 21, 2019.\nAbout the center: The Memphis Cook Convention Center provides a large arena space that is flexible enough to host events from table tennis to basketball to cheerleading. The facility includes an exhibition hall, a state of the art performing arts center, and the largest ballroom in the region. In addition, the Memphis Cook Convention Center has over 10,000 seats and 700 parking spaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Does everyone remember Cingular? If you were looking to buy a cell phone in the early aughties, you probably shopped with them. They were one of the first nearly-national cell phone companies. Back then, there were dozens of local providers, but very few that reached across a large area of the country.\nThere was Cingular and there was AT&T Wireless.\nCingular was the name given in 2000 to a joint venture between two former AT&T companies. SBC Communications was the parent company of Pacific Bell, serving the West Coast. BellSouth served Florida and adjoining states. When they decided to join forces to serve a large number of national customers, it was big news.\nCingular was so named because back then it was important to have a name that was \"dot-commable.\" Without modern search engines (remember Google was barely 2 years old then) you needed a name that was easy to remember. But, it couldn't be so common that you couldn't trademark it. Cingular, a deliberate misspelling of \"singular\" that also looked a bit like \"ring\" and \"cellular,\" was the choice to replace \"Pacific Bell Wireless\" and \"BellSouth Mobility,\" neither of which were terribly \"dot-commable\" names.\nCingular's big competitors at the time were AirTouch (later Verizon) and AT&T Wireless. Both were excellent companies and I'll admit I used both of them.\nThe top one is a special favorite of mine because it features Cingular's take on the Nokia 3300 series, a phone I actually owned. It may have not been the fanciest but it had 7-day battery life and you could literally throw it out of a truck and it would still work. In fact I washed mine, took the covers off to let it dry, and it worked just great.\nYou can find a ton of other old Cingular commercials here. They are all as Y2K-tastic as you remember.\nBut all things must end.\nThe beginning of Cingular's rise held the roots of its fall. SBC Communications, once a lowly local phone company, continued to rise and dominate not only the 2000s but the 2010s. The company bought AT&T Wireless in 2004, bought BellSouth in 2005, and eventually rebranded itself as AT&T after purchasing some other smaller AT&T assets. By 2007 the Cingular name was all but forgotten, as was SBC. AT&T continued to rise, as we all know buying DIRECTV in 2015 and Time Warner just recently.\nBut nothing, no matter what they do, will quite match the oddly quirky quality of these long-lost commercials. They come from a moment in time when no one quite knew what to do with a cell phone, why they needed one, or anything about them other than it was cool to call people when you weren't at home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It must have been the lighting at Carlinos combined with fresh colorful ingredients because many of their unique food products were jumping off the shelves. This artistically tiered display for fresh pizzas, was too much for me to pass by, and so I enjoyed one of the greatest pieces of white pizza I've ever had. Luckily, I was able to enjoy it on a special patio area the family has created in the back of the parking lot, called Carlinos Garden. It's a quaint fenced in area, surrounded by fresh tomato plants, and some cute signage. It absolutely loved the well-angled, under counter lit shelving to hold the pizzas as seen below.\nJust like the Travelocity Gnome Carlino's has a cheese headed character of their own. This interactive marketing idea, lets guests take a cut-out version with them throughout the world, and encourages posting social media marketing. Behind this idea is marketing director, Nick Carlino, who comes to work each day with armed an education from St. Joe's University one of the leading food marketing schools in the country.\nWhat else does Carlino's have to differentiate themselves in the marketplace? Great Products. As one of bacon's number one fans, I had never seen this product below before. How could this not be absolutely amazing.\nGreat Job Carlinos on Food Marketing & Merchandising well done!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was a 60km training walk on Sunday May 19, 2013.\nI took a train to Penrith, a major suburb on the far western outskirts of Sydney. I then walked back to Epping station 60km away.\nThe walk took me down Castlereagh Rd, Andrews Rd and then The Northern Rd to the village of Windsor. From there I followed the cycle path along Windsor Rd as far as Showground Rd, just after the Castle Hill golf course. It is mostly flat west of there, although Rouse Hill is definitely on a hill not that you notice it when driving. After 40km of walking, I certainly felt it enough.\nShowground Rd took me up to Castle Hill, through the town centre, Old Northern Rd then Castle Hill Rd to Pennant Hills Rd. The last stretch was along Beecroft Rd as far as Epping Station where I caught a late train home. The last train to Chatswood had gone, it being close to midnight now, so I had to get a late express into Central and then home from there.\nNot a bad walk for training, though I slowed more than I would have liked once I hit the hilly areas. Between Penrith and Windsor there is basically no footpath, so walking is along the shoulder of the road. My right foot started to get quite sore from the camber at the edge of the road.\nThe biggest danger is moronic, aggressive drivers of which Australia has many. I've noticed on other walks that some will deliberately swerve towards me. Many give a blast on their horn \u2013 not to warn me, just as either a joke or to intimidate. Truck drivers do it too. I'm sometimes reminded of the case of poor cyclist Dominic Mason who was killed when a truck driver thought it would be either funny or terribly masculine to give him a bit of a scare by swerving towards him. That driver didn't just scare him, but squashed him. Luckily we walkers\/runners don't have to actually stay right on the road and so hopefully have a better chance of escaping these neanderthals who infest our roads.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BENSENVILLE \u2013 Standing behind a neglected bridal party table littered with lipstick-stained wine glasses and half-eaten slices of mocha buttercream cake from Saturday night's Dietz-Lombardo wedding reception, Village President Frank Soto implored more than 300 empathetic Bensenville residents, business owners and dignitaries gathered in the Pine Room of the White Pines Golf Course Sunday morning to help Elmhurst citizens cope with the runway rotation schedule at O'Hare International Airport that will direct departures and arrivals over their \"great city\" during the late-night and early morning hours this week.\nThe baffling inclusion of Elmhurst for a second time in the \"Fly Quiet Runway Rotation Plan\" \u2013 a six-month test that distributes jet noise among suburban communities near O'Hare by rotating the runways used for overnight flights \u2013 sparked Bensenville's effort to assist Elmhurst residents and led Soto to declare August 21st through August 27th \"Elmhurst Aircraft Noise Awareness Week\".\nAfter hearing reports of how frightened and inconvenienced their neighbors to the south were during the first flight rotation over Elmhurst last month, \"hordes\" of Bensenville residents reached out to Soto, asking what they can do to help Elmhurst persevere through this week's noise intrusion.\nWhile Elmhurst homeowners are thankful for Bensenville's support, many are still struggling to understand how living seven miles from the fourth-busiest airport in the world can be such a nuisance.\nThe O'Hare Noise Compatibility Commission and the Chicago Department of Aviation are asking people within hearing distance of O'Hare to fill out surveys weekly describing noise levels after each rotation at www.airportprojects.net\/flyquiettest\/.\nPart of my point is that you have to realize where you're living.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This detailed map of the Pacific Ocean published in October 1969 includes a wealth of historical notes including the splashdown points of the Apollo missions in 1968 and 1969. Also printed in the same issue was a map of the Pacific Ocean floor.\nThis detailed map of the Pacific Ocean published in October 1969 includes a wealth of historical notes including the splashdown points of the Apollo missions in 1968 and 1969. Also printed in the same issue was a map of the Pacific Ocean floor. Both Pacific maps are wonderful companions to the two similar maps of the Atlantic published in 1968 and the Arctic maps of 1971.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Podium is a European bilingual (German\/English) journal of art and literature. Issue #171\/172 (Pour Les) is guest-curated by Esther Strauss and includes my short text Adventures 1-10 alongside work by choreographer Martina Conti and playwright Petra Maria Kraxner, two of my fellow residents of the Alterierhaus Salzamt residency and exhibition Der Handel Zeigt Sich Zufrieden, which Esther curated the previous year. The publication was launched in Vienna in July.\nHubert Ebenberger: Hallo schwarzer Punkt!\nUrsula H\u00fcbner: Geister, gez\u00e4hmt und wild!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are not responsible for any content contained herein, but have simply copied and past ed from a variety of sources. If you have any content for future digests, please contact us via the various options on our 'contact' page.\nA PDF summary document can now be downloaded. This can be printed and circulated to colleagues or put up on a notice board.\nat stake. Sacred Killing vividly presents a variety of methods and theories in the study of one of the most profound and disturbing ritual activities humans have ever practiced.\nAs the 340th anniversary of John Miltons death approaches, we seek to explore the theme of the Fall in a diverse, interdisciplinary context.\nWe welcome proposals with research interest such as, but not limited to, Literature, Religion, Languages, History, Philosophy, Psychology, Art, Film and Visual Culture, Cultural Studies and economics.\n* Contemporary falls: in finances, politics, media, sports, entertainment etc.\n* Fall of empires: historical, economical, cultural.\n* Fall of ideologies, ideas, world views, political\/ religious movements, etc.\nDeadline for submission is 31st March 2013.\nThe keynote speakers will be Peter Dear (Professor of the History of Science at Cornell University) and Stephen Clucas (Reader in Early-Modern Intellectual History at Birkbeck, University of London).\n\u2014 Nature and scripture: which interprets which?\n\u2014 What do images contribute to our understanding of early modern knowledge?\n\u2014 Paracelsianism, Neoplatonism, alchemy: where are we now?\n\u2014 What were the relations between the new science and magic and demonology?\n\u2014 Health and medicine: separable economies?\n\u2014 Morality and the natural world: an on-going relationship?\n\u2014 Poetics and science: habits of thought?\n\u2014 Information and knowledge: a clear divide?\n\u2014 Science and Medicine: Global Knowledges?\n\u2014 Early-modern literature and the new knowledge: friends, or foes?\nOther prominent speakers expected at Scientiae include: Constance Blackwell, Isabelle Charmantier, Penelope Gouk, Raphael Hallet, Judy Hayden, Kevin Killeen, Sachiko Kusukawa, Vivian Nutton, Brian Ogilvie, Stephen Pender, Claire Preston, Jennifer Rampling, Anna Marie Roos and Richard Serjeantson.\nThe conference lanaguages are English and French.\nImportant notice: Organisers of Thematic Sessions (STS) and Working Groups (WGT) and Presenters of papers have to be mambers of the International Society for the Sociology of Religions (ISSR). Each participant may only present one paper at the conference.\nCFP: The Quran and Islamic Tradition in Comparative Perspective unit of the ISBL welcomes proposals for both individual papers and pre-arranged panels at the international meeting at the University of St. Andrews, Scotland, July 7-11, 2013.\nWe especially welcome papers of a theoretical or methodological nature that explore the ramifications of the comparative study of the Bible and Jewish and Christian tradition alongside the Quran and Islamic tradition.\nProposals for panels or individual papers can be submitted online at http:\/\/www.sbl-site.org\/meetings\/Internationalmeeting.aspx.\nThe deadline for submission of proposals is February 1, 2013. Please note that membership in the Society of Biblical Literature is required in order to submit a paper proposal.\nThe aim of this colloquium is to explore how authoritative texts, culture heroes, and authors were invoked ritually for cursing, protection, and divination in the ancient and late antique Near Eastern and Mediterranean world. The speakers represent a wide range of specializations in ancient ritual practice, including Egyptian, Near Eastern, Greek, Roman, Jewish, and Christian materials.\nOr post a booking form (attached) and a cheque payable to 'Inform' to Inform, Houghton St., London WC2A 2AE. (Inform@lse.ac.uk; 020 7955 7677).\nIt would be our pleasure to invite representatives of archives as well as other individuals interested in folklore archiving to join the network by contacting the co-ordinator Ave Gor\u0161i\u010d by e-mail at <avetupits@folklore.ee> by December 1, 2012. Your suggestions concerning the archive network are warmly welcome.\nParticipants of the round table: Ave Gor\u0161i\u010d (Estonia), Risto J\u00e4rv (Estonia), Anu Korb (Estonia), Svetlana Kosyreva (Russia), Kati Mikkola (Finland), Mari Sarv (Estonia), Janika Oras (Estonia), Rabindranath Sarma (India), Lina Sokolovait\u0117 (Lithuania), Svetlana Tsonkova (Bulgaria), Ergo-Hart V\u00e4strik (Estonia).\nIn the era of digital revolution and under the circumstances of economic depression, folklore archives in different countries face and share similar problems. The need for a more intense cooperation in the field of folklore archiving was underlined at the round table of the 85th anniversary conference of the Estonian Folklore Archives in Tartu on September 24\u201325, 2012, which brought together archivists and researchers from Estonia, Finland, Latvia, Lithuania and Russia. Participants of the round table suggested launching an international network of folklore archives that would bring together both institutions (representatives of folklore archives) and individuals whose research is related to folklore archives.\nWebsite with links to participating archives and researchers, providing preliminary information in a common foreign language (English) on archives, researchers\/archivists and their topics, and about the accessibility of digitised collections of different institutions.\nNetwork meetings and online groups to discuss possibilities for joint financing, cooperation in the field of collecting campaigns, technical upgrading, etc.\nJoint seminars, conferences and panels in international conferences.\nNewsletter in English, published and disseminated electronically.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While this does not pertain directly to the Delta SkyClub I wanted to share this story anyway. As a Delta GM member I deceided to take advantage of the new China Southern Sky Pearl Lounge in Guangzhou, China. For those who are not aware as of last September China Southern is now a part of SkyTeam. I opted to use the showers in the club as I still had another leg to fly to Bangkok. The shower room BTW was out of this world, very nice with every ammentity a flyer would need (listerine, exfoilating cleanser, cucumber and green tea shampoo) and a huge glass shower, granite lavatory and 1 peice Kohler toilet.\nAfter I got out of the shower I was looking into the mirror over the lavatory sink and noticed a red light in the center lower part of the mirror from inside the mirror. I looked closer and could easily see the outline of a 6\" x 6\" monitor (maybe camera) above the red light inside the mirror. It then became obvious that this was a 2 way mirror.\nNow this really doesn't bother me, truthfully I expect this sort of thing from certain countries I visit but as a SkyTeam member one would think that there are certain practices an airline would have to follow in order to wear this badge. It makes much more sense to put a camera outside the entrance to the shower room to see who comes and goes, but even then this was a private shower that the staff had to unlock for me to gain entrance. It's not like anybody could just walk into the shower room without the staff knowing.\nAnyway that's my story, it happened February 7th 2012.\nI too, would think that the airlines (international in nature) need to uphold some general best practices.\nCould you describe a bit more the 6x6 monitor? I ask because what you describe could be consistent with a mirror heater (designed to keep a face-sized portion of the mirror steam-free).\nI'm thinking there is some other explanation here.\nCould it have been a motion detector sensor to turn off the lights and water when no one is in the room?\nI have no idea what this means; there is no correlation between being in Sky Team and the quality of the lounges; Delta is an obvious example with horrible, overcrowded, poor stocked day care centers. You really think KE\/CZ\/etc enjoy associating themselves with that? Or do you think they just accept this is part of partnering with (awful) US airlines?\nThis is the most logical explanation to me (at least I would like to believe it more than the alternative). The mirror heater mats are very common, and some have LED lights. There are all kinds of them, 6x6, 12x12, built-in, glue-on, etc.\nAnd if there were indeed a 2-way mirror\/surveillance device there, wouldn't it be more logical to install it in a more clandestine way and without LED lights?\nWhile this does not pertain directly to the Delta SkyClub I wanted to share this story anyway. As a Delta GM member I deceided to take advantage of the new China Southern Sky Pearl Lounge in Guangzhou, China. For those who are not aware as of last September China Southern is now a part of SkyTeam.\nBecause of the reasons posted by you, the more appropriate forum is the SkyTeam forum, which is now the new home for this thread.\nOr do you think they just accept this is part of partnering with (awful) US airlines?\nAll three alliances have to accept that they have to partner with (awful) US member carriers.\nFirstly it was in all likelihood a heater or a motion sensor.\nSecondly, CZ has been a member of Skyteam for at least 5 years if not more.\nIt could be one of those new built-in TV inside the mirror. The red light is there to show Standby status. I have seen it in some St Regis and other high end properties. If that is the case, there should be a TV remote available somewhere.\nCould this be a tv? Was at a hotel - name now escaping me - with a small TV behind the mirrow in the bathroom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For a full obituary of the author, Grandizo Munis, see Revolutionary History Vol.2 No.2, Summer 1969, which also published 3 articles by Munis, and a personal memoir of him by Ernest Rogers.\nIn 1962 an alliance between the principle underground organisations in Spain was signed. The following comment is from ALARMA (see end footnote).\nTo the younger generation, overwhelmed by the daily routines of the dictatorship, there is perhaps an attractive promise here. But nothing is further from reality. The stated declaration of the United Nations protects no more than the right of men to submit to the exploitation and political dictates of capitalism. It would be difficult for it to be otherwise since the UN is an association of the property-owning states of world capital. And, incidentally, these same rights, agreed in the declaration, are a dead letter in the majority of the countries of the western bloc and the so-called neutrals, as they are in all the states of the Eastern bloc. But let us suppose for the moment that, tomorrow, these rights were fully implemented in Spain. Would they thus solve the problems of a society that has been established for more than 30 years? Would they, hell! As far as the rights which are necessary and immediately achievable, the stated declaration hardly represents those agreed by the regulations of a prison for condemned men. Thus the third point of the agreement is limited to creating a rational capitalist society.\nHowever it is worth the toil involved to note the very deliberate formulation of this point in the agreement \"To re-establish civil liberties UNTIL the full enjoyment is reached \u2026 (etc.)\". That is to say the signatories, even if they had the power to do so, would not grant the full normal rights and liberties of bourgeois legality, but setting themselves up as the tutors of a people supposedly overcome by pettiness, incapacity, indifference etc, they would simply define, according to their own views, what such \"liberty\" means. At the most the pact is merely a promised 'Constitution'.\nIt is of course quite natural that the so-called \"Sindical de Trabajadores Vascos\" (Basque Workers Union \u2013 Eds) should \"rise in revolt\" in this sort of way, because, being just a clerico-paternalistic organisation, it considers itself called upon to ward off a new revolution with volleys of harsh words. And neither is it a wonder that the UGT tends to the same sort of thing. For decades now the \"Socialist\" Parties have adopted all the values of existing society \u2013 exploitation included. The new feature is that the CNT now makes an act of contrition and also bows down before the same \"legality\". But the novelty is only relative because, after the first few months of the civil war, it was they who permitted the Stalinists to get away with the destruction of revolutionary conquests. The signing of the pact gives a dominant character to bourgeois tendencies, and from its birth, this will give it an organic stability.\nFor many years we have maintained that a revolution based on trade union organisations is an impossibility, and that is irrespective of whether they are anarchist or marxist. All trade unionism becomes, sooner or later, useful to capital, and in the course of society's development it has shown a great partiality to State Capitalism: that is to say with the same thing that the enemies of the working class call \"Socialism\". Humiliating themselves before the UGT and the little men of the crucifix who lead the Basque union, the CNT has reached the end of its evolutionary cycle. Now it can be no more than an another body within capitalist legality. Thus the revolutionary defence of the economic interests of the working class will need new forms of organisation: directly elected committees in the work-places. By this means, the working class can achieve the expropriation of capital and the abolition of wage labour, a task that, if nothing else, is incompatible with Trade Union organisation (including the best that can be imagined).\nPoint no.4 of the Agreement promises in vain \"to oppose any other anti-democratic regime\" that tries to succeed the Franco one. But world forces will not be stopped or diverted by words. The dictatorial tendencies of Spanish capitalism, until now embodied in the clergy and army, are a necessity imposed by their particular characteristics within a concrete historical framework. This tendency can find new outlets in State Capitalism, which has already given proof of its reactionary efficiency in so many countries, but cannot be abolished except by the death of capitalism itself. And if the Pact in question reveals itself to be ineffective in limiting the misconduct of the clergy and army, it would not represent much of an obstacle to a State Capitalist dictatorship. It is impossible to move to the future without ending the economy of capital and wage labour.\nThe same sort of ideas has also given birth to the so-called \"Frente de Fuerzas Democraticas\" (Front of Democratic Forces) These democratic forces are partly composed of ex-falangists that now call themselves, \"monarchists\" and \"Christian Democrats\", and partly of adherents of the Socialist Party. Like other similar organisations built in the past, it is probable that the \"Frente\" does not reach any circles outside (inside? ERC) Spain, though most of the aforementioned string along with prominent people inside Spain who, at the same time, have their own fish to fry. The one thing they will not do is to rally the people against the regime, and even less to encourage the revolt of the oppressed. Their clear intention is the opposite: to guarantee that the succession to Franco takes place takes within the existing social order.\nThese pseudo-democratic pact-makers promise to submit the question of the kind of regime that would have to be established in Spain to the popular will. They abuse the words. They will promise to hold elections, whether very monarchist or very republican, which are only two of the many forms of the same capitalist society. A real choice of regime has to be based on the property system: socialism or capitalism, precisely establishing that state property is no less capitalist than the individual variety. But in making this choice, the labouring masses find themselves in a plain position of inferiority, since all resources and means of influence are in the hands of their opponents. Neither the gentlemen of the \"Frente\", nor the worshippers of Stalin, ever called upon them to take this sort of decision.\nThe concrete problems to decide are these: either the army and the professional police or their dissolution and the arming of the working class; either an economy based on capitalism and wage labour or an economy based on the workers of production and distribution; either a parliamentary democracy or a government based on workers' committees. But, as in 1936, these and other decisive questions will have to be resolved by working class struggle.\nFootnote. The editorial board wish to point out that they do not necessarily agree with all the views and ideas expressed in the above article. Since however the article is a) informative and b) the considered assessment of a Spanish grouping on the political situation in Spain, they hope that its appearance in English will prove useful. ALARMA (organ of the group known as 'Fomento Obrero Revolutionario') is published in France, but also circulates illegally in Spain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're thinking \"I need to call a Local Oral Surgery Service Near Me\" in Oak Creek, WI then you're in luck. JustLocalBusiness.com Simple Business Search provides the most up to date business listings, phone numbers and information about Plumber services, contractors and companies in Oak Creek, WI.\nDenis P. Lynch, D.D.S., Ph.D.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Hart Taxis will close down at 4am on Christmas Day and our offices will reopen at 7am Boxing Day.\nAny early morning jobs before 7am on 26th December will still continue to be carried out.\nHart Taxis would like to wish all our customers old and new a fantastic Christmas and prosperous 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SPP Round 2 Applications \u2013 Due Date Reminder!\nDEEL is conducting a second round of selection of families for open Seattle Preschool Program slots. There are openings at the two new Seattle Public School sites \u2013 Original and New Van Asselt, and the Hoa Mai Vietnamese Dual Language Program. We'll also be filling slots opened up form Round 1. Just a reminder \u2013 Applications are due to DEEL by end of the day Monday, September 7!\nParents will be notified of the selection results by September 14.\nPlease email preschool@seattle.gov with any questions.\n1. A Vietnamese-English dual language program at the Hoa Mai Vietnamese Bilingual Preschool in Mount Baker.\n2. Two NEW Seattle Public School classrooms (located at the Old and New Van Asselt Schools) approved by the Seattle School District Board on August 19.\n3. Openings made available within Round 1 assigned classrooms.\nIMPORTANT! DEEL received three times as many applications as we had openings for our first round of SPP selection. Families not selected during the first round have the option of being included in the second round without needing to reapply.\nRound 2 applications are due by the end of the day September 7. Parents will be notified of the selection results by September 14.\nDEEL will send out notifications to families on Friday, August 21, about whether or not their children have been selected to enroll for fall 2015 Seattle Preschool Program (SPP) classes. Families will receive notification via e-mail and telephone. Please check your e-mails and voice mail on Friday.\nSeattle Preschool Program (SPP) Application Deadline Extended!\nThe deadline for completing the SPP application form has been extended from July 31 to Monday, August 10, 5:00 PM. Families who apply by the deadline will be entered into the first round of selections for preschool enrollment.\nTo make the application accessible to all families, DEEL will be making application forms available in 14 different languages. As soon as they are posted on our site, we will send an updated notice.\nThe first round of selections will occur the week of August 17, and families will be notified of their status by Friday, August 21.\nYou can access the SPP application form here, an application information sheet, and the tuition calculator.\nFor more information about applying for SPP, contact the Department of Education and Early Learning at preschool@seattle.gov or call (206) 684-3942.\nThe completed SPP application form is due by Friday, July 31, in order to be considered for the first round of selections for SPP.\nThe first round of selections will occur the week of August 3, and families will be notified of their status by Monday, August 10.\nToday Mayor Ed Murray and Council President Tim Burgess announced the SPP Providers selected for Year 1 (2015-16). Read the press release or visit our website!\nThe Families & Education Levy partners with schools across our city to fund the professional development of the RULER curriculum.\nTake a look at this latest video from the Seattle Channel's Our City, Our Schools series to learn more about how this curriculum is making a positive impact on students and their success.\nHow do students' emotions impact their academic achievement?\nSeattle Public Schools is using an approach known as RULER to help students and teachers Recognize, Understand, Label, Express and Regulate emotions.\nHost Brian Callanan interviews RULER co-creator Marc Brackett and profiles the RULER program at Graham Hill Elementary, South Shore K-8 and Olympic Hills Elementary for a comprehensive look at how teaching emotional smarts can help boost academic success. This segment is part of the Our City, Our Schools special series, which offers an up-close look at Seattle Families and Education levy-funded programs designed to increase student success in school. Gain more insight on your feelings with the Yale Center for Emotional Intelligence's Mood Meter App \u2014 mentioned in the show.\nCongratulations to our Levy-funded schools who won the Washington Achievement Award!\nEvery year the State's Office of Superintendent of Public Instruction selects schools who are top performers in the State, and 10 of those top performers are Seattle Levy-funded schools.\nWe are so proud of the great work happening in Seattle Schools!\nOverall Excellence \u2013 Awarded to the top 5% of all WA elementary, middle and high schools (based on three years of data).\nHigh Progress \u2013 Top 10% of schools making progress in reading and math combined performance for all students (based on three years of data).\nSpecial Recognition \u2013 Reading and Mathematics Growth \u2013 Top 5% of schools with a 3-year average Median Student Growth Percentiles in reading and\/or math.\nThe Department of Education and Early Learning has a new blog!\nThanks for checking out our blog. We are excited to keep you up to date on what's happening at DEEL.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems like everyone has a cute dog in Los Angeles. Dogs are (wo)man's best friend and are proven to make you happier and live longer. Whether you can't get a pet or can't get enough of other people's pets, check out these 6 awesome places for animal lovers in LA.\nRescue cats and coffee await you at Crumbs and Whiskers! Sip on your beverage of choice while cuddling with rescue cats. If you happen to fall in love with one, you can apply to adopt!\nReal Housewife of Beverly Hills Lisa Vanderpump began Vanderpump Dogs in 2016 after founding Stop Yulin Forever, a program which rescues dogs from the Yulin meat festival in China. Play with rescue pups, volunteer your time, and shop the merchandise in this pink paradise, you'll never want to leave!\n1,000 acre Saddlerock ranch and vineyard is home to zebras, camels, and even a giraffe! Many animals head to the ranch for \"retirement\" after performing in Hollywood. Feed them snacks (provided by Malibu Wine Safaris), sip on wine, and enjoy your time in heaven!\nLA's Wildlife Learning Center is home to Lola, a two-toed sloth who appeared on the Ellen show with Kirsten Bell and Zeus, a blind owl among many other animals.\nTen Alaskan Timber wolves are the stars of the show at the Shadowland Foundation, a non-profit that teaches Los Angelenos about the contributions of wolves. Meet and greet the wolves \u2013 they're very friendly!\nThe Dog Cafe is a halfway house for rescues looking for their forever home. Make a reservation, grab a drink, and connect with a canine. Even if you don't end up adopting, you're helping to socialize the pups! That's a win-win.\nReady to learn more about animals in LA? Check out these five purrfect reasons to visit Crumbs and Whiskers!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rod Stewart: Steroid 'addiction' shrank my manhood! - New York Daily News Rod Stewart: Steroid 'addiction' shrank my manhood!\nRod Stewart: Steroid 'addiction' shrank my manhood!\nHe may no longer deserve the name Rod Stewart.\nThe 68-year-old rocker revealed that while some guys have all the luck \u2014 he certainly didn't after continual steroid use shrank his... Maggie May.\n\"I let myself down on tours in the late 80s when I was addicted to steroids. In those days we didn't have in-ear monitors and the band kept getting louder and drunker, and I kept blowing my voice out,\" Stewart told Mojo Magazine.\n\"The steroids will take down the swelling in any membrane \u2014 including your k--b-- \u2014 and it's what you do when you're in a bit of a pinch and need to do a show and you can't sing,\" the 68-year-old rocker told Mojo magazine, using a British euphemism for his microphone and amps.\nThe former Small Faces frontman found there was a big cost involved with using the drugs. While it saved his voice, it gave him reasons to sing the blues.\n\"One night, on stage in Sheffield, I thought I was in the kitchen with my mum because the steroids had eaten a hole in my stomach,\" he told Mojo. \"I was bleeding internally and hallucinating. What the audience must have thought?\"\nStewart has overcome the side-effects, though \u2014 the notorious rock 'n' roll Lothario is married to model Penny Lancaster, 42, and has eight children, ranging in ages from two to 50, from four different women.\nThough he called steroids \"the most horrible drugs in the world,\" Stewart did add that he still uses them about once a year \"if I'm really struggling.\n\"It gets you through the show,\" he told the magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a used and still in overall good playing condition Godin ACS that blends classical guitar concepts with elements of solid body design. There are some chips and dings on the top of the body, none of which affect playability nor tone. The incredibly comfortable neck from the Multiac Nylon is used here with a maple body & cedar top to achieve our most affordable synth access guitar ever. Although the body is chambered, response from the maple top is more typical of a solid body design than that of a traditional acoustic guitar. This design results in an instrument that is virtually free of feedback, making it easy to use even when the band gets loud. The engine in both the ACS-SA & ACS-SA Slim includes individual transducer saddles powered by a customized preamp system from the RMCTM Pickup Company. This system not only produces superb amplified sound, but also produces a hexaphonic output through a 13-pin connector enabling direct access to RolandTM GR-Series guitar synthesizers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqrwb b/data_all_eng_slimpj/shuffled/split2/finalzzzaqrwb
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+{"text":"Print design is still in high demand since print is still a viable form of media. Not every designer knows how to design for print. Save yourself a lot of headaches by hiring someone like me who has been doing print design for fifteen years.\nNewspapers. I also design newspapers and magazines. You can see the newspapers I design each month for Local Umbrella Media here.\nThese are samples of various logo designs I have created. Logo design has changed a lot with services such as Fiver that give people access to cheap design. Logos that I design start at around $250 and average around $1,000. Branding requires much more time and a much bigger budget and it involves much more than just a logo design. If you would like to know the difference please contact me for a free consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hosting a party is fun but it's important to have fun appetizers to munch on before the main meal is ready. These recipes are sure to be a hit with the guests while they mingle!\nMy idea of the perfect tea party recipe involves a huge burst of flavor in a teeny tiny bite. Or two teeny tiny bites. I thought of some of the flavors I have fallen in love with recently and incorporated them into two delicious tartlet recipes. If a tea party is on your agenda, you will want to remove the plastic food from your table and make these!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Victoria Marks, an Alpert Award winner and Guggenheim Fellow, has been practicing knowing and unknowing, making dances for stage and film, for the past 34 years. Marks' creative work migrates between choreo-portraits for individuals who don't identify as dancers, and dances for and with dancers that fuel her inquiries into movement. Since 1995, Marks has been a Professor of Choreography at UCLA. Upcoming projects include a Fulbright Distinguished Scholar fellowship for new work with Bogota's \"Concuerpos\" Dance Company, a Rauschenberg Residency for the creation of a new dance film, and \"Desire on Campus\" made for UCLA Sorority and Fraternity students. Marks has received grants, fellowships and awards including two Fulbrights for Choreography, and with filmmaker Margaret Williams, the Grand Prix in the Video Danse Festival (1995\/96), the Golden Antenae Award from Bulgaria, the IMZ Award for best screen choreography and the Best of Show in the Dance Film Association's Dance and the Camera Festival.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take care of yourself. You've invested in your wardrobe, but don't forget the rest: grooming. We've partnered with top, internationally renowned brands and products such as Baxter, Rudy's and Marvis. Work these into your regime, so you can look and feel fresh from head to toe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The February Issue Of Prima Is Out Now!\nLisa Riley chats to Prima about her 12-stone weight loss, the possibility of having kids and her new book, The Honesty Diet.\nThe February issue of Prima is out now with the inspirational Lisa Riley as our cover star. She chats to Prima about her amazing 12-stone weight loss, her hopes and dreams for 2018, and her brand new book, the Honesty Diet.\nLisa's also sharing the secrets to her weight loss success, plus exclusive recipes from her new plan for you to try at home.\nAnd, if you're hoping to make a fresh start in 2018, we've got all the advice you need to get started, covering everything from getting fit to improving your finances.\nIf you're already dreaming about this year's big getaway, then make sure you check out our pick of 2018's hottest holiday destinations, from Bristol to Buenos Aires.\nA bit closer to home, we've also got top tips from interiors expert Sarah Beeny on how to declutter and get organised, as well as bright and bold homeware picks to update your living space.\nPlus, we've got genius style tips for dressing with more confidence, top tips on beating your phobias and advice on protecting your heart health.\nWe want to know what you love about Prima! Have you found some useful tips to save you time and money, or tried one of our recipes which has been a big hit with your family? Or perhaps you've made one of our stylish patterns or created a beautiful craft project, and have photos you'd love to share of your makes with other readers.\nSimply email us at prima@hearst.co.uk with 'Prima Letters' in the subject line, or write to Prima Letters, 33 Broadwick St, London W1F 0DQ. There's a beautiful bouquet of flowers worth \u00a325 for the star letter!\nWhat are you waiting for? Get your copy NOW! Or why not get Prima delivered directly to your door? Subscribe here!\nThe March issue of Prima is out now!\nThe October issue of Prima is out now!\nThe January issue of Prima is out now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasmlw b/data_all_eng_slimpj/shuffled/split2/finalzzzasmlw
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Cosmo Police Galivan is simply the epitome of \"badass\". In Jair's own words, \"This game is a nifty platform RPG that's sort of like a cross between Metroid and Strider, but cooler.\" In my own words, \"Konami completely ripped off this game when they were making the gameplay engine for Castlevania: Symphony of the Night.\" Anyway, if those descriptions don't help you, the gist of the game is that you run around in a side-scrolling environemnt, kicking the crap out of bad guys with sword, and finding items so that you may advance farther \u2013 items that make you jump higher, better weapons, etc. All while you level up in a standard RPG expereince system. It's a really fun game, and you're really missing out if you don't find the ROM and try it out.\nJair's translation is simply lovely, and I think that everyone can stand and applaud him for such a fine job. No really, it's great! Really great!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a private, family owned wholesale\/retail fuel company. We supply commercial, industrial and residential customers with the following products at great discounted prices.\nHy Grade Oil offers residential and commercial fuel services in the Clifton, NJ area.\ni've been an oil customer of hygrade's for years. great service and always reasonable prices. highly recommended.\nPosted by Guest09531 on December 06, 2010. Brought to you by superpages.\nPosted by jaimelynn412 on November 03, 2009. Brought to you by yellowpages.\nHy-Grade Propane can be found at River Rd 301. The following is offered: Propane. The entry is present with us since Sep 9, 2010 and was last updated on Nov 14, 2013. In Clifton there are 1 other Propane. An overview can be found here.\nWe are a private, family owned wholesale\/retail fuel company. We currently own and operate Hy-Grade Propane and Hy-Grade Oil which is located in Clifton, NJ. We also own and operate retail gas stations and C-Stores. We are currently selling almost 35 Million Gallons per year. We are dedicated in providing excellent service at discounted prices.\nPosted on November 03, 2014. Brought to you by yellowmoxie.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The highly critical CIA Inspector General's Report about the CIA's interrogation tactics was withheld from the public. In 2008, a highly redacted version was released only after a lawsuit was filed by the American Civil Liberties Union. A slightly less redacted version was released (Aug., 2009). The IG report is highly critical of the agency's detention and interrogation activities. It questions the efficacy, ethics and legality of CIA's harsh methods which were expressly forbidden and banned by federal law. These include physical torture such as the application of pressure to the arteries on the sides of a detainee's neck resulting in near loss of consciousness; mock executions brandishing a handgun and power drill; waterboarding methods that diverged radically from SERE training, methods that had not been approved by the DoJ.\nThe CIA used waterboarding in a way that had not been approved by DoJ, calling into question the legality of the technique inasmuch as DoJ's basis for approval was pulled out from under. The waterboarding of detainees was NOT similar to how it was used in SERE military training; differences included duration, frequency, the amount of water used, and the obstruction of detainees' air passages.\nThe IG repeatedly brought abuses and violations to the attention of Attorney General John Ashcroft who saw nothing wrong with waterboarding a detainee 119 times. Whereas federal law bans the use of torture and expressly forbids threatening a detainee with \"imminent death,\" the IG report makes clear that the gross abuse of prisoners \u2013 i.e., the use of torture \u2013 had the full backing of AG Ashcroft and the Justice Department.\nThe harsh interrogation program, begun in 2002, was poorly managed; some interrogators were poorly trained, poorly informed, and used unauthorized techniques. An example cited: \"During the four days the individual was detained, an Agency independent contractor, who was a paramilitary officer, is alleged to have severely beaten the detainee with a large metal flashlight and kicked him during interrogation sessions.\" The detainee died in custody.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All high quality mobile apps are available for free download. 10,000+ users downloaded Silhouette Free latest version on 9Apps for free every week! And now, this app adds more functions. This hot app was released on 2014-08-08. Don't wait any longer\u2013 the application will shock you.\nHere comes the big news! This top Personalization app is just 1.5M. What calls for special attention is that massive of people use this app and they really enjoy it. 9Apps also provides other hot Personalization apps(games) for android mobile phone. 9Apps provides massive new Apps and Games, free download and no need for root!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mini laptops & netbooks are portable laptops that come more handy when compared to the regular laptop. They are device that can perform the majority of the functions of a desktop or laptop, but is extremely mobile. Unlike a desktop computer, mini laptops are perfect for those who are constantly on the go and are in need of a mobile PC. Whether as a student or business executive netbooks will serve you a lot. Here at Konga you will find a large selection of mini laptops and notebooks from top brands like Lenovo, HP, Dell, Asus and many others at affordable prices. If you have been dreaming of buying a laptop Konga present you the opportunity to actualize your dream with our array of mini laptops and netbooks at budget friendly prices. No matter which road you choose to travel, take technology wherever you go with a mini laptop.\nNetbooks look like miniature laptops, with screens rarely exceeding 10 or 12 inches. Netbooks are primarily mobile Web browsing tools which offer connectivity to the Internet and the most pertinent productivity tools, such as word processing, calculations and the like, they are actually excellent budget computing options, educational tools and mobile workstations. A netbook's familiar layout and relatively advanced functionality make it an ideal mobile workstation. Netbooks are an economical and productive means to educate students and move businesses forward. No matter which road you choose to travel, take technology wherever you go with a mini laptop on konga.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasnib b/data_all_eng_slimpj/shuffled/split2/finalzzzasnib
new file mode 100644
index 0000000000000000000000000000000000000000..b1d700ee1342f35117f36c991ddb8180f4bb9858
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"TRY US FIRST - WE WANT YOUR PART EXCHANGE SEE OUR WEBSITE FOR DETAILS, FINANCE AVAILABLE UP TO 5 YEARS WITH NO DEPOSIT!!! \u00a31170 OF FACTORY FITTED OPTIONS! APPEARANCE & CONVENIENCE PACK! 1 Previous Owner, Service History, MOT till Oct 19, Tax Available, PAS, Climate, Cruise, Panoramic Glass Roof with Electric Blind, Xenons, Multi-Function Steering Wheel, Bluetooth, Front & Rear Parking Aids, CD Sound System, Electric\/Heated Seats, E\/Windows, R\/C\/Locking, Alarm\/Immobiliser, Aluminium Roof Rails, 17\" Mulit Spoked Alloys, HPi Clear, Px Welcome, Fully Colour Coded in Caribou Metallic with Black Half Leather Interior, EXCEPTIONAL RARELY AVAILABLE GRAND C-MAX 7 SEATER PETROL WITH TOP SPEC AND IN GREAT COLOUR, FIRST TO SEE\/DRIVE WILL BUY!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was at a local pub when this couple started going at it by the side of the bar. I hope you don't promote public displays of affection. What are your thoughts?\nOne quick question Michael; do you like extra starch on your dry cleaning? In reality we agree there is a line one should be aware of in public. This also depends on the venue. You said you were at a pub. Was there no were more isolated for these two to go? Our opinion, there is always a way to get someone to a more private setting. PDA is not for most. One last question Michael, did they look like this??\nThis is right up there with the guy that lives with all the dolls. Good find, +2.\nBunny to the stuffed animal, \"do you feel it now?\"\nor was that what my ex asked me?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In search of a novel Mother's Day gift that gets your kids involved in the making process? Check out these fun DIY Building Brick Frames we made with our Peel 'n Stick Baseplates and a simple dollar store frame hack!\nSome pictures of your adorable kiddos!\nAs a sharp knife is required, we definitely recommend that Dad or another adult is in charge of making the frames. All that's required is to measure up the pieces of baseplate and cut them. We decided to go for a width of two studs as this all fitted together nicely. When you cut, be sure to cut against a line of bricks and to follow the same groove each time. Then fold the baseplate along the groove until it snaps and you can cut off the attached backing.\nOnce you've got the necessary four strips of baseplate, one by one peel off the backing and stick them in place. Don't push down too hard to begin with as you may need to reposition the strips a little so that they all line up.\nOnce all of the strips are stuck down firmly, just check to make sure that there are no sharp edges left exposed. If there are, you can use the craft knife to carefully file them off.\nAnd there you have it- the frames are constructed and ready for your kids to start decorating!\nWe put out an inviting mixture of bricks and mini figures and encouraged our kids to get creative when designing a special frame for Mom. We also put their pictures in the frames so that they could understand the gift in its entirety.\nThey could not wait, and started adding bricks and mini figures to their frames straight away.\nOur kids loved being able to customize their photo frames for Mom. We made sure that we captured just how much they enjoyed creating for Mom so that she can be made to feel extra special!\nWhat more could any Mom ask for than a hand crafted keep sake of her little angels?!\nBest of all, your kids can re-design their photo frames as often as they like... and Mom can update the pictures as often as she likes too!\nWould your kids love this project? Tag us with your great Mother's Day building brick ideas, @creativeqt, on Facebook and Instagram!\nCreative QT designs quality + innovative toys that declutter homes and inspire creative play. Founded by parents of five, Adam and Dana Sue Hinkle, Creative QT's vision is to empower parents and encourage a culture of families that Make Time Together. All products are designed to enrich families' lives through active, creative play and play based learning. Creative QT products are laboratory tested for compliance with CPSC requirements and are free of lead, cadmium and phthalates so you can play with confidence. So, go ahead \u2026 today is the day: be your kid's hero.\n\"LEGO\u00ae is a trademark of the LEGO Group of companies which does not sponsor, authorize or endorse this site.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Initiation of cell migration requires morphological polarization with formation of a dominant leading pseudopodium and rear compartment. A molecular understanding of this process has been limited, due to the inability to biochemically separate the leading pseudopodium from the rear of the cell. Here we examine the spatio-temporal localization and activation of cytoskeletal-associated signals in purified pseudopodia directed to undergo growth or retraction. Pseudopodia growth requires assembly of a p130Crk-associated substrate (CAS)\/c-CrkII (Crk) scaffold, which facilitates translocation and activation of Rac1. Interestingly, Rac1 activation then serves as a positive-feedback loop to maintain CAS\/Crk coupling and pseudopodia extension. Conversely, disassembly of this molecular scaffold is critical for export and down regulation of Rac1 activity and induction of pseudopodia retraction. Surprisingly, the uncoupling of Crk from CAS during pseudopodium retraction is independent of changes in focal adhesion kinase activity and CAS tyrosine phosphorylation. These findings establish CAS\/Crk as an essential scaffold for Rac1-mediated pseudopodia growth and retraction, and illustrate spatio-temporal segregation of cytoskeletal signals during cell polarization.\nDirected cell movement or chemotaxis is exhibited during wound healing, angiogenesis, embryonic development, and immune function (Lauffenburger and Horwitz, 1996). This process is highly conserved, as prokaryotes and eukaryotes from Dictyostelium discoideum to human leukocytes exhibit the ability to sense and move in the direction of a chemoattractant (Parent and Devreotes, 1999; Jin et al., 2000; Servant et al., 2000). Recent evidence indicates that when eukaryotic cells encounter a chemoattractant gradient they respond by local activation and amplification of signals on the side facing the gradient (Parent et al., 1998; Meili et al., 1999; Jin et al., 2000; Servant et al., 2000). These signals facilitate localized actin polymerization leading to membrane protrusion in the direction of the gradient. The protrusion of a dominant leading pseudopodium (or lamellipodium) marks the first sign of morphological polarity with establishment of an anterior and posterior compartment, and occurs independently of cell body translocation or chemotaxis (Lauffenburger and Horwitz, 1996; Parent and Devreotes, 1999). Once a dominant pseudopodium is formed, cell movement commences in the direction of the gradient as the cell undergoes a cycle of membrane extension at the front and contraction at the rear.\nRecent work has shown that integrin-cytoskeletal linkages, chemokine receptors, and actin regulatory proteins are spatially regulated in migratory cells (Ridley et al., 1992; Schmidt et al., 1993; Manes et al., 1999; Nobes and Hall, 1999; Eddy et al., 2000). Adhesive signals by integrins may also spatially localize to extending pseudopodia where they fine tune and maintain directional growth while suppressing retraction and detachment mechanisms (Smilenov et al., 1999; Kiosses et al., 2001; Laukaitis et al., 2001). Significant progress has also been made in determining the intracellular organization in migratory cells of pleckstrin homology (PH)*-domain proteins, integrins, and the small GTPase Rac1 using GFP technology (Meili et al., 1999; Jin et al., 2000; Servant et al., 2000; Kraynov et al., 2001; Laukaitis et al., 2001). However, this work is typically done in individual cells, preventing molecular and biochemical evaluation of chemotactic signals and their spatio-temporal association with regulatory proteins and scaffolds. This is due to the lack of cellular material available for analysis, as well as the inability to specifically isolate the pseudopodium and cell body for biochemical comparison using previous techniques. The biochemical purification of the pseudopodium and cell body is necessary to understand the molecular detail of entire signaling networks and spatio-temporal mechanisms of regulation including protein translocation, activation\/phosphorylation, and formation of complex multiprotein scaffolds. In this work we provide evidence for purification of pseudopodia induced to undergo growth or retraction, and demonstrate the spatial, dynamic assembly of the CAS\/Crk\/Rac signaling complex that controls this process.\nDirectional cell migration or chemotaxis requires cells to sense the direction and proximity of a chemoattractant. This requires activation of localized signals and actin polymerization on the cell membrane facing the gradient. On the other hand, random migration or chemokinesis on the other hand is persistent cell movement in the absence or presence of a uniform concentration of chemokine (Lauffenburger and Horwitz, 1996). To determine whether lysophosphatidic acid (LPA) (Fukushima et al., 2001) and insulin induced directed or random cell migration, NIH 3T3 and COS-7 cells were examined for cell migration through 8.0-\u03bcm porous membranes in the presence of either a gradient or uniform concentration of these chemokines. Only cells exposed to an LPA gradient were induced to migrate, indicating that LPA is a true chemoattractant and does not facilitate random cell migration (Fig. 1). LPA-induced chemotaxis was dose dependent and reached a maximum with 100\u2013200 ng\/ml. In contrast, insulin promoted random migration, but was a poor chemoattractant for both cell types (Fig. 1).\nLysophosphatidic acid is a potent chemoattractant for NIH 3T3 and COS-7 cells. (a) NIH 3T3 cells were examined for cell migration for 3 h in 8.0-\u03bcm porous Boyden chambers containing different concentrations of LPA or insulin placed in the bottom, top, or top and bottom compartments. The number of migratory cells per microscopic field (200\u00d7) on the underside of the membrane was counted as described in Materials and methods. (b) COS-7 cells were examined for cell migration as described in panel a. Each point represents the mean \u00b1 SEM of three triplicate migration chambers of three independent experiments.\nWhen cells respond to a chemoattractant gradient they extend pseudopodia in the direction of the chemokine before cell translocation. The formation of a dominant leading pseudopodium establishes cell polarity and the future direction of chemotaxis. These observations prompted us to determine whether cells would polarize by extending pseudopodia through 3.0-\u03bcm porous membranes toward a chemoattractant gradient, independent of cell body translocation. Indeed, cells exposed to a gradient, but not a uniform concentration of LPA, extend pseudopodia through small pores specifically in the direction of the gradient (Fig. 2, a\u2013c). As expected, cells exposed to a gradient of insulin failed to polarize and extend pseudopodia through the pores (unpublished data). Using a confocal microscope sequentially focused at the upper and lower membrane surface, we determined that >90% of the cells polarize by extending pseudopodia in response to an LPA gradient (unpublished data). Extension of pseudopodia through 3.0-\u03bcm pores was first detected 10\u201315 min after exposure to the LPA gradient and proceeded linearly for 60\u201390 min (Fig. 2, a and b). Formation of the LPA gradient under these conditions was steep and linear for at least 3 h (Fig. 2 d). In addition, nuclei were not detected (DAPI staining) on the lower surface of the membrane, but were restricted solely to the cell body on the upper surface, indicating that only pseudopodia protrude through the pores (unpublished data). Most importantly, pseudopodia extension was prevented in cells expressing dominant negative mutants of the small GTPase Rac1 (RacN17) or Cdc42 (Cdc42N17) (Ridley et al., 1992) (Fig. 2, e and f). Rac and Cdc42 are well established to control cell polarity through regulation of actin protrusive processes at the leading front of migratory cells (Allen et al., 1998; Nobes and Hall, 1999; Etienne-Manneville and Hall, 2001). Together, these findings indicate that cells establish polarity by extending leading pseudopodia through 3.0-\u03bcm pores toward a gradient of LPA in a Rac and Cdc42 dependent manner.\nCells extend pseudopodia through 3.0-\u03bcm pores toward an LPA gradient, but not a uniform concentration of LPA. (a) NIH 3T3 cells were allowed to attach to fibronectin coated 3.0- \u03bcm porous membranes for 2 h. Pseudopodia extension was then examined for various times in the absence (NT) or presence of LPA (100 ng\/ml) in the bottom, top, or top and bottom compartments. Pseudopodia protein on the underside of the membrane was determined as described in Materials and methods. Each point represents the mean \u00b1 SEM of three triplicate membranes of three independent experiments. (b) COS-7 cells were examined for pseudopodia protrusion as described in panel a. (c) Confocal images of NIH 3T3 pseudopodia protrusion on the undersurface of a 3.0-\u03bcm porous membrane in response to LPA (100 ng\/ml) as a gradient (LPA bottom) or uniform concentration (LPA top and bottom). Cells were labeled with cell tracker green and then fixed at the indicated times to visualize protruding pseudopodia on the undersurface of the membrane. Bar, 15 \u03bcm. (d) Diffusion of 3H-LPA (100 ng\/ml) from the lower chamber to the upper chamber through a 3.0-\u03bcm porous membrane was measured at the times indicated. Each point represents the mean counts per minute (CPM) \u00b1 SEM of three triplicate chambers of three independent experiments. (e) COS-7 cells transfected with dominant negative Rac1 (Rac N17), Cdc42 (Cdc42N17), or the empty vector (mock) were examined for pseudopodia formation toward an LPA gradient for 60 min as described in (a). Each bar represents the mean \u00b1 SEM of three triplicate membranes of three independent experiments. An aliquot of cells used for the pseudopodia assay was also lysed and Western blotted for exogenous Rac and Cdc42 expression. RacN17 and Cdc42N17 are myc- and HA-tagged, respectively, and thus show reduced mobility in SDS-PAGE relative to endogenous proteins (Wt). (f) An aliquot of cells transfected as in panel e was examined for cell adhesion to collagen-coated dishes for 30 min in the presence or absence (NT) of LPA (100 ng\/ml) as described in Materials and methods. Results are reported as cells attached per microscopic field (200\u00d7). Each bar represents the mean \u00b1 SEM of five fields of three triplicate wells of three independent experiments.\nGiven that the cells polarized toward a chemoattractant gradient, removing the gradient should induce pseudopodia retraction and loss of cell polarity. To investigate this possibility, cells were induced to extend pseudopodia toward an LPA gradient for 60 min. The gradient was then removed from the lower chamber and pseudopodia retraction was examined for various times. Loss of the LPA gradient was sufficient to reverse the polarized phenotype and induce pseudopodia retraction (Fig. 3, a and b). Confocal imaging and time-lapse determination revealed that pseudopodia retraction began within 5 min of removing the gradient and proceeded for 2 h (Fig. 3 c). No new pseudopodia extensions were observed during this time period, and >80% of the cells displayed pseuodopodia retraction under these conditions (unpublished data). As expected, restoring the LPA gradient to these cells inhibited the retraction process and induced pseudopodia growth (unpublished data). Therefore, cells retract their pseudopodia protrusions and lose cell polarity upon removal of the chemoattractant gradient.\nPseudopodia retract upon removal of an LPA gradient. (a) Pseudopodia extension of NIH 3T3 cells through 3.0-\u03bcm pores toward an LPA gradient was examined for the indicated times, or the LPA gradient was removed after 60 min and pseudopodia were allowed to retract for the indicated times (LPA Removed). Total pseudopodia protein on the underside of the membrane was determined as described in Materials and methods. Each point represents the mean \u00b1 SEM of three triplicate membranes of three independent experiments. (b) Pseudopodia were allowed to extend for 60 min (time 0) toward an LPA gradient as described in panel a. The gradient was then removed and pseudopodia were fixed at the indicated times. Cells were labeled with cell tracker green (CTG) to visualize retracting pseudopodia on the undersurface of the membrane by confocal microscopy. Bar, 15 \u03bcm. (c) NIH 3T3 cells labeled with CTG were allowed to extend pseudopodia through 3.0-\u03bcm pores toward an LPA gradient for 60 min. The LPA gradient was then removed (time 0) and the pseudopodia allowed to retract for the indicated times. Time-lapse images of retracting pseudopodia were taken with a confocal microscope focused at the pore\u2013membrane interface on the undersurface of the polycarbonate membrane. See Video 1 (available at http:\/\/www.jcb.org\/cgi\/content\/full\/jcb.200111032\/DC1) for a movie illustrating pseudopodia retraction. Bar, 15 \u03bcm. (d) COS-7 cell pseudopodia extension toward an LPA gradient was determined in the presence or absence (NT) of function-blocking antibodies to \u03b1v\u03b25 and \u03b21 integrins (25 \u03bcg\/ml) for the indicated times. Percent pseudopodia growth is the amount of pseudopodia protein on the undersurface of the membrane induced by cells exposed to an LPA gradient relative to cells in the absence of LPA. (e) Retraction of COS-7 cell pseudopodia were determined as described in panel a in the presence or absence (NT) of antibodies to \u03b1v\u03b25 and \u03b21 integrins (25 \u03bcg\/ml) for the indicated times. Each point represents the mean \u00b1 SEM of three triplicate membranes of three independent experiments.\nRecent evidence indicates that integrin adhesion receptors play a critical role in facilitating and maintaining directional growth of pseudopodia on extracellular matrix (ECM) proteins (Bailly et al., 1998; Kiosses et al., 2001; Laukaitis et al., 2001). To investigate whether integrins were necessary for pseudopodia extension on a collagen substrate, COS-7 cells were allowed to attach to collagen-coated membranes for 2 h and then exposed to an LPA gradient in the presence of function-blocking antibodies to \u03b21 integrins, which facilitates attachment of these cells to collagen (Cho and Klemke, 2000). The anti-\u03b21 antibody prevented pseudopodia growth on collagen, whereas control antibodies to the vitronectin receptor \u03b1v\u03b25 present on these cells did not (Fig. 3 d). Importantly, the \u03b21 blocking antibodies specifically prevented pseudopodia extension and did not cause detachment of the cell body from the substratum (unpublished data). The anti-\u03b21 antibodies did not alter pseudopodia retraction, indicating that formation of new adhesion contacts were not necessary for the retraction process per se (Fig. 3 e). These findings demonstrate that integrins are necessary for pseudopodia growth on the ECM, and provide additional evidence that new pseudopodia do not extend through the pores after the gradient is removed, as this would depend on new integrin contacts.\nThe above findings indicate that pseudopodia growth and retraction is a dynamic process involving changes in focal adhesions and the actin cytoskeleton. To directly examine cytoskeletal components as well as complex signaling pathways that control cell polarity, we sought to biochemically separate the cell into its leading pseudopodium and cell body for protein analysis. Our findings that cells polarize by extending pseudopodia through 3.0-\u03bcm pores, only in the direction of a chemoattractant gradient, presented us with a unique opportunity to separate and purify these structures. Cells were allowed to extend pseudopodia for 60 min toward a chemoattractant gradient or the gradient was removed and cells were allowed to retract for 30 min. At these times, the chemoattractant gradient as well as pseudopodia growth and retraction are linear (Fig. 2). The cell body on the upper surface of the membrane was manually removed and the pseudopodia on the undersurface extracted with detergent. The cell body was purified in a similar manner except that pseudopodia on the lower surface were manually removed and the cell body on the upper surface extracted with detergent. The total profile of proteins isolated from the cell body and pseudopodium was normalized and then analyzed by one-dimensional SDS-PAGE. As expected, abundant proteins were similar in the cell body and pseudopodium, indicating that the samples were normalized appropriately (Fig. 4 a). However, nuclear histones were clearly absent from the pseudopodium, providing additional evidence for the selective purification of this structure (14\u201318- and 30-kD proteins) (Fig. 4 a).\nBiochemical characterization of cytoskeletal-regulatory proteins in growing and retracting pseudopodia. (a) NIH 3T3 cells were allowed to extend pseudopodia toward an LPA gradient (100 ng\/ml) for 60 min (growth), or the LPA gradient was removed and pseudopodia allowed to retract for 30 min (retraction). Proteins (10 \u03bcg) isolated from the cell body on the upper membrane surface or pseudopodia (Pseudo) on the lower membrane surface were resolved by one-dimensional SDS-PAGE and GelCode Staining (Pierce Chemical Co.) as described in Materials and methods. Total cellular proteins were also isolated from cells attached to culture dishes either not treated (NT) or treated with a uniform concentration of LPA for 60 min (growth control). Cells were also treated with a uniform concentration of LPA for 60 min and then washed and proteins isolated after 30 min of retraction (retraction control). Arrowheads indicate nuclear histone proteins which are absent in purified pseudopodia. (b) Proteins isolated as described in panel a were analyzed by Western blotting using antibodies to the indicated proteins. Whole cell represents total cellular protein isolated from untreated cells attached to fibronectin coated dishes for 90 min. (c) Proteins prepared from NIH 3T3 cells as described in panel a were examined for GTP-bound activated Rac and Cdc42 using the p21-binding domain of PAK, which selectively binds Rac- and Cdc42-GTP. Rho-GTP was detected using the Rhotekin Rho binding domain. GTP bound (active) or total protein (total) in the corresponding whole-cell lysates was detected by Western blotting as described in Materials and methods. Densitometry was used to determine the ratio of GTP bound Rac, Cdc42, and Rho to the total GTPase protein present in an aliquot of the same protein lysates used for the activity assay. Fold increase represents the change in GTPase activity in the cell body and pseudopodial fractions relative to basal GTPase activity present in untreated whole cells (NT). (d) NIH 3T3 cells were either held in suspension for 30 min (sus) or allowed to attach for 2 h to either fibronectin coated culture dishes, or the upper surface of a 3.0-\u03bcm porous membrane coated with fibronectin. Whole cells on culture dishes or pseudopodia in the growth and retraction phase were isolated as described in panel a and analyzed for tyrosine phosphorylation by Western blotting with anti-phosphotyrosine antibodies. Blots treated with ECL reagent were exposed to film for 30 and 90 s. Arrowheads indicate proteins with increased phosphotyrosine in purified pseudopodia. Asterisk shows proteins with increased phosphotyrosine in retracting pseudopodia. (e) Total proteins isolated as described in (d) were analyzed by Western blotting with phosphorylation site\u2013specific antibodies to FAK at tyrosine's 397 (autophosphorylation, c-src and PI3K binding sites), 576, and 577 (catalytic activation sites). Blots were stripped and reprobed with antibodies to FAK protein to evaluate the level of FAK protein present in the lysates. (f) Proteins isolated as described in (a) were either analyzed by Western blotting with phosphorylation-site specific antibodies to tyrosine 783 of human PLC\u03b3-1 or immunoprecipitated with antibodies to CAS and then immunoblotted with antiphosphotyrosine antibodies. NT is total cellular protein isolated from untreated cells attached to fibronectin coated dishes for 120 min. Arrows indicate tyrosine phosphorylated proteins of 85 (p85), and 70 kD (p70) that coimmunprecipitate with CAS specifically in growing, but not retracting pseudopodia. Arrowhead shows a tyrosine-phosphorylated protein of 60 kD that coimmunoprecipitates with CAS in the cell body, but not pseudopodia of polarized cells. IgH is the immunoglobulin heavy chain.\nNext, we examined the expression of several cytoskeletal proteins previously associated with pseudopodia structures in cells. Caldesmon, dynamin II, paxillin, and filamin were all substantially increased in the pseudopodium (Fig. 4 b) (Ridley et al., 1992; Helfman et al., 1999; Ohta et al., 1999; McNiven et al., 2000; Laukaitis et al., 2001). In contrast, the actin-severing protein gelsolin was restricted to the cell body proper and was not present in the pseudopodium, whereas extracellular-regulated kinase (ERK)2 did not show a spatial change in polarized cells (Azuma et al., 1998). Importantly, Rho, Rac, and Cdc42 showed significantly increased activity in the extending pseudopodium compared with the cell body (Fig. 4 c). Associated with the increased GTPase activity was increased Rho and Rac, but not Cdc42 protein levels, in the pseudopodium. On the other hand, retracting pseudopodia showed decreased Rho, Rac, and Cdc42 activity, as well as decreased Rho and Rac protein levels. However, whereas Rho activity was clearly decreased during retraction, a notable amount of Rho activity remained in the pseudopodium under these conditions (Fig. 4 c). There was also a small increase in Rho activity in the body of retracting cells. Rho may play an important role in both growth and retraction processes through its ability to regulate Rac and Cdc42 activity as well as the actin\/myosin contractile machinery (Schmitz et al., 2000). Pseudopodia isolated after 15 and 90 min of growth or 15 and 90 min of retraction showed identical results (unpublished data). It is important to note that during LPA-induced pseudopodia growth there is no change in Rho, Rac, or Cdc42 activity in the cell body relative to the nontreated whole cell group. Thus, it is clear that the cell body on the upper surface does not simultaneously extend pseudopodia, as this would also lead to increased GTPase activity in the upper compartment. Together, these findings demonstrate the biochemical purification of pseudopodia and the spatial segregation of cytoskeletal regulatory proteins in polarized cells.\nEstablishment of a leading pseudopodium requires spatial regulation of actin polymerization and formation of focal adhesions, which is associated with tyrosine phosphorylation of cellular proteins (Lauffenburger and Horwitz, 1996; Aplin et al., 1998). Therefore, we examined the phosphotyrosine protein profile in the cell body and pseudopodium of cells polarized towards a chemoattractant gradient. Phosphoproteins of 220, 125\u2013130, 100, and 70 kD were present in purified pseudopodia and either absent or dephosphorylated in the cell body (Fig. 4 d). Importantly, these proteins were dephosphorylated in suspension cells, indicating that their phosphorylation is dependent on cell adhesion to the ECM. In contrast, phosphoproteins of 74 and 28 kD were present in the cell body and either absent or dephosphorylated in the pseudopodium. Phosphotyrosine proteins appeared similar under growth and retraction conditions except for a prominent 40-kD protein that was phosphorylated only in pseudopodia undergoing retraction, but not growth, suggesting a possible role for this protein in the retraction process (Fig. 4 d).\nAnalysis of the focal adhesion\u2013associated proteins FAK, CAS, and PLC\u03b3-1 revealed that these p120\u2013130-kD phosphoproteins were strongly activated\/phosphorylated during pseudopodia growth as well as retraction, although there may be a small decrease in FAK 576 and 577 phosphorylation during retraction (Fig. 4, e and f). Importantly, the relative levels of FAK, CAS, and PLC\u03b3-1 were similar in the pseudopodium and cell body of polarized cells during growth and retraction. Thus, the protein level and tyrosine phosphorylation of these signals do not change significantly (Fig. 4, e and f), even though pseudopodium growth\/retraction involves dynamic focal contact remodelling and actin cytoskeletal changes. These findings suggest that general dephosphorylation or loss of focal adhesion\u2013associated signals is not the primary mechanism responsible for pseudopodium retraction and detachment from the substratum. Thus, the process of pseudopodia retraction is not simply the reverse of the signaling processes that mediate growth. This is significantly different than cell detachment from the ECM, which is accompanied by complete dephosphorylation and inactivation of FAK, CAS, and PLC\u03b3-1 (Aplin et al., 1998).\nWhat role then does tyrosine phosphorylation of these and other cytoskeletal regulatory proteins play during pseudopodia growth and retraction? One possibility is that these proteins assemble specific signaling scaffolds consisting of unique kinases, substrates, and effector proteins that regulate this process. To investigate this possibility, we examined the formation of a FAK\/CAS\/Crk protein complex, as the spatio-temporal assembly of this scaffold and its role in mediating cell migration are poorly understood (Klemke et al., 1998). Interestingly, Crk strongly associated with CAS in growing but was significantly reduced in retracting pseudopodia (Fig. 5 a). The interaction was specific to the pseudopodium, as CAS isolated from the cell body of polarized cells did not show Crk binding. In contrast, FAK, which binds to the src-homology (SH)3 domain of CAS, showed no difference in binding to CAS under these conditions (Fig. 5 a). In addition, we observed several tyrosine-phosphorylated proteins that coimmunoprecipitated with CAS in growing, but not retracting pseudopodia (Fig. 4 f). In contrast, a 60-kD phosphoprotein associated with CAS only in the cell body, but not the pseudopodium. Denaturation of cellular proteins before CAS immunoprecipitation prevented the association of these proteins with CAS, indicating that the interaction is specific and not related to antibody cross-reactivity (unpublished data). Thus, CAS\/Crk complexes assemble and disassemble in a highly spatial manner during pseudopodia extension and retraction.\nCAS\/Crk coupling and Rac activity regulate pseudopodia growth and retraction. (a) NIH 3T3 cells attach to fibronectin coated 3.0-\u03bcm porous membranes were allowed to extend pseudopodia toward a LPA gradient (100 ng\/ml) for 60 min (growth), or the LPA gradient was removed and the pseudopodia allowed to retract for 30 min (retraction). Proteins isolated from the cell body or pseudopodia under growth or retraction conditions were prepared as described in Fig. 4 a. Total cellular protein (NT) was also isolated from cells attached to culture dishes in the absence of LPA. CAS or Crk was then immunoprecipitated and Western blotted for associated Crk or CAS, respectively. Also, FAK association with CAS was examined in the CAS immunoprecipitates by Western blotting with FAK specific antibodies. (b) COS-7 cells were transfected with the empty vector (Mock) or the vector encoding either CAS with its substrate domain truncated (CAS-SD) or Crk with a mutated SH2 domain (Crk-SH2). Cells were then examined for pseudopodia extension in response to a LPA gradient (100 ng\/ml) or left untreated for 60 min. Pseudopodia protein was determined as described in Materials and methods. (c) COS-7 cells were transfected with the empty vector (Mock) or vectors encoding CAS and Crk. Cells were then examined for pseudopodia extension in response to a LPA gradient (100 ng\/ml) or not treated for 60 min. Pseudopodia protein on the underside of the membrane was determined as described in Materials and methods. (d) COS-7 cells transfected as in panel c were allowed to extend pseudopodia toward an LPA gradient (100 ng\/ml) for 60 min (growth), or the LPA gradient was removed and the pseudopodia allowed to retract for 30 min (retraction). Pseudopodia protein was determined as described in Materials and methods. The bottom panel shows CAS\/Crk coupling in purified pseudopodia undergoing growth (60 min) or retraction (60 min) from cells transfected as described in panel c. (e) Cell body and pseudopodia proteins prepared from cells treated as described in panel c were examined for activated Rac or total Rac protein in whole cell lysates as described in Fig. 4 c.\nCell adhesion to the ECM promotes Crk binding via its SH2 domain to phosphotyrosine residues present in the substrate domain of CAS, whereas cell detachment causes complete dephosphorylation of CAS and disassembly of CAS\/Crk complexes (Vuori et al., 1996; Klemke et al., 1998). Therefore, it is surprising that the overall level of protein and tyrosine phosphorylation of CAS does not change during pseudopodium retraction and detachment from the ECM, but Crk binding does. This suggests that assembly and disassembly of CAS\/Crk in the pseudopodium is tightly regulated through phosphorylation\/dephosphorylation of a specific subset of tyrosine residues present in CAS. Alternatively, phosphorylation of the regulatory tyrosine 221 of Crk, which prevents CAS\/Crk coupling in cells, may regulate this process (Kain and Klemke, 2001). However, we did not detect significant changes in tyrosine phosphorylation of Crk during pseudopodia growth and retraction (unpublished data). Therefore, tyrosine phosphorylation of the substrate domain of CAS may be a critical event involved in CAS\/Crk coupling and pseudopodium formation. To directly test this possibility, we examined pseudopodia formation in cells expressing CAS with its substrate domain deleted (CAS-SD). CAS-SD or Crk with a mutated SH2 domain (Crk-SH2) serve as dominant negative proteins preventing CAS\/Crk coupling and Rac activation in motile cells (Klemke et al., 1998). Expression of CAS-SD or Crk-SH2 prevented pseudopodia extension in response to an LPA gradient (Fig. 5 b). These findings demonstrate that CAS\/Crk coupling is necessary for cell polarization and extension of a leading pseudopodium.\nTo investigate whether CAS\/Crk coupling could enhance pseudopodia extension, we exogenously expressed full-length CAS and Crk in cells. However, these cells did not show a significant change in pseudopodia extension in response to an LPA gradient, suggesting that these molecular signals are not limiting to the process of gradient sensing and chemotaxis and that an additional component(s) is required to mediate this process (Fig. 5 c). This is different from haptotaxis migration, as exogenous CAS\/Crk coupling in cells is sufficient to facilitate strong migration as well as pseudopodia extension (unpublished data) toward an adhesive gradient of ECM protein in the absence of a soluble chemoattractant (Klemke et al., 1998). Surprisingly, whereas increased CAS\/Crk coupling in these cells did not impact pseudopodia extension, it did prevent membrane retraction, when the gradient was removed (Fig. 5 d). In this case, CAS\/Crk complexes were not disassembled, and the activation and translocation of Rac into pseudopodia were persistent (Fig. 5, d and e). These findings suggest that sustained Rac activity sends a positive feedback signal to upstream kinases\/phosphatases that regulate CAS\/Crk coupling and pseudopodium extension. To investigate this possibility, we examined CAS\/Crk coupling in cells expressing dominant negative RacN17. Inhibition of Rac activity in cells significantly decreased CAS\/Crk coupling, which was independent of changes in FAK and PLC\u03b3-1 activity (Fig. 6, a and b). Moreover, the inhibition of CAS\/Crk coupling was independent of changes in overall CAS tyrosine phosphorylation. Expression of activated RacQ61L in cells did not alter CAS\/Crk coupling, indicating that this complex was maximally activated in these cells (Fig. 6 a). Therefore, Rac can operate upstream to specifically regulate assembly of CAS\/Crk complexes in cells. This supports the idea that Rac activation serves as a positive feedback signal to modulate CAS\/Crk coupling in migratory cells. Together, our findings demonstrate that assembly of a CAS\/Crk complex is important for pseudopodia extension, whereas disassembly facilitates retraction through deactivation of Rac and its export from this cellular structure. A model depicting the role of CAS\/Crk coupling in regulating Rac-mediated pseudopodium dynamics in chemotactic cells is shown in Fig. 7 and discussed below.\nRac activity regulates assembly of CAS\/Crk complexes in cells. COS-7 cells were transfected with CAS and Crk along with either myc-tagged dominant negative (RacN17) or dominant positive (RacQ61L) Rac1. Cells were then lysed in detergent (whole cell lysate) and examined for either (a) assembly of CAS\/Crk complexes as described in Fig. 5 a; or (b) changes in FAK and PLC\u03b3-1 activity as described in Fig. 4, e and f. Blots were stripped and reprobed with antibodies to FAK and PLC\u03b3-1 to confirm equal protein loading.\nProposed model for the role of CAS\/Crk coupling in regulation of Rac-mediated pseudopodial dynamics. (Left) Model illustrating the basic steps of cell polarization and chemotaxis on the ECM. (Step 1) Stationary cells are attached to the underlying ECM through integrin receptors (squares). Cells are then exposed to a soluble gradient of growth\/factor or chemokine (Step 2). This activates cell surface chemattractant receptors leading to activation and amplification of Rac signaling events on the side facing the gradient. (Step 3). Rho, Cdc42, and Rac then regulate localized actin dynamics as well as force requirements leading to membrane protrusion in the direction of the gradient. This process is independent of actual cell body translocation or chemotaxis and marks the first sign of morphological polarity with establishment of a dominant leading pseudopodium and posterior compartment. Importantly, evidence indicates that the initial protrusion of a pseudopodium at the cell surface is independent of integrins and the ECM contacts. However, integrins do play a critical role in pseudopodial dynamics by tethering the extending membrane to the substratum, which initiates the molecular coupling of CAS and Crk. CAS\/Crk coupling in turn mediates Rac activity leading to sustained and directional pseudopodium growth. A pseudopodium that does not attach to the ECM rapidly retracts back to the cell body. (Step 4) Once a dominant pseudopodium is formed, cell movement commences in the direction of the gradient as the cell undergoes repeated cycles of membrane protrusion, adhesion to the ECM, and CAS\/Crk\/Rac activation. Importantly, our findings suggest that sustained Rac activity then provides a positive feedback signal to maintain CAS\/Crk coupling and membrane extension at the leading front as the cell moves on the ECM (Fig. 7, right panel; Discussion). On the other hand, loss of the chemoattractant gradient or inappropriate contact with the ECM terminates pseudopodium extension, turning off CAS\/Crk\/Rac activity and chemotaxis. Therefore, in this model integrin ligation events at the leading front of the extending membrane cooperate with chemoattractant receptors to fine tune and maintain directional growth, while suppressing retraction mechanisms through regulation of CAS\/Crk and Rac.\nWe provide several lines of evidence for the biochemical purification of pseudopodia. First, >90% of cells polarize by extending pseudopodia through 3.0-\u03bcm pores in the direction of a chemoattractant gradient, whereas cells exposed to a uniform concentration of chemokine do not. Importantly, pseudopodium extension was directional and proceeded in a linear manner, independent of cell body translocation, which allowed us to differentially isolate this structure from the cell body. Second, cytoskeletal regulatory proteins previously associated with pseudopodia-like structures were present in protein extracts using this fractionation method (Fig. 4). In contrast, nuclear histones associated with the cell body region of polarized cells were not present in the pseudopodial extract. Finally, and most importantly, Rac and Cdc42 activity were increased in extending, but not retracting pseudopodia. Moreover, Rac and Cdc42 activity were necessary for this response as expression of a dominant negative forms of these proteins in cells prevented pseudopodia extension. Rac and Cdc42 are well documented to facilitate actin-based protrusive mechanisms leading to membrane extension and polarity in migratory cells (Ridley et al., 1992; Nobes and Hall, 1999). Together, these findings demonstrate that cells polarize by extending pseudopodia through 3.0-\u03bcm pores in the direction of a chemoattractant, and that it is possible to purify these structures for biochemical analysis.\nAn important aspect of this model is the ability to differentially control pseudopodia growth and retraction. Whereas nonpolarized cells in culture have been observed to extend and retract exploratory pseudopodia, the addition of a uniform concentration of chemoattractant only increases these random events, with little or no cell polarization (i.e., extension of one dominant pseudopodium). In our system, we were able to induce in a synchronous manner a leading pseudopodium in >90% of the cells. We found that removing the chemoattractant gradient also caused a synchronous loss of cell polarity and retraction of pseudopodia in the majority of cells. This allowed us to differentially monitor biochemical changes during pseudopodium growth as well as retraction. This aspect of the model is particularly attractive for studying components that regulate actin polymerization\/depolymerization, myosin contractility, and adhesive signals that facilitate membrane attachment\/detachment from the ECM. In addition, we show that it is possible with this model to quantify as well as visualize pseudopodia in living cells using confocal microscopy and time-lapse imaging. Therefore, analysis of signaling dynamics of growing and retracting pseudopodia in live cells is also possible with this system.\nWe demonstrate that CAS and Crk are specifically assembled during pseudopodia growth and then disassembled during retraction. Surprisingly, this occurred without apparent change in the overall level of CAS tyrosine phosphorylation or FAK activation, which is an upstream activator of CAS (Vuori et al., 1996; Tachibana et al., 1997). One explanation is that Crk couples only to a specific subset of phosphotyrosine residues present in the substrate domain of CAS, which changes during pseudopodia growth and retraction. CAS\/Crk association is mediated through the binding of the SH2 domain of Crk to phosphotyrosine residues present in the substrate domain of CAS (Matsuda et al., 1993). In fact, there are 15 tyrosine residues in this region of CAS that correspond to potential SH2 binding motifs, 9 of which conform to the Crk SH2 recognition sequence (YD(V\/T)P (Klemke et al., 1998). Alternatively, regulation may occur through serine phosphorylation of CAS (Ma et al., 2001) or phosphorylation of the regulatory tyrosine 221 of Crk, which prevents CAS\/Crk coupling in cells (Kain and Klemke, 2001). However, the latter is unlikely as we did not detect a significant change in Crk tyrosine phosphorylation. The upstream and downstream components that modulate the assembly\/disassembly of this molecular scaffold in the pseudopodium are not yet clear, but likely candidates include c-src, PTP-PEST, and PTP-1B (Garton et al., 1996; Liu et al., 1996; Vuori et al., 1996).\nThe active translocation of Rac to the pseudopodium also implies that there is an import\/export mechanism that controls Rac transport to the different poles of migratory cells. It is unlikely that CAS and Crk are directly involved in this process, as these proteins do not appear to translocate in and out of growing and retracting pseudopodia. Rather, it is more likely that assembly of CAS\/Crk is important for maintaining localized Rac activity within the pseudopodium. For example, it may be that when cells encounter a chemoattractant gradient, Rac (and associated regulatory proteins) translocates to the side of the membrane facing the gradient as previously described for PH domain proteins like Akt (Servant et al., 2000) (Fig. 7). The localized Rac activity then induces actin polymerization leading to protrusion of a pseudopodium from the cell surface and recruitment of high-affinity integrins to this region (Kiosses et al., 2001). Interestingly, protrusion of the pseudopodium from the membrane surface is independent of integrin ligation and attachment to the ECM (Bailly et al., 1998). This clearly places Rac activation and actin protrusive mechanisms as an early response upstream of integrin ligation and CAS\/Crk assembly. However, pseudopodia attachment to the ECM is critical to stabilize the protrusive structure, as protruding membranes that do not contact the ECM readily retract back to the cell body (Bailly et al., 1998). Therefore, it is likely that during pseudopodia attachment to the substratum, integrin activation and focal complex formation play a central role in driving assembly of CAS\/Crk, leading to sustained Rac activation and persistent membrane protrusion. Our findings suggest that the persistent Rac activity then serves as a positive input to maintain a high level of CAS\/Crk coupling at the leading front (Fig. 7). In contrast, withdrawal of the chemotactic signal terminates membrane extension and focal adhesion formation resulting in uncoupling of the CAS\/Crk\/Rac signaling module leading to pseudopodium retraction. Interestingly, the disassembly of this complex and membrane retraction occur independently of changes in FAK and PLC\u03b3-1 activity, and CAS tyrosine phosphorylation. This suggests that the Rac feedback pathway specifically regulates CAS\/Crk coupling and does not generally block upstream integrin signaling processes. The Rac feedback loop may target a phosphatase that dephosphorylates a specific tyrosine residue(s) in the substrate domain of CAS, which facilitates Crk binding. Apparently, integrin and cytoskeletal signals continue to play a prominent role during pseudopodium retraction by assembling and regulating new protein scaffolds that mediate this process. However, it is clear that pseudopodium retraction is not the simple reversal of the adhesive-signaling processes that facilitate growth.\nIn summary, we purified pseudopodia in the process of growth or retraction and biochemically examined the spatial localization and activation of adhesion and cytoskeletal-associated signals that control this process. We demonstrate that pseudopodia extension and retraction require Rac activation\/deactivation, respectively, through spatial assembly\/disassembly of a CAS\/Crk protein scaffold. Our findings provide a biochemical understanding of how cytoskeletal and adhesive signals control morphological changes necessary for cell migration, and illustrate the dynamic spatio-temporal complexity of signal regulation during cell polarization.\nThe expression plasmid pUCCAGGS containing full-length as well as mutant forms of the c-Crk cDNA were constructed as described previously (Matsuda et al., 1993; Tanaka et al., 1993). The pEBG expression plasmid containing wild-type CAS or CAS with an in-frame deletion of its substrate domain (CAS-SD) has been described previously (Mayer et al., 1995). Myc-tagged dominant negative Rac1 (N17), dominant positive Rac1 (Q61L), and HA-tagged dominant negative Cdc42 (Cdc42N17) in pcDNA3 have been described previously (Klemke et al., 1998). Antibodies to Crk, CAS, and paxillin, were from Transduction Laboratories. Antibodies to myc (9E10), ERK2, and FAK were from Santa Cruz Biotechnology. Antibodies to phosphotyrosine (monoclonal antibody 4G10), Rho, Rac1, Cdc42, and the GTPase activity kits for these proteins were from Upstate Biotechnology. Goat anti\u2013rabbit and anti\u2013mouse antibodies were from Bio-Rad Laboratories. Antibodies to gelsolin, filamin, the \u03b21 integrin subunit, integrin \u03b1v\u03b25, caldesmon, and QCM migration kit were from Chemicon International. Phosphospecific antibodies to FAK tyrosine 397, 576, 577, and PLC\u03b3-1 tyrosine 783 were from Biosource International. Anti-dynamin II antibodies were provided by Dr. Sandra Schmid (The Scripps Research Institute, La Jolla, CA). LPA and anti-vinculin antibodies were from Sigma-Aldrich. 3H-LPA (50 Ci\/mmol) was from NEN Life Science Products, Inc. Cell tracker green was from Molecular Probes, Inc. COS-7 cells were from American Type Culture Collection. Mouse NIH3T3 cells were provided by Dr. Tony Hunter (The Salk Institute, La Jolla, CA).\nPseudopodia extension was monitored using a pseudopodia assay kit (ECM 650; Chemicon International) or Costar chambers. Serum-starved cells (75,000) were placed into the upper compartment of a chamber (6.5 \u03bcm) equipped with a 3.0-\u03bcm porous polycarbonate membrane coated on both sides with an optimal amount of ECM protein (5 \u03bcg\/ml fibronectin or collagen type I). Cells were allowed to attach and spread on the upper surface of the membrane for 2 h, and then stimulated with LPA (Sigma-Aldrich), insulin, or buffer only, which was placed in the lower chamber to establish a gradient or placed in the upper and lower chamber to form a uniform concentration. Cells were allowed to extend pseudopodia through the pores toward the direction of the gradient for various times. To initiate pseudopodia retraction, the chemoattractant was removed or an equivalent amount of chemoattractant placed in the upper chamber to create a uniform concentration. The cell body on the upper surface was manually removed with a cotton swab and the total pseudopodia protein only on the undersurface determined using BCA and a microprotein assay system (Pierce Chemical Co.). It is also possible to stain pseudopodia with 1% crystal violet in 2% ethanol and then elute the dye with 10% acetic acid, which can be measured in an ELISA plate reader (OD 600) for comparison to a standard curve as previously described (Klemke et al., 1998).\nTo specifically isolate proteins from growing pseudopodia, 1\u20131.5 \u00d7 106 cells were induced to form pseudopodia for various times as described above or not treated using a pseudopodia isolation kit (ECM 660; Chemicon International) or chambers from Costar (24 mm 3.0-\u03bcm pores). Cells were rinsed in excess cold PBS and either rapidly fixed in 100% ice-cold methanol (Western blotting of whole-cell lysates) or not fixed (for GTPase activity and immunoprecipitation assays). We found that the protein profile of pseudopodia proteins from fixed and unfixed cells is identical under these conditions as determined by silver staining and SDS-PAGE (unpublished data). Cell bodies on the upper membrane surface were manually removed with cotton swab and pseudopodia on the undersurface scrapped into lysis buffer (100 mM Tris, pH 7.4, 5 mM EDTA, 150 mM NaCl, 1 mM sodium orthovanadate, protease inhibitors [cocktail tablet; Boehringer Mannheim Corp.]) containing the appropriate detergent for Western blotting of whole-cell lysates (1% SDS), CAS\/Crk coupling (1% Triton X-100), or GTPase activity assays (Triton X-100 according to the manufacturer's recommendation; UBI). Cell bodies were purified in a similar manner except that pseudopodia on the undersurface were removed and the cell body on the upper surface scraped into lysis buffer and detergent. Retracting pseudopodia were induced for various times by removing or placing chemoatttractant in the upper chamber to create a uniform concentration as described above. We typically obtain 25\u201330 \u03bcg of pseudopodia protein from each 24-mm well stimulated with chemokine for 30\u201360 min. This represents \u223c4\u20135% of the total cellular protein. Cell culture, transient transfections and efficiency determinations, Western blotting, chemotaxis, and adhesion assays were performed at least three times, and a representative image is shown as previously described (Klemke et al., 1998). We routinely obtain transfection efficiencies in COS-7 and NIH 3T3 cells of 80 and 70%, respectively.\nCells were labeled with cell tracker green according to the manufacturer's recommendation (Molecular Probes). Pseudopodia retraction was initiated by removing chemoattractant from the lower chamber as described above. The upper compartment (6.5 \u03bcm) was then fitted to an Attofluor cell chamber containing a glass coverslip (Molecular Probes) and placed into a stage heater (20\/20 Technologies) on a Zeiss Axiovert 100TV microscope. A BioRad 1024 confocal microscope was used to capture time-lapse images of pseudopodia on the undersurface of the membrane every 60 s with a Zeiss 633 Achostigmat lens and LaserSharp software (Bio-Rad Laboratories). QuickTime and Adobe photoshop software were used to prepare time-lapse movies and representative montage images for each time frame assembled for comparison. Immunofluorescent staining and confocal imaging of cells were performed as described (Cheresh et al., 1999).\nVideo 1 (available at http:\/\/www.jcb.org\/cgi\/content\/full\/jcb.200111032\/DC1) shows time-lapse images of pseudopodia retraction through 3.0-\u03bcm pores as depicted in Fig. 3 C.\nWe thank Drs. H. Hirai (University of Tokyo, Tokyo, Japan), T. Hunter, P. Devreotes (Johns Hopkins University, Baltimore, MD), M. Matsuda (Osaka University, Osaka, Japan), B. Mayer (University of Connecticut Health Center, Farmington, CT), K. Vuori (The Burnham Institute, La Jolla, CA), D. Cheresh, and M. Schwartz (The Scripps Research Institute, La Jolla, CA) for providing reagents and advice concerning this project. We also thank Dr. K. Spencer (The Scripps Research Institute) for helpful advice and assistance with confocal microscopy and preparation of movies.\nThis work was supported by National Institute of Health grants CA 78493-01, BCRP 6KB0046. This manuscript is number 14289-IMM from The Scripps Research Institute.\n\u21b5* Abbreviations used in this paper ECM, extracellular matrix; ERK, extracellular-regulated kinase; FAK, focal adhesion kinase; LPA, lysophosphatidic acid; PH, pleckstrin homology; SD, substrate domain; SH, src-homology.\nAllen, W., D. Zicha, and G. Jones. 1998. A role for Cdc42 in macrophage chemotaxis. J. Cell Biol. 141:1147\u20131157.\nAplin, A., A. Howe, S. Alahari, and R. Juliano. 1998. Signal transduction and signal modulation by cell adhesion receptors: the role of integrins, cadherins, immunoglobulin-cell adhesion molecules, and selectins. Pharm. Rev. 50:197\u2013263.\nAzuma, T., W. Witke, T. Stossel, J. Hartwig, and D. Kwiatkowski. 1998. Gelsolin is a downstream effector of Rac for fibroblast motility. EMBO J. 17:1362\u20131370.\nBailly, M., L. Yan, G. Whitesides, J. Condeelis, and J. Segall. 1998. Regulation of protrusion shape and adhesion to the substratum during chemotactic responses of mammalian carcinoma cells. Exp. Cell Res. 241:285\u2013299.\nCheresh, D., J. Leng, and R. Klemke. 1999. Regulation of cell contraction and membrane ruffling by distinct signals in migratory cells. J. Cell Biol. 146:1107\u20131116.\nCho, S., and R. Klemke. 2000. Extracellular-regulated kinase activation and CAS\/Crk coupling regulate cell migration and suppress apoptosis during invasion of the extracellular matrix. J. Cell Biol. 149:223\u2013236.\nEddy, R., L. Pierini, F. Matsumura, and F. Maxfield. 2000. Ca21-dependent myosin II activation is required for uropod retraction during neutrophil migration. J. Cell Sci. 113:1287\u20131298.\nEtienne-Manneville, S., and A. Hall. 2001. Integrin-mediated activation of Cdc42 controls cell polarity in migrating astrocytes through PKCz. Cell. 106: 489\u2013498.\nFukushima, N., I. Ishii, J. Contos, J. Weiner, and J. Chun. 2001. Lysophospholipid receptors. Annu. Rev. Pharmacl. Toxicol. 41:507\u2013534.\nGarton, A., A. Flint, and N. Tonks. 1996. Identification of p130cas as a substrate for the cytosolic protein tyrosine phosphatase PTP-PEST. Mol. Cell Biol. 16:6408\u20136418.\nHelfman, D., E. Levy, C. Berthier, M. Shtutman, D. Riveline, I. Grosheva, A. Lachish-Zalait, M. Elbaum, and A. Bershadsky. 1999. Caldemon inhibits nonmuscle cell contractility and interferes with the formation of focal adhesions. Mol. Biol. Cell. 10:3097\u20133112.\nJin, T., N. Zhang, Y. Long, C. Parent, P. Deverotes. 2000. Localization of the G protein bg complex in living cells during chemotaxis. Science. 287:1034\u20131036.\nKain, K., and R. Klemke. 2001. Inhibition of cell migration by Abl family tyrosine kinases through uncoupling of Crk-CAS complexes. J. Biol. Chem. 276:16185\u201316192.\nKiosses, W., S. Shattil, N. Pampori, and M. Schwartz. 2001. Rac recruits high-affinity integrin avb3 to lamellipodia in endothelial cell migration. Nat. Cell Biol. 3:316\u2013320.\nKlemke, R., J. Leng, R. Molander, P. Brooks, K. Vuori, and D. Cheresh. 1998. CAS\/Crk coupling serves as a \"molecular switch\" for induction of cell migration. J. Cell Biol. 140:961\u2013972.\nKraynov, V., C. Chamberlain, G. Bokoch, M. Schwartz, S. Slabaugh, and K. Hahn. 2001. Localized Rac activation dynamics visualized in living cells. Science. 290:333\u2013337.\nLauffenburger, D., and A. Horwitz. 1996. Cell migration: A physically integrated molecular process. Cell. 84:359\u2013369.\nLaukaitis, C., D. Webb, K. Donais, and A. Horwitz. 2001. Differential dynamics of a5 integrin, paxillin, and a-actinin during formation and disassembly of adhesions in migrating cells. J. Cell Biol. 153:1427\u20131440.\nLiu, F., D. Hill, and J. Chernoff. 1996. Direct binding of the proline-rich region of protein tyrosine phosphatase 1B to the src homology 3 domain of p130cas. J. Biol. Chem. 271:31290\u201331295.\nMa, A., A. Richardson, E. Schaefer, and J. Parsons. 2001. Serine phosphorylation of focal adhesion kinase in interphase and mitosis: a possible role in modulating binding to p130cas. Mol. Biol. Cell. 12:1\u201312.\nManes, S., E. Mira, C. Gomez-Mouton, R. Lacalle, P. Keller, J. Labrador, and C. Martinez-A. 1999. Membrane raft microdomains mediate front-rear polarity in migrating cells. EMBO J. 18:6211\u20136220.\nMatsuda, M., S. Nagata, K. Tanaka, K. Nagashima, and T. Kurata. 1993. Structural requirement of CRK SH2 region for binding to phosphotyrosine containing proteins. J. Biol. Chem. 268:4441\u20134446.\nMayer, B., H. Hirai, and R. Sakai. 1995. Evidence that SH2 domains promote processive phosphorylation by protein-tyrosine kinases. Curr. Biol. 5:296\u2013305.\nMcNiven, M., L. Kim, E. Krueger, J. Orth, H. Cao, and T. Wong. 2000. Regulated interactions between dynamin and the actin-binding protein cortactin modulate cell shape. J. Cell Biol. 151:187\u2013198.\nMeili, R., C. Ellsworth, S. Lee, T. Reddy, H. Ma, and R. Firtel. 1999. Chemoattractant-mediated transient activation and membrane localization of Akt\/PKB is required for efficient chemotaxis to cAMP in Dictyostelium. EMBO J. 18:2092\u20132105.\nNobes, C., and A. Hall. 1999. Rho GTPases control polarity, protrusion, and adhesion during cell movement. J. Cell Biol. 144:1235\u20131244.\nOhta, Y., N. Suzuki, S. Nakamura, J. Hartwig, and T. Stossel. 1999. The small GTPases RalA targets filamin to induce filopodia. Proc. Natl. Acad. Sci. 96:2122\u20132128.\nParent, C., and P. Devreotes. 1999. A cell's sense of direction. Science. 284:765\u2013770.\nParent, C., B. Blacklock, W. Froehlich, D. Murphy, and P. Devreotes. 1998. G protein signaling events are activated at the leading edge of chemotactic cells. Cell. 95:81\u201391.\nRidley, A., H. Paterson, C. Johnston, D. Diekman, and A. Hall. 1992. The small GTP-binding protein Rac regulates growth factor-induced membrane ruffling. Cell. 70:401\u2013410.\nSchmidt, C., A. Horwitz, D. Lauffenburger, and M. Sheetz. 1993. Integrin cytoskeletal interactions in migrating fibroblasts are dynamic and asymmetrical, and regulated. J. Cell Biol. 123:977\u2013991.\nSchmitz, A., E. Govek, B. Bottner, and L. Van Aelst. 2000. Rho GTPases: Signaling, migration, and invasion. Exp. Cell Res. 261:1\u201312.\nServant, G., O. Weiner, P. Herzmark, T. Balla, J. Sedat, and H. Bourne. 2000. Polarization of chemoattractant receptor signaling during neutrophil chemotaxis. Science. 287:1037\u20131040.\nSmilenov, L., A. Mikhailov, Jr. R. Pelham, E. Marcantonio, and G. Gundersen. 1999. Focal adhesion motility revealed in stationary fibroblasts. Science. 286:1172\u20131174.\nTachibana, K., T. Urano, H. Fujita, Y. Ohashi, K. Kamiguchi, S. Iwata, H. Hirai, and C. Morimoto. 1997. Tyrosine phosphorylation of Crk-associated substrates by focal adhesion kinase. J. Biol. Chem. 272:29083\u201329090.\nTanaka, S., S. Hattori, T. Kurata, K. Nagashima, Y. Fukui, S. Nakamura, and M. Matsuda. 1993. Both the SH2 and SH3 domains of Crk protein required for neuronal differentiation of PC12 cells. Mol. Cell. Biol. 13:4409\u20134415.\nVuori, K., H. Hirai, S. Aizawa, and E. Ruoslahti. 1996. Induction of p130CAS signaling complex formation upon integrin-mediated cell adhesion: a role for src family kinases. Mol. Cell. Biol. 16:2606\u20132613.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fax: 508-655-2763 School Hours: 8:35 a.m. - 3:00 p.m.\nEarly Release Hours: 8:35 a.m. - 12:00 p.m.\nPine Hill School's core values of respect and concern for others, personal responsibility, and excellence in learning are promoted throughout the school community. Noteworthy features of the school include a warm and friendly atmosphere, strong parent\/guardian involvement, a high degree of fiscal community support, a talented and dedicated faculty, and a commitment to favorable class sizes.\nPine Hill School has 20 classes with approximately 383 students in kindergarten through grade five. The average class size at Pine Hill School is 19 students. All students are transported to school on buses.\nThe curriculum of the Pine Hill School is designed to promote problem solving, thinking strategies, and the development of creative abilities while emphasizing the mastery of basic skills. Integration of the subject areas is stressed in order to help students make meaningful associations in their learning. Grade level teachers meet weekly to discuss curriculum and student issues. The curriculum is designed to maximize opportunities for a wide range of abilities. An integrated classroom experience allows all children to explore and expand upon various aspects of the curriculum based on interest, motivation, and aptitude.\nThe classroom curriculum is supported by a strong specialist program, which includes music, art, library, health, and physical education. Teaching and learning are also enhanced by a computer specialist who works with students and teachers in the classroom as well as in the computer lab. Instrumental lessons are an after-school option for students beginning in grade four. A band program during the school day is available for fifth graders.\nA well-developed program for students with special needs includes tutorial services, language remediation, related services and an integrated preschool classroom. Services focus on maintaining students within the classroom setting whenever feasible. A school counselor is available to work in all Pine Hill classrooms and with individual or small groups of students as well as parents\/guardians.\nThe Pine Hill School is dedicated to on-going professional development in order to continuously enhance and strengthen the skills and abilities of the faculty. Students are dismissed at noon two Wednesdays a month to provide time for workshops, curriculum development, and parent\/guardian conferences. Regular reports about professional development activities are contained in the monthly Pine Hill School newsletter.\nThe Community School Association (CSA), which is comprised of parents\/guardians, community members, and school personnel, seeks to provide constructive support for the school and maintain open lines of communication. Throughout the year, the CSA engages in several fund raising efforts to benefit the school. Each year the CSA Enrichment Committee previews and obtains a host of special performers, presentations, and resident artist programs to expand and enrich the school curriculum. The educational program at the Pine Hill School is supplemented by an extensive volunteer program organized by the CSA. Volunteers work on special projects, in the classrooms, dining room, and library. For more information, go to the Pine Hill website at www.doversherborn.org.\nThe Student Council offers students their first experience with a representative form of government. Two students from each class in grades three through five are elected to represent their classmates. Elections are held twice a year to maximize opportunities for students to participate. During weekly meetings with the principal and the student council advisor, students share their thoughts and concerns regarding the enhancement of the school climate and work on service projects to benefit the school and the community.\nThe Dover Sherborn Education Fund (DSEF), a non-profit educational foundation, allocates resources to improve curriculum, services, and programs above and beyond that which is supported in the annual school budgets of the Dover Sherborn Public Schools. For more information, go to the Pine Hill website at www.doversherborn.org.\nStudents completing fifth grade at Pine Hill School continue on to the Dover Sherborn Regional Middle School for grades six, seven, and eight. Students attend the Dover Sherborn Regional High School for grades nine through twelve. Transition planning occurs throughout the spring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasyuh b/data_all_eng_slimpj/shuffled/split2/finalzzzasyuh
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+{"text":"This report covers the objectives and the status of a long-range program to develop techniques for assessing the resource potential of liquid-dominated geothermal systems. Field studies underway in northern Nevada comprise a systematic integrated program of geologic, geophysical, and geochemical measurements, necessary to specify a drilling program encompassing heat flow holes, deep calibration holes, and ultimately, deep test wells. The status of Nevada field activities is described. The areas under study are in a region characterized by high heat flow where temperatures at depth in some geothermal systems exceed 180 C. Areas presently being examined include Beowawe Hot Springs in Whirlwind Valley. Buffalo Valley Hot Springs, Leach Hot Springs in Grass Valley, and Kyle Hot Springs in Buena Vista Valley. Geologic studies encompass detailed examinations of structure and lithology to establish the geologic framework of the areas. The geothermal occurrences are characterized by zones of intense fault intersection, which furnish permeable channelways for the introduction of meteoric water into regions of high temperature at depth.\nWollenberg, H.A.; Asaro, F.; Bowman, H.; McEvilly, T.; Morrison, F.; Witherspoon, P.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Todd, Chairman of the ODCS school board, joined the school board in 2010. He and his wife, Sue, have been married for over 20 years, and have three sons: their eldest is a graduate and two currently attend Open Door Christian Schools. The Wrights reside in Avon and are members of Church of the Open Door. They are active in various church ministries as well as coaching for ODCS. Todd currently works as a Territory Manager for Action Fabricators. Previously, he held the position of General Manager of North American Commercial Operations for Prayon Incorporated (international chemical manufacturer). Todd and his wife are both graduates of Taylor University. Todd received his Bachelor of Science in Business Systems from Taylor, and an MBA from Washington University in St. Louis.\nBrian joined the ODCS school board in 2006. He and his wife Kendra live in Oberlin and attend North Olmsted Friends Church. Brian and Kendra have two daughters who have graduated from ODCS. Their oldest daughter graduated from Wheaton College (Illinois) and is currently enrolled at Marshall Law School in Cleveland. Their second daughter is a sophomore at Wheaton College. Their son currently attends ODCS. Brian is employed by Titanium Asset Management Corp. where he serves as the Chief Executive Officer and is a member of the board of directors. Additionally, he serves as the CEO of Titanium's subsidiary Boyd Watterson Asset Management LLC. Titanium is a $9.5 billion multi-strategy investment management firm that maintains investment operations in Milwaukee, Chicago, Cleveland, Charlotte, and Sarasota. Brian obtained an MBA from Case Western Reserve University and is a Chartered Financial Analyst (CFA). He received a Bachelor of Arts in English and Political Science from Cleveland State University.\nKevin and his wife Connie (Van Wingerden) have been married since 2004 and have 3 children, 2 of whom are school-age and attend ODCS. Their experience with Open Door has been a bit special in that Connie graduated from the school and taught 3rd grade here for 8 years. Kevin is a product of the public school system having attended a large school system in Lima, Ohio and two public universities. He graduated with a bachelor's of science degree in biology and a minor in chemistry from the University of Toledo in 1998 and from Wright State University School of Medicine in 2002. He has since become a big believer in the value of Christian education as a means to reinforce what is taught at home. Kevin is a family physician at the Cleveland Clinic and serves as the medical director for the Cleveland Clinic Strongsville Family Health and Ambulatory Surgery Center. The Hopkins family stepped out in faith in 2014 to follow God's leading and help plant a new church in North Ridgeville, Ohio. Westside Community Church has started out small but is committed to helping make disciples of Jesus Christ by building relationships with our neighbors and loving them as ourselves.\nAlex and his wife Mari have been married for 15 years and have 4 children currently enrolled at ODCS. They returned from a 3-year mission to El Salvador in 2016 and Alex is the Executive Director of Church of the Open Door. Prior to moving to Central America, Alex was partner of Filtrexx International, an environmental management company. Alex has a BA from Kent State University and has studied marketing and business management.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yodariquo and Aspro give first hand impressions of GoldenEye 007 (Wii), Dragon Quest IV (DS), Crackdown (360) and Too Human (360) before getting into the news of the week.\n54:46 Nintendo has QA Approval? Since When?\nDoes culture play a part in game development?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Keep your pigeons bacterial and fungal-free in a natural way!\nRopa-B Liquid 10% is made from the best quality oregano oil that is extracted from our own oregano fields. Ropa-B liquid helps to protect against many bacteria such as salmonella, pasteurella and streptococcal. Many forms of fungi, parasites and coccidia are not immune to the rapid and purifying effect of carvacrol and thymol, two substances found in essential oregano oil. Scientific research has shown that these powerful substances tackle harmful microorganisms such as bacteria, parasites and fungi.\n1,5 ml per liter of drinking water, 3-4 days a week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Founded in Bulgaria in 2002, EGT (Euro Games Technology) is well known in the gaming world for developing, producing and distributing a range of top quality products. EGT was launched with just one slot to its name \u2013 the aptly named Gladiator. From those humble beginnings, the firm now employs 800+ people, and fans in over 60 countries worldwide use and enjoy the products and games it produces. Click here to read more.\nDespite such impressive growth, EGT maintains its original base in Sofia, Bulgaria's capital. Having got a foot in the door, EGT chipped away at the Bulgarian games market, before expanding to Russia, neighbouring Romania and beyond to tackle the wider Eastern European market.\nAround this time, the firm's newer games were finding a way into casinos in the region, and in 2005, EGT launched an innovative system which allowed multiple machines to work from just one network. This was groundbreaking in the gaming world, and propelled the company even further up the ladder to success. New markets opened up, including prizes like Macau, and in 2008, EGT released yet another unique design \u2013 the slots known as Vega Vision, which gave customers more choices without changing terminals.\nEGT was not the only company who delayed making inroads in the world of online gameplay, but the wait was definitely worth it, as these days the company is pretty famous for producing video slots with amazing graphics and themes which can be enjoyed by laptop or mobile players on various devices. Meanwhile, of course, EGT continued to work the other aspects of their business, producing top class blackjack and bingo terminals, along with touch screens used in live dealer casino games.\nEGT's output is so consistently popular, as well as being wide-ranging, that is pretty much impossible to pick out particular games and products which could be labeled as its 'most popular', so what follows is by no means a definitive answer to such a question.\nIn terms of casino games, its progressive jackpot games, such as 4 Happy Hits, are enjoyed by thousands, with eye-catching 3D graphics and multi-leveled jackpot prizes. Lady's Cards and Diamond Life enjoy similar success. Online players aren't forgotten here either, with the Burning Hot, Flaming Hot and 40 Super Hot Trilogy blazing a trail in the virtual world.\nTouch screen technology development is another string to the bow of EGT, with outstanding work such as Luxury Touch Table Live Roulette [LTTR] cleverly blending the actual (in the form of a real roulette wheel), with the virtual \u2013 players place wagers via a touch screen.\nHaving established such a huge presence across the entire spectrum of the gaming world, it will be exciting to see where the dedication and imagination of the EGT team will take them in the future. One thing is sure, their loyal fan base and the entire world of gaming technology will be waiting with bated breath to find out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauvhk b/data_all_eng_slimpj/shuffled/split2/finalzzzauvhk
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index 0000000000000000000000000000000000000000..aacb036a1af059ca36d7a31ba97df1b4bbf90b84
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+{"text":"\" These new products are now added to OES's portfolio of technology solutions that not only eliminate part defects from entering the automotive supply chain, but will now provide a high level of error proofing and production traceability\" says Michael Reeve, VP of product development.\nOES works closely and collaboratively with manufacturers, machine suppliers, representatives and distributors to bring practical leading and effective technology solutions that prevent part defects from entering the automotive supply chain. OES holds 9 US patents and two patents pending.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When interviewing Jermyn Corbyn . Andrew Marr on Sunday 23 rd. Interviewed Jermyn Corbyn for total for 19 minutes He spent 11 of those minutes questioning Corbyn about anti-Semitism repeating for example the allegations that Corbyn had gone to a cemetery in Tunisia to pay homage at the graves of members of the black September movement. Raising once again the anti-sematic cartoon which surely has been shown and discussed many times in the past months.\nAh, a ghastly new troll.\nThe BBC is utter scum. Just like this 'professor'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The major focus of this site is to look at long term macro economic shifts in the global economy and how that will affect different investment classes and choices.\nHowever, while the larger picture is fairly clear, the question I am most often asked is, \"what is going to happen next?\" \"What is going to happen right now?\"\nWhile I believe this question is less important than the big picture, I'll take some time here to talk about the short term picture I see and what I am personally doing with investment decisions.\nI believe we are heading through a period of short term disinflation where assets across the board are going to fall in price. In response to this event, central banks around the world will bring the nuclear option that will either re-inflate many markets, or break the financial system completely.\nSo what I am doing to prepare for this?\nThe first thing I have been doing is raising cash. I am not selling any of the assets I already known, but I am not investing new money into the assets I like because I am waiting for a better price.\nAs an investor during these periods of fear in the markets you want to understand what is a fundamentally strong investment, one that could fall in price due to coming liquidations. In times like this investors may need to sell anything to stay alive, creating opportunities for the prepared.\nI watch the price of these assets on a daily basis. I then wait for them to go on sale, similar to a housewife watching the prices at the grocery store.\nSo what do I recommend in the short term?\nTry to create your own shopping list and decide which investments are fundamentally strong and which are fundamentally weak.\nGood luck shopping. It's going to be a bumpy road ahead and hopefully many of the best assets will continue to go on sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mung beans are known by several names in the Eastern world, especially in India and China. Some other names for mung beans are green soy, moog dal, mung and golden gram. Mung beans are traditional used in Bangladeshi, Indian and Chinese cuisine. They are an excellent source of protein. Sprouts of mung beans are a source of vitamin C. There are several substitutes for mung beans when cooking.\nTrying to find a substitute for mung beans in cooking, many people substitute with pigeon peas or snow peas. Cooking with these provides a sturdy substitute to cooking with mung beans. They are commonly found in many grocery stores.\nMung beans sprouts can also be substituted in cooking as well. Sunflower sprouts and soybean sprouts are a good substitute for mung bean sprouts because of their bean-like flavor. The sprouts are mainly used in stir fries and provide a crunchy texture to Asian cuisine.\nSonya Kanti has more than five years of experience writing professionally. Currently in medical school, she focuses her writing on health-related issues. She's written for the \"PITT MED\" medical magazine published by the University of Pittsburgh Medical School. She has a Bachelor of Arts degree in political science with a minor in biochemistry from Loyola University Chicago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since 2004, Qorvus has completed many custom-engineered projects and complex turn-key solutions for mission-critical applications, brought to us by our VAR partners. These projects range from technical feasibility studies and stand-alone solar-powered wireless surveillance systems for use on public airport runways, to remote-access wireless hi-res camera systems installed on multiple 50-ton die-gripper systems at major automotive manufacturing plants, and remote-access camera and pressure-wash control systems installed on the cutting face of tunnel-boring equipment.\nGenerally the first step in our relationship consists of one or two on-line or in-person meetings followed up by a proposal outlining deliverables, schedule, and costs. Our services are also available on a hourly basis ranging from $125 to $225 per hour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Video was huge in 2016 - whether it was live-streamed or not - and all signs point towards an even bigger year for video in 2017.\nVideo creation is now easier than ever, with tools popping up left and right to help even absolute beginners create beautiful, compelling videos.\nIn this post, we'll look at some of the best tools to help you create beautiful videos for your marketing campaigns.\nAs noted in the introduction, video has grown a lot over recent years. Bombarded with content options, audiences are looking for variety, and the connective capacity of video has given publishers new ways to differentiate themselves and stand out from the noise.\nVideo is also an amazing asset for improved conversion rates. Studies have found that just by including a video on your landing page, you can increase your conversion rate by up to 80%, while adding a video to your email can increase your click-through rate by a staggering 200-300%. People who watch video are also 64% more likely to buy a product online - and the impressive stats in video's favor continue on and on.\nVideo also performs great in all kinds of formats. Whether it's a social media video, an ad, or a presentation video, users love watching videos online (a third of all our online activity is spent watching videos).\nSo, as you can see, video should be an essential part of your 2017 marketing strategy; it not only increases conversion rates, but it also improves engagement on social media and your website and blog and it can help you stand out from the crowd.\nAs a business, there are all kinds of videos that you can create in order to boost your results.\nHere are six tools to help you create better video content.\nRendrFX is a video creation tool for websites, social media and other marketing purpose. And even better, you can get started using it for free.\nVideo creation is extremely easy with RendrFX - all you have to do is add your own content (videos, images, audio, text, colours, graphics) and use their professionally designed motion graphics templates. There are over 350 templates to choose from, and you'll also have access to Videoblock's collection of over half a million stock media files, which include videos, photos, graphics and audio, all at no extra cost.\nAnimoto is another great, easy to use video creator, that's perfect for businesses, as well as photographers and regular users who want to create beautiful videos for their families.\nThey've only recently introduced their marketing video builder, but it works great. While there isn't that big of an offering on templates, you can very easily add the media you want and record your voice over it. You also have complete control over how you want to optimize the text from your videos and you can create video collages or split screens with videos or photos shared along your text.\nWideo is another tool designed to help you create marketing videos as easily as possible.\nTheir video creator is very easy to use - you can start with one of their templates, or simply drag and drop the elements you want to use into the video builder.\nIt's also easy to create animations with Wideo, even if you have no experience whatsoever in design, and there are plenty of customizations that you can make to keep you creating original videos for a while.\nBiteable is an online animation video creator that can be used by complete beginners.\nIn order to get started with your very own video, you can use one of their templates, or some of the hundreds of different styles of animated scenes, photos or live action videos. You can also add your own content to make your video unique, such as text, photos, colors and sound. They also provide a selection of music clips that you can add to your video.\nWith Biteable, you can create all kinds of videos, such as presentations, infographic videos, logo animations, slideshows, ads for your business and more.\nPowtoon is another tool for creating animated videos - it provides ready-made templates which you can easily customize with their drag and drop feature, making it very easy to create an animated video or presentation in just minutes.\nYou can create videos for all kinds of applications, including for marketing purposes, and the results are fun and engaging, making them great for social media, email and promotional purposes.\nKizoa is a movie maker and video editor with lots of extra features to help you create engaging videos. They offer dozens of customizable templates that you can choose from, or alternatively, you can start your own video from scratch.\nYou can then add all kinds of fun stuff to your videos, such as one of hundreds of effects and animations, text or animated text, as well as GIFs - and if you're out of GIFs, you can use their GIF-creator to create a new one on the spot.\nAlthough Kizoa is mostly aimed at people who want to create videos for personal use, you can still create some beautiful videos for your social media and other marketing channels as well.\nVideo is going to be huge in 2017, with more businesses using it to promote themselves and diversify their content offering. One of the best ways of standing out from the crowd will be with unique, engaging content, so get started with your first videos now in order to kick-off the next year with a bang.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Opposing Illegal Immigration Is Not \"Racist\"\nI guess you aren't aware then that Mexicans and other Latinos hold the highest quotas for legal immigration into our country by far? So yes, those quotas are specifically for them. I disagree that many anti-illegals are racists and objecting to illegal immigration isn't being anti-Hispanic.\nRegarding your question: \"Wouldn't it be simpler to say 'get here, get a job, keep that job for a year and welcome to America'?\"\nAnswer: No. Just imagine how many people would show up, if that's all it took. We would be overwhelmed (even worse than we are now) with functionally illiterate, unskilled people. Do you know what that does? It displaces unskilled Americans in the workforce. Plus it depresses wages to the point where jobs that would at least propel people into the lower middle class quickly disappear.\nNot all illegals are Hispanic. I don't care where in the world an illegal alien comes from. Nor do I care if they got here from sneaking across a border or overstaying a visa. They all have one thing in common, and that is this---they have no business being here and need to leave.\nBTW, I'm half-Hispanic and over the years I've taken abuse from reconquista types (La Raza members or those whose thinking is in line with the Brown Berets). According to them, if I don't support open borders and illegal immigration, I'm a \"racist\" and a \"traitor\" who should be ashamed. If I challenge them on that thinking, I don't get an answer, just more verbal abuse.\nI have neighbors who came here legally from South America. They are against illegal immigration. The wife even told me that she doesn't like seeing signage in Spanish because, in her words, \"In America, we speak English\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below you will find past presentations for parent workshops we have run at Kensington Avenue Primary School for 2018\/19.\nIf you were unable to attend a particular workshop or wish to go through a presentation again please take the opportunity to look through this section. If there is anything you don't understand or would like any further help, please ask your child's class teacher.\nWorkshop presentations will usually be uploaded on the same day the workshop took place so check back if ever you miss one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Editor's note: Friday's events at the Barrett & O'Connor Washington Center focused on helping professional athletes transition out of their playing careers, new ways of involving citizens on decisions about science and policy, and a roundtable on redesigning design education. Find the blog roundup here, with links to the other day's highlights.\nThe week is over, but the grand opening isn't \u2014 events continue next week at the new Barrett & O'Connor Washington Center in our nation's capital.\n\"The McCain Institute at ASU: How Do We End Terrorism?\" 5-7 p.m. Monday, March 19 \u2014 Former Homeland Security Advisor (and McCain Institute Trustee) Fran Townsend, former Director of the US National Counter-Terrorism Center Nick Rasmussen, and special guests describe the evolution of global terrorism since 9\/11and discuss long-term approaches for overcoming it.\nCGEST Reception, 5:30-7 p.m. Tuesday, March 20 \u2014 This event hosted by the Center for Gender Equity in Science and Technology convenes leading African-American women in STEM to provide a forum for women to strategize and build coalitions, continue discourse from previous gatherings that have led to grant-funded projects, share job announcements and explore opportunities to support and lead interagency functions.\nNearly two dozen Washington-area design professionals joined Jason Schupbach, director of The Design School in the Herberger Institute for Design and the Arts, to share their ideas about the future of design on Friday afternoon. Drawn from government, business, industry associations and education, this roundtable discussion to gather insights and make plans for a school redesign is one of four Schupbach and his team have organized in cities around the country, including San Francisco, New York and Los Angeles, in addition to Washington, D.C.\n\"How can design education be more relevant?\" read the main screen. \"Where is design going?\" read a white board soon jammed with sticky-note responses. Over the next year, the school will take what is learned in the roundtables, expand input from a wider public online, engage in an internal dialogue and make plans to transform The Design School's curriculum and how best to shape the designers of the future.\nHere, Schupbach talks about some of the questions the school is facing.\nIn the world of science and emerging technology, it can seem sometimes that the latest research stays within the realm of policymakers and scientists.\nBut there are growing numbers of people in those communities who think the knowledge \u2014 and the ideas for how we move forward \u2014 should belong to everyone.\nA four-hour event Friday at the Barrett & O'Connor Washington Center \u2014 hosted by ASU's School for the Future of Innovation in Society, School of Earth and Space Exploration, and the Consortium for Science, Policy and Outcomes, in partnership with the Museum of Science, Boston \u2014 brought in a variety of members of the Washington science policy community, media and academic organizations.\n\"Your role today is as a citizen, not as a member of the inside-the-Beltway 'privileged class,'\" CSPO co-director Daniel Sarewitz told the crowd before they were divided into teams and charged with the day's discussion mission.\nIn 2014, CSPO led a pilot study using Participatory Technology Assessment (pTA) to elicit informed and diverse citizen views and help inform decisions at NASA about its Asteroid Initiative.\nOn Friday, participants at the ASU center went through the same process, focusing on planetary defense (that is, defending the planet against asteroid collisions \u2014 though one NASA representative in attendance did tell this writer in an aside that \"nothing is headed our way today\" \u2014 or anytime soon).\nAfter spending about an hour deliberating at their tables with the interactive decision boards, the teams presented their choices to the room. Most chose option 3 (the U.S. leading an international partnership) though a few went with option 2 (creating a new U.S. office of planetary defense).\nJust as had been the response in CSPO's pilot citizen-input study, participants were enthusiastic after the process and full of questions and comments, including some questioning whether we should be spending money on planetary defense at all \u2014 especially \"when climate change is an issue as well,\" one person said.\nThe goal of such citizen forums is to both engage the public in issues of science, technology and policy, and to seek the creativity of people outside the usual voices heard in science debate.\nLed by Kenneth Shropshire, CEO of the Global Sport Institute and Adidas Distinguished Professor of global sport, a panel of experts discussed how education can help athletes transition out of their playing careers and onto paths to long-term success.\nThe panelists of the Global Sport Initiative discussed financial incentives, racial disparities and how higher education can better prepare student-athletes for life after their playing careers end. The panel was moderated by GSI CEO Ken Shropshire (left) and included Martin Carlsson-Wall of the Stockholm School of Economics and Arthur McAfee, the senior vice president of player engagement for the NFL (with microphone).\nFollowing the panel, Shropshire spoke with Jacques McClendon, director of player engagement for the Los Angeles Rams and former NFL and University of Tennessee football player.\nMcClendon said the transition from student-athlete to star-athlete and back to just plain student can be rough for many.\nIt comes down to motivating student-athletes to finding and believing in their second path.\n\u2022 New America: This is a Washington-based interdisciplinary think tank and civic enterprise that strives to understand problems and opportunities facing the United States. ASU and New America projects include Future Tense, an exploration of emerging technologies and their transformative effects, and the Future of War, an examination of the changing nature of war.\n\u2022 Council on Competitiveness: ASU is an active member of the Council on Competitiveness, which is a nonpartisan, nongovernmental organization in Washington, D.C., that focuses the debate on competitiveness by bringing together business, labor, academic and government leaders to evaluate economic challenges and opportunities. ASU President Michael Crow serves as the group's university vice chairman.\n\u2022 U.S. Chamber of Commerce \u2014 U.S. Israel Business Initiative: ASU is an active member of the U.S.-Israel Business Initiative, which is part of the U.S. Chamber. It is the only Washington-based national program focused on advancing the business partnership between the United States and Israel. The initiative is focused on making the alliance into one of the world's strongest innovation-based commercial relationships, and it is an important voice, advocate and platform for the bilateral business relationship between these two countries.\n\u2022 Commercial Spaceflight Federation: ASU is an active associate member of the federation, works to promote the development of commercial human space flight, pursue greater safety and share best practices with members. It is focused on preserving American leadership in aerospace through technology innovation and inspiring careers in science and engineering.\n\u2022 APLU: The Association of Public and Land-Grant Universities is a Washington-based association with 237 members. It is a research, policy and advocacy organization dedicated to strengthening and advancing the work of public universities in the U.S., Canada and Mexico. ASU is very active in this association and is engaged with many of the association's councils and commissions. APLU's agenda is focused on three themes: increasing degree completion and academic success, advancing scientific research and expanding engagement.\nAllen Morrison, CEO and director general at the Thunderbird School of Global Management at ASU, speaks about combining values and education to better prepare our leaders.\nThis morning's panel was hosted by the Global Sport Institute. What's that?\nTo start \u2014 the singular \"sport\" in the name is on purpose. It involves organized athletics, yes, but also looks at fitness, education, sociology, technology and other issues. It's part of the Global Sport Alliance, a partnership between ASU and adidas aimed at shaping the future of sport and amplifying sport's positive impact on society.\nGSI supports and translates complex research to broad audiences and is led by Kenneth L. Shropshire, an international expert at the intersection of sports, business, law and society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My Dream Drive in a PAGANI ZONDA!\nWe've gone this far, driven by the winds of all times.\nPagani, an Argentinean engineer transplanted into the heart of Italian exotic car country, worked for a time at Lamborghini before branching out on his own.\nPagani Zonda Cinque. imunchshrooms 02\/17\/16. Dream car, from a renaissance man.\nPagani Huayra \u2014 $1.3M. This is an Italian supercar which has a twin-turbo Mercedes-AMG V12, six-litre engine which produces 730 horsepower and 740 lb.\n... the special editions of the Huayra that will inevitably be produced by Pagani will be able to replicate the raw insanity and drama that the Cinque has.\nPagani Sport Car at Dreamcars Asia 2005 Expo, Malaysia.\nIt depends almost exclusively on trust. Here your word carries more weight than a written contract. And if you don't honor it, you are finished.\nPagani Zonda R in the USA!!! 1 of 15 Worldwide!!!\nI know that it's literally impossible to get my hands on one but i'll still try.\nSo now they're back at it again (probably in partnership with Pagani, it seems).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As a farmer, I have always known that everything I produce is dependent on the soil I grow my crops in and the rain that falls on them. Without these natural resources, we would fail. However, it is not just farmers that soils are important to. They are the cornerstone for the survival of life on earth and, as former American President Roosevelt appreciated, the key to the prosperity of whole nations and civilizations.\nEver since mankind turned from hunter gatherer to farmer, we have had the sad record of destroying our soils, which led to the demise of empires. Whether this being the Ancient Greeks, where ancient monuments are now surrounded by arid bedrock, in what was once fertile farm land. Or the fall of the Roman Empire, which spread out north and south from Italy, in the attempt to bring food in from new agricultural lands. Or more recently the fall of the Soviet Union, as the population got fed up with constantly queuing for essential staple foods, where in reality they were not producing enough food to feed themselves.\nIn nature crops and animals grow naturally quite well, but they are a bit randomly spread out. As mankind started to domesticate crops and animals for their needs, they started clearing areas of land to produce more food in a given area. The land was cultivated by the plough, originally a small implement pulled by an ox or donkey, today it is much larger and pulled by tractors. But the principle is the same. The plough turns over and breaks up the soil surface to create a seed bed to plant crops in. The advantages are that it provides soils free from weeds, provides good conditions and soil structure for plants to grow in. It also gives a nutritional boost to the plants as bacteria breakdown minerals for the plants to feed off.\nOver time the disadvantages of ploughing however outweigh the advantages. The freshly disturbed surface of the earth is very fragile, especially when it rains, with soil erosion being particularly noticeable on slopes. As rain drops hit the soil surface, water drains down-hill into streams, rivers and eventually into seas and oceans. But the water takes the fertile soil particles with them, which in time can remove the soil completely; think of those ancient Greek monuments standing on top of rocky outcrops. But it still happens today as this satellite image of the UK clearly shows. Now whilst I am all for us exporting more products to our friends in Europe, I am not sure we want to be sending them our fertile soil.\nAlso, whilst ploughing creates a lovely loose structure for seeds to germinate in, the action of ploughing exposes the soil to oxygen in the atmosphere. This gets all the soil bacteria really excited, similar to a young child being given sugary sweets, running around really quickly, before collapsing in a heap on the floor once the sugar rush is over. The bacteria 'run around' in this high oxygen atmosphere, giving lots of nutrition to the crop, but it is short lived as the bacteria eat up the store of carbon in the soil, respiring as they do so, and thus release the stored carbon to the atmosphere as carbon dioxide. This has the double negative effect, of gradually reducing the nutritional content of the soil and increasing the global warming effects of increased CO2 in the atmosphere.\nHaving learnt much on soil quality over the years, I took the decision not to plough on our farm back in 1998. Admittedly, my incentive was not just to save the planet, but also to save money as ploughing is an expensive operation. Although it was known my crop yields could reduce a little, hopefully this would be more than compensated by reduced costs. When I stopped ploughing, I made lots of mistakes as yields reduced dramatically for a short period, as well as weed pressure increasing. Our neighbours thought I was a bit daft \u2013 they may be right on that one. But over the years, I have learnt an awful lot as the system has improved. The theory of not ploughing is that naturally plant roots and creatures like worms improve soil structure. The bacteria and other micro fauna improve the soil health and biology, converting old plant residues and mineral content of the soil into plant food. So by not ploughing, the soil structure and organic matter content gradually improves year on year and, the carbon from the soil is not released to the atmosphere as CO2. I continue to learn from my experiences as we are making a healthy environment for the plants to grow in, but this is the underlying theory.\nDelegation from China, including Deputy Director General of the Ministry of Agriculture and the Chinese delegate for the United Nations Environment team.\nToday, soil health is something that governments around the world are realising is important and we should be doing something to improve our most important natural asset. I was especially proud recently when, along with LEAF, I hosted a delegation from China which included; The Deputy Director General of the Ministry of Agriculture and the Chinese delegate for the United Nations Environment team. They were very interested in what we are achieving with our soil health on the farm.\nPlenty of commentators accuse agriculture of being a huge cause of global warming from CO2 emissions, however I can show from what we are doing on our farm, agriculture can play a major role in reducing CO2 emissions when looking after our soils.\nThe graph shows how the soil on our fields is improving in quality, by increasing in soil organic matter. To put this another way, we are actively absorbing CO2 out of the atmosphere and locking it into our soils. If this is repeated around the world, the benefits could be enormous; it has been estimated that agriculture could reduce global CO2 emissions by between 10% to potentially 30%. The graph also shows the increase in Cation Exchange Capacity (C.E.C). In basic terms as the CEC increases, there is more nutrition within the soil for the plants to grow healthily.\nSo, in the space of a few years of trying to work with nature, the results clearly show we are creating a healthier, more nutritious environment for crops to grow in. Not only is this good for the soil and the wider environment, but it is good for us also, as healthy soils grow healthy crops and healthy crops create healthy food for us to enjoy eating.\nLearn more about our environmental credentials here and where to buy our LEAF Marque Cold Pressed Rapeseed Oil here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beautiful home with an OPEN FLOOR PLAN built 2016. Public space downstairs is open and airy, powder room down. Kitchen with stainless appliances, granite counters, tile backsplash. 3 beds\/2.5 baths. Upstairs loft, laundry room upstairs. Master bedroom has 2 linen closets AND walk in closet. Abundant closet space up & down! Tankless water heater, sink R\/O, & water softener. Ring doorbell. Paver driveway and patio. Two car garage w\/storage racks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Romney's distorted 47% comments about Americans who don't pay income taxes are a favorite talking point for many conservative Republicans, who often use them as evidence that America's expenses are paid by a relatively small portion of the populace, while the rest \u2014 in Romney's words \u2014 don't \"take personal responsibility\" for themselves. What is missing from the discussion is an identification of who makes up this 47%.\nAccording to the nonpartisan Tax Policy Center, 53.6% of Americans pay a portion of their income in federal income taxes, and 46.4% don't. Of those who don't, 61% (28.3% of the population) pay payroll taxes, such as Social Security and Medicare, but have enough deductions and tax credits that their federal income tax liability has shrunk to zero. Basically, 81.9% of the population is gainfully employed and provides some measure of their income to the federal government. As for the rest, 10.3% of the population, 22% of non-income tax payers are elderly and likely retired. An additional 6.9% are non-elderly but have incomes below $20,000 poverty line. It's not as though the rest of the 47% (actually, 46%) are doing especially well because almost 90% of this group consists of households making less than $50,000 per year, and 80% make less than $30,000. The remainder is mostly composed of higher-income households that benefit from significant tax credits for children and education.\nAbout 162,000 Americans among the top 10% of earners avoid paying any federal income tax. This includes approximately 3,000 people in the top 0.1%, a group who make more than $2,000,000 per year. A large part of their low tax rate, The New York Times' Bruce Bartrlett suggested, lies in the fact that many \u2014 like Romney \u2014 derive most of their income from capital gains, which are taxed at only 15%. To further reduce their liability, many are able to offset their taxes because of losses that they took in previous years. Alternately, some invest in tax-free municipal bonds or take advantage of a slew of other tax loopholes. Willard Mitt Romney is part of this group.\nAs Ezra Klein explained in an article recently published in The Washington Post, many tax cuts for the poor were enacted to make tax cuts for the rich \"more politically palatable.\" The tax breaks passed into law in Reagan's second term and Bush's tenure were accompanied by large increases in the number of households that were exempted from paying taxes. In Reagan's last few years in office, the percentage of households not paying taxes jumped by more than 10%. Under Bush, it increased by more than 25%.\nThe clips from the private Romney fundraiser provide astounding insight into a candidate who has closely guarded his true feelings as he tries to find an approach that works with the small percentage of voters he believes are still persuadable. But that's not all that the video revealed.\nIn the video, Romney also commented about on a two-state solution to the Israeli-Palestinian crisis. Romney said that the Palestinians did not want peace and only sought the destruction of Israel, and that he would not actively pursue the peace process but he would instead seek to \"kick the ball down the field,\" and that he had paid no real attention when a former secretary of state had told him that peace might be possible in the Middle East. Romney's comments represent a drastic break with US policy which has supported a two-state solution since the Clinton administration.\nOn Wednesday, October 3, 1012, Romney and President Obama will meet in their first televised debate. Romney is a skilled debater, having participated in19 debates during the Republican primaries. President Obama, while an articulate speaker, has not debated in four years and may be a bit \"rusty.\" However, the President's biggest debate asset is the fact that he has been engrossed in all of the issues for the past 3-1\/2 years. Romney's biggest debate challenge is that he must be specific and accurate in his responses, otherwise his campaign is dead.\nPosted on October 2, 2012 October 3, 2012 Author adminCategories What's Up!\nIt's really a nice and useful piece of information. I am glad that you just shared this helpful info with us. Please keep us up to date like this. Thanks for sharing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"La IPA\/saison de la s\u00e9rie \"\u00e9pisode\" de Bor\u00e9al que j'appr\u00e9cie le moins. Tout de m\u00eame agr\u00e9able offre une amertume plus florale et finale dry.\nEarned the International Women's Day (2019) badge! Earned the The Great White North badge!\nEarned the Bar Explorer (Level 7) badge! Earned the Trip to the Farm badge!\nEarned the Middle of the Road (Level 3) badge! Earned the Brewery Pioneer (Level 3) badge! Earned the Trip to the Farm badge!\nEarned the Middle of the Road (Level 26) badge! Earned the Birthday Brew badge! Earned the Trip to the Farm (Level 4) badge!\nBonne bi\u00e8re saison\/IPA tr\u00e8s florale et terreuse. Toutefois sans plus.\nBien citronn\u00e9e avec une amertume agr\u00e9able. \"Light\" malgr\u00e9 le 7%, bi\u00e8re de soif d'\u00e9t\u00e9! rondeur un peu sucr\u00e9e \u00e0 mi chemin avec finale s\u00e8che.\nEarned the Photogenic Brew (Level 47) badge! Earned the For the Can (Level 9) badge! Earned the Trip to the Farm (Level 6) badge!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cadgwith certainly provides a dramatic location and backdrop for gig racing. Towering rocks, fast tides and a shingle beach add to the many things the organisers and crews need to think about. Flushing & Mylor were well represented this year with Mens Vets, Mens A & B, and Ladies A & B crews.\nA strong start from Mens Vets in the first race of the day saw them pushing well up the first leg holding 2nd or 3rd place towards the mark. However a crew member started to not feel quite right and the crew sensibly decided to pull out of the race. Mens-A in Penarrow focused together well had an excellent race coming second to Falmouth after a very hard fought race with Padstow's Vixen. Padstow defended their inside line on the course well and it was only in the last 10 strokes that Penarrow was able to nudge ahead.\nLadies-A dug in hard on the first leg with a great start but the inshore berth, although out of the tide for the first leg, means that you have to get ahead of many other boats by the first mark. A good row overall and with two young bow rowers to take 8th place. Mens-B 7th and Ladies-B an excellent 4th rounded off this top afternoon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavlym b/data_all_eng_slimpj/shuffled/split2/finalzzzavlym
new file mode 100644
index 0000000000000000000000000000000000000000..91888ffdc280b23f4b1d52cf1435d3e61e373727
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavlym
@@ -0,0 +1,5 @@
+{"text":"Integrated with Inventory, Process Planning\/MRP, Maintenance Management and Accounts Payable, the Purchasing Module creates Purchase Requisitions and Purchase Orders based to replenish inventory based on requirements and\/or min\/max inventory levels. Approval process of Requisitions is included. Blanket Purchase Orders are also supported.\nReceiving of Purchase Orders with Supplier Returns is used for 3 way matching of Supplier Invoices. Matched and Approved invoices are then available in Accounts Payable for supplier payment.\nSupplier Performance is also included.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Haven't found the right listing? There are 9 other Roof Contractors in Plymouth. Click here!\nNor'east Services can be found at 212 Carver Rd . The following is offered: Roof Contractors . In Plymouth there are 9 other Roof Contractors. An overview can be found here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Savings with Rocky Mountain Chocolate Factory coupon codes and promo codes in April 2019. Today's top Rocky Mountain Chocolate Factory coupon: Get 25% Off Selected Favorites.\nExpires: 12\/10\/18 Details: Tap offer to replicate the voucher code. Remember to paste code when you take a look at. Online just.\nExpires: 03\/12\/18 Details: Get Free Shipping on $75 or even more.\nGet Free of charge Common Delivery on $75 or even more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robert Walters recruits for IT sales roles across a range of industries. Our consultants have the expertise to assist you in securing your next career move.\nView the latest IT sales jobs that we are currently recruiting for below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(CNS): The RCIPS Air Operations Unit carried out two emergency medical evacuations from Little Cayman in less than 48 hours this week after two women in two separate incidents needed to be airlifted to the George Town hospital on Grand Cayman. The first was on Tuesday evening, when a woman who had been diving had decompression issues; and the second came early Thursday morning, when a woman sustained a head injury after crashing her scooter. Both victims are undergoing treatment.\nOn Tuesday, 27 November, at around 5:55pm, the Air Operations Unit had collected the woman who had decompression problems after her dive. It took the chopper around 45 minutes to get her into the hands of medical personnel on Grand Cayman who were awaiting the arrival.\nThen, around 3:30am on Thursday the unit received an urgent call for the helicopter to transfer the injured woman from Little Cayman after the collision, which had happened about an hour previously while she was on her way home.\nMedical staff and equipment were taken on board the chopper, which left Grand Cayman around 4:45am and returned from Little Cayman with the injured woman around 6:15am. The helicopter landed on the Cricket Pitch at Huldah Avenue, where the patient was immediately handed over to the EMS personnel awaiting their arrival and taken directly to the hospital for further medical treatment.\nThe current police helicopter is fulfilling a number of roles, from search and rescue to tracking criminals on the run, but it will soon be joined by a second machine, after the government confirmed recently that CIG has partnered with the UK to purchase a new purpose-built chopper to help with search and rescue, medical emergencies and regional disaster relief.\nI completely agree with the chopper services between the Islands in which is good in emergencies in life and death . But what is the use of having Medical facilities on the Islands and don't use them , like in this diving accident on Little Cayman. Is Cayman Brac alot closer to Little Cayman than Grand Cayman ? Did Cayman Brac have the facility that was needed ? Why did the patient have to be flown back to GC ?\nMaybe someone should explain why the nearest facility was not utilized in this case . Is this a case where a human life is played against the value of money .\nIf she had decompression sickness, she would have needed to go in the decompression chamber and that is only on grand cayman.\n10:09pm , what is the Hospital on Cayman Brac ? And isn't there a decompression chamber of the east side of that Hospital in the Brac ? Please update me it's been a long time that I have been to Brac .\nBoth injured had to get to a medical facility fast.\nDive victim needed needed to be in the decompression chamber in \u2013 Grand Cayman.\nRoad accident victim needed best medical facility \u2013 Grand Cayman.\nAppears the airfields were closed, and never seen a plane land on the cricket pitch. Or for that matter on a helipad at Brac hospital. Ever seen a plane land at Health City. Probably best idea is too have a multifunction helicopter, or is that too obvious.\nKudos to RCIPS and the Medical team! Nothing more precious than life!\nIn most places, the chopper would not be the story.\nWhy did they fly the woman with decompression issues to Grand Cayman? Isn't there a decompression chamber at the eastern end of Faith Hospital on Cayman Brac?\nReally makes you wonder why all the money was spent on the decompression chamber if it isn't used when needed.\nTo 1:25 PM \u2013 yes there is a hyperbaric chamber at the southeasternmost point of Faith Hospital. It has been there for years and years \u2013 understand that it is not officially run by HSA. Maybe HSA can provide some clarification to the public.\nAnonymous 1: 25 pm , in this case it looks like the decompression chamber was built for bragging rights , or just for the Cayman Brac divers only . But it sure looks funny and funky to me too . I think that issue come down to the way of how government is developing the Islands for convenience verses money . Meaning that the Brac can have a decompression chamber built but can't except patients because it would deprive government entity of money or someone else.\nThis is why it's a good idea to have a chopper in Cayman.\nHow come, years ago, when a certain doctor based on Cayman Brac (who had his own plane) was not offered some sort of deal to carry patients for inter island service? Seems like it would be a good thing if there are people like this doctor who come supplement\/assist, instead of having to rely upon the police helicopter travelling over from Grand Cayman. Common Sense?\nIf you knew anything about aviation, using the words \"doctor\" and \"his own plane\" in the same statement would do more than raise a few eyebrows with what you just proposed !\nCould a fixed-wing craft have performed the medevacs to the same standard or better (more space) for the same or less overall cost? \u2013 All these incidents proved is that we occasionally need an air ambulance, not what the best air ambulance is.\nWhat a stupid, moronic, idiotic thing to say \u2013 waste gas \u2013 if it was one of your loved ones, family members, friends would you say the same thing? Oh and news flash, most of us damn expats\/foreigners\/visitors have to pay for such services \u2013 not fortunate enough to have CINICO.\nThank you 1:31PM for stating the obvious \u2013 we don't expect that the Cayman Islands Government is going to eat the cost of providing urgent airlift out of Little Cayman \u2013 just like we all paid to relax in Little Cayman, we will pay our medical expenses \u2013 so 11:02AM can rest assured that his\/her government is not wasting gas on us.\nReally? How cold hearted can you get! Like another poster said \u2013 how about if it was you or one of your loved ones who needed the service. I for one, am proud that we have the facility and can possibly save lives.\nOr is this just badly expressed?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavrcl b/data_all_eng_slimpj/shuffled/split2/finalzzzavrcl
new file mode 100644
index 0000000000000000000000000000000000000000..0c46cdd95d6954f2f1e35683fc6bd4d88345e42e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavrcl
@@ -0,0 +1,5 @@
+{"text":"The KLIM Optics grant true protection in front of screens. These glasses protect your eyes from the harmful blue light emitted by screens.\nVery light and extremely comfortable, the structure of the KLIM Optics is made of TR90, a next generation material which is very popular in high-end glasses.\nHigh quality. Our glasses filter 86% of the blue light \u2014 way higher than the industry average. The lenses are German made. The frame is partly made of TR90, a very robust but light material. The glasses only weigh 20g, you won't feel them on your nose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Empires have risen and receded, the world is no longer flat, but the simple nutmeg still grows on Siau Island, indifferent to its place in the grand historical story. Life sways to simple rhythms, governed by tides, trade winds and customs refined over many generations.\nOne Degree's newest film invites you to spend a leisurely afternoon exploring the secrets of Siau Island's treasured spice. Meet farmer Erasmus Rompah as he tends his nutmeg grove, checking the delicate apricot shapes that decorate each branch as they slowly ripen into bright yellow fruit. Erasmus' world is bigger than Siau; there is a hint of history and cosmopolitan adventures in his story. In younger days, he sailed throughout the Far East on cargo ships that carried Indonesian lumber. After 11 years at sea, he returned to Siau to tend the family's land, and soon to marry.\nHis parents named him for Desiderius Erasmus, the famous Dutch renaissance theologian. But his friends have long called him Thomas, based on another great theological reference: In life and in business, he wants to see it to believe it. It is a philosophy that dovetails beautifully with One Degree's passion for transparency, capturing stories with lens and pen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I want to donate a used textbook to the Textbooks for Change program. Where can I drop off my donation?\nYear-round, students can drop off their used textbooks in the Textbooks for Change donation bin located in the outer lobby of the Gerstein Science Information Centre (top of the set of stairs, left side), during their opening hours.\nThe U of T Libraries has partnered with Textbooks for Change, a social venture that provides affordable and accessible educational material to students locally and abroad. Postsecondary textbook donations are either sent to universities in Africa, or sold at a lower price to North American students with the proceeds being used to fund student-led impact initiatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take control of the food you put in your body!\nBasil, Romaine, Carrot & Tomato.\nWelcome to Garden to Plate, a website dedicated to assisting beginning gardeners with beautiful yet informative graphics on starting their own garden. This project is a collaboration between a teacher and graphic designer who both share a passion for sharing useful information in a simple and engaging way.\nWe take great care to provide you with both a visually stimulating experience while offering the most accurate information as simply as possible. Please, take some time to explore our website and our love for gardening and healthy eating. If you enjoy what you see, please take a moment and check out our products page and help support our cause!\nWow! We are excited to finally be bringing this idea live after (literally!) years of effort. To finally have our website go live and share our journey with you is truly a historic moment for us. Garden to Plate is a collaboration between a teacher (Rob) and graphic designer (Eric) in Buffalo, NY \u2013 a place we see undergoing its own sustainable food transformation. We hope to share with the world some of the things happening locally as well as bring ideas back to our community!\nEven more, we want to use our creative skills to provide meaningful information in a new, useful, and engaging way for the sustainable food movement.\nContact us if you are interested in placing an order!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To register your game on Steam, first enter the exact email where you receive your RimWorld download links. The email must be exactly what you typed in when you purchased the game, with the same letters capitalized.\nThis system registers your copy of the game directly on your Steam account. It does not give a separate Steam key. Legally, you own one copy of the game for your own use only. This system just lets you access it in several ways, for your convenience.\nIf you bought with PayPal, use the email of your PayPal account. If you bought with Stripe, use the email you typed into Stripe when you purchased the game.\nTo have your DRM-free download link resent to you, enter the exact email where you receive your RimWorld download links. The email must be exactly what you typed in when you purchase the game, with the same letters capitalized.\nIf you don't receive anything in an hour, or don't know your purchase email, send a message to [email protected] and Michael will help you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavuno b/data_all_eng_slimpj/shuffled/split2/finalzzzavuno
new file mode 100644
index 0000000000000000000000000000000000000000..b21500737eee2f6a1f7559f51dd381fcd0b6e4e2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavuno
@@ -0,0 +1,5 @@
+{"text":"I was feeling quite fancy after roasting my cherry tomatoes, so I raided my fridge to see what I could quickly throw together to make it into a meal. And boy did this turn out goooood. The cucumber slices makes this super refreshing and cooling.\nSuper fast and easy to put together. Simply spiralized cucumber + cube avocado + fresh bocconcini cheese + roasted cherry tomatoes + olive oil + balsamic vinaigrette + salt & pepper to taste and voila!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SERTRES SA, domiciled at La Rambla 130 4-1\u00aa A08002 Barcelona , phone-number +34 93 3026631, with C.I.F. A582245879 registered in the Barcelona Mercantile Registry, volume 7789, book 7051, Secci\u00f3n 2\u00aa, folio 98, page number 90471, registration 1\u00aa (Hereinafter \"MOKA\").\nThis legal notice regulates access and use of the website service www.moka.barcelona (Hereinafter, the \"Web Site\") MOKA provides Internet users interested in its services and content related to the restaurant services (Hereinafter \"Users\").\nMOKA may change at any time and without notice, the design, layout and\/or configuration of the Website, and any or all services and add new services.\nIn compliance with the provisions of Law 15\/1999, of December 13 of Personal Data Protection (LOPD, for its Spanish abbreviation), by this notice, MOKA informs the user alias (\"User\") of the personal data that requests, so that users can freely and voluntarily whether they wish to provide MOKA personal data (hereinafter \"Personal Data\") that may be required or to be obtained from the users during the application registration or information on the website operated by MOKA hosted in the URL www.moka.barcelona or services that can be contracted through the portal.\nThe Personal Data collected will be automatically processed and incorporated into the relevant files for MOKA and registered with the Spanish Agency for Data Protection, being MOKA owner and responsible for the file (hereinafter, the \"File\"). The data provided can be used to inform our products, news and \/ or promotions. To this end, MOKA provide the user with adequate technical resources, prior, can access this notice on the Data Protection Policy or any other relevant information and give their consent so that in MOKA proceed to treat your Personal Information.\nIn general, the provision of services does not require prior subscription or registration of users. However, MOKA conditions the use of some services to the prior completion of the registration of user, selecting the identifier (ID or login) and the password that the user agrees to keep and use with due diligence.\nMOKA assigned the identifier selected by the user as long as there has previously been selected by another user. Using the password is not transferable, not being permitted to transfer, even temporarily, to third parties. As such, the user must take the necessary measures for the custody of the password selected by him, avoiding the use of it by others. Consequently, the user is the sole responsible for the use of the chosen password, with the complete indemnity of MOKA. In the event that the User knows or suspects that third parties are using their password, they must communicate the facts to MOKA as soon as possible.\nSimilarly, the User's responsibility to keep all information provided to MOKA constantly updated so that it responds at every moment to the situation of the User. In any case, the User shall be solely responsible for any false or inaccurate statements made and the damage caused to MOKA or others for the information provided.\nUnder the anti-spamming policy MOKA User agrees not to use and collect data from the distribution lists that can be accessed through the information and services contained on the website for the realization promotional activities or publicity purposes and to send commercial communications of any kind and through any medium not previously requested or consented to by MOKA and \/ or stakeholders. The user who intentionally or culpably breaches any of the foregoing obligations liable for all damages caused.\nAlso, MOKA not guarantee the availability, continuity or infallibility of the operation of the Website, and accordingly excludes, to the fullest extent permitted by law, any liability for damages of any kind that may result from the lack of availability or continuity of operation of the Website and the services provided in the same, as well as errors in the access to different web pages or those from which, where appropriate, such services are rendered.\nMOKA excludes any liability for damages of any kind that may result from the services provided by third parties through this Website as well as the media to enable them to manage service requests, specifically, without limitation not limited to: for acts of unfair competition and illegal advertising as a result of the provision of services by third parties through the website, as well as the lack of truthfulness, accuracy, completeness, errors, defects, relevance and \/ or timeliness of content transmitted, distributed, stored, received, obtained, made available or accessible through the services provided by third parties through this website.\nMOKA only uses technical cookies, for customization and data analysis, which could be own or third-party, and in any case these cookies will treat any personal data or identify any browsing habits for advertising purposes. Anyways, please note that you can enable or disable these cookies by following the instructions of your browser.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those of you who are feeling quick-witted enough will have noticed that the first 3 of my 10 above give a rough translation of Doddiscombsleigh (I apparently wasn't wrong about the 'whole sentence' thing).\nAll you need to add is a postcode and you would have a perfectly serviceable postal address. If you lived on the Isles of Scilly by the way, that address would almost certainly get your Christmas card to the right house!\nB2, Dartmoor\u2026 ('Dart' by the way, means 'oak').\nOne of the very few things I learned in Geography lessons back in my schooldays was that a valley was open at each end and a comb, combe, coomb, cwm (Welsh) was closed at one end. The only other thing I remember learning was that stalaCtites hang from the roof like iCicles and stalaGmites grow on the Ground.\nI hope the Doddi isn't for storing Kens. The Diddy people might object.\nIt is amazing isn't it, which snippets of information come back to you. I wonder what happens to the ones we forget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Body Bliss. Dip into divine moisture with our ultra-nourishing Body Butter. Richly comforting Aloe & Shea Butter envelop your body in lush hydration as Cocoa Butter & Avocado Oil restore radiance. Delicate floral infusions of Sugared Pastille soothe as they smooth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tradecom takes pride in being a contributing member of our community. We believe in business that is responsible and that allows us to build a better future for the world. In everything we do, we take ownership and a long-term stance that ensures our commitment to ourselves, our clients and our families.\nAs part of our 30th Anniversary celebrations in 2016 and in support of the nation-wide #Givingweek initiative, some of the causes we support include Children and Education, Disability Improvement and Special Needs Care.\nTradecom actively seeks out a cause that we can support on a sustainable basis and reviews our partnership on a yearly basis to see where we can do better.\nTradecom supports the charities our staff believe in and does a corporate matching for the donations each staff makes (for every dollar donated to an Institution of a Public Character (IPC), Tradecom does a 1-for-1 matching on a yearly basis).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawaot b/data_all_eng_slimpj/shuffled/split2/finalzzzawaot
new file mode 100644
index 0000000000000000000000000000000000000000..d49c328fa54d0edcee59463df4ddeeb3b6bab16f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawaot
@@ -0,0 +1,5 @@
+{"text":"CHENGDU, Jan. 2 (Xinhua) \u2014 Trains made nearly 180 round trips between Chengdu, capital of southwest China's Sichuan Province and Lodz, Poland, in the last two and a half years.\nBEIJING, July 1 (Xinhua) \u2014 International cooperation in production capacity offers huge opportunities for participating in a country's economy. China and Europe, at different stages of development and with their respective strengths and demands, have a lot to share in this regard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can I set white using a non-neutral color?\nDiscussion in 'Image Processing' started by Klorenzo, May 11, 2014.\nHi, I have several pictures of an event shot under a few different lights (natural, artificial, etc.). I did not take a picture of a gray card in all of these situations.\nI know that I can use any white\/gray element for an approximate WB but I already have a good starting point with the auto wb and I want to do something more precise and uniform across different pictures.\nSo now I know that I have a difference of 400K in the two pictures and I can take the temperature of the second picture and set it to the first minus 400. From a couple of tests I think this does not work.\nIs there any trick like this that I could use? Or something to make a quick \"color comparison\" of the dress color in different photos?\nI'm not sure how it's done, but I read that Sports Illustrated knows all the colors of the NFL uniforms and uses that info to do color correction, so it must be possible.\nYou can use pretty much any colour, but you're not really setting 'white balance' but calibrating to a specific colour. If you are using Lightroom, you can copy the colour adjustments of one shot and paste that to all related shots and then do whatever you want with the other settings.\nThe Kelvin scale used in the White Balance control is not linear, its a geometric progression. A 400k shift in the neighborhood of 3000k is a vastly different shift than a 400k shift in the 5500k neighborhood. Also, choosing a strong color (e.g. red dress) is a very poor strategy for several reasons.\nYou should choose the most neutral color that is in both the \"good\" image and the \"bad\" image you are trying to correct. You need to sample the RGB values for that object in the \"good\" image and then adjust the color in the \"bad\" image so that you get the same sampled RGB values. In LR, you can sample the values by reading the numbers displayed below the Histogram while the cursor is over the chosen target. In PS, you can set several sample points and can control the size of the sample (number of pixels in the sampled array).\nYou should choose the most neutral color that is in both the \"good\" image and the \"bad\" image you are trying to correct. You need to sample the RGB values for that object in the \"good\" image and then adjust the color in the \"bad\" image so that you get the same sampled RGB values.\nIs there a simple strategy to \"convert\" the RGB difference in a Kelvin difference? I do not expect a direct \"mathematical\" mapping, but maybe some \"tricks\".\nSuppose I sample a white something in the \"good\" picture and it measures 220, 230, 240 and the second one it is 210, 220, 250. Now I can see the second one is shifted towards the blue, but how much? What about the green difference?\nI think I could just try small corrections in WB and tint trying to make them reasonably close. It looks like a lot of work.\nIt becomes difficult to do because you are adjusting not just colour temperature, but colour hue as well. I had to do this with a wedding I once took in a church that had the most abominable lighting that I've ever encountered, arc lamps, daylight coming through huge multi-coloured leadlight windows and incandescent lighting as well. All I could really do was set white balance on the bride's dress and then spend hours doing fine adjustments trying to keep the dress white, but with detail, while trying not to make the groom's dark blue suit go completely black. I hate wedding photography, I really, really, hate wedding photography.\nthere is a small tool avaliable in LR that can help you pick the neutral color (like grey) and it automatically sets the WB for the rest of the picture. That can be a good starting point.\nBut it is true you'd have to do it for each individual picture (or copy paste the settings for the pictures taken under same\/similar light).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your price for a Mindsteps Workshop depends on the scope of your program\u2014 we develop a customized program based on your school or district's specific challenges and budget. Regardless of the focus or scope of the workshop we devise for you, we want you to feel confident in your investment and focus on content rather than cost. That's why our workshops are always quoted as a flat rate rather than as an hourly fee.\nWe take the time to understand your specific pain points, staff challenges and district mandates. This conversation is always free. From there, we develop a written proposal outlining several investment options. There's no risk, no cost and no commitment to see if we can help address your challenges, so why not schedule a call to tell us about your school?\nWhat's included with a Mindsteps workshop?\nOur prices are all-inclusive\u2014 there are no hidden costs. All workshop materials, planning sessions and travel are included in your investment.\nIs there a workshop capacity limit or minimum requirement?\nMindsteps recommends no more than 150 educators in a workshop to maintain the interactive experience we provide. For groups larger than 150, we recommend considering a keynote presentation.\nWorkshops are typically booked a minimum of two months in advance of the presentation date.\nAfter our initial conversation, we will send you a written proposal for your review with multiple investment options within 36 hours. From there, we work with you to detail a program, engage decision makers and finalize logistics. We are your partner on the program throughout the planning process so when the workshop day arrives, there are no surprises \u2014 only inspired, excited participants.\nAll our workshops start with a conversation to better understand your situation. Just give us a few details and we'll set up a time to talk. From there, we provide a proposal and get a date on the calendar that works for you.\nAre there supplemental resources available for post-workshop professional development?\nWe know mastery is a journey. We offer additional materials for both teachers and administrators that directly support our workshops and enable them to continue their learning. These materials are at an additional cost, however are NOT required. All sponsoring schools and districts, as well as participating teachers and administrators, are eligible for a discount on all Mindsteps products as part of an engagement with us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cater with style and ease.\nAll of your catering needs on a silver platter. From crockery to cutlery to kitchen equipment, our collection of quality catering supplies is guaranteed to make your delectable menu look even more divine!\nAll the essentials to simplify your next catering event\u2013no matter how big or small.\nSet the scene for the perfect festival, fundraiser, or sporting event with a collection guaranteed to draw in the crowds. Whether your guests will be covered in mud from a weekend-long music festival or dressed to the nines to raise money at a charity gala, The Main Event is Complete with everything you'll need to stage an engaging and memorable celebration.\nDress to impress for a whimsical garden party. With enchanting lighting and all white d\u00e9cor this collection features an elegant balance of glitz and glam. Whether your glass is full of the finest tea or the bubbliest champagne the secret garden is assured to be a magical affair.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Hyundai Grand i10 | HYUNDAI - NEW THINKING. NEW POSSIBILITIES.\nthe made for India sedan boasts of style, quality, space, comfort, safety and convenience.\nTangible beauty in all facets.\nBrilliant on the road but gentle on the eyes.\nmost important factors in choosing a car.\nenhances the sporty character of the front.\nHyundai Grand i10 is bound to catch your eye at the very first glance.\nAssertive and enticing with a subtle charm all around.\nlooks and projects powerful brand presence.\nillumination and enhances the overall front profile.\nprofile of Grand i10 and easy to operate.\nfoldable mirror offers great combination of comfort and convenience.\nensure high visibility for added safety.\nThe sleek and sporty roof antenna enhances looks and provides seamless connectivity.\nThe interiors of Grand i10 can be best described as stylish yet practical.\nThe bright and airy cabin with its quality feel and carefully crafted materials set the scene for many urban adventures.\ncomfortable for drivers of all stature.\nand informative at the same time.\na storage space for glasses and small knick-knacks.\nDriving in traffic is a breeze with 4-speed automatic transmission available in selected gasoline variants of Grand i10.\nfor the driver and passengers in the event of frontal impact.\nwith the driver and passenger airbags in case of front impact.\nto ensure proper visibility of trailing traffic.\nwelding techniques for body robustness.\nand innumerable hi-tech features Grand i10 offers you.\nto enhance rear passenger comfort with instant cooling.\nwith various controls just at your fingertips.\nand avoids distraction while driving.\ncabin temperature for better comfort.\nthe risk of a possible collision.\nstart and stop of car with just a touch.\nessential papers and offers build in bottle holder.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaydsq b/data_all_eng_slimpj/shuffled/split2/finalzzzaydsq
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+{"text":"Parents: Need a night out?\nsnacks, and enjoy social interaction with other kids! Please feed kids before coming, snacks will be provided, but not a meal.\nParents Night Out runs from 6pm to 9pm once per month and dates will vary.\nCheck our Calendar for upcoming dates or contact our Family Director for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jake Burton Carpenter's little company has come a long way since founding Burton Snowboards out of a Vermont barn in 1977. What was once a hobby has become a leading-industry manufacturer of snowboards and apparel. How popular and respected have Burton Snowboards become? The company has only provided the outfits for the US Olympic Snowboard team in 2006 and 2010. And, the company is returning for a third consecutive run.\nBurton recently unveiled the gear for Team USA, which will be worn during the 2014 Winter Olympics in Sochi, Russia from Feb. 7 to 23, 2014. Just as the company did with riders during the games in Torino and Vancouver, Burton received feedback from the athletes on what they would prefer to wear. Burton also was inspired by American heritage \u2013 mainly handcrafted patchwork.\nThe highlight of the Team USA uniforms are the one-of-a-kind red, white and blue quilt jackets. The quilt-pattern jackets were based off of a vintage quilt discovered at an antique fair of all places. The quilt was taken back to Burton's headquarters in Vermont where it was deconstructed and re-constructed by a veteran Vermont quilt maker in order to get it ready to replicate onto the outerwear fabrics. The end-result was a patriotic vintage quilt and flag print jacket that is bound to turn heads while up in the mountains.\nThe uniforms also include corduroy pants, gloves, mitts, beanies and cozy wool\/silk blend competition pull-over fleeces. And, perhaps best of all for the athletes, Burton worked with the U.S. Army Natick Soldier RD&E Center to develop the all-new DRYRIDE Vaporshell laminate. This technology created a waterproof and breathable fabric that can stand up to any weather condition. Perfect for battling the elements offered by the frigid Russian winter.\nWhile there isn't a release date, you can grab these uniforms for yourself in the near feature. Just in case you want to get all 'Merica' will tearing up the slopes this winter. Just keep up with Burton via facebook.com\/burtonsnowboards, twitter.com\/burtonsnowboard and @burtonsnowboard on Instagram.\nCheck out the collection below and let us know if you'd be willing to sport that unique quilt and flag print jacket. Honestly, we kinda wish that every member of Team USA would be outfitted with this gear during the 2014 Winter Olympics.\nThis entry was posted in Winter and tagged 2014 winter olympics, active gear, burton snowboards. Bookmark the permalink.\nLove the jacket. When & where can we get one?\nI can't understand for the life of me why in the world they would've chosen this design. They are horribly ugly and make all of our competitors look like hobos! Not a great way to represent the USA!\nI can't wait to buy these gloves!\nCan't wait for this stuff to be released! I will definitely sport that!\nCan't wait for these to release!!! Deffinetly gonna pick up a jacket and pants!!!\nWhen are these going on general sale.\nI live in the UK hope I can get one imported.\nI would defiantly buy a set of these snowboarding uniforms for me and my wife for our winter holidays. We live in Guernsey but love traveling around America.\nHell yea. I want one!!! The uniform is awesome!!\nAbsolutely would buy it. Can't wait.\nShaun White just put his up for sale on eBay.\nI would DEFINITELY love to get one of these jackets! They're awesome and just my style!\nI am looking forward to wearing that \"unique quilt and flag print jacket.\" Hope to see it available SOON.\nCan't wait til they are available to purchase. I definitely want one.\nThese uniforms are off the hook! Any idea if and when they will be available for purchase?\nI want the jacket. Love it.\nWant the jacket!! Love it!!!\nI just spoke to a burton rep and she said the jackets will not be mass produced.\nWhen and where can I purchase the jacket and accessories? Already have everything picked out that I want. Probably have to save up for the jacket but I can start collecting. lol Gotta have this!!\nI know the official Burton site is selling the gloves, beanies, t-shirts and hoodies. I have not seen the jackets or pants yet.\nGreat looking jacket!! Would love to be able to get my hands on one!\nThose jackets are so tough!! Are they up for sale?!\nCan the jacket please be for sale already!!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Place the 2 and 1\/2 cups of water in the pot and bring to a simmer (not a rolling boil).\nAdd the 2 cups of oil and either 1\/4 lb. of dry flower medicine or trimmings. Do not stir, as the moving water will do this for you.\nCover and turn the heat down to medium-low. You do not want to cook the water off because this helps to keep your product from burning, and thus wasting the THC.\nAfter 20 minutes, turn down the heat all the way to low.\nAfter an additional 40 minutes (1 hour total) turn off the heat and remove the pressue cooker (or crockpot) from the heated surface. The remaining matter in the pot should look like wet mashed up lawn clippings with much of the liquid remaining.\nNext, scoop the matter in your crock pot or pressure cooker into your grape press or press pot. While you are doing this, ensure that you spread the matter evenly in the press, so as to get the maximum pressing ability. Pour any remaining liquid into the press, as this is where the majority of the THC is concentrated.\nReminder: Foods prepared with infused cannabis oil can provide long-lasting, effective relief for many conditions and symptoms. When medicating with edible products, remember that a little goes a long way and the effects can take up to an hour to be noticeable. Start with a very small portion (1-inch square brownie, for example, or one small cookie) and wait to feel the effects before ingesting more. Keep out of the reach of children and pets.\nPlace the whole milk, whipping cream, vanilla, and trimmings\/dried flower medicine in a large pot and heat on low\/simmer. This will ensure that you do not OVERCOOK the medicine which will remove its potency.\nSimmer for 2 hours on low.\nNext, use a cheesecloth or fine strainer (like a coffee filter) to strain the milk once it's done simmering to remove plant material.\nRefrigerate up to 48 hours.\nFreeze any unused portion in ice cube trays to enjoy later.\nReminder: Foods prepared with infused cannabis milk can provide long-lasting, effective relief for many conditions and symptoms. When medicating with edible products, remember that a little goes a long way and the effects can take up to an hour to be noticeable. Start with a very small portion (1-inch square brownie, for example, or one small cookie) and wait to feel the effects before ingesting more. Keep out of the reach of children and pets.\nPour into a glass and top with fresh strawberries, kiwi, and orange slices to dress it up.\nReminder: Foods prepared with infused cannabis half-and-half can provide long-lasting, effective relief for many conditions and symptoms. When medicating with edible products, remember that a little goes a long way and the effects can take up to an hour to be noticeable. Start with a very small portion (1-inch square brownie, for example, or one small cookie) and wait to feel the effects before ingesting more. Keep out of the reach of children and pets.\nPreheat oven to 375 degrees. Line a light-colored baking sheet with parchment paper.\nIn a medium bowl, beat both kinds of butter with an electric mixer on medium speed until fluffy (about 20 seconds).\nAdd sugar, then beat for an additional 30 seconds.\nAdd egg yolk and vanilla, then beat until combined.\nIn a separate bowl, combine flour, salt, baking soda, and baking powder. Stir to combine.\nSlowly add the flour mixture to the butter mixture and then beat until combined.\nAdd chocolate chips and stir in.\nPlace 12 dough balls evenly spaced on the cookie sheet.\nBake for 8-10 minutes or until golden brown at the edges.\nAllow the cookies to cool on a wire rack for 5 minutes and then enjoy.\nNote: Be sure to use a gram scale when using infused butter, potency always varies with each batch, however, WCM consistently tests ours at @ 9-8 mg THC\/ gram.\nReminder: Foods prepared with canna-butter can provide long-lasting, effective relief for many conditions and symptoms. When medicating with edible products, remember that a little goes a long way and the effects can take up to an hour to be noticeable. Start with a very small portion (1-inch square brownie, for example, or one small cookie) and wait to feel the effects before ingesting more. Keep out of the reach of children and pets.\nMake a well in the center of the dry ingredients and pour in the wet ingredients, mix until smooth.\nHeat an oiled frying pan over medium-high heat and place 1\/4 cup of batter on the pan per pancake. Brown on both sides.\nPlace all Baker's Mix in an oven-safe bag (such as a Reynold's Turkey Bag) and close with oven tie.\nDecarboxylate (bake to activate the THC) Baker's Mix in the oven at 240 degrees for 30 mins. Be sure to agitate and move product around in the oven bag every 10 minutes, to avoid any hot spots in the Baker's mix (be careful, it's very hot!).\nThen put all ingredients in a crockpot or double boiler.\nHold at a constant temp of 180 degrees and allow to steep for 2-4 hrs, stirring intermittently.\nThen turn off heat and let the butter cool.\nStrain\/press out in 3 sheets of cheesecloth over a strainer, or alternatively in a French press. This will remove the plant material and leave with pure infused butter.\nPour into mini loaf pans or butter molds of your choice.\nAfter it sets up, butter may be kept frozen for up to 6 months in an airtight container (clearly labeled as a medicated product).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Louis Rom\u00e9goux is a Sheffield-born Anglo-French folk musician, who in 2010 swapped the steel city for the rolling hills to the south of Graz.\nOver the last 15 years Louis has toured extensively throughout Europe, playing festivals to critical acclaim and gaining a reputation as an impressive live performer in the clubs and bars of central Europe and beyond. Drawing on influences from both sides of the channel, as well as a few from across that larger Atlantic pond, Louis sound captures the essence of folk in its raw, free flowing form. Encompassing ever-expanding boundaries, the fluidity and freedom of Louis music keep his live performances fresh and exciting every time.\nIn his time as a solo and band musician Louis has gigged with the likes of John Smith, Teddy Thompson, Slow Club, Denis Jones, John Fairhurst, Liz Green, The Reverend, Ed Prosek and more. His recent festival appearances include Riba Rocks (Spain), Sound of the Forest (Germany), plus an Italy tour and an appearance at the legendary Ronnie Scott's in London in June 2013. He also sang in a Sheffield-based band called Watch This Fire Spread and was both a boy chorister and then later a choral scholar with Sheffield Cathedral Choir. In 2017 & 2018 Louis worked with the Viennese composer Alfred Polansky on \"Songs between heaven and hell - the Malcolm Lowry project\", a collection of Malcom Lowry's poetry set to music by Polansky. A series of concerts followed and the album was received to great acclaim in the Austrian press.\nHe has self-released three EPs and one mini-album. His new mini-album \"Solo\" (released 2016), plus his last two EPs 'Letters' (released spring 2013) and 'Milou' (released spring 2014) are both available on iTunes, Spotify, GooglePlay and Amazon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul Soulellis (b. 1968) is a New York-based artist and designer, maintaining his studio at NEW INC at the New Museum. He teaches at Rhode Island School of Design.\nHe founded his research-based graphic design and publishing studio Counterpractice in 2014. Previously, he ran Soulellis Studio for 12 years, producing award-winning work for clients like Cornell University, TED, Waterworks, Esri, The Rockefeller Foundation and The Municipal Art Society of New York.\nPaul is the founder of Library of the Printed Web, a physical archive devoted to web-to-print artists' books, zines and other printout matter. The on-going project has gathered international attention, was featured twice in the New Yorker, and was shown at The Book Affair at the opening of the 55th Venice Biennale.\nIn 2014, Paul began publishing Printed Web, a semi-annual \"exhibition in print\" devoted entirely to web-to-print art and discourse. Printed Web #3 will launch at Offprint London at the Tate Modern in May 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"New Blood and Plunder Gear!\nThe new Blood and Plunder blisters as well as the \"No Peace Beyond The Line\" book are now here at GCmini.com! The blisters range from Dutch commanders to militia Calvary. The new book is an entity in itself. While you can't use this book alone, it must be used in conjunction with the rule book, it does add a lot to the game. Such as new rules to refine and add complexity to gameplay as well as going the extra mile and giving rules for campaign settings. In which your unit can progress and be upgraded over time. There's also a lengthy section detailing the factions, their units, and their abilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a graphic designer and art director.\nWant to get in touch? Cool, but just to be clear, I don't tweet, birds do, you can follow me on, Flickr, and I'm also on LinkedIn (because who isn't?).\nIf you prefer more traditional means of communication you can always send me an email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's the honest instruction for business success provided by 60 of the largest U.S. corporations that, according to a Wall Street Journal analysis, \"parked a total of $166 billion offshore last year\" shielding more than 40 percent of their profits from U.S. taxes. They all do it, including Microsoft, GE and pharmaceutical giant Abbott Laboratories. Many, like GE, are so good at it that they have avoided taxes altogether in some recent years.\nBut they all still expect Uncle Sam to come to their aid with military firepower in case the natives abroad get restless and nationalize their company's assets. We still have a blockade against Cuba because Fidel Castro more than a half century ago dared seize an American-owned telephone company. During that same period, we have consistently intervened to maintain the lock of U.S. corporations on the world's resources, continuing to the present task of making Iraq and Libya safe for our oil companies.\nAmerica's multinational corporations still need the Navy to protect shipping lanes and the Commerce Department to safeguard U.S. copyrights. They also expect the Federal Reserve and Treasury Department to intervene to provide bailouts and cheap money when the corporate financial swindlers get into trouble, like GE, which almost went aground when its GE Capital financial wing got caught in the great banking meltdown.\nThey want a huge U.S. government to finance scientific breakthroughs, educate the future workforce, sustain the infrastructure and provide for law and order on the home front, but they just don't feel they should have to pay for a system of governance, even though it primarily serves their corporate interests. The U.S. government exists primarily to make the world safe for multinational corporations, but those firms feel no obligation to pay for that protection in return.\nOracle increased its foreign holdings by one-third, including new subsidiaries in low-tax Ireland, and thereby was able to add a cool $272 million to the company's bottom line by avoiding U.S. taxes. Abbott estimates that it saved $1.6 billion in U.S. taxes through its operations in more than a dozen countries. By moving $8.1 billion of its profits overseas, Abbott was able to claim a pretax loss on its U.S. operations. Johnson & Johnson, another health industry giant, has almost all of its cash\u2014$14.8 billion out of $14.9 billion\u2014abroad, yet still claims to be a U.S. company.\nOne of the longtime leaders in offshore tax avoidance has been that once-American-as-apple-pie company GE, which in a more innocent time hired Ronald Reagan to advertise its wares. Now GE has nearly two-thirds of its jobs abroad, avoided U.S. taxes in the previous two years and has $108 billion stashed overseas.\nTwo years ago, President Obama appointed GE CEO Jeffrey Immelt to chair his Jobs Council, despite the fact that Immelt had cut his company's U.S. workforce by a fifth. GE's expertise is no longer in appliance manufacturing, a division Immelt has tried to shed, but rather in financial manipulation.\nGE Capital was a leader in the financial scams that still haunt the U.S. economy, and Immelt has been most effective in lobbying Washington politicians to rig the tax laws to benefit his and other multinational corporations. He has created some jobs, but unfortunately, they are abroad, along with his company's untaxed profits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Izvornik: XVIII. Dani psihologije u Zadru Sa\u017eeci radova \/ Penezi\u0107, Z., \u0106ubela Adori\u0107, V., Ombla, J., Sli\u0161kovi\u0107, A., Sori\u0107, I., Valerjev, P., Vuli\u0107-Prtori\u0107, A. (ur.). - Zadar : Odjel za psihologiju, Sveu\u010dili\u0161te u Zadru , 2012. (ISBN: 978-953-7237-34-9).\nMjesto i datum: Zadar, Hrvatska, 24. - 26. 05. 2012.\nThe underlying idea of the family development perspective is that family in whole undergoes continuous changes in number and structure of family roles as well as in quality of family relations as an expected result of developmental challenges. The aim of this research is to describe the changes in ways married couples express love in different stages of the family development. The total sample consisted of 456 married couples from Serbia, and 302 married couples from Croatia. In both samples the duration of marriage was from 1 month to 57 years. Around 20% of couples in both Serbia and Croatia have no children. The rest of the samples were, according to the age of the oldest child, divided into categories describing family development stages (families with young child, families with preschool age child, families with school age child, families with adolescent, launching children, families with adult child, families in later life-aging families). The Ways of Showing Love Scale (Hui\u0107, Kamenov & Jugovi\u0107, 2010) measuring four expressive (\"communal orientation\/sacrifice\" ; \"emotional openness and support\" ; \"physical affection\" ; \"verbal affection\/gifting\") and two instrumental factors (\"domestic instrumentality\" ; \"public instrumentality\")was administered. The results indicate significant differences in overall ways of showing love in different family development stages in both Serbian and Croatian sample. Also, both men and women behave differently when expressing love in different stages of family development. This trend was observed in all measured expressive factors and public instrumentality, whereas domestic instrumentality tends to be more gender specific and stable across family development. In all stages of family development ways of showing love are significant correlates of marital satisfaction. However, significant differences have been observed in the ways that expressive and instrumental ways of showing love correlate to marital satisfaction in both Serbian and Croatian couples. Both developmental and cultural differences are discussed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to St. John's Lutheran Church in Montpelier OH, where we have faithfully proclaimed the love of Jesus Christ to the people of our surrounding area for over 150 years! Our region has seen many changes through the years, but two wonderful truths have not changed - the blessings of God in Jesus our Savior, and the warmth and friendliness of the church family here at St. John's.\nFor us, worship is not a \"spectator sport\". We will put you to work singing, reading, and participating in our services. Maybe you will be working on our mission trips. Perhaps teaching our youth is a good fit for you. There are many ways to serve at St. John's.\nA staffed nursery is available at the 10 a.m. service.\nOther services are held throughout the year as well including: Wednesday Lenten Services; Ash Wednesday and Maundy Thursday; Easter Sunrise; Mid-week Advent Services; and Christmas Eve Service, just to name a few.\nWe love having visitors and look forward to meeting you and worshiping together.\nA Lenten supper meal will be offered starting at 6 pm on April 3 and 10.\nPotluck meal beginning at 6:00 p.m.\nWorship Service at 7:30 p.m.\nPraise and Worship Service at 8:00 a.m.\nEaster breakfast served 8 - 10 a.m.\nTraditional Worship Service at 10:00 a.m.\nPastor Paul began serving at St. John's in 2011. A native Ohioan, Pastor Paul is a graduate of Capital University and Trinity Lutheran Seminary. He is married to Joy, who is a private music teacher. They have two adult children, and three grandsons. In his spare time, Paul enjoys tennis, music, gardening, watching college basketball, and cheering for the Cleveland Indians (he has even seen the Indians lose in eight different cities!).\nChurch council has implemented a policy about weather cancellations for St. John's events. Worship services will only be cancelled if Williams County is under a Level Three snow emergency. Cancellations of other church activities will be decided by the leaders of those activities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..6e512ab1b292e2271b38f07b67c227842bfa81e6
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazjsr
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+{"text":"Here, we have the power, the right, of choice. We choose every aspect of our reality. Here, the victim and the perpetrator each choose each other. For one reason or another, they find each other. We give our permission. We welcome these experiences with out words, our behaviours and our thoughts. Even the innocent, seemingly spiritual notion that \"someone has to be a martyr\" this is an invitation. The idea that darkness has to have a victim, creates that dual experience. This is not said to blame victims, but to free them. The violent and the meek are two halves of one whole. Represented for example in the planetary energies of Mars and Venus. The warlike compulsion and the loving compulsion. How are these two apparent oppposites resolved? We make a choice. We wed the two, to empower the One. When we choose love and light all falsehood falls away. We are bigger than these wars, bigger than these petty racial and social biases. We martyr ourselves in attempts to distance ourselves from \"that enemy.\" We polarize as the opposite and thus create an invitation, because opposites attract. Balance must always be attained.\nSaying you are not a thing, makes that thing manifest to test your words\/thoughts for truth. Are you? Or aren't you? Both are true. Within you is the capacity for that way of being. Acceptance leads to freedom from fear. Simply be what you are, do not worry about what you are not. There are no enemies.\nThe masculine and the feminine are key to understanding and transcending the disharmonies we suffer. Beyond religion, beyond dogmas. Every being is a being of light, a part of God and is to be saved. Union brings healing. Simply speak the truth. Be the truth, perpetuate only love. But always mindful that one extreme creates the other. A victim creates her own attacker. So simply be. Without allowing fear to create shadows.\nWe do not wish to suffer. We suffer because we are in disunion. We suffer because we think ourselves unworthy. We feel separate in a sea of never ending unicity... and so we die of thirst in this never ending sea of God. God is creativity creating. Without fear, without good or bad. Without duality. Duality is a limitation 'here' but this is not the only here. We are unlimited in creativity and can shatter this box which contains us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MPS hand controls are available from Access Options in California. These unique disabled driving devices enable easier driving with a disability. Access Options offers sales, installation, and maintenance for MPS (Monarch) hand controls from each of their locations in Sunnyvale and Watsonville, CA.\nThe Monarch Mark IA hand driving control incorporates the most popular method of operation, and undoubtedly the least fatiguing available. Through mechanical linkage, the brakes are applied by the forward motion away from the driver toward the brake pedal or dashboard. Again with mechanical linkage, the gas or accelerator is applied by a downward movement toward the driver's lap and at a right angle to the brake. Designed with a flat handle and accelerator rod that fits behind the dash panel makes it the least intrusive Hand Control on the market today. This quality hand control comes with our easy to use accelerator lock out that when engaged locks out the accelerator function of the hand control so drivers using the pedals cannot use the hand control.\nThe MPS Push\/Rock Hand Control is the newest addition to our line up of quality driving aids. This control enables the driver to apply the accelerator and brakes by hand. When the upright handle is pushed forward toward the brake pedal the brakes are applied. When the upright handle is rocked rearward toward the user the accelerator is applied. It also comes with Accelerator lock-out feature for added safty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wakol PU 225 Adhesive is a two-part adhesive suitable for engineered and solid wood flooring. Ideal for use with wide plank flooring where a solid firm bond is required. You can use it for underfloor heating system.\nPour hardener into the adhesive and mix with an electric whisk stirrer for 3 minutes until fully mixed.\nMixing ratio: 8 parts adhesive to 1 part hardener.\nWakol PU 225 Adhesive has an open time of approximately 50 minutes. Please note, we recommend you to wear gloves during handling because it turns the hands black.\nWakol PU 225 comes in a 6.75 kg tub, component A and a hardener - 0.75 kr, component B.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Walmart is facing controversy for selling a shirt with an offensive phrase on their website.\nThe $18.99 T shirt bearing the message \"Rope. Tree. Journalist. SOME ASSEMBLY REQUIRED\" was listed on Walmart.com through third-party seller Teespring.\nHanes Ultimate X-Temp Crew Undershirt 3-pk.\nInstead, I suggest this shirt from queer-run We Should All Be Mirandas, which donates 15% of proceeds to Nixon's campaign.\nWe eventually made a turn for Ocean City Airport and entered on a three mile forty-five degree approach for left down wind runway one-four.\nI'm a chick so I guess I'll be able to pull off that Roses hoodie that most are saying they can't! You're a hellova guy if you can tho!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He is already one hour late for our meeting.\nI sometimes stay up late on the weekend.\nMy girlfriend is sometimes late.\nI'm sorry (that) I'm late.\nI'm really sorry for keeping you waiting.\nI promise that I won't be late again.\nDid I keep you waiting?\nI'm going to be about 20 minutes late.\nI was late for our date.\nI have never been late for school.\nThe train is running late.\nThe train is a little behind schedule.\nThe train was far behind schedule.\nThe project is running behind schedule.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazohd b/data_all_eng_slimpj/shuffled/split2/finalzzzazohd
new file mode 100644
index 0000000000000000000000000000000000000000..023ef286d1222134d66742045e709fdeeebf08c3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazohd
@@ -0,0 +1,5 @@
+{"text":"We are world-class organisers of B2B exhibitions & conferences. Organisers of International Security Expo \u2013 the global security showcase that makes a real difference and now, the fastest growing security event worldwide.\nWith 20 years of management experience and a committed team on hand, we produce large-scale exhibitions supported by award winning conferences, workshops, master classes, seminars and live demonstration programmes that run alongside.\nWe aim to produce an annual event that delivers industry specific content, that formulates key debates, that initiates ideas and that delivers better understanding and security whether it be through technology, a public body, a private organisation, through political understanding or the police and other security services.\nIt is always difficult when putting an event together to get the right balance of activity to generate sufficient interest \u2013 from a show organisers perspective the business book says do just enough to get the footfall through the door and thereby attract enough exhibitors to make it worthwhile. Well we are the International Security Expo team and we ripped that book up - The 2018 Expo received 11,753 International Visitors, 76 Officially Hosted Countries with 361 International Delegates joined by 300+ Exhibitors.\nIf you are interested in a career at Nineteen Group and think you would be a good fit for us, we would be happy to discuss future opportunities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'd buy no.trust, dis.trust, and mis.trust, but then again, maybe I'm just being cynical.\nI'm all for holding banks to higher standards, but it isn't going to work if you're charging 150k for a domain. How many are going to pay that when other TLDs are available?\nEven if it were cheaper, it still isnt going to work. If users are too stupid to think therealhsbc.geocities.com is their bankimg login page (or more likely, don't even glance at the address bar) then it doesn't matter whether it's hsbc.trust or hsbc.honestguvitsreallyus.\nGood luck with that. Banks follow their own internal practises and they generally have enough business that they can tell people like the PCI-DSS enforcers where to go. They also really do not want domain-name proliferation which gets people used to the idea of multiple domain suffixes for them. That just leads to easier phishing. There's also little marketing use as few people will know how much it costs or why that should translate into security. A registrar telling a bank how to be secure? I don't think so.\nThere might be a margin from cybersquatting 'ingodwe'.\nWould you be interested in a .itspracticallystealing domain for ONLY US$100,000?\n150k you say? For a virtual asset you don't control. Doesn't sound very appealing.\nthat you're just taking the piss with this and not being serious.\nYes ultra secure, because the price will stop someone starting the domain yourbank.trust.verify.account.fake (replace .fake with any real TLD) from people that fall from phishing scams.\nDoes any domain name inspire trust because that's what the domain name implies?\nThe first time one of these domain names gets hacked, and they will... You lose that trust forever.\nThis is where their business model fails.\nI don't think it has any trust to begin with.\nIt sounds like you're trying to hard to make people trust you.\nIt took me a moment but have a beer and an upvote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"E Ink keeps unveiling fresh technology and this time it came up with the JustWrite tech, used for writing on digital paper. They usually handle stuff like black and white screens on eReaders like the Kindle.\nThe new technology promises a \"natural writing experience\" with close to zero latency. The concept was shown during the the Connected Ink 2018 event in Tokyo, taking place this week. The JustWrite film involves a durable and bendable, plus light plastic screen, which can be reshaped as you please.\nThis means you could even employ a JustWrite display with holes in it. The device will be available in any size up to 3 feet in width, meaning it could be used for blackboard purposes in schools. It enables a pen and paper style experience, without requiring a TFT backplane.\nThere's also an optional digitizer, but you can also use with just a stylus. The core implementation fits in the size of an A4 sheet, sans the bulky TFT backplane. It needs very little electricity and has very low latency. JustWrite also works with pens, brushes, markers and stamps.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This pool table has been in the family for several decades. It was first in a pool hall in Masontown, WV before my great grandparents purchased in the 1950s. The pool table is similar to the Brunswick Twentieth Century pool table. It has a Mahogany finish & lightning tubular pocket ball return. The slate is a 3 piece, 1 inch thick. It is 8 ft long, 4 ft wide. Only thing it needs is new felt. I'm trying to sell it, but I'm not sure of its worth. I don't want to sell it for too much\/too little.\nNice table. It's hard to tell from the photos, but it looks to be in need of a restoration.\nI would value your Brunswick 20th Century pool table at around $500 give or take. The fact is that there are a lot of older Brunswick pool tables on the market, many in pristine condition. This one does not appear to be in great shape.\nThat, plus the fact that this was a lower-end Brunswick pool table (e.g. constructed with laminates and veneers) means a lower value.\nAlso consider that, on top of restoration costs, someone will also have to pay another ~$600 to disassemble, move, set-up, level, and re-cloth the table.\nHowever, a fully-restored version of your table could fetch $2000 to $3000 depending on the quality of the restoration.\nThe pool table has been restored, it just needs new felt.\nOh nice. Can we actually get a look at the restored version of the pool table you want us to value?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Host Introduction VPSFX founded with the aim to provide enterprise hosting and server services at reasonable prices. Since it's inception, VPSFX has been committed to entrust customers with effective & easy to use web hosting services backed by unmatched 24\/7 quality technical support and guaranteed server uptime. They combine quality service and support with unparalleled response time and focus on giving customers exactly what they need. They mainly invest in their infrastructure to ensure that VPSFX provides top notch service for business clients. All of their virtual private servers come with a 100Mb interface and 1TB per month of total transfer. Cloud servers are built on powerful multi-core machines with dedicated ECC RAM and resources.\nWhat we like: Sales assistance received via email was really good. Sales agent responded all our questions in detail within just few minutes. Response via other resources was also good.\nWhat needs improvement: Would like to see live chat 24x7. It will be good if additional support resources are considered like phone.\nWhat we like: VPSFX technicians reinstalled mySql successfully without loss of any data.\nWhat needs improvement: Would like to see live chat and phone options.\nWhat we like: Performance of VPS server was smooth, steady and fast till the end of our subscription with VPSFX. Also, the downtime was negligible.\nVPSFX core focus is to provide their customers with a more personal approach rather than them being counted as numbers and being treated differently. Their target market is the VPS, they've done a lot of research into the industry and managed to pull off what they believe to be the best service available. They are commitment to provide the best service and high-performance equipment at affordable prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbaoas b/data_all_eng_slimpj/shuffled/split2/finalzzzbaoas
new file mode 100644
index 0000000000000000000000000000000000000000..15c3d53822593c816fc1b187740a9cd8db08585b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbaoas
@@ -0,0 +1,5 @@
+{"text":"John Quinn auctioneers is offering for sale of No 13 Ashfield Road, Greenfields, Newcastle, a four bedroom semidetached residence in pristine condition.\nThe property, which is within walking distance of the university and hospital, comprises a sitting room, superb fully fitted kitchen, four bedrooms, two which have beautiful en suites, and main bathroom with bath, shower, and whb. The property has off street parking to the front and fully tarmacadamed driveway. The house is fully double glazed and has oil fired central heating.\nJohn Quinn says that the property is ready to walk straight into and is situated in one of Newcastle's most popular and desirable residential areas.\nThe property has a BER of F on BER cert no 109725408.\nTo arrange a viewing contact John Quinn at 091 569174.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'s WTWC, an NBC affiliate. Monday was his first day, and he made the move from Panama City's ABC affil, WMBB where he handled production for commercials, etc. Jason's always been drawn to telling stories through a lens \u2013 he started making films with the family camcorder. Once he got his degree from Univ of West Florida, he's been involved in long and short form projects \u2013 and happy about it. Well Dunne! Jason. I know you'll enjoy working with CSD STEVE LEMON, who himself is involved in a project \u2013 a documentary about his father's spy plane.\nPenn State's KELLIE GRUTKO has been named VP of Marketing for Comcast Spotlight. She'll do the branding, promos, communications and corporate creative stuff. Well Dunne!\nThings are all atwitter with Turner's NBA coverage. Read about it here.\nWGN in Chicago is getting a makeover. Read about it by clicking here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a party, there are always a ton of empty bottles and cans lying around. I'm sure your first intuition is to crack the guy lying on your couch upside his skull with a the 40 he stole from you last night. Since that might end up landing you in jail, we figured it would be a good idea to put together ideas to keep you out of trouble. The redneck down the street thinks they are good for shooting, while others like to make art or furniture out of them. It's time for you to check out our awesome list of ways to utilize your empty 40s and aluminum beer cans.\nIf you are going for the ultra classy look in your pad, a beer chandelier is the way to go. This awesome light fixture will spice up that bar in your basement, and will be the centerpiece of your beer bottle collection. Believe me, your future wife will be impressed. For more class, make sure this is directly above a makeshift beer pong table. When you get tired of the look, just drink some more and replace the bottles!\nWhat is your favorite from the list above? What else have you done with your empty beer cans and bottles? Let us know in the comments below!\nthat sniper video is pretty sick! i seriously can't believe he hit a can that far away. i think ill have to play some beer can basketball next weekend \u2013 looks like an awesome idea\u2026what do you think i could use for a backboard and net?\nI think you could just use a piece of cardboard and a trash can to make the \"backboard\" and \"net\". Probably the easiest combo of what you'll have lying around!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our bookstore is currently closed while we undergo a redesign to better serve our members. For your convenience the books will soon be offered on OxfordClub.com along with the other membership benefits.\nWe'll be sure to announce our grand opening!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The dream numerous people to are now living a house inside own had began to realize with the advent of home loan services. In the olden days people saved money to acquire some property. Obvious the trend followed all over the world, but today this trend changed. People can buy property or any house without having any means along with own. This is simply because they can avail loans to climb the property ladder and own a house of their own. They need to choose a package that will fit their money situation. Of course any kind of time time this situation can change. Means positivity . decide to buy some house inside your do not have funds you can take the help your property buyer to get hold of house for in your own.\nThese agents or dealers will allow you avail loans at the financial institutions or banks. This method for Fourth Avenue Residences you to get a mortgage. The features of such loans taken will vary for different everyday people. The size of mortgage loan may depend around size of house and the associated with the house the actual reason bought. In the urban areas the price of of property is high. In the rural areas sneakers area of property may cost significantly less. Also the maturity with the loan and procedure of paying could vary for each property dealings at time of buying. Consideration considerable variation on other characteristics when choosing any property far too. The whole point is you get a loan or financial help buy a house which you can pay in installments actually period of a few years until is actually very all paid up.\nThis way your property buyer will in order to to buy a house, which discover own and are living it without making payments on the full amount. Shell out for it in monthly installments once you would pay rent for any house that you enjoy rented. The number of rent may be a little less however the amount payable as monthly installments in a position to a little high. Whatever the amount not only do you it will be decided on the cornerstone of your financial conditions prevailing at the time of buying a. And you can choose the size of your house and property influenced by what you have enough money for at present additionally think you can afford to pay previously future. Actually no one can predict foreseeable future but based within your salary you may make some plan to spend for the house in monthly finance payments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbpck b/data_all_eng_slimpj/shuffled/split2/finalzzzbbpck
new file mode 100644
index 0000000000000000000000000000000000000000..d651bc1ab24300d8301384d714545d3f01cda3e5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbpck
@@ -0,0 +1,5 @@
+{"text":"Make a 'steel' on the perfect band for your event in Carmarthenshire, Carmarthenshire by booking one of our 10 Steel Drum Bands near you!\nRecent review: \"Amazing! Would highly recommend. All the guests gave great feed back and they were really lovely guys! Thanks again and maybe we'll see you again sometime.\"\nWedding Reception at the Druidstone Hotel in a marquee. Need a band for a couple of hours, then will have a dj.\n50th \/ 18th birthday party in a field, Marquee, on a farm. Sat 16th July. St Davids, Pembrokeshire. 70 guests approx.\nLooking for a covers band to perform at wedding reception. the wedding is the first of the venue and may potentially lead to publicity due to the newness of the venue in the Bristol area.\nA 30th birthday party of 50-60 people at a venue in Clifton, Bristol. Would be looking for a 1.5-2 hour set of classic and modern funk\/soul.\nOutdoor wedding but we do have several marquees, one of which is a catering tent. we would like a sit down plated lunch for 42 members of our immediate family. I am a chef myself and would like the option to be able to write a menu and liase with your head chef to see what is viable.\nHi Eugene, I emailed yesterday we do really like the Edge, who you suggested, however we think their play set might be a bit too contemporary - good for us, perhaps less ideal for the older generation. Hence why we are looking around. The Magic Tones look very good. In terms of details our wedding is a marquee wedding.\nWe are looking for a small jaz band for our wedding reception. Possibly able to cover Carl Porter and other 1920s style, also \"The Way You Look Tonight\" (Fred Astaire) is 'our song' so they would need to have this in their repertoire.\nHello, we are looking for a live duo to play at our tipi wedding in Backwell this summer. We would ideally like someone to play outside for an hour for welcome drinks and set a louge\/laid back vibe and then play again in the evening (1-2hours) to start the party - more contemporary, like you have on the website to listen to. You have a great voice! Are you available and how much would this cost? I sent you an email via another website but wanted to go via two channels, just incase! Thank you!\nSmall birthday party\/housewarming at home in Cornwall. Require good duo acoustic act. Party will be outdoors weather permitting but act can be positioned in sun room which has bi-folding doors and electricity.\nindoors - a show to be performed when a band have their half time break. 20-30 mins. its a birthday party.\nI work for Lloyds banking group and my self and a group of colleagues are holding a family fun day to raise money for BBC Children in need. The event is being held on Sunday 7th August 12-4.30 pm at The Moretons in Bredon, Tewkesbury. It is an outdoors location. I was wondering if you would be available for a couple of hours to come along and do some magic. However as it is a charity event we are looking to do this for as little cost as possible. Any support or help you would be able to offer us would be greatly appreciated.\nI am getting married in St Fagans at 2:30pm and having afternoon tea and photos. Would like the evening do in a tipi for about 80 guests. Would like a live band and buffet style dinner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was nice to have a 3 day weekend, but I am looking forward to doing some great lessons with my scholars. It's a crazy week with field day for grades 4th-6th and some field trips in there, as well.\nIf you haven't heard, our Little Free Library is now open in the school lobby. You can check out our website for more information. We hope the school district puts it in its permanent location soon!\nThis week we will continuing working on the relationship between multiplication and division through creating charts. This week's math game will be student choice. Please return it in the math game folder on Friday.\nThis week, we will continue chapter 2 in the math book, which is working with fractions, decimals and percents. There is no class on Friday due to field day.\nOn Tuesday we will read a story called \"Underwater Hockey\" and the students will retell the story. Due to field day we will not have class on Wednesday and Thursday they have a field trip. On Friday we will read \"Who Stole King Cole\" and complete some comprehension activities. Please continue working on the fluency packet. Directions are on my website. Please practice the fluency packets at least twice a week. The more your child practices this packet, the more quickly your child will progress.\nOn Tuesday we will read a story called \"Underwater Hockey\" and the students will retell the story. Due to field day we will not have class on Wednesday and Thursday they have a field trip. On Friday we will read \"Who Stole King Cole\" and complete some comprehension activities.\nOn Tuesday, we will be reading a story called \"Underwater Hockey.\" The students will be drawing pictures to go along with the story. On Wednesday and Thursday we will be working on a story from Six Way Paragraphs called \"Abe's favorite story.\" On Wednesday, we will discuss vocabulary from the passage. On Thursday the students will discuss their answers to questions that they completed for homework and compare their support for the answers. On Friday there is no class due to field day.\nThis week, we will review unit 6. This unit focuses on the adding suffix s to words. For example, dog + s = dogs. When reading a word in isolation with a suffix, we first read the baseword (dog) and then read the word with the suffix (dogs). Our goal is to be charting the students' fluency 2-3 times per week. Be sure to be working on the assigned probe each week. We will continue with fluency probes this quarter. We have moved on to different probes. This week's probe is 7.\nTuesday and Wednesday we will read a passage, \"Firefighters Save Shepherd from Sinkhole,\" and learn how to identify the problem and solution. On Tuesday we will read the passage together and talk about key words. On Wednesday the students will share their answers to the passage and we will discuss what happened in detail. Also they will watch a short clip from the firefighters rescuing the shepherd. On Thursday there will be no class due to field day. On Friday the students will be working on Read Naturally.\nThis week we continue working on the topic of kindness as the topic of our writing. We will be using the basic strategies of Bold Beginnings to develop different styles of topic sentences in reference to kindness. This will give them a variety of topic sentences to choose from and will help them start brainstorming the direction they would like their assignment to head in. They will be developing the topic of what is kindness into paragraphs. Students will have a choice to enter their finished assignment into the Be Kind contest. We will also continue to work with Four Level Grammar. This is a program that is being used throughout the school to immerse children in grammar. We will spend 5-10 minutes a day on grammar.\nOn Tuesday, we will complete the dictation part of the lesson for substep 1.4. I hope that we will be able to move on next week. Due to the field day, specials schedules have shifted. I will not have the students the other days of the week. Please continue working on the fluency packet that was sent home a couple of weeks ago. Directions are on my website. Please practice the fluency packets at least twice a week. The more your child practices this packet, the more quickly your child will progress. Please do not skip this important part of homework.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After making waves during its multiple releases in 2010, the Jordan Brand is looking to make another noise this 2011 with the release of a new colourway of the Jordan Flight 45 High that is already hitting strides in the market.\nThe Jordan Flight 45 High, which was made to pay homage to Michael Jordan's return in the NBA after a two-year hiatus, will come out in White\/Varsity Red-Black-Cement Grey colourway that was clearly derived from its famous sole color.\nAlthough we've already seen impressive colourways of this shoe before, this new edition is a must buy especially for those who are looking for an elegant basketball sneaker that would also suit your lifestyle needs!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Which financial institutions do you do business with?\nRemind to AGF Investments that it finances Earth Day and the World Wildlife Fund, and that the mining investments in Goldcorp and Tahoe Resources are not in coherence with those supports.\n*all the investments amounts cited have been taken from Comprehensive Ownership Detail Report of both mining companies, dating from Sep. 30, 2013.\nLet your financial advisor know that you are concerned about the impact of Goldcorp and Tahoe Resources' activity in Guatemala. Ask if they are up-to-date regarding the situation.\nAsk what ethical principles your financial institution follows when making investment decisions.\nMake it clear that you are unwilling to hold stock in Goldcorp or Tahoe Resources.\nPropose divestment. There are other interesting, ethical and profitable investment alternatives available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The difference between the approach of historians and hagiographers is well known. The historian describes the life of a person, his biography, while the hagiographer writes a description of his holiness. The short span of time separating us from the New Martyrs allows us the opportunity to see this difference.\nNo book on the history of the Russian Church in the twentieth century can be complete without including the name of St. Athanasius (Sakharov), Bishop of Kovrov, a catacomb bishop who reunited with the Moscow Patriarchate in 1945. We will not, however, discuss the historical events in which he participated, but rather how holiness exists in time \u2013 all the more so, since we are considering the twentieth century, which is nearly contemporary.\nIcon of St. Athanasius (Sakharov) the Confessor, Bishop of Kovrov. The icon was painted in 2006 for the Church of the Protection of the Theotokos in the Butyrskaya prison from donations from the parishioners of the St. Nicholas Church in Biryulyovo.\nThe beginning of the Church-wide veneration of the New Martyrs and Confessors of Russia allowed us the entirely unique experience of witnessing the attainment of holiness. Not long ago, our lives were separated from the earthly lives of the saints by centuries. While reading about early Christian martyrs, the temporal distance deprived us of the ability to relive their struggles. In the lives of the saints we studied historical details, searching for moral lessons while enjoying the exquisite style of Byzantine hagiography. We could not do one thing: experience for ourselves the fear and pain of a person thrown into an arena teeming with wild beasts.\nIt has been centuries since people forgot how to read the lives of saints as eyewitness accounts. Not incidentally, literary historians have noted a fundamental abstraction in hagiography and the refusal of authors to specify historical details. D.S. Likhachev once pointed out that the authors of the lives of Russian saints wrote descriptively and without political terminology about very specific details. Instead of \"burgomaster\" [posadnik] they wrote of \"a certain nobleman\" or of \"an elder,\" and instead of \"Prince,\" they wrote of \"a ruler of that land.\" The names of episodic characters were eliminated and replaced with the descriptive \"a man,\" \"a certain maiden,\" etc. In practice, glorification by the Russian Orthodox Church took place after a considerable time. However, the glorification of the New Martyrs and Confessors made it impossible to leave out historical specificity.\nThe new saints are practically our contemporaries. They are the contemporaries of our grandparents or great-grandparents, and so their lives belong to a time we know about not only from books, but also from the stories of our older generations. In these accounts, the abstract words \"ruler of the city\" cannot appear, simply because the portraits of these rulers hung in the schools our parents attended. And children with red scarves [i.e., Young Pioneers], whose descendants we are, brought these rulers flowers on the anniversaries of the revolution.\nSergei Sakharov was born into a family with a traditional, provincial way of life. As a child he went to church joyously and was very fond of solemn hierarchal church services, and at home played \"church\" in bishop's robes constructed from his mother's shawl; he imitated the service, by censing, blessing, and so on. As he matured he had the ability to bypass, and perhaps even to ignore, the temptations of his times. Very indicative in this respect are the years the saint spent studying at the seminary in Vladimir. His memories of seminary life were always very warm. An historian, however, might reconstruct the seminary life of those years as anything but idyllic.\nIn 1903, when the future saint studied there, the seminary in Vladimir had a secret cell of the Social Democratic Party, which broke up after one of its organizers was killed by the careless handling of explosives. An underground seminary congress gathered there in the summer of 1905. In December of the same year the seminarians went on strike.\nCharacteristically, Sergei Sakharov, who was a witness of these events, did not talk about them at all. One gets the impression that social life and political passions did not particularly interest him. Remembering his seminary years, he often spoke of Archbishop Nicholas (Nalimov) of Vladimir, a strict ascetic who was an expert on the divine services. It seems that the rest passed by unnoticed and did not have much significance for the saint.\nAfter seminary he studied at the Moscow Theological Academy (here Sergei Sakharov was tonsured a monk with the name of Athanasius). He also taught and participated in the Local Council, after which the monk Athanasius returned to the Vladimir Diocese.\nHe became a bishop in 1921. Before his consecration to the episcopacy the future hierarch was summoned to the GPU and threatened with repression if he agreed to become a bishop. Before the Revolution, being a bishop meant having many significant social benefits, but by 1920 a hierarch was guaranteed persecution and deprivation.\nSt. Athanasius was arrested for the first time just seven months after his consecration. By his own count, over the years he spent only two years, nine months, and two days in diocesan service, yet \"in bonds and bitter work\" (i.e., prison and exile) he spent twenty-one years, eleven months, and twelve days.\nWe know much about prisoner life. In reading the letters of St. Athanasius from the camps and the memories of people who saw him there, one is struck by his incredible ability to ignore the inhumane living conditions and by his strength to remain a monk in the face of it all.\nTo comment on this \"Christmas portrait,\" it should be added that this letter was written by a fifty-one-year-old man with heart disease, who could barely breath when walking, but had to walk daily five kilometers to his workplace. The work consisted of unloading firewood, logging, snow removal, and more. Any Church-related items were not permitted in camp. Everything was taken away, not only books but even a hand-painted Easter egg received by mail. Despite official permission for imprisoned priests to keep their long hair, the hierarch was forced to cut his hair.\nThe retired bishop spent the last years of his life in the city of Petushki, Vladimir region. He was denied political rehabilitation. During the Khrushchev period former Soviet party functionaries arrested during the purges were politically rehabilitated. Church leaders were still considered enemies of the people, so the authorities did not question the correctness of their condemnation.\nSt. Athanasius spent the remainder of his life after his return from the camps in Petushki. The house where the saint reposed has been preserved. Pious parishioners turned this house into a small museum. In the room are his personal items: a bed, desk, icons, embroidered vestments, and a wooden Panagia, carved and painted by him.\nIn 1958, the prosecutor of the Vladimir region wrote that since Bishop Athanasius had for over thirty years not concealed that he was \"a believer and servant of the Church, and so could not agree with the atheistic authorities in matters of religious views and worship,\" his sentence was therefore fair and there was no reason for rehabilitation.\nThe bishop, during his last years, could serve only at home in his cell \u2013 which had its advantages, not only because of the widespread reduction of church services, which greatly distressed Bishop Athanasius, but because only in his cell could he pray before icons of the last Tsar and his family and of Patriarch Tikhon, both of whom he greatly venerated. Only where the saint was devoid of prying eyes could he strictly fast and pray for Russia each year on November 7 and 8(Soviet holidays dedicated to the October Revolution).\nSt. Athanasius died on October 28, 1962. His last words were \"Prayer will save you all!\" He was buried in the cemetery of the Presentation in Vladimir, adjacent to the barbed wire fence of the Vladimir prison that he repeatedly had to visit. After his glorification in the autumn of 2000, the saint's relics were transferred to the Monastery of the Nativity of the Mother of God, of which he had been superior in 1920. For a long time the monastery housed the Vladimir Cheka. The procession with the relics went along the same path that Bishop Athanasius followed when led from prison for interrogation.\nHere we are attempting to speak of a saint, not of a historical figure. Therefore, we do not write about St. Athanasius' participation in the Local Council of 1917-1918, nor about his struggles against the Renovationists, nor about his break with Metropolitan Sergei (Stragorodsky), and then, after the Second World War, his reunification with the Moscow Patriarchate, nor of his many years as a corrector of liturgical books. We are not interested in him as an historical person, but rather in how sanctity triumphs over historical time.\nunto the salvation of our souls.\nWhen these troparia were being written, the Bolshevik war against the Russian Church was just beginning and few believed that this persecution would only increase. An historian analyzing the trends of these times might say that things could have evolved differently. Yet holiness is a victory over time and the holy hagiographer can afford to write what the historian would call an anachronism.\n The State Political Directorate was the secret police of the Russian Soviet Federative Socialist Republic (RSFSR) and the Soviet Union from 1922 until 1934. Formed from the Cheka, the original Russian state security organization, on February 6, 1922, it was initially known under the Russian abbreviation GPU\u2013short for \"Gosudarstvennoye Politicheskoye Upravlenie of the NKVD of the RSFSR\" (\u0413\u043e\u0441\u0443\u0434\u0430\u0440\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0435 \u041f\u043e\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0423\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u041d\u041a\u0412\u0414 \u0420\u0421\u0424\u0421\u0420). Its first chief was the Cheka's former chairman, Felix Dzerzhinsky.\n Rehabilitation (Russian: \u0440\u0435\u0430\u0431\u0438\u043b\u0438\u0442\u0430\u0446\u0438\u044f, transliterated in English as reabilitatsiya or reabilitacija) in the context of the former Soviet Union, and the Post-Soviet states, was the restoration of a person who was criminally prosecuted without due basis to the state of acquittal.\n Mikhail Efimovich Gubonin (June 24, 1907 Moscow province \u2013 October 9, 1971 Moscow) \u2014 Russian artist, church historian, archivist, author of several literary works.\n Cheka (Chrezvychaynaya Komissiya, Extraordinary Commission) was the first of a succession of Soviet state security organizations. It was created by decree on December 20, 1917, by Vladimir Lenin and subsequently led by aristocrat-turned-communist Felix Dzerzhinsky. After 1922, the Cheka underwent a series of reorganizations into bodies whose members continued to be referred to as \"Chekisty\" (Chekists) until the late 1980s.\n The Renovationist Church (Russian: Obnovlencheskaya Tserkov), a reform movement supported by the Soviet government, was a federation of several reformist church groups that took over the central administration of the Russian Orthodox church in 1922 and for over two decades controlled many religious institutions in the Soviet Union.\n Seeking to convince Soviet authorities to stop the campaign of terror and persecution against the Church, Metropolitan Sergei sought means of peaceful reconciliation with the government. On July 29, 1927, he issued his infamous Declaration, in which he professed the absolute loyalty of the Russian Orthodox Church to the Soviet Union and to its government's interests. The Declaration, albeit well intentioned, sparked immediate controversy among Russian churchmen, many of whom (including many notable and respected bishops in prisons and exile) broke communion with Sergei. Later, some of these bishops reconciled with him, but St. Athanasius and many other bishops remained in opposition to the \"official Church\" until the election of Patriarch Alexei I in 1945.\n Boris Aleksandrovich Turaev (Born July 24 [Aug. 5 O.S.], 1868 \u2013 July 23, 1920, in Petrograd (St. Petersburg). Russian Orientalist. Founder of the Russian school of the history and philology of the ancient East. Seminal figure in Russian Egyptology and Ethiopian studies. Academician of the Russian Academy of Sciences (1918; corresponding member, 1913).\nTranslated from Russian by Sophia Moshura.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I'm officially confused: The new iDrive in the 5 and 6 Series??\nI'm officially confused: The new iDrive in the 5 and 6 Series??\nFor the past weeks, BMW has been reinforcing the fact that the new iDrive Navigation System will not make its way into the 5 and 6 Series, but it will be available in the 1, 3 or 7 Series. Top level BMW executives confirmed that for now, no Car Infotainment Computer in the 5er and 6er.\nBut\u2026.yesterday, in a press release issued before the Paris Auto Show that starts next month, BMW dropped the bomb, but maybe it was a surprise just for us.\nThe new generation of BMW iDrive standing out in particular through the newly designed Controller including direct selection buttons, an optimised menu structure and innovative display technology, is being introduced in parallel both in the new BMW 7 Series and in the new BMW 3 Series, and is also featured as of autumn 2008 in the BMW 6 Series, the BMW 5 Series, and the BMW 1 Series.\nThe M5 Saloon, M5 Toruing, M6 Coup\u00e9 and M6 Convertible also come as standard with the new generation of BMW iDrive. Extra safety, in turn, is offered on all four models by the new exterior mirrors covering an even larger area of vision. And last but not least, a new metallic color included in the range is Carbon Black.\nI do remember seeing the online configurator for the 2009 models, and the new iDrive was definitely being featured there, but since it was never mentioned in the 09 ordering guides released by BMW, I figured that it was just an error.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We actually spent hundreds of millions more on candy and costumes than on deciding who will be the president.\nAmericans are often told that there's too much money in politics \u2014 specifically that we spend too much on campaigns. But that claim doesn't hold up when you consider that in our elections, we are choosing the commissioners and council members, mayors, governors, members of Congress and president who lead the local, state and federal governments of the most powerful country in the world.\nDuring the 2012 presidential election cycle, the Federal Election Commission reports that candidates, political parties and political action committees raised and spent a little more than $7 billion. That is a relatively small amount when compared to other types of spending.\nCandidates spend on radio and TV ads, mail brochures, and personal appearances just as American companies spend to advertise their products and services. The candidates are advertising their views and opinions on the public policy issues we face as a nation, and how they propose to solve those problems and lead us to a better quality life. And political parties and PACs engage in similar advertising spending with their public support or opposition to candidates.\nIn 2014, total U.S. advertising spending was just more than $141 billion, according to Kantar Media. So the amount spent in the 2012 election was only about 5 percent of the total advertising dollars spent in the United States to sell everything from cars to diapers. In 2014, the National Retail Federation estimates that Americans spent $7.4 billion on Halloween. So we actually spent hundreds of millions more on candy and costumes than on deciding who will be the president of the United States.\nVirtually all of this money was publicly reported. America today has the most complex and comprehensive reporting requirements in our history. Candidates, political parties and political action committees are required by law to report all of the money they raise and spend on politics, and that information is easily available on the website of the FEC and equivalent state agencies.\nEven nonprofit advocacy organizations, whose major purpose is social welfare and not politics, have to report spending on political ads to the FEC when they support or oppose a particular federal candidate based on an issue important to their members.\nThe myth of \"dark money\" supposedly corrupting our politics is just that \u2014 a myth. First of all, the amount spent on direct political ads by nonpolitical nonprofits like the NRA or Planned Parenthood is a tiny percentage of total election spending. Of the more than $7 billion spent in the 2012 election, the FEC reports that only $300 million was spent by nonpolitical organizations on independent political expenditures. That is only 4.3 percent of the total.\nAnd do you really need to know who the donors are to the NRA or Planned Parenthood to judge the credibility of an ad they run about a particular candidate? Of course not. The real purpose of those who want to force such organizations to reveal all of their donors is to intimidate and harass contributors whose support is vital to organizations the critics don't like.\nFears of \"foreign money\" influencing our elections are also baseless. Federal law bars foreign companies and individuals from participating in American elections. They cannot contribute to candidates, and they cannot engage in independent political spending. In fact, when the U.S. Supreme Court threw out the federal ban on independent political spending by companies and unions as a fundamental violation of the First Amendment in the Citizens United decision in 2010, it specifically noted that the ban on foreign companies was not affected.\nCorruption is against the law. When government officials accept bribes or misuse campaign funds, they are prosecuted. Just look at the successful ABSCAM prosecution conducted by the FBI and the U.S. Justice Department in the 1970s and early 1980s. Or look at the conviction of former Congressman Jesse Jackson Jr. and his wife in 2013 for fraudulently using campaign funds for personal expenses.\nBut that type of bribery and corruption should not be confused with legitimate campaign donations made by Americans to the candidates and political parties that they favor and associate with based on their values, principles, ideas and character. We donate to the candidates we like and who we believe will do the things we think need to be done when they are elected to office. That is not corruption \u2014 that is the democratic process at work.\nHans A. von Spakovsky is a senior legal fellow at The Heritage Foundation and a former commissioner on the Federal Election Commission. Along with John Fund, he is the co-author of \"Who's Counting? How Fraudsters and Bureaucrats Put Your Vote at Risk\" and \"Obama's Enforcer: Eric Holder's Justice Department.\" He wrote this for InsideSources.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Currencies: Pataka (MOP) = 100 abo.\nMacao was established December 20, 1999 after the abolition of the Portuguese colony of Macau. Macao Special Administrative Region or Macao - an autonomous territory in the People's Republic of China. The former Portuguese colony. Macao - the port, a major financial center. Development of light industry, gambling brings the state budget to 40%. Macao is a little area of ??gambling in Hong Kong. Developed for tourism as the country makes a profit.\nThe population of half a million people. The official languages ??of Chinese and Portuguese are also used by Cantonese.\nIn circulation are banknotes in denominations of 1,000, 500, 100, 50, 20 and 10 patacas, coins of 10, 5, 2 and 1, Pataki, as well as in 50, 20 and 10 of the ABO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Malcolm Dent Digital Media Ltd is the name of the company that we work under. I, together with Carol my wife and Chris my son film and produce Travel and commercial DVD formats around the world. We also duplicate in all formats for our many worldwide clients in both PAL and NTSC, and design and host websites for clients, including streaming video.\nNever have I been more envious of BBC cameramen with enormous zoom lenses as when I am trying to capture my subject on a 12 to 1 zoom capability. I would trade most of the useless extras that manufacturers seem convinced that the purchaser wants, for a more powerful optical zoom lens.\nWhile having dinner with Justin the manager, he mentioned that he needed a video of Pinewood to help attract people to his delightful resort. I explained that I only had a small camera with me, but the place looked so good that I was sure I could produce enough to help him.\nLittle did I realise then what a course of events would lead from that dinner. I sent the rest of our holiday as I usually do, getting up at sunrise to film the shore creatures, then travelling to various points of interest to gather material for the video. We then returned to England and I edited the film and sent it back to Kenya for his verdict.\nDuring our second filming trip to Kenya, I linked my home town of Gosport in Hampshire through my Rotary Club with a project run by the Mnarani Hotel, using local people and paying them to produce desks for the local school. Many hotel guests sponsored desks which had their name carved on them.\nHappily, he was delighted and ordered 500 copies to distribute to his agents around the world.\nWhat made our video stand out was the attention to detail with the wildlife, as I would spend hours trying to film a crab coming out of it's sandy hide-hole on the beach, or remain motionless by a tree trying to capture a Humming bird (I have wasted more video on Humming birds than on any other subject, and never been happy with the end result).\nSo, I had finished the project, had a happy client, but I was left in a void, for I felt that the video would be a useful selling tool, but to whom? Unusually for me I had a flash of inspiration. We had arranged the holiday with Tropical Places, who had looked after us through their Kenya rep extremely well, so instead of sending them a thank-you note, I sent the video. You can imagine my amazement when the CEO rang me up to congratulate me on the video and to ask if we would like to visit some of their locations and make films for them. Over the next 5 years we travelled around the Caribbean making videos of the Islands, using the same format that had worked so well in Kenya.\nThey just arranged all travel and accommodation, and it was left to me to film and edit as I saw fit and then give them the final edited video, which, apart from very minor changes was just as I had finished it.\nCarol accompanied me on several of the trips as I insisted on 2 weeks per location in case of overcast or inclement weather, and when underwater footage was essential, my son Chris came so that we could split our time and double the footage.\nWe shot on average 12 hours of DV footage on our VX1000\/2000 cameras and I cannot praise them highly enough. Small enough not to attract attention but high quality enough to give sparkling results, it is the still pictures from our video footage that populate these webs.\nFor the future, we are awaiting our next journey of adventure with our travel partners with anticipation, as they have broadened our horizons to the world through our video work, and we look forward to many more years providing the public with informative, honest appraisals of holidays around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Skap\u00e1n\u00ea talks about the future of Big Data and Data Processing, followed by a demo of OpenIO SDS and Grid For Apps for image Recognition with Google TensorFlow.\nAmita Potnis from IDC explains why high performance object storage will change the storage landscape.\nInternet Initiative Japan uses OpenIO object storage for the all-flash infrastructure of their Secure MX Service.\nA whiteboarding session illustrating OpenIO SDS internal components and how they work.\nA practical and cost-effective object storage infrastructure based on SoYouStart dedicated ARM servers and OpenIO SDS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Industry experts will teach attendees to differentiate between LAFD Chief's Regulation 4 and Title 19, and how to prepare for required testing and inspections.\nMake sure your building is Reg 4 and Title 19 compliant.\nFind out what is required for the annual and 5-year testing and inspections of the fire protection systems and equipment.\nUnderstand what The Compliance Engine is and how it works.\nLearn Best Practices to prepare your building and tenants for Reg 4 and Title 19 testing and inspections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PHONE TOUCH (UK) LIMITED is a Private Limited Company from HILLINGDON and has the status: Active.\nPHONE TOUCH (UK) LIMITED was incorporated 15 years ago on 29\/04\/2003 and has the registered number: 04748490. The accounts status is TOTAL EXEMPTION FULL and accounts are next due on 31\/12\/2019.\nRT OFFICE CLEANING SERVICES LIMITED UXBRIDGE Dissolved... TOTAL EXEMPTION SMALL 96090 - Other service activities n.e.c.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renault Duster was launched in India since mid-2012, and has remained very helpful in sailing its brand peacefully in the country. The Delhi Auto Expo 2014 gave it a fresh lease of life with an Adventure Edition on the list; and to be honest, it's only the cosmetic makeover of the existing Duster, which is planted on the top-of-line trim with 110bhp variant. However, we found out the French carmaker is revving up Adventure Edition hard during the TVC, and it seems they wanted to get it over the Nissan's Terrano for retaining their hold in the category as soon as possible. So, the events like Desert Odyssey in Rajasthan and other such under the hat of 'Gang of Dusters' are the answers to keep the Renault's compact SUV aspirants in momentum always, and also to pep them with what tricks they can do with their vehicles, just like the Mahindra with the Thar (but Thar is really a capable off-roader) in off-roading.\nHowever, after the Delhi Auto Expo 2014, we got a chance to drive the Duster Adventure on the tarmac of India's capital city Delhi, but for a very shorter distance.\nAt that time weren't been able to take photos, as it was near to the restricted photography area we mean VVIPs residences, so our crew decided of not risking our time in clicking some shots there, as residents of Delhi are not like Bombay, we mean not so understanding. Starting from the front, or you can say the whole of exteriors, it has blackened alloys and headlamps, plastic guard all around the vehicle, and nothing else to elaborate. Hope so, those who had went to the Renault showroom prior reading to our review, must be knowing those said cosmetic gimmicks are available in the form of accessories; it's no exclusivity buying this Adventure Edition as an whole package. But despite adding those bells and whistles Duster Adventure not seems to be a complete makeover, something or the other had always felt unfulfilled.\nMoved inside, the broad pricing daylight of Delhi has made us realize, the cabin was not subtle with the outer paint shade. Inside was draped with black in the main theme, with fluorescent contrasting bits sprinkled all around. Hopefully, that color scheme maybe looking best on the Duster in black paint shade, but the Moss paint shade on outer was greeted as a stranger to them. I must say that, the fit and finish were upto the mark, but weren't helping a lot either.\nDrive was as same as the existing 110bhp Duster.\nHence, those who were fixed choosing between the flashy yet muscular brat Nissan Terrano and the sophisticated lad Renault Duster, they now have an option to go sophisticated as well as wild at the same time in Duster Adventure Edition. Priced at 11.7 lakh (ex-showroom, Delhi) it is almost near the neck of Nissan Terrano top-end trim. Let's see how it is going to help the Duster crazy Indians, but we are waiting to try our hands on the Duster facelift that is speculated to come to India somewhere by 2015.\nRenault Duster is been just rolled out with facelift. And we also reported about the pickup variant of same doing test rounds on Romania yesterday. Claiming more of the excitement, some source now released a video on internet which shows off the Renault Duster of 2014 Dakar Rally.\nDakar Rally is one of the most extreme rallies on the planet. And to bear its effect a team from Argentina has built up a 'monstrous Duster'. The vehicle is laden with number of add-ups, suspension kit, a modified body and ultimately a \"V6\". But there are no confirmations about the technical specifications of Duster.\nRenault Duster has been successful in the past Dakar rallies too; we bet this new Duster for 2014 Dakar will be no lesser than 300hp.\nRenault Duster facelift is rolled out with LED headlamps, new front grille, and other cosmetic treats but the French has forgotten to load an automatic transmission on it.\nMeanwhile, do not expect the Renault making this devious Duster rolling into production. It is a custom built by one of the trams for 2014 Dakar Rally.\nGoing to get the Nissan's version of Duster very soon, Renault had started testing the Duster facelift to go boomerang on it by next year or so. Duster had debuted in international market by 2010 and came to India just last year itself. In spite of being termed for this so short time, Duster had did great jobs for the organization wherever it was rolled out. But now it is in the schedule for many of the markets to receive its facelift, who are quite bored looking the same for long three years. And here for them we present, the new \"Duster facelift\" that is recently spied testing in Southern Europe going under the heat test, which is expectedly to go previewing 2013 Frankfurt Auto Show.\nThe posted photos are slightly camouflaged, so only minute details can be revealed. The front is almost covered, where most of the changes are going to take place.\nRear of the vehicle is expected to be same, but the interiors will receive updates as brief as possible. In order to remain cost effective, there will be no more of the heavy pegs comparison for the world's cheapest SUV, but hopefully the quality that it is going to ally here for the new generation will be of ample satisfaction.\nSo till the new French wine arrives to Frankfurt paddock, we can wait and bite our nails annoyingly for the cost of our most loved yet cheapest off-roader to remain in the respective plethora itself. Hopefully, it will not be hiked, because the Nissan is also planning a one with niche interiors that will be rolled out by the end of this year, so Duster can now enjoy those unaffected overwhelming sale numbers too.\nRenault is very much popular for its low cost brand Dacia in Europe, and SUV Duster as the most prominent face worldwide.\nThe Renault's SUV Duster is also a bread butter of this French carmaker here. Though, nowadays it is doing better than competitors but rivalries in SUV segment are getting so harder than one would have thought of, that's why automakers needs to be at a core of dynamism for maintaining the striking balance. To get a clear-cut pie of market share, Renault is preparing to do with a facelift of its world's cheapest SUV at Geneva Motor Show this year.\nHowever, presently Ford is grossing all the media hype with EcoSport, but is still on a shaky ground about pricing, that's because, it is the only factor which had laid them back in past launches.\nDuster seems in full swing as crossed the 400,000 production mark recently.\nBiting more of crispiness, facelifted French offering is expected to sport new front grille, refreshed headlamps, tweaked interiors and exclusively, a bulge of more equipments. It is also heard that Logan MCV will too be sharing the same Swiss paddock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Use these prepaid calling cards for calls from Germany to Germany Frankfurt. Below rates are available only to calls to Germany Frankfurt from Germany. You can use these cards to make call from Germany Frankfurt to another destination. Look reates here or use our search form (below) for cheapest calling rates search.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rather than the usual menu holder in restaurant tables, how about an advertising player with charging port? Choose to display static printed graphic or have a 7 inch screen (or dual screen) that play any of your advertising media. This table top display not only plays your advertisement, it also allows your customer to charge their mobile phone while having their meal or drinks.\nUse our free amazon cloud server to upload your contents anytime, anywhere! The player runs on Android 4.4.2 and you can install any apps of your choice. Play video files or scrolling menu or promotion images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Baking soda isn't just for baking anymore. It's been used for years as a cheap, natural remedy for heartburn, teeth whitening, calluses, and so much more.\nBut now, scientists may have found another new amazing use for this household substance: combating autoimmune disease.\nPreliminary research from the Medical College of Georgia at Augusta University published in the Journal of Immunology shows that baking soda, which is found in almost every home, may promote an anti-inflammatory environment beneficial in the face of inflammatory diseases like autoimmunity.\nThese mesothelial cells contain microvilli, little \"fingers\" that sense invaders in the body and alert the organs to produce an immune response. Basically, the baking soda told the spleen to calm down that response.\nTo understand the importance of this further, let's look at further connections between the immune system and inflammation.\nAutoimmunity refers to when the immune system turns against its own healthy tissues and cells, leading to a negative immune response, or autoimmune disease.\nThese types of immune responses cause harmful inflammation in the body. At the same time, lifestyle factors that promote inflammation, such as a diet full of refined sugars and carbs, may be the root of some inflammatory autoimmune conditions.\nThis is why reducing inflammation and cooling down the immune response can provide relief.\nIn the case of this study, it was found that after two weeks of drinking the baking soda-water solution, there were changes in certain immune cells called macrophages. These cells help \"clean up\" debris from dead or damaged cells and are often seen early on in an immune response.\nThe researchers began to see less M1 cells, which promote inflammation, and more M2 cells, which reduce inflammation.\nThey also found these anti-inflammatory effects were happening in other areas too, including in the kidneys and blood circulating through the body.\nIn addition, there was a positive shift in other immune cells prevent the immune system from creating a response. This was found to last at least four hours in humans after the baking soda was consumed.\nKeto diets rich in healthy fats and very low in inflammatory carbs and sugars can help regulate inflammation in your body, which is why many with autoimmune disease turn to the Keto Zone diet for relief.\nTo learn more about the benefits of the Keto Zone for autoimmune disease and overall health, pick up a copy of the book here.\nA common dosage recommendation is adding a fourth of a teaspoon of baking soda to a glass of water. This can help to reduce stomach acid, but it's important to remember that not all indigestion is caused by an overproduction of acid so if you still have symptoms two weeks later, speak to your doctor.\nBaking soda is a great tool for cleaning and removing stains, but there are many health benefits associated with the use of baking soda, too. Sodium bicarbonate is sometimes used as a supplement because it provides dietary bicarbonate. When taken orally, baking soda can raise serum levels of bicarbonate.\nBaking soda is known to help to neutralize acid and improve pH balance in the body. Baking soda is often used internally to quell digestive dismay such as acid reflux or heartburn. When these complaints are due to overconsumption of acidic foods or a generally acidic state of the body, slowly drinking some baking soda in water can help to neutralize the acid and get your body's pH back to a better place.\nBaking soda has been shown to kill off bacteria including Streptococcus mutans, which is a type of bacteria associated with tooth decay. Baking soda is also effective against various fungal groups including yeasts, dermatophytes and molds that cause skin and nail infections in humans.\nResearch reveals that baking uses include the promotion of kidney health. A clinical study published in the Journal of the American Society of Nephrology looked at the effects of sodium bicarbonate on 134 patients with chronic kidney disease (CKD) and low blood bicarbonate levels.\nWhat did they find? The subjects who supplemented with bicarbonate tolerated it well and were significantly less likely to experience rapid progression of their kidney disease. Additionally, there were less patients that developed end-stage renal disease (ESRD) in the bicarbonate group compared to the control group. Overall, the researchers conclude, \"This study demonstrates that bicarbonate supplementation slows the rate of progression of renal failure to ESRD and improves nutritional status among patients with CKD.\nAccording to the CDC, urinary tract infections (UTIs) are one of the most common infections and as Mayo Clinic points out, women have a higher risk of developing a UTI than men.\nOverall, baking soda appears to be an easy, inexpensive way to improve UTI symptoms accompanied by acidic urine with little to no unwanted side effects.\nA scientific article published in 2013, titled \"Practical considerations for bicarbonate loading and sports performance,\" points out that studies to date have demonstrated that taking sodium bicarbonate before exercise (also known as bicarbonate loading) may have \"a moderate positive effect\" on athletic performance that includes one to seven minutes of sustained strenuous exercise. In addition, sodium bicarbonate may also be helpful for prolonged physical activity involving intermittent or sustained periods of high-intensity training.\nAnother small clinical study of eight healthy male subjects found consuming baking soda before intermittent cycling improved their sprint performance.\nChemotherapy side effects as make the list of baking soda uses for health. If you or someone you know has gone through chemotherapy, then you probably already know how bad the side effects of this cancer treatment can be. For example, undesirable changes to the mouth and throat can occur in some patients.\nRinsing with a baking soda mixture daily can help to improve these unwanted side effects. Combine a fourth of a teaspoon of baking soda, an eighth of a teaspoon of sea salt with one cup of warm and rinse your mouth three times per day. Each time, follow the baking soda salt mixture with a rinse of just plain warm water.\nThese are just some of the many possible health benefits of baking soda! In the following section, you'll learn about even more health uses and benefits of this amazing natural remedy.\nInsufficient enzyme activity in your system also causes other negative effects on your body. Because enzymes play essential roles in all of the systems of your body (in addition to the digestive system), you need enzymes to keep all of your organs, muscles, and tissues functioning. Your body, however, has only a limited number of enzymes to do all of these jobs.\nThe layers of probiotics in your intestines actually work with your immune system by preventing bad bacteria\u2014such as salmonella, fungi, Candida, and other nasty disease-causing organisms\u2014from taking hold in your intestines and making you sick. In fact, probiotics are your first line of defense against disease.\nI became ill. Nothing seemed to be working right in my body. As I put stuff in it, it came right back out. Within about a three-week period I lost 20 pounds, was terribly dehydrated and so weak I could barely get out of bed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delve into the wonderful world of blinds with these images of vertical blinds, metal blinds, wood blinds, and faux wood blinds in San Antonio, TX. And, view how our custom blinds can look in your house from these sample images. Then, contact us at 210-876-0377 to schedule your free in-home design consultation today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This song is which album releted plz tell me i am needed this song i like it .it wonder full song.\nI think it is a point of view. I usually meet people who rather say what they suppose others want to hear. Good and well written! I will come back to your site for sure!\nThank you a lot for your great article. I have been searching for such content for a really long time. Not everything is completely clear to me, but it is definitely interesting and worth reading.\nThank you a lot for your great post. I have been searching for such information for a very long time. Not everything is fully clear to me, but it is definitely interesting and worth reading.\nregion without having to encounter large crowds at ports along the way.\nfor revisiting. I wonder how much effort you put to create this kind of fantastic informative web site.\nthis. ill be checking your rss feed so i don't miss out the good things! again, terrific web-site remember to keep this up! Please pardon me if my english is not good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Color:Multicolored Enjoy a rich provincial flavor in this stylish accent table. Built in a fetching half-round shape, its ornate sculpting and accent detail ratchet up the decadence factor. Rich brown offers a finish that's both warm and versatile. Traditional turned legs with an attached bottom shelf expand its utility and visual elegance. Make this accent table a fixture in an entryway or a main living space.\nWe sell a big quantity of Entry Table with Curved Front and Inlay Shelf Brown in our GRTi LLC, so that you may buy the best option for you.\nIf you don't want to purchase the new Entry Table with Curved Front and Inlay Shelf Brown by big price, you can order product from our shop.\nIf you want to order Entry Table with Curved Front and Inlay Shelf Brown in the United States and other countries and get the products of best quality and relatively low price, you can make the good choice by visiting our shop. You do not need to search for store, because everything that you may need as a customer is easy to find in the catalog of products on the website of GRTi LLC. We offer the best Entry Table with Curved Front and Inlay Shelf Brown and the best service to the residents all over the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yates Scott shares his long time recipe for homemade vanilla extract. He hasn't used anything else for years. He says it makes the world of difference in baking.\n* I think it would make a great homemade gift as well.\n5-6 good quality vanilla beans used per 1 cup of alcohol.\n2. Put your vanilla beans in your glass bottle or canning jar and cover with vodka. Smaller batches might have to fold your vanilla beans in half. Always make sure the vanilla beans are submerged in alcohol.\n3. Close the jar or bottle and store in a cool, dry place for 8 weeks. Give the bottle a shake once or twice every week or so. As you use the vanilla, simply add in more liquor to replace what you have used. If you feel you need to, then simply replace the vanilla beans to strengthen the flavor of your extract again. Usually I switch them out after about 6 months if i'm baking a lot...if not, they last a long time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It's now been about 10 years since Lehman Brothers filed for bankruptcy, which was the largest in U.S. history and is widely considered to be the start of the 2008 financial crisis and Great Recession.\nAlthough a decade has passed, the financial crisis had a deep impact on millions of Americans. In fact, many Americans still behave differently with their finances and feel differently about certain things than they did before the crisis. Here are five lingering effects of the financial crisis as revealed in a recent NerdWallet survey.\nAccording to the survey, nearly three-fourths of Americans say that they've changed their financial habits as a result of the financial crisis. Even 10 years later, 46% of Americans say they're more cautious about spending, and 38% say that they avoid debt as much as possible as a direct result of the crisis.\nTo be clear, this is a good lasting effect of the financial crisis. In too many cases, spending habits and Americans' attitudes toward debt had become dangerously carefree in the years leading up to the crisis.\nSeventy-six percent of Americans say that the financial crisis is responsible for financial worries they still have today. For example, about one-third of Americans say they worry more about their lack of an emergency fund. Experts generally suggest that Americans should try to build up six months' worth of expenses in a readily accessible place. Recent data indicates that about 41% of Americans couldn't come up with $400 without going into debt, so it's no wonder this is a common worry.\nIn addition to worrying about their emergency fund, many Americans say that they now worry more about their retirement savings (or lack thereof) as well as their credit card debt.\nThere has always been a significant amount of people who are scared of stocks. The stock market crash of 1929 left lingering fears and more recently, many people lost money or knew someone who got burned investing in dot-com stocks in the late 1990s and early 2000s.\nHowever, the financial crisis produced such a wide-reaching stock market plunge that it scared even more Americans away from stocks. At the market's March 2009 low point, the S&P 500 had shed more than 56% of its value.\nEven so, it's important for all Americans to realize that a diverse portfolio of stocks remains the best way to create wealth over the long run. The stock market has historically delivered total returns of about 10% annually over periods of a few decades.\nStill not convinced? Consider this: If you had invested $10,000 in an S&P 500 index fund on Oct. 5, 2007, the market's peak before the financial crisis and arguably the worst possible time to invest in recent history, your investment would be worth more than $23,400 today (including reinvested dividends). In other words, not only would you have recovered all of your losses, but you would have more than doubled your original investment.\nAlmost half of Americans (49%) surveyed say that they're less trusting of national banks than before the crisis. In some ways, this certainly makes sense. After all, as a whole, the banks were behaving badly and ended up needing a taxpayer-funded bailout.\nOn the other hand, while there's certainly the case to be made that banks can't be trusted to always do the right thing when it comes to risk management, there's no reason not to trust banks with your money in checking and savings accounts. FDIC insurance guarantees that up to $250,000 in deposits are safe per depositor, per bank.\nHere's one that's not surprising. In fact, I was more surprised that only 62% of Americans say that their perception of buying a home has changed as a result of the financial crisis. After all, the mortgage industry now is practically unrecognizable in many ways compared to back then. I (thankfully) never took advantage of one, but I clearly remember being offered NINJA (no income, no job, no assets) loans on several occasions.\nSpecifically, many Americans say they do not trust mortgage lenders to act in their best interest, despite reforms such as the Truth in Lending Act, and many more don't trust real estate agents. And nearly one-third of Americans say that their definition of \"home affordability\" has changed as a result of the crisis.\nThe bottom line is that even a decade later, the financial crisis still affects Americans in several ways. Some of these are certainly good \u2014 for example, I already mentioned how Americans as a whole needed to get a bit more conservative when it came to taking on debt.\nHowever, some of the effects of the financial crisis could be doing the younger generations a disservice \u2014 specifically the fear of the stock market. Too many millennials believe that cash is the best place for long-term savings, and this could rob them of hundreds of thousands of dollars throughout their lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Immulab was formed following the acquisition of Seqirus IH by Paragon Care. Paragon Care is an ASX listed Australian company that specialises in diagnostics, acute, primary and aged healthcare. The heritage, expertise and over 70 years of experience in the manufacture of IVD's has transferred through to the new entity \u2013 Immulab. Paragon Care is committed to investing in Immulab to ensure it realises and exceeds its full potential showcasing \u2013 innovation, outstanding quality and superior service.\nThe Immulab business specialises in the production of Reagent Red Blood Cells (RRBCs), monoclonal blood grouping reagents and ancillary products for immunohaematology laboratories, and we are the exclusive Australian and New Zealand distributor of Immucor\u00ae automated analysers. We also distribute and support the Immucor range of reagents, BioArray Genotyping and Blood Bank automation.\nOur team is focused on providing high quality, reliable products and technical excellence for all of our customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High-quality Amplifier dust cover for BUGERA 333 XL INFINIUM AMP.\nHigh-quality Amplifier dust cover for BUGERA 6262 AMP.\nHigh-quality Amplifier dust cover for BUGERA V 22 COMBO.\nHigh-quality Amplifier dust cover for BUGERA V 5 COMBO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The project's feature is triple glazing units with two Low E glasses provided the highest energy efficient rate and light transmission. Both outer and inner panes are laminated which eliminates any optical distortions on the fa\u0441ade without compromising of its safety level.\nGlass-BIPV product: facade panels with integrated PV cells. in total there are about 56.35 kWp installed on this power plant. Bespoke appearance was achieved by tailor made bus bars which were covered with black 3M tape and became virtually invisible on completely homogeneous black modules.\nDouble glazed stepped units for structural glazing.\nThe glazing is performed by double glazing units of high safety level: both outer and inner panes are laminated and heat strengthened.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our first home on Sixth Lake in Inlet, NY was a small, (under 3 acres), parcel of land on which to manage horses. However, for 18 years we managed up to 4 horses and six goats rather creatively and efficiently on our tiny property. During that time, I could never have imagined another place on Earth that I would rather be than on our beloved Lakeview Farm. Rod had always envisioned a larger and more appropriate homestead on which to manage livestock, but it never had occurred to me that the time would come when we would indeed leave the home that had realized my childhood dreams and granted all of my wishes.\nIn today's excerpt from Finding My Way to Moose River Farm, I come to terms with the fact that its time to let go of Lakeview Farm and dare to dream bigger.\nI don't know when the epiphany occurred but suddenly it became clear to me what Rod and I needed to do. It was time to leave Lakeview Farm and begin another phase of our lives with the animals. As clear as this revelation was to me, was the lack of knowing where this phase would begin. What did it look like and how would we go about achieving it?\nI shared my thoughts with Rod when I returned home, (from visiting a friend), a few days later. He had always dreamed of living on a larger property with land on which to manage and create. He seemed to be waiting for me to prepare myself to say goodbye to Lakeview Farm. Although, he had lovingly restored and doubled the house in size, it was not his dream house. Rod had no attachment to Sixth Lake either. He viewed it as an eighteen year investment that would provide us with the capital to develop a larger piece of land. As the more sentimental of the pair, I saw Lakeview Farm as my dream come true. It had given me what I always wanted, horses right outside my kitchen window.\nMy Dream Barn started with 2 stalls that I could see vividly from the kitchen window. The flower box under Promise's window added the perfect touch of color every summer.\nThe view of Sixth Lake from our dining room window.\nThe 'barn side' of our house was brimming with color every summer.\nEasau had his picture taken on the lakeside lawn.\nOn hot days, Noah could be found burrowed in the cool wet sand of our beach.\nSpike enjoyed 'scuba diving' in our tub.\nThe newly excavated riding ring, (1988), complete with newly constructed jumps enabled me to begin my riding lesson business. (In the distance, Ludie investigates the sand footing.).\nLudie and Noah snooze in the sun room while Casey preens her feathers.\nOlivia, Holly and Christopher strolled down to the lake every morning.\nLouise, Christopher, Noah and Ludie hanging out on the lawn.\nRachel and Hannah watch the goings on from their secure paddock.\nAn Evening at Moose River Farm was presented as a riding recital last Monday night. Nine young riders demonstrated their equestrian skills for an audience of about thirty spectators. In addition, the 2012 MRF Quadrille rode an encore performance of their drill from Hoofbeats in the Adirondacks 2012 earlier in July. Carin Mei also returned from Hoofbeats to informally demonstrate the upper level Dressage abilities of her two Iberian geldings, Traquejo and Voltaire.\nUnder a perfect summer evening sky, the horses and riders showed off an array of exercises that have helped establish their balance in the saddle.\nPhotos by Don Allen and Michele deCamp.\nJean welcomes the crowd and gets the program rolling.\nThe 2012 MRF Quadrille Team rode to selected music from Grease.\nCourtney and Jessica rode this one out on account of injuries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Body-armor manufacturer Ceradyne, which has plants in Lexington, said Tuesday it plans to close its manufacturing plant in Bazet, France, in a quest to cut costs.\nCalifornia-based Ceradyne said it will transfer the Bazet plant's industrial ceramics production to a plant in Kempten, Germany. That portion of the company produces ceramics that end up as part of new construction projects, which have slowed considerably since the recession began.\nCeradyne plans to close the plant by the end of the year and cut its 95-person work force. The company noted the possibility of that during its earnings report in April, saying then that it has cut its U.S. work force by 33 percent since the start of 2008. In Lexington, it has laid off 125 temporary and full-time workers, reducing its head count to 200.\nThe company said the French plant's closing and severance packages will cost between $11 million and $13 million. It expects the majority of these charges to be accrued during the second quarter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The incident took place near Nagan Chowrangi when unknown armed men opened fire at an unregistered vehicle that resembled police mobile.\nWounding the two men and a minor girl sitting inside the vehicle seriously, the assailants fled the scene. The injured were immediately whisked to hospital, however they all succumbed to deadly wounds on the way.\nThe deceased men were identified as Abid and Ateeq, and the girl as Tasbia.\nPolice said that Abid was a city warden.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This online casino is powered by Microgaming and this means that players will only find high-quality games when they access the website of Wild Jack. This software provider is known for creating top-of-the-notch slot machines and table games, so your experience with this casino will be amazing, for sure.\nWild Jack Casino got a license from Malta Gaming Authority which means they have the legal right to operate in many regions, so it is totally safe to play casino games on their website if you don't live in one of the restricted countries, of course. To avoid any potential problem, before you create an account in this gambling place you need to read the terms and conditions carefully and to check the list of restricted countries to be sure you are able to legally place bets here.\nAlthough it doesn't offer as many payment services as other casinos, Wild Jack manages to cover the needs of a big number of casino games players by allowing them to use the most popular systems for deposits and withdrawals.\nPeople who speak at least one of these languages will be able to understand every single word that's written on the pages of this online casino.\nHigh rollers get a special treatment when playing casino games in this gambling place and their benefits are not limited to the online medium. The brand that handles the VIP program of Wild Jack is Fortune Lounge, which offers players all kinds of great rewards, from exquisite dinners to vacations in beautiful resorts. Among the benefits that are exclusively offered to VIPs, this gambling place also includes special tournaments, bigger match bonuses and other incentives.\nWhile the tournaments are only available to members of the VIP club, other promotions apply for all types of players. From time to time, lucky people who play casino games in Wild Jack can win a fine dining experience worth of $15k. Pay attention to the 'promotions' page, as all the special promotions are listed there when they become available. You'll be able to get free credits, cashback or reward points which will help you win more without spending more money.\nIf you have any question or issue related to your experience with this gambling place, you can get in touch with a representative by either using the live chat option or by calling them or sending an e-mail. Before you decide to get in touch with them, take a look at the FAQ page, as this might solve your problem faster.\nWith great games provided by a leading software developer, many payment methods accepted and lots of other positive characteristics, this online gambling place is a good choice for any type of casino players. Give it a try and see for yourself!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've just made Queries easier than they've ever been.\nNow even the most beginner of users can set up a brand search in as little as 90 seconds and be confident it's bringing in the same great quality data you're used to.\nThat means everyone in your business can benefit from social insights easier and faster than ever before.\nThe new Query Wizard does all the hard work for you. You simply answer a few basic questions about the brand you want to track and it'll build an accurate search for you. No need to remember operators or work out where the brackets go.\nWe've even included a library of exclusion options so you can remove types of mentions you may not be interested in such as job, real estate and second-hand adverts, profanity or competitions \u2013 and these are available in English, German, Spanish, Portuguese (Brazilian), and French. You simply tick which you want to exclude and they'll be removed automatically.\nYou get all the power of Brandwatch without the complexity of Boolean, and can have a brand search set up in seconds \u2013 even if you've never used Brandwatch before.\nYou might be wondering if this means we're banishing Boolean forever, but we're still big fans \u2013 and still using it behind the scenes to build your Query. We've always been proud to offer the most flexible Query builder on the market, with 23 different Boolean operators on hand to craft the most relevant, accurate search possible. But we recognise that this flexibility and freedom can be a bit overwhelming for new or more casual users.\nThe new Wizard is designed to give you the best of both worlds \u2013 power coupled with simplicity.\nTry it out and see for yourself \u2013 simply click on 'new' in the Query section of the platform and the Query Wizard will guide you through.\nOf course, for those like me who do get excited by a cunningly-placed Near\/nf operator, you can still get in up to your elbows in all the Boolean operators you desire by choosing to build Queries from scratch in the Query Editor \u2013 perfect for more sophisticated searches such as purchase intent or market research Queries.\nIt doesn't stop here either \u2013 we're planning on adding more and more different Query types to the Wizard so you can go beyond just brand searches in the future.\nThis update is part of a wider mission. We believe Brandwatch is the deepest, most flexible platform on the market and we know expert users and data analysts love it. But that's not enough for us.\nWe want everyone across every business to benefit from the value social intelligence can bring, but one of the biggest obstacles is the perceived complexity of analytics platforms from time-poor and new users.\nWe believe the key to making social intelligence more accessible is to reduce the time to insight and make it easy to get to for even the most beginner of users. That means we want anyone to be able to log into Brandwatch and find something that delights within the first 15 seconds \u2013 be it a new insight, a beautiful design, or a feature that feels good to use.\nWe've already released some of our first steps towards this \u2013 such as the Use Case Dashboards for instant insights, and our library of ready-made projects in Insights Central \u2013 and now, of course, the Query Wizard.\nOver the next 12 months and beyond you'll notice an ongoing series of new features and updates all designed to make getting insights quicker and easier for everyone. We're excited; we hope you are too.\nIf you're a client of Brandwatch, you can see the new Query Wizard in action and learn more about Queries in our Training Spotlight sessions \u2013 sign up here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The American Society of Cinematographers (ASC) has announced it's nominees for the 2018 ASC Student Heritage Awards competition, and for the second year in a row Chapman University has a nominee in each of the competitions categories.\nThese awards are designed to inspire the next generation of cinematographers and to help them pursue their dreams. They also celebrate the memory of the Society's most extraordinary members. Each year, the ASC Student Heritage Awards are re-named in honor of esteemed ASC members. Many ASC Student Heritage Award winners have gone on to have successful careers in filmmaking, and several have been invited to be ASC members themselves.\nEligible students must be in undergraduate or graduate school or have graduated within the past year. A jury of ASC members will choose the winners, who will be announced at the ASC Student Heritage Awards celebration event on Saturday, October 13, 2018, at the ASC Clubhouse in Hollywood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Bhubaneswar: The Centre will develop three mega power projects of 4000MW each in Orissa through the competitive tariff prices bidding route, the Union power ministry and the state government agreed.\nOrissa had formed three shell Companies Bedbahal Thermal Power Co, Marthapur Thermal Power Co Ltd and Santarapur Thermal Power Co Ltd under the government-owned Orissa Hydro Power Corp (OHPC).\nThese shell Companies were formed to take advantage of the Union government's policy to allot coal blocks to state-owned shell Companies. The Union power ministry, however, had been insisting that the state should allow the Centre to take over the shell Companies and develop the projects.\nThe issue was discussed at a meeting between chief minister Naveen Patnaik and Union power secretary Anil Razdan.\nIt was decided that the Centre will develop the mega power projects which are separate from the ultra mega power projects (UMPPs) and share half the power generated with the state at the bidding price. The state will provide the land, water, infrastructure and administrative support.\nThe state's energy minister, SN Patro, said the three projects will bring in an investment of Rs 50,000 crore.\nThese projects are not to be confused with the Ultra Mega Power Projects (UMPP) allotted to the state.\n'We are close to start the work on the Orissa UMPP,' the union power secretary told reporters.\nHe said the state has identified land for the project in Sundergarh district. Once the Central Electricity Authority (CEA) gives its clearance, the state would start the bidding process.\nFor the UMPP, the state will be allocated one-third of the power at generation cost.\nRazdan was here to review the progress made in implementation of the Rajiv Gandhi Grameen Vidyut Yojana (RGGVY) under the Eastern Region Power Grid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are 37 top-rated Remediation Services in your area and 43 to avoid.\nBusiness Description: We have been in the asbestos, mold, lead and water & fire restoration for over 30 years. We are fully licensed and insured in all fields of work that we do. We currently except credit cards to help those who prefer to use a credit card. We handle all size projects including small ones in which other companies don't like to handle. Our prices are very competitive but our quality is second to none. We are currently running a promotion of 15% off any project greater than $1,000.00 before discount.\nBusiness Description: Founded in 1989 as a radon mitigation firm serving western NY, Mitigation Tech has grown to become a statewide recognized leader in the design and construction of systems that manage all forms of soil vapor intrusion. Our employees have provided mitigation solutions for radon gas, volatilized chlorinated solvents and other unwanted vapors at over 3,500 residential, commercial and government locations, representing over 8,000,000 sq. ft. of depressurized sub-slabs.\nWe were very impressed with Leahe Development Inc for our mold problem in our attic. After contacting another well known company first and having them not follow through I called Eric at Leahe Development. He was very knowledgeable and provided excellent customer service. Eric came out and sprayed on the day he promised and did an amazing job on our attic. He was very thorough. I highly recommend this company. I am attaching before and after photos for reference.\nAs far as it has gone, it's been excellent. There was no charge. As far as the presentation, it was very professional. Something that I really personally liked was the fact that there is no pressure. They mainly gave out information and left it up to us to contact them.\nJohn was professional and thorough during the inspection. We trusted him immediately.\nJohn looked over every inch of our house and found minor issues and problems we would never have thought to look for. He discovered that the GFCI circuit on the outlets in the kitchen was not functioning - which is something one wants to know, given the proximity of the outlets to the sink! He found signs that the attic isn't getting enough air circulation, which is likely the cause of a minor leak in the roof of the garage.\nWhen John finished the inspection, he compiled an exhaustive report of his findings and took the time to sit with us to explain the results and his suggestions for remediating them.\nWe would absolutely work with John again in the future and will recommend him to our friends.\nA-Plus Cleaning & Restoration Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"We Rented to the Lenins,\" is a sort of 'human interest' anecdote by a Swiss cobbler and his wife who rented space in their house in Zurich to Lenin and his wife, Krupskya, in 1917. As the editors note, this unusual piece for PR contains 'naive and shrewd comments' about the Lenins' domestic life in exile. It also gives us a portrait of Lenin's wife, Krupskya, who was his colleague, carer, and a serious writer-revolutionary herself.\n\"IN ADDITION TO being Lenin's wife which \u2013 by the way, was not accidental \u2013 Krupskaya was an outstanding personality in her devotion to the cause, her energy and her purity of character. She was unquestionably a woman of intelligence. It is not astonishing, however, that while remaining side by side with Lenin, her political thinking did not receive an independent development. On far too many occasions, she had had the opportunity to convince herself of his correctness, and she became accustomed to trust her great companion and leader. After Lenin's death Krupskaya's life took an extremely tragic turn. It was as if she were paying for the happiness that had fallen to her lot.\nNext Article Anatomy of the Popular Front. Sidney Hook. PR, vol. 6, no.3, Spring 1939.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We like to simplify artistic language at the Redlin Art Center. So we've compiled a list of common industry art terms you may need to know.\nWhat is an artist proof? In offset reproductions, artist's proofs are additional prints not included in, but of the same quality as, the regular edition. In original prints, artist's proofs are the first prints pulled, which are the truest prints in the edition because the plates and screens have not yet been worn down. Artist's proofs are distinguished by the abbreviation AP and are numbered separately. They usually represent 10% of an edition and are slightly more expensive than prints in the regular edition.\nWhat is a canvas print? A reproduction in which an image is printed directly onto canvas. View canvas prints.\nWhat is a canvas transfer? A reproduction in which inks are chemically lifted off a piece of paper and applied to a piece of canvas.\nWhat is a certificate of authenticity? A warranty card or statement of authenticity of a limited-edition print that records the title of the work, the artist's name, the edition size and the print's number within the edition, the number of the artist's proofs, and the release date. It is a guarantee that the edition is limited and that the image will not be published again in the same form.\nWhat is a dealer? Galleries, collectible shops or individuals who carry and sell artwork. Authorized dealers are those who, by signed agreement, carry and sell the artwork represented by certain print publishers.\nWhat is a distributor? A person or company responsible for marketing and selling prints and supplying prints to galleries. Sometimes the publisher and the distributor are the same entity.\nWhat is a limited-edition print? A reproduction of an original work of art that is signed and sequentially numbered by the artist. The total number of prints is fixed or limited by the artist or the publisher.\nWhat is an open-edition print? A reproduction of an original work of art that is sometimes signed by the artist. The number of prints published is not predetermined.\nWhat is an original painting? A one-of-a-kind image created by an artist that often sells for several thousands of dollars. Over 100 of Terry Redlin's original oil paintings are on display in the Redlin Art Center and can be viewed in the Collection.\nWhat is the secondary market? An unofficial network of dealers and individuals who buy and sell prints above the issue price after a publisher sells out of an image.\nWhat is the secondary market value? The reported price for a sold-out limited edition is set by supply and demand. Unlike retail prices, secondary market prices can vary from one source to another.\nWhat does signed and numbered mean? Limited-edition prints that have been signed and sequentially numbered by the artist. The artist's signature is usually found in one of the lower corners of the print and is accompanied by a number that looks like a fraction. The top number indicates the number of the print, and the bottom number indicates the total number of prints in the edition.\nWhat does sold-out mean? When all prints in a limited edition have been sold, the edition is sold-out and prints must be bought and sold on the secondary market.\nWhat is a time-limited edition? An edition whose size is established by the number of orders a publisher receives during a set period of time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bhuria (Sunil Shetty) a contract labourer finds a sacred Rudraksh, possessed by Raavan, the demon king and thereby accquires supernatural powers. He tries to establish his rule over the universe but is stopped by Varun (Sanjay Dutt), a yogic who stands in his way. Varun is helped by his Father Pandit Ved Pujan (Kabir Bedi) and a researcher called Doctor Gayathri (Bipasha Basu) in his mission. But will this be enough for Varun to stop Bhuria? The ultimate battle between good and evil begins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I was browsing YouTube the other day, and I came across some video where an English teacher talks about common English errors made by foreign English speakers.\nThe only thing that these videos achieve is the following: THEY FREAK ENGLISH STUDENTS OUT!\nImagine now for a second that you're trying to learn some other practical skill, let's say \u2013 dancing.\nWhat do you think are the chances of you learning to dance successfully if you're constantly BOMBARDED WITH NEGATIVITY?\nThat's right \u2013 next to none!\nThen why are all these teachers hell-bent on talking about mistakes non-stop? To be honest with you, I haven't got a clue!\nThe only thing I can say in relation to this is that they haven't been learning to speak in English themselves for the simple reason that they're all native English speakers and they've learned to speak their native language within their family setting as children.\nI, on the other hand, struggled with my English fluency for a long, long time before I realized what I was doing wrong and I strongly believe it gives me an edge over any native English speaking teacher.\nI have a first-hand experience in what it feels like when you're dealing with negativity as an English student, and I know for a fact that the only way to fluency is by focusing on what you CAN say in English instead of being so focused on MISTAKES!\nHey,,,a million thanks to you Robby sir..\na hats off to you and your precious contribution to this sphere.I believe that if one make his weakness as his strength,he becomes invincible.You are invincible.The way you present your experience and turn them in wisdom pearls .It is commendable.I run short of words to appreciate you.\nI am running my institute where I teach my students communication skills in English.You know I also see eye to eye with you .I wish to exchange a lot of words with you if you give me any platform where i can connect with you,I shall feel myself fortunate.you are requested to tell me please.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are so many little things I'm grateful for\u2013so many that are large, but so many small gifts.\nI'm thankful for the freedom I enjoy that I generally take for granted.\nI'm thankful for my new kale starts. They're beautiful, and I can't wait until the grow up and I can eat them!\nI'm grateful for the freedom to say no when I feel like it. It's such a privileged to turn down jobs that I simply don't want to do because they don't fit my interests or just don't feel right.\nThankful for ginger chicken (YUM!!!) and roasted broccoli for dinner.\nIf I won the breads I would make French toast first. I miss good french toast and haven't made any that was truly good since I had to stop eating gluten.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mostly Spoiler free, but beware.\nOn the morning of the day I was going to see this movie, I don't think I've ever been as nervous about sitting down in a cinema in my life. I hadn't read a single review about this movie. I stopped watching YouTube videos about it weeks ago. I was going in relatively clean. But as I woke up on the morning we were set to see it, I panicked. What if it wasn't any good?\nI'm a massive MCU fan. I've been waiting for this film ever since I saw Thanos grin at the end of the original Avengers movie. Could it possibly live up to the expectations I've been building up all these years? The answer is yes. Its not just great, but its probably the best movie Marvel have ever made.\nIf you're not up to date with the MCU, Infinity War has no mercy. It throws you in at the end of Thor Ragnarok with almost no backstory. Characters appear on screen constantly with no introduction. Usually I would argue against this, but this is not a stand alone movie. Marvel have changed the game. This is the culmination of ten years of world building, where they did the heavy lifting when it came to developing characters years ago. Infinity war is about the pay off the fans deserve for helping this grow into a blockbuster machine. To be fair the movie does go in to some effort of explaining who Thanos is and why he's doing what he's doing. And honestly, that's all you need at a base level. Knowing everything about previous MCU movies is just an added bonus for nerds like me.\nLike all Avenger movies Stark is the heart of the film. Going back to the first Iron Man, Robert Downey Jnr has always been at the centre of the MCU. He such a great actor and gets better with every movie. He can do funny and sarcastic, but when it comes down to it he can also nail the emotional side of the character. Another standout is Thor. With Ragnarok, Hemsworth showed that he had great comedy chops. In Infinity War he's still super funny but also shows us a vulnerable side we've never seen from Thor before.\nOf course all of the great action and comedy would mean nothing if the bad guy wasn't up to scratch. In Thanos the MCU has found its best villain yet. Always criticized for its uninteresting and lifeless villains, Thanos along with Black Panther's Killmonger, break the mould. Thanos is as bad as they come. He commits genocide like its nothing. He will kill anyone who gets in his way to finding the Infinity Stones. But still Marvel have found a way to have the audience have empathy for him. He is an absolute psychopath, yet the Russo brothers still find a way the audience to feel that although his methods are horrific, he had a point about certain things. He just happens to believe that wiping out half the universe, will cause the universe to re balance it self. For this act of mass genocide he expects the universe will be grateful. Mad Titan indeed.\nSpeaking of the Russo brothers, what the directors have achieved with this movie is astonishing. Yes you may hope that some characters would have more screen time, but the way they have balanced the insane amount of characters on screen with all the story that needs to be told really is an amazing feat.\nMild spoilers past this point.\nInfinity War is filled with amazing action, suspense and laughs. But its the last third that will last with you even after you have left the theatre. It sounds hyperbolic but the last psrt of this film is as good an ending you will see all year. It can never be said that Disney don't have the guts to take risks. Its one of the bravest endings ever in movie history, let alone the MCU. Avengers Infinity War is an absolutely breath taking piece of movie making. An epic film with Marvel at the height of their powers, and an ending that will have you emotional for days.\nOrder this amazing movie right here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have always been fascinated by the subconscious mind in my psychotherapy work. And my studies in \"Family of Origin Therapy\" assisted my understanding and therapy with clients profoundly. From this work I perfected a form of personal family of origin analysis interpretation and explanation to assist others.\nI spent years working in psychotherapy and many more years working with hypnotherapy and hypnosis to assist others in their therapy, recovery and understanding of themselves. This personal family of origin interpretation and analysis is a blueprint for this subconscious understanding.\nYou will also have the opportunity to have your own personal family of origin interpretation, your personal symbols interpreted, your tree drawing interpretation, your mandala drawing interpretation as well as the forensic graphology interpretation of your own handwriting.\nAll of these analyses coupled together will give you powerful information about yourself, your life, your goals and your personality.\nRemember, the interpretation needs to be done in this correct form, using these valid techniques in order to be truthful information.\nRoles within families can and do powerfully affect the ability to interface with the world. These roles can effect health, well-being, and relationships, especially if the person moves through life with a sense of denial or no information about how to positively live the birth order role . If this role is not working, the behavior can be altered in a more positive manner. Totally shifting a role, thinking there will be no repercussions, can be devastating, however.\nOne of the more visible and, at the time, interesting Presidents of the US, was a middle child of a very large affluent family. He was a sickly child who longed for his mother's attention and nurturing. What he got was a very absent mother and was left in the care of a nanny most of the time. His father was competitive and lived from the philosophy of a person being expendable within the family if they did not adequately play their role within the family system. This child got to see that his acting-out sister was labeled as different and given a lobotomy to make her more compliant. She was also sent away from the family and lived in an institution until her death.\nThis middle child had made up his mind to play the role of the college-level professor and writer. This profession would suit his nature which was that he liked books, time to himself, disheveled ways of dressing, and the limelight only when he chose it. He was headed in this direction until the most awful thing imaginable happened to him.\nHis oldest brother had been groomed to be the President of the United States by his father. His older brother was handsome, loved to dress well , extroverted, loved the limelight, and liked to lead. His older brother had unfortunately taken on characteristics of the father also in that he liked living on the edge of danger. The older brother chose a very dangerous flight with explosives during World War II and the plane he was flying exploded killing him in the process.\nWhat happened next? The middle child, without time to grieve his older brother or his sister who had the lobotomy, was bumped up into limelight position. All of a sudden, his clothes had to be perfect, one of his books was cleaned up by a ghost writer (never substantiated but suspected) and won a Pulitzer. His choice of women to date was also now monitored. No longer could he openly choose the large breasted, slightly tarnished looking women he preferred. He needed a mate who would be acceptable to the world.\nWhat happened to this man who moved in the blink of an eye from lost child\/middle child to limelight\/first born child? He apparently excelled for a while. He was actually intellectually brighter than the original first born. When groomed properly, he could hold his own with the original first born in the looks department. He put on a good show when the competitiveness showing was needed, but where his older brother loved to be competitive, this man would have preferred to read or write a good book. He kept a competitive pace with his father watching constantly, however.\nHe found a suitable and quite presentable woman to date. They eventually married with the world watching and his father orchestrating. Things looked as if everything would work for a while.\n2. He cheated openly, but the media protected him. The ones he cheated with were the illicit, \"love- of- the- gutter\" types. His father had had that shadow also . He mimicked that shadow even though he didn't have the huge fortune to cover his tracks which his father had. Things became harder for the father to keep things covered-up for this son.\n4. He and his brother took on dangerous types to try to prosecute. Ironically, information has come out recently that the family fortune may have been linked to the bootlegging controlled by these types. Their openly going after this group, may have been a form of subconscious rebellion against the father.\n5. At age 47, annihilation came in the form of violence through assassination and the role was over.\nThe shift from middle to first born role was devastating to this life that ended at a young age. Autopsy reports show that he had advanced kidney disease. Psychologically, kidney often represents: Am I safe in my world? Ironically, the other middle child of this large family who was lobotomized, lived to be quite elderly in the institution. She had literally lost her ability to process the world through emotion. She had become a robot emotionally.\nThis first born son had separation anxiety at age six. His sister, who looked like a twin to him, was taken away for a year for treatment for leukemia. The thinking at that time was to protect children by not telling them all information. This first born and his brothers were shuffled around their small town for a year, staying with friends. They were separated from each other so that the friends would only have one child in the home. Their mother stayed with the sister getting treatment in New York at the hospital. The father stayed at his family apartment in New York and came to the small town rarely to check on his other children.\nWhen discussing this situation as adults, the oldest is still angry as witnessed by his description. The middle played mascot with his comment and said: \"At least we weren't sent to a kennel.\" The second middle stayed invisible with no comment because there had been scandal twice already in his life . The youngest child, Marvin, has remained out of the limelight totally, and Doro was not born when all of this was happening. The only two children living at the time of their sister's death was six year old first born and a very young toddler, the third child born at the time.\nThe sister, Robin, did not make it and died from leukemia without the children being prepared for what was happening. She died one year later. During that year the children had been separated from their parents, their physical home, from all siblings, kept in the dark about what was happening, and then presented with the death of their sister.\nFirst born gets into teenage and young adult alcoholism.\nHe is a C student at two Ivy League colleges where his Dad and Grandfather had been Phi Beta Kappa.\nHe floats through life and relationships until fairly late in his young adulthood. He meets a young woman, who is an only child, and she does an intervention on his alcoholism. They marry.\nHe cleans up at that point and begins to play first born rather than the rebellious scapegoat he had played during adolescence and young adulthood.\nHis next brother: had the good looks of his father. Loved to dress well and was Phi Beta Kappa at the Ivy League schools of his father and grandfather. He, however, married a woman not chosen by the family, and she was of another ethnic group. He played the role of middle child almost to the letter.\nThe first born is angry when talking about the situation. The child next to him tries to diffuse with mascot behavior.\nA youngest boy child is never written about, photographed rarely, and the media never really zero in on him at all. He has managed to stay basically invisible .\nThe boy next to the second born son was the closest to his mother. He had learning problems which she attempted to address. As an adult he was involved in one business scandal and one personal scandal. He looked different than the other children and stood out as the only blond.\nA sister was born after the death of the first sister with little written about her life except that she attempts to intervene in the lives of her nieces and nephews by providing help, especially at times of crisis. She has been divorced and remarried during her brother's two terms in office. Not much other media attention has been given to her.\nWhen it came time to choose a political candidate to promote which child was chosen?\nWas it the Phi Beta Kappa Child with the Ivy League education who dressed well and personally stayed out of trouble ?\nNo, the first born was who had cleaned up was chosen. Reason: role lines are usually very rigid without conscious acceptance of what is happening within the system.\nThe media is saying the marriage has had problems. The addiction may be back. Obviously, health has suffered and severe aging has happened to this first born. The elders still cover and deny facts. The child next to him has gained substantial weight, supported the first born in many ways during the tenure, had a child who had severe addiction problems, and has gotten out of the limelight for now. His wife also had legal problems which were pretty severe.\nThe other youngest male child has stayed invisible through it all never really making it to media attention. The world does not really easily know his name or face. Ironically, he was the handsome child of the group.\nThe third born son at the time of the separation anxiety when their sister was dying: rode out his personal scandal and has stayed away from media attention during the last years.\nThis situation shows what can happened through separation anxiety and magical thinking of children caused by the separation anxiety. Also it shows how roles can be held rigidly and some of the repercussions of playing out family roles when perhaps the person may not really want to play that particular role at all.\nThe family of origin information is a strong predictor. For those seeking subconscious information, the family of origination evaluation can show the role they have played, any renegotiations to another role that may have happened with or without repercussions, and how to understand the role within the family system. Understanding can be pivotal to having a healthier lifestyle. With understanding, pitfalls can be avoided, communication can be assisted, and health can also be assisted. It is the old philosophy of the truth shall set you free.\nThe first born, next born, and last born all have role characteristics. People can play a role in the family, a modification within society, and can pick and choose characteristics that are beneficial in all interactions. The key is to understand what is happening and what the buttons are which bring on automatic reactions.\nHaving been trained in family of origin therapy, I can usually spot the role a person is playing very quickly. Most of the time the role follows birth order , but not always. Sometimes, people refuse a certain role and it goes to someone else in the family to play. Also, lots of years between births can alter the traditional role.\nAt a gathering a few years ago, pictures were being taken of some siblings in attendance. The child of one of these siblings wanted to capture the event in pictures (she was the oldest daughter of one of the people throwing the party). She wanted the picture to reflect birth order of her family. The second born son who was a mature adult balked like a child when asked to move from number one position. He literally reverted to childhood and took a whine tone to his voice by saying: \"I am the one who acts the most like the first born.\" He reluctantly, very reluctantly, was moved to the birth order position in the picture taking line. His button, by the way, is recognition and power. It is dangerous for that to be hidden in the subconscious because a manipulator could use that in a dangerous manner. It was obvious that he never realized that he fell into the whining child role in front of family and guests present to help celebrate.\nIt was quite interesting, recently to see a birth order role played out in the national media. The young child of a man in the national news recently wanted to preserve the family history of the day by taking her own camera, which is a young child's camera, and taking pictures of her family all the way to the last leaving of the White House for balls that night. Her parents and sibling were all respecting her position in the family as family historian by posing for her. It brought tears to my eyes to see this young child playing her role. Here she was at a young age, with her camera, diligently doing her job which was to preserve and protect her family.\nThe picture below of this young family speaks volumes. If you will get out informal pictures and look, you can see positions within a family of origin. The oldest daughter here is positioned in a position of importance with her head leaning towards her Dad. Many write-ups in the paper and magazines talked of her preserving the memory of this eventful day for this family with her camera. She is obviously the first born and playing a first born positive role. There are many pitfalls which she can fall into with this role connected to health, relationship, and life in general. However, at her young age she is the preserver of history with her effort and camera. See the picture below and get out some of your family pictures and look at them to see how they literally paint a picture of family dynamics.\nThe picture below is of an only child. You can see that all attention goes directly to this child which can be good and bad. She doesn't have anyone with which to experience things, but also all energy does come to her. If the energy is all peace and calm, things go well. However, if not, resilience has to be developed in order to survive. Also, parents have a tendency to triangle an only between them in times of conflict. Similar characteristics show up in the lives of only children and first born children as evidenced by both of these preserving the history of their family. Since she was a little older she had a video camera.\nOrder your own personal family of origin analysis and with any other analysis ordered (Tree, Mandala, or Handwriting), you will assist the information you obtain by having an evaluation of your Family of Origin position. As stated earlier, having knowledge of the role you are playing with the positives and pitfalls will powerfully assist you in life and your well-being.\nyou order the analysis and then a questionnaire will be emailed to you for you to complete. After you complete the questionaire a detailed analysis with recommendations will then be emailed back to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Red, Green, and Yellow are the colors of the three most popular Thai curry pastes.\nA balanced diet is important in winter. Several vitamins and minerals (like zinc) are found in salads and vegetables as well as in herbs and sprouts.\nTea with ginger is also popular against impending colds.\nIn winter, we also swear by our wide selection of curry dishes. They are delicious and excellent heartwarming dishes.\nBe Creative - Thai food in the Winter time!\nHerbs & Spices are a \"must have\" in todays' kitchen.\nWith the right spices and curry pastes can be done in a snap. Curry is the basis of some of the world's most fundamental and flavorful dishes, from blindingly hot, chili-fueled Thai dishes to hearty, slow-burning Indian masalas. Curry paste is most often associated with the deeply spicy and fragrant flavors of Thai food.\nCooking like in Thailand! Organic Thai curry pastes.\nAll ingredients from organic farming.\nAt a purchase of 10 packs of our organic curry pastes.\nThe spiciness and flavours of Thai cuisine are well known. As with other Asian cuisines, balance, detail and variety are of great significance to Thai chefs.\nThai people refer to dishes that are known as \"Thai curries\" in the Western world as \"kaeng\". The first Thai dictionary from 1873 CE defines kaeng as a watery dish to be eaten with rice and utilizing shrimp paste, onions or shallots, chillies, and garlic as essential ingredients. Coconut milk is not included in this definition and many Thai curries, such as kaeng som and kaeng pa, do not feature it. Curries in Lanna (northern Thai) cuisine, with only a few exceptions, do not use coconut milk due to coconut palms not growing well, if at all, in the climate of the Thai highlands. The spiciness of Thai curries depends on the amount and kind of chilli used in the making of the paste. Even within one type of curry the spiciness can differ widely.\nCurries are eaten in combination with rice, the long grained jasmine rice in central and southern Thailand and sticky rice in northern and northeastern Thailand, and with noodles such as khanom chin (fermented rice noodles). Certain curries can also be eaten with roti, the Thai version of the Indian-style fried flat bread from Malaysia called roti canai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Urban Alliance is more than jobs, outreach and housing. Our desire is to empower others to bring transformation to their lives. Please take a moment to watch this video of three individuals who have been positively impacted through the initiatives of Urban Alliance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a social worker and therapist , and I attended one of your workshops a couple years ago. I have a question for you related to a client I am currently treating. This client is a man in his mid 20s and his father died when my client was in his mid-teens. His father died from complications from drinking too much. My client worries he should not be grieving anymore and also worries he will forget his father. The worry that he will forget his father is quite distressing for him. Can you provide some general guidance on how I can work with this fear that he will forget his father? Additionally, if you can recommend any articles to read on this topic that would also be wonderful.\nMy brief response to your case-based query is that the premature death of your client's father and his son's fear of forgetting him speaks to our common need to conserve a bond with the deceased rather than to \"let them go.\" In our contemporary, secular age, rituals of continuity and transition once provided by spiritual and cultural practices of remembrance often are radically condensed or relinquished altogether, and we are left needing to invent them in quite personal ways for ourselves. In this, therapy can provide assistance by joining with clients in imagining how we might keep our loved ones stories alive\u2014especially their preferred stories of proud or close moments shared with our clients, rather than only their stories of brokenness, conflict, or absence. Of course, the painful parts of the story\u2014including in this case the destructive drinking\u2014must also be acknowledged candidly, but I would join the client in seeking to restore to memory and perhaps to some form of public telling the preferred stories as well. You might find a variety of resources helpful in envisioning this, from professional books like Klass and Steffen's \"Continuing bonds in bereavement\" published this year by Routledge to sensitive popular movies like \"Coco,\" which traces a young boy's efforts to embrace imperfect family ties while also finding his own unique way in the world.\nAll of this is to say that nearly anything, any practice, that honors our loved one helps restore a constructive bond, and in this sense helps as well with our grief by restoring connection in the form that is sustainable now. Finding and displaying on one's computer or phone's home screen a picture of your client and father together (rather than the father alone), sharing stories about times with dad in person or in online memorials, performing small acts of kindness such as paying in advance for the order of the next person in the drive-up line at Starbucks and doing so in the father's memory, or undertaking some form of legacy project, such as constructing a scrapbook of shared memories or contributing time to a cause that might be relevant to the father (such as speaking about problems of substance abuse on campus) could all serve this function of constructing and honoring an enduring bond with dad. Such meaningful actions take a strong stand against forgetting, and remind us that in a life that in which we sometimes exercise far too little control over events, in other ways choice and agency are fully available to us\u2014including in how we continue to love someone in his absence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted on Fri, Feb 17, 2012 : 10:22 p.m.\nThe words of Huron High School boys basketball coach Waleed Samaha echoed off the walls of the school's Riverdome.\nVisiting Saline had just scored to cut the River Rats' lead to 31-28 with 2:04 to go in the first half and Samaha wasn't happy.\n\"How about making them do what we want?\" asked Samaha angrily during the timeout, referring to his team's defense.\nThat timeout sparked an 18-3 run that gave Huron all the cushion it would need Friday night to cruise to a 67-51 win on senior night.\n\"Senior nights are just funny. Too much emotion going in,\" Samaha said. \"That's what I attribute that to.\"\nIt didn't help that Saline hit three 3-pointers in a 75-second span to take an early 9-3 lead.\n\"We were matching them blow-for-blow,\" Saline coach Matt Seidl said. \"We were playing about as well offensively as we have in a while.\"\nThe Hornets led 18-16 after the first quarter.\n\"They did a great job,\" Samaha said. \"We were kind of hoping Saline would miss a few, but they hit everything early.\"\nAfter the timeout, Saline scored three points in the next eight minutes of game play, being held without a field goal. Huron pushed the three-point lead to eight at halftime and 12 eighty-nine seconds into the second half.\n\"That three points to eight, that was a big difference,\" Seidl said. About the bad start to the third quarter, he said: \"Now it's 12 and now we're kind of in a chase mode. About half the game plan is gone.\"\nFor Huron (11-5, 7-2 Southeastern Conference), Kendall Thomas tied his career high with 23 points.\n\"I was hot, but the game just came to me,\" Thomas said.\nMike Lewis scored 18 points despite checking himself out of the game for a stretch in the third-quarter, dry-heaving as he dealt with an illness, according to Samaha. Andre Bond scored 14.\n\"For us to beat them, they need to cooperate a little bit,\" Seidl said. \"And they didn't.\"\nSophomore K.C. Borseth scored 14 points for Saline (7-9, 4-8), which has lost nine of its last 10 games.\n\"These guys have really done everything I've asked,\" Seidl said. \"That's why actually I'm getting frustrated, because I want them to be rewarded for all they've done.\"\nSat, Feb 18, 2012 : 2:07 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"52cc 2.3HP Gas Powered Ice Post Hole Digger W\/ 6 Ice Dril Bit w\/Sharp Blades. Lightweight and easy to start, this one-man post hole digger is ideal for installing fence posts, decks, planting trees and shrubs, ice fishing, and more.\nIncluded 150mm (aprx 6 inches) wide drill 1M deep Ice Bit with sharp Blades as shown on picture. Pull Start Engine Displacement:52cc Rated output power:1700W Drilling rotational speed: 170rpm Two-cycle oil \/Gasoline mixing ratio:1:25 Manual Instruction: Download. Please make sure you order the correct item as orders cannot be altered after the order is paid for. 5-8 days to your door, we do not guarantee transit time. Excluded AK, HI, PR, APO, GUAM, VI, or PO Box. The flat rates quoted in the table will apply to most destinations in the countries\/states listed.\nThe item \"52cc 2.3HP Gas Powered Ice Post Hole Digger With 6 Ice Dril Bit withSharp Blades\" is in sale since Sunday, January 27, 2013. This item is in the category \"Business & Industrial\\Heavy Equipment, Parts & Attachments\\Heavy Equipment Attachments\\Post Hole Diggers\". The seller is \"econoestore\" and is located in Rowland Heights, California. This item can be shipped to United States, Canada.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Create two cards for only $15.00. All materials included, please bring along your craft box.\nWe design three new cards every week.\nYou can also choose a card to make from a large variety cards that were already taught in the class. New techniques will be taught every week and the cards that are made in these workshops are of very high quality.\nAll the ladies enjoy the company of each other and become friends. We also share new ideas in an very relaxed environment. The classes are informal and everyone work on their own, following the instructions provided. Help is always available if you get stuck.\nCard kits, which includes instructions and all the materials you will need, are available to buy @ $6.50 if you want to create more cards at home.\nThis is a very popular craft activity with more than 20 ladies attending the two workshops weekly. Please visit the calendar on the home page to find a time that suits you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkmek b/data_all_eng_slimpj/shuffled/split2/finalzzzbkmek
new file mode 100644
index 0000000000000000000000000000000000000000..1a791b3d242819b917d1b82536231459f9549c5e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkmek
@@ -0,0 +1,5 @@
+{"text":"So the new tagline on our site is \"Pets+People=Priceless\", I thought this post was a perfect example of why I chose that.\nSome photos just capture that magical moment, the look of love on a pet's face when they just know everything is right in the world. I thought I would share these images from a recent shoot, \"the look\" says it all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is\u200b a\u200b dream come true for\u200b many those who had the wish to\u200b go round the world but couldn't go because of\u200b very costly airfare. But now with the help of\u200b cheap flights and\u200b low airfare it\u200b is\u200b possible for\u200b them to\u200b enjoy the trip of\u200b the world. Previously only few rich people can afford traveling in\u200b Air flights but now a\u200b days there are massive discounts on\u200b all flights around the world to\u200b increase the chance for\u200b ordinary people to\u200b fly in\u200b aircrafts. There are also schemes like cheap airfare deals for\u200b students and\u200b aged citizens by some traveling organizations to\u200b give them more chances to\u200b explore the world.\nCheap Airfare helps those people a\u200b lot who really can't afford traveling in\u200b flights but have the wish in\u200b their mind to\u200b roam around the world. if\u200b someone travels to\u200b some place of\u200b tourism or\u200b holiday destinations at\u200b off peak season\u200b he can easily take the advantage of\u200b the low airfare or\u200b cheap airfare. Different Airline companies are offering cheap airfares on\u200b flights to\u200b a\u200b wide range of\u200b destinations. Again\u200b due to\u200b some reason\u200b different Airline companies are decreasing their airfare in\u200b a\u200b regular manner and\u200b also providing with many interesting offers and\u200b discounts. Governments of\u200b different countries are also accepting this\u200b good move by the Airline companies as\u200b they are giving chances to\u200b lots of\u200b people to\u200b travel in\u200b flights. if\u200b there is\u200b any cancellation\u200b in\u200b flight tickets at\u200b last moment then Airline companies offer those seats to\u200b other passenger in\u200b a\u200b quite discounted price or\u200b even sometimes you\u200b can bargain\u200b with the traveling agents for\u200b more discounts. this\u200b is\u200b considered as\u200b one of\u200b their good business policies. But in\u200b such cases if\u200b you\u200b miss the flight you\u200b have to\u200b loose the hope of\u200b getting any refund.\nBesides the Airline companies there are lots of\u200b traveling agencies who are offering fare at\u200b much lower cost than normal during off peak seasons and\u200b also giving discounts in\u200b some special cases. Cheap tourism Package and\u200b business tours are also been conducted by travel agents at\u200b a\u200b very low price. There are also various online travel agents who are ready to\u200b provide you\u200b with ultimate satisfaction\u200b in\u200b traveling from one place to\u200b another and\u200b one can easily confirm his booking for\u200b traveling through these agents just by the Internet. Some agents even can give you\u200b ticket at\u200b normal price during the high peak seasons but the problem is\u200b that you\u200b might have to\u200b face difficulty in\u200b order to\u200b find out if\u200b they have the specific dates for\u200b you\u200b or\u200b not which you\u200b want.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it possible for a restaurant to do one dish too well? I'm starting to think so.\nSometimes, no matter how deep, how tempting and how well-crafted the rest of the menu is, I'm so hooked on one standout specialty that I simply can't bring myself to try anything else.\nHere are 10 dishes so good that I've mentally ordered them the moment I walk in the door.\nChicken-Fried Steak at Texaz Grill, 6003 N. 16th St., Phoenix, (602) 248-7827. Sure, the T-bone and filet look good. But the he-man-size chicken-fried steak is a proven winner, tender enough to cut with a fork, encased in crisp batter and bathed with thick, peppery country gravy. Great skin-on mashed spuds alongside, too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We love book themed crafts and activities so I am so thrilled about today's post because it is filled with the cutest ideas including Chicka Chicka Boom Boom, The Rainbow Fish, The Hungry Caterpillar, Pinkalicious Pete the Cat, and more\u2026 !\nThey are all created by the Kids Craft Stars, who are a bunch of talented crafty ladies (that I am so excited to be a part of) who challenge each other with a monthly theme, this month being Children's Books. You can follow the Kids Craft Stars on Pinterest and Instagram to make sure you catch all the cute crafts our team comes up with!\nthese are fantastic! We love pairing a craft with a book, pinning!\nHi Susen, Thanks so much!!!\nWhat a great resource for some really amazing books!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Authorized Uses: As a solvent: Cellulose coatings; Synthetic resin coatings; Shellac coatings; Other natural resin coatings; Other coatings; Cellulose plastics; Non-cellulose plastics, including resins; Photographic film and emulsions; Transparent sheeting; Explosives; Cellulose intermediates and industrial collodions; Soldering flux; Adhesives and binders; Proprietary solvents; Solvents and thinners (other than proprietary solvents or special industrial solvents); Solvents, special (restricted sale); Polishes; Inks (not including meat branding inks); Stains (wood, etc.); Shampoos; Soap and bath preparations; Cellulose compounds (dehydration); Sodium hydrosulfite (dehydration); Other dehydration products; Petroleum products; Processing pectin; Processing other food products; Processing crude drugs; Processing glandular products, vitamins, hormones, and yeasts; Processing antibiotics and vaccines; Processing medicinal chemicals (including alkaloids); Processing blood and blood products; Miscellaneous drug processing (including manufacture of pills); Processing dyes and intermediates; Processing perfume materials and fixatives; Processing photographic chemicals; Processing rosin; Processing rubber (latex); Processing other chemicals; Processing miscellaneous products; Disinfectants, insecticides, fungicides, and other biocides; Embalming fluids and related products; Sterilizing and preserving solutions; Industrial detergents and soaps; Cleaning solutions (including household detergents); Photoengraving and rotogravure dyes and solutions; Other dye solutions; Miscellaneous solutions (including duplicating fluids). As a raw material: Ethyl acetate; Ethyl chloride; Other ethyl esters; Ethylamines; Dyes and intermediates; Acetaldehyde; Other aldehydes; Ethyl ether; Other ethers; Ethylene dibromide; Ethylene gas; Xanthates; Fulminate of mercury and other detonators; Drugs and medicinal chemicals; Other chemicals. As a fuel: Automobile and supplementary fuels; Airplane and supplementary fuels; Rocket and jet fuels; Proprietary heating fuels; Other fuel uses. As a fluid: Scientific instruments; Brake fluids; Cutting oil; Refrigerating uses; Other fluid uses; Proprietary anti-freeze. Miscellaneous uses: Product development and pilot plant uses (own use only); Specialized uses (unclassified). This product is for further commercial manufacturing, laboratory or research use, and may be used as an excipient or a process solvent for pharmaceutical purposes. It is not intended for use as an active ingredient in drug manufacturing nor as a medical device or disinfectant. Appropriate\/legal use of this product is the responsibility of the user.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkuxn b/data_all_eng_slimpj/shuffled/split2/finalzzzbkuxn
new file mode 100644
index 0000000000000000000000000000000000000000..63d26eb7872a47e72bbaac92c31972c7f9eb7e1b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkuxn
@@ -0,0 +1,5 @@
+{"text":"MOUNTAIN VIEW, CA -- (Marketwire) -- 05\/04\/09 -- Employers anticipate hiring 22 percentfewer new college graduates from the class of 2009 than they hired from theclass of 2008, according to the National Association of Colleges andEmployers(1), which is why eHealthInsurance, today, released eight tips forcollege graduates that have yet to secure full-time employment and are inneed of affordable healthinsurance.\nWhile students under 25 years of age are often covered through theirparents' health insurance plan or a university plan, many carriers will notallow dependent children over 18 to stay on parents' plans after theygraduate. In 2007, the Census Bureau estimated that there were nearly 8million uninsured young adults (ages 18-24), making them the largestsegment of the uninsured population. That problem is likely to becompounded with the rise in unemployment and lack of new jobs resultingfrom the current economic recession.\n1. Get the low-down on your hometown: If you live in Massachusetts, NewYork, New Jersey, Maine, or Vermont then you live in a \"Guaranteed Issue\"state, which means that you can't be denied medical coverage forpre-existing health conditions. Go online, search for the best plan you canfind and buy. If you don't live in a \"Guaranteed Issue\" state then you needto be aware of how any pre-existing health conditions you have could affectyour ability to qualify for coverage. Remember that health insurance isregulated and sold on a state level. If you are moving to another stateafter graduating, you'll need to buy health insurance in that state. Youcan talk to an agent licensed in your state by calling eHealthInsurancetoll free: 1-800-977-8860.\n3. Talk to an agent: If you have a pre-existing health condition, it's agood idea to talk to a licensed agent in your state that can help youfigure out whether or not your pre-existing health problems would excludeyou from qualifying for an individual policy in your state. You can talk toan agent licensed in your state by calling eHealthInsurance toll free:1-800-977-8860. It's also a good idea to talk to your doctor about anypre-existing conditions you might have. Try to get a sense of how much careyou might require and how much coverage you might need.\n4. Shop 'til you drop: One of the best places to search for policies iswith an online broker. By going to the web you're likely to find thelargest number of plans available. By law, every broker must sell policiesat the same price, so you can't find a better deal by switching brokers.eHealthInsurance.com is the largest online broker in the United States.eHealth is licensed to sell insurance in all 50 states and the District ofColumbia and it has more than 10,000 policies available from over 180carriers.\n5. Maintain your independence: If you're thinking that you'll change jobsfrequently, or that you're more likely to freelance and work independently,why tie your health insurance to your job? It might make more sense to getan individual health insurance plan that will stay with you regardless ofwhere your career takes you. Then, if a prospective employer offers youhealth insurance benefits, you can decline their coverage and negotiate ahigher salary. According to J.D. Power and Associates' 2009 NationalHealth Insurance Plan Study(SM), people with private health insurance weremore satisfied with their plans than those with insurance from a smallemployer (50 or fewer employees). Before you switch, ask prospectiveco-workers if they like the group plan. If it's not better, and it's notcheaper, why change?\n6. You can never be too safe: Be careful to protect your privateinformation. Some online brokers simply aggregate leads and send them tolive brokers who may call you directly. To avoid excessive telemarketingcalls, control the process by using an online broker likeeHealthInsurance.com that doesn't sell your private information to otherbrokers and lets you to search for quotes anonymously. If you have anyquestions that you can't answer on your own, eHealthInsurance.com also hasa call center staffed by licensed agents that don't make bigger commissionsby selling you more expensive plans. You'll get straightforward, unbiasedadvice from an agent licensed in your state.\n8. Not ready for a long-term commitment? Have a fling: If you know you'llhave a job within a few months after graduation, short-term healthinsurance plans may be a good option for you. Short-term health insurancetypically lasts for six months, has less stringent qualifications and canbe a great way to prevent a health insurance gap before you start that newjob. Note that these are not a good long-term solution: short-term plansare more limited in benefits and can often only be renewed once.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Date: noon April 24-noon April 28, 2005.\nLast year's 1st Conference on Foundations of Nanoscience(FNANO04) in April, 2004 had an significant impact on the emerging fields of Nanoscience and Self-assembly -- for the first time it brought many of the leading Nanoscientists and researchers working in a wide variety of areas of Self-assembly in the same place to present invited talks.\nThe upcoming 2nd Conference on Foundations of Nanoscience will have a modified format, with many contributed posters and open discussion periods to allow for attendee interaction, as well as invited talks by distinguished Nanoscientists.\nSUBMISSION DEADLINE: extended to March 1, 2005 (this is now a firm deadline).\nSUBMISSION REQUIREMENTS: If you wish to present a talk or poster at FNANO05, you need to submit by March 1, 2005 an extended abstract of at least one page in PDF format. Submitted papers can be either an abstract or a paper, ranging in length between one page to 12 pages, at the author's option. Even if your talk is invited, you still need to submit at least a one page abstract by this date.\nA hard copy of this year's FNANO05 Conference Proceedings will again provided to all registered FNANO05 Conference attendees and subsequently available from the electronic publisher Sciencetechnica.\nDEADLINE for Uploading ACCEPTED PAPERS (for invited talks & accepted submitted papers and posters) for Publication in the Conference Proceedings: April 1, 2005 at the Website: http:\/\/fnano05.cs.duke.edu\/submit\/.\nConference Proceedings Paper Format Instructions: http:\/\/www.cs.duke.edu\/~reif\/FNANO\/instructions.html .\nObtaining Extra or Electronic Copies of FNANO Conference Proceedings: Both printed and electronic (DC-ROM and download) versions of this year's FNANO05 Proceedings will be available in a few weeks for purchase from the electronic publisher sciencetechnica.com .\nObtaining Last Years FNANO Conference Proceedings at 50% Discount: As a participant at this year's FNANO conference, you are entitled for a limited time period starting April 23 to purchase both printed and electronic (DC-ROM and download) versions of last year's FNANO04 Proceedings at a 50% DISCOUNT off the retail price at the web site sciencetechnica.com\/2004special .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gran had lived a lot of years and raised sons and grandsons, and then along came Amy Catherine. Gran's joy knew no bounds. The little girl had come along after five boys and her mother was glad for Gran's help. Gran made quilts for her grandchildren, but this girl's quilt would be special, containing embroidered pictures of her young life--pictures that Amy had drawn. Amy was a modern girl for her era, and at that time, a woman's job was in the home, but there was no money to be made there. Amy managed an answer of her own. She'll tell you about. It has joys and agonizing tears, but that is just part of life.\nJohnny wanted to marry Amy but girls are difficult to understand, and impossible to lead. He finally decided Amy must be allowed to go her own way, and he would lead her in that direction if he could just get ahead of her. Johnny proved his worth, though, in a battle with the Mississippi Central Railway that was trying to... well, it's in the book and Johnny should get to tell it in his own way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2015, Newfoundland and Labrador had the most rapidly aging population in the country \u2013 which when combined with high rates of youth out-migration, declining birth rates, and an increasing number of people moving from rural parts of the province to more urban centres, means that the province is facing an unprecedented population challenge. Without intervention, this trend will have a drastic impact on the economy, governance, and overall quality of life for the people of the province. Planning for this change and developing strategies to adjust and adapt to the change is paramount.\nThis report focuses specifically on Labrador. Funded by the International Grenfell Association (IGA), it provides a set of population projections for the IGA regions of Labrador and the Northern Peninsula of Newfoundland and Labrador West for the period 2016-2036. These projections provide a basis for further research into the implications of demographic change in this region. Results from three projection models are presented. These models are: The Natural Survival Model (NS), the Historical (Cyclic) Survival Model (HS), and the Replacement Survival Model (RS).\nThe NS, HS and RS models help provide insight to how population will change in terms of both the total number of people living in a region as well as the resulting age structure. The NS model indicates the capacity of a region to grow by natural replacement by accounting for regional fertility and death rates, though does not consider migration. This model also identifies regions whose age structure combined with its fertility and death rates can or cannot maintain their populations without in-migration. The HS model results provide insight into the age structure of populations if past migration trends continue into the future. For some regions in Labrador this may be likely, given that over multiple census periods there has been very little, if any, in-migration and significant outmigration of younger cohorts without replacement. The result, a decreasing and rapidly aging population, is particularly significant in those regions that will decline even where a significant labour force replacement factor is built in as indicated in the results from the RS model.\nLabrador: For Labrador as a whole the NS model indicates that births would exceed deaths over the projection period and, excluding migration, there is an internal propensity for growth. However, when migration trends are factored in, the overall population is projected to decline by 8% between 2016 and 2036. To maintain the current workforce population to 2036 at least 50% of that workforce would need to be replaced, primarily through reduced out-migration or higher rates of in-migration.\nHowever, within Labrador there are significant differences between regions and so this overall picture may be misleading, hence the need to consider individual regions within Labrador.\nLabrador North Coast: The large number of people in the younger age cohorts indicates a natural propensity to maintain the population. However, when past migration trends are factored in, the population is projected to decrease slightly between 2016 and 2036 under a \"Medium Scenario\" case. To ensure that the workforce population is maintained, replacement success, through population retention and\/or in-migration, would need to be in the order of 50%.\nCentral Labrador: The NS model shows a population increase over the projection period. When migration trends are included the population would increase slightly to 2031 and then decline to 2019 levels by 2036. Theoretically the workforce in this region could be maintained without any replacement strategy.\nLabrador South Coast: The NS model predicts a population decline of 4% between 2016 and 2036. When past migration trends are included the population would decrease by 33% by 2036 under the Medium Scenario case. This is a region with a rapidly aging population. In 2011 the average age was 41, and by 2036 it is projected to be 55. To maintain the workforce at current levels a 50% replacement success is required, which would have to occur primarily through in-migration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Central Bank of the Bahamas has released (24th January) their fourth banknote which is part of their current family of banknotes and incorporates CRISP technology. In keeping with its launch strategy for the CRISP Evolution (CE) family of banknotes which began in 2016 with the issue of the $10 denomination in 2016, the half dollar denomination was introduced into circulation and is the first complete re-design of this banknote denomination since 1984.\nThe predominantly grey note includes shades of green, blue, coral, lilac, and red, and measures 156 by 67 millimetres. The face includes a portrait of HM Queen Elizabeth II placed to the right half along with the signature of the governor of the Central Bank of The Bahamas. The text Central Bank of The Bahamas is placed below a representation of a map of the Bahamian Islands in the centre of the note. To the left of the Queen's portrait is the declaration These notes are legal tender under the Central Bank of The Bahamas Act 2000 for payment of any amount Fifty Cents, which is seen in a vertical paragraph. An image of a strongback flower is placed in the centre design of the note.\nThe back of the notes is designed utilising a vertical arrangement and includes a vignette of Sister Sarah in the straw market. The vignette is flanked above by the numeral $\u00bd that appears in the upper left and lower right corners, while the words Fifty Cents are in the upper right quadrant. Just below the vignette of Sister Sarah is the coat of arms of the Commonwealth of the Bahamas along with the words Central Bank of The Bahamas.\nWatermark: The image of Queen Elizabeth II, as seen on the banknote, and numeral $\u00bd, are visible when the banknote is held against a light and are located towards the far left when viewed from the front side.\nSolid security thread: A thread which appears as a solid line rather than just appearing on the surface of the note in segments, as it is shown on the reverse.\nColour-shifting ink application: On the face of the banknote is a lizard shape that changes colour from metallic gold to green as the note is tilted.\nIridescent print: \u00ad There is a subtle, lustrous golden band across the back of the note which includes repeating images of the letters CBB, $1\/2, and a basket weave pattern.\nThe Central Bank of the Bahamas has also announced they are preparing for the release of a new version of the $50 banknote in 2019. New versions of the $3 banknotes are also slated for issue in 2019. For additional information about this banknote and others issued by the Central Bank of the Bahamas, please visit their website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblkyt b/data_all_eng_slimpj/shuffled/split2/finalzzzblkyt
new file mode 100644
index 0000000000000000000000000000000000000000..30fb066d717c52484d9b5295d5691665a5ccfbc5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblkyt
@@ -0,0 +1,5 @@
+{"text":"Back to home page. I have not used any similar programs, to my knowledge. If you'd like to get the additional items you've selected to qualify for this offer, close this window and add these items to your cart. Back to home page. The item may be missing the original packaging, or in the original packaging but not sealed.\nThanks for finding it for me!!\nContact the seller \u2013 opens in a new window or tab and request a shipping method to your location. VikingmanIbm pc camera 33l4889 14, Learn More \u2013 opens in a new window or tab Any international shipping and import charges are paid in part to Pitney Bowes Inc.\nBuy only this item Close this window. Did your message disappear? Has been in storage for about camerq years. Community Forum Software by IP.\nAdd to watch list Remove from watch list. An ibm pc camera 33l4889 occurred, please try again. Covers your purchase price and original shipping. For 3l34889 information, see the Global Shipping Program terms and conditions \u2013 opens in a new window or tab No additional import charges on delivery Import charges: Downloads View all categories Upload file New files since last visit Files from past 7 days. This item may be a floor model or store return that has been used. Please enter a valid ZIP Code.\nSomeone please help me! Wonder why I have to type seventy one more characters to leave ibm pc camera 33l4889 required entry after I've answered the question.\nGet an immediate offer. Discussion in caemra Hardware ' started by wraMar 14, George Howard Sat, 22 Dec Thanks for ANY info.\nPlease igm in to reply. Hutch Tue, 24 Nov Select a valid country. Follow on-screen instruction to finish the setup. Interest will be charged to your account from the purchase date if the balance is not paid in full within 6 months. Learn More \u2013 ibm pc camera 33l4889 in a new window or tab Any international shipping is paid in part to Pitney Bowes Inc.\nGet an immediate offer. Shipping cost cannot be calculated. Read the Forums FAQ.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This crazy cultural experience that Brazil calls Carnival is seriously something completely out of this world!There are huge blocos happening 24\/7. And I am not exaggerating when I say 24\/7!lol Anywhere in the city you will find a bloco!I've been going to the ones in Ipanema cuz it is right by the beach and I can't ever get enough of the beach lol Yesterdays day bloco had over 120,000 people!Carnival in Rio is enjoyed by everyone!People of all ages!Young single people and families alike!I have even seen elderly people chilling on the beach enjoying life. This city is all about that, enjoying life and being positive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coriander, due to its rich aroma because of its essential oils, aid proper digestion.\nThe most common use of coriander seed is in curry powders, where it is the bulkiest constituent, often rough ground in India to give a crunchy texture. The seeds can be likewise used in stews and soups.\nThey blend well with smoked meats and game and feature in traditional English black pudding recipes and Italian mortadella sausage. Coriander is an ingredient of garam masala, pickling spices and pudding spices and is used in cakes, breads and other baked foods. Sugared comfits made from the seeds are a traditional sweetmeat and breathe sweetener.\nIt enhances fish dishes and, with other spices, may form a delicious coating for spiced fish or chicken, rubbed into the scored flesh and grilled. Coriander complements chili and is included in many chili recipes, such as harissa, the hot North African red pepper sauce. It may be added to cream or cottage cheese.\nCoriander, due to its rich aroma because of its essential oils, apart from being an excellent appetizer, helps to support proper digestion of food. Puffed coriander can also be chewed to help freshen breath.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CONTEST RULES: NO PURCHASE OR PAYMENT OF ANY KIND IS NECESSARY TO ENTER OR WIN THIS CONTEST. THIS IS A SKILL CONTEST.\nPurchase will not improve odds of winning. Void where prohibited. Sponsored by International Vitamin Corporation, 500 Halls Mill Road, Freehold, NJ 07728. Winners will be selected by a panel of 5 judges, who will judge all entries on the basis of originality, creativity, and the ability to provide details on how B- 100 facilitated a life change of some kind. Decisions of the judges are ;nal. One grand prize of a trip for 2 to NYC valued at $5,000 will be awarded, twenty ( 20) $50 Costco Cash Cards will be awarded as second place prizes, thirty ( 30) bottles of Kirkland Signature B- 100, a $14.39 retail value, will be awarded as third place prizes. Employees of Costco and the Sponsor and their af;liates and families are not eligible to enter or win. To enter, submit a 100 or less word testimonial, typed or neatly hand written, by regular mail to The Costco Connection, KS B- 100 Contest, PO BOX 34088, Seattle, WA 98124-1088. Entry MUST include your name, daytime phone number and\/or e-mail address. Please staple all pages of your entry together with your name and phone number at the top of each numbered page. Limit one entry per household. Entries must be received by 11: 59 P.M. PST on July 1, 2011. Winners will be chosen on or around August 10, 2011, and noti;ed by phone or e-mail as soon as administratively feasible thereafter. Except where prohibited by law, Contest entry constitutes consent to use winner's name and entry for purposes of advertising and promotion, without further compensation or consent. By entering, all entrants grant to Costco and Sponsor a royalty-free, perpetual license to use their entries for any purpose whatsoever. Entrant waives all claims against Costco Wholesale, the Sponsor, their employees and agents for injuries, damage or losses related to the Contest and\/or prizes.\n*Entrants will be responsible for all taxes related to the prizes. All prizes must be claimed within 30 days of noti;cation. If not claimed by such date, the prize will be forfeited. Grand prize winner will receive 2 economy class air-tickets to NYC, with a 3 day\/2 night stay in a 4-star hotel (room charges are for accomodations only). A $2000 gift card will be provided to cover additional expenses for the entire trip. All expenses beyond that amount will be covered by the winner. Travel plans will need 45 days advance notice and are subject to availability. Blackout dates may apply. No substitution or transfer of prizes except at the sole discretion of the sponsor. Contest valid for US Residents only (except Puerto Rico). All reservations are ;nal \u2013 No cancellations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baripada: Over 40 school students were taken ill after consuming bhog during Saraswati puja celebrations in Rangamathia-based primary school under Baripada block in Mayurbhanj district, Monday.\nAccording to sources, the schoolchildren had consumed food during a feast organised on the occasion of Saraswati puja celebrations.\nHowever, a few hours later, the students complained of uneasiness and nausea. Following this, the school authorities rushed them to Baripada district headquarters hospital.\nThe doctors have informed the school authorities that the students fell ill due to food poisoning. The condition of the students is stated to be stable, sources said.\nSimilarly, 19 girls of a private marketing firm at Baxi Chowk in Jharsuguda were taken ill Sunday after consuming stale food from their mess.\nAs per sources, soon after having food, the girls complained of stomach pain and some of them started vomiting and were rushed to the District Headquarters Hospital.\nThe doctors also said they fell ill due to food poisoning. It is suspected that the food was made early and served late causing food poisoning, added the sources.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblqxq b/data_all_eng_slimpj/shuffled/split2/finalzzzblqxq
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+{"text":"Explain what Zewei's article from the week 4 reading was all about.\nThis assignment is a take-home essay consisting of 3 questions, 2 pages total, to test knowledge and assimilation of the course objectives. Please exclusively use the course materials to support each answer. No outside sources are permitted.\nBased on Steinberg's article from the week 5 reading, in what ways is NGO influence reflected in Israel?\nBased on the Cragg, Arnold, and Muchlinski article from the e-reserves (\"Human Rights and Business\"), when did \"business and human rights\" become an international topic? What explains the delay?\nPrevious Previous post: Investigate professional journals in the field of fire prevention of interest to you.\nNext Next post: Develop a branding strategy for your product that covers the brand name, logo, slogan, and at least one (1) brand extension.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am so excited to have you here with us, and I want to congratulate you on taking the first steps toward becoming a firefighter without wasting time, effort and money!\nWhich stage(s) of the recruitment process have you passed?\nRIGHT NOW - I'm an action taker!\nNo. I have no ability to afford up to $95 p\/wk to invest in getting your systems & help.\nAfter submitting your application you will be directed to schedule your free Application Call. One of my hand picked team members will be in touch at your booked time, so make sure to have your phone ready and be in a quiet place where you can get the most out of the call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All information enclosed in these terms and conditions is given to the very best of our knowledge and experience, and may be updated or changed subject to the discretion of the directors of Scorpio Chauffeur Cars Ltd.\nAll vehicles and their chauffeurs are licensed by their respective local authority, and strictly adhere to all rules and regulations that are dictated by that authority and prevailing at that time. Details of these licences are available from the Company on request.\nThe \"Company's discretion\" shall be decided by the Directors of the Company at all times, and their decision(s) will be fully binding in all matters.\nOnce the Hirer has made a booking and this has been confirmed, the price quoted to the Hirer will not be subject to alteration unless there is any deviation from details of the booking. However, the Company will not accept responsibility for any errors or inaccuracies resulting from information provided by the Hirer, but will if any such information is bought to our attention, investigate immediately and act upon those findings as soon as practical.\nThe company can only undertake journeys for the Hirer that have been pre-booked and for which written confirmation has been received from the Company. Bookings will only be confirmed once a written (email or text) request has been sent from the Hirer to the Company. Any special instructions must be confirmed in writing by e-mail direct to the Company at the time of booking. We understand that mistakes and misunderstandings can happen, or that plans change prior to the start of a journey, so the use of e-mail for booking and confirmation provides an essential safeguard for both the Company and the Hirer.\nIf no confirmation is received within 24 hours of the Hirer's booking, it is the responsibility of the Hirer to contact the Company.\nThe Company's pricing policy can, upon request, be supplied prior to opening an account with the Company. The Company will notify all account holders of any changes to our pricing policy, in writing, 1 calendar month prior to them becoming active.\nThe company will provide the Hirer with a written quote at the time of booking, but it is the Hirer's responsibility to agree these prices with the Company before confirming any booking. The Company regrets that we are unable to retrospectively change any prices after the journey has been commenced.\nOur minimum rate for any booking is \u00a340.\nWhere applicable, we will add any additional charges incurred on your behalf to the amount chargeable to the Hirer. This may include, but is not limited to, London Congestion Charging, Car parking or Road Tolls.\nAny booking that falls between the hours of 01.00 and 05.00 will be charged at the standard rate plus 50%, unless otherwise agreed in writing by the Company.\nEaster Sunday, Christmas Day, Boxing Day, New Year's Eve and New Year's Day will be charged at the standard rate plus 100%.\nAll prices and quotes will be subject to VAT at the current rate.\nIf the Company is notified that a booking is cancelled 3 \u2013 6 hours before the start of the journey, 50 % of the total fare will be charged.\nIf the Company is notified that a booking is cancelled 3 hours or less before the start of the journey, 100 % of the total fare will be charged.\nIf a passenger fails to turn up at the pre-arranged time and place within 1 hour and no contact has been made, you will be charged the full fare for the entire booking, including any waiting time incurred.\nIf a booking is cancelled as our driver arrives, or whilst the driver is waiting in the case of a wait & return booking, the full fare for the entire booking will be charged, including any \"un-used\" element of the booking.\nIn the case of multiple vehicle bookings, major sporting or social events (such as Ascot, Wimbledon or Goodwood) or whole day bookings, the Company will charge 100% of the fare if we receive less than 48 hours' notice and 50% of the fare if we receive less than 5 days' notice.\nAll of the above cancellation conditions assume the driver is not already en-route. If that is the case, the full fare will be charged.\nAll journey costs and any other monies due to the Company by the Hirer will, unless prior arrangements have been agreed to in writing, be charged to the Hirers' main account. Payments can be made by major credit card, online bank transfer, cash or cheque.\nFor those Hirers with a monthly credit account facility, the Company will issue an invoice during the first week of each month. Our payment terms offer 15 days from the date of the invoice. If these terms are not met, the Company reserves the right to apply a charge that includes interest, late payment charge and if incurred, costs for debt recovery in line with the Late Payment of Commercial Debts (Interest) Act 1998.\nThe Company credit facilities are offered at the discretion of the directors and maybe withdrawn at any time without prior notice.\nUnless previously agreed in writing, NO payments should be made to our driver.\nIn case of London Heathrow and London Gatwick Airports, the company will monitor the flight status for your inbound flight to ensure that the driver arrives to meet you as your plane lands. We allow 45 minutes waiting (commencing at the confirmed actual time of landing). Any additional waiting time is charged at our standard rate for Waiting Time. Parking costs are charged at the same rate incurred by the driver.\nIn the case of London City, Luton, Stansted and Southampton airports, the Company will dispatch the driver in time to arrive at the scheduled landing time. We will be monitoring the flight status, but in the case of some short haul flights it may not have taken off prior to the driver setting off. In this instance, the 45 minute waiting time commences at the Scheduled landing time, and any further waiting will be charged as above.\nUnless notified of any changes to the booked pickup time by the passenger(s) in advance, the driver will arrive at the nominated meeting place at the scheduled time. We allow 30 minutes waiting (commencing at the scheduled time of arrival). Any additional waiting time is charged at our standard rate for Waiting Time. Parking costs are charged at the same rate incurred by the driver.\nALL \u2013 The company will request a mobile contact number for the lead passenger at the time of booking and will text this number on arrival at the pick-up point (terminal, concourse etc) to advise the passenger of their position. The company will not be responsible for extended waiting or parking charges that may be incurred in the event that the passenger neglects to turn their phone on for any reason.\nThe Company reserves the right at any time to change, replace or renew the vehicle(s) or chauffeur(s) booked or advertised in order to maintain the Company's high standard. The replacement vehicle or chauffeur would be of the same type and standard as originally booked and wherever possible, this will be duly notified to the Hirer.\nAll our vehicles have a no smoking policy.\nThe consumption of food or food products within the vehicle(s) is not permitted. The supply of bottled water within each car may only be consumed in the vehicle and must not be removed without the Chauffeur's consent.\nThe Hirer shall be fully responsible and liable for any damage caused both inside and outside the vehicle(s) or to our chauffeur by the Hirer or any members of his\/her party, howsoever caused. This includes incitement to any third party, which results in damage to the vehicle or its contents. Where the vehicle requires valeting as a result of the Hirer soiling the interior, a fixed charge of \u00a3350 will be applied to cover the cost of both the valeting and the vehicle being out of service for the duration.\nThe Hirer will agree to be liable for the total retail costs of repair including VAT, and the Company will determine the location of the repairer. In addition, the Hirer will be liable to pay to the Company a fixed daily rate (determined by the Company) while the vehicle or chauffeur is out of commission for such restoration, plus any further incurred losses i.e. lost bookings. Where the Hirer has made payment by credit card for the hire, the Company reserves the right to deduct any costs for damage described above from the credit card given.\nPlease treat our Chauffeurs with respect. The driver is responsible for the safety of the Vehicle and its occupants. Any Passenger whose conduct the driver reasonably believes to be drunken and disorderly, threatening, abusive, dangerous or in breach of any statutory regulation may be removed from a Vehicle or prevented from boarding. In this event no refund will be given and indeed additional charges may be applied.\nIn the event of any extra requests or alterations to the booking (i.e additional pick-up points, change of venues, extended waiting etc. causing extra mileage and time) not made at the time of confirming the booking, the Driver will endeavour to meet these requirements where possible. If such alternations will prevent the driver from continuing onto his next booking, the Company reserves the right to decline such requests or to attempt to source a replacement vehicle to continue \"as directed\" by the passenger(s). Any such alterations would in most cases involve extra charges being levied on the original quotation.\nSevere and unexpected adverse weather conditions.\nSevere and unavoidable traffic congestion such as road traffic accidents or unexpected road closures.\nVehicle breakdown or accident occurring with the passenger on board.\nIn the event of vehicle breakdown or accident during the journey, the Company will make every effort to source another vehicle and driver to transfer the passenger into such that the journey can be continued. In this instance, all reasonable additional costs incurred in completing the journey will be borne by the Company.\nWhere the Company is aware of possible adverse conditions PRIOR to commencing the journey, we reserve the right to either cancel the booking, advise an alternate time frame for the journey or supply a different Vehicle \/Driver to undertake the booking.\nThe Passenger(s) shall remain responsible at all times for their Luggage & Personal Items and shall ensure that all their Luggage & Personal Items are loaded into the Vehicle prior to commencement of the Service, and unloaded from the Vehicle upon completion of the Service. The Company accepts no responsibility for any loss of \/ damage to Luggage & Personal Items or consequential losses arising as a result of Luggage & Personal Items which are not loaded to or unloaded from the vehicle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love writing about little baby steps in personal finance. Like how to invest your first $1000 or how to earn $100 in a day legally. This is another baby step guide. As an investor, you might want small exposure in metals, for diversity's sake.\nWithout enough money, today, it's still possible to invest in the future of gold and silver. This post is about finding ways for you to invest your first $100 in metal future.\nLet's start with an anecdotal story.\nThere are two people who built houses, one built his on a sandy foundation and the other of a rocky foundation.\nNow, who do whose house do you think withstood the storms of life? In simple words, what am trying to say is that it is good to make the right choices especially when it comes to investment matters.\nSolid investment decisions would involve having exposure to a mix of every investment type.\nSome people think that investment needs a lot of cash in order to reap the benefits of something which is not really true.\nGenerally speaking, it is wise to diversify your investments as much as possible. Buying gold bars and coins if you have that much to invest. Usually, the value of gold rises when the stock market falls.\nInvesting in gold is really easy. There are no forms to fill out, no special terms and conditions to understand and no need to wait for your application to be approved.\nPlus, it is usually really easy to sell. So, when you need to release some funds quickly you can easily do so.\nHowever, you do need to bear in mind the fact that the price of precious metals fluctuates. So, if you do need to sell it quickly when the price is low you can potentially be left out of pocket.\nInvesting in gold and silver is one of the best decisions you can make. Someone may ask, why invest in silver and gold?\nLet's start with silver which is more volatile with fewer cost barriers and it is also a quick profit maker.\nOn the other hand, gold has for many years been used as a medium of exchange and is known to withstand inflation. It is also very expensive to purchase.\nTherefore rushing to invest in these two precious metals is tantamount to playing with fire. You need to have a strategy that will work out for you.\nThere are various ways in which you can invest in both gold and silver even with less than 100 dollars.\nGold and Platinum are definitely the most precious metals on earth (if you consider radioactive materials out of the equation) and that explains why gold products are so expensive.\nIf you are a new investor and probably want gold in your investment list and you don't have huge amounts of money to buy gold, then you should try investing in small denomination gold coins, in order to be on the safe side.\nAs much as it would be foolish to think that you cannot completely lose money through this means, it is usually all about relativity and even compared to other precious metals, gold is often more stable.\nWith your little cash, you can, therefore, purchase smaller proportions of gold also known as ETFs and patiently allow it to stay there as its value appreciates. As a beginner, you need to exercise a lot of patience if you really want to reap big.\nDo not be carried away by greed and quickly start trading on your gold because you might end up making huge losses.\nInvesting in silver is a little bit riskier compared to investing in gold. You should note that silver is highly volatile and as much as its price might rise sharply most of the time, it also has the ability to significantly decline.\nTherefore as a new investor in silver, you should not rush things rather you should use it as a speculation vehicle.\nAn adage goes that it only fools who test the depth of a river with both legs.\nThis basically means that you should make your move with absolute caution and caution here means avoiding investing in leveraged silver or putting your wife and kids in line in the name of silver investments.\nThe most advisable thing to do is to take advantage of the barriers of small prices in the silver markets to purchase several ounces of silver ETFs and then mark the time when the prices rise in order to reap good profits.\nWhy most people get disappointed in these kinds of investments is that they are always in a hurry to dispose of their gold or silver. Remember this is not a short-term investment that is if you want to make good money.\nIt's like purchasing treasury bills or bonds from the government. You need to be patient enough to be able to get the most out of it.\nIn addition, you can as well decide to invest in gold and silver small jewelry which is currently a booming business especially in the fashion industry.\nWhat you need to do is identify a supply whom you can negotiate with to ensure that you buy with wholesale price and sell them at the retail price in order to make some profit.\nIn summary, there is nothing as important as making a worthwhile investment. For this case, investing in gold and silver is very much possible even at less than $100.\nHowever, you have to be a little bit careful when putting your money on silver. But for gold, you can buy as much as you want since it is very stable and often appreciates in value.\nNevertheless, don't be in a rush to sell them off, learn the art or patience when trading in gold if you want to reap high.\nReaders, for diversification purpose you should always find various alternative investments. If you don't have any exposure to metal then perhaps you can gauge the water by investing that $100 you have saved last month. Why not start now?\nThanks for sharing this best way with us how to invest in gold less than $100. Yes, It is really a good thing that most of the people curious to know that how much minimum amount must be needed to invest in gold.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A special blend to help kids with focus and concentration.\nYounger children often struggle to pay attention as they have relatively short attention spans. Help your child maintain focus at home or at school with this special blend for children that can help support brain function and improve concentration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblska b/data_all_eng_slimpj/shuffled/split2/finalzzzblska
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+{"text":"Whether you're a beginner exerciser or a longtime gym-goer, the push-up can be a seriously challenging move. It requires technique, upper-body strength, and total-body coordination to master. There's also the intimidation factor\u2014the name alone can conjure up bad bootcamp class experiences or memories from grade school fitness tests.\nIf, like many folks, you struggle to perform push-ups correctly (perhaps, like yours truly, you avoid them altogether), there's now a tool that can help: the Sling Shot Push Up.\nThis assisted movement device, created by American record-holding pro powerlifter Mark Bell, is part of the Sling Shot product line that was originally created for powerlifters and is now becoming more mainstream. The @mbslingshot Instagram account, the official account for all Sling Shot products, has more than 181K followers. Search #mbslingshot on Instagram and you'll see more than 50,000 posts.\nBut how, exactly, does this giant rubber-band-like device assist with and improve your push-ups? And can anyone do a push-up by simply slipping it on? We chatted with a Sling Shot employee and an unaffiliated certified strength and conditioning specialist to learn more.\nFirst, here's a little history and context on this specialized device.\nThrough his years of powerlifting, Bell \"constantly injured his pecs [the chest muscles],\" Steven Granzella, elite powerlifter and general manager at Mark Bell's Sling Shot, tells SELF. In an effort to prevent further damage to these muscles while also continuing to train, he invented the OG Sling Shot product in 2010, marketed as \"a bench press accessory tool\" to help lifters add weight to their bench press and reinforce proper technique, among other purposes. Today, there are four additional models of the Sling Shot geared toward slightly different goals, including the newest one\u2014the Push Up\u2014released in March 2018.\nCompared to other Sling Shot products, which are designed to assist more experienced weightlifters, the Push Up tool requires zero prior skills or strength in order to use it, says Granzella. It can help someone that struggles with doing a single push-up, plus those who are already able to perform the move correctly, but want to increase the number of reps they can do at a time.\nThere are several ways that the tool can help you improve your push-up skills.\nThe Sling Shot Push Up, as mentioned, assists users through this difficult portion of the move. As you lower down, you load up the tool with your bodyweight, essentially creating a rubber band-like effect that then \"slingshots\" you back up to the starting position. The deeper you go into the movement, the more assistance you receive. Said assistance can then help users identify particular parts of the movement where they need to create and hold more tension.\n\"As you start to descend into a push-up [while wearing the device], it is pretty much impossible to do this incorrectly,\" says DiSalvo.\nThe device is fairly simple to put on and use.\nWith the logo facing outward, slide the circles over your biceps one at a time, placing them a couple inches above each elbow. Note that depending on your arm size, the circles may feel tight or slightly loose\u2014this variance doesn't matter, explains Granzella. What does matter, when it comes to fit, is how the chest piece stretches across your body. If it's too loose, you won't get as much assistance at the bottom of the push-up; too tight, and you'll experience too much tension. Check out the sizing chart on the product website to learn more about sizing.\nFrom here, you're ready to attempt a push-up.\nIf you can't yet perform an unassisted push-up, start by putting on the device and attempting a few reps. Over time, build up the number of reps you attempt, and when you feel strong and confident enough, remove the device and attempt an unassisted push-up.\nIf you are already able to do several unassisted push-ups in a row, DiSalvos recommends doing a set with the device, resting, and then trying a set without. Some people, he says, will notice an \"immediate carryover effect\" of improved form and ability when doing the move sans tool. From there, incorporate the tool into your training as an occasional supplement\u2014not an every-single-time thing. \"If you are trying to build up a lot of push-up strength, you want to program in unassisted push-ups too,\" says DiSalvo. On that note, the better your push-up form becomes, the less you should use the tool, he adds.\nYou can also use this tool to assist your bench presses and dips, says Granzella. Simply put it on as you would for a push-up and perform your bench press or dip as you normally would.\nLastly, though the tool, in general, is very safe\u2014\"We haven't seen anyone get hurt using device, as long as they put it on correctly,\" says Granzella\u2014as with any type of movement, if you start to feel any pain, especially if it's in your joints, stop what you're doing, says DiSalvo.\nThe bottom line: Push-ups can be challenging both physically and mentally, and this tool, as odd as it may look, can help on both counts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brand New and supplied in sealed retail packs. Fresh stocks always assured. Built in flash so ideal for indoor use and perfect for weddings, parties as well as holidays, outdoor pursuits, walking, hiking and festivals etc.\nEach camera is loaded with a 27 exposure Superia X-TRA 400 35mm film that delivers sharp, brilliant colour prints. Built in battery to power the built in flash unit. Leave your expensive camera and risks behind.\nPowerful flash system extends the flash range to 14 feet Continuous flash recharges in 5 seconds Pop-up flash-ready indicator light.\nLarge and clear optical viewfinder makes it fast to use and easy to see and frame the picture. Great for young and old alike. If it gets dropped you wont have to worry.\nLarge and legible exposure counter makes it easy to check the number of remaining exposures.\nGreat to take to parties, events, concerts, festivals, birthdays, Christmas Parties and other social occasions.\nThe camera has 27 exposures of Fujicolor 400 ISO film giving truly excellent results whether outdoors, indoors or at night, with its powerful built-in 3 metre range flash.\nA disposable camera is a disposable camera, what more can I say.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alcudia (Puerto Alcudia) is one the most popular resorts in Majorca located on the north east corner of Majorca in Spain. Weather in Alcudia is excellent for the beach holiday lovers. Alcudia enjoys an great Mediterranean climate of hot summers and mild winters. August is the warmest month with an average temperature of 29 \u00b0C and January is the coolest month with an average temperature of 15 \u00b0C. With an average temperature of 31 degrees, Alcudia is the perfect holiday town to enjoy the sunshine and summer holidays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TradCatKnight: Traditionalists accepting the Council?\nImagine if the \"conservatives\" also known as the false right repented of their errors to join the Resistance. Imagine if all the delusional banter of ...let us all \"get together\" (who attend a Latin Mass) and put side doctrinal differences (not strategic) would just fade away. Imagine a Church which once again loves the Truth and attempts to re-establish Its' boundaries in principle. Imagine a restoration of \"clear thinking\" to conquer the often cowardice compromise. Now you know why we must continue fight and cannot join in with the pseudo traditionalists. Vatican II was the modernist's new playbook, the New Mass was their expression of worship. How do you say we ought join in with others embracing either the Council (to whatever degree) or the New Mass?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At a hearing before the Senate Banking Committee today, FHFA Director Mel Watt highlighted the need for housing finance reform, noting that Congress must decide the future of Fannie Mae and Freddie Mac, including whether or not the enterprises should receive backing from the federal government.\nWatt cautioned that the conservatorships \"are not sustainable and they need to end as soon as Congress can chart the way forward.\" He added that swift action is needed, as the capital buffer currently in place under the terms of the Senior Preferred Stock Purchase Agreements with the Treasury Department has been scaled back significantly and will reduce to zero on Jan. 1. \"At that point, neither enterprise will have the ability to weather any loss it experiences without drawing further on taxpayer support,\" Watt said.\nIn his opening statement, committee Chairman Mike Crapo (R-Idaho) raised concern about the risk taxpayers face under the current arrangement. \"A housing finance system dependent on two government-sponsored enterprises in perpetual conservatorship is not a sustainable solution,\" Crapo said. \"Taxpayers today bear too much risk, and the government plays too big a role in the mortgage market.\" He added that transferring risk away from the government into the private sector and away from taxpayers was \"essential,\" and urged FHFA to continue exploring options for risk transferring credit risk and incentivizing private sector participation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblykq b/data_all_eng_slimpj/shuffled/split2/finalzzzblykq
new file mode 100644
index 0000000000000000000000000000000000000000..5430f379b2d25b5f8893a19e3c1157e74a4653d0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblykq
@@ -0,0 +1,5 @@
+{"text":"One of the thrills of purchasing a new house or renovating your current one is creating a floor plan that works for you. Kitchen renovations can make a huge difference in meeting your family's needs, but there's no \"one size fits all\" solution to laying out what is certainly one of the busiest \u2014 and most appliance-heavy \u2014 rooms in any home.\nPerhaps the most often used appliance in modern kitchens today is the microwave. From heating dinner in less than 60 seconds to defrosting meats, the microwave has made itself indispensable in most kitchens.\nHowever, if you cook often or the square footage in your kitchen is restricted, counter space is probably at a premium. Whether you choose a built-in or countertop version, strategically locating your microwave can make your kitchen more functional, as well as more aesthetically pleasing.\nTraditionally, microwaves have perched right above the stove. However, many homeowners are now using that prime real estate for the installation of large decorative vent hoods. But above the stovetop is still a great option for many people \u2014 the microwave remains easily accessible and provides venting functions as well as additional lighting for your cooktop. Just make sure you don't place it too far out of reach for any regular users in your household, such as younger children and elderly relatives.\nIf you love the convenience of a microwave but find the actual boxiness of it to be a bit unsightly, consider a drawer microwave. While drawer microwaves can cost quite a bit more than standard models, they are both discreet and sleek-looking.\nCountertop ovens are both very accessible and easy to relocate. Renovating around a countertop unit can be as easy as simply moving it from one end of the counter to another. Nevertheless, countertop models still must be vented, so be careful about pushing your microwave flush against the backsplash. As always, make sure to follow manufacturer's venting instructions carefully.\nWhy not put one oven next to another? If you have a separate cooktop and built-in wall oven, making room for a microwave above the latter can free up space elsewhere in your kitchen. Your microwave can even be integrated into your kitchen redesign so that it resembles a second oven. Depending on the arrangement of your kitchen, you may need to work with a contractor to complete such an installation.\nMany contemporary homeowners prefer the look of an appliance garage (or dedicated appliance cabinet) and like being able to tuck their stand mixers, food processors, coffee makers, etc. neatly out of sight. An appliance cabinet can be the perfect place to house a microwave, especially a smaller model. Remember, however, that proper venting is still a must. Such cabinets may also require additional electrical work in order to make sure adequate power is available to the appliances stored there.\nLastly, you can opt to have a spot for your microwave custom built into your existing cabinetry. Planning an island for your new kitchen? A raised area here is a popular spot for the microwave in many modern kitchens. Or maybe you want to keep the microwave close to the pantry or refrigerator. Just make sure the microwave is placed at a level that works for everyone in your household. After all, it's not like your microwave is seldom-used.\nSpeaking of microwaves, did you know that built-in microwaves are covered by American Home Shield? An American Home Shield policy provides you with access to trusted, expert contractors who take pride in making your kitchen repairs as painless as possible. And built-in microwaves are covered under our Appliance Plan. To learn more, contact us today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"President-elect Donald Trump has formally announced Exxon-Mobil Corp chairman and CEO Rex Tillerson as his nomination to serve as US secretary of state today.\nTrump on Tuesday morning called Tillerson 'one of the truly great business leaders of the world' in a Tweet promoting the pick, which has already drawn controversy on Capitol Hill.\n'His tenacity, broad experience and deep understanding of geopolitics make him an excellent choice for Secretary of State. He will promote regional stability and focus on the core national security interests of the United States,' Trump said.\nBut before the nomination got announced it was already drawing concern from GOP Senators Lindsey Graham, John McCain, and Marco Rubio, Russia hawks who are wary of closer ties to Russian president Vladimir Putin.\nThe official statement mentions Russia in Tillerson's bio.\n'In January 1998, he was promoted to vice president of Exxon Ventures (CIS) Inc. and president of Exxon Neftegas Limited. In those roles, he was responsible for Exxon's holdings in Russia and the Caspian Sea as well as the Sakhalin I consortium operations offshore Sakhalin Island, Russia,' according to the statement.\nIt was a brief reference to the part of Tillerson's corporate bio that could pose the greatest danger to his successful confirmation.\nHe has had multiple contacts with Putin as he negotiated energy deals in Russia, got awarded a medal of friendship by the Russian strongman, and is captured on video toasting champagne glasses with Putin after inking a deal.\nTillerson was awarded Russia's Order of Friendship after making a deal with state-owned oil firm Rosneft.\nTrump transition spokesman Jason Miller defended Tillerson under tough questioning on CNN Tuesday morning.\n'Not only is he figuratively a boy scout, he's literally a boy scout,' Miller said, talking about Tillerson's time as an Eagle Scout and his past presidency of the Boy Scouts of America.\nMcCain has forecast his 'concerns' about the Tillerson nomination.\nOpponents can't filibuster Tillerson's nomination because of a rules change pushed through by Democrats. But if three Republicans opposed it, the nomination would go down.\nThe Trump transition was aware of the concerns, but elected to push through the Tillerson announcement quickly. The Trump transition announced Monday night Trump would delay an announcement about how he would organize his business during his presidency and instead focus on forming his government.\nTrump, who teased the State announcement last night, met with Tillerson twice during the past week and pointed to the CEO's deep relations with Moscow as a selling point.\nBut he could face a battle over the appointment, with Tillerson's relationship with Vladimir Putin likely to be troublesome among several Republican senators.\nIn a statement, Trump said: 'His tenacity, broad experience and deep understanding of geopolitics make him an excellent choice for Secretary of State.\nTillerson said he shared Trump's 'vision for restoring the credibility of the United States' foreign relations and advancing our country's national security'.\nThe President-elect's statement said: 'Rex Tillerson's career is the embodiment of the American dream.\n'Through hard work, dedication and smart deal making, Rex rose through the ranks to become CEO of ExxonMobil, one of the world's largest and most respected companies.\nAs ExxonMobil's head, Tillerson maintained close ties with Russia and was awarded the Order of Friendship by President Vladimir Putin - a sticking point among several Republican senators who find his cozy relationship troublesome.\nAfter the initial favorite, former New York Mayor Rudy Giuliani withdrew himself from consideration, Tillerson emerged as a frontrunner several weeks into the deliberation.\nTrump picked Tillerson, 64, after the Texan was backed by several Republican establishment figures including former Secretary of State James Baker, former Secretary of State Condoleeza Rice and former Defense Secretary Robert Gates, the transition official said.\nRomney and other contenders such as former CIA director David Petraeus and Republican Senator Bob Corker of Tennessee, all received calls on Monday informing them of Trump's decision to nominate Tillerson, the New York Times reported.\nRomney shared a statement on his Facebook page that read: 'It was an honor to have been considered for Secretary of State of our great country. My discussions with President-elect Trump have been both enjoyable and enlightening.\n'I have very high hopes that the new administration will lead the nation to greater strength, prosperity and peace,' he added.\nRomney, the 2012 Republican presidential candidate, was a vocal critic of Trump, although the two appeared to make up as the president-elect considered him to head the State Department.\nBut some of Trump's senior advisers warned that tapping Romney for the job would anger the president-elect's loyal supporters.\nTrump was not familiar with Tillerson until he was recommended by former defense secretary Robert Gates and James Baker III, who worked as secretary of state under George H. W. Bush, the Times reported.\nBoth Trump's son-in-law Jared Kushner, and his chief strategist Stephen Bannon backed Tillerson, who hit it off with Trump, according to the Times.\nNBC News first reported Tillerson would be chosen as secretary of state, citing two sources close to the transition team. Fox News also reported the ExxonMobil executive as Trump's pick on Monday.\nHe added that it was 'a great advantage' that Tillerson knew 'many of the players', and did 'massive deals in Russia'.\nIn 2011, ExxonMobil signed a deal with Rosneft, Russia's largest state-owned oil company, for joint oil exploration and production. Since then, the companies have formed 10 joint ventures for projects in Russia.\nBut the energy deal was put on hold when the US and European allies imposed sanctions against Russia for annexing Crimea.\nExxonMobil reportedly vowed to resume the agreement once sanctions were lifted \u2013 and Tillerson has already spoken out against trade penalties on Russia, claiming they harm ordinary people in the country.\nTillerson, who is expected to retire from ExxonMobil next year, also has tens of millions of dollars of his pension tied to the oil firm, presenting possible conflicts of interest.\nSupporters, however, claim he is a skilled negotiator who has vital experience dealing with world leaders.\n'He has had more interactive time with Vladimir Putin than probably any other American with the exception of Henry Kissinger,' John Hamre, a deputy defense secretary to Bill Clinton told the Wall Street Journal.\nTillerson's close ties to the Kremlin, however, is a worrisome sign to many, especially after the CIA's conclusion that Russia interfered in the US elections with the goal of seeing Trump in the White House.\nRepublican Senator John McCain, along with majority leader Mitch McConnell, have voiced their support for a congressional investigation into Russia's involvement.\n'You want to give the president of the United States the benefit of the doubt because the people have spoken. But Vladi\u00admir Putin is a thug, a bully and a murderer, and anybody else who describes him as anything else is lying,' McCain said.\nSenator Marco Rubio also tweeted his displeasure about the potential pick on Sunday.\n'Being a 'friend of Vladimir' is not an attribute I am hoping for from a #SecretaryOfState,' Rubio wrote, initialing the tweet to ensure Americans knew it was coming directly from the senator's mouth.\nIf Tillerson is nominated, climate change could be another divisive issue. ExxonMobile is under investigation by the New York Attorney General's Office for allegedly misleading investors, regulators and the public on what it knew about global warming.\nOn Monday, Trump's senior adviser Kellyanne Conway was forced to defend Tillerson against claims he's too friendly with Russia, saying: 'It's not like he's pounding down vodka with Vladimir Putin at the local bar'.\n'We know we don't have a good relationship with many countries around the world, including Russia,' she insisted.\n'If, as secretary of state, Rex Tillerson or whomever the president-elect chooses ... if he can go ahead and improve relationships and advance American interests and advance the Trump doctrine, then we should all welcome that,' Conway added.\nTillerson emerged as Trump's leading candidate after former New York Mayor Rudy Giuliani formally withdrew from consideration for secretary of state.\nEx-Beijing ambassador Jon Huntsman, former U.S. Ambassador to the United Nations John Bolton, Congressman Dana Rohrabacher, have also been considered for the role.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Soroptimist's of San Diego are excited to announce tickets are available for the 8th Annual Wine in the Wilderness event. Click on the photo below to visit the Wine in the Wilderness page on our website to purchase tickets or to sign up as a sponsor.\nIt will be great to see you all again, we look forward to meeting your family and friends as well. Remember, sharing is caring. Please share this post on your favorite social media pages to show your support.\nKeep up to date with the latest news and event information. Please contact us if you have any questions, would like to attend an event, or would like to know more about being a part of the difference we make.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Electrical tool for bending slats, needed during the construction of wooden models such as HMS Victory, Sansima Trinida and many others. Bender for slats can also be applied in the construction of aircraft dioramas, etc.\nWorking principle is very simple, just soak a slat in warm water and then lay it on the wooden block for slats profiling and form the desired shape with a heated tip. Block for profiling slats has two additional semi-circular holes to be able if necessary for form a slat shaped \"U\". If you want to bend the thick slats should do it progressively in order not to break the slats.\nThe whole set is packed in a practical wooden box whose upper flap serves as a handle to the tip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What we all wish, particularly for the children in the Middle East on both sides, is peace and for peace to come as soon as possible.\nInstead of this, unfortunately, in the Palestinian areas we have recurring conflicts, sometimes armed conflicts with all their ugliness. Children are the most defenseless suffering party in all conflicts - and for Palestinian children it is even more so.\nBecause, sadly, beyond the inevitable suffering in any war zone, Palestinian children have become pawns of their political and military leaders. Inside their schools, children are subjected to hate-speech, defamatory indoctrination and glorification of death and suicide bombings. Outside the schools, they are forced and coerced into participating in acts of war and terror, and are used as human shields. Several Palestinian minors have been indoctrinated, trained and sent on suicide missions, among them also the developmentally challenged and other helpless minors.\nAll these are in blatant violation of international law; are actually war crimes. Led today, not surprisingly, by the de-facto Palestinian leadership, the Gazan Hamas regime itself.\nIsraeli children have always been deliberately targeted by Palestinian terror. They want to harm Israelis where it hurts most; to kill and terrorize their beloved children; like the sniper killing of a 10 month-old baby, Shalhevet Paz, or the point blank extermination of the four daughters (age 3 to 11) of Tali Hatuel, or by indiscriminate rocket shelling of civilian targets, schools and kindergartens in Sderot. After a total 14 Israeli civilians killed (incl. three 4-years-olds) by Kassam rockets in Sderot (an equivalent of 600 rocket victims in the US, proportionally!) children there had to learn in school-buildings fortified like concrete bunkers, and be alert for the air-raid alarms which sounded whenever a Kassam rocket was launched, several times daily, leaving them 15 seconds to run to shelters and take all protective measures.\nTo counterbalance their bad name for targeting of Israeli children, Palestinians 'invented' Pallywood, staging and manufacturing movies with lies and libels against Israelis 'killing' Pal children, like the Al-Durra myth.\nJewish love of children is proverbial, sometimes ridiculed and self-ridiculed for its exaggerations. One can see a similar trend in Israeli Arab families. There is a hope that this attitude will also 'infect' Israel's Arab neighbors. Because, as Golda Meir said once: \"There will be peace in Middle-East when the Arabs start to love their children more than they hate us\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmyhz b/data_all_eng_slimpj/shuffled/split2/finalzzzbmyhz
new file mode 100644
index 0000000000000000000000000000000000000000..00f8d25da84804ac97d25ea0612b450466650e03
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmyhz
@@ -0,0 +1,5 @@
+{"text":"Arrival in Chennai followed by assisted transfer to the hotel pre-booked for your stay. On reaching the hotel, check into our room and retire for the day. Overnight stay in Chennai.\nOn the 2nd day, after breakfast, embark on a sightseeing tour of Chennai covering Fort St. George, San Thome Cathedral and some museums housed with a fine collection of stone sculptures and bronze sculptures. In the afternoon, leave by road for Mahabalipuram. On reaching Mahabalipuram, check into the hotel for overnight stay.\nPost a scrumptious breakfast in the morning, embark on a sightseeing tour of Mahabalipuram covering the 7th century architectural marvels- Arjuna's Penance, Five Rathas and Shore Temple. Later, return back to the hotel for dinner and overnight stay.\nToday, in the morning, take a road trip to Tanjore. On reaching Tanjore, venture out of your hotel and visit the famous Brihadewswara Temple, a famous UNESCO World Heritage Site. Stay overnight in Kancheepuram.\nToday, in the morning, you will be transferred to the international airport\/railway station to board flight\/train for onward journey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're reading this, you already know the importance of color in your world. The palette of colors that surrounds you is full of magic, memories and feelings. And color can enhance a space in ways otherwise not possible.\nAs an architectural color consultant, I can help you understand and select paint color schemes that reflect your needs and taste, while getting you past the fear of choosing the wrong color. No more safe beige!\nWhether you're a home-owner, property manager, architect or real-estate agent, we'll work together to blend what you like with what will enhance your property. I work within the architectural context, considering all surrounding elements so that color is not an afterthought, but an integral part of the form.\nA consultation can be as simple as selecting new paint colors for one room, or as complex as creating a totally new interior\/exterior environment of colors.\nI have worked on historic properties, new construction, renovations and homes both large and small. I'll introduce you to a world that goes FAR beyond the tiny paint chips found in retail paint departments. We'll create your ideal color space by working together.\nPainting can be expensive, disruptive and time consuming. Why not make sure you do it right, and get what you want? Call a professional - Colors by Zoe!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The case of the addiction to the drugs are now growing too fast right now. It is very important to note that this thing that can be noticed easily due to the fact that this whole story are being experienced by the victims that is being reported from all different parts of the world. That clearly being said, it can show us that the drugs and alcoholic beverages are being taken and consumed by many people in the entire world and that is the main reason why the earth is now becoming a harsh and a very dangerous place to reside and live in a peaceful way. That is why it must not be tolerated by anyone as time has already come when larger or the bigger problems or issues with drugs and alcohol may need more serious efforts to be required to curb the oppressing situation.\nThe governments around the globe from almost all the nations in the world are considering to treat this matter and other related issues with great seriousness and needs to be solved as soon as possible. The initiatives which might be being taken seem to be satisfactory. The drug addicts and their family contributors are required to make sure the ultimate components to deliver a better and high-quality remedy for them. The Addiction Treatment Center in Stuart Florida and programs are running now in almost different parts of the world and the good thing is that most of these rehabilitation centers can be seen to be run by those government sectors and the Non-government Organizations (NGOs).\nThere is this certain Christian Drug Rehab in Florida program and facilities that is very essential and also right there available at the Christian alcoholic and also the drug rehab center which is now very authentic, legit and therefore we can say that it is more efficient and effective for the drug addicts who are suffering some serious and sorts of alcohol and drug addiction. The physical features and characteristics of this kind of Christian drug rehab centers are enumerated or listed and few of this are only provided and this are the major ones.\nThe major thing that can be appreciated about this rehab is that it focuses majority onto the high quality of the treatment and at the same time it is very fantastic to know that it aims for the good and long lasting recovery of the patients in the rehab center. The medical practitioners like the doctors and medical staff like the nurses and aids are very good and polite and at the same time very efficient in their respective task. The medical staffs like the doctor are now well trained to do the care and they are aiming to improve the condition of the addicts and to live back to his or her previous activities and usual day to day works.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Abubilla Music Foundation is our charitable wing, which we use to support inspirational music projects from around the world \u2013 projects which actively preserve and promote a community's cultural music heritage while also discovering and nurturing new, emerging talent. In a fast paced world, it's easy for the origins of music to be lost or forgotten.\nTo read about the mission of the Abubilla Music Foundation, click here.\nTo find out about the administration and trustees of the AMF click here.\nScroll down to view our current projects and latest news.\nSinging Wells is a project with the preservation of East Africa's music heritage at its very heart. While working on this project we will be supporting Ketebul Music in Nairobi as they embark on a mission to record traditional East African music in villages throughout the region. The team from Abubilla Music will be providing advice and expertise for the Ketebul Music sound engineers while they travel throughout the region undertaking field recordings. The Abubilla Music Foundation is helping to provide financial support for essential equipment and services which Ketebul Music needs to make sure the project achieves its objectives.\nFirst and foremost Meninos do Morumbi is a band, albeit a huge one. When you talk to its founder, Flavio Pimenta, he makes clear that everyday he is nurturing the 'soul of the band', made up of 100's of kids that need to learn how to make their way on their own in the world. But this Sao Paolo organisation is far more; it gives children and teenagers an alternative to street life by teaching them drumming, singing and dance, plus providing them with traditional schooling.\nOur work with Meninos do Morumbi will start with funding from the Abubilla Music Foundation to help them develop their website and digital marketing capabilities. We have worked closely with technicians in Brazil to populate the site with content, and thus ensure that valuable time with the band is not taken away from Flavio or his immediate team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rebel Hair Studio is committed to making you feel and look your best. Our team of cutting and color experts are dedicated to professional development and strive to always stay on top of the newest trends. The Rebel team attends in-salon education and travel globally to train with the leading colorists, artists and educators in our industry. This means we can offer our clients a consistently superior service.\nWe are a salon of artists who work together as a team. This philosophy allows you to request any designer anytime.\nBelow is a representation of the services we offer. Prices vary depending on stylist so call today to find the best fit for you!\nOur conditioning treatments are uniquely designed for each client's needs.\nPlease inform our staff if you are using Retin-A or other similar products.\nTo ensure that your next appointment is at a convenient date and time for you, we recommend that you schedule your next visit prior to leaving the salon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnwnk b/data_all_eng_slimpj/shuffled/split2/finalzzzbnwnk
new file mode 100644
index 0000000000000000000000000000000000000000..d9a529f539a35dd52b7951dc087e7552574276c9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Manufacturer & Supplier of Grass Pavers. Our product range also comprises of Railing Planter, FRP Planters and Stainless Steel Planters.\nOwing to our highly-advanced infrastructure unit and immense knowledge, we are engaged in offering a sophisticated range of Grass Pavers. It is thoroughly examined by our skilled team of professionals to ensure its unbeatable quality. It is manufactured with fine quality of raw material and high-end technology, which conforms to predefined guidelines of quality. This paver is exclusively designed to render maximum space to grass cover as well as supporting vehicle movement. Our clients can avail it at industry leading prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A classic yet modern heavyweight basketweave in either neutral or fashion hues. Our Respect collection is made of 100% recycled material - a favourite of design-savvy, conscious consumers. Leftover cutouts from the fashion industry are transformed into a high-quality, sustainable fabric without using a drop of water or dye. A structured two-tone fabric with a tactile finish. Designed by Bemz. Woven in Europe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To get on our contact list, or to make a new application, please see below.\nWe contact everyone on the list in late June, and accept applications during the month of July. Assignments are made the first week of August. Applications received on or after August 1 are considered following the assignments of those who applied within the initial application timeframe.\nIf you have previously participated or completed the application, you need only send an email (metcom@gci.net) to confirm your intent to participate this year. In your email include any new contact information or product descriptions, and provide three preferred locations.\nWe give preference to those who have products not commonly available in Juneau and who convey an intent to help promote your presence at the Market through social media and\/or conventional advertising.\nNew applicants may use the on-line application below or request by email or post card that an application form be mailed.\nUpon receipt of your application (email, on-line, or by USPS mail) we will promptly reply by email to verify that your request has been received. If you do not receive an email reply, please contact us again. There are a few applicants who don't have email \u2013 we will contact by telephone if necessary.\nPlease see FAQs for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For this \"Sobriety & Addiction Recovery\" Healing Gemstone collection, we have hand selected 5 healing stones: Blue Tiger's Eye, Amethyst, Black Onyx, Hematite, and Carnelian. The set includes one of each of these along with a bag.\nOne of the best stones for staying sober, fighting addiction, and sticking to diet and exercise; Blue Tiger's Eye motivates us to try new things and set aside our fears to accomplish our dreams. It enhances physical strength, and vitality. It improves feelings of self-worth, confidence, and personal power. Blue Tiger's Eye allows us act with integrity and actualize our true inner potential.\nAmethyst is the number one best crystal to us for achieving sobriety and fighting addiction. The original Greek name \"Amethysos\" literally means \"Not Drunk\" in Greek. Amethyst crystals. It is a calming stone that soothes the soul and gives you patience, persistence, and positivity. It also deepens understanding of oneness and the connection to the divine.\nBlack Onyx helps us remove our masks and understand who we are by helping us see the darkness within us before seeing the light. It shows us the way to our personal strength by helping us make new perceptions and insights about ourselves that transform us from the inside out. It assists us in overcoming our fears and achieving our full potential. It removes self-doubt, anxiety, and everyday stress. It provides centering and balance.\nHematite is the \"Gemstone of Grounding.\" It provides us with stability and a deep connection to the earth and it's many inhabitants. It neutralizes negative energy and improves our personal strength. It improves our personal strength and it improves our problem solving skills and makes us more productive and organized. It is a powerhouse crystal that will motivate you to keep your life on the right track.\nWarmth, passion, and energy\u2026 Carnelian is the \"Gemstone of Motivation.\" It provides an extremely empowering and positivity force that sets good things in motion and rejuvenates the mind, body, and spirit. Carnelian enhances motivation, determination, and courage. It provides good luck by attracting prosperity, resources and success. It protects against internal or external anger, resentment and abuse. It boosts metabolism, rejuvenates life force by promoting blood circulation, It assists with physical training and exercise.\nFor this \"Sobriety & Addiction Recovery\" Healing Gemstone collection, we have hand selected 5 healing stones: Blue Tiger's Eye, Amethyst, Black Onyx, Hematite, Carnelian. The set includes one of each of these along with a bag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last week we were delighted to collect an amazing big file of historical photographs of the River Hull for Open Bridges: A River Full Of Stories. Thanks to Angus Young of the Hull Daily Mail who has contributed the Mail's archive of River Hull photos to the project. The pictures range from the turn of the 19th century to the 1980s. Some were filed by the Hull Daily Mail as far back as the 1940s and some have not or rarely been published since. It was exciting to open the box and find scenes of the river showing it packed with working barges, aerial scenes taken from high buildings, and people at work on the River to illustrate the fascinating stories we've heard at the memory sharing sessions to life. We have spent the week photographing them and tidying them up ready to use for the project. We're looking forward to using them to illustrate the River Hull stories we have gathered during the memory sharing sessions, but here are a few just to get started.\nAll photographs from the Hull Daily Mail used with permission. We'll credit the original photographer where we can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpaby b/data_all_eng_slimpj/shuffled/split2/finalzzzbpaby
new file mode 100644
index 0000000000000000000000000000000000000000..9fea6436ef63ec9fc43c08ce259383017f83d94e
--- /dev/null
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+{"text":"1. Minimize Omega's exposure and risk on projects.\n2. Ensure project documents are complete, current, and stored appropriately..\n3. Keep Project Manager well informed of status of project and serve as liaison between Project and Procurement\/Sales\/Assembly\/Logistics\/QA departments.\n4. Work with the Service team in project utions; providing technical documents to enable them to perform testing, commissioning , repair and maintenance tasks locally \/ overseas, on-shore and offshore.\n5. Advise Service Engineers on the system setup and functionality during IAT and FAT stages.\n6. Be responsible for the tools and parts issued under his care.\n7. Observe and implement the company EHS policy; ensure compliance with all safety policies, practices and procedures at all times.\n8. Work within project deadlines and prioritizing tasks time to time as scheduled.\n9. Perform site testing and commissioning as required.\n10. Any other duties assigned by immediate Superior.\n1. 3 years of experience in marine, shipyard, petrochemical, power plant, oil & gas or other harsh industries.\n2. Lead and ute project work plans and revises as appropriate to meet changing needs and requirements.\n3. Lead team to meet daily schedules on operational aspects of all projects and related activities.\n4. Effectively apply Omega methodology throughout the project ution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"China is a fascinating country with rich history. It is one of the top tourist destinations. It offers diverse culture and traditions that date back thousands of years. It also offers a modern view of major cities which have glitter no less than the cities in USA or Europe.\nTo visit China, US citizens need visa. However if you are a US citizen stopping in China for less than 72 hours on your visit to some other county, you can get visa at major airports. Visa photos for China are can be 33 x48mm or 2x2 inch depending on the China consulate in the USA. So please read your visa application instructions and select correct size on your order so we can provide you accurate photos.\nThe 2x2 inch photos for China visa have a bit larger head size requirements than that in USA passport photos. For 33mm x 48mm sized photos and detailed requirements, here are sample China Visa photos.\n1) Passport \u2013 You should submit your original passport that is valid for at least another 6 months with at least one blank visa page and a photocopy of the passport's information\/photo page.\n2) Visa Application Form \u2013 You should submit the Visa Application Form of the People's Republic of China which is truthfully completed and signed. You should submit the truthfully completed and signed Supplementary Visa Application Form if you are seeking to work (Z Visa) or study (X Visa) in China, if you are a citizen of a third country, or if someone else traveling with you shares the same passport with you.\n3) Photo \u2013 Please affix one color photo on the Application Form. The photo should be recent, front view, in 48mm x33mm size without head covering.\nChinese Passport Photos must be 48mm x 33mm. Head height is between 28mm - 33mm. Head width is between 21mm - 24mm. A white or light blue background is required for Chinese Passport Photos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"EDMONTON, ALBERTA--(CCNMatthews - Sept. 7, 2005) - Titan Trading Analytics Inc. (TSX VENTURE:TTA) (OTCBB:TITAF) (\"TTA\") is pleased to announce the formation of a wholly owned subsidiary, Titan Trading USA, LLC. Titan Trading USA is a Delaware LLC licensed to do business in New York and Florida. The formation of Titan Trading USA is a major step forward for the parent company, representing its first corporate venture into the United States and foreign countries.\nOne of the goals of Titan Trading USA is to market and distribute TTA's order processing system, TOPS. TOPS is an automated trading system developed over the past two and a half years by TTA and Cignal Technologies of Rhode Island.\nWith the formation of Titan Trading USA, the company will have strategic alignment with seasoned advisors and brokerage firms. TTA is also in the final stages of registering Titan Trading USA, LLC with the Securities and Exchange Commission as a Registered Investment Advisor. Upon registration, Titan Trading USA will be entitled to trade client accounts and earn a percentage of proceeds.\nThe home office for Titan Trading USA is located on 3rd Avenue in Midtown Manhattan.\nThe TSX Venture Exchange does not accept responsibility for the adequacy and accuracy of this release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Italian design can only come from those who live and breathe the passion and style for flawless flare. A trait only inherited by birth and practice, the eye only sees superb mastery through generations of watch-makers. From the fine lines and curves to the intricate detail and craftsmanship, Enrico Margaritelli shares his family legacy with the world one face at a time. The CT Scuderia watch matched for the world of elite sport-racing.\nDesigned by Enrico Margaritelli, the CT Scuderia brings to life the visions and ideas procured on the drafting boards of visionary designers.\nAs a third generation Italian watch-maker, Enrico Margaritelli's impressive career includes collaborations with Fossil as head designer, as well as for Emporio Armani watch collections. Enrico comes from a family of passionate watch-makers. His biggest inspiration, his grandfather, Ariodante Margaritelli, was a dedicated watch maker in Parma, Italy where he created highly technical precision watch instruments for the Military during World War I and II.\nThe clean lines of CT Scuderia were envisioned by Enrico's enthusiasm for experimenting with unusual shapes, lines, and patterns. The unique feature of the crown and pushers positioned at 12 o'clock characterize the watch as a stop-watch. Its generous 46 mm size case can be separated from the strap to be used as a neck watch\/stop-watch alone. The interchangeable perforated leather strap is the same leather utilized in luxury sportscars, handmade in Switzerland. The CT Scuderia is of the first watch houses to acquire RONDA quartz movement 3520D, which will be produced in limited quantities in 2012.\n\u25b6 Ultra soft-silicon straps are also available interchangeable.\n\u25b6 Case construction is a piece of Hand Made Industrial Art: 6 o'clock horns are apply by screws and different than the one use at 12th o'clock as must be go around the pushers and the crown.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Evermine Coupon Codes 2019: Up To $22 Off & Free Shipping.\nUse these new Evermine promo codes to save on your current purchase. Our exclusive Evermine coupon codes are updated frequently, so please bookmark this page! Also, check out other jaw dropping discounts for your favorite brands and online stores!\nPopular Evermine free shipping coupon codes: Redeeem this special discount offer while it lasts. $2.99 Flat Rate Shipping. Check out the latest Evermine promotional coupons and save today.","meta":{"redpajama_set_name":"RedPajamaC4"}}